A Diffusion Approximation Based on Renewal Processes with Applications to Strongly Biased Run–Tumble Motion

We consider organisms which use a renewal strategy such as run–tumble when moving in space, for example to perform chemotaxis in chemical gradients. We derive a diffusion approximation for the motion, applying a central limit theorem due to Anscombe for renewal-reward processes; this theorem has not...

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Bibliographic Details
Published in:Bulletin of mathematical biology 2016-03, Vol.78 (3), p.556-579
Main Author: Thygesen, Uffe Høgsbro
Format: Article
Language:English
Subjects:
Cell Biology
Chemotaxis - physiology
Computer Simulation
Diffusion
Life Sciences
Mathematical and Computational Biology
Mathematical Concepts
Mathematics
Mathematics and Statistics
Models, Biological
Monte Carlo Method
Motion
Movement - physiology
Original Article
Stochastic Processes
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Summary:We consider organisms which use a renewal strategy such as run–tumble when moving in space, for example to perform chemotaxis in chemical gradients. We derive a diffusion approximation for the motion, applying a central limit theorem due to Anscombe for renewal-reward processes; this theorem has not previously been applied in this context. Our results extend previous work, which has established the mean drift but not the diffusivity. For a classical model of tumble rates applied to chemotaxis, we find that the resulting chemotactic drift saturates to the swimming velocity of the organism when the chemical gradients grow increasingly steep. The dispersal becomes anisotropic in steep gradients, with larger dispersal across the gradient than along the gradient. In contrast to one-dimensional settings, strong bias increases dispersal. We next include Brownian rotation in the model and find that, in limit of high chemotactic sensitivity, the chemotactic drift is 64 % of the swimming velocity, independent of the magnitude of the Brownian rotation. We finally derive characteristic timescales of the motion that can be used to assess whether the diffusion limit is justified in a given situation. The proposed technique for obtaining diffusion approximations is conceptually and computationally simple, and applicable also when statistics of the motion is obtained empirically or through Monte Carlo simulation of the motion.
ISSN:0092-8240
1522-9602
DOI:10.1007/s11538-016-0155-3