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Macroscopic equations for bacterial chemotaxis: integration of detailed biochemistry of cell signaling

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Abstract

Chemotaxis of single cells has been extensively studied and a great deal on intracellular signaling and cell movement is known. However, systematic methods to embed such information into continuum PDE models for cell population dynamics are still in their infancy. In this paper, we consider chemotaxis of run-and-tumble bacteria and derive continuum models that take into account of the detailed biochemistry of intracellular signaling. We analytically show that the macroscopic bacterial density can be approximated by the Patlak–Keller–Segel equation in response to signals that change slowly in space and time. We derive, for the first time, general formulas that represent the chemotactic sensitivity in terms of detailed descriptions of single-cell signaling dynamics in arbitrary space dimensions. These general formulas are useful in explaining relations of single cell behavior and population dynamics. As an example, we apply the theory to chemotaxis of bacterium Escherichia coli and show how the structure and kinetics of the intracellular signaling network determine the sensing properties of E. coli  populations. Numerical comparison of the derived PDEs and the underlying cell-based models show quantitative agreements for signals that change slowly, and qualitative agreements for signals that change extremely fast. The general theory we develop here is readily applicable to chemotaxis of other run-and-tumble bacteria, or collective behavior of other individuals that move using a similar strategy.

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Notes

  1. An exception is the reaction rate for the dissociation of the complex CheYp-CheZ. We take it to be \(8.45 \,\hbox {s}^{-1}\) instead of \(10 \, \hbox {s}^{-1}\) as in Table 1 of Xin and Othmer (2012). This change was necessary to reproduce the numerical results in Xin and Othmer (2012). Personal communications reveal that this discrepancy was due to a typo of Xin and Othmer (2012).

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Acknowledgments

CX would like to thank Professor Hans Othmer and Dr. Radek Erban for insightful discussions and comments on the paper. CX would like to dedicate this paper to Professor Hans G. Othmer’s 70th birthday.

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  1. Department of Mathematics, Ohio State University, Columbus, OH, 43210, USA

    Chuan Xue

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Correspondence to Chuan Xue.

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CX is supported by the National Science Foundation in the United States through grant DMS-1312966. CX is also supported by the Mathematical Biosciences Institute at the Ohio State University as a long-term visitor.

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Xue, C. Macroscopic equations for bacterial chemotaxis: integration of detailed biochemistry of cell signaling. J. Math. Biol. 70, 1–44 (2015). https://doi.org/10.1007/s00285-013-0748-5

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