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Abstract

Traction forces developed by most cell types play a significant role in the spatial organisation of biological tissues. However, due to the complexity of cell-extracellular matrix interactions, these forces are quantitatively difficult to estimate without explicitly considering cell properties and extracellular mechanical matrix responses. Recent experimental devices elaborated for measuring cell traction on extracellular matrix use cell deposits on a piece of gel placed between one fixed and one moving holder. We formulate here a mathematical model describing the dynamic behaviour of the cell-gel medium in such devices. This model is based on a mechanical force balance quantification of the gel visco-elastic response to the traction forces exerted by the diffusing cells. Thus, we theoretically analyzed and simulated the displacement of the free moving boundary of the system under various conditions for cells and gel concentrations. This model is then used as the theoretical basis of an experimental device where endothelial cells are seeded on a rectangular biogel of fibrin cast between two floating holders, one fixed and the other linked to a force sensor. From a comparison of displacement of the gel moving boundary simulated by the model and the experimental data recorded from the moving holder displacement, the magnitude of the traction forces exerted by the endothelial cell on the fibrin gel was estimated for different experimental situations. Different analytical expressions for the cell traction term are proposed and the corresponding force quantifications are compared to the traction force measurements reported for various kind of cells with the use of similar or different experimental devices.

Adaptation trajectories during adhesion and spreading affect future cell states

“…30 min on collagen and 4 hours on fibrin. After 8.5 h the motion of the fluorescent beads ceased, indicating mechanical equilibrium between the cell traction forces and the elastic resistance of the protein gels, possibily in combination with remodelling of the gels 26 , 33 , 34 . Within this time frame, most cells remained within the window of recording.…”

Application of Sensing Techniques to Cellular Force Measurement

“…They also have the advantage that the measured forces of cells embedded in a 3-D matrix are more physiologically relevant to cells in vivo that reside in a 3-D tissue environment. However, these methods do not measure CTFs per se [35]. Meanwhile, cells are heterogeneous and the forces that they generate vary in a wide range.…”

A model for one-dimensional morphoelasticity and its application to fibroblast-populated collagen lattices

“…An alternative framework, which addresses this issue, but which is only valid for small displacement gradients, was developed by Barocas et al (1995) who replaced the Kelvin-Voigt constitutive law in the Moon-Tranquillo model with a Maxwell constitutive law. In contrast, Ferrenq et al (1997) retained the linear Kelvin-Voigt constitutive law, but restricted their focus to situations in which the displacement gradient is small and linear theory is valid. A different approach, that used the theory of mixtures to describe the interaction between the fibrous lattice and the permeating fluid medium, was the biphasic model of collagen lattice contraction developed by Barocas and coworkers Tranquillo, 1994, 1997).…”

“…3.2, it is often appropriate to treat contracting FPCLs as 1-D bodies. Indeed, Ferrenq et al (1997) constructed a model of FPCL contraction in 1-D Cartesian coordinates and various authors have assumed radial symmetry to develop 1-D descriptions of FPCL contraction (Moon and Tranquillo, 1993;Ramtani et al, 2002;Ramtani, 2004). We hence derive a one-dimensional model, based on the following equation (derived explicitly in Appendix A) that is appropriate for describing the mechanical behaviour of morphoelastic solids with small effective strain…”