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A Mathematical Description of Bacterial Chemotaxis in Response to Two Stimuli

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Abstract

Bacteria are often exposed to multiple stimuli in complex environments, and their efficient chemotactic decisions are critical to survive and grow in their native environments. Bacterial responses to the environmental stimuli depend on the ratio of their corresponding chemoreceptors. By incorporating the signaling machinery of individual cells, we analyze the collective motion of a population of Escherichia coli bacteria in response to two stimuli, mainly serine and methyl-aspartate (MeAsp), in a one-dimensional and a two-dimensional environment, which is inspired by experimental results in Y. Kalinin et al., J. Bacteriol. 192(7):1796–1800, 2010. Under suitable conditions, we show that if the ratio of the main chemoreceptors of individual cells, namely Tar/Tsr, is less than a specific threshold, the bacteria move to the gradient of serine, and if the ratio is greater than the threshold, the group of bacteria moves toward the gradient of MeAsp. Finally, we examine the theory with Monte Carlo agent-based simulations and verify that our results qualitatively agree well with the experimental results in Y. Kalinin et al. (2010).

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Acknowledgements

The authors would like to thank Professor Eduardo Sontag for sharing the Matlab codes for one-dimensional space (used in Aminzare and Sontag (2013)) and Professor Hans Othmer for helpful discussions. This work is partially supported by the University of Iowa Old Gold Fellowship and Simons Foundation (712522) to ZA. The authors would also like to thank the anonymous referee who provided valuable suggestions and comments to improve this work.

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  1. Department of Mathematics, State University of New York at New Paltz, New York, NY, USA

    Jeungeun Park

  2. Department of Mathematics, University of Iowa, Iowa City, IA, USA

    Zahra Aminzare

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  1. Jeungeun Park
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Correspondence to Zahra Aminzare.

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Appendix

Appendix

Table 1 Parameters used in intracellular signaling pathway of E. coli
Full size table

A brief description of Monte Carlo simulation: In a one-dimensional (respectively, two-dimensional) channel, we locate an ensemble of 100,000 agents in the center of the channel \(x=200\) (respectively, \((x,y) = (200, 800)\)) at time \(t=0\). At each time step, the individuals choose a direction +1 or -1 (respectively, \((\cos (\theta ), \sin (\theta ))\), \(\theta \in [0, 2\pi )\)) at random, and move in that direction with a constant speed \(\nu >0\). At each time step, the internal dynamics of each individual are computed by Euler method. At the end of each time step, we choose a number between 0 and 1 randomly and compare the number with the probability of change from run to tumble in interval of length dt, namely \(\lambda (a) dt\). If the turn occurs, the cell moves in the opposite direction with a probability of 0.5 (respectively, rotates by \(\theta \in [0, 2\pi )\), where \(\theta \) is chosen at random). If a cell is located outside the spatial domain, we relocate the cell by imposing reflecting boundary conditions.

See Tables 1 and 2.

Table 2 Input data used in numerical simulation
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Park, J., Aminzare, Z. A Mathematical Description of Bacterial Chemotaxis in Response to Two Stimuli. Bull Math Biol 84, 9 (2022). https://doi.org/10.1007/s11538-021-00965-6

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