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* Departments of Chemical and Biomolecular Engineering and Bioengineering, University of Pennsylvania, Philadelphia, Pennsylvania
Correspondence: Address reprint requests to Daniel A. Hammer, 120 Hayden Hall, 3320 Smith Walk, Dept. of Bioengineering, University of Pennsylvania, Philadelphia, PA 19104. Tel.: 215-573-6761; E-mail: hammer{at}seas.upenn.edu.
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ABSTRACT |
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INTRODUCTION |
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,2) to cancer metastasis (3). In these processes and others, cells are first captured from the bloodstream to the blood vessel wall by molecular interactions. This initial tethering is followed by rolling on the endothelial cells of the vessel and then firm adhesion, allowing cells to migrate to their intended location in the tissue (1,4,5). In neutrophils, the rolling is mediated largely by selectins, and firm adhesion is mediated by integrins. We are interested in the mechanisms leukocytes use to tether and roll via selectins on the vascular wall.
To identify the important molecular parameters of cell rolling, our lab has developed a computational tool called Adhesive Dynamics (6). In Adhesive Dynamics, bonds stochastically form and break according to probabilities determined by the forward and reverse reaction rates of the bonding molecules. A variety of rate laws are available to characterize the coupling of force and dissociation. A force balance is required to determine the net motion of the particle and molecular positions are updated from the particle kinematics. Thus, Adhesive Dynamics is an ideal method to couple mechanical and chemical effects in adhesion.
In the original version of Adhesive Dynamics, the cell was idealized as a hard sphere with rigid microvilli (6). With the advent of cell-free rolling, in which rolling can be recreated with molecules attached to hard spheres, many of the recent versions of Adhesive Dynamics have simulated rolling with molecularly-coated hard spheres (7–12). For example, Adhesive Dynamics has successfully been used to map out a state diagram for a hard sphere model of the cell with adhesion molecules randomly distributed about its surface, linking different dynamics of adhesion to mechanochemical properties of the adhesion molecules. It was determined that experimentally measured values of the mechanochemical properties of selectins self-consistently fell within the rolling regime of the state diagram (9). Furthermore, Adhesive Dynamics with rigid spheres could simulate the transition from rolling to firm adhesion by including both selectins and integrins, which mediate firm adhesion, on the model cell surface (7).
Although these versions of Adhesive Dynamics have been adequate to recreate cell-free rolling, it has been a limitation that the deformation of microvilli has been omitted, specifically for simulating leukocyte motion. Microvilli have several important effects on adhesion. First, they promote molecular presentation because they are long and protrude from the cell surface. Immunogold-labeling has shown that L-selectin and P-selectin glycoprotein ligand-1 (PSGL-1), which mediate the initial tethering and rolling in an adhesion event, are predominantly clustered on the microvilli tips to facilitate adhesive contact with the surface (13,14).
Secondly, the microvilli are known to stretch and tether under force, as observed (15,16) and measured by Shao and co-workers. They studied the deformation of microvilli under force by micropipette aspiration techniques and developed equations to describe the rheology of microvilli under a given force. They found that a microvillus can exist in two distinct regimes. First, under small pulling forces, microvilli stretch as Hookean springs as the axial actin within the protrusions stretches. Beyond a threshold level of force, the cell membrane pulls away from the cytoskeleton allowing the microvilli to form long cylindrical membrane tethers with viscous properties. The deformation of the microvilli can be quite substantial, easily on the order of cell diameters, as differential interference contrast videomicroscopy has shown in tethering experiments of neutrophils on P-selectin-coated glass (17). A long lever arm made possible by a stretched microvillus allows a greater component of the bond force to resist shear flow (see Fig. 2). This means a smaller total bond force is required to maintain adhesion, and that bonds can last longer. Thus, microvillus deformation may play a considerable role in controlling the dynamics of adhesion, as anticipated, but not fully investigated, by Shao and co-workers (15) using static mechanical energy balances.
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,18–23). By exploring the effect of microvillus deformability on adhesion, we remarkably find that microvillus rheology is ideally suited to leukocyte rolling dynamics, being adequately, but not overly, compliant to support neutrophil rolling. |
MODEL |
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,24). The novel addition is to allow the microvilli to deform. As illustrated in Fig. 1, the cell is represented by a hard sphere with deformable cylindrical protrusions representing the microvilli. The microvilli are randomly distributed and their density can be varied. Although the microvillus lengths are heterogeneous on real cells (13), this model assumes a homogeneous population of microvilli with equivalent equilibrium lengths of 0.3 µm (15).
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). Measurements of the density of PSGL-1 molecules or P-selectin binding sites on neutrophils range from 48 PSGL-1/µm2 to 124 P-selectin binding sites/µm2 (25–28). Several authors have noted that the effective density of adhesion molecules is smaller than the total, owing to crowding or orientational effects (29,30). Thus, the number we use is the effective total number per microvillus tip.
Properties of bonds
We model the bonds between adhesion molecules on the cell and substrate as Hookean springs, so the chemical force on the cell due to each bond, f, is calculated by
 | (1) |
is the bond spring constant, z is the stressed length of the bond, and 
is the equilibrium length of the bond. In this model, the unstressed bond length is 0.1 µm, equal to the sum of the extracellular lengths of P-selectin and PSGL-1, two adhesion molecules that mediate rolling (31).
The dissociation rate, kr, is force-dependent according to the following model from Bell (32),
 | (2) |
is the unstressed dissociation rate constant, 
is the reactive compliance, and kBT is the thermal energy. The reactive compliance is a parameter with units of length that describes the degree to which force facilitates bond breakage—a smaller value of reactive compliance means that the bond is less susceptible to breakage by force. To satisfy the Boltzmann distribution at equilibrium (33), the forward rate, kf, must depend on separation distance according to
 | (3) |
is the unstressed association rate, i.e., the forward reaction rate in the case that the new bond would be of its equilibrium length (z = ).
Cell motion
To simulate cell rolling, motion was discretized and integrated forward in time with the Euler method using a time-step of 10–7 s. This time-step allowed us to capture the fast motions of the springlike bonds and microvilli, but a smaller time-step did not significantly alter the results. However, in checking for bond formation and breakage, the time-step required for accuracy in the results was only 10–6 s. The use of two different time-steps maintained accuracy in the program while also permitting shorter run times. At each time-step for cell motion, the following steps were performed:
External forces on the cell include: 1), gravity and buoyancy; 2), a generic repulsive force; and 3), chemical forces and torques. Because the cell is slightly denser than the fluid in the simulation, gravity promotes close apposition of the cell to the surface to allow bonding. The generic repulsive force was derived from a phenomenological potential that represents the effects of electrostatic repulsion, solvent exclusion, and polymer compression when the cell and surface approach each other (34). It takes the form
 | (4) |
is the length scale of the repulsive force on the order of the glycocalyx thickness. Because the length scale is small compared to the separation distance between the plane surface and cell body, the protruding microvilli in the contact area are the predominant cell structures responsible for the repulsion that prevents the cell from going through the planar surface in the simulation. Finally, the chemical forces due to all bonds are calculated from Eq. 1.
In Step II of the simulation, bonds are checked for formation and breakage according to their respective probabilities, Pf and Pr, given by
 | (5) |
 | (6) |
t is the time-step for checking for bond formation and breakage used in the simulation (6,24). When checking for bond formation, only molecules of a reasonable proximity to the reactive surface are considered. The contact area, shown in the closeup in Fig. 1, is defined as the area of the cell closest to the substrate where molecules have a practical probability to bind. For molecules outside of the contact area, the probability of bond formation is vanishingly small, so it is assumed to be zero and not checked. At each time-step, unbound molecules on all microvilli in the contact area are tested for bond formation. Any free molecules on bound microvilli, which may be outside of the contact area, are checked as well. Also, all bound molecules are tested for bond breakage. The positions of only the bound microvilli are stored from step to step, saving memory and computational time. The positions of unbound microvilli in the contact area are randomly assigned to check for bond formation at each time-step.
A Monte Carlo technique is used to determine the formation and breakage of each molecule to be tested. A random number is chosen between 0 and 1, and this number is compared to the probabilities given in Eqs. 5 and 6. If the random number is less than the probability, the event will happen.
As has been shown by Chang and Hammer (35), the unstressed forward rate is a function of the relative motion of the cell with respect to the substrate and the density of ligand on the substrate surface. Since the relative motion of the cell is significant only when the cell is unbound, this amounts to two distinct on-rates—i.e., the on-rate in the convective limit, 
when the cell is unbound; and the on-rate in the diffusive limit, 
when the cell is bound by at least one bond. The values of these two unstressed association rates are estimated from the following expressions, which were adapted from the derivations of Chang and Hammer (35):
 | (7) |
 | (8) |
ligand is the surface density of ligand on the substrate. Also, b is half of the mean distance between ligands on the substrate and Pe = |
|a/D is the Peclet number based on the relative velocity, ||, between the cell and substrate surfaces.
After all the appropriate molecules are checked for bond formation or breakage at a given time-step, the position of the cell is updated for the motion that occurred during that time-step in Step III of the simulation. In this model, Stokes flow is assumed since the Reynolds number for the flow and particle is small. Furthermore, since the microvilli are small compared to the sphere radius and sparse compared to their diameter, we have ignored their effect on the hydrodynamic force (36). Therefore, Goldman and co-workers' (37) solution to the problem of a sphere translating and rotating near a planar wall in shear flow is used to predict the translational and rotational velocities of the cell under the external forces calculated in Step I of the simulation method by constructing a mobility matrix for the motion of the particle. This matrix has been explicitly stated in previous Adhesive Dynamics articles (see, for example, (6) and (24)). It is assumed that these velocities hold for the duration of the time-step, so the position of the cell and all of the bound microvilli on its surface are appropriately updated.
Finally, in Step IV, knowing the positions of the microvilli on the cell surface (updated in the previous step) and the bond on the planar surface (fixed in time), the lengths of the bound microvilli are updated. It is assumed that 1), the microvilli are free to bend at their attachment to the cell surface so a microvillus and its bonds are always in a straight line (38); and 2), there is force balance between the bonds and microvillus. As mentioned in the Introduction, the micropipette experiments revealed a biphasic behavior for the microvilli (15)—an extension regime with an elastic response to force at small forces and a tether regime with a plastic response at larger forces, as follows:
 | (9) |
 | (10) |
mv is the microvillus spring constant and 
is the equilibrium length of the homogeneous microvilli. The value Ftrans is the force at which a microvillus transitions from the extension regime to the tether regime. Experiments have shown that microvilli transition over a range of forces, depending on the degree of association between the membrane and cytoskeleton. However, in this model, one value in the middle of the range is assumed. In Eq. 10, F0 is the yield force related to the degree of association of the cytoskeleton and membrane. The parameter kteth is an effective surface viscosity of the membrane and dL/dt is the rate of elongation of the microvillus tether. Thus, when the cell is moving slowly and the microvillus is growing very slowly in the tether regime, the force transmitted to the bond will be small, approaching F0, allowing bonds to last longer. When a microvillus is in the extension regime, the position of the microvillus tip at the next time-step is determined by balancing the forces of springs in series. When a microvillus is in the tether regime, however, the Euler method is used with Eq. 10 to determine the new microvillus length that will balance the bond force on that microvillus. With the end of Step IV, the time is updated and the process is repeated until the end of the simulation.
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RESULTS |
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12, never dropping below one bond, so this cell is always bound to the surface. The velocity, therefore, is consistently below the hydrodynamic free-stream velocity, Vh. However, spikes in velocity occur as bonds at the back-edge of the cell, which provide much of the resistance to the shear forces, break.
State diagram
Hammer and co-workers showed that the dynamic state of adhesion is controlled by the Bell model parameters of dissociation rate and reactive compliance (9). Using the parameters listed in Table 1, we simulated a number of cells while varying the Bell model parameters at a fixed microvillus compliance. Each run started with the cell close enough to the plane to initiate bonding and continued for 20 s. At each set of parameters, 5–15 cells were simulated—more at lower velocities, where rolling velocity fluctuated relatively more—to determine an average rolling velocity. The behavior of the cell was observed and four states were defined as follows:
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bonds last longer, allowing the cell to be more intimately bound to the surface, hence the firmly bound region at lower values of the dissociation rate.
However, at large values of reactive compliance, it is difficult to get rolling for any value of 
Instead, simulations at these large values show either a firmly bound cell, a fast moving cell, or a landing cell—a cell that suddenly switches from traveling near Vh to being firmly bound. This behavior occurs because the large reactive compliance gives rise to bonds that are very susceptible to breakage by force. Therefore, if a bond forms, it usually breaks before a second bond can form to sustain rolling. However, if the dissociation rate is low enough, it is possible for a second bond to form before the first one breaks. Once a second bond forms, the force per bond is reduced so that the bonds do not break. But, since the dissociation rate has to be low for the second bond to form, even more bonds form and the cell becomes firmly adherent. At higher this "landing" never occurs.
Finally, at a small range of intermediate values of the reactive compliance, rolling behavior is tunable by varying either the reactive compliance or the dissociation rate. It is in this range where most experimental values of the Bell model parameters for adhesion molecule pairs that mediate rolling lie (9). However, this intermediate area of the state diagram is not as prominent in this study as it was for the nondeformable model of Chang and co-workers (9). In contrast to what was previously observed for hard sphere rolling, the state diagram of Fig. 4 is dominated by a region where rolling behavior is determined primarily by dissociation rate.
Fig. 5 illustrates how the rolling velocity varies with the unstressed dissociation rate. Each line represents a different value of the reactive compliance. The larger reactive compliances (to the left in Fig. 5) show a sharp transition from firmly adherent to no-adhesion. As reactive compliance decreases, the curves show a more moderate transition from low to high velocity as dissociation rate increases. For small reactive compliances, the curves collapse onto each other, which is seen in the state diagram as the region where alone determines rolling behavior. At = 0 Å, bonds are considered ideal, i.e., force does not influence the rate of bond breakage. As reactive compliance becomes very small and close to zero, further decreasing of does not significantly change a bond's response to applied force.
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and = 0.3 Å). These Bell model parameters fall within the rolling regime on the state diagram. Also, they are similar to values for the P-selectin-PSGL-1 bond, which is known to support rolling, as determined by tethering experiments (16,39–42).
In the following section, the average velocity of an individual cell is calculated as the total distance traveled during the last half of the simulation divided by that length of time to ensure that a steady-state velocity is calculated. The average rolling velocity at a given set of parameters is determined by averaging the velocities of a number (5–15) of cells. The average velocity as a function of shear rate is compared for different parameters.
Simulations of cell detachment have shown that the membrane peeling velocity decreases if receptors on the membrane are clustered (43). To explore the effect of receptor clustering in our model, the total number of selectin molecules on the cell surface was kept constant whereas the density of microvilli was adjusted. Since the receptors are localized to the microvillus tips, this changed the number of molecules per microvillus, or the clustering, which had a significant effect on rolling velocity. Fig. 6 A shows that cells with more highly clustered molecules tend to roll more slowly and are more likely to become firmly bound. When there are more molecules on a microvillus tip, more will bind on average and rebinding can occur (44). It is difficult for a given microvillus with many bonds to detach since all of the bonds must be broken at once. So, even though the more highly clustered case has a smaller surface density of microvilli, the much higher propensity for a given microvillus to stay attached by at least one bond leads to slower rolling.
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Since recent simulations have used a spring constant of = 105 pN/µm (7,10,12), we used that value for the base case simulations (see Table 1). However, values of a general bond spring constant estimated by various groups span orders of magnitude from 10 to 106 pN/µm (34,45). Atomic force microscopy has yielded more precise values of a bond spring constant on the order of 104 pN/µm for the P-selectin/PSGL-1 bond (46) and BSA/Ab-BSA bond (47). Accordingly, we explored the effect of bond spring constant in the range of 104–105 pN/µm on rolling velocity. Fig. 6 C shows that smaller values of the spring constant, i.e., more compliant bonds, lead to slower rolling. At 
= 700 s–1, the highest shear rate tested (data not shown), the largest bond strain was 7% for = 105 pN/µm and 71% for = 104 pN/µm. A bond strain of 71% is reasonable, considering that the extended conformation of a residue in a selectin ligand molecule is 90% longer than the relaxed conformation (48). Not only are the more compliant bonds more easily stretched, which creates a longer lever arm to resist flow and allows new bonds to form in the meantime, but the association and dissociation rates are affected through their force-dependence.
Additionally, we looked at details of the rolling behavior of cells with different microvillus densities and the same number of selectin molecules on each microvillus tip, as in Fig. 6 B. At a shear rate of 180 s–1, we compared the variance in rolling velocity and pause times of cells with microvillus densities of 4, 6, and 8 µm–2. We found, as expected, that the variance in rolling velocity decreases as the number of microvilli increases. This trend is shown in Fig. 7 A, where the variance averaged over all cells with a given microvillus density is plotted as a function of the average number of microvilli bound at that density. The average velocity of the cells also decreases as the number of microvilli increases (Fig. 7 A, inset). Note that the prescribed microvillus density represented by each point increases as the number of bound microvilli increases.
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33 frames/s. It is interesting that the average duration of pauses does not significantly change with microvillus density, suggesting that it is a property of the microvilli themselves determining the pause time. However, the number of pauses per cell increases as the microvillus density increases, which leads to a larger fraction of time spent paused and therefore a smaller rolling velocity. Pause-time distributions for each microvillus density are given in Fig. 7 C. These histograms reveal that the distributions of pause times are similar, with a small number of long pauses lasting nearly 6 s, yet there are fewer total pauses at lower microvillus densities.
Microvillus parameter exploration
For the analysis of microvillus rheology, the parameters describing the deformation of the microvilli were varied. The base case model parameter values used in this study are listed in Table 1, and were estimated from micropipette measurements of microvillus stretching performed by Shao and colleagues (15,49).
Surface viscosity of the membrane
From Eq. 10, kteth is the effective surface viscosity of the membrane tether that occurs because of slip between the membrane and cytoskeleton, slip between membrane bilayers, and membrane flow (50,51). We varied kteth over three orders of magnitude to study the role of tether viscosity on rolling velocity. Fig. 8 illustrates that kteth has a large effect on the magnitude of rolling velocity as a function of shear rate. At low shear rates (Fig. 8 A), the velocity of cells with less viscous tethers rises more sharply with shear rate than that of cells with more viscous tethers. However, at high shear rates, the opposite is true. Shown in Fig. 8 B, less viscous tethers lead to a more moderate rise in velocity with shear rate than more viscous tethers.
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10 pN s/µm, very close to the physiological value of kteth of 11 pN s/µm measured by Shao and co-workers (15).
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= 180 s–1, Fig. 10 gives the cell displacement, instantaneous velocity, and number of bonds as functions of time for kteth = 0.1 and 100 pN s/µm. In each case, the number of bonds is similar (Fig. 10, C and F), but the cell motion differs. The trajectory of the cell with low-viscosity tethers in Fig. 10 A (average velocity 25 µm/s) has periods of fast, yet still hindered, motion, seen as segments of large slope. These fast motions occur as the microvilli stretch rapidly. The trajectory also has periods of little motion when the lever arms reach sufficient lengths to balance the force of flow. The instantaneous velocity in Fig. 10 B correspondingly has long periods of high velocity with some pauses in between. The trajectory of the cell with high-viscosity tethers in Fig. 10 D (average velocity 6 µm/s) is very different. There are brief periods of fast motion as microvilli at the back edge of the cell break, but generally the motion is slow since the viscous microvilli stretch slowly, as demonstrated by the instantaneous velocity in Fig. 10 E. Thus, the fluctuation in rolling velocity can be affected by the microvillus dynamics.
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,51,52). Accordingly, we examined effects of varying both F0 and Ftrans on rolling velocity as a function of shear rate. At high shear rates, it is clear from Fig. 11 A that a higher degree of association of the membrane and cytoskeleton, which makes it more difficult to pull a tether, leads to slower rolling. This may contradict intuition that tethering allows cells to roll more slowly. Instead, regardless of the parameter values, the microvilli still stretch as springs in the extension regime so the benefits of a long lever arm are still apparent. However, when the microvillus force reaches Ftrans, it suddenly tethers and, since the microvillus length is changing at that time, the force jumps up by the term kteth x dL/dt in Eq. 10. When Ftrans is small, tethering happens earlier and the jump in force is felt earlier, causing bonds to break faster, which results in faster rolling.
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mv, is important when the microvillus is in the extension regime (under smaller forces). Fig. 11 B makes it apparent that stiffer microvilli lead to faster rolling. This is an expected outcome since stiffer microvilli do not stretch as much, resulting in shorter lever arms and bonds that cannot resist the force of flow as well.
Although it is clear that increasing the microvillus spring constant increases rolling velocity, it is interesting that a two-order-of-magnitude increase in the spring constant results in only a severalfold increase in rolling velocity. Despite the fact that microvilli with a larger mv cannot stretch as much in the extension regime, they can still tether. Microvilli at the back edge of the cell, where most resistance to flow is localized, moderate the effect of stiffer microvilli as they are more likely to be in the tether regime.
Microvillus deformability
As discussed in the last sections, stiffer microvilli lead to somewhat faster rolling. Also, increasing the degree of association between the membrane and cytoskeleton was found to actually decrease the rolling velocity. However, it has been hypothesized that deformation of microvilli greatly stabilizes rolling. To study the effects of microvillus deformation in general, three parameters were adjusted. To reduce the deformability of the microvilli, the microvillus spring constant was increased to make the microvilli stiffer in the extension regime. In addition, these stiffer microvilli were assigned larger values of the parameters F0 and Ftrans to ensure that stiffer microvilli also had a reduced ability to tether, thereby rendering them less deformable overall. The base case uses mv = 43 pN/µm, F0 = 45 pN, and Ftrans = 45 pN. For simplicity, the values of all three parameters were set to the same numerical value (with different units). The simulation results in Fig. 11 C for different sets of the three parameters show that microvillus deformation is in fact important to a cell's ability to roll slowly in shear flow. Clearly, more compliant microvilli lead to slower rolling, as has been postulated (15).
Comparison to experiment
Finally, we were interested in the ability of the new model with microvillus deformability to recreate experimental data of in vitro cell rolling. In Fig. 12, rolling velocity versus shear rate for the model using the base case parameters is plotted against published rolling velocity data from various labs (1,27,39,42,53–55). The comparison demonstrates that the model matches experimental data at low shear rates, < 200 s–1, both showing a nearly linear increase in velocity with shear rate. Since the model does not include deformability of the whole cell body, it is not expected to be accurate at high shear rates where the cell deforms enough to affect rolling velocity (18,19,22,55).
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) reported the position of a neutrophil as a function of time as it rolled on a surface of 10 sites/µm2 P-selectin at a shear rate of 150 s–1. The trajectory is characterized by periods of motion intermixed with pauses, which appear as plateaus in the distance versus time. Chen and Springer (48) show a similar trajectory for a neutrophil rolling on 200 sites/µm2 E-selectin at a shear rate of 100 s–1. A closeup of the distance-versus-time trajectory of a representative rolling cell from Adhesive Dynamics is shown in Fig. 13 along with the experimental trajectories of Schmidtke and Diamond. Trajectories from both sources show repeated plateaus characteristic of rolling with microvillus stretching. Initially, when a microvillus is short, the lever arm is small and the force of flow is not resisted well, so the cell moves relatively quickly. As the microvillus stretches, its angle allows a larger resistance to the fluid force and the cell moves more slowly, and the displacement is independent of time. As bonds break, this process is repeated. Thus, the inclusion of deformable microvilli in Adhesive Dynamics leads to the characteristic pause dynamics of neutrophil rolling seen in experiments.
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DISCUSSION |
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). We created a state diagram for adhesion in Bell parameter space and found four adhesion states—firmly bound, landing, rolling, and no-adhesion. Landing was observed only for large reactive compliance.
We also explored properties of the model by observing rolling velocity versus shear rate as we varied parameters. Beginning with some basic model properties, we simulated the effects of receptor clustering, the total number of receptors on the cell, and the bond spring constant, and found anticipated results. We were surprised by two effects of the tether viscosity on adhesion. First, rolling velocity showed a minimum at an intermediate value of the tether viscosity (see Fig. 9) of 10 pN s/µm, very close to the actual reported value of this parameter (11 pN s/µm; see (15)). This result suggests neutrophil rheology has evolved to the most efficient mechanics to sustain rolling. Second, we found that kteth can affect the fluctuations in the rolling velocity. Since fluctuations may be a method for the cell to control ligation times and signaling dynamics, the control of fluctuations by cell mechanics suggests a more complex mechanochemical control of cell response.
Finally, we compared our simulation results to published experiments and found that the rolling velocities agreed for shear rates <200 s–1. Furthermore, simulation trajectories had a shape characteristic of microvillus stretching, as seen in high-resolution experiments (17,48), where plateaus in motion suggest regions in which the microvilli are being stretched.
Our results indicate that the deformation of microvilli significantly influences the rolling behavior of leukocytes, an observation also seen in experiments (16). First, a comparison of the state diagram generated in this article to that for an Adhesive Dynamics model without microvilli (9) suggests that deformable microvilli decrease the impact of the reactive compliance on adhesion at small values of the reactive compliance. We reason that this is due to the decreased force required from the bonds when the microvilli can stretch and tether. Since reactive compliance is a measure of the sensitivity of bonds to applied force, it has a smaller influence on rolling when the force felt by the bonds is smaller. In addition, the previous Adhesive Dynamics model by Chang and co-workers (9) required a greater density (80 mol/µm2 versus 16 mol/µm2 in this study) of bonding molecules on the cell surface and a larger unstressed on-rate (84 s–1 versus 3 s–1 in the diffusion limit in this study) to achieve rolling behavior. As discussed, Fig. 11 C shows that as microvilli become less deformable, the cell's ability to roll diminishes. Therefore, microvillus deformability contributes to the stability of leukocyte rolling.
Results in this article indicate that a cell could tune its rolling behavior by modulating its mechanics. For example, as seen in Fig. 9, the minimum in rolling velocity at a given shear rate occurs at a smaller value of kteth as the shear rate increases. This suggests that a cell could employ a shear-thinning effect to help stabilize its rolling velocity over a range of shear rates. In fact, it has been reported that tethers pulled from passive neutrophils at a force >100 pN, and therefore larger velocity, are nearly threefold less viscous than tethers pulled at a smaller force and smaller velocity (50). Furthermore, Drury and Dembo (56) reported that a model of micropipette aspiration of neutrophils required shear-thinning to reproduce experimental data over a range of conditions. Another adjustable feature that may ideally modify cell rolling is the adhesion energy between the membrane and cytoskeleton. Once a neutrophil is activated, the membrane becomes more tightly bound to the cytoskeleton due to an increased number of integral membrane proteins, thereby increasing F0 and Ftrans (57). In support of this idea, tether rheology measurements performed on activated neutrophils show that the yield force, F0, is nearly twice that of a passive neutrophil (50,51). The increase in adhesion energy is a possible mechanism to slow down a cell once it becomes activated to promote firm adhesion. More subtly, through controlling microvillus mechanics, the cell may be able to control the dynamics of receptor ligation and the resulting signaling from receptors, which may influence the probability that a cell would stop at a site of inflammation.
In the Adhesive Dynamics model outlined in this article, we used the Bell model to represent the effect of force on the rate of bond breakage. It has been determined that the Bell model holds for the P-selectin/PSGL-1 bond at high rates of force loading (>100 pN/s) on the bond (58), and others have reported that physiological loading rates on adhesion molecules are 100–10,000 pN/s (59). It may apparently be concluded that the Bell model is acceptable for an analysis of leukocyte rolling (9,60). However, the tethering experiments often used to determine the Bell model parameters require a number of approximations. Considering the deformation of the microvilli, it is likely that there are errors in estimating the force on a bond and the lever arm. In addition, the viscous deformation of the microvilli, which gives a diminishing force on the bonds as the stretching rate slows, may result in a small loading rate on the bonds for which the Bell model is not appropriate. Instead, the P-selectin/PSGL-1 bond could have biphasic breakage properties. Marshall and co-workers found catch-bond behavior at low forces and slip-bond behavior at higher forces (61). Their plots of bond lifetime versus force suggest that Bell parameters fit to high forces only would predict bond lifetimes significantly longer than those measured experimentally. In addition, Evans and co-workers have reported that there may be two dissociation pathways for the P-selectin/PSGL-1 bond, and one of the pathways appears to allow bonds to break faster at low forces (62). We hypothesize that incorporating a biphasic model for bond breakage by force into Adhesive Dynamics, and including microvillus deformation, could possibly explain the dynamics of tethering seen in pause-time experiments. Conversely, the inability to account for microvillus deformation when using tethering measurements to determine Bell model parameters for individual molecules may lead to spurious values for the molecular constants themselves. The model presented above should be used to interpret tethering results and determine accurate parameters for adhesion molecules.
Though the model in this article is useful to predict the effects of microvillus deformability on leukocyte rolling, there are a number of features still missing from the model that may improve the overall accuracy of simulations. First of all, some details of the microvilli were neglected in this model. Microvillus lengths are heterogeneous (13), so it could be predominantly the longest microvilli that interact and bind with the substrate (36). Furthermore, tether length may be limited by the excess membrane on the cell available for elongation. Once the membrane reservoir is exhausted, there is an abrupt exponential increase in force (52), which could cause bonds to break, thereby restricting tether lengths to reasonable values. Features to consider in future models include the possibility that receptors may be extracted from the membrane (32). Although cells would retain enough adhesion molecules to sustain rolling (1), it is possible that some tethers may detach by receptor extraction since the cytoskeleton is not present to anchor the molecules (49).
Finally, larger-scale considerations would help make a more complete model of cell rolling. In addition to the deformability of microvilli, the entire body of the cell is deformable under the force of flow, which allows the cell to take on a more hydrodynamic form when subjected to high shear rates. This deformation also increases the contact area, so more adhesion molecules can bind to the substrate and slow the cell down as the shear rate increases. The incorporation of cell deformability into Adhesive Dynamics could elucidate the role of cellular features in the plateau of rolling velocity with shear rate. Several other groups have begun to address these macroscopic effects, largely omitting microvilli and using simplified versions of the cell's macrorheology (19–21,23). The final model of leukocyte rolling would be one that combines all of these features.
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Submitted on October 6, 2004; accepted for publication April 27, 2005.
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The fewer receptors a cell has, the more quickly it rolls and the lower the shear rate at which the cell flies away from the substrate surface. (C) Different values of the bond spring constant, 
