Incorporating Cellular Stochasticity in Solid–Fluid Mixture Biofilm Models
Abstract
:1. Introduction
2. Solid-Fluid Mixture Model of a Biofilm Spreading on an Agar/Air Interface
2.1. Mass Balance
2.2. Driving Forces
2.2.1. Stresses in the Solid and the Fluid
2.2.2. Interaction and Inertial Forces
2.3. Equations of Motion
2.4. Final Equations
2.5. Motion of the Air–Biofilm Interface
2.6. Motion of the Agar/Biofilm Interface
3. Incorporating Cellular Behavior
3.1. Cellular Automata and Dynamic Energy Budget
- We set an initial distribution of N bacteria characterized by their energies , volumes , damage , hazard , acclimation , and attached EPS volume , .
- Each bacterium is initially classified as normal, surfactin producer, EPS producer, or inert. Bacteria are distributed in the tiles of the grid. The empty space around them is filled with water and dissolved substances. In this way, we may compute the volume fractions of biomass and fluid in each tile , as well as the osmotic pressure . The pressure is obtained from (13) with and from (29).
- We compute stationary solutions of the Equation (31) for by a relaxation numerical scheme. All except the equation for are solved using the grid defining with no flux boundary conditions. The equation for is solved in the biofilm–agar domain, that is, , imposing continuity of concentrations and fluxes at the agar–biofilm interface and no flux boundary conditions at the air–biofilm interface.
- We update the bacterial status checking whether normal bacteria become surfactin or EPS producers, whether any of them deactivates or dies, and whether they divide, with the probabilities assigned to each situation. In case a bacterium divides, we reallocate the newborn cell.
- In the resulting biofilm configuration , we compute the volume fractions of biomass and fluid in each tile. This also provides the osmotic pressure The fluid pressure is obtained from (13), the displacements from (13), and from (29). The solid velocities are approximated by . Then, the fluid velocity is given by (9).
- We yet need to take into account water absorption from agar. To do so, we solve in the biofilm/agar system. Alternatively, we can solve only in the biofilm, using at the biofilm/agar interface and for boundary conditions, where h and R are reference values for the biofilm height and radius. Then, we revise the biofilm configuration, creating water tiles with probability and shifting the contains of the neighbouring tiles. This provides the final biofilm configuration , that is, the occupied tiles, their contents, the bacterial status and fields, as well as the values of the continuous fields at each tile.
3.2. Balance Equation Approach
4. Discussion
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A
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Carpio, A.; Cebrián, E. Incorporating Cellular Stochasticity in Solid–Fluid Mixture Biofilm Models. Entropy 2020, 22, 188. https://doi.org/10.3390/e22020188
Carpio A, Cebrián E. Incorporating Cellular Stochasticity in Solid–Fluid Mixture Biofilm Models. Entropy. 2020; 22(2):188. https://doi.org/10.3390/e22020188
Chicago/Turabian StyleCarpio, Ana, and Elena Cebrián. 2020. "Incorporating Cellular Stochasticity in Solid–Fluid Mixture Biofilm Models" Entropy 22, no. 2: 188. https://doi.org/10.3390/e22020188
APA StyleCarpio, A., & Cebrián, E. (2020). Incorporating Cellular Stochasticity in Solid–Fluid Mixture Biofilm Models. Entropy, 22(2), 188. https://doi.org/10.3390/e22020188