Integrin Clustering Is Driven by Mechanical Resistance from the Glycocalyx and the Substrate
[['Cell adhesion to the ECM is mediated by a family of heterodimeric surface receptors called integrins [1] .', 'In addition to their function as mechanical anchors , integrins also participate in signal transduction and thereby regulate important cell behaviors , such as differentiation , motility , survival , and morphogenesis [2] , [3] .', 'To signal , integrins assemble laterally in the membrane and recruit structural and signaling proteins to form a clustered adhesion complex .', 'In addition to their signaling function , assembled adhesion complexes also physically link the cell cytoskeleton to the ECM and transmit traction forces necessary for mechanical cell processes , such as motility and cell shape changes [4]–[7] .', 'Both the physical and chemical properties of the ECM influence integrin adhesion complex assembly [3] , [8]–[15] .', 'The density of matrix ligands and their affinity for integrin receptors determines the number , size and distribution of integrin complexes in the cell membrane [8] , [12] , [15] , [16] .', 'Integrin clustering is especially sensitive to ligand spacing , as nanometer differences in the average spacing between ligands dictates whether or not integrins assemble into large adhesion complexes , such as focal adhesions [8] , [16] .', 'Matrix rigidity also regulates integrin function , as stiff matrices promote the assembly of large integrin complexes ( focal adhesions ) while compliant matrices support the assembly of small point-like integrin structures if any at all [13] , [14] .', 'Since integrin clustering is functionally linked to signal transduction and cell behavior , matrix-regulated adhesion assembly serves as a key sensory process that enables a cell to interrogate and respond to its extracellular environment .', 'Current theory holds that the adhesion complex is embedded with molecular sensors that mediate response to matrix properties .', 'Possibilities include protein switches that undergo tension-dependent conformational changes ( [17]–[20] and reviewed in [21] ) , as well as multivalent adaptor proteins whose incorporation into the adhesion complex are predicted to depend on factors such as matrix ligand density , matrix stiffness , and cell contractility [22]–[25] .', 'Although receiving less attention , the integrin-ligand interaction itself could also be sensitive to the physical and chemical properties of the ECM .', 'When both a receptor and its ligand are tethered , the kinetics and thermodynamics of complex formation depends on the intrinsic chemistry of the interaction; the distance the molecules must stretch to reach each other; and , theoretically , the compliance of the materials the molecules are tethered to [26]–[28] .', 'This suggests that integrins could function as sensors if their aggregation is linked to bond formation with ligand .', 'How could integrin-ligand interaction drive integrin assembly ?', 'One popular hypothesis holds that ligand interaction induces large allosteric changes in integrins that extend to their intracellular domains ( reviewed in [1] , [29] ) .', 'These changes in conformation could facilitate the recruitment of intracellular adaptor or signaling proteins that crosslink and cluster integrins [22] , [30] .', 'Other possibilities , however , likely exist .', 'Following interaction with matrix-immobilized ligands , for example , integrins can assemble into complexes in a matrix-dependent manner prior to the recruitment of intracellular proteins [10] , [31]–[33] .', 'Consistent with this observation , a chemo-mechanical basis for how receptor-ligand interactions can drive receptor clustering independent of intracellular interactions has been described theoretically .', 'When membranes possess two or more receptors of different lengths and chemical affinities , or possess large non-specific repellers ( i . e . large proteoglycans or glycoproteins ) , the receptors tend to phase separate into clustered or ring-like structures upon interaction with a substrate or another membrane [34]–[37] .', 'In essence , receptors aggregate due to a competition between receptor-mediated adhesion and non-specific repulsion that resists adhesion .', 'Integrin clustering could therefore naturally depend on the factors that control adhesion , including ligand chemistry , matrix stiffness , and cell stiffness , and also on the factors that mediate repulsion , such as the physical properties of the glycocalyx .', 'Hence , integrins may be able to respond to matrix properties without the necessity of auxiliary sensor proteins in the adhesion complex .', 'To explore if the glycocalyx can mediate integrin clustering independent of intracellular adaptors and if this clustering is responsive to the chemical and physical parameters that define the ECM , we developed a computational model of integrin-ligand interaction that includes a mechanical description of the cell-ECM interface .', 'The model is based on the simulation algorithm called Adhesive Dynamics [38]–[45] , which was originally devised to study the chemo-mechanics of receptor-mediated cell adhesion under shear flow [46] .', 'Adhesive Dynamics models integrin-ligand bonds as Hookean springs , which allows the distance-dependent kinetic rates of bond formation and rupture to be calculated with a model developed by Bell and co-workers [26] , [47] .', 'In this work , Adhesive Dynamics was expanded to include a lattice spring model ( LSM ) of the cell-ECM interface .', 'The LSM utilizes a defined lattice of nodes with interconnecting springs to calculate the elastic behavior of solid materials [48] , [49] .', 'Integration of the Adhesive Dynamics and LSM algorithms enables integrin dynamics , including force-dependent bond formation and rupture , to be explored in the context of a deformable cell-ECM interface .', 'Using the newly developed computational technique , we evaluate the relationship between integrin clustering , cell and glycocalyx mechanics , and the chemistry and mechanics of the matrix , and in doing so , predict that integrins themselves are responsive to matrix properties .'], ['Stress and strain in the interface were calculated with the LSM numerical method .', 'The LSM is a computationally-efficient mesoscopic approach frequently used in fracture mechanics that utilizes a system of regularly spaced nodes and interconnecting harmonic springs to model the mechanical behavior of solids ( reviewed in [49] ) .', 'When the node lattice , arrangement of spring connections , and spring constants are chosen correctly , the large-scale behavior of the LSM directly maps onto linear elasticity theory [48] .', 'The LSM is numerically equivalent to a finite element model that has simple linear elements [48]; however , we employ the LSM methodology over the more commonly used finite element method for two primary reasons .', 'First , the integrin-ligand bonds are described by discrete springs [46] , and these springs can be easily incorporated into the LSM .', 'Second , the LSM avoids computationally expensive remeshing algorithms , which a finite element method would need to call upon each time bond formation or rupture occurred in the interface .', 'To implement the LSM , a node and spring model was constructed for both the membrane/cortex plate and the ECM substrate , as shown in Figure 1B .', 'Nodes were placed on an initially cubic lattice and all nearest {1 0 0} and next-nearest {1 1 0} neighbor nodes were connected by Hookean springs , each having the same spring constant .', 'In response to stress , springs could pivot freely and the nodes could undergo translational movements that minimized the potential energy of the spring network .', "A system configured in this manner behaves as an isotropic elastic solid that has a fixed Poisson's ratio ν\u200a=\u200a1/4 and an adjustable Young's modulus: ( 2 ) where Δx is the LSM lattice node spacing and σ is the Hookean spring constant [57] , [58] .", 'If Δx is small compared to the length scale of interest , the spring system approximates an elastic continuum .', 'The actin cortex and ECM , however , are not continuous on the protein-length scale , which is relevant to integrin-ligand interaction .', 'To better reflect the micro-architecture of cell-ECM interface , we used an LSM lattice spacing of 20 nm , which is on the order of the size of a matrix protein or cytoskeletal filament .', "Changes in the lattice spacing by an order of magnitude , though , were not expected to alter the qualitative nature of our results if the spring constants were also adjusted to maintain the Young's moduli .", 'In all simulations unless otherwise noted , a 1 . 4 µm×1 . 4 µm area of the cell membrane was simulated .', 'A 40-nm thick membrane/cortex plate and a 400-nm thick ECM substrate spanning this area were constructed using a 70×70×3 and a 70×70×21 node LSM , respectively ( Figure 1B ) .', 'The springs of each LSM were assigned a spring constant ( Table 1 ) that achieved the desired material rigidity ( Equation 2 ) .', 'The harmonic potential ( Equation 1 ) between the membrane and substrate ,', 'i . e .', 'the glycocalyx , was added to the model by incorporating additional linear springs between the ECM-substrate and membrane-spring networks .', 'To add the springs , the plate and substrate networks were aligned and each node in the top surface of the substrate network was connected by a Hookean spring to the node directly above it in the bottom surface of the plate network ( Figure 1B ) .', 'The equilibrium spring length of these connections was set equal to the desired thickness of the glycocalyx .', 'The spring constant of the connections , σg , was related to the effective compressibility of the glycocalyx layer ( See Equation 1 ) by the following expression: ( 3 ) Bonds were modeled as Hookean springs and were added to the LSM by connecting the desired node in the top surface of the matrix LSM with a node in the bottom surface of the membrane/cortex LSM .', 'Likewise , a bond was removed by removing the appropriate spring from the model .', 'The deformations in the LSM caused by bond formation were calculated by relaxing the entire spring network to mechanical equilibrium .', 'The potential energy stored in the LSM was given by ( 4 ) where the summation i is over all nodes in the system , the summation j is over all nodes connected to node i , |rij| is the distance between node i and j , and σij and lij are the spring constant and equilibrium length of the spring connecting node i and', 'j . The system energy was minimized when the vector sum of forces on each node that can undergo translation was zero , which was achieved by iteratively solving the following system of equations ( 5 ) For relaxation , periodic boundary conditions were applied to the LSM nodes forming the lateral sides of the substrate and membrane/cortex networks .', 'Under this condition , which was implemented to limit finite-size effects , the material strain induced by a stress at one side of the network propagates in a mirror-like fashion on the opposite side of the network .', 'Initially , ligand binding sites and integrin receptors were distributed uniformly and randomly in the cell-ECM interface .', 'Nodes on the top surface of the substrate LSM were selected at random and designated as ligand binding sites until the desired ligand density was achieved ( Figure 1B ) .', 'Since the lattice spacing was 20 nm , the maximum ligand density was 2500 #/µm2 , which is approximately the saturating density for large ECM proteins , such as fibronectin or collagen , absorbed on flat substrates , such as tissue culture plastic or glass slides [59] , [60] .', 'Integrin receptors were placed randomly on the bottom surface of the membrane/cortex plate , but unlike the ECM ligands , the positions of free integrins ( not bound to ligand ) were not limited to sites of LSM nodes .', 'Three integrin states were described in the model that reflect the major conformational states integrins are known to adopt: “inactive” ( low-affinity ) , “active” ( high-affinity ) , and ligand occupied [61] .', 'Although inactive integrins can bind soluble ligands in in vitro binding assays [62] , [63] , when locked in the inactive conformation through molecular engineering and expressed on the cell surface , integrins ( αIIbβ3 and αvβ3 ) do not bind tethered ligands [64] .', 'We thus made the assumption that only active integrins can bind ligand to reflect the relatively low probability of bond formation between matrix-tethered ligands and inactive integrins on the cell surface .', 'Four integrin reactions were therefore modeled in our simulation: activation of inactive integrins , deactivation of active integrins , bond formation between active integrin and ligand , and bond dissociation .', 'In addition , integrin “hop” reactions were included to describe the diffusive movements of unbound integrins .', 'The conversion between active and inactive integrin states was described by simple transition rates , ka and ki , which describe , respectively , the rate of conformational change from the inactive to active and active to inactive states .', 'In the cell , the dynamic equilibrium between active and inactive integrin states depends on a variety of factors , including divalent cations , cell signaling , and intracellular integrin binding partners such as talin .', 'ka and ki in this model can be viewed as phenomenological parameters that take all these influences into account .', 'The distance-dependent rates of bond formation and rupture were calculated according to the equations formulated by Bell and co-workers [26] , [47] .', 'As mentioned , integrin-ligand bonds were modeled as Hookean springs .', "For such a bond , the reverse reaction rate in the Bell model takes the form of: ( 6 ) where is the unstressed intrinsic dissociation rate , F is the force on the bond , and γ is an empirically measured quantity with units of length describing the bond's sensitivity to force [26] , [28] .", "The bond force was calculated using Hooke's law based on the equilibrium extension of the bond as determined by Equation 5 .", 'Since the force on each bond could be unique , an individual rupture rate for each bond was calculated .', 'Association rates were calculated for each active integrin and ligand in close proximity .', 'Integrin ligand binding partners that were separated by a lateral cutoff distance greater than 10 nm in the xy-plane were assigned an association rate of exactly zero .', 'For pairs that were within the cutoff , the bond formation rate directly followed from the Boltzmann distribution for affinity [65] and was given by: ( 7 ) where is the unstressed intrinsic association rate and ΔE is the minimum mechanical potential energy change resulting from bond formation [47] .', 'To calculate F and ΔE for a specific pair of binding partners , a bond spring was temporally connected to the desired ligand site , and the system was relaxed to equilibrium .', "ΔE was then calculated according to Equation 4 and F was determined by Hooke's law .", 'Diffusion of unbound integrins ( inactive and active unbound ) was modeled by hop reactions in the plane of the membrane ( bottom surface of LSM plate ) .', 'As originally proposed by Elf and Ehrenberg [66] , the hops were of discrete length Δℓ and occurred along the four directions defined by the positive and negative x- and y- axes .', 'The rate for a specific integrin to undergo a hop reaction was given by: ( 8 ) where D is the diffusion coefficient for integrins in the membrane ( 10−10 cm2/s ) .', 'The length Δℓ that we used in the simulation is 5 nm , which is on the order of the diameter of the integrin molecule .', 'Periodic boundaries were employed for receptor diffusion to limit finite-size effects .', 'The time evolution of the system was simulated by kinetic Monte Carlo according to the Gillespie algorithm [67] .', 'For a given chemical and mechanical state of the system , the Gillespie algorithm determined the reaction that occurred next and the time that elapsed until that reaction occurred .', 'Reactions were selected through random number sampling of a probability distribution constructed based on the kinetic rates of all possible reactions .', 'The system ultimately was evolved through an iterative process of calculating the reaction rates for the current system state , selecting the next reaction , executing the reaction , updating the rates , and repeating ( Figure 2 ) .', 'To determine the next reaction and the variable time step once the reaction rates were calculated , two random numbers ran1 and ran2 were generated from a uniform probability distribution between 0 and 1 .', 'The next reaction , μ , was selected according to: ( 9 ) where is the rate constant for a particular reaction involving a specific integrin and the summation i is over all possible reactions .', 'The time that elapsed between the last reaction and the newly selected reaction was given by: ( 10 ) After the next reaction and τ were determined , the selected reaction was executed .', 'Either an integrin was moved by Δℓ in a randomly selected direction ( hop reaction ) , the activity state of the integrin was flipped ( activation or deactivation reaction ) , or a bond was incorporated or removed from the LSM at the appropriate ligand site ( bond formation or dissociation ) .', 'The simulation time was then incremented by τ , the spring network was relaxed back to mechanical equilibrium using Equation 5 , and the reaction rates were again calculated .', 'This procedure was repeated until the desired simulation time elapsed ( Figure 2 ) .', 'The mechanical energy minimization defined by Equation 5 was computationally expensive .', 'Two main optimizations were thus implemented to reduce the frequency of calls to the minimization algorithm and increase its efficiency .', 'First , energy minimums were stored to memory upon calculation to avoid repeatedly minimizing the same configuration of integrin bonds .', 'Second , smaller sub-systems of springs and nodes were minimized , as opposed to the entire spring network .', 'Since strain induced by an integrin-ligand bond vanished with sufficient distance from the bond , the total system potential energy minimum could be computed by minimizing a smaller sub-region surrounding the bond .', 'The distance from a bond at which the strain vanished depended on the physical parameters defining the system , and hence sub-system size was optimized for a particular set of matrix , membrane/cortex , glycocalyx , and bond parameters .', 'Typical sub-systems ranged from 400–800 nm in dimension .', 'For minimization with Equation 5 , nodes at the boundary of a sub-system were constrained to their current location to implement the vanishing strain boundary condition .', 'A small number of simulations were executed on rigid matrix substrates that had reaction interfaces spanning a membrane area greater than 1 . 4 µm×1 . 4 µm .', 'Solutions for these larger systems were obtained by approximating the distance-dependent rate of integrin-ligand bond formation .', 'Specifically , the minimum change in system potential energy , ΔE , and the equilibrium force on the integrin ligand bond , F , necessary to compute the bond formation rate were estimated based on the equilibrium separation distance between the unbound integrin and ligand .', 'This approximation avoided the necessity of repeatedly minimizing the system energy to calculate bond formation rates .', 'In order to estimate ΔE and F , the dependence of ΔE and F on equilibrium integrin-ligand separation distance was first determined .', 'To do so , integrin-ligand bonds were randomly and sequentially added to a model cell-ECM interface .', 'For each integrin-ligand bond added , the equilibrium integrin-ligand separation distance before binding and the equilibrium bond force and change in system potential energy after binding were recorded .', 'Plots of F versus initial separation distance and ΔE versus initial separation distance squared were each well-fit by quadratic equations ( Figure S1 ) , at least over the range of physical parameters utilized in this work .', 'Because bonds were added to the system randomly , the relationships did not depend on a specific configuration of bonds .', "The relationships were dependent , however , on the model's physical parameters , and thus force and energy relationships were determined for each combination of physical parameters examined .", 'During simulation of integrin dynamics , the curve-fits were used to estimate ΔE and F as a function of equilibrium integrin-ligand separation distance to calculate the bond formation rates .', 'For best-estimate system parameters ( See below ) , the average errors in approximating ΔE and F based on curve fits were 6 . 3% and 3 . 5% , respectively , corresponding to an average error in bond formation rate of 3 . 7% according to Equation 7 .', 'Results from simulations of integrin dynamics with estimated ΔE and F were not statistically different from those in which rates were calculated by directly minimizing the system energy ( Figure S2 ) .', 'Table 1 lists the parameters that were used in the simulations unless otherwise noted .', 'The dynamic integrin parameters were based on those reported for fibronectin and the α5β1 integrin .', 'Other possible integrin parameters , however , were also considered to extend the relevance of the model results to other types of integrins and cell surface receptors .', 'For α5β1 , the kinetic rates of the integrin-ligand interaction , the force-dependence of the interaction , the mobility of the integrin in the membrane , and the density of integrin on the cell surface have been reported [62] , [63] , [68]–[70] .', 'The rates of integrin activation and deactivation , however , have not been measured experimentally .', 'Based on experimental reports of the equilibrium distribution of inactive and active integrins on the cell surface [30] , [64] , the free energy of conformational change was approximated to be 2–3 kbT [44] .', 'Considerations of molecular diffusion rates provide an upper limit of ∼1 s for the large structural movement that occurs during activation [71] .', 'We thus used estimates of 0 . 5 s−1 and 5 s−1 for the activation and deactivation rates , respectively , although other possibilities were evaluated .', 'The springs comprising the membrane/cortex plate were assigned a Hookean constant that achieved the experimentally measured flexural rigidity ( bending modulus ) of the actin cortex , 1×10−19 N·m [72] , [73] .', 'The Hookean constant was related to the flexural rigidity , I , of the plate by: ( 11 ) where h is the thickness of the plate .', 'The Hookean constants of the ECM substrate springs were varied according to Equation 1 to achieve elastic moduli in the physiological range of 102–105 Pa [13] , [74] .', 'Since cellular experiments are typically conducted on extremely rigid non-deformable glass or plastic substrates , we also constructed non-deformable substrates in our model by assigning an arbitrarily large spring constant of 1000 pN/nm .', "This approximates a substrate with a Young's modulus of roughly 0 . 1 GPa ( for comparison , glass or tissue culture plastic is ∼1 GPa [13] ) .", 'The thickness of the glycocalyx is reported to be approximately 40–50 nm [52] and up to 100 nm for certain cell types such as endothelial cells [53] .', 'In this model , a best estimate of 43 nm was used for the glycocalyx spring length ,', 'i . e .', 'its thickness , but other values were considered .', 'While the stiffness of the glycocalyx has not yet been measured directly , estimates are available .', 'Agrawal and Radhakrishnan estimated the glycocalyx stiffness by fitting simulations of nano-particle adhesion on the cell surface to analogous experimental data .', 'Based on these results we estimated σg to be 0 . 02 pN/nm [56] .', 'This estimate is in good agreement with purely theoretical estimates calculated by considering the statistical mechanics of chain molecules anchored to a surface [47] .', 'Like the glycocalyx thickness , additional possibilities for σg were explored .', 'To analyze the extent of integrin clustering , a two-dimensional point pattern analysis of the integrin membrane positions projected onto the xy-plane was constructed .', "The analysis was performed using Ripley's K-function [75] , [76] , which measures the extent to which a point pattern deviates from a random Poisson distribution and is given by: ( 12 ) where the summations i and j are over all integrin point positions , A is the projected area of the membrane , s is the sampling radius , and Wij ( s ) is exactly equal to one if the distance between points i and j is less than s and zero otherwise .", 'Periodic boundaries were utilized in the calculation of Wij ( s ) .', 'To facilitate the interpretation of the statistic , these data were transformed [77] into the following form: ( 13 ) For a point pattern with complete spatial randomness , R ( s ) has an expected value of zero , and if the points are clustered R ( s ) has a positive value .', 'The maximal value of R ( s ) and the radius at which the function is maximal provide a measure of the degree of clustering and the cluster size respectively .', 'To analyze the degree of cooperativity in integrin-ligand binding interactions , Hill plots of the steady state bond fraction versus ligand density were constructed .', 'The plots were fit to a version of the Hill equation that also accounts for the possibility of ligand depletion: ( 14 ) where U is the bond fraction , R is the total integrin receptor density , L is the total ligand density , Kd is the bond dissociation constant and nHill is the Hill coefficient .', 'The model was fit to the Hill plots for both Kd and nHill with non-linear least squares regression .'], ['Integrin-ligand binding rates are dependent on the distance the molecules must stretch to reach each other .', 'By inducing mechanical deformations , adhesive bond formation could modify these distances and therefore be cooperative due to mechanical coupling .', 'In order to determine how the cell membrane/cortex deforms during binding , we calculated the equilibrium deformations that were induced by the addition of integrin-ligand bonds into our mechanical model of the cell , glycocalyx , and matrix .', 'A single integrin bond between the cell and a rigid ECM substrate caused a highly localized deformation that extended laterally in the plane of the membrane approximately 150 nm from the bound site ( Figure 3 ) .', 'The placement of additional bonds in the deformed region pulled a significantly larger area of the cell into closer proximity with the matrix substrate ( Figure 3 ) .', 'We thus imagined that new bond formation would be most favorable nearby existing bonds , since the distance between integrins and ligands would be reduced in this area , and that bond formation would become increasingly favorable as additional bonds accumulated together and induced larger deformations .', 'To test if bond formation was indeed cooperative , we ran simulations of integrin dynamics on rigid ECM substrates of varying ligand density and constructed Hill plots of the steady-state bond fraction ( Figure 4A ) .', 'Since the thickness of the glycocalyx determines the initial distance between integrin and ligand partners , the effective thickness of the glycocalyx ( lg−lb ) was also varied in an attempt to manipulate cooperativity .', 'Hill plots were constructed from these simulation results and were fit to a form of the Hill equation which accounts for low ligand density ( Equation 14 ) .', 'The best-fit Hill coefficients were greater than one , indicating cooperative integrin binding , and increased with enhanced glycocalyx thickness ( Figure 4B ) .', 'Cooperative integrin-ligand interactions resulted in a clustered pattern of integrin bonds , as can be seen Figure 4E , which shows integrin positions after 30 minutes of simulation on rigid substrates ( L - 2500 #/µm2 ) .', 'With increasing glycocalyx thickness and hence more cooperative integrin-ligand interactions , integrin clusters became fewer in number , larger in size , and more densely packed with integrins ( Figure 4E ) .', 'To quantify the extent of clustering , we preformed a point-pattern analysis on the steady-state integrin positions by calculating the maximum of the transformed Ripley K-function , R ( s ) .', 'Maximum values greater than zero indicate that the integrins are clustered and the magnitude of the value is related to the degree of integrin clustering .', 'Our point-pattern analysis demonstrated that the degree of clustering increased with enhanced glycocalyx thickness ( Figure 4C ) and was proportional to the level of cooperativity , as indicated by the Hill coefficient ( Figure 4D ) .', 'Kinetically , integrin clusters typically formed within tens of seconds to minutes of simulated time .', 'Figure 5 shows the chemo-mechanical evolution of the integrin system for best-estimate parameters ( Table 1 ) on a rigid matrix ( L - 2 , 500 #/µm2 ) .', 'As Figure 5 demonstrates , new bonds formed rapidly in regions of the cell-ECM interface deformed by prior bonds and formed slowly in regions devoid of bonds .', 'The bonds began to form after approximately a ten second delay , at which point the rate of bond formation accelerated until saturation was reached after approximately 50 seconds ( Figure 6A ) .', 'The statistical measure of integrin clustering , max R ( s ) , exhibited a similar kinetic profile to that of the bond fraction , indicating that clustering was primarily driven by bond formation ( Figure 6B ) .', 'While integrin clustering was primarily driven by the initial binding of integrins to the matrix , integrins continued to condense in the clusters over a much slower time-scale due to bond rearrangements occurring through repeated cycles of bond breakage and reformation ( See white arrows – Figure 5B; See also the slow upward rise in Ripley statistic – Figure 6B ) .', 'Hence , integrin clustering was biphasic and characterized by an initial fast bond formation and clustering step , followed by a slow bond rearrangement and condensing process .', 'Since integrin clustering required both integrin-ligand adhesion and cell-ECM repulsion , we mapped the relationship between integrin-ligand affinity and glycocalyx-mediated cell-ECM repulsion .', 'To do so , we ran simulations in which the glycocalyx-thickness and chemical-affinity parameter space was systematically varied .', 'As shown in Figure 7 , clustering depended strongly on receptor-ligand affinity and the effective thickness of the glycocalyx .', 'For high-affinity interactions but relatively thin glycocalyxes , the majority of integrin receptors were bound but not clustered ( Compare Figure 7A and 7B ) .', 'When the receptor-ligand affinity was relatively low , integrin receptors were neither bound nor clustered .', 'If the glycocalyx was relatively thick compared to the integrin bond length and the receptor-ligand interaction was of sufficient affinity , however , integrins bound ligand and assembled into clusters .', 'Integrin clustering was particularly sensitive to variations in glycocalyx thickness and bond length , as small changes of five to ten nanometers in the effective thickness of the glycocalyx could switch the integrin system from clustered to unclustered or vice versa .', 'While integrins are able to switch between activity states , this property was not essential for integrin clustering .', 'For best estimate parameters , integrins clustered if they were constitutively maintained in the active , ligand-binding conformation or instead were allowed to switch between inactive and active states ( data not shown ) .', 'Receptor density was modified to control the number of active receptors available for binding .', 'Although clusters were smaller and less frequent in number for lower initial densities ( Figure S4 ) , integrins generally clustered when the initial receptor density was high or low .', 'Increased integrin bond stiffness , however , generally enhanced clustering ( Figure 8 and S3 ) .', 'Integrin clustering in our simulations thus was controlled by integrin bond length ( Figure 7 , 8 , and S3 ) , bond stiffness ( Figure 8 and S3 ) , and affinity for ligand ( Figure 7 ) , which all depend or are predicted to depend on integrin activation state .', 'These results suggest a functional link between integrin conformation , glycocalyx properties , and integrin clustering .', 'For physiologically-relevant parameters , int
['Introduction', 'Model', 'Results', 'Discussion']
['Integrins have emerged as key sensory molecules that translate chemical and physical cues from the extracellular matrix ( ECM ) into biochemical signals that regulate cell behavior .', 'Integrins function by clustering into adhesion plaques , but the molecular mechanisms that drive integrin clustering in response to interaction with the ECM remain unclear .', 'To explore how deformations in the cell-ECM interface influence integrin clustering , we developed a spatial-temporal simulation that integrates the micro-mechanics of the cell , glycocalyx , and ECM with a simple chemical model of integrin activation and ligand interaction .', 'Due to mechanical coupling , we find that integrin-ligand interactions are highly cooperative , and this cooperativity is sufficient to drive integrin clustering even in the absence of cytoskeletal crosslinking or homotypic integrin-integrin interactions .', 'The glycocalyx largely mediates this cooperativity and hence may be a key regulator of integrin function .', 'Remarkably , integrin clustering in the model is naturally responsive to the chemical and physical properties of the ECM , including ligand density , matrix rigidity , and the chemical affinity of ligand for receptor .', 'Consistent with experimental observations , we find that integrin clustering is robust on rigid substrates with high ligand density , but is impaired on substrates that are highly compliant or have low ligand density .', 'We thus demonstrate how integrins themselves could function as sensory molecules that begin sensing matrix properties even before large multi-molecular adhesion complexes are assembled .']
['Critical cell decisions , including whether to live , proliferate , or assemble into tissue structures , are directed by cues from the extracellular matrix , the external protein scaffold that surrounds cells .', 'Integrin receptors on the cell surface bind to the extracellular matrix and cluster into complexes that translate matrix cues into the set of instructions a cell follows .', 'Using a newly developed model of the cell-matrix interface , in this work we detail a simple yet efficient mechanism by which integrins could “sense” important matrix properties , including chemical composition and mechanical stiffness , and cluster appropriately .', 'This mechanism relies on mechanical resistance to integrin-matrix interaction provided by the glycocalyx , the slimy sugar and protein coating on the cell , as well as the stiffness of the matrix and the cell itself .', 'In general , the resistance alters integrin-ligand reaction rates , such that integrin clustering is favored for many physiologically relevant conditions .', 'Interestingly , the mechanical properties of the cell and ECM are altered in many prevalent diseases , such as cancer , and our work suggests how these mechanical perturbations might adversely influence integrin function .']
['cell biology/cell signaling', 'biophysics/theory and simulation', 'cell biology/extra-cellular matrix', 'computational biology', 'cell biology/cell adhesion']