A chemo-mechanical free-energy-based approach to model durotaxis and extracellular stiffness-dependent contraction and polarization of cells

Abstract

We propose a chemo-mechanical model based on stress-dependent recruitment of myosin motors to describe how the contractility, polarization and strain in cells vary with the stiffness of their surroundings and their shape. A contractility tensor, which depends on the distribution of myosin motors, is introduced to describe the chemical free energy of the cell due to myosin recruitment. We explicitly include the contributions to the free energy that arise from mechanosensitive signalling pathways (such as the SFX, Rho-Rock and MLCK pathways) through chemo-mechanical coupling parameters. Taking the variations of the total free energy, which consists of the chemical and mechanical components, in accordance with the second law of thermodynamics provides equations for the temporal evolution of the active stress and the contractility tensor. Following this approach, we are able to recover the well-known Hill relation for active stresses, based on the fundamental principles of irreversible thermodynamics rather than phenomenology. We have numerically implemented our free energy-based approach to model spatial distribution of strain and contractility in (i) cells supported by flexible microposts, (ii) cells on two-dimensional substrates, and (iii) cells in three-dimensional matrices. We demonstrate how the polarization of the cells and the orientation of stress fibres can be deduced from the eigenvalues and eigenvectors of the contractility tensor. Our calculations suggest that the chemical free energy of the cell decreases with the stiffness of the extracellular environment as the cytoskeleton polarizes in response to stress-dependent recruitment of molecular motors. The mechanical energy, which includes the strain energy and motor potential energy, however, increases with stiffness, but the overall energy is lower for cells in stiffer environments. This provides a thermodynamic basis for durotaxis, whereby cells preferentially migrate towards stiffer regions of the extracellular environment. Our models also explain, from an energetic perspective, why the shape of the cells can change in response to stiffness of the surroundings. The effect of the stiffness of the nucleus on its shape and the orientation of the stress fibres is also studied for all the above geometries. Along with making testable predictions, we have estimated the magnitudes of the chemo-mechanical coupling parameters for myofibroblasts based on data reported in the literature.

A Chemomechanical Model for Nuclear Morphology and Stresses during Cell Transendothelial Migration

“…1, b and c). We adopted our recently developed chemomechanical model (21) to describe the stress-dependent actomyosin activity (Fig. 1 e), which is mediated by mechanosensitive signaling pathways such as the Src-family kinases, ROCK, and myosin light chain kinase, as shown in Fig.…”

“…1 d; refer to the Supporting Material for details). Here, we applied our recently published model (21) to introduce the stiffness-dependent recruitment of the contractile machinery (Fig. 1 e; refer to the Supporting Material for detailed descriptions) that accounts for the influence of both intracellular (for example, signal pathways) and extracellular cues (ECM modulus and deformation).…”

“…2 a) and a stiffer endothelium also impedes transmigration. Although the modulus of the ECM ðm t Þ has little influence on the resistance force, it has a strong effect on the actomyosin contractile forces: at the same chemomechanical coupling level, a softer ECM induces lower levels of cellular contractile force (21,26,27), which may not be sufficient for the cells to overcome the resistance force. Therefore, it is less likely for the cell to transmigrate when the ECM is soft (Fig.…”

Modeling the two-way feedback between contractility and matrix realignment reveals a nonlinear mode of cancer cell invasion

“…Because each myosin motor behaves like a force "dipole," the volume averaged density of the motors, or the contractility is treated as a symmetric tensor, ρ ij (25). In the quiescent state, that is, in the absence of external mechanical forces, the attachment of the myosins to the cytoskeleton has an intrinsic turnover rate that is determined by the molecular binding and unbinding of myosins to the cytoskeleton, leading to a steady-state contractility that is isotropic.…”

Interplay of Platelet Contractility and Elasticity of Fibrin/Erythrocytes in Blood Clot Retraction

“…Mathematical models have been used to assess the feedback effect that matrix stiffness has on the mechanics of active contractile cells (26) and to describe morphological changes and mechanical responses in an active elastic material (27). Given the structural similarities between previously described contracting microtissue and contracting blood clots, we chose to couple the basis of these aforementioned mathematical models with what is known about the individual components of clots to, for the first time to our knowledge, couple active contractile mechanics with a viscoelastic matrix (Fig.…”