a model for cd2/cd58-mediated adhesion strengthening

11
Annals of Biomedical Engineering, Vol. 33, No. 4, April 2005 (© 2005) pp. 483–493 DOI: 10.1007/s10439-005-2504-5 A Model for CD2/CD58-Mediated Adhesion Strengthening JIN-YU SHAO, 1 Y AN YU, 1 and MICHAEL L. DUSTIN 2 1 Department of Biomedical Engineering, Washington University, Saint Louis, MO; and 2 Department of Pathology, New York UniversitySchool of Medicine, New York, New York (Received 31 May 2004; accepted 30 October 2004) Abstract—Stable cell adhesion is vital for structural integrity and functional efficacy. Yet how low affinity adhesion molecules such as CD2 and CD58 can produce stable cell adhesion is still not com- pletely understood. In this paper, we present a theoretical model that simulates the accumulation of CD2 and CD58 in the contact area of a Jurkat T lymphoblast and a CD58-containing substrate. The cell is assumed to have a spherical shape initially and it is allowed to spread gradually on a circular substrate. Mobile CD2 and CD58 can diffuse freely on both the cell and substrate. Their binding in the contact area is controlled by first-order kinetics. The contact area grows linearly with the total number of CD2/CD58 bonds. Cellular deformation and cytoskeleton involvement were not considered. This time-dependent moving-boundary problem was solved with the Crank–Nicolson finite difference scheme and the variable space grid method. Our simulated results are in rea- sonable agreement with the experimental observations. The role of diffusion becomes more and more prominent during the contact area increase, which is not sensitive to the kinetic rate constants tested in this study. However, it is very sensitive to the dissociation equilibrium constant and the concentrations of CD2 and CD58. Keywords—Kinetics, Diffusion, Lymphocyte, Moving boundary, Equilibrium constant, Receptor–ligand bonds. NOMENCLATURE α rate of increase in the contact area (µm 2 /molecule) ε 112 , dimensionless parameters defined in the p bc , p bs text A C instantaneous contact area (µm 2 ) A instantaneous concentration of mobile CD2 (molecules/µm 2 ) A I instantaneous concentration of immobile CD2 (molecules/µm 2 ) B instantaneous concentration of mobile CD58 (molecules/µm 2 ) B I instantaneous concentration of immobile CD58 (molecules/µm 2 ) Address correspondence to Jin-Yu Shao, PhD, Department of Biomed- ical Engineering, Washington University in St. Louis, Campus Box 1097, Rm 290E UA Whitaker Hall, One Brookings Drive, St. Louis, MO 63130- 4899. Electonic mail: [email protected] B T bound CD58 concentration at equilibrium (molecules/µm 2 ) C A0 initial concentration of mobile CD2 (molecules/µm 2 ) C AI initial concentration of immobile CD2 (molecules/µm 2 ) C B0 initial concentration of mobile CD58 (molecules/µm 2 ) C BI initial concentration of immobile CD58 (molecules/µm 2 ) C instantaneous bond concentration due to mobile CD2 and CD58 (molecules/µm 2 ) C I instantaneous bond concentration due to immobile CD2 (molecules/µm 2 ) C II instantaneous bond concentration due to immobile CD58 (molecules/µm 2 ) D A diffusion coefficient of CD2 (µm 2 /s) D AC diffusion coefficient of CD2 in the contact region (µm 2 /s) D B diffusion coefficient of CD58 (µm 2 /s) D BC diffusion coefficient of CD58 in the contact region (µm 2 /s) F free CD58 concentration at equilibrium (molecules/µm 2 ) f A fractional mobility of CD2 f B fractional mobility of CD58 f mA percentage of bonds due to mobile CD2 f mB percentage of bonds due to mobile CD58 K d 2D dissociation equilibrium constant (molecules/µm 2 ) k f 2D forward rate constant for mobile CD2 and CD58 (µm 2 /s) k r 2D reverse rate constant for mobile CD2 and CD58 (s 1 ) k I f 2D forward rate constant for immobile CD2 and mobile CD58 (µm 2 /s) k I r 2D reverse rate constant for immobile CD2 and mobile CD58 (s 1 ) k II f 2D forward rate constant for mobile CD2 and immobile CD58 (µm 2 /s) k II r 2D reverse rate constant for mobile CD2 and immobile CD58 (s 1 ) 483 0090-6964/05/0400-0483/1 C 2005 Biomedical Engineering Society

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Page 1: A Model for CD2/CD58-Mediated Adhesion Strengthening

Annals of Biomedical Engineering, Vol. 33, No. 4, April 2005 (© 2005) pp. 483–493DOI: 10.1007/s10439-005-2504-5

A Model for CD2/CD58-Mediated Adhesion Strengthening

JIN-YU SHAO,1 YAN YU,1 and MICHAEL L. DUSTIN2

1Department of Biomedical Engineering, Washington University, Saint Louis, MO; and 2Department of Pathology,New York University School of Medicine, New York, New York

(Received 31 May 2004; accepted 30 October 2004)

Abstract—Stable cell adhesion is vital for structural integrity andfunctional efficacy. Yet how low affinity adhesion molecules suchas CD2 and CD58 can produce stable cell adhesion is still not com-pletely understood. In this paper, we present a theoretical modelthat simulates the accumulation of CD2 and CD58 in the contactarea of a Jurkat T lymphoblast and a CD58-containing substrate.The cell is assumed to have a spherical shape initially and it isallowed to spread gradually on a circular substrate. Mobile CD2and CD58 can diffuse freely on both the cell and substrate. Theirbinding in the contact area is controlled by first-order kinetics. Thecontact area grows linearly with the total number of CD2/CD58bonds. Cellular deformation and cytoskeleton involvement werenot considered. This time-dependent moving-boundary problemwas solved with the Crank–Nicolson finite difference scheme andthe variable space grid method. Our simulated results are in rea-sonable agreement with the experimental observations. The roleof diffusion becomes more and more prominent during the contactarea increase, which is not sensitive to the kinetic rate constantstested in this study. However, it is very sensitive to the dissociationequilibrium constant and the concentrations of CD2 and CD58.

Keywords—Kinetics, Diffusion, Lymphocyte, Moving boundary,Equilibrium constant, Receptor–ligand bonds.

NOMENCLATURE

α rate of increase in the contact area(µm2/molecule)

ε1−12, dimensionless parameters defined in thepbc, pbs text

AC instantaneous contact area (µm2)A instantaneous concentration of mobile CD2

(molecules/µm2)AI instantaneous concentration of immobile

CD2 (molecules/µm2)B instantaneous concentration of mobile CD58

(molecules/µm2)BI instantaneous concentration of immobile

CD58 (molecules/µm2)

Address correspondence to Jin-Yu Shao, PhD, Department of Biomed-ical Engineering, Washington University in St. Louis, Campus Box 1097,Rm 290E UA Whitaker Hall, One Brookings Drive, St. Louis, MO 63130-4899. Electonic mail: [email protected]

BT bound CD58 concentration at equilibrium(molecules/µm2)

CA0 initial concentration of mobile CD2 (molecules/µm2)CAI initial concentration of immobile CD2

(molecules/µm2)CB0 initial concentration of mobile CD58

(molecules/µm2)CBI initial concentration of immobile CD58

(molecules/µm2)C instantaneous bond concentration due to mobile CD2

and CD58 (molecules/µm2)CI instantaneous bond concentration due to immobile

CD2 (molecules/µm2)CII instantaneous bond concentration due to immobile

CD58 (molecules/µm2)DA diffusion coefficient of CD2 (µm2/s)DAC diffusion coefficient of CD2 in the contact region

(µm2/s)DB diffusion coefficient of CD58 (µm2/s)DBC diffusion coefficient of CD58 in the contact region

(µm2/s)F free CD58 concentration at equilibrium

(molecules/µm2)fA fractional mobility of CD2fB fractional mobility of CD58fmA percentage of bonds due to mobile CD2fmB percentage of bonds due to mobile CD58Kd 2D dissociation equilibrium constant

(molecules/µm2)kf 2D forward rate constant for mobile CD2 and CD58

(µm2/s)kr 2D reverse rate constant for mobile CD2 and CD58

(s−1)kI

f 2D forward rate constant for immobile CD2 andmobile CD58 (µm2/s)

kIr 2D reverse rate constant for immobile CD2 and

mobile CD58 (s−1)kII

f 2D forward rate constant for mobile CD2 andimmobile CD58 (µm2/s)

kIIr 2D reverse rate constant for mobile CD2 and

immobile CD58 (s−1)

483

0090-6964/05/0400-0483/1 C© 2005 Biomedical Engineering Society

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484 SHAO et al.

N1 total number of intervals in the contact regionN2 total number of intervals in the noncontact

substrateN3 total number of intervals in the noncontact cell

surfaceNtA total number of CD2 molecules per cellNtB total number of CD58 available on the substrate

for each cellNb total number of bonds in the contact area at

equilibriumRC0 initial radius of the contact area (µm)RS radius of the cell (µm)RC instantaneous radius of the contact area (µm)RSUB radius of the substrate (µm)r, θ, ϕ spherical coordinate systemr, θ, z cylindrical coordinate systemSb contact area at equilibrium (µm2)Scell cellular surface area (µm2)SSUB available substrate area for each cell (µm2)t time (s)

INTRODUCTION

Specific and stable cell adhesion is a crucial step ina myriad of biological processes including inflammation,cell locomotion, and cancer metastasis. It is well-knownthat specific cell adhesion is mediated by molecular bondsbetween receptors and ligands. To maintain its versatility,a cell must have many types of receptor constitutively ex-pressed on its surface. As a result, the concentration of acertain type of membrane receptor is usually small and,even with large forward reaction rate constants, not manybonds can form shortly after a cell contacts another cellor substrate. However, only a few receptor–ligand bondsare usually not strong enough to support stable adhesion,which is required for the ensuing signal transduction, cellmigration, and other cellular responses.24 To achieve stableadhesion, a cell can increase its receptor–ligand affinity,9,25

upregulate its receptor expression,23,24 or redistribute morereceptors and ligands into the contact area.4,14,18 The lattermechanism, which is the focus of this paper, is involved inthe adhesion between T lymphocytes and antigen present-ing cells (APCs).

T lymphocyte adhesion to APCs results in the formationof the immunological synapse, where different receptor–ligand pairs occupy distinct geometric regions in the con-tact area between these two cells. The immunologicalsynapse formation involves proteins such as lymphocytefunction-related antigen-1 (LFA-1), intercellular adhesionmolecule-1 (ICAM-1), T cell receptor (TCR), peptide-major histocompatibility complex (pMHC), LFA-2 (CD2),and LFA-3 (CD58).11,16 In the synapse, short TCR/pMHCand CD2/CD58 bonds tend to accumulate in the centralregion, while long LFA-1/ICAM-1 bonds prefer the pe-

riphery. Several theoretical models related to the synapseformation have been developed. Coombs et al. only con-sidered the TCR/pMHC interaction in the synapse,6 whileothers considered all molecules involved and predicted thespatiotemporal evolution of the synapse.3,5,17,22 In all thesemodels, a constant contact area in the synapse was as-sumed. However, it has been shown experimentally that,when CD2-expressing Jurkat T lymphoblasts were allowedto settle on a CD58-incorporated lipid bilayer substrate,both CD2 and CD58 would accumulate in the contactarea, which would increase simultaneously with the num-ber of CD2 and CD58 molecules inside the contact area.4,14

The same phenomenon of molecular accumulation in thecontact area was also observed in the adhesion betweencell-size lipid vesicles bearing dinitrophenyl(DNP)-lipidhaptens and rat basophilic leukemia cells that expressFcε receptors bound with anti-DNP IgE.18 Although thedetailed experimental data are available on CD2/CD58-mediated adhesion strengthening, to our knowledge, notheoretical understanding of this phenomenon has beenattempted. Here, we present a model that illustrates howvarious parameters such as diffusion coefficients and ki-netic rate constants influence this adhesion strengtheningprocess mediated by CD2 and CD58.

CD2 is an adhesion molecule expressed constitutivelyon T lymphocytes and its most potent ligand in humansfound to date is CD58. The three-dimensional (3D) affin-ity between CD2 and CD58, measured when at least oneof them is in solution, is low (from 9 to 22 µM at 37◦Cand 0.5 µM at 25◦C), but their reverse reaction rate con-stant is quite high (≥4 s−1 at 37◦C).12,26 The interac-tion between CD2 and CD58 is governed by their two-dimensional (2D) kinetics since both of them are confinedto 2D cellular membrane surfaces. For receptor–ligand in-teractions in general, although their 2D reverse reactionrate constant might be comparable to their 3D counter-part, their forward reaction rate constant is definitely dif-ferent from their 3D value.10,19–21 It is possible to con-vert a 3D affinity constant to its 2D counterpart with theheight of the confinement region (σ ), which can be calcu-lated by dividing the 2D dissociation equilibrium constant(Kd) by the 3D Kd.2,12,14 However, this calculation makes anumber of assumptions that have not been experimentallytested.

By analyzing the protein and bond concentrations atthe equilibrium state of CD2/CD58-mediated adhesionstrengthening, the 2D Kd for CD2/CD58 binding can becalculated to be around 1.1/µm2.8,12 The 2D kinetic ratesof CD2/CD58 binding are still unknown, but photobleach-ing studies suggest that their interactions remain dynamicin the contact area such that bound receptors and ligandsdo dissociate and diffuse apart.8 Overall, the accumula-tion process is limited by diffusion. However, whether andhow receptor–ligand kinetics has any impact on the contactarea increase are still unknown. A theoretical model for

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Simulation of Adhesion Strengthening 485

CD2/CD58-mediated adhesion strengthening will help usbetter understand its molecular mechanism, thus leadingus to improved understanding of in vivo processes, inter-pretation of in vitro assays, and analysis and design ofbiotechnological processes.

MATHEMATICAL MODEL

The adhesion of a single T lymphoblast to a CD58-incorporated substrate and its ensuing strengthening, i.e.,the experiment conducted by Dustin et al.,12,14 will be sim-ulated. It is not our intention to model the formation of theimmunological synapse although the CD2/CD58 interac-tion is involved in the synapse formation. In our model, thecell is spherical initially. During the adhesion strengthening,the cell shape outside the contact area is assumed to remainspherical, as suggested by the electron micrograph shownin Fig. 1(a). This is likely due to the excess membranematerials stored in the microvilli as seen in Fig. 1(a). Asa result, cellular deformation and cytoskeletal involvementwere considered minimal during the whole process and ne-glected in the model. Also shown in Fig. 1(a) is the flat cellmembrane in the contact region. Consequently, the contactregion is assumed to be flat as shown in our model geometry[Fig. 1(b)]. The cell is modeled as a sphere with constantvolume, from which θmax was determined, and the substrateis modeled as a circular area with the contact region at thecenter. The spherical coordinate system (r, θ, ϕ) is selectedfor the spherical region of the cell (only θ is used in themodel) and the cylindrical coordinate system (r, θ, z) isselected for the contact region of the cell and substrate, aswell as the noncontact region of the substrate (only r is usedin the model).

In each of these four regions, the concentrations of CD2and CD58 are modeled by reaction-diffusion equations. Thereaction between CD2 and CD58 is modeled by first-orderkinetics. It is assumed that there is no convection on boththe cell and substrate. The initial contact area is dependenton the molecular lengths and cellular topography. Usingthe data obtained by Dustin et al.,12 we plotted the contactarea (AC) and the total number of bonds in the contactarea (Nb) at equilibrium and found a linear relationship(Fig. 2):

AC = 7.6 + 1.0 × 10−3 Nb. (1)

Therefore, it is assumed that the size of the contact area islinearly correlated with the number of bonds in the contactregion, i.e.,

π R2C = π R2

C0 + αNb = π R2C0

+ α

∫ RC

0(C + CI + CII) 2πr dr, (2)

where RC0 is the initial contact radius and α = 1.0 × 10−3

µm2. For details of the governing equations, initial condi-

FIGURE 1. Model geometry. (a) An electron micrograph show-ing a Jurkat T lymphocyte adherent to a glass-supported pla-nar lipid bilayer containing CD58. (b) Schematic representa-tion of the model geometry (not drawn to scale). RC is theinstantaneous radius of the contact region. RS is the instan-taneous radius of the spherical region of the cell. θmax is theangle shown in the diagram and it represents the interfacebetween the contact and spherical region of the cell. RSUB isthe radius of the substrate region. RSUB can be determined bythe cell density on the substrate, i.e., πR2

SUB is equal to thetotal substrate area divided by the total number of cells on thesubstrate.

tions, boundary conditions, nondimensionalization, param-eter values, and solution procedures, see Appendices A, B,and C.

RESULTS

Dynamics of CD2, CD58, and CD2/CD58 Bondsduring the Adhesion Strengthening

As observed by Dustin et al.,12,14 our simulation showedthat the contact area between the cell and substrate in-creased as more and more CD2 and CD58 diffused into thecontact region. During this process, the characteristic timeof reaction is much shorter than the characteristic time of

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486 SHAO et al.

FIGURE 2. The linear correlation between the contact area (AC)and the total number of bonds in the contact area (Nb). Datafrom Dustin et al.12 A linear fit through the five points yields:AC = 7.6 + 1.0 × 10−3Nb (correlation coefficient = 0.99913).

diffusion, so the concentrations of CD2 and CD58 in thecontact region decreased dramatically and rapidly at the be-ginning. Shown in Figs. 3(a), 3(b), and 3(c) are how the con-centrations of mobile CD2, mobile CD58, and CD2/CD58bonds changed over time, respectively, for a typical case:CA0 = 41/µm2, CB0 = 36/µm2, and Kd = 1.1/µm2. It isobvious that large concentration gradients occurred aroundthe interface between the contact and noncontact regions[Figs. 3(a) and 3(b)]. CD58 diffuses almost 10 times fasteron the substrate than CD2 does on the cell surface. Conse-quently, their concentration changes showed different pat-terns as illustrated in Figs. 3(a) and 3(b). In the contactregion, the concentration of mobile CD2 first decreasedto a very low level due to the fast reaction in the contactregion, and then increased slowly to about 70% of the initialconcentration (steady state) as more and more CD2 diffusedinto the contact region. Although the concentration of mo-bile CD58 in the contact region also decreased first, it thenincreased rapidly due to the faster diffusion of CD58 onthe substrate. Afterwards, it decreased again to about 55%of the initial concentration (steady state) as more and moreCD2 accumulated slowly in the contact region and formedmore bonds with CD58.

As shown in Fig. 3(c), there was a rapid increase in theradius of the contact region at the beginning as if the initialcontact radius were larger than one (a supplemental videothat shows the whole strengthening process is available bycontacting the authors at [email protected] or onlineat http://biomed.wustl.edu/faculty/shao/movie2.htm). Dur-ing the latter strengthening phase, most of the increase inthe contact radius took place before t = 150. Before thewhole process reached the steady state where a uniformbond distribution was reached, the bond concentration wasalways larger at the periphery of the contact region. Atthe steady state, the contribution from immobile CD2 andCD58 to the contact area increase was only about 4.5%.

FIGURE 3. Dynamics of the concentrations of mobile CD2, mo-bile CD58, and CD2/CD58 bonds. All variables shown here aredimensionless (see Appendix B for their scales). (a) The con-centration of mobile CD2 vs. the cellular peripheral distance(L) at six chosen time points. L is defined as the distance fromthe center of the contact region to a certain point on the cellsurface and can be calculated from L = r + RS(θmax − θ). (b)The concentration of mobile CD58 vs. radius (r) at six cho-sen time points. (c) The radius of the contact region vs. time.The color bar shows different concentrations of CD2/CD58bonds.

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Simulation of Adhesion Strengthening 487

While keeping Kd at 1.1/µm2, we increased the 2D rateconstants to 10 times or decreased them to one-fourth oftheir values used in this simulation (see Appendix C). Wefound almost no difference in the overall growth of thecontact area although a slightly different growth rate wasobserved at the beginning. Therefore, the overall processwas dictated by Kd and it was insensitive to the rate con-stants in the vicinity of the rate constants used here. Unlesswe build an extremely slow reaction (not very realistic)into our model, different reaction rate constants wouldessentially have no effect on the growth of the contactarea. Consequently, to enable us to calculate the 2D rateconstants of CD2/CD58 reaction from this type of experi-ment, the contact area increase or the molecular bond con-centration has to be precisely measured in the first fewseconds.

Dependence of AC on the Concentration of CD58

If more CD58 is incorporated into the lipid bilayer onthe substrate, more increase in the contact area would beexpected. As in the experiment done by Dustin et al.,12 weused five different concentration of mobile CD58 in oursimulation: CB0 = 9, 18, 36, 72, and 144 molecules/µm2.As shown in Fig. 4(a), at larger concentrations of CD58,more computational time was needed for the computation toreach the steady state and more contact area was generated.The overall increase in the contact area was very sensitiveto the concentration of CD58. Although our results showedthe same trend as the values obtained by Dustin et al.,12 theyare consistently lower at corresponding concentrations ofCD58 [Fig. 4(b)]. This is probably because the increase inthe contact area is equally sensitive to the concentration ofCD2. If 110/µm2 (an upper limit) was used as the total con-centration of CD2 on the cell surface (CA0 = 77/µm2),12 amuch larger contact area, 64.39 µm2, would be generatedat CB0 = 144/µm2 [Fig. 4(b)].

Dependence of AC on Kd

If Kd is decreased, i.e., the affinity between CD2 andCD58 is stronger, more bonds would form. Consequently,more CD2 and CD58 would diffuse into the contact re-gion, and the contact area would become larger. Figure 5(a)shows how the contact area would grow over time atCB0 = 36/µm2 and five different values of Kd. The forwardreaction rate constant was kept constant when Kd was var-ied. At smaller Kds, more CD2 and CD58 will diffuse intothe contact region, thus requiring more computational timefor these cases to reach the steady state. Figure 5(b) showsthe comparison of our results with the values obtained byDustin et al. at Kd = 0.4/µm2, five different concentrationsof CD58, and two different concentrations of CD2.12 It isclear that the contact area is very sensitive to Kd. If thetotal number of CD2 can be accurately measured in the

FIGURE 4. Dependence of AC on the concentration of CD58.(a) The growth of AC over time at five different concentrationsof mobile CD58. (b) Comparison between the data from Dustinet al.12 and our simulated final contact area at Kd = 1.1/µm2,five different concentrations of mobile CD58, and two differentconcentrations of mobile CD2. All error bars show the standarderrors of the mean.

experiment, the value of Kd can be found by fitting ournumerical results to the experimental values.

DISCUSSION

In this paper, we have developed a theoretical model tosimulate the adhesion strengthening between a T cell anda substrate coated with CD58, which resembles an APCsurface. The model, which is based on diffusion theoryand chemical reaction kinetics, includes all the parametersthat have been found to date to play a role in this process.We did not attempt to simulate the immunological synapseformation, but rather focused on one aspect of the synapseformation, i.e., CD2/CD58-mediated adhesion strengthen-ing and contact-area increase. In a way, this approach issimilar to the one used by Coombs et al.,6 who evaluatedthe roles of serial engagement and kinetic proofreading inTCR internalization by considering TCR-pMHC bindingalone in the synapse. The results that were obtained from

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488 SHAO et al.

FIGURE 5. Dependence of AC on Kd. (a) The growth of AC overtime at five different values of Kd. (b) Comparison between thedata from Dustin et al.12 and our simulated final contact areaat Kd = 0.4/µm2, five different concentrations of mobile CD58,and two different concentrations of mobile CD2. All error barsshow the standard errors of the mean.

our model are in reasonable agreement with the data fromthe experiment done by Dustin et al.12,14 We found thatthe bond concentration was always higher at the periph-ery of the contact region before the whole process reachedthe steady state. This could be due to two reasons: 1) weassumed that CD2/CD58 bonds are not diffusible in thecontact region, probably through its interaction with CD2-associated protein at a forward reaction rate constant of3.0 × 103 M−1·s−1 and a reverse reaction rate constant of3.8 × 10−4 s−1 (Kd = 130 nM),13 and 2) free CD2 andCD58 always reach the area close to the contact bound-ary first. Even if CD2/CD58 bonds can move around inthe contact region, the bond concentration would remainhigh at the periphery of the contact region because of theaforementioned second reason. However, the concentrationdifference along the radial direction might be too smallto be measurable with existing imaging technology. Oncea CD2/CD58 bond is formed, its contribution to the con-tact area increase will not depend upon where it is in thecontact area. Therefore, we do not expect that bond mo-bility will greatly influence the contact-area increase. In

our simulation, we also predicted that the concentrationof free CD58 would decrease, then increase, and decreaseagain in the contact region. Since only the total concentra-tion of CD58 was measured in the experiment by Dustinet al.,12,14 whether this prediction is correct remains to beverified.

Scatchard analysis cannot be directly applied to theadhesion-strengthening experiment because the total num-ber of receptor and the total number of ligand are not con-stant in the contact area. Consequently, Zhu-Golan anal-ysis, which is an improved version of Scatchard analysisthat takes account of variable receptor and ligand numbersin the contact area, was employed by Dustin et al.12 Weperformed Zhu-Golan analysis of our simulated data whereKd = 1.1/µm2 was used, and obtained a value of 1.1/µm2

for the 2D Kd.12 This result shows that Zhu-Golan analysisis reasonable, although it did not consider the immobilefraction of the molecules involved. CD2 and CD58 are mo-bile and the contact area is relatively small compared withthe cell surface area or the substrate, so the contributionfrom the mobile fractions of CD2 and CD58 to the adhesionstrengthening is dominant. When the immobile fraction ofCD2 and CD58 is large or the contact area is not small com-pared with the cell surface area or the substrate, Zhu-Golananalysis likely needs to be modified for the calculation ofKd (Appendix D).

The model established here is quite robust althoughlong computational time is needed for the cases where theconcentration of CD58 is large, the dissociation equilib-rium constant is small, or the characteristic time of re-action becomes too long or too short. For these cases,parallel computing or more powerful computers will bemuch more effective. It should not be difficult to extendthis model to other systems like CD2/CD48 interactions.However, some limitations do exist. One limitation of thecurrent model is that the mechanics of the cellular de-formation was not considered. Another limitation is theabsence of active participation of the cytoskeleton in theprocess of adhesion strengthening. Although these fac-tors might not be involved in the CD2/CD58-mediatedadhesion strengthening, if they are included in the fu-ture model, it might allow us to simulate more compli-cated systems such as focal adhesion development. Moreimportantly, it might allow us to combine this modelwith other models of synapse formation and develop amore comprehensive model for T lymphocyte adhesion toAPCs.

APPENDIX A: GOVERNING EQUATIONS,INITIAL CONDITIONS, ANDBOUNDARY CONDITIONS

Both mobile and immobile CD2 are present on the cellsurface. For mobile CD2 in the noncontact region of the

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Simulation of Adhesion Strengthening 489

cell (spherical),

∂ A

∂t= DA

1

R2S sin θ

∂θ

(sin θ

∂ A

∂θ

), (A.1)

A = CA0 @ t = 0 and 0 ≤ θ ≤ θmax, (A.2)

∂ A

∂θ= 0 @ θ = 0, (A.3)

where A is the instantaneous concentration of mobile CD2,t is time, DA is the diffusion coefficient of CD2 in thenoncontact region, RS is the radius of the cell, and CA0 isthe initial concentration of mobile CD2.

For mobile CD2 in the contact region of the cell (flat andcircular),

∂ A

∂t= DAC

1

r

∂r

(r

∂ A

∂r

)− ∂C

∂t, (A.4)

A = CA0 @ t = 0 and 0 ≤ r ≤ RC, (A.5)

∂ A

∂r= 0 @ r = 0, (A.6)

where DAC is the diffusion coefficient of CD2 in the contactregion, C is the bond concentration, and RC is the instan-taneous radius of the contact region. For immobile CD2,their diffusion coefficient is zero, but they can still reactwith mobile CD58. Their concentration, AI, is governed by

∂ AI

∂t= −∂CI

∂t, (A.7)

AI = CAI = CA01 − fA

fA@ t = 0 and 0 ≤ r ≤ RC,

(A.8)

where CI is the bond concentration due to immobile CD2and fA is the fractional mobility of CD2.

CD58 also has two fractions: mobile and immobile. Formobile CD58 in the contact region of the substrate (flat andcircular),

∂ B

∂t= DBC

1

r

∂r

(r

∂ B

∂r

)− ∂C

∂t, (A.9)

B = CB0 @ t = 0 and 0 ≤ r ≤ RC,

(A.10)

∂ B

∂r= 0 @ r = 0, (A.11)

where B is the instantaneous concentration of mobile CD58,DBC is the diffusion coefficient of CD58 in the contactregion, and CB0 is the initial concentration of CD58 in thecontact region. For immobile CD58 in the contact region,we have

∂ BI

∂t= −∂CII

∂t, (A.12)

BI = CBI = CB01 − fB

fB@ t = 0 and 0 ≤ r ≤ RC,

(A.13)

where BI is the concentration of immobile CD58 on thesubstrate, CII is the bond concentration due to immobileCD58, and fB is the fractional mobility of CD58.

For mobile CD58 in the non-contact region of the sub-strate (flat and circular),

∂ B

∂t= DB

1

r

∂r

(r

∂ B

∂r

), (A.14)

B = CB0 @ t = 0 and RC ≤ r ≤ RSUB,

(A.15)

∂ B

∂r= 0 @ r = RSUB, (A.16)

where DB is the diffusion coefficient of CD58 and RSUB isthe radius of the substrate.

For the reactions in the contact region,

∂C

∂t= kf AB − krC, (A.17)

∂CI

∂t= kI

f AI B − kIrCI, (A.18)

∂CII

∂t= kII

f ABI − kIIr CII, (A.19)

C = CI = CII = 0 @ t = 0 and 0 ≤ r ≤ RC,

(A.20)

where kf , kr, kIf , kI

r , kIIf , and kII

r are the two-dimensionalforward and reverse rate constants. The diffusion of thereceptor–ligand bond is neglected. Since receptor–ligandbindings are usually diffusion-limited on the cell surface,we assumed that

kIf = kf

DBC

DAC + DBC, (A.21)

kIIf = kf

DAC

DAC + DBC, (A.22)

kIr

kIf

= kIIr

kIIf

= kr

kf= Kd. (A.23)

At the interface between the contact and noncontact re-gion, the concentration and flux should be continuous. Forthe continuity of flux on the cell, we have

DAC∂ A

∂r

∣∣∣∣r=RC

= − DA1

RS

∂ A

∂θ

∣∣∣∣θ=θmax

. (A.24)

For the continuity of flux on the substrate, we have

DBC∂ B

∂r

∣∣∣∣r=R−

C

= DB∂ B

∂r

∣∣∣∣r=R+

C

. (A.25)

APPENDIX B: NONDIMENSIONALIZATION

Next, we will nondimensionalize our equations, bound-ary conditions, and initial conditions using CA0 as the scale

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490 SHAO et al.

for concentration, the initial radius of the contact region(RC0) as the scale for length, and R2

C0/DA as the scale fortime, which is 33.6 s. The following are the dimension-less equations, boundary conditions, and initial conditions(for convenience, dimensionless variables are representedby their original symbols including A, AI, B, BI, C, CI,CII, RC, r, and t). Note that θ itself is a dimensionlessvariable.

For mobile CD2 in the noncontact region of the cell,

∂ A

∂t= ε1

1

sin θ

∂θ

(sin θ

∂ A

∂θ

), (B.1)

A = 1 @ t = 0 and 0 ≤ θ ≤ θmax, (B.2)

where

ε1 = R2C0

R2S

. (B.3)

The boundary condition at θ = 0 remains the same.For mobile CD2 in the contact region of the cell,

∂ A

∂t= ε2

1

r

∂r

(r

∂ A

∂r

)− ∂C

∂t, (B.4)

A = 1 @ t = 0 and 0 ≤ r ≤ RC, (B.5)

where

ε2 = DAC

DA. (B.6)

The boundary condition at r = 0 remains the same. Forimmobile CD2 in the contact region, Eq. (A.7) will remainthe same, but Eq. (A.8) would become

AI = 1 − fA

fA@ t = 0 and 0 ≤ r ≤ RC. (B.7)

For mobile CD58 in the contact region of the substrate,

∂ B

∂t= ε3

1

r

∂r

(r

∂ B

∂r

)− ∂C

∂t, (B.8)

B = CB0

CA0@ t = 0 and 0 ≤ r ≤ RC, (B.9)

where

ε3 = DBC

DA. (B.10)

The boundary condition at r = 0 remains the same. Forimmobile CD58, Eq. (A.12) will remain the same, but Eq.(A.13) would become

BI = CB0

CA0

1 − fB

fB@ t = 0 and 0 ≤ r ≤ RC.

(B.11)For mobile CD58 in the noncontact region of the

substrate,

∂ B

∂t= ε4

1

r

∂r

(r

∂ B

∂r

), (B.12)

B = CB0

CA0@ t = 0 and RC ≤ r ≤ RSUB

RC0,

(B.13)

where

ε4 = DB

DA. (B.14)

The boundary condition at r = 0 remains the same.For the reactions in the contact region,

∂C

∂t= ε5 AB − ε6C, (B.15)

∂CI

∂t= ε7 AI B − ε8CI, (B.16)

∂CII

∂t= ε9 ABI − ε10CII, (B.17)

where

ε5 = kfCA0 R2C0

DA, (B.18)

ε6 = kr R2C0

DA, (B.19)

ε7 = kIfCA0 R2

C0

DA, (B.20)

ε8 = kIr R2

C0

DA, (B.21)

ε9 = kIIf CA0 R2

C0

DA, (B.22)

ε10 = kIIr R2

C0

DA. (B.23)

The initial condition [Eq. (A.20)] remains the same.At the interface between the contact and noncontact

region,

ε2∂ A

∂r

∣∣∣∣r=RC

= −ε11∂ A

∂θ

∣∣∣∣θ=θmax

, (B.24)

ε3∂ B

∂r

∣∣∣∣r=R−

C

= ε4∂ B

∂r

∣∣∣∣r=R+

C

, (B.25)

where

ε11 = RC0

RS. (B.26)

For the contact area increase [Eq. (2)],

R2C = 1 + ε12

∫ RC

0(C + CI + CII)2r dr, (B.27)

where

ε12 = αCA0. (B.28)

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Simulation of Adhesion Strengthening 491

APPENDIX C: PARAMETER VALUES ANDNUMERICAL SOLUTION

Since RC = RC(t), this is a moving boundary problem.All the equations were discretized with a combination of thevariable-space-grid scheme of the forward front-trackingmethod and the Crank–Nicolson scheme of the finite differ-ence method.7,15 The variable-space-grid scheme requiresthe time derivative to be modified accordingly.7 In the con-tact region of the cell and substrate, as well as the noncontactregion of the substrate,

δr = RC(t)

N1or δr = (RSUB/RC0) − RC(t)

N2,

(C.1)

(∂ Xi/∂t) j = (∂ Xi/∂r )t (dr/dt) j + (∂ Xi/∂t)r ,

(C.2)

where N1 and N2 are the total number of intervals in thecontact and noncontact region respectively, i refers to the ithnodal point, j refers to the jth grid line, and X represents A,AI, B, BI, C, CI, or CII. In the noncontact region of the cell,

δθ = θmax(t)

N3, (C.3)

(∂ Ai/∂t) j = (∂ Ai/∂r)t (dθ/dt) j + (∂ Ai/∂t)θ ,

(C.4)

where N3 is the total number of intervals in this region.Because of the moving boundary nature of this problem,

a boundary condition is needed for C, CI, and CII at theinterface between the contact and noncontact region, i.e.,r = RC(t). At any time instant and r = RC(t), the CD2 andCD58 molecules at this location will have no time to reactwith each other because this is always the newly formedcontact point. Therefore, the following boundary conditionis employed:

C = CI = CII = 0 @ r = RC. (C.5)

However, this does create a sharp gradient as time pro-gresses and the bond concentration increases in the contactregion. To solve this problem, we used very dense meshesbetween r = 0.9999RC and r = RC.

The Jurkat T cell line has a high level of CD2 expres-sion. The total number of mobile CD2 molecules per cell isestimated to be around 47143, calculated with Zhu-Golananalysis and the fractional mobility of 0.7.12 On average, thesurface area of a Jurkat T cell is ∼800 µm2, so the initialconcentration of mobile CD2 is ∼41/µm2 and the initialconcentration of immobile CD2 is 18/µm2. Five differenttotal initial concentrations of CD58 on the substrate (13,25, 50, 100, and 200/µm2), which were used in the ex-periment by Dustin et al.12 were used in this computation.Correspondingly, with the fractional mobility at 0.72, theinitial concentrations of mobile CD58 in the planar bilayer

on the substrate are 9, 18, 36, 72, and 144/µm2 and theinitial concentrations of immobile CD58 are 4, 7, 14, 28,and 56/µm2. DBC, which is 0.13 µm2/s, is slightly smallerthan DB, which is 0.59 µm2/s.14 This is likely due to the re-tardation to CD58 diffusion caused by the molecules in thecontact area. We assumed that DAC/DA = DBCDB with thevalue of DA at 0.072 µm2/s.14 We calculated RSUB to be ∼16µm from the cell density on the substrate.14 The value of 1.1molecules/µm2 was adopted for the 2D Kd of CD2/CD58interactions at room temperature. The forward reaction rateconstant of 4.16/µm2 was estimated from the Bell theory(2π DA + 2π DB).1 Correspondingly, the reverse reactionrate constant was calculated to be 4.576/s, which is close toits 3D counterpart.10,26

The computation was done with the following parame-ters: N1 = 11000, N2 = 10000, and N3 = 10000. The ini-tial dimensionless timestep was 10−5 and increased grad-ually to 10−4. To ensure that these timesteps and spatialintervals are small enough, we tested our program withsmaller timesteps and spatial intervals and obtained al-most identical results. When we doubled and halved DA

and DB intentionally, we found that it took less and moretime to reach the steady state, respectively, while the finalcontact area remained the same. On a 3.2 GHz PentiumPC, it took approximately 30 h to reach t = 300 (steadystate for most cases) for the case where CA0 = 41/µm2 andCB0 = 144/µm2. In summary, all the parameters used inthe computation are listed in Table 1.

APPENDIX D: MODIFICATION TOZHU-GOLAN ANALYSIS

If the immobile fraction of CD2 or CD58 is large, Zhu-Golan analysis needs to be modified for the calculation ofKd. In Zhu-Golan analysis, the bound CD58 concentration(BT) and the free CD58 concentration (F) at equilibriumare related by12

BT

F= NtA fA

KdScell− pbc BT

Kd, (D.1)

TABLE 1. Parameters of the Model

Symbol Value Symbol Value

CA0 41/µm2 kf 4.16 µm2/sCB0 9–144/µm2 kr 4.58/sDA 0.072 µm2/s kI

f 3.70 µm2/sDAC 0.016 µm2/s kI

r 4.58/sDB 0.59 µm2/s kII

f 0.456 µm2/sDBC 0.13 µm2/s kII

r 4.576/sfA 0.7 N1 11000fB 0.72 N2 10000RS 7.98 µm N3 10000RC0 1.56 µm α 0.001 µm2

RSUB 16.1 µm

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492 SHAO et al.

where NtA is the total number of CD2 molecules percell, Kd = kr/kf , pbc = Sb/Scell, Sb is the contact area atequilibrium (µm2), and Scell is the cellular surface area(µm2). However, this analysis ignored the fact that someCD58 molecules were also immobile and the total numberof CD58 molecules in the contact area was not constant.Besides, if the immobile fraction of CD2 is large or thecontact area is large compared with Scell, this analysis maynot apply. In this appendix, we propose another analysisthat takes account of all these factors.

The reactions between CD2 and CD58 can be describedby Eqs. (A.17), (A.18), and (A.19). Consequently, at equi-librium, we should have [Eq. (A.23)]

C

AB= CI

AI B= CII

ABI= 1

Kd. (D.2)

Therefore,

BT = C + CI + CII = 1

Kd{AB + AI B + A(F − B)}

(D.3)With mass conservation, B, A, and AI can be determined asfollows

B = NtB fB

SSUB− Nb fmB

SSUB= CB0 − pbs fmB BT, (D.4)

A = NtA fA

Scell− Nb fmA

Scell= CA0 − pbc fmA BT, (D.5)

BI = NtB(1 − fB)(Sb/SSUB) − Nb(1 − fmB)

Sb

= CBI − (1 − fmB)BT, (D.6)

AI = NtA(1 − fA)(Sb/Scell) − Nb(1 − fmA)

Sb

= CAI − (1 − fmA)BT, (D.7)

where NtB is the total number of CD58 available on thesubstrate for each cell, SSUB is the available substrate areafor each cell, fmB is the percentage of bonds due to mobileCD58, fmA is the percentage of bonds due to mobile CD2,and pbs = Sb/SSUB.

Since F = B + BI, fmB can be solved from Eqs. (D.4)and (D.6) as

fmB = C + CI

BT= F + BT − (NtB/SSUB)

BT (1 − pbs). (D.8)

Moreover, fmA can be solved from Eqs. (D.2) and (D.4) as

fmA = C + CII

BT= 1 − fmB

1 − (CB0 − pbs fmB BT)/F. (D.9)

Therefore, fmB and fmA can be calculated for each ex-periment and used for further analysis. From Eqs. (D.2),(D.4), (D.5), (D.7), and (D.9), the following equations canbe obtained

fmA BT

F= CA0

Kd− pbc fmA BT

Kd, (D.10)

fmB BT

CB0 − pbs fmB BT= CA0 + CAI

Kd

− BT[pbc fmA + (1 − fmA)]

Kd.

(D.11)

Either Eqs. (D.10) or (D.11) will allow us to calculate Kd

by linear regression. If fA = fB = 1 (all CD2 and CD58molecules are mobile), fmA = fmB = 1 and F = B. BothEqs. (D.10) and (D.11) will become Eq. (D.1) (Zhu-Golananalysis).

AUXILIARY MATERIALS

For auxiliary material, please contact the authorsat [email protected] or visit online http://biomed.wustl.edu/faculty/shao/movie2.htm. This material includesa movie showing the increase in the contact area overtime (t = 0 to 200) and the dynamics of the CD2/CD58bond concentration during this process [corresponding toFig. 3(c)].

ACKNOWLEDGMENTS

This work was supported by the National Institutes ofHealth grants R01 HL069947 (JYS), R21 RR017014 (JYS),R01 AI043542 (MLD), and the Irene Diamond Foundation(MLD).

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