Free energy analysis of cell spreading

Abstract

In this study we present a steady-state adaptation of the thermodynamically motivated stress fiber (SF) model of Vigliotti et al. (2015). We implement this steady-state formulation in a non-local finite element setting where we also consider global conservation of the total number of cytoskeletal proteins within the cell, global conservation of the number of binding integrins on the cell membrane, and adhesion limiting ligand density on the substrate surface. We present a number of simulations of cell spreading in which we consider a limited subset of the possible deformed spread-states assumed by the cell in order to examine the hypothesis that free energy minimization drives the process of cell spreading. Simulations suggest that cell spreading can be viewed as a competition between (i) decreasing cytoskeletal free energy due to strain induced assembly of cytoskeletal proteins into contractile SFs, and (ii) increasing elastic free energy due to stretching of the mechanically passive components of the cell. The computed minimum free energy spread area is shown to be lower for a cell on a compliant substrate than on a rigid substrate. Furthermore, a low substrate ligand density is found to limit cell spreading. The predicted dependence of cell spread area on substrate stiffness and ligand density is in agreement with the experiments of Engler et al. (2003). We also simulate the experiments of Théry et al. (2006), whereby initially circular cells deform and adhere to "V-shaped" and "Y-shaped" ligand patches. Analysis of a number of different spread states reveals that deformed configurations with the lowest free energy exhibit a SF distribution that corresponds to experimental observations, i.e. a high concentration of highly aligned SFs occurs along free edges, with lower SF concentrations in the interior of the cell. In summary, the results of this study suggest that cell spreading is driven by free energy minimization based on a competition between decreasing cytoskeletal free energy and increasing passive elastic free energy.

Thermodynamic Modeling of the Statistics of Cell Spreading on Ligand-Coated Elastic Substrates

“…Such a competition allows for the identification of a low free-energy state, and it is shown that the predicted cell areas and SF alignments in these configurations are similar to reported experimental measurements (14). However, in the study of McEvoy et al (13), a single low-energy spread state is identified. This deterministic approach neglects the fact that cells display a fluctuating response to their microenvironment in terms of observables such as SF alignment and spread area.…”

“…under the constraint that the cell maintains a homeostatic state. In previous studies, the importance of considering the system free energy in the interpretation of cell-spreading behavior has been recognized (13,29). McEvoy et al (13) identified low (or minimal) free-energy cell spread states within a limited phase space of axisymmetric configurations.…”

“…The FA model proposed here is a modification to the model of McEvoy et al (13) in which adhesion is assumed to be via integrins that exist in a single state. These integrins form complexes by binding to ligands that have a density N H per unit area on the surface of the elastic substrate.…”

“…However, it has been demonstrated that complexes cannot support a force greater than a critical value F max (21)(22)(23). Direct enforcement of the condition that no complex force exceeds F max at the cell-substrate interface would require an iterative adjustment of spread state (as implemented for simplified microstates by McEvoy et al (13)) and is therefore excessively computationally expensive in the context of the Monte Carlo simulations required for sampling the homeostatic ensemble. Here, we use the alternative approach of a penalty scheme to ensure that a very small number of spread states contain complexes with forces F > F max .…”

“…Typically, previous models have assumed the spread state as the reference configuration and simulate a single deterministic outcome (6)(7)(8)(9)(10)(11)(12). McEvoy et al (13) recently implemented a framework whereby an initially unadhered cell deforms to a range of possible spread states, and the system free energy is computed for each configuration. It is demonstrated that cell spreading entails a competition between the increasing elastic free energy because of the stretching of passive cell components and the decreasing cytoskeletal free energy as contractile proteins assemble to form SFs.…”