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Three-Dimensional Cellular Automaton for Modeling of Self-Similar Evolution in Biofilm-Forming Bacterial Populations

Author

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  • Samvel Sarukhanian

    (Laboratory of Mathematical Modeling of Complex Physical and Biological Systems, Amur State University, Ignatyevskoye Shosse, 21, 675027 Blagoveshchensk, Russia)

  • Anna Maslovskaya

    (Laboratory of Mathematical Modeling of Complex Physical and Biological Systems, Amur State University, Ignatyevskoye Shosse, 21, 675027 Blagoveshchensk, Russia)

  • Christina Kuttler

    (Department of Mathematics, School of Computation, Information and Technology, Technical University of Munich, Boltzmannstraße 3, 85747 Garching, Germany)

Abstract

Bacterial populations often form colonies and structures in biofilm. The paper aims to design suitable algorithms to simulate self-similar evolution in this context, specifically by employing a hybrid model that includes a cellular automaton for the bacterial cells and their dynamics. This is combined with the diffusion of the nutrient (as a random walk), and the consumption of nutrients by biomass. Lastly, bacterial cells divide when reaching high levels. The algorithm computes the space-time distribution of biomass under limited nutrient conditions, taking into account the collective redistribution of nutrients. To achieve better geometry in this modified model approach, truncated octahedron cells are applied to design the lattice of the cellular automaton. This allows us to implement self-similar realistic bacterial biofilm growth due to an increased number of inner relations for each cell. The simulation system was developed using C# on the Unity platform for fast calculation. The software implementation was executed in combination with the procedure of surface roughness measurements based on computations of fractional dimensions. The results of the simulations qualitatively correspond to experimental observations of the population dynamics of biofilm-forming bacteria. Based on in silico experiments, quantitative dependencies of the geometrical complexity of the biofilm structure on the level of consumed nutrients and oxygen were revealed. Our findings suggest that the more complex structure with a fractal dimension of the biofilm boundaries (around 2.6) corresponds to a certain range of nutrient levels, after which the structure degenerates and the biofilm homogenizes, filling the available space provided and tending towards a strictly 3D structure. The developed hybrid approach allows realistic scenario modeling of the spatial evolution of biofilm-forming bacterial populations and specifies geometric characteristics of visualized self-similar biofilm bacterial structures.

Suggested Citation

  • Samvel Sarukhanian & Anna Maslovskaya & Christina Kuttler, 2023. "Three-Dimensional Cellular Automaton for Modeling of Self-Similar Evolution in Biofilm-Forming Bacterial Populations," Mathematics, MDPI, vol. 11(15), pages 1-18, July.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:15:p:3346-:d:1207011
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    References listed on IDEAS

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    4. Wang, Hao & Garcia, Pablo Venegas & Ahmed, Shohel & Heggerud, Christopher M., 2022. "Mathematical comparison and empirical review of the Monod and Droop forms for resource-based population dynamics," Ecological Modelling, Elsevier, vol. 466(C).
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