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A stochastic model for the movement of a white blood cell

Part of: Limit theorems Physiological, cellular and medical topics

Published online by Cambridge University Press:  01 July 2016

Silvia Heubach  and
Joseph Watkins
Silvia Heubach*
Affiliation:
California State University, Los Angeles
Joseph Watkins*
Affiliation:
University of Arizona, Tucson
*
* Postal address: Department of Mathematics and Computer Science, California State University, 5151 State University Drive, Los Angeles, CA 90032, USA.
** Postal address: Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA.
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Abstract

We present a stochastic model for the movement of a white blood cell both in uniform concentration of chemoattractant and in the presence of a chemoattractant gradient. It is assumed that the rotational velocity is proportional to the weighted difference of the occupied receptors in the two halves of the cell and that each of the receptors stays free or occupied for an exponential length of time. We define processes corresponding to a cell with 2nß + 1 receptors (receptor sites). In the case of constant concentration, we show that the limiting process for the rotational velocity is an Ornstein–Uhlenbeck process. Its drift coefficient depends on the parameters of the exponential waiting times and its diffusion coefficient depends in addition also on the weight function. In the inhomogeneous case, the velocity process has a diffusion limit with drift coefficient depending on the concentration gradient and diffusion coefficient depending on the concentration and the weight function.

Keywords

RANDOM MOTILITYCHEMOTAXISBOUND AND FREE RECEPTORSDIFFUSION APPROXIMATIONORNSTEIN—UHLENBECK PROCESS

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 92C99: None of the above, but in this section
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 27 , Issue 2 , June 1995 , pp. 443 - 475
Copyright
Copyright © Applied Probability Trust 1995 

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