Mathematical model for the effects of adhesion and mechanics on cell migration speed

Abstract

Migration of mammalian blood and tissue cells over adhesive surfaces is apparently mediated by specific reversible reactions between cell membrane adhesion receptors and complementary ligands attached to the substratum. Although in a number of systems these receptors and ligand molecules have been isolated and identified, a theory capable of predicting the effects of their properties on cell migration behavior currently does not exist. We present a simple mathematical model for elucidating the dependence of cell speed on adhesion-receptor/ligand binding and cell mechanical properties. Our model can be applied to propose answers to questions such as: does an optimal adhesiveness exist for cell movement? How might changes in receptor and ligand density and/or affinity affect the rate of migration? Can cell rheological properties influence movement speed? This model incorporates cytoskeletal force generation, cell polarization, and dynamic adhesion as requirements for persistent cell movement. A critical feature is the proposed existence of an asymmetry in some cell adhesion-receptor property, correlated with cell polarity. We consider two major alternative mechanisms underlying this asymmetry: (a) a spatial distribution of adhesion-receptor number due to polarized endocytic trafficking and (b) a spatial variation in adhesion-receptor/ligand bond strength. Applying a viscoelastic-solid model for cell mechanics allows us to represent one-dimensional locomotion with a system of differential equations describing cell deformation and displacement along with adhesion-receptor dynamics. In this paper, we solve these equations under the simplifying assumption that receptor dynamics are at a quasi-steady state relative to cell locomotion. Thus, our results are strictly valid for sufficiently slow cell movement, as typically observed for tissue cells such as fibroblasts. Numerical examples relevant to experimental systems are provided. Our results predict how cell speed might vary with intracellular contractile force, cell rheology, receptor/ligand kinetics, and receptor/ligand number densities. A biphasic dependence is shown to be possible with respect to some of the system parameters, with position of the maxima essentially governed by a balance between transmitted contractile force and adhesiveness. We demonstrate that predictions for the two alternative asymmetry mechanisms can be distinguished and could be experimentally tested using cell populations possessing different adhesion-receptor numbers.

Polarization and Movement of Keratocytes: A Multiscale Modelling Approach

“…An overarching recent objective has been to start combining diverse aspects of cell motility into a framework of a spatial cell. Lauffenburger (1989);DiMilla et al (1991); Gracheva and Othmer (2004) have studied one-dimensional cross-sections with predefined polarity, to analyse the role of cell-substrate adhesion, cytoskeleton force generation and actin-myosin. Introducing forces involved with protrusion and retraction together with filamentturnover, Bottino et al (2002); Rubinstein et al (2005) have developed 2D descriptions of a crawling nematode sperm cell and a lamellipod, respectively.…”

How cellular movement determines the collective force generated by the Dictyostelium discoideum slug

“…DiMilla et al (1991) analysed a one-dimensional (1D) model in which a cell consisted of discrete subunits, each with an elastic spring, dash-pot and contractile element connected in parallel. Interaction with the substrate was modeled by additional Maxwell elements at the front and rear.…”

Damped and persistent oscillations in a simple model of cell crawling

“…Though the cell is a threedimensional system, its properties are assumed to be uniform in the direction perpendicular to its motion and parallel to the substrate. This one-dimensional model retains the basic ideas of earlier models [3,4,7] but in a minimal form. The most important contributions of this study are: (i) a simple free-body analysis of the fundamental force balances involved in cell crawling; (ii) mathematical expressions for the effects of physical parameters on cell crawling speed and stability; (iii) a hypothesis for the mechanism of oscillatory cell locomotion characterized by alternating contraction at the rear of the cell and extension at the leading edge.…”

“…In a seminal paper on modelling cell migration, DiMilla et al [4] describe a viscoelastic cell subject to internal cytoskeletal forces and dynamic interaction with the substrate. The study explicitly considers cells in mammalian tissue and blood, such as leukocytes or fibroblasts.…”