DYNAMIC PHOTOSYNTHETIC RESPONSE OF SOYBEANS:
MODEL DEVELOPMENT AND ELEVATED CO2 EXPERIMENTS
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
I would like to thank everyone who has helped me in my research
program. I have been very fortunate, having been surrounded by com-
petent and cooperative people. In particular, I want to thank Dr. Jim
Jones who has been amazingly patient and helpful throughout my Ph.D.
program. Also I want to thank Drs. Hartwell Allen, Ken Boote and
Thomas Humphreys who have all been very generous with their time despite
their busy schedules. In addition, I want to thank Dr. G.L. Zachariah,
chairman of my committee and Drs. D. Buffington, C. Hsieh, and R. Irey
who served as members of my supervisory committee.
I would also like to express my appreciation to Bill Campbell,
Paul Lane and Kelton Johns whose skills were essential to the success
of the experimental phase of my research. Also a special message of
gratitude must go to Klaus Heimburg, Yung Le Morgan and my other friends
who went to such great lengths keeping my spirits elevated yet humble.
Finally, I do want to express my appreciation to Laura, my wife,
and to Ralph and Arlen Jones, my parents, for their excellent support
on every level. I hope that everyone who has been involved with me
during this project will somehow benefit by that association.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS . . ... ..
LIST OF TABLES . . .
LIST OF FIGURES . . .
ABSTRACT . . . .
INTRODUCTION . . . .
BIOCHEMICAL PHOTOSYNTHESIS MODEL . .
The Light Reactions . .
Dark Reactions: Calvin Cycle . .
Dark Reactions: Phosphate Translocator System. .
Dark Reactions: Chloroplastic Starch Pathway .
Dark Reactions: Cytoplasmic Sucrose Pathway .
Simplifying Assumptions . .
Differential Equation Development . .
Carboxylation, Photorespiration and Enzyme Kinetics
CO2 and 0? Concentration in the Stroma .
Membrane Transport Considerations .. ..
Competitive Inhibition of Sucrose Formation .
Postscript on RuBP Carboxylase-Oxygenase and RuBP
Postscript on Photosynthesis-Respiration Roles of
Chloroplastic PGA . .
MODEL RESULTS. . . .
The Chloroplastic Carbon Dioxide Balance, [C02o
Time Rate of Change of Starch . .
The Chloroplastic Inorganic Phosphate Balance, [Pl]
The Cytoplasmic Sucrose Balance, [SUCROSE] .
Real World Scenarios . .
Direct Manipulation; Sucrose Feeding .
Direct Manipulation; Selective Shading .
Temperature Manipulation . .
CO2 Concentration Manipulation . .
EXPERIMENTAL METHOD . . .
Procedure . . .
Physical Characteristics . .
Controlled Environmental Factors . .
Soil Respiration . .
Water . . .
TABLE OF CONTENTS (contd.)
EXPERIMENTAL METHOD (contd.)
EXPERIMENTAL RESULTS . .
Controls. . .
Carbon Balance. . .
Transpiration . .
Plant Growth Parameters .
Variations in Photosynthetic Rate
Physiological Responses .
SUMMARY AND CONCLUSIONS. .
APPENDIX 1--ABBREVIATIONS LISTING. ..
APPENDIX 2--GLOSSARY OF TERMS. .
APPENDIX 3--UNITS LISTING. .
APPENDIX 4--SYMBOL LISTING .
APPENDIX 5--PROBABLE ERROR ANALYSIS OF
BIBLIOGRAPHY . .
BIOGRAPHICAL SKETCH. . .
CO2 MASS BALANCE
LIST OF TABLES
1 Calvin cycle reactions .. . 17
2 Chloroplastic starch cycle reactions ... 20
3 Cytoplasmic sucrose pathway reactions. .. 23
4 Biochemical photosynthesis model equations ...... 44
5 Key assumptions used in model development. .. 48
6 In vivo biochemical parameters used in model evaluation. 59
7 Mesophyll resistance values from the literature. 60
8 Comparison of in vivo and in vitro biochemical parameters 64
9 Chloroplastic concentrations of key metabolites in
light and dark . . 67
10 Measured starch accumulation parameters. ... 69
11 Starch accumulation as a function of inorganic phosphate
concentration. . . ... .. 73
12 Measured chloroplastic-cytoplasmic inorganic phosphate
interactions . . 74
13 Adenosine diphosphate formation as a function of inorganic
phosphate levels . .... 82
14 Final above ground biomass . ... 124
15 Morning-afternoon photosynthetic data. ... 126
16 Regression analysis of photosynthesis data from Chambers
1 (Low-High) and 2 (High-High) . .. 128
17 Regression analysis of photosynthesis data from Chambers
3 (Low-Low) and 4 (High-Low) . ... 130
18 Specific leaf weight measurements. . ... 137
19 Comparison of specific leaf weight and soluble carbo-
hydrates in Chambers 1 and 4 . .. 140
LIST OF TABLES (contd.)
20 Vertical distribution of leaf nitrogen .. 142
21 Vertical distribution of leaf chlorophyll. 145
22 Vertical distribution of leaf inorganic phosphate. 146
LIST OF FIGURES
1 Photosynthesis and source-sink balance schematic .... 3
2 Calvin cycle schematic . ... 15
3 Phosphate translocator system schematic. .. 18
4 Chloroplastic starch cycle schematic .. 21
5 Cytoplasmic sucrose pathway schematic. ... 24
6 PGA-DHAP pathway schematic . .... .28
7 Simplified PGA-DHAP pathway. . ... 28
8 DHAP-G6P pathway schematic . .. 29
9 Simplified DHAP-G6P pathway schematic. ... 29
10 Simplified overall photosynthesis schematic. .. 31
11 Carboxylase-oxygenase pathway schematic. .. 38
12 Final photosynthesis schematic . ... 47
13 Modelled [CO2]-photosynthesis-[RuBP] response curves 62
14 Measured and modelled [C02]-photosynthesis response
curves . . . 63
15 Modelled -photosynthesis response curves .. 65
16 Measured and modelled starch accumulation response 70
17 Time courses of modelled inorganic phosphate-starch
response . .. 72
18 Measured and modelled inorganic phosphate-starch response 76
19 Comparison of chloroplastic and external concentrations
of inorganic phosphate . ... 77
20 Comparison of carbon partitioning pathways ...... 79
21 Measured and modelled inorganic phosphate-transport
response . . ... ...... .80
22 Modelled adenosine diphosphate-inorganic phosphate
response . . ... ...... .83
23 Measured chloroplastic PGA-inorganic phosphate response. 85
24 Measured and modelled sucrose concentration-[EXPORT]
response . . 87
25 Control chamber schematic. . 97
26 Control system schematic .. . 100
27 Diurnal temperature control. . ... 109
28 Time courses of PAR, [CO2] and CER on April 21 .. 110
29 Time courses of PAR, [CO2] and CER on April 22 ...... .11l
30 Photosynthetic light response curves; Chambers 1 (Low-
High) and 2 (High-High). . .. 113
31 Photosynthetic light response curves; Chambers 3 (Low-
Low) and 4 (High-Low). . . ... 114
32 Time courses of dark respiration rates . .. 116
33 Time courses of transpiration rates. . .. 117
34 Time courses of water use efficiency . .. 119
35 Time courses of individual leaf areas. . 120
36 Time courses of leaf area index. . ... 121
37 Time courses of canopy leaf mass. . ... 122
38 Morning and afternoon photosynthetic light response;
Chambers 1 (Low-High) and 4 (High-Low) .. 125
39 Time courses of morning CER at 500 pE/m**2/sec .. 134
40 Time courses of afternoon CER at 500 pE/m**2/sec 135
41 Diurnal specific leaf weights; Chambers 1 (Low-High) and
4 (High-Low) . . 139
42 Comparison of carbohydrate levels and specific leaf
weight . . ... .141
43 Vertical nitrogen distribution; Chambers 1 (Low-High)
and 4 (High-Low) . . 144
Abstract of Dissertation Presented to the Graduate
Council of the University of Florida in Partial
Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
DYNAMIC PHOTOSYNTHETIC RESPONSE OF SOYBEANS:
MODEL DEVELOPMENT AND ELEVATED CO2 EXPERIMENTS
Chairman: Dr. G.L. Zachariah
Major Department: Mechanical Engineering
A biochemical photosynthesis model has been developed and experi-
ments at the whole plant level have been conducted in a holistic in-
vestigation of photosynthetic rate controls. The model was based on
known biochemical pathways and emphasized the roles of starch, sucrose
and inorganic phosphate in the feedback control of CO2 fixation. The
experimental work was designed to investigate short term rate response
to a sudden change in CO2 concentration and long term physical adaptation
to distinctive environmental CO2 levels. In the experimental procedure
different source-sink balances were established for each treatment and
plant response was measured. In the model the mechanisms for source-
sink feedback control of the biochemical CO2 fixation process were
developed. The overall purpose of the work has been to demonstrate and
clarify the role played by source-sink relations in the feedback regu-
lation of photosynthesis.
Comparison of model predictions with measurements from the litera-
ture have shown the model's individual response equations to behave well.
Overall the model demonstrated the central role of inorganic phosphate
as a regulator of the Calvin cycle via its affect on the ATP/ADP ratio
and as a regulator, partitioning CO2 between starch and sucrose syn-
thesis. In turn, the concentration of inorganic phosphate was modelled
to depend inversely on cytoplasmic sucrose level which was modelled to
depend on export. By functionally linking the export of sucrose from
the cytoplasm to sink demand, the model qualitatively described the
photosynthetic inhibition and enhancement observed in a variety of
The experimental results showed soybeans (Glycine max c.v. Bragg)
to respond to elevated CO2 levels by increasing net photosynthesis and
respiration in the short term and increasing leaf area and biomass in
the long term. Close analysis of photosynthetic light response found
morning enhancement and/or afternoon inhibition of CO2 fixation in
plants exposed to high [C02]. In the canopy switched to high CO2, the
magnitude of afternoon inhibition was suppressed by rapidly enhanced
sink strength. Reduced CO2 fixation corresponded to higher levels of
soluble carbohydrates in the leaves. Taken together these results
supported the proposed interdependence between leaf sucrose levels and
The experimental work showed the association between source-sink
balancing and photosynthetic rate while the biochemical model demon-
strated a linkage mechanism. The differences in detail of the bio-
chemical and whole plant levels prevented direct quantitative compari-
sons between model and experimental results. Nevertheless the experi-
mental and model results were qualitatively consistent and this work
represented a necessary effort at a holistic explanation of photosynthetic
All living systems in the biosphere are based directly or indirectly
on sunlight, carbon dioxide and water. Photosynthesis is the process
common to green plants that combine these elements into energy-rich
carbohydrates. Within plant systems the carbohydrates produced by
photosynthesis have two functions: first, as substrate for plant tissue
synthesis and second, as energy for the synthesis reactions. As some
of the carbohydrate building blocks are combined with nitrogen, sulfur
and other elements into plant tissue, others are respired releasing
required reaction energy.
Photosynthesis occurs in plant leaves while tissue synthesis and
respiration use carbohydrates throughout the plant structure. Carbo-
hydrates are translocated from the production source in the leaves to
consumption sinks through the phloem which directly connects the photo-
synthetic leaf material to active reaction sites.
More specifically, the chain of events that provides carbohydrates
for plant growth begins in microscopic organelles called chloroplasts
which are suspended in the cytoplasm of photosynthetic cells. Carbon
dioxide diffuses from the atmosphere to the chloroplasts where it is
chemically reduced in a process called the Calvin cycle, driven with
energy supplied by light. The assimilated CO2 is either converted to
sucrose in the cytoplasm, or to starch in the chloroplast, both of which
are carbohydrates. A schematic representation of this system is given
in Figure (1). The translocation system begins with the export of a
three carbon phosphorylated compound dihydroxyacetonee phosphate, DHAP)
from the chloroplast to the cytoplasm, where it is converted to sucrose,
which moves into the phloem for distribution throughout the plant.
The balance between CO2 fixation in the chloroplast and carbohydrate
export from the cytoplasm is important to photosynthetic feedback control,
but is poorly understood.
The balance between carbohydrate sources and sinks can be disrupted
by changing environmental conditions. When this happens, photosynthesizing
cells must adjust to the new circumstances. If photosynthetic rate is
increased or if transport to the phloem is decreased then excessive
amounts of carbohydrates may accumulate in the photosynthetic cells.
When this occurs starch is reversibly formed in the chloroplasts, storing
fixed carbon until it can be mobilized and shipped to the cytoplasm for
conversion to sucrose and export to the rest of the plant. If starch
accumulation continues past a certain point, cells can be physically
damaged [Noble and Craig, 1973]. It is postulated that before damage
occurs, the plant, under normal conditions, adapts by either decreasing
its rate of photosynthetic supply of carbohydrates, or by increasing
its rate of sucrose translocation and utilization.
Changes in the rate of starch accumulation have been observed in
response to many situations. For instance, when plants have been
switched from atmospheres with ambient levels of CO2 to enriched CO2
conditions, starch accumulation and photosynthesis increased dramati-
cally but while starch levels continued to rise after several hours,
photosynthetic rates declined [Mauney et al., 1979]. In another
example, plants acclimated to warm nights were found to have increased
\ \ yDHAP
LEAVES ---- TRANSPO
Figure 1. Photosynthesis and source-sink balance schematic. PAR is
photosynthetically active radiation. All abbreviations are defined in
Appendix 1. Terms are defined in Appendix 2.
starch concentrations and decreased photosynthetic rates following
exposure to low night time temperatures.
In the case of high [C02], the plant's mature leaves exported more
carbohydrates than they had in low [C02], but not enough to balance
the increased supply. Consequently, the excess was stored as starch.
In the case of low night temperatures, carbohydrates normally used
in night time growth and maintenance processes were not used due to
temperature dependent reductions in process rates. On the following day
reduced sink demand caused carbohydrate levels in the photosynthetic
cells to increase.
Observations such as these are often explained conceptually in
terms of imbalances in the source-sink relationship. Whether the car-
bohydrate levels increase due to high CO2 fixation or reduced sucrose
export, the system balances itself by reducing photosynthesis. On the
other hand, when sink strength is increased and carbohydrate levels in
the photosynthetic cells decline, maximum photosynthetic rates are
observed to increase [Thorne and Koller, 1974]. Source-sink balancing
is superficially a straightforward and satisfying explanation for
alterations in photosynthetic response. Unfortunately, the details of
the proposed source-sink mechanism are not well understood.
The photosynthetic system consists of complex and interrelated
processes all of which serve to control the overall CO2 fixation rate.
In simple terms, every plant's physical and chemical environment is
continuously changing. In turn, compound levels and process rates
which determine photosynthetic rate within the plant are also changing.
In this sense, plant photosynthesis is dynamic.
Macroscale variations in photosynthetic capacity are a function of
nutrient availability, water supply, temperature, sunlight, 02 concen-
tration and CO2 concentration. At the cellular level, photosynthetic
capacity is a function of enzyme availability, enzyme activity and
substrate concentrations. All of these factors combine to establish
individual reaction rates, the products of which are substrates for
subsequent reactions. The integrated sum of the various reactions
control particular process rates. Finally, whole plant parameters and
biochemical processes are functionally coupled by transport systems some
of which are passive and some of which are active. In both cases con-
centration gradients and transport resistances are determining factors
in the mass transport rates of substances from the environment to the
biochemical reaction sites of photosynthesis. The sum of these inter-
actions determines the instantaneous rate of CO2 fixation.
This brief holistic description of photosynthesis emphasizes that
response should be considered at both the whole plant and biochemical
levels of plant organization. Modellers began trying to relate external
environmental conditions to photosynthesis as a biochemical process
thirty years ago [Rabinowitch, 1951]. Since then, a steady stream of
increasingly biochemical models has been developed [Chartier, 1970;
Charles-Edwards and Ludwig, 1974] leading to the biochemical models
proposed by Peisker  and more recently by Farquahr et al. .
One of the main criteria which these modellers have set is the
adequate simulation of various photosynthetic light response curves.
Usually, results from experimental work on photosynthesis are presented
graphically in photosynthetic light response curves which plot carbon
dioxide exchange rate (CER) against light level. Because of the very
strong dependence of CO2 assimilation on light level, the graphical
relationship between them has long been considered a fundamental measure
of response, whether research is on whole plant canopies or reconstituted
The fundamental importance of the photosynthetic light response
curve (PLRC) is further enhanced by the similar shape of most plots
generated for a wide variety of plants, which implies very similar
underlying mechanisms. Several distinct functional equations have been
proposed which generate curves of the appropriate shape [Thornley, 1976].
However, the most commonly derived equations are variations of the
rectangular hyperbola. This particular equation is scientifically
satisfying because of its theoretical basis in enzyme kinetics, where
it is called a Michaelis-Menten response curve. Because of the cyclical
enzymatic pathways which photosynthesis follows, it seems quite natural
that the rate of CO2 uptake should have a Michaelis-Menten form. The
goal for modellers has been to relate light via an assumed biochemical
pathway to CO2 uptake in such a way that external parameters could be
used to represent the variations in photosynthesis in response to light.
Photosynthesis models such as those described above are currently
used in conjunction with other crop system models to predict growth
rates and yield. These models concentrate on the uptake of CO2 in
response to light level while ignoring how or why fixed carbon is par-
titioned between starch and sucrose. As a result, they work well under
normal conditions but are generally inadequate for describing response
to unusual circumstances.
It is hypothesized that partitioning between stored chloroplastic
starch and cytoplasmic sucrose is central to the feedback control of
photosynthesis. As noted in the examples of low night temperatures and
increased CO2 levels, starch accumulation is often reported in associa-
tion with reduced photosynthetic response. To model this relationship,
starch and sucrose cannot be simply divided into unrelated pools.
Synthesis of starch and sucrose must respond functionally to mechanisms
which prescribe how fixed carbon is to be partitioned. One of the main
goals of this research has been to develop a model based on such mechanisms.
To accomplish this goal a detailed photosynthesis model has been
developed, based on five specific biochemical pathways:
1. the light reactions
2. the Calvin cycle,
3. the chloroplastic starch cycle,
4. the cytoplasmic sucrose pathway, and
5. the phosphate translocator system.
The purpose of the developed model is to consider the hypothesized roles
that starch and sucrose play in the feedback control of CO2 fixation.
This is done by identifying the possible interactions among the five
pathways that could limit photosynthesis and control partitioning. From
this perspective it becomes possible to investigate the complex rela-
tionship between biochemical dynamics and the ambient environment. Such
a detailed model can also be used to suggest how the photosynthetic
system might respond in the long term to different prevailing environments.
At the whole plant level field experiments were conducted concurrently
with the development of a biochemical photosynthesis model. The experi-
mental work was designed to investigate short term response to a
sudden change in CO2 concentration and long term adaptation to distinctive
environmental CO2 levels. All of the treatments were exposed to constant
moderate air temperatures under well-fertilized and well-watered conditions.
The experiments were conducted under natural sunlight which varied, but
all treatments were exposed equally to these changes.
To assess the plant-environment interactions four general classes
of data are required: (1) external parameters such as temperature,
quantum flux density and CO2 concentration which define the environment;
(2) gas exchange rates such as transpiration, daytime CO2 exchange and
night time C02 exchange which define water use,photosynthesis and
respiration rates; (3) whole plant parameters such as height, leaf area
and biomass which are indicators of adaptive response; and finally,
(4) physiological parameters such as specific leaf weight, chlorophyll
levels and nitrogen levels which are indicators of biochemical system
response. The short and long-term adaptive response of plants to a
particular sequence of prevailing environments can be characterized in
terms of these four data sets.
At the biochemical level the experimental results can be applied to
questions concerning substrate levels and process rates on the micro-
scale. For instance, the data can suggest how C02 level effects chloro-
phyll concentrations and enzyme levels (as a function of nitrogen) as
well as CO2 fixation process rates. On a larger scale the data can
indicate how source-sink balancing is affected by CO2 concentration
through the measurements of diurnal specific leaf weight and by the
instantaneous measurements of photosynthetic light response. Finally,
at the whole plant level, biomass accumulation and leaf area are direct
measures of the integrated adaptive response of whole plants to different
The experimental work was designed to provide the data necessary
to answer these questions. With these data, the hypothesis that plants
adaptively respond to different prevailing environmental concentrations
of CO2 can be tested. Furthermore, the adjustments that occur at the
whole plant level and the biochemical level can be considered separately,
to determine whether the adaptation of the whole plant is consistent with
biochemical level response.
Combining the experimental results with the biochemical photosynthesis
model, a qualitative explanation of short-term feedback controlled
response and long-term whole plant adaptation to prevailing CO2 concen-
trations is proposed. This proposed mathematical-conceptual model pro-
vides a framework for both a short-term model to explain inhibition and
enhancement of photosynthetic light response and a long-term model of
adaptation to differing environments.
The overall purpose of this research is based on the concept of
photosynthetic adaptation which has been defined as environmentally
induced adjustments in physiology, anatomy and morphology that allow a
plant to improve photosynthetic efficiency in a new environment [Bjorkman
and Berry, 1973]. Stated another way, adaptation enables whole plants
to maximize photosynthetic productivity under locally prevailing envi-
ronmental conditions [Tooming, 1970]. The purpose of this research is
to gain a more complete understanding of photosynthesis as a dynamic
process at both the whole plant and biochemical levels.
It is postulated that plant canopies will adaptively respond to
different prevailing environmental concentrations of carbon dioxide. In
particular, this project has been designed to describe and explain the
responses of soybean canopies grown continuously in different carbon
dioxide concentrations during their vegetative stage of growth and the
subsequent short and long-term canopy responses to a step change in CO2
levels. The experimental objective of the research has been to grow
and monitor soybean canopies in four computer controlled, closed environ-
mental chambers. A parallel theoretical objective has been to develop a
biochemical level model of photosynthesis. Finally, the third objective
of the research has been to qualitatively relate the whole plant experi-
mental observations and the biochemical model within the conceptual
framework of source-sink balancing.
The long range goal of this research is to devise a physically based
dynamic model of photosynthesis complete with feedback controls. Such a
model could be used with other sub-system models to describe any whole
plant system. In turn individual plant models are the basis of crop
models which have increasingly wide application.
BIOCHEMICAL PHOTOSYNTHESIS MODEL
Photosynthesis is commonly modeled [Lehninger, 197C] as a process
in which carbon dioxide (C02) and water (H20) are chemically combined
in the presence of light quanta (nhv) to form glucose (C6H1206) and
oxygen (02). In equation form this is expressed as:
6 002 + 6 H20 + nhv C6H1206 + 6 02 + heat (1)
Note that mass is conserved explicitly in the equation stoichiometry,
whereas energy is implicitly conserved by equating light input (nhv) to
the chemical energy in glucose (C6H1206) and the energy degraded to
heat. It is useful to rewrite equation (1), replacing the light quanta
term (nhv) with an associated free energy change (AG).
6 CO2 + 6 H20 C6H1206+602
AG= 686 kcal/mole (2)
This emphasizes that in the formation of glucose, light is not a sub-
strate, but rather, indirectly supplies the energy to drive this
"uphill" reaction. Finally, a third equation can be written to make
the obvious point that carbonated water exposed to sunlight will not
produce glucose and oxygen. CO2 and water must pass through an exten-
sive series of biochemical cycles and pathways before being transformed
into glucose; therefore, equation (1) might be written once again:
6CO2+6H20 + nhv b C6H1206+602 + heat (3)
In the following sections, some of the details of the photosynthetic
blackbox will be discussed, simplified and condensed into a mathematical
The Light Reactions
When photosynthesis is considered in detail, equation (1) is seen
to incorporate two chemical processes which are coupled into the total
system by which glucose is formed. Light is required in an initial
process in which photosynthetically active radiation (PAR) is converted
into chemical energy via excitation of chlorophyll and accessory pigment
molecules. These initial reactions are called the Light Reactions. PAR
is electromagnetic radiation with wavelength between 400 and 700 nm. It
is usually measured with a quantum sensor and typical units are micro-
Einsteins/m**2/sec. PAR quanta are absorbed by a diverse group of pig-
ments located in the chloroplasts of photosynthesizing cells. The most
commonly known and abundant pigment is chlorophyll, of which there are
several kinds that differ slightly in structure and absorption spectrum.
Chlorophyll is the main light absorbing pigment in green plants. When
plants are exposed to PAR, quanta are absorbed, causing high energy
electrons to escape from the excited pigment molecules. Some of these
electrons fall back to ground state and the chlorophyll molecules give
up their captured quanta as fluorescence and heat. Others leave the
chlorophyll completely and enter an electron-carrier pathway, flowing
down an energy gradient from one carrier to the next. When an electron
moves through this transport system, it loses potential energy at each
transfer between carriers. At certain of the transfers in the chain,
the potential drop is partially conserved by driving the energy requiring
phosphorylation of adenosine diphosphate (ADP) to adenosine triphosphate
(ATP). Two quanta are absorbed for each electron to move through the
complete pathway. In this manner light-induced electron flow is con-
verted to chemical bond energy. The process is called photophosphory-
lation and can be represented by the following equation:
ADP + P + nhv ATP + heat, (4)
where P is inorganic phosphate.
Only part of the photoinduced potential is used to produce ATP. Most
of the conserved electrochemical energy goes to the last acceptor in the
chain, which is the oxidized form of nicotinamide adenine dinucleotide
phosphate (NADP) which receives the electron as well as a proton and is
accordingly reduced as the final step in the light reactions, This pro-
cess of using photoinduced electron flow to yield a reduced product is
called photoreduction. Equation (4) can be expanded to represent the
overall light reaction, including both photophosphorylation and photo-
reduction as follows [Lehninger, 1973]:
2 ADP + NADPox + 2 P + 2 hv + H20
NADPred + 2 ATP + 02 + heat. (5)
With this more detailed understanding, equation (1) can be rewritten
again to clarify the relationship between PAR as an energy source and
the glucose formation equation. It is
6 NADPred + 6 H20 + 12 ATP + 6 CO2
SC6H1206 + 6 NADPox + 12 ADP + 12 P + 6 02 (6)
Equation (6) describes an overall process which has been only briefly
outlined. More complete descriptions of the light reactions are avail-
able in articles by Rabinowitchand Godvinjee  and Zelitch .
Dark Reactions: Calvin Cycle
The biochemical pathway to sucrose following the light reactions is
referred to as the Dark Reaction. A central portion of this process is
the Calvin Cycle, which occurs in the chloroplast, along with the light
reactions. The first step in the Calvin cycle is the reduction of CO2
(the carboxylation reaction). Specifically, the reaction combines C02,
H20 and ribulose-l,5-bisphosphate (RuBP) in the presence of the enzyme
ribulose-1,5-bisphosphate carboxylase (RuBPc) to form 2 molecules of
3-phosphoglyceric acid (PGA). In equation form it is
H20 + CO2 + RuBP --- 2 PGA AGO = -8.4 kcal/mole (7)
Equation (7) is an exergonic or downhill reaction having a negative
standard free energy change [Bassham, 1971] and, therefore, requires no
energy or reducing power to proceed. Although not directly required in
equation (7), the light reaction's products (ATP and NADPred) drive the
complex sequence of enzyme catalyzed reactions called the Calvin cycle,
which regenerates RuBP [Bassham, 1971].
In the second reaction of the Calvin cycle, ATP is directly required
for the phosphorylation of PGA, producing the high energy phosphate com-
pound 1,3-biphosphoglyceric acid (BPGA):
PGA + ATP -* BPGA + ADP. (8)
The reducing power generated by the light reaction is utilized in the
subsequent step as follows:
BPGA + NADPred GAP + P + NADPx, (9)
where GAP is glyceraldehyde-3-phosphate. The only other reaction in
FMP EMP XMP
GAP -SDBP )
BPGA ADP R
\ PGA C02
Figure 2. Calvin cycle schematic. RuBP, ribulose-1,5 -bisphosphate, RuP;
ribulose-5-phosphate; XMP xylulose-5-phosphate; SDMP, sedoheptulose-l-
phosphate; SDBP, sedoheptulose-1,7-bisphosphate; EMP erythrose-4-phosphate;
FMP, fructose-6-phosphate; FBP, fructose-1,6-phosphate; DHAP, dihydrox-
yacetone phosphate; GAP, glyceraldehyde-3-phosphate; BPGA, 1,3-phospho-
glyceric acid; PGA, 3-phosphoglyceric acid. All abbreviations are listed
alphabetically in Appendix 1 (Based on Bassham ).
the Calvin cycle directly using the products of the light reaction is
step 13 in Table (1), in which ribulose-5-phosphate (RuP) is phosphory-
lated to RuBP:
RuP + ATP RuBP + ADP. (10)
The preceding three equations summarize the direct interaction
between the Calvin cycle and the light reactions, while equation (7)
represents the all important link between the microscale biochemical
pathways and the macroscale carbon exchange rates.
Dark Reactions: Phosphate Translocator System
Both the light reactions and the Calvin cycle occur in chloroplasts
while the final steps in the dark reactions take place in the cytoplasm,
requiring that the carbon fixed in equation (7) be exported through the
outer chloroplastic membrane. The primary export product is dihydroxy-
acetone phosphate (DHAP) [Heber, 1974], which is produced in reversible
equilibrium with GAP as the fourth step in the Calvin cycle. For this
pathway to function continuously, cytoplasmic phosphate must be imported
to the chloroplast in direct proportion to the exported DHAP. The
exchange is part of the phosphate translocator system. DHAP is a crucial
intermediate which is balanced among three pathways: export for use as
substrate or energy in the cytoplasmic dark reactions, continuation in
the Calvin cycle as substrate for regenerating RuBP, or to storage as
starch in the chloroplast. A schematic is given in Figure (3) to
clarify these relationships. The other portion of the system imports
PGA from the cytoplasm in exchange for chloroplastic inorganic phosphate
[Kelly et al., 1976]. The importance of maintaining phosphate balances
is clear, considering its role in energy transfer and storage. Once
Table 1. Calvin cycle reactions (H20 not shown).
6 CO2 + 6 RuBP
12 PGA + 12 ATP
12 BPGA + 12 NADPred
3 GAP + 3 DHAP
2 FMP + 2 GAP
2 EMP + 2 DHAP
2 SMP + 2 GAP
6 RuP + 6 ATP
Reactions are taken
Triose phosphate isomerase
from Lenhinger .
! BPGA + 12 ADP
! GAP + NADPox
FMP + 3 P
XMP + 2 EMP
SMP + 2 P
RMP + 2 XMP
RuBP + 6 ADP
I \ _
Figure 3. Phosphate translocator system schematic. Outlines the
membrane exchange mechanism by which fixed carbon is exported from
chloroplasts to cytoplasm for sucrose synthesis. All abbreviations
are identified in Appendix 1 (Based on Heber ).
P P NADPox
DHAP is in the cytoplasm, it can be utilized as a substrate in the
sucrose pathway or it can be oxidized and dephosphorylated by the
reverse reactions of equations (9) and (10). In this way the energy
equivalents of ATP and NADPred are transported to the cytoplasm. When
used for energy transfer, the DHAP is converted to PGA, which can be
transported back to the chloroplast [Herold and Walker, 1979].
Dark Reactions: Chloroplastic Starch Pathway
In the chloroplast, the starch pathway is cyclical, moving from
DHAP to starch, and later being reconverted to DHAP. The intermediate
steps have been worked out in detail and are presented in Table (2).
The cycle actually moves through steps (5) and (6) of the Calvin cycle
in which fructose-1,6-bisphosphate (FBP) is irreversibly converted to
fructose-6-phosphate (FMP), which enters the starch pathway, forming
glucose-6-phosphate (G6P). The rate limiting step in starch formation
is the reaction in which ATP combines with glucose-l-phosphate (GIP) to
form ADP-glucose. This reaction is not only promoted by high levels of
ATP, but also by high levels of PGA, and is inhibited by high levels of
inorganic phosphate [Kaiser and Bassham, 1979]. The return starch
mobilization reactions follow the same essential pathway, except that
the rate limiting step is the energy requiring conversion of FMP to
FBP. This reaction obtains energy and a phosphate group from ATP, while
simultaneously being inhibited by high levels of ATP.
The system behavior described acts as a regulator encouraging starch
storage during the day and release at night. Also note that energy units
are required in both the formation and breakdown of starch. A schematic
of this cycle is given in Figure (4).
Table 2. Chloroplastic starch cycle reactions.
DHAP + GAP
GlP + ATP
Starch + P
FMP + ATP
Dekinase FMP + P
Pyrophosphorylase ADP-glucose + P
Amylose synthetase Starch
Glucan phosphorylase G1P
Phosphohexoisomerase FMP + ADP
Aldolase DHAP + GAP
are based on Lehninger  and Kelly et al.
FM --P G6P
Figure 4. Chloroplastic starch cycle schematic. Outlines the starch
storage and remobilization mechanism by which chloroplast can store fixed
carbon. All abbreviations are listed in Appendix 1 (Based on Kelly et
Dark Reactions: Cytoplasmic Sucrose Pathway
The other carbon balancing pathway is to the cytoplasm, where DHAP
is the initial substrate leading to sucrose formation. In a series of
steps similar to those in starch formation, DHAP is converted to glucose-
1-phosphate (GlP). At this point, sucrose formation diverges from the
starch pathway. Instead of the common energy compound ATP, G1P combines
with uridine triphosphate (UTP) to form UDP-glucose. This compound
reacts with FMP to form sucrose-6-phosphate (SMP), which directly yields
sucrose. One important aspect of this pathway is that the reaction rate
of the SMP conversion to sucrose may be subject to product inhibition
[Hawker, 1967]. Sucrose is the primary sugar compound exported from
photosynthesizing cells. If the export of sucrose is less than its
synthesis, then cytoplasmic levels will increase. When this happens,
the high concentration of sucrose potentially inhibits the enzyme that
dephosphorylates SMP, causing a buildup of its concentration and a cor-
responding decrease in the level of inorganic phosphate. This pathway
is detailed in Table (3), and a schematic is presented in Figure (5).
Another point to consider is that uridine triphosphate (UTP) formation
is driven by ATP, and that the roles of these compounds in carbohydrate
formation are very similar.
In review, equation (1) is seen to represent substrate and energy
fluxes into the photosynthetic blackbox on the left hand side and product
fluxes out of the system on the right hand side. To better understand
this equation, the blackbox has been described in terms of five sub-
1. the light reaction
2. the Calvin cycle
3. the phosphate translocator system
4. the chloroplastic starch cycle
5. the cytoplasmic sucrose pathway
Table 3. Cytoplasmic sucrose pathway reactions.
Step Substrate Enzyme Product
1 DHAP + GAP Aldolase FBP
2 FBP Dekinase FMP + P
3 FMP Phosphohexoisomerase G6P
4 G6P Phosphoglucomutase G1P
5 GIP + UTP Pyrophosphorylase UDPG + PP
6 UDPG + FMP Phosphosynthetase SMP + UDP
7 SMP Phosphatase Sucrose + P
Note: PP is an abbreviation for pyrophosphate. Reactions are based on
Lehninger  and Kelly et al. .
SUCROSE UTP G
Figure 5. Cytoplasmic sucrose pathway schematic. Abbreviations are
defined in Appendix 1 (Based on Lehninger  and Kelly et al.
From the schematics and equations given to describe these processes,
the fluxes in equation (1) are plainly visible. The light reactions use
captured quanta (nhv) to drive the hydrolysis of water (H20), which sup-
plies electrons for use in the reduction of NADP and simultaneous forma-
tion of ATP. Oxygen (02) is liberated as a by-product of hydrolysis.
These interactions are described in equation (5). Carbon dioxide (CO2)
enters the system reacting with RuBP to form the first products in the
Calvin cycle, according to equation (7). Fixed carbon is transported
from chloroplast to cytoplasm where sucrose is formed and exported from
the cell. Some fixed carbon is temporarily stored in the chloroplast as
starch and later is transported to the cytoplasm, where it also forms
sucrose. The model development emphasizes the feedback regulation of
the carbon flow paths into and out of the photosynthetic cell.
As can be seen from the preceding tables and schematics, there are
a large number of intermediates involved in each subsystem. In the model
being developed, the concentration of intermediates fluctuates in response
to net flow. It is assumed that keeping track of each intermediate pool
is not necessary because of the serial nature of the processes. To aid
in determining which intermediates should be retained, mediating enzymes
have been grouped according to their characteristics as given by Kelly
et al. , Heber  and Lehninger . Essentially, all bio-
chemical reactions on the cellular level are catalyzed enzymatically, each
enzyme may or may not be inhibited or promoted by any other compound or ion.
Enzyme, product and substrate groups are also unique in their degree of bio-
chemical reversibility, which will vary with pH and other factors. In short,
reactions can range from very simple reversible flow between two proportionate
pools to irreversible reactions which are highly sensitive to a range of
inhibitors and promoters. For purposes of modelling, it is essential to
sort out the reactions which may be crucial rate limiting and path
selecting points from those that are not. Four criteria have been
devised to sort out the significant interactions in each subsystem.
The first simplifying criterion is that nonallosteric enzymes can be
ignored and that reversible path reactions between a substrate pool and
a product pool can be condensed into a single representative tank. An
excellent example of this situation is the fourth step in the Calvin
cycle, in which DHAP and GAP quickly reach an equilibrium because of
the continuous availability of active enzyme [Lehninger, 1970].
DHAP GAP AG=1.0 kcal/mole (11)
In modelling this portion of the Calvin cycle, no distinction needs to
be made between these triose phosphates; the presence of one implies
the presence of the other in some approximately constant ratio. Since
DHAP has a significant role in path selection, the representative
storage pool is referred to as DHAP.
The second criterion extends the conditions of the first to include
nonallosteric reactions which require energy or reducing units. The
second Calvin cycle reaction fits these requirements.
PGA + ATP + BPGA + ADP AGo=4.5 kcal/mole (12)
In the chloroplast exposed to light, ATP levels are high and equation (12)
can be expected to have a net flow to the right, while in the dark, the
flow will reverse direction (see postscript on PGA).
The third Calvin cycle reaction is similar to the second, requiring
reduced NADP to proceed from BPGA to GAP. The reactions described so far
under the first and second criteria are modelled schematically in
Figure (6), using "Energy Language Symbols" developed by H.T. Odum .
There are four predominate symbols used: the tank which symbolizes the
concentration of a metabolite in the reaction medium; the interaction
symbol which relates substrates and/or allosteric effectors; the circle
which represents an unlimited flow source, and the interaction arrow,
which shows the explicit path taken (see Appendix IV).
Looking at Figure (6), BPGA and GAP can be consolidated into a flow
path between PGA and DHAP which is moderated by the levels of ATP and
NADPred. The process can be further simplified by assuming that reducing
power and energy units come from the same source in a reasonably constant
ratio. Hence, if ATP is available, then NADPred should also be available.
The condensed model is shown schematically in Figure (7).
A third criterion treats the class of nonallosteric reactions which
have products. An example is the dephosphorylation of FBP to form FMP
and inorganic phosphate, P, which occurs in the Calvin cycle as well as
the cytoplasmic sucrose pathway. This process can be modelled schemati-
cally as in Figure (8). The assumption is that tanks can be condensed,
but the product P cannot be ignored. FBP can be absorbed into the DHAP
tank and FMP can be absorbed into the G6P tank according to the first
criterion, while P must flow to an inorganic phosphate tank. The simpli-
ifed schematic version is in Figure (9).
The fourth criterion dealswith allosteric rate limited reactions
by controlling flow between pools with elements that sense reaction
inhibitors and/or promoters. In the starch forming reaction, step 6 in
Table (2), the enzyme is inhibited by inorganic phosphate (P) and pro-
moted by the intermediate, PGA [Kaiser and Bassham, 1979]. So, even
Figure 6. PGA-DHAP pathway schematic.
in Table (1).
Explicit equations are listed
Figure 7. Simplified PGA-DHAP pathway. All abbreviations are listed
in Appendix 1. (Symbols are defined in Appendix 4.)
DHAP-G6P pathway schematic.
Figure 9. Simplified DHAP-G6P pathway schematic. All abbreviations are
defined in Appendix 1. (Symbols are defined in Appendix 4.)
Explicit equations are in
though these compounds are not substrates in the reaction, they must be
included in some form to control the reaction.
The application of these criteria to the five subsystems results in
the simplified model shown schematically in Figure (10). As shown below,
the schematic can be used to generate differential equations by doing
simple mass flow balances into and out of each tank.
Differential Equation Development
The interaction symbol between tanks indicates a simple algebraic
process involving the concentrations of reactants and a rate constant.
The level of PGA1 in the chloroplast is expected to vary according to
d[PGA1] = 2k [RuBP] [C02] + k2 [PGA2] [Pl] k3 [PGA1] [ATPl]. (13)
The first term on the right hand side of equation (13) represents
an addition to the PGA1 pool resulting from the reduction of CO2 which
is represented as a function of the concentrations of CO2 and RuBP as
moderated by the rate constant k1. The 2 is a numerical constant stoich-
iometrically required to maintain the system's carbon balance. The
second term is the phosphate translocator flow of PGA from the cytoplasm
to chloroplast as a function of [PGA2] and [Pl], multiplied by the rate
constant, k2. The third term is the Calvin cycle forward flow to DHAP,
a function of [PGA1] and available energy units, [ATP1]. Notice that
the third term embodies the assumptions outlined in Figures (6) and (7).
Figure 10. Simplified overall photosynthesis schematic. In the inter-
action symbols the k values are reaction rate constants. The divisor
sign implies inhibition. Numbers following intermediates separate
chloroplastic (1) from cytoplasmic (2) pools. All abbreviations are
defined in Appendix 1. (Symbols are defined in Appendix 4.)
Other equations derived from the schematic are
dt = k3 [PGA1] [ATP1] + 2k4 [STARCH] [Pl]
k10 [DHAPl] [ATP1] k5 [DHAP1] [P2]
k6 [DHAP1] [PGA1] [ATP1]/[P1]
d[STARCH] -= k [DHAP1] [PGAl] [ATP1]/[PI]
k4 [STARCH] [Pl]
dt = k5 [DHAPI] [P2]
k7 [DHAP2] [ADP2]
d[PGA2] = k7 [DHAP2] [ADP2] -
- k8 [DHAP2] [ATP2]/[SUCROSE]
d[SCROSE]= k8 [DHAP2] [ATP2]/[SUCROSE] [EXPORT]
dt 3/5k DHAP P1 k [RBP]
d[RuBP] = 3/5ki0 [DHAP1] [ATP1] kI [RuBP] [C02].
In addition to DHAP and PGA, separate tanks in chloroplast and
cytoplasm are maintained for ATP, ADP, and P. Their differential
= k9 [LIGHT] [ADP] [P1]
k3 [PGA1] [ATP1] k4 [STARCH] [P1]
k6 [DHAP1] [PGA1] [ATP1]/[Pl]
k10 [DHAP1] [ATP1]
d[P1 = k5 [DHAP1] [P2] + 2k6 [DHAP1] [PGA1] [ATP1]/[P1]
dt 5 6
k4 [STARCH] [PI] k2 [PGA2] [P1]
k9 [LIGHT] [ADP] [PI]
k7 [DHAP2] [ADP2]
- k8 [DHAP2] [ATP2]/[SUCROSE]
d -2 = k2 [PGA2] [P1] + 2k8 [DHAP2] [ATP2]/[SUCROSE]
k5 [DHAP1] [P2].
Some terms require numerical constants in order to maintain the system's
carbon and phosphate balances.
Most of the terms in these equations are concentration driven in
the sense that higher substrate or cofactor levels increase the proba-
bility of a reaction. Allosteric effects should be distinguished; in
equation (14), the last term, PGA1, behaves like the substrate DHAP1,but
its effect is to promote the enzyme which catalyzes the reaction. In the
same term, P1 acts to allosterically inhibit the process rate. The
algebraic form of the inhibitory effect is arbitrary in this equation.
Different responses could be obtained with different algebraic arrange-
ments. Without in vivo response curves, a very simple feedback response
has been chosen.
In the schematic given in Figure (10), the components of equation (1),
which flow into and out of the blackbox, are each treated differently.
This reflects certain assumptions about the paths which each input and
output to the photosynthesizing system must follow. The source of the
PAR is represented schematically as a circle rather than a tank, since
the absorption of quanta has no effect on the rate of quanta arriving at
the active site. In contrast, CO2 is a tank placed inside the chloroplast,
implying that concentration of available CO2 is affected by the rates of
CO2 arrival and exit at the active site. Water is ignored, since it is
always present in abundance for chemical reaction, even when the plant is
under water stress. Oxygen which is liberated in the chloroplast is also
ignored because it has only a minor impact on the high levels of 02
already present due to diffusion from the atmosphere. Finally, sucrose
(carbohydrate) is exported according to some arbitrary function with a
dependence on sucrose concentration.
The most glaring shortcoming of the model developed so far involves
the first step in the Calvin cycle. This is the rate limiting reaction
for the entire system. As such, it must be considered in more detail.
A second major omission from the model is the competitive oxy-
genase reaction, photorespiration, which is closely related to the first
Calvin cycle step, the carboxylase reaction.
Carboxylation, Photorespiration and Enzyme Kinetics
As described in equation (1), the first reaction in the Calvin cycle
is the fixation of CO2, which can be abbreviated for purposes of a
Michaelis-Menten style analysis to the following
CO2 + RuBP 2 PGA. (26)
Water has been eliminated since it is always at saturation levels
and does not affect the reactions rate. In competition with this reac-
tion is photorespiration, which can be written as
02 + RuBP -7 3/2 PGA + 1/2 CO2. (27)
Equation (27) is not strictly chemically correct as written. Actually,
phosphoglycolate is a product of this reaction that goes on to produce
CO2 and PGA, which are products of interest in this model. Both of
these reactions utilize the same enzyme, called ribulose bisphosphate
carboxylase (RuBPc) in equation (26) and ribulose bisphosphate oxygenase
(RuBPo) in equation (27). In addition, both reactions require the same
second substance, RuBP, which is often assumed to be at saturating con-
centrations [Jensen et al., 1978], allowing it to be ignored, like water.
With this assumption, both reactions above can be approximated by first
order processes which can be analyzed with enzyme kinetic theory.
Consider the reactions to be occurring in a well mixed medium, with
concentrations of carbon dioxide substrate, [C02] oxygen substrate,
 ribulose bisphosphate substrate [RuBP] and activated ribulose
bisphosphate carboxylase-oxygenase enzyme (E). Assume that the enzyme
reacts with either CO2 or 02 as a function of the activated enzyme's
relative affinity for the two substrates and the temperature adjusted
relative concentrations of the two substrates. Under the assumption that
RuBP can be ignored because it is at saturation levels, the carboxylation
reaction equations are
E + C02= ECO2 (28)
ECO2 --- PGA,
where C1, C2 and C5 are reaction rate constants, and ECO2 is the enzyme
substrate complex. These equations can be manipulated to yield a
Michaelis-Menten equation of the following form:
Vc = C02 + (29)
where Vc is the rate of substrate CO2 utilization, Vcmax is the maximum
rate of substrate CO2 utilization and Kc is the concentration of sub-
strate CO2 that will sustain the reaction at 1/2 the maximum rate. The
corresponding equation for the oxygenase reaction is
Vo = +02J + Ko (30)
where the variables are analogous to those in the preceding equation.
Given that both of these "first order" reactions use the same enzyme and
assuming that neither substrate affects the affinity of the enzyme for
the other, they can be treated as a classic case of competitive inhi-
bition. The resulting equation for carboxylation which includes this
competitive effect is
Vco = Vcomax [C2 (31)
LCOJ2 + Kc (1 + LO[2/K)
If RuBP is not assumed to be at saturating levels and is included in
the analysis, the carboxylation equations must be expanded to include
an intermediate reaction as follows:
E + CO2 ECO
2 C 2
ECO2 + RuBP 4 ECO2RuBP
ECO2RuBP C E + PGA (32)
where C2 and C4 are reaction rate constants and ECO2RuBP is a second
stage substrate-enzyme complex. Assuming that each substrate combines
with its specific site without either substrate affecting the enzyme's
affinity for the other, then the carboxylation equation is
( [O2] [RuBP]
Vcr = Vcrmax C 2 + K [RuBP] + K(33)
[CO2 + K ) RuBPJ + Kr
As before, parallel development yields the following oxygenase equation:
Vor = Vormax( [021 RuBPI (34)
 + K0 I[RuBP] + Kr
These relationships describe the flow rates among the reactant and
product pools as shown in Figure (11).
Figure 11. Carboxylase-oxygenase pathway schematic. This schematic
models the competition by 02 and CO2 for RuBP. The numbers represent
stoichiometric balances for each path and the new interaction
symbol represents a Michaelis-Menten function.
From the equations developed, it is postulated that the rate of
photosynthesis is a function of the substrate concentrations within the
stroma. The carboxylase reaction is regulated by the concentration of
CO2 at the reaction site, while the oxygenase reaction is a function of
02 concentration and both processes depend on the RuBP level. Since both
pathways draw on the available supply of RuBP, the steady state level is
lowered, which effectively depresses both reaction rates [Canvin, 1978].
The effect of this analysis on the model is to change the functional
form of the flows among the tanks involved in the first Calvin cycle
reaction. This involves the differential equations for PGA1 and RuBP,
which are numbers (13) and (19), respectively. Instead of the simple
algebraic product of concentrations, equations (33) and (34) are used
to model flows from the RuBP tank and to the PGA tank. The new balance
dPGA 2 Vcrmax [C02] V [RuBP]
dt + KC,[RuBP]+ Kr
3 (  [RuBP]
2 x  + Ko RuBPJ + Kr
+ k2 [PGA2] [Pl]- k3 [PGA1] [ATP1] (35)
d[RuBP = k10 [DHAPI] [ATPl]
Vcrmax [CO] + Kc)j [RuBP]
([02 ) + K) [RuBP] + Kr)
Vormax 102 K [RBP K (36)
21+ [RRuB]1 + K i"
Note that in equation (35), the first term on the left hand side is
multiplied by a factor of two. This is consistent with the stoichiometry
of equation (7), in which one mole of RuBP is oxidized by one mole of
CO2 to yield 2 moles of PGA. In the oxygenase reaction, the factor is
3/2. It remains to consider the pools of CO2 and 02 in the stromal
compartment [Hall and Bjorkman, 1975; Bruin et al., 1970].
CO, and 0, Concentration in the Stroma
Ambient levels of CO2 are unaffected by the flux to an individual
leaf and can be treated as an unlimited source, while the carboxylation
reaction in the leaf's chloroplast acts as a sink. The effect is to
create a concentration gradient between the ambient source and each
chloroplastic tank. Between source and sink, there are resistances,
some of which are more constant than others. The primary resistance
to CO2 flux in terrestrial C3 plants such as soybeans, is caused by the
stomates, which respond to a variety of factors, such as light, leaf
water potential and CO2 concentration. For stomatal response in this
model, water stress is ignored, light level is treated as an "opening"
switch, and CO2 concentration does not affect resistance. Under these
assumptions, the daytime cellular CO2 concentration, [C02]i,is treated
as a constant for any given ambient CO2 level [Wong et al., 1979]. The
diffusion of CO2 from the intercellular space, through the cell wall,
through the cytoplasm, through the chloroplastic membrane and into the
stroma, is the process of interest. In some previous models, this
process has been approximated by assuming a CO2 concentration of zero
at the active site [Chartier and Prioul, 1976], and solving for a
mesophylll resistance," which is a conglomerate of various membrane,
photorespiration and carboxylation effects. In this model, the reduction
of CO2 is assumed to take place in the well mixed fluid compartment
of the chloroplast, the stroma, which has an average CO2 concentration
[CO2] The resistance to diffusive transport between the intercellular
space and the stroma is the sum of the two membranes and the intervening
cytoplasmic fluid, which is defined as membrane resistance, Rm. This
process is modeled as simple diffusion transport, using the equation
CER = C Rm- (37)
Using estimated values for Rm [Charles-Edwards and Ludwig, 1974],[C02]
can be defined in terms of CER, which is the carbon dioxide exchange rate.
A second source of stromal CO2 is the oxidation of phosphoglycolate
(PGly) shown in Figure (11). According to work done by Bruin et al. ,
one mole of oxygenated RuBP produces 3/2 moles of PGA and 1/2 mole of CO2.
The input to the stromal CO2 tank due to photorespiration can be written
as a fraction of equation (34). The rate of export from the CO2 tank is
equal to the carboxylation rate in equation (33). Using these equations
as well as equation (37), a differential balance for the stromal CO2
concentration can be written as follows:
d[CO2] [C02]i [C02]
+ Vormax  [RuBP]
2 [02 + K LRuBP] + Kr
V [CO 2] [RuBP] (38)
Vcmx C021 + Kc /LRuBPJ + Kr38
The oxygen concentration at the reaction site within the chloroplast
is a function of diffusion from an unlimited ambient reservoir where
its absolute concentration is 700 times greater than [C02]. In addition,
02 is being supplied in the thylakoids, where the light reaction is
splitting water faster than the oxygenase reaction can remove it.
Deviations from equilibrium with air levels of oxygen caused by the last
two processes are assumed to be damped out by the diffusion process.
Oxygen concentration is assumed to be in constant equilibrium with ambient
levels (21% at 1 atmosphere) [Sinclair et al., 1977].
Membrane Transport Considerations
During analysis of model results, a two-substrate Michaelis-Menten
function describing the exchange of DHAP1 for P2 was found to correspond
to empirical results much better than a first order function (see Figure
21 ). Therefore, the term
k5 [DHAP1] [P2]
used in equations (14) and (16) has been replaced by the following:
Vdpmax [DHAP1] [P2]+
dpax [DHAP1] + d [P2] + Kp
There is ample theoretical basis for modelling membrane transport with a
Michaelis-Menten function [Epstein and Hagen, 1952].
A second assumption has been made simply for the sake of convenience.
The modelled exchange of P1 for PGA2 has been found to be a relatively
small term on a functional basis. Further, direct measurements of this
particular process have not been made. Thus, with no direct data on
process rate or how it may be altered by chloroplastic or cytoplasmic
conditions, the term is too arbitrary to remain in the model. The term
k2 [PGA2] [PI]
is assumed to be small and without recognized control significance;
therefore it has been dropped from equations (13), (17), (22), and (25).
Competitive Inhibition of Sucrose Formation
As with membrane transport, the formation of sucrose was not well
modelled as a first order reaction. During model analysis it was found
that sucrose formation could be realistically modelled as a Michaelis-
Menten function of [DHAP2] competitively inhibited by [SUCROSE]. There-
fore, the term
k8 [DHAP2] [ATP2]/[SUCROSE]
used in equations (16) and (18) has been replaced by the following:
[DHAP2] + Kd2 (1 + [SUCROSE]/Ks) "
The basis for modelling sucrose formation this way is provided by the
results obtained by Hawker . Those results indicated that the
conversion of sucrose phosphate to sucrose is regulated by competitive
inhibition of the mediating enzyme by sucrose itself. Since information
on [ATP2] is so limited, it has not been included in the new term.
The modelled system of equations as altered by the assumptions in
the preceding three sections is listed in Table (4). A final schematic
representation of the model as described by the system of equations is
given in Figure (12). In addition, a listing of major assumptions
involved in the model's development are listed in Table (5).
During model development many explicit and implicit assumptions
were made. In the following postscripts two important central assumptions
are considered in detail. The first is an explicit assumption about
Table 4. Biochemical photosynthesis model equations.
d[PGAI] 2 Vcrmax
+ 3 Vormax
[C02] + Kc
[ 02] Ko
Lo2l + Ko
[RuBPJ + Kr]
LRuBPJ + Kr
k3 [PGA1] [ATPl]
d[DHAPl] = k3 [PGA1] [ATP]+ 2k4 [STARCH] [Pl]
k10 [DHAP1] [ATP1] k6 [DHAP1] [PGA1] [ATP1]/[P1]
- Vdpmax [DHAP1] + Kd
[P2] + Kp]
d[STARCH = k6 [DHAP1] [PGA1] [ATP1]/[PI]
d- k STARCH
k4 [STARCH] [PI]
d[DHAP2] = Vdpmax
S [DHAP] ) [P2]
[DHAP1] + Kd [P2] + Kp
k7 [DHAP2] [ADP2]
LDHAP2J + Kd2 (1 + LSUCROSEJ/Ks)
dt : k7 [DHAP2] [ADP2]
LDHAP2J + Kd2 (1 + LSUCROSEJ/Ks)
Table 4. (contd.)
d tRuBP kl [DHAP1] [ATP1]
Vcrmax [CO 2 [RuBP]
V [C02] + Kc [RuBP + Kr
( [ 02] [RuBP]
Vormax 02] + Ko RuBP] + Kr
= kg [LIGHT] [ADP1] [P1]
k3 [PGA1] [ATP1] k4 [STARCH] [P1]
k6 [DHAP1] [PGA1] [ATP1]/[PI]
k10 [DHAP1] [ATP1]
d[P11 = Vdpmax [DHAP] K
dt (DHAPI + Kd
[P2] + Kp
+ 2k6 [DHAP1] [PGA1] [ATP1]/[P1]
k9 [LIGHT] [ADPI] [P1] k4 [STARCH] [P1]
dt2 = k7 [DHAP2] [ADP2]
[DHAP2] + Kd2 (1 + [SUCROSE]/Ks)
Table 4. (contd.)
d[P2] 2 Vdsmax [DHAP2] (46)
dt [DHAP2] + Kd2 (1 + [SUCROSE]/Ks)
Vd [DHAP] ) [P2]
Vdpmax [DHAP1 + Kd) LP2] + Kp
dt ([C02]i [C02])/Rm (38)
+ 1 Vonnax  [RuBP]
2  + Ko [RuBPj + Kr
( [C02] 2 [RuBP]
Vcrmax LCO2J + Kc [RuBPJ + Kr
Figure 12. Final photosynthesis schematic. This is a schematic repre-
sentation of the equations in Table (4) and the assumptions in Table (5).
The new rounded interaction symbol indicates a Michaelis-Menten response.
All abbreviations are defined in Appendix 1.
Table 5. Key assumptions used in model development.
1. Intermediate biochemical compounds that are not at path selecting
points and do not allosterically affect enzyme activity levels can
2. RuBP carboxylase-oxygenase is assumed to have a constant activity
level which can limit overall CO2 fixation rate.
3. Starch and sucrose formation pathways are regulated by allosteric
control of enzyme activity levels which can be functionally expressed
as concentrations of inhibitors and enhancers.
4. Concentration of CO2 in the intercellular space is a constant
function of external CO2 concentration.
5. Concentration of 02 in the chloroplast is assumed to be in equilibrium
with ambient 02 level which is constant at a partial pressure of
6. The dark reactions all take place in the stroma of the chloroplast
which is treated as a well-mixed reaction tank with uniform
the enzyme kinetics of RuBPc-o and the second is an implicit assumption
about the thermodynamically uphill nature of the Calvin cycle. These
discussions help to demonstrate the ever-present perils of modelling
as well as the insights to be gained from modelling.
Postscript on RuBP Carboxylase-Oxygenase and RuBP Concentrations
Enzyme kinetic theory which was used in the last section to derive
the Michaelis-Menten style relationships for photosynthesis and photo-
respiration relies implicitly on certain assumptions about the behavior
of an enzyme.
In the carboxylase reaction described in equation (32), it was
assumed that RuBPc had a constant activity level over the range of cir-
cumstances for which the analysis applies. More precisely, the well-
stirred reaction tank (stroma) is assumed to have a fixed quantity of
enzyme, which is either tied up as an enzyme-substrate complex, or is
available for reaction. The available portion is expected to have a
certain affinity for the various substrates which is unaffected by
conditions inside the reaction medium. Recent work suggests that these
assumptions may not be applicable to RuBPc-o.
It is well established that the activity level of RuBPc-o, measured
in vitro, is highly variable [Bassham et al., 1978]. Until recently,
these measurements gave Michaelis-Menten constants which were consistently
too high to account for associated photosynthetic rates. It has now been
established that, if the enzyme is preincubated in the presence of CO2
and Mg++, then the activity levels in vivo reaction rates can be obtained
[Bahr and Jensen, 1978]. Even under these circumstances, the high
activity rate can only be maintained for a short period of time.
Based on their work, Jensen et al. , have suggested that the
enzyme exists within the intact chloroplasts in a range of active states
which are regulated by the pH and Mg++ ion concentration. The chloro-
plast can be divided into two compartments, the stroma and thylokoid.
When exposed to light, a proton gradient is established between these
compartments which raises the pH in the stroma. In counter flow to the
hydrogen ions moving out, Mg++ ions move from the thylakoid to the
stroma. Thus, during the day, the enzyme's activity is enhanced and in
the dark it is deactivated. In support of this hypothesis, Bassham
et al.  found that a stable pool of RuBP is maintained by chloro-
plast in the dark. This is surprising in view of the extremely negative
standard free energy change associated with the carboxylation reaction
(AGO = -8.4 kcal/mole). If the enzyme were active, the reaction would
certainly proceed rapidly and irreversibly. Although this evidence
seems compelling, very recent work by Robinson et al.  indicates
that, in vivo, the activity level changes only slightly. Clearly, the
issue is controversial.
This raises the question of whether or not the postulated dependence
of photosynthesis on the concentrations of CO2, RuBP and 02 is only
partially true. Does the changing pH of the reaction medium alter the
enzyme's turnover rate enough to affect the photosynthetic light response
curve? Do the assumptions put forth in the various biochemical-empirical
models actually predict biochemical observations? An example may clarify
For a C3 plant under ambient levels of CO2 (- 320 vpm), Wong et al.
 have found that, regardless of light level, the intercellular
level of CO2 is constant (, 250 vpm). Also, assume that R which is
the CO2 diffusion resistance associated with cell walls, membranes,
cytoplasm and chloroplast stroma, is between 1 and 5 sec/cm [Tenhunen
et al., 1977]. Then, rewriting equation (37), the stromal concentration
of CO2 can be solved for any given carbon dioxide exchange rate, CER.
[CO2] = [C02]i CER Rm (44)
Let CER1 = 1 g/m**2/hr,
CER2 = 5 g/m**2/hr and
Rm = 2 sec/cm.
Then [C02] 1 = .28 gCO2/m**3%192 vpm
[CO2] 2 = .06 gCO2/m**3\41 vpm.
The estimated stromal CO2 concentration decreases by a factor of
4.67 for a five fold increase in the carbon exchange rate (CER). Assume
that the carboxylation reaction rate, Vcr, changes in the same proportion
as CER, then using standard chemical kinetics to estimate the relative
concentrations of RuBP
[RuBP]l Vcrl [C02] 2
[RuBP]2 Vcr2 [C02]l (45)
The calculation yields 47.9 times more RuBP in case 2 than in case 1.
Using enzyme kinetics as in equation (33) would yield an even larger
This analysis seems reasonable, at saturating light levels CO2
becomes limiting while ATP and NADPred levels are high, allowing for
very rapid replacement of RuBP used, thus RuBP levels are high while
CO2 levels are low and vice versa. This conclusion is completely incon-
sistent with the biochemical literature. Jensen et al.  found that
the RuBP pool sizes in isolated chloroplasts were approximately the same
under 25 microEinsteins/m**2/sec and 800 microEinsteins/m**2/sec, even
though the carbon exchange rate was 5 times higher under high light.
Bassham et al.  found that the RuBP pool size did increase with
decreasing [CO2] in reconstituted spinach chloroplasts. [RuBP] concen-
tration increased by 1/3 when CO2 was decreased from 202 vpm to 116 vpm.
Neither of these experiments were done on whole leaves, but the dif-
ference in results observed and predicted with the equations above is
At present, the mechanistic nature of the enzyme RuBPc-o is not
well enough understood to be included in the model. When sufficient
data are available,it can be included by functionally adjusting Kc, Kr,
Ko, Vermax and Vormax to light level, or any other relevant parameter.
In the meantime, this exercise will illustrate the potential of modelling
biochemical pathways of photosynthesis, but unresolved complexities
should serve as a reminder of the limitations of this model.
Postscript on Photosynthesis-Respiration Roles of Chloroplastic PGA
Control over the photosynthetic rate may be directly related to the
dual purposes served by the first few reactions of the Calvin cycle. The
reversible nature of these reactions was briefly discussed in applying
simplifying assumptions to equations (16). The direction of these
processes is a function of straight forward thermodynamic considerations.
In this model, the direction of this sequence of chemical reactions is
not in question, since "forward" reactions predominate during photosynthesis.
However, the thermodynamic requirements that determine the net direction
of a reaction are manifested in the concentrations of the reactants. The
resulting dynamic balance among the various biochemical intermediates
has been postulated to be a central control mechanism regulating photo-
synthetic rate [Walker and Robinson, 1978]. The following discussion
describes the thermodynamics involved and demonstrates how equation (8)
The first product of photosynthetic CO2 reduction is 3-phosphoglyceric
acid (PGA), which, through three subsequent steps yields DHAP at the
expense of one unit each of ATP and NADPred. This process is shown
schematically in Figure (5) and the specific equations are numbers 1
through 4 in Table (1). The requirement for light generated energy
units in this sequence implies that the pathway is "uphill." Recog-
nizing that the reactions are reversible and that the reverse "downhill"
reactions form part of the oxidative glycolytic pathway [Kelly et al.,
1976], the situation becomes even more intriguing.
During the daytime, when absorbed light energy is abundant, the
chloroplasts are autotropic, photosynthetically forming energy rich
carbohydrates, some of which are exported to the heterotrophic cytoplasm
and some of which are stored in the chloroplasts as starch. In the
dark, the chloroplasts, like the rest of the cell, are heterotrophic
and must obtain energy by breaking down carbohydrates. Thus, in the
light the "uphill" energy pathways predominate, while at night, the
"downhill" energy releasing paths are activated. PGA and DHAP are both
intermediates in both the "uphill" Calvin cycle and the "downhill"
The choice of which pathway is active can be shown by rewriting
PGA + ATP tBPGA + ADP AGo = +4.5 Kcal (8)
For a reaction to occur, the free energy change (AG) must be negative.
Such reactions are termed exergonic. Chemical reactions with a positive
free energy change are termed endergonic and will not occur without
energy input. Qualitatively, the free energy change is the fraction of
total energy change which is available to do work as the system proceeds
toward equilibrium, in accord with the second law of thermodynamics. For
this reaction, the free energy change (AG) is a function of the law of
chemical equilibrium (or the mass action law) which simply states that
in a system at chemical equilibrium, the concentrations of reactants
and products will be such that the following expression holds:
Keq [PGA] ADP
PKe GAJ LATPJ *
where Keq is the equilibrium constant.
Free energy change is related to the above relationship as follows:
AG + AGO + RT In [BPGA] [ADP]
AG + AG + RT n [PGA LATP '
where R is the universal gas constant, T is absolute temperature and AG
is the standard free energy change. At equilibrium conditions for a
given temperature, free energy is minimized (entropy is maximized),
allowing for no further change in free energy (AG = 0). Therefore, the
standard free energy is expressed as
AGo = -RT In (Keq).
The net reaction direction, therefore, depends on the concentrations
of products and reactants at a particular time. Since the standard free
energy for this reactions a positive 4.5 Kcal, the balance of concen-
trations must be such that
RT In < [ADP< 4.5 Kcal/mole
if the forward reaction is to proceed. At 250 C the required concen-
tration balance can be calculated for the Calvin cycle to predominate;
the ratio of reactants to products must be
The ratio ATP/ADP is reported to be in the range of 1 to 10 [Heber, 1974];
therefore, the ratio of PGA to BPGA is expected to range between 2000
and 200. Thus, the chloroplastic levels of BPGA under lighted conditions
are expected to be very low [Walker and Robinson, 1978]. This conclusion
is consistent with the relative abundance of data on the PGA concentra-
tion and the almost total lack of data on BPGA levels.
In summary, when chloroplasts are exposed to light, the Calvin cycle
pathway providing RuBP for CO2 reduction is enzymatically activated,
rapidly leading to production of PGA. As the concentration of PGA
rapidly increases and the balance of ATP to ADP responds to the balance
of light levels and bioenergetic requirements, the "forward reaction" is
quickly established and photosynthesis proceeds. Concentrations and
flow between tanks quickly reaches a quasi-steady state which is a func-
tion of all the factors affecting the balance of intermediates such as
light driven regeneration of ATP. These various factors have been
incorporated into the model and are analyzed in the following results
The biochemical photosynthesis model outlined in Figure (12) and
Table (4) is functionally complete. Given a complete set of data, it
would be possible to do computer simulations without further assumptions.
However, the data set required would be extensive, precise and subject
to wide variations from one set of conditions to another. In an effort
to circumvent this problem and to evaluate the model's behavior, a
series of simple flow situations have been posed, based on partial data
sets and supplemental assumptions. Measured substrate and product
fluxes and concentrations of intermediates given in the literature have
been used to analyze specific tanks within the model for consistency
with the real world.
This section emphasizes the feedback interdependence of the entire
system, which expresses itself in control of the photosynthetic rate.
At the biochemical level a steady flow system is established in response
to external conditions, such that intermediate concentrations are adjusted,
partitioning between starch storage and sucrose is delineated and net
carbon uptake is fixed. The analysis centers on the flows of CO2 into
and out of the modelled system. The analysis begins with the modelled
CO2 balance, followed by the starch balance, the partition regulating
phosphate balances and,finally, the sucrose balance. Working forward
along the photosynthetic carbon pathway through each of these key tanks,
the response of each balance equation is evaluated. By considering how
the modelled tank balances compare to measured concentrations under
differeing steady flow situations, the model's usefulness can be tested.
The Chloroplastic Carbon Dioxide Balance, [CO 2
The CO2 concentration balance in the stroma of the chloroplast is
described in the model by equation (38):
d[CO2] [Co02i [C02] 1 [02 [RuBP (38)
dt Rm + 2 vormax Vr [0a ] + K [RuBP] + Kr
S[ [C02] \/ [RuBP]
Ccrmax 2 + Kc [RuBP] + Kr)
where the right hand terms represent fluxes due to diffusion, photo-
respiration and photosynthesis, respectively. Under steady flow condi-
tions, the chloroplastic CO2 concentration is in steady state and
equation (38) can be written as follows:
[C02]i [C02] [RuBP] Vcrmax [C02] 1 Vormax 
Rm \[RuBP + Kr [ LC2] + Kc  + K
The equation has been rearranged so that each side equals the net flux
of ambient CO2 into the chloroplasts, referred to as net photosynthesis
or the carbon dioxide exchange rate (CER). Thus the following two equa-
tions can be written as:
CER = Rm (37)
CER = [RuBP] crmax [C02 1 Vormax [0,] (48)
S[RuBP] + K rL [CO2 + Kc LOJ + K- (4
Equation (48) involves eight biochemical parameters on the right hand
side, all of which have been measured and reported under a variety of
circumstances with variable credibility. The parameter values to be used
in the following analysis are listed in Table (6). It is useful to
recognize parameters in Table (6) as two separate groups. The first
group includes Ko, Vormax, Kc, Vcrmax and Kr, which are assumed constant,
although that depends entirely on the constancy of the activity of the
enzyme, RuBPc-o. The second group, , [C02] and [RuBP], is assumed
to vary, although under real world conditions,  is approximately
By rearranging equation (48), the values in Table (6) can be used
to solve for [C02] as follows:
[ [RuBP] + Kr 1 Vormax \]
1m c [ LRuBPJ 1 [2 2 + K /
Ec02] / [RUBPJ +Kr 1 Vorax [0
Vcrmax C [RuBP] + 2 [ [02J + K0
= 128 vpm
For the specific experimental treatment presented by Heldt et al. ,
the [C02] is 128 vpm or 4.3 pM dissolved CO2. As a check on the value,
equation (47) can be rearranged to solve for Rm.
Rm = ([C02]i [C02]) / CER = 1.84 sec/cm
The value obtained compares well with Rm values found by various inde-
pendent researchers. Several values are presented in Table (7) for
Table 6. In vivo biochemical parameters used in model evaluation.
Biochemical In Vivo Value Reference
[C02]a 320 vpm Wong et al. 
[C02]i 230 vpm Wong et al. 
Kc 230 vpm Bahr and Jensen 
Vcrmax 300 umol C02/mg chl/hr Farquahr et al. 
 210 mbar Sinclair et al. [197 ]
Ko 330 mbar Farquahr et al. 
Vormax 80 Pmol 02/mg chl/hr Kent and Young 
[RuBP] 280 vM Heldt et al. 
Kr 30 PM Bassham et al. 
Note: This table contains recent approximation-measurements of the
various parameters in equations (47) and (48). The term in vivo
refers to the functioning living plant; such values are not necessarily
constant when biochemical systems are reproduced outside the functioning
plant, in vitro. Abbreviations are in Appendix 1.
Table 7. Mesophyll resistance values from the literature.
Mesophyll Crop Reference
6.8 sec/cm Bean Chartier et al. 
2.3 sec/cm Wheat Ku and Edwards 
2.9 sec/cm Cotton Jones and Slayter 
Note: This table contains suggested Rm
determined by independent researchers.
with the value obtained using Table (6)
values for C3 plants as
These values compare favorably
in vivo parameters.
As a first analysis of the model sensitivity, a series of CER vs
[C02] response curves at various constant RuBP concentrations have been
generated while  is constant at 210 mbars. Since both RuBP and CO2
are substrates in the carboxylation reaction, it is expected that to
increase either one will increase photosynthetic rate. As shown in
Figure (13), the modelled responses are consistent with reported CER
responses to CO2 concentrations.
In a more specific test of the model's sensitivity, equation (48)
is used to generate CER from 6 pairs of RuBP and CO2 concentrations for
comparison with data obtained by Bassham et al.  in experimental
work with reconstituted spinach chloroplasts. When the values in Table
(6) are used, the results are greatly shifted. Recognizing that RuBP
c-o activity is certainly reduced in vitro, values for Vcrmax and Vormax
were lowered to 140 Pmol C02/mg chl/hr and 28 imol02/mg chl/hr, and Kc
increased to 400 vpm. With the modified biochemical constants, the
comparison between modelled and measured values is quite good.
The characteristic behavior of the simulated CER response to [C02]
is consistent with the in vitro data of Bassham et al. . In addi-
tion, the magnitude of the simulated response was similar to those results.
Simulated and measured values are compared in Figure (14).
The effect of photorespiration can be examined by varying the level
of 02 for different fixed values of [C02] and [RuBP]. Results are
graphically presented in Figure (15). The trends predicted for changing
02 values are consistent with general observations. Higher levels of
oxygen increase the rate of photorespiration and decrease the carbon
dioxide exchange rate.
The results of the modelled initial fixation of carbon are generally
consistent with empirically observed behavior. Moving forward through
= 400 AM
= 200 iM
[RuBP] = 100 VM
[RuBP] = 50 vM
Figure 13. Modelled [CO2]-photosynthesis-[RuBP] response curves.
is modelled in units of mg CO2 fixed/dm**2 leaf area/hour at four
different fixed concentrations of RuBP. [0 ] is assumed to be in
equilibrium with an atmospheric partial pressure of 210 mbar.
I 1 I I I
1 / Measured values
Modelled, K = 400 vpm
20. o Modelled, K = 230 vpm
I I I I I
200 400 600 800 1000 1200
Figure 14. Measured and modelled [CO ]-photosynthesis response curves.
Graphs compare measured values from Bassham et al.  with values
from the model. The measured values were obtained in vitro. Therefore,
in vitro rate constants listed in Table (8) were used to obtain the model
values. The difference between the curves is in the K values. For the
dashed curve Kc equals 400 vpm. The other curve has a Kc value of 230 vpm.
Table 8. Comparison of in vivo and in vitro biochemical parameters.
chemical In Vivo Values In Vitro Values
Kc 230 vpm 400 vpm
Vcrmax 300 pmol C02/mg chl/hr 140 ymol C02/mg chl/hr
Ko 330 mbar 330 mbar
Vormax 80 pmol 02/mg chl/hr 37 Imol 02/mg chl/hr
Kr 30 30
Note: In vitro values are primarily from Bassham et al.  and
in vivo values are from Table (6). All abbreviations are defined in
[CO2] = 400 vpm
X ; -
S[CO] = 200 vpm
[CO2] = 100 vpm -
Q? '~ ~- ------- --
50. RuBPI constant
o [RuBP] variable
5t 0 0 I0 20 210
Figure 15. Modelled -photosynthesis response curves. For the solid
lines [RuBP] is constant at 280 vM. For the dashed line [RuBP] is
280 PM for the left most value and 50 vM for the right most value. Since
increased  increases the competition for RuBP its level is expected
to decline as in the dashed lines.
the model, fixed carbon is partitioned between paths leading to export and
to storage in the starch tank. Although it represents a relatively
small fraction of carbon fixed, starch formation rate and concentration
are highly visible barometers of shifting biochemical balances within
photosynthetic cells. As such, it is the next model compartment for
Time Rate of Change of Starch
As an initial test of the starch balance predicted by the model in
equation (15), fluxes to the starch tank under light and dark conditions
have been considered. Since starch has been widely observed to accumu-
late during the day and to be remobilized in the dark [Heldt et al.,
, equation (15) should show a net flux into the starch tank under
lighted conditions and net export in the dark. Rewriting equation (15)
d[STARCH] = k6 [DHAP1] [ATP1] [PGA1] / [Pl]
k4 [Pl] [STARCH] (15)
the first term on the right hand side is the functional rate of starch
formation and the second term is the rate of mobilization and export of
starch. There are five variables and two constants to be considered in
solving for the starch flux, d[STARCH]/dt. Although these variables
are dynamic, changing rapidly in response to external conditions, they
tend to have fundamentally different values under light and dark condi-
tions. Table (9) lists relative values suggested from the literature
for some of the variables.
With these values, the rate of starch formation term would be a
factor of 20 lower in the dark than in the light, while the starch
Table 9. Chloroplastic concentration of key metabolites in light and
Metabolite Light Dark Reference
[PGA1] 4.0 mM 1.6 mM Kaiser and Bassham 
[DHAP1] .4 mM .4 mF Kaiser and Urbach 
[PI] 3.0 mM 12.0 mM Kaiser and Bassham 
[ATP1] 2.0 1.0 Heber [1974
Note: Recently measured-approximated concentrations of metabolites
important to starch formation and remobilization. The values for
ATPI are relative numbers only.
mobilization term would be 4 times higher in the dark than in the light.
Equation (15) predicts that, in the light, starch formation will predomi-
nate, while in the dark, starch will be exported. This result is con-
sistent with observed empirical behavior.
If the variables [DHAP1], [ATP1], [PGA1] and [Pl] are assumed to be
constant, the starch balance can be written as
d[STARCH] = A B [STARCH] (49)
where A=k6 [ATP1] [DHAP11 [PGA1] / [Pl] and B = k4 [P1]. This equation
can be integrated to yield
[STARCH] = A (l-e-Bt)/B (50)
which is valid as long as A and B are constant. Under steady flow con-
ditions, intermediates are at least roughly in steady state. In experi-
mental work done by Upmeyer and Koller , soybean seedlings were
grown under constant conditions: saturating light, 300 vpm CO2, 250 C
and 60% relative humidity. Artificial lighting was switched on for 16
hours and off for 8 hours every day. Their data indicated that, from four
hours after the lights were switched on until twelve hours later, the CO2
exchange rate (CER) was constant at 34 mg/dm2/hr. These conditions
approximate a steady flow state; therefore, the variables A and B can
be assumed constant.
Measurements of starch flux and concentration taken directly from
Upmeyer and Koller's data can be plugged into equations (49) and (50)
to solve for A and B. Values are given in Table (10).
The predicted values of [STARCH] based on Table (10) values of A and
B are plotted graphically with the experimentally measured values in
Table 10. Measured starch accumulation parameters.
Hours Under d[STARCH] [ST
Constant Light dtTARCH
4 4.25 mg/dm**2/hr 25.5 mg/dm**2
12 .87 mg/dm**2/hr 46.0 mg/dm**2
A = 8.45 mg/dm**2/hr
B = .16 hr-1
Note: Values are from graphical data presented by Upmeyer and Koller
. These values are used to calculate parameters A and B from
o Measured values
Hours at Constant Light
Figure 16. Measured and modelled starch accumulation response. Measured
values are from Upmeyer and Koller .
Since the level of chloroplastic inorganic phosphate [P1] is
proportional to starch mobilization and inversely proportional to
starch formation, it is the primary functional mechanism affecting the
starch accumulation rate. To determine how changes in [Pl] affect
[STARCH] a proportionate family of constants was calculated.
The ratio A/B is numerically equivalent to the maximum predicted
concentration of starch. The curves resulting from the five sets of
constants are given in Figure (17).
The general trend predicted by the set of curves in Figure (17) is
consistent with observations that starch accumulation is increased as
inorganic phosphate levels are decreased. In experimental work com-
paring starch levels in phosphate deficient plants and phosphate rich
plants, starch concentrations were as much as 10 times higher in the
plants deprived of phosphate [Herold et al., 1976]. The starch levels
predicted and graphed in Figure (17) mimic these observations, ranging
over a factor of approximately 10.
A more realistic analysis of the model's starch balance must con-
sider the other variables [DHAP1], [PGA1] and [ATP1] in addition to
inorganic phosphate [Pl]. The regulatory role played by the precise
mix of these variables can be explored in more detail by using a set of
data in which the chloroplastic level of [Pl] was held at four differing
quasi-steady state levels. Heldt et al.  manipulated [Pl] levels
in a suspension of chloroplasts by controlling the level of inorganic
phosphate in the medium external to the chloroplasts. This is equiva-
lent to controlling [P2] in the model. Experiments were short term, 10
minutes and all external variables such as light, temperature and pre-
treatment of chloroplasts were the same for each [P2]. Pertinent data
from the experiments are listed in Table (12).
[P1]r = .67
[P1]r = .80
_Pl r = 1.00
[Pl]r = 1.00
HOURS AT CONSTANT LIGHT
Figure 17. Time courses of modelled inorganic phosphate-starch response.
[P1]r = 1.33
20' [P1]r = 2.00
2 4 6 8 10 12
HOURS AT CONSTANT LIGHT
Figure 17. Time courses of modelled inorganic phosphate-starch response.
[P1]r is the relative concentration of chloroplastic inorganic phos-
phate. It is equal to the normalizing ratio [P1]n/[P1]3 given in Table 11.
Starch accumulation as a function of
n [P1]n/[P1]3 An Bn An/Bn
mg/dm**2/hr hr mg/dm**2
1 2.00 4.22 .33 12.8
2 1.33 6.34 .22 28.9
3 1.00 8.45 .16 51.3
4 .80 10.56 .13 80.2
5 .67 12.67 .11 115.5
Note: This table demonstrates how changes in the relative value of [PI]
effect the starch formation parameters A and B. Relative values of [P1]
are obtained by normalizing [Pl]n by [P1]3.
Measured chloroplastic-cytoplasmic inorganic phosphate
Mm 1 2 3 4
[P2] .96 .43 .19 .08
[Pl] 9.60 7.00 4.00 2.20
[PGA2] .004 .009 .016 .024
[PGA1] 2.90 6.00 6.90 8.90
[DHAP1] .17 .33 .40 .25
[RuBP] 3.20 3.70 4.10 4.90
umol C02/mg chl/hr
[STARCH] .30 1.40 7.90 7.70
CER 91.3 108.40 113.90 59.70
Note: Metabolite and rate responses to changes in the cytoplasmic level
of inorganic phosphate. Values are from Heldt et al.. No data on
ATP levels were given. Abbreviations are in Appendix 1.
Assuming that in this short experiment starch levels are very low,
the mobilization term can be neglected. Therefore, starch accumulation
is a function only of the starch formation rate:
d t [S =k6 [ATPl] [DHAPl] [PGA1] / [P] (51)
Values predicted by this equation are compared with measured values in
Figure (18). (No values for [ATPl] were given, so for convenience of
scale, assume that k6* [ATP1] is constant and equals 7.7.)
For the various steady flow situations outlined, the model correctly
mimics qualitative behavior and corresponds reasonably well with absolute
values. From this analysis, the most obvious feature of the modelled
starch tank to emerge is the central regulating role played by chloro-
plastic levels of inorganic phosphate. To further explore the inorganic
phosphate control mechanism as it affects starch accumulation as well as
carbon export, the [PI] tank is treated in the following section.
The Chloroplastic Inorganic Phosphate Balance, [P1]
In the experimental work summarized by the data in Table (12), dif-
ferent levels of P1 were maintained by manipulating the inorganic phosphate
concentrations in the medium, external to the chloroplasts. This was
equivalent to adjusting the cytoplasmic inorganic phosphate concentra-
tion, [P2]. The correspondence between [P1] and [P2] is quite strong,
as can be seen in Table (12), and graphically in Figure (19).
This strong proportional dependence should be reflected in equation
(44), which is the chloroplastic inorganic phosphate balance equation:
d[Pl] = Vdpmax [DHAPl] [P2]
dt ([DHAPl] + Kd)([P2] + Kp2) +2k6 [DHAP1] [PGA1] [ATP] / [Pl]
- k9 [LIGHT] [ADP1] [P1] k4 [STARCH] [P1] .
I I I I
2 4 6 8 10
Figure 18. Measured and modelled inorganic phosphate-starch response.
Graph compares measured and modelled rates of starch formation as a
function of chloroplastic levels of inorganic phosphate, [PI]. Starch
accumulation rates are in units of Imol CO2 fixed as starch/mg
I I I I
2 4 6 8 10
Figure 19. Comparison of chloroplastic and external concentrations of
inorganic phosphate. Based on data from Heldt et al. .
In equation (44), the modelled dependence of [P1] for [P2] is in the
first term on the right hand side. This term represents the export of
DHAP1 from the chloroplast in strict counter exchange for inorganic
phosphate from the cytoplasm, P2. The second term on the right hand
side is the flux to the P1 pool, resulting from the net dephosphorylation
of glucose phosphate in the accumulation of starch. Comparing the rates
of starch accumulation with CO2 assimilation rates in Table (12), it is
clear that much more fixed carbon was being exported (term 1) than was
being stored as starch (term 2). This relationship is illustrated
graphically in Figure (20). Numerically, in the most extreme case with
[Pl] equal to 2.2 mM in treatment 4, starch accumulation was approxi-
mately 13% of the total carbon dioxide fixed and sucrose export was 87%
of CO2 fixed.
In other experimental work, the maximum rates of starch buildup are
cited as being from 10% to 20% under normal conditions [Herold and
Walker, 19791. Based on these data, the first term on the right hand
side of equation (44) is the primary process supplying inorganic phosphate
to the chloroplast. Therefore, the strong measured functional dependence
of [P1] on [P2] is also a predominant feature of the modelled [Pl] balance.
In the discussion above, the modelled correspondence between the first
term on the right hand side and export of DHAP1 to the sucrose tank are
equated. This can be expressed as
[TRANSPORT] = Vdpmax( [DHAP] [P2] (52)
LDHAP1J + K [P2] + K
where [TRANSPORT] equals the time rate of DHAP1 transport to the cytoplasm.
Equation (52) can be tested directly, using the data in Table (12), along
with assumed values for Kdl, Kp and Vdpmax. Results are shown in Figure (21).
E Sucrose export
2 4 6 8 10
Figure 20. Comparison of carbon partitioning pathways. Compares total
CO2 fixation rate with the rates of sucrose export and starch accumula-
tion at various measured levels of chloroplastic inorganic phosphate,
[P1]. Based on data from Heldt et al. . CER is given in units
of micromoles C02 fixed/mg chlorophyll/hr.
o Measured transport
o Michaelis-Menten modelled transport
A First order modelled transport
I I I I I I
.2 .4 .6 .8 1.0 1.2
Figure 21. Measured and modelled inorganic phosphate-transport response.
Graphs compare the amount of fixed carbon transported from chloroplast
to cytoplasm as a function of [P2]. Modelled curve constants were
adjusted to intersect at the right most point. Measured values are
from Heldt et al. .
For a comparison, a first order relationship is also tested and illus-
trated in Figure (21). The first order function has the following form:
[TRANSPORT] = K5 [DHAP1] [P2] (53)
The modelled term based on two substrate Michaelis-Menten kinetics
clearly fits the observed data better than the first order relationship.
Under steady flow conditions, such that CO2 uptake (CER) equals carbo-
hydrate production over a given period of time, concentrations of the
various cyclical intermediates are very nearly in steady state. With
this assumption, the [Pl] balance can be analyzed in steady state, which
can be written as follows:
[Vdpmax [D+HAKd] [P2] p 2(k6 [DHAPl] [PGAl] [ATPl] / [Pl]
[DHAPI] + K [ H[P2] + K 6
k4 [STARCH] [Pl]) = k9 [LIGHT] [ADP1] [P1] (54)
In equation (54), the first term equals export from the chloroplast to
the sucrose tank, the second term is the net accumulation of starch
within the chloroplast. Taken together, the terms on the left hand
side represent total carbohydrate formation rate which, in steady flow,
must equal the carbon dioxide exchange rate. The right hand side of the
equation is simply the rate of [ATP1] formation.
Using data from Table (12), equation (54) can be solved for the term
kg [LIGHT] [ADP1] at different values of [P1]. Since [LIGHT] is constant
in all four treatments, differences in the term reflect changing values
of [ADP1]. Results for the four treatments described in Table (12) are
listed in Table (13) and graphed in Figure (22).
The results show an increase in [ADP1]as [Pl] decreases. If the sum
of [ADP1] and [ATP1] is assumed constant, then [ATP1] levels must decline
Adenosine diphosphate formation as a function of inorganic
Value 1 2 3 4
[Pl] mM 9.6 7.0 4.0 2.2
CER mol C02/mg chl/hr 91.3 108.4 113.9 59.7
kg [LIGHT] [ADP1] 9.5 15.5 28.5 27.2
Note: This table uses values from Table
levels of ADP1 that result from changing
constant in each treatment.
(12) to evaluate the relative
P1 levels. [LIGHT] and k are
Figure 22. Modelled adenosine diphosphate-inorganic phosphate response.
Based on data from Heldt et al. . [P1] is the chloroplastic con-
centration of inorganic phosphate. Adenosine diphosphate is a function
of [LIGHT] which was constant in the experimental work and the rate
constant, kg. Therefore, changes in the kg [LIGHT] [ADP1] term reflect
changes in the modelled concentration of A1P1.
as [ADP1] increases and the [ATP1] / [ADP1] ratio must therefore decrease
as [Pl] decreases.
In general terms, the relationship found in Figure (22) is consis-
tent with data presented by Kaiser and Urbach , which detail the
interactions between [Pl], [ATPl] and [ADP1]. In essence, they showed
that when internal levels of Pl were reduced, the [ATP1] / [ADP1] ratio
responded immediately by decreasing dramatically. They further found
that the decline in [ATP1] / [ADP1] could be quickly reversed by
increasing the external supplies of inorganic phosphate, equivalent to
increasing [P2] in the model.
Further indirect evidence for the [ATPl] / [ADP1] ratio's dependence
on [P1] and [PGA1] is shown in Figure (23).
From Figures (22) and (23), the overall relationship that emerges
is a direct correspondence between [Pl] and [ATP1] and an inverse pro-
portionality between [P1] and [PGA1]. As explained in the model develop-
ment section, the concentration of PGA1 is thermodynamically regulated.
For the forward cyclic PGA1 reaction to proceed against a relatively
large positive standard Gibbs' free energy change, the ratio in the
AGO + RTIn ([BPGA] [ADP]
A [PGA] [ATPJ]'
must be very large. This constraint couples [Pl] levels directly to
[PGA1] through its effect on the [ATP1] / [ADPl] ratio.
This explains the control path between chloroplasts and cytoplasm.
[P1] is closely dependent on [P2] and by regulating the level of ATP1,
[Pl] can affect both the concentrations of intermediates and individual
reaction rates as was shown in Figure (19).
' l 4"
2 4 6 8 10 12
[PGA1 ], mM
Figure 23. Measured chloroplastic PGA-inorganic phosphate response.
Based on data from Heldt et al. .
The following section considers how the cytoplasmic inorganic
phosphate levels are regulated within the sucrose export system.
The Cytoplasmic Sucrose Balance, [SUCROSE]
Photosynthate that does not go to the starch tank flows to the cyto-
plasmic sucrose tank, from which it is exported to the rest of the
plant. This balance is modelled by equation (43):
d[SUCROSE] Vdsmax [DHAP2] (
dt [DHAP2] + Kd2(l + [SUCROSE] / Ks)
where [EXPORT] equals the time rate of sucrose export from the cytoplasm.
Under steady flow conditions, the sucrose tank is in steady state with
imports from the chloroplasts just offsetting exports to the rest of
the plant. In steady state, equation (18) can be simplified to the
[EXPORT] = Vdsmax [DHAP2] (55)
[DHAP2J + Kd2 ( + SUCROSE] / Ks)
Equation (55) has two variables and three constants on the right hand
side. For purposes of analysis, assume that [DHAP2] is normalized by
some average concentration such that, at an average concentration
[DHAP2] equals 1. Further assume that Vexmax is approximately equal
to the maximum rate of CO2 fixation, Vcrmax, which has been approximated
in Table (6) to equal 300 pmol C02/mg chl/hr. Kd2 and Ks have been
assumed to equal 1.5 and 100 mM respectively, based partially on Hawker
. The responsiveness ofequation (55) can be tested with these
values. Figure (24) illustrates the effect on [EXPORT] of varying
S120 [DHAP2] =2.0
S80 'o. Measured
S". [DHAP2] =1.0-.
[DHAP2] = .5
I0 I I
160 200 300
Figure 24. Measured and modelled sucrose concentration-[EXPORT] response.
Measured values (0) are from Hawker . Modelled values (.) are
modelled for different normalized DHAP2 concentrations. [EXPORT] is
given in units of pmol CO2 fixed and exported/mg chlorophyll/hour.
[SUCROSE] while [DHAP2]n is held constant, the dashed line is from data
taken by Hawker , who did pioneering work on feedback inhibition
of sucrose formation.
Real World Scenarios
Much experimental work has been done on the relationships between
photosynthesis, starch and sucrose. A question of particular interest
is how partitioning between these photosynthetic end products and their
respective concentrations might functionally affect photosynthetic
rate. In researching this question, experimentalists have caused many
different techniques to alter levels of starch and sucrose in a wide
variety of plant materials and in settings ranging from laboratory to
Some of the more common methods of changing leaf carbohydrate levels
are manipulations of diurnal temperature regimes, ambient CO2 concentra-
tions and by manipulating the source-sink balance of carbohydrates. In
each of these methods, enough work has been done to suggest certain
general response patterns, although there are numerous exceptions which
appear irreconcilable with the general trend. Direct manipulation of
source-sink balance fits into the model more simply than temperature
or CO2 manipulation and will be considered first.
Direct Manipulation; Sucrose Feeding
In two specific experiments [Moore et al., 1974; Habeshaw, 1973],
sucrose was directly applied to photosynthetic plant material. In both
cases, increased levels of external sucrose depressed photosynthetic
rate and increased carbohydrate concentrations in leaves. A large
fraction of the increased carbohydrate was in the form of starch. A
typical scenario can be applied to the model as follows.
The experimental plants are grown in some consistent environment,
resulting in a particular balance between the sucrose exported from
photosynthetic cells and the substrate and energy requirements of
plant's growing points. By direct external application of sucrose,
the concentration in the region around photosynthetic cells increases
and apparent demand for sucrose decreases (perhaps by disrupting a dif-
fusion gradient). This results in decreased export and cytoplasmic
sucrose levels rise. High concentrations of sucrose cause [P2] to
decrease, which can subsequently decrease [P1] enough to reduce photo-
synthesis. Low levels of P1 are also directly linked to increased
levels of PGAI, which promote the accumulation of starch.
Direct Manipulation; Selective Shading
Another type of direct source-sink manipulation is the experimental
work done by Thorne and Koller , in which experimental plants grown
in full light were completely shaded except for one leaf (the "source
leaf"). Control plants remained unshaded. The "source leaf" on the
experimental plant and a comparable leaf on a control plant were moni-
tored for starch level, sucrose level, photosynthetic rate and inorganic
After eight days, the source leaves from the shaded plants had much
lower levels of starch, higher levels of sucrose, higher levels of
inorganic phosphate, and higher photosynthetic rates than did the source
leaves from the unshaded control plants.
In terms of the model, this experiment poses an interesting problem.
The sink demand was much higher for the source leaves on shaded plants
than on the unshaded plants. According to the model, this should have
decreased the cytoplasmic SUCROSE levels, increasing [P2] and [Pl],
and decreasing [PGA1], all of which would facilitate photosynthesis and
mobilization of starch.
All of these things happened, except [SUCROSE] increased steadily
from 1% to 3% dry weight after nine days. This apparent inconsistency
between the model and empirical results deserves close attention.
A central assumption in discussing the feedback biochemical controls
of the sucrose system is that cytoplasmic inorganic phosphate levels are
constant (in the short term). In this experiment, inorganic phosphate
levels were six times higher in the shaded plant's source leaves than
in the control plant after eight days of treatment. This means that both
sucrose and inorganic phosphate levels increased significantly at the
same time, which cannot be explained by the short term assumptions used
in the model as derived. This apparently paradoxical situation could
result simply from increased import of phosphate (perhaps being mined
from the shaded leaves which would have high levels available).
With higher levels of P2, larger pools of phosphorylated sucrose
antecedents can be maintained without affecting photosynthetic rates
via control of Pl concentration. From Figure (24), it is clear that
export rates are increased by high concentrations of DHAP2, which rep-
resents the pool of phosphorylated sucrose antecedents.
For the shaded plants in the Koller and Thomas experiment sink
demand was high. Even so, the source leaf maintained relatively higher
levels of sucrose than the control. This gradient promoted relatively
rapid export, which was evident in the higher CO2 exchange rates found
for the shaded plant's source leaves. At normal phosphate levels, the
high sucrose concentration maintained in these leaves would reduce avail-
able inorganic phosphate, causing starch formation and reducing