Protein polymerization in cells is an essential process in a wide range of biological
phenomena. The coordinate polymerization of proteins controls the correct
running of the cell, regulating its rigidity, activating its migration or taking part
in the mitosis process. In cell migration, the actin-myosin interaction and the
actin polymerization itself generate mechanisms that favor the cell advance (see
Ananthakrishnan et al. (2007)). Cell migration is crucial in the development of
biological processes such as chemotaxis, cancer metastasis or tissue regeneration.
Understandig what mechanical factors are responsible of this cell movement
oers the possibility of developing new techniques to, for example, control the
advance of invasive tumor cells. In-vitro works allow considering cell migration
as a process regulated mainly by the actin laments polymerization inside the
cell (Pravincumar et al. (2012)). Cell migration is considered as a mechanical
cycle of three phases: protrusion -due to actin polymerization-, adhesion and
contraction -due to the actin myosin interaction-. Increasing the knowledge
of the actin polymerization process, of the actin-myosin interactions and their
eects on the mechanics of cell is crucial to understand the cell migration process
(Larripa et al. (2006), Zheltukhin et al. (2011)).
The aim of this work is to obtain a one-dimensional mathematical model to
simulate the behavior of the dierent proteins inside the cell in order to pre-
dict its migration. Due to the complexity of this process, we will focus on the
interstitial migration, that is, the cell movement through micro
uidic channels.
We will propose a system of coupled reaction-diusion equations to simulate
the actin and myosin concentrations in order to model the actin polymerization
process together with the myosin contractile eect in the cell. Moreover, actin
and myosin concentrations are a key factor in the cell migration process since
they aect directly to the mechanical deformability of the cell. In this work,
we will couple the previous equations with the mechanical response of the cell,
computing its deformation and stresses. Cells will be considered as viscoelastic
materials, including the contraction and protrusion eects due to myosin and
actin concentrations. In order to carry out the numerical simulation of the pro-
posed model, we will use a suitable spatial-temporal integration scheme that
let us compute the suered deformation coupled with the protein concentration
variation and study the eect of the actin and myosin concentrations on the
cytoskeleton deformation.

References
R. Ananthakrishnan, A. Ehrlicher. The forces behind cell movement. Interna-
tional Journal of Biological Sciences 3 (2007) 303-317.
K. Larripa, A. Mogilner. Transport of a 1D viscoelastic actin-myosin strip of
gel as a model of a crawling cell. Physica A 372 (2006) 113-123.
P. Pravincumar, D.L. Bader, M.M. Knight. Viscoelastic cell mechanics and
actin remodelling are dependent on the rate of applied pressure. PLoS ONE 7
(2012) e43938. doi:10.1371/journal.pone.0043938
S. Zheltukhin, R. Lui. One-dimensional viscoelastic cell motility models. Math-
ematical Biosciences 229 (2011) 30-40.