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* Department of Chemical and Biomolecular Engineering, The Johns Hopkins University, Baltimore, Maryland 21218; and Department of Mechanical Engineering, University of Maryland Baltimore County, Baltimore, Maryland 21250
Correspondence: Address reprint requests to Konstantinos Konstantopoulos, PhD, Dept. of Chemical and Biomolecular Engineering, The Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218. Tel.: 410-516-6290; Fax: 410-516-5510; E-mail: kkonsta1{at}jhu.edu.
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INTRODUCTION |
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). According to this model, free-flowing leukocytes first tether and roll on the layer of activated endothelial cells, then stop, flatten, and squeeze between endothelial cells into the underlying tissues. The initial tethering and rolling of leukocytes on activated endothelium are mediated primarily by the selectins, E-, P- and L-selectin. However, due to the distinct kinetics of expression of the selectins and/or their respective ligands, different selectins participate at different time points during the inflammatory process. Several in vivo studies suggest that P-selectin binding to P-selectin-glycoprotein-ligand-1 (PSGL-1) mediates the very earliest leukocyte rolling during an inflammatory process (Dore et al., 1993; Mayadas et al., 1993), consistent with its rapid, inducible expression (within seconds to minutes) to the plasma membrane of stimulated endothelium.
The rolling phenomenon has been reconstituted in cell-free systems, where PSGL-1-bearing microspheres roll on P-selectin-coated substrates under dynamic flow conditions (Park et al., 2002; Rodgers et al., 2000; Yago et al., 2002). The ability of selectin-ligand pairs to mediate rolling interactions in shear flow is attributed to their fast association (kon) and dissociation (koff) rates as well as their capacity to form high-strength tethers under rapid loading (Alon et al., 1995; Hanley et al., 2003, 2004). Although the dynamics of adhesion is primarily controlled by the physical chemistry of adhesion molecules (Chang et al., 2000; Hammer and Apte, 1992), cellular features such as cell deformability and morphology can affect the rolling behavior. Indeed, recent experimental studies have indicated that polymorphonuclear (PMN) leukocyte rolling is significantly slower and relatively smoother than that of microspheres coated with a matched density of PSGL-1 molecules (Park et al., 2002; Yago et al., 2002). Along these lines, formaldehyde-fixed PMNs display reduced cell deformability and faster rolling on P-selectin-coated surfaces in shear flow compared to untreated PMNs (Park et al., 2002). In contrast, increased PMN deformability, conferred upon treatment with actin cytoskeleton disrupting cytochalasins, results in lower rolling velocities (McCarty et al., 2003).
Theoretical predictions of the kinetics of receptor-ligand interactions under external force have been carried out using primarily the Bell (1978) and Hookean spring models (Dembo, 1994). These models have been adopted to estimate the unstressed dissociation rate constant and susceptibility of bond rupture under force in perfusion assays (Alon et al., 1995; Schmidtke and Diamond, 2000), and force-spectroscopy experiments (Evans et al., 2001; Hanley et al., 2003, 2004; Tees et al., 2001). Moreover, these kinetic models have been used in computational studies pioneered by Hammer and co-workers (Chang and Hammer, 1996, 2000; Chang et al., 2000; Hammer and Apte, 1992; King and Hammer, 2001) in which cell rolling on selectin-coated substrates was simulated with the adhesive dynamics algorithm. In this algorithm, the cell is idealized as a hard sphere and rolling results from a balance of forces and torques on the cell due to hydrodynamic shear and receptor-ligand bonds (Chang and Hammer, 1996, 2000; Hammer and Apte, 1992). The incorporation of the Monte Carlo method in this algorithm to simulate the stochastic nature of receptor-ligand interactions has enabled it to successfully reproduce the experimentally observed "stop and go" motion of rolling cells. However, the major limitation of the adhesive dynamics algorithm is its inability to account for cell deformation that has been experimentally shown to critically affect the rolling behavior of a cell.
Other computational models such as the elastic ring (Dong et al., 1999; Dong and Lei, 2000) and compound drop (N'Dri et al., 2003) models, designed to account for cell deformability during rolling in shear flow, have two major drawbacks. First, the kinetics of receptor-ligand binding is represented by deterministic relationships, and thus, the models are incapable of capturing the jerky or "stop and go" cell rolling pattern. Second, the aforementioned models have been limited to a two-dimensional (2-D) representation of cell rolling. An additional limitation of the elastic ring model (Dong et al., 1999; Dong and Lei, 2000) stems from the fact that nonphysical constraints are imposed on peeling length limits as well as initial cell shape. On the other hand, the cell shapes predicted by the compound drop model at large deformations deviate from those observed experimentally (N'Dri et al., 2003). These deviations may be the direct outcome of modeling cell membrane as surface tension interface of a liquid drop rather than an elastic membrane, as recently suggested (Marella and Udaykumar, 2004).
To overcome the above-mentioned limitations of all previously described models, we developed a more realistic three-dimensional (3-D) model simulating receptor-mediated rolling of a deformable cell on a selectin-coated surface in a linear shear field. The model parameters were chosen to represent PSGL-1-mediated PMN rolling on a P-selectin-decorated planar surface. To this end, the immersed boundary method (IBM) (Peskin and McQueen, 1989) was used to simulate motion of an elastic capsule near a plane in a linear shear field. The IBM, originally developed to simulate blood flow in a mammalian heart (Peskin and McQueen, 1989), has found widespread application in the simulation of macro- and microscale biological systems (Bottino, 1998; Fogelson, 1984), and has been adapted by one of us to model 3-D deformation of an erythrocyte ghost in a linear shear field (Eggleton and Popel, 1998). In this study, the IBM was coupled to the Hookean spring model to simulate the force-dependent kinetics of receptor-ligand interactions (Dembo, 1994), whereas the stochastic behavior of P-selectin-PSGL-1 bond formation and breakage was simulated by the Monte Carlo method (Hammer and Apte, 1992; King and Hammer, 2001).
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COMPUTATIONAL METHOD |
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); and 3), the Monte Carlo method for simulating the formation and breakage of receptor-ligand bonds with kinetic rate constants based on the Hookean spring model (Dembo, 1994; Hammer and Apte, 1992).
Immersed boundary method
A 2-D representation of IBM was recently adopted in the viscous drop model (N'Dri et al., 2003) to simulate receptor-ligand-mediated cell rolling. We provide herein a brief description of IBM adapted to simulate the 3-D motion of an elastic capsule in shear flow near a planar surface.
The motion of a Newtonian fluid inside and outside the elastic capsule is governed by the continuity (Eq. 1) and the Navier-Stokes (Eq. 2) equations:
 | (1) |
 | (2) |
and µ are the density and viscosity of the fluid, respectively; and u(x) and p(x) are the velocity and pressure, respectively, at fluid grid nodes x(x,y,z). F(x) is the total force acting on each of the fluid grid nodes comprising the force exerted by the plane, elastic capsule, and receptor-ligand bonds. The inertial term in Eq. 2 is neglected because the Reynolds number for flow of microscopic particles under physiological flow conditions is very small. The stationary plane over which fluid flow occurs is modeled as a network of nodes interconnected by stiff springs similar to a previous IBM implementation simulating flow over a solid surface (Lai and Peskin, 2000). A linear shear field is generated by applying a constant force along a plane parallel to and far from the stationary surface.
At the beginning of each time step t, restoring forces due to the displacement of nodes in the elastic membrane and the stationary plane as well as those due to the stretching of bonds are calculated. These forces, denoted by F(X) and located at immersed boundary nodes X, are distributed to the nearby fluid grid nodes x (Fig. 1 A), using an appropriately chosen function Dh(X – x).
 | (3) |
-function,
 | (4) |
-functions given by
 | (5) |
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) to give the fluid velocity u(x) for the time step, t + 
t. The "no slip" condition at the elastic membrane surface and the stationary plane are enforced by moving their respective nodes with the local fluid velocity. The velocity U(X) of a membrane node X is the sum of the velocities at the fluid grid nodes x (Fig. 1 B), weighted by the discrete -function.
 | (6) |
At the end of each time step, the position of the nodes of the elastic membrane and the plane are updated using the relation,
 | (7) |
This leads to deformation of the elastic elements, and the restoring forces are recalculated and added to the N-S equation at the beginning of the next time step. The process is repeated to march forward in time. We implemented an algorithm for improved volume conservation (Peskin and Printz, 1993; Rosar, 1994), which limited systematic volume loss from the capsule to <3% over the entire period of the simulation.
Membrane constitutive equation
The cell is approximated as a spherical capsule with an elastic membrane in an initial strain-free state, which is discretized into flat triangular elements. The neo-Hookean membrane material is assumed to be incompressible and initially isotropic so that the strain energy, W, can be expressed as a function of only the in-plane principal stretch ratios, 
1 and 2 (Eggleton and Popel, 1998),
 | (8) |
; Eggleton and Popel, 1998).
Monte Carlo simulation of receptor-ligand interactions
According to the Hookean spring model, the forward and reverse rate constants for receptor-ligand interactions under external force are given by Dembo (1994)
 | (9) |
 | (10) |
and 
are the forward and reverse rate constants at an equilibrium distance , kf, and kr are the forward and reverse rate constants at a distance (xm – ) from equilibrium; kb is the Boltzman constant whereas T is the absolute temperature; and 
and ts represent the spring constants in the bound and transition state, respectively. The force acting on the bond is given by
 | (11) |
The stochastic nature of receptor-ligand interactions are included using Monte Carlo simulation. In a time interval t, the probability (Pb) that a receptor will bind is (Hammer and Apte, 1992)
 | (12) |
)
 | (13) |
At each time step, the probabilities of bond formation and breakage are compared to random numbers (Pran1 and Pran2) between 0 and 1. Pb > Pran1 indicates bond formation whereas Pr > Pran2 indicates bond rupture. PSGL-1 molecules are assumed to be concentrated on tips of PMN microvilli (Moore et al., 1995). The time of receptor-ligand bond formation and breakage as well as the number of bonds per cell and per microvillus were recorded for every time step. A time step of 10–6 s was used to simulate cell rolling for a period of 1 s. In the kinetic model of receptor-ligand interactions, the microvilli are modeled as solid cylinders that do not deform under force. It is also to be noted that IBM is used to simulate the motion of an elastic capsule with a smooth membrane and does not account for the roughness due to microvilli.
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RESULTS |
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).
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; Smith et al., 2002), by recording the average length in the direction of flow to height (L/H) ratio of the cell. The average L/H displayed a marked increase with shear rate for the cell with the most compliant membrane (Eh = 0.3 dyn/cm; Fig. 3). In contrast, the increase in average L/H with increasing shear rate was less pronounced for cells with stiffer membranes (Fig. 3). The membrane stiffness values of 0.3 and 1.2 dyn/cm were chosen to match the extent of PMN deformation previously observed in vivo (Damiano et al., 1996; Smith et al., 2002) for relatively low levels of shear stress typical in venous circulation (Turitto, 1982). The highest membrane stiffness value of 3.0 dyn/cm was selected to approximate the rolling behavior of a rigid sphere. The aforementioned selected membrane stiffness values fall within the range of previously published values for surface tension used in a 2-D viscous drop model of cell rolling (N'Dri et al., 2003).
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). As shown in Fig. 5 B, the variance in velocity increased with increasing membrane stiffness over the entire range of shear rates examined, thereby indicating relatively uniform rolling for the more compliant cells. We next estimated the effect of membrane stiffness on the average rolling velocity of the cells. It is noteworthy that at the low shear rate of 100 s–1, the average rolling velocity did not vary significantly with the cell membrane stiffness (Fig. 6 A). In contrast, pronounced differences were observed at higher shear rates. In particular, the average rolling velocity for the cell with a stiffer membrane (Eh = 3.0 dyn/cm) increased appreciably with shear rate from 
1 µm/s at 100 s–1 to 15 µm/s at 400 s–1 (Fig. 6 A). To the contrary, only a modest increase in the average velocity of more compliant cells was noted with increasing shear (Fig. 6 A). We also investigated whether the spring constant for receptor-ligand bonds affects the average rolling velocity. As can be seen in Fig. 6 B, no significant change in the average rolling velocity is detected when fractional spring slippage, f defined as ( – ts)/, was varied from 0.01 to 0.02 for shear rate values of 100 and 400 s–1, a finding that is consistent with previous theoretical predictions (Chang et al., 2000; N'Dri et al., 2003).
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DISCUSSION |
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In our simulations, the increase in the deformation index (L/H) with shear rate was significantly higher in cells with a more compliant membrane compared to those with a stiffer membrane. The L/H values for the more compliant membrane (Eh = 0.3 dyn/cm) show very good agreement with those recorded during in vivo leukocyte rolling where L/H increased from 1.1 to 1.2 upon increasing shear rate from 100 to 400 s–1 (Damiano et al., 1996; Smith et al., 2002). Moreover, the cell-substrate contact area estimates for a cell with a membrane stiffness value of 0.3 dyn/cm compare well with previously published in vivo estimates where the contact area increased from 19.6 to 35.8 µm2 when the shear rate was increased from 100 to 400 s–1 (Firrell and Lipowsky, 1989). In agreement with experimental data (Park et al., 2002; Yago et al., 2002), we observe that the average rolling velocity increases with increasing shear rates. Moreover, our computational model predicts that cells with a stiffer membrane roll relatively faster than those with a more compliant membrane only at higher shear rates. In particular, at a shear stress of 3.2 dyn/cm2 (shear rate = 400 s–1), the rolling velocities for cells with membrane stiffness values of 0.3 and 3.0 dyn/cm are in good agreement with experimentally observed values for PMNs (5 µm/s) and PSGL-1-bearing microspheres, respectively (15 µm/s) (Yago et al., 2002). In accord with in vitro flow-based assays, rolling of more compliant cells is relatively stable compared to cells with stiffer membranes that show sharper variations in the instantaneous rolling velocity (Park et al., 2002; Ramachandran et al., 2004).
It is noteworthy that increase in shear rate did not significantly affect the average number of bonds per microvillus for a given membrane stiffness value. However, this number decreased with increasing membrane stiffness. Taking into account the cross-sectional area of the microvillus tip, the length of the receptor-ligand bond, and the density of P-selectin on the substrate (Table 1), we estimated that at most 10 P-selectin molecules are available for binding to PSGL-1 molecules on a single microvillus. Because the number of PSGL-1 molecules per microvillus is much larger (50/microvillus), our simulations indicate that receptor-ligand binding during PMN rolling is limited by selectin density. These results suggest that PMN recruitment to sites of inflammation under physiologically relevant flow conditions may be controlled by the level of selectin expression on the activated endothelium.
Our theoretical predictions reveal that the average bond lifetimes for cells with a membrane stiffness of 0.3 dyn/cm decreased from 0.36 to 0.2 s when shear rate was increased from 200 to 400 s–1. These values are in the range of experimentally observed tether lifetimes for PMNs in shear flow. For instance, average tether lifetimes of PMNs on platelet P-selectin decreased from 0.6 to 0.15 s when shear rate was increased from 100 to 250 s–1 (Schmidtke and Diamond, 2000). In a separate study, the transient interactions of untreated PMNs, gluteraldehyde-fixed PMNs, and PSGL-1-bearing microspheres on P-selectin averaged 0.317, 0.215, and 0.076 s, respectively, at a wall shear stress of 0.5 dyn/cm2 (shear rate was 62.5 s–1) (Park et al., 2002). It is interesting that the bond lifetimes for cells with membrane stiffness values of 1.2 and 3.0 dyn/cm are closer to those measured experimentally for gluteraldehyde-fixed PMNs and PSGL-1-coated microspheres interacting with P-selectin surfaces (Park et al., 2002).
An elastic membrane was used in our simulations because it was recently shown to better approximate the PMN surface as compared to surface tension in a viscous drop model of the leukocyte (Marella and Udaykumar, 2004). However, a limitation of our three-dimensional model is that the viscosity of the fluid inside the capsule is the same as that of the suspending fluid. Therefore, membrane stiffness values (Eh = 0.3–3.0 dyn/cm), used in this study, were chosen to obtain cell deformation observed in previous experimental studies (Smith et al., 2002), and are in the same range as the surface tension values used in the 2-D viscous drop model of PMN rolling (N'Dri et al., 2003). However, the membrane stiffness values used in this study to compensate for the absence of viscoelastic resistance from the cytoskeleton are one order of magnitude higher than the previously reported cortical tension values (0.03 dyn/cm) estimated from PMN aspiration experiments (Drury and Dembo, 2001; Hochmuth et al., 1993). A more comprehensive viscous drop model was recently proposed, which treated the cytoplasm as a two-phase system comprising of a solvent and a polymer network (Herant et al., 2003). However, the model has not been adapted to simulate 3-D nonaxisymmetric PMN rolling on a selectin-coated plane. Another important feature not accounted for in all previous simulations of PMN rolling including those presented in this study is microvillus viscoelasticity, which has been shown to affect the force experienced by receptor-ligand bonds (Shao et al., 1998). Therefore, our model in its present form is unable to simulate the thin cylindrical membrane tethers pulled during PMN rolling (Schmidtke and Diamond, 2000; Park et al., 2002). A recent study showed that the P-selectin-PSGL-1 bond exhibits a transition from a "catch" bond (koff decreases with increasing applied force on the bond) to a "slip" bond (koff increases with increasing applied force) in the shear stress range of 0.2–0.4 dyn/cm2 (Marshall et al., 2003). However, the parameter values for the Hookean spring model for the "catch-slip" transition are not known in this regime and thus were not incorporated in our model of PMN rolling.
In conclusion, we have used a 3-D computational model to demonstrate that cell deformation affects the kinetics of receptor-ligand interactions and the number of bonds, and consequently PMN rolling on a selectin-coated surface. Our simulations are the first to capture most of the features relevant to selectin-mediated PMN rolling on the activated endothelium under physiological flow conditions. Future work will be focused on extending the model to include biomechanical elements of the cytoplasm and subsequent chemotactic modulation of cytoskeletal viscoelastic properties during active and passive PMN deformation.
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ACKNOWLEDGEMENTS |
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This work was supported by the National Science Foundation (grant BES0093524).
Submitted on August 3, 2004; accepted for publication October 7, 2004.
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REFERENCES |
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Y. V. Pereverzev and O. V. Prezhdo Dissociation of Biological Catch-Bond by Periodic Perturbation Biophys. J., July 15, 2006; 91(2): L19 - L21. [Abstract] [Full Text] [PDF]
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K. E. Edmondson, W. S. Denney, and S. L. Diamond Neutrophil-Bead Collision Assay: Pharmacologically Induced Changes in Membrane Mechanics Regulate the PSGL-1/P-Selectin Adhesion Lifetime Biophys. J., November 1, 2005; 89(5): 3603 - 3614. [Abstract] [Full Text] [PDF]
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M. Abkarian and A. Viallat Dynamics of Vesicles in a Wall-Bounded Shear Flow Biophys. J., August 1, 2005; 89(2): 1055 - 1066. [Abstract] [Full Text] [PDF]
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K. E. Caputo and D. A. Hammer Effect of Microvillus Deformability on Leukocyte Adhesion Explored Using Adhesive Dynamics Simulations Biophys. J., July 1, 2005; 89(1): 187 - 200. [Abstract] [Full Text] [PDF]
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,•), which represent the moving immersed boundary. (A) At a fluid node (x), the restoring force exerted due to deformation of elastic membrane elements is the sum of forces at all immersed boundary nodes (•) inside the circle with radius, 2h (h is the grid spacing), weighted by a discrete 
