In this work, we built a new model to study integrin adhesion and clustering that couples the chemistry of bond formation with the mechanics of a composite, layered material representing the cell membrane/cortex, glycocalyx, and ECM. The biology incorporated into the model was basic and included only integrin activation/deactivation and association/dissociation reactions. Despite the simplicity of the molecular interactions, when coupled to the mechanics of the system, our model exhibited complex integrin adhesion behaviors that match those reported in the experimental literature. These behaviors can be explained by one simple principle: when deformations in the cell membrane or ECM accompany bond formation, the distance-dependent kinetic rates for other potential integrin-ligand binding interactions are modified. In essence, integrin bonds pull the cell membrane and ECM substrate into closer proximity and new bonds form more readily in these deformed regions. We showed that for realistic model parameters, clustering was sensitive to both the physical and chemical properties of the matrix, suggesting a simple yet efficient mechanism by which integrin adhesions sense matrix properties.
Integrin clustering in our model was driven by the interplay between integrin-mediated adhesion and glycocalyx-mediated cell-ECM repulsion. While a relationship between integrin-ligand affinity and integrin clustering has been suggested 
, we now show that the thickness and stiffness of the glycocalyx may regulate this relationship. Indeed, we found that manipulating glycocalyx thickness/stiffness parameters while maintaining the intrinsic integrin-ligand affinity can switch the integrin system from an unclustered state to a clustered state or vice-versa. Similarly, changes in integrin-ligand affinity could also induce a switch in integrin clustering state depending on glycocalyx parameters. Furthermore, changes in integrin length, such as the structural extension that occurs during activation, could change the effective thickness of the glycocalyx to also modulate integrin clustering. In general, high-affinity integrin-ligand interactions in the context of a relatively thick or stiff glycocalyx promoted integrin clustering. A glycocalyx too thick or rigid, however, impeded bond formation and thereby prevented clustering. These results suggest the glycocalyx is a potent regulator of integrin system behavior and signaling. Such a relationship may be extremely important in diseases such as breast cancer, in which 95% of the cells have modified glycocalyx composition or structure and in which integrin clustering is functionally-linked to loss of tissue homeostasis and the development of a malignant phenotype 
Our model provides one explanation for the exquisite sensitivity integrins exhibit in response to variations in matrix-ligand density 
. In cellular experiments on rigid ligand-coated substrates, integrins cluster when the average intermolecular ligand spacing is less than or equal to 58 nm, but not when it is greater than or equal to 73 nm 
. These results have fueled the notion that cells posses molecular “rulers” that mediate this chemo-sensory process. Our model suggests that the ruler might actually be the cell membrane and associated actin cortex rather than a specific molecule, such as an adhesion plaque protein. In order for integrins to cluster, we found that the average spacing between ligand molecules had to be less than the lateral width of the membrane/cortex deformation induced by an integrin bond. If the deformation was too small relative to the ligand spacing, integrin-ligand binding was not cooperative and integrins did not cluster. For our best-estimate mechanical parameters, the width of cell deformation induced by an integrin bond (150 nm) was on the order of the experimentally-measured ligand spacing at which the unclustered-to-clustered integrin transition occurs experimentally. Moreover, when best-estimate mechanical parameters were utilized in simulations of integrin dynamics, we found that an average intermolecular ligand spacing of 71 nm was necessary to drive integrin clustering in our model, which is in excellent agreement with experimental results. The width of the cell surface deformation was primarily determined by the ratio of the glycocalyx stiffness to membrane/cortex thickness, and hence this ratio controlled the threshold ligand density required for integrin clustering. We thus propose that the integrin adhesion system may be intrinsically sensitive to ligand density and that this sensitivity may be tuned by the mechanical properties of the cell and glycocalyx.
We also found that integrin clustering was responsive to matrix stiffness. On progressively more compliant substrates, the rate of integrin-ligand bond formation was increasingly fast due to the enhanced flexibility of the ligand binding site. Consequently bond formation was not cooperative on highly compliant substrates, since new bonds could readily form in the interface regardless of proximity to pre-existing bonds. After evaluating a range of matrix stiffnesses, we determined that integrin clustering in our model requires a substrate with a Young's modulus of at least 2000 Pa, at which point the extent of clustering increases with the logarithm of substrate stiffness until maximum clustering is achieved around 100,000 Pa. These results agree well with cellular experiments conducted on ECM-functionalized hydrogels of tunable stiffness, on which integrins assemble into larger and more numerous adhesions on matrices above 1000 Pa 
. Furthermore, cell behaviors correlated with integrin clustering, such as cell spreading, demonstrate an incremental response to increases in matrix stiffness between approximately 1000 Pa to 50,000 Pa, which is again in agreement with the sensitivity range for integrin clustering predicted in this work 
. While integrin-mediated matrix mechano-sensing has been assumed to require actomyosin contractility to generate matrix probing forces and adhesion plaque proteins to respond to these force (reviewed in 
), our model would suggest that integrin themselves can respond to matrix stiffness in one manner independent of myosin or plaque proteins.
Experimentally-observed features of integrin clustering, such as its sensitivity to matrix properties, were recapitulated in our model without the incorporation of cytoskeletal adaptor proteins into the model. Indeed, for best-estimate parameters, the kinetic profiles of integrin bond formation and clustering simulated by our model recapitulate the short delay in integrin bond formation observed experimentally when the cell first contacts the ECM, as well as the fast rate of de novo
integrin adhesion assembly and clustering observed in cells 
. This does not suggest, however, that cytoskeletal interactions are insignificant. Many lines of experimental evidence clearly demonstrate that cytoskeletal interactions regulate the size and signaling activity of integrin adhesion structures (reviewed in 
). We envision that integrin-cytoskeletal interactions could synergize with the mechanically-coupled integrin-ligand interactions described in this work to drive a more robust integrin clustering response with heightened sensitivity to matrix properties or with additional levels of regulation. Our model, however, does offer an explanation for how integrins can cluster prior to recruiting cytoskeletal adaptor proteins, as has been observed in time lapse studies of adhesion complex assembly 
. Similar to the kinetics of integrin assembly in these time-lapse studies, integrins in our model spontaneously clustered on rigid substrates in tens of seconds to minutes even though cytoskeletal interactions were not included in the model. Provocatively, since clustering was sensitive to matrix properties, our results suggest that integrins may begin to sense matrix properties prior to the assembly of more advanced adhesion structures, such as focal complexes and focal adhesions 
It is well-documented that force promotes integrin adhesion complex assembly, which raises the question of whether cytoskeletal forces would influence the myosin-independent integrin clustering described in this work. In our model, integrins cluster because one bond pays a portion of the energy penalty associated with compressing the glycocalyx for the next integrin to complex a nearby ligand site. Myosin-driven forces on integrin bonds could actively pull the cell and ECM into closer spatial proximity, and hence pay this energy penalty 
. In the context of our integrin clustering model, these force-driven deformations should enhance integrin bond formation and aggregation to possibly achieve states of integrin cluster size or density that would otherwise be unlikely. Similarly, exogenously applied forces to the cell, such as fluid shear forces in the vasculature, could induce deformations in the cell-ECM interface that modify integrin clustering response. Therefore, both endogenous contractile forces and exogenous applied forces could influence integrin distribution through a mechanism similar to that proposed in this work.
Many aspects of integrin clustering described by our model are justified experimentally. For example, reports have demonstrated that receptor-ligand interactions are distance-dependent 
and that the cell and ECM are in closest proximity at sites containing integrin adhesions 
. Perhaps some of the best support of the model is provided by studies with biomimetic lipid vesicles. When lipid vesicles functionalized with adhesion molecules and a repulsive brush border are brought in contact with a complimentary solid surface, receptor-ligand bonds cluster despite the simple chemistry of the vesicle system 
. Since the repulsive brush border is required for patterned bond formation, these studies suggest that adhesive bond clustering results from the interplay between adhesion and repulsion, as our model predicts.
Several novel predictions stemming from our model, however, must still be validated experimentally. This includes determining if matrix rigidity controls integrin clustering by altering kinetic rates of bond formation, evaluating if cell and glycocalyx stiffness controls the relationship between integrin clustering and ligand density, and determining if the glycocalyx is indeed a potent regulator of integrin function and clustering. Testing these predictions should provide significant insight into how cell adhesions sense and respond to their ECM environment.
In conclusion, we showed how the coupling between the chemistry of bond formation and the mechanics of the cell and glycocalyx may drive integrin clustering in a matrix-dependent manner. Our results suggest a mechanism by which integrins function as sensors of matrix rigidity and chemistry.