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

J. C. Dallon; Lynnae C. Despain; Emily J. Evans; Christopher P. Grant

doi:10.3934/dcdsb.2020129

Article Contents

2020, Volume 25, Issue 12: 4839-4852. Doi: 10.3934/dcdsb.2020129

This issue Previous Article A new weak solution to an optimal stopping problem Next Article Using automatic differentiation to compute periodic orbits of delay differential equations

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
^* Corresponding author

Received: September 30, 2019

Revised: December 31, 2019

Early access: April 2020

Published: December 2020

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Abstract

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.

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Mathematics Subject Classification: Primary: 60J25, 60J75; Secondary: 92C17.

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Figure 1. 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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