3D Computational Modeling and Simulation o

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Computers in Biology and Medicine 38 (2008) 738753www.intl.elsevierhealth.com/journals/cobm

3D computational modeling and simulation ofleukocyte rolling adhesion and deformation

Vijay Pappu, Prosenjit Bagchi

Department of Mechanical and Aerospace Engineering, Rutgers University, The State University of New Jersey, 98 Brett Road, Piscataway, NJ 08854, USA

Received 2 November 2007; accepted 3 April 2008

AbstractA 3D computational fluid dynamic (CFD) model is presented to simulate transient rolling adhesion and deformation of leukocytes over a

P-selectin coated surface in shear flow. The computational model is based on immersed boundary method for cell deformation, and stochastic

Monte Carlo simulation for receptor/ligand interaction. The model is shown to predict the characteristic stop-and-go motion of rolling

leukocytes. Here we examine the effect of cell deformation, shear rate, and microvilli distribution on the rolling characteristics. Comparison

with experimental measurements is presented throughout the article. We observe that compliant cells roll more stably, and have longer pause

times due to reduced bond force and increased bond lifetime. Microvilli presentation is shown to affect rolling characteristics by altering the

step size, but not pause times. Our simulations predict a significant sideway motion of the cell arising purely due to receptor/ligand interaction,

and discrete nature of microvilli distribution. Adhesion is seen to occur via multiple tethers, each of which forms multiple selectin bonds, but

often one tether is sufficient to support rolling. The adhesion force is concentrated in only 13 tethered microvilli in the rear-most part of a

cell. We also observe that the number of selectin bonds that hold the cell effectively against hydrodynamic shear is significantly less than the

total adhesion bonds formed between a cell and the substrate. The force loading on individual microvillus and selectin bond is not continuous,

rather occurs in steps. Further, we find that the peak force on a tethered microvillus is much higher than that measured to cause tether extrusion.

2008 Elsevier Ltd. All rights reserved.

Keywords: Computational fluid dynamics; Immersed boundary method; Fluid structure interaction; Microcirculation; Leukocyte; Selectin; Receptors; Microvilli

1. Introduction

Adhesion of circulating leukocytes to vascular endothelium

is a key event in inflammatory response [1]. The process,

often called adhesion cascade, involves multiple steps that begin

with initial arrest or tethering of leukocytes to the endothelium,

followed by slow rolling of the cells. Subsequently, leukocytesfirmly adhere and spread over the endothelium, and then trans-

migrate to the sites of inflammation. Extensive studies in the

past have shown that the tethering and rolling are mediated

by three types of adhesion molecules, P-, E-, and L-selectins,

which bind to their respective ligands with high affinity [24].

P-selectin-glycoprotein-ligand-1 (PSGL-1) is a common lig-

and that is known to bind to all three selectins. Flow chamber

Corresponding author. Tel.: +1 7324453656.

E-mail address: [email protected] (P. Bagchi).

0010-4825/$- see front matter 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compbiomed.2008.04.002

studies have also shown tethering and rolling of leukocytes over

selectin-coated surfaces [58].

Analysis of leukocyte trajectory revealed that a rolling cell

does not flow continuously, but rather it moves in a stop-and-

go manner [4,5,9,10] due to random formation and breakage of

receptor/ligand bonds. Selectin bonds are known to have high

association and dissociation rates. A threshold shear is requiredto mediate rolling [11,12]. However, above the threshold shear,

cells roll stably with relatively less variation in rolling veloc-

ity as shear rate increases 20 folds [8,10]. Chen and Springer

[10] hypothesized an automatic braking system in which

receptor/ligand bonds increase with increasing shear to stabi-

lize rolling.

Microvilli, which are protrusions from the cell surface, also

plays a critical role in rolling [1315]. L-selectins and PSGL-1

are concentrated on the tips of microvilli. When a leukocyte

comes in proximity to the endothelium, bond formation is
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V. Pappu, P. Bagchi / Computers in Biology and Medicine 38 (2008) 738753 739

facilitated due to high accessibility of microvilli tips. Concen-

tration of L-selectins and PSGL-1 on microvilli tips also sug-

gests that selectin bonds are likely to form in clusters, rather

than distribute uniformly over the entire cell/substrate contact

zone.

While circulating leukocytes maintain a spherical shape,

rolling leukocytes are known to deform to tear-drop shapesat higher shear [8,16,17]. Firrell and Lipowsky [16] ob-

served nearly 140% increase of WBC length as the shear rate

increased from 50 to 800 s1. Cell deformability may affect

rolling in several ways [9,13,18]. Upon initial tethering, the

cell/substrate contact area becomes flat allowing formation of

newer microvilli tethers to further stabilize the rolling. Adhe-

sion via multiple microvilli would reduce the force on individ-

ual selectin bonds and prolong bond lifetime. Cell deformation

may also reduce the hydrodynamic drag, and hence the bond

force. Comparison of rolling characteristics of ligand-coated

microspheres, and fixed and normal leukocytes suggested that

normal cells roll more smoothly with longer pauses compared

to microspheres or fixed leukocytes [7,9]. The bond force was

also estimated to be significantly lower in case of an untreated

leukocyte than that on a microsphere.

Among several computational models, the adhesive dynam-

ics simulation (ADS) pioneered by Hammer and co-workers

made a significant contribution to theoretical understanding of

leukocyte rolling [1922]. In ADS, leukocytes are modeled

as rigid spheres, and the receptor/ligand interaction is sim-

ulated by stochastic Monte Carlo simulation. Recent works

by the same group have incorporated microvilli deformation

within the framework of ADS [23]. Deformation of an adher-

ent leukocyte was considered in two-dimensions by Dong and

co-investigators by modeling a leukocyte as a viscous liquiddrop surrounded by an elastic ring [8,24,25]. NDri et al. [26]

modeled leukocytes as 2D compound liquid drops to study the

effect of cell nucleus on deformation. The role of viscoelas-

ticity, and microvilli extension during leukocyte adhesion and

rolling were considered in a recent 3D model developed by

Khismatullin and Truskey [27,28]. However, the characteris-

tic stop-and-go motion of a rolling leukocyte, as observed in

vitro and in vivo, was not reported by Dong and co-workers,

NDri et al., and Khismatullin and Truskey.

Recently, Jadhav et al. [29] developed a 3D model for rolling

leukocytes by coupling cell deformation with stochastic simu-

lation of receptor/ligand interaction. Their model was able toreplicate the stop-and-go motion of leukocytes. Such com-

putational tools can be used to gain deeper insights into the

biomechanics of cell rolling and adhesion. In this article, we

present a similar 3D computational fluid dynamic (CFD) model

to simulate rolling adhesion of deformable leukocytes over a

P-selectin coated surface in a shear flow. Our computational

model is based on the immersed boundary method (IBM) for

cell deformation, and stochastic Monte Carlo simulation of

receptor/ligand interaction. Our model has been able to predict

the characteristic stop-and-go motion of rolling leukocytes.

Using the model, we address three specific questions in this

article: (i) How does deformation affect the rolling character-

istics of cells? (ii) How does distribution of microvilli affect

cell rolling? (iii) How is the adhesion force distributed in the

cellsubstrate contact zone?

2. Computational methodology

The flow configuration is described in Fig. 1. The adhesive

rolling motion of deformable leukocytes in shear flow over aP-selectin coated planar surface is considered. Shear rate is var-

ied as 100, 300 and 500 s1. The initial shape of the leuko-

cyte is spherical with diameter 8 m. Microvilli, each of length

350 nm, are distributed randomly over the cell surface. Bonds

are allowed to form in the microvilli tips. We consider two dif-

ferent microvilli distributions, Nmv =21 and 155, where Nmv is

the number of total microvilli. We consider deformation of the

cell, but not of microvilli. Further parameters of the problem

are listed in Table 1.

The computational modeling is based on the IBM [3032]

which is particularly suitable for the present study, as the leuko-

cyte is modeled as a compound liquid drop surrounded by hy-

perelastic membrane (discussed later). In IBM, a single set of

equations is used to solve the fluid motion interior and exterior

of the cell. The fluid motion is governed by the continuity and

NavierStokes equations as

u = 0, (1)

ju

jt+ u u

= p + (u + (u)T), (2)

where u is the fluid velocity, is the density, p is the pressure,

and is the viscosity. The cell surface is then recognized by a

source-like term F added to the r.h.s. of Eq. (2). For a rolling

cell in shear flow, the force on the cell surface can arise fromtwo contributions: fe the elastic force due to cell deformation,

and fa the adhesive force due to bond formation between the

cell and the substrate. The source term F is related to fe and

fa as

F(x, t ) =

jS

[fe(x, t) + fa(x

, t )](x x) dx. (3)

Here x is a point in the flow domain, x is a point on the

cell surface jS, and is the delta function which vanishes

everywhere except at the membrane. Models for computation

of fe and fa are described later.

The NavierStokes equations are discretized on a fixed

Eulerian grid, and the cellplasma interface is tracked in a

Lagrangian manner by a set of marker points distributed on the

cell surface (Fig. 1). The equations are first solved to obtain

the fluid velocity and pressure in the entire flow domain. The

velocity of the cell membrane is then obtained by interpolating

the fluid velocity as

u(x) =

S

u(x)(x x) dx, (4)

where S denotes the entire flow domain. Cells are then

advected as

dx

dt = u(x

). (5)


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740 V. Pappu, P. Bagchi / Computers in Biology and Medicine 38 (2008) 738753

leukocyte

Nucleus

shear flow

Eulerian grid

Y

X

X

Y

Z

cell surface

Microvillus

Selectin bondsPlate

Fig. 1. (A) Schematic of leukocyte rolling over a selectin-coated surface in a shear flow. A 2D slice is considered to show the Eulerian grid used to discretize

the flow domain. (B) Actual 3D cell showing the Lagrangian mesh on the surface. The discrete locations of microvilli are shown by . (C) Schematic of one

microvillus and selectin bonds.

Table 1

Parameter values used in simulations

Parameter Value Source

Shear rate 100, 300, 500 s1

Leukocyte diameter 8m

Membrane stiffness (Eh) 2.6, 0.9, 0.3 dyn/cm [29]

Microvillus length 0.35m [14,43]

Number of microvillus (Nmv) 21, 155 [14]

Number of ligands 50/microvilli [29]

Receptor site density 144/m2 [6]

Selectin bond length (l0) 0.1m [48]

Spring constant (kb) 1 pN/nm [48]

Transition spring constant (kts) 0.99 pN/nm [6]

Unstressed forward rate (k0f ) 3.30 s1 [49]

Unstressed reverse rate (k0r ) 3.7 s1 [6]

Distribution of the forces from cell surface to the surround-

ing fluid grid (Eq. (3)), and interpolation of the fluid veloc-

ity onto the cell surface (Eq. (4)) involve a 3D delta function

which is constructed by multiplying three 1D delta functions as

(x x) = (x x)(y y)(z z). For numerical imple-

mentation, a discretized representation of the -function is used

as [31]

D(x x) =1

643

3i=1

1 + cos

2(xi x

i )

for |xi x

i |2, i = 1, 2, 3,

D(x x) = 0 otherwise, (6)

where is the Eulerian grid size. The above representation

approaches the analytical delta function as approaches zero.

The discrete delta function is so constructed that the distribu-

tion of the surface forces, or the interpolation for the surfacevelocity, is performed over a sphere of diameter equal to four

Eulerian grid points surrounding each Lagrangian node. In dis-

crete form, the integrals in Eqs. (3) and (4) are written as

F(xj) = i D(xj xi )f(x

i ), (7)

u(xi ) = jD(xj xi )u(xj), (8)

where i and j represent the Lagrangian and Eulerian grid points,

respectively.

We model leukocytes as compound liquid drops surrounded

by thin hyperelastic membranes. The membrane is assumed to

follow the neo-Hookean law. The strain energy function for themembrane is then given by

W =Eh

6(21 +

22 +

21

22 3), (9)

where E is the modulus of elasticity, h is the membrane thick-

ness, and 1 and 2 are the principal stretch ratios. Note that

the neo-Hookean model does not strictly represent a leukocyte

membrane. Models which closely resemble large deformation

of leukocytes during micropipette aspiration exist in the litera-

ture [33,34], and can be implemented in IBM. We choose the

neo-Hookean model as it is easy to implement, yet can predict

cell deformation under rolling conditions [29]. Three values of


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Eh, 0.3, 0.9, and 2.6 dyn/cm, are considered in the simulations

following Jadhav et al. [29].

The cell surface is discretized using triangular elements

(Fig. 1). We use a finite element model to compute the elastic

force generated on the cell surface due to deformation [35].

In this model, the elastic force fe is obtained at three nodes of

each element by differentiating the strain energy function Wwith respect to the nodal displacement v as

fe =jW

j1

j1

jv+

jW

j2

j2

jv. (10)

The main idea is that a general 3D deformation of the membrane

can be reduced to a 2D problem by assuming that individual

triangular element on the membrane remains flat even after

deformation, and that the membrane force remains invariant

under a rigid body rotation. This assumption still allows large

deformation of a leukocyte in the model. The resultant force feat a membrane node is the vector sum of the forces exerted by

all elements surrounding that node.Formation of receptor/ligand bonds between a leukocyte and

the substrate is simulated using stochastic Monte Carlo method

[19,29]. Bonds are assumed to behave as stretched springs under

force loading following a Hookean model [36]. The probability

of formation of a new bond, and that of breakage of an existing

bond, in a time interval t, are given by

Pf = 1 exp(kft ) (11)

and

Pr = 1 exp(krt ), (12)

respectively, where kf and kr are the forward and reverse reac-tion rates which are computed as

kf = k0f exp

kts(l l0)2

2KBT

(13)

and

kr = k0r exp

(kb kts)(l l0)

2

2KBT

, (14)

where k0f and k0r are the unstressed reaction rates, kb is the

spring constant, kts is the transition state spring constant, l and

l0 are the stretched and unstretched lengths of a bond, KB isthe Boltzmann constant, and T is the absolute temperature.

Values of the parameters are given in Table 1. At a given time

instance, two random numbers N1 and N2, between 0 and 1,

are generated. A new bond is allowed to form if Pf > N1, and

an existing bond is allowed to break if Pr > N2 [19,29]. Force

in each bond fb is then obtained as

fb = kb(l l0). (15)

The adhesion force fa is the vector sum of the forces arising

from all bonds formed in a microvillus tip.

The NavierStokes equations are discretized spatially using

a second-order finite difference scheme, and temporally using

a two-step time-split scheme. In this method the momentum

equation is split into an advectiondiffusion equation and a

Poisson equation for the pressure. The body-force term is re-

tained in the advectiondiffusion equation. The nonlinear terms

are treated explicitly using a second-order AdamsBashforth

scheme, and the viscous terms are treated semi-implicitly us-

ing the second-order CrankNicholson scheme. The resultinglinear equations are inverted using an ADI (alternating direc-

tion implicit) scheme to yield a predicted velocity field. The

Poisson equation is then solved to obtain pressure at the next

time level. Using the new pressure, the velocity field is cor-

rected so that it satisfies the divergence-free condition. The cell

surface is also advected using a second-order AdamsBashforth

scheme. Details of the time-step scheme are given in Bagchi

and Balachandar [37]. Computational domain is a rectangular

box with the longest axis aligned parallel to the flow direction

(X). The domain is assumed periodic in the X and Z directions.

Typical Eulerian resolution used in the flow solver is 320 points

in X, and 120 points each in the Y and Z directions. Typical

Lagrangian mesh used to discretize the cell surface consists of

1280 triangular elements.

3. Validation of numerical method

We first validate our IBM code against published results on

cell deformation. For this purpose, we consider deformation

of a spherical capsule as a model cell placed in a linear shear

flow as shown in Fig. 2. A capsule is a liquid drop surrounded

by an elastic membrane. In a shear flow a capsule deforms into

an ellipsoidal shape. The deformation can be expressed using a

dimensionless parameter D =(LB)/(L +B), where L and B

are the major and minor axis of the ellipsoid in the plane of theshear (Fig. 2A). Time-history of D for various dimensionless

shear rate is shown in Fig. 2B and compared with that obtained

by boundary integral simulation by Ramanujan and Pozrikidis

[38]. Excellent agreement between the two simulations is

observed.

Sensitivity of our results to the Eulerian and Lagrangian res-

olutions is shown in Fig. 2C (and, in inset 2D) by considering

three test simulations at different resolutions: (i) 803 Eulerian

grids and 1280 Lagrangian elements, (ii) 1203 Eulerian grids

and 1280 Lagrangian elements, and (iii) 1203 Eulerian grids

and 5120 Lagrangian elements. No significant difference is

observed between the three test cases.We also keep track of the cell volume during the simulations.

The change in the cell volume is less than 0.1% from its

initial volume. The projection method used here for flow solver

satisfies the mass (or, volume) conservation up to 1014 at

every grid point in the computational domain.

4. Results

Snapshots of a rolling leukocyte obtained from a simulation

at 500 s1 and Eh = 2.6dyn/cm are presented in Fig. 3. Shown

here is a sequence of initial tethering, cell deformation, and

tether breakage. In Fig. 3A, initial bonds are just formed on

microvillus 1. The cell shape is nearly spherical at this time.


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B

L

Undeformed Capsule

Deformed Capsule

00 2 4 6 8

0.2

0.4

0.6

0.8

0.2

0.025

0.05

0.1

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

3.5 4 4.5

0.38

0.39

0.4

Fig. 2. Validation of IBM code. (A) Schematic of a spherical capsule deforming in shear flow. (B) Deformation index D versus time. : present results;

Ramanujan and Pozrikidis [38]. Results are shown for various values of dimensionless parameter a/2Eh where a is cell diameter and is shear rate.

(C) Grid resolution test. , 803 Eulerian points and 1280 Lagrangian elements; 1203 Eulerian points and 1280 Lagrangian elements;

1203 Eulerian points and 5120 Lagrangian elements. (D) Inset showing the closeup.

Upon initial tethering, the cell rotates clockwise due to the fluid

torque (Fig. 3B). The rolling motion temporarily stops, and

the cell deforms to make a flat contact area with the substrate

(Fig. 3C). As a result, more microvilli become available for

bond formation, such as microvillus 2 in Fig. 3C. Sub-

sequently, microvillus 1 breaks, cell rolling commences

(Fig. 3D), and the contact area decreases. The cell is eventually

tethered via microvillus 2 (Fig. 3E). The contact area in-

creases again, and microvilli 3 and 4 also become available

for bond formation. Fig. 3F shows the breakage of microvillus

2 followed by cell rolling. In Fig. 3G, the cell is shown to

tether again by microvillus 4.

The rolling sequence of a more compliant cell (Eh =

0.3dyn/cm) is presented in Fig. 4. Fig. 4A shows the initial

arrest of the cell by tethering of microvillus 1, followed by

deformation of the cell into a tear-drop shape in Figs. 4BC.

The contact area increases significantly, and four microvilli

become available for bond formation. However, the cell is

anchored only by microvillus 1 located in the sharp corner

formed at the trailing edge. In Fig. 4D, microvillus 1 breaks,

and the rear end of the cell retracts. Figs. 4E and F show

formation of a new tether via microvillus 2, followed by cell

spreading.

Figs.3 and 4 show that during rolling adhesion, the cell shape

deviates significantly from its spherical shape. Average contact

area is shown in Fig. 5A. It increases with increasing shear

rate and decreasing membrane stiffness. The present results

show excellent agreement with the in vivo measurements by

Firrell and Lipowsky [16]. We also compute a dimensionless

parameter called deformation index, L/H, where L is the end-

to-end length along the flow direction, and H is the height of the

cell (Fig. 5B). The ratio increases with increasing shear rate and

decreasing membrane stiffness. The qualitative trend and the

range of values are in agreement with the in vivo measurements

of Damiano et al. [17] and Firrell and Lipowsky [16] (shown in

Fig. 5B), and 3D computational modeling of Jadhav et al. [29].

Next we consider the trajectory of a rolling leukocyte.

Fig. 6 shows the axial (x) displacement of the cell w.r.t. time.

Cell motion is characterized by a series of steps during which

the cell rolls, and pauses during which the cell is adherent. Our

results show the role of shear-rate and membrane stiffness in

modulating the steps and pauses. Effect of varying shear rate at

a constant Eh = 0.3dyn/cm is illustrated by plots AC which

show shorter pause times and more frequent steps with increas-

ing shear. Effect of varying membrane stiffness at a constant

shear is illustrated by plots CE, which show decreasing step

sizes and longer pause times with decreasing stiffness. The

instantaneous rolling velocity corresponding to the trajectories

is also shown in the same figure (right panel). The stochastic

nature of the cell motion is evident here by fluctuations in cell


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1

A

1

B

X

Y

Z

YX

Z

1

2

C

2 D

2

3

4E

4F

4 G

Fig. 3. Sequence of a rolling leukocyte at 500 s1 shear rate, Eh = 2.6dyn/cm, and Nmv = 21. The shear flow and the cell movement are from left to right.

The left panel shows sideview, and the right panel shows bottomview. Lagrangian mesh on the cell surface is also shown. In the bottomview, microvilli

forming bonds are marked by numbers 1, 2 etc.

velocity. The fluctuations increase with increasing shear rate

and membrane stiffness implying that a leukocyte rolls less

stably at higher shear rate and membrane stiffness.

The average pause time of a rolling leukocyte obtained from

our simulations is shown in Fig. 7A. The average pause time

varies between 0.04 and 0.6 s. It strongly depends on shear

rate, and decreases with increasing shear rate. The pause time

depends strongly on membrane compliance at low shear rate,

but weakly at higher shear rate. It increases with increasing

membrane compliance. In Fig. 7A, we compare our computed

pause times with the experimental data of Smith et al. [6],

and find reasonable agreement. Our simulations also predict

that the average pause time does not depend on the microvilli

distribution given by Nmv (not shown).

The average step size of a rolling leukocyte obtained from

our simulations is shown in Fig. 7B. The step size strongly


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1

X

Y

Z

1

1

2

3

4 C

1

2

B

XY

Z

1

2

A

1

2

3

4D

2

3

4E

2

3

4F

Fig. 4. Same as in Fig. 3 except Eh = 0.3dyn/cm.

depends on Nmv. For Nmv = 155, the average step size is in the

range 0.2.0.6m, and it does not strongly depend on shear rate

and membrane stiffness. For Nmv = 21, the step size increases

significantly due to more sparse distribution of microvilli. The

step size also decreases with decreasing membrane stiffness

and increasing shear rate. For Nmv = 21, the step size depends

strongly on the membrane stiffness at low shear, but weakly on

shear rate at low stiffness.


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shear rate (1/s)

100 200 300 400 50010

20

30

40

contactaream

2

shear rate (1/s)

deformationindex(L/H)

100 200 300 400 5001

1.2

1.4

1.6

1.8

2

Firrell &Lipowsky

Fig. 5. Average contact area (A) and deformation index (B). Dash lines are best fit curve obtained by Firrell and Lipowsky [16] based on in vivo measurements;

solid lines are present results. Eh = 2.6dyn/cm, Eh = 0.9dyn/cm, Eh = 0.3dyn/cm.

Average rolling velocity is shown in Fig. 8. It increases with

increasing shear rate, increasing membrane stiffness, and de-

creasing Nmv. At Nmv = 21, the average rolling velocity ranges

from about 4 to 112 m/s, whereas at Nmv = 155, it ranges

from 3 to 11 m/s. At higher Nmv, and lower shear, the rolling

velocity does not change significantly with varying membrane

stiffness; but it depends strongly on membrane stiffness at lower

Nmv and higher shear. In the figure, we compare our results

with three experimental measurements, Kim and Sarelius [4],

Yago et al. [9], and Ramachandran et al. [39], which show rea-

sonable agreement. Our results at Nmv = 21 are in agreement

also with in vivo data (30.

50m/s) of Firrell and Lipowsky[16].

As a quantification of the stochastic nature of cell rolling,

we compute RMS (root-mean-square) of instantaneous rolling

velocity in Fig. 10A. The RMS of the axial velocity shows

a strong dependence on shear rate, membrane stiffness, and

microvillus distribution. The RMS increases with increasing

shear rate, and membrane stiffness. On the contrary, it decreases

with increasing number of microvilli. At Nmv = 155, the RMS

does not change appreciably w.r.t. shear rate and membrane

stiffness, though it shows an increasing trend w.r.t. these pa-

rameters. At Nmv = 21, several-folds increase in the RMS is

observed at higher membrane stiffness, and higher shear rate.

Hence, the cell rolls more stably with the denser population of

microvilli, and at lower shear rate and membrane stiffness.

Interestingly, our simulations predict that during the rolling

motion, a leukocyte can undergo a significant sideway move-

ment. Fig. 9 shows the instantaneous lateral velocity of the cell

for a representative case. The lateral velocity, similar to the

axial velocity, also shows fluctuations which increase with in-

creasing shear rate and membrane stiffness. The RMS of the

lateral velocity fluctuations is shown in Fig. 10B for Nmv = 21

which shows similar trend as that of the axial velocity RMS.

The lateral velocity RMS increases with increasing shear rate,

and membrane stiffness, and decreasing number of microvillus.

For Nmv = 21, the RMS of the lateral velocity is comparable

to, though lower than, that of the axial velocity. At higher

Nmv = 155, the RMS of the lateral velocity is found to be sig-

nificantly low (not shown).

Total adhesion force between the cell and the substrate is

shown in Fig. 11A as a function of shear rate, membrane stiff-

ness, and microvilli distribution. The adhesive force obtained

in our simulations ranges from about 100 to 750 pN for shear

rates 100.500s1. In comparison, the adhesive force estimated

by House and Lipowsky [40], based on in vivo measurements,

ranges from 110 to 7610 pN for shear rates 200.2500 s1. The

adhesive force increases with increasing membrane stiffness,

partly due to increased hydrodynamic drag on less compliantcells. The adhesive force also increases with increasing shear

rate and increasing microvilli population.

It is also of interest to examine how the adhesive force is dis-

tributed within the cell-surface contact area. This is shown in

Fig. 11B for a representative case at 500 s1, Eh = 0.3dyn/cm,

and Nmv = 155. In our simulations, bonds are assumed to form

at the microvilli tips only. Hence the distribution of the adhe-

sive force is not continuous, rather discrete. In Fig. 11B, we

show the 3D distribution of the microvilli over the cell surface

in the contact area. Together Figs. 11B and d show how the

adhesive force is distributed among all 20 microvilli that are

present in the contact area. Nearly 90% of the total adhesion

force is concentrated in three microvilli (marked by arrows in

Fig. 11D) which form tethers in the rear end of the cell. The

force on each of these tethered microvilli ranges from 200 to

400 pN, while that on each of the remaining 17 microvilli is in

the range 020 pN. The maximum force on a tethered microvil-

lus is shown in Fig. 11C as a function of shear rate, membrane

stiffness, and microvilli distribution. This force is obtained

before a tether breaks away from the substrate. It ranges from

about 80 to 420 pN.

The average number of microvilli that form tethers, and the

total number of microvilli available within the contact area are

shown in Fig. 12. For Nmv = 155, the number of microvilli

within the contact area increases with increasing shear rate


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Time (s)

Displacement(micron)

1 2 3

10

15

20

25

C

B

A

Time (s)

Displacement(micr

on)

0.1 0.2 0.3 0.4 0.5

20

30

40

50

60

E

D

C

Velocity(micron/

s)

0 0.5 1 1.5 2 2.5

0

100

200

0 0.5 1 1.5 2 2.5

0

200

400

600

800

0 0.2 0.4 0.6

0

200

400

600

800

0 0.2 0.4 0.6

0

200

400

600

800

Time (s)

0 0.2 0.4 0.6

0

500

1000

Fig. 6. Axial displacement (left figures) and axial velocity (right figures) of a rolling leukocyte. A, B, and C show the effect of shear rate which varies as

100, 300, and 500 s1, respectively, at a constant Eh = 0.3dyn/cm. C, D, and E show the effect of membrane stiffness which varies as Eh = 0.3, 0.9, and

2.6dyn/cm, respectively, at a constant shear rate of 500s1.

and membrane compliance due to increased deformation of the

cell. For the most compliant cell considered here at 500 s1

shear rate, nearly 35 microvilli are available in the contact area.

However, only five of them form tethers. The number of teth-

ers increases slightly with increasing shear rate and membrane

compliance. For Nmv = 21, the average number of tethers is

even lower, and in the range of 12.

We next examine the average number of selectin bonds

formed between a cell and the substrate in Fig. 13. Note that we

assume 50 PSGL-1 ligands on each microvillus tip (Table 1).

In contrast, we observe that only about half of them, at most,

forms bonds. The force on these bonds ranges from 0 to

100 pN (discussed later). However, even less than half of these

bonds carry a force > 10 pN. Thus we consider a bond to be


10/16


shear rate (1/s)

pausetim

e(s)

0 100 200 300 400 500

0

0.2

0.4

0.6

0.8

Smith et al

shear rate (1/s)

stepsize(m

icron)

100 200 300 400 5000

1

2

3

4

5

Fig. 7. (A) Average pause time, and (B) step distance as functions of shear rate, microvilli distribution, and membrane stiffness. Eh = 2.6, Eh = 0.9, and

Eh = 0.3dyn/cm. Solid lines represent Nmv = 21 and dash line Nmv = 155. represent in vitro data of Smith et al. [6]. Pause time does not depend on

Nmv, hence only data for Nmv = 155 are shown.

X

X** *

shear rate (1/s)

100 200 300 400 500 6000

5

10

15

shear rate (1/s)

a

veragevelocity(micron/s)

100 200 300 400 5000

20

40

60

80

100

120

Kim &Sarelius

Fig. 8. Average rolling velocity for (A) Nmv = 21 and (B) Nmv = 155. Present results: Eh = 0.3, Eh = 0.9, Eh = 2.6dyn/cm. Experimental

results: - - - - Kim and Sarelius [4], Yago et al. [9], X Ramachandran et al. [39].

stretched when the force on it is > 5 pN. The average number

of total bonds ranges from 200 to 500, and agrees well with

that estimated by Jadhav et al. [29]. Remarkably, however, the

number of stretched bonds lies in the range 2080.Next we examine the history of the force on individual

microvilli. A representative case at 500 s1, Eh = 0.3dyn/cm,

and Nmv =21 is considered in Fig. 14A. The microvilli are iden-

tified by numbers 1, 2 etc. The location of these microvilli

on the cell surface was shown earlier in Fig. 4. Strikingly,

our simulations show that the force on a microvillus devel-

ops in steps. As an illustration, consider microvillus 4 which

comes in contact with the substrate and forms adhesion bonds

at around t= 0.03 s. However, the bonds on microvillus 4 are

not stretched until t= 0.13 s as the cell is tethered by microvilli

2 and 3. Hence the force acting on microvillus 4 remains

less than 10 pN. At t = 0.13 s, microvillus 2 breaks. As the

cell begins to roll, bonds in microvilli 3 and 4 are stretched,

and the forces on them sharply increase. Subsequently, the cell

is tethered by microvillus 3, and the force on it increases to

about 350 pN. The force on microvillus 4 also increases to

80 pN due to bond stretching. At aroundt =

0.175 s, microvil-lus 3 breaks. The cell rolls again, and the bonds in microvillus

4 are stretched further, and the force jumps to 280 pN.

We next examine the force history in individual

P-selectin/PSGL-1 bonds in Fig. 14B. Formation and breakage

of individual bonds occur throughout the lifetime of a tether.

When one bond breaks, forces on the remaining bonds in the

cluster increase in a step-like manner. It also implies that mul-

tiple bond breakage, rather than a single event, is necessary

for a microvillus tether to break. We also note that for a few

bonds, the peak force on individual bond is as high as 50 pN,

while for others the force remains below 10 pN. It again sug-

gests that, though multiple bonds are present, not all of them

are effective in tethering the cell.


11/16


The average bond rupture force, and the bond lifetime are

shown in Figs. 15A and B. The rupture force varies in the

range of 2080 pN, and it increases with increasing shear

rate and membrane stiffness. The bond lifetime, which corre-

lates well with the cell pause times shown earlier, decreases

with increasing shear and membrane stiffness. Dependence

of bond lifetime on membrane stiffness is stronger at lowshear.

5. Discussion

3D computational modeling and simulation are presented

to study adhesive rolling of deformable leukocytes in a shear

flow. We study the effect of cell deformation, shear rate, and

microvilli distribution on the rolling characteristics. The simu-

lations show the transient deformation of a tethered leukocyte

in to a tear-drop shape as observed in previous experiments

(Fig. 4). Computed cell deformation agrees well with earlier in

vivo and in vitro measurements (Fig. 5). The average rolling ve-

locity increases with increasing shear and membrane stiffness,

and also agrees well with previous experimental measurements

(Fig. 8).

The sequence of cell shape presented in Figs. 3 and 4 sug-

gested that the cellsubstrate contact area of a rolling leukocyte

varies with timeit increases when the cell is adherent, and

Time (s)

0.1 0.2 0.3 0.4 0.5

-200

0

200

400

E

Fig. 9. Sideway velocity (m/s) of a rolling leukocyte. The case shown is

the same as in Fig. 6E.

shear rate (1/s)

RMSvelocity(micron

/s)

100 200 300 400 5000

50

100

150

200

250

shear rate (1/s)

100 200 300 400 5000

50

100

150

200

250

Fig. 10. RMS velocity fluctuation of (A) axial and (B) sideway motion. Nmv = 21 and - - - - - Nmv = 155. Eh = 0.3, Eh = 0.9, and Eh = 2.6dyn/cm.

For (B) Nmv = 155 does not give any significant sideway motion, and hence data not shown.

decreases when the cell is rolling, in agreement with in vitro

observation of Dong et al. [8]. Previous in vivo and in vitro

studies suggested that rolling leukocytes become flattened

against the substrate upon initial tethering. Our results in Figs. 3

and 4 suggest that upon initial tethering, the cell rotates about

the tether followed by flattening of the cell surface which

occurs very rapidly, and often within a fraction of a second.One distinct characteristics of leukocyte rolling observed in

vivo and in vitro is that the cells do not roll continually, but

rather in a stop-and-go manner. Previous computational models

that considered deformation of adherent leukocytes did not re-

port such intermittent motion [2428]. The stop-and-go motion

is due to formation and breaking of receptor/ligand bonds, and

can only be predicted by stochastic simulation. Coupling cell

deformation with stochastic bond kinetics is relatively new in

the context of modeling of leukocyte rolling. Our IBM simula-

tion has been able to predict the stop-and-go motion of a leuko-

cyte. Our simulations predict that the fluctuations in rolling ve-

locity are reduced, and hence the rolling motion stabilizes, with

increasing cell deformability, in agreement with earlier exper-

imental observations [7,9].

Our simulations predicted that a leukocyte can undergo a

significant sideway motion during rolling. The sideway motion

is purely due to the stochastic nature of bond formation, and

discrete nature of microvilli presentation. A microvillus can

form an initial tether that may be oriented at a non-parallel angle

with the shear flow, causing the cell to move sideway once an

earlier tether breaks. However, once the cell is tethered by the

new microvillus, it quickly orients itself parallel to the flow.

Not only the stochastic formation and breakage of selectin

bonds, also the presentation of the microvilli affects the fluc-

tuating motion of a leukocyte. By considering two differentmicrovilli distributions, we show that the average rolling veloc-

ity and fluctuations are higher for the sparse distribution. The

higher rolling velocity is caused by increasing step size which

seems to correlate with the inter-microvilli distance. Presence

of microvilli in our model allows formation of discrete bond

clusters rather than uniform distribution of bonds over the


12/16


shear rate (1/s)

force(pN)

100 200 300 400 5000

200

400

600

800

shear rate (1/s)

force(pN)

100 200 300 400 5000

100

200

300

400

500

Flow

Y X

Fig. 11. (A) Average total adhesion force, (B) distribution of adhesion force among all bound microvilli in the cell/substrate contact area, (C) average peakforce on individual tethered microvilli, and (D) distribution of bound microvilli over the cell surface. The arrows indicate three microvilli forming tethers.

Nmv = 21 and - - - - - Nmv = 155. Eh = 2.6 and Eh = 0.3dyn/cm.

entire cell/substrate contact zone. In vivo measurement by

Zhao et al. [42] obtained step size 2 m, while in vitro mea-

surements by Alon et al. [46] for L-selectin mediated rolling

estimated 3.4m. These data, together with our prediction

of 0.5.4m step size for P-selectin mediated rolling sug-

gested that the step-distance is independent of the nature of

receptorligand complex, and is dependent on the microvilli

distribution. The pause time between successive rolling steps

is observed to depend not only on shear rate, but also on cell

deformability. On the contrary, it did not depend on microvilli

distribution.

We found that leukocyte adhesion is via multiple tethers,

varying from 2 to 5 in numbers. These numbers are, however,

significantly less than the number of microvilli, often up to 35,

which are available within the cell/substrate contact area, and

are observed to form bonds. Increasing the shear rate by a factor

of five only doubled the number of tethered microvilli from an

average value of 2.5 to 5. Though multiple tethers are observed

in most of our simulations, we also observe that a single tether

is often sufficient to support rolling. We examined how the

adhesion force is distributed among all microvilli that are bound

to the substrate. Though nearly all microvilli in the contact

area contain bonds, significant adhesive force is concentrated

in 13 tethered microvilli located in the rear-most part of the

cell.

The peak force on a tethered microvillus predicted by

our simulations is significantly above the force required for

tether extrusion (61 pN) measured by Shao et al. [43]. Our

results, therefore, predict that the microvilli in the rear end

of the cell would develop tether extrusion, while those in the

remaining part of the contact area would at most undergo

elastic tension. Although we have not considered deformation

of a microvillus in our model, such a consideration would

reduce the tether force [23,43,44]. In that case, tether ex-

trusion may occur only at higher shear rate and membrane

stiffness.

We observe that multiple, rather than single, selectin bonds

form per microvillus. When a tether microvillus breaks and

pulls from the substrate, several bonds, varying from 9 to 14,

break simultaneously. We also observe that during a pause in the

rolling motion, individual bonds within the tethered microvillus

can break. But breakage of individual bond does not lead to the


13/16


shear rate (1/s)

100

300

500

0

10

20

30

40

shear rate (1/s)10

0300

500

0

10

20

30

40

shear rate (1/s)

100

300

500

0

10

20

30

40

shear rate (1/s)

100

300

500

0

10

20

30

40155 microvilli

Eh=0.3 dyn/cm

155 microvilli

Eh=2.6 dyn/cm

21 microvilli

Eh=2.6 dyn/cm

21 microvilli

Eh=0.3 dyn/cm

Fig. 12. Average number of total bound microvilli (open bars) and tethered microvilli (solid bars) as functions of shear rate, cell compliance, and microvilli

distribution.

breakage of the tether, as other bonds quickly stretch to share

the load. Simultaneous breakage of multiple bonds can only

cause a tether to retract.

The facts that multiple bonds form per microvillus, and

the pause time of rolling motion was found to depend on cell

compliance, are noteworthy in the context of experimental es-

timation of bond dissociation rates. Bond dissociation rates are

often measured based on the pause times obtained from leuko-

cyte rolling in flow chamber. However, cell deformation cannot

be easily controlled in such experiments. Assuming single bond

dissociation, the Bell model [45] is fit to the measurements to

obtain the bond dissociation rate. Direct measurements using

atomic force microscopy or laser trap yielded different values

than that obtained from flow chamber experiments [3,11,47].

Our results suggest that two possible reasons for discrepancy

are cell deformation and formation of multiple bonds per

tether.

Our simulations predicted the average number of total

bonds in the range 100500, and it increases with increas-

ing shear rate and membrane compliance due to increased

cell-surface contact area resulting in more bound microvilli.

However, we observe that only a few bonds residing in the

tethered microvilli are significantly stretched. The average

number of stretched bonds is in the range of 2080 which

increases with increasing shear rate, due to increasing num-

ber of tethers, but does not depend on membrane compliance

due to the fact that the number of tethers did not increase ap-

preciably with increasing cell deformability. Per microvillus,

only 914 bonds are predicted to be significantly stretched,

which agrees well with the measurements of Chen and

Springer [10].

The maximum and average rupture force of a P-selectin/

PSGL-1 bond was estimated to be 100 and 60 pN, respectively,

which is in the same range as measured recently by Marshal

et al. [48] using atomic force microscopy. It is also in the

same range as that obtained by Schmidtke and Diamond ([41],

86172 pN at 100250 1/s). We observe that bond force in-

creases with increasing shear and membrane stiffness. Depen-

dence of bond force on cell deformability is also in agreement

with previous in vitro studies using untreated neutrophil, and

ligand-coated microbeads [7]. The bond force was measured to

be 500 pN for microbeads and 124 pN for neutrophil at a shear


14/16


shear rate (1/s)

Bonds

100 200 300 400 5000

100

200

300

400

500

Fig. 13. Average total bonds (solid lines) and stretched bonds (dash lines).

Eh = 2.6 and Eh = 0.3dyn/cm.

Time (s)

force

(pN)

0.05 0.1 0.15 0.2 0.250

100

200

300

400

500

1

2

3

3

5

4

Time (s)

force

(pN)

0.7 0.8 0.9 1 1.1 1.2 1.3 1.40

10

2030

40

50

60

70

Fig. 14. (A) Force history of a few microvilli and (B) force history of a few bonds within a particular microvillus. In (A), microvilli are indicated by number

1, 2, etc. which are same as shown earlier in Fig. 4. In (B) arrows indicate step increase in bond force in response to breaking of another bond.

shear rate (1/s)

Bondpeakforce(pN)

100 200 300 400 5000

20

40

60

80

100

shear rate (1/s)

Bondlifetime(s)

100 200 400300 500

0

0.2

0.4

0.6

0.8

1

Fig. 15. (A) Peak bond force and (B) bond life time. Eh = 2.6 and Eh = 0.3dyn/cm.

rate 100 s1 (see also Smith et al. [6]). Significantly reduced

bond force for neutrophil compared to that for microspheres

was attributed to microvilli elongation. In contrast, microvilli

extension was not considered in our model. Our result, there-

fore, suggests that cell deformation helps reducing the bond

force and prolong bond lifetime. Hydrodynamic shear and cell

deformation have competing effects on bond dissociation. Forceon individual bond increases, and hence bond lifetime de-

creases, with increasing shear. Despite rapid bond dissociation

at high shear, leukocyte rolling is stabilized by cell deformation

via two pathways. First, increased cell/substrate contact area

causes more microvilli to be accessible for bond formation. In-

creased number of tethers with increasing shear alleviates the

force on individual bond. The second mechanism is the reduc-

tion of the hydrodynamic drag on a deformed cell which also

reduces bond force and hence the dissociation rate.

In conclusion, we presented a 3D model to simulate a rolling

leukocyte in shear flow. Our model predicted the stop-and-go

motion of leukocytes. The major results are: (i) compliant cells

roll more stably, and have longer pauses due to reduced bond

force and increased bond lifetime, (ii) microvilli presentation

affects rolling characteristics by altering the step size, (iii)

adhesion force is concentrated in only 13 tethered microvilli

in the rear-most part of a cell, (iv) number of effective bonds

is much less than total adhesion bonds, (v) adhesion is via


15/16


multiple tethers, each of which forms multiple selectin bonds,

but often one tether is sufficient to support rolling, (vi) force

loading on individual microvillus and selectin bond is not con-

tinuous, rather occurs is steps, (vii) peak force on a tethered

microvillus is much higher than that measured to cause tether

extrusion. Thus, both cell deformation and microvillus defor-

mation occur simultaneously during cell rolling, and need tobe considered in future computationalmodels.

Conflict of interest statement

None declared.

Acknowledgments

This research is partially supported by a Busch Biomedical

Grant from Rutgers University, and NSF Grant BES-0603035.

Computational support from National Center for Supercomput-

ing Applications at Urbana, IL is acknowledged. Authors thank

R.M. Kalluri and Sai Doddi for their help.

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Prosenjit Bagchi received his PhD from the University of Illinois at Urbana-Champaign in 2002. Currently he is an assistant professor in Mechanical andAerospace Engineering at Rutgers University.

Vijay Pappu is a graduate student in Mechanical and Aerospace Engineeringat Rutgers University.

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