  Many animal cells change their shape depending on the stiffness of the
substrate on which they are cultured: they assume small, rounded shapes in soft
ECMs, they elongate within stiffer ECMs, and flatten out on hard substrates.
Cells tend to prefer stiffer parts of the substrate, a phenomenon known as
durotaxis. Such mechanosensitive responses to ECM mechanics are key to
understanding the regulation of biological tissues by mechanical cues, as it
occurs, e.g., during angiogenesis and the alignment of cells in muscles and
tendons. Although it is well established that the mechanical cell-ECM
interactions are mediated by focal adhesions, the mechanosensitive molecular
complexes linking the cytoskeleton to the substrate, it is poorly understood
how the stiffness-dependent kinetics of the focal adhesions eventually produce
the observed interdependence of substrate stiffness and cell shape and cell
behavior. Here we show that the mechanosensitive behavior of single-focal
adhesions, cell contractility and substrate adhesivity together suffice to
explain the observed stiffness-dependent behavior of cells. We introduce a
multiscale computational model that is based upon the following assumptions:
(1) cells apply forces onto the substrate through FAs; (2) the FAs grow and
stabilize due to these forces; (3) within a given time-interval, the force that
the FAs experience is lower on soft substrates than on stiffer substrates due
to the time it takes to reach mechanical equilibrium; and (4) smaller FAs are
pulled from the substrate more easily than larger FAs. Our model combines the
cellular Potts model for the cells with a finite-element model for the
substrate, and describes each FA using differential equations. Together these
assumptions provide a unifying model for cell spreading, cell elongation and
durotaxis in response to substrate mechanics.
