The Impact of Elastic Deformations of the Extracellular Matrix on Cell Migration

Abstract

The mechanical properties of the extracellular matrix, in particular its stiffness, are known to impact cell migration. In this paper, we develop a mathematical model of a single cell migrating on an elastic matrix, which accounts for the deformation of the matrix induced by forces exerted by the cell, and investigate how the stiffness impacts the direction and speed of migration. We model a cell in 1D as a nucleus connected to a number of adhesion sites through elastic springs. The cell migrates by randomly updating the position of its adhesion sites. We start by investigating the case where the cell springs are constant, and then go on to assuming that they depend on the matrix stiffness, on matrices of both uniform stiffness as well as those with a stiffness gradient. We find that the assumption that cell springs depend on the substrate stiffness is necessary and sufficient for an efficient durotactic response. We compare simulations to recent experimental observations of human cancer cells exhibiting durotaxis, which show good qualitative agreement.

Keywords: Cell migration; Durotaxis; Mathematical modeling; Stochastic simulation.

Fig. 1

Plot of the displacement function u in the case of a substrate with constant stiffness (left) and linearly increasing stiffness (right). Cell size $20\mathrm{\mu }$m, on domain $[-1,1]$ mm, with $C=1$ kPa and $C(X)=1+0.8X$ kPa, respectively, ${\alpha }_1={\alpha }_2=100$ N/mm (Color figure online)

Fig. 2

Illustration of the cell on the undeformed ECM (Lagrangian description) and the corresponding cell on the deformed ECM (Eulerian description) (Color figure online)

Fig. 3

Cartoon of the steps of cell migration. (i) Initial state of cell on an undeformed substrate when the cell exerts no force. (ii) The cell exerts force so the substrate deforms. (iii) The cell updates the position of an adhesion site on the deformed substrate. (iv) We find the corresponding new position in the Lagrangian description, by relaxing the forces exerted by the cell. (v) Again the cell exerts force so that the substrate deforms. We are now back in the same situation as in (ii) (Color figure online)

Fig. 4

Plot of the average cell speeds for (a) and average ECM displacements (b), for varying substrate stiffness. The cell spring coefficients are equal and constant (Color figure online)

Fig. 5

Plot of the average cell speeds for (a) and average ECM displacements (b), FMI (c) and average cell position (d) for varying cell spring coefficients $\alpha$. The stiffness function with a gradient is shown in (d) (Color figure online)

Fig. 6

The cell migration speed of a migrating cell in a plastic EMC of varying stiffness (Color figure online)