A non local model for 2D cell migration in response to mechanical stimuli

Cell migration is one of the most important and most studied phenomena in biology and plays a fundamental role for many physiological and pathological processes such as morphogenesis, wound healing and tumorigenesis. In the body many cells, such as fibroblasts, migrate individually by adhering to a protein substrate known as the extracellular matrix (ECM). There are many mechanisms affecting cell motion. For instance, it is well known that cells sense the concentration gradients of particular chemicals and migrate attracted or repelled by them (chemotaxis). In recent years, researchers have performed experiments demonstrating that cells can also migrate in response to mechanical stimuli of the substrate to which they adhere: motion toward regions with higher stiffness is called durotaxis, while motion guided by the stress or the deformation of the substrate itself is called tensotaxis. Unlike chemotaxis, these migratory processes are not yet fully understood from a biological point of view, which makes the investigation of mathematical models suitable for their reproduction even more important. This Thesis is placed exactly in this context. In fact, its goal is to appropriately modify a mathematical model proposed by Colombi et al. (A. Colombi, S. Falletta, M. Scianna, L. Scuderi (2021), An integro-differential non-local model for cell migration and its efficient numerical solution, Mathematics and Computers in Simulation, Vol. 180, 179-204) that deals with single-cell migration in response to chemotactic signals, to account for mechanical cues. In the case of durotaxis the cell moves by changing its direction of polarization according to the different stiffness of the substrate. In the case of tensotaxis, the substrate on which the cell moves is appropriately deformed and the cell polarizes and migrates in response to proper scalar measures of the substrate strain or stress. From the mathematical point of view the equations of motion of the cell are nonlocal integro-differential equations, with integral terms that are in charge of describing the nonlocal evaluation of the considered mechanical cue by the cell. Their evaluation calls for explicit numerical methods, as well as appropriate quadrature techniques to account for possible singularities of the integrand functions. In addition, it must be taken into account that the integrand functions may be defined only pointwise, and therefore their interpolation at quadrature nodes is necessary. The equations are solved with a Matlab code by partitioning the bidimensional substrate with a uniform square mesh, eventually deformed in the case of tensotaxis. The mechanical stimulus to be integrated in the equations of motion is known experimentally in the case of durotaxis, while it is derived by the solution of the mechanical problem for the substrate in the case of tensotaxis. Specifically, the mass and momentum balance equations for the substrate are defined neglecting the mutual mechanical interaction between the cell and the substrate. The latter is modeled either as a linear elastic solid or as a hyperelastic Yeoh’s solid. Then, the mechanical problem is solved with the software Comsol Multiphysics and the quantities of interest are imported into the Matlab code that numerically implements the equations of motion. In both cases the equations of motion of the cell are solved by simulating different experimental setups found in the literature and the numerical simulations show a qualitative agreement with the experimental observations.