Contour Models of Cellular Adhesion

Abstract

The development of traction-force microscopy, in the past two decades, has created the unprecedented opportunity of performing direct mechanical measurements on living cells as they adhere or crawl on unifrom or micro-patterend substrates. Simultaneously, this has created the demand for a theoretical framework able to decipher the experimental observations, shed light on the complex biomechanical processes that govern the interaction between the cell and the extracellular matrix and offer testable predictions. Contour models of cellular adhesion, represent one of the simplest and yet most insightful approach in this problem. Rooted in the paradigm of active matter, these models allow to explicitly determine the shape of the cell edge and calculate the traction forces experienced by the substrate, starting from the internal and peripheral contractile stresses as well as the passive restoring forces and bending moments arising within the actin cortex and the plasma membrane. In this chapter I provide a general overview of contour models of cellular adhesion and review the specific cases of cells equipped with isotropic and anisotropic actin cytoskeleton as well as the role of bending elasticity.

Force propagation between epithelial cells depends on active coupling and mechano-structural polarization

“…This so called anisotropic tension model predicts elliptical arcs and a position dependent line tension in the fiber [13]. A comprehensive summary of the different types of existing contour models can be found in [14].…”

Mechanical interplay between cell shape and actin cytoskeleton organization

“…In this article we investigated the spatial organization of cells adhering on a rigid substrate at a discrete number of points. Our approach is based on a contour model for the cell shape [8,[23][24][25][26] coupled with a continuous phenomenological model for the actin cytoskeleton, inspired by the physics of nematic liquid crystals [62]. This approach can be carried out at various levels of complexity, offering progressively insightful results.…”

“…where t is time, ξ t is a (translational) friction coefficient, Σout and Σin are the stress tensors on the two sides of the cell boundary and F cortex is the stress resultant along the cell contour [8,22,23,25,26,56]. The temporal evolution of the cell contour is then dictated by a competition between internal and external bulk stresses acting on the cell boundary and the contractile forces arising within the cell cortex.…”

“…In our previous work [22] we have investigated the shape and traction forces of adherent cells [23] characterized by a highly anisotropic actin cytoskeleton. Using a contour model of cellular adhesion [8,[23][24][25][26], we demonstrated that the edge of these cells can be accurately approximated by a collection of elliptical arcs obtained from a unique ellipse, whose eccentricity depends on the degree of anisotropy of the contractile stresses arising from the actin cytoskeleton. Furthermore, our model quantitatively predicts how the anisotropy of the actin cytoskeleton determines the magnitudes and directions of traction forces.…”

Stress fibers orient traction forces on micropatterns: A hybrid cellular Potts model study

“…Several types of mathematical models have been developed to interpret such experimental findings and predict the shapes of adherent cells [39]. The simplest among these are two-dimensional contour models [10,16,[46][47][48][49][50], in which the cell is completely described by a closed curve that represents the cell contour. For adherent cells with a limited number of discrete adhesion sites, for instance cells on micropillar arrays, the contour consists of a set of connected arcs, hereafter referred to as "cellular arcs", which connect two consecutive adhesion sites.…”