Stability of Heterogeneous Parallel-Bond Adhesion Clusters Under Static Load

Stability of heterogeneous parallel-bond adhesion clusters under static load

Anil K. Dasanna, Gerhard Gompper, and Dmitry A. Fedosov∗ Theoretical Physics of Living Matter, Institute of Biological Information Processing and Institute for Advanced Simulation, Forschungszentrum Jülich, 52425 Jülich, Germany

Adhesion interactions mediated by multiple bond types are relevant for many biological and soft matter systems, including the adhesion of biological cells and functionalized colloidal particles to various substrates. To elucidate advantages and disadvantages of multiple bond populations for the stability of heterogeneous adhesion clusters of receptor-ligand pairs, a theoretical model for a homogeneous parallel adhesion bond cluster under constant loading is extended to several bond types. The stability of the entire cluster can be tuned by changing densities of different bond populations as well as their extensional rigidity and binding properties. In particular, bond extensional rigidities determine the distribution of total load to be shared between different sub- populations. Under a gradual increase of the total load, the rupture of a heterogeneous adhesion cluster can be thought of as a multistep discrete process, in which one of the bond sub-populations ruptures first, followed by similar rupture steps of other sub-populations or by immediate detachment of the remaining cluster. This rupture behavior is qualitatively independent of involved bond types, such as slip and catch bonds. Interestingly, an optimal stability is generally achieved when the total cluster load is shared such that loads on distinct bond populations are equal to their individual critical rupture forces. We also show that cluster heterogeneity can drastically affect cluster lifetime.

I. INTRODUCTION they act synergistically [28], the exact roles of different receptor-ligand pairs remain largely unknown. Adhesion interactions via receptor-ligand bonds are es- Biological cells often interact with a substrate through sential for many biological and soft matter systems. Ex- a number of localized adhesion sites called focal adhe- amples include cell adhesion [1–4] and migration [5, 6], sions [3], which can be thought of as localized clusters of synapse formation [7, 8], adhesion of lipid vesicles [9–11] adhesive bonds under applied stress. Similarly, leukocyte and drug-delivery carriers [12, 13] to a substrate. Such adhesion can be approximated through adhesive interac- adhesive interactions depend on the properties of recep- tions of several discrete clusters, since PSGL-1 proteins tors and ligands (e.g. density, kinetic rates, mobility) and are primarily located at the cell’s microvilli tips [29]. Fur- the characteristics of adhered particles (e.g. size, shape, thermore, PfEMP-1 receptors are positioned at cytoad- deformability). For instance, binding/dissociation rates herent knobs, representing discrete adhesion clusters on of receptors and their mobility together with membrane the surface of malaria-infected erythrocytes [30]. The constraints strongly affect the formation of immunologi- first simple theoretical model for local adhesion clusters cal synapse characterized by a highly organized pattern with a single bond population was proposed by Bell [31], of receptor proteins [8, 14]. In addition to bond rates who used a mean-field approach to describe the stabil- and mobility, membrane/substrate deformation and ap- ity of a parallel adhesion-bond cluster of fixed size under plied stresses play an important role in the nucleation constant loading. Later, the original model has been ex- of bond domains [15] and their growth and distribution tended to dynamic loading [32, 33] and generalized to a [10, 16–18]. stochastic model for parallel bond clusters [34–36], which Bond-mediated adhesion interactions often involve shows that the cluster lifetime is always finite and in- more than one type of receptor-ligand pairs with dis- creases exponentially with the number of bonds within tinct intrinsic properties. For instance, leukocytes be- the cluster. Note that these models are applicable to fore extravasation first bind to and roll at an endothelial local adhesion clusters with a fixed number of adhesive cell layer, then show a firm adhesion at the surface [19– sites (i.e. without receptor mobility). Furthermore, they 21]. This process is facilitated by the ability of P-selectin can be used to quantify the adhesion of functionalized glycoprotein (PSGL-1) at the surface of leukocytes to rigid particles used in self-assembled functional materi-

arXiv:2005.00264v2 [physics.bio-ph] 22 Sep 2020 bind to both selectin and integrin molecules expressed als [37, 38] or for drug delivery [12, 13]. at endothelial cells. Another example is the adhesion of In the theoretical model for a homogeneous adhesion malaria-infected red blood cells to the endothelium, in bond cluster, each bond can form with a constant on-rate order to avoid their removal in the spleen [22–24]. Here, κon and rupture with an off-rate κoff (F ) which depends Plasmodium falciparum erythrocyte membrane receptor on the applied force F . Note that the ratio κon/κoff rep- (PfEMP-1) can bind to multiple ligands (e.g. CD36, resents a Boltzmann factor related to the energy change ICAM-1, and CSA molecules) at the surface of endothe- due to bond formation, and therefore, it characterizes lial cells [25–27]. Even though it is hypothesized that binding strength. The original model by Bell [31] considered a so-called slip bond with κoff = κ0 exp (F/f d), where κ0 is the unstressed off-rate and and f d is a char- ∗ [email protected] acteristic force scale (typically a few pN), such that the 2

ment. Finally, we employ master equation to compute F the lifetime of heterogeneous clusters for various parameters, and show that the heterogeneity can drastically aﬀect cluster lifetime.

k1 k2 II. METHODS AND MODELS

A. Parallel bond cluster model κ0 γ1 γ2

We start with the original model for parallel slip-bond cluster under a constant loading [31, 34]. The system contains Nt adhesion sites, where N(τ) Nt (τ is time) ≤ FIG. 1. Heterogeneous parallel bond cluster with two differ- bonds or bound springs with an extensional rigidity k and on 0 ent bond populations indicated by blue and red colors. γ1 a dimensionless rebinding rate γ = κ /κ can stochasti- and γ2 are rebinding rates of the 1st and 2nd bond types, re- cally form under an external force F . From the stability spectively. Similarly, k1 and k2 are the corresponding spring analysis [31], with the assumption that each spring shares 0 0 rigidities. κ is a reference off-rate whose inverse 1/κ sets a the same force F/N, there exists a critical force f c below basic timescale. which the cluster equilibrates to an average number of bonds N and above which the cluster is unstable and dissociates,h i i.e. N = 0. The critical force f c and the crit- bond lifetime decreases with increasing F [39]. Some ical number Nc of springs for slip-bond cluster are given biological bonds may behave differently, so-called catch by [31, 35] bonds, such that their lifetime increases first with increasing F until a certain threshold, and then decreases f c pln(γ/e) with increasing F similar to a slip bond. The catch- t pln c t (1) d = N (γ/e), N = N , bond behavior was first predicted theoretically [40] and f 1 + pln(γ/e) later discovered for leukocytes experimentally [41]. The parallel bond-cluster model for slip bonds has also been where pln(a) is the product logarithm function which x adapted to the case of catch bonds [42, 43]. solves the equation x e = a. In this article, we extend the homogeneous parallel- We generalize this model to a heterogeneous cluster bond-cluster model to multiple bond populations and with two different bond populations, characterized by the show that the stability and lifetime of a heterogeneous extensional rigidities k1 and k2 and the rebinding rates on 0 on 0 cluster can be tuned by changing the fractions of differ- γ1 = κ1 /κ and γ2 = κ2 /κ , see Fig. 1. Despite the ent bond populations and their extensional rigidity and fact that unstressed off-rates of distinct slip-bond popu- binding properties. We use both stochastic simulations lations can be different, the rebinding rates γ1 and γ2 are and the mean-field approach to construct critical rupture- defined here using a reference κ0 off-rate, whose inverse force diagrams for a number of relevant parameters. Un- 1/κ0 sets the basic timescale in the system. The total der a gradual increase of the applied load, the dissociation number Nt of adhesion sites is assumed to be constant, of a heterogeneous bond cluster can well be described and ρ determines a fraction of type-1 adhesion sites, such t t t t by a multistep discrete process, which starts with the that N1 = ρN and N2 = (1 ρ)N . We also define r − rupture of one of the bond sub-populations and contin- a spring-rigidity ratio k = k1/k2 and a rebinding ratio r 0 ues with similar rupture steps of other sub-populations γ = γ1/γ2. At any time τ = tκ , the applied force is or shows a sudden detachment of the remaining cluster. r This cluster-dissociation behavior is qualitatively inde- F = N1k1∆x + N2k2∆x = (N1k + N2) k2∆x, (2) pendent of involved bond types, including slip and catch bond sub-clusters. To maximize the critical rupture force where ∆x is the extension of bound springs. f1 and f2 of a heterogeneous cluster for fixed bond kinetics and are the forces acting on the corresponding populations of sub-cluster fractions, the distribution of loads on distinct bond types 1 and 2, given by bond sub-clusters has to be such that individual loads on the sub-clusters are equal to their critical rupture forces. N F kr N F 1 2 (3) f1 = r , f2 = r , The load balance within a heterogeneous cluster is con- N1k + N2 N1k + N2 trolled by the ratio of bond extensional rigidities. Our r results show that a strong load disbalance (i.e. major- such that f1 + f2 = F . Clearly, k directly controls the ity of the total force is applied on a single sub-cluster) distribution of forces between the two bond populations. is generally disadvantageous for overall cluster stability, For the case of γr = 1 and kr = 1, the heterogeneous clus- because the bond sub-cluster carrying the load majority ter becomes identical to the homogeneous bond cluster quickly ruptures without significant stability enhance- considered previously. 3

B. Mean-ﬁeld approximation (a) The average number of bonds is governed by two non- dimensionalized rate equations: 102

oﬀ ) dN1 κ1 t τ

= N + (N N )γ , (4) ( dτ − 1 κ0 1 − 1 1 off N dN2 κ2 = N + (Nt N )γ . (5) r r dτ − 2 κ0 2 − 2 2 k = 1, γ = 5 kr = 1, γr = 1 For a cluster with two slip-bond populations, r r off 0 d off k = 5, γ = 1 κ1 (f1/N1) = κ1 exp [f1/(N1f1 )] and κ2 (f2/N2) = 0 d d d κ2 exp [f2/(N2f2 )]. f1 and f2 are the two force scales 0 5 10 15 20 25 d d d which are assumed to be the same, f1 = f2 = f . τ Equations (4) and (5) are coupled via the forces f1 and (b) r r f2 in Eq. (3) and are used to deduce the average number k = 1, γ = 5 N = N + N of bonds and critical force f c of the kr = 1, γr = 1 h i h 1 2i 100 entire cluster. Note that further on all quantities with kr = 5, γr = 1 force dimensions are implicitly normalized by f d. i N h C. Master equation 50

An extension of the mean-ﬁeld approach is a one-step, f c two-variable master equation for this system [44]. If Pi,j is the probability of i type-1 bonds and j type-2 bonds, 0 then the master equation is 30 50 70 90 F

dPi,j i+1,j i,j+1 i−1,j = r Pi ,j + r Pi,j + g Pi− ,j+ dτ 1 +1 2 +1 1 1 h i FIG. 2. Behavior of a heterogeneous cluster with two slip- i,j−1 i,j i,j i,j i,j bond populations. (a) Evolution of N(τ) = N (τ)+N (τ) for g2 Pi,j−1 r1 + r2 + g1 + g2 Pi,j, (6) 1 2 − F = 50, ρ = 0.3, and Nt = 200. Trajectories from stochastic i,j off 0 i,j off 0 simulations are shown by colored lines, the numerical solution where r1 = i κ1 (f1/i)/κ and r2 = j κ2 (f2/j)/κ i,j i,j of Eqs. (4) and (5) by black lines. The critical force for a t r r c are the reverse rates and g1 = γ1(N1 i) and g2 = uniform cluster with k = 1 and γ = 1 is f ' 55.7. (b) t − γ2(N2 j) are the rebinding rates for type-1 and type-2 Average number hNi of bonds as a function of the applied bonds,− respectively. Then, the average numbers of type-1 force F for different γr and kr. Lines correspond to the mean- and type-2 bonds can be computed as field approximation, symbols to stochastic simulations. The arrows indicate the corresponding f c values. X X i = iPi,j, j = jPi,j. (7) h i h i i,j i,j This equation can be simplified to Cluster lifetime Ti,j corresponds to the time required i,j i,j for the cluster to dissociate completely, i.e. time for 1 = g (Ti ,j Ti,j) + g (Ti,j Ti,j) − 1 +1 − 2 +1 − i, j 0, 0 (no bonds). The equation for Ti,j of i,j i,j { } → { } + r (Ti−1,j Ti,j) + r (Ti,j−1 Ti,j) . (9) a heterogeneous cluster can be derived similarly to the 1 − 2 − approach for parallel cluster with a single bond type t t This recursive relation generates N1 N2 algebraic equa- [45], and then solved numerically, following the recur- tions, which are solved by inverting× the corresponding sive scheme for the lifetime of a cluster with catch and coefficient matrix [44, 45]. The computation also requires slip bonds [44]. For i type-1 bonds and j type-2 bonds, appropriate boundary conditions for on- and off-rates for after a time interval ∆t, the cluster jumps to one of the both bond types and the state 0, 0 acts as an absorbing four neighboring states ( i + 1, j , i 1, j , i, j + 1 , { } boundary condition, i.e. T0,0 = 0. or i, j 1 ) or it remains{ at the} state{ − i,} j ,{ which} is described{ − as} { }

i,j i,j i,j D. Stochastic simulations Ti,j ∆t = g ∆t Ti ,j + g ∆t Ti,j + r ∆t Ti− ,j − 1 +1 2 +1 1 1 i,j + r2 ∆t Ti,j−1 (8) Another approach for analyzing cluster stability is di- i,j i,j i,j i,j rect stochastic simulations using the Gillespie’s algo- + 1 g ∆t g ∆t r ∆t r ∆t Ti,j. − 1 − 2 − 1 − 2 rithm [46]. The heterogeneous system is described by 4 four rate equations, two for each type of bonds, repre- large τ. For kr = 1 and γr = 5 (blue line), the cluster is senting their association and dissociation as very stable because the critical force is much larger than F = 50, which is evident from Fig. 2(b), where the av- κon O 1 C , (10) erage number N of bonds is presented as a function of 1 −−→ 1 h ri r off F for different γ and k . In contrast, the cluster with κ1 r r C1 O1, (11) k = 5 and γ = 1 quickly dissociates at F = 50, as it −−→on significantly exceeds the critical force. The differences in κ2 O2 C2, (12) N between stochastic simulations (symbols) and deter- −−→ h i c off ministic solutions (lines), as F approaches f in Fig. 2(b), κ2 C2 O2, (13) characterize the fraction of simulations where cluster dis- −−→ sociation has occurred within the total simulation time. where O denotes an open state and C a closed state. Note that the cluster lifetime is always finite, but it can on off on off Here, κ1 , κ1 , κ2 , and κ2 are the reaction rates. be much larger than the total time of stochastic simula- The algorithm follows by generating two independent tions when the applied force is considerably smaller than random numbers ξ1 and ξ2 that are uniformly distributed f c. at the interval [0, 1]. Then, ξ1 is employed to define the time dτ at which next reaction occurs, while ξ2 is used to choose which reaction occurs next. The time dτ is given A. Critical force and stability enhancement by

1 1 Dissociation of a heterogeneous cluster can be thought dτ = log , (14) of as a multistep process. For two bond populations, as α ξ 1 the applied force F is increased, one of the sub-clusters dissociates first, followed by the detachment of the other. where α = α1 + α2 + α3 + α4 is the combined propensity function with separate propensity functions Thus, depending on how F is shared between two sub- clusters which is controlled by kr [see Eq. (3)], there exist α = γ (Nt N ), (15) two possibilities 1 1 1 − 1 off 0 c c c α = N κ (f /N )/κ , (16) (i) f1 = f ,N1 = N & f2 f , (19) 2 1 1 1 1 1 1 ≤ 2 t c c c α = γ (N N ), (17) (ii) f2 = f2 ,N2 = N2 & f1 f1 , (20) 3 2 2 − 2 ≤ off 0 c c c c α4 = N2 κ2 (f2/N2)/κ . (18) where f1 , f2 , N1, and N2 are the corresponding critical forces and numbers of bonds of the two sub-clusters For ξ [0, α /α) the first reaction [Eq. (10)], for ξ separately. In the first case, Eqs. (19) and (5) with two 2 ∈ 1 2 ∈ [α1/α, (α1 + α2)/α) the second reaction [Eq. (11)], for unknowns F and N2 become the third reaction ξ2 [(α1 + α2)/α, (α1 + α2 + α3)/α) Nc F kr [Eq.∈ (12)], and for the fourth 1 c (21) ξ2 [(α1 + α2 + α3)/α, 1] c r = f1 , reaction [Eq. (13)] is chosen,∈ respectively. Following this N1k + N2 t off 0 algorithm, N1(τ) and N2(τ) are advanced in time until ((1 ρ)N N2)γ2 = N2 κ2 (f2/N2)/κ , (22) the entire cluster detaches (i.e. N = 0 and N = 0) or − − 1 2 from which the applied force F c required to initially dis- when a pre-defined maximum number of simulation steps sociate the first sub-cluster can be computed. In the is reached. second case, Eqs. (20) and (4) with two unknowns F and N1 are given by III. RESULTS Nc F 2 c (23) r c = f2 , N1k + N2 We first consider a heterogeneous cluster with two slip- (ρNt N )γ = N κoff (f /N )/κ0, (24) bond populations denoted as slip-slip bond cluster. For − 1 1 1 1 1 1 0 0 0 from which the applied force c required to initially dis- simplicity, we assume κ1 = κ2 = κ . Figure 2(a) shows F typical evolution of N(τ) = N1(τ)+N2(τ) for several slip- sociate the second sub-cluster can be calculated numeri- slip bond clusters with various kr and γr, where F = 50, cally. ρ = 0.3, Nt = 200. The case of kr = 1 and γr = 1 cor- After one of the sub-clusters has initially dissociated responds to a homogeneous cluster for which f c 55.7. under the force F c, the other sub-cluster can immediately Even though F < f c, the stochastic trajectory (red' line) rupture, if F c is larger than or equal to its individual shows a complete cluster dissociation due to fluctuations critical force, or detachment of the remaining sub-cluster in N and the condition of N(τ) = 0 for simulation ter- requires a force that is larger than F c. This condition mination. Note that cluster dissociation occurs more fre- can be taken into account through the requirement that quently when the applied force is approaching f c. The the critical force f c for rupturing the entire cluster must c c,m c c corresponding solution of deterministic Eqs. (4) and (5) necessarily satisfy f f1,2 = max(f1 , f2 ). Thus, we ≥ , , shown by the black line converges to a constant N for obtain f c = F c if F c f c m, and f c = f c m otherwise. ≥ 1,2 1,2 5

r Surprisingly, kopt for a slip-slip bond cluster depends only on the rebinding rates, and is independent of ρ. The (a) r dashed line in Fig. 3(a) represents kopt and separates the f c map into two regions. Region on the right side from the dashed line corresponds to Eq. (19), where the ﬁrst sub-cluster dissociates ﬁrst. Consequently, the region on the left side corresponds to Eq. (20), where the second r sub-cluster is initially ruptured. Noteworthy, kopt is a weakly increasing function of γr, indicating that large or small values of kr (or strongly disproportionate load sharing between sub-clusters) are disadvantageous for the stability of entire cluster. It is also interesting to consider the quantity

f c f c χ = c c = c,m , (27) max(f1 , f2 ) f1,2 (b) which describes the effect of a weaker sub-cluster (i.e. the c c sub-cluster with a smaller critical force min(f1 , f2 )) on c c c f in comparison with the critical force max(f1 , f2 ) of the strongest sub-cluster. Figure 3(b) shows the stability enhancement χ by a weaker sub-cluster as a function of kr and γr. It can be shown analytically (see Appendix A) that the maximum possible enhancement is χmax = 2, r r which is located on the kopt line at a γ value determined c c by the equality f1 = f2 . Thus, maximum enhancement of f c by a weaker sub-cluster is achieved when critical forces of individual sub-clusters are equal. Furthermore, large or small values of γr and kr (compared to unity) generally result in χ 1, and therefore, nearly no stability enhancement by the≈ addition of a weaker sub-cluster. FIG. 3. Stability characteristics of a slip-slip bond cluster. (a) f c map for different kr and γr. (b) Cluster stability enhance- c c,m r 1.0 ment χ = f /f1,2 by a weaker sub-cluster for various k and r r τs γ . The dashed lines show kopt values from Eq. (26), which c r correspond to a maximum f for a fixed γ . Here, ρ = 0.3 τc and Nt = 200. 0.8

Figure 3(a) shows the critical force map as a function 0.6 of kr and γr for a slip-slip bond cluster with ρ = 0.3. c ,

r c s

Note that for any fixed γ , there exists a maximum f , τ c which corresponds to the special case with f1 = f1 & c 0.4 f2 = f2 . Then, the ratio c c r f1 f1 N1k = c = c (25) 0.2 f2 f2 N2 allows the calculation of optimal kr values for cluster stability. For a slip-slip bond cluster, we obtain 0.0 0.0 0.5 1.0 1.5 2.0 2.5 0 r 1 + pln(γ1/e) f kopt = 0 , (26) 1 + pln(γ2/e) 0 0 FIG. 4. Comparison of lifetimes of single catch (τc) and slip where γ = γ κ0/κ0 and γ = γ κ0/κ0. Thus, for a 0 off 0 0 0 0 1 1 1 2 2 2 (τs) bonds defined as κ /κ . Here, κs /κ = 0.02, κc /κ = r c 0 0 fixed γ , the largest f is achieved when the forces on 20, β = 0.2, and κ2/κ = 1. individual bond sub-clusters, which is controlled by kr, c c are equal to the corresponding critical forces f1 and f2 . 6

B. Catch-slip bond cluster using Eq. (19) or Eq. (20), depending on whether kr is r c larger or smaller than kopt. This computation of f first c c To study the effect of a catch-bond sub-population on requires the calculation of f1 and N1 of the catch-bond cluster stability, we consider a catch-slip bond system sub-cluster, which is performed by solving a system of ˙ where the type-1 bond population is represented by catch the rate equation and its derivative, i.e. N1 = 0 and ˙ r r c bonds and type-2 by slip bonds. The off-rate of catch dN1/dN1 = 0. Thus, for each pair of k and γ , f1 and c r bonds is given by [47] N1 are first pre-computed in order to find the optimal k value as kr = (f c/f c)/(Nc /Nc ), which is then used to d d opt 1 2 1 2 off 0 f1/(N1f β) 0 −f1/(N1f β) κ1 (f1/N1) = κs e + κc e , (28) determine whether the condition in Eq. (19) or Eq. (20) has to be considered. Figure 5 presents critical force c 0 0 f where κs and κc are unstressed off-rates of the slip and and stability enhancement maps of a catch-slip bond off χ catch contributions to κ1 , and β is a non-dimensional cluster as a function of kr and γr. Both f c and χ maps quantity that alters the characteristic force scale (i.e. resemble the corresponding contour plots for the slip-slip d d f1 = f β). To illustrate differences between catch and bond cluster in Fig. 3. Therefore, the multistep rupture slip bonds, Fig. 4 shows the lifetimes of single bonds de- behavior discussed for slip-slip clusters above is also di- 0 off fined as κ /κ . Lifetime of the slip bond τs is a mono- rectly applicable for catch-slip bond clusters. tonically decreasing function of force f, while for the catch bond, τc first increases and then decreases, representing catch and slip parts. C. Effect of bond-population fraction

(a) Figure 6 shows the critical force f c and stability enhancement χ of a slip-slip bond cluster for two fractions (ρ = 0.1 and ρ = 0.5) of the first bond population. For ρ = 0.1, f c values are smaller than those for ρ = 0.3 in Fig. 3, while for ρ = 0.5, f c values are larger than for ρ = 0.3. However, the overall structure of the f c and χ maps remains qualitatively similar for all ρ values. Note that the value of χmax = 2 is also independent of ρ, but r its position on the kopt line changes with ρ. For example, r r χmax lies at γ 82 for ρ = 0.1, at γ 3.5 for ρ = 0.3, and at exactly γ'r = 1 for ρ = 0.5, which' corresponds to a homogeneous cluster. r For a fixed γ , the optimal fraction ρopt, such that χmax r c c lies at γ , can be found from the equality f1 = f2 that for a slip-slip bond cluster yields (see Appendix A) (b) 0 pln(γ2/e) ρopt = 0 0 . (29) pln(γ1/e) + pln(γ2/e) Thus, the fraction ρ can also be tuned to control f c and χ, which is illustrated in Fig. 7 for different combinations of kr and γr. For the same γr, two curves with different kr values (red and blue curves) give rise to maximum stability enhancement at the same fraction because ρopt is independent of kr, which is consistent with no dependence r r of kopt on ρ in Eq. (26). However, only k values close to r kopt approach χmax = 2. In case of three different bond populations, χmax = 3 (see Appendix A), suggesting that χmax is equal to the number of bond populations.

FIG. 5. Cluster stability for a catch-slip bond cluster. (a) D. Cluster lifetime f c map and (b) cluster stability enhancement χ by a weaker sub-cluster for different kr and γr values. The dashed lines r correspond to kopt calculated numerically. Here, ρ = 0.3, For a single bond with vanishing rebinding rate, its life- t N = 200, and other parameters are the same as in Fig. 4. time T is simply T = 1/κoff . In the mean-field description, when the applied force is less than f c, the cluster The critical force of a catch-slip bond cluster is com- lifetime is infinite. However, stochastic fluctuations may puted similarly to that of the slip-slip bond cluster by result in cluster dissociation within a finite time. The 7

ρ = 0.1 ρ = 0.5 (a) (b)

FIG. 6. [(a) & (b)] Critical force f c and [(c) & (d)] stability enhancement χ of a slip-slip bond cluster for different bond fractions [(a) & (c)] ρ = 0.1 and [(b) & (d)] ρ = 0.5 with Nt = 200. Note that the ranges of kr and γr for ρ = 0.1 are different r from those for ρ = 0.5 and ρ = 0.3 in Fig. 3. The dashed lines indicate kopt values from Eq. (26), which correspond to a maximum f c for a fixed γr. lifetime of a heterogeneous cluster can be obtained from that direct stochastic simulations do not permit the cal- the master equation (6), and also directly from stochastic culation of T for large Nt. Figure 8 also shows that the simulations. Figure 8 presents lifetimes of slip-slip bond lifetime for Nt = 200 (magenta color) rapidly increases clusters for different model parameters, with an excellent when the applied force becomes smaller than the critical agreement between the results from Eq. (6) and stochas- force. tic simulations for Nt = 10. Clearly, cluster lifetimes are finite even when F < f c. T increases drastically r with increasing γ for a given ρ. Furthermore, the clus- IV. DISCUSSION AND CONCLUSIONS ter lifetime strongly increases with increasing ρ for a fixed γr > 1. Differences in T for various kr can be pronounced In this study, we have investigated the stability of het- when the applied force F is close to f c, since a change erogeneous bond clusters under a constant load using the in the load distribution controlled by kr can make an ini- theoretical model for parallel bond cluster. One of our tially stable cluster unstable and vice versa. However, main results is that the rupture of heterogeneous bond for F f c and F f c, the effect of kr on T is expected clusters usually occurs as a multistep discrete process. As to be negligible, as load re-distribution will have no sig- the load on a cluster with two bond populations is gradu- nificant effect on whole cluster stability or rupture. Note ally increased, one of the sub-clusters eventually ruptures that we have employed a relatively small value of Nt = 10 when the individual load on this sub-cluster exceeds its because the lifetime increases exponentially with Nt, so critical rupture force. After the first bond sub-cluster 8

180 (a) r r k = 1.2, γ = 3.0 104 r r 160 k = 5.0, γ = 6.0 r r k = 0.5, γ = 3.0 103 140 102 120

c 1

f 10 T 100 100 80 ρ = 0.5, γr = 6 1 10− ρ = 0.3, γr = 12 60 ρ = 0.3, γr = 6 2 r 10− ρ = 0.1, γ = 1 40 N t = 200 0.0 0.2 0.4 0.6 0.8 1.0 10 3 − 2 1 0 ρ 10− 10− 10 F/N t (b) 2.0 kr = 1.2, γr = 3.0 r kr = 5.0, γr = 6.0 FIG. 8. Lifetime of a slip-slip bond cluster for diﬀerent γ r t 1.8 kr = 0.5, γr = 3.0 and ρ with k = 1 and N = 10. Circle symbols represent stochastic simulations and solid lines are obtained using the master equation (6). The dashed line marks the critical force r t 1.6 for ρ = 0.1 and γ = 1. The magenta line is for N = 200 and γr = 1. Above the critical force, T ∼ exp (−F/Nt). χ

1.4 described above is based on the mean-field approximation. In reality, stochastic fluctuations in the number 1.2 of bonds of different sub-populations can affect the behavior of a heterogeneous cluster. For instance, the sequence, in which different bond sub-populations are rup- 1.0 0.0 0.2 0.4 0.6 0.8 1.0 tured as predicted by the mean-field theory, can be al- ρ tered. Furthermore, the step-like dissociation of various bond sub-clusters in the mean-field approximation is

c partially smoothed through the stochastic fluctuations in FIG. 7. Effect of bond fraction ρ on (a) the critical force f bond numbers, as illustrated in Fig. 2(b). Note that these and (b) the stability enhancement χ of a slip-slip bond cluster for several γr and kr values. Here, Nt = 200. differences between the stochastic behavior of a cluster and its mean-field approximation are only expected when the applied force is sufficiently close to the critical force has dissociated, the total applied load F fully acts on for cluster rupture. Similarly, spontaneous rupture of the second remaining sub-cluster, resulting in either its a bond sub-cluster can unexpectedly occur if the dis- immediate rupture if F is larger than its critical rupture tributed load on that sub-cluster is close to its individual force or a further increase of the applied load is required rupture force. to dissociate the second sub-cluster. This rupture be- An interesting conclusion from our study is that the havior of a two-population cluster can straightforwardly multistep rupture process of a heterogeneous cluster is be generalized to the dissociation of bond clusters with qualitatively independent whether only slip or catch bond many sub-populations. The sequence of sub-cluster rup- populations are considered or their mixture. Note that in ture in this process depends on the distribution of total case of catch bonds, in addition to the effect of stochas- load among bond sub-clusters and their individual crit- tic fluctuations already discussed, such fluctuations may ical rupture forces. For fixed fractions of different bond lead to a sub-cluster rupture at nearly zero load, since populations, the load distribution is directly controlled the lifetime of catch bonds without applied forces can by the ratios kr of bond extensional rigidities, while the be small. Therefore, catch-bond sub-populations are ex- sub-cluster rupture forces are governed by bond kinetic pected to enhance cluster stability primarily under a non- rates. Furthermore, bond-population fractions within a zero load. heterogeneous cluster affect both the load distribution Our analysis shows that there exists an optimal spring- r and the sub-cluster critical forces. rigidity ratio kopt (or optimal load balancing among bond The discrete multistep process of cluster dissociation sub-clusters) that results in the maximum critical force 9 for a given γr and ρ. The optimal load distribution is focal adhesion sites in order to characterize their adhe- achieved when the individual loads on different bond sub- sion stability. As an example, it can be applied to bond clusters are equal to their corresponding critical forces, clusters located at the villi of leukocytes [29] or at the ad- such that all sub-clusters rupture at the same time. This hesive knobs of malaria-infected erythrocytes [30]. Fur- means that the maximum possible rupture force of the thermore, it is useful to estimate adhesion stability of max Pm c whole cluster is simply f = i fi , where m is the functionalized colloidal particles used in functional ma- number of different bond populations. Thus, for fixed terials [37, 38] or for drug delivery by micro- and nano- critical forces of bond sub-clusters, we can define a range carriers [12, 13]. Note that the application of this model of possible critical forces of a heterogeneous cluster as to such systems requires some knowledge about involved c,m c max c,m m c f f f where f = maxi (fi ) is the maxi- receptor-ligand pairs, including bond kinetics and densi- mum≤ of critical≤ forces of all bond sub-clusters. The load ties. Thus, the presented model can be used for the quan- r c max r distribution at kopt leads to f = f , while k 0 or tification of local cluster-adhesion measurements and bet- kr results in f c f c,m. Therefore, a strong→ load ter understanding of the role of different bond popula- disbalance→ ∞ (kr 1 or→kr 1) has generally no advan- tions within heterogeneous adhesion clusters. tages for overall cluster stability, because one of the bond populations carries the majority of the load and gets easily ruptured without significant stability enhancement. The upper bound on possible critical forces of a hetero- ACKNOWLEDGMENTS geneous cluster discussed above provides an intuitive ex- planation for the maximum stability enhancement χmax We would like to thank Ulrich S. Schwarz (Heidelberg by a weaker sub-cluster. Starting from a first bond sub- University, Germany) for insightful and stimulating dis- c cluster with its critical rupture force f1 , we add the sec- cussions. ond bond population that is weaker than the first pop- c c max c c ulation, i.e. f2 f1 . To maximize f = f1 + f2 of ≤ c c max the whole cluster, we select f2 = f1 , so that f = c c Appendix A: Optimal fraction ρ and maximum 2f1 = 2f2 and χmax = 2 for the cluster with two bond populations. In other words, the addition of a second stability enhancement bond population that is weaker than the first population r can at most double the critical rupture force of the whole For a fixed γ , the optimal fraction ρopt of the first cluster. Clearly, this argument can be generalized to m bond population of a slip-slip bond cluster, such that the r bond sub-populations, such that 1 χ χmax where maximum stability enhancement χmax lies exactly at γ , ≤ ≤ c c χmax = m. Note that such a maximum in χ may not can be found from the equality f1 = f2 as easily be achieved in biological systems, as it requires si- multaneous regulation of multiple parameters, including t 0 t 0 ρN pln(γ1/e) = (1 ρ)N pln(γ2/e) intrinsic properties and densities of different bond popu- − pln(γ0 /e) lations. ρ = 2 . (A1) ⇒ opt pln(γ0 /e) + pln(γ0 /e) In addition to γr and kr for controlling f c of a heteroge- 1 2 neous cluster, the fraction ρ of different bond populations r can be tuned to alter the critical force and lifetime of the Here, we can also compute γopt corresponding to χmax r for a fixed ρ. cluster. Interestingly, kopt for a slip-slip bond cluster in The value of for a slip-slip bond cluster can be Eq. (26) is independent of ρ, while the position of χmax χmax with respect to γr is strongly affected by ρ. Note that found analytically as follows. For simplicity, we assume 0 0 0 0 0 the ability to adjust density fractions of different bond that κ1 = κ2 = κ , so that γ1 = γ1 and γ2 = γ2. First r r r populations has direct biological relevance as cells can of all, χmax lies at the kopt line, as kopt represents k c r regulate receptor density, while γr and kr are intrinsic values that correspond to the maximum f for fixed γ c,m r properties of bond populations within the cluster. and f1,2 is independent of k . Therefore, we restrict further analysis of to the r line, on which c, Finally, it is important to discuss limitations of the χ kopt f1 = f1 c , c, and c . presented theoretical model. This model is obviously too N1 = N1 f2 = f2 N2 = N2 r r c c simple to quantitatively describe whole-cell adhesion, as For γ < γopt, f1 < f2 which implies that it does not consider cell deformation and possible recep- c c c r c tor and ligand mobilities. In such cases, more sophisti- r r f f N1kopt + N2 cated models or simulations need to be applied, as it has χ γ < γopt = c,m = c = c f , f N been done for modeling immunological synapse [8, 14] or 1 2 2 2 ρ pln(γ /e) the distribution and dynamic growth of different bond = 1 + 1 . (A2) 1 ρ pln(γ /e) domains [10, 16–18]. The presented model considers het- − 2 erogeneous bond clusters in which receptor densities and r r population fractions are nearly conserved over time. We Note that χ γ < γopt is a monotonically increasing r r r r c c expect that this model is applicable to highly localized function of γ1 = γ γ2 or γ . For γ > γopt, f1 > f2 , 10

r leading to N2k2F f2 = r r , (A9) N1k1 + N2k2 + N3 c c c r c r r f f N1kopt + N2 χ γ > γopt = c,m = c = c r f1,2 f1 N1kopt N F f = 3 , (A10) 1 ρ pln(γ /e) 3 N kr + N kr + N = 1 + − 2 . (A3) 1 1 2 2 3 pln ρ (γ1/e) r r where k1 = k1/k3 and k2 = k2/k3. Similarly, we define r and r . Then, the point where c r r γ1 = γ1/γ3 γ2 = γ2/γ3 f1 = χ γ > γopt is a monotonically decreasing function of c c is considered, as it corresponds r r f2 = f3 = f1 = f2 = f3 γ . The monotonic behavior of χ along the kopt line for to , which can be shown by arguments similar to r χmax these two cases of γ proves that χmax is achieved exactly those for a cluster with two bond populations. Fractions at the γr value, where f c = f c. Then, if we plug the opt 1 2 of the three populations are 1 ρ2 ρ3, ρ2, and ρ3, where expression for ρopt from Eq. (A1) into Eq. (A2) or (A3), − − ρ2 + ρ3 1. we obtain that ≤ c c c c c c Then, χmax = f /max(f1 , f2 , f3 ) = f /f3 is calculated as χ = 2 (A4) max Nc kr Nc kr 1 opt1 2 opt2 χmax = 1 + c + c (A11) for a slip-slip bond cluster with two bond populations. N3 N3 1 ρopt ρopt pln(γ /e) ρopt pln(γ /e) To generalize the result for a slip-slip bond cluster with = 1 + − 2 − 3 1 + 2 2 , two distinct populations, a cluster with three different ρopt3 pln(γ3/e) ρopt3 pln(γ3/e) slip bond populations is considered. For simplicity, we where assume that κ0 = κ0 = κ0 = κ0, so that γ0 = γ , γ0 = γ , 1 2 3 1 1 2 2 1 + pln(γ /e) 1 + pln(γ /e) and γ0 = γ . In the mean-field description, the average kr = 1 , kr = 2 , 3 3 opt1 opt2 number of bonds is determined by 1 + pln(γ3/e) 1 + pln(γ3/e)

dN d 1 f1/(N1f ) t pln(γ1/e)pln(γ3/e) = N1 e 1 + (N N1)γ1, (A5) 1 ρopt2 = , dτ − − f(γ1, γ2, γ3)

pln(γ1/e)pln(γ2/e) dN2 f / N f d = N e 2 ( 2 2 ) + (Nt N )γ , (A6) ρopt3 = , dτ − 2 2 − 2 2 f(γ1, γ2, γ3)

and f(γ1, γ2, γ3) = pln(γ1/e)pln(γ2/e) + pln(γ1/e)pln(γ3/e) + pln(γ2/e)pln(γ3/e). By plug- dN3 f /(N f d) t ging and into Eq. (A11), we obtain that = N e 3 3 3 + (N N )γ , (A7) ρopt2 ρopt3 dτ − 3 3 − 3 3 χmax = 3 (A12) t t t t where N1 + N2 + N3 = N . Similarly to the case with two sub-clusters, the force balance results in for a slip-slip bond cluster with three diﬀerent bond populations. This argument can be extended to m diﬀerent r sub-populations sharing the same load, whose maximum N1k1F f1 = r r , (A8) stability enhancement is equal to m. N1k1 + N2k2 + N3

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