A computational model of cell migration coupling the growth of focal adhesions with oscillatory cell protrusions

Cell migration is a highly integrated process where actin turnover, actomyosin contractility, and adhesion dynamics are all closely linked. In this paper, we propose a computational model investigating the coupling of these fundamental processes within the context of spontaneous (i.e. unstimulated) cell migration. In the unstimulated cell, membrane oscillations originating from the interaction between passive hydrostatic pressure and contractility are sufficient to lead to the formation of adhesion spots. Cell contractility then leads to the maturation of these adhesion spots into focal adhesions. Due to active actin polymerization, which reinforces protrusion at the leading edge, the traction force required for cell translocation can be generated. Computational simulations first show that the model hypotheses allow one to reproduce the main features of fibroblast cell migration and established results on the biphasic aspect of the cell speed as a function of adhesion strength. The model also demonstrates that certain temporal parameters, such as the adhesion proteins recycling time and adhesion lifetimes, influence cell motion patterns, particularly cell speed and persistence of the direction of migration. This study provides some elements, which allow a better understanding of spontaneous cell migration and enables a first glance at how an individual cell would potentially react once exposed to a stimulus.     

Prestress and Adhesion Site Dynamics Control Cell Sensitivity to Extracellular Stiffness

This study aims at improving the understanding of mechanisms responsible for cell sensitivity to extracellular environment. We explain how substrate mechanical properties can modulate the force regulation of cell sensitive elements primarily adhesion sites. We present a theoretical and experimental comparison between two radically different approaches of the force regulation of adhesion sites that depends on their either stationary or dynamic behavior. The most classical stationary model fails to predict cell sensitivity to substrate stiffness whereas the dynamic model predicts extracellular stiffness dependence. This is due to a time dependent reaction force in response to actomyosin traction force exerted on cell sensitive elements. We purposely used two cellular models, i.e., alveolar epithelial cells and alveolar macrophages exhibiting respectively stationary and dynamic adhesion sites, and compared their sensitivity to theoretical predictions. Mechanical and structural results show that alveolar epithelial cells exhibit significant prestress supported by evident stress fibers and lacks sensitivity to substrate stiffness. On the other hand, alveolar macrophages exhibit low prestress and exhibit sensitivity to substrate stiffness. Altogether, theory and experiments consistently show that adhesion site dynamics and cytoskeleton prestress control cell sensitivity to extracellular environment with an optimal sensitivity expected in the intermediate range.     

Influence of type I collagen surface density on fibroblast spreading, motility, and contractility

We examine the relationships of three variables (projected area, migration speed, and traction force) at various type I collagen surface densities in a population of fibroblasts. We observe that cell area is initially an increasing function of ligand density, but that above a certain transition level, increases in surface collagen cause cell area to decline. The threshold collagen density that separates these two qualitatively different regimes, approximately 160 molecules/ microm(2), is approximately equal to the cell surface density of integrin molecules. These results suggest a model in which collagen density induces a qualitative transition in the fundamental way that fibroblasts interact with the substrate. At low density, the availability of collagen binding sites is limiting and the cells simply try to flatten as much as possible by pulling on the few available sites as hard as they can. The force per bond under these conditions approaches 100 pN, approximately equal to the force required for rupture of integrin-peptide bonds. In contrast, at high collagen density adhesion, traction force and motility are limited by the availability of free integrins on the cell surface since so many of these receptors are bound to the surface ligand and the force per bond is very low.     

A stochastic description of Dictyostelium chemotaxis

Chemotaxis, the directed motion of a cell toward a chemical source, plays a key role in many essential biological processes. Here, we derive a statistical model that quantitatively describes the chemotactic motion of eukaryotic cells in a chemical gradient. Our model is based on observations of the chemotactic motion of the social ameba Dictyostelium discoideum, a model organism for eukaryotic chemotaxis. A large number of cell trajectories in stationary, linear chemoattractant gradients is measured, using microfluidic tools in combination with automated cell tracking. We describe the directional motion as the interplay between deterministic and stochastic contributions based on a Langevin equation. The functional form of this equation is directly extracted from experimental data by angle-resolved conditional averages. It contains quadratic deterministic damping and multiplicative noise. In the presence of an external gradient, the deterministic part shows a clear angular dependence that takes the form of a force pointing in gradient direction. With increasing gradient steepness, this force passes through a maximum that coincides with maxima in both speed and directionality of the cells. The stochastic part, on the other hand, does not depend on the orientation of the directional cue and remains independent of the gradient magnitude. Numerical simulations of our probabilistic model yield quantitative agreement with the experimental distribution functions. Thus our model captures well the dynamics of chemotactic cells and can serve to quantify differences and similarities of different chemotactic eukaryotes. Finally, on the basis of our model, we can characterize the heterogeneity within a population of chemotactic cells.     

Force measurements on single molecular contacts through evanescent wave microscopy

We introduce a new method to apply controlled forces on single molecules. The motion of a micron-sized bead attached to a solid surface through a single molecular contact is tracked by evanescent wave microscopy as a force is exerted through a flow. We report measurements of the streptavidin-biotin bond rupture force obtained with this technique. We also obtain detailed measurements of the balance of forces involved in detaching an adhering bead with a flow. A small lateral force translates into a much bigger normal force on the attachment point. This effect is relevant for the interpretation of common cell adhesion assays.     

Stabilizing to disruptive transition of focal adhesion response to mechanical forces

Strong mechanical forces can, obviously, disrupt cell–cell and cell–matrix adhesions, e.g., cyclic uniaxial stretch induces instability of cell adhesion, which then causes the reorientation of cells away from the stretching direction. However, recent experiments also demonstrated the existence of force dependent adhesion growth (rather than dissociation). To provide a quantitative explanation for the two seemingly contradictory phenomena, a microscopic model that includes both integrin–integrin interaction and integrin–ligand interaction is developed at molecular level by treating the focal adhesion as an adhesion cluster. The integrin clustering dynamics and integrin–ligand binding dynamics are then simulated within one unified theoretical frame with Monte Carlo simulation. We find that the focal adhesion will grow when the traction force is higher than a relative small threshold value, and the growth is dominated by the reduction of local chemical potential energy by the traction force. In contrast, the focal adhesion will rupture when the traction force exceeds a second threshold value, and the rupture is dominated by the breaking of integrin–ligand bonds. Consistent with the experiments, these results suggest a force map for various responses of cell adhesion to different scales of mechanical force.     

Depolymerization-driven flow in nematode spermatozoa relates crawling speed to size and shape

Cell crawling is an inherently physical process that includes protrusion of the leading edge, adhesion to the substrate, and advance of the trailing cell body. Research into advance of the cell body has focused on actomyosin contraction, with cytoskeletal disassembly regarded as incidental, rather than causative; however, extracts from nematode spermatozoa, which use Major Sperm Protein rather than actin, provide at least one example where cytoskeletal disassembly apparently generates force in the absence of molecular motors. To test whether depolymerization can explain force production during nematode sperm crawling, we constructed a mathematical model that simultaneously describes the dynamics of both the cytoskeleton and the cytosol. We also performed corresponding experiments using motile Caenorhabditis elegans spermatozoa. Our experiments reveal that crawling speed is an increasing function of both cell size and anterior-posterior elongation. The quantitative, depolymerization-driven model robustly predicts that cell speed should increase with cell size and yields a cytoskeletal disassembly rate that is consistent with previous measurements. Notably, the model requires anisotropic elasticity, with the cell being stiffer along the direction of motion, to accurately reproduce the dependence of speed on elongation. Our simulations also predict that speed should increase with cytoskeletal anisotropy and disassembly rate.     

Enforced unbinding of biomembranes whose mutual adhesion is mediated by a specific interaction

The enforced unbinding of a biomembrane from a rigid substrate whose adhesion is mediated by a specific interaction has been studied theoretically. We argue that the unbinding takes place via motion of the adhesion rim. We account for the release of the elastic energy (stored in the membrane curvature) in the binding-unbinding process and obtain the dissociation constant at the rim. We further deduce an equation of motion for the rim. The solution exhibits an initial phase describing a slow motion followed by a regime of rapid motion and finally breaking of the adhesion. We show that the unbinding force depends on the rate of force application as F(*) approximately equal (F')(beta). When small forces are applied, beta=1/2, while in the case of large forces, beta=1/3.     

Characteristics of trajectory in the migration of Amoeba proteus

Summary. We investigated the behavior of migration of Amoeba proteus in an isotropic environment. We found that the trajectory in the migration of A. proteus is smooth in the observation time of 500-1000 s, but its migration every second (the cell velocity) on the trajectory randomly changes. Stochastic analysis of the cell velocity and the turn angle of the trajectory has shown that the histograms of the both variables well fit to Gaussian curves. Supposing a simple model equation for the cell motion, we have estimated the motive force of the migrating cell, which is of the order of piconewton. Furthermore, we have found that the cell velocity and the turn angle have a negative cross-correlation coefficient, which suggests that the amoeba explores better environment by changing frequently its migrating direction at a low speed and it moves rectilinearly to the best environment at a high speed. On the other hand, the model equation has simulated the negative correlation between the cell velocity and the turn angle. This indicates that the apparently rational behavior comes from intrinsic characteristics in the dynamical system where the motive force is not torquelike.     

Force-induced focal adhesion translocation: effects of force amplitude and frequency

Vascular endothelial cells rapidly transduce local mechanical forces into biological signals through numerous processes including the activation of focal adhesion sites. To examine the mechanosensing capabilities of these adhesion sites, focal adhesion translocation was monitored over the course of 5 min with GFP-paxillin while applying nN-level magnetic trap shear forces to the cell apex via integrin-linked magnetic beads. A nongraded steady-load threshold for mechanotransduction was established between 0.90 and 1.45 nN. Activation was greatest near the point of forcing (<7.5 µm), indicating that shear forces imposed on the apical cell membrane transmit nonuniformly to the basal cell surface and that focal adhesion sites may function as individual mechanosensors responding to local levels of force. Results from a continuum, viscoelastic finite element model of magnetocytometry that represented experimental focal adhesion attachments provided support for a nonuniform force transmission to basal surface focal adhesion sites. To further understand the role of force transmission on focal adhesion activation and dynamics, sinusoidally varying forces were applied at 0.1, 1.0, 10, and 50 Hz with a 1.45 nN offset and a 2.25 nN maximum. At 10 and 50 Hz, focal adhesion activation did not vary with spatial location, as observed for steady loading, whereas the response was minimized at 1.0 Hz. Furthermore, applying the tyrosine kinase inhibitors genistein and PP2, a specific Src family kinase inhibitor, showed tyrosine kinase signaling has a role in force-induced translocation. These results highlight the mutual importance of force transmission and biochemical signaling in focal adhesion mechanotransduction. mechanotransduction; endothelial cell; paxillin; viscoelastic model     

Theory of force regulation by nascent adhesion sites

The mechanical coupling of a cell with the extracellular matrix relies on adhesion sites, clusters of membrane-associated proteins that communicate forces generated along the F-Actin filaments of the cytoskeleton to connecting tissue. Nascent adhesion sites have been shown to regulate these forces in response to tissue rigidity. Force-regulation by substrate rigidity of adhesion sites with fixed area is not possible for stationary adhesion sites, according to elasticity theory. A simple model is presented to describe force regulation by dynamical adhesion sites.     

Area and Geometry Dependence of Cell Migration in Asymmetric Two-State Micropatterns

Microstructured surfaces provide a unique framework to probe cell migration and cytoskeletal dynamics in a standardized manner. Here, we report on the steady-state occupancy probability of cells in asymmetric two-state microstructures that consist of two fibronectin-coated adhesion sites connected by a thin guidance cue. In these dumbbell-like structures, cells transition between the two sites in a repeated and stochastic manner, and average dwell times in the respective microenvironments are determined from the cell trajectories. We study the dynamics of human breast carcinoma cells (MDA-MB-231) in these microstructures as a function of area, shape, and orientation of the adhesion sites. On square adhesive sites with different areas, we find that the occupancy probability ratio is directly proportional to the ratio of corresponding adhesion site areas. These asymmetries are well captured by a simple model for the stochastic nonlinear dynamics of the cells, which reveals generic features of the motion. Sites of equal area but different shape lead to equal occupancy if shapes are isotropic (e.g., squared or circular). In contrast, an asymmetry in the occupancy is induced by anisotropic shapes like rhombi, triangles, or rectangles that enable motion in the direction perpendicular to the transition axis. Analysis of the two-dimensional motion of cells between two rectangles with orthogonal orientation suggests that cellular transition rates depend on the cell polarization induced by anisotropic micropatterns. Taken together, our results illustrate how two-state micropatterns provide a dynamic migration assay with distinct dwell times and relative cell occupancy as readouts, which may be useful to probe cell-microenvironment interactions.     

Mathematical model for the effects of adhesion and mechanics on cell migration speed.

Migration of mammalian blood and tissue cells over adhesive surfaces is apparently mediated by specific reversible reactions between cell membrane adhesion receptors and complementary ligands attached to the substratum. Although in a number of systems these receptors and ligand molecules have been isolated and identified, a theory capable of predicting the effects of their properties on cell migration behavior currently does not exist. We present a simple mathematical model for elucidating the dependence of cell speed on adhesion-receptor/ligand binding and cell mechanical properties. Our model can be applied to propose answers to questions such as: does an optimal adhesiveness exist for cell movement? How might changes in receptor and ligand density and/or affinity affect the rate of migration? Can cell rheological properties influence movement speed? This model incorporates cytoskeletal force generation, cell polarization, and dynamic adhesion as requirements for persistent cell movement. A critical feature is the proposed existence of an asymmetry in some cell adhesion-receptor property, correlated with cell polarity. We consider two major alternative mechanisms underlying this asymmetry: (a) a spatial distribution of adhesion-receptor number due to polarized endocytic trafficking and (b) a spatial variation in adhesion-receptor/ligand bond strength. Applying a viscoelastic-solid model for cell mechanics allows us to represent one-dimensional locomotion with a system of differential equations describing cell deformation and displacement along with adhesion-receptor dynamics. In this paper, we solve these equations under the simplifying assumption that receptor dynamics are at a quasi-steady state relative to cell locomotion. Thus, our results are strictly valid for sufficiently slow cell movement, as typically observed for tissue cells such as fibroblasts. Numerical examples relevant to experimental systems are provided. Our results predict how cell speed might vary with intracellular contractile force, cell rheology, receptor/ligand kinetics, and receptor/ligand number densities. A biphasic dependence is shown to be possible with respect to some of the system parameters, with position of the maxima essentially governed by a balance between transmitted contractile force and adhesiveness. We demonstrate that predictions for the two alternative asymmetry mechanisms can be distinguished and could be experimentally tested using cell populations possessing different adhesion-receptor numbers.     

van der Waals forces influencing adhesion of cells

Adhesion molecules, often thought to be acting by a ‘lock and key’ mechanism, have been thought to control the adhesion of cells. While there is no doubt that a coating of adhesion molecules such as fibronectin on a surface affects cell adhesion, this paper aims to show that such surface contamination is only one factor in the equation. Starting from the baseline idea that van der Waals force is a ubiquitous attraction between all molecules, and thereby must contribute to cell adhesion, it is clear that effects from geometry, elasticity and surface molecules must all add on to the basic cell attractive force. These effects of geometry, elasticity and surface molecules are analysed. The adhesion force measured between macroscopic polymer spheres was found to be strongest when the surfaces were absolutely smooth and clean, with no projecting protruberances. Values of the measured surface energy were then about 35 mJ m⁻², as expected for van der Waals attractions between the non-polar molecules. Surface projections such as abrasion roughness or dust reduced the molecular adhesion substantially. Water cut the measured surface energy to 3.4 mJ m⁻². Surface active molecules lowered the adhesion still further to less than 0.3 mJ m⁻². These observations do not support the lock and key concept.     

A mathematical analysis of focal adhesion lifetimes and their effect on cell motility

By analyzing the distributions of focal adhesion (FA) lifetimes from different cell types, we found that a gamma distribution best matched the experimental distributions. In all but one case, it was a unimodal, non-symmetric gamma distribution. We used a mathematical model of cell motion to help understand the mechanics and data behind the FA lifetime distributions. The model uses a detach-rate function to determine how long an FA will persist before it detaches. The detach-rate function that produced distributions with a best-fit gamma curve that closely matched that of the data was both force and time dependent. Using the data gathered from the matching simulations, we calculated both the cell speed and mean FA lifetime and compared them. Where available, we also compared this relationship to that of the experimental data and found that the simulation reasonably matches it in most cases. In both the simulations and experimental data, the cell speed and mean FA lifetime are related, with longer mean lifetimes being indicative of slower speeds. We suspect that one of the main predictors of cell speed for migrating cells is the distribution of the FA lifetimes.     

A stochastic model for adhesion-mediated cell random motility and haptotaxis

The active migration of blood and tissue cells is important in a number of physiological processes including inflammation, wound healing, embryogenesis, and tumor cell metastasis. These cells move by transmitting cytoplasmic force through membrane receptors which are bound specifically to adhesion ligands in the surrounding substratum. Recently, much research has focused on the influence of the composition of extracellular matrix and the distribution of its components on the speed and direction of cell migration. It is commonly believed that the magnitude of the adhesion influences cell speed and/or random turning behavior, whereas a gradient of adhesion may bias the net direction of the cell movement, a phenomenon known as haptotaxis. The mechanisms underlying these responses are presently not understood.A stochastic model is presented to provide a mechanistic understanding of how the magnitude and distribution of adhesion ligands in the substratum influence cell movement. The receptor-mediated cell migration is modeled as an interrelation of random processes on distinct time scales. Adhesion receptors undergo rapid binding and transport, resulting in a stochastic spatial distribution of bound receptors fluctuating about some mean distribution. This results in a fluctuating spatio-temporal pattern of forces on the cell, which in turn affects the speed and turning behavior on a longer time scale. The model equations are a system of nonlinear stochastic differential equations (SDE's) which govern the time evolution of the spatial distribution of bound and free receptors, and the orientation and position of the cell. These SDE's are integrated numerically to simulate the behavior of the model cell on both a uniform substratum, and on a gradient of adhesion ligand concentration.Furthermore, analysis of the governing SDE system and corresponding Fokker-Planck equation (FPE) yields analytical expressions for indices which characterize cell movement on multiple time scales in terms of cell cytomechanical, morphological, and receptor binding and transport parameters. For a uniform adhesion ligand concentration, this analysis provides expressions for traditional cell movement indices such as mean speed, directional persistence time, and random motility coefficient. In a small gradient of adhesion, a perturbation analysis of the FPE yields a constitutive cell flux expression which includes a drift term for haptotactic directional cell migration. The haptotactic drift contains terms identified as contributions from directional orientation bias (taxis).     

Simulation of cell rolling and adhesion on surfaces in shear flow. Microvilli-coated hard spheres with adhesive springs.

The adhesion of cells to ligand-coated surfaces in viscous shear flow is an important step in many physiological processes, such as the neutrophil-mediated inflammatory response, lymphocyte homing, and tumor cell metastasis. This article describes a calculational method that allows simulation of the interaction of a single cell with a ligand-coated surface. The cell is idealized as a microvilli-coated hard sphere covered with adhesive springs. The distribution of microvilli on the cell surface, the distribution of receptors on microvilli tips, and the forward and reverse reaction between receptor and ligand are all simulated using random number sampling of appropriate probability functions. The velocity of the cell at each time step in the simulation results from a balance of hydrodynamic, colloidal, and bonding forces; the bonding force is derived by summing the individual contributions of each receptor-ligand tether. The model can simulate the effect of many parameters on adhesion, such as the number of receptors on microvilli tips, the density of ligand, the rates of reaction between receptor and ligand, the stiffness of the springs, the response of springs to extension, and the magnitude of hydrodynamic stresses. By varying these parameters, the model can successfully recreate the entire range of expected and observed adhesive phenomena, from completely unencumbered motion, to rolling, to transient attachment, to firm adhesion. Also, the model can provide meaningful statistical measures of adhesion, including the mean and variance in velocity, rate constants for cell attachment and detachment, and the frequency of adhesion. We find a critical modulating parameter of adhesion is the fractional spring slippage, which relates the extension of a bond to its rate of breakage; the higher the slippage, the faster the breakage for the same extension. Changes in the fractional spring slippage can radically change the adhesive behavior of a cell. We show that stiffer springs will only serve to increase adhesion if the fractional slippage remains small. In addition, our simulations emphasize the importance of reaction rates between receptor and ligand, rather than affinity, as being the key determinant of adhesion under flow. These results suggest reaction rates and response to stress of adhesion molecules must be independently measured to understand how adhesion is controlled at the molecular level.     

Adhesive Dynamics Simulation of G-Protein-Mediated Chemokine-Activated Neutrophil Adhesion

To reach sites of inflammation, a blood-borne neutrophil first rolls over the vessel wall, becoming firmly adherent on activation, and then transmigrates through the endothelium. In this study, we simulate the transition to firm adhesion via chemokine-induced integrin activation. To recreate the transition from rolling to firm adhesion, we use an integrated signaling adhesive dynamics simulation that includes selectin, integrin, and chemokine interactions between the cell and an adhesive substrate. Integrin bonds are of low affinity until activated by chemokine binding to G-protein coupled receptors on the model cell. The signal propagates within the cell through probabilistic diffusion and reaction of the signaling elements to induce the high-affinity integrins required for firm adhesion. This model showed that integrins become progressively active as cells roll and interact with chemokines, leading to a slight slowing before firm adhesion on a timescale similar to that observed in experiments. Increasing the density of chemokine resulted in decreases in the rolling time before stopping, consistent with experimental observations. However, a limit is reached where further increases in chemokine density do not increase adhesion. We found that the timescale for integrin activation correlated with the time to stop. Further, altering parameters within the intracellular signaling cascade that changed the speed of integrin activation, such as effector activation and dissociation rates, correspondingly affected the time to firm adhesion. For all conditions tested, the number of active integrin bonds at the point of firm adhesion was relatively constant. The model predicts that the time to stop would be relatively independent of selectin or integrin density, but strongly dependent on the shear rate because higher shear rates limit the intrinsic activation rate of integrins and require more integrins for adhesion.     

Simulation of cell rolling and adhesion on surfaces in shear flow

The adhesion of cells to ligand-coated surfaces in viscous shear flow is an important step in many physiological processes, such as the neutrophil-mediated inflammatory response, lymphocyte homing, and tumor cell metastasis. This article describes a calculational method that allows simulation of the interaction of a single cell with a ligandcoated surface. The cell is idealized as a microvilli-coated hard sphere covered with adhesive springs. The distribution of microvilli on the cell surface, the distribution of receptors on microvilli tips, and the forward and reverse reaction between receptor and ligand are all simulated using random number sampling of appropriate probability functions. The velocity of the cell at each time step in the simulation results from a balance of hydrodynamic, colloidal, and bonding forces; the bonding force is derived by summing the individual contributions of each receptor-ligand tether. The model can simulate the effect of many parameters on adhesion, such as the number of receptors on microvilli tips, the density of ligand, the rates of reaction between receptor and ligand, the stiffness of the springs, the response of springs to extension, and the magnitude of hydrodynamic stresses. By varying these parameters, the model can successfully recreate the entire range of expected and observed adhesive phenomena, from completely unencumbered motion, to rolling, to transient attachment, to firm adhesion. Also, the model can provide meaningful statistical measures of adhesion, including the mean and variance in velocity, rate constants for ceil attachment and detachment, and the frequency of adhesion. We find a critical modulating parameter of adhesion is the fractional spring slippage, which relates the extension of a bond to its rate of breakage; the higher the slippage, the faster the breakage for the same extension. Changes in the fractional spring slippage can radically change the adhesive behavior of a cell. We show that stiffer springs will only serve to increase adhesion if the fractional slippage remains small. In addition, our simulations emphasize the importance of reaction rates between receptor and ligand, rather than affinity, as being the key determinant of adhesion under flow. These results suggest reaction rates and response to stress of adhesion molecules must be independently measured to understand how adhesion is controlled at the molecular level.     

The forward rate of binding of surface-tethered reactants: effect of relative motion between two surfaces

The reaction of molecules confined to two dimensions is of interest in cell adhesion, specifically for the reaction between cell surface receptors and substrate-bound ligand. We have developed a model to describe the overall rate of reaction of species that are bound to surfaces under relative motion, such that the Peclet number is order one or greater. The encounter rate between reactive species is calculated from solution of the two-dimensional convection-diffusion equation. The probability that each encounter will lead to binding depends on the intrinsic rate of reaction and the encounter duration. The encounter duration is obtained from the theory of first passage times. We find that the binding rate increases with relative velocity between the two surfaces, then reaches a plateau. This plateau indicates that the increase in the encounter rate is counterbalanced by the decrease in the encounter duration as the relative velocity increases. The binding rate is fully described by two dimensionless parameters, the Peclet number and the Damk?hler number. We use this model to explain data from the cell adhesion literature by incorporating these rate laws into "adhesive dynamics" simulations to model the binding of a cell to a surface under flow. Leukocytes are known to display a "shear threshold effect" when binding selectin-coated surfaces under shear flow, defined as an increase in bind rate with shear; this effect, as calculated here, is due to an increase in collisions between receptor and ligand with increasing shear. The model can be used to explain other published data on the effect of wall shear rate on the binding of cells to surfaces, specifically the mild decrease in binding within a fixed area with increasing shear rate.