Cell focal adhesions are micrometer-sized aggregates of proteins that anchor the cell to the extracellular matrix. Within the cell, these adhesions are connected to the contractile, actin cytoskeleton; this allows the adhesions to transmit forces to the surrounding matrix and makes the adhesion assembly sensitive to the rigidity of their environment. In this article, we predict the dynamics of focal adhesions as a function of the rigidity of the substrate. We generalize previous theories and include the fact that the dynamics of proteins that adsorb to adhesions are also driven by their coupling to cell contractility and the deformation of the matrix. We predict that adhesions reach a finite size that is proportional to the elastic compliance of the substrate, on a timescale that also scales with the compliance: focal adhesions quickly reach a relatively small, steady-state size on soft materials. However, their apparent sliding is not sensitive to the rigidity of the substrate. We also suggest some experimental probes of these ideas and discuss the nature of information that can be extracted from cell force microscopy on deformable substrates.
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