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Langmuir 2009, 25, 1540-1546
How Focal Adhesion Size Depends on Integrin Affinity Tong Zhao,†,⊥ Ying Li,‡,⊥ and Aaron R. Dinner*,†,§,| Department of Chemistry, Department of Physics, James Franck Institute, and Institute for Biophysical Dynamics, The UniVersity of Chicago, Chicago, Illinois 60637 ReceiVed September 4, 2007 Understanding how the thermodynamics and kinetics of integrin receptor binding and clustering impact the formation of focal adhesions is important for understanding the mechanisms cells use to sense and respond to physical cues in their environment. Cells on chemically well-defined surfaces were observed to have distributions of focal adhesions shifted toward smaller sizes when presented with higher affinity ligands (Kato, M.; Mrksich, M. Biochemistry 2004, 43, 2699). In this paper, we account for this trend with a simple model in which integrins are treated as particles on a lattice, and their stochastic dynamics are simulated with a kinetic Monte Carlo algorithm. How the trend depends on force-coupled growth, membrane fluctuations, and heterogeneity of receptor-ligand interactions is analyzed. Predictions are made for substrates in which the ligands presented can vary in either space or time, so that the model can be validated experimentally.
1. Introduction Cells reside in an insoluble network of proteins, the extracellular matrix (ECM). Adhesion to the ECM profoundly influences many cellular processes, including proliferation, differentiation, and survival.1,2 Adhesion is most commonly mediated by the integrin family of integral membrane proteins,3,4 which laterally assemble into micrometer-scale clusters known as “focal adhesions” (FAs). FAs recruit multiple kinase and phosphatase proteins1,5-8 in addition to cytoskeletal proteins that link them to contractile filaments.9,10 In this way, they are thought to act as hubs that bidirectionally translate between mechanical forces and chemical signals to enable the cell to sense and respond to its physical environment.11-13 Experiments suggest that FAs are regulated at three stages: (1) formation of integrin clusters,14,15 (2) force-induced anisotropic growth of focal adhesions (FAs),16,17 and (3) regulated disassembly of FAs.12,18,19 Most theoretical attention to date has focused on the second aspect. Several models that relate chemical * To whom correspondence should be addressed. E-mail: dinner@ uchicago.edu. Tel: (773) 702-2330. Fax: (773) 834-5250. † Department of Chemistry. ‡ Department of Physics. § James Franck Institute. | Institute for Biophysical Dynamics. ⊥ Supported by an Army Multi-University Research Initiative grant and the National Science Foundation.
(1) Giancotti, F. G.; Ruoslahti, E. Science 1999, 285, 1028–1032. (2) Bershadsky, A.; Balaban, N. Q.; Geiger, B. Annu. ReV. Cell DeV. Biol. 2003, 19, 677–695. (3) Hynes, R. O. Cell 1992, 69, 11–25. (4) Hynes, R. O. Cell 2002, 110, 673–687. (5) Clark, E. A.; Brugge, J. S. Science 1995, 268, 233–239. (6) Miyamoto, S.; Akiyama, S. K.; Yamada, K. M. Science 1995, 267, 883– 885. (7) Burridge, K.; Chrzanowska-Wodnicka, M. Annu. ReV. Cell DeV. Biol. 1996, 12, 463–518. (8) Schoenwaelder, S.; Burridge, K. Curr. Opin. Cell Biol. 1999, 11, 274–286. (9) Choquet, D.; Felsenfeld, D. P.; Sheetz, M. P. Cell 1997, 88, 39–48. (10) Felsenfeld, D. P.; Schwartzberg, P. L.; Venegas, A.; Tse, R.; Sheetz, M. P. Nat. Cell Biol. 1999, 1, 200–206. (11) Lauffenburger, D. A.; Horwitz, A. F. Cell 1996, 84, 359. (12) Webb, D. J.; Parsons, J. T.; Horwitz, A. F. Nat. Cell Biol. 2002, 4, E97. (13) Ridley, A. J.; Schwartz, M. A.; Burridge, K.; Firtel, R. A.; Ginsberg, M. H.; Borisy, G. Science 2003, 302, 1704. (14) Nobes, C. D.; Hall, A. Cell 1995, 81, 53–62. (15) Izzard, C. Cell Motil. Cytoskeleton 1998, 10, 137–142. (16) Riveline, D.; Zamir, E.; Balaban, N. Q.; Schwarz, U. S.; Ishizaki, T.; Narumiya, S.; Kam, Z.; Geiger, B.; Bershadsky, A. D. J. Cell Biol. 2001, 153, 1175.
and elastic energies at the level of a single FA have been introduced to account for anisotropic growth in the direction of an applied force.20-22 While integrin switching between highand low-affinity states has been considered in a model for contractility-coupled FA formation,23 the effects of varying the intrinsic affinity of the integrins for the ECM have not been investigated theoretically. Recently, however, Kato and Mrksich observed that the immobilized ligands presented to a cell impact both the size and distribution of FAs. Cells on substrates presenting cyclic (high affinity) RGD ligands had almost twice as many FAs (231 ( 30 FAs/cell) than did those on linear (low affinity) RGD substrates (121 ( 49 FAs/cell). Less intuitively, the former FAs were of a smaller median size (0.78 µm2) than the latter (1.14 µm2). This observation suggests that the FA size distribution is governed by kinetic factors in addition to thermodynamic ones. In this paper, we introduce a simple model to study how FA size depends on the interplay of the diffusion of integrins, their affinity for a substrate, their ability to switch between high- and low-affinity conformations,24-26 and their interactions with other molecules in FAs and contractile filaments. Integrins are treated as particles on a lattice, and their stochastic dynamics are simulated with a kinetic Monte Carlo algorithm. The model shows how the experimentally observed trends27 result from a competition between initial nucleation of clusters and subsequent coarsening. A sensitivity analysis is performed for model parameters controlling the form of the receptor-ligand interactions, mem(17) Balaban, N. Q.; Schwarz, U. S.; Riveline, D.; Goichberg, P.; Tzur, G.; Sabanay, I.; Mahalu, D.; Safran, S.; Bershadsky, A.; Addadi, L.; Geiger, B. Nat. Cell Biol. 2001, 3, 466. (18) Westhoff, M. A.; Serrels, B.; Fincham, V. J.; Frame, M. C.; Carragher, N. O. Mol. Cell. Biol. 2004, 24, 8113. (19) Ezratty, E. J.; Patridge, M. A.; Gundersen, G. G. Nat. Cell Biol. 2005, 7, 581. (20) Nicolas, A.; Geiger, B.; Safran, S. A. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 12520. (21) Shemesh, T.; Geiger, B.; Bershadsky, A. D.; Kozlov, M. M. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 12383. (22) Besser, A.; Safran, S. A. Biophys. J. 2006, 90, 3469. (23) Deshpande, V. S.; Mrksich, M.; McMeeking, R. M.; Evans, A. G. J. Mech. Phys. Sol. 2008, 56, 1484–1510. (24) Xiong, J. P.; Stehle, T.; Diefenbach, B.; Zhang, R.; Dunker, R.; Scott, D. L.; Joachimiak, A.; Goodman, S. L.; Arnaout, M. A. Science 2001, 294, 339. (25) Carman, C. V.; Springer, T. A. Curr. Opin. Cell Biol. 2003, 15, 547. (26) Xiao, T.; Takagi, J.; Coller, B. S.; Wang, J. H.; Springer, T. A. Nature 2004, 432, 59–67. (27) Kato, M.; Mrksich, M. Biochemistry 2004, 43, 2699.
10.1021/la8026804 CCC: $40.75 2009 American Chemical Society Published on Web 01/08/2009
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entropic forces arising from membrane pinning, and force-induced assembly, we consider the additional equilibrium: k+i
two nearest-neighbor bound integrins {\} k-i
two associated integrins (2)
Figure 1. Schematic illustration of the model system. (a) Top view of the square lattice where integrins (solid squares) diffuse and collide occasionally. (b) Side view of the system: integrins interact with substrate ligands (Eb) and two bound integrins that are nearest-neighbors form a “bond” (Ei).
brane fluctuations, and the influence of forces transmitted through stress fibers. On the basis of the mechanistic insights gained, we then explore substrates in which the ligands presented can vary in either space or time; these studies yield predictions that can be used to validate the model experimentally.
2. Model 2.1. Model with a Static Membrane. The model treats integrins explicitly and all other features of the cell and its environment implicitly. Integrins are restricted to sites on a lattice that represents a portion of the basal membrane of the cell (Figure 1). At most, a single integrin is allowed at each lattice site. The linear dimension for an integrin headgroup is 12-15 nm,3 but the lattice spacing must also account for the size of proteins that cross-link integrins in FAs, such as talin (∼60 nm). Adding these numbers, the lattice spacing (a) is 60-75 nm, although this particular choice is not essential for the results. The results presented here are for a 100 × 100 lattice, which corresponds to an area on the order of 1% of the contact area for a fibroblast spread on a surface. Taking the total number of integrins on such a cell (assuming a total surface area of ∼5000 µm2) to be 104-105,28 we expect 70-1100 integrins in the membrane patch considered. We fix the integrin number at 100 throughout the simulations. Periodic boundary conditions are employed to minimize finite size effects. Integrins can diffuse, switch between inactive (low affinity) and active (high affinity) states, and bind and unbind the substrate. Active and inactive integrins that are not bound diffuse with the same rates, and this process is modeled as a hopping reaction to nearest-neighbor sites on the lattice that are unoccupied. Switching and binding are described by the dynamic equilibria k+a
k+b
k-a
k-b
inactive integrins {\} active integrins {\} bound integrins (1) with the equilibrium constants k+a/k-a ) exp(-Ea) and k-b/k+b ) exp(-Eb), where Ea (Eb) is the free energy of integrin activation (binding) in units of kBT, kB is Boltzmann’s constant, and T is temperature. To account for the heterogeneity of integrin affinities the binding energy is normally distributed around Eb with standard deviation σ. The influence on affinities from membrane fluctuations is discussed later. To represent the adaptor proteins that cross-link the cytoplasmic domains of integrins in FAs (e.g., talin,29-32 as mentioned above), (28) Bell, G. I.; Dembo, M.; Bongrand, P. Biophys. J. 1984, 45, 1051–1064. (29) Critchley, D. Curr. Opin. Cell Biol. 2000, 12, 133–139. (30) Tadokoro, S.; Shattil, S. J.; Eto, K.; Tai, V.; Liddington, R. C.; de Pereda, J. M.; Ginsberg, M. H.; Calderwood, D. A. Science 2003, 302, 103–106.
with the equilibrium constant k-i/k+i ) exp(-Ei). The energy Ei favors aggregation. Based on the observation that the local density of integrins bound to the ECM is proportional to the magnitude of the force density,17 we factor the energy Ei into a constant part, representing the spontaneous interaction energy E0i , and a varying part, representing the force-induced energy ∆Ei. For the latter, we model the force on an FA in a manner analogous to ref 33. Each pair of FAs is connected by a stress fiber, which is assumed to disassemble at a constant rate but assemble at a bab| rate proportional to the sizes of FAs with which it connects: |F Fab is the force on a stress fiber connecting the ∝ NaNb, where b a-th and b-th FAs, and Na and Nb are their respective numbers of integrins. We consider integrins to be part of the same cluster if they are nearest neighbors, and we count a cluster as an FA if it contains more than two integrins. This form assumes a quasiequilibrium, which is reasonable because the creation and destruction of stress fibers is expected to be slow in comparison with the FA dynamics. Force-induced growth contributes to the association of two bound nearest-neighbors in an FA at a rate proportional to the density of the net force on that FA: ∆k+i ∝ |Σbb Fab|/Na. Summing over all stress fibers and taking the dissociation rate k-i to be constant, the interaction energy for integrins in the a-th FA becomes
[ [
Ei ) ln ) ln
[
0 k+i + ∆k+i k-i 0 k+i
k-i
+
] ]
∆k+i k-i
0
) ln eEi + ∆f Ei0
|
) ln[e + ∆f|
∑ b bFab| Na
]
∑ Nb · be ab|]
(3)
b
The second term in each pair of square brackets corresponds to the varying interaction energy, where ∆f is an adjustable parameter that describes the extent to which the nucleation of stress fibers and integrins affect each other, and b eab is the unit vector pointing from the center of the a-th to the b-th FA. The lattice is periodic, and we use the minimum image convention to calculate the directions of stress fibers and forces on them. In summary, the total energy of a configuration of the lattice is
E)-
N
N
l)1
l)1
∑ Eip〈lm〉 - ∑ Eb(l)nl + Ea∑ ml 〈lm〉
(4)
where nl ∈ 0, 1 (ml ∈ 0, 1) is the occupation number of bound (active) integrins at lattice site l, and p〈lm〉 ∈ 0, 1 is a variable that reports whether there is an association between the nearestneighbor pair 〈lm〉. It is important to note that the association (31) Garcia-Alvarez, B.; de Pereda, J. M.; Calderwood, D. A.; Ulmer, T. S.; Critchley, D.; Campbell, I. D.; Ginsberg, M. H.; Liddington, R. C. Mol. Cell 2003, 11, 49–58. (32) Nayal, A.; Webb, D. J.; Horwitz, A. F. Curr. Opin. Cell Biol. 2004, 16, 94–98.
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energy Ei varies from pair to pair, depending on the net force on FAs in which the integrins are located. We use a kinetic Monte Carlo procedure to simulate the dynamics of the model. At each step, we select between diffusion (hopping), activation, binding/unbinding, and nearest-neighbor association/dissociation with likelihoods νd/νa/νb/νi ) 0.83:0.08: 0.08:0.005. These likelihoods are chosen for consistency with experimentally estimated time scales. For a selected reaction category, a random sweep over all the possible reactions in the current configuration is performed. Hopping of free integrins to a randomly chosen vacant nearest-neighbor site is always accepted. Given that integrins diffuse with 10-4 < D0 < 1 µm2/ s,33,34 each Monte Carlo step corresponds to a time interval of ∆t ) 0.8a2/4D0 = 10 ms (D0 ) 0.1 µm2/s and a ) 70 nm). Every 1000 MC steps, we update the force on all FAs and the interaction energies of integrins. This time interval is faster than the time scale of stress fiber assembly (several minutes33), so it serves as a good approximation to monitoring the forces continuously. The update breaks detailed balance, but this is reasonable in this context since the system of interest is out of equilibrium. We choose the remaining parameters as follows. We take the activation energy to be Ea ) 3. This value is consistent with an earlier model.23 The binding energy varies with the types and density of ECM ligands and integrin receptors. Typical unbinding rates range from 0.01 to 1 s-1, and the binding rate is of the order of s-1.35-38 We take the equilibrium constant following ligand-receptor encounter to be that of integrin-R5β1 with fibronectin, which is measured to be 0.01,37 and study Eb between 0 and 10kBT. The standard deviation for this energy is chosen to be σ ) Eb/3. As for the nearest-neighbor interaction, we assume a spontaneous association energy E0i in the same range as Eb, 0-9kBT, and consider two cases separately: one where Ei is weakly affected by the force on FAs (∆f ) 2) and one where it is strongly affected (∆f ) 40). 2.2. Model of a Fluctutating Membrane. To explore the effects of membrane fluctuations on the size distribution of FAs, we model the membrane as a thin fluid sheet that lies above the surface on which the cell rests. The membrane is parametrized by the separation between it and the ECM: z(x, y, t), where x and y are the coordinates defining the plane of the surface. The membranefree energy includes the bending elastic energy, the surface tension, and an additional potential responsible for pinning the membrane at adhesion sites:39,40
γ F) 2
∫ dx∫
∫ ∫
κ dy[∇z(x, y, t)]2 + dx dy[∇2z(x, y, t)]2 + 2 λ dx dyC(x, y, t)[z(x, y, t) - z0]2 (5) 2
∫ ∫
where γ is the surface tension, κ is the bending rigidity, C(x, y, t) is the concentration of receptor-ligand complexes, and λ is the curvature of the energy well for receptor-ligand binding. The (33) Novak, I. L.; Slepchenko, B. M.; Mogilner, A.; Loew, L. M. Phys. ReV. Lett. 2004, 93, 268109. (34) Jacobson, K. A.; Moore, S. E.; Yang, B.; Doherty, P.; Gordon, G. W.; Walsh, F. S. Biochim. Biophys. Acta 1997, 1330, 138. (35) Bruinsma, R. Biophys. J. 2005, 89, 87–94. (36) Taubenberger, A.; Cisneros, D. A.; Friedrichs, J.; Puech, P, H.; Muller, D. J.; Franz, C. M. Mol. Biol. Cell 2007, 18, 1634–1644. (37) Li, F.; Redick, S. D.; Erickson, H. P.; Moy, V. T. Biophys. J. 2003, 84, 1252–1262. (38) Thoumine, O.; Kocian, P.; Kottelat, A. Eur. Biophys. J. 2000, 29, 387– 408. (39) Qi, S. Y.; Groves, J. T.; Chakraborty, A. K. Proc. Natl. Acad. Sci. USA 2001, 98, 6548–6553. (40) Gov, N.; Zilman, A. G.; Safran, S. Phys. ReV. Lett. 2003, 90, 228101. (41) Zamir, E.; Geiger, B. J. Cell Sci. 2001, 114, 3583–3590. (42) Geiger, B.; Bershadsky, A.; Pankov, R.; Yamada, K. Nat. ReV. Mol. Cell Biol. 2001, 2, 793–805.
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last term accounts for the energy penalty that arises from the deviation of the local distance z(x, y, t) from the natural length of integrin-substrate complexes, z0. In other words, a local concentration of bound integrins stabilizes a membrane separation of z0. For consistency bewteen the membrane and integrin parts of the model, the distance between the membrane and ECM feeds back to influence the binding kinetics. As in a previous model,39 we assume the equilibrium contant follows a simple Gaussian distribution centered at the natural length of an integrin complex with a standard deviation ε
k+b k0+b - [z(x, y, t) - z0]2 ) e 2ε2 k-b k0
(6)
-b
In other words, the binding energy for an integrin can be written as a sum of two parts:
Emb ) E0b -
[z(x, y, t) - z0]2 2ε2
(7)
where Emb represents the binding energy after including membrane fluctutations, and Eb0 is the maximum value of the binding energy for this integrin that normally distributes with a mean of Eb and a standard deviation σ. The overall heterogenity in the binding energy arises from two sources: the intrinsic affinity difference of receptors described by the standard deviation σ and the membrane fluctuation described by the standard deviation . The surface tension and bending rigidity of cell membrane are chosen according to the measured value for Dictyostelium discoideum cells,39 γ ) 0.22kBT/(nm)2 and κ ) 391kBT. The natural length of integrin-substrate complexes is chosen to be the average distance between the cell membrane and the ECM,41 z0 ) 15 nm. The curvature of the energy well for receptor-ligand binding is chosen to be of the same order as the spring constant of the complex, λ ) 50kBT/(nm)2.36,42 We perform a sensitivity analysis for ε since its value is unknown; unless otherwise specified, we take it to be ε ) 5 nm. We simulate the membrane dynamics with a Langevin equation:
∂z δF ) -M + ζ dt δz
(8)
where M is a phenomenological constant that determines the membrane response to free energy changes and ζ is a Gaussian white noise with amplitude √2MkBT. The results are insensitive to the value of M over a large range of values; unless otherwise specified, we take it to be M ) 5 × 10-4 nm2/s. Numerically, we further coarse-grain the integrin lattice with a unit length scale of 60-75 nm to obtain a lattice with a length scale of 240-300 nm and simulate the membrane dynamics at the latter resolution. We start with a uniform distance z ) z0. At each Monte Carlo step, integrins choose among allowed reactions; when an integrin attempts to bind, the energy is determined by eq 7. At the same time, we calculate the free energy change δF from eq 5 and use a first-order integrator to solve eq 8. Each time step corresponds to 10 ms. The local concentrations of integrin complexes are given by the number of bound integrins in a cell of the large length scale lattice. For the parameters given above, the major contribution to δF/δz in eq 8 comes from the pinning energy in regions where there are bound integrins.
3. Results and Discussion In this section, we show that the model reproduces observed trends for substrates presenting different RGD peptides27 and
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Figure 3. Average cluster sizes as a function of the affinity for the surface (Eb) and the spontaneous pairwise interaction (Ei0). Energies are in units of kBT. Figure 2. Cumulative integrin cluster size distribution for Eb ) 6 (black) and 8 (red) kB; Ei0 ) 4kBT. Unless otherwise specified, all averages are calculated at 108 MC steps in 30 or more independent simulations with ∆f ) 2 and σ ) Eb/3 and a static membrane.
then set that behavior in a broader context by fully exploring the space of parameters. The calculations make clear that the distribution of FA sizes results from a competition between the thermodynamics of integrin receptor binding and the kinetics of integrin cluster coarsening. Predictions for the behavior on substrates in which the ligand varies (in time or space) are then presented to enable validation of the model. 3.1. Basic Model. In the experiments of interest, Kato and Mrksich quantified the FA size distribution with cumulative histograms (see Figure 6 of ref 27) on surfaces presenting either cyclic or linear RGD peptide. Soluble cyclic peptide was 22 times more effective than linear at reducing cell attachment to one-half in inhibition assays in which cell adhesion was measured as a function of the concentration of competing soluble ligand. Here we study Eb ) 6 and 8kBT, which corresponds to a 7-fold difference in the affinity of individual receptors for the surface (Figure 2). Larger variations in affinity are discussed subsequently. Consistent with the experimental observations, the median of the distribution shifts to smaller values as the affinity increases (higher Eb). In going from Eb ) 6 to Eb ) 8kBT, the total number of clusters increased by 60% and the average size decreased by 52%, again in agreement with the behavior observed for cells on surfaces presenting linear (low affinity) and cyclic (high affinity) RGD. The model thus reproduces the known trends in FA size and number as affinities vary over ranges comparable to those studied experimentally. To set these results in a broader context, we varied Eb from 0 to 10 kBT and E0i from 0 to 9 kBT. The average cluster size is shown as a function of Eb and E0i in Figure 3. Above a threshold value (E0i ≈ 3kBT), stable clusters form and vary nonmonotonically in size with Eb (Figure 4). As E0i increases, integrin clustering is enhanced and the number of free integrins is reduced. Because increases in Eb and E0i both act in this way, increases in E0i shift the crossover toward smaller Eb. Below the threshold in E0i , no stable clusters form. The absence of clusters results from the fact that the tendency for entropically driven mixing overwhelms that for energetically favorable aggregation. The physical origin for the observed biphasic behavior at E0i g 3kBT is a competition between the thermodynamic propensity of binding and the kinetic mobility of integrins. To see this, we set E0i ) 0 and ∆f ) 0, and plot the fraction of bound integrins as a function of Eb in Figure 5. It increases monotonically with the substrate affinity and saturates for Eb g 4kBT. These data are to be compared with the decreasing mobility of integrins as the substrate affinity increases. To see the effect of kinetic hindrance
that results completely from the substrate affinity changes, we calculate the diffusion constant from D ) 〈x2(t)〉/4t for all t. Consistent with the monovalency that results for E0i ) 0 and ∆f ) 0, D obeys Arrhenius scaling and decreases exponentially as the affinity of integrins for the substrate increases when Eb g Ea and most integrins are activated (Figure 5). The physical picture that emerges is the following. When the affinity of integrins for the substrate is low, few integrins are bound, so clusters fail to nucleate and grow. When the affinity is high, integrins readily bind and form small clusters; however, integrins must unbind to diffuse, and clusters are thus hindered from coalescing. As a result, at finite times, the average cluster size exhibits a maximum as a function of affinity (Eb ≈ 4kBT in Figure 4). It is important to note that, for the model considered, this result is due to the fact that we study the system at finite times; two large clusters will always be accessed eventually for E0i g 3kBT because such a configuration will maximize the force density. In cells, integrins are introduced and removed from the membrane, leading to a steady-state balance and a finite FA size. The model considered here could readily be extended to include such effects, but the basic physics is expected to be the same. The results above are obtained in the case that the assembly of stress fibers and the nearest-neighbor association of integrins are coupled to a moderate extent (∆f ) 2). We further study the case of strong interplay by increasing ∆f to 40. This parameter changes the association energy in a nonlinear way. An increase in Ei is more significant for a small base energy (E0i ), e.g., with an average FA size of 40 integrins, Ei is increased by 4kBT if E0i ) 0kBT and by 0.4kBT if E0i ) 8kBT. The biphasic dependence of FA size on Eb holds at ∆f ) 40, but the E0i threshold for stable FAs is lowered (Figure 6). The cooperativity between FAs and stress fibers offers a positive feedback mechanism that stabilizes FA formation: an increase in FA sizes promotes stress fiber assembly, which in turn imposes larger forces on FAs and leads to their further growth. Indeed, when the coupling is strong, the force-induced clustering of nearest-neighbor integrins is sufficient to trigger FA growth even if spontaneous association is disfavored (E0i e 0). For the standard deviation of binding energy σ, an increase in this value gradually decreases the average size of FAs (Figure 7). We attribute this trend to the fact that more integrins are either unbound because they have low Eb or bound too tightly to coalesce because they have high Eb. Results in the presence of membrane fluctuations are qualitatively the same as in their absence. As the average binding energy Eb increases, the average cluster size varies nonmonotonically (Figure 8 (left)). Nevertheless, there are quantitative effects. In particular, the peak in the curve shifts to the right because the membrane fluctutations decrease the effective binding energy of integrins Ebm as in eq 7. To get the same extent of
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Figure 4. Top left: A cut of the 2D map in Figure 3 along Ei0 ) 4kBT. Top right: the average FA number under the same Ei0. Bottom: typical configurations of the system at different values of Eb as labeled.
Figure 5. Calculated effective diffusion constant (left) and the total number of bound integrins (right) with Ei0 ) 0. D0 ) 18.07a2/s is the bare integrin diffusion constant without substrate binding. Using the lattice constant a ) 70 nm, we obtain D0 ) 0.088 µm2/s, slightly smaller than the free integrin diffusion constant in the membrane specified earlier (0.1 µm2/s), due to volume exclusion. Figure 7. Average cluster size as a function of the standard deviation σ of the binding energy; Eb ) 4kB and Ei0 ) 4kB.
Figure 6. Average cluster sizes as a function of the affinity for the surface (Eb) and the spontaneous pairwise interaction (Ei0) under a strong interaction between FAs and stress fibers ∆f ) 40. Energies are in units of kBT.
clustering as in the absence of membrane fluctutations, a larger Eb is required. These results are consistent with previous studies where membrane deformation imposed energy penalties and prevented the growth of FAs.43 The separation between the membrane and the ECM fluctuates around z0 with an amplitude of 20% (Figure 8 (right)). As mentioned in defining the model, we perform parameter sensitivity analyses for the standard deviation of the binding equilibrium constant ε and the response coefficient M. As the standard deviation ε decreases, integrins become more sensitive to the difference between the local separation z and average distance z0, and a larger Eb is (43) Wei, Y. J. Langmuir 2008, 24, 5644–5646.
consequently required for integrins to cluster (Figure 9 (left)). As the response coefficient M changes over orders of magnitude, the average cluster size varies only slightly (Figure 9 (right)). 3.2. Alternate Substrates. The studies above show that high affinity ligands promote nucleation, while low affinity ligands promote coarsening. In this section, we investigate how variations in either space or time in the ligand affinities presented impact the cluster sizes observed in simulations as a means of providing predictions for further validation of the model. 3.2.1. Dynamic Substrates. Substrates that change their affinity dynamically can now be made.44 To simulate such experiments, we change Eb at a defined time (t ) 1.7 × 107 MC steps in the simulations shown, Figure 10). We find that, upon switching the affinity, the coarsening rate (the slope) changes immediately to that characteristic of the new affinity. The lack of memory suggests that subsequent changes will have like effects. The resulting size distribution will thus depend strongly on the sequence of affinities experienced up to the observation time. 3.2.2. Patterned Substrates. To investigate the effects of spatial variations, we simulate cluster formation for different patterns of high and low affinity ligands with a fixed proportion of the former (25%). As shown in Figure 11, FAs generally form in high affinity regions. When these regions are large (compared (44) Yousaf, M. N.; Houseman, B. T.; Mrksich, M. Angew. Chem., Int. Ed. 2001, 40, 1093. (45) Tan, J. L.; Tien, J.; Pirone, D. M.; Gray, D. S.; Bhadriraju, K.; Chen, C. S. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 1484–1489.
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Figure 8. Effect of membrane fluctuations. Left: the average cluster size as a function of the average binding energy Eb at a spontanesous association energy of Ei0 ) 4kBT. Results for fluctuating (black solid) and a static (red dashed) membranes are compared. Right: the distance between the membrane and ECM (z) at a representative time. The size of one lattice site is 240-300 nm; x and y index the sites of the lattice. Black points represent integrins. Parameters are as specified in the text.
Figure 9. Parameter sensitivity analysis at E0i ) 4kBT. Left: The standard deviation ε is varied, 20 (black), 10 (red), and 5 nm (green). No FAs are observed for ε ) 1 and 2.5 nm. Right: The response coefficient M is varied, 5 × 10-5 (black), 5 × 10-4 (red), and 5 × 10-2 (green) nm2/s.
Figure 10. Effects on cluster size of dynamically switching substrate affinity. The total simulation time is 5 × 107 MC steps. The substrate affinity is switched from low Eb ) 5kBT (high Eb ) 8kBT) to high (low) at 1.7 × 107 MC steps. Data on the control substrates without kinetic switching are also shown. Ei0 ) 4kBT.
to the clusters), the clusters are entirely within them, while, when these regions are small, the clusters span them. Interestingly, in the former case, the FAs tend to localize to edges, consistent with experimental observations (for example, see ref 45). This feature of FAs has been ascribed to force-induced growth (treadmilling).33 Although our model incorporates force-induced growth in a similar way, the phenomenon also appears when this feature is not considered (∆f ) 0, data not shown), which suggests that the kinetics of penetrating regions contributes to edge
Figure 11. Typical configurations on patterned substrates. For ease of visualization, we show a ∼50 × 50 sublattice for each configuration. A fixed fraction of 25% high affinity ligands (gray sublattice, Eb ) 8kBT) are distributed in different patterns among the low affinity ligands (white background, Eb ) 5kBT). Integrins are indicated with black stars. Ei ) 4kBT0. Left: random, 1 × 1 and 1 × 2 (top to bottom); right: 2 × 2, 5 × 5, and 25 × 25 (top to bottom).
localization when adhesive islands are on subcellular length scales. Even though the coverage level of high affinity ligands is exactly the same in each case, the average cluster size is observed to vary with the island size. For large enough islands (5 × 5 and 25 × 25), FAs grow on the island edges. The average cluster size is larger than that for the uniform high affinity substrate (compare Figure 11 with Figure 4 for Eb ) 8kBT), which suggests that the low affinity channels promote
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clustering by facilitating diffusion. For the 1 × 1, 2 × 1, and 2 × 2 patterns, receptors must break favorable interactions with the surface and neighbors to further coarsen once clusters fill the islands. If we consider the spontaneous association energy only, there are energy barriers of 8, 13, and 18 kBT for breaking away from an existing cluster the same size as a high affinity island in 1 × 1, 2 × 1, and 2 × 2 panels of Figure 11, respectively. By the same token, the spacing between islands will impact the distribution of cluster sizes as well. The behavior observed for a random distribution, which has roughly the same probability of having two high affinity sites adjacent to each other as the 2 × 1, is consistent with these ideas.
4. Conclusions We have shown how a competition between the thermodynamics of integrin binding and the kinetics of cluster coarsening can give rise to the observed dependence of FA size on substrate
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affinity. Understanding how the tendencies for integrins to bind (controlled through Eb in the model) and cluster (controlled through Ei) impact FA dynamics is important for understanding the mechanisms cells use to sense and respond to physical cues in their environment. Simulations in which the affinities of the ligands presented varied in either space or time were used to illustrate how appropriately designed substrates could be used to manipulate FA dynamics by means other than traditional genetic approaches. Such experiments are now feasible and could lead to novel means to probe fundamental molecular dynamics in cells and harness the cytoskeletal machinery for engineering purposes. Acknowledgment. We thank Milan Mrksich for helpful discussions and a critical reading of the manuscript. LA8026804
