A bulk-surface moving-mesh finite element method for modelling cell migration pathways

Cell migration is an ubiquitous process in life that is mainly triggered by the dynamics of the actin cytoskeleton and therefore is driven by both mechanical properties and biochemical processes. It is a multistep process essential for mammalian organisms and is closely linked to development, cancer invasion and metastasis formation, wound healing, immune response, tissue differentiation and regeneration, and inflammation. Experimental, theoretical and computational studies have been key to elucidate the mechanisms underlying cell migration. On one hand, rapid advances in experimental techniques allow for detailed experimental measurements of cell migration pathways, while, on the other, computational approaches allow for the modelling, analysis and understanding of such observations. Here, we present a computational framework coupling mechanical properties with biochemical processes to model two–dimensional cell migration by considering membrane and cytosolic activities that may be triggered by external cues. Our computational approach shows that the numerical implementation of the mechanobiochemical model is able to deal with fundamental characteristics such as: (i) membrane polarisation, (ii) cytosolic polarisation, and (iii) actin-dependent protrusions. This approach can be generalised to deal with single cell migration through complex non-isotropic environments, both in 2- and 3-dimensions. Author summary When a single or group of cells follow directed movement in response to either chemical and/or mechanical cues, this process is known as cell migration. It is essential for many biological processes such as immune response, embryogenesis, gastrulation, wound repair, cancer metastasis, tumour invasion, inflammation and tissue homeostasis. However, aberrant or defects in cell migration lead to various abnormalities and life-threatening medical conditions [[1][1]–[4][2]]. Increasing our knowledge on cell migration can help abate the spread of highly malignant cancer cells, reduce the invasion of white cells in the inflammatory process, enhance the healing of wounds and reduce congenital defects in brain development that lead to mental disorders. In this study, we present a computational framework that allows us to couple mechanical properties with biochemical signalling processes to study long time behaviour of single cell migration (either directed or random). The novelty is that the evolution law for the velocity (also known as the flow or material velocity) is described by a biomechanical force balance model posed inside the cell and this in turn is driven by the actomyosin spatiotemporal model (following the classical theory of reaction-diffusion) which is responsible for force generation as described in many experimental works [[2][3], [5][4], [8][5], [10][6], [11][7]]. Hence, our modelling approach is based on a new mathematical formalism of bulk-surface partial differential equations coupled with a novel adaptive moving-mesh finite element method to allow for significant cell deformations during migration. The approach set premises to study cell migration through complex non-isotropic environments, thereby giving biologists a predictive tool for modelling cell migration. ### Competing Interest Statement The authors have declared no competing interest. [1]: #ref-1 [2]: #ref-4 [3]: #ref-2 [4]: #ref-5 [5]: #ref-8 [6]: #ref-10 [7]: #ref-11