© Copyright 2001 American Chemical Society
JULY 10, 2001 VOLUME 17, NUMBER 14
Letters Hydrodynamic Collisions Suppress Fluctuations in the Rolling Velocity of Adhesive Blood Cells Michael R. King, Stephen D. Rodgers, and Daniel A. Hammer* Departments of Chemical Engineering and Bioengineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104 Received February 13, 2001. In Final Form: May 10, 2001 The slow rolling motion of white blood cells transiently adhering to the interior surface of venules under shear is simulated numerically. Molecular bond breakage events cause a noisiness in rolling velocity over time scales longer than the lifetime of an individual bond. It is shown that the hydrodynamic interactions between particles produce a smoother rolling behavior, by modulating the cell’s motion during the brief forward jumps following a bond breakage event. Good agreement is found with flow chamber experiments using adhesive spherical beads.
The interplay between transient chemical bonding and hydrodynamic forces in the microcirculation controls the eventual fate of blood cells. Spherical white blood cells (neutrophils) reversibly bind to the interior surface of venules through a family of adhesion molecules known as selectins and translate at an average velocity much lower (0.1-50%) than that of freely suspended particles near the wall.1 This enables the bound neutrophil to communicate with the layer of endothelial cells that line venules through a cell signaling cascade,2 at times causing the neutrophil to firmly deposit at a site of injury as needed. Disruption of the normal function can lead to inflammatory diseases or interfere with lymphocyte/stem cell homing.3 Obtaining an accurate picture of cell rolling in vivo is complicated by the large number of chemical species involved, with at least three selectin molecules and five nonexclusive binding partners, each exhibiting quantitative differences in binding kinetics.2 Furthermore, physical parameters are constantly varying, such as the wall shear * To whom correspondence should be addressed. Department of Bioengineering. E-mail:
[email protected]. (1) Lawrence, M. B.; Springer, T. A. Cell 1991, 65, 859. (2) Ebnet, K.; Vestweber, D. Histochem. Cell Biol. 1999, 112, 1. (3) Springer, T. A. Annu. Rev. Physiol. 1995, 57, 827.
rate and the local surface coverage of adhesion molecules as cells become activated or deactivated. These complexities have been effectively dealt with by recreating the rolling phenomena in a cell-free system, replacing the endothelium with an immobilized layer of a single adhesion molecule and perfusing this surface with either neutrophils4 or carbohydrate-coated spherical beads.5 The term “rolling” as applied to blood cells contacting the interior surface of venules is used in the biomedical literature to describe the slow translational motion of cells without slip, as specific chemical adhesion provides precisely the necessary torque to prevent sliding. Controlled flow chamber experiments have revealed that the dynamics of rolling is inherently noisy, displaying timevarying fluctuations that generally scale with the average velocity. It is hypothesized that brief pauses in motion are integral in the transition to firm adhesion, mediated by another family of molecules called integrins2 which have characteristically slower binding kinetics. The noisiness of cell rolling arises from the relatively small number (4) Goetz, D. J.; El-Sabban, M. E.; Pauli, B. U.; Hammer, D. A. Biophys. J. 1994, 66, 2202. (5) Brunk, D. K.; Goetz, D. J.; Hammer, D. A. Biophys. J. 1996, 71, 2902.
10.1021/la010234b CCC: $20.00 © 2001 American Chemical Society Published on Web 06/09/2001
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of bonds in the contact zone between cell and substrate, a view supported by numerical simulation and the observation that rolling variance is increased by a reduction in site density.5 The disconnect between these recent quantitative studies of cell rolling and realistic physiologic behavior is the concentration of cells or beads used. The volume fraction used in in vitro experiments is often kept low by design to avoid the complicating effects of cell collisions, non-Newtonian rheology, and imaging difficulties, while whole blood is composed of about 45% cells by volume. Despite the chaotic nature of suspension flow, evidence supports the claim that rolling is supported at much higher shear rates in vivo and furthermore that the motion is smoother and with fewer pauses.6 This dichotomy of the anticipated and observed effects of cell collisions, given the important biological role of neutrophil rolling, motivates one to consider the effect of particle density on rolling from a rigorous fluid mechanical perspective. In this letter, we report on a numerical simulation of cell rolling that includes hydrodynamic interactions between multiple particles in suspension. The current study is motivated by experimental observations of cellfree rolling at higher bead concentrations than have been previously considered, which demonstrate that binary collisions alone can effect a qualitative change in rolling dynamics. A calculational algorithm referred to as adhesive dynamics is used to model adhesive interactions.7 Each molecular bond shared between the model spherical cell and the bounding planar wall is tracked as it moves through the near-contact region. The sum of the forces and torques exerted by the bonds on the cell at each instant in time affects the overall motion of the cell. A Monte Carlo simulation of the formation and dissociation of these bonds is performed which incorporates the force-dependent binding kinetics. While the random placement of adhesion molecules and stochastic nature of unbinding can be expected to produce fluctuations in motion, there is an emergent noisiness to rolling that manifests itself over time scales longer than the lifetime of an individual bond. The adhesion molecules are modeled as linear springs and assumed to be aligned vertically upon formation. One model commonly used to describe the kinetics of single biomolecular bond failure is due to Bell8
kr ) k0r exp(r0F/kbT)
(1)
which relates the rate of dissociation kr to the magnitude of the force on the bond, F. Typical values for the unstressed off-rate k0r and reactive compliance r0 are 2 s-1 and 0.4 Å for P-selectin binding with P-selectin ligand-1 (PSGL-1). The rate of formation directly follows from the Boltzmann distribution for affinity kf/kr ) (kf0/k0r ) exp[-σ|xb - λ|2/ 2kbT], where σ is the Hookean spring constant and |xb λ| is the deviation bond length. The expression for the binding rate must also incorporate the effect of the relative motion of the two surfaces. Chang and Hammer9 calculated the effective rate of collision of surface-tethered reactants in relative motion when the Peclet number Pe ) (radius of receptor)(relative velocity)/(lateral diffusivity) is nonzero and showed that the on-rate exhibits a first-order dependence on Pe. The result is that the probability of a bond forming is proportional to the slip velocity between (6) Damiano, E. R.; Westheider, J.; Tozeren, A.; Ley, K. Circ. Res. 1996, 71, 1122. (7) Hammer, D. A.; Apte, S. M. Biophys. J. 1992, 62, 35. (8) Bell, G. I. Science 1978, 200, 618. (9) Chang, K.-C.; Hammer, D. A. Biophys. J. 1999, 76, 1280.
Figure 1. (a) Schematic diagram of the multiparticle adhesive dynamics simulation. Far from the cells, the external flow is a linear shear flow, but the motion of the cells is coupled through a complex disturbance flow. A layer of surface roughness elements is placed outside of the hydrodynamic radius a, to account for the fact that cells (or beads) are not mathematically smooth spheres. (b) A 2-D cross section of the contact area between surfaces. During rolling, the upstream edge of the contact area is lifted away from the wall while the downstream edge is brought closer to the wall, presenting new reactive surface area.
the cell and plane. The solution algorithm is as follows: (1) all unbound molecules in the contact area are tested for formation against the probability Pf ) 1 - exp(-kf∆t); (2) all of the currently bound molecules are tested for breakage against the probability Pr ) 1 - exp(-kr∆t); (3) the external forces and torques on each cell are summed; (4) the mobility calculation is performed to determine the rigid body motions of the cells; (5) cell and bond positions are updated according to the kinematics of particle motion. See Figure 1a for a schematic diagram of the simulation. Unless firmly adhered to a surface, white blood cells can be effectively modeled as rigid spherical particles, as is evidenced by the good agreement between bead versus cell in vitro experiments.5 Typical values of physical parameters yield Reynolds numbers Re ) γ˘ a2/ν ) O(10-3), where γ˘ ∼ 100 s-1 is the shear rate, a ) 5 µm is the particle radius, and ν ) 1 cS is the kinematic viscosity of the suspending fluid. Thus, inertia can be neglected and fluid motion is governed by the Stokes equation
µ∇2u ) ∇p
∇‚u ) 0
(2)
where u is the velocity, µ is the fluid viscosity, and p is the local pressure. No-slip boundary conditions hold at the particle surfaces and at z ) 0, the position of the planar wall. The multiparticle problem is considerably more complex than the case of an isolated sphere, for which closed form solutions are available. We use a technique called the completed double-layer boundary integral equation method (CDL-BIEM), as well described in a text by Kim and Karrila.10 Applying the standard boundary (10) Kim, S.; Karrila, S. J. Microhydrodynamics: Principles and Selected Applications; Butterworth-Heinemann: Stoneham, MA, 1991; Chapters 14-17.
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element method to the Stokes flow problem produces a Fredholm integral equation of the first kind, which is generally ill-conditioned. By posing the mobility problem in terms of a compact double layer operator and completing the range with the velocity field resulting from a known distribution of point forces and torques placed inside each particle, one can derive a fixed point iteration scheme for solving the integral representation of eq 2,
u(x) - u∞(x) ) uRC(x) + ISK(x - ξ)‚φ(ξ) dS(ξ)
(3)
where u∞ is the ambient velocity in the absence of particles, uRC is a “range completing” velocity generated by point forces and torques that accounts for the fact that the illbehaved single layer integral has been discarded, K is the double layer operator, and φ is the unknown double layer distribution. After reducing the spectral radius of the corresponding discretization, eq 3 is found to converge rapidly. The presence of the wall is treated by incorporating the singularity solutions corresponding to a point force near a plane.11 The large separation of length scales between the deviation bond lengths and the particle radii requires very small time increments (∆t ∼ 10-7 s). To speed the calculation, a coarse discretization is used that does not resolve the particle-particle and particle-plane lubrication forces, which are added from known solutions as “external” forces. A steric layer is added to the sphere and wall surfaces as a model of surface roughness, with adhesion molecules placed outside of this layer. At distances shorter than the equilibrium bond length of 30 nm, the surfaces exert an electrostatic repulsive force12 of the form Frep ) F0τe-τ/ (1 - e-τ). In the simulation, values of τ ) 5 Å and F0 ) 103 N were used. Roughness values were w ) 50 nm for the wall and s ) 175 nm for the spheres. An intrinsic on-rate of kf0 ) 500 s-1 and a molecular spring constant of σ ) 100 dyn/cm were used. Gravitational force was included, with the positively buoyant beads having a density of 1.05 g/cm3. Figure 1b shows the scale and orientation of a typical instantaneous arrangement of 21 bonds between the sphere and planar surfaces, relative to the curvature of the sphere. Streptavidin-coated microspheres of radius 5.4 µm were covered with sialyl Lewisx (sLex) through a sLex-PAAbiotin linkage. Sialyl Lewisx is the functional carbohydrate domain presented by many selectin-binding ligands such as PSGL-1. The beads were then suspended in a phosphate-buffered saline (PBS)/1% bovine serum albumin (BSA) solution. Polystyrene slides were incubated with soluble P-selectin and later washed with PBS and 2% BSA to block nonspecific adhesion. The substrate is then placed in the well of a parallel plate flow chamber. Flow is driven by a syringe pump, and the system is imaged from below using an inverted phase contrast microscope equipped with a high-speed video recorder. Surface coverages were 90 molecule/µm2 of sLex and 180 molecule/µm2 of P-selectin, densities which support slow rolling motion at the shear rate of 100 s-1. During an experiment, a dilute mixture (0.1 vol %) of sLex coated and uncoated beads is injected into the flow chamber, where the beads gradually settle and many start to roll adhesively. The upper portion of Figure 2 shows a representative example of the streamwise translation of the adherent particle obtained from bead experiments. The gray trajectory corresponds to a 1 s interval during which the particle (11) Phan-Thien, N.; Tullock, D.; Kim, S. Comput. Mech. 1992, 9, 121. (12) Bell, G. I.; Dembo, M.; Bongrand, P. Biophys. J. 1984, 45, 1051.
Figure 2. Trajectory of a rolling adhesive bead, while isolated (in gray) or undergoing binary collisions (in black). Inset: Translational velocities, averaged over 0.1 s.
Figure 3. Top view of the relative trajectories of binary collisions. The experimental trajectories are reflected above the symmetry plane ∆y ) 0, and the simulated trajectories are reflected below. The initial elevation of particle 2 for these five simulations (∆z0/a), in order of increasing ∆yf (top to bottom), is 1.0, 1.0, 0.5, 1.0, and 0.0.
is isolated. The darker trajectory in Figure 2 is of that same particle during a 1 s interval when it is successively undergoing collision with freestream particles. During the time interval of Figure 2, the experimental particle undergoing collision experiences five nearfield contact interactions (see Figure 3) and several farfield interactions where there is no physical contact between cells. Depending upon the upstream coordinates prior to interaction, the faster, unbound particle is observed to either go around the rolling particle while remaining coplanar with it or move up-and-over the bound particle. One striking difference between the two displayed trajectories lies in the velocity variance. This is most easily seen in a plot of the velocity averaged over intervals of 0.1 s, inset in Figure
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2. Velocity fluctuations on this time scale are biologically relevant insofar as they may enable slower-binding integrins to take hold of a rolling leukocyte. Note that the presence of nearby particles decreases the variance by an order of magnitude and in this particular example has eliminated any observation of pauses in motion. The second result demonstrated by Figure 2 is that the average rolling velocity is essentially unchanged by the particle collisions. Simulations corresponding to the above-described experiments were carried out to determine whether the adhesive dynamics formulation coupled with multiparticle hydrodynamic calculations could reproduce the observed phenomena. A rolling cell (particle 1) is initiated with a steady-state bond configuration. A second, nonadherent particle (particle 2) is then placed 5 radii upstream and given some displacement in the y- and z-directions. Integration in time is carried out, with particle 2 periodically returned to an upstream position when it has reached a point 5 radii downstream of particle 1. Particle collisions generally last about 0.05-0.2 s, depending on the initial position of particle 2. Several of the resulting interactions are given in the lower half of Figure 3. Symmetry across the ∆y ) 0 plane was assumed, with experimental trajectories reflected above the plane and simulated trajectories reflected below the plane to permit a sideby-side comparison. Two of the experimental trajectories are incomplete: either a different freestream particle was being tracked during a portion of the interaction or in one case occurring at the end of the experiment. In Figure 3, the discontinuity in the slope of a trajectory indicates when physical contact is made between particles. During physical contact, electrostatic repulsion and lubrication forces between the surfaces prevent particle overlap and particle 2 is displaced around the bound particle 1. Despite the reversibility of simple Stokes flow problems, the nonhydrodynamic interparticle force introduces irreversibility to this interaction: during approach, electrostatic repulsion pushes the particles apart, while during separation there is no nonspecific attraction to resist the separation of the particles induced by fluid motion. This is a well-known effect in suspension flow that gives rise to such phenomena as diffusion down concentration gradients and the random walk of noncolloidal spheres.13 Good agreement is found between the experimental and simulated trajectories projected into the plane, as shown in Figure 3. Of perhaps greater interest in the study of selectinmediated rolling is the effect these particle-particle collisions have on the dynamics of a rolling particle. The lower part of Figure 2 shows the simulated version of the experiment. Two-particle simulations with ∆yinit ) 0.1 radii and ∆zinit ) 1 radii are presented along with those performed for an isolated adherent particle. Note that like bead experiments, numerical simulations also demonstrate that the main effect of such collisions is twofold: to suppress fluctuations in rolling velocity that occur over time scales on the order of 0.1 s, while leaving the average rolling velocity essentially unchanged. An order of magnitude reduction in velocity variance is observed, both in experiment and simulation. During a 1 s simulation of isolated cell rolling, approximately 600-800 bond breakage events occur. With the average number of bonds at about 20, this places the average lifetime of any one bond at 1/30 s. This suggests that velocity fluctuations that occur over tenths of seconds arise not because of a single stray or persistent bond but (13) Da Cunha, F. R.; Hinch, E. J. J. Fluid Mech. 1996, 309, 211.
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Figure 4. (a) The translational (〈|U|〉) and slip (〈|U - ωa|〉) velocities of a rolling cell directly following bond breakage, averaged over all breakage events. The cell momentarily moves at a velocity approaching the freestream, until the remaining bonds catch the cell and cause deceleration. (b) The change in the number of bonds directly after bond breakage (discrete curves), shown along with ∆Nb averaged over all breakage events including those that do not result in reformation (continuous curve). Note that the net change in bond number is positive, a consequence of the enhancement in formation rate due to slip. Each nonhorizontal line segment represents a change ∆Nb ) 1, and many different realizations of bond formation after bond breakage are superimposed. In some cases, one bond forms, while in other cases several bonds are formed. For example, in one case (broken line A-G) the particle formed three bonds in a stepwise manner (A-B, C-D, E-F); in another example, the particle formed one bond after breakage (H-J).
rather through some cooperative mechanism. At the trailing edge of the contact area, each bond briefly takes its turn supplying most of the force resisting the shear flow, before inevitably failing. Once this bond breaks, the cell jumps forward while rapidly decelerating as the remaining bonds stretch. When one recalls that the rate of binding for tethered reactants is proportional to the relative velocity between surfaces, a mechanism acting to destabilize the equilibrium number of bonds becomes apparent. A single breakage event often leads to multiple formation events, hereafter called “jump formation”. The instantaneous slip velocity following bond breakage, averaged over all such events during an isolated cell simulation 〈|U - ωa|〉, is plotted in Figure 4a. Figure 4b shows the change in the number of bonds over the same time interval, with the average change shown in bold. Although the majority of breakage events result in no formation of additional bonds, the occasional jumps in Nb driving the cell to pause or slow are readily seen. Importantly, bond breakage often leads to bond reformation rather than additional bond breakage. Two particular breakage events are highlighted in Figure 4b, showing examples of a triple or single reformation occurrence. The upper graph of Figure 5a shows the instantaneous number of bonds during the simulation of isolated cell
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Figure 5. (a) Instantaneous number of bonds during the rolling simulations of Figure 2 while isolated (top) or undergoing binary collisions (bottom). The relative streamwise position of the freestream particle is superimposed in gray. (b) A histogram of the change in the number of bonds when the freestream particle is relatively far as defined by ∆x < -15 µm or ∆x > 10 µm (in white), as compared to ∆Nb during nearfield collisional interaction defined by -15 e ∆x e 10 µm (in black). Note that the average change in Nb is statistically different from zero in both cases.
rolling presented in Figure 2. Note the rapid jumps in Nbond of 10 or more that occur. Fluctuations in rolling velocity do not arise simply from the random placement of adhesion molecules and the stochastic binding kinetics, in which case the fluctuations in Nbond would appear more symmetric in time. Jump formation drives the system away from equilibrium, eventually leading to an observable pause or lowered cell velocity. This correlation between the number of bonds and velocity can be readily seen by comparison with Figure 2. Between jump formations, bond number gradually returns to the equilibrium value. In the lower graph of Figure 5a, the number of bonds during the multiparticle rolling simulation of Figure 2 is plotted along with the relative position of the approaching particle. While Nbond exhibits fluctuations equal in magnitude to the case of an isolated cell, each peak corresponds to an individual collision. The correlation between individual particle collisions and a transient increase in the number of bonds is quantified in Figure 5b. Over the 18 collisions depicted in Figure 5a, the total change in Nb over the close interaction between particles, defined as -15 e x2 - x1 e 10 µm, was calculated. This
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is compared to the change in Nb during the rest of the simulation, defined as ∆x > 10 µm or ∆x < -15 µm. The average change in Nb during a nearfield cell-cell interaction is +3.44 ( 1.28, while the average change in Nb during the time between nearfield collisions is -3.17 ( 0.85. These average values of ∆Nb are plotted in Figure 5b along with the 2σ error in the mean, showing both to be statistically different from zero. We may conclude from this that the fluctuations in the number of bonds exhibited in the lower plot of Figure 5a are positively correlated with individual cellular collisions. This mechanism allows the bound cell to resist velocity variations that would be caused by the faster colliding particle. The time-averaged number of bonds in the presence of the second particle is 22, higher than Nbond ) 19 for the isolated particle. Thus, rather than destabilizing rolling, the mechanism by which multiple bonds can be formed directly after a bond breakage event helps a bound cell resist being convected away by freestream particles. If the second particle is positioned so as to cause a disturbance flow and/or contact force that would accelerate the bound cell, the next bond breakage event will more likely result in the formation of many more bonds. Conversely, when another particle is far upstream or downstream, often reducing the hydrodynamic drag experienced by the bound cell, jump formation will be less likely. This study, in particular the demonstration that the presence of nearby particles can smooth the rolling behavior of adherent particles, represents one step toward gaining a more realistic picture of leukocyte rolling in vivo. Clearly, in a concentrated suspension of cells there are other complicating hydrodynamic effects that may be simulated using this methodology: interactions between rolling or between rolling and firmly adherent cells, ternary and higher collisions, and far-field interactions. A complete understanding of cell rolling must include interactions between nearby rolling particles, considering that neutrophil-neutrophil recruitment tends to align rolling cells into linear trains. Alternatively, it is wellknown that white blood cells are marginalized to the nearwall regions of microvessels, while the more deformable red cells can aggregate into a fast-moving core near the center axis. While an individual red cell must be treated as a nonspherical, highly deformable particle, the effect of red cells on cell rolling dynamics may prove to be well approximated by simulations featuring a bumpy wall or monolayer of rigid spheres. While it is tempting to introduce the non-Newtonian behavior of concentrated blood in the form of bulk rheological properties such as elasticity and elevated viscosity, care must be taken when considering the proper local environment for such calculations, considering that red cells and platelets will be depleted from approaching surfaces in a complex manner. Acknowledgment. This work was funded by the National Institutes of Health, Grant No. HL18208, and an NIH National Research Service Award to M. King (F32 HL10353). LA010234B
