Citation
Enhancement of D-lactate production in a continuous culture of a mutant escherichia coli through periodic operation

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Title:
Enhancement of D-lactate production in a continuous culture of a mutant escherichia coli through periodic operation
Creator:
Rodin, Jonathan Ben, 1966- ( Dissertant )
Svoronos, Spyros ( Thesis advisor )
Westermann-Clark, Gerald ( Reviewer )
Koopman, Ben ( Reviewer )
Ingram, Lonnie ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1992
Language:
English
Physical Description:
vii, 175 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Aeration ( jstor )
Aerobic conditions ( jstor )
Anaerobic conditions ( jstor )
Biomass ( jstor )
Flasks ( jstor )
Lactates ( jstor )
Modeling ( jstor )
Parametric models ( jstor )
Productivity ( jstor )
Property reversion ( jstor )
Chemical Engineering thesis Ph. D
Dissertations, Academic -- Chemical Engineering -- UF
Escherichia coli ( lcsh )
Lactic acid -- Metabolism ( lcsh )
Microbial metabolites ( lcsh )
City of Gainesville ( local )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )
theses ( marcgt )

Notes

Abstract:
In some biological systems, the environmental conditions that are optimal for microbial growth differ from the optimal conditions optimal for producing a desired metabolite. If production of this metabolite were the process objective, one could continuously operate a reactor system at the optimal production conditions. However, for a given reactor volume, changing the conditions periodically could increase overall production of the desired metabolite. This is possible since, due to higher growth rates under the optimal growth conditions, one could operate the system at significantly higher flowrates and, thus, obtain more product A system involving E. coli mutant LCB898 was used as a model system. Under anaerobic conditions this bacterium will produce large amounts of d-lactic acid, whereas under aerobic conditions, this bacterium will grow faster. The possibility of increasing total lactate production by cycling dissoved oxygen was investigated. Before any optimization work could be done, an adequate model for describing the behavior of this system under both steady-state and transient conditions had to be developed and tested. Such a model was developed using batch and continuous data and then tested by comparison with shifts between conditions. A method for determining the optimal waveform for the proposed cycling was developed by extending previous work by Lyberatos and Svoronos. The method involved Carleman linearization of the model equations around a steady state and subsequent development of a term for the performance measure. The system studied oscillated between purely aerobic and anaerobic metabolisms with no intermediate conditions. Thus, an "imaginary" steady state of intermediate metabolism had to be used for linearization. In numerical simulation of the determined optimal cycling, significant improvement over strictly anaerobic operation was found. Experimental verification of this was performed and improvement, though not as significant as theoretical predictions would indicate, was found. Additionally it was found that the mutant was probably reverting to a form where little d-lactic acid was produced. Cycling of dissolved oxygen apparentiy helps delay this reversion.
Thesis:
Thesis (Ph. D.)--University of Florida, 1992.
Bibliography:
Includes bibliographical references (leaves 171-174).
Additional Physical Form:
Also available on World Wide Web
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Jonathan Ben Rodin.

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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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ENHANCEMENT OF D-LACTATE PRODUCTION IN A
CONTINUOUS CULTURE OF A MUTANT ESCHERICHIA COLI
THROUGH PERIODIC OPERATION













By


JONATHAN BEN RODIN


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA

1992


UJ~aSIT'I : FURIDA LI












ACKNOWLEDGMENTS


I would like to thank my committee chairmen, Professors Spyros Svoronos and

Gerasimos Lyberatos, for providing guidance above and beyond the call of duty. They are

appreciated for being there to help and encourage me during the bad times when the project

was going down seemingly blind alleys.

There are other faculty members who have truly been of great assistance. I would

like to thank my other committee members first, Professors Gerald Westermann-Clark,

Seymour Block, Ben Koopman, and Lonnie Ingram. Dr. Ingram is especially appreciated

for all of his very useful advice without which this project would not have been completed.

He has also been very generous with use of his laboratory and equipment. In addition to

my committee members, Professors Bitsanis and Crisalle are thanked for all of their

guidance.

Several graduate students have helped me out during my research project. Three

deserve special mention, Jeff Mejia, Pratap Pullammanappallil, and Christina Stalhandske

for all the technical assistance they provided. If only one student was to be mentioned, Jeff

Mejia would be the one. He is the one who initially suggested the reversion problem with

this system. If not for his advice, I would still be in the dark as to the cause of all of the

problems associated with continuous anaerobic operation. Christina's help was

instrumental in the work performed on the aerobic characterization of bacterial growth.

Pratap, besides being one of my closest friends during my years in Gainesville, has been

an indispensable resource of information and help with my project. Additionally, Jeffrey

Harmon has lent a helping hand on several occasions. Lastly, in addition to those







mentioned above, I have been very fortunate to have had numerous other friends among the

graduate students in this department.

Several undergraduates from this department have assisted me at various times

during the duration of this project. I have been very fortunate to have had these people

work with me. The following undergraduates have worked as laboratory assistants or

completed small research projects with me: Dawn Mackland, Mike Hinson, Reann

Soodeen, Craig Moates, Erik Dunmire, and John Walker. Among these students, Dawn,

Craig, and Mike have each shown dedication above and beyond what was expected from

them. Like the graduate students that have gone through this department, I have been

fortunate to call several of the undergraduates, again in addition to those mentioned above,

friends.

This section would not be complete without mentioning Mr. Tracy Lambert, the

department's maintenance specialist. He has been of invaluable help to me during this

project.

Finally, I would like to thank my parents, sister, brother, and sister-in-law.

Without their love, support, and encouragement, I probably would not have been able to

endure this whole ordeal. My brother was also generous with allowing me free use of the

facilities and equipment at his business, Gallery Graphics.













TABLE OF CONTENTS

ACKNOWLEDGEMENTS .................................................................. ii

A B ST R A C T .................................................................................... vi

CHAPTERS

1 INTRODUCTION ........................................ ............. 1

2 THEORETICAL METHODS ............................................. 4

2.1 Overview ........................................... ......... 4
2.2 Determination of Optimal Steady-state Operation ............ 5
2.3 A New Method of Determining Optimal Periodic
Pulsing ............................ ......... ......... ......... 7
2.3.1 Carleman Linearization ......................... 7
2.3.2 Performance Measure Calculation .............. 10
2.4 Model Fitting with Nonlinear Least Squares Methods ....... 15

3 EXPERIMENTAL METHODS ......................................... 19

3.1 Organism Description ........................................... 19
3.2 Analytical Methods ............................................ 21
3.2.1 Biomass and Cell Number Determination ..... 21
3.2.2 Glucose Analysis ............................... 25
3.2.3 d-Lactate Measurement ........................ 26
3.2.4 Other Analyses .................................. 28
3.3 Feed Medium Composition ................................... 30
3.4 Feed Preparation ................................................ 32
3.5 Experimental Operation ......................................... 38
3.5.1 Operational Conditions ........................ 38
3.5.2 Shake Flask Experimental Procedure ......... 40
3.5.3 Reactor Experimental Procedure ............... 42
3.5.3.1 System Description ............ 42
3.5.3.2 System Startup and Operation 44

4 ANAEROBIC GROWTH OF E. COLI LCB898 ...................... 48

4.1 Background ...................................... ............ 48
4.2 Batch Growth ...................................... ......... 51
4.3 Continuous Growth ............................................. 65
4.4 Modeling .......................................................... 73
4.4.1 Presentation of Model ........................... 73
4.4.2 Model Parameter Fitting ....................... 74








5 AEROBIC GROWTH OF E. COL LCB898 ........................

5.1 Background ...................................................
5.2 Experimental and Modelling Results Introduction ............
5.3 Batch Growth .................................................
5.4 Continuous Growth ..........................................
5.5 M odeling ......................................................
5.5.1 Presentation of Model ........................
5.5.2 Model Parameter Fitting .....................

6 THE EFFECTS OF SHIFTS IN AERATION OF E. COLI LCB898 ..

6.1 Background ...................................................
6.2 Development of the Combined Aerobic-Anaerobic Model ...
6.3 Testing of the Model .........................................

7 EFFECTS OF AERATION CYCLING ON LACTATE
PRODUCTIVITY OF E. COLI LCB898 .............................

7.1 Background ...................................................
7.2 Theoretical Investigation into Cycling .....................
7.3 Experimetal Confirmation of Lactate Productivity
Optimization Results .........................................

8 REVERSION OF E. COL LCB898 AND A POSSIBLE NEW
METHOD OF AVOIDANCE OF REVERSION .......................

9 CONCLUSIONS ........................................................


84

84
85
87
87
95
95
101

113

113
114
116


122

122
123

135


151

163


APPENDIX

MATHEMATICS PROGRAMS FOR COMPUTATION OF
CARLEMAN LINEARIZATION MATRICES .................................. 165

LIST OF REFERENCES ..................................................................... 171

BIOGRAPHICAL SKETCH ................................................................ 175












Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


ENHANCEMENT OF D-LACTATE PRODUCTION IN A
CONTINUOUS CULTURE OF A MUTANT ESCHERICHIA COLI
THROUGH PERIODIC OPERATION

By

Jonathan Ben Rodin

December, 1992



Chairman: S. Svoronos
Cochairman: G. Lyberatos
Major Department: Chemical Engineering


In some biological systems, the environmental conditions that are optimal for

microbial growth differ from the optimal conditions optimal for producing a desired

metabolite. If production of this metabolite were the process objective, one could

continuously operate a reactor system at the optimal production conditions. However, for a

given reactor volume, changing the conditions periodically could increase overall

production of the desired metabolite. This is possible since, due to higher growth rates

under the optimal growth conditions, one could operate the system at significantly higher

flowrates and, thus, obtain more product. A system involving E. coli mutant LCB898 was

used as a model system. Under anaerobic conditions this bacterium will produce large

amounts of d-lactic acid, whereas under aerobic conditions, this bacterium will grow faster.

The possibility of increasing total lactate production by cycling dissoved oxygen was

investigated.







Before any optimization work could be done, an adequate model for describing the

behavior of this system under both steady-state and transient conditions had to be

developed and tested. Such a model was developed using batch and continuous data and

then tested by comparison with shifts between conditions.

A method for determining the optimal waveform for the proposed cycling was

developed by extending previous work by Lyberatos and Svoronos. The method involved

Carleman linearization of the model equations around a steady state and subsequent

development of a term for the performance measure. The system studied oscillated

between purely aerobic and anaerobic metabolisms with no intermediate conditions. Thus,

an "imaginary" steady state of intermediate metabolism had to be used for linearization.

In numerical simulation of the determined optimal cycling, significant improvement

over strictly anaerobic operation was found. Experimental verification of this was

performed and improvement, though not as significant as theoretical predictions would

indicate, was found. Additionally it was found that the mutant was probably reverting to a

form where little d-lactic acid was produced. Cycling of dissolved oxygen apparently helps

delay this reversion.












CHAPTER 1
INTRODUCTION




Chemical processes are usually operated in one of three different manners, batch,

fed-batch, or continuous, each with its own advantages and disadvantages. Batch

operation is advantageous when small quantities of a product are desired. This type of

operation is perhaps the simplest since it only involves charging the reactor with the

appropriate reactants at the start of the process and removal of the products at its

completion. Unfortunately, this type of operation includes frequent downtimes where the

reactor is being either charged or purged, and production is thus temporarily stopped. Fed-

batch operation, where feed is added continuously but nothing is being withdrawn, is also

advantageous when small quantities of a product are desired. This type of operation is

optimal, for example, when it is desired to keep reactant concentration low. Fed-batch

operation has the same downtime problems that batch operation does. Continuous

operation is typically used when large quantities of a product are desired. It has advantages

over the batch-type processes in that no downtimes for charging or purging the reactor are

necessary. However, higher control and instrumentation costs are incurred. When large

quantities of product are desired, continuous operation is the usual method of choice.

When continuous processes are used, they are usually operated, after start-up

transients die out, in a steady-state manner. This involves keeping the process variables

constant. This type of operation is relatively easy to model and control. However, optimal

operation may involve taking the reacting volume through multiple steps of processing, for

example, operating the system in a neutral environment for an amount of time, and then in

an acidic environment. The simplest solution to this type of problem for a two-step process







would be to have two tanks in series, where one has a particular set of environmental

conditions maintained in it, and the other has a different set of environmental conditions

maintained. However, this increases the total volume and instrumentation (and thus cost)

of the system. An alternative approach is periodic operation. This involves only one tank,

but the environmental conditions are manipulated with respect to time. In general chemical

systems small time constants are the rule. As a result, conditions in the reactor would have

to be changed frequently to observe a significant improvement over steady-state operation.

This would lead to substantial control costs. However, in biological systems the time

constants are relatively large. Thus conditions may not have to be changed rapidly.

Therefore, biological systems may be suitable for periodic operation.

Periodic operation of continuous culture systems has been investigated by other

workers [e.g. 1-8] and it has been found to be useful in achieving desired process goals.

For example, it has been used for enhancing the production of yeast in continuous cultures

[7] and for solving the problem of plasmid stability in continuous recombinant cultures [8].

To establish appropriate operating conditions, kinetic models are required that adequately

describe the transient behavior of the culture being investigated. Such models have been

developed previously [e.g. 9-11].

Biological reactors can be operated to achieve one of the following process

objectives: utilization of nutrient (e.g., wastewater treatment), biomass production,

production of a particular metabolite, or conversion of one chemical substance to another

(bioconversion). The productivity of a continuous culture is determined by two variables,

number and state of cells. To achieve high productivity or substrate utilization, it is

desirable to have a large number of cells in the optimal state for the particular objective. A

system can be manipulated to change the state of the cells through control of environmental

conditions. However, in microbial growth processes, the optimal cell state for maximum

growth of the cells is generally not the same as the optimal state for production of a desired

metabolite. A solution to this problem is cycling environmental conditions (and thus cell







state) in the reactor. A large number of cells producing the desired metabolite could result.

Previous efforts in cycling of conditions will be described later in this work.

As an example, an E. coli mutant (LCB898), which produces d-lactic acid in large

amounts, was examined in this work. Under aerobic growth conditions this organism

produces high amounts of biomass but insignificant amotints of d-lactic acid. Under

anaerobic conditions it produces significant amounts of d-lactic acid and less biomass. If

the goal is to maximize lactate production per reactor volume per time, one could simply

operate the system under strictly anaerobic conditions. Alternatively, cycling dissolved

oxygen level (and sometimes residence time) may improve productivity over that achieved

by strict anaerobic operation. This improvement, in the case of constant residence time, is

achieved by operating the system at higher flowrates. The higher flowrates are allowed by

the aerobic (faster growth) portion of the cycle. Improvement by cycling is explained

further in the chapter on cycling.

An unexpected problem that occurred while investigating this process was that of

reversion of the genotype of E. coli LCB898 under anaerobic continuous operation. This

reversion led the organism into a state where little d-lactic acid was produced. The

proposed cycling of aeration also appears to be a new method for avoiding reversion or

mutation of organisms under continuous operation.

This dissertation reports theoretical and experimental investigations on aeration

cycling for E. coli LCB898. First, the general theoretical problem of optimizing a system

undergoing cycling of environmental conditions will be formulated. A description of the

particular microbial system under anaerobic and aerobic conditions, including modeling

results, will then be given. Transient behavior of the system will then be discussed,

followed by cycling predictions and experimental results. The question of reversion delay

will subsequently be addressed. Conclusions will then be presented.












CHAPTER 2
THEORETICAL METHODS



2.1 Overview


An important job of the engineer is to determine the best way to operate a system.

This usually will be the way that maximizes profitability within safety limits. It can involve

anything from deciding how best to place workers on an assembly line to finding the

optimal control setting for the temperature of a chemical reactor. The engineer will usually

attack this problem by devising a mathematical formulation of the system and then use one

or more optimization methods to design the best operation. The formulation will involve a

statement of some sort of performance criterion to be optimized, along with descriptions of

various equality and/or inequality constraints on the system. Examples of the performance

criterion include maximizing the number of automobiles produced or minimizing the

amount of byproduct from a reactor. Examples of constraints are the number of workers

available at any given time or the maximum operating temperature of a reactor. Two kinds

of constraints usually encountered are equality constraints, where some quantity of the

system must always be equal to some other quantity, and inequality constraints, where

some quantity of the system is bounded within certain valuess. The optimization method

used will be selected by the engineer based on several factors including the system under

study, available computing power, and others. Much work has been devoted to finding

such optimization methods [e.g. 12-14].

A class of optimization problems exists where the equality constraints are not

algebraic equations but differential equations. These are called optimal control problems

[13, page 364]. Frequently, for this type of system, the engineer will want to vary the







control settings with time to maximize the productivity. Many methods have been

developed for handling this kind of problem and one will be discussed later. Others can be

found elsewhere [e.g. 15,16].

Continuous-flow reactor systems are typically operated at optimal steady-state

conditions. However, several workers [17-23] have looked into operating systems at

nonconstant conditions by cycling control variables around the optimal steady-state control

settings and found improvement for certain performance criteria. Sometimes, though, there

may be a need to examine systems where there is no true optimal steady-state control

setting to cycle around [e.g. 8], and an imaginary intermediate state, only existing

mathematically, may be used to design the best cyclic operation. As an example of this

type of system one could imagine a reactor whose air pumps can be either on or off, with

the on setting resulting in complete aeration. A mathematical model of the system may

show that intermediate aeration is optimal for production. Then the engineer must decide

how to best operate the air pumps to get optimal production.

In this chapter, the problem of determining optimal steady-state operation will be

addressed first. The method used to determine optimal square-wave cycling will then be

discussed, and finally a short description of some of the model-fitting techniques will be

given.



2.2 Determination of Optimal Steady-state Operation

The typical system examined is one described by a set of differential equations:


dxi fi(xl,x2,...n;U,U2,...Ur) (1)
dt


i=l, 2,...n







where xi= state variable i

fi= function describing the rate of change of variable i

n= number of state variables

ui= control variable

r= number of control variables

t= time



Steady states for such a system are found by setting the right-hand side of equation

1 equal to a zero vector and solving (by algebraic manipulation or by use of numerical

methods [24]) for the values of xi. The optimum steady-state operating conditions (i. e. the

optimal control settings) for such a system can be found by treating the resulting steady-

state model equations as equality constraints, expressing necessary inequality constraints

and stating a performance measure. Since the equations become algebraic at steady state,

many methods of optimization can be used. If the steady-state model equations can be

solved to yield an explicit expression of the performance measure as a function solely of the

control variables (and if inequality constraints do not come into play), then classical

theories of determining the optimal control settings involving setting the partial derivatives

with respect to each control variable to zero and solving for the settings can be used [12].

Frequently an explicit expression cannot be found and other methods have to be used.

One can find the optimum steady-state settings by performing, when possible, a

simple numerical search for the control values giving the maximum (or minimum) value of

a stated performance measure, but it will frequently be necessary to use a different method

when the system increases in complexity. As stated before, many methods exist. A few of

them include quasi-Newton and conjugate gradient methods. These methods are described

elsewhere [12,13].








2.3 A New Method of Determining Optimal Periodic Pulsing

This new method involves Carleman linearization of a general nonlinear system,

expression of an explicit formula of the performance measure for pulsed cycling of the

linearized system, and then a search, using the explicit performance measure, for the

optimal settings for the system.



2.3.1 Carleman Linearization

Carleman linearization is a method of describing a general nonlinear system of first

order differential equations in a linearized form. It was first introduced in 1932 [25] and

has been applied to various problems in nonlinear system dynamics and control [17,26-

34]. A brief overview of this method is presented here. This description was composed

using earlier descriptions [17,35].

A restatement of equation 1 would be as follows:


x=f(x,u) ; f(0,0)= (2)



with xe Rn a vector of deviation state variables and ue Rm a vector of deviation control

variables from a nominal steady state. This system is defined to have n dimensions. It is

assumed that the functions f are differentiable up to order r at Q. Taylor expansion is then

performed on system (2) and the monomials of order up to r are introduced as new

variables. These monomials are then differentiated and terms of order up to r are retained.

The result is a linear system in the new variables and is called the nh order Carleman

linearization. For example, the system


xi = -xl +3x2 + x2
S 2(3)
x2 = -xI + 4u




8


where the manipulated variable u is treated as a parameter, would have the following

monomials approximated as variables wi.


XI W1
x2 W2
X2 W3 (4)
x2 W4
_x w5



The second order Carleman linearization for the system in equation 3 is


-1 3 0 0 1 0
0 0 -1 0 0 4u
w= 0 0 -2 6 0 w+ 0 (5)
4u 0 0 -1 3 0
0 8u 0 0 0 0



The computation of the necessary partial derivatives for expressions of higher order

Carleman linearization can be tedious, and, in the case of elaborate differential equations,

can lead to errors. A new program written in Mathematica [36] has been developed for

computing the necessary Carleman matrices and vectors and is presented in Appendix A,

along with an associated program used to compute necessary Kronecker products (the

Kronecker package offered by Mathematica is not usable for this type of problem). This

program, using the capabilities of Mathematica, does not require the user to provide

necessary derivatives; it computes the derivatives analytically. A description of the

algorithm used in the program was given by Lyberatos and Svoronos [17], but will also be

presented here.

Let 0 represent Kronecker multiplication. The Taylor series expansion about

x = 0 can be expressed as









x= Alkxk] + Alo (6)
k=l


where x[k] = x x 0 ... 0 x (k terms)

and 0 represents Kronecker multiplication

e.g. [27, page 11] if A is an mxn matrix and B is a pxq matrix then
allB a12B ... alnB

B= a21B a22B .." a2nB
AOB=

amlB am2B .-. amnB



The resulting matrix has the dimensions mp x nq.

The following system then comprises the rh order Carleman approximation to the

original system


A1,1 AI,2 A1,3 ... Ai,r A1,'
A2,0 A2.1 A2,2 --- A2,r-1 0
[2 0 A3,0 A3,1 A3,r-2 0
x =w= w+ (7)
S 0 0 A4,0o A4,r-3

xIr]
0 0 -'" Ar,o Ar 0



where Ai,j = In G Ai_1,j + A,j 0 Ini-1

Ig=gxg Identity Matrix



The above system will have a dimensionality of n+n2+n3+...+nr. Since the vector

w will contain monomial redundancies (e.g. xlx2 and x2xl), the system should be

simplified. This is done by eliminating rows that correspond to the same monomial and

adding the corresponding columns together. The Carleman system will then have a






r min(n,r)
dimensionality of n+j-lCj or equivalently Y rCj nCj where mCq is the number of
j=1 j=1

combinations of m objects taken q at a time mCq = [14,23].
q!(m q)!I



2.3.2 Performance Measure Calculation

In this section we will develop a method to determine an explicit performance

measure for a Carleman linearized system when we cycle our controls on a system between
two fixed settings. Consider the waveform in figure 1. In this figure uS and up are the two

control settings available, and ui represents a fixed intermediate (though not necessary

realistic) control setting. These settings can be vectors or scalars. 6 and p represent the

vector or scalar deviations of the two respective control settings from ui. T represents the

period of the cycle, and E represents the fraction of the period spent at one of the control

settings (ug).

This problem is very similar to one looked at by Lyberatos and Svoronos [17]. In

that work, they looked at square-wave cycling around an optimal steady state. The

deviations from that state were allowed to vary. The intermediate state mentioned above is

analogous to the optimal steady state that Lyberatos and Svoronos examined. Additionally,

in the present problem, the deviations from the intermediate are fixed. Thus, the previous

mathematical development that they performed can be used to a large extent here with only

minor changes. The main parts of their derivation will also be given here.

Again, the problem to be examined is (in deviation variables)


x = f(x,u); f(0,o) = 0 (8)


where f(x,u) is analytic in x at 0 for all admissible u vectors (or scalars). The u vectors

(scalars) are treated as staying constant for either part of the cycle. Taylor








KI- T -*|
K- eT 41)


Section of control waveform being analyzed


u"


SUi



up


Figure 1.






expansion around x-0 is performed on this system and the nh order Carleman linearization
is obtained


I S(5)w + z(5) t e [nT, (n + E)T)
S S(p)w + z(p) t [(n + E)T, (n + 1)T) (9)



The performance measure under cyclic conditions can be represented as the
following:


I T
S=- oP(x,u)dt (10)
T


where J=time-averaged performance measure

T=period of the cycle
P=instantaneous performance measure


It is assumed that P(x, u) is analytic in x at 0, in which case it can be linearized.. This is

done by Taylor series expansion around x-0 and is cast in terms of the Carleman

coordinates w. The performance measure then takes the representation


J= I [ro () + r'(u)w(t)]dt (11)

where represents the matrix transpose.

In the derivation that follows, the following identities, which apply to any invertible
n x n matrices and were proved by Lyberatos and Svoronos [17], will be needed


(In AB)-A = A(In BA)-1 (12)


(In AB)- A(In BA)-'B = In


(13)








(In A)(In BA)- (In B) = (In B)(In AB)-'(In A)(14)


Now, equation 9 can be integrated to give the following ultimate periodic solution:


Se(5)(t-nT) w -I es(8)(t-nT)]S(8)-^z(6)

for t E [nT,(n + e)T)


eS(p)(t-(n+E)T)WE


(15)


-[I es(P)(1-n+E)T) s(p y-1z(P)
for t E [(n + E)T, (n + 1)T)


D=eS(5)ET


(16)


R=eS(P)(1-E)T


wo and we can be expressed as:


S= -[I- RD]-[[R RD]S()- z(8) + [I R]S(p)'z(p)] (17)


w = -[I- DR]-[[I- D]S()-' z(8)+ [D DR]S(p) z(p)] (18)


From (15) it can be seen that


(19)


wE = Dwo [I- D]S()-'z(6)

and
wo = Rw -[I R]S(p)z()


(20)


Defining






Using equation 15, the cycle average performance measure given in equation 11 can
be expressed as


J(T,e,5)= ir'(8) [eS()ET -] S()-'wO +
T-
Ir'(p)[eS(p)(1-E)T_ I]S(p)- wE_
Ts
Tr'(8S)[()T + I es()E'T]S(5)-2 z() (21)
S'(p)[(p)(1- e)T + I eS()(-E)T]S(p)-2z(p) +

ro()E + ro(P)(1 -E)


Using that S(8)-1 commutes with D and S(p)-1 commutes with R along with
equations 19 and 20, equation 21 can be rewritten as:


J(T,E,8) = r()S()l wE 0w ETz(8)] +
T -
r'(p)S(p)- [w w -(1 E)Tz(p)] + (22)
T -
ro()E + ro(p)(l E)


Using equations 12-14, equation 17 and equation 18 the following is obtained:


wE w0 = (I- R)(I- DR)-(I D)

[S(p)-' (p)- S()- z(5)] (23)

Finally, from equations 22 and 23 follows







J(T, ,)= -[r'(p)S(p)-' -r'()S(8)-I

(I- R)(I-DR)-(I- D) [s( p) (p) S(8)' ()]
T
+-ro(8) r'()S(8 () (24)

+(1- E)[ro(p) r'(p)S(p)z()]

This is the form of the cyclic average performance measure that is the most useful.

2.4 Model Fitting with Nonlinear Least Squares Methods
This topic is not directly related to the above discussion which led to an expression
for the performance measure, but this fitting, used in some parts of this work, is an
optimization technique and belongs in a chapter discussing the theory behind the overall
project. The type of problem that is being considered here is one where data are being
collected from some experiment and parameters for a model have to be determined. One
can manipulate the data in some fashion, such as semilog or log-log plotting, to find
necessary parameters, but, if no apparent manipulation exists for the proposed model, then
some other method must be used. One can attack such a problem by varying the
parameters of a model and determining how good the fit is to the data with those guessed
parameters. One can use methods of optimization to find the best way to vary the
parameters. The method of choice was Levenberg-Marquardt optimization due to its wide
use in prepackaged computer programs such as MATLAB and Kaleidagraph. A
description of the problem being examined, along with a short description of Levenberg-
Marquardt optimization, is appropriate here. A further description of the method of
optimization used can be found in numerical analysis texts such as Numerical Recipes [24].
Let us consider an experiment where data are taken at several times during the run.
In figure 2, an example experiment where three different types (x 1, x2, and x3) of data are
collected is shown.







In an experiment like that shown in figure 2, the experimenter collected all three

different types of data at the same instant. This type of collection is preferable for later

computational purposes, but is not necessary with the method to be described. In other

words, if one of the data types is difficult to collect simultaneously with the other data

types, then the following analysis still applies, but the computational effort may be

increased.

Let it be assumed that a general model for a system like that shown in figure 2 is

given in equation 2, but the vector of ordinary differential equations is also a function of

parameters. In other words


x=((x, ) ; x(0)= x (25)



where p=the vector of parameters for the model



For such a system, the parameter vector p and, occasionally, the initial condition

vector Xo (or just some parts of either of these vectors) must be determined. For any set of

guessed parameters and/or the initial conditions, the model equations can be integrated,

either analytically or by numerical methods such as Runge-Kutta integration, to show the

predictions for that set of guesses. The model predictions for this problem would be

computed for each time instant that data is available for comparison.

A performance measure for "goodness" of model fit then can be described as


nsp ndt ,2
P(Xd,,q)= nsp ndt Xdi(j)-i(j)) (26)
j=1 i=1

where P=performance measure
(j)
Xd. =one type of data point at time instant j (d is a vector of all data)

x )=corresponding model prediction for xd, (x is a vector of predictions)


qi=weight of one type of measurement (Q is the vector of the weights)














0 0

0 xl
O x2
A x3


0 A
0= r -I


0 1 2 3 4
Time


5 6 7


Figure 2. An example experiment where, at any time point, three different types of
data (xl, x2, and x3) are collected.







nsp=number of sampling points

ndt=number of data types



Our goal is to minimize the stated performance measure. The weighting factors q

serve two purposes. First, they can be used if one type of measurement is more "trusted"

than another. For example, one may give more weight to a simple measurement of

temperature taken with a thermistor than to a viscosity measurement taken with a poor

viscometer. This use of the weighting factor is going to be subject to the good judgment of

the experimenter and should be handled with caution. The other, more important, purpose

of the weighting factor is as a normalization constant. Alternative forms of the performance

measure, such as summing the logarithms of the squared residuals instead of the actual
2
squared residuals, (Xd i )2, can also be useful for normalizing.

Once the performance measure is expressed in the form of equation 26, the

optimization method of Levenberg and Marquardt can be used. This method is an elegant

combination of steepest gradient and inverse Hessian methods of optimization The

general algorithm involves use of steepest descent methods far from the minimum, and

then, as the minimum becomes more closely approximated, a smooth transition to the

inverse Hessian methods [24, page 523-524]. A prepackaged program was used in this

work.












CHAPTER 3
EXPERIMENTAL METHODS




3.1 Organism Description

The organism used in this project was Escherichia coli strain LCB898. The original

culture was obtained from Dr. L.O. Ingram, Department of Microbiology and Cell Science,

University of Florida. The genotype is thrl leu6 ton A21 str lac Y1 sup E44pfll [37). The

important aspect of this organism is its mutation in the pfl gene causing lack of expression

of that gene. This is the gene for the production of pyruvate formate-lyase (pfl), an enzyme

which is primarily responsible for the conversion of pyruvate to acetyl-CoA and format

under anaerobic conditions.

The pertinent biochemical pathways for this organism are shown in figure 3 which

was prepared based on the diagram in Pascal [38]. It points out the following important

features: lack of pfl activity [37], anaerobic inhibition of the pyruvate dehydrogenase

(PDH) pathway [38,39], and anaerobic induction of the d-lactate dehydrogenase (LDH)

pathway. Further descriptions of this organism's metabolism will be given in later

chapters.

The culture was maintained on plates of rich broth agar with the following

composition in deionized water [40]: 10 g/1 tryptone 5 g/l sodium chloride lg/l yeast

extract and 15 g/1 agar. The agar was prepared by mixing the ingredients together in

water, heating the solution to near boiling while stirring, and then pouring approximately

12 ml aliquots of the molten agar into individual 16X125 mm tubes. These tubes were

capped and autoclaved at 1210C for 30 minutes. Subculturing was performed on a monthly







Glucose

2NAD 2ADP


2NADH 2ATP

2 Pyruvate


Anaerobic-
Anaerbi- LDH -2 D-Lactate



Aerobic
2NAD

PDH

I 2ADH


2 Acetyl-CoA

Aerobic
Aerobic


Pertinent Biochemical Pathways in E. coli LCB898.


2 Formate


4--


Figure 3.


2NADH 2NADI







basis with the freshly inoculated plates being incubated at 370C in a Fisher Model 255D

incubator for 24 hours then being stored in a refrigerator.


3.2 Analytical Methods



3.2.1. Biomass and Cell Number Determination

The amount of biomass in the system was determined by two different methods,

cell counting and spectrophotometric turbidity (actually absorbance) measurement. The cell

counting involved serial dilution of a culture aseptically withdrawn from the culture vessel.

The dilutions were made in 8.5 g/1 solution of sodium chloride in water[41, p. 434] and the

agar used had the same composition as the rich broth agar used in culture storage.

Preparations involved pouring several 9 ml aliquots of the saline solution into 16X125 mm

tubes and pouring several 99 ml aliquots into bottles. Additionally, some empty capped

tubes were also prepared. These were all autoclaved for approximately 30 minutes.

Sterile, disposable, individually-wrapped borosilicate glass pipets (Fisher pipets) were

used in this procedure. A sample procedure for serial dilution and colony counting of a

culture is given in the two following paragraphs. The serial dilution procedure should be

performed under a laminar flow hood.

This sample procedure is designed for experiments where cell counts are expected

to be between thirty million and three hundred million cells/ml. Since it is desired to dilute

to 30-300 cells/ml, it is necessary to dilute samples 100000X, one million X, and ten

million X for later counting. Obviously this method can be modified for other expected cell

counts. Approximately 15 minutes before a sample is to be taken, three agar tubes are

placed in a boiling water bath in order to melt the tube contents. Immediately before the

measurement, the tubes are withdrawn from the water bath and stored in the laminar flow

hood, along with a propipettor, a pack of sterile gloves, three labeled sterile disposable

Fisher petri dishes, an appropriate number of pipets, dilution bottles, and tubes. A fresh







pipet should be used after each dilution step. The sample is removed from the culture (if

from a chemostat, the effluent sidearm tube (to be described later) is flamed and

approximately 10 ml are allowed to flow out into a presterilized empty tube; if from a

flask, the cap is removed under the laminar flow hood and a sample is aseptically pipetted

out) and one ml is pipetted from the tube or culture flask into a ninety nine ml saline bottle.

This bottle is then well shaken. At this point the original culture has been diluted 100 X. A

one ml sample is then aseptically transferred from the first dilution bottle to another ninety

nine ml saline bottle. Again, this bottle is well shaken. The culture sample has now been

diluted 10000 X. A one ml sample is then aseptically transferred to a 9 ml sample tube and

the tube is well shaken. 2 ml of the 100000X diluted sample is then taken. One ml is put

into the next 9 ml sterile tube and the other milliliter is pipetted onto the appropriately

labeled petri dish. One tubeful of agar, when it becomes lukewarm, is then poured into the

petri dish and the dish is subsequently mildly swirled in order to provide a more even

distribution of colony forming units. The new 1 million X dilution tube (now a 10 ml tube

with the one ml of sample added) is handled in the same way as the previous 100000X

tube. Lastly, the ten million X dilution tube has one ml transferred to the appropriate petri

dish. The agar in the petri dishes is allowed to solidify. The dishes are then inverted and

incubated at 370C for 24 hours.

Upon sufficient incubation, the petri dishes are taken out of the incubator and are

individually placed on a Quebec darkfield colony counter. A hand-held colony counter is

used to ensure accounting. A marking pen is used in order to make dots under each colony

appearing on the plate so that no colony gets counted twice. When counting, care must be

taken to count colonies on the edge of the petri dish and to check if some of the colonies are

growing directly underneath another colony. The results of each plate count are recorded,

and the cell counting procedure is finished.

Biomass concentration was measured spectrophotometrically using a Milton Roy

Spectronic 20D spectrophotometer. The procedure for this measurement is relatively







simple. The spectrophotometer is first set for operation at 550 nm. Deionized water is then

added to a clean cuvet, the cuvet is put into the sample chamber, and zeroing is performed.

The cuvet is then removed and the water is poured and then shaken out of the cuvet. Some

sample is then added and the cuvet is swirled. This sample is used to eliminate the effects

of residual water. At this point, the first amount of sample is poured out and at least three

ml of fresh sample are added. The cuvet is then placed into the sample chamber again and

an absorbance reading is then taken and recorded. If the absorbance reading is above 0.4,

appropriate dilution is performed. For example, in more dense samples, three ml of

deionized water would be added to one ml of sample for a 1/4 dilution. During the aerobic

operation experiments, 1/16 dilutions were necessary towards the end of the batch runs and

during continuous steady states.

In order to correlate the spectrophotometric absorbance reading to an actual dry

mass concentration, a calibration curve has to be obtained. A large (approximately 200 ml)

sample is taken from the reactor (at the end of a batch run). Eighty ml each of 75%, 25%,

and 12.5% dilutions are made in three separate beakers. The spectrophotometric

absorbances of each of these samples is then measured. In more turbid samples, when the

absorbance readings are significantly above 0.4, the diluted samples are used for

computation of the absorbance. Fifty ml of each of these dilutions are pipetted into

appropriately labeled centrifuge tubes. The tubes are then placed into a Precision Universal

Centrifuge set at 2000 rpm for 60 minutes. The supernatant is then decanted, the remaining

contents are washed with approximately 5 ml of deionized water, and the tubes are

centrifuged for 20 more minutes. The supernatant was pipetted off and the remaining

contents are then emptied into preweighed and labeled petri dishes for drying. These

dishes are dried at 1050C for 48 hours and the contents are weighed. In this manner, a

calibration curve was prepared. The calibration curve is shown in figure 4.






1.000
0.900
o0.800-
0.700-
S0.600
Z 0.500
U 0.400-
I 0.300
0.200-
0.100
0.000 .
0.000 0.300 0.600 0.900 1.200 1.500
Absorbance @ 550 nm

Figure 4. Dry Mass Calibration Curve. The linear fit shown is
Dry Mass Concentration (g/1)=0.565*Absorbance
with a squared regression coefficient of 0.998







3.2.2 Glucose Analysis

The device used for glucose concentration determination was an Analytical

Research Model 110 Glucose Monitor. The analyzer's main component is the

electrochemical sensor on which an immobilized glucose oxidase membrane is mounted.

This enzyme catalyzes the reaction between glucose and oxygen to produce hydrogen

peroxide. Hydrogen peroxide is detected by the sensor and an electrical signal proportional

to glucose concentration is produced [42, p. 52]. The analyzer's pumps will take a small

(approximately 1.5 ml) aliquot of the sample through a port and pass most of it by the

membrane, leaving only a small plug of fluid for the actual analysis. This analysis is

performed in approximately three minutes, at which time almost all the glucose will have

reacted. Usually three aliquots are measured and the results are averaged. The following

paragraph gives a brief description of the actual procedure.

In the description that follows, one cycle is defined as the time between glucose

analyzer samplings. The glucose analyzer is operated as follows. Glucose calibration

solutions of appropriate concentrations (operator's judgment) are prepared and allowed to

dissolve for at least 2 hours. Fifty percent dilutions of calibrations should also be

prepared. For example, if calibration is to be made with a 2 g/1 glucose solution, a 1 g/

glucose solution should also be prepared. Additionally, the glucose analyzer is switched

from idle to blank for 2 hours before calibrations are to be done. After these two hours the

calibration tube is placed in 100 ml of fresh water and the switch is set to "Cal". For the

next 15 minutes, water is allowed to pump through the system. After this 15 minutes, the

glucose analyzer is zeroed by setting the zero dial so that the peak readout during the

portion of the cycle between the ready light indicator coming on and the following sampling

reads ".000" g/1. One more cycle is observed to check for appropriate zeroing. At this

point, calibration of the glucose analyzer is performed. The sample tube is placed in the

appropriate calibration solution. The switch is then set to "Sample". Three cycles are

allowed to follow and the calibration is completed by setting the cal dial so that the peak






value of glucose read during the portion of the cycle between the ready light turning on and

the following sampling reads out the calibration value of the glucose solution. This

calibration is then checked with a 50% solution of the calibrator solution by putting the

sample tube into the appropriate calibrator solution. Three samples are taken and

calibration is checked by reading the appropriate peak value. If calibration is appropriate,

samples can than be measured for glucose content. Four samplings are taken, with the last

three peaks being recorded as data. Recalibration is performed once every hour that

sampling is done. If the calibration stays relatively accurate, a calibration check with 50%

calibration solution need not be completed. If glucose analysis results are below 1/2 of the

top calibration value, recalibration is performed at one half the calibration value.



3.2.3 d-Lactate Measurement

The measurement of d-lactate was based on a modification of Sigma procedure 816-

UV, which was designed for measurement of I-lactate. The major modification to this

method was the use of d-lactic dehydrogenase (Sigma L-2395) instead of 1-lactic

dehydrogenase, provided with the original Sigma kit. The principle of this test is explained

below.

In the metabolism of E. coli and most other chemoheterotrophic bacteria, pyruvate

is converted by lactate dehydrogenase into lactate. This, however, is a reversible reaction.

In other words, the same enzyme can be used to convert lactate into pyruvate. The

conversion of d-lactate to pyruvate will be accompanied by a reduction of one molecule of

NAD+ into NADH. NADH shows strong spectrophotometric absorbance at 340 nm,

whereas NAD+ does not absorb at this wavelength. Thus, using the indirect method of

spectrophotometrically measuring NADH in a mixture, the amount of d-lactate can be

determined. The problem of backconversion of the pyruvate into d-lactate is handled by

adding hydrazine to the mixture. Hydrazine reacts with the pyruvate and forms a complex

that d-lactic dehydrogenase cannot convert back into d-lactate.







The experimental method involved preparation of a calibration curve of d-lactate

concentration against spectrophotometric absorbance at 340 nm. The spectrophotometer

used was a Milton Roy Spectronic 20D. A solution of approximately 200 mg/l d-lactic acid

(Sigma L0625) or the Lithium salt (Sigma LI000) in deionized water (approximately in this

case means that the experimenter knows the exact concentration within experimental

accuracy, but it is not necessarily exactly 200 mg/1 in concentration) was prepared when

calibration was performed. When the lithium salt was used, adjustments were made for the

weight of a lithium atom as opposed to a hydrogen atom in the free acid form. Dilutions of

approximately 87.5%, 75%, 62.5%, 50%, 37.5% and 25% were made of this solution.

Additionally, a solution of 2.50 g NAD+ (Sigma N7004) and 500 ml Glycine buffer

(Sigma 826-3) added to 1 liter of water was prepared. The mixture will be referred to as

"NAD solution" from this point. Since the NAD solution must be made immediately before

the lactate measurement, the actual amount prepared would depend on the number of lactate

samples to be analyzed. One would prepare 3 ml of the NAD solution (2 ml of water, 1 ml

of glycine, and 5 mg of NAD+) per sample, plus at least one extra 3 ml solution for

preparation of a blank. 2.8 ml aliquots of NAD solution were pipetted into the appropriate

number of labeled test cuvets plus one blank cuvet. Each labeled test cuvet had 0.2 ml of

the corresponding full strength or diluted calibration solution added and mixed .

Additionally, the blank cuvet had 0.2 ml of deionized water added and mixed. The 340 nm

spectrophotometric absorbances of each test cuvet against the blank were then measured

This value, which will be referred to as the zero absorbance, was used to compensate for

cuvet-to-cuvet variability.

Sixty units of d-lactic dehydrogenase were then added to each of the test cuvets and

the blank cuvet. The cuvets were then incubated at 370C for 30 minutes. The new

absorbances against the blank were then measured, and the differences between the new

absorbances and the corresponding zero absorbances were then calculated and plotted

against the corresponding test lactate concentrations. The curve obtained is shown in figure






5, along with the results of a least squares fit to the data. Although the substance measured

is referred to as d-lactate throughout this work, the term d-lactate is actually slightly

inaccurate. D-lactic acid is the substance actually being measured. Frequently these The

relatively high correlation coefficient indicates that the assumption of a linear relationship in

this range of test lactate concentrations is satisfactory.

The procedure for measuring unknown lactate concentrations is essentially the same

as that for measuring the net absorbances of the calibration solutions. The key differences

will only be described. First, the unknowns must be diluted however many times to where

their lactate concentration is in the range between 50 and 200 mg/1. The amount of dilution

is usually based on previous methods, but for the first experiments this had to determined

by trial and error. Second, this dilution is also useful in diluting the effects of any residual

biomass or other substances in the filtered samples on the spectrophotometric readings.

Any remaining residuals would be taken into account by the zero absorbance measurement.

These residuals will not affect the net absorbance measurements as the enzyme used is

specific for d-lactate. Last, least, and most obvious, the calibration curve is used to

determine unknown concentrations, as opposed to the preparation of the curve when

calibrating.



3.2.4. Other Analyses

Other measurement methods were used in this research. These will only be

mentioned briefly as they were only seldomly used. These include amino acid analysis and

ethanol analysis. The amino acid measurements were made by an outside laboratory

(Interdisciplinary Center for Biotechnology Research, University of Florida, Gainesville,

Florida) on an amino acid analyzer. These amino acid measurements were used solely to

determine whether or not leucine or threonine nutritional limitations were encountered.

Ethanol concentrations, when measured, were determined using Sigma kit 332-UV.








220.000
^200.000
180.000
E I
,. 160.000
0
o
*= 140.000-
-
1 120.000
100.000-
U 80.000-
60.000-
4
] 40.000-
*6 0nnnn -


0.000
0.0


0
0

0


0-
00


" I '... I" . . I '
0.200 0.400 0.600
Test Absorbance @340 nm


d-Lactate Calibration Curve. The linear fit shown is
d-Lactate Concentration (mg/l)=260.144*Absorbance
with a squared regression coefficient of 0.998


Figure 5.


0.800







3.3 Feed Medium Composition

Many factors had to be considered in the design of the feed composition and

preparation. The first consideration was whether to use a complex medium such as Luria

broth or a minimal medium. A glucose minimal medium was chosen as the probability of

interference of feed components with measurements is lower. Once a minimal medium was

chosen, other considerations had to be taken into account. These included pH, buffering,

nitrogen requirements, trace minerals and metals, genetic deficiencies, substrate amount,

and interactions between these components during heat sterilization. These interactions

must be considered when deciding which component solutions to autoclave in the same

flask. Ideally, each component solution should be autoclaved separately. However, in

order to maintain sterility during the mixing process, there should be as few separate flasks

as possible. Thus, given the considerations described below, a design was chosen

inbetween these two extremes. A general consideration given was the separation of

inorganic from organic components in order to avoid production of toxic byproducts during

autoclaving. Finally, all the ingredients were prepared in deionized water.

The medium should be buffered with target pH 7. Buffering lowers the amount of

base needed to maintain pH during the experiments. M9 medium [41, p. 431 and 43, p.

A.3] satisfies this requirement. Additionally the nitrogen and some of the trace mineral

requirements are satisfied by the use of M9 medium. M9 medium includes sodium

phosphate dibasic, potassium phosphate monobasic, sodium chloride, ammonium chloride,

calcium chloride and magnesium sulfate. The exact amounts used will be given later.

Miller [41, p. 431] suggests separate autoclaving of calcium chloride and of magnesium

sulfate from the rest of the salts.

Other trace metals were added to the medium as suggested elsewhere[40,44,45].

These included selenium oxide, hydrated ferrous sulfate, hydrated ammonium molybdate,

and hydrated manganese sulfate The amounts used will be given later. It was suggested

these should be sterilized by filtration, but sterility was a major concern in this work, so







these compounds were autoclaved together in solution. Bridson and Brecker [46]

suggested autoclaving metals separately from phosphates in order to avoid precipitation.

Genetic deficiencies of E. coli LCB898 had to be accounted for in the medium

formulation. The genotype mentioned in the organism description indicates requirements

for threonine, leucine, and thiamine. Additionally, as the pyruvate-formate lyase gene is

mutated, acetate may be required to satisfy some biosynthetic requirements of the cell.

Several batch experiments were performed under both aerobic and anaerobic conditions in

order to determine the amounts of these chemicals required to insure glucose limitation of

growth. Amino acid analysis was performed on a sample of the batch at the point where

growth was no longer seen to see if any residual amino acids were left in solution. The

final amounts chosen were those that allowed glucose limited growth up to at least a

concentration of 4 g/l glucose in the medium. This was indicated by the cessation of

growth when glucose became exhausted. Miller [41, p. 431] suggests separate

autoclaving of the amino acids and vitamin from the M9 salts, and Bridson and Brecker[46]

suggest separate autoclaving of the amino acids from the carbohydrates to avoid Maillard

reactions. Thus the amino acids and vitamin solutions were autoclaved together separately

from all other components, and, to avoid any other possible feed reactions, the acetate was

also separately autoclaved.

The final feed component to be discussed is the growth limiting substrate of

glucose. Glucose (Sigma) was used as the limiting substrate because of its relatively

straightforward measurement on the glucose analyzer previously mentioned. A main feed

concentration of 4 g/1 was decided on for several reasons. A feed glucose concentration too

low would make batch growth measurements difficult, as the analyzer available is

somewhat inaccurate at measuring glucose concentrations below 100 mg/1, and d-lactate

measurements below 25 mg/l are also of questionable accuracy. Another reason to avoid

low glucose concentrations is the desirability of visible turbidity in the reactor system. For

example, in a batch growth experiment, the onset of visible turbidity serves as a marker for






the beginning of more frequent measurements. This onset is still well before glucose

exhaustion and indicates the approximate point where glucose, d-lactate, and biomass

concentrations begin to measurably deviate from the starting values. On the other hand

high glucose concentrations also are not beneficial. First of all, autoclaving of high glucose

concentration solutions will lead to increased carmelization of the feed glucose. Also,

higher glucose concentrations represent higher biomass concentration. During aerobic

growth aeration may become insufficient at higher biomass concentrations. Buffering,

amino acid addition, and other feed components would have to be increased in

concentration. Finally, thick growth may cause other experimental problems such as

increased effluent tube wall growth, high amounts of base addition to maintain pH, and

more difficult sampling and cleanup. Thus, an intermediate glucose concentration of 4 g/

was picked. However, any glucose concentration from approximately 3 g/l to 10 g/1 would

also have satisfied the above criteria. Glucose solutions were autoclaved separately from

all other components to avoid all possible cross reactions (e.g. Maillard reactions). As

glucose concentration was the one feed component measured during all experiments, this

was the component that was most important to keep "pure".

The final feed composition used is given in Table 1.



3.4 Feed Preparation

The feed medium was prepared in three different configurations as follows: shake

flask, reactor batch, and continuous feed. Each will be discussed separately, and the flask

grouping listings in table 1 will be used. The term "2.7X concentrated" will hereafter be

used to refer to concentration higher than that listed in table 1 by a factor of 2.7. For

example, a 2.7X concentrated solution of flasking group 1 would be a solution of 10.8 g/1

glucose in water. One stock solution was used in all three configurations, a 100X







Table 1. Feed medium recipe with flasking divisions

Feed Ingredient Amount added per liter of

deionized water

Glucose 4 g

KH2PO4 3 g

NaCI 0.5 g

NH4C1 1 g

Na2HPO4 6 g

Threonine 0.5 g

Leucine 0.5 g

Thiamine 5 mg

CH3COONa-3H20 1.66 g

MgSO4 0.24 g

CaC12 11.96 mg

SeO2 1.1 mg

FeSO4*7H20 27.8 mg

(NH4)6Mo7*4H20 1.765 mg

MnSO4-H20 1.69 mg


Flasking Division


1

2

2

2

2

3

3

3

4

5

6

7

7

7

7


Note: For example flask 2 would include Na2HPO4, KH2PO4, NaCI, and NH4C1






concentrated solution of flasking group 7. One liter of this was prepared when necessary.

Ten milliliters of this solution were used for every liter of feed solution.

When shake flasks were prepared, stock solutions were used. The stock solutions

prepared were 100 ml bottles of 25X concentrated solutions of flasking groups 2-6.

Glucose solution was freshly prepared for each shake flask. In order to illustrate the

preparation of a shake flask, the preparation of the usual 250 ml amount of flask medium

will be given. 10 ml of each of the group 2-6 bottles are pipetted into separate tubes and

the tubes are capped. 2.5 ml of the group 7 flask are also pipetted into a tube. Finally,

197.5 ml of deionized water is poured into a shake flask and 1 g of glucose is added. The

flask is then capped with a paper towel and foil and tied with a string. All of these are then

autoclaved for 25 minutes. Finally, upon cooling of the ingredients, the ingredients are

poured together into the shake flask under a laminar flow hood.

The medium preparation for startup batches is described next. Although the startup

medium descriptions in this section do not appear to "add up" to the concentrations given in

Table 1, when the actual startup procedures are considered later, the final startup

composition does "add up" to the correct medium. Two liters of 1.4 X concentrated

flasking group 2 salts are prepared and poured into the chemostat and then autoclaved

within the reactor. A 56 ml aliquot of the flasking group 7 stock solution is pipetted into a

glass flask. Four additional glass flasks with 150 ml each of deionized water are set aside.

To each of those flasks 5.6 times the mass listed in table 1 for one flasking group is added,

for groups 3,4,5, and 6. These flasks are capped with paper towel and foil and then tied

with string. The glucose solution is prepared by adding 22.4 g of glucose to 944 ml of

water in the main feed flask, which is shown in figure 6. The main feed flask preparation

is as follows. A new Supor filter is placed in the Fisher 47 mm filter holder. The

connections are as follows: the flask outlet tube leads to its own filter holder, which leads

to the needle for puncturing into reactor. The needle is wrapped in foil wrap. Additionally,

there is a tube for a sterile nitrogen inlet into the flask for replacing the emptied fluid. The

















































Magnetic Stirrer


Main Batch Feed Flask Diagram


Copper
wires

Metal
clamp

Copper wire assembly on
stopper.


Figure 6.







nitrogen introduced passes through a Bacti-Vent air filter. The rods going through the

rubber stopper are made of glass. The main feed flask and the glass flasks are autoclaved

for 30 minutes. After these cool down, all of the flasks are placed under a laminar flow

hood. The metal clamp on the main feed flask is loosened and the contents of the glass

flasks are added. The metal clamp is then retightened and the feed is ready to be added to

the reactor, as will be described later in the description of reactor startup.

Finally, the continuous feed medium preparation will be described. Sixteen liters of

continuous feed medium were made in any single batch, thus sixteen times the amount of

each flasking group to be added per liter is added to separate amounts of water. The

procedure for a single batch is described. In three large flasks, 3.5 liter aliquots of

deionized water are added, along with separate additions of 16X of the Table 1 masses for

flasking groups 1, 2, and 3. For example, in the glucose flask, 64 g of glucose are added

to 3.5 liters of water. The group 7 flask has 160 ml of the appropriate stock solution added

and then filled with deionized water to 3 liters. The acetate flask has 26.56 g of

NaAcetate*3H20 added to 500 ml of water. The calcium chloride flask has 192 mg of

CaCI2 added to 250 ml of water and the last flask has 3.94.3 g of MgSO4*7H20 added to

250 ml of water. Finally, the feed carboy, diagrammed in figure 7 is prepared as follows.

Two fresh .22 [t Supor filters are placed in the appropriate filter holders, along with a fresh

Bacti-Vent air filter for sterile nitrogen introduction. The carboy is then autoclaved empty

and uncapped. One and a half liters of deionized water are then poured into the carboy and

it is capped. The tubes are then clamped, the needle and air filter are covered with foil

wrap, and the whole apparatus (except the N2 bag and the magnetic stirrer) is autoclaved

for 30 minutes. All of the flasks are covered with paper towel and foil wrap and then tied

with string. They are also autoclaved for 30 minutes. The cap of the carboy is removed

and the contents of the other flasks are added under a laminar flow hood.








0.22 p filters
/.,,


Air
Filter


Needle
to
reactor


Quick
Disconnect


Magnetic
Stirrer


Figure 7. Continuous Feed Carboy Diagram


Magnetic
SStirbar







3.5 Experimental Operation



3.5.1 Operational Conditions

The temperature that the experiments were operated at was 370C. This temperature was

chosen as it is the normal optimal temperature for growth of E. coli. The other major

environmental variable that was held constant was pH. It was suggested [40] that

somewhat acidic pH's gave higher batch yields of d-lactic acid. As pH 7 is the optimal pH

for E.coli growth, the possibility of cycling pH in addition to aeration was examined.

Preliminary experiments were performed early in this investigation to examine the effects of

pH on the anaerobic batch yield of d-lactate on glucose. The results will be described

briefly. These experiments were performed before any aerobic experiments were and,

thus, a slightly different medium formulation was used as shown in table 2. It should be

emphasized that the pH effects experiment was the only one described in this work using

the table 2 recipe. All other experiments used the main recipe given in Table 1.

The use of shake flasks will be described later. A starter flask with an initial medium

composition described in table 2 in 500 ml of deionized water was inoculated, placed in a

37C AO constant temperature shaker bath and kept there until the flask contents appeared

turbid. Three test flasks were prepared during the growth phase of the starter flask cultures

with 150 ml of medium in each. The medium ingredients of these three flasks were the

same as those in Table 2, but the concentrations of each were set so that, upon dilution with

100 ml of liquid, they were the same as those given in Table 2, except that the amounts of

potassium phosphate mono- and dibasic were varied to give the desired pH values.






Table 2. First Medium Recipe

Glucose 3.0 g/l
K2HPO4 4.9 g/

KH2PO4 3.0 g/1

NaAcetate 0.2 g/1
(NH4)2SO4 0.1 g/1
CaC12 0.2 g/1

MgSO4.7H20 0.1 g/1
FeS04-7H20 0.05 g/1

L-Threonine 0.05 g/1

L-Leucine 0.05 g/l

Thiamine 0.005 g/I\







When the starter flask reached the appropriate turbidity, the three test flasks were

each inoculated with 100 ml of the starter culture. Spectrophotometric absorbances, pH

values and lactic acid concentrations were then measured at half-hour intervals. The results

of this experiment are given in table 3. The large increase in lactate concentration in the pH

7 flask over a 5 hour period, along with the large increase in absorbance over a 26 hour

period, would indicate that the operating pH should be kept at a value around seven. An

additional benefit in choosing constant pH operation is that shifts in pH are difficult.

Buffering is generally desirable in microbial systems, but buffering would require higher

base or acid amounts to be added to cause a shift. High addition of these solutions have a

diluting effect on the culture and thus will interfere with measurements. This problem,

though, may possibly be overcome by use of gases such as C02 and N2 instead of acid or

base additions to manipulate pH.



3.5.2 Shake Flask Experimental Procedure

The shake flasks were prepared as described in the section on feed preparation.

Further procedural details will be given here. In a starter culture, after the contents were

poured together under a laminar flow hood, a small inoculum was taken off of the culture

storage agar dish with a sterile loop and then transferred into the combined medium flask.

In other types of shake flasks, liquid inocula may be used instead of the agar culture.

When liquid inocula were used, only small amounts (-1 ml) were usually added. After

inoculation, the flask was recapped with the paper towel/foil wrap cap and tied. The flask

was then placed in an AO shaker bath set at 370C. If the shake flask culture was to be used

for reactor inoculation, it was usually left in the bath for approximately 12 hours. For yield

experiments, the flasks would be left in for longer periods. The usual shake flask volume

used was 250 ml.




41


Table 3. Effects of pH on Lactate Production


Initial pH 7 6.4 5.85

Initial Lactate Conc. 105 110 110
(mg/1)
Initial Absorbance 0.186 0.186 0.186
(550 nm)

Lactate Conc. after 5 206 151 119
hours (mg/1)

550 nm Absorbance 0.200 0.205 0.175
after 5 hours

pH after 5 hours 6.98 6.4 5.87


Lactate Conc. after 2900 1340 292
26 hours (mg/1)

550 nm Absorbance 0.49 0.37 0.22
after 26 hours

pH after 26 hours 5.24 5.26 5.38







3.5.3 Reactor Experimental Procedure



3.5.3.1 System Description

The reactor experimental procedures will now be described. The reactor, a

Bioengineering KLF 2000, was used for all of the batch and continuous experiments. A

diagram of the reactor system is shown in figure 8. In this figure the long dashed lines

represent measurements for the chemostat control unit, and the dotted lines represent

control outputs. Except where mentioned later, the reactor volume was always maintained

at 2 liters. If the system is in continuous mode, a load cell is used to determine the system

weight. A peristaltic pump maintains a constant effluent flowrate. The pumping rate is

calibrated by collection of the fluid in a graduated cylinder. When a small drop in weight is

detected the control unit activates the influent pump until the reactor is back up to its

operating weight. Only small deviations in the level were allowed. The pH value of the

system was measured with an Ingold Ag/AgCl pH electrode and controlled by a

Bioengineering M7832N pH controller. This pH control maintained constant pH by

controlling pumps for previously autoclaved 1 M HCI and NaOH solutions prepared

separately. The constant 370C temperature was maintained by a ptl00 temperature sensor,

a Bioengineering K54450 controller, and an 800 watt heater. Agitation for the reactor was

set at 700 rpm, and a baffle cage within the reactor helped insure good mixing. Dissolved

oxygen was monitored with a Cole-Parmer polarographic electrode and a Cole-Parmer

Model 5513 dissolved oxygen meter. When anaerobic conditions were necessary, filtered

(as shown in figure 8) Alphagaz oxygen-free nitrogen was bled over the top of the culture

at the rate of approximately 30 ml/min. When aeration was necessary, an Air Cadet pump

was used. It pumped air from underneath a UV hood, through a filter apparatus identical to

the same as that used for nitrogen, and through a sparging tube with the outlet bent

underneath the bottom rotor blade within the reactor. Two and a half vvm was the air

flowrate obtainable with this pump into the two liter reactor. When periodic switching of





























Freezer
Section


Refrigerator


Feed
Carboy


Peristaltic
Pump

Chemostat


I:f
I:

I:


43



Flowrate pH
controller controller
I -
Agitation Temperatur
control controller

Chemostat
Control
UUnit







IAcid






Base

Bioengineering
Filter
Air vent filter
type.3 pm


Air Cadet
"-I I Air Pump

JCell
fluent UV Hood
dearm Ar
Inlet


--


Experimental Reactor Setup


Figure 8.






the aeration was performed, a slow flow of nitrogen was continuously maintained over the

top of the culture so as to maintain a positive pressure (which helps prevent outside

contamination). Switching of conditions during the aeration cycling was performed by

simply plugging the Air Cadet pump into an X-10 wall module and setting switch times on

an X-10 computer interface (X-10 (USA) Inc. 185A LeGrand Ave. Northvale, NJ). The

gas was released through a Bioengineering gas outlet apparatus. This had a cooling jacket,

which was maintained at about 100C in order to minimize evaporation of culture volume.

This cooling was accomplished by continuous pumping of cooled Prestone antifreeze

through a Hotpoint refrigerator and freezer using a Manostat pump set on its lowest

pumping speed. The end of the effluent gas hose was placed in a dilute Betadine solution

in order to help prevent contamination.



3.5.3.2 System Startup and Operation

The following startup procedure was used for all batch and continuous

experiments. It should be emphasized that all continuous runs were started as batch runs.

The only variations were whether anaerobic or aerobic procedures were going to be used.

The cycling runs were all started up under aerobic conditions. Prior to startup, the effluent

tube, acid, base, and inlet gas filters were autoclaved. A batch feed was also prepared.

Additionally, a starter shake flask culture was prepared and inoculated 12 hours prior to

reactor inoculation. The chemostat was filled with the appropriate salts as described in the

batch feed description to the 2000 ml level. The topcap and bottom attachment rings were

secured in place. The pH electrode was precalibrated to pH 7 and 4 and inserted into the

topcap. The reactor was now ready for autoclaving.

The autoclaving of the reactor was done in situ. Here, the pH electrode was

pressurized to 30 psi by connecting it to an air cylinder, the stirrer was set to 800 rpm, the

gas outlet was opened, and temperature set-point of the chemostat was changed to 121 0C.

When the temperature reached 990C, the gas outlet was closed. The temperature was






allowed to reach 1210C, and was kept there for 30 minutes. After 121C was held for 30

minutes, the temperature set point was changed to 1040C. When reactor temperature

reached 1040C, the first thing that was done was to turn on the pump for the gas outlet

apparatus reflux coolant. The reactor pressure was raised by using either air from the Air

Cadet pump or nitrogen from a cylinder. The gas outlet was also opened immediately. The

reactor temperature set-point was then changed to 370C in preparation for the actual

experiment.

After the reactor cooled down to 370C, the effluent, acid, and base tubes were each

placed through peristaltic pumps and then appropriate connections were made to the reactor

using aseptic technique. The effluent tube was immediately clamped to avoid loss of

reactor liquid. The main feed flask (described in the batch feed preparation) was then

connected to the reactor. Eight hundred milliliters were then pumped into the reactor for a

total volume of 2.8 liters. The reactor conditions were then set to pH 7, 370C, and 700 rpm

agitation. Finally, 20 ml of inoculum were taken by a sterile syringe from the starter flask

under a laminar flow hood and then injected through one of the reactor seals. The reactor at

this point was prepared and inoculated. A batch run was thus begun.

Immediately after inoculation a sample was aseptically taken through the effluent

sidearm aseptically into a previously autoclaved and capped tube. Serial dilution was

immediately performed and the remaining sample had its absorbance measured and then

was centrifuged for ten minutes in a Fisher Centrific 228 centrifuge. After centrifugation

the sample was filtered using a syringe and MSI Magna Nylon 66.22 p filters and then

heat shocked for five minutes in boiling water. Finally, the sample was allowed to cool to

room temperature and then stored in a freezer for later glucose and lactate analysis.

Frequent measurements were taken during batch runs. The acid and base levels

were monitored and recorded. Upon the absorbance values reaching above 0.4,

appropriate dilution with deionized water was performed in order to obtain a measurable

absorbance. The batch measurements were performed until absorbance stopped increasing.






Up to this point, the batch and continuous experiments were performed in a synonymous

manner (except for the obvious need for preparation of a continuous feed carboy during the

batch start of a continuous run). If a run was intended to be strictly batch, measurements

were continued for several hours into the stationary phase. If a run was intended to be a

continuous run, the system was switched into a continuous mode before the end of

exponential phase. The point of switching was usually about one hour before the expected

end of exponential phase. This end point was estimated using absorbance measurements

and comparing them with previous batch results.

The following procedure was used to switch the reactor from the initial batch mode

to a continuous mode. The feed carboy was connected to the reactor after its tube was led

through the influent peristaltic pump. Subsequently the reactor volume was lowered to 2

liters by draining through the effluent tube. The weight set point was entered into the

chemostat control unit, the effluent flowrate was set, and the reactor was then in a

continuous mode.

Measurements were taken in the same fashion as in the batch runs. They were

taken at least three times on a daily basis, but usually the frequency was much higher.

Additional considerations during continuous operation were daily monitoring of the base

reservoir, checking of the tubing and overall system condition, contamination testing,

effluent disposal, and feed carboy preparation and changing.

Contamination testing was usually performed by the following two methods:

preparation of a shake flask deficient in the appropriate amino acids and inoculation with

reactor contents, and microscopic examination of a Gram-stained sample of the reactor

contents. In the former method, an additional control flask was prepared with the

appropriate amino acids. The two flasks were seeded with identical volumes of reactor

volume. They were then examined for growth after overnight shaker bath incubation. If

the deficient flask showed significant growth, then the reactor was declared contaminated.

This method is somewhat dubious, though, as the deficient flask could be selective for




47


leucine and threonine revertants. Thus, the primary method of contamination testing was

the mentioned microscopic examination. If the slide appeared to have only red rods, the

continuous operation was declared successful to that point.

When operating in continuous mode a new feed carboy had to be prepared daily.

The feed was changed by switching the quick connect fitting at the end of each carboy tube

while all ends were immersed in rubbing alcohol. The quick connect change in alcohol,

along with the second filter placed between this connect and the reactor inlet, helped insure

sterility.












CHAPTER 4
ANAEROBIC GROWTH OF E. COLI LCB898




4.1 Background

Under anaerobic environmental conditions, where no alternate electron acceptors

such as nitrate, fumarate, or sulfate, are available, Escherichia coli uses fermentation as its

pathway for energy production. Fermentation, as defined by Brock [47, p.802], is a group

of catabolic reactions producing ATP in which organic compounds serve as both primary

electron donor and ultimate electron acceptor. When compared to aerobic or anaerobic

respiration, fermentation is not a very efficient method of producing ATP and, thus,

overall cell biomass [48, p. 54]. In order to understand fermentation, the reactions

involved are briefly described. The important reactions are diagrammed in figure 9 (this

diagram was drawn with the help of Neidhardt and Brock [49, p. 153 and 47, p. 126]).

First, the metabolic pathway common to both aerobic and anaerobic metabolism, pyruvate

formation, will be examined. After this, fermentative pyruvate dissimilation will be

considered.

The pathways of glucose degradation to pyruvate shown in figure 9 are the

Embden-Meyerhof-Pamas (EMP) pathway and the pentose-phosphate pathway. Typically,

in E. coli grown anaerobically on glucose, 92-95% of the glucose will be degraded by the

EMP pathway and 5-8% will be degraded by the pentose-phosphate pathway [50,51]. The

common first reaction to both pathways is the phosphorylation of glucose to glucose 6-

phosphate. After this point, the split between the two pathways occurs. The pentose-

phosphate pathway's major roles are formation of pentose phosphates for nucleotide






Glucose
ATP

ADP
Glucose 6-phosphate



Fructose 6-phosphai


NADP+ NADPH


> ADP
Fructose 1,6-diphosphate
(Fructose 1,6-diphosphate
splits into two
Glyceraldehyde 3-phosphates)

Glyceraldehyde 3-phosphate --
I- NAD+


NADH


1,3-Diphosphoglycerate
ADP

ATP
3-Phosphoglycerate


2-Phosphoglycerate


EMP
Pathway


----------------------I1

6-Phospho-
6-Phospho- glucnate I
gluconolactone NADP

,NADPH
Pentose
5-phosphate


I
Erythrose
4-phosphate

Pentose-Phosphate
Pathway


ATP ADP
Succinate Succinyl-CoA
'NNADl

F -NADH
Fumarate


Aspartate


NAD+

I LNADH NH4 a- Ketoglutarate
Phosphoenolpyruvate Oxaloacetate
ADP NADPH
ATP pfl f- NADP
ATPy upv Acetyl c Citrate--.- )> Isocitrate
Pruvat7 CoA aolic Acetyl-P Acetate

S dh Formate Acetaldehyde Ethanol
NAD+4 CO +H NADH NAD+, CoA NADH NAD+
d-Lactate C +2

Figure 9. Main Anaerobic Biochemical Pathways in E. coli LCB898


I--






biosynthesis and NADPH generation [48, p.31]. The EMP pathway will be described in

the following paragraph.

During pyruvate formation, a net yield of 2 moles of ATP per mole of glucose is

obtained by substrate-level phosphorylation. Additionally, 2 moles of NADH, the main

source of reducing power for biosynthesis, are generated. Some of the EMP pathway

intermediate metabolites also serve as biosynthetic precursors. These include fructose 6-

phosphate and phosphoenolpyruvate. The point at which aerobic and anaerobic

metabolisms differ is the degradation of pyruvate.

In wild-type E. coli cells, pyruvate is normally dissimilated under anaerobic

conditions by two pathways with no additional ATP generation, one being catalysed by

pyruvate formate-lyase (pfl), the other by d-lactate dehydrogenase (ldh) [52, p.151]. Pfl is

inactive under aerobic conditions [53]. The products of the pfl degradation include

format, acetate, ethanol, C02, and H2 [49, p. 163]. The sole product of the lactate

dehydrogenase pathway is d-lactic acid. In anaerobic wild-type E. coli K12 batches [54],

only traces of d-lactic acid are produced. This would indicate that the Idh pathway is not

normally used.

In E. coli LCB898, a K12 mutant, a mutation exists in the gene responsible for

production of pyruvate-formate lyase, thus closing off that pathway for pyruvate

dissimilation [37,38,55]. High yields of d-lactic acid from glucose therefore are expected

in this mutant. An additional consequence is that when growth is anaerobic in a minimal

medium, the addition of acetate may be required since acetyl-CoA cannot be produced

without the action of pyruvate formate-lyase.

To summarize, two points must be reiterated. The first is that when glucose is

processed through the EMP pathway, a net yield of 2 ATP molecules for every glucose

molecule degraded is observed. During aerobic metabolism, which is to be described later,

oxidative phosphorylation can also be employed. It will be shown that the net ATP yield

per glucose is much higher when the additional phosphorylation is performed. Again, ATP






yield is directly proportional to growth yield. Thus, relatively low biomass yields under

anaerobic conditions are expected. The second main point is that E. coli LCB898 should

show high yields of d-lactate on glucose.


4.2 Batch Growth

In all of the following discussed batch results, time 0 represents the point at which

inoculation of the reactor was inoculated. The results for one of the anaerobic batch runs,

to be designated anaerobic batch run 1, are shown in figures 10 and 11. This was one of

the preliminary runs to help determine a final feed composition. It is clear that biomass

concentration stopped increasing well before glucose in the system was exhausted from the

results shown in figure 10. Any glucose consumed after 46 hours was strictly being used

for maintenance. The results shown in figure 11 indicate that the glucose was largely being

converted into lactate. In this experiment only 50 mg/l of each amino acid and no metals

were used in the medium. It was hypothesized (and later confirmed with amino acid

analysis) that threonine was exhausted at the point of entering stationary phase.

In the next batch experimental run, anaerobic batch run 2, the amino acid

concentrations were doubled, but metals were still not added to the medium. The results

for this experiment are shown in figures 12 and 13. Again, as can be seen in figure 13, the

bacteria appeared to enter a stationary growth phase before glucose was exhausted. Amino

acid analysis showed excess threonine and leucine. Other work with this organism

[40,44,45] was then reexamined and metals were then added to the final formulation of the

medium.

In figures 14-17 all of the experimental results for a batch run with sufficient amino

acids and metals added to the medium are shown. This run is designated anaerobic batch

run 3. At 27 hours, the point of glucose exhaustion, growth had essentially stopped, as

can be seen in figures 14-16. During this growth phase, as shown in figure 17, lactate was

being produced in what appears to be a growth-associated manner. Lactate concentration








4500-

4000

~3500
E
c 3000
o
S2500

E 2000
o
0 1500
o
1000

500
nI


A AA


10


20 30 40 50
Time (hours)


Figure 10. Anaerobic batch run 1. Glucose and biomass concentration against time.


I-


140


S OS


A
oLX

ti o-


f\


-120

100

80

60

-40

20

n


A


I


- - -









3000 140


3000-


10 20 30
Time (hours)


40 50


Figure 11. Anaerobic batch run 1. Lactate and biomass concentration against time.


2500

E



0 1500

S1000

500
*J500


* Lactate Concentration
A Biomass Concentration



A *
A
SA.



..




4AA .. A


zX


0


- .. .. . .. II


-120

-100

-80

-60

-40

-20

rA


-


-


140


0n












4500- -350
-4000- ** A
% A300
E 3500 <------
S3000 -. A 250

2500 -200

o :A -150 c.
1500
S000lo 100 g
U : A
S500- A -50
0 A o. 0
0 5 10 15 20 25 30 35
Time (hours)

Figure 12. Anaerobic batch run 2. Glucose and biomass concentration against time.









4000- 350
3500 Lactate Concentration A *
3500 300
A Dry Mass Concentration 0
5 3000 A
E A -250
I 2500
-200 C
J 2000 A
8 A -150
6 1500- .
5 -100
S1000 A

500 A -50

0- A -A0
0 5 10 15 20 25 30 35
Time (hours)


Figure 13. Anaerobic batch run 2. Lactate and biomass concentration against time.








300.000


250.000 S


a 200.000
o

C)
S150.000


100.000-


50.000-


0.000-A
0 5 10 15 20 25 30 35
Time (hours)
Figure 14. Anaerobic batch run 3. Biomass concentration against time.









7.000E+ 11

6.000E+11 *
*

5.000E+11
*
6 4.000E+11

3.000E+11 *
z

6 2.000E+11-

1.000E+1 1-

O.00E+O .
0 5 10 15 20 25 30 35
Time (hours)
Figure 15. Anaerobic batch run 3. Cell number concentration against time.










4000.000

3500.000

S3000.000-
E *
CS
o 2500.000 *

5 2000.000-
0
U 1500.000-
o S
1000.000- *

500.000-
*
0.000- i
0.000 5.000 10.000 15.000 20.000 25.000
Time (hours)
Figure 16. Anaerobic batch run 3. Glucose concentration against time.


30.000 35.000
30.000 35.000








3000.000
*

2500.000-


2000.000
S
CS
= 1500.000-
C.)

8 1000.000-


500.000


0.000- ., 1. 1
0.000 5.000 10.000 15.000 20.000 25.000 30.000 35.000
Time (hours)

Figure 17. Anaerobic batch run 3. Lactate concentration against time.






did continue to increase beyond 27 hours, but this may be explained by a slow release of

the lactate from the cells. The low lactate concentration seen at 27 hours was probably due

to experimental error in the measurement, as the previous point indicated a lactate

concentration approximately 300 mg/l higher. Appropriate values for growth parameters

for this run will be given later in this chapter. Approximately 2 hours after entering the

stationary phase, the lactate measurements shown in figure 17 showed an interesting

response. The lactate showed a sharp dip with a subsequent increase. Ordinarily this may

have been simply dismissed as experimental error. A later experimental run was performed

under the same conditions as anaerobic batch run 3, to be designated anaerobic batch run 4,

and its results are shown in figures 18-21. The lactate results for this run are shown in

figure 21. The previously mentioned dip was also seen here. This dip was not of major

importance to this work as it occurs during stationary phase. This work primarily involved

exponential phase growth in continuous operation. However, possible explanations for it

are appropriate. The simplest hypothesis is experimental error. This hypothesis is not

probable though as it has been seen more than once. Another is that the cells may have

been growing diauxically on some other substance in the medium such as acetate, leucine,

or threonine. This would have been an unusual diauxy, though, as biomass concentration,

shown in figures 14 and 18, did not seem to increase during the period of glucose

exhaustion. The cells may have been using one of the listed substances in the medium for

maintenance while ingesting d-lactate along with this substance and later releasing the

lactate for some unknown reason. Still another hypothesis is that some of the cells may

have been genetically reverting to a pfl+ form (to be discussed further in a later chapter).

The properties of the revertant are not completely understood and this could have been

some effect of the revertant's growth alongside with the non-revertants. Again, whatever

the true explanation for this behavior may be, it was not of primary importance to this work

as most of the work performed in this project was on exponential phase growth.










300-



250-



200-



150

Q

5 100



50

I


I
0



Figure 18.


*
**
**
*




S
*
*

*

*
*
0
0.
**

S..
*
* S S


5 10 15 20 25 30 35
Time (hours)

Anaerobic batch run 4. Biomass concentration against time.


II-.









9E+1 1

-8E+11

7E+11

6E+11

~ 5E+11

0 4E+11

-~ 3E+11
E
Z 2E+11

U 1E+11


0 5 10 15 20 25 30
Time (hours)
Figure 19. Anaerobic batch run 4. Cell number concentration against time.










4000

3500

%3000

o 2500

5 2000



0
o
U 1500

-1000

500

0


0.000


5.000 10.000 15.000 20.000 25.000
Time (hours)


30.000 35.000


Figure 20. Anaerobic batch run 4. Glucose concentration against time.


0 S
0 *


-0





0
-0
- .







,*****
1 1 1 1 1


40.000











3500



3000-



E 2500-

0
S2000



| 1500-
o -

S1000



500-


I


0 0


0
0

00
0

0


* *


0 5 10 15 20


Time (hours)

Figure 21. Anaerobic batch run 4. Lactate concentration against time.


25 30 35


||


I I I I I I I -


~ -


I . . .


v







4.3 Continuous Growth

All continuous anaerobic experiments were run with the standard feed medium

shown in table 1. In all of the following discussions of continuous results, time 0

represents the point at which the reactor was inoculated. The point of switch from batch to

continuous operation will be mentioned for each run. It should be reemphasized that all of

the continuous runs were started as anaerobic batch runs. The switching point was chosen

on the basis of reactor turbidity so as to avoid washout and glucose exhaustion. The

results for one of the anaerobic continuous experiments, anaerobic continuous run 1, are

shown in figures 22-25. This experiment was performed at a dilution rate of 0.164 hr1.

(Dilution rate is defined as the ratio of flowrate to reactor volume. In reactor engineering

terminology, this represents the reciprocal of the residence time.) The switch to continuous

operation in this run was done at 83 hours. In figure 22, it is seen that a steady state,

defined as where the state variables of the system are unchanging, was seen at

approximately 108 hours. Data collected past 130 hours show surprising results and these

will be presented and discussed in Chapter 8. Some of the results of another continuous

anaerobic experiment (with dilution rate of 0.17 hr-1), anaerobic continuous run 2, are

shown in figures 26 and 27. The increased number of data taken at the switch point show

the smooth change in system condition from batch to continuous operation. In this

experiment, a steady state was seen at approximately 44 hours.

Values for the averaged apparent steady-state values for various dilution rates of cell

number, biomass, lactate and glucose concentrations are given in table 4. These values

were taken from the shown continuous experiments along with other continuous anaerobic

experiments. The results show low values for residual glucose concentration and high

conversions to lactate. The biomass concentration is also consistently low. Surprisingly,

the biomass concentration seemed to rise with dilution rate, which is not in agreement with

normal Monod behavior. These increases were not very large though. In contrast to

biomass, glucose did follow the expected Monod behavior of increasing with dilution rate.










300


250-


200-
o


150-
s
C
U -
S100-
0 -


20.00


S I ' I
40.00 60.00 80.00
Time (hours)


1 I 120.
100.00 120.00


Figure 22. Anaerobic continuous run 1. Biomass concentration against time.


I


*gg *







0
0


0


L


L


0.00


140.00


I I --


. I,









8.00E+ 11

,7.00E+11

S6.00E+11

o
5.00E+11-

84.00E+11-

3 3.00E+11
E
z 2.00E+11

S1.00E+11

O.OOE+O **
0 20 40 60 80 100 120 140
Time (hours)
Figure 23. Anaerobic continuous run 1. Cell number concentration against time.

















E



C
0


U
v
o






0
C-.
O

o


0 20 40


60 80
Time (hours)


120


Figure 24. Anaerobic continuous run 1. Glucose concentration against time.


*

3500- *
*
3000-


2500 0


2000


1500


1000-


500-

n- *


u


140


I I -


' '


k/














* *


4000-


3500-


S3000-
E
2500-


g 2000-


Q 1500-


c 1000-
-.-


500-

n-


*
*
*


a *O


20 40


Figure 25. Anaerobic continuous run


60 80 100 120
Time (hours)

1. Lactate concentration against time.


S

S
S


140


0.




70



300


250
E*

200
o
S *
5 150-


|100-
E 0

50-

0
0.00 10.00 20.00 30.00 40.00 50.00 60.00
Time (hours)
Figure 26. Anaerobic continuous run 2. Biomass concentration against time.











4000-


3500


S3000-


8 2500-


5 2000
U

oo
S1500


O 1000-


500

n--


0 0
0
Os

0

Og



S

0


0
00
S
00
0
S


0.00 10.00 20.00 30.00 40.00 50.
Time (hours)

Figure 27. Anaerobic continuous run 2. Lactate concentration against time.


60


00


.00







Table 4. Apparent Anaerobic Continuous Steady States

Dilution Rate Biomass Cell Number Glucose

Concentration Concentration Concentration

(hr-1) (mg/1) (cells/) (mg/1)

.17 283

.164 248 7.65x1011 136

.13 240 6.7xl011 86


Lactate

Concentration

(mg/1)
3402

3214

3397







4.4 Modeling



4.4.1 Presentation of Model

The following model was proposed to describe this system under batch anaerobic

conditions during exponential growth


= maxanaerobic (27)
Ks,anaerobic + S


1 Pmax,anaerobicS (28)
s= ----x (28)
Yx/s,anaerobic Ks,anaerobic + S


p = max,anaerobic (29)
Ks,anaerobic + S



where q= time derivative of q

x=biomass (dry mass) concentration in the reactor

s=substrate (glucose) concentration in the reactor

p=product (d-lactate) concentration in the reactor

lrmax, anaerobic-maximum growth rate under anaerobic conditions

Ks, anaerobic=saturation parameter under anaerobic conditions

Yx/s, anaerobic=yield of biomass on substrate under anaerobic conditions
(=growth-associated lactate production parameter

The first model equation, equation 27, shows simple Monod dependence of biomass

growth rate on glucose concentration. Equation 28, representing the time derivative of

glucose concentration, was chosen with the assumption that all glucose consumed during

exponential growth was consumed for the purpose of producing biomass. The third

equation, equation 29, gives the time derivative of d-lactate concentration in such a system.







It was assumed that the d-lactate production under exponential growth conditions was

strictly anaerobically growth-associated.

A continuous form of this model is as follows


x =maxanaerobicS
Ks,anaerobic + S


1 L-max,anaerobicS
s = x + D(sp s) (31)
Yx/s,anaerobic Ksanaerobic + S


p = max,anaerobicS p
p = a x Dp (32)
Ks,anaerobic + s


Flowrate
where D=dilution rate= -
Volume

sF=feed substrate (glucose) concentration

The above continuous form of the model was developed for a continuous stirred tank

reactor under conditions of perfect mixing. No biomass or product was introduced in the

feed, so no feed lactate or biomass was accounted for in this model. Inlet substrate was

accounted for in the substrate equation. The equations were simply extensions of equations

27-29 with additional terms for dilution of the biomass, substrate, and product out of the

reactor.



4.4.2 Model Parameter Fitting

The batch run measurements were the primary ones used for parameter

computation. The primary justification for this was that the anaerobic continuous runs

described previously were not performed until the ending of this work. Also, they showed

problems with reversion, which will be described in a later chapter, and measured steady

states were difficult to find. The parameters used in later modeling will be given here,

along with justification for their determination. Improved values for model parameters are






also given, but these were not the ones used in designing the later cycling operations still to

be described.

The first parameters to be discussed are the maximum growth rate, .Lmax, anaerobic,

and the saturation constant, Ks, anaerobic. These parameters would usually be determined

by use of a Lineweaver-Burke plot [56, p. 106] of continuous data. In these types of plots,

a large number of continuous steady states are necessary. Under steady-state conditions,

the time derivatives of the continuous form of the model are all equal to zero. By

manipulation of the steady-state version of 30, the following relation can be obtained


1_ Ks.anaerobic 1 1 (33)
D [ max,anaerobic Sss [ max,anaerobic



where sss=steady-state residual glucose concentration in the reactor

A plot could then be made of reciprocal dilution rate against reciprocal residual glucose

concentration. A least-squares fit would then be made to the data. The intercept would be

equivalent to the reciprocal of the maximum anaerobic growth rate, and the slope would be

equal to the saturation parameter divided by the maximum anaerobic growth rate.

Unfortunately, due to the reversion problems mentioned above, it was very difficult to

obtain continuous steady-state data. Equation 33 can be used on one data point to find one

of the two unknown parameters if the other can be satisfactorily estimated. It can be seen

that at a dilution rate equal to half of the maximum growth rate, the residual substrate in the

reactor is equal to Ks, anaerobic. In anaerobic batch run 3 the dry-mass growth rate was

0.190 hr1 and in anaerobic batch run 4 it was 0.23 hr1. These were obtained by simple

exponential fitting of the biomass data during exponential phase. For example, the data

used to calculate anaerobic batch run 3's growth rate was the dry mass data taken between

16 and 27 hours. The correlation coefficient for the fit (R2) was greater than 0.99. It was

assumed that these growth rates were at the maximum, as the glucose concentration was

very high during most of the duration of the runs. Were this not the case, the R2 would not







have been so close to 1. For the sake of consistency, anaerobic batch run 3 was the run

that served as the source of the data used for fitting model parameters. The reason why this

run was chosen over anaerobic batch run 4, was that the run 4's growth rate of 0.23 hr1

was significantly higher than that seen in other experiments performed. Most experiments,

including anaerobic batch run 1 and 2, indicated growth rates of the culture having values

between 0.19 and 0.20 hr-1. Therefore, the final value of I.max, anaerobic used was 0.190

hr1. The two Ks, anaerobic parameters then calculated using equation 33 for the anaerobic

continuous steady states given in table 4 are then 22.4 and 40.4 mg/1 for dilution rates

0.164 and 0.13 hr-1, respectively. An average Ks, anaerobic of 31.4 mg/1 would then be the

apparent value to be used. For several of the simulations described later the anaerobic

continuous experiments had not yet been performed. For these, a Ks, anaerobic value of 98

mg/l, based on preliminary continuous data using the feed medium listed in table 2, was

used for several of the later simulations in this dissertation. Use of the revised parameter

value of 31.4 mg/1 is suggested for future work with this model.

The next parameter to be determined was the yield of biomass on glucose under

anaerobic conditions, Yx/s, anaerobic. Using equations 30 and 32 and the assumption of

steady state the following relation for continuous operation can be stated



xss=Yx/s, anaerobic(SF-Sss) (34)


where xss=steady-state biomass concentration in the reactor
Using this relation, yield values of .064 and .061 mg biomass were indicated for dilution
mg glucose

rates 0.164 and 0.13 hr1, respectively. Anaerobic batch run 3 was the base run used and

the biomass yield was computed for this using a material balance argument which is very

similar to equation 34.

x=Yx/s(so-s) (35)







where so=initial glucose concentration in the batch



In figure 28, a plot of biomass against consumed substrate is shown. The line fit to the

data and forced through zero is also exhibited. The slope of this line gives the value of

Yx/s, anaerobic determined by this experiment The fit indicates a yield value of
0.063 mg biomass. However, as can be seen from the large amount of scatter around the
mg glucose

line, this was not a very good fit. The correlation coefficient for this curve, R2, was only

0.90. Since the correlation coefficient was so low, the yield used in calculation was

computed by simply averaging the biomass concentrations found in the stationary phase

and dividing the average by the initial glucose concentration of 4000 mg/l. The yield
mg biomass
computed in this manner was 0.068 mg s. This was the value used in the modeling
mg glucose
work. The fit value of 0.063 mg biomass turned out to be in good agreement with the
mg glucose

later continuous results, and, thus, this result should be used in the future.

The final parameter to be discussed for the anaerobic model is the anaerobic growth

associated lactate production parameter, a. Using equations 30, 32, and the steady-state

assumption, the following continuous steady-state relationship holds
a = PSS (36)
Xss



where pss=steady-state product (d-lactate) concentration
mg d lactate
The continuous runs indicate values for a of 12.02, 12.95, and 14.15 mg d for
mg biomass

dilution rates of 0.17, 0.164, and 0.13 hr1, respectively. The variability in these values is

quite large and leaves some of the d-lactate measurements in question. The determination

of a for a batch run is done by using equation 27 and equation 29 to obtain








300.000-


250.000


e 200.000
0


8 150.000-


S100.000-


50.000-


0.000
0


Figure 28. An


S













0


500


aerobic batch


1000 1500 2000 2500 3000 3500 4000
Glucose consumed (mg/1)

run 3. Biomass concentration against glucose consumed.







p = ax (37)



Subsequent integration yields

p=ox+po+oxo (38)



where po=initial d-lactate concentration

Xo=initial biomass concentration

The slope of a plot of biomass against d-lactate concentration during exponential phase for

a batch run is equivalent to the a parameter. This plot for anaerobic batch run 3 is shown

in figure 29, along with the results of a linear least squares fit to the data. The problem

with this method is that, due to the scatter of the d-lactate values seen in figure 29, the final

value is very dependent on where exponential phase is declared to begin. For example, if

all of the data that were used for the computation of anaerobic batch run 3's growth rate

was used, an a parameter of 8.64 mg/mg (with a correlation coefficient, R2, of only 0.88)

would be the determined value. This correlation coefficient was unacceptably low. If the

last data point, due to its questionable reliability, was ignored the a parameter value

increased to 9.64. If the only points considered on this figure were those between 100 and

250 mg/1 biomass, an a parameter of 16.7 mg/mg would be found. Due to all of this

scatter, a batch value for a had to be determined by another batch run. Anaerobic batch run

4, in spite of its high growth rate, was then examined and used to obtain an estimate of a.

The results of the corresponding lactate against biomass curve are presented in figure 30,

along with linear least squares fitting results. The data showed a much higher degree of

linearity than that seen for anaerobic batch run 3. The a parameter fit for these data was

10.8 with a correlation coefficient, R2, of 0.98. This was the value used in later

simulations. Upon comparison of this value with those given by the continuous runs, this









2500


-2000-


1500-

0 1000
*


2 500-


0
0 50 100 150 200 250 3(
Biomass Concentration (mg/1)
Figure 29. Anaerobic batch run 3. Lactate concentration against biomass concentration.




81


value for a would underpredict lactate concentration by approximately 20%. Increased

values of a should be considered in future use of the model.

The parameter values are summarized in table 5. The suggested revised a

parameter was based on averaging the results for the determined o parameters of the three

continuous experiments. The large differences in updated parameter values for yield,

saturation and a parameters suggested that use of the model in designing any operation

must be checked for sensitivity to these parameters.










3000

-2500

S2000

1500

S1000-

5 500-


0 50 100 150 200 250
Biomass Concentration (mg/1)

Figure 30. Anaerobic batch run 4. Lactate concentration against biomass concentration.









Table 5. Anaerobic Model Parameter Values

Parameter Value used in later work

Plmax, anaerobic .19 hr-

Ks, anaerobic 98 mg/1

Yx/s, anaerobic .068 mg/mg
at 10.77 mg/mg


Possible improvement

.19 hr-1

31.4 mg/1

.063 mg/mg

13 mg/mg












CHAPTER 5
AEROBIC GROWTH OF E. COLI LCB898




5.1 Backgound

Under aerobic conditions most facultative aerobes, such as E. coli, will adjust their

metabolism to take advantage of available oxygen and increase its production of biomass.

The first part of aerobic metabolism of glucose is production of pyruvate. This part

was described in the previous chapter on anaerobic growth. The only major difference to

be mentioned here is that 25% of the glucose will enter the pentose-phosphate pathway

under aerobic conditions [50,51]. Where the major difference between fermentation and

aerobic respiration on glucose lies, is in the fate of pyruvate. The aerobic metabolism of

glucose for a wild-type E. coli is shown in figure 31 [49, p. 155, 45]. The aerobic

metabolism of E. coli LCB898 shouldn't deviate significantly from this as no significant

aerobic metabolism genes are mutated in its genome. Pyruvate dehydogenase, which is

only produced aerobically [39], is the enzyme responsible for aerobic conversion of

pyruvate to acetyl-CoA. After it is formed, acetyl-CoA is sent through the TCA cycle. In

the TCA cycle only one ATP molecule (per molecule of pyruvate) is produced by substrate-

level phosphorylation, specifically the conversion of succinyl-CoA to succinate [48, p. 54].

All of the other ATP molecules are generated by oxidative phosphorylation, where the

energy is produced by the transfer of electrons from NADH, NADPH, and FADH2 to

oxygen [49]. As can be seen in figure 31, from each turn of the cycle two molecules of

NAD+, one of NADP+, and one of FAD+ are reduced, with one additional NAD+

reduction occurring in the conversion of pyruvate to acetyl-CoA. In E. coli, two molecules

of ATP can be generated for each molecule of NADH or NADPH, and one molecule of






ATP can be produced for each molecule of FADH2 oxidized [48, p. 43]. To sum up, ten

molecules of ATP can be produced, when the TCA cycle is employed, for every molecule

of pyruvate processed. When one adds, per glucose molecule, the two ATP molecules

generated by fermentation, the 2 ATP's that can be generated from the 2 molecules of

NADH produced (and not needed in d-lactate formation) and the TCA generated ATP's

from the two pyruvate molecules formed, 26 molecules of ATP can be produced for every

glucose molecule, as opposed to the synthesis of just two molecules of ATP for each

glucose molecule strictly fermented.

Of course, these ATP yields are theoretical. Fermentative efficiency can be as high

as 50%, as opposed to the TCA cycle efficiency of 39% [47, p. 134]. Nonetheless, even

though pure fermentation has higher efficiency, the aerobic metabolism will give much

higher amounts of ATP per glucose molecule consumed than will pure fermentation. Since

growth requires ATP for energy, aerobic growth should show higher biomass yields on

glucose.

There are two main products of aerobic growth of E. coli on glucose, specifically

biomass and CO2 [57, p.802]. Three CO2 molecules are produced by the TCA degradation

of each pyruvate molecule, and thus 6 CO2 molecules are produced for every glucose used

strictly for catabolism.



5.2 Experimental and Modeling Results Introduction

Most of the aerobic experiments were performed in conjunction with a master's

thesis project performed by Christina Stalhandske [58]. The experiments were jointly

performed by this author and Stalhandske and the modeling work, while contained in that

work, was mostly performed by this author. Thus, most of the results, both experimental

and modeling, are taken from this thesis.







ATP

ADP
Glucose 6-phosphate



Fructose 6-phosphat


NADP+ NADPH


----- ---

I Tb
1 )I


4 ) ADP
Fructose 1,6-diphosphate
(Fructose 1,6-diphosphate
splits into two
Glyceraldehyde 3-phosphates)

Glyceraldehyde 3-phosphate ----
+NA EMP
r EMP


NADH


Pathway


-------- ------------

6-Phospho-
6-Phospho- gluc nate
gluconolactone NADP

NADPH
f Pentose
5-phosphate


Erythrose
4-phosphate

Pentose-Phosphate
Pathway


v AIF ADU
1,3-Diphosphoglycerate
ADP Succinate < Suc
FAD +
ATP FADH 2
3-Phosphoglycerate 2 4
SFumarate
J TCA
Cycle
2-Phosphoglycerate Malate
NAD

V NADH N '- Ketogl
PhosphoenolpyruvateOxaloacetate
Oxaloacetate
ADP

APyruva Acetyl nitrate Isocitrate
Pyruva CoAli LCB898

Figure 31. Main Aerobic Biochemical Pathways in E. coli LCB898


cinyl-CoA



NADI

-NAD+


I


utarate

NADPH
NADP+


|


I /






5.3 Batch Growth

In all of the following discussed batch results, time 0 represents the point at which

the reactor was inoculated. The results for one of the aerobic batch runs, to be designated

aerobic batch run 1, are shown in figures 32-35. This was one of the preliminary runs to

help determine a final feed composition. As can be seen from figure 32, glucose was not

exhausted when biomass concentration stopped rapidly increasing at 14 hours. Biomass

did increase slightly after 14 hours, but not in an exponential manner. Glucose did

continue to drop, but it is hypothesized that this glucose was largely used for maintenance.

Amino acid analysis showed that threonine was exhausted before glucose The lactate and

ethanol concentrations of each sample were measured to see if they were produced in a

significant amount. These quantities turned out to be negligible. Thus, in the following

strictly aerobic batch and continuous runs ethanol and lactate measurements were not

measured regularly.

The results of a batch run with sufficient amino acids aerobic batch run 2, are

shown in figures 36-38. This time the cells grew exponentially until glucose was

exhausted, which indicated that glucose was the limiting substrate. It appeared that 15

hours after starting the run, cell number and biomass concentration stopped increasing in an

exponential manner. Further increases in dry mass can be attributed to diauxy on

remaining amino acids or acetate present.



5.4 Continuous Growth

All continuous aerobic experiments were operated with the standard feed medium.

In all of the following discussed continuous results, time 0 represents the point at which the

reactor was inoculated. The point of switch from batch to continuous operation will be

mentioned. It should be reemphasized that all of the continuous runs were started as

aerobic batch runs. The switching point was determined on the basis of turbidity and

chosen so as to avoid washout and glucose exhaustion.








5-

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0


* Biomass Concentration

A Glucose Concentration


A


*


AI

AA


) 5 10 15 20 25 30 2
Time (hours)


Figure 32. Aerobic batch run 1. Glucose and biomass concentration against time.


AAA
AI
AI
A A




89


3.00E+12


S2.50E+12 -
I .
g 2.00E+12
0
g 1.50E+12-


S1.00E+12
z S
v 5.00E+11


O.OOE+O --
0 5 10 15 20 25 30 35
Time (hours)
Figure 33. Aerobic batch run 1. Cell number concentration against time.










.IV -
*


25-



10-



5-



0-
5
O 0O



5 *
*

0 __--__-______________


T T .
15 20
Time (hours)


Figure 34. Aerobic batch run 1. Lactate concentration against time.


O l . Ia















E
" 4O
C


30
0
U
zO
LL 2


u 3 1U 15 20 25
Time (hours)

Figure 35. Aerobic batch run 1. Ethanol concentration against time.




92



3500

3000-


N 2500
0 0
, 2000

1500-

S1000

*
s looo-

500-

0 a- AA O 6O0
0 5 10 15 20 25 30 35
Time (hours)


Figure 36. Aerobic batch run 2. Biomass concentration against time.








9.00E+12

8.00E+12

7.00E+12

g 6.00E+12-

S5.00E+12
u
o 4.00E+12

S3.00E+12

S2.00E+12

1.00E+12

0.OOE+0 --
0


Figure 37. Aerobic


*
*







*




*
*



1 ---* I- 1 l 1 1 1 1 1 1 I I.. I I I I- 1 1 I . .
5 10 15 20 25 30
Time (hours)

batch run 2. Cell number concentration against time.




Full Text
ENHANCEMENT OF D-LACTATE PRODUCTION IN A
CONTINUOUS CULTURE OF A MUTANT ESCHERICHIA COL1
THROUGH PERIODIC OPERATION
By
JONATHAN BEN RODIN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1992
UEÜV3ÍSUY OF FLORIDA HEMS'®

ACKNOWLEDGMENTS
I would like to thank my committee chairmen, Professors Spyros Svoronos and
Gerasimos Lyberatos, for providing guidance above and beyond the call of duty. They are
appreciated for being there to help and encourage me during the bad times when the project
was going down seemingly blind alleys.
There are other faculty members who have truly been of great assistance. I would
like to thank my other committee members first, Professors Gerald Westermann-Clark,
Seymour Block, Ben Koopman, and Lonnie Ingram. Dr. Ingram is especially appreciated
for all of his very useful advice without which this project would not have been completed.
He has also been very generous with use of his laboratory and equipment. In addition to
my committee members, Professors Bitsanis and Crisalle are thanked for all of their
guidance.
Several graduate students have helped me out during my research project. Three
deserve special mention, Jeff Mejia, Pratap Pullammanappallil, and Christina Stalhandske
for all the technical assistance they provided. If only one student was to be mentioned, Jeff
Mejia would be the one. He is the one who initially suggested the reversion problem with
this system. If not for his advice, I would still be in the dark as to the cause of all of the
problems associated with continuous anaerobic operation. Christina’s help was
instrumental in the work performed on the aerobic characterization of bacterial growth.
Pratap, besides being one of my closest friends during my years in Gainesville, has been
an indispensable resource of information and help with my project. Additionally, Jeffrey
Harmon has lent a helping hand on several occasions. Lastly, in addition to those
u

mentioned above, I have been very fortunate to have had numerous other friends among the
graduate students in this department.
Several undergraduates from this department have assisted me at various times
during the duration of this project. I have been very fortunate to have had these people
work with me. The following undergraduates have worked as laboratory assistants or
completed small research projects with me: Dawn Mackland, Mike Hinson, Reann
Soodeen, Craig Moates, Erik Dunmire, and John Walker. Among these students, Dawn,
Craig, and Mike have each shown dedication above and beyond what was expected from
them. Like the graduate students that have gone through this department, I have been
fortunate to call several of the undergraduates, again in addition to those mentioned above,
friends.
This section would not be complete without mentioning Mr. Tracy Lambert, the
department’s maintenance specialist. He has been of invaluable help to me during this
project.
Finally, I would like to thank my parents, sister, brother, and sister-in-law.
Without their love, support, and encouragement, I probably would not have been able to
endure this whole ordeal. My brother was also generous with allowing me free use of the
facilities and equipment at his business, Gallery Graphics.

TABLE OF CONTENTS
page
ACKNOWLEDGEMENTS ii
ABSTRACT vi
CHAPTERS
1 INTRODUCTION 1
2 THEORETICAL METHODS 4
2.1 Overview 4
2.2 Determination of Optimal Steady-state Operation 5
2.3 A New Method of Determining Optimal Periodic
Pulsing 7
2.3.1 Carleman Linearization 7
2.3.2 Performance Measure Calculation 10
2.4 Model Fitting with Nonlinear Least Squares Methods 15
3 EXPERIMENTAL METHODS 19
3.1 Organism Description 19
3.2 Analytical Methods 21
3.2.1 Biomass and Cell Number Determination 21
3.2.2 Glucose Analysis 25
3.2.3 d-Lactate Measurement 26
3.2.4 Other Analyses 28
3.3 Feed Medium Composition 30
3.4 Feed Preparation 32
3.5 Experimental Operation 38
3.5.1 Operational Conditions 38
3.5.2 Shake Flask Experimental Procedure 40
3.5.3 Reactor Experimental Procedure 42
3.5.3.1 System Description 42
3.5.3.2 System Startup and Operation . 44
4 ANAEROBIC GROWTH OF E. COU LCB898 48
4.1 Background 48
4.2 Batch Growth 51
4.3 Continuous Growth 65
4.4 Modeling 73
4.4.1 Presentation of Model 73
4.4.2 Model Parameter Fitting 74
iv

5 AEROBIC GROWTH OF E. COU LCB898 84
5.1 Background 84
5.2 Experimental and Modelling Results Introduction 85
5.3 Batch Growth 87
5.4 Continuous Growth 87
5.5 Modeling 95
5.5.1 Presentation of Model 95
5.5.2 Model Parameter Fitting 101
6 THE EFFECTS OF SHIFTS IN AERATION OF E. COU LCB898 .. 113
6.1 Background 113
6.2 Development of the Combined Aerobic-Anaerobic Model ... 114
6.3 Testing of the Model 116
7 EFFECTS OF AERATION CYCLING ON LACTATE
PRODUCTIVITY OF E. COU LCB898 122
7.1 Background 122
7.2 Theoretical Investigation into Cycling 123
7.3 Experimetal Confirmation of Lactate Productivity
Optimization Results 135
8 REVERSION OF E. COU LCB898 AND A POSSIBLE NEW
METHOD OF AVOIDANCE OF REVERSION 151
9 CONCLUSIONS 163
APPENDIX
MATHEMATICA PROGRAMS FOR COMPUTATION OF
CARLEMAN LINEARIZATION MATRICES 165
LIST OF REFERENCES 171
BIOGRAPHICAL SKETCH 175
v

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
ENHANCEMENT OF D-LACTATE PRODUCTION IN A
CONTINUOUS CULTURE OF A MUTANT ESCHERICHIA COLI
THROUGH PERIODIC OPERATION
By
Jonathan Ben Rodin
December, 1992
Chairman: S. Svoronos
Cochairman: G. Lyberatos
Major Department: Chemical Engineering
In some biological systems, the environmental conditions that are optimal for
microbial growth differ from the optimal conditions optimal for producing a desired
metabolite. If production of this metabolite were the process objective, one could
continuously operate a reactor system at the optimal production conditions. However, for a
given reactor volume, changing the conditions periodically could increase overall
production of the desired metabolite. This is possible since, due to higher growth rates
under the optimal growth conditions, one could operate the system at significantly higher
flowrates and, thus, obtain more product. A system involving E. coli mutant LCB898 was
used as a model system. Under anaerobic conditions this bacterium will produce large
amounts of d-lactic acid, whereas under aerobic conditions, this bacterium will grow faster.
The possibility of increasing total lactate production by cycling dissoved oxygen was
investigated.
vi

Before any optimization work could be done, an adequate model for describing the
behavior of this system under both steady-state and transient conditions had to be
developed and tested. Such a model was developed using batch and continuous data and
then tested by comparison with shifts between conditions.
A method for determining the optimal waveform for the proposed cycling was
developed by extending previous work by Lyberatos and Svoronos. The method involved
Carleman linearization of the model equations around a steady state and subsequent
development of a term for the performance measure. The system studied oscillated
between purely aerobic and anaerobic metabolisms with no intermediate conditions. Thus,
an “imaginary” steady state of intermediate metabolism had to be used for linearization.
In numerical simulation of the determined optimal cycling, significant improvement
over strictly anaerobic operation was found. Experimental verification of this was
performed and improvement, though not as significant as theoretical predictions would
indicate, was found. Additionally it was found that the mutant was probably reverting to a
form where little d-lactic acid was produced. Cycling of dissolved oxygen apparently helps
delay this reversion.
vii

CHAPTER 1
INTRODUCTION
Chemical processes are usually operated in one of three different manners, batch,
fed-batch, or continuous, each with its own advantages and disadvantages. Batch
operation is advantageous when small quantities of a product are desired. This type of
operation is perhaps the simplest since it only involves charging the reactor with the
appropriate reactants at the start of the process and removal of the products at its
completion. Unfortunately, this type of operation includes frequent downtimes where the
reactor is being either charged or purged, and production is thus temporarily stopped. Fed-
batch operation, where feed is added continuously but nothing is being withdrawn, is also
advantageous when small quantities of a product are desired. This type of operation is
optimal, for example, when it is desired to keep reactant concentration low. Fed-batch
operation has the same downtime problems that batch operation does. Continuous
operation is typically used when large quantities of a product are desired. It has advantages
over the batch-type processes in that no downtimes for charging or purging the reactor are
necessary. However, higher control and instrumentation costs are incurred. When large
quantities of product are desired, continuous operation is the usual method of choice.
When continuous processes are used, they are usually operated, after start-up
transients die out, in a steady-state manner. This involves keeping the process variables
constant. This type of operation is relatively easy to model and control. However, optimal
operation may involve taking the reacting volume through multiple steps of processing, for
example, operating the system in a neutral environment for an amount of time, and then in
an acidic environment. The simplest solution to this type of problem for a two-step process
1

2
would be to have two tanks in series, where one has a particular set of environmental
conditions maintained in it, and the other has a different set of environmental conditions
maintained. However, this increases the total volume and instrumentation (and thus cost)
of the system. An alternative approach is periodic operation. This involves only one tank,
but the environmental conditions are manipulated with respect to time. In general chemical
systems small time constants are the rule. As a result, conditions in the reactor would have
to be changed frequently to observe a significant improvement over steady-state operation.
This would lead to substantial control costs. However, in biological systems the time
constants are relatively large. Thus conditions may not have to be changed rapidly.
Therefore, biological systems may be suitable for periodic operation.
Periodic operation of continuous culture systems has been investigated by other
workers [e.g. 1-8] and it has been found to be useful in achieving desired process goals.
For example, it has been used for enhancing the production of yeast in continuous cultures
[7] and for solving the problem of plasmid stability in continuous recombinant cultures [8].
To establish appropriate operating conditions, kinetic models are required that adequately
describe the transient behavior of the culture being investigated. Such models have been
developed previously [e.g. 9-11].
Biological reactors can be operated to achieve one of the following process
objectives: utilization of nutrient (e.g., wastewater treatment), biomass production,
production of a particular metabolite, or conversion of one chemical substance to another
(bioconversion). The productivity of a continuous culture is determined by two variables,
number and state of cells. To achieve high productivity or substrate utilization, it is
desirable to have a large number of cells in the optimal state for the particular objective. A
system can be manipulated to change the state of the cells through control of environmental
conditions. However, in microbial growth processes, the optimal cell state for maximum
growth of the cells is generally not the same as the optimal state for production of a desired
metabolite. A solution to this problem is cycling environmental conditions (and thus cell

3
state) in the reactor. A large number of cells producing the desired metabolite could result.
Previous efforts in cycling of conditions will be described later in this work.
As an example, an E. coli mutant (LCB898), which produces d-lactic acid in large
amounts, was examined in this work. Under aerobic growth conditions this organism
produces high amounts of biomass but insignificant amounts of d-lactic acid. Under
anaerobic conditions it produces significant amounts of d-lactic acid and less biomass. If
the goal is to maximize lactate production per reactor volume per time, one could simply
operate the system under strictly anaerobic conditions. Alternatively, cycling dissolved
oxygen level (and sometimes residence time) may improve productivity over that achieved
by strict anaerobic operation. This improvement, in the case of constant residence time, is
achieved by operating the system at higher flowrates. The higher flowrates are allowed by
the aerobic (faster growth) portion of the cycle. Improvement by cycling is explained
further in the chapter on cycling.
An unexpected problem that occurred while investigating this process was that of
reversion of the genotype of E. coli LCB898 under anaerobic continuous operation. This
reversion led the organism into a state where little d-lactic acid was produced. The
proposed cycling of aeration also appears to be a new method for avoiding reversion or
mutation of organisms under continuous operation.
This dissertation reports theoretical and experimental investigations on aeration
cycling for E. coli LCB898. First, the general theoretical problem of optimizing a system
undergoing cycling of environmental conditions will be formulated. A description of the
particular microbial system under anaerobic and aerobic condidons, including modeling
results, will then be given. Transient behavior of the system will then be discussed,
followed by cycling predictions and experimental results. The question of reversion delay
will subsequently be addressed. Conclusions will then be presented.

CHAPTER 2
THEORETICAL METHODS
2.1 Overview
An important job of the engineer is to determine the best way to operate a system.
This usually will be the way that maximizes profitability within safety limits. It can involve
anything from deciding how best to place workers on an assembly line to finding the
optimal control setting for the temperature of a chemical reactor. The engineer will usually
attack this problem by devising a mathematical formulation of the system and then use one
or more optimization methods to design the best operation. The formulation will involve a
statement of some sort of performance criterion to be optimized, along with descriptions of
various equality and/or inequality constraints on the system. Examples of the performance
criterion include maximizing the number of automobiles produced or minimizing the
amount of byproduct from a reactor. Examples of constraints are the number of workers
available at any given time or the maximum operating temperature of a reactor. Two kinds
of constraints usually encountered are equality constraints, where some quantity of the
system must always be equal to some other quantity, and inequality constraints, where
some quantity of the system is bounded within certain value(s). The optimization method
used will be selected by the engineer based on several factors including the system under
study, available computing power, and others. Much work has been devoted to finding
such optimization methods [e.g. 12-14],
A class of optimization problems exists where the equality constraints are not
algebraic equations but differential equations. These are called optimal control problems
[13, page 364], Frequently, for this type of system, the engineer will want to vary the
4

5
control settings with time to maximize the productivity. Many methods have been
developed for handling this kind of problem and one will be discussed later. Others can be
found elsewhere [e.g. 15,16].
Continuous-flow reactor systems are typically operated at optimal steady-state
conditions. However, several workers [17-23] have looked into operating systems at
nonconstant conditions by cycling control variables around the optimal steady-state control
settings and found improvement for certain performance criteria. Sometimes, though, there
may be a need to examine systems where there is no true optimal steady-state control
setting to cycle around [e.g. 8], and an imaginary intermediate state, only existing
mathematically, may be used to design the best cyclic operation. As an example of this
type of system one could imagine a reactor whose air pumps can be either on or off, with
the on setting resulting in complete aeration. A mathematical model of the system may
show that intermediate aeration is optimal for production. Then the engineer must decide
how to best operate the air pumps to get optimal production.
In this chapter, the problem of determining optimal steady-state operation will be
addressed first. The method used to determine optimal square-wave cycling will then be
discussed, and finally a short description of some of the model-fitting techniques will be
given.
2.2 Determination of Optimal Steady-state Operation
The typical system examined is one described by a set of differential equations:
^L = f¡(x1,x2,...xn;u1,u2,...ur) (1)
i = 1, 2,...n

6
where x¡=
fi=
n=
Ui=
r=
state variable i
function describing the rate of change of variable i
number of state variables
control variable
number of control variables
t= time
Steady states for such a system are found by setting the right-hand side of equation
1 equal to a zero vector and solving (by algebraic manipulation or by use of numerical
methods [24]) for the values of x¡. The optimum steady-state operating conditions (i. e. the
optimal control settings) for such a system can be found by treating the resulting steady-
state model equations as equality constraints, expressing necessary inequality constraints
and stating a performance measure. Since the equations become algebraic at steady state,
many methods of optimization can be used. If the steady-state model equations can be
solved to yield an explicit expression of the performance measure as a function solely of the
control variables (and if inequality constraints do not come into play), then classical
theories of determining the optimal control settings involving setting the partial derivatives
with respect to each control variable to zero and solving for the settings can be used [12],
Frequently an explicit expression cannot be found and other methods have to be used.
One can find the optimum steady-state settings by performing, when possible, a
simple numerical search for the control values giving the maximum (or minimum) value of
a stated performance measure, but it will frequently be necessary to use a different method
when the system increases in complexity. As stated before, many methods exist. A few of
them include quasi-Newton and conjugate gradient methods. These methods are described
elsewhere [12,13].

7
2.3 A New Method of Determining Optimal Periodic Pulsing
This new method involves Carleman linearization of a general nonlinear system,
expression of an explicit formula of the performance measure for pulsed cycling of the
linearized system, and then a search, using the explicit performance measure, for the
optimal settings for the system.
2.3.1 Carleman ' mearization
Carleman linearization is a method of describing a general nonlinear system of first
order differential equations in a linearized form. It was first introduced in 1932 [25] and
has been applied to various problems in nonlinear system dynamics and control [17,26-
34], A brief overview of this method is presented here. This description was composed
using earlier descriptions [17,35].
A restatement of equation 1 would be as follows:
£ = f(x>u) ; f(0,0) = 0 (2)
with x_e Rn a vector of deviation state variables and ue Rm a vector of deviation control
variables from a nominal steady state. This system is defined to have n dimensions. It is
assumed that the functions f are differentiable up to order r at Q. Taylor expansion is then
performed on system (2) and the monomials of order up to r are introduced as new
variables. These monomials are then differentiated and terms of order up to r are retained.
The result is a linear system in the new variables and is called the rth order Carleman
linearization. For example, the system
• 2
xi = -Xj +3x2 + x2
• 2
X2 = -Xj + 4u
(3)

8
where the manipulated variable u is treated as a parameter, would have the following
monomials approximated as variables w¡.
*1
Wi
*2
2
w2
*1
—
w3
xlx2
w4
1
X
tO N
1
_w5_
(4)
The second order Carleman linearization for the system in equation 3 is
"-1
3
0
0
1 '
'O'
0
0
-1
0
0
4u
0
0
-2
6
0
w +
0
4u
0
0
-1
3
0
0
8u
0
0
0
0 _
(5)
The computation of the necessary partial derivatives for expressions of higher order
Carleman linearization can be tedious, and, in the case of elaborate differential equations,
can lead to errors. A new program written in Mathematica [36] has been developed for
computing the necessary Carleman matrices and vectors and is presented in Appendix A,
along with an associated program used to compute necessary Kronecker products (the
Kronecker package offered by Mathematica is not usable for this type of problem). This
program, using the capabilities of Mathematica, does not require the user to provide
necessary derivatives; it computes the derivatives analytically. A description of the
algorithm used in the program was given by Lyberatos and Svoronos [17], but will also be
presented here.
Let <8> represent Kronecker multiplication. The Taylor series expansion about
x = 0 can be expressed as

9
x - IAikxW + A10
(6)
k=l
where x^ = x 0 x 0 ... 0> x (k terms)
and 0 represents Kronecker multiplication
e.g. [27, page 11] if A is an mxn matrix and B is a pxq matrix then
f aHB a^B ••• a1nB ^
32jB 322B •”
A 0 B =
lln
a2nB
AmlB ^m2B
The resulting matrix has the dimensions mp x nq.
The following system then comprises the rth order Carleman approximation to the
original system
•
X
Al,l
Aj 2
A 1,3 •"
Al.r
A 1,0
A2,o
A 2j
A 2,2
A2,r-1
0
*12]
X 1 J
•
= w =
0
A 3,0
A3,]
A3,r-2
w +
0
x[r]
0
0
O
< ...
A4,r-3
l
0
0
Ar0
Ar,]
0
where A¡j = ln 0 Ai—1 j + Ajj 0 In¡_i
Ig=gxg Identity Matrix
(7)
The above system will have a dimensionality of n+n2+n3+...+nr. Since the vector
w will contain monomial redundancies (e.g. xjx2 and X2xj), the system should be
simplified. This is done by eliminating rows that correspond to the same monomial and
adding the corresponding columns together. The Carleman system will then have a

10
dimensionality of X n+j-i^j or equivalently
j=i
combinations of m objects taken q at a time
min(ntr) _ ^
X rCj nCj where mCq is the number
j=i
r , j
mCQ = — [14,23]-
V ’ q'(m-q)'J
of
2.3.2 Performance Measure Calculation
In this section we will develop a method to determine an explicit performance
measure for a Carleman linearized system when we cycle our controls on a system between
two fixed settings. Consider the waveform in figure 1. In this figure u§ and up are the two
control settings available, and u¡ represents a fixed intermediate (though not necessary
realistic) control setting. These settings can be vectors or scalars. 5 and p represent the
vector or scalar deviations of the two respective control settings from u¡. T represents the
period of the cycle, and £ represents the fraction of the period spent at one of the control
settings (us).
This problem is very similar to one looked at by Lyberatos and Svoronos [17]. In
that work, they looked at square-wave cycling around an optimal steady state. The
deviations from that state were allowed to vary. The intermediate state mentioned above is
analogous to the optimal steady state that Lyberatos and Svoronos examined. Additionally,
in the present problem, the deviations from the intermediate are fixed. Thus, the previous
mathematical development that they performed can be used to a large extent here with only
minor changes. The main parts of their derivation will also be given here.
Again, the problem to be examined is (in deviation variables)
£ = £(x,u); f(0,0) = 0 (8)
where f (x,u) is analytic in x at 0 for all admissible u vectors (or scalars). The u vectors
(scalars) are treated as staying constant for either part of the cycle. Taylor

11
K T >\
Figure 1. Section of control waveform being analyzed
>

12
expansion around x=0 is performed on this system and the nh order Car lemán linearization
is obtained
* ÍS(5)w + z(5) te[nT, (n + e)T)
w — <
— }s(p)w + z(p) t 6 [(n + e)T, (n+ 1)T)
(9)
The performance measure under cyclic conditions can be represented as the
following:
J=YloTp(x,u)dt (10)
where J=time-averaged performance measure
T=period of the cycle
P=instantaneous performance measure
It is assumed that P(x. u) is analytic in x at Q, in which case it can be linearized.. This is
done by Taylor series expansion around x=0 and is cast in terms of the Carleman
coordinates w. The performance measure then takes the representation
J = Y^[r°(M) + i(-)-(t)ldt (n)
where ' represents the matrix transpose.
In the derivation that follows, the following identities, which apply to any invertible
n x n matrices and were proved by Lyberatos and Svoronos [17], will be needed
(ln-AB)"1A = A(In-BA)-1 (12)
(ln-AB)“1-A(In-BA)-,B = In
(13)

13
(In - A)(In - BA)_1(In - B) = (In - B)(In - AB)-1(In - A)(14)
Now, equation 9 can be integrated to give the following ultimate periodic solution
w(t) =
,S(S)(t-nT)w0 _
»S(p)(t-(n+e)T)we _
-eS(6)(t-nT)]S(5)-lz(5)
for t e [nT,(n + e)T)
I - es(p)(‘-(n+e)T)]s(P)-i z(P)
for t e [(n + e)T,(n + 1)T)
(15)
Defining
D=eS(5)eT
R=eS(p)(l-e)T
(16)
w° and we can be expressed as:
w° = -[I - RD]_1[[R - RD]S(8)_1z(6) + [I - R]S(p)_1z(p)] (17)
wE = -[I - DR]"1 [[I - D]S(5)_1 z(6) + [D - DR]S(p)_1 z(p)] (18)
From (15) it can be seen that
w£ = Dw° — [I — D]S(6) ^(S) (19)
and
w° = Rwe - [I -R]S(p)_1z(p)
(20)

14
Using equation 15, the cycle average performance measure given in equation 11 can
be expressed as
J(T,e,5) = ^r'(8)[es(5)£T - l]s(5)_1 w° +
^r’(p)[eS(p)(1_£)T -l]s(p)_1w£ -
ir'(5)[s(5)eT+I-eS(6)£T]s(5)-2z(6)- (21)
~l'(p)[s(p)(l - e)T +1 - eS(p)(1-£)T]s(p)-2z(p) +
ro(S)e + r0(p)(l-e)
Using that S(8)'J commutes with D and S(p)'1 commutes with R along with
equations 19 and 20, equation 21 can be rewritten as:
J(T,e,8) = ^r’(8)S(8)_1[we - w° -eTz(8)] +
ir’^^fw0 - wE - (1 - e)Tz(p)] + (22)
r0(8)e + r0(p)(l-e)
Using equations 12-14, equation 17 and equation 18 the following is obtained:
we - w° = (I- R)(I - DR)-1 (I - D)
[s(p)_1£(p) - sis)-1z(s)] (23)
Finally, from equations 22 and 23 follows

15
J(T,e,S) = -[r’(p)S(p)_1 - r'(8)S(8)_l
(IrR)(1-pR)-(.-D)[s(p)-,_z(p)-s(S)-t(5)]
+e[r0(8)-r’(5)S(8)-1s(8)] (24)
+(1 - e)[ro(p) - I'(p)s(pr‘s(p)]
This is the form of the cyclic average performance measure that is the most useful.
2.4 Model Fitting with Nonlinear Least Squares Methods
This topic is not directly related to the above discussion which led to an expression
for the performance measure, but this fitting, used in some parts of this work, is an
optimization technique and belongs in a chapter discussing the theory behind the overall
project. The type of problem that is being considered here is one where data are being
collected from some experiment and parameters for a model have to be determined. One
can manipulate the data in some fashion, such as semilog or log-log plotting, to find
necessary parameters, but, if no apparent manipulation exists for the proposed model, then
some other method must be used. One can attack such a problem by varying the
parameters of a model and determining how good the fit is to the data with those guessed
parameters. One can use methods of optimization to find the best way to vary the
parameters. The method of choice was Levenberg-Marquardt optimization due to its wide
use in prepackaged computer programs such as MATLAB and Kaleidagraph. A
descripdon of the problem being examined, along with a short description of Levenberg-
Marquardt optimization, is appropriate here. A further description of the method of
optimization used can be found in numerical analysis texts such as Numerical Recipes [24],
Let us consider an experiment where data are taken at several times during the run.
In figure 2, an example experiment where three different types (x 1, x2, and x3) of data are
collected is shown.

16
In an experiment like that shown in figure 2, the experimenter collected all three
different types of data at the same instant This type of collection is preferable for later
computational putposes, but is not necessary with the method to be described In other
words, if one of the data types is difficult to collect simultaneously with the other data
types, then the following analysis still applies, but the computational effort may be
increased.
Let it be assumed that a general model for a system like that shown in figure 2 is
given in equation 2, but the vector of ordinary differential equations is also a function of
parameters. In other words
(25)
where p=the vector of parameters for the model
For such a system, the parameter vector p and, occasionally, the initial condition
vector xo (or just some parts of either of these vectors) must be determined. For any set of
guessed parameters and/or the initial conditions, the model equations can be integrated,
either analytically or by numerical methods such as Runge-Kutta integration, to show the
predictions for that set of guesses. The model predictions for this problem would be
computed for each time instant that data is available for comparison.
A performance measure for "goodness" of model fit then can be described as
(26)
where P=performance measure
(j)
xd =one type of data point at time instant j (xj is a vector of all data)
x-^^corresponding model prediction for xd (x is a vector of predictions)
qi=weight of one type of measurement (q is the vector of the weights)

o 17
1Ü|
t â–¡
(
¿ A
O
O 1 2
3 4
Time
O
A
n.
5
O
o Xl
â–¡ x2
A x3
á
7
Figure 2. An example experiment where, at any time point, three different types of
data (xl, x2, and x3) are collected.

18
nsp=n umber of sampling points
ndt=number of data types
Our goal is to minimize the stated performance measure. The weighting factors q
serve two purposes. First, they can be used if one type of measurement is more "trusted"
than another. For example, one may give more weight to a simple measurement of
temperature taken with a thermistor than to a viscosity measurement taken with a poor
viscometer. This use of the weighting factor is going to be subject to the good judgment of
the experimenter and should be handled with caution. The other, more important, purpose
of the weighting factor is as a normalization constant. Alternative forms of the performance
measure, such as summing the logarithms of the squared residuals instead of the actual
squared residuals, (xd_ -x¡) , can also be useful for normalizing.
Once the performance measure is expressed in the form of equation 26, the
optimization method of Levenberg and Marquardt can be used. This method is an elegant
combination of steepest gradient and inverse Hessian methods of optimization . The
general algorithm involves use of steepest descent methods far from the minimum, and
then, as the minimum becomes more closely approximated, a smooth transition to the
inverse Hessian methods [24, page 523-524], A prepackaged program was used in this
work.

CHAPTER 3
EXPERIMENTAL METHODS
3.1 Organism Description
The organism used in this project was Escherichia coli strain LCB898. The original
culture was obtained from Dr. L.O. Ingram, Department of Microbiology and Cell Science,
University of Florida. The genotype is thr\ leu§ ton A21 str lac Y1 sup E44 pfl\ [37]. The
important aspect of this organism is its mutation in the pfl gene causing lack of expression
of that gene. This is the gene for the production of pyruvate formate-lyase (pfl), an enzyme
which is primarily responsible for the conversion of pyruvate to acetyl-CoA and formate
under anaerobic conditions.
The pertinent biochemical pathways for this organism are shown in figure 3 which
was prepared based on the diagram in Pascal [38]. It points out the following important
features: lack of pfl activity [37], anaerobic inhibition of the pyruvate dehydrogenase
(PDH) pathway [38,39], and anaerobic induction of the d-lactate dehydrogenase (LDH)
pathway. Further descriptions of this organism's metabolism will be given in later
chapters.
The culture was maintained on plates of rich broth agar with the follow'ing
composition in deionized water [40]: 10 g/1 tryptone , 5 g/1 sodium chloride , lg/1 yeast
extract, and 15 g/1 agar. The agar was prepared by mixing the ingredients together in
water, heating the solution to near boiling while stirring, and then pouring approximately
12 ml aliquots of the molten agar into individual 16X125 mm tubes. These tubes were
capped and autoclaved at 121 °C for 30 minutes. Subculturing was performed on a monthly
19

20
Glucose
2NAD+->
'sr
2NADH
2ADP
2ATP
\T
2 Pyruvate
I
Aerobic
4-
Anaerobic
2NADH 2NAD
LDH
\2 D-Lactate
\
Aerobic
\
\
â–  2NAD
PDH
>2N
ADH
/
/
Figure 3. Pertinent Biochemical Pathways in E. coli LCB898.

21
basis with the freshly inoculated plates being incubated at 37°C in a Fisher Model 255D
incubator for 24 hours then being stored in a refrigerator.
3.2 Analytical Methuds
3 2.1 Biomass and Cell Number Determination
The amount of biomass in the system was determined by two different methods,
cell counting and spectrophotometric turbidity (actually absorbance) measurement The cell
counting involved serial dilution of a culture aseptically withdrawn from the culture vessel.
The dilutions were made in 8.5 g/1 solution of sodium chloride in water[41, p. 434] and the
agar used had the same composition as the rich broth agar used in culture storage.
Preparations involved pouring several 9 ml aliquots of the saline solution into 16X125 mm
tubes and pouring several 99 ml aliquots into bottles. Additionally, some empty capped
tubes were also prepared. These were all autoclaved for approximately 30 minutes.
Sterile, disposable, individually-wrapped borosilicate glass pipets (Fisher pipets) were
used in this procedure. A sample procedure for serial dilution and colony counting of a
culture is given in the two following paragraphs. The serial dilution procedure should be
performed under a laminar flow hoed.
This sample procedure is designed for experiments where cell counts are expected
to be between thirty million and three hundred million cells/ml. Since it is desired to dilute
to 30-300 cells/ml, it is necessary to dilute samples 100000X, one million X, and ten
million X for later counting. Obviously this method can be modified for other expected cell
counts. Approximately 15 minutes before a sample is to be taken, three agar tubes are
placed in a boiling water bath in order to melt the tube contents. Immediately before the
measurement, the tubes are withdrawn from the water bath and stored in the laminar flow
hood, along with a propipettor, a pack of sterile gloves, three labeled sterile disposable
Fisher petri dishes, an appropriate number of pipets, dilution bottles, and tubes. A fresh

22
pipet should be used after each dilution step. The sample is removed from the culture (if
from a chemostat, the effluent sidearm tube (to be described later) is flamed and
approximately 10 ml are allowed to flow out into a presterilized empty tube; if from a
flask, the cap is removed under the laminar flow hood and a sample is aseptically pipetted
out) and one ml is pipetted from the tube or culture flask into a ninety nine ml saline bottle.
This bottle is then well shaken. At this point the original culture has been diluted 100 X. A
one ml sample is then aseptically transferred from the first dilution bottle to another ninety
nine ml saline bottle. Again, this bottle is well shaken. The culture sample has now been
diluted 10000 X. A one ml sample is then aseptically transferred to a 9 ml sample tube and
the tube is well shaken. 2 ml of the 100000X diluted sample is then taken. One ml is put
into the next 9 ml sterile tube and the other milliliter is pipetted onto the appropriately
labeled petri dish. One tubeful of agar, when it becomes lukewarm, is then poured into the
petri dish and the dish is subsequently mildly swirled in order to provide a more even
distribution of colony forming units. The new 1 million X dilution tube (now a 10 ml tube
with the one ml of sample added) is handled in the same way as the previous 100000X
tube. Lastly, the ten million X dilution tube has one ml transferred to the appropriate petri
dish. The agar in the petri dishes is allowed to solidify. The dishes are then inverted and
incubated at 37°C for 24 hours.
Upon sufficient incubation, the petri dishes are taken out of the incubator and are
individually placed on a Quebec darkfleld colony counter. A hand-held colony counter is
used to ensure accounting. A marking pen is used in order to make dots under each colony
appearing on the plate so that no colony gets counted twice. When counting, care must be
taken to count colonies on the edge of the petri dish and to check if some of the colonies are
growing directly underneath another colony. The results of each plate count are recorded,
and the cell counting procedure is finished.
Biomass concentration was measured spectrophotometrically using a Milton Roy
Spectronic 20D spectrophotometer. The procedure for this measurement is relatively

23
simple. The spectrophotometer is first set for operation at 550 nm. Deionized water is then
added to a clean cuvet, the cuvet is put into the sample chamber, and zeroing is performed.
The cuvet is then removed and the water is poured and then shaken out of the cuvet. Some
sample is then added and the cuvet is swirled. This sample is used to eliminate the effects
of residual water. At this point, the first amount of sample is poured out and at least three
ml of fresh sample are added. The cuvet is then placed into the sample chamber again and
an absorbance reading is then taken and recorded. If the absorbance reading is above 0.4,
appropriate dilution is performed. For example, in more dense samples, three ml of
deionized water would be added to one ml of sample for a 1/4 dilution. During the aerobic
operation experiments, 1/16 dilutions were necessary towards the end of the batch runs and
during continuous steady states.
In order to correlate the spectrophotometric absorbance reading to an actual dry
mass concentration, a calibration curve has to be obtained. A large (approximately 200 ml)
sample is taken from the reactor (at the end of a batch run). Eighty ml each of 75%, 25%,
and 12.5% dilutions are made in three separate beakers. The spectrophotometric
absorbances of each of these samples is then measured. In more turbid samples, when the
absorbance readings are significantly above 0.4, the diluted samples are used for
computation of the absorbance. Fifty ml of each of these dilutions are pipetted into
appropriately labeled centrifuge lutes. The tubes are then placed into a Precision Universal
Centrifuge set at 2000 rpm for 60 minutes. The supernatant is then decanted, the remaining
contents are washed with approximately 5 ml of deionized water, and the tubes are
centrifuged for 20 more minutes. The supernatant was pipetted off and the remaining
contents are then emptied into preweighed and labeled petri dishes for drying. These
dishes are dried at 105°C for 48 hours and the contents are weighed. In this manner, a
calibration curve was prepared. The calibration curve is shown in figure 4.

Dry Mass Concentration (g/1)
24
Figure 4. Dry Mass Calibration Curve. The linear fit shown is
Dry Mass Concentration (g/l)=0.565*Absorbance
with a squared regression coefficient of 0.998

25
3.2.2 Glucose Anal\si;>
The device used tor glucose concentration determination was an Analytical
Research Model 110 Glucose Monitor. The analyzer's main component is the
electrochemical sensor on which an immobilized glucose oxidase membrane is mounted
This enzyme catalyzes the reaction between glucose and oxygen to produce hydrogen
peroxide. Hydrogen peroxide is detected by the sensor and an electrical signal proportional
to glucose concentration is produced [42, p. 52], The analyzer's pumps will take a small
(approximately 1.5 ml) aliquot of the sample through a port and pass most of it by the
membrane, leaving only a small plug of fluid for the actual analysis. This analysis is
performed in approximately three minutes, at which time almost all the glucose will have
reacted. Usually three aliquots are measured and the results are averaged. The following
paragraph gives a brief description of the actual procedure.
In the description that follows, one cycle is defined as the time between glucose
analyzer samplings. The glucose analyzer is operated as follows. Glucose calibration
solutions of appropriate concentrations (operator's judgment) are prepared and allowed to
dissolve for at least 2 hours. Fifty percent dilutions of calibrations should also be
prepared. For example, if calibration is to be made with a 2 g/1 glucose solution, a 1 g/1
glucose solution should also be prepared. Additionally, the glucose analyzer is switched
from idle to blank for 2 hours before calib^tions are to be done. After these two hours the
calibration tube is placed in 100 ml of fresh water and the switch is set to "Cal". For the
next 15 minutes, water is allowed to pump through the system. After this 15 minutes, the
glucose analyzer is zeroed by setting the zero dial so that the peak readout during the
portion of the cycle between the ready light indicator coming on and the following sampling
reads ".000" g/1. One more cycle is observed to check for appropriate zeroing. At this
point, calibration of the glucose analyzer is performed. The sample tube is placed in the
appropriate calibration solution. The switch is then set to "Sample". Three cycles are
allowed to follow and the calibration is completed by setting the cal dial so that the peak

26
value of glucose read during the portion of the cycle between the ready light turning on and
the following sampling reads out the calibration value of the glucose solution. This
calibration is then checked with a 50% solution of the calibrator solution by putting the
sample tube into the appropriate calibrator solution. Three samples are taken and
calibration is checked by reading the appropriate peak value. If calibration is appropriate,
samples can than be measured for glucose content. Four samplings are taken, with the last
three peaks being recorded as data. Recalibration is performed once every hour that
sampling is done. If the calibration stays relatively accurate, a calibration check with 50%
calibration solution need not be completed. If glucose analysis results are below 1/2 of the
top calibration value, recalibration is performed at one half the calibration value.
3.2,3 d-Lactate Measurement
The measurement of d-lactate was based on a modification of Sigma procedure 816-
UV, which was designed for measurement of 1-lactate. The major modification to this
method was the use of d-lactic dehydrogenase (Sigma L-2395) instead of 1-lactic
dehydrogenase, provided with the original Sigma kit The principie of this test is explained
below.
In the metabolism of E. coli and most other chemoheterotrophic bacteria, pyruvate
is converted by lactate dehydrogenase into lactate. This, however, is a reversible reaction.
In other words, the same enzyme can be used to convert lactate into pyruvate. The
conversion of d-lactate to pyruvate will be accompanied by a reduction of one molecule of
NAD+ into NADH. NADH shows strong spectrophotometric absorbance at 340 nm,
whereas NAD+ does not absorb at this wavelength. Thus, using the indirect method of
spectrophotometrically measuring NADH in a mixture, the amount of d-lactate can be
determined. The problem of backconversion of the pyruvate into d-lactate is handled by
adding hydrazine to the mixture. Hydrazine reacts with the pyruvate and forms a complex
that d-lactic dehydrogenase cannot convert back into d-lactate.

27
The experimental method involved preparation of a calibration curve of d-lactate
concentration against spectrophotometric absorbance at 340 nm. The spectrophotometer
used was a Milton Roy Spectronic 20D. A solution of approximately 200 mg/1 d-lactic acid
(Sigma L06251 or the Lithium salt (Sigma LI000) in deionized water (approximately in this
case means that the experimenter knows the exact concentration within experimental
accuracy, but it is not necessarily exactly 200 mg/1 in concentration) was prepared when
calibration was performed. When the lithium salt was used, adjustments were made for the
weight of a lithium atom as opposed to a hydrogen atom in the free acid form. Dilutions of
approximately 87.5%, 75%, 62.5%, 50%, 37.5% and 25% were made of this solution.
Additionally, a solution of 2.50 g NAD+ (Sigma N7004) and 500 ml Glycine buffer
(Sigma 826-3) added to 1 liter of water was prepared. The mixture will be referred to as
"NAD solution" from this point. Since the NAD solution must be made immediately before
the lactate measurement, the actual amount prepared would depend on the number of lactate
samples to be analyzed. One would prepare 3 ml of the NAD solution (2 ml of water, 1 ml
of glycine, and 5 mg of NAD+) per sample, plus at least one extra 3 ml solution for
preparation of a blank. 2 8 ml aliquots of NAD solution were pipetted into the appropriate
number of labeled test cuvets plus one blank cuvet Each labeled test cuvet had 0.2 ml of
the corresponding full strength or diluted calibration solution added and mixed .
Additionally, the blank cuvet had 0.2 ml of deionized water added and mixed. The 340 nm
spectrophotometric absorbances of each test cuvet against the blank were then measured
This value, which will be referred to as the zero absorbance, was used to compensate for
cuvet-to-cuvet variability.
Sixty units of d-lactic dehydrogenase were then added to each of the test cuvets and
the blank cuvet. The cuvets were then incubated at 37°C for 30 minutes. The new
absorbances against the blank were then measured, and the differences between the new
absorbances and the corresponding zero absorbances were then calculated and plotted
against the corresponding test lactate concentrations. The curve obtained is shown in figure

28
5, along with the results of a least squares fit to the data. Although the substance measured
is referred to as d-lactate throughout this work, the term d-lactate is actually slightly
inaccurate. D-lactic acid is the substance actually being measured. Frequently these The
relatively high correlation coefficient indicates that the assumption of a linear relationship in
this range of test lactate concentrations is satisfactory.
The procedure for measuring unknown lactate concentrations is essentially the same
as that for measuring the net absorbances of the calibration solutions. The key differences
will only be described. First, the unknowns must be diluted however many times to where
their lactate concentration is in the range between 50 and 200 mg/1. The amount of dilution
is usually based on previous methods, but for the first experiments this had to determined
by trial and error. Second, this dilution is also useful in diluting the effects of any residual
biomass or other substances in the filtered samples on the spectrophotometric readings.
Any remaining residuals would be taken into account by the zero absorbance measurement.
These residuals will not affect the net absorbance measurements as the enzyme used is
specific for d-lactate. Last, least, and most obvious, the calibration curve is used to
determine unknown concentrations, as opposed to the preparation of the curve when
calibrating.
3.2,4. Other Analyses
Other measurement methods were used in this research. These will only be
mentioned briefly as they were only seldomly used. These include amino acid analysis and
ethanol analysis. The amino acid measurements were made by an outside laboratory
(Interdisciplinary Center for Biotechnology Research, University of Florida, Gainesville,
Florida) on an amino acid analyzer. These amino acid measurements were used solely to
determine whether or not leucine or threonine nutritional limitations were encountered.
Ethanol concentrations, when measured, were determined using Sigma kit 332-UV.

d-Lactate Concentration (mg/1)
29
Test Absorbance @340 nm
Figure 5. d-Lactate Calibration Curve. The linear fit shown is
d-Lactate Concentration (mg/l)=260.144*Absorbance
with a squared regression coefficient of 0.998

30
3.3 Feed Medium Composition
Many factors had to be considered in the design of the feed composition and
preparation. The first consideration was whether to use a complex medium such as Luna
broth or a minimal medium. A glucose minimal medium was chosen as the probability of
interference of feed components with measurements is lower. Once a minimal medium was
chosen, other considerations had to be taken into account. These included pH, buffering,
nitrogen requirements, trace minerals and metals, genetic deficiencies, substrate amount,
and interactions between these components during heat sterilization. These interactions
must be considered when deciding which component solutions to autoclave in the same
flask. Ideally, each component solution should be autoclaved separately. However, in
order to maintain sterility during the mixing process, there should be as few separate flasks
as possible. Thus, given the considerations described below, a design was chosen
inbetween these two extremes. A general consideration given was the separation of
inorganic from organic components in order to avoid production of toxic byproducts during
autoclaving. Finally, all the ingredients were pre^ared in deionized water.
The medium should be buffered w ith target pH 7. Buffering lowers the amount of
base needed to maintain pH during the experiments. M9 medium [41, p. 431 and 43, p.
A.3] satisfies this requirement. Additionally the nitrogen and some of the trace mineral
requirements are satisfied by the use of M9 medium. M9 medium includes sodium
phosphate dibasic, potassium phosphate monobasic, sodium chloride, ammonium chloride,
calcium chloride and magnesium sulfate. The exact amounts used will be given later.
Miller [41, p. 431] suggests separate autoclaving of calcium chloride and of magnesium
sulfate from the rest of the salts.
Other trace metals were added to the medium as suggested elsewhere[40,44,45].
These included selenium oxide, hydrated ferrous sulfate, hydrated ammonium molybdate,
and hydrated manganese sulfate . The amounts used will be given later. It was suggested
these should be sterilized by filtration, but sterility was a major concern in this work, so

31
these compounds were autoclaved together in solution. Bridson and Brecker [46]
suggested autoclaving metals separately from phosphates in order to avoid precipitation.
Genetic deficiencies of E. coli LCB898 had to be accounted for in the medium
formulation. The genotype mentioned in the organism description indicates requirements
for threonine, leucine, and thiamine. Additionally, as the pyruvate-formate lyase gene is
mutated, acetate may be required to satisfy some biosynthetic requirements of the cell.
Several batch experiments were performed under both aerobic and anaerobic conditions in
order to determine the amounts of these chemicals required to insure glucose limitation of
growth. Amino acid analysis was performed on a sample of the batch at the point where
growth was no longer seen to see if any residual amino acids were left in solution. The
final amounts chosen were those that allowed glucose limited growth up to at least a
concentration of 4 g/1 glucose in the medium. This was indicated by the cessation of
growth when glucose became exhausted. Miller [41, p. 431] suggests separate
autoclaving of the amino acids and vitamin from the M9 salts, and Bridson and Brecker[46]
suggest separate autoclaving of the amino acids from the carbohydrates to avoid Maillard
reactions. Thus the amino acids and vitamin solutions were autoclaved together separately
from all other components, and, to avoid any other possible feed reactions, the acetate was
also separately autoclaved.
The final feed component to be discussed is the growth limiting substrate of
glucose. Glucose (Sigma) was used as the limiting substrate because of its relatively
straightforward measurement on the glucose analyzer previously mentioned. A main feed
concentration of 4 g/1 was decided on for several reasons. A feed glucose concentration too
low would make batch growth measurements difficult, as the analyzer available is
somewhat inaccurate at measuring glucose concentrations below 100 mg/1, and d-lactate
measurements below 25 mg/1 are also of questionable accuracy. Another reason to avoid
low glucose concentrations is the desirability of visible turbidity in the reactor system. For
example, in a batch growth experiment, the onset of visible turbidity serves as a marker for

32
the beginning of more frequent measurements. This onset is still well before glucose
exhaustion and indicates the approximate point where glucose, d-lactate, and biomass
concentrations begin to measurably deviate from the starting values. On the other hand
high glucose concentrations also are not beneficial. First of all, autoclaving of high glucose
concentration solutions will lead to increased carmelization of the feed glucose. Also,
higher glucose concentrations represent higher biomass concentration. During aerobic
growth aeration may become insufficient at higher biomass concentrations. Buffering,
amino acid addition, and other feed components would have to be increased in
concentration. Finally, thick growth may cause other experimental problems such as
increased effluent tube wall growth, high amounts of base addition to maintain pH, and
more difficult sampling and cleanup. Thus, an intermediate glucose concentration of 4 g/1
was picked. However, any glucose concentration from approximately 3 g/1 to 10 g/1 would
also have satisfied the above criteria. Glucose solutions were autoclaved separately from
all other components to avoid all possible cross reactions (e.g. Maillard reactions). As
glucose concentration was the one feed component measured during all experiments, this
was the component that was most important to keep "pure".
The final feed composition used is given in Table 1.
3.4 Feed Preparation
The feed medium was prepared in three different configurations as follows: shake
flask, reactor batch, and continuous feed. Each will be discussed separately, and the flask
grouping listings in table 1 will be used. The term "2.7X concentrated" will hereafter be
used to refer to concentration higher than that listed in table 1 by a factor of 2.7. For
example, a 2.7X concentrated solution of flashing group 1 would be a solution of 10.8 g/1
glucose in water. One stock solution was used in all three configurations, a 100X

33
Table 1. Feed medium recipe with flasking divisions
Feed Ingredient
Amount added per liter of
deionized water
Flasking Division
Glucose
4 g
1
KH2PO4
3 g
2
NaCl
0.5 g
2
NH4CI
1 g
2
Na2HP04
6 g
2
Threonine
0.5 g
3
Leucine
0.5 g
3
Thiamine
5 mg
3
CH3COONa«3H20
1.66 g
4
MgS04
0.24 g
5
CaCl2
11.96 mg
6
SeCL
1.1 mg
7
FeS04»7H20
27.8 mg
7
(NH4)5Mo7»4H20
1.765 mg
7
MnS04*H20
1.69 mg
7
Note: For example flask 2 would include Na2HPC>4, KH2PO4, NaCl, and NH4CI

34
concentrated solution of flasking group 7. One liter of this was prepared when necessary.
Ten milliliters of this solution were used for every liter of feed solution.
When shake flasks were prepared, stock solutions were used. The stock solutions
prepared were 100 ml bottles of 25X concentrated solutions of flasking groups 2-6.
Glucose solution was freshly prepared for each shake flask. In order to illustrate the
preparation of a shake flask, the preparation of the usual 250 ml amount of flask medium
will be given. 10 ml of each of the group 2-6 bottles are pipetted into separate tubes and
the tubes are capped. 2.5 ml of the group 7 flask are also pipetted into a tube. Finally,
197.5 ml of deionized water is poured into a shake flask and 1 g of glucose is added. The
flask is then capped with a paper towel and foil and tied with a string. All of these are then
autoclaved for 25 minutes. Finally, upon cooling of the ingredients, the ingredients are
poured together into the shake flask under a laminar flow hood.
The medium preparation for startup batches is described next. Although the startup
medium descriptions in this section do not appear to "add up" to the concentrations given in
Table 1, when the actual startup procedures are considered later, the final startup
composition does "add up" to the correct medium. Two liters of 1.4 X concentrated
flasking group 2 salts are prepared and poured into the chemostat and then autoclaved
within the reactor. A 56 ml aliquot of the flasking group 7 stock solution is pipetted into a
glass flask. Four additional glass flasks with 150 ml each of deionized water are set aside.
To each of those flasks 5.6 times the mass listed in table 1 for one flasking group is added,
for groups 3,4,5, and 6. These flasks are capped with paper towel and foil and then tied
with string. The glucose solution is prepared by adding 22.4 g of glucose to 944 ml of
water in the main feed flask, which is shown in figure 6. The main feed flask preparation
is as follows. A new Supor filter is placed in the Fisher 47 mm filter holder. The
connections are as follows: the flask outlet tube leads to its own filter holder, which leads
to the needle for puncturing into reactor. The needle is wrapped in foil wrap. Additionally,
there is a tube for a sterile nitrogen inlet into the flask for replacing the emptied fluid. The

35
Figure 6.
Main Batch Feed Flask Diagram

36
nitrogen introduced passes through a Bacti-Vent air filter, The rods going through the
rubber stopper are made of glass. The main feed flask and the glass flasks are autoclaved
for 30 minutes. After these cool down, all of the flasks are placed under a laminar flow
hood. The metal clamp on the main feed flask is loosened and the contents of the glass
flasks are added. The metal clamp is then retightened and the feed is ready to be added to
the reactor, as will be described later in the description of reactor startup.
Finally, the continuous feed medium preparation will be described. Sixteen liters of
continuous feed medium were made in any single batch, thus sixteen times the amount of
each flashing group to be added per liter is added to separate amounts of water. The
procedure for a single batch is described. In three large flasks, 3.5 liter aliquots of
deionized water are added, along with separate additions of 16X of the Table 1 masses for
flashing groups 1, 2, and 3. For example, in the glucose flask, 64 g of glucose are added
to 3.5 liters of water. The group 7 flask has 160 ml of the appropriate stock solution added
and then filled with deionized water to 3 liters. The acetate flask has 26.56 g of
NaAcetute»3H20 added to 500 ml of water. The calcium chloride flask has 192 mg of
CaCl2 added to 250 ml of water and the last flask has 3.943 g of MgSG4»7H20 added to
250 ml of water. Finally, the feed carboy, diagrammed in figure 7 is prepared as follows.
Two fresh .22 |i Supor filters are placed in the appropriate filter holders, along with a fresh
Bacti-Vent air filter for sterile nitrogen introduction. The carboy is then autoclaved empty
and uncapped. One and a half liters of deionized water are then poured into the carboy and
it is capped. The tubes are then clamped, the needle and air filter are covered with foil
wrap, and the whole apparatus (except the N2 bag and the magnetic stirrer) is autoclaved
for 30 minutes. All of the flasks are covered with paper towel and foil wrap and then tied
with string. They are also autoclaved for 30 minutes. The cap of the carboy is removed
and the contents of the other flasks are added under a laminar flow hood.

37
Figure 7. Continuous Feed Carboy Diagram
Magnetic
Stirrer

38
3.5 Experimental Operation
3.5.1 Operational Condition^
The temperature that the experiments were operated at was 37°C. This temperature was
chosen as it is the normal optimal temperature for growth of E. coli. The other major
environmental variable that was held constant was pH. It was suggested [40] that
somewhat acidic pH's gave higher batch yields of d-lactic acid. As pH 7 is the optimal pH
for E.coli growth, the possibility of cycling pH in addition to aeration was examined.
Preliminary experiments were performed early in this investigation to examine the effects of
pH on the anaerobic batch yield of d-lactate on glucose. The results will be described
briefly. These experiments were performed before any aerobic experiments were and,
thus, a slightly different medium formulation was used as shown in table 2. It should be
emphasized that the pH effects experiment was the only one described in this work using
the table 2 recipe. All other experiments used the main recipe given in Table 1.
The use of shake flasks will be described later. A starter flask with an initial medium
composition described in table 2 in 500 ml of deionized water was inoculated, placed in a
37°C AO constant temperature shaker bath and kept there until the flask contents appeared
turbid. Three test flasks were p-epared during the growth phase of the starter flask cultures
with 150 ml of medium in each. The medium ingredients of these three flasks were the
same as those in Table 2, but the concentrations of each were set so that, upon dilution with
100 ml of liquid, they were the same as those given in Table 2, except that the amounts of
potassium phosphate mono- and dibasic were varied to give the desired pH values.

39
Table 2. First Medium Recipe
Glucose
3.0 g/1
k2hpo4
4.9 g/1
kh2po4
3.0 g/1
NaAcetate
0.2 g/1
(NH4)2S04
0.1 g/1
CaCl2
0.2 g/1
MgSO4‘7H20
0.1 g/1
FeS04»7H20
0.05 g/1
L-Threonine
0.05 g/1
L-Leucine
0.05 g/1
Thiamine
0.005 g/I\

40
When the starter flask reached the appropriate turbidity, the three test flasks were
each inoculated with 100 ml of the starter culture. Spectrophotometric absorbances, pH
values and lactic acid concentrations were then measured at half-hour intervals. The results
of this experiment are given in table 3. The large increase in lactate concentration in the pH
7 flask over a 5 hour period, along with the large increase in absorbance over a 26 hour
period, would indicate that the operating pH should be kept at a value around seven. An
additional benefit in choosing constant pH operation is that shifts in pH are difficult
Buffering is generally desirable in microbial systems, but buffering would require higher
base or acid amounts to be added to cause a shift. High addition of these solutions have a
diluting effect on the culture and thus will interfere with measurements. This problem,
though, may possibly be overcome by use of gases such as CO2 and N2 instead of acid or
base additions to manipulate pH.
3.5.2 Shake Flask Experimental Procedure
The shake flasks were prepared as described in the section on feed preparation.
Further procedural details will be given here. In a starter culture, after the contents were
poured together under a laminar flow hood, a small inoculum was taken off of the culture
storage agar dish with a sterile loop and then transferred into the combined medium flask.
In other types of shake flasks, liquid inocula may be used instead of the agar culture.
When liquid inocula were used, only small amounts (~1 ml) were usually added. After
inoculation, the flask was recapped with the paper towel/foil wrap cap and tied. The flask
was then placed in an AO shaker bath set at 37°C. If the shake flask culture was to be used
for reactor inoculation, it was usually left in the bath for approximately 12 hours. For yield
experiments, the flasks would be left in for longer periods. The usual shake flask volume
used was 250 ml.

41
Table 3. Effects of pH on Lactate Production
Initial pH
7
Initial Lactate Cone.
(mg/1)
105
Initial Absorbance
(550 nm)
0.186
Lactate Cone, after 5
hours (mg/1)
206
550 nm Absorbance
after 5 hours
0.200
pH after 5 hours
6.98
Lactate Cone, after
26 hours (mg/1)
2900
550 nm Absorbance
after 26 hours
0.49
pH after 26 hours
5.24
6.4
5.85
110
110
0.186
0.186
151
119
0.205
0.175
6.4
5.87
1340
292
0.37
0.22
5.26
5.38

42
153 Reactor Experimental Procedure
^ 5.3.1 System Description
The reactor experimental procedures will now be described. The reactor, a
Bioengineering KLF 2000, was used for all of the batch and continuous experiments. A
diagram of the reactor system is shown in figure 8. In this figure the long dashed lines
represent measurements for the chemostat control unit, and the dotted lines represent
control outputs. Except where mentioned later, the reactor volume was always maintained
at 2 liters. If the system is in continuous mode, a load cell is used to determine the system
weight A peristaltic pump maintains a constant effluent flowrate. The pumping rate is
calibrated by collection of the fluid in a graduated cylinder. When a small drop in weight is
detected the control unit activates the influent pump until the reactor is back up to its
operating weight. Only small deviations in the level were allowed. The pH value of the
system was measured with an Ingold Ag/AgCl pH electrode and controlled by a
Bioengineering M7832N pH controller. This pH control maintained constant pH by
controlling pumps for previously autoclaved 1 M HC1 and NaOH solutions prepared
separately. The constant 37°C temperature was maintained by a ptlOO temperature sensor,
a Bioengineering K54450 controller, and an 800 watt heater. Agitation for the reactor was
set at 700 rpm, and a baffle cage within the reactor helped insure good mixing. Dissolved
oxygen was monitored with a Cole-Parmer polarographic electrode and a Cole-Parmer
Model 5513 dissolved oxygen meter. When anaerobic conditions were necessary, filtered
(as shown in figure 8) Alphagaz oxygen-free nitrogen was bled over the top of the culture
at the rate of approximately 30 ml/min. When aeration was necessary, an Air Cadet pump
was used. It pumped air from underneath a UV hood, through a filter apparatus identical to
the same as that used for nitrogen, and through a sparging tube with the outlet bent
underneath the bottom rotor blade within the reactor. Two and a half vvm was the air
flowrate obtainable with this pump into the two liter reactor. When periodic switching of

43
' r
i ;
Feed
Carboy
1 :
1 i
i ;
V
)
Flowrate
pH
controller
controller
Agitador»
control
Temperature
controller
Chemostat
Control
Unit
Figure 8.
Experimental Reactor Setup

44
the aeration was performed, a slow flow of nitrogen was continuously maintained over the
top of the culture so as to maintain a positive pressure (which helps prevent outside
contamination). Switching of conditions during the aeration cycling was performed by
simply plugging the Air Cadet pump into an X-10 wall module and setting switch times on
an X-10 computer interface (X-10 (USA) Inc. 185A LeGrand Ave. Northvale, NJ). The
gas was released through a Bioengineering gas outlet apparatus. This had a cooling jacket,
which was maintained at about 10°C in order to minimize evaporation of culture volume.
This cooling was accomplished by continuous pumping of cooled Prestone antifreeze
through a Hotpoint refrigerator and freezer using a Manostat pump set on its lowest
pumping speed. The end of the effluent gas hose was placed in a dilute Betadine solution
in order to help prevent contamination.
3.5.3.2 System Startup and Oper.itior
The following startup procedure was used for all batch and continuous
experiments. It should be emphasized that all continuous runs were stuned as batch runs.
The only variations were whether anaerobic or aerobic procedures were going to be used.
The cycling runs were all started up under aerobic conditions. Prior to startup, the effluent
tube, acid, base, and inlet gas filters were autoclaved. A batch feed was also prepared.
Additionally, a starter shake flask culture was prepared and inoculated 12 hours prior to
reactor inoculation. The chemostat was filled with the appropriate salts as described in the
batch feed description to the 2000 ml level. The topcap and bottom attachment rings were
secured in place. The pH electrode was precalibrated to pH 7 and 4 and inserted into the
topcap. The reactor was now ready for autoclaving.
The autoclaving of the reactor was done in situ. Here, the pH electrode was
pressurized to 30 psi by connecting it to an air cylinder, the stirrer was set to 800 rpm, the
gas outlet was opened, and temperature set-point of the chemostat was changed to 121 °C.
When the temperature reached 99°C, the gas outlet was closed. The temperature was

45
allowed to reach 121 °C, and was kept there for 30 minutes. After 121 °C was held for 30
minutes, the temperature set point was changed to 104°C. When reactor temperature
reached 104°C, the first thing that was done was to turn on the pump for the gas outlet
apparatus reflux coolant The reactor pressure was raised by using either air from the Air
Cadet pump or nitrogen from a cylinder. The gas outlet was also opened immediately. The
reactor temperature set-point was then changed to 37°C in preparation for the actual
experiment.
After the reactor cooled down to 37°C, the effluent, acid, and base tubes were each
placed through peristaltic pumps and then appropriate connections w ere made to the reactor
using aseptic technique. The effluent tube was immediately clamped to avoid loss of
reactor liquid. The main feed flask (described in the batch feed preparation) was then
connected to the reactor. Eight hundred milliliters were then pumped into the reactor for a
total volume of 2.8 liters. The reactor conditions were then set to pH 7, 37°C, and 700 rpm
agitation. Finally, 20 ml of inoculum were taken by a sterile syringe from the starter flask
under a laminar flow hood and then injected through one of the reactor seals. The reactor at
this point was prepared and inoculated. A batch run was thus begun.
Immediately after inoculation a sample was aseptically taken through the effluent
sidearm aseptically into a previously autoclaved and capped tube. Serial dilution was
immediately performed and the remaining sample had its absorbance measured and then
was centrifuged for ten minutes in a Fisher Centrific 228 centrifuge. After centrifugation
the sample was filtered using a syringe and MSI Magna Nylon 66 .22 p. filters and then
heat shocked for five minutes in boiling water. Finally, the sample was allowed to cool to
room temperature and then stored in a freezer for later glucose and lactate analysis.
Frequent measurements were taken during batch runs. The acid and base levels
were monitored and recorded. Upon the absorbance values reaching above 0.4,
appropriate dilution with deionized water was performed in order to obtain a measurable
absorbance. The batch measurements were performed until absorbance stopped increasing.

Up to this point, the batch and continuous experiments were performed in a synonymous
manner (except for the obvious need for preparation of a continuous feed carboy during the
batch start of a continuous run). If a run was intended to be strictly batch, measurements
were continued for several hours into the stationary phase. If a run was intended to be a
continuous run, the system was switched into a continuous mode before the end of
exponential phase. The point of switching was usually about one hour before the expected
end of exponential phase. This end point was estimated using absorbance measurements
and comparing them with previous batch results.
The following procedure was used to switch the reactor from the initial batch mode
to a continuous mode. The feed carboy was connected to the reactor after its tube was led
through the influent peristaltic pump. Subsequently the reactor volume was lowered to 2
liters by draining through the effluent tube. The weight set point was entered into the
chemostat control unit, the effluent flowrate was set, and the reactor was then in a
continuous mode.
Measurements were taken in the same fashion as in the batch runs. They were
taken at least three times on a daily basis, but usually the frequency was much higher.
Additional considerations during continuous operation were daily monitoring of the base
reservoir, checking of the tubing and overall system condition, contamination testing,
effluent disposal, and feed carboy preparation and changing.
Contamination testing was usually performed by the following two methods:
preparation of a shake flask deficient in the appropriate amino acids and inoculation with
reactor contents, and microscopic examination of a Gram-stained sample of the reactor
contents. In the former method, an additional control flask was prepared with the
appropriate amino acids. The two flasks were seeded with identical volumes of reactor
volume. They were then examined for growth after overnight shaker bath incubation. If
the deficient flask showed significant growth, then the reactor was declared contaminated.
This method is somewhat dubious, though, as the deficient flask could be selective for

47
leucine and threonine revertants. Thus, the primary method of contamination testing was
the mentioned microscopic examination. If the slide appeared to have only red rods, the
continuous operation was declared successful to that point
When operating in continuous mode a new feed carboy had to be prepared daily.
The feed was changed by switching the quick connect fitting at the end of each carboy tube
while all ends were immersed in rubbing alcohol. The quick connect change in alcohol,
along with the second filter placed between this connect and the reactor inlet, helped insure
sterility.

CHAPTER 4
ANAEROBIC GROWTH OF E. COLI LCB898
4.1 Background
Under anaerobic environmental conditions, where no alternate electron acceptors
such as nitrate, fumarate, or sulfate, are available, Escherichia coli uses fermentation as its
pathway for energy production. Fermentation, as defined by Brock [47, p.802], is a group
of catabolic reactions producing ATP in which organic compounds serve as both primary
electron donor and ultimate electron acceptor. When compared to aerobic or anaerobic
respiration, fermentation is not a very efficient method of producing ATP and, thus,
overall cell biomass [48, p. 54]. In order to understand fermentation, the reactions
involved are briefly described. The important reactions are diagrammed in figure 9 (this
diagram was drawn with the help of Neidhardt and Brock [49, p. 153 and 47, p. 126]).
First, the metabolic pathway common to both aerobic and anaerobic metabolism, pyruvate
formation, will be examined. After this, fermentative pyruvate dissimilation will be
considered.
The pathways of glucose degradation to pyruvate shown in figure 9 are the
Embden-Meyerhof-Pamas (EMP) pathway and the pentose-phosphate pathway. Typically,
in E. coli grown anaerobically on glucose, 92-95% of the glucose will be degraded by the
EMP pathway and 5-8% will be degraded by the pentose-phosphate pathway [50,51]. The
common first reaction to both pathways is the phosphorylation of glucose to glucose 6-
phosphate. After this point, the split between the two pathways occurs. The pentose-
phosphate pathway's major roles are formation of pentose phosphates for nucleotide
48

49
y)>AD?
Glucose 6-phosphate
Glucose
r
ATP
'NADP+ NADPH
1 V/
6-Phospho-
gluconolactone
6-Phospho-
gluconate
Fructose 6-phosphate
ATP
r*
ADP
Fructose 1,6-diphosphate
(Fructose 1,6-diphosphate
splits into two
Glyceraldehyde 3-phosphates)
Glyceraldehyde 3-phosphate ^
NAD+
>¡>NADH
1,3-Diphosphoglycerate
k,adp
EMP
Pathway
ATP
3-Phosphoglycerate
I
2-Phosphoglycerate
Phosphoenolpyruvate
ADP^I
NADP
)> NADPH
yf
Pentose
5-phosphate
U
Erythrose
4-phosphate
Pentose-Phosphate
Pathway
Succinate
)^>NAD+
^-NADH
Fumarate
t
Aspartate
NAD+
^-NADH
Oxaloacetate
ATP ADP
V ±
->• Succinyl-CoA
a- Ketoglutarate
I
Anabolic
—> Citrate
t
NADPH
.+
NAD+^v
d-Lactate
NADP
Isocitrate
Acetyl-P >• Acetate
Acetaldehyde Ethanol
NADH NAD*, Co A NADH* NAD+
Figure 9. Main Anaerobic Biochemical Pathways in E. coli LCB898

50
biosynthesis and NADPH generation [48, p.31]. The EMP pathway will be described in
the following paragraph.
During pyruvate formation, a net yield of 2 moles of ATP per mole of glucose is
obtained by substrate-level phosphorylation. Additionally, 2 moles of NADH, the main
source of reducing power for biosynthesis, are generated. Some of the EMP pathway
intermediate metabolites also serve as biosynthetic precursors. These include fructose 6-
phosphate and phosphoenolpyruvate. The point at which aerobic and anaerobic
metabolisms differ is the degradation of pyruvate.
In wild-type E. coli cells, pyruvate is normally dissimilated under anaerobic
conditions by two pathways with no additional ATP generation, one being catalysed by
pyruvate formate-lyase (pfl), the other by d-lactate dehydrogenase (ldh) [52, p. 151]. Pfl is
inactive under aerobic conditions [53]. The products of the pfl degradation include
formate, acetate, ethanol, CO2, and H2 [49, p. 163]. The sole product of the lactate
dehydrogenase pathway is d-lactic acid. In anaerobic wild-type E. coli K12 batches [54],
only traces of d-lactic acid are produced. This would indicate that the ldh pathway is not
normally used.
In E. coli LCB898 a K12 mutant, a mutation exists in the gene responsible for
production of pyruvate-formate lyase, thus closing off that pathway for pyruvate
dissimilation [37,38,55]. High yields of d-lactic acid from glucose therefore are expected
in this mutant An additional consequence is that when growth is anaerobic in a minimal
medium, the addition of acetate may be required since acetyl-CoA cannot be produced
without the action of pyruvate formate-lyase.
To summarize, two points must be reiterated. The first is that when glucose is
processed through the EMP pathway, a net yield of 2 ATP molecules for every glucose
molecule degraded is observed. During aerobic metabolism, which is to be described later,
oxidative phosphorylation can also be employed. It will be shown that the net ATP yield
per glucose is much higher when the additional phosphorylation is performed. Again, ATP

51
yield is directly proportional to growth yield. Thus, relatively low biomass yields under
anaerobic conditions are expected. The second main point is that E. coli LCB898 should
show high yields of d-lactate on glucose
4,2 Batch Growth
In all of the following discussed batch results, time 0 represents the point at which
inoculation of the reactor was inoculated. The results for one of the anaerobic batch runs,
to be designated anaerobic batch run 1, are shown in figures 10 and 11. This was one of
the preliminary runs to help determine a final feed composition. It is clear that biomass
concentration stopped increasing well before glucose in the system was exhausted from the
results shown in figure 10. Any glucose consumed after 46 hours was strictly being used
for maintenance. The results shown in figure 11 indicate that the glucose was largely being
converted into lactate. In this experiment only 50 mg/1 of each amino acid and no metals
were used in the medium. It was hypothesized (and later confirmed with amino acid
analysis) that threonine was exhausted at the point of entering stationary phase.
In the next batch experimental run, anaerobic batch run 2, the amino acid
concentrations were doubled, but metals were still not added to the medium. The results
for this experiment are shown in figures 12 and 13. Again, as can be seen in figure 13, the
bacteria appeared to enter a stationary growth phase before glucose was exhausted. Amino
acid analysis showed excess threonine and leucine. Other work with this organism
[40,44,451 was then reexamined and metals were then added to the final formulation of the
medium.
In figures 14-17 all of the experimental results for a batch run with sufficient amino
acids and metals added to the medium are shown. This run is designated anaerobic batch
run 3. At 27 hours, the point of glucose exhaustion, growth had essentially stopped, as
can be seen in figures 14-16. During this growth phase, as shown in figure 17, lactate was
being produced in what appears to be a growth-associated manner. Lactate concentration

52
Biomass Concentration (mg/1)

53
3000
^ 2500-
E
^ 2000-
o
£
g 1500-
o
c
o
« 1000-
2
o
J H
J 500-
0^.
0
• Lactate Concentration
A Biomass Concentration
20 30 40
Time (hours)
Figure 11. Anaerobic batch run 1. Lactate and biomass concentration against time.
Biomass Concentration (mg/1)

54
350
0
O
<3
2
VI
Vi
n
o
3
O
a
3
cr
p.
o
3
3
Figure 12. Anaerobic batch run 2. Glucose and biomass concentration against time.

55
4000
3500-
!§3000-
3 -
o 2500-
« :
g 2000-
c
Ó 1500
O
§ 1000-
.3
500
o 4-
o
• Lactate Concentration
A Dry Mass Concentration
A
AA *
-300
rrrjTTrrjT
5 10
Tlr
15
20
i-rryr
25
350
-250 2
p
Vi
Vi
n
o
3
o
a
3
3
c.
o
3
3
£
-200
-150
-100
-50
0
30 35
Time (hours)
Figure 13. Anaerobic batch run 2. Lactate and biomass concentration against time.

Dry Mass Concentration (mg/1)
56
300.000
250.000
200.000
150.000
100.000
50.000
0.000
0 5 10 15 20 25 30 35
Time (hours)
Figure 14. Anaerobic batch run 3. Biomass concentration against time.

Cell Number (Cells/1)
57
7.000E+11-]
6.000E+11 -
5.000E+11 -
4.000E+11-
3.000E+11-
2.000E+11 -
l.OOOE+11 -
O.OOOE+O -* ’iii—i—[-
0 5
1 I"'1 1
15
f â–  | i â–  i i i | i i
25 30
Time (hours)
Figure 15. Anaerobic batch run 3. Cell number concentration against time.
'f T
35

Glucose Concentration (mg/1)
58
Figure 16. Anaerobic batch run 3. Glucose concentration against time.

Lactate Concentration (mg/1)
59
3000.000
2500.000-
2000.000-
1500.000-
1000.000-
500.000-
0.000-
• •
• •
0.000 5.000 10.000 15.000 20.000 25.000 30.000 35.000
Time (hours)
Figure 17. Anaerobic batch run 3. Lactate concentration against time.

60
did continue to increase beyond 27 hours, but this may be explained by a slow release of
the lactate from the cells. The low lactate concentration seen at 27 hours was probably due
to experimental error in the measurement, as the previous point indicated a lactate
concentration approximately 300 mg/1 higher. Appropriate values for growth parameters
for this run will be given later in this chapter. Approximately 2 hours after entering the
stationary phase, the lactate measurements shown in figure 17 showed an interesting
response. The lactate showed a sharp dip with a subsequent increase. Ordinarily this may
have been simply dismissed as experimental error. A later experimental run was performed
under the same conditions as anaerobic batch run 3, to be designated anaerobic batch run 4,
and its results are shown in figures 18-21. The lactate results for this run are shown in
figure 21. The previously mentioned dip was also seen here. This dip was not of major
importance to this work as it occurs during stationary phase. This work primarily involved
exponential phase growth in continuous operation. However, possible explanations for it
are appropriate. The simplest hypothesis is experimental error. This hypothesis is not
probable though as it has been seen more than once. Another is that the cells may have
been growing diauxically on some other substance in the medium such as acetate, leucine,
or threonine. This would have been an unusual diauxy, though, as biomass concentration,
shown in figures 14 and 18, did not seem to increase during the period of glucose
exhaustion. The cells may have been using one of the listed substances in the medium for
maintenance while ingesting d-lactate along with this substance and later releasing the
lactate for some unknown reason. Still another hypothesis is that some of the cells may
have been genetically reverting to a pfl+ form (to be discussed further in a later chapter).
The properties of the revertant are not completely understood and this could have been
some effect of the revertant's growth alongside with the non-revertants. Again, whatever
the true explanation for this behavior may be, it was not of primary importance to this work
as most of the work performed in this project was on exponential phase growth.

61

Cell Number Concentration (Cells/1)
62
9E+11
8E+11
7E+11
6E+11
5E+11
4E+11
3E+11
2E+11
1E+11
0E+0
0 5 10 15 20 25 30 35 40
Time (hours)
Figure 19. Anaerobic batch run 4. Cell number concentration against time.

63
4000
3500 A
13000-
| 2500-
S
S 2000-
o
c
<3 1500-
L>
i/i
O
3 1000
O
• • •
•• •
500-
T I I T' I !-!
I "F I 'P"| I I 1 f
r I I —T“f 1 T"7 1 I "I I I
0
0.000 5.000 10.000 15.000 20.000 25.000 30.000 35.000 40.000
T
T
Time (hours)
Figure 20. Anaerobic batch run 4. Glucose concentration against time.

64

65
4.3 Continuous Growth
All continuous anaerobic experiments were run with the standard feed medium
shown in table 1. In all of the following discussions of continuous results, time 0
represents the point at which the reactor was inoculated. The point of switch from batch to
continuous operation will be mentioned for each run. It should be reemphasized that all of
the continuous runs were started as anaerobic batch runs. The switching point was chosen
on the basis of reactor turbidity so as to avoid washout and glucose exhaustion. The
results for one of the anaerobic continuous experiments, anaerobic continuous run 1, are
shown in figures 22-25. This experiment was performed at a dilution rate of 0.164 hr1.
(Dilution rate is defined as the ratio of flowrate to reactor volume. In reactor engineering
terminology, this represents the reciprocal of the residence time.) The switch to continuous
operation in this run was done at 83 hours. In figure 22, it is seen that a steady state,
defined as where the state variables of the system are unchanging, was seen at
approximately 108 hours. Data collected past 130 hours show surprising results and these
will be presented and discussed in Chapter 8. Some of the results of another continuous
anaerobic experiment (with dilution rate of 0.17 hr1), anaerobic continuous run 2, are
shown in figures 26 and 27. The increased number of data taken at the switch point show
the smooth change in system condition from batch to continuous operation. In this
experiment, a steady state was seen at approximately 44 hours.
Values for the averaged apparent steady-state values for various dilution rates of cell
number, biomass, lactate and glucose concentrations are given in table 4. These values
were taken from the shown continuous experiments along with other continuous anaerobic
experiments. The results show low values for residual glucose concentration and high
conversions to lactate. The biomass concentration is also consistently low. Surprisingly,
the biomass concentration seemed to rise w ith dilution rate, which is not in agreement with
normal Monod behavior. These increases were not very large though. In contrast to
biomass, glucose did follow the expected Monod behavior of increasing with dilution rate.

Dry Mass Concentration (mg/1)
66
300
250-
200-
150-
100-
50-
0
• •
• • •
• ••
T
—'—I—1—1—'—I—1—1—1 "I '1 ■■—1—I—1—r~*"'7 1—1
0.00 20.00 40.00 60.00 80.00 100.00 120.00
Time (hours)
Figure 22. Anaerobic continuous run 1. Biomass concentration against time.
140.00

Cell Number Concentration (Cells/1)
67
8.00E+1W
7.00E+11-
6.00E+11
5.00E+11-;
4.00E+11 -
3.00E+11-
2.00E+11 -
1.00E+11
0.00E+0
*—I 1—« J*“l 1 1 *| *1
«•
II' [ I I I
1 I â– ' 1
0 20 40 60 80 100 120
Time (hours)
Figure 23. Anaerobic continuous run 1. Cell number concentration against time.
140

68
4000-1-
3500-
*§> 3000-
E
s-/
o 2500-
o
e
E 2000-
O
c
1500-
O
J 1000-
o
500-
Time (hours)
Figure 24. Anaerobic continuous run 1. Glucose concentration against time.

69
4000
'S,
E,
c
©
â– a
g
c
©
o
c
o
U
2
o
«
3500-
3000-
2500-|
2000^
1500
1000
500-
0-
0
• •
*—i r—+-
• •
—I—i 1 1 1
20 40 60 80
Time (hours)
100
120
Figure 25. Anaerobic continuous run 1. Lactate concentration against time.
140

Biomass Concentration (mg/1)
70
-i—i—i—i—|—i—i—i—i—| i i famm\ i r-1 r | » i rT"| i i i i
0.00 10.00 20.00 30.00 40.00 50.00 60.00
Time (hours)
Figure 26. Anaerobic continuous run 2. Biomass concentration against time.

71
4000
3500-
§3000-
E
§ 2500-
â– a
£
g 2000-
o
c
U 1500-
B
3
8 1000-
500-
0.00
• •
• •
• •
i 11 i i i
10.00
20.00
30.00
40.00
50.00
Time (hours)
Figure 27. Anaerobic continuous run 2. Lactate concentration against time.
60.00

72
Table 4. Apparent Anaerobic Continuous Steady States
Dilution Rate
Biomass
Cell Number
Glucose
Lactate
Concentration
Concentration
Concentration
Concentration
(hr1)
(mg/1)
(cells/1)
(mg/1)
(mg/1)
.17
283
3402
.164
248
7.65xlOn
136
3214
.13
240
6.7X1011
86
3397

73
4.4 Modeling
4,4.1 Presentation of Model
The following model was proposed to describe this system under batch anaerobic
conditions during exponential growth
^ _ M-rnax,anaerobic^
K -L c
^s.anaerobic 3
* 1 1-^max.anaerobic^
Y IC +«
1 x/s,anaerobic ^s,anaerobic ~ s
(28)
(27)
* _ l^max,anaerobic^
IX — IX /v
K
s, anaerobic
+ S
(29)
where q= time derivative of q
x=biomass (dry mass) concentration in the reactor
s=substrate (glucose) concentration in the reactor
p=product (d-lactate) concentration in the reactor
Umax, anaerobic^1^'11111111 growth rate under anaerobic conditions
Ks, anaerobic=saturation parameter under anaerobic conditions
Yx/s, anaerobic=yield of biomass on substrate under anaerobic conditions
a=growth-associated lactate production parameter
The first model equation, equation 27, shows simple Monod dependence of biomass
growth rate on glucose concentration. Equation 28, representing the time derivative of
glucose concentration, was chosen with the assumption that all glucose consumed during
exponential growth was consumed for the purpose of producing biomass. The third
equation, equation 29, gives the time derivative of d-lactate concentration in such a system.

74
It was assumed that the d-lactate production under exponential growth conditions was
strictly anaerobically growth-associated.
A continuous form of this model is as follows
x =
Fmax,anaerobic^ _
K -L C
^s,anaerobic 3
(30)
s = —
1
M-max,anaerobic^
Y • K . -L c
1 x/s,anaerobic â– rvs,anaerobic T 3
x + D(sp -s)
(31)
* Fmax,anaerobic* x
r K . Q r
^s,anaerobic a
(32)
, Flowrate
where D=dilution rate=
Volume
sp=feed substrate (glucose) concentration
The above continuous form of the model was developed for a continuous stirred tank
reactor under conditions of perfect mixing. No biomass or product was introduced in the
feed, so no feed lactate or biomass was accounted for in this model. Inlet substrate was
accounted for in the substrate equation. The equations were simply extensions of equations
27-29 with additional terms for dilution of the biomass, substrate, and product out of the
reactor.
4,4,2 Model Parameter Fitting
The batch run measurements were the primary ones used for parameter
computation. The primary justification for this was that the anaerobic continuous runs
described previously were not performed until the ending of this work. Also, they showed
problems with reversion, which will be described in a later chapter, and measured steady
states were difficult to find. The parameters used in later modeling will be given here,
along with justification for their determination. Improved values for model parameters are

75
also given, but these were not the ones used in designing the later cycling operations still to
be described.
The first parameters to be discussed are the maximum growth rate, JJ-max anaerobic,
and the saturation constant, Ks, anaerobic- These parameters would usually be determined
by use of a Lineweaver-Burke plot [56, p.106] of continuous data. In these types of plots,
a large number of continuous steady states are necessary. Under steady-state conditions,
the time derivatives of the continuous form of the model are all equal to zero. By
manipulation of the steady-state version of 30, the following relation can be obtained
D
K 1
â– lvs, anaerobic 1
+ â– 
1
M-max,anaerobic ^ss M-max.anaerobic
(33)
where sss=steady-state residual glucose concentration in the reactor
A plot could then be rru'de of reciprocal dilution rate against reciprocal residual glucose
concentration. A least-squares fit would then be made to the data. The intercept would be
equivalent to the reciprocal of the maximum anaerobic growth rate, and the slope would be
equal to the saturation parameter divided by the maximum anaerobic growth rate.
Unfortunately, due to the reversion problems mentioned above, it was very difficult to
obtain continuous steady-state data. Equation 33 can be used on one data point to find one
of the two unknown parameters if the other can be satisfactorily estimated. It can be seen
that at a dilution rate equal to half of the maximum growth rate, the residual substrate in the
reactor is equal to Ks> anaerobic- In anaerobic batch run 3 the dry-mass growth rate was
0.190 hr1 and in anaerobic batch run 4 it was 0.23 hr1. These were obtained by simple
exponential fitting of the biomass data during exponential phase. For example, the data
used to calculate anaerobic batch run 3’s growth rate was the dry mass data taken between
16 and 27 hours. The correlation coefficient for the fit (R2) was greater than 0.99. It was
assumed that these growth rates were at the maximum, as the glucose concentration was
very high during most of the duration of the runs. Were this not the case, the R2 would not

76
have been so close to 1. For the sake of consistency, anaerobic batch run 3 was the run
that served as the source of the data used for fitting model parameters. The reason why this
run was chosen over anaerobic batch run 4, was that the run 4's growth rate of 0.23 hr1
was significantly higher than that seen in other experiments performed. Most experiments,
including anaerobic batch run 1 and 2, indicated growth rates of the culture having values
between 0.19 and 0.20 hr-1. Therefore, the final value of |!max, anaerobic used was 0.190
hr1. The two Ks, anaerobic parameters then calculated using equation 33 for the anaerobic
continuous steady states given in table 4 are then 22.4 and 40.4 mg/1 for dilution rates
0.164 and 0.13 hr'1, respectively. An average Ks> anaerobic of 31.4 mg/1 would then be the
apparent value to be used. For several of the simulations described later the anaerobic
continuous experiments had not yet been performed. For these, a Ks, anaerobic value of 98
mg/1, based on preliminary continuous data using the feed medium listed in table 2, was
used for several of the later simulations in this dissertation. Use of the revised parameter
value of 31.4 mg/1 is suggested for future work with this model.
The next parameter to be determined was the yield of biomass on glucose under
anaerobic conditions, Yx/S, anaerobic- Using equations 30 and 32 and the assumption of
steady state the following relation for continuous operation can be stated
xss=Yx/s, anaerobic(sF'sss) (34)
where xss=steady-state biomass concentration in the reactor
Using this relation, yield values of .064 and .061 —g biomass werg ¡Seated for di]ution
mg glucose
rates 0.164 and 0.13 hr1, respectively. Anaerobic batch run 3 was the base run used and
the biomass yield was computed for this using a material balance argument which is very
similar to equation 34.
x=Yx/s(So'S)
(35)

77
where So=initial glucose concentration in the batch
In figure 28, a plot of biomass against consumed substrate is shown. The line fit to the
data and forced through zero is also exhibited. The slope of this line gives the value of
Yx/s, anaerobic determined by this experiment The fit indicates a yield value of
0.063 m§ biomass ^Qwever^ as can ^ seen from ^g j^gg amount of scatter around the
mg glucose
line, this was not a very good fit. The correlation coefficient for this curve, R2, was only
0.90. Since the correlation coefficient was so low, the yield used in calculation was
computed by simply averaging the biomass concentrations found in the stationary phase
and dividing the average by the initial glucose concentration of 4000 mg/1. The yield
mg biomass
computed in this manner was 0.068
mg glucose
. This was the value used in the modeling
work. The fit value of 0.063 biomass ^j^g^ out t0 ^ ¡n g0(Kj agrément with the
mg glucose
later continuous results, and, thus, this result should be used in the future.
The final parameter to be discussed for the anaerobic model is the anaerobic growth
associated lactate production parameter, a. Using equations 30, 32, and the steady-state
assumption, the following continuous steady-state relationship holds
(X = ^ss
(36)
where pss=steady-state product (d-lactate) concentration
The continuous runs indicate values for a of 12.02, 12.95, and 14.15
mg d - lactate
mg biomass
for
dilution rates of 0.17,0.164, and 0.13 hr1, respectively. The variability in these values is
quite large and leaves some of the d-lactate measurements in question. The determination
of a for a batch run is done by using equation 27 and equation 29 to obtain

Dry Mass Concentration (mg/1)
78
Figure 28. Anaerobic batch run 3. Biomass concentration against glucose consumed.

79
p = ax (37)
Subsequent integration yields
p=ax+p0+ax0 (38)
where p0=initial d-lactate concentration
Xo=initial biomass concentration
The slope of a plot of biomass against d-lactate concentration during exponential phase for
a batch run is equivalent to the a parameter. This plot for anaerobic batch run 3 is shown
in figure 29, along with the results of a linear least squares fit to the data. The problem
with this method is that, due to the scatter of the d-lactate values seen in figure 29, the final
value is very' dependent on where exponential phase is declared to begin. For example, if
all of the data that were used for the computation of anaerobic batch run 3's growth rate
was used, an a parameter of 8.64 mg/mg (with a correlation coefficient, R2, of only 0.88)
would be the determined value. This correlation coefficient was unacceptably low. If the
last data point, due to its questionable reliability, was ignored the a parameter value
increased to 9.64. If the only points considered on this figure were those between 100 and
250 mg/1 biomass, an a parameter of 16.7 mg/mg would be found. Due to all of this
scatter, a batch value for a had to be determined by another batch run. Anaerobic batch run
4, in spite of its high growth rate, was then examined and used to obtain an estimate of a.
The results of the corresponding lactate against biomass curve are presented in figure 30,
along with linear least squares fitting results. The data showed a much higher degree of
linearity than that seen for anaerobic batch run 3. The a parameter fit for these data was
10.8 with a correlation coefficient, R2, of 0.98. This was the value used in later
simulations. Upon comparison of this value with those given by the continuous runs, this

80
Figure 29. Anaerobic batch run 3. Lactate concentration against biomass concentration.

81
value for a would underpredict lactate concentration by approximately 20%. Increased
values of a should be considered in future use of the model.
The parameter values are summarized in table 5. The suggested revised a
parameter was based on averaging the results for the determined a parameters of the three
continuous experiments. The large differences in updated parameter values for yield,
saturation and a parameters suggested that use of the model in designing any operation
must be checked for sensitivity to these parameters.

82
Figure 30. Anaerobic batch run 4. Lactate concentration against biomass concentration.

83
Table 5. Anaerobic Model Parameter Values
Parameter
Value used in later work
Possible improvement
Pm ax. anaerobic
.19 hr1
.19 hr1
Ks, anaerobic
98 mg/1
31.4 mg/1
Yx/s, anaerobic
.068 mg/mg
.063 mg/mg
a
10.77 mg/mg
13 mg/mg

CHAPTER 5
AEROBIC GROWTH OF E. COLI LCB898
5.1 Backgound
Under aerobic conditions most facultative aerobes, such as E. coli, will adjust their
metabolism to take advantage of available oxygen and increase its production of biomass.
The first part of aerobic metabolism of glucose is production of pyruvate. This part
was described in the previous chapter on anaerobic growth. The only major difference to
be mentioned here is that 25% of the glucose will enter the pentose-phosphate pathway
under aerobic conditions [50,51]. Where the major difference between fermentation and
aerobic respiration on glucose lies, is in the fate of pyruvate. The aerobic metabolism of
glucose for a wild-type £. coli is shown in figure 31 [49, p. 155, 45]. The aerobic
metabolism of E. coli LCB898 shouldn't deviate significantly from this as no significant
aerobic metabolism genes are mutated in its genome. Pyruvate dehydogenase, which is
only produced aerobically [39], is the enzyme responsible for aerobic conversion of
pyruvate to acetyl-CoA. After it is formed, acetyl-CoA is sent through the TCA cycle. In
the TCA cycle only one ATP molecule (per molecule of pyruvate) is produced by substrate-
level phosphorylation, specifically the conversion of succinyl-CoA to succinate [48, p. 54],
All of the other ATP molecules are generated by oxidative phosphorylation, where the
energy is produced by the transfer of electrons from NADH, NADPH, and FADH2 to
oxygen [49]. As can be seen in figure 31, from each turn of the cycle two molecules of
NAD+, one of NADP+, and one of FAD+ are reduced, with one additional NAD+
reduction occurring in the conversion of pyruvate to acetyl-CoA. In E. coli, two molecules
of ATP can be generated for each molecule of NADH or NADPH, and one molecule of
84

85
ATP can be produced for each molecule of FADH2 oxidized [48, p. 43]. To sum up, ten
molecules of ATP can be produced, when the TCA cycle is employed, for every molecule
of pyruvate processed. When one adds, per glucose molecule, the two ATP molecules
generated by fermentation, the 2 ATP's that can be generated from the 2 molecules of
NADH produced (and not needed in d-lactate formation) and the TCA generated ATP's
from the two pyruvate molecules formed, 26 molecules of ATP can be produced for every
glucose molecule, as opposed to the synthesis of just two molecules of ATP for each
glucose molecule strictly fermented.
Of course, these ATP yields are theoretical. Fermentative efficiency can be as high
as 50%, as opposed to the TCA cycle efficiency of 39% [47, p. 134]. Nonetheless, even
though pure fermentation has higher efficiency, the aerobic metabolism will give much
higher amounts of ATP per glucose molecule consumed than will pure fermentation. Since
growth requires ATP for energy, aerobic growth should show higher biomass yields on
glucose.
There are two main products of aerobic grow th of E. coli on glucose, specifically
biomass and CO2 [57, p.802]. Three CO2 molecules are produced by the TCA degradation
of each pyruvate molecule, and thus 6 CO2 molecules are produced for every glucose used
strictly for catabolism.
5 2 Experimental and Modeling Result-» Introduction
Most of the aerobic experiments were performed in conjunction with a master's
thesis project performed by Christina Stalhandske [58]. The experiments were jointly
performed by this author and Stalhandske and the modeling work, while contained in that
work, was mostly performed by this author. Thus, most of the results, both experimental
and modeling, are taken from this thesis.

86
>¡>ADP
Glucose 6-phosphate
Glucose
r
ATP
!NADP+ NADPH
!
i
~i
n
6-Phospho-
gluconolactone
6-Phospho
gluconate
Fructose 6-phosphate
ATP
^-NADP+
NADPH
Pentose
5-phosphate
Fructose 1,6-diphosphate
(Fructose 1,6-diphosphate
splits into two
Glyceraldehyde 3-phosphates)
Glyceraldehyde 3-phosphate 4-
NAD+
^ NADH
1,3-Diphosphoglycerate
^ADP
Y FAD
ATP
3-Phosphoglycerate
i
2-Phosphoglycerate
Phosphoenolpyruvate
ADP^I
ATP
Pyruvate
Succinate 4
FADH
4
Fumara te
4
Malate
NAD+"X
NADH '
Oxaloacetate
Succinyl-CoA
A
TCA
Cycle
^>NADH
^-NAD+
a- Ketoglutarate
Acetyl
CoA
NADPH
NADP
â– > Citrate
Isocitrate
Figure 31. Main Aerobic Biochemical Pathways in E. coli LCB898

87
5.3 Batch Growth
In all of the following discussed batch results, time 0 represents the point at which
the reactor was inoculated. The results for one of the aerobic batch runs, to be designated
aerobic batch run 1, are shown in figures 32-35. This was one of the preliminary runs to
help determine a final feed composition. As can be seen from figure 32, glucose was not
exhausted when biomass concentration stopped rapidly increasing at 14 hours. Biomass
did increase slightly after 14 hours, but not in an exponential manner. Glucose did
continue to drop, but it is hypothesized that this glucose was largely used for maintenance.
Amino acid analysis showed that threonine was exhausted before glucose . The lactate and
ethanol concentrations of each sample were measured to see if they were produced in a
significant amount. These quantities turned out to be negligible. Thus, in the following
strictly aerobic batch and continuous runs ethanol and lactate measurements were not
measured regularly.
The results of a batch run with sufficient amino acids , aerobic batch run 2, are
shown in figures 36-38. This time the cells grew exponentially until glucose was
exhausted, which indicated that glucose was the limiting substrate. It appeared that 15
hours after starting the run, cell number and biomass concentration stopped increasing in an
exponential manner. Further increases in dry mass can be attributed to diauxy on
remaining amino acids or acetate present
5.4 Continuous Growth
All continuous aerobic experiments were operated with the standard feed medium.
In all of the following discussed continuous results, time 0 represents the point at which the
reactor was inoculated. The point of sw’itch from batch to continuous operation will be
mentioned. It should be reemphasized that all of the continuous runs were started as
aerobic batch runs. The switching point was determined on the basis of turbidity and
chosen so as to avoid washout and glucose exhaustion.

88
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0 5 10 15 20 25 30 35
Time (hours)
Figure 32. Aerobic batch run 1. Glucose and biomass concentration against time.
• •
“I—I—I-*—p » >
Biomass Concentration
Glucose Concentration
""{ M| i i -i | T i i t r i r i ""T~ ^ t

Cell Number Concentration (Cells/1)
89
3.00E+12
2.50E+12
2.00E+12
1.50E+12
1.00E+12
5.00E+11
0.00E+0
0 5 10 15 20 25 30 35
Time (hours)
Figure 33. Aerobic batch run 1. Cell number concentration against time.

90
30
25-
20-
£
c
o
•a
s
g 15-
u
c
6
a io-
3
u
«
-J
5-1
0-
• •
T
5
—i 1 1 1 1 r-
15 20
Time (hours)
0
10
25
Figure 34. Aerobic batch run 1. Lactate concentration against time.
30
35

91
60-T
50-
E
o 20-
§ ;
•5
W
10-
0 - '"i i ■— i i | ii i i j
0 5 10
t—r
~i T 1~* r â–  1
15 20
1 l r
25
Time (hours)
Figure 35. Aerobic batch run 1. Ethanol concentration against time.
~r i-r
30
35

92
3500
3000 H
/—s
*§>
¿2500-
c
o
i 2000-
c
u
(J
J 1500-1
C/5
3
^ loco¬
es
Q
500 -1
0
0
I T I
. . ) —1 ■ 1 " T —1 ■ | T ! T" T " |
10 15 20 25
Time (hours)
30
Figure 36. Aerobic batch run 2. Biomass concentration against time.
r t 1
35

Cell Number Concentration (Cells/1)
93

94

95
The results for one of the aerobic continuous experiments, aerobic continuous run 1, are
shown in figures 39-41. This experiment was performed at a dilution rate of 0.335 hr1.
The switch to continuous operation was done at 13.5 hours. In figure 39, it is seen that a
steady state, defined as where the state variables of the system are unchanging, is reached
at approximately 59 hours.
Values for the averaged apparent steady-state values for various dilution rates of cell
number, biomass, glucose, and cell number concentrations are given in table 6. These
values were taken from the shown experiments along with other continuous anaerobic
experiments. All of the lactate measurements for these runs only showed trace amounts.
The results show low values for residual glucose concentration. The biomass
concentration is also consistently high. The biomass concentration dropped, the residual
glucose concentration increased and the average cell size increased with increasing dilution
rate. This behavior of the biomass and glucose concentration is in agreement with standard
Monod behavior [59,11, p.101].
5.5 Modelinti
5.5.1 Presentation ofMuJv.i
The following model was proposed to describe this system under batch aerobic
conditions during exponential growth
(39)
^s, aerobic + ^
S = -
(40)

Dry Mass Concentration (mg/1)
96
2500.000
2000.000
1500.000
1000.000
500.000
0.000
0 10 20 30 40 50 60 70 80
Time (hours)
Figure 39. Aerobic continuous run 1. Biomass concentration against time. D=.335 hr1

Cel! Number Concentration (Cells/1)
97
8.00E+12
7.00E+12
6.00E+12
5.00E+12
4.00E+12
3.00E+12
2.00E+12
1.00E+12
O.OOE+O
0 10 20 30 40 50 60 70 80
Time (hours)
Figure 40. Aerobic continuous run 1. Cell number concentration against time. D=.335 hr1

98
5000-1
4500 -j
•
»
'Z- 4000-
•
•§> :
••
¿3500^
•
c
â– a 3000 A
b
5 2500-j
c
& 2000-
v
O 1500-
3
b looo-
500-j
0-^
—1 1 1 r-T 1 1 1 1 I I 1 1 1 1—1* 1 T-l 1—1 ^ -1 1 1—1 1—1—1 1—P ■! *1 P‘ I *1 "T- ' | 1—
* T * I j 1 I I I j I 1 1 1 | 1 1 1 I J 1 1 I 1 1 1 1 1 I 1 1 1 I 1 ] I I l I
0 10 20 30 40 50 60 70 80
Time (hours)
Figure 41. Aerobic continuous run 1. Glucose concentration against time. D=.335 hr1

99
Table 6. Apparent Aerobic Continuous Steady States
Dilution Rate
(hr1)
Biomass
Concentration
(mg/1)
Cell Number
Concentration
(cells/1)
Glucose
Concentration
(mg/1)
Per Cell
Mass
(mg)
.263
2200
1.2xl013
70
l^xlO-M
.45
1920
4.9xl012
203
3.92xl0-10
.335
2000
6.7xl012
93
2.99xl0-10

100
n = ; n
K + <:
^s, aerobic 3
l^max, aerobic^
(41)
where q= time derivative of q
x=biomass (dry mass) concentration in the reactor
s=substrate (glucose) concentration in the reactor
n=cell number concentration in the reactor
Umax, aerobic-ma*imum growth rate under aerobic conditions
Ks aerobic=saturation parameter under aerobic conditions
Yx/s, acrobic=yield of biomass on substrate under aerobic conditions
The first model equation, equation 39, shows simple Monod dependence of biomass
growth rate on glucose concentration. Equation 40, representing the time derivative of
glucose concentration, was chosen with the assumption that all glucose consumed during
exponential growth was consumed for the purpose of producing biomass. Equation 41 is
similar to 39, and was based on the assumption of constant cell mass.
A continuous form of this model is as follows
M-rnax, anaerobic^
(42)
s = -
max,anaerobic^
X + D(Sp - s)
(43)
Y K +«
1 x/s,anaerobic ^s,anaerobic ' 3
max, anaerobic^
(44)
where D=dilution rate=
Flowrate
Volume
SF=feed substrate (glucose) concentration

101
The above continuous form of the model was developed for a continuous stirred tank
reactor under conditions of perfect mixing. No biomass was introduced in the feed, so no
feed biomass is accounted for in this model. Inlet substrate was accounted for in the
substrate equation. The equations were simply extensions of Equations 39-41 with
additional terms for dilution of the biomass and substrate out of the reactor.
5 Model Parameter Fitting
Methods similar to those used for determining the anaerobic model parameters
could have been used for the aerobic model parameters as well. However, these methods,
except for the yield determination, were not used here. The project described in the thesis
of Stalhandske [58] involved nonlinear least squares methods that were described in the
prior chapter on theoretical methods. The parameters from her thesis were the ones used in
later simulations. In this thesis an alternate model, originally described elsewhere
[11,61,62], was also presented and modeling results were also given for this model. This
model will only be briefly presented. This alternate model, designated as the RLS model,
is represented by the following equations
(45)
(46)
n = : n
K +7
^s, aerobic '
M^max,aerobic^
(47)
z = a(s - z)
(48)

102
where v=parameter incorporating both geometric and maximum growth rate
factors
K s, aerobic» Ks a=adaptability parameter
z=weighted average of previous substrate concentrations
This model was derived by considering cellular surface area-to-volume ratio and normal
Michaelis-Menten kinetics. Basically, in equation 45, (n/x)1^ is proportional to surface
area-to-volume ratio for the cell and s/K's>aerobjc+s is proportional to specific glucose
uptake per unit surface area of the cell. Thus v represents a lumped proportionality factor
for the geometric and maximum growth rate terms. This model predicts instantaneous
change in biomass specific growth rate (actually it predicts an overshoot to the new steady
state), but a more gradual, monotonic change in number of cells specific growth rate. A
chemostat form of this model, similar to that given in Equations 42-44, is presented below.
For the sake of simplicity, this form was based on the assumption of instantaneous
adjustment of n to changing substrate conditions. This would mean that a would have
infinite value.
^s,aerobic S V X
(49)
H max, aerobic^
n = n -
^s, aerobic ^
(50)
(51)

103
The yield was computed by plotting biomass against glucose consumed, using the data
from aerobic batch run 2 with the method previously described in the anaerobic growth
chapter. The results of this are shown in figure 42, along with a linear least squares fit
forced through zero. Only the exponential growth biomass points are shown and used in
the fit The slope of this line is the determined yield for this experiment. This yield was
found to be 0.493 m§ biomass a corre]at¡on coefficient, R2, of 0.99. The yield
mg glucose
used in the later simulations had a slightly inaccurate value of 0.486 used. The 0.493 value
should be the one considered for future use. All of the other model parameters were
determined by the use of aerobic batch run 2's exponential phase data (all but the last 2
points are shown). In addition to the parameters, starting amounts of biomass and cell
number concentrations were fit. The results of these cune fits are shown in figures 43-45.
The only preset model parameters were yield and initial glucose concentration. In the batch
simulations, for the sake of simplicity, the adaptability parameter, a, was assumed to have
infinite value. The biomass plot shows the RLS model to give slightly better fits than the
Monod model does. The Monod model does give a somewhat better fit of the residual
glucose concentration results, but the RLS model gives significantly better predictions for
the cell number data, especially towards the end of the run. Finally the sum of squared
residuals for the RLS model is about 25% lower than that of the Monod model, indicating a
slightly better overall fit by the RLS model. The lower sum of squared residuals is not
surprising as two additional parameters were available for fitting. The model parameters in
the Stalhandske thesis [58] are given in table 7.
Under steady-state continuous operation, equations 42-44, 49, and 51 give
xss — ^x/s, aerobic (^F — Sss) (52)

Biomass concncntratíon (mg/1)
104
Figure 42. Aerobic batch run 2. Biomass concentration against glucose consumed.

biomass concentration (mg/ml)
105
time (hrs)
Figure 43. Aerobic batch run 2. Biomass concentration against time. Dashed line
represents Monod model fit, and solid line represents RLS model fit.

number of cells (cells/ml)
106
xlOv
Figure 44. Aerobic batch run 2. Cell number concentration against time. Dashed line
represents Monod model Fit, and solid line represents RLS model fit.

glucose concentration (mg/ml)
107
time (hrs)
Figure 45. Aerobic batch run 2. Glucose concentration against time. Dashed line
represents Monod model fit, and solid line represents RLS model fit.
18

108
Table 7. Aerobic Model Parameters
Model Parameter/Starting
Monod Model Fit
RLS Model Fit
Condition
Umax, aerobic(hr'l)
.6129
.6481
Ks.aerobic (mg/1)
147.7
108.2
, mg' .
^ cell1/3hr
N/A
5.79x10‘4
K s, aerobic (mg/l)
N/A
897
Initial cells (cells/1)
1.12xl09
5.54xl08
Initial biomass (mg/1)
.312
.0354
X residuals
.028
.021

109
DK
^ss
s,aerobic
f* max,aerobic ^
^ss
f k' +s V
D—5 — xss (for the RLS model)
V
vs
ss y
(53)
(54)
nss = xss ~ (for the Monod model) (55)
xo
where xss=steady-state biomass concentration in the reactor
sss=steady-state glucose concentration in the reactor
nss=steady-state cell number concentration in the reactor
n0=initial cell number concentration in the batch calculation
x^initial biomass concentration in the batch
Equation 55 simply indicates constant cell size. These relations were used in Stalhandske's
thesis[ 58] to compare the model predictions to experimental data. The results are shown in
tables 8-10. The RLS model gives superior predictions of glucose concentration at all
dilution rates, and a somewhat better biomass prediction at the .335 hr1 dilution rate.
However, its cell numbers predictions are very poor.

110
Table 8. Aerobic model comparison with continuous data for D=.263 hr1
Model Used
Dry mass (mg/1)
Glucose (mg/1)
Number of cells/1
Data
2200
70
1.2 x 1013
Monod
1900
111
6.7 x 1012
RLS
1900
74
4.0 x 1014

Ill
Table 9. Aerobic model comparison with continuous data for D=.45 hr1
Model Used
Dry mass (mg/1)
Glucose (mg/1)
Number of cells/1
Data
1920
203
1.2 x 1013
Monod
1700
408
6.2 x 1012
RLS
1810
246
8.4 x 1013

112
Table 10. Aerobic model comparison with continuous data for D=.335 hr1
Model Used
Dry mass (mg/1)
Glucose (mg/1)
Number of cells/1
Data
2000
93
6.7 x 1012
Monod
1900
178
6.7 x 1012
RLS
1900
116
2.4 x 1014

CHAPTER 6
THE EFFECTS OF SHIFTS IN
AERATION ON E. COLILCB898
6.1 Background
The effects of shifts in aeration on some species have been studied by other
workers [e.g. 62,63]. Shifts in aeration condition for £. coli specifically have also been
examined [e.g. 50,64,65]. There are differences between the physiology of the cells under
each aerobic and anaerobic conditionsf 50,65] including what metabolic enzymes and other
proteins are present. Smith and Neidhardt[ 50.65] indicated that these changes manifest
themselves in a smooth transition of growth rates of wild-type E. coli K12 when sharp
shifts were performed from anaerobic to aerobic conditions, with some delay (25 minutes)
in the adjustment to the final grow th rate. This delay may or may not be applicable to the
mutant E. coli. Pyruvate processing is one of the key steps in £. coli glucose metabolism.
Pyruvate dehydrogenase is the enzyme responsible for conversion of pyruvate to acetyl-
CoA under aerobic conditions. Production of this enzyme may be inducible by pyruvate
under aerobic conditions[65]. Since the E. coli mutant may have a different amount of
intracellular pyruvate than normal £. coli K12 cells under anaerobic conditions, the
adjustment to aerobic conditions may be different than that of the strain described in the
work of Smith and NeidhardL
In the other type of shift, aerobic to anaerobic conditions, Smith and Neidhardt [50]
showed a complete lag in bacterial growth for 20 minutes before aerobic growth started.
However, this observation may or may not be applicable to the organism under study as
pyruvate formate lyase (pfl) is a key anaerobic metabolic enzyme in normal £. coli, which
is absent in this mutant
113

114
6.2 Development of the Combined Aerobic-Anaerobic Model
A simple model to describe E. coli LCB898 under both aerobic and anaerobic
conditions was developed for later use in developing cycling strategies. The model chosen
was a simple blending of the anaerobic and Monod aerobic models to describe biomass,
glucose, and product responses to variable aeration conditions. The Monod aerobic model,
in spite of its inferior predictive capability compared to the previously described RLS
model, was used for the sake of overall model simplicity. The individual models were
described earlier and only the special aspects of the combination model will be discussed.
The batch equations are as follows
x =
M-max, aerobic » joc | l1 max, anaerobic % ^
A
K + <;
^s,aerobic ~ s
^s,anaerobic ^
SX
(58)
s =
1
l1 max, aerobic
k *x/s,aerobic aerobic ~ 3
i 1-^ max, anaerobic
^x/s, an aerobic ^s, anaerobic ^
• doc
'(1 - doc)
SX
(59)
* ^ max, anaerobic /i j \
P = 0CT"^ — *(l-doc) (60)
^s, anaerobic S
where q= time derivative of q
x=biomass (dry mass) concentration in the reactor
s=substrate (glucose) concentration in the reactor
p=product (d-lactate) concentration in the reactor
Ibnax, anaerobic=maximum growth rate under anaerobic conditions
Ks, anacrobic=saturation parameter under anaerobic conditions
Yx/s, anaerobic=yield of biomass on substrate under anaerobic conditions
a=growth-associated lactate production parameter

115
Umax, aerobic-maximum growth rate under aerobic conditions
Ks aerob¡c=saturation parameter under aerobic conditions
Yx/s, aerobic=yield of biomass on substrate under aerobic conditions
Ko=oxygen saturation parameter
doc=aeration switching parameter
A chemostat form of this model, developed for the same conditions as the previously
described continuous models, is given by the following equations
x =
1^ max,aerobic
s, ^s, aerobic ^
1^ max, anaerobic
K -L c
^s,anaerobic T a
• doc +
\
(1 - doc)
sx - Dx
/
(61)
s =
1
1^ max,aerobic
Y K + c
k Ax/s,aerobic ^s,aerobic ~ 5
1 Umax, anaerobic
• doc -
• (1 - doc) sx + D(sf - s)
Y^/s,anaerobic ^s,anaerobic ^ ) (62)
p = tt^max.anaerobicSX # _ doc) _ Dp (63)
^s, anaerobic ^
where SF=feed substrate (glucose) concentration
D=dilution rate=flowrate/volume
Cell number concentrations were not modeled here in order to minimize the number of
equations.
The parameter doc has physically allowable values of 1 or 0, with 1 denoting
aerobic metabolism and 0 anaerobic. In later work it will be convenient to consider
imaginary culture states of intermediate doc values. The model has the assumption of no
lactate production or consumption under aerobic conditions. It also shows instantaneous
adjustment between aeration conditions. This may not be a very good assumption when

116
switching from aerobic to anaerobic conditions considering the observations of Smith and
Neidhardt where a lag phase was observed. However, no such lag is indicated by the
experimental results discussed below, or the cycling results of Chapter 7.
The model parameters used are reviewed in table 11.
6.3 Testing of the Model
An experiment was performed to investigate the effect of a shift between aerobic
and anaerobic conditions on the mutant. An aerobic continuous steady state for a dilution
rate higher than strict anaerobic operation would allow was established, after which
aeration was removed. This type of experiment was performed since analysis of a shift
from a steady state is relatively straightforward. The starting conditions of a continuous
aerobic culture are then well established and time-invariant. Shifts from transient culture
conditions, for example aeration shifts in batch cultures, are far more difficult to analyze, as
the starting conditions would not be well established.
This experiment was conducted by continuing aerobic continuous run 1. Again,
this was run at a dilution rate of 0.335 hr'1-, a dilution rate well below the maximum
aerobic growth rate, but one which would lead to biomass washout under anaerobic
conditions. The results of this experiment, along with model predictions, are shown in
figures 46-48. The first points of each of these figures represent the aerobic steady state.
The points represent data and the curves represent the model predictions. The biomass
curve indicated a smooth approach towards the expected washout. The biomass model
predictions were in excellent agreement with the experimental data. The glucose response
showed initial low values, due to the large amount of biomass that was still in the reactor,
but washout of the biomass led to increasing glucose concentration at 75 hours. The
glucose predictions were very accurate until washout was approached. The lactate
response showed a maximum at 75 hours. The lactate concentration rose very quickly after
the switch, but began to drop when biomass washout began to take effect. Again, the
model did a satisfactory job. The model and its anaerobic parameters thus seemed useful

117
for later cycling studies, at least as far as description of aerobic to anaerobic shifts are
concerned.

118
Table 11. Model parameter values
Parameter
Value
Pmax, anaerobic
.19 hr1
K.s, anaerobic
98 mg/1
Yx/.s, anaerobic
.068 mg/mg
a
10.77 mg/mg
Mrnax, aerobic
.6129 hr1
Ks,aerobic (mg/1)
147.7
Yx/s, aerobic
.486 mg/mg

Biomass concentration (g/1)
119
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
°70
\ 4-
80
90
100
Time (hrs)
110
120
Figure 46. Aerobic to anaerobic shift experiment. Biomass concentration against
time. Solid line represents model predictions, pluses represent data
130

Glucose concentration (g/1)
120
4
3.5
3 -
2.5 -
2-
1.5-
1 -
*
+
**-*â– '+
80
90
100
Time (hrs)
110
120
Figure 47. Aerobic to anaerobic shift experiment. Glucose concentration against
time. Solid line represents model predictions, pluses represent data

Lactate concentration (g/1)
121
2.5
2
1.5
1
t
0.5 -
°7(f 80 90
100
Time (hrs)
TÍÓ
120
130
Figure 48. Aerobic to anaerobic shift expenment. Lactate concentration against
time. Solid line represents model predictions, pluses represent data

CHAPTER 7
EFFECTS OF AERATION CYCLING ON
LACTATE PRODUCTIVITY OF E.COLI LCB898
7.1 Background
In previous chapters it was shown that E. coli LCB898 has higher growth rates
and yields under aerobic conditions than under anaerobic conditions. It was also shown
that d-lactate is only produced significantly under anaerobic conditions. This behavior
lends itself to examining the possibility of nonconstant aeration of the reactor under
continuous operating conditions. Cycling of air into an E. coli LCB898 bioreactor is
examined in this chapter.
Cycling of operating conditions in a bioreactor has been investigated by other
workers. The effects of periodic operation of feed substrate [1,2,5,6] and residual
substrate compositions [11] on cellular composition have been studied. Cycling of reactor
dilution rate has been used to increase biomass productivity of yeast [66] and plasmid
stability of recombinant E. coli K12 [67],
In order to investigate aeration cycling, a method had to be used to determine the
optimal cycling waveform. In other words, the period of cycling and the fraction spent
under aerobic and anaerobic conditions had to be determined. In addition, the best constant
dilution rate with which to operate the reactor had to be found. A theoretical method
involving Carleman linearization, previously described in the theoretical methods chapter,
was used on the previously described combined aerobic-anaerobic model. Aeration cycling
can be coupled with more complicated flow strategies, such as simultaneous cycling of
dilution rate, variable reactor volume and others. Unfortunately, for these strategies, the
122

123
theoretical method used, which approximates the system equations by a Taylor series
expansion required too high an expansion order to be practical.
For the sake of later comparison, the optimum model predicted steady-state lactate
productivity under anaerobic conditions was determined first. The model predictions for
lactate productivity against dilution rate are shown in figure 49. The determined optimal
lactate productivity was 408 mg/(l»hour)with the optimum being found at a dilution rate of
.161 hr-1. The corresponding experimentally determined anaerobic steady-state optimum
lactate productivities were 527 mg/(Miour) at a dilution rate of 0.164 hr1, and 578
mg/(l*hour) at a dilution rate of 0.17 hr1. As these steady states should be approximately
the same, the average of these two productivities were used as the optimal value for later
comparison. The average value was 553 mg/(l»hour).
7.2 Theoretical Investigation into Cycling
For the cycling considered, the system is run at a constant dilution rate, with air
being turned on and off. This proposed control waveform is shown in figure 50. The
measure of performance considered was average lactate productivity over a cycle. The
optimization variables were period, fraction spent under each set of conditions, and dilution
rate need to be determined. The fraction spent under aerobic conditions, referred to as the
aerobic fraction, is symbolized by e in the figure. The period is symbolized by T, and the
dissolved oxygen condition, 0 for anaerobic, 1 for aerobic, is symbolized by doc.
According to the model the optimal cycling period is zero. Fast cycling between
two values is equivalent to using a constant average value. In other words, if fast cycling
between two control settings is performed, the system being controlled will only "see” the
average of the two control settings. In the system under study, infinitely fast cycling can
then be investigated by setting the doc variable to the aerobic fraction. For example, if an
aerobic fraction of 0.3 was simulated with infinitely fast (zero) period, the combined
aerobic-anaerobic model would have its doc value set to 0.3. Steady-state solutions of the
model were found numerically for various dilution rates and aerobic fractions. The

Lactate productivity (g/l/hr)
124
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Dilution Rate (1/hr)
Figure 49. Anaerobic model-predicted lactate productivity against dilution rate.

125
Figure 50. Proposed dissolved oxygen cycling waveform.

126
corresponding values for the lactate productivity, defined by dilution rate multiplied by
lactate concentration, are presented in table 12. Lower aerobic fraction results are only
given up to certain dilution rates, because above these rates washout of biomass occurred.
The optimum shown in this table is found at an aerobic fraction of 0.45 and a dilution rate
of 0.30 hr1. The value for the lactate productivity at that point was 573.9 mg/(l»hour). A
more careful search was performed near that point, and the optimal zero period conditions
found were aerobic fraction of 0.45 and dilution rate of 0.31 hr1, with a lactate
productivity of 576.2 mgAThour).
The effects of increasing period were then examined, as large periods were deemed
desirable for three reasons: (1) the actual experimental implementation becomes easier, (2)
equipment wear is lessened, and (3) the model is not applicable during the time the cells
switch metabolism. The Carleman-linearization-based method previously described as
used with deviation variables from the imaginary (since doc is not 0 or 1) steady state: doc
0.45, dilution rate 0.31 hr-1, biomass concentration 597.3 mg/1, glucose concentration
587.9 mg/1, and lactate concentration 1858.8 mg/1. In other words, the zero period optimal
solution was the steady state used as basis for the Carleman linearization.
The results for several orders of Carleman linearization are given in table 13. The results of
Carleman order 2 and 3, which should be more accurate than order 1 results, indicate that
the decrease in lactate productivity is insignificant up to approximately 2 hours, and, as the
3rd order results indicate, productivity only drops 3% from the optimum at 5 hours. The
experiments were conducted with period of 5 hours.
The Carleman-linearization-based method was also used to recreate table 12. This
was performed to determine the usefulness of this procedure for predicting conditions far
away from the point of linearization. The results for a third order Carleman linearization
around the zero period steady state for a dilution rate of 0.31 hr'1 and an aerobic fraction of
0.45 are shown in table 14. As can be seen in this table, the Carleman linearization did a
poor job of predicting lactate productivities at the higher aerobic fractions. Also, washout

127
Table 12. Zero period lactate productivities (mg/(l»hour) for various conditions predicted
by Newton-Raphson solution of model
Dilution rate (hr1)
Aerobic
Fraction
0.1
0.16
0
285.0
407.6
0.1
276.1
421.8
0.2
265.1
410.4
0.3
251.9
391.8
0.4
236.1
367.7
0.45
227.1
353.6
0.5
217.1
337.9
0.6
193.7
300.9
0.7
164.1
254.3
0.8
125.8
194.1
0.9
73.9
113.4
1.0
0
0
0.2
0.25
0.3
463.7
489.8
459.4
475.3
548.6
323.1
448.6
534.6
571.2
431.9
518.1
573.9
413.0
497.3
560.9
367.8
444.7
509.7
310.5
375.7
433.1
236.5
285.7
329.9
137.7
165.9
191.3
0
0
0
0.35 0.4 0.5
463.3
551.6
548.8
483.6
477.8
492.4
367.0
391.9
192.4
213.3
230.4
222.1
0
0
0

128
was not adequately predicted for high dilution rate, low aerobic fraction, conditions, as can
be seen clearly in the first two rows of table 14. When comparison to the table 12 results
were made, good predictions of most sub-washout low aerobic fraction productivities were
apparent. Good predictions were also seen near the optimum, which was to be expected as
this was the point around which linearization was made.
The suggested improvements in parameters shown in table 5 were also investigated.
The results of the zero period optima found for different conditions are shown in table 15.
Differing anaerobic parameters sometimes slightly changed the optimum operating
conditions, but none of these resulted in optimal lactate productivities significantly different
from those for the original-parameter optimal conditions.
Simulations, using Runge-Kutta integration of the model equations, for dilution rate
of 0.31 hr1, 5 hour period, and aerobic fraction of 0.45 are shown in figures 51-53. The
cycles shown were started off aerobically. The glucose concentration response showed
wide swings between about 200 mg/1 and 1000 mg/1. The switch to anaerobic conditions is
clearly seen in the biomass and lactate plots. Biomass immediately stopped increasing and
lactate immediately started increasing at the shift. Corresponding comments apply at the
shift from anaerobic to aerobic conditions. The glucose response did not show switches as
clearly. During anaerobic conditions glucose initially decreases as the cells consume it at a
higher rate than under aerobic conditions, but eventually glucose starts increasing due to
cell washout. The switch to aerobic conditions, at every 5 hours, is seen by a sharp
increase of slope in the glucose concentration. The glucose response to the shift to
anaerobic conditions is similar to that seen in the biomass and lactate curves.
Finally, the optimal periodic lactate productivity of 576.2 mg/(l*hour) was
approximately 40 % higher than the optimal anaerobic steady-state productivity of 407.7
mg/(Miour).

129
Table 13. Carleman predictions for average lactate productivity (mg/(l»hour)) over a cycle
at a dilution rate of 0.31 hr'1 and an aerobic fraction of 0.45 as a function of Carleman
order and period.
Carleman order
Period (hours)
1
2
3
.00001
576.2
576.2
576.2
.001
576.2
576.2
576.2
.1
576.5
576.2
576.2
1
577.3
575.9
575.9
2
582.3
574.7
574.9
5
584.7
560.5
564.3
6
588.7
550.8
553.9
7.5
595.6
531.4
516.5

130
Table 14. Zero period lactate productivities (mg/(l*hour) for various conditions predicted
by third order Carleman linearization of model
Dilution rate Ow1)
Aerobic
Fraction
0.1
0.16
0
295.0
404.4
0.1
292.8
423
0.2
284.8
421
0.3
271.6
402.8
0.4
253.3
374.5
0.45
242.8
359.1
0.5
232.2
345.8
0.6
216.5
338
0.7
220.5
388.3
0.8
253.4
622.6
0.9
352.3
-1437.8
1.0
0
0
0.2
0.25
0.3
407.6
176.1
-1798.7
469.1
518.6
573.9
490.6
507.2
95.0
481.4
548.3
464.8
451.8
535.6
571.4
434.3
518.6
573.9
419
501.0
561.7
415.9
495.1
536.7
528
706.4
624.5
2109.3
-803.2
-191.0
-289.5
-63.1
181.2
0
0
0
0.35 0.4 0.5
471.1
554.3
549.6
474.9
433.0
554.1
22.5
31.0
438.0
-137.5
-25.9
205.1
0
0
0

131
Table 15. Optimum Conditions as a Function of Anaerobic Parameter Values, and Lactate
Productivity Values for Dilution rate of 0.31 hr1 and aerobic fraction 0.45
a
Ks, anaerobic
Yx/s, anaerobic
Optimum
Opt Lactate
Lac. Prod.
(mg/mg)
(mg/1)
(mg/mg)
Conditions
Productivity
(mg/l/hr)
at D=.31,
£=.45
10.77
31.4
.068
e=.45,
612.3
612.3
D=.31
10.77
31.4
.063
e=48.
577.3
577.2
D=.32
10.77
98
.063
£=.48,
544.4
544.1
D=.32
13
31.4
.068
£=.45,
739.1
739.1
D= 31
13
98
.068
£=.45,
695.5
695.5
D=.31
13
98
.063
£=.46,
656.9
656.8
D=.31
13
31.4
.063
£=.46,
696.7
696.7
D=.31

Biomass concentration (g/1)
132
l
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 5 10 15 20 25
Time (hrc)
Figure 51. Simulations of effects of periodic operation (D=.31,e=.45) on biomass
concentrauon.

Lactate concentration (g/1)
133
Time (hrs)
Figure 52. Simulations of effects of periodic operation (D=.31,e=.45) on lactate
concentration.

Glucose concentration (p/1 >
134
1.4
1.2
1
Ü.8
0.6
0.4
0.2
0
0 5 10 15 20 25
Time (hisj
Figure 53. Simulations of effects of periodic operation (D=.31,e=.45) on glucose
concentration.

135
7,3 Experimental confirmation of lactate productivity optimization results
Experiments were performed to confirm an increase of productivity by using
cycling. Two periodic operation runs were performed. Both of these experiments were
conducted with a dilution rate of 0.31 hr*, an aerobic fraction of 0.45 and a period of 5
hours. They were initiated as aerobic batches and switched to continuous operation and
cyclic aeration when biomass concentration was approximately 800 mg/1, the theoretically
predicted value. Each run's results will be described separately.
All of the results for periodic run 1 are shown in figures 54-56. The switch to
aeration cycling and continuous operation was performed at 13 hours. These figures are
confusing as they present all of the data collected. It is more useful to examine figures 57-
59, which only show the results for the switch points between aeration conditions. The
biomass results showed a smooth approach to what appears to be a cyclic steady state at
about 93 hours. The measured biomass concentrations were significantly higher than the
model-predicted values. The glucose measurements showed a decrease, followed by a
peak, then again followed by a decrease to what appears to be a cyclic steady state at 103
hours. The glucose measurements were significantly lower than the model-predicted
values. The lactate concentration measurements showed a smooth approach to an apparent
cyclic steady state at about 93 hours. The lactate concentration measurements were only
slightly higher than the model-predicted values. Unfortunately, the experiment ended
prematurely due to unforeseen happenings (a line clogged) and without collection of data
for an entire control cycle. However, data were taken for the duration of the anaerobic
portion of one of the later cycles and the lactate concentration during the aerobic part of the
cycle can be accurately estimated since practically no lactate is produced under aerobic
conditions.
The following relation describes the effect of lactate concentration during aerobic
continuous operation

Biomass concentration (mg/1)
136
1400
1200
1000
800
600
400
200
0
Figure 54. Periodic Run 1. Experimental results of effects of periodic operation
(D=.31,e=.45) on biomass concentration.
~ ^ i i j i I i— »| — r- | | j ■ t I' | 1 i l | l l 1 ] i 1 I T
0 20 40 60 80 100 120
Time (hours)
140

Lactate concentration (mg/I)
137
3000 —j
2500-
2000-
•
-
•
1500-
•••
1000-
-
• •
500-
•
0-
1
0 20
n—'—'—'—i—'—'—1—i—r
40 60 80
Time (hours)
1 ] r-r—!—| 1
100 120
Figure 55.
Periodic Run 1. Experimental results of effects of periodic operation
(D=.31x= 45) on lactate concentration.

138
Figure 56. Periodic Run 1. Experimental results of effects of periodic operation
(D=.31,e=.45) on glucose concentration.

Biomass concentration (mg/1)
139
1400
1200
1000
800
600
400
200
0
0 20 40 60 80 100 120 140
Time (hours)
Figure 57. Periodic Run 1. Experimental results of effects of periodic operation
(D=.31,e=.45) on biomass concentration. Only the shift points are shown.

Lactate concentration (mg/1)
140
3000
2500
2000
1500
1000
500
0
0 20 40 60 80 100 120 140
Time (hours)
Figure 58. Periodic Run 1. Experimental results of effects of periodic operation
(D=.31,e=.45) on lactate concentration. Only the shift points are shown.

141
4500
4000
^3500
f 3000
c
1 2500
c
i 2000
c
o
§ 1500
u
Ü 1000
500
0
0 20 40 60 80 100 120 140
Time (hours)
Figure 59. Periodic Run 1. Experimental results of effects of periodic operation
(D=.31.e=.45) on biomass concentration. Only the shift points are shown.

142
p = -Dp
where p=lactate concentration
D=dilution rate
which can be integrated to
P = P0e'Dt
where p0=original lactate concentration
t=time
(64)
(65)
Equation 65 was used with p0 being set to the lactate concentration at the end of the
previous anaerobic phase and D being set to the appropriate dilution rate of 0.31 hr1. A
least squares linear fit to the anaerobic lactate production data was also made, with the
results being shown in figure 60. In order to determine the mean value, the anaerobic line
and the aerobic exponential curve were then integrated separately. The area under the
straight line was divided by 2.75 hours, which was the anaerobic cycle time, and the area
under the exponential curve was divided by 2.25 hours. The average lactate concentration
determined by this method was 1976 mg/1, which indicated a lactate productivity of 613
mg/(Miour).
The biomass and lactate results for periodic run 2 are shown in figures 61 and 62.
The switch to aeration cycling and continuous operation was performed at 13 hours.
Again, these figures were confusing as they present all of the data collected. Similar
switchpoint-only plots to those for periodic run 1 are shown in figures 63 and 64. The
biomass results showed a smooth approach to what appears to be a cyclic steady state at
about 60 hours. The lactate concentration showed a smooth approach to an apparent cyclic
steady state also at about 60 hours. Three entire cycles were measured in this experiment,
with the lactate values shown in figure 65. The second lactate peak was then looked at to
obtain an average lactate productivity over a cycle. This was performed by the same
procedure as used in the previous run, except that the aerobic cycle was fit to an

Lactate concentration (mg/1)
143
Figure 60. Periodic Run 1. Experimental results of effects of periodic operation
(D=.3l,e=.45) on lactate concentration for one cycle. The first line
represents a linear fit to the data, and the second curve represents model-
predicted lactate dropoff.

Biomass concentration (mg/1)
144
Figure 61. Periodic Run 2. Experimental results of effects of periodic operation
(D=.31,£=.45) on biomass concentration.

Lactate concentration (mg/1)
145
3000
2500
2000
1500
1000
500
0
0 10 20 30 40 50 60 70 80 90 100
Time (hours)
Figure 62. Periodic Run 2. Experimental results of effects of periodic operation
(D=.31x=.45) on lactate concentration.

Biomass concentration (mg/1)
146
1400
1200
1000
800
600
400
200
0
0 10 20 30 40 50 60 70 80 90 100
Time (hours)
Figure 63. Periodic Run 2. Experimental results of effects of periodic operation
(D=.31,e=.45) on biomass concentration. Only the shift points are shown.

Lactate concentration (mg/1)
147
3000
2500
2000
1500
1000
500
0
0 10 20 30 40 50 60 70 80 90 100
Time (hours)
Figure 64. Periodic Run 2. Experimental results of effects of periodic operation
(D=.3i,£=.45) on lactate concentration. Only the shift points are shown.

Lactate concentration (mg/1)
148
3000
2500-
2000-
1500-
.• *
1000-
500-
0 — — 1 T 1 ’ | ' ' ■ I | ’ !
60 65 70
T_r
75
i i i 1 i' i i i i | i
80 85
Time (hours)
1-T-| I I"!
90 95
Figure 65.
Periodic Run 2. Experimenta] results of effects of periodic operation
(D=.3Le=.45) on lactate concentration. Only three cycles are shown.
100

149
exponential curve instead of being estimated. The results of these fits are shown in figure
66. The average lactate concentration determined for this run was 1898 mg/1, giving a
lactate productivity of 588 mg/(l»hour).
The average of these lactate productivities is 600 mg/Ohour). This indicates
approximately an 8.5% improvement over the experimentally determined 553 mg/I. While
this is not as large an improvement as the model predicted, it must be considered that the
model, based on batch anaerobic data does underpredict anaerobic continuous steady-state
lactate productivity. It can be argued then that the cycling lactate productivity should have
been higher too, but it can also be argued that not enough is known about the transient
behavior of E. coli LCB898 and microorganisms in general under aeration cycling to
accurately describe metabolite production under these conditions. Nonetheless, an
improvement by cycling was seen. The more interesting improvement is actually seen in
the next chapter on reversion avoidance.

Lactate concentration (mg/1)
150
Time (hours)
Figure 66. Periodic Run 2. Experimental results of effects of periodic operation
(D=.31,e=45) on lactate concentration for one cycle. The first line
represents a linear fit to the data, and the second curve represents an
exponential fit to the data.

CHAPTER 8
REVERSION OF E. COLI LCB898 AND
A POSSIBLE NEW METHOD OF AVOIDANCE OF
REVERSION
Mutation frequently occurs in microbial cells grown under continuous steady-state
reactor conditions. Examples of mutation in continuous culture have been reported by
several workers [e.g. 68-77]. The probable cause of mutation in several of these reports
was genetic adaptation of the culture to substrate-limited conditions. As one example,
Novick and Szilard [69], in continuous culture experiments with E. coli B/l, found that a
mutant arose which they designated B/l/f. This mutant showed growth five times as fast
as the original strain under tryptophan-limited conditions. They argued that this increased
growth rate gave the B/l/f cells a selective advantage over the B/l cells under the
tryptophan-limited conditions seen in their continuous culture experiments, and, in time,
the B/l/f cells must displace the original B/l cells. Another example of mutation under
continuous conditions was found in the work of Ruijter [73]. Here it was found that
certain continuous operations would induce fast mutation of Salmonella typhimurium to a
form with improved enzymatic systems for glucose uptake under glucose-limited
conditions.
A specific kind of mutation is that of reversion of a previously mutated culture to its
original wild-type form. An example of this was found in the work of Kiss and
Stephanopoulos[77]. There, reversion of an L-lysine producing Corynebacterium
glutamicum mutant to its original nonproducing form was investigated. The lysine
producing mutant was auxotrophic for threonine, whereas the revertant was not. This gave
the revertant an advantage over the mutant, and, in every continuous culture experiment that
151

152
was run, the revertant eventually took over with the consequence that the lysine
concentration in their cultures fell to near zero.
As will be shown later in this chapter, reversion during anaerobic continuous
operation may have been taking place in our system. The mutant, E. coli LCB898, may
have been reverting to a pfl+ form. A likely consequence of this reversion would be very
low lactate production by the revertant. The cycling of aeration described in the previous
chapter has shown promise for delaying this reversion. A possible explanation will be
given later.
The biomass, glucose, and lactate measurements for anaerobic continuous run 1 are
shown in figures 67-69. These figures were formatted differently than those shown in
Chapter 4, in that the time scale has been adjusted to time after the start of continuous
operation; they also include later data points. The biomass and lactate figures show the
previously described continuous steady state up to about 40 hours after the switch. After
this apparent steady state, some sort of change seems to have occurred where the biomass
increased and the lactate decreased. Very low lactate concentrations were already seen at 80
hours. The unusual response in glucose and biomass concentrations seen at about 100
hours were due to a system upset where a feed tube was pinched off and reactor volume
dropped. A new biomass steady state appears, ignoring the upset, to have been established
at 100 hours. The glucose concentration stays low during the entire expeñment, again with
the exception of the upset.
The biomass and lactate results for anaerobic continuous run 2 are shown in figures
70 and 71. These figures also have been reformatted with more data included in a similar
fashion to figures 67-69. The biomass and lactate results show the previously described

153
450
400
§350
E
= 300
c
V
o 200
U
I 150
S 100
50
0
0 20 40 60 80 100 120 140
Time (hours)
Figure 67. Anaerobic continuous run 1. Biomass concentration against
time after switch to continuous.

154
4000 q
3500-
^3000 4
5
_
-
C
o
2500 4
ca
fa
-
c
V
2000-
o
_
o
-
u
1500-

-
-
Ü
ca
1000 4
-
500 4
04
40 60 80 100
Time after switch to continuous (hours)
Figure 68. Anaerobic continuous run 1. Lactate concentration against
time after switch to continuous.

155
4000 -q—
3500-
% 3000 -
c
! 2500-
4^
C3
fa
g 2000-
^ 1500-
t/>
C
= 1000-
o :
500-
0- —
0
Figure 69.
• • •
T—i—T—T—T—r
20 40
120
Time after switch to continuous (hours)
Anaerobic continuous run 1. Glucose concentration against
time after switch to continuous.
140

156
400
5 200 ^
I»,
C/i
CT3
o 100 A
0.00
I—i r t—¡—|—i—r—-—¡ t ¡ i—!—| i r
10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00
Time after switch to continuous (hours)
Figure 70. Anaerobic continuous run 2. Biomass concentration against
time after switch to continuous.

157
4000
3500-
^ 3000â– 
E
g 2500-
CZ
| 2000-
o
c
£ 1500
0>
c5
% 1000 2
500-
0 —f 1 1 I ] 1 ! I I 1 | ' I ■ I 1 1 ! I 1 ! I I 1 1 I ! ! ! [— r -I 1- T ] 1—T
0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00
Time after switch to continuous (hours)
Figure 71. Anaerobic continuous run 2. Lactate concentration against
time after switch to continuous.
80.00

158
continuous steady state up to about 30 hours after the switch to continuous operation, then,
like in the previously described experiment, some sort of change occurred where the
biomass increased and the lactate decreased. In this run, lactate concentration was very low
at 60 hours. In order to test for reversion, gas production tests were performed. The
principle behind the test was that E. coli LCB898 should show little or no gas production
due to anaerobic glucose catabolism, as no CO2 and H2 production from pyruvate
degradation by exclusive action of lactate dehydrogenase is possible. The tests were
performed as follows: Two small samples of the reactor contents were taken at 60 hours,
spread on to agar plates, and then the plates were incubated overnight. Ten colonies were
selected from each of the agar plates and inoculated separately into ten test tubes with 13 ml
each of 10 g/1 tryptone, 5 g/1 NaCl, 1 g/1 yeast extract, and 1 g/1 glucose solution, along
with an inverted Durham tube in each tube. Another small sample was taken at 75 hours
and a similar procedure was performed. Additionally sample cultures of the stock culture
were also similarly processed. All twenty of the 60 hour samples showed gas production,
and eight of the ten 75 hour samples showed gas production. The samples from the stock
culture showed no gas production. This appeared to indicate that pyruvate formate-lyase
mutation had reverted to an active form. This hypothesis could explain the unusual results
from anaerobic continuous runs 1 and 2 and other continuous runs. (The same type of
behavior was seen in every anaerobic continuous experiment run.) The pyruvate formate-
lyase active revertant appeared to be able to use glucose more efficiently for biomass
production, as indicated by the increasing biomass concentrations seen. However, little
lactate is seen. The likely explanation for this reversion is that the low glucose
concentration seen in a steady-state continuous reactor would provide a selective pressure
for those cells that can use the available glucose most efficiently, as the revertants seem to
do.
This apparent reversion obviously lowered the lactate productivity after relatively
short periods of time. The cycling experiments, besides showing an improvement in lactate

159
productivity, showed promise in delaying, or perhaps avoiding, this reversion. This was
an unexpected additional benefit of the periodic operation. The cycling experimental results
for lactate concentration are shown in figures 72 and 73, along with the glucose results for
periodic run 1 in figure 74. Only the results at the switch points were shown in these
figures, and they were also reformatted for showing the results only after the switch to
continuous operation. The experiments were not ended due to reversion. They were ended
only due to experimental procedural problems such as feed blockage. Periodic run 1 still
showed high lactate productivity at 115 hours, and periodic run 2 still showed high lactate
productivity' at 85 hours. The glucose results for periodic run 1 showed that average
glucose concentration was higher than that for strict anaerobic operation. It appeared that
periodic operation helped avoid reversion. A likely explanation as follows. The aerobic
portion of the cycle, likely, provided no selective advantage for revertants. In addition, the
average glucose concentration was considerably higher than that for strict anaerobic
operation, thus lowering the pressure for revertant selection.
The delay or avoidance of reversion by periodic operation shows potential for other
applications. The work shown here is preliminary, and further investigations of this
phenomenon with E. coli LCB898 or other microorganisms should be performed. This
idea could be extended to other types of cycling, such as pH, dilution rate, or inlet feed
composition. If some organism was showing undesirable mutation in a reactor condition
under one set of conditions (for example, acidic or anaerobic) but little or no reversion
under another set of conditions (for example, neutral pH or aerobic), then periodic cycling
of these conditions might be a useful operating strategy to avoid overall culture reversion.

Lactate concentration (mg/1)
160
3000
2500
2000
1500
1000
500
0
0 20 40 60 80 100 120
Time after switch to continuous (hours)
Figure 72. Periodic run 1. Lactate concentration against time after switch to
continuous. Only shift points are shown.

Lactate concentration (mg/1)
161
3000
2500
2000
1500
1000
500
0
0 10 20 30 40 50 60 70 80 90
Time after switch to continuous (hours)
Figure 73. Periodic run 2. Lactate concentration against time after switch to
continuous. Only shift points are shown.

Glucose concentration (mg/1)
162
4500
4000
3500
3000
2500
2000
1500
1000
500
0
0 20 40 60 80 100 120
Time after switch to continuous (hours)
Figure 74. Periodic run 1. Glucose concentration against time after switch to
continuous. Only shift points are shown.

CHAPTER 9
CONCLUSIONS
In some biological systems the optimal environmental conditions for producing of a
desired metabolic product differ from the conditions that are optimal for cell growth. If
producing this metabolite were the objective, one could continuously operate a reactor
system at the optimal steady-state production conditions. For a given reactor volume,
however, periodically switching conditions could increase overall metabolite production.
This is possible since, due to higher growth rates under the optimal growth conditions, one
could operate the system at significantly higher flowrates and, thus, obtain higher
productivity.
A system involving E. coli mutant LCB898 was used as a model system. This
mutant lacks the enzyme pyruvate-formate lyase and as a result under anaerobic growth
conditions practically all of its pyruvate is converted to d-lactate. Thus it produces large
amounts of d-lactic acid under oxygen-poor conditions, whereas it grows at a considerably
higher rate under aerobic conditions when almost no lactate is produced. It was
experimentally determined that the aerobic maximum specific growth rate is 0.61 hr'1 as
compared to 0.19 hr~l for anaerobic conditions. The possibility of increasing total lactate
productivity by cycling of aeration investigated.
A method for determining the optimal waveform for the proposed cycling was
developed by extending previous work by Lyberatos and Svoronos. The method involves
Carleman linearization of the system equations around the optimal steady state and
subsequent development of an expression for performance measure as a function of period,
amplitude and pulse shape. This method was adapted to the present system. However, in
163

164
this system, one control variable was aeration which was either on or off with no
intermediate values. Thus, an “imaginary” steady state had to be used for linearization.
Since the optimization method requires availability of a dynamic model, such a model was
developed using the results of both batch and continuous experiments.
The Carleman-based-method was applied to the above described model in order to
determine the optimal cycling conditions. The results were very encouraging since a
significant improvement in productivity over the optimal steady state (40%) was predicted.
Experimental testing of the calculated cycling strategy, however, showed a considerably
lower improvement.
Perhaps the most exciting (and accidental) discovery in this work was a possible
new method to help delay or avoid mutation or reversion of microorganisms to a nonuseful
form in a continuous steady-state bioreactor. This method essentially involves cycling
between revertant selective and nonselective conditions. Only preliminary work has been
performed, and further investigations are strongly suggested.
V

APPENDIX
MATHEMATICA PROGRAMS FOR COMPUTATION
OF CARLEMAN LINEARIZATION MATRICES
On the following pages program listings for two Mathematica programs developed
to compute the Carleman linearization of a system of first order differential equations such
as that shown in (2). The first program, Kronecker, is an auxiliary program used by the
main program, CarlN, to figure Kronecker products. The Mathematica utility program for
computing Kronecker products, Outer, did not compute the products correctly, so this
auxiliary program was necessary. The second program listed, CarlN, is the main program
to compute the Carleman linearization of the system. The program was based on the
algorithm described in Chapter 2.
In order to use these programs, they must be installed into an active session of
Mathematica. These programs have been tested on both Macintosh and PC systems. The
following is the format of an execute statement for this program:
CarlN[cn,list of ode's]
where cn=Carleman order
list of ode's= the list of first order ordinary differential equations for a
system to be linearized where the state variables are cast as
deviation variables from some steady state and are referred to
as y[i]
For example, the input statement for the second order linearization of the system
given in (3) (assuming the variables are in deviation form) was as follows:
CarlN[2,{-y[l]+3*y[2]+y[2]A2,-y[l]A2+4*u}].
The output is stored in the following arrays:
165

166
CC= Contracted Carleman matrix (e.g. the 5X5 matrix in (5))
zvec= Associated Carleman vector (e.g. the 5X1 vector in (5))
wvec the list of approximated variables (e.g. the left-hand-side vector in (4))

167
Kronecker::usage="Kronecker[ml,m2] figures the Kronecker product of two matrices or
vectors ml,m2"
Kronecker/: Kronecker[ml_List, m2_List] := Block[{atv,btv,ctv,dtv,av,bv,cv,dv,jv,
kv,nrm 1 ,ekv,ncm 1 ,m Ia,nrm2,fkv,ncm2,m2a,iav,jav,rtp,ctp},
jv=Dimensions[ml];nrml=jv[[l]];ekv=TensorRank[ml]; Ifjekv <
1.1,ncml=l;mla=Outer[Times,ml,{ 1}], ncml=jv[[2]];mla=ml];
kv=Dimensions[m2];nrm2=kv[[l]];fkv=TensorRank[m2]; Ififkv <
1.1,ncm2=l;m2a=Outer[Times,m2,{ 1}], ncm2=kv[[2]];m2a=m2];qkv3=l;
Do[Do[qkm3[qkl 1 ,qkl2]=m 1 a[[qkl 1 ,qkl2]]*m2a,
{qkl2,ncml }],{qkll,nrml}]; m3=Table[{ },{nrml*nrm2}];
Do[Do[qktml=qkm3[qkvl,qkv2];Do[dope=qkl3-qkv3+l;
m3[[qkl3]]=Flatten[Append[m3[[qkl3]], qktml[[dope]]]]; ,{qkl3,qkv3,qkv3+nrm2-
1,1 }],{qkv2,ncml}]; qkv3=qkv3+nrm2,{qkvl ,nrml}];]
CarlN::usage="CarlN[cn,f] figures the cn order Carleman linearization of an n-dimensional
system where f=the list of n equations that describe the system. The variables should be in
deviation form.
The equation list should be as follows f={fl,f2,..fn}. The varibles should be in the form
y[l],y[2], etc.
Output includes wvec=Carleman coordinate vector CC=the contracted Carleman matrix
zvec=the associated Carleman vector wvec=the variable order of the Carleman matrix"

168
CarlN/: Car)N[cn_Integer,f_List] := Block[{a,il,jl, kl,ll,ml,nl,ol,pl,ql,rl,sl,
yv,lvec,gg,gto,vlt,bsv 1 ,bsv2,bsv3,bsv4,ul,tl, gs,tma,tmb,hll,tmc,tmd,ill,wvt,ccdim,wvf,
tpmat,tme,tmh,tmg,tmf, test, 111, mil, pmat2,pmat3,
pmat,uccdim,tvy,tvx,tvv,tmi,tmii,tmij,tvw,TURCC,
UCC,tvs,tvta,tvu,ptva,tvuu,tvt,URCC,ffv,A10m, ffvv},
n=Length[f];yv=Table[0,{n},{ 1}]; Do[yv[[il,l]]=y[il],{il,n)];vlist[l]=yv;
lvec=Table[0,{cn}];lvec[[l]]=Length[vlist[l]];
(* Following big Do statement forms
vlists and initializes uncontracted submatrices of Taylor portion of Carleman matrices *);
Do[Kronecker[yv,vlist[(gl-l)]];vlist[gl]=m3;lvec[[gl]]= Length[vlist[gl]],{gl,2,cn,l}];
Do[A 1 [il]=Table[0,{ n}, {lvec[[il]] }];gg=Al [il];
Do[Do[gg[[pl,ql]]=f[[pl]],{ql,lvec[[il]]}],{pl,n}];Al[in=gg,{il,cn}];
(* Determination of all uncontracted Alx matrices for l<=x<=cnn *);
Do[gto=Al[ml];vlt=vlist[ml]; Do[Do[Do[gto[[nl,ol]]= D[gtoffnl,ol]],{y[rl],
Exponent!v]tf[o], 1 ]],y[rl]]}],{rl,n}] ,{ol,lvec[[ml]]}] ,{nl,n}]; Al[ml]=gto (l/(ml!))
,{ml,cn}]; Do[bsv4=A 1 [tl];bsv 1 =Dimensions[bsv4];
Do[Do[Do[bsv4[[bsv2,bsv3]]=bsv4[[bsv2,bsv3]]/. y[ul]->0,{ul,n}], {bsv3,bsvl[[2]J}],
{bsv2,bsv 1[[1]]}];A[1 ,tl]=bsv4,{ tl.cn}];
(* Determination of A10 submatrix *); Al[0]=f;gs=f; Do[Do[gs[[jl]]=
gs[[jl]] /• y[kl]->0,{kl,n}] ,{jl,n}]; A1 [0]=gs;A[ 1,0]=A 1 [0];
(* Determination of Remaining Uncontracted Submatrices *);

169
Do[tma=IdentityMatrix[n]; tmb=IdentityMatrix[nA(hll-l)];Do[ Kronecker[tma,A[hll-l,ill]]
;tmc=m3; Kronecker[A[l,ill],tmb];tmd=m3; A[hll,ill]=tmc+tmd,{ill,0,cn-hll+l,l}],
{hll,2,cn,l}];
(* Formation of Contracted variable list *);
wvt=Table[0,{cn}];ccdim=0;Do[wvt[[jll]]= vUst[jll];ccdim=ccdim+Binomial[n+jll-l,jll]
,{jll,cn} ];wvt=Flatten[wvt];
wvf=Union[wvf|;tpmat=Table|0,{ccdim)];pmat=tpmat;
Do[tpmat[[kll]l=Position[wvt,wvf[fkll]]], {kll,ccdim}];
Do[tme=Length[tpmat[[mll]]];tmh={}; Do[tmg=tpmat[[mll,lll]];
tmf=Length[tmg];If[tmf {lll,tme}];pmat[[mll]]=tmh,{mll,ccdim}]; pmat=Map[Flatten,pmat,{ 1}];
tmi=Table[tva,{tva,ccdim}]; pmat2=pmat[[tmi,{ 1 )]];pmat3=Flatten[pmat2];
pmat2=Flatten(Sort|pmat2]]; ;wvec=wvtf|pmat2]];wvec=OuterlTimes,wvec,{ 1}];
(* Formation of Uncontracted Carleman matrix *); uccdim=0;Do[uccdim=uccdim +
nAnll,{nll.cn}]; UA[l]=Transpose[A[l,l]];UA[l]={UA[l]};
Do[UA[l]=Append[UA[l],Transpose!A[l,tvz]]] ,{tvz,2,cn,l }];UA[l]=Flatten[UA[l],l];
UA[l]=Transpose|UA[l]]; If[cn>l,
Do[UA[tvy]=Transpose[A[tvy.O]]; UA[tvy]={UA[tvy]}; Do[UA[tvy]=Append[UA[tvy],
Transpose[A[t\'y,tvx]]], {tvx,cn-tvy+l}]; UA[tvy]=Flatten[UA[tvy],l];
UA[tvy]=Transpose[UA[tvy]]; tvv=Dimensions[UA[tvy]];
Ifltvv[[2]] tmi=Table[0,{ tvv[[ 1 ]] J,{uccdim-tvv[[21]} ];
tmii=Transpose[tinij;tmij={Transpose[UA[tvy]]};

170
UA[tvy]=Prepend[tmij,tmii]; UA[tvy]=Flatten[UA[tvy],l];
UA[tvy]=Transpose[UA[tvy]],else=4];
.{tvy,2,cn,l }],else=2];
UCC={A[1]}; If[cn>l,
Do[UCC=Append[UCC,UA[tvw]],{tvw,2,cn,l}] ,else=3]; UCC=Flatten[UCC,l];
(* Column Contraction of Matrix *); TURCC=Table[0,{ccdim),{uccdim}];
Do[tvta=pmat3[[tvu]];tvs=Length[pmat[[tvu]]]; ptva=Flatten[Position[pmat2,tvta]][[l]];
Do[tvuu=pmat|[tvu,tvt]]; TURCC[[ptva]]=Transpose[UCC][[tvuu]]+
TURCC[[ptva]],{tvt,tvs}],{tvu,ccdim}];
URCC=Transpose[TURCC];
(* Row (and Final!) Contraction of Matrix *);
CC=Table[0,{ccdim),{ccdim} ]; Do[CC[[ffv]]=URCC[[pmat2[[ffv]]]], {ffv,ccdim)];
(* Final form of z vector *);
zvec=Table[ 0,{ ccdim},{1}];
A10m=Outer[Times,A[l,0J,{ 1}]; Do[zvec[[ffvv]]=A10m[[ffvv]],{ffvv,n}]]

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BIOGRAPHICAL SKETCH
Jonathan Ben Rodin was bom on December 6,1966, in Lincoln, Nebraska. Until
he reached the age of twenty, he lived in various cities of the Midwest, including Lincoln,
Nebraska, and the Iowa cities of Sioux City and Iowa City. In Iowa City he finished his
undergraduate chemical engineering degree at the University of Iowa. He then went to the
University of Florida for his graduate work. His future plans are currently undetermined.
175

I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
jf A- ^
yyrc
Spiros A. Svoronos, Chairman
Associate Professor of Chemical
Engineering
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
Gerasimds Lyberatos, Cochairman
Associate Professor of Chemical
Engineering
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
'C/s—
Gerald B. Westermann-Clark
Associate Professor of Chemical
Engineering
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy. yn
'U-
/Seymour S. Block
. Professor of Chemical Engineering

I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
Ben Koopman
Professor of Environmental
Engineering Sciences
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy. y
a
W
Lonnie O. Ingram
Professor of Microbiology and Cell
Science
This dissertation was submitted to the Graduate Faculty of the College of
Engineering and to the Graduate School and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
December, 1992
—
/
^ ¿2— ■ t J ^
- Winfred M. Phillips
Dean, College of Engineering
Madelyn M. Lockhart
Dean, Graduate School