Taxis Equations for Amoeboid Cells
Radek Erban and Hans Othmer
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Submission date: 02. Feb. 2006
published in: Journal of mathematical biology, 54 (2007) 6, p. 847-885
DOI number (of the published article): 10.1007/s00285-007-0070-1
Keywords and phrases: amoeboid cells, microscopic models, direction-sensing, aggregation, chemotaxis equation, velocity jump process
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The classical macroscopic chemotaxis equations have previously been derived from an individual-based description of the tactic response of cells that use a "run-and-tumble" strategy in response to environmental cues [R. Erban and H. Othmer, From individual to collective behaviour in bacterial chemo- taxis, SIAM Journal on Applied Mathematics 65 (2004), no. 2, 361 - 391.; From signal transduction to spatial pattern formation in E. coli: A paradigm for multi-scale modeling in biology, Multiscale Modeling and Simulation 3(2005), no. 2, 362 - 394.]. Here we derive macroscopic equations for the more complex type of behavioral response characteristic of crawling cells, which detect a signal, extract directional information from a scalar concentration field, and change their motile behavior accordingly. We present several models of increasing complexity for which the derivation of population-level equations is possible, and we show how experimentally-measured statistics can be obtained from the transport equation formalism.We also show that amoeboid cells that do not adapt to constant signals can still aggregate in steady gradients, but not in response to periodic waves. This is in contrast to the case of cells that use a "run-and-tumble" strategy, where adaptation is essential.