### Abstract

Spatially distributed populations of various types of bacteria often display intricate spatial patterns that are thought to result from the cellular response to gradients of nutrients or other attractants. In the past decade a great deal has been learned about signal transduction, metabolism, and movement in E. coli and other bacteria, but translating the individual-level behavior into population-level dynamics is still a challenging problem. However, this is a necessary step because it is computationally impractical to use a strictly cell-based model to understand patterning in growing populations, since the total number of cells may reach 10 ^{12}-10 ^{14} in some experiments. In the past phenomenological equations such as the Patlak-Keller-Segel equations have been used in modeling the cell movement that is involved in the formation of such patterns, but the question remains as to how the microscopic behavior can be correctly described by a macroscopic equation. Significant progress has been made for bacterial species that employ a "run-and-tumble" strategy of movement, in that macroscopic equations based on simplified schemes for signal transduction and turning behavior have been derived [R. Erban and H. G. Othmer, SIAM J. Appl. Math., 65 (2004), pp. 361-391; R. Erban and H. G. Othmer, Multiscale Model. Simul., 3 (2005), pp. 362-394]. Here we extend previous work in a number of directions: (i) we allow for time-dependent signals, which extends the applicability of the equations to natural environments, (ii) we use a more general turning rate function that better describes the biological behavior, and (iii) we incorporate the effect of hydrodynamic forces that arise when cells swim in close proximity to a surface. We also develop a new approach to solving the moment equations derived from the transport equation that does not involve closure assumptions. Numerical examples show that the solution of the lowest-order macroscopic equation agrees well with the solution obtained from a Monte Carlo simulation of cell movement under a variety of temporal protocols for the signal. We also apply the method to derive equations of chemotactic movement that are governed by multiple chemotactic signals.

Original language | English (US) |
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Pages (from-to) | 133-167 |

Number of pages | 35 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 70 |

Issue number | 1 |

DOIs | |

State | Published - 2009 |

### Keywords

- Chemotaxis equations
- Diffusion approximation
- Pattern formation
- Transport equations
- Velocity-jump processes