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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
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.

Keywords

reaction-diffusion equations
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), 395–554.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), 13719–13723.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), 277–285.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), 6572–6577.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), 18891–18896.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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