# American Institute of Mathematical Sciences

December  2020, 25(12): 4839-4852. doi: 10.3934/dcdsb.2020129

## A continuous-time stochastic model of cell motion in the presence of a chemoattractant

 Department of Mathematics, Brigham Young University, Provo, UT 84602-6539, USA

* Corresponding author

Received  October 2019 Revised  January 2020 Published  April 2020

We consider a force-based model for cell motion which models cell forces using Hooke's law and a random outreach from the cell center. In previous work this model was simplified to track the centroid by setting the relaxation time to zero, and a formula for the expected velocity of the centroid was derived. Here we extend that formula to allow for chemotaxis of the cell by allowing the outreach distribution to depend on the spatial location of the centroid.

Citation: J. C. Dallon, Lynnae C. Despain, Emily J. Evans, Christopher P. Grant. A continuous-time stochastic model of cell motion in the presence of a chemoattractant. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4839-4852. doi: 10.3934/dcdsb.2020129
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Panel a) and c) depicts the state space $\mathit{x_e}$ given as an example. The circle represents the cell centroid. The black "x"s represent attached I-sites (labeled 0 and 1) and the red "x" denotes the detached I-site (labeled 2). The sets $A$ and $B$ represent the example sets used in the measure for case 1 and case 2. Panels a) and b) are for case 1 where I-site 1 dettaches. Panel a) is the initial configuration and panel b) is the new state. Likewise, panels c), and d) are for cases 2 where I-site 2 attaches. The sets $\bar{A}$ and $\bar{B}$ are the sets $A$ and $B$ transformed by $S^{-1}\circ F^{-1}$. In panel a), $\bar{A} = A$ and $\bar{B} = B+(.5, .5)$. In panel c), $\bar{A} = A-(.5, .5)$ and $\bar{B} = 3B-(1.5, 1.5)$ and their intersection contains $\mathbf{ \pmb{\mathsf{ η}}} = (1, 0)$. In panel d), I-site 2 has attached at location $\mathbf {x}_2 = \mathbf{c}+\mathbf{ \pmb{\mathsf{ η}}} = (.5, .5)+(1, 0)$
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