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5 - Reaction–diffusion Equations

from Part I - Physical Tools

Published online by Cambridge University Press:  12 December 2024

Thomas Andrew Waigh
Affiliation:
University of Manchester
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Summary

Introduces some simple reaction-diffusion equations to describe pattern formation in bacterial cells and biofilms including anomalous wave fronts, Turing patterns and the French flag model.

Type
Chapter
Information
The Physics of Bacteria
From Cells to Biofilms
, pp. 52 - 57
Publisher: Cambridge University Press
Print publication year: 2024

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References

Suggested Reading

Edelstein-Keshet, L., Mathematical Models in Biology. SIAM Classics: 2005. Excellent pedagogic account of reaction-diffusion equations.CrossRefGoogle Scholar
Hillen, T.; Leonard, I. E.; van Rossel, H., Partial Differential Equations: Theory and Completely Solved Problems. FriesenPress: 2009. A good place to learn some of the mathematics relevant to reaction-diffusion equations.Google Scholar
Mendez, V.; Fedotov, S., Reaction-transport Systems: Mesoscopic Foundations, Fronts and Spatial Instabilities. Springer: 2010. A challenging mathematical book that considers how to generalize reaction-diffusion equations to include the anomalous transport kinetics that are common in biology.CrossRefGoogle Scholar
Pismen, L., Active Matter Within and Around Us. Springer: 2021. Contains an interesting account of reaction-diffusion equations relevant to biophysics.CrossRefGoogle Scholar

References

Fall, C. P.; Marland, E. S.; Wagner, J. M.; Tyson, J. T., Computational Cell Biology. Springer: 2003.Google Scholar
Hillen, T.; Leonard, I. E.; van Rossel, H., Partial Differential Equations: Theory and Completely Solved Problems. Friesen Press: 2019.Google Scholar
Ben-Jacob, E.; Cohen, I.; Levine, H., Cooperative self-organization of microorganisms. Advances in Physics 2010, 49 (4), 395554.CrossRefGoogle Scholar
Keener, J.; Sneyd, J., Mathematical Physiology. Springer: 2009.CrossRefGoogle Scholar
Dieterle, P. B.; Amir, A., Diffusive wave dynamics beyond the continuum limit. Physical Review E 2021, 104 (1), 014406.CrossRefGoogle ScholarPubMed
Blee, J. A.; Roberts, I. S.; Waigh, T. A., Spatial propagation of electrical signals in circular biofilms. Physical Review E 2019, 100 (5-1), 052401.CrossRefGoogle ScholarPubMed
Akabuogu, E. U.; Martorelli, V.; Krasovec, R.; Roberts, I. S.; Waigh, T. A., Emergence of ion-channel mediated electrical oscillations in E. coli biofilms. eLife 2023, to appear.Google Scholar
Hennes, M.; Bender, N.; Cronenberg, T.; Welker, A.; Maier, B., Collective polarization dynamics in bacterial colonies signify the occurrence of distinct subpopulations. PLOS Biology 2023, 21 (1), e3001960.CrossRefGoogle ScholarPubMed
Cross, M.; Greenside, H., Pattern Formation Dynamics in Nonequilibrium Systems. Cambridge University Press: 2009.CrossRefGoogle Scholar
Palsson, E.; Lee, K. J.; Goldstein, R. E.; Franke, J.; Kessin, R. H.; Cox, E. C., Selection for spiral waves in the social amoebae Dictyostelium. Proceedings of the National Academy of Sciences of the United States of America 1997, 94 (25), 1371913723.CrossRefGoogle ScholarPubMed
Duran-Nebreda, S.; Pla, J.; Vidiella, B.; Pinero, J.; Conde-Pueyo, N.; Sole, R., Synthetic lateral inhibition in periodic pattern forming microbial colonies. ACS Synthetic Biology 2021, 10 (2), 277285.CrossRefGoogle ScholarPubMed
Karig, D.; Martini, M.; Weiss, R., Stochastic Turing patterns in a synthetic bacterial population. Proceedings of the National Academy of Sciences of the United States of America 2018, 115 (26), 65726577.CrossRefGoogle Scholar
Alon, U., An Introduction to Systems Biology: Design Principles of Biological Circuits. 2nd ed. CRC Press: 2020.Google Scholar
Pismen, L., Active Matter Within and Around Us: From Self-propelled Particles to Flocks and Living Forms. Springer: 2021.CrossRefGoogle Scholar
Asally, M.; Kittisopikul, M.; Rue, P.; Du, Y.; Hu, Z.; Cagatay, T.; Robinson, A. B.; Lu, H.; Garcia-Ojalvo, J.; Suel, G. M., Localised cell death focuses mechanical forces during 3D patterning in a biofilm. Proceedings of the National Academy of Sciences of the United States of America 2012, 109 (46), 1889118896.CrossRefGoogle Scholar
Erban, R.; Chapman, S. J., Stochastic Modelling of Reaction-diffuion Processes. Cambridge University Press: 2020.Google Scholar
Mendez, V.; Fedotov, S.; Horsthemke, W., Reaction-transport Systems: Mesoscopic Foundations, Fronts and Spatial Instabilities. Springer: 2012.Google Scholar
Waigh, T. A.; Korabel, N., Heterogeneous anomalous transport in cellular and molecular biology. Reports on Progress in Physics 2023, 86 (12), 126601.CrossRefGoogle ScholarPubMed

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