Recent experiments showed that engineered Escherichia coli colonies grow and self-organize into periodic stripes with high and low cell densities in semi-solid agar. The stripes develop sequentially behind a radially propagating colony front, similar to the formation of many other periodic patterns in nature. These bacteria were created by genetically coupling the intracellular chemotaxis pathway of wild-type cells with a quorum sensing module through the protein CheZ. In this paper, we develop multiscale models to investigate how this intracellular pathway affects stripe formation. We first develop a detailed hybrid model that treats each cell as an individual particle and incorporates intracellular signaling via an internal ODE system. To overcome the computational cost of the hybrid model caused by the large number of cells involved, we next derive a mean-field PDE model from the hybrid model using asymptotic analysis. We show that this analysis is justified by the tight agreement between the PDE model and the hybrid model in 1D simulations. Numerical simulations of the PDE model in 2D with radial symmetry agree with experimental data semi-quantitatively. Finally, we use the PDE model to make a number of testable predictions on how the stripe patterns depend on cell-level parameters, including cell speed, cell doubling time and the turnover rate of intracellular CheZ.
Bibliographical noteFunding Information:
MT was supported by National Science Foundation of China 91330203 (http://www.nsfc.gov.cn) and the RGC of the HKSAR under grant 202713 during her visits to HKBU(http://www.ugc.edu.hk/eng/rgc/). CX was supported in part by US National Science Foundation grant No. DMS 1312966 and National Science Foundation CAREER Award 1553637 (https://www.nsf.gov). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. MT would like to thank Prof. Leihan Tang for many insightful discussions during her visits to HKBU. CX and MT would like to thank the Mathematical Bioscience Institute, Ohio State University for support as long-term visitors.
© 2018 Xue et al. http://creativecommons.org/licenses/by/4.0/