- Email: [email protected]
Modeling the kinetics of cell membrane spreading on substrates with ligand density gradient
ARTICLE IN PRESS
Journal of Biomechanics 41 (2008) 921–925 www.elsevier.com/locate/jbiomech www.JBiomech.com
Short communication
Modeling the kinetics of cell membrane spreading on substrates with ligand density gradient Alireza S. Sarvestani, Esmaiel Jabbari Biomimetic Materials and Tissue Engineering Laboratories, Department of Chemical Engineering, Swearingen Engineering Center, University of South Carolina, Rm. 2C11, Columbia, SC 29208, USA Accepted 3 November 2007
Abstract An analytical model is developed for the effect of surface gradient in ligand density on the adhesion kinetics of a curved elastic membrane with mobile receptors. The displacement and speed of spreading at the edge of adhesion zone as well as the density profile of receptors along the membrane are predicted as a function of time. According to results, in the diffusion-controlled regime, the front edge displacement of adhesion zone and the rate of membrane spreading decreased with increasing the ligand density in a certain direction. Furthermore, the displacement of the edge of the adhesion zone did not scale with the square root of time, as observed on substrates with uniform ligand density. r 2007 Elsevier Ltd. All rights reserved. Keywords: Cell membrane; Spreading; Kinetics; Ligand density
1. Introduction Cell spreading on substrates is a dynamic process involving non-covalent association between membrane receptors on the cell surface and complementary ligands on the substrate. The following limiting regimes can be defined for the displacement kinetics at the edge of the adhesion zone (Boulbitch et al., 2001): (1) reaction-controlled regime, in which the formation and breaking of cell–substrate contacts are controlled by the rate of reversible reaction between the ligands (L) and receptors (R) L þ R2LR:
(1)
(2) diffusion-controlled regime, in which the mobile receptors are recruited from the regions on the membrane far away from the adhesion zone (Brochard-Wyart and de Gennes, 2002; Freund and Lin, 2004; Shenoy and Freund, 2005). At low receptor concentrations, the characteristic time for receptor diffusion is longer than that for the ligand–receptor reaction, and hence cell spreading is Corresponding author. Tel.: +1 803 777 8022; fax: +1 803 777 0973.
E-mail address: [email protected] (E. Jabbari). 0021-9290/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2007.11.004
mediated by diffusion of mobile receptors. At high receptor concentrations, the rate of spreading of the adhesion zone is controlled by the rate of ligand–receptor association. Cell migration experiments demonstrate that cell–substrate interactions increase with ligand density up to a critical value, above which cell density and speed of migration reach either a plateau value or decrease (DiMilla et al., 1993; Huttenlocher et al., 1996; Maheshwari et al., 1999; Maheshwari et al., 2000). Other experimental results with fibroblasts on substrates with RGD ligand density gradient (Harris et al., 2006) show higher cell adhesion in the direction of increasing RGD density. However, saturation effect is observed above a certain density of adhesive ligands where increasing the RGD density no longer improves cell adhesion. In the present work, we propose a model to examine the effect of gradient in surface ligand density on the kinetics of membrane adhesion. In the reaction-controlled regime (see Eq. (1)), increasing the ligand density should promote the membrane–substrate binding (Boulbitch et al., 2001). The situation, however, is quite different when adhesion is diffusion-controlled (i.e., at high ligand concentration). Therefore, we focus on the case of cell spreading and
ARTICLE IN PRESS 922
A.S. Sarvestani, E. Jabbari / Journal of Biomechanics 41 (2008) 921–925
adhesion on gradient substrates in the diffusion-controlled regime. The model predicts that in this regime, the displacement and rate of spreading of the edge of adhesion zone decreases with increasing ligand density gradient.
(Boulbitch et al., 2001; Freund and Lin, 2004) qc qj ¼ qt qr
for r4aðtÞ.
(3)
For roaðtÞ, we have c(r, t) ¼ ceq(r) and j(r, t) ¼ 0. The conservation of membrane receptors requires that Z aðtÞ Z 1 d ceq ðrÞ dr þ cðr; tÞ dr ¼ 0, (4) dt 0 aðtÞ
2. Theory The cross-section of the edge of adhesion zone between the membrane of a large vesicle and the substrate is depicted schematically in Fig. 1. The position along the cell membrane is described by an arc length coordinate, r. The membrane–substrate adhesion zone represents a straight line moving along the horizontal axis x, whose position compared with the origin, r ¼ 0, is denoted by a(t). The ligand density is taken to be constant for xp0 and changes linearly with horizontal distance from x ¼ 0 for all x40 region. The equilibrium density of ligand–receptor pairs is ( for rp0; c0 ceq ðrÞ ¼ (2) c0 þ br for 0proaðtÞ;
from which one can deduce _ þ baðtÞaðtÞ _ þ j þ ¼ 0, ðc0 cþ ÞaðtÞ
(5)
where it is assumed that c(r, t)-cN and j(r, t)-0 as r-N. The total free energy for r40 region of the membrane, after normalization by kBT (kB is the Boltzmann constant and T is the absolute temperature), can be written as (Freund and Lin, 2004) Z aðtÞ ceq ðrÞ 1 2 F ðtÞ ¼ ceq ðrÞgb þ ceq ðrÞ ln þ Bk0 dr c1 2 0 Z 1 cðr; tÞ cðr; tÞ ln dr. ð6Þ þ c0 aðtÞ
where c0 and b are constants and their values are determined by the ligand density on the surface (Fig. 1). The edge of the advancing zone at t ¼ 0 is assumed to be at r ¼ 0. The distribution of receptors along the membrane is also shown schematically in Fig. 1. The density of receptors in the free membrane part, far away from the adhesion zone, is assumed to be cN. In the immediate vicinity of the edge of adhesion zone, r ¼ a+, the local receptor density is c+, which is less than cN. This difference in ligand density induces a chemical potential gradient that drives the diffusion of receptors from more dense (far away) to less dense regions in the vicinity of adhesion zone. If c(r, t) and j(r, t) represent the receptor density and flux for r4aðtÞ, respectively, we have
where gb is the interaction energy of a receptor–ligand pair, B is the bending modulus per unit width, and k0 is the stress-free curvature of the membrane. Differentiation of Eq. (6) with respect to time yields c0 þ baðtÞ 1 _ F_ ðtÞ ¼ ðc0 þ baðtÞÞ ln ðc0 þ baðtÞÞgb þ Bk20 aðtÞ c1 2 Z 1 cþ _ þ cþ ln ð7Þ c_w dr, aðtÞ c1 aðtÞ where w ¼ ln(c(r)/cN)+1 is the local chemical potential. mobile receptors
membrane
x a(t) c(r,t)
substrate with ligand density gradient
c0 + a(t) c0
c c+
r
Fig. 1. Sketch of the interface between the membrane and ligand gradient substrate (upper) and density profile of the mobile receptors along the membrane (lower).
ARTICLE IN PRESS A.S. Sarvestani, E. Jabbari / Journal of Biomechanics 41 (2008) 921–925
The integral term in Eq. (7) can be written as Z 1 c_w dr ¼ j þ wþ QðtÞ,
1.0
(8)
aðtÞ
and m is the mobility parameter. In Eq. (8) it is assumed that the mean receptor speed is proportional to the negative of chemical potential gradient, where m is the proportionality constant. Consequently, the net flux of receptors is
τ = 0.4 τ = 0.6
0.6
τ = 1.0
0.4
c+ / c∞
0.2
qw . (10) qr For a closed system, the free energy decreases with a rate equal to that of dissipation. Hence,
0.0 0.0
jðr; tÞ ¼ mc
c0 þ baðtÞ 1 ðc0 þ baðtÞÞgb þ Bk20 cþ 2 þ cþ c0 baðtÞ ¼ 0.
τ = 0.2
τ = 0.8 c(,) / c∞
where Q(t) is the rate of energy dissipation due to diffusion of membrane receptors, given by Z 1 2 qw QðtÞ ¼ mc dr, (9) qr aðtÞ
0.8
923
0.5
1.0 (distance)
1.5
2.0
1.0
ðc0 þ baðtÞÞ ln
1 c0 c0 Bk20 =c0 þ 1 þ ln þ þ ¼ 0. 2 c0 c0
(13)
Eq. (13) provides the initial condition for solving Eq. (12) for the fist time interval Dt. Using Eqs. (5) and (11), the calculated values of c(r, Dt) can be used to determine the length of adhesion zone, a(Dt), and the speed of membrane spreading at the edge of adhesion zone, _ aðDtÞ. For the second time interval Dtptp2Dt, Eq. (12) can be solved numerically with the boundary condition cþ ðt ¼ DtÞ ¼ cðaðDtÞ; DtÞ where cðaðDtÞ; DtÞ is calculated from previous time interval. This iterative procedure is repeated up to any arbitrary adhesion time. 3. Results The receptor density profiles within the free part of the membrane at different times are shown in Fig. 2. The time
τ = 0.4
0.6
0.2
τ = 1.0
0.50 0.48 0.46 0.20
0.0 0.0
τ = 0.6 τ = 0.8
0.52
0.4 c+ / c∞
qc q2 c ¼ m 2 for r4aðtÞ, (12) qt qr subjected to the boundary conditions c(N, t)=cN and c (r, 0)=cN for r4aðtÞ. Contrary to the case of uniform ligand density (Boulbitch et al., 2001; Freund and Lin, 2004), there is no simple analytical solution to Eq. (12) for the case of non-uniform ligand density. This equation, however, can be solved numerically by using the method of finite difference. The value of c+(t ¼ 0+), for the first time interval Dt, can be evaluated using the analytical solution of uniform ligand density. As shown by Freund and Lin (2004), c0þ ¼ cþ ðt ¼ 0þ Þ can be obtained by solving the following equation:
τ = 0.2 c(,) / c∞
Eqs. (3) and (10) lead to the following diffusion equation for the distribution of surface density of the ligands
gb
0.8
ð11Þ
0.5
1.0 (distance)
0.40 1.5
2.0
Fig. 2. Normalized density profile of the receptors along the membrane interacting with a substrate with (a) uniform density of ligands (b ¼ 0) and (b) linearly increasing ligand density (b ¼ 10). The inset in (b) shows the variation of c+/cN, the normalized concentration of receptors at the edge of the advancing zone with adhesion time. The time and distance are represented by dimensionless quantities t ¼ tmk20 and r ¼ rk0, respectively. Parameter values are assumed to be c0/cN ¼ 2.0 and gnb ¼ 1:0.
and distance are represented by dimensionless quantities t ¼ tmk20 and r ¼ rk0, respectively. The ratio of equilibrium (rp0) and asymptotic (r-N) concentration of receptors, c0/cN, and reduced bond strength gnb ¼ gb 1 2 2 Bk0 =c0 are the two dimensionless system parameters in our study. In the diffusion-controlled regime, the equilibrium concentration of ligand–receptor pairs on the substrate should be larger than the asymptotic concentration of receptors to induce a diffusional flux, i.e., c0 =c1 41. In addition, adhesion occurs only if the ligand–receptor energy of adhesion is larger than the membrane elastic energy (Freund and Lin, 2004). Hence, gb is a non-negative parameter.
ARTICLE IN PRESS A.S. Sarvestani, E. Jabbari / Journal of Biomechanics 41 (2008) 921–925
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Figs. 2(a) and (b) compare the receptor density profiles at different time points, for uniform ligand density (b ¼ 0) and linearly increasing ligand density (b ¼ 10) on the substrate, respectively. As expected, c+, the concentration of receptors at the edge of the advancing zone, is constant when ligand density is uniform, irrespective of the adhesion time. For ligand gradient substrates, however, c+ is not constant and increases with time as the local ligand density at the edge of the advancing zone increases (see the inset of Fig. 2(b)). The increase in ligand density at the edge of adhesion zone reduces the chemical potential gradient in the membrane which, in turn, reduces the driving force for the diffusion of receptors (see Eq. (10)). The effect of ligand gradient slope on the variation of displacement at the edge of adhesion zone, aðtÞ ¼ k0 aðtÞ, and the speed of spreading, daðtÞ=dt, is shown by Figs. 3(a) and (b), respectively. The results show that at any instant,
0.7 0.6 () ∼
()
0.5
=0
0.4
=5
0.3
= 10
0.2 0.1 0.0 0.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
(time) 0.7 0.6
=0
d() d
0.5 0.4 =5
0.3 0.2 0.1 0.0 0.0
= 10 0.2
0.4 (time)
Fig. 3. The effect of gradient in ligand density on (a) displacement of the edge of adhesion zone and (b) speed of membrane spreading on the substrate. The time and distance are represented by dimensionless quantities t ¼ tmk20 and r ¼ rk0, respectively. Parameter values are the same as in Fig. 2.
the edge displacement and speed of spreading are lower on the gradient substrate (b40) compared to those on the uniform substrate (b ¼ 0). This effect became more pronounced by increasing the gradient. Again, this observation can be attributed to the reduction in the driving force for diffusion of membrane receptors as the magnitude of c+ is increased with increasing equilibrium density of ligand–receptor pairs, ceq. In addition, the results show that when b40, the edge displacement does not follow the square-root law with time (i.e., at1/2), contrary to the case of b ¼ 0 (Boulbitch et al., 2001; Freund and Lin, 2004). According to the model predictions, for t41.0, the value of a(t) asymptotically approaches 0.44 and 0.37 for b ¼ 5 and 10, respectively, while it increases monotonically for b ¼ 0. The main conclusion of the model is that, in diffusioncontrolled regime, the membrane spreading speed decreases with increasing ligand density on the substrate. The kinetics of membrane adhesion on uniform ligand conjugated surfaces has been studied experimentally, but to our best knowledge, there is no experimental study on the kinetics of membrane spreading on ligand gradient surfaces. A number of studies have measured the spatial distribution of fibroblasts on substrates with gradient density of fibronectin or RGD (Mei et al., 2005; Harris et al., 2006). The results of these studies demonstrate that the variation of cell distribution along the gradient is biphasic; that is, cell density increases toward the increasing direction of gradient and above a certain ligand concentration, it decreases or remains constant. This biphasic cell density distribution can be explained by the fact that above a certain ligand concentration on the gradient, the mechanism of cell spreading switches from reaction-controlled to diffusion-controlled regime. Therefore, according to the predictions of presented model, increasing the ligand concentration reduces the rate of cell–substrate association which, in turn, can inhibit the cell migration. Similar reasoning can be used to explain the biophasic dependence of the density and motility of fibroblasts on substrates coated with different densities of fibronectin or collagen IV (Huttenlocher et al., 1996; Maheshwari et al., 1999; Maheshwari et al., 2000). In the presented model, it is assumed that an unlimited number of free receptors are available within aðtÞoro1 region. In a more realistic situation, the number of receptors available to contribute to adhesion is finite and membrane spreading eventually comes to a halt. Under this condition, the depletion effect of receptor–ligand pairs becomes important and should be taken into account by changing the configurational entropy function in the chemical potential (Smith et al., 2006). Finally, it should be emphasized that the kinetics of membrane spreading, as described here, is only a primary event in cell adhesion. Coupling of cytoskeleton to the receptors and consequently, segregation of ligand–receptor pairs (ligand clustering) and the formation of discrete adhesion domains are the events which eventually control the cellular adhesion (Bruinsma et al., 2000).
ARTICLE IN PRESS A.S. Sarvestani, E. Jabbari / Journal of Biomechanics 41 (2008) 921–925
Conflict of interest statement The authors declare that they have no proprietary, financial, professional, or other personal interest of any nature or kind that in any product, service and/or company related to the results or outcomes of the manuscript titled ‘‘Modeling the Kinetics of Cell Membrane Spreading on Substrates with Ligand Density Gradient’’ by Alireza S. Servestani and Esmaiel Jabbari. Acknowledgments This work was supported by grants from the AO (Arbeitsgemeinschaft Fur Osteosynthesefragen) Foundation (AORF Project 05-J95) and the Aircast Foundation. References Boulbitch, A., Guttenberg, Z., Sackmann, E., 2001. Kinetics of membrane adhesion mediated by ligand–receptor interaction studied with a biomimetic system. Biophysical Journal 81, 2743–2751. Brochard-Wyart, F., de Gennes, P.G., 2002. Adhesion induced by mobile receptors: dynamics. Proceedings of the National Academy of Sciences 99, 7854–7859. Bruinsma, R., Behrisch, A., Sackmann, E., 2000. Adhesive switching of membranes: experiment and theory. Physical Review E 61, 4253–4267.
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DiMilla, P.A., Stone, J.A., Quinn, J.A., Albelda, S.M., Lauffenburger, D.A., 1993. Maximal migration of human smooth muscle cells on fibronectin and type IV collagen occurs at an intermediate attachment strength. Journal of Cell Biology 122, 729–737. Freund, L.B., Lin, Y., 2004. The role of receptor mobility in spontaneous adhesive contact and implications for cell adhesion. Journal of the Mechanics and Physics of Solids 52, 2455–2472. Harris, B.P., Kutty, J.K., Fritz, E.W., Webb, C.K., Burg, K.J.L., Metters, A.T., 2006. Photopatterned polymer brushes promoting cell adhesion gradients. Langmuir 22, 4467–4471. Huttenlocher, A., Ginsberg, M., Horwitz, A., 1996. Modulation of cell migration by integrin mediated affinity and cytoskeletal interactions. Journal of Cell Biology 134, 1551–1562. Maheshwari, G., Wells, A., Griffith, L.G., Lauffenburger, D.A., 1999. Biophysical integration of effects of epidermal growth factor and fibronectin on fibroblast migration. Biophysical Journal 76, 2814–2823. Maheshwari, G., Brown, G.L., Lauffenburger, D.A., Wells, A., Griffith, L.G., 2000. Cell adhesion and motility depend on nanoscale RGD clustering. Journal of Cell Science 113, 1677–1686. Mei, Y., Wu, T., Xu, C., Langenbach, K.J., Elliott, J.T., Vogt, B.D., Beers, K.L., Amis, E.J., Washburn, N.R., 2005. Tuning cell adhesion on gradient poly(2-hydroxyethyl methacrylate)-grafted surfaces. Langmuir 21, 12309–12314. Shenoy, V.B., Freund, L.B., 2005. Growth and shape stability of a biological membrane adhesion complex in the diffusion mediated regime. Proceedings of the National Academy of Sciences 102, 3213–3218. Smith, A.-S., Lorz, B.G., Seifert, U., Sackmann, E., 2006. Antagonistinduced deadhesion of specifically adhered vesicles. Biophysical Journal 90, 1064–1080.
Journal of Biomechanics 41 (2008) 921–925 www.elsevier.com/locate/jbiomech www.JBiomech.com
Short communication
Modeling the kinetics of cell membrane spreading on substrates with ligand density gradient Alireza S. Sarvestani, Esmaiel Jabbari Biomimetic Materials and Tissue Engineering Laboratories, Department of Chemical Engineering, Swearingen Engineering Center, University of South Carolina, Rm. 2C11, Columbia, SC 29208, USA Accepted 3 November 2007
Abstract An analytical model is developed for the effect of surface gradient in ligand density on the adhesion kinetics of a curved elastic membrane with mobile receptors. The displacement and speed of spreading at the edge of adhesion zone as well as the density profile of receptors along the membrane are predicted as a function of time. According to results, in the diffusion-controlled regime, the front edge displacement of adhesion zone and the rate of membrane spreading decreased with increasing the ligand density in a certain direction. Furthermore, the displacement of the edge of the adhesion zone did not scale with the square root of time, as observed on substrates with uniform ligand density. r 2007 Elsevier Ltd. All rights reserved. Keywords: Cell membrane; Spreading; Kinetics; Ligand density
1. Introduction Cell spreading on substrates is a dynamic process involving non-covalent association between membrane receptors on the cell surface and complementary ligands on the substrate. The following limiting regimes can be defined for the displacement kinetics at the edge of the adhesion zone (Boulbitch et al., 2001): (1) reaction-controlled regime, in which the formation and breaking of cell–substrate contacts are controlled by the rate of reversible reaction between the ligands (L) and receptors (R) L þ R2LR:
(1)
(2) diffusion-controlled regime, in which the mobile receptors are recruited from the regions on the membrane far away from the adhesion zone (Brochard-Wyart and de Gennes, 2002; Freund and Lin, 2004; Shenoy and Freund, 2005). At low receptor concentrations, the characteristic time for receptor diffusion is longer than that for the ligand–receptor reaction, and hence cell spreading is Corresponding author. Tel.: +1 803 777 8022; fax: +1 803 777 0973.
E-mail address: [email protected] (E. Jabbari). 0021-9290/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2007.11.004
mediated by diffusion of mobile receptors. At high receptor concentrations, the rate of spreading of the adhesion zone is controlled by the rate of ligand–receptor association. Cell migration experiments demonstrate that cell–substrate interactions increase with ligand density up to a critical value, above which cell density and speed of migration reach either a plateau value or decrease (DiMilla et al., 1993; Huttenlocher et al., 1996; Maheshwari et al., 1999; Maheshwari et al., 2000). Other experimental results with fibroblasts on substrates with RGD ligand density gradient (Harris et al., 2006) show higher cell adhesion in the direction of increasing RGD density. However, saturation effect is observed above a certain density of adhesive ligands where increasing the RGD density no longer improves cell adhesion. In the present work, we propose a model to examine the effect of gradient in surface ligand density on the kinetics of membrane adhesion. In the reaction-controlled regime (see Eq. (1)), increasing the ligand density should promote the membrane–substrate binding (Boulbitch et al., 2001). The situation, however, is quite different when adhesion is diffusion-controlled (i.e., at high ligand concentration). Therefore, we focus on the case of cell spreading and
ARTICLE IN PRESS 922
A.S. Sarvestani, E. Jabbari / Journal of Biomechanics 41 (2008) 921–925
adhesion on gradient substrates in the diffusion-controlled regime. The model predicts that in this regime, the displacement and rate of spreading of the edge of adhesion zone decreases with increasing ligand density gradient.
(Boulbitch et al., 2001; Freund and Lin, 2004) qc qj ¼ qt qr
for r4aðtÞ.
(3)
For roaðtÞ, we have c(r, t) ¼ ceq(r) and j(r, t) ¼ 0. The conservation of membrane receptors requires that Z aðtÞ Z 1 d ceq ðrÞ dr þ cðr; tÞ dr ¼ 0, (4) dt 0 aðtÞ
2. Theory The cross-section of the edge of adhesion zone between the membrane of a large vesicle and the substrate is depicted schematically in Fig. 1. The position along the cell membrane is described by an arc length coordinate, r. The membrane–substrate adhesion zone represents a straight line moving along the horizontal axis x, whose position compared with the origin, r ¼ 0, is denoted by a(t). The ligand density is taken to be constant for xp0 and changes linearly with horizontal distance from x ¼ 0 for all x40 region. The equilibrium density of ligand–receptor pairs is ( for rp0; c0 ceq ðrÞ ¼ (2) c0 þ br for 0proaðtÞ;
from which one can deduce _ þ baðtÞaðtÞ _ þ j þ ¼ 0, ðc0 cþ ÞaðtÞ
(5)
where it is assumed that c(r, t)-cN and j(r, t)-0 as r-N. The total free energy for r40 region of the membrane, after normalization by kBT (kB is the Boltzmann constant and T is the absolute temperature), can be written as (Freund and Lin, 2004) Z aðtÞ ceq ðrÞ 1 2 F ðtÞ ¼ ceq ðrÞgb þ ceq ðrÞ ln þ Bk0 dr c1 2 0 Z 1 cðr; tÞ cðr; tÞ ln dr. ð6Þ þ c0 aðtÞ
where c0 and b are constants and their values are determined by the ligand density on the surface (Fig. 1). The edge of the advancing zone at t ¼ 0 is assumed to be at r ¼ 0. The distribution of receptors along the membrane is also shown schematically in Fig. 1. The density of receptors in the free membrane part, far away from the adhesion zone, is assumed to be cN. In the immediate vicinity of the edge of adhesion zone, r ¼ a+, the local receptor density is c+, which is less than cN. This difference in ligand density induces a chemical potential gradient that drives the diffusion of receptors from more dense (far away) to less dense regions in the vicinity of adhesion zone. If c(r, t) and j(r, t) represent the receptor density and flux for r4aðtÞ, respectively, we have
where gb is the interaction energy of a receptor–ligand pair, B is the bending modulus per unit width, and k0 is the stress-free curvature of the membrane. Differentiation of Eq. (6) with respect to time yields c0 þ baðtÞ 1 _ F_ ðtÞ ¼ ðc0 þ baðtÞÞ ln ðc0 þ baðtÞÞgb þ Bk20 aðtÞ c1 2 Z 1 cþ _ þ cþ ln ð7Þ c_w dr, aðtÞ c1 aðtÞ where w ¼ ln(c(r)/cN)+1 is the local chemical potential. mobile receptors
membrane
x a(t) c(r,t)
substrate with ligand density gradient
c0 + a(t) c0
c c+
r
Fig. 1. Sketch of the interface between the membrane and ligand gradient substrate (upper) and density profile of the mobile receptors along the membrane (lower).
ARTICLE IN PRESS A.S. Sarvestani, E. Jabbari / Journal of Biomechanics 41 (2008) 921–925
The integral term in Eq. (7) can be written as Z 1 c_w dr ¼ j þ wþ QðtÞ,
1.0
(8)
aðtÞ
and m is the mobility parameter. In Eq. (8) it is assumed that the mean receptor speed is proportional to the negative of chemical potential gradient, where m is the proportionality constant. Consequently, the net flux of receptors is
τ = 0.4 τ = 0.6
0.6
τ = 1.0
0.4
c+ / c∞
0.2
qw . (10) qr For a closed system, the free energy decreases with a rate equal to that of dissipation. Hence,
0.0 0.0
jðr; tÞ ¼ mc
c0 þ baðtÞ 1 ðc0 þ baðtÞÞgb þ Bk20 cþ 2 þ cþ c0 baðtÞ ¼ 0.
τ = 0.2
τ = 0.8 c(,) / c∞
where Q(t) is the rate of energy dissipation due to diffusion of membrane receptors, given by Z 1 2 qw QðtÞ ¼ mc dr, (9) qr aðtÞ
0.8
923
0.5
1.0 (distance)
1.5
2.0
1.0
ðc0 þ baðtÞÞ ln
1 c0 c0 Bk20 =c0 þ 1 þ ln þ þ ¼ 0. 2 c0 c0
(13)
Eq. (13) provides the initial condition for solving Eq. (12) for the fist time interval Dt. Using Eqs. (5) and (11), the calculated values of c(r, Dt) can be used to determine the length of adhesion zone, a(Dt), and the speed of membrane spreading at the edge of adhesion zone, _ aðDtÞ. For the second time interval Dtptp2Dt, Eq. (12) can be solved numerically with the boundary condition cþ ðt ¼ DtÞ ¼ cðaðDtÞ; DtÞ where cðaðDtÞ; DtÞ is calculated from previous time interval. This iterative procedure is repeated up to any arbitrary adhesion time. 3. Results The receptor density profiles within the free part of the membrane at different times are shown in Fig. 2. The time
τ = 0.4
0.6
0.2
τ = 1.0
0.50 0.48 0.46 0.20
0.0 0.0
τ = 0.6 τ = 0.8
0.52
0.4 c+ / c∞
qc q2 c ¼ m 2 for r4aðtÞ, (12) qt qr subjected to the boundary conditions c(N, t)=cN and c (r, 0)=cN for r4aðtÞ. Contrary to the case of uniform ligand density (Boulbitch et al., 2001; Freund and Lin, 2004), there is no simple analytical solution to Eq. (12) for the case of non-uniform ligand density. This equation, however, can be solved numerically by using the method of finite difference. The value of c+(t ¼ 0+), for the first time interval Dt, can be evaluated using the analytical solution of uniform ligand density. As shown by Freund and Lin (2004), c0þ ¼ cþ ðt ¼ 0þ Þ can be obtained by solving the following equation:
τ = 0.2 c(,) / c∞
Eqs. (3) and (10) lead to the following diffusion equation for the distribution of surface density of the ligands
gb
0.8
ð11Þ
0.5
1.0 (distance)
0.40 1.5
2.0
Fig. 2. Normalized density profile of the receptors along the membrane interacting with a substrate with (a) uniform density of ligands (b ¼ 0) and (b) linearly increasing ligand density (b ¼ 10). The inset in (b) shows the variation of c+/cN, the normalized concentration of receptors at the edge of the advancing zone with adhesion time. The time and distance are represented by dimensionless quantities t ¼ tmk20 and r ¼ rk0, respectively. Parameter values are assumed to be c0/cN ¼ 2.0 and gnb ¼ 1:0.
and distance are represented by dimensionless quantities t ¼ tmk20 and r ¼ rk0, respectively. The ratio of equilibrium (rp0) and asymptotic (r-N) concentration of receptors, c0/cN, and reduced bond strength gnb ¼ gb 1 2 2 Bk0 =c0 are the two dimensionless system parameters in our study. In the diffusion-controlled regime, the equilibrium concentration of ligand–receptor pairs on the substrate should be larger than the asymptotic concentration of receptors to induce a diffusional flux, i.e., c0 =c1 41. In addition, adhesion occurs only if the ligand–receptor energy of adhesion is larger than the membrane elastic energy (Freund and Lin, 2004). Hence, gb is a non-negative parameter.
ARTICLE IN PRESS A.S. Sarvestani, E. Jabbari / Journal of Biomechanics 41 (2008) 921–925
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Figs. 2(a) and (b) compare the receptor density profiles at different time points, for uniform ligand density (b ¼ 0) and linearly increasing ligand density (b ¼ 10) on the substrate, respectively. As expected, c+, the concentration of receptors at the edge of the advancing zone, is constant when ligand density is uniform, irrespective of the adhesion time. For ligand gradient substrates, however, c+ is not constant and increases with time as the local ligand density at the edge of the advancing zone increases (see the inset of Fig. 2(b)). The increase in ligand density at the edge of adhesion zone reduces the chemical potential gradient in the membrane which, in turn, reduces the driving force for the diffusion of receptors (see Eq. (10)). The effect of ligand gradient slope on the variation of displacement at the edge of adhesion zone, aðtÞ ¼ k0 aðtÞ, and the speed of spreading, daðtÞ=dt, is shown by Figs. 3(a) and (b), respectively. The results show that at any instant,
0.7 0.6 () ∼
()
0.5
=0
0.4
=5
0.3
= 10
0.2 0.1 0.0 0.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
(time) 0.7 0.6
=0
d() d
0.5 0.4 =5
0.3 0.2 0.1 0.0 0.0
= 10 0.2
0.4 (time)
Fig. 3. The effect of gradient in ligand density on (a) displacement of the edge of adhesion zone and (b) speed of membrane spreading on the substrate. The time and distance are represented by dimensionless quantities t ¼ tmk20 and r ¼ rk0, respectively. Parameter values are the same as in Fig. 2.
the edge displacement and speed of spreading are lower on the gradient substrate (b40) compared to those on the uniform substrate (b ¼ 0). This effect became more pronounced by increasing the gradient. Again, this observation can be attributed to the reduction in the driving force for diffusion of membrane receptors as the magnitude of c+ is increased with increasing equilibrium density of ligand–receptor pairs, ceq. In addition, the results show that when b40, the edge displacement does not follow the square-root law with time (i.e., at1/2), contrary to the case of b ¼ 0 (Boulbitch et al., 2001; Freund and Lin, 2004). According to the model predictions, for t41.0, the value of a(t) asymptotically approaches 0.44 and 0.37 for b ¼ 5 and 10, respectively, while it increases monotonically for b ¼ 0. The main conclusion of the model is that, in diffusioncontrolled regime, the membrane spreading speed decreases with increasing ligand density on the substrate. The kinetics of membrane adhesion on uniform ligand conjugated surfaces has been studied experimentally, but to our best knowledge, there is no experimental study on the kinetics of membrane spreading on ligand gradient surfaces. A number of studies have measured the spatial distribution of fibroblasts on substrates with gradient density of fibronectin or RGD (Mei et al., 2005; Harris et al., 2006). The results of these studies demonstrate that the variation of cell distribution along the gradient is biphasic; that is, cell density increases toward the increasing direction of gradient and above a certain ligand concentration, it decreases or remains constant. This biphasic cell density distribution can be explained by the fact that above a certain ligand concentration on the gradient, the mechanism of cell spreading switches from reaction-controlled to diffusion-controlled regime. Therefore, according to the predictions of presented model, increasing the ligand concentration reduces the rate of cell–substrate association which, in turn, can inhibit the cell migration. Similar reasoning can be used to explain the biophasic dependence of the density and motility of fibroblasts on substrates coated with different densities of fibronectin or collagen IV (Huttenlocher et al., 1996; Maheshwari et al., 1999; Maheshwari et al., 2000). In the presented model, it is assumed that an unlimited number of free receptors are available within aðtÞoro1 region. In a more realistic situation, the number of receptors available to contribute to adhesion is finite and membrane spreading eventually comes to a halt. Under this condition, the depletion effect of receptor–ligand pairs becomes important and should be taken into account by changing the configurational entropy function in the chemical potential (Smith et al., 2006). Finally, it should be emphasized that the kinetics of membrane spreading, as described here, is only a primary event in cell adhesion. Coupling of cytoskeleton to the receptors and consequently, segregation of ligand–receptor pairs (ligand clustering) and the formation of discrete adhesion domains are the events which eventually control the cellular adhesion (Bruinsma et al., 2000).
ARTICLE IN PRESS A.S. Sarvestani, E. Jabbari / Journal of Biomechanics 41 (2008) 921–925
Conflict of interest statement The authors declare that they have no proprietary, financial, professional, or other personal interest of any nature or kind that in any product, service and/or company related to the results or outcomes of the manuscript titled ‘‘Modeling the Kinetics of Cell Membrane Spreading on Substrates with Ligand Density Gradient’’ by Alireza S. Servestani and Esmaiel Jabbari. Acknowledgments This work was supported by grants from the AO (Arbeitsgemeinschaft Fur Osteosynthesefragen) Foundation (AORF Project 05-J95) and the Aircast Foundation. References Boulbitch, A., Guttenberg, Z., Sackmann, E., 2001. Kinetics of membrane adhesion mediated by ligand–receptor interaction studied with a biomimetic system. Biophysical Journal 81, 2743–2751. Brochard-Wyart, F., de Gennes, P.G., 2002. Adhesion induced by mobile receptors: dynamics. Proceedings of the National Academy of Sciences 99, 7854–7859. Bruinsma, R., Behrisch, A., Sackmann, E., 2000. Adhesive switching of membranes: experiment and theory. Physical Review E 61, 4253–4267.
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