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MODELING AND MEASUREMENT OF THE ROLE OF MACROMOLECULAR BINDING IN THE ATTACHMENT OF A BROWNIAN PARTICLE TO A SURFACE By HUILIAN MA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2004 Copyright 2004 by Huilian Ma To my dearest daughter, Siyu ACKNOWLEDGEMENTS 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. ChangWon 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 help. 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, PeggyJo 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! TABLE OF CONTENTS ACKNOWLEDGEMENTS ............................ iv ABSTRACT ......... ....... .................. viii CHAPTER 1 INTRODUCTION ................... ............ 1 2 ADHESION OF BIOLOGICAL PARTICLES TO SURFACES ....... 4 2.1 Particle Surface Properties .................. .......... 4 2.2 Interaction Forces Involved During Adhesion and Adhesion Mechanisms 5 2.2.1 NonSpecific Interactions and Attachment Mechanisms .. 6 2.2.2 Specific ReceptorLigand 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 3 ATTACHMENT KINETICS OF BIOLOGICAL PARTICLES TO SURFACES .. ... ..... .. ... ...................... .. 21 3.1 ParticleSurface 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 4 APPLICATION IN ATTACHMENT OF S. AUREUS TO FIBRINOGEN COATED SURFACES ........ ...................... 62 4.1 Staphylococcus aureus: Structure and Characteristics ........ 62 4.2 FibrinogenClumping Factor Interaction ................ 63 4.3 Model Parameter Estimations ... ............... ... 64 4.4 Model Predictions and Comparison with Experimental Data ..... 74 5 EFFECT OF PARTICLE CURVATURE ON ATTACHMENT ........ 85 5.1 Effect of Curvature on ParticleSurface 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 FirstPassage Time Method . . 90 5.2.3 Discussions and Suggestions . ..... 93 5.2.4 Extension to Other Irregular Surfaces . .... 98 6 EXPERIMENTAL VALIDATIONS USING OPTICAL TRAPPING 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 7 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK 117 APPENDIX 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 MODELING AND MEASUREMENT OF THE ROLE OF MACROMOLECULAR BINDING IN THE ATTACHMENT OF A BROWNIAN PARTICLE TO A SURFACE By Huilian Ma August 2004 Chairman: Richard B. Dickinson Major Department: Chemical Engineering Attachment of cells or biological particles to surfaces through the formation of receptorligand bonds or other macromolecular bridges is important in many phys iological, biomedical, biotechnological, and environmental processes. Along with the wellcharacterized 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 saddletransition 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 fibrinogencoated 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 threedimensional optical trapping along with evanescent wave light scattering techniques to measure the at tachment dynamics of streptavidincoated beads to biotinylated PEG (polyethylene glycol) surfaces. CHAPTER 1 INTRODUCTION Attachment of biological particles to surfaces (or other particles) is often me diated by both colloidal forces and receptorligand (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 wellcharacterized colloidal forces such as vad der Waals attraction, elec trical doublelayer interactions, and steric stabilization, the discrete bonds formed between a particle and a surface through surfacebound 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 timescale comparable to the timescale 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 twostep process [8, 9, 10, 11]: (i) attraction to a weak secondary energy minimum created by nonspecific 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 fibrinogencoated 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 ratelimited 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 receptorligand bond, it can be surmised that attachment required formation of only a few adhesincollagen 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 infectionresistant 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 fibrinogencoated surfaces and the direct ex perimental testing of this model using threedimensional 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 fibrinogencoated 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 threedimensional optical trapping along with evanescent wave light scattering techniques to measure the attachment dynam ics of streptavidincoated 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. CHAPTER 2 ADHESION OF BIOLOGICAL PARTICLES TO SURFACES 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, cellsurface adhesion can be exploited for biotechnological purposes, such as affinitybased 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 crosslinked 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 nonspecific interactions (which are usually defined as the types of interactions that do not involve cell surface binding molecules), as discussed below. Therefore, from a physicochemical 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 receptorligand 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 nonspecific and specific interactions and then describe the adhesion mechanism within each category. 2.2.1 NonSpecific Interactions and Attachment Mechanisms Nonspecific interactions are defined as interactions between a particle and a surface (or another particle) that do not involve surfacebound macromolecules, namely, that are not biochemically specific but do act to increase or decrease the overall strength of the interactions. The common operating nonspecific 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 particlesubstrate 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 surfaceexpressed macro molecules as well as glycocalyx that usually is negatively charged due to the present La) Convect on D ffus on D ffus on Boundary Layer Longer Ranue Forces Attachment 'I ,Detachment Partce Surface Lb) k,(z) k,Iz) or k,(z) ; z ~ktz 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. Longrange interaction forces govern the rate of attachment and shortrange 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 receptorligand 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 cellcell or cellsubstratum (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 nonspecific 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 nonspecific forces, Bongrand and Bell [16] calculated the magnitudes of these forces for the case of cellcell 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 cellcell 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 nonspecific longrange interactions between bacterial cells and surfaces is I 10 <52 0 60 100 (41 10' 106 105 103 102 10 160 200 250 300 s(A) steric \ electrostatic 0 50 100 150 200 250 s(A) Figure 2.2: Three major nonspecific 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] STERIC ELECTROSTATIC VAN DER WAALS I__ l i C4 U, *i15~ 1o I  g a . r_ DLVO (DerjaguinLandauVerweyOverbeek, [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 polymerinduced steric forces [19] have been introduced to the DLVO theory, which formed the socalled extended DLVO theories. Since it predicts a potential energy profile as a continuous function of separation distance by using various systemspecific 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 receptorligand binding interactions are not taken into account at all, even though to some extent it is able to successfully explain nonspecific mechanism of microbial adhesion. 2.2.2 Specific ReceptorLigand Interactions and Attachment Mechanisms Despite the repulsive constraints described above, specific receptorligand 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 receptorligand 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 receptorligand binding are Lifshitzvan der Waals (LW), electrostatic (EL) and Lewis acidbase (AB) (or, electron acceptorelectron donor) interactions [17]. These are the same forces that give rise to the nonspecific interactions de scribed earlier. Although the physical nature of the forces underlying specific in teractions is the same as that of nonspecific interactions, receptorligand binding is often described in terms of chemical reaction kinetics. The rationale for apply ing this framework comes from physical characteristics of these weak, noncovalent specific binding interactions. First, the interaction range between receptor and lig and binding domains is very short (~ 1 nm) compared to nonspecific 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 "locktokey" 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 nonspecific interactions, which have been successfully described by forcedistance 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 forcedistance relationship is lumped into the kinetic rate and equilibrium constants. TwoDimensional Kinetics. Contrasted to three dimensional (3D) mo tions of free molecules in solutions, the surfacebound receptors and complementary ligands are limited to two dimensional (2D) 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 2D/2D 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 receptorligand bonds have not yet been possible when two reacting molecules are linked to surfaces. A more common approach is to first measure the M 3D affinity (determined when both receptor and ligand molecules diffuse freely in free solutions) and then convert it to a 2D affinity using a parameter, which, in general, is on the order of the size of the receptorligand complex molecules [20, 21]. A method to convert 3D binding kinetic rate affinity constants to 2D ones is also developed in Chapter 4. Recently, measurements of 2D 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 NbBT)' (2.1) Table 2.1: Units for measurement of receptorligand interactions where ko (in s1) 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 receptorligand bonds formed. An alternative approach assumes the receptorligand bond behaves like a Hookean spring, as proposed by Dembo and coworkers [25, 26]. In this case, affinity will decrease as a receptorligand bond moves away from its equilibrium position according to a Boltzmann distribution shown below 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 straindependence 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 receptorligand bonds. As a result of this low bond number, smallscale 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 stopandgo types of motions with highly fluctuated velocities [29, 30]. Even in the wellcontrolled 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 welldefined, 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 receptorligand 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, timedependent, 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 subfamily 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 hospitalacquired infections, and this pair interaction serves as the model experimental system in Chapter 4 by which the effect of MSCRAMM length on cellsurface 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 cellcell (or cellsurface) adhesion. The models they developed are to study the adhesion between two freely deformable cells with surfacebound 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, cellcell separation dis tance and number of bonds as a function of both receptor properties and nonspecific 'Here, the pressure should use surface pressure, because binding reactions between two reacting species take place in a twodimensional 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 receptormediated cell adhesion to a ligandcoated 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 timecourse 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 nonzero 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. CozensRoberts 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 invivo 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 timedependent 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 steadystate 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 nonspecific forces, such as van der Waals, electrostatic, steric stabilization, and the biochemically spe cific receptorligand binding interactions. Thus, particlesurface 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 realtime measurement under welldefined 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 steadystate flux of particles over this boundary is proportional to the particle concentration at this boundary by applying firstorder 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] dc, 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 receptorligand bonds. Then, Dembo et al. [26] combined cell deformation mechanics with receptorligand 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 receptorcoated beads from a ligandcoated 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 receptorligand 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 micropipetpressurized 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 nN/nm. CHAPTER 3 ATTACHMENT KINETICS OF BIOLOGICAL PARTICLES TO SURFACES 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 receptorligand 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 timescale comparable to the timescale 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 ParticleSurface 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 particlesurface system under consideration, which is a sim ple case for receptormediated 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 surfacebound molecules available for binding in the contact area to form n bonds between the particle and the surface. The total particlesurface inter action energy at the interface is the sum of the energy of n bonds and the nonspecific 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 surfaceindependent 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 cellcell 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) = oebz + 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 receptorligand 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 particlesurface 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, 1 E(z) = o + ~Yb(Z z0)2, (3.6) where to = e(zo) and yb is bond stiffness. As for the real receptorligand 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 FokkerPlanck 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 "O c 30 0 " 25 saddle 20 Z 1520 Z 15 secondary minimum 10 separatrix 5 0 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 nm1 Repulsive decay length Co 1 kBT Unstrained bond energy 7b 0.002 kBT/nm2 Bond stiffness zo 40 nm Unstrained bond length 7, 8x104 kBT/nm2 Optical trap stiffness zp 200 nm Optical trap center kfo 1000 s1 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 103 kBT/nm2 Potential coefficient a3 106 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 StokesEinstein 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 Jna1) (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, Jnn1, is given by Jnn = kr(z)np(n, z,t) + kf(z)[N (n l)]p(n 1, z,t), (3.11) where k,(z) and kf(z) are the rate constants of bond dissociation and formation, respectively. 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 noflux boundary conditions in the ndimension are Jn (0, z) = 0 and Jn1 (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 firstpassage 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 firstpassage 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 firstpassage 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 lefthand 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 righthand 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 ' dn 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 firstpassage 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 FokkerPlanck 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 FokkerPlank 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). (3.25) The operator E changes n to n + 1 and therefore ( to 4 + N1/2, such that E = 1+ N/ + N .. a0 2 a2 0 (N1/2)m (3.26) m a (3.26) m=O 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 N1/2. Substitution of Eq. (3.26) into Eq. (3.25) to change the variable n to ( yields [58, 60]: S(N1/2)m [N n(z) N2] p(, z, t) Otp(, z, t) = zJ + ki(z) m! a~ [N n ) N' m= 1 c (N1/2 m am +k(z) m! 8m [neq)+ N/2] p(, z,t). (3.27) m=l Rearranging Eq. (3.27) yields (0 1) Nim/2 am r [m ((, z) ap(, z, 0) = . J. + m m a (z) 'Na / p(, z,t), (3.28) m= 1 n ~mN1 with 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 N1/2 vanish. Therefore, by truncating Eq. (3.28) at m = 2 and collecting the terms of 0(1) and 0(N'12), we get a FokkerPlanck 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 1e(/k~ r Thus, the number of bonds at equilibrium is ee(z)/kfT n,(z) = N e) (3.32) 1+ ee(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) = NleA((,z)/kBT, (3.35) 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 FokkerPlanck 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) (3.37) 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) with 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 FokkerPlanck 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 FokkerPlanck equation, i.e., Eq. (3.38), using the values of its key parameters at the saddle point. We first define parameter kT(3.40) e (3.40) AA' 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) becomes 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) where A A((,z) aAy) = A (3.43) AA B(, z) (,) AA (3.44) D(z) X(y) = ) (3.45) D(z,) (() (z) (3.46) tD(z,)AA 7 kBT (3.47) k12kT and S= 12 (z) (3.48) D(z,) 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 twodimensional FokkerPlanck 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 FokkerPlanck equation using MFPT approach to evaluate the particle attachment rate at lownoise, higherenergybarrier limit. MFPT at HighEnergy Barrier and LowNoise 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 FokkerPlanck 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 pseudostationary 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 socalled "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]) faapP,(_, 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 firstpassage 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 weaknoise 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 firstpassage time irrespective of whether Q is an attractive domain and whether the noise is weak. However, only in these latter cases will the mean firstpassage 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 sharplypeaked 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) J.an SNexp(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) with A (=A) eN ln [1 + K,(yA)], (3.59) AA S82a^(y) 2ao(,y) A= AX a A CA (3.60) __a__,_8) a2cQy) B ACA J AA 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 (e1le 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]: 1/2 Ps N1 exp(as/e) det (3.63) Pr~~"P(" idktOliJ 1 where 1s a(s + k, ys + 5y) = as + af,~j~6y + O(((,y)3), (3.64) as ) eNIn [1 + K,(ys)], (3.65) as A and 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] ()= eU/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(saA)/E (3.73) A+ det aA where A+ = (d + a) + (yd a)2 + 47ybc (3.74) 2 with a = &2c Y) (3.75) oy2 Cs, s b 82a(,y) (3.76) c 9= ) (3.77) 9 "S,,YS, and d = (3.78) a2 Es,ys Switching back to the time scale using Eq. (3.47), one obtains for the actual mean firstpassage time 12kBT 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 eA(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: S(3.81) 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) PA 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 firstpassage 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 firstpassage 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 ratelimiting regimes are observed from this plot. One is bindingratelimited. 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 diffusionratelimited. 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. U)ii 2 3 4 5 log(kyf (s1) Figure 3.3: A plot of the mean firstpassage 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 3.43.7, the mean firstpassage 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 firstpassage 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 firstpassage 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 10 8  MFPT 6 **.*.*** Numerical S4 2 0 2 I 6 7 8 9 10 11 12 Energy barrier, AA (kB7) Figure 3.4: The mean firstpassage 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 20 30 10 minimum 40 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. 16 14 MFPT 1..2.... Numerical 12 & 10 S8 6' 4 2 8 10 12 14 16 Energy barrier, AA (kBT) Figure 3.6: The mean firstpassage 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). 0 T20 minimum $ 10 20 30 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 particlesurface 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 51 10 8 . ..  MFPT *.,. .......* Numerical 6 *. 4 2 0 20 30 40 50 60 Unstrained bond length, zo (nn# Figure 3.8: The mean firstpassage 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. M 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. 7 6 Bond stiffness, l (kBT/nm2) Figure 3.10: The mean firstpassage 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 5 S10 SDecreasing bond stiffness 15 20 25 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 firstpassage 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 60 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. 6.0 5.5 5.0 w 4.5 4.0 3.5 3.0 2.5 2.0 4 3 2 1 0 Bond energy, kBT Figure 3.14: The mean firstpassage time is plotted as a function of unstressed bond energy. 20 40 I 2 kBT 50 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 firstpassage 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 ratelimiting regime), but becomes independent of kfo at faster binding (i.e., the diffusion ratelimiting 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. 3.7 3.6 3.5 IQ 3.4 3.3 3.2 100 200 300 400 500 Particle radius, R (nm) Figure 3.16: The mean firstpassage 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. 2 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 ratelimiting 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; dashdotted line AA = 7kBT. CHAPTER 4 APPLICATION IN ATTACHMENT OF S. AUREUS TO FIBRINOGEN COATED SURFACES 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 fibrinogencoated 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 hospitalacquired 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, grampositive 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 Ctermini prefer to attach to the cell membrane, whiles hydrophilic Ntermini 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 FibrinogenClumping 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 serineaspartic 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 (SerAsp) 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 fibrinogenbinding 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 Rregion by genetically engineering mutants with Rregion of reduced length. Also, under the welldefined flow conditions, our research group have demonstrated that the length of this Rregion promotes adhesion of S. aureus to fibrinogencoated surfaces; in other words, the relatively larger length of Rregion 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 fibrinogenclumping factor interactions is around 9.9x109 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 12residue 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 factorfibrinogen interaction are described. For a basic model of receptorligand 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 fibrinogenbinding proteins that bind fibrinogen simultaneously and also each fibrinogen molecule is a dimer and has the potential to bind two copies of each bacterial fibrinogenbinding molecule. Thus, this reported value for dissociation rate constant will not be used in current study. Membrane and Cell Wall Spanning Domain I  SerAsp Repeat Domain FibrinogenBlnding I Domain "Stalk" region Figure 4.1: Illustration of S. aureus and cell surfaceexpressed 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 fibrinogenbinding domain. Table 4.1: Parameters used in the adhesion of S. aureus to fibrinogencoated surface Symbol Parameter Estimated value Unit Reference k 3D unstressed forward 7563.6 /Ms1 [21] binding rate constant 1.256x 101 nm'/(#.s) k,3 3D unstressed reverse 3857.6 s1 = o K rate constant KN 3D unstressed dissociation 0.51 pjM [72] rate constant 3.07x 10 #/nm ko 2D unstressed forward 377.3 pM'nm s1 Eq. (4.16) binding rate constant 6.265x 108 nm'/(#s) ko 2D unstressed reverse 3857.6 s1 = fk K rate constant KN 2D unstressed dissociation 10.2 p/Mnm 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 is d[C) 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 twostep 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 factorfibrinogen interaction. Usually, experimental measurements of binding kinetic rate constants are con ducted when both reacting components are in soluble forms (denoted as 3D 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 fibrinogencoated surfaces, where both reacting molecules are attached to the surfaces (i.e., 2D binding kinetics). So in what follows, we shall derive a relationship between 3D and 2D kinetics. Let's first study 3D binding kinetics. If both receptor and ligand molecules are in extracellular medium, that is, they are freely to move in threedimensional ta intrinsic reaction step transport step k, 4 ko "receptoriligand 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. ' with 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 reactionlimited 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 = 106 cm2/s (which is usually in the range of 105 ~ 107 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 1011 cm3/s 7563.6 [Ms'1. (Note, here we have used Avogadro's number to convert the units for kf to pM's1.) Therefore, kIff = KDkOk = 0.51 pM x 7563.6 pM's1 = 3857.6 s1. 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 fibrinogencoated surface, movements for receptor and ligand molecules along the direction perpendicular to the surfaces are not allowed except Brownian motions for the nongrafted ends. Clearly, 2D binding kinetics differ greatly from 3D 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 receptorligand 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 receptorligand complex formed be n, then the balance equation for species n can now be written as dn 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) nq(z) (4 Rearranging the above equation leads to neq(z) ( [L2D (4.7) 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 2D kinetic rate constants and experimentally measured 3D 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, 2D binding kinetics differs from 3D binding kinetics only in the vertical movements (i.e., z direction) of ligand molecules. For the case of receptorligand binding interactions confined within the contact region, the effective concentration of the free ends of ligand molecules, which is now distancedependent, exhibits a Gaussian distribution with positions centered at zo. Again, zo here is the unstressed receptorligand 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 through 2 = kBT (4.11) 7b 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, 2D forward binding rate constant is related to 3D 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 receptorligand 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 U7 2kT fo a 07r where k2D is the unstressed forward binding rate constant for 2D binding kinetics. So, this relationship provides a way to estimate 2D forward binding rate constants from measured 3D 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 V z ... "A ' Z Ligands iY Y Y , Bronian motions Brownian motions Y ... ... .... Rey Y Y Re eptors Figure 4.3: Illustration of receptorligand binding interaction confined between the particle and substrate surfaces (which are assumed to be flat, uniform with constant areas). Receptorligand 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 receptorligand binding complexes (or other macromolecular bonds) are subjected to fluid shear stress with corresponding shear rate ranging from 40 to 2000 s1. 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 fibrinogencoated 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) 2D dissociation rate constant is related to 3D one through the following equation kK' = = = KD V (4.18) 0fo fo Now, we switch back to discussions on estimation of receptorligand 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 factorfibrinogen bridges. As described earlier, the Rregion in the clumping factor molecule con sists of only alternating serine and aspartic acid amino acid residues. Assuming this dipeptide repeat region is ahelical [73], one can estimate the length of the Rregion 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 Rregion ("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 Rregion "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 ahelix. 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 /M1nm's1 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 receptorligand 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 bimolecular reaction between receptor and ligand into monomolecular 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 forcedistance 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 receptorligand 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 forcedistance 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 forcedistance 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 forcedistance curve between S. aureus and 3casein 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 factorfibrinogen 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 fibrinogencoated 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 firstpassage 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 fibrinogencoated 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 0.5 0.4 0.3  0.2 4*4 0.1  0AL A A 0.1 *r Jump 0.2 ' 0.3 0 100 200 300 400 500 Separation distance (nm) Figure 4.4: Equilibrium forcedistance profiles measured by optical trapping tech nique for interaction between S. aureus and fibrinogencoated 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] 2 S force = 1.38eM4x Z R2 = 0.8923 0. oL L 0.5 0L .... t. , 0* 0 100 200 300 400 Separation distance (nm) Figure 4.5: Fitting of the measured equilibrium forcedistance profile between S. au reus and 3casein 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 forcedistance curve to extract values for fitting parameters. 0 Surface repulsion s 100 _ 0 I Total potential 150 Binding interaction 200 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 factorfibrinogen Table 4.2: The effect of stalk length on attachment of S. aureus to fibrinogencoated 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 2.34x104 2.46 x104 2.45x10 2.28x104 1.74x104 zo nm 40 50 60 70 100 AA kBT 0.5 0.3 0.2 0.15 0.05 k+ + m/min 1.20x104 7.20x105 4.19x105 2.30x10 3.23x106 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 fibrinogencoated 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 factorfibrinogen binding interac tion dominate and cause the particle to attach. Figure 4.7 illustrates the dependence of the mean firstpassage 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 81 0.025 0.02 0.015 0.01 0.005 0 20 40 60 80 100 120 140 160 180 200 Separation distance, 101, z (nm) Figure 4.7: The mean firstpassage 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 simulations. energy wells away from the surface to promote attachment, as demonstrated in Figure 3.9. The predicted results for the mean firstpassage 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 increases. 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 fibrinogencoated surface is really diffusionlimited. Therefore, the mean firstpassage 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. 2.8 2.1 1.4 I 0.7 0 I" I * I 1 II 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 fibrinogencoated 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  0.08 ( 0.06 0.04 0.02 0 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 fibrinogencoated surface. The calculations are done using mean firstpassage time method as described in Chapter 3. The parameters used in the calculations are listed in Table 4.1. CHAPTER 5 EFFECT OF PARTICLE CURVATURE ON ATTACHMENT 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 firstpassage time method to obtain particle attachment rate constant that considers curvature effects of the particle is described. 5.1 Effect of Curvature on ParticleSurface 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) = ee(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 nonspecific 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 nonspecific 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 (5.6) 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 7Tr/2 8tp( z, t) = 9, J [Jne+1.n Jnoo1] 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 Jneno1 = kr(z, 0)nop(no, z, t) + kf(z, 0) [No (no 1)] p(ne 1, z, t), (5.9) where the angledependent (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 1Ci'.kBT. The average number of bonds formed (namely, the bond number at equilibrium) at angle 0 region is ee(z,()/kBT 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 FirstPassage 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, with A A(, z) a (5, y)= AA (5.19) AA B )(520) S(, y) AA (5.20) D(z) X() = (5.21) D(z,) ' (y,) (z,) (5.22) 7o = 12l1(z' O) (5.23) D(z,)' and tD(z,)AA 7 = k (5.24) 12 kBT At lownoise highenergybarrier limit, the extension of the mean firstpassage time formalism from twodimensional to multidimensional FokkerPlanck equation is readily realizable. Here, the detailed derivations for multidimensional case are not reiterated (please refer to the derivations for 2D case in Chapter 3 for more information), only the final results for the dimensionless mean firstpassage time, 7,, and the deposition rate constant of the particle from fluid suspensions are shown below Sde 0, s1/2 I =eto I e(asaA)/E (5.25) A det 'a and k+ = D(z,) a AA+ det a 1/2 e(SaA)/V (5.26) 27T2kBT 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 subpopulations which are grouped based on angle 0) at the saddle point 