Directional cell motility plays a key role in many biological processes like morphogenesis, inflammation, wound repair, angiogenesis, immune response, and tumor metastasis. Cells respond to the gradient in surface ligand density by directed locomotion towards the direction of higher ligand density. Theoretical models which address the physical basis underlying the regulatory effect of ligand gradient on cell motility are highly desirable. Predictive models not only contribute to a better understanding of biological processes, but they also provide a quantitative interconnection between cell motility and biophysical properties of the extracellular matrix (ECM) for rational design of biomaterials as scaffolds in tissue engineering. In this work, we consider a one-dimensional (1D) continuum viscoelastic model to predict the cell velocity in response to linearly increasing density of surface ligands on a substrate. The cell is considered as a 1D linear viscoelastic object with position dependent elasticity due to the variation in actin network density. The cell-substrate interaction is characterized by a frictional force, controlled by the density of ligand-receptor pairs. The generation of contractile stresses is described in terms of kinetic equations for the reactions between actins, myosins, and guanine nucleotide regulatory proteins. The model predictions show a reasonable agreement with experimentally measured cell speeds, considering biologically relevant values for the model parameters. The model predicts a biphasic relationship between cell speed and slope of gradient as well as a maximum limiting speed after a finite migration time. For a given slope of ligand gradient, the onset of the limiting speed appears at longer times for substrates with lower ligand gradients. The model can be applied to the design of biomaterials as scaffolds for guided tissue regeneration as it predicts an optimum range for the slope of ligand gradient.
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