Modeling and measurement of the role of macromolecular binding in the attachment of a Brownian particle to surface


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Modeling and measurement of the role of macromolecular binding in the attachment of a Brownian particle to surface
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ix, 141 leaves : ill. ; 29 cm.
Ma, Huilian, 1971-
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Chemical Engineering thesis, Ph. D   ( lcsh )
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Thesis (Ph. D.)--University of Florida, 2004.
Includes bibliographical references.
Statement of Responsibility:
by Huilian Ma.
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Copyright 2004


Huilian Ma

To my dearest daughter, Siyu


First and foremost I would like to thank my advisor, Dr. Richard Dickinson,

for his consistent guidance, encouragement and support throughout the whole project.

His motivation, dedication and passion towards research as well as his integrity and

optimism constantly inspired me all these years!

I also give my thanks to Dr. Anthony Ladd, Dr. Ranganathan Narayanan,
Dr. Timothy Anderson, Dr. Oscar Crisalle, Dr. Fan Ren, Dr. Chang-Won Park and

Dr. Ben Koopman for their understanding and encouragement for my continuation

of my doctoral research here. Without their help, my success in conducting this

research work would not have been possible.

I am very grateful for the support of graduated and current research group

members. Special thanks go to Dr. Aaron Clapp, Dr. Jonah Klein, Jamaica Prince,

Jeffrey Sharp, and Murali Rangarajan for their inspiring and helpful discussions, as

well as their patience in teaching me the optical trapping and evanescent wave light

scattering techniques. In addition, I have enjoyed so much the joyful laboratory

environments created by my fellow students mentioned above as well the most recent
lab members including Luzelena Caro, Colin Sturm, and Kimberly Interliggi. This

work could not have been progressed smoothly without their encouragement and

My stay in this department all these years would be the most eventful time
period of my life. And I myself have gone through an unexpected and difficult
transition during this time period. I would like to express my heartfelt thanks to

all the faculty, staff, and graduates in this department for their support and help
during the difficult times. In particular, many thanks go to Shirley Kelly, Nancy

Krell, Peggy-Jo Daugherty, Deborah Sanoval, Nora Infante, Andrea Weatherby, and
all other caring staff too numerous to name here.

I gratefully acknowledge the financial assistance and support received from
several funding sources, including the National Science Foundation, the NSF Particle
Engineering Research Center, the Department of Chemical Engineering at University
of Florida, and especially the support of my advisor who has always assured adequate
resources in the form of experimental materials and funding.
Finally and most importantly, I am so grateful to my family for their uncon-
ditional and unfailing love, care and encouragement throughout my whole life and
especially during my overseas study period. I could not find adequate words to ex-

press my feelings towards them. I just want to tell my dearest mom and dad, brother
and sister that I love them all!


ACKNOWLEDGEMENTS ............................ iv

ABSTRACT ......... ....... .................. viii


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


2.1 Particle Surface Properties .................. .......... 4
2.2 Interaction Forces Involved During Adhesion and Adhesion Mechanisms 5
2.2.1 Non-Specific Interactions and Attachment Mechanisms .. 6
2.2.2 Specific Receptor-Ligand Interactions and Attachment
Mechanisms ............................ 10
2.3 Techniques for Measuring Particle Attachment ............. 17
2.3.1 Adhesion Kinetics ..... .................... 18
2.3.2 Adhesion Strength .......................... 19

.. ... ..... .. ... ...................... .. 21

3.1 Particle-Surface Interaction Energy ................... 22
3.2 Probability Flux ................... ......... 25
3.3 Mean First Passage Time Approach .................... 29
3.3.1 Numerical Method .......................... 31
3.3.2 Analytical Approximations ................ .... 32
3.4 Results and Discussions ........................... 42

COATED SURFACES ........ ...................... 62
4.1 Staphylococcus aureus: Structure and Characteristics ........ 62
4.2 Fibrinogen-Clumping Factor Interaction ................ 63
4.3 Model Parameter Estimations ... ............... ... 64
4.4 Model Predictions and Comparison with Experimental Data ..... 74


5.1 Effect of Curvature on Particle-Surface Interaction Energy ...... 85
5.2 Effect of Curvature on Dynamic Process ................. 88

5.2.1 van Kampen's System Size Expansion ............. 89
5.2.2 Mean First-Passage Time Method . . 90
5.2.3 Discussions and Suggestions . ..... 93
5.2.4 Extension to Other Irregular Surfaces . .... 98

TECHNIQUE ................................. 99
6.1 Materials and Methods ........................... 99
6.2 Experimental Setup ........................... 101
6.3 Data Analysis and Measurement .................. ..103
6.3.1 Evanescent Wave Light Scattering ............... 104
6.3.2 Brownian Motions of a Particle in a Potential Well ... 104
6.3.3 Calibration of Optical Trap . . ... 106
6.3.4 Measurement of Equilibrium and Viscous Forces ........ 107
6.3.5 Experimental Measurements and Discussions .......... 108



MATLAB CODES ................................... 121

REFERENCES ................ .................... 135

BIOGRAPHICAL SKETCH ............................. 141

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



Huilian Ma

August 2004

Chairman: Richard B. Dickinson
Major Department: Chemical Engineering

Attachment of cells or biological particles to surfaces through the formation

of receptor-ligand bonds or other macromolecular bridges is important in many phys-

iological, biomedical, biotechnological, and environmental processes. Along with the

well-characterized colloidal forces, the binding kinetics, which is often not taken into

account in colloidal theories (e.g., in DLVO theory), also plays a significant role in
determining the rate of particle attachment. A probabilistic model is presented here

for studying the attachment of a rigid biological particle to a surface when both

Brownian motions and binding kinetics are important. From this model, we com-

pute the mean time required for the particle moving from an unattached state (e.g.,
the second energy minimum) over an energy barrier at a saddle-transition point to
an attached state from an energy landscape determined by both colloidal and bind-
ing interactions. Based upon the physical and molecular properties of the binding

species, the approach advocated here provides an analytical approximation for the
mean transition time from the second energy minimum as well as the deposition rate

constant for the general case where neither binding nor particle diffusion is necessarily
rate limiting. Application of this model in the adhesion of Staphylococcus aureus to
fibrinogen-coated surfaces is also described and model prediction results are compared
with experimental measurements from our research group. In addition, experiments
for a direct testing of model predictions are described using three-dimensional optical
trapping along with evanescent wave light scattering techniques to measure the at-
tachment dynamics of streptavidin-coated beads to biotinylated PEG (polyethylene
glycol) surfaces.


Attachment of biological particles to surfaces (or other particles) is often me-

diated by both colloidal forces and receptor-ligand (or other macromolecular) binding

interactions. This attachment process is relevant to many fields such as physiological,
biomedical, biotechnological, and environmental fields. Examples include recognition

and adhesion of cells in the body to extracellular matrix proteins for proper functions

[1], infections caused by bacterial adhesion to implanted device surfaces [2], affinity-
based separation of cell populations [3], and particle removal by filtration. Along
with the well-characterized colloidal forces such as vad der Waals attraction, elec-
trical double-layer interactions, and steric stabilization, the discrete bonds formed

between a particle and a surface through surface-bound macromolecules also play

very significant roles in determining the rate of particle attachment to a surface.
While models have been developed to predict the deposition rate of particles

to a surface under the influence of equilibrium colloidal forces, as accounted for in

DLVO theory [4, 5, 6, 7], models that account for the role of macromolecular bridg-
ing in attachment have been limited. One challenge is that macromolecular bridge

formation and dissociation may occur on a time-scale comparable to the time-scale
of particle diffusion (which is more important for smaller sized particles, e.g., particle
size of micron or submicron here) near a surface, such that the forces involved in
bridging may not be assumed to be at equilibrium. Attachment of bacteria to a sur-
face is often considered to be a two-step process [8, 9, 10, 11]: (i) attraction to a weak
secondary energy minimum created by non-specific colloidal interactions, followed by

(ii) strong adhesion by the formation of specific binding interaction between cell-

surface macromolecules ("adhesins") and complimentary ligands on the substratum
to form an "irreversible" attachment (i.e., the probability of spontaneous detachment

is negligibly small. In this view, the affinity and kinetics of the binding interactions
as well as the lengths of binding molecules are important parameters in determin-
ing whether or not attachment will be achieved during the time the bacterium is
reversibly associated with the surface. For example, Hartford et al. [12] showed

that attachment of Staphylococcus aureus (S. aureus) to fibrinogen-coated surfaces
depended strongly on the length of a putative stalk region on an adhesin expressed

on bacterial surface. Mascari and Ross [13] observed that the attachment rate of S.
aureus to collagen depended upon the density of bacterial adhesins, which suggested
that the deposition was rate-limited by the formation of specific bonds with collagen.

They also showed that about 10pN shear force on a bacterium was sufficient to begin
to diminish the attachment rate. Since 10 pN is typically required to accelerate disso-

ciation of a single receptor-ligand bond, it can be surmised that attachment required

formation of only a few adhesin-collagen bonds while the bacterium was close enough

to the surface for bonds to form. Thus, predictions of deposition rate by diffusion

over an energy barrier [14] only cannot account for such observations.
The goal of the work described in this thesis is to have a fundamental un-
derstanding of the attachment of biological particles to surfaces when both particle
diffusion and macromolecular binding kinetics are important. This understanding

may aid in enhancing or reducing particle attachment according to actual practical
applications by modifying particle surface or corresponding substrate properties, for

example, in designing infection-resistant biomaterials to prevent bacterial infection
of implanted intravascular and biomedical devices.
This dissertation describes the development of a dynamic model for studying
the attachment of biological particles to a surface that accounts for both colloidal

forces and macromolecular binding interactions. In addition, application of this model

in the adhesion of bacterium S. aureus to fibrinogen-coated surfaces and the direct ex-

perimental testing of this model using three-dimensional optical trapping techniques
are also described.

This dissertation is organized as follows. Chapter 2 gives an overview on the
current stage of knowledge with respect to biological properties and physics underly-

ing the adhesion mechanisms of particles to a surface and the common measurement

techniques used to study adhesion. In Chapter 3, a dynamic model is developed

to predict the mean transition time for a particle from the unattached state (e.g.,
the secondary energy minimum) to the attached state as well as the deposition rate

constant when both Brownian motions and macromolecular binding interactions are
important. The application of the model developed in Chapter 3 in the adhesion
of S. aureus to fibrinogen-coated surfaces is described in Chapter 4 and results ob-
tained from the model are compared with experimental measurements by other re-

search group members. Chapter 5 attempts to examine how the surface curvature

of particles affect attachment rate. In Chapter 6, experiments to evaluate the model
predictions in Chapter 3 are described using three-dimensional optical trapping along

with evanescent wave light scattering techniques to measure the attachment dynam-

ics of streptavidin-coated beads to biotinylated PEG (polyethylene glycol) surfaces.

Chapter 7 concludes this thesis with the major accomplishments and results and also
gives recommendations for future work.


Adhesion of biological particles to a surface is often mediated by receptor-
ligand or other macromolecular binding interactions. For example, the ability of

cells to recognize and adhere specifically to other cells or to extracellular tissue ma-
trix proteins is critical to many physiological processes. Cells flowing through the

body's circulatory systems (blood and lymph) adhere to vessel endothelia in partic-

ular organs for proper functions [1]. Also, cell-surface adhesion can be exploited for
biotechnological purposes, such as affinity-based separation of cell populations [3]. In
addition, microbial adhesion to surfaces plays a very important role in a variety of
fields covering different aspects of nature and human life, such as water treatment,

food industry, marine science, and most importantly biomedical fields. For instance,

adhesion of bacteria to human tissue surfaces or implanted biomedical device sur-
faces is an initial step in the pathogenesis of infection [2]. Therefore, having a better

understanding of adhesion mechanism helps to either enhance or reduce adhesion

according to practical needs by modifying molecular structures and properties of
particle surfaces and corresponding substrata.

2.1 Particle Surface Properties

The basic surface structure for all living cells shows great similarities across
all species and genera. This consensus structure, termed the fluid mosaic model,
consists of a continuum bilayer membrane of lipids punctuated by proteins of varying
penetration. The basic building blocks of all membranes are the phospholipids. Most
of the lipids of the membrane show free lateral motions if they are not associated

with proteins. Proteins embedded in the lipid matrix carry out most of the specific

processes associated with the membrane.

Gram positive prokaryotes have a relatively thick cell wall, consisting mainly

of peptidoglycan layer (i.e., a rigid polysaccharide layer cross-linked with lipopro-

teins and other polysaccharides) that surrounds the membrane. In gram negative
bacteria, there is a second bilayer outside the peptidoglycan. Thus the long range

interactions between bacterial cell surfaces and substrates or other bacteria may be

also influenced by these cell wall effects. The cell wall is absent in animal cells, but a

complex negatively charged layer, known as glycocalyx composed of polysaccharides

often attached to the lipids and proteins of the membrane, extends up to about 150

nm away from the cell surface. Likewise, these high molecular weight molecules con-

tained in glycocalyx are also important for cellular adhesion, and can induce great

morphological changes in cell shape and motility.

The adhesion of cells (e.g., bacteria) to a surface is a result of an interplay

between specific interaction (which involves in the stereochemical participation of

cell surface receptors and the complementary binding molecules such as ligands) and

non-specific interactions (which are usually defined as the types of interactions that

do not involve cell surface binding molecules), as discussed below. Therefore, from a

physico-chemical viewpoint, the real functional surface of biological particles such as
cells is a tangled mat of negatively charged polysaccharides and proteins dotted with

specific recognition sites.

2.2 Interaction Forces Involved During Adhesion and Adhesion Mechanisms

To better understand the adhesion mechanism of biological particles to sur-

faces, it is essential that the interaction forces involved throughout the entire adhesion
process can be identified and characterized.
The transition of a particle from a fluid suspension ("free" or "unattached"
state hereinafter) to attachment usually involves the following steps, as illustrated in

Figure 2.1(a). The particle first must arrive at the vicinity of the surface via fluid

convection or motility (for cells). Then it must cross the diffusion boundary layer

near the surface through Brownian motions to reach the region where the long range
interaction forces between the particle and the surface become significant enough to

overcome the driving forces resulted from Brownian motions (~ 1 pN). Finally, the

particle must be able to resist any dislodging forces (for example, fluid shear) and to

stay attached to the surface through the formation of receptor-ligand bonds. Since
the processes of transport and attachment take place in series, the slower of these

steps limits the overall deposition rate of cells to a surface. Below, we shall discuss the

major forces involved during the attachment process according to the classification of
these forces into non-specific and specific interactions and then describe the adhesion
mechanism within each category.
2.2.1 Non-Specific Interactions and Attachment Mechanisms

Non-specific interactions are defined as interactions between a particle and

a surface (or another particle) that do not involve surface-bound macromolecules,
namely, that are not biochemically specific but do act to increase or decrease the
overall strength of the interactions. The common operating non-specific interactions

include DLVO (i.e., attractive van der Waals and repulsive electrostatic interactions),

hydration, steric, and hydrophobic interactions. Among these, three major ones are

important for the typical biological adhesion processes: electrostatic forces, steric
stabilization, and van der Waals or electrodynamic forces. All of them are present in

the adhesion process, but each is dominant at a different particle-substrate separation
distance. The relationship between these forces and separation distance has been
characterized in many papers [15, 16, 17].

As mentioned earlier, the surface of biological particles such as cells consists
of a lipid bilayer containing ligands and/or other embedded surface-expressed macro-
molecules as well as glycocalyx that usually is negatively charged due to the present

La) Convect on

D ffus on D ffus on

Longer Ranue
Forces Attachment 'I ,Detachment

Partce Surface




; z

Substrate Surface

Figure 2.1: Critical steps involved in particle attachment and illustration of specific
binding interactions. (a) Critical steps in particle attachment: convective transport,
diffusion, attachment and resistance to detachment. Long-range interaction forces
govern the rate of attachment and short-range interaction forces govern the strength
of attachment that resists detachment from the surface [18]. (b) Illustration of the
reversible processes of formation and dissociation for a single bond which could be a
receptor-ligand pair or a macromolecule bound to a bare surface. kf(z) is the forward
binding rate constant and k,(z) is the reverse dissociation rate constant. Here a
uniform interaction region at the interface between the particle and the substrate is
assumed. Also we assume that the bonds formed do not affect the availability of
binding property and the activity of the binding sites on the substrate surface.

sialic acid residues. For cell-cell or cell-substratum (if negatively charged too) ad-

hesion, the bringing together of two negatively charged surfaces leads to an overall

repulsive electrostatic force between them as a result of the overlapping electric dou-
ble layers surrounding the cell or the substrate. The glycocalyx consists of polymers
in a hydrated environment. As this polymer coat approaches to a surface, this layer

is compressed and some of the water molecules are pushed out. A repulsive force
termed steric stabilization results because of the steric compression of the polymer

chains and also because of the osmotic tendency of water molecules to return. The

other non-specific forces, van der Waals forces, ubiquitous and usually attractive,

arise from the charge interaction of polarizable molecules, including molecules with

no net charges, on the cell surface as well as in the solvent.
Using a simplified physical model for cell membranes and mathematical de-

scriptions of these three non-specific forces, Bongrand and Bell [16] calculated the
magnitudes of these forces for the case of cell-cell adhesion. Their results are shown
in Figure 2.2, where the interaction potential per unit area between the two like cells

and also the force per unit area required to separate these two cells are plotted as
a function of separation distance. Here, the interaction energy is obtained by inte-

grating the force from a given distance to infinity. At small separation distances,

repulsive potentials dominate the overall cell-cell interactions; as the separation dis-
tance increases, repulsive potentials fall off (at distance on the order of 200 A) and

the attractive van der Waals forces act to bring cell surfaces into proximity and thus
to increase the likelihood of cellular adhesion.
For smaller (micron or submicron) sized particles, the effects of Brownian
motions on them are more significant. So, the first predictive models for particle
attachment were drawn from colloidal physics. The most cited theory used to quan-
tify the non-specific long-range interactions between bacterial cells and surfaces is



0 60 100








160 200 250 300


\ electrostatic

0 50 100 150 200 250

Figure 2.2: Three major non-specific forces present in cellular adhesion: electrostatic,
steric and van der Waals forces. (a) Each interaction potential is plotted as a function
of separation distance; (b) Interaction forces between two cells are plotted versus
separation distance. Absolute values are shown in both plots. Source: [16]




I__ l i


1o I

-- g a

. r_

DLVO (Derjaguin-Landau-Verwey-Overbeek, [4, 5]) theory [11]. Classic DLVO the-
ory accounts for electrostatic repulsion and van der Waals attraction; later on, other
factors such as hydrophobic interactions [7] and polymer-induced steric forces [19]
have been introduced to the DLVO theory, which formed the so-called extended
DLVO theories. Since it predicts a potential energy profile as a continuous function
of separation distance by using various system-specific parameters (e.g., the Hamaker
constant, surface charge, solution ionic strength, particle size, etc.), the DLVO theory
does provide a conceptual framework for interpreting particle attachment; that is, a
particle must pass over an energy barrier to become attached to a surface, which shall
be employed in the derivation of our model in Chapter 3. However, the DLVO the-
ory fails to fully elucidate the mechanism of particle attachment because the specific
receptor-ligand binding interactions are not taken into account at all, even though
to some extent it is able to successfully explain non-specific mechanism of microbial
2.2.2 Specific Receptor-Ligand Interactions and Attachment Mechanisms

Despite the repulsive constraints described above, specific receptor-ligand in-
teractions, as illustrated in Figure 2.1(b), are the favored explanation for the ability
of microbes or biological particles to adhere and recognize the corresponding binding
components on other particles or surfaces. In what follows, the main characteristics of
the receptor-ligand interactions are discussed and some specific adhesion mechanisms
are described afterwards.
Characteristics of Specific Interactions

Specific Forces and Reaction Kinetics. The primary physical forces that
are involved in receptor-ligand binding are Lifshitz-van der Waals (LW), electrostatic
(EL) and Lewis acid-base (AB) (or, electron acceptor-electron donor) interactions

[17]. These are the same forces that give rise to the non-specific interactions de-

scribed earlier. Although the physical nature of the forces underlying specific in-

teractions is the same as that of non-specific interactions, receptor-ligand binding

is often described in terms of chemical reaction kinetics. The rationale for apply-

ing this framework comes from physical characteristics of these weak, non-covalent

specific binding interactions. First, the interaction range between receptor and lig-

and binding domains is very short (~ 1 nm) compared to non-specific interactions.

Also, receptors and corresponding binding molecules must be properly oriented to-

ward each other for bonding to occur. As a result, the interactions between them

are often described by "lock-to-key" type, in other words, only when the receptor

and ligand molecules are in proper stereochemical orientations, will binding reaction

occur; otherwise, these two molecules behave like the other one does not exist even

at very close distance. Therefore, unlike the non-specific interactions, which have

been successfully described by force-distance relationship, for specific interactions,

however, it is difficult to relate the separation distance between two particle surfaces

to the binding molecular sites. The advantage of using the kinetic description is that

the information contained in the force-distance relationship is lumped into the kinetic

rate and equilibrium constants.

Two-Dimensional Kinetics. Contrasted to three dimensional (3-D) mo-

tions of free molecules in solutions, the surface-bound receptors and complementary

ligands are limited to two dimensional (2-D) movements. Table 2.1 shows the typ-

ical units for each species as well as for the binding rate constants. In the current

work, we shall deal primarily with 2-D/2-D case. To determine values for the affin-

ity and kinetic rate constants, it is essential to know (or measure) the number of

bonds formed between the two approaching surfaces. However, direct experimental

measurements of receptor-ligand bonds have not yet been possible when two reacting

molecules are linked to surfaces. A more common approach is to first measure the


3-D affinity (determined when both receptor and ligand molecules diffuse freely in
free solutions) and then convert it to a 2-D affinity using a parameter, which, in
general, is on the order of the size of the receptor-ligand complex molecules [20, 21].
A method to convert 3-D binding kinetic rate affinity constants to 2-D ones is also
developed in Chapter 4. Recently, measurements of 2-D affinity [22, 23] and binding
rate constants [24] have begun and solid biophysical measurements are required to
fully elucidate this relationship.
Coupling of Kinetics and Mechanics. Since both receptors and ligands
(or other binding macromolecules and the complementary binding sites) are anchored
at particle surfaces, besides thermal agitations, the crosslinks formed between them
are usually subjected to a dislodging force, such as fluid shear, that tends to alter
binding kinetic rates, and thus alter the strength and the lifetime of these bonds as
well. There are several models to account for the effect of external forces on binding
kinetics. The most commonly used one was developed by Bell in 1978 [20]. Drawing
from the kinetic theory of the strength of solids, it assumes that although the forward
rate constant should not be affected by the applied force, the reverse rate constant,
and thus the affinity, will vary exponentially with force f

( af1)
k,(f) k exp Nb-BT)' (2.1)

Table 2.1: Units for measurement of receptor-ligand interactions

where ko (in s-1) is the reverse rate constant in the absence of force, a is a length
parameter defined as the bond interaction distance, f is the applied force and Nb
is the number of receptor-ligand bonds formed. An alternative approach assumes
the receptor-ligand bond behaves like a Hookean spring, as proposed by Dembo and
coworkers [25, 26]. In this case, affinity will decrease as a receptor-ligand bond moves
away from its equilibrium position according to a Boltzmann distribution shown

K (z) = Kqo exp (- 2kB ) (2.2)

where % is the mechanical spring constant of the bond and zo is the unstressed bond
length. From the strain-dependence of the transition state theory between bound
state and free state, Dembo et al. suggested two exponential laws for kr (z) and
kf (z), but required their ratio to satisfy the above equation with the knowledge
of Kq (z) = kf (z) /k, (z). Evans and Ritchie [27] placed the relationship between
reverse rate constant and bond force on a more rigorous foundation by deriving it from
Kramers' theory [28] for the escape of thermally agitated particles from a potential
well tilted by an applied force. Under greatly simplified conditions, their results were
found to be a combination of power law and exponential model between reverse rate
and applied force, and also kf is assumed to be constant.
Stochastic Characteristics and Probabilistic Kinetic Framework. Over
the past decade, it has been recognized that cellular adhesions are often mediated via
a surprisingly small number of receptor-ligand bonds. As a result of this low bond
number, small-scale adhesion becomes stochastic in nature and many experimental
observations have shown this property. For example, in the flow chamber assay, a
moving cell interacting with a stationary surface is observed to undergo stop-and-go
types of motions with highly fluctuated velocities [29, 30]. Even in the well-controlled
micropipette experiment, when a pair of cells (or engineered biological particles) is
brought together in any single test, adhesion events still occur randomly, despite all

the other experimental conditions are kept identical [23, 24]. In addition, any sin-

gle measurement for adhesion lifetime or detachment force also lacks deterministic

value. Regardless of this randomness, the occurrence of bond formation or dissocia-
tion events has a certain likelihood. Although any single measurement is of little use,

a collection of many measurements can reveal a well-defined, although still highly
scattered, distribution, which corresponds to the probability density for the occur-

rence of any particular value. Such randomness is not caused by measurement errors

but is a manifestation of the stochastic nature inherent in the chemistry of receptor-
ligand binding, which becomes more significant when the number of bonds per cell is
small. This calls for a probabilistic description of the receptor-ligand binding kinetics.

Although the probabilistic theory for kinetics of small systems has been known
for quite a long time [31], not until 1990 was it applied to cellular or microbial
adhesions [32, 33]. The idea is that the number of bonds that an adherent cell may
have is a discrete, time-dependent, random variable that fluctuates significantly. To

describe the state of the system requires a probability vector because any positive

number of bonds could associate with an adhesion. Each possible scenario has a

defined likelihood, given by a component of the probability vector. This is in contrast

to the deterministic description that uses only a single scalar for the averaged number

of bonds. The law of conservation of mass is thus replaced with the conservation of

probability density for the low number of bonds. The probability of the cell adhesion
states evolves due to the probability influxes and effluxes to and from these states.
Specific Adhesion Mechanisms

As described earlier, cells express surface associated proteins ("adhesins") that
can bind specifically to complementary proteins expressed by the host, and the pres-
ence of these binding interactions can have a significant effect on the adhesion of
cells to surfaces. Most bacterial pathogens produce MSCRAMMs (microbial surface

components recognizing adhesive matrix molecules), which are a sub-family of ad-

hesins that react specifically with extracellular matrix molecules. For example, the

specific binding interaction between the MSCRAMM "clumping factor", expressed

by bacterium Staphylococcus aureus, and the extracellular matrix protein fibrinogen

is associated with hospital-acquired infections, and this pair interaction serves as the

model experimental system in Chapter 4 by which the effect of MSCRAMM length

on cell-surface interaction forces and particle deposition rate constants is evaluated

and compared with the predictive results using our dynamic model from Chapter 3.

Also, many biomemtic experimental systems are constructed to carry out the relevant

experiments in vitro to avoid in vivo complexity. One of such exemplary systems is to

utilize the high specificity and affinity between biotin and (strept)avidin molecules,

which is investigated in Chapter 6 using the optical trapping technique for a direct

testing of our dynamic model in Chapter 3.

Several theoretical models to account for such specific molecular interactions

are briefly introduced below. Bell and coworkers pioneered thermodynamic equilib-

rium approaches to cell-cell (or cell-surface) adhesion. The models they developed

are to study the adhesion between two freely deformable cells with surface-bound

receptors (mobile or immobile) [25, 34, 35). They formulated an equation for the

change in Gibbs free energy of a closed system containing two such deformable cells
which changes from a state of no interaction to a state of adhesion. This equation

allows them to calculate the equilibrium state by minimizing the Gibbs free energy at

constant temperature and pressure 1 to find the contact area, cell-cell separation dis-

tance and number of bonds as a function of both receptor properties and non-specific
'Here, the pressure should use surface pressure, because binding reactions between two reacting
species take place in a two-dimensional contact region. Minimization of the Gibbs free energy should
be performed under constant surface pressure instead of external pressure of the fluid surrounding
the cell membranes, which is not expected to have much effect on surface thermodynamics.

interactions. However, in their models, the cell membrane deformation energy is ne-
glected in the minimization of free energy; and also the kinetic process involved in

reaching adhesive equilibrium is not taken into account at all.

Developing dynamic models for cellular adhesion allows us to investigate the
equilibrium state of the cell as well as the kinetics of the formation of that state.

In many physiological environments, cells are flowing through the body's circulatory

systems. Adhesion of cells to corresponding surfaces under such conditions may not
reach the equilibrium state before the cells are able to form enough bonds to resist

detachment forces. Therefore, kinetic analysis of cell adhesion may provide useful
information on adhesion mechanism under these circumstances. Hammer and Lauf-

fenburger [36] developed a dynamic model for the receptor-mediated cell adhesion
to a ligand-coated surface to study the outcome of a cell transiently encountering

the surface under fluid shear flow. The kinetic species balances were used to study
the time-course of bond number density as well as surface density for free receptors.
After a given time period (should be long enough for binding reactions to occur),

if bond number density is zero, the cell will not adhere to the surface during the
encounter; but if bond number density approaches to a non-zero steady state value,

the cell must become stably attached. Although such a deterministic model is useful

and successful in explaining many experimental observations, it predicts either no
adhesion or all adhesion ("none or all") results, and thus fails to predict the inter-

mediate cases. Cozens-Roberts and coworkers extended the deterministic dynamic
model by Hammer and Lauffenburger to a probabilistic framework, in which they cal-
culated the probability that a certain number of bonds between a cell and a surface
exists at any given time [32, 33, 37]. As described earlier, the probabilistic nature of
binding kinetics is inherent in the system involving relatively small number of react-
ing molecules, such as for the case of receptor molecules over the cell surface. The

key difference from the earlier model is that the binding kinetics is cast in proba-

bilistic rather than deterministic form. Applications of these models in real living

cells are still limited because of the complexity of in-vivo cell systems. In addition,

a combination of mechanical and dynamic approaches was also used for studying

cell attachment and detachment [26]. Dickinson developed a dynamic model to pre-
dict the attachment rate constant of a rigid Brownian particle to a surface mediated
by both colloidal forces and macromolecular binding interactions [38]. Considering
Brownian motions (for micron sized particles) and binding as two coupled stochastic

processes, the model derived a time-dependent probability density equation to de-

scribe the probability for the particle of having a certain bond number at a certain

separation distance from the surface. Then, the steady-state probability flux of par-
ticles to a surface was examined as a function of the binding parameters, such as the
surface density of binding molecules, the rate constants of binding and dissociation,
and the effective length of the bonds. Analytical solutions were obtained for the

deposition rate constant for the limiting cases of slow or fast binding relative to the

rate particle diffusion. However, the intermediate case where both processes are simi-
larly fast required numerical solution. In Chapter 3, similar theoretical framework as
Dickinson's model but a new approach is used to obtain the deposition rate constant

of biological particles to a surface (please also see Ref. [39]).

2.3 Techniques for Measuring Particle Attachment

As described earlier, adhesive interaction between a biological particle (or a
cell) and a surface is a result of net contributions from both non-specific forces,
such as van der Waals, electrostatic, steric stabilization, and the biochemically spe-
cific receptor-ligand binding interactions. Thus, particle-surface reaction kinetics
depends upon the particle and surface properties, the medium composition, and ex-
ternal forces, such as depositional and hydrodynamic forces. Due to complexity from

physiological conditions, in vitro assays are typically used for studying transient be-

haviors of particle or cell adhesion (although in vivo experiments have also been

performed with implanted grafts [40, 41]). Experiments relevant to cellular adhesion

can be roughly classified into two categories: adhesion kinetics and adhesion strength.

Below we describe in brief the common techniques used in each category.
2.3.1 Adhesion Kinetics

Particle adhesion kinetics can be studied through kinetic analysis of attach-
ment as well as detachment measurements. In general, attachment kinetic experi-

ments are performed by incubating the biological particles (or cells) and surface for

a variable amount of time (the attachment time) and then subjecting the particles

to a constant external force for a given amount of time. These experiments provide

data on the percentage of adherent particles before and after exposure to the force

as a function of the attachment time. In general, detachment kinetic experiments

are performed by incubating the particles and the surface for a given amount of time

and then exposing the particles to a constant external force for a various amount of

time. The common methods used to exert force on the cells are: micromanipulation,

centrifugation, and hydrodynamic shear.

In micromanipulation, the particle is held in a micropipette and brought into

contact with an affinity surface; the particle is then withdrawn from the receiving

surface after a specified amount of time by increasing the suction pressure of the

micropipette. In fact, this assay has also been used to generate useful information

on the adhesive force between a biological particle and a surface and also on the

mechanical properties of the cell membrane [42]. In centrifugation, particles are
allowed to settle onto a receiving surface under the influence of gravity for a period
of time and then centrifuged. Hydrodynamic shear assays can be divided into the

following categories: flow between parallel plates, flow between a rotating disc and a

stationary disc, flow between a rotating cone and a stationary disc, and axisymmetric

flow between parallel discs. If the hydrodynamic fluid shear is used as a means for

external force application, accurate interpretation of experimental measurement data

requires to quantify the influence of fluid flow, which includes transport and wall

shear effects. Thus, flow cells for direct visualization and real-time measurement

under well-defined flow conditions have been developed [2, 43].

To demonstrate how to extract adhesion kinetics information from these exper-

imental measurements, an equation for obtaining particle attachment rate constants

is given below using parallel flow chambers along with automated video microscopy

systems, which measures the time course of the changes in the surface concentration

of particles [43]. Within the diffusive boundary layer distance, the steady-state flux

of particles over this boundary is proportional to the particle concentration at this

boundary by applying first-order kinetics. This flux is equal to the rate of accumu-

lation of particles on the surface, thus, one can obtain the deposition rate constant,
k+, from the following relationship [2]
S= k+c, (2.3)

where c, is the attached particle density on the surface; and c is the concentration of

particles at the boundary layer distance along the flow direction.
2.3.2 Adhesion Strength

The strength of particle adhesion is a critical determinant in many physiologi-

cal processes, for example, in cell locomotion where bonds must be formed and broken

in a tightly controlled manner. The measurement of adhesion strength is achieved

by relating the mechanically applied tension strength on the cell membrane to ad-
hesive energy, which shares similarity with the mechanics of peeling of an adhesive

tape. Corresponding theoretical models to support this method were first developed
by Evans [44, 45, 46], in which the forces due to "pulling" on cell membrane are as-

sumed to be in balance with the resistance by receptor-ligand bonds. Then, Dembo
et al. [26] combined cell deformation mechanics with receptor-ligand binding kinetics

to develop a dynamic and mechanical approach to cell attachment and detachment.
This model is capable of describing the relationship between the transient behavior

of contact zone spreading or shrinking and cell binding properties. Later, Kuo and
Lauffenburger [47] used the radial flow chamber to measure the critical shear stress for

detachment of receptor-coated beads from a ligand-coated substratum as a function
of receptor density.
Over the past decade, more and more research groups have been focusing on

studying the strength of a single receptor-ligand bond. Most direct measurements of
single bond strength have been performed with four types of ultrasensitive probes:

the atomic force microscope (AFM) [48, 49], where force is sensed by deflection of
a thin silicon nitride cantilever; the biomembrane force probe (BFP) [42, 50], where
force is sensed by axial displacement of a glass microsphere glued to the pole of
a micropipet-pressurized membrane capsule; the laser optical tweezer (LOT) [51],
where force is sensed by displacement of a microsphere trapped in a narrowly focused
beam of laser light; and the dynamic force spectroscopy (DFS), which is similar to
BFP while thermal fluctuations of probes are also accounted for in data analysis
[52, 53, 54]. These probes are able to detect forces in the range of < 1 pN/nm to 1


As well known, real cells are of complex shapes and structures. To have a

complete understanding of cellular adhesion requires that not only the whole cell

be investigated in attempts to model the biological behavior of the cell surface, but

also that the functions and behaviors of each individual component inside the cell

be examined. Therefore, the combined knowledge from many different fields will be

needed to depict a full picture of the cell adhesion, such as the biological, molecular,

quantitative behaviors of the cell, and so on. In this chapter here, to make com-

plex materials simple, we shall study the attachment mechanism of a rigid biological

particle, instead of a real cell, to a surface through receptor-ligand or other macro-

molecular binding interactions. The chemistry of the system is ignored, in an analogy

to the treatment of polymers as long chains by Professor de Gennes [55] 1. In this

way, we are hoping that the underlying physics for this simple system will help shed

some light into the complex adhesion mechanism of actual cellular systems.

From the description in Chapter 2, attachment of a free particle (or a cell)

from a fluid suspension to a surface usually involves two major competing processes:

(i) Brownian motions of a particle, which is more significant for smaller particles

(e.g., on the order of micron or submicron size); and (ii) interaction forces between

the particle and the surface. Which one of these two processes limits the overall

attachment rate of particles to surface is of great interests for both fundamental

researches and practical applications. In fact, in many cases, macromolecular bond
1Perhaps Prof. de Gennes's greatest attribute is to make complex materials appear simple. In
this treatment of polymers, the chemistry of the polymer is ignored so that one can look at the
underlying physics for this polymer by treating it as a long chain.

formation and dissociation may occur on a time-scale comparable to the time-scale of

particle diffusion near a surface, such that the forces involved in binding may not be

assumed to be at equilibrium, as treated in DLVO theory. Therefore, in this chapter,
we shall develop a dynamic model to predict particle attachment rate constant for a

general case when the specific binding forces cannot be assumed at equilibrium.

3.1 Particle-Surface Interaction Energy

Let us first start with thermodynamic analysis of the interactions between
a rigid biological particle and a surface in the presence of binding macromolecules.

Figure 3.1 illustrates the particle-surface system under consideration, which is a sim-

ple case for receptor-mediated cell adhesion. The particle is covered uniformly with

binding macromolecules (or receptors), and the substratum is coated with comple-

mentary binding sites (or ligands) in great excess, hence, the surface density for the
binding sites can be treated as constant. Upon close approach to the surface, a flat

and constant contact area for macromolecular binding at the interface between the

particle and the surface is assumed throughout the whole attachment period. Here,

we shall focus on the initial adhesion for this rigid particle to the surface (in other

words, the spreading and deformation of real cells are not considered). Suppose there
are N identical surface-bound molecules available for binding in the contact area to

form n bonds between the particle and the surface. The total particle-surface inter-

action energy at the interface is the sum of the energy of n bonds and the non-specific
interactions, O(z), which is dependent on the separation distance, z, of the particle

from the surface, as illustrated in Figure 3.1. Here O(z) includes electrostatic re-
pulsions, van der Waals attractions, steric stabilization, hydrophobic forces, and any
other surface-independent body interactions such as gravitation. The energy of every
single bond, e(z), is also a function of z due to stretching and/or compression of the
bond away from its equilibrium position. The Helmholtz free energy at the interface,

A(n, z), is then given by

A(n,z) = (z) +ne(z) TS(n) (3.1)

= (z) + ne(z) kBTln N ()! (3.2)

where the entropy, S, is determined by the degeneracy of the system with n bonds
at N binding macromolecules:

S = kBIn N. (3.3)
sn!(N n)!
Upon using the Stirlings approximation for large N and n, the free energy in Eq.
(3.2) becomes 2

A(n, z) = O(z) + n(z) + (N n)kBTn (N ) + nkBTln (N). (3.4)

In general, analysis here can be applied for any reasonable functions O(z) and
e(z), provided that the energy maximum in ((z) lies within the interaction domain
of the bridging macromolecules. According to the study of cell-cell adhesion by
Bongrand and Bell [16], the attractive van der Waals forces between cells are negligible
and the repulsive energy barrier arises mainly from a combination of two effects: (i)
electrostatic double layer repulsion due to the negatively charged cell surfaces [15],
and (ii) the steric stabilization effect as a result of the presence of a hydrated polymer
layer on cell membranes (e.g., glycocalyx) [56]. As a first approximation, O(z) is
assumed to consist of a phenomenological equation to account for both electrostatic
repulsion and steric stabilization force and a superimposed parabolic potential around
the center, zp:
4(z) = oe-bz + p(Z zp)2, (3.5)

where qo represents the baseline repulsive potential, the parameter b is the inverse
decay length for repulsion. As shown in the following sections, the functional form
2The unstressed bond energy, co, for typical reversible specific receptor-ligand binding interaction
is usually about several kBT, and bond number n is relatively small compared to N, so, the entropic
contribution to the overall binding energy is dominant, as indicated in Eq. (3.4).


Figure 3.1: A schematic of particle-surface interaction through macromolecular bind-

ing. The bonds illustrated here are treated as ideal springs with bond energy increas-

ing parabolically when being stretched or compressed away from their equilibrium

position, Zo.

of 0 (z) is not essential to the analysis, as long as there are no net repulsive forces
around the secondary energy minimum, see Figure 3.2. The energy of bonds and
the kinetic rate constants are determined by the physical and chemical properties of
the specific binding macromolecules. Here for simplicity, the bonds formed in the
interface are treated as ideal springs with energy increasing quadratically when being
strained or stressed away from their minimal energy bond length, zo. Thus,

E(z) =- o + ~Yb(Z z0)2, (3.6)

where to = e(zo) and yb is bond stiffness. As for the real receptor-ligand bonds,

the ideal "spring model" used here might not be a good one, however, if any better
models for these bonds are developed, they can simply replace Eq. (3.6).
Shown in Figure 3.2 is a theoretical contour plot of A(n, z) as functions of bond
number n and separation distance z based upon the above representative expressions
for 0 (z) and e (z) using the parameter values listed in Table 3.1. Two minima
exist in the plot: a primary minimum with n > 0 corresponding to an attached

particle with multiple bonds, and a secondary minimum at larger separation distance
with n = 0 where essentially only long range colloidal forces dominate. To attach
to the surface, a particle has to move from the secondary (2) minimum, over the
transition saddle point, and into the primary (10) energy minimum. In the following
section, we derive a Fokker-Planck equation to predict the probability flux of the

particle from the unattached state (20 energy minimum) to the attached state (10
energy minimum) using an energy landscape determined by both colloidal and binding
interaction forces.

3.2 Probability Flux

Let us first analyze the forces acting on the particle. These forces include the
deterministic force in z direction, Az a fluctuating force (namely, the thermal
force that results in Brownian fluctuations of the particle), and the bridging force to

- primary minimum

C 35
c 30
"- 25
Z 1520
Z 15 secondary minimum


50 100 150 200 250 300 350 400
Separation distance, z (nm)

Figure 3.2: A hypothetical contour plot of the thermodynamic free energy, A(n, z),
as a function of bond number, n, and separation distance, z. To attach to a surface,
a particle must move from the secondary energy minimum, over the transition saddle
point and into the primary minimum. The separatrix boundary which passes through
the saddle point and separates the two energy minima is shown by the solid line.

10 11\111111

Table 3.1: Base parameter values

Symbol Base value Parameter

N 50 Number of binding molecules
0o 90 kBT Baseline repulsive potential
b 0.02 nm-1 Repulsive decay length
Co -1 kBT Unstrained bond energy
7b 0.002 kBT/nm2 Bond stiffness
zo 40 nm Unstrained bond length
7, 8x10-4 kBT/nm2 Optical trap stiffness
zp 200 nm Optical trap center
kfo 1000 s-1 Forward binding rate constant
A 0.5 nm Bond length at tradition state
R 300 nm Particle radius
L 1 pm Boundary layer thickness
ao 0.5 kBT Baseline polynomial potential
al 0 kBT/nm Potential coefficient
a2 10-3 kBT/nm2 Potential coefficient
a3 -10-6 kBT/nm3 Potential coefficient
a4 0 kBT/nm4 Potential coefficient

form n bonds. The deterministic force consists of particle inertial forces and hydrody-
namic drag forces. Assuming inertial forces are negligible compared to hydrodynamic
drag forces at low Reynolds number, the drift velocity of the particle in z direction
is the deterministic force divided by the hydrodynamic drag coefficient, 6(z), (for all
the results shown later, the functional form of 6(z) for a sphere descending normally
to a solid surface as solved by Brenner is used [57]); and 6(z) can then be determined
using Stokes-Einstein equation by the diffusion coefficient of the particle, D(z):

V(z;n) 1 8A(n,z) (37)
V(z;n) = (3.7)
6(z) 8z
D(z) OA(n, z)
kT (3.8)
kBT z '
where V(z; n) is the drift velocity of the particle in z direction.
Next we consider the dynamic process of particle attachment in terms of the
probability flux into the primary energy minimum on the energy landscape shown in
Figure 3.2. Assuming Markov processes in both variables z and n (i.e., the future
statistics depend only on the present, not their history), the balance equation for the
particle is

Otp(n, z, t) = -dJz (Jn+-1n Jn-a-1) (3.9)

where p(n, z, t) is the joint probability density of the particle being located at distance
z with n bonds at time t. Jz, the probability flux J in the z direction, is given by

D(z) 8A(n, z)
Jz = kT z p(n, z, t) D(z)p(n, z, t). (3.10)

The primary assumptions underlying Eq. (3.10) are the validity of Eq. (3.7) (viscous
drag force proportional to velocity) and a fast relaxation of velocity fluctuations
relative to the time scale of interest, which is usually true for the particle size within
the colloidal domain. And the probability flux in the discrete n dimension, Jn-n-1,
is given by

Jn-n-- = -kr(z)np(n, z,t) + kf(z)[N (n- l)]p(n 1, z,t),


where k,(z) and kf(z) are the rate constants of bond dissociation and formation,


The boundary conditions for Eq. (3.9) are determined by the physical lim-
itations on n and z and on how an "attached" particle is defined. Because z > 0,

the no flux boundary condition at z = 0 is J, (n, 0) = 0. Similarly, since 0 <
n < N, the no-flux boundary conditions in the n-dimension are Jn-- (0, z) = 0

and Jn--1 (N, z) = 0. Furthermore, the bonds are assumed not be able to ex-

ist at separation distances larger than the diffusive layer thickness L, such that

J, (n > 0, L) = 0.
In the following section, the mean first-passage time approach to solve Eq.
(3.9) to get the attachment rate of the particle to a surface is described.

3.3 Mean First Passage Time Approach

As mentioned earlier, to attach to a surface, a particle must move from the
unattached state (i.e., the secondary minimum as shown in Figure 3.2), cross over
the energy barrier on the transition state and fall into the primary potential well to

become attached. Suppose the potential energy in the attached state is sufficiently

lower than that at transition state, the relaxation time for the particle from the

transition state to attachment is negligible, therefore, once the time required for the
particle from the unattached to the transition state is calculated, the rate constant

of particle attachment to the surface can then be deduced. In what follows, the mean
first-passage time method is described in brief to obtain the mean transition time.

Consider a homogeneous Markov process within a system Q which contains a
single energy minimum (e.g., the region of the secondary energy minimum as illus-

trated in Figure 3.2). The first-passage time formalism determines the mean time
T(n, z) required for a particle starting at [n, z] E f to exit through a specified bound-
ary 8ft (in our case here, the separatrix containing the saddle point between the two
energy minima, also indicated in Figure 3.2). The probability that at time t the

particle is still in 0 is

f p(n', z', tn, z, 0) dn'dz' = G(n, z, t), (3.12)

which essentially is the probability Prob(T > t), where T is the time the particle
leaves the system. The boundary conditions are

G(n, z, t) = 1, [n, z] E O-
= 0, [n, z]E 8~ (3.13)

The mean first passage time, T(n, z), is given by

T (n,z) = T) =- tdG(n,z,t)

= G(n,z, t)dt, (3.14)

where the last term is obtained using integration by parts. Since the system is time
homogeneous, one can use the backward equation for Eq. (3.9), that is [58]

t tp(n',z', tin, 0) (, z, tn, z, 0) + D(z) (n', t n, z, 0)
kBT Oz 3z2
+kf (z) (N n) [p (n + 1', z', tin + 1, z, 0) p (n', z', tn, z, 0)]

+k (z) n [p(n 1', z', tin 1, z, 0) p (n', z', tn, z, 0)]. (3.15)

In view of Eq. (3.12), one gets

D(z) 8A(n, z) 82
OtG(n, z, t) D ) A(n, z) G(n, z, t) + D(z)-2 G(n, z, t)
kBT Oz j
+kf (z) (N n) [G(n + 1, z, t) G(n, z, t)]

+k, (z) n [G(n 1, z, t) G(n, z, t)]. (3.16)

Integrating Eq. (3.16) over (0, oo), one gets for the left-hand side of this equation

SOtG(n, z, t)dt = G (nz, oo) G (n, z, 0) = -1, (3.17)
Z, ,0

where we have used the fact G(n, z, oo) = 0, as t oo; then one obtains the
following expression by using Eq. (3.14) for the right-hand side

D(z) 8A(n, z) 0T(n, z) ) T(n, z)
L+T(n, z) = =+ D(z)
kBT Oz az +z2
+kf(z)(N n) [T(n + 1, z) T(n, z)] + k,(z)n [T(n 1, z) T(n, z)]

= -1, (3.18)

with the boundary condition

T(n, z) = 0, [n, z] E 80 separatrix. (3.19)

Because of the difficulty in analytically solving Eq. (3.18), next, we seek both numer-
ical method and analytical approximations and then compare the results obtained
from both methods.
3.3.1 Numerical Method

Finite difference method is used to numerically solve Eq. (3.18) subjected
to the boundary conditions in Eq. (3.19) [59]. The code is written in MatLab
and attached in Appendix. The separatrix boundary which separates the two energy
minima and goes through the saddle point, as illustrated in Figure 3.2, is dynamically
determined by simulating into which energy well the particle would go starting from
point (n, z) E 1 based on their deterministic equations:

dz D(z)9A (n,z)
dt kBT Oz '
S= kf (z)(N n) k(z) n. (3.21)

Clearly, the position of the separatrix is directly related to velocity, diffusion, kinetic
rate constants of the particle as well as potential energy. Any factor that could
influence these parameters, such as particle size, bond energy, forward binding rate
constants, would change separatrix location. Therefore, the separatrix should be
determined first for every data set prior to the run of the program to obtain the mean

first-passage time T(n, z) in Eq. (3.18). The same boundary conditions as those for
Eq. (3.9) apply here also, which are converted in terms of T(n, z) as follows:

[T(n,z)-T(n- l,z)]ln=o = 0,

[T(n+ l,z) T(n,z)] In=N = 0,

9T (n, z) zL = 0,

T(n, z) = 0, [n, z] E O( separatrix. (3.22)

3.3.2 Analytical Approximations

In this section, an analytical approach is described to obtain the mean first-
passage time as well as particle attachment rate constant from the dynamic model
in Eq. (3.9). First a Fokker-Planck equation is derived for the probability density
of a particle with respect to bond number and separation distance from a surface,
using an appropriate free energy landscape determined by the colloidal and binding
energies. This equation is then used in the MFPT approximation to predict the
probability flux over the saddle transition state from the secondary energy minimum
corresponding to the unattached state.
Derivation of a Fokker-Plank Equation

Usually the receptor number over the cell surface is on the order of 103 ~ 10'
[16]. Thus, it is reasonable to assume that there will be plenty of binding molecules
(or receptors) available in the contact region between the particle and the surface.
So below we shall examine Eq. (3.9) under this conation. In the large N limit,
the number of bonds formed between the particle and the surface fluctuates around
its equilibrium value with a width of order N/2 [60]. As previously mentioned,
in order to attach to a surface, a particle must cross over an energy barrier at a
saddle point. Thus, the number of bonds formed at the saddle point gives us a good
reference regarding to the fluctuations in bond number. Therefore, a new variable (

is introduced so that
n = neq() N1/2, (3.23)

where z, is the separation distance for the saddle point from the surface, and nq(z,)
is the equilibrium bond number at saddle. Then an operator E is defined by its effect
on an arbitrary function f(n):

Ef(n) = f(n + 1),

E-'f(n) = f(n- 1). (3.24)

By so doing, the probability conservation equation for the particle, Eq. (3.9), can be
rewritten as

Otp(n, z, t) = -9.J, + kf(z) (E- 1) (N n) p(n, z, t) + Ck(z) (E 1) np(n, z, t).
The operator E changes n to n + 1 and therefore ( to 4 + N-1/2, such that

E = 1+ N-/ + -N- ..
a0 2 a2
0 (N-1/2)m (3.26)
m a (3.26)
Therefore, the discrete variable n is now approximated as a continuous fluctuating
variable (, and the change for ( can be written in powers of small parameter N-1/2.
Substitution of Eq. (3.26) into Eq. (3.25) to change the variable n to ( yields [58, 60]:

S(-N-1/2)m [N- n(z) N2] p(, z, t)
Otp(, z, t) = zJ + ki(z) m! a~ [N n ) N'
m= 1
c (N-1/2 m am
+k(z) m! 8m [neq)+ N/2] p(, z,t). (3.27)
Rearranging Eq. (3.27) yields

(0 1) Ni-m/2 am r [m ((, z)
ap(, z, 0) = -. J. + -m m a (z) 'Na / p(, z,t), (3.28)
m= 1 n ~mN1

ac (z) = kz N 1 + (-1)mk,(z) (3.29)

/ (, z) = [k(z) (-1)mkr(z)i (3.30)

For the large N limit, only the terms of order 0(1) or higher order of N112
will survive and all the other terms with higher powers of N-1/2 vanish. Therefore,
by truncating Eq. (3.28) at m = 2 and collecting the terms of 0(1) and 0(N'12), we
get a Fokker-Planck equation for the probability density in terms of z and (

9tp(, z, t) = -J.
+ {N112 [(k,(z)+k()) [kf(z) ( kf( + +[k k()],}p((,z,t)

+I k(z) + [k(z) kf(z)] p(e,z,t). (3.31)

Since each binding macromolecule will be in either free or bound state with binding
energy either 0 or e (z), respectively, the probability of such binding molecules being
in bound state is 1-e(/k~ r Thus, the number of bonds at equilibrium is
n,(z) = N e) (3.32)
1+ e-e(z)/kBT'
and the equilibrium constant is then given by

Keq (z e (z) e(z)/kBT (3.33)
KeqN(Z) neq(z)

In view of Eqs. (3.32) and (3.33), Eq. (3.31) now becomes

8tp(, z,t) = -aEJz
+ a N"2kr (z) Ke(z) Keq(z) + k(z) [1 + Kq(z)] p(~,z,t)
( I 1 + Keq (Z)
+1 Keq(z) + Keqq(z,) C'
S q((z) 2 p(, z, t). (3.34)
2 1 + Keq,(z,) s r
(Here, again we have used Kq,(z) = k(.) The stationary probability is known as

p,,(, z) = N-le-A((,z)/kBT,


with N being the normalization constant and A (, z) is obtained by replacement of
n with ( in Eq. (3.4) as

A((,z) = O(z) + [neq(z.) + N1/2] (z)

+ [N neq(Z) N2I'(] kBTln N n,(z,) N1/2C

+ [n,(z,) + N/2C] kBTin [n,(z)+ N1/2 (3.36)

Therefore, for the Fokker-Planck dynamics in Eq. (3.34), the drift term in C dimension
is not derivable from the potential A(,, z) ("potential conditions" are not satisfied
here [61, 62]), but rather from another potential which is expressible as the line
integral of the drift and diffusion terms in C dimension:
B(, z) 2N1/2kTKeQ(Z) Keq(z) C + kT,(z)] [1 + 2 (z),
Keq(z,) + Keq(z) Keqg(Z) + Keq(Z)
where C(z) is only a function of z. Combination of Eqs. (3.10), (3.34), (3.36) and
(3.37) leads to
08 D(z) bA(C, z) 8
Otp(, z, =t) z kBT =z Z) p(z, t) + D(z) p(A, z, t)
8 (z) 8B( ,z) (3.38)
[I [(kZT p( z, t) + (z) p(9,z, ) (3.38)
1 K, q(z) + K() q(z,)
q(z) = -k(z) (3.39)
2 1 + Keq (z)
defining the effective diffusion coefficient in the C dimension.
Dimensionless Fokker-Planck Equation

Since a particle must pass over an energy barrier at a saddle point to attach to
a surface, the properties of the saddle point which separates the two energy minima
are very critical for determining the rate of particle attachment. Below, we scale the
above Fokker-Planck equation, i.e., Eq. (3.38), using the values of its key parameters
at the saddle point. We first define parameter
e- (3.40)

where AA = A(0, = 0, z,) A(,~, z,) is the barrier height determined by the free
energy at the saddle, [(~ = 0, z,], with respect to a reference energy minimum, [6m, zm]
(e.g., the secondary energy minimum of Eq. (3.36)); and e represents the relative
noise strength. Then we introduce a length scale 1 = 1z zm\, which is the distance
between the saddle and the reference minimum, and define

S= (3.41)

Using the newly defined parameters in Eqs. (3.40) and (3.41), now Eq. (3.38)

o Ba((, y) 0 r
9P(, Y, 7r) = [X() p(, y,r7)+ x(y) p(, y,)
p y, r) Oy Oy O

+7 () [p(, y,P ) + (,y (() p((,y,r)], (3.42)

A A((,z)
aAy) = A (3.43)
B(, z)
(,) AA (3.44)
X(y) = ) (3.45)

(() (z) (3.46)

7- kBT (3.47)

S= 12 (z) (3.48)
In a brief summary for the previous several sections, by approximating the
discontinuity of the bonds formed between the particle and the surface as a continu-
ously fluctuating variable ( in the limit of large number of available binding macro-
molecules, a two-dimensional Fokker-Planck equation can be derived to describe the

probability flux of the particle to the surface based upon a proper free energy land-
scape determined by the colloidal and binding interactions. Next we shall solve the
Fokker-Planck equation using MFPT approach to evaluate the particle attachment
rate at low-noise, higher-energy-barrier limit.
MFPT at High-Energy Barrier and Low-Noise Limit

Similar to the derivation for Eq. (3.18), it is straightforward to show that
[58, 61]
L+r(, y) = -1, [,y] E 0 0f, (3.49)

where L+ is the backward Fokker-Planck operator

L+ = -X(Y) ) + x(y) 2
Oy y + y2
0o((, y) 0 02
-7(() + e-((y) 9, (3.50)

and the corresponding boundary condition is

T(, y) 0, [, y] E 05. (3.51)

In the limit of weak noise (i.e., e is small) and relatively high energy barrier, a
realization of the process [C(7), y(7)] will reside in Q for a long enough time to assume
a pseudo-stationary distribution before the thermal noise drives it over the boundary
&0. Hence, r(C,y) becomes independent of [C,y] (r(C,y) Tp) practically everywhere
within 2, except for very thin boundary near 0P. Therefore, one may write

Tr(,y) = 7pf( ,y), (3.52)

where the so-called "form function" f((,y) satisfies the following conditions

f((,y) = 1 [,y] E 0 0f (3.53)

f(C,y) 0 [C,y] E 2.

Multiplying Eq. (3.49) with the stationary probability in Eq. (3.35) and integrating
over Q 0f, one obtains by use of the Gaussian theorem and Eq. (3.52) [63, 64])

fa-apP,(_, y)d/dy
7 = (3.54)
E PI(E. ZY) [x(y) 2fLL)y +-y((y)8'(3. *4)dn
where dn denotes the outward normal to the boundary 092. So now the asymptotic
mean first-passage time is expressed in terms of the stationary distribution ps(, y)
and of the gradient of form function f(C,y) on 8~. As of the form function, plugging
Eq. (3.52) into Eq. (3.49) and since 7,p e'/' [58, 65] for the weak-noise limit neglect
the inhomogeneity and write
L+f(6,y) = 0, (3.55)

with the matching conditions as stated in Eq. (3.53). Indeed, Eq. (3.54) is a valid
expression for the mean first-passage time irrespective of whether Q is an attractive
domain and whether the noise is weak. However, only in these latter cases will the
mean first-passage time be essentially independent of the starting point, and the
precise location of the source will not matter. Thus, solving Eq. (3.54) only requires
evaluation of the volume and the line integrals in the equation and solving Eq. (3.55)
for the form function f(,,y).
For weak noise, the sharply-peaked stationary probability density at the stable
stationary points of the drift field allows one to evaluate both integrals in Eq. (3.54)
in Gaussian approximations. The volume integral in the numerator in Eq. (3.54) is
dominated by the linear neighborhood of the energy minimum [CA, YA], and yields in
leading order in noise strength e

PA -= p.a (,y)ckdy (3.56)

SN-exp(-QA/e) det -1/2, (3.57)

where PA is the probability of the particle being in the linear neighborhood of the
secondary energy minimum; the quantities aA and the matrix aj are determined by

local quadratic expansion of a(Y, y) in the vicinity of [CA, YA], namely

a(CA + C, YA + 6y) = A + aSy + O((,y)3), (3.58)


A (=A) eN ln [1 + K,(yA)], (3.59)
S82a^(y) 2ao(,y)
A= AX a A CA (3.60)
__a__,_8) a2cQy)
Clearly, if the 20 potential well dynamics of the particle can not be assumed as

parabolic, the population within domain 0 differs from the Gaussian approximation
in Eq. (3.57). For instance, for many cases of the attachment of a particle to a
surface from colloidal suspensions, the particle experiences the conventional colloidal
interactions, which cannot be assumed quadratic, even at low noise limit. However,
the actual population within domain 0 can be corrected by a prefactor for PA which
now assumes a temperature dependence. For example, suppose we approximate the
colloidal potential with a polynomial around the secondary minimum:

S(z) = A + al (z m) + a2 (z )2 +a3(z- m)3 + a4 (z- m)4. (3.61)

If we denote the integral in the numerator of Eq. (3.54) for such suspension cases as

PAS, we find the correction

P S et ) a) e25/12 (-a1-/e lzs PA (3.62)
S2E 1214a ,"a1 (e-1le- e )PA. (3

Essentially, the treatment of the saddle point is the same as that of the po-
tential well except that at the saddle there is only one unstable direction whose
corresponding eigenvalue of matrix as is negative. Hence, the stationary probability
in the denominator of Eq. (3.54) is given by making quadratic approximation of

a (C, y) in the linear neighborhood of the saddle point [cs = 0, ys]:
Ps N-1 exp(-as/e) det (3.63)
Pr~-~"P(" idktOliJ -1

a(s + k, ys + 5y) = as + af,~j~6y + O(((,y)3), (3.64)

as ) eNIn [1 + K,(ys)], (3.65)
as A


S L sLs s,s (3.66)

In order to determine the gradient of the form function f(C,y), one chooses
the following ansatz which already satisfies the conditions in Eq. (3.53) [66, 67, 68]

()= e-U/du. (3.67)

From Eqs. (3.50) and (3.55) one obtains in leading order in noise strength e a first-
order partial differential equation for p((,y):

Oa(C,y)8p ()Op 0( ) ap2
X(T) '((y) ~ ex(y) + e-((Y) p = 0. (3.68)

In the vicinity of the saddle that contributes significantly to the integral in the de-
nominator in Eq. (3.54), one finds that the solution of Eq. (3.68) is

p = (D,-/A+)-'/ r, (3.69)

where r is the unstable direction of the drift field at the saddle point, D,, is the
diffusion matrix in this unstable direction, and A+ is the positive eigenvalue of

L92p(C y) 192apa^y)
Is,ys Css
Combining Eqs. (3.67) and (3.69) one gets for the line integral in Eq. (3.54) [63]

fP(C,) (9f) Of + ^7(() O dn -A+Ps/(7E). (3.71)

Hence, from Eqs. (3.54), (3.57), (3.63) and (3.71), the final result for the dimension-
less MFPT is

p =- (PA/Ps) (3.72)
r de s| 1/2
S det e(s-aA)/E (3.73)
A+ det aA

A+ = (d + a) + (yd a)2 + 47ybc (3.74)

a = &2c Y) (3.75)
oy2 Cs, s

b- 82a(,y) (3.76)

c 9= ) (3.77)
9 -"S,,YS,

d = (3.78)
a2 Es,ys
Switching back to the time scale using Eq. (3.47), one obtains for the actual mean
first-passage time
T = zs)AA (3.79)

Since at the top of the saddle point the particle has equal probability of going
to either side [69], the attachment rate constant, k, of the particle to a surface will
be given by
1 D(z,)AA +Ps .
2T, 12kBT 27rPA (
Therefore, the rate obtained here is governed by the deterministic dynamics of the
particle at the saddle, stretched by the relative frequency of finding the particle at
the saddle compared to at the potential well.

Derivation of Deposition Rate Constant

In this section, we shall relate the particle attachment rate constant, k, with
the deposition rate constant of the particle, k+, which shares much broader appli-
cations. Suppose the probability of a particle being at the onset of the diffusion
boundary layer, as shown in Figure 3.1 (a), is P(L) oc e-A(CL,YL)/kBT, the deposition
rate constant of the particle is defined as the probability flux over the saddle barrier,
j, divided by P(L) in the steady state:

k P(L)

When examining Eqs. (3.54) and (3.80) for k, one finds out that k is actually the
ratio of the flux crossing over the energy barrier to the probability of the particle
being within the potential well region, namely

k = (3.82)

Combination of Eqs. (3.81) and (3.82) together yields

PA D(z,)AA A+Ps
k+ = k L (3.83)
P(L) 12PkBT 2rP(L)

From this equation, one sees that the deposition rate constant is governed by the de-
terministic dynamics of the particle at the saddle, stretched by the relative frequency
of finding the particle at the saddle compared to at the source which is located at the
startpoint of the diffusive boundary layer, and that k+ is independent of the actual
shape and location of the secondary energy minimum.

3.4 Results and Discussions

Quantitative predictions require the functional forms for the forward and re-
verse rate constants, k1(z) and k, (z) respectively. Here, we adopt the dynamic
models proposed by Dembo et al. [26] for the strain dependence of the transition
state energy between the bound and free states. The functional forms of the rate

constants with respect to z are

kr(z) = kro exp L ( zo) (3.84)

kf(z) = kfo exp [A#(z zo) exp [- (z zo)2 (3.85)

where k,. = kfoexp(eo/kBT) and A is the bond length at the transition state. As
one can see, combination of these two equations yields back to Eq. (2.2).

Using the parameter values assumed in Table 3.1, we show some general pre-
diction results, and also compare the mean first-passage times obtained using the
MFPT formalism (i.e., Eq. (3.79)) with those by the numerical method, Eq. (3.18),
and demonstrate their dependency on the binding parameters as well. In Figure 3.3,
T is plotted as a function of kfo at fixed Keq(zo) = kfo/kro. One can see that the mean
first-passage time, T, decreases linearly with the forward binding rate constant, kfo,
for slower binding, but becomes independent of kfo for faster binding compared to the
rate of the particle diffusion near the surface. Two rate-limiting regimes are observed
from this plot. One is binding-rate-limited. In this regime, a particle may approach
to a distance very close to the surface where binding is energetically favored, but the
binding reaction is too slow to allow attachment. The particle actually feels repulsive
surface forces and fluctuates away from the surface by Brownian motions. The other
one is diffusion-rate-limited. In this regime, binding is fast such that binding reaction
can be assumed always near equilibrium. Thus, the attachment really depends on
how fast a particle can diffuse over the energy barrier to become attached, which is
similar to the case of particle deposition without macromolecular binding in Ref. [14].
Therefore, not only the affinity but also the kinetic rate constants play an important
role in determining particle attachment. For the intermediate case where both Brow-
nian motions and binding processes are similarly fast, our analytical approximations
given by Eq. (3.79) predict smooth transition between these two limiting regimes.
Also, the results from the two methods are consistent with each other.


2 3 4 5

log(kyf (s1)

Figure 3.3: A plot of the mean first-passage time, T, vs. the forward binding rate
constant, kfo, at fixed K,(zo). As the plot shows, T decreases linearly with kfo for
slower binding, but becomes independent of kfo for faster binding. The solid line is
from MFPT approach and the dotted line from numerical method. The MFPT ap-
proach is a good analytical approximation for the corresponding numerical approach.

In Figures 3.4-3.7, the mean first-passage time, T, is plotted versus the energy

barrier height, AA, which is varied by altering the baseline repulsive potential, Oo,
for the case of the imposed trapping potential in Figure 3.4, and by adjusting the
coefficient, a3, in the case of particle attachment from the colloidal suspensions in Eq.

(3.61) in Figure 3.6. In both MFPT and numerical methods, ln(T) increases linearly
with AA with slope of 1, indicating the mean first-passage time is exponentially de-

pendent on the height of a sufficiently sharply peaked energy barrier. In both plots,
the results from two methods match very well except that at lower AA some discrep-
ancies occur. These discrepancies are partly due to the fact that the free energies at

the saddle as well as in the secondary minimum spread out and become less sharply
distributed with declining energy barrier, as demonstrated in Figures 3.5 and 3.7,
respectively; hence, the Gaussian approximations as treated in the MFPT approach

in the neighborhoods of these spots start to show their drawbacks. Furthermore, the
number of bonds that could be formed near the saddle region becomes less and less
with decreasing AA, so the discrete nature of bonds grows more and more significant

and should be considered rather than the continuous approximation in the MFPT
treatment. From here, one can also see that the mean first-passage time approach

can be essentially applied for any reasonable function O(z).

Next, we shall examine the dependence of T on the properties of binding
macromolecules. For illustration purpose, Eq. (3.5) is used for O(z). In Figures 3.8-

3.13, T is plotted versus various binding parameters such as unstrained bond length
zo in Figure 3.8, bond stiffness % in Figure 3.10 and the number of macromolecules
available for binding, N, in Figure 3.12. As the plots show, T decreases greatly
with bond length, increases dramatically with bond stiffness, while the availability of
binding molecules at the interface tends to decrease T. From the equilibrium energy
distance profiles shown in Figure 3.9, one can see that increasing bond lengths lead
to the decrease in the height of energy barrier, which, as a result, decreases T as


6 **.*.*** Numerical




-2 I
6 7 8 9 10 11 12

Energy barrier, AA (kB7)

Figure 3.4: The mean first-passage time, T, is plotted against the energy barrier at
the saddle point, AA, which is varied by varying the baseline repulsive potential, Oo.
The solid lines are from the MFPT approach, Eq. (3.79) and the dotted lines from
numerical solutions, Eq. (3.18).

I20 minimum




10 minimum

50 100 150 200 250 300 350 400
Separation distance, z (nm)

Figure 3.5: This plot shows the minimal energy paths, A(neq(z), z), for different 4o
in Figure 3.4. The lesser Gaussian distributions of the free energies at the saddle
and/or the secondary minimum for lower AA account for the discrepancies in MFPT
and numerical methods.


1..2.... Numerical

& 10




8 10 12 14 16

Energy barrier, AA (kBT)

Figure 3.6: The mean first-passage time, T, is plotted against the energy barrier at
the saddle point, AA, which is varied by adjusting the coefficient a3 for the particle
attachment from colloidal suspensions. The solid lines are from the MFPT approach,
Eq. (3.79) and the dotted lines from numerical solutions, Eq. (3.18).

T20 minimum

$ -10


10 minimum

-40 I I I I
50 100 150 200 250 300 350 400
Separation distance, z (nm)

Figure 3.7: This plot shows the minimal energy paths, A(n,(z), z), for different a3
in Figure 3.6. The lesser Gaussian distributions of the free energies at the saddle
and/or the secondary minimum for lower AA account for the discrepancies in MFPT
and numerical methods.

displayed in Figures 3.4 and 3.6; also, larger bond lengths shift the primary energy
wells away from the surface, which helps for the binding molecules to extend over
the energy barrier to become attached. Similarly, if the stiffness of bond increases,
the energy barrier increases too, as shown in Figure 3.11; thus, stiffer or less flexible
binding molecules are less easier to climb over the energy barrier to attach. When
the number of binding molecules in the contact area is increased, the height of energy
barrier is only slightly decreased, so T decreases slightly too, though the depth of
primary energy minimum drops considerably, see Figure 3.13. From here, one can
clearly conclude that it is the height of the energy barrier at the saddle transition

state that governs the overall attachment rate of particles to a surface. In all the
plots shown here, the MFPT method is in excellent agreement with the numerical
solutions. Only when N becomes much smaller, does the continuous treatment of
the discrete bonds in MFPT approach start to break up, thus deviations between the
two methods begin showing up.
The mean first passage time, T, is also plotted as a function of unstressed

bond energy, eo, in Figure 3.14. T increases monotonically with co. This can
also be explained by the increasing energy barrier with the increasing bond en-

ergy, shown in Figure 3.15. eo is directly related to the affinity of bonds through
K = exp(-eo/kBT). As this plot shows, if the bond energy is very high, corre-
sponding to low affinity, then attachment is not thermodynamically favorable. So,
lower bond energy serves to lower the effective energy barrier to attachment.
The effect of particle size on attachment is examined as well. Since we have
assumed a uniform contact area between the particle and the surface, and this contact
region remains unchanged during attachment, the change in particle radius would not
alter the overall particle-surface interaction energy, as long as all the other binding
parameters including number of binding molecules within the contact area are fixed.
When T is plotted versus particle radius, R, one finds out that T increases slightly



8 .
.. -- MFPT
*.,. .......* Numerical
6 *.




20 30 40 50 60

Unstrained bond length, zo (nn#

Figure 3.8: The mean first-passage time, T, is plotted against unstrained bond length,
zo. In the plot, the solid line is from MFPT approach and the dotted line from
numerical method. As the plot shows, longer bond length allows the molecules to
cross the energy barrier to attach the surface, thus to decrease T.


0 50 100 150 200 250
Separation distance, z (nm)

300 350 400

Figure 3.9: This plot shows the minimal energy paths, A(n,(z),z), for different
unstrained bond length, zo, in Figure 3.8. The energy barrier at the saddle point
decreases as the unstrained bond length increases.



Bond stiffness, l (kBT/nm2)

Figure 3.10: The mean first-passage time, T, is plotted as a function of bond stiffness,
b. In the plot, the solid line is from MFPT approach and the dotted line from nu-
merical method. As the plot shows, lower bond stiffness allows the binding molecules
to cross the energy barrier to attach the surface, thus to decrease T.

I 0 Decreasing energy barriers

SDecreasing bond stiffness



-30 I I I I I
0 50 100 150 200 250 300 350 400
Separation distance, z (nm)

Figure 3.11: This plot shows the minimal energy paths, A(nq(z),z), for different
bond stiffness %y in Figure 3.10. The energy barrier at the saddle point decreases
with decreasing bond stiffness.

40 60 80 100

Number of binding molecules in contact area, N

Figure 3.12: The mean first-passage time, T, is plotted vs. the number of binding
molecules available in the contact area, N. In the plot, the solid line is from MFPT
approach and the dotted line from numerical method. As the plot shows, larger num-
ber of binding macromolecules tends to increase probability of particle attachment
to the surface, thus to decrease T.

P r/

Increasing number of binding
-50 molecules in contace area, N
0 50 100 150 200 250 300 350 400
Separation distance, z (nm)

Figure 3.13: This plot shows the minimal energy paths, A(n,(z),z), for different
number of binding molecules in the contact area as shown in Figure 3.13. The energy
barrier at the saddle point decreases with increasing N.




w 4.5





4 -3 -2 -1 0
Bond energy, kBT

Figure 3.14: The mean first-passage time is plotted as a function of unstressed bond


-40 I
-2 kBT

-60 V

0 50 100 150 200 250 300 350 400
Separation distance, z (nm)

Figure 3.15: The minimal energy path, A(n,(z), z), is plotted for different values of
the unstressed bond energy, eo as shown in Figure 3.14. Lower bond energy tends to
lower the effective energy barrier to attachment.

with R in Figure 3.16. This could be due to the slower Brownian motions for larger
size of particles. The diffusion process for larger particles slows down, which would
increase the time needed for attachment. In fact, the assumption of a flat and uniform
contact area between the particle and the surface is an oversimplification of the actual

reacting system, especially for a rigid particle where surface deformation is not easy.
From our intuition, the distribution of binding molecules over the curved particle
surface would have an effect on the interaction energy, and thus the attachment rate.
This aspect will be further described in Chapter 5.

The deposition rate constant, k+, is plotted as a function of the forward bind-
ing rate constant, kfo in Figure 3.17 at varied energy barriers at the saddle point

using the mean first-passage time approach. The energy barrier AA is varied by
altering the value of the coefficient a3 in Eq. (3.61). In all the plots shown here, k+
increases linearly with kfo at slower binding (i.e., the binding rate-limiting regime),
but becomes independent of kfo at faster binding (i.e., the diffusion rate-limiting
regime). The intermediate region where both binding and diffusion are similarly fast

can be also approximated analytically from MFPT approach. Once again, the plots
show that k+ increases with decreasing AA.




IQ 3.4


100 200 300 400 500

Particle radius, R (nm)

Figure 3.16: The mean first-passage time, T, is plotted as a function of particle
radius, R. As the plot shows, T increases with particle size, though the equilibrium
energy barrier at the saddle transition state remains unchanged based on the uniform
contact area assumption. Longer time is needed to attachment for larger particle as
a result of slowing down of particle diffusion over the energy barrier.


-5** .=12 kt

-3- *M---

4'.5 .AA=.. kBT
Sr ............... AA=10 kBT

-.-.... A=9kBT
-6 -...- .. AM=7kBT

1 2 3 4 5 6

log(kf (-1)

Figure 3.17: The deposition rate constant, k+, is plotted as a function of the forward
binding rate constant, kfo, at different energy barrier height which is varied by varying
the coefficient a3 in Eq. (3.61). The results shown here are from the mean first-
passage time method. In all the plots, k+ increases linearly with kfo at binding rate-
limiting regime, but reaches a plateau and becomes independent of kfo at diffusion
rate-limiting regime. One can also see that the higher the energy barrier, the lower
the deposition rate constant becomes. The following symbols were used: solid line -
AA = 12kBT; dotted line AA = lOksT; broken line AA = 9kBT; dash-dotted
line AA = 7kBT.


In previous chapter, we have developed a dynamical model to estimate the

attachment rate constant of particles to a surface analytically as well as numerically.

Also in that chapter, we used the parameter values assumed in Table 3.1 to illustrate
how to predict the attachment rate constant (or, time to attachment). The model

prediction results from both analytical approximations and numerical computations

are consistent with each other, and also agree qualitatively with the findings in Ref.

[38]. Further evaluations of this model require modeling real biological adhesion sys-

tems. In this chapter, we shall use parameter values (either measured experimentally
or estimated from other reasonable models) to predict the deposition rate constants

of Staphylococcus aureus to fibrinogen-coated surfaces from our dynamic model, then
compare model prediction results with experimental measurement data obtained from

our research group.

4.1 Staphylococcus aureus: Structure and Characteristics

Staphylococcus aureus (S. aureus) is a highly pathogenic bacterial strain that

causes nearly half of hospital-acquired infections and is involved in many device-
centered infections. In addition, it becomes more resistant to many types of antibiotic
treatments, so prevention of infection from it becomes even more crucial.
S. aureus is a nearly spherical, gram-positive bacterium with diameter ranging
from 0.7 ~ 1.2 /pm, with its SEM images shown in Figure 4.1(a). On its surface, S. au-

reus expresses a family of specific protein adhesins (termed MSCRAMMs, microbial

surface components recognizing adhesive matrix molecules) that mediates its adher-

ence to plasma or extracellular matrix proteins, such as collagen [13], fibronectin,
Protein A and fibrinogen [70, 71, 72]. Each of these cellular MSCRAMMs interacts
uniquely with their corresponding matrix proteins. But, these molecules share cer-
tain common genetic and structural features. For instance, hydrophobic amino acid
residues near the C-termini prefer to attach to the cell membrane, whiles hydrophilic

N-termini can extend the binding regions of these MSCRAMMs over cell membrane
to a distance for interacting with the complementary proteins from plasma or extra-
cellular region.

4.2 Fibrinogen-Clumping Factor Interaction

One type of adhesins that S. aureus expresses is called clumping factor A
(ClfA), which is the cell surface protein responsible for binding to fibrinogen. Clump-
ing factor derives its name from the fact that visible clumps of bacteria are observed
in concentrated bacterial suspensions containing fibrinogen. It is a 92 kDa protein
(i.e., 933 amino acid residues) that consists of three different domains: a membrane

and cell wall spanning domain, a serine-aspartic acid dipeptide repeat domain ("the

R region") and the fibrinogen binding domain at the end of the molecule, as shown in
Figure 4.1 (b). The dipeptide (Ser-Asp) repeat region is of 308 amino acid residues
in length for wild type [71]. It has been postulated that this hydrophilic dipeptide

repeat region is able to extend the fibrinogen-binding domain to promote its ability to
interact with its environments, even appreciably far from the cell wall. Hartford et al.
[12] have shown that adhesion of S. aureus depends upon the length of this R-region
by genetically engineering mutants with R-region of reduced length. Also, under the
well-defined flow conditions, our research group have demonstrated that the length of
this R-region promotes adhesion of S. aureus to fibrinogen-coated surfaces; in other

words, the relatively larger length of R-region corresponds to an enhanced rate of at-

tachment [73, 74, 75]. The relative detailed descriptions of their experimental results

are given in the last section of this chapter.

Fibrinogen is a large protein (340 kDa, ~ 40 nm) that is made up of six

dimeric polypeptide chains, two of each alpha, beta and gamma chains. The affinity

of fibrinogen to S. aureus is very strong. It is reported that the binding dissociation

constant for fibrinogen-clumping factor interactions is around 9.9x10-9 M using

radiolabeled fibrinogen method [76] 1. Also, using radiolabeled fibrinogen fragments

and corresponding monoclonal antibodies, the investigators have identified that the

specific region of interaction on the fibrinogen molecule to be the last twelve amino

acid residues of the gamma chain [77]. A more detailed description of properties and

preparation for this interacting 12-residue polypeptide is given in Ref. [74].

4.3 Model Parameter Estimations

In order to model the attachment of S. aureus to a surface through binding

interaction between clumping factor and fibrinogen, we need to find out reasonable

values for all the relevant parameters that are listed in Table 4.1. Below, a number of

important parameters regarding to this specific clumping factor-fibrinogen interaction

are described.

For a basic model of receptor-ligand binding, consider the case in which a

monovalent receptor R binds reversibly to a monovalent ligand L in one step to form

a receptor/ligand complex C, without any other processes modifying this interaction:

R+L -~ f C, (4.1)
'The value for KD of 9.9 nM was determined for the binding of soluble fibrinogen to S. aureus
Newman cells. Newman cells could express several fibrinogen-binding proteins that bind fibrinogen
simultaneously and also each fibrinogen molecule is a dimer and has the potential to bind two
copies of each bacterial fibrinogen-binding molecule. Thus, this reported value for dissociation rate
constant will not be used in current study.

Membrane and
Cell Wall Spanning
I- -

Ser-Asp Repeat Domain

I Domain

"Stalk" region

Figure 4.1: Illustration of S. aureus and cell surface-expressed clumping factor pro-
tein. (a) Electron micrograph of 1 p/m silica particle and S. aureus bacteria with
diameter of about 730 nm. Source: [74] (b) A schematic of clumping factor that
consists of three distinctive domains: a membrane and cell wall spanning domain, a
dipeptide repeating domain and a fibrinogen-binding domain.

Table 4.1: Parameters used in the adhesion of S. aureus to fibrinogen-coated surface

Symbol Parameter Estimated value Unit Reference
k 3-D unstressed forward 7563.6 /M-s-1 [21]
binding rate constant 1.256x 101 nm'/(#.s)
k,3 3-D unstressed reverse 3857.6 s-1 = o K
rate constant
KN 3-D unstressed dissociation 0.51 pjM [72]
rate constant 3.07x 10- #/nm
ko 2-D unstressed forward 377.3 pM-'nm- s-1 Eq. (4.16)
binding rate constant 6.265x 108 nm'/(#s)
ko 2-D unstressed reverse 3857.6 s-1 = fk K
rate constant
KN 2-D unstressed dissociation 10.2 p/M-nm Eq. (4.18)
rate constant 6.155x 10- #/nm2
Yb bond stiffness 0.06 pN/nm Eq. (4.11)
0.015 kBT/nm _
zo unstressed bond length 40 nm [75, 74]
o2 variance of bond fluctuations 64 nm2
eo unstressed bond energy -4.4 kBT Eq. (4.20)
[L]2D number density for ligands 0.0005 #/nm2 [20]
N number of receptors 50
in contact area

again, where kf is the forward binding rate constant and kr is the reverse rate con-

stant. The equation describing the time rate of change of the receptor/ligand complex

dt = kf [R] [L] k, [C], (4.2)

where [R], IL] and [C] are the concentrations for receptor, ligand and their binding

complex molecules, respectively.

As well known, however, the binding of two molecules in chemical reaction
kinetics is really a two-step process, requiring first molecular transport of each in-

dividual molecular species with transport rate constant kt before intrinsic chemical

reactions or binding interactions can occur, as shown in Figure 4.2. Usually at cellu-

lar and subcellular length scales, diffusive transport dominates convective transport,

so that we can consider the transport mechanism to be molecular diffusion. The
chemical or binding reaction itself is then characterized by the intrinsic association

rate constant k,, and intrinsic dissociation rate constant koff. Thus, the measured
values for kf and k, in kinetic experiments are really the combination rate constant

including both the transport and reaction effects. Below, we shall estimate the contri-

butions of each individual rate constant to the overall rate constants for the clumping

factor-fibrinogen interaction.

Usually, experimental measurements of binding kinetic rate constants are con-
ducted when both reacting components are in soluble forms (denoted as 3-D binding
kinetics hereinafter). Attempts to measure two dimensional binding kinetics have just

begun recently. As of now, there are very limited measurement data reported in liter-
ature regarding to adhesion kinetics of S. aureus to fibrinogen-coated surfaces, where
both reacting molecules are attached to the surfaces (i.e., 2-D binding kinetics). So
in what follows, we shall derive a relationship between 3-D and 2-D kinetics.

Let's first study 3-D binding kinetics. If both receptor and ligand molecules
are in extracellular medium, that is, they are freely to move in three-dimensional

ta intrinsic reaction step
transport step k,

4 ko

encounter complex"

Figure 4.2: Separation of the overall binding or dissociation event into two steps. The
intrinsic binding step is characterized by rate constants k,, and koIf that are deter-
mined by the molecular properties of receptor and ligand molecules. The transport
step is characterized by rate constant kt which is influenced by diffusion and geomet-
ric considerations. The intermediate step where receptor and ligand are close enough
but have yet formed the bond is termed as receptor/ligand "encounter complex".
Source: [21]

directions, it has been shown that [78]

k- i + (4.3)
kt + k.o \ kt kr. '

kt = 47rDs, (4.4)

where the diffusion coefficient D is sum of the receptor and ligand diffusivities, i.e.,
D = DR + DL; and s is the encounter radius assuming the origin of a spherical
coordinate system is placed at the center of the receptor molecule. Generally, for
receptors and ligands in free solutions, the experimentally measured value for kf is
more close to reaction-limited binding regime, so that kf ~ kc,,, under the condition
of k, < kt. Here, however, for a very crude estimation, we would use kt = 47rDs
to approximate k,,. Assuming D = 10-6 cm2/s (which is usually in the range of
10-5 ~ 10-7 cm2/s for species of cellular or subcellular length scales) and s = 10

nm (which is generally 1 10 nm), then kf = k,, 1.256 x 10-11 cm3/s 7563.6
[M-s-'1. (Note, here we have used Avogadro's number to convert the units for kf
to pM-'s-1.) Therefore, kIff = KDkOk = 0.51 pM x 7563.6 pM-'s-1 = 3857.6 s-1.
Here, K3D = 0.51 0.19 jpM is obtained from Ref. [72] through measurement of
binding of soluble clumping factor to immobilized fibrinogen using a surface plasmon
resonance sensor.

The above analysis is applicable for cases when both receptor and ligand
molecules are in free solutions. In fact, however, when both of them are anchored to
surfaces as for the adhesion of S. aureus to fibrinogen-coated surface, movements for
receptor and ligand molecules along the direction perpendicular to the surfaces are
not allowed except Brownian motions for the non-grafted ends. Clearly, 2-D binding
kinetics differ greatly from 3-D case. In what follows, symbols 3D and 2D are used
as superscripts or subscripts to distinguish between these two scenarios.

Suppose ligand molecules (or complementary binding sites) on the substrate
surface greatly outnumber receptor molecules over the particle surface, then the num-
ber density for ligand molecules, [L]2o, could be assumed as a constant, and thus,
the receptor-ligand binding reaction reduces to the simplest case of macromolecular
binding reaction that has been examined in Chapter 3. In other words, the occur-
rence of binding between receptor and ligand molecules can be interpreted as that a
receptor molecule changes from its free state to bound state. Let the total number
of receptor molecules available for binding within the contact region be N, and the
number of receptor-ligand complex formed be n, then the balance equation for species
n can now be written as

dt = kf(z)(N n)[L]2D k,(z)n. (4.5)

At equilibrium, one has

k,(z) [N n,(z)] [L]2D K ()
ky(z) n-q(z) (4

Rearranging the above equation leads to

neq(z) ( [L2D
N n.(z) K( (z)
Combination of the above equation with Eq. (3.33) in Chapter 3 yields

ne(z) e(z)kBT [L]2D (4.8)
N nq(z) KD(Z)

The bond energy, e(z), is then given by

e(z) = -kBTln ([L]2 (4.9)

We shall return to this equation for evaluation of the unstressed bond energy eo shortly
after we derive a relationship between 2-D kinetic rate constants and experimentally
measured 3-D values.
Assuming each ligand molecule on the substrate surface is capable of sampling
laterally all the corresponding receptors on the particle surface without restrictions,
and the free end of ligand molecule (where the binding domain is located at) can
undergo unhindered Brownian motions, although the other end of the molecule is
clamped at the surface, as illustrated in Figure 4.3. Based on this assumption, 2-D
binding kinetics differs from 3-D binding kinetics only in the vertical movements (i.e.,
z direction) of ligand molecules. For the case of receptor-ligand binding interactions
confined within the contact region, the effective concentration of the free ends of
ligand molecules, which is now distance-dependent, exhibits a Gaussian distribution
with positions centered at zo. Again, zo here is the unstressed receptor-ligand bond
length, or zo can also be interpreted to represent the equilibrium "stalk" length
of clumping factor molecules expressed on S. aureus surface. In other words, the
probability of finding the binding ends of ligand molecules at distance z can be
described by a normal distribution as follows:

p() = exp (z )2 (4.10)
V27 exp 2kBT

where a is the width of Gaussian distribution and is related to bond stiffness %b
2 = kBT (4.11)
Let the surface density of ligand molecules on the particle surface be [L]2D, the
effective concentration of the free ends of ligand molecules in the contact region,
which now depends upon the distance, is given by

[L]3D(z) = [L]2Dp(z). (4.12)

Therefore, 2-D forward binding rate constant is related to 3-D one through the use
of Eq. (4.12) as follows

kD (z) [L]2D = kD [L)3D(z), (4.13)

= k3D [L]2Dp(z), (4.14)
k 3D
-= [L]2D exp [2 (Z zo2] (4.15)
o 2kBT

Assuming the strain dependence of the transition state between a bound state and a
free state for receptor-ligand bonds, like the dynamic model developed by Dembo et
al. [26], one sees that
k3D [ Ik 3D
kD(z)- exp (z )2 k2D (4.16)
S U-7 2kT fo a 0-7r

where k2D is the unstressed forward binding rate constant for 2-D binding kinetics.
So, this relationship provides a way to estimate 2-D forward binding rate constants
from measured 3-D values, provided that we know o, which is directly related to the
properties of the bonds formed between the bacterium and the substrate and will be
addressed shortly.
Usually, detachment kinetics is much more complicated than attachment ki-
netics. Recently, numerous studies involved in probing single molecular bond kinetics
under applied forces [27, 54] suggest that the dissociation rate constant be influenced


z ...

"A '



Y Y ,

-Bronian motions
Brownian motions

... ... ....

Rey Y

Re eptors

Figure 4.3: Illustration of receptor-ligand binding interaction confined between the
particle and substrate surfaces (which are assumed to be flat, uniform with constant
areas). Receptor-ligand bonds formed are treated as springs with binding energy
increasing parabolically when being pulled/compressed away from their equilibrium
position zo. The free end of ligand molecules undergoes unhindered Brownian mo-
tions, which indicates that the effective concentration of ligand molecules within the
contact region exhibits a Gaussian distribution.


by many factors, such as the energy landscape that the bonds are subjected to, the
magnitude and direction of the external forces applied, the loading rate of external
forces, and the like. The whole dissociation process is a dynamic process, given the
fact that the bond can dissociate spontaneously under zero force. Bell [20] estimated
that the strength of a single bond, denoted as F here, is approximately

F 1.7 (67r) R2,, (4.17)

where r, is the shear stress applied and R is the radius of the bacterium. Under phys-
iological conditions, the receptor-ligand binding complexes (or other macromolecular
bonds) are subjected to fluid shear stress with corresponding shear rate ranging from
40 to 2000 s-1. Thus, the force that a single bond experiences is roughly within the
range of 0.1 ~ 6 pN, which is relatively weak. In addition, since we are focusing on
the attachment of S. aureus to fibrinogen-coated surface, for simplicity, the reverse
binding rate constant is assumed not to be affected by external stress; in other words,
kr is treated as a constant here. Therefore, in view of Eq. (4.16) 2-D dissociation
rate constant is related to 3-D one through the following equation

kK' = = = KD V (4.18)
0fo fo
Now, we switch back to discussions on estimation of receptor-ligand bond
energy from binding kinetics. In view of Eqs. (4.9), (4.12) and (4.18), one sees that

O = -ikBTln ([L]2D (4.19)

= -kBTln( [L]2D (4.20)

Next, we are going to evaluate the stiffness of clumping factor-fibrinogen
bridges. As described earlier, the R-region in the clumping factor molecule con-
sists of only alternating serine and aspartic acid amino acid residues. Assuming this
dipeptide repeat region is a-helical [73], one can estimate the length of the R-region

in nanometers based upon the number of residues contained in this region. Then,

we need to make a reasonable approximation about the stiffness of the fibrinogen-
clumping factor bond, which is assumed to contain mainly the R-region ("stalk"
region) of varied lengths from the clumping factor molecule. Statistical mechanical

theory for semiflexible chains in semidilute solutions is used here to estimate the
stiffness of these bonds. Suppose that a semiflexible R-region "stalk" of length L can

be treated as a Hookean spring, then its stiffness, Yb, is given by [79]

kT7b = (4.21)

where Ap is the "stalk" persistence length, which could not be found in literature

even though the dipeptide "stalk" repeat region is assumed to be a-helix. As a first
attempt, the persistence length for actin filament2 would be used here for a very

coarse approximation. So, Ap is assumed to be around 200 nm. Then the variance
of "stalk" bond fluctuations is obtained from Eq. (4.11), and thus, a- ~ 8 nm. In

turn, kD = 377.3 /M-1nm-'s-1 from Eq. (4.16) and Kg = 10.2 /M.nm from Eq.

(4.18). Suppose that the number density of ligand molecules on the particle surface,

[L]2D, is about 0.0005 #/nm2 (which corresponds to about 45 nm of spacing between
the grafted ends of ligand molecules [20]), then the unstrained receptor-ligand bond

energy is given as eo = -4.4 kBT by using Eq. (4.20). The estimated values for these

parameters are listed in Table 4.1.

4.4 Model Predictions and Comparison with Experimental Data

In order to use the dynamic model developed in Chapter 3 (or the correspond-
ing MatLab codes attached in Appendix) to calculate the deposition rate constant

of S. aureus to a surface covered with fibrinogen, a few more parameters are needed

and discussed here. The size of the contact area on the bacterial surface, which is
assumed to be flat and uniform, is estimated to be about 10 percent of the whole
2Actin filaments belong to another major research area in our research group.

surface area of S. aureus with diameter of about 730 10 nm as shown in Figure

4.1(a). The number of receptors on the substrate available in binding in this contact

region is taken as N = 50, which corresponds to a smaller surface density than that of

ligand molecules on S. aureus surface. To convert the bi-molecular reaction between

receptor and ligand into mono-molecular one under the assumption of excess ligand

molecules, the forward binding rate constant is actually the product of kD and [L]2D,

which gives us a value of 3.13x 105 s-.

A series of equilibrium force measurements between a S. aureus bacterium
and various substrates have been conducted in our research group using the optical

trapping technique, which is described in details in Chapter 6. Figure 4.4 shows two

force-distance profiles for a S. aureus cell with deleted "stalk" region compared to
that with genetically engineered "stalk" length of about 46 nm from Ref. [74]. For a

bacterium with longer "stalk" region, attraction occurs at a distance far away from

the surface and allows the bacterium to overcome repulsions for attachment. The

interactions present here include van der Waals attractions, electrostatic and steric

repulsions as well as receptor-ligand binding interactions. Since there is always a
strong attraction occurring between fibrinogen and clumping factor molecules with

sufficient long "stalk" region expressed on the surface of S. aureus, which is indicated

by the "jump" of the bacterium to the surface to adherence in Figure 4.4, direct

fitting to such measured equilibrium force-distance curves to extract parameters turns

out to be not easy. In retrospect, it would have been good to have also measured
certain isolated repulsive as well as attractive force-distance profiles so as to fit them
separately to obtain better fitting parameters. Since such force measurements are

not currently available, as an approximation, the measured equilibrium force-distance
curve between S. aureus and 3-casein coated surface from Ref. [74] (where specific

binding interactions are absent) is used here and severed as a target fit curve to get

reasonable values for repulsive forces. For the purpose of simplicity, the repulsive

potential is assumed to consist of only electrostatic and steric repulsions that are

characteristic of having an exponential decay with decay length of about 27.5 nm,

which is obtained by extracting the experimental data from Ref. [74] and by fitting

them with a phenomenological expression for repulsive potential as shown in Figure

4.5. And the baseline repulsive potential is found to be about 9.2 kBT.

Using the parameter values shown in Table 4.1, the total interaction energy

at equilibrium, along with the decomposed surface repulsive potential and attractive

binding interaction, is plotted as a function of separation distance in Figure 4.6 for

a S. aureus cell with "stalk" length of 40 nm. From the plot, one can see that

the attraction resulted from clumping factor-fibrinogen binding reaction dominates

over repulsions over the entire characteristic interaction domain of S. aureus to the

surface, assuming here that such binding interactions are allowed to reach equilibrium.

The energy barrier at the saddle transition region that prevents bacterial cells from

attachment is much less evident, which is akin to the equilibrium force measurement

results shown in Figure 4.4. Thus, one would speculate that the deposition rate

constant (or time to attachment) for a S. aureus to a fibrinogen-coated surface is

governed by particle diffusion when far away from the surface; and then when the

bacterium is near the surface, the strong attraction between clumping factor and

fibrinogen molecules will pull the particle into the primary energy minimum and lead

to attachment. The mean first-passage time approach developed in Chapter 3 might

not be applicable for this particular case, because particle attachment rate is not

determined mainly by the energy barrier (less than 3 kBT as shown in Figure 4.5)

and the dynamic properties of bacterial particle at the saddle transition region. To

demonstrate the validity of the above speculation, the effect of "stalk" length on the
deposition rate constants of S. aureus cells to fibrinogen-coated surface is illustrated
in Table 4.2. From this table, one can see that k+ first slightly increases and then

decreases greatly as zo increases, which is quite contrary to what would be expected



0.3 -

0.2 4*4

-0.1 -

*r Jump
-0.2 '

0 100 200 300 400 500

Separation distance (nm)

Figure 4.4: Equilibrium force-distance profiles measured by optical trapping tech-
nique for interaction between S. aureus and fibrinogen-coated surface. Triangles:
bacterial cells with deleted stalk; Circles: bacterial cells with stalk length of about
46 nm. For a bacterium with longer stalk length, the bacterium will jump to the
surface and become attached at a distance much larger than its actual stalk length.
Source: [74]


S force = 1.38e-M4x
Z R2 = 0.8923

L 0.5

0L .... t. ,
0 100 200 300 400

Separation distance (nm)

Figure 4.5: Fitting of the measured equilibrium force-distance profile between S. au-
reus and 3-casein coated surface from Ref. [74] with a phenomenological expression
for repulsion that is assumed to consist of electrostatic repulsion and steric stabiliza-
tion, namely, O(z) = o exp(-bz), which leads to do/dz = 4o(-b)exp(-bz), and the
latter is used to fit the the measured force-distance curve to extract values for fitting

-0 Surface repulsion

s -100 _
0 I Total potential


Binding interaction

50 100 150 200 250 300
Separation distance, z (nm)

Figure 4.6: The total interaction energy at equilibrium, along with the decomposed
surface repulsive potential and attractive binding interaction, is plotted as a function
of separation distance for a S. aureus cell with "stalk" length of 40 nm.

to happen, namely, longer zo tends to lower the energy barrier at saddle and thus
promote attachment, as already demonstrated in Figure 3.8 in Chapter 3 and also
in Figure 4.8 below. Therefore, we may conclude that the analytical MFPT method
can not be utilized here to compute k+. However, the dynamic model developed in
Chapter 3 still applies to the current case, and numerical simulations with their basis
on the model are performed below to calculate again the effect of zo on k+.
The MatLab codes used to calculate the time to attachment (or the depo-
sition rate constant) for S. aureus to a surface through clumping factor-fibrinogen

Table 4.2: The effect of stalk length on attachment of S. aureus to fibrinogen-coated
surface. The calculations were done by the analytical MFPT method and the param-
eter values are listed in Table 4.1.

zo nm 5 10 15 20 30
AA kBT 1.9 1.5 1.3 1.0 0.7
k+ Apm/min 2.34x10-4 2.46 x10-4 2.45x10- 2.28x10-4 1.74x10-4
zo nm 40 50 60 70 100
AA kBT 0.5 0.3 0.2 0.15 0.05
k+ + m/min 1.20x10-4 7.20x10-5 4.19x10-5 2.30x10- 3.23x10-6

binding interactions are similar to those in Appendix, but with key parameter values

listed in Table 4.1. From the above description, for the attachment of S. aureus to

fibrinogen-coated surface based on the energy landscape as illustrated in Figure 4.6,

particle diffusion determines attachment rate until the particle approaches near the

surface where strong attractions due to clumping factor-fibrinogen binding interac-

tion dominate and cause the particle to attach. Figure 4.7 illustrates the dependence

of the mean first-passage time, Tp(z), as a function of separation distance for a S.

aureus bacterium with "stalk" length of 40 nm. One can see that T,(z) decreases

monotonously with the distance as the bacterium moves towards the surface, which

is consistent to the diffusive transport mechanism before the bacterium reaches the

saddle point.

The deposition rate constant, k+, is plotted as a function of "stalk" length in

Figure 4.8, where k+ is extracted from Eq. (2.3) in Chapter 2 using parallel plate

flow chamber assay to measure the attachment kinetics of S. aureus with genetically

engineered "stalk" of varied length under a simple shear flow [75]. From the graph,
one sees that a larger "stalk" length will allow the bacterial cell to effectively extend

over the energy barrier at saddle and then to adhere to the surface. The underlying

physical interpretation of this observation is that longer "stalk" length tends to lower
the energy barrier at saddle, and at the same time shifts the positions of primary







0 20 40 60 80 100 120 140 160 180 200
Separation distance, 10-1, z (nm)

Figure 4.7: The mean first-passage time, Tp(z), is plotted as a function of separation
distance for a S. aureus bacterium with stalk length of about 40 nm from numerical

energy wells away from the surface to promote attachment, as demonstrated in Figure

3.9. The predicted results for the mean first-passage time from numerical simulations

are shown in Figure 4.9. One can see from the plot that as "stalk" length, z0,

increases, T first decreases linearly with zo then approaches to zero 3. Likewise, k+

first increases linearly with zo and then also reaches to a plateau as "stalk" length


3Something is not correct here. Since there is no obvious energy barrier on the energy landscape,
and also binding interaction between clumping factor and fibrinogen is very fast, attachment of
S. aureus to fibrinogen-coated surface is really diffusion-limited. Therefore, the mean first-passage
time method calculated should approach the asymptote determined by particle diffusion, rather
than approach to zero. Further investigation of this point will be addressed in a letter to this thesis.



1.4 -I


0 I"
I -----* I 1-- ----I-I---- Ir*-
0 50 100 150 20 250 300 350

Length of stalk region (residues)

Figure 4.8: The deposition rate constant, k+, is plotted versus stalk length for the
attachment of S. aureus to fibrinogen-coated surface in a flow field measured by
PPFC (parallel plate flow chamber) assay. Diamonds: bacterial cells with varied
stalk length; Squares: bacterial cells with deleted stalk. Source: [75]

0.1 -


( 0.06-



0 100 200 300 400
Stalk length, (nm)

Figure 4.9: The deposition rate constant, k+, is plotted versus stalk length for the
attachment of S. aureus to fibrinogen-coated surface. The calculations are done using
mean first-passage time method as described in Chapter 3. The parameters used in
the calculations are listed in Table 4.1.


For real cells, a roughly flat contact zone might be formed upon close approach

to other cells or substrata as a result of the hydrodynamic pressure that would build

up in the contact region as well as the deformability of cell membranes. Thus, an

assumption of a uniform contact zone might not deviate too much from the actual

situations for the early stage of cellular adhesion. In many other adhesion cases,

however, the surface of biological particles is not flat and also is somewhat rigid,

less deformable. Therefore, accurate predictions on deposition rate constants of such

particles to surfaces necessitate a more accurate account for the effects of particle

curvature on attachment. Here, following the analysis in Chapter 3, a derivation

of mean first-passage time method to obtain particle attachment rate constant that

considers curvature effects of the particle is described.

5.1 Effect of Curvature on Particle-Surface Interaction Energy

The spherical and rigid biological particle under consideration is illustrated in

Figure 5.1. The separation distance between the particle surface and the substrate is

given by z + R (1 cos 0), where 0 is the angle from the nearest point on the particle

surface to the substrate, and R is the radius of the particle. Similarly as in Chapter 3,

for the purpose of simplicity, the bonds formed between the particle and the surface
are also treated to behave like ideal springs, thus, the bond energy, e(z, 0), with a

quadratic dependence on the separation distance, is given by

(, 0) = o + [z + R (1 cos 0) zo]2 O < 0 < (5.1)
2 2
= 00oo, otherwise

K% de

Figure 5.1: A schematic for attachment of a spherical particle to a surface, where the
positions of binding molecules over the particle surface are described by angle 8.

where eo is the unstrained bond energy and zo is the unstrained bond length. Here,
we have assumed that the bonds cannot be formed when 0 > 1, that is, the bonds
are more rigid and less flexible such that they are not able to bend themselves along
the curved particle surface. For any individual binding molecule, it will be in either
bound or free state with binding energy of either e(z, 0) or 0, respectively. Let the
partition function of a free molecule be 1, and that of a single bound macromolecule
be Zb, which is given by [80, 81]

Zb(0) = e-e(zO)/lkT. (5.2)

If the binding molecules over the particle surface are assumed to be independent,
indistinguishable and immobile, the total canonical partition function, Z, for the
particle is [80, 81]
,r/2 N!
Z = No ) b(O) (5.3)
0 (No ne)!9no!
where No, equal to 27rR2p sin OdO, is the number of the binding molecules on the
particle surface at an incremental dO zone at angle 0 no is the number of the
bonds formed at this same region (shown as shaded area in Figure 5.1), and p is
the surface density of the binding macromolecules (Note: the distribution of these
binding molecules over the particle surface could be uniform or angle dependent).
Hence, the total Helmholtz free energy that includes non-specific and specific binding
interactions is then given by

A(e, z) = (z)- kBTlnZ (5.4)
S () + n2 (z,0) kBT n No! ] dO, (5.5)
aJO (No no)!
where 4(z) represents all the non-specific interactions such as electrostatic, van de
Waals and steric interactions, no is a vector with varying 0, and Eq. (5.5) is obtained
using Eq. (5.3). Next, applying the Sterling's approximations for large No and no 1,
'In order to use Sterling's approximation, No and no here should be relatively large. If we
approximate In(n!) nln(n) n, then when n > 5, it turns out to be a roughly accurate estimate.

the above equation becomes

A (n, z) = (z) + F2 [n (z, 0) + (No no) kBT In No no + nekT In dO.
Jo [N N e

5.2 Effect of Curvature on Dynamic Process

The binding macromolecules at different positions on the particle surface react
independently with the complementary binding sites on the substrate. It is reasonable
to assume Markovian processes for all variables no and z, thus, the equation for the
conservation of probability densities is
8tp( z, t) = -9, J [Jne+1-.n Jno-o-1] dO, (5.7)

where p(ne, z, t) is the joint probability density of particle with bond number dis-
tribution as no at separation distance z at time t. The treatment used here is that
every new ne (that is, the bonds formed at a differing angle 0) is added onto the
probability balance equation as a new dimension. Likewise, Jz, the probability flux
J at the z direction, is given by

D(z) OA (To, z)
Jz = k- 9A ( zp(n z, t) D(z)8,p(O, z, t), (5.8)

and Jn,,o,-1, the probability flux in no dimension, is then given by

Jne--no-1 = -kr(z, 0)nop(no, z, t) + kf(z, 0) [No (no 1)] p(ne 1, z, t), (5.9)

where the angle-dependent (i.e., distance dependent) kr(z,0) and kf(z,0) are the
bond dissociation and formation rate constants, respectively. We can see that Eq.
(5.7) is difficult to be solved analytically. So, next we shall simplify it through the
use of van Kampen's system size expansion, and then use MFPT method to obtain
analytical approximations for the resultant equation.

5.2.1 van Kampen's System Size Expansion

As described earlier, each individual binding macromolecule on the particle

surface is in either bound or free states with binding energy of e (z, 0) or 0, respec-

tively, and so, the probability of finding a binding molecule being in bound state is

1-Ci'.kBT. The average number of bonds formed (namely, the bond number at
equilibrium) at angle 0 region is

n 1z, e) = No i e-(z,O)/kT' (5.10)

and the corresponding affinity at the same region is then given by

Kn (z, 0) e-(z,)/ksT. (5.11)
Ne neq(z, )

Similarly, by defining

no = ne z, 0) + N (5.12)

where z, stands for the separation distance for particle being at the saddle point 2,

and using van Kampen's system size expansion [60] for Eq. (5.7) in the limit of all

large No's, one gets

OtP(,zt) = -9^Jz

(z1) Ker (z,,O) + Keg (z, )) K2
+ i k a (z, 0) N.12+K. Keq(z,) + [1 + K, (z, 0)] e} p(e, z, t)d0

+ 2k(z, 0) Kq () p(o, z, t)dO. (5.13)
0 2 1 + Keq (z, 0) 08
2How to define and find such a saddle point is very important. From mathematical point of
view, a saddle point is a critical point at which the gradient is zero and the discriminant is less
than zero. Here, we use the free energy, Eq. (5.6) or Eq. (5.14) as a target function to define
the saddle point. The requirement that the gradient of free energy function at saddle point is zero
leads to that the particle is in chemical equilibrium. For a spherical particle under current study,
simultaneous reaching to chemical equilibrium for all the binding molecules at different curved
regions over the particle surface is not trivial, though theoretically possible, because the kinetic
binding and dissociation processes for molecules located at more distal regions on the particle
surface might be too slow to allow for such an equilibrium to be realized. More comments on this
aspect are entailed in the following Discussions and Suggestions Section.

Replacement of no with e for the free energy in Eq. (5.6) leads to

A( z) = O(z) + n(zn,, ) + N6 e] e (z, 0) dO
Jonq(Z, 0) _
+ /o [No ne(z,, 0)- NJ /2 ] keT n e- (z,, ) N/21] d
f7r12 N11260(za, +
+ ] [weq(z.s,) 6 sTn nzN NJ')> dO. (5.14)

Yet again, the drift terms in Ce dimensions cannot be derived from the potential
A(o, z) but rather from another potential, denoted as B(,, z), which is expressible
as the line integral of drift over diffusion terms in Ze dimensions:
fZ) i/2 /2 Keq (z, 0) K (z, 0)
B(, z) = 2 NO k
J0 Keq (zs, 0) + K,(z, 0)
+kBT[1 + Keq(s,O)] [1 + K(z,0)]} dO
K,(z., ) + K,(z, 0)
+C(z), (5.15)

where C(z) depends upon the separation distance z only. In view of Eqs. (5.8),
(5.14) and (5.15), the above equation becomes
a D(z) aA ( z) a
8tp(,z, t) = z [Dk A p(~, z, t) + D(z) -p(-, z,t) (5.16)
+tp(Tzt) fz/9 (z
+ k(z,)T aB (z)p(We, z, t) + rl(z, 0) p(4B, z, t) dO,
Jo dge kBIT 64o d
where the effective diffusivity in & dimensions is:

1 Kq (,,0) + K(z, )(5.17)
r7(z, 0) = K k,.(z, +) (5.17)
2 1 + Keq (Z,, 0)

5.2.2 Mean First-Passage Time Method

Now using the parameters defined in Eqs. (3.40) and (3.41), Eq. (5.16) can
be made dimensionless as follows

p([ y, ~) = Y)) p(e, y, t) + ex(y) p(, y, t) (5.18)

fo c/ 8o (
+ t ~- ~o o 9 o~)8 o' o v(Se,y,t) + F.'YeC(Yo)gP(eYt) dO,

A A(, z)
a (5, y)= AA (5.19)

B )(520)
S(, y) AA (5.20)

X() = (5.21)
D(z,) '

(y,)- (z,) (5.22)

7o = 12l1(z-' O) (5.23)
7 = k (5.24)
12 kBT

At low-noise high-energy-barrier limit, the extension of the mean first-passage
time formalism from two-dimensional to multi-dimensional Fokker-Planck equation
is readily realizable. Here, the detailed derivations for multi-dimensional case are
not reiterated (please refer to the derivations for 2-D case in Chapter 3 for more
information), only the final results for the dimensionless mean first-passage time, 7,,
and the deposition rate constant of the particle from fluid suspensions are shown
Sde 0, s1/2
I =eto I e(as-aA)/E (5.25)
A det 'a
k+ = D(z,) a AA+ det a -1/2 e-(S-aA)/V (5.26)
where A+ is the positive eigenvalue of drift matrix (it is not a 2 x 2 matrix any more,
instead (mo + 1) x (me + 1), where me is the number of sub-populations which are
grouped based on angle 0) at the saddle point