A new combined differential-discrete cellular automaton approach for biofilm modeling: Application for growth in gel beads

Microbe-cellulose hydrogels as a model system for particulate carbon degradation in soil aggregates

Particulate carbon (C) degradation in soils is a critical process in the global C cycle governing greenhouse gas fluxes and C storage. Millimeter-scale soil aggregates impose strong controls on particulate C degradation by inducing chemical gradients of e.g. oxygen, as well as limiting microbial mobility in pore structures. To date, experimental models of soil aggregates have incorporated porosity and chemical gradients but not particulate C. Here, we demonstrate a proof-of-concept encapsulating microbial cells and particulate C substrates in hydrogel matrices as a novel experimental model for soil aggregates. Ruminiclostridium cellulolyticum was co-encapsulated with cellulose in millimeter-scale polyethyleneglycol-dimethacrylate (PEGDMA) hydrogel beads. Microbial activity was delayed in hydrogel-encapsulated conditions, with cellulose degradation and fermentation activity being observed after 13 days of incubation. Unexpectedly, hydrogel encapsulation shifted product formation of R. cellulolyticum from an ethanol-lactate-acetate mixture to an acetate-dominated product profile. Fluorescence microscopy enabled simultaneous visualization of the PEGDMA matrix, cellulose particles, and individual cells in the matrix, demonstrating growth on cellulose particles during incubation. Together, these microbe-cellulose-PEGDMA hydrogels present a novel, reproducible experimental soil surrogate to connect single cells to process outcomes at the scale of soil aggregates and ecosystems.

Increasing aggregate size reduces single-cell organic carbon incorporation by hydrogel-embedded wetland microbes

Microbial degradation of organic carbon in sediments is impacted by the availability of oxygen and substrates for growth. To better understand how particle size and redox zonation impact microbial organic carbon incorporation, techniques that maintain spatial information are necessary to quantify elemental cycling at the microscale. In this study, we produced hydrogel microspheres of various diameters (100, 250, and 500 μm) and inoculated them with an aerobic heterotrophic bacterium isolated from a freshwater wetland (Flavobacterium sp.), and in a second experiment with a microbial community from an urban lacustrine wetland. The hydrogel-embedded microbial populations were incubated with 13C-labeled substrates to quantify organic carbon incorporation into biomass via nanoSIMS. Additionally, luminescent nanosensors enabled spatially explicit measurements of oxygen concentrations inside the microspheres. The experimental data were then incorporated into a reactive-transport model to project long-term steady-state conditions. Smaller (100 μm) particles exhibited the highest microbial cell-specific growth per volume, but also showed higher absolute activity near the surface compared to the larger particles (250 and 500 μm). The experimental results and computational models demonstrate that organic carbon availability was not high enough to allow steep oxygen gradients and as a result, all particle sizes remained well-oxygenated. Our study provides a foundational framework for future studies investigating spatially dependent microbial activity in aggregates using isotopically labeled substrates to quantify growth.

Cooperation in bioluminescence: understanding the role of autoinducers by a stochastic random resistor model

Quorum sensing is a communication mechanism adopted by different bacterial strains for the regulation of gene transcription. It takes place through the exchange of molecules called autoinducers. Bioluminescence is an emergent threshold phenomenon shown by some bacteria strains. Its precise relationship to quorum sensing is a debated topic, particularly regarding the role of the different autoinducers used by bacteria. In this paper, assuming a direct relationship between bioluminescence and quorum sensing, we investigate the role of multiple autoinducers in the bioluminescence response of Vibrio harveyi, considered as a model bioluminescent strain, due to its quorum sensing circuitry involving an array of three different autoinducers. Experiments on mutants of this bacterium, obtained by suppression of one or more autoinducers, reveal their relative non-trivial relevance and cooperative interaction patterns. The proposed analysis is implemented on a regular lattice, whose nodes represent microbial entities equipped with charges, which represent the ability to up/down regulate the gene expression. Quorum sensing results from a Coulomb-type field, produced by the charges. In analogy with random resistor network models, the lattice is permeated by an effective current which accounts for the amount and distribution of the charges. We propose that the presence of different autoinducers correspond to a different up/down regulation of gene expression, i.e., to a different way to account for the charges. Then, by introducing a modulation of the charge dependence into the current flowing within the network, we show that it is able to describe the bioluminescence exhibited by V. harveyi mutants. Furthermore, modulation of the charge dependence allows the interactions between the different autoinducers to be taken into account, providing a prediction regarding the data obtainable under specific growth conditions.

A novel model for biofilm initiation in porous media flow

Bacteria often form biofilms in porous environments where an external flow is present, such as soil or filtration systems. To understand the initial stages of biofilm formation, one needs to study the interactions between cells, the fluid and the confining geometries. Here, we present an agent based numerical model for bacteria that takes into account the planktonic stage of motile cells as well as surface attachment and biofilm growth in a lattice Boltzmann fluid. In the planktonic stage we show the importance of the interplay between complex flow and cell motility when determining positions of surface attachment. In the growth stage we show the applicability of our model by investigating how external flow and biofilm stiffness determine qualitative colony morphologies as well as quantitative measurements of, e.g., permeability.

Microbial fuel cell compared to a chemostat

Microbial Fuel Cells (MFCs) represent a green and sustainable energy conversion system that integrate bacterial biofilms within an electrochemical two-electrode set-up to produce electricity from organic waste. In this review, we focus on a novel exploratory model, regarding "thin" biofilms forming on highly perfusable (non-diffusible) anodes in small-scale, continuous flow MFCs due to the unique properties of the electroactive biofilm. We discuss how this type of MFC can behave as a chemostat in fulfilling common properties including steady state growth and multiple steady states within the limit of biological physicochemical conditions imposed by the external environment. With continuous steady state growth, there is also continuous metabolic rate and continuous electrical power production, which like the chemostat can be controlled. The model suggests that in addition to controlling growth rate and power output by changing the external resistive load, it will be possible instead to change the flow rate/dilution rate.

Approximate Models of Microbiological Processes in a Biofilm Formed on Fine Spherical Particles

This paper concerns the dynamical modeling of the microbiological processes that occur in the biofilms that are formed on fine inert particles. Such biofilm forms e.g. in fluidized-bed bio-reactors, expanded bed biofilm reactors and biofilm air-lift suspension reactors. An approximate model that is based on the Laplace–Carson transform and a family of approximate models that are based on the concept of the pseudo-stationary substrate concentration profile in the biofilm were proposed. The applicability of the models to the microbiological processes was evaluated following Monod or Haldane kinetics in the conditions of dynamical biofilm growth. The use of approximate models significantly simplifies the computations compared to the exact one. Moreover, the stiffness that was present in the exact model, which was solved numerically by the method of lines, was eliminated. Good accuracy was obtained even for large internal mass transfer resistances in the biofilm. It was shown that significantly higher accuracy was obtained using one of the proposed models than that which was obtained using the previously published approximate model that was derived using the homotopy analysis method.

Modeling interspecific competition of the microbial community during anaerobic digestion based on cellular automata and ADM1

Interspecific competition for substrate and space gives rise to considerable variation in biomass distribution within the microbial community. To study microbial community in depth, we used several research methods as sampling and analytical measurements, and developed a cellular automata (CA) model that would facilitate a description of the microbial growth process based on Anaerobic Digestion Model No. 1 (ADM1) of the International Water Association (IWA). Using the CA model, we aimed to determine whether interspecific competition occurs among acidogens, acetogens and methanogens, and to examine the influence of interspecific competition on the spatial structure of microbial communities. We found that acetogens and methanogens competed for core space, resulting in a multi-layer structure. Butyrate-degrading acetogens increased in number, resulting in inhibition of propionate-degrading acetogens. Hydrogenotrophic methanogens showed stronger competitive advantage than acetotrophic methanogens. The simulation showed that the multi-layer structure of the microbial community was formed by interspecific competition.

Modeling of Symbiotic Bacterial Biofilm Growth with an Example of the Streptococcus-Veillonella sp. System

We present a multi-dimensional continuum mathematical model for modeling the growth of a symbiotic biofilm system. We take a dual-species namely, the Streptococcus-Veillonella sp. biofilm system as an example for numerical investigations. The presented model describes both the cooperation and competition between these species of bacteria. The coupled partial differential equations are solved by using an integrative finite element numerical strategy. Numerical examples are carried out for studying the evolution and distribution of the bio-components. The results demonstrate that the presented model is capable of describing the symbiotic behavior of the biofilm system. However, homogenized numerical solutions are observed locally. To study the homogenization behavior of the model, numerical investigations regarding on how random initial biomass distribution influences the homogenization process are carried out. We found that a smaller correlation length of the initial biomass distribution leads to faster homogenization of the solution globally, however, shows more fluctuated biomass profiles along the biofilm thickness direction. More realistic scenarios with bacteria in patches are also investigated numerically in this study.

Coupled flow and biomass-nutrient growth at pore-scale with permeable biofilm, adaptive singularity and multiple species

In this paper we describe a coupled model for flow and microbial growth as well as nutrient utilization. These processes occur within and outside the biofilm phase formed by the microbes. The primary challenge is to address the volume constraint of maximum cell density but also to allow some microbial presence outside the contiguous biofilm phase. Our model derives from the continuum analogues of the mechanism of cell shoving introduced in discrete biomass models, and in particular from the models exploiting singular diffusivity as well as from models of variational inequality type which impose explicit constraints. We blend these approaches and propose a new idea to adapt the magnitude of the diffusivity automatically so as to ensure the volume constraint without affecting the reactions; this construction can be implemented in many variants without deteriorating the overall efficiency. The second challenge is to account for the flow and transport in the bulk fluid phase adjacent to the biofilm phase. We use the Brinkman flow model with a spatially variable permeability depending on biomass amount. The fluid flow allows some advection of the nutrient within the biofilm phase as well as for the flow even when the pores are close to being plugged up. Our entire model is monolithic and computationally robust even in complex pore-scale geometries, and extends to multiple species. We provide illustrations of our model and of related approaches. The results of the model can be easily post-processed to provide Darcy scale properties of the porous medium, e.g., one can predict how the permeability changes depending on the biomass growth in many realistic scenarios.

Mechanics of biofilms formed of bacteria with fimbriae appendages

Gram-negative bacteria, as well as some Gram-positive bacteria, possess hair-like appendages known as fimbriae, which play an important role in adhesion of the bacteria to surfaces or to other bacteria. Unlike the sex pili or flagellum, the fimbriae are quite numerous, with of order 1000 fimbriae appendages per bacterial cell. In this paper, a recently developed hybrid model for bacterial biofilms is used to examine the role of fimbriae tension force on the mechanics of bacterial biofilms. Each bacterial cell is represented in this model by a spherocylindrical particle, which interact with each other through collision, adhesion, lubrication force, and fimbrial force. The bacterial cells absorb water and nutrients and produce extracellular polymeric substance (EPS). The flow of water and EPS, and nutrient diffusion within these substances, is computed using a continuum model that accounts for important effects such as osmotic pressure gradient, drag force on the bacterial cells, and viscous shear. The fimbrial force is modeled using an outer spherocylinder capsule around each cell, which can transmit tensile forces to neighboring cells with which the fimbriae capsule collides. We find that the biofilm structure during the growth process is dominated by a balance between outward drag force on the cells due to the EPS flow away from the bacterial colony and the inward tensile fimbrial force acting on chains of cells connected by adhesive fimbriae appendages. The fimbrial force also introduces a large rotational motion of the cells and disrupts cell alignment caused by viscous torque imposed by the EPS flow. The current paper characterizes the competing effects of EPS drag and fimbrial force using a series of computations with different values of the ratio of EPS to bacterial cell production rate and different numbers of fimbriae per cell.

Modeling of Biofilm Growth on Fine Spherical Particles with the Use of Cellular Automata: The Influence of Cell Death and Lysis on the Biofilm Structure

The paper concerns the modeling of heterogeneous biofilm growth on fine spherical particles of such biofilm forms as, e.g., fluidized-bed bioreactors. Three discrete mathematical models based on cellular automata theory were proposed. The double-substrate kinetics of biomass growth, biomass displacement, internal and external mass transfer resistances, death and lysis of microbiological cells and biofilm detachment were taken into account. It was shown that there are no significant qualitative and quantitative differences between biofilm growth on flat and spherical particles of different radii. Computer simulations were compared with experimental observations. Qualitative and quantitative agreement areachieved if both cell death and lysis aretaken into consideration and a proper algorithm of biomass displacement is used. The value of the bacteria lysis rate coefficient was estimated.

Effective nitrogen removal from ammonium-depleted wastewater by partial nitritation and anammox immobilized in granular and thin layer gel carriers

This study investigates the effect of physicochemical conditions on the partial nitritation and anammox treatment by immobilized ammonia oxidizers under ammonium-deplete conditions. The impact of oxygen and temperature was studied by measuring the activity of immobilized aerobic and anaerobic ammonia-oxidizing organisms (Ammonia-oxidizing bacteria (AOB) and archaea (AOA), and Anammox bacteria) embedded in polyvinyl alcohol - sodium alginate (PVA-SA) beads and in thin layer poly-ethylene glycol hydrogels. Beads and flat hydrogels were incubated in a fluidized bed reactor (FBR) and in two flow cells, respectively. Both systems were fed with synthetic wastewater (15 mg N-NH₄⁺/L) at different temperatures (20 °C and/or 30 °C) and different dissolved oxygen (DO) concentrations (0.1, 0.3, 0.5 and/or 1 mg/L) over 152 and 207 days, respectively. The FBR system had a maximum removal rate of 1.7 g-N/m³/d at 0.1 mg O₂/L, corresponding to 80% removal efficiency, while a high aerobic ammonia-oxidizing activity but a partial oxygen inhibition of Anammox bacteria were observed at higher DO concentrations. In both flow cells, nitrogen removal efficiency was highest (80%) at 30 °C and 1 mg O₂/L while removal was less favorable at lower DO and lower temperature. Our results indicate a potential use of hydrogel beads for an energy efficient technology with reduced aeration demand for treating low ammonia wastewater, while layered hydrogels are a possible first step for biological treatments of wastewater using tangential flow. In addition, we provide blueprint drawings of the flow cells, which may be used to 3D-print the apparatus for other applications.

Biofilm Structure Promotes Coexistence of Phage-Resistant and Phage-Susceptible Bacteria

Encounters among bacteria and their viral predators (bacteriophages) are among the most common ecological interactions on Earth. These encounters are likely to occur with regularity inside surface-bound communities that microbes most often occupy in natural environments. Such communities, termed biofilms, are spatially constrained: interactions become limited to near neighbors, diffusion of solutes and particulates can be reduced, and there is pronounced heterogeneity in nutrient access and physiological state. It is appreciated from prior theoretical work that phage-bacteria interactions are fundamentally different in spatially structured contexts, as opposed to well-mixed liquid culture. Spatially structured communities are predicted to promote the protection of susceptible host cells from phage exposure, and thus weaken selection for phage resistance. The details and generality of this prediction in realistic biofilm environments, however, are not known. Here, we explore phage-host interactions using experiments and simulations that are tuned to represent the essential elements of biofilm communities. Our simulations show that in biofilms, phage-resistant cells-as their relative abundance increases-can protect clusters of susceptible cells from phage exposure, promoting the coexistence of susceptible and phage-resistant bacteria under a large array of conditions. We characterize the population dynamics underlying this coexistence, and we show that coexistence is recapitulated in an experimental model of biofilm growth measured with confocal microscopy. Our results provide a clear view into the dynamics of phage resistance in biofilms with single-cell resolution of the underlying cell-virion interactions, linking the predictions of canonical theory to realistic models and in vitro experiments of biofilm growth.IMPORTANCE In the natural environment, bacteria most often live in communities bound to one another by secreted adhesives. These communities, or biofilms, play a central role in biogeochemical cycling, microbiome functioning, wastewater treatment, and disease. Wherever there are bacteria, there are also viruses that attack them, called phages. Interactions between bacteria and phages are likely to occur ubiquitously in biofilms. We show here, using simulations and experiments, that biofilms will in most conditions allow phage-susceptible bacteria to be protected from phage exposure, if they are growing alongside other cells that are phage resistant. This result has implications for the fundamental ecology of phage-bacteria interactions, as well as the development of phage-based antimicrobial therapeutics.

Hybrid Model of Bacterial Biofilm Growth

Bacterial biofilms play a critical role in environmental processes, water treatment, human health, and food processing. They exhibit highly complex dynamics due to the interactions between the bacteria and the extracellular polymeric substance (EPS), water, and nutrients and minerals that make up the biofilm. We present a hybrid computational model in which the dynamics of discrete bacterial cells are simulated within a multiphase continuum, consisting of EPS and water as separate interacting phases, through which nutrients and minerals diffuse. Bacterial cells in our model consume water and nutrients in order to grow, divide, and produce EPS. Consequently, EPS flows outward from the bacterial colony, while water flows inward. The model predicts bacterial colony formation as a treelike structure. The distribution of bacterial growth and EPS production is found to be sensitive to the pore spacing between bacteria and the consumption of nutrients within the bacterial colony. Forces that are sometimes neglected in biofilm simulations, such as lubrication force between nearby bacterial cells and osmotic (swelling) pressure force resulting from gradients in EPS concentration, are observed to have an important effect on biofilm growth via their influence on bacteria pore spacing and associated water/nutrient percolation into the bacterial colony.

NUFEB: A massively parallel simulator for individual-based modelling of microbial communities

We present NUFEB (Newcastle University Frontiers in Engineering Biology), a flexible, efficient, and open source software for simulating the 3D dynamics of microbial communities. The tool is based on the Individual-based Modelling (IbM) approach, where microbes are represented as discrete units and their behaviour changes over time due to a variety of processes. This approach allows us to study population behaviours that emerge from the interaction between individuals and their environment. NUFEB is built on top of the classical molecular dynamics simulator LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator), which we extended with IbM features. A wide range of biological, physical and chemical processes are implemented to explicitly model microbial systems, with particular emphasis on biofilms. NUFEB is fully parallelised and allows for the simulation of large numbers of microbes (10⁷ individuals and beyond). The parallelisation is based on a domain decomposition scheme that divides the domain into multiple sub-domains which are distributed to different processors. NUFEB also offers a collection of post-processing routines for the visualisation and analysis of simulation output. In this article, we give an overview of NUFEB’s functionalities and implementation details. We provide examples that illustrate the type of microbial systems NUFEB can be used to model and simulate.

Individual-based Models (IbM) are one of the most promising frameworks to study microbial communities, as they can explicitly describe the behaviour of each cell. The development of a general-purpose IbM solver should focus on efficiency and flexibility due to the unique characteristics of microbial systems. However, available tools for these purposes present significant limitations. Most of them only facilitate serial computing for single simulation, or only focus on biological processes, but do not model mechanical and chemical processes in detail. In this work, we introduce the IbM solver NUFEB that addresses some of these shortcomings. The tool facilitates the modelling of much needed biological, chemical, physical and individual microbes in detail, and offers the flexibility of model extension and customisation. NUFEB is also fully parallelised and allows for the simulation of large complex microbial system. In this paper, we first give an overview of NUFEB’s functionalities and implementation details. Then, we use NUFEB to model and simulate a biofilm system with fluid dynamics, and a large and complex biofilm system with multiple microbial functional groups and multiple nutrients.

Extracellular DNA and Type IV Pilus Expression Regulate the Structure and Kinetics of Biofilm Formation by Nontypeable Haemophilus influenzae

Biofilms formed in the middle ear by nontypeable Haemophilus influenzae (NTHI) are central to the chronicity, recurrence, and refractive nature of otitis media (OM). However, mechanisms that underlie the emergence of specific NTHI biofilm structures are unclear. We combined computational analysis tools and in silico modeling rooted in statistical physics with confocal imaging of NTHI biofilms formed in vitro during static culture in order to identify mechanisms that give rise to distinguishing morphological features. Our analysis of confocal images of biofilms formed by NTHI strain 86-028NP using pair correlations of local bacterial densities within sequential planes parallel to the substrate showed the presence of fractal structures of short length scales (≤10 μm). The in silico modeling revealed that extracellular DNA (eDNA) and type IV pilus (Tfp) expression played important roles in giving rise to the fractal structures and allowed us to predict a substantial reduction of these structures for an isogenic mutant (ΔcomE) that was significantly compromised in its ability to release eDNA into the biofilm matrix and had impaired Tfp function. This prediction was confirmed by analysis of confocal images of in vitro ΔcomE strain biofilms. The fractal structures potentially generate niches for NTHI survival in the hostile middle ear microenvironment by dramatically increasing the contact area of the biofilm with the surrounding environment, facilitating nutrient exchange, and by generating spatial positive feedback to quorum signaling.

NTHI is a major bacterial pathogen for OM, which is a common ear infection in children worldwide. Chronic OM is associated with bacterial biofilm formation in the middle ear; therefore, knowledge of the mechanisms that underlie NTHI biofilm formation is important for the development of therapeutic strategies for NTHI-associated OM. Our combined approach using confocal imaging of NTHI biofilms formed in vitro and mathematical tools for analysis of pairwise density correlations and agent-based modeling revealed that eDNA and Tfp expression were important factors in the development of fractal structures in NTHI biofilms. These structures may help NTHI survive in hostile environments, such as the middle ear. Our in silico model can be used in combination with laboratory or animal modeling studies to further define the mechanisms that underlie NTHI biofilm development during OM and thereby guide the rational design of, and optimize time and cost for, benchwork and preclinical studies.

Computational Systems Biology in Cancer: Modeling Methods and Applications

In recent years it has become clear that carcinogenesis is a complex process, both at the molecular and cellular levels. Understanding the origins, growth and spread of cancer, therefore requires an integrated or system-wide approach. Computational systems biology is an emerging sub-discipline in systems biology that utilizes the wealth of data from genomic, proteomic and metabolomic studies to build computer simulations of intra and intercellular processes. Several useful descriptive and predictive models of the origin, growth and spread of cancers have been developed in an effort to better understand the disease and potential therapeutic approaches. In this review we describe and assess the practical and theoretical underpinnings of commonly-used modeling approaches, including ordinary and partial differential equations, petri nets, cellular automata, agent based models and hybrid systems. A number of computer-based formalisms have been implemented to improve the accessibility of the various approaches to researchers whose primary interest lies outside of model development. We discuss several of these and describe how they have led to novel insights into tumor genesis, growth, apoptosis, vascularization and therapy.

Towards an integrated understanding of how micro scale processes shape groundwater ecosystem functions

Micro scale processes are expected to have a fundamental role in shaping groundwater ecosystems and yet they remain poorly understood and under-researched. In part, this is due to the fact that sampling is rarely carried out at the scale at which microorganisms, and their grazers and predators, function and thus we lack essential information. While set within a larger scale framework in terms of geochemical features, supply with energy and nutrients, and exchange intensity and dynamics, the micro scale adds variability, by providing heterogeneous zones at the micro scale which enable a wider range of redox reactions. Here we outline how understanding micro scale processes better may lead to improved appreciation of the range of ecosystems functions taking place at all scales. Such processes are relied upon in bioremediation and we demonstrate that ecosystem modelling as well as engineering measures have to take into account, and use, understanding at the micro scale. We discuss the importance of integrating faunal processes and computational appraisals in research, in order to continue to secure sustainable water resources from groundwater.

A mechanistic Individual-based Model of microbial communities

Accurate predictive modelling of the growth of microbial communities requires the credible representation of the interactions of biological, chemical and mechanical processes. However, although biological and chemical processes are represented in a number of Individual-based Models (IbMs) the interaction of growth and mechanics is limited. Conversely, there are mechanically sophisticated IbMs with only elementary biology and chemistry. This study focuses on addressing these limitations by developing a flexible IbM that can robustly combine the biological, chemical and physical processes that dictate the emergent properties of a wide range of bacterial communities. This IbM is developed by creating a microbiological adaptation of the open source Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). This innovation should provide the basis for “bottom up” prediction of the emergent behaviour of entire microbial systems. In the model presented here, bacterial growth, division, decay, mechanical contact among bacterial cells, and adhesion between the bacteria and extracellular polymeric substances are incorporated. In addition, fluid-bacteria interaction is implemented to simulate biofilm deformation and erosion. The model predicts that the surface morphology of biofilms becomes smoother with increased nutrient concentration, which agrees well with previous literature. In addition, the results show that increased shear rate results in smoother and more compact biofilms. The model can also predict shear rate dependent biofilm deformation, erosion, streamer formation and breakup.

Continuum and discrete approach in modeling biofilm development and structure: a review

The scientific community has recognized that almost 99% of the microbial life on earth is represented by biofilms. Considering the impacts of their sessile lifestyle on both natural and human activities, extensive experimental activity has been carried out to understand how biofilms grow and interact with the environment. Many mathematical models have also been developed to simulate and elucidate the main processes characterizing the biofilm growth. Two main mathematical approaches for biomass representation can be distinguished: continuum and discrete. This review is aimed at exploring the main characteristics of each approach. Continuum models can simulate the biofilm processes in a quantitative and deterministic way. However, they require a multidimensional formulation to take into account the biofilm spatial heterogeneity, which makes the models quite complicated, requiring significant computational effort. Discrete models are more recent and can represent the typical multidimensional structural heterogeneity of biofilm reflecting the experimental expectations, but they generate computational results including elements of randomness and introduce stochastic effects into the solutions.

Reducing discrepancies between 3D and 2D simulations due to cell packing density

Modelling all three spatial dimensions is often much more computationally expensive than modelling a two-dimensional simplification of the same system. Researchers comparing these approaches in individual-based models of microbial biofilms report quantitative, but not qualitative, differences between 2D and 3D simulations. We show that a large part of the discrepancy is due to the different space packing densities of circles versus spheres, and demonstrate methods to compensate for this: the internal density of individuals or the distances between them can be scaled. This result is likely to be useful in similar models, such as smoothed particle hydrodynamics.

Influence of Nutrient Availability and Quorum Sensing on the Formation of Metabolically Inactive Microcolonies Within Structurally Heterogeneous Bacterial Biofilms: An Individual-Based 3D Cellular Automata Model

The resistance of bacterial biofilms to antibiotic treatment has been attributed to the emergence of structurally heterogeneous microenvironments containing metabolically inactive cell populations. In this study, we use a three-dimensional individual-based cellular automata model to investigate the influence of nutrient availability and quorum sensing on microbial heterogeneity in growing biofilms. Mature biofilms exhibited at least three structurally distinct strata: a high-volume, homogeneous region sandwiched between two compact sections of high heterogeneity. Cell death occurred preferentially in layers in close proximity to the substratum, resulting in increased heterogeneity in this section of the biofilm; the thickness and heterogeneity of this lowermost layer increased with time, ultimately leading to sloughing. The model predicted the formation of metabolically dormant cellular microniches embedded within faster-growing cell clusters. Biofilms utilizing quorum sensing were more heterogeneous compared to their non-quorum sensing counterparts, and resisted sloughing, featuring a cell-devoid layer of EPS atop the substratum upon which the remainder of the biofilm developed. Overall, our study provides a computational framework to analyze metabolic diversity and heterogeneity of biofilm-associated microorganisms and may pave the way toward gaining further insights into the biophysical mechanisms of antibiotic resistance.

A mechanistic Individual-based Model of microbial communities

Accurate predictive modelling of the growth of microbial communities requires the credible representation of the interactions of biological, chemical and mechanical processes. However, although biological and chemical processes are represented in a number of Individual-based Models (IbMs) the interaction of growth and mechanics is limited. Conversely, there are mechanically sophisticated IbMs with only elementary biology and chemistry. This study focuses on addressing these limitations by developing a flexible IbM that can robustly combine the biological, chemical and physical processes that dictate the emergent properties of a wide range of bacterial communities. This IbM is developed by creating a microbiological adaptation of the open source Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). This innovation should provide the basis for "bottom up" prediction of the emergent behaviour of entire microbial systems. In the model presented here, bacterial growth, division, decay, mechanical contact among bacterial cells, and adhesion between the bacteria and extracellular polymeric substances are incorporated. In addition, fluid-bacteria interaction is implemented to simulate biofilm deformation and erosion. The model predicts that the surface morphology of biofilms becomes smoother with increased nutrient concentration, which agrees well with previous literature. In addition, the results show that increased shear rate results in smoother and more compact biofilms. The model can also predict shear rate dependent biofilm deformation, erosion, streamer formation and breakup.

A Mixed-Culture Biofilm Model with Cross-Diffusion

We propose a deterministic continuum model for mixed-culture biofilms. A crucial aspect is that movement of one species is affected by the presence of the other. This leads to a degenerate cross-diffusion system that generalizes an earlier single-species biofilm model. Two derivations of this new model are given. One, like cellular automata biofilm models, starts from a discrete in space lattice differential equation where the spatial interaction is described by microscopic rules. The other one starts from the same continuous mass balances that are the basis of other deterministic biofilm models, but it gives up a simplifying assumption of these models that has recently been criticized as being too restrictive in terms of ecological structure. We show that both model derivations lead to the same PDE model, if corresponding closure assumptions are introduced. To investigate the role of cross-diffusion, we conduct numerical simulations of three biofilm systems: competition, allelopathy and a mixed system formed by an aerobic and an anaerobic species. In all cases, we find that accounting for cross-diffusion affects local distribution of biomass, but it does not affect overall lumped quantities such as the total amount of biomass in the system.

Branching instability in expanding bacterial colonies

Self-organization in developing living organisms relies on the capability of cells to duplicate and perform a collective motion inside the surrounding environment. Chemical and mechanical interactions coordinate such a cooperative behaviour, driving the dynamical evolution of the macroscopic system. In this work, we perform an analytical and computational analysis to study pattern formation during the spreading of an initially circular bacterial colony on a Petri dish. The continuous mathematical model addresses the growth and the chemotactic migration of the living monolayer, together with the diffusion and consumption of nutrients in the agar. The governing equations contain four dimensionless parameters, accounting for the interplay among the chemotactic response, the bacteria-substrate interaction and the experimental geometry. The spreading colony is found to be always linearly unstable to perturbations of the interface, whereas branching instability arises in finite-element numerical simulations. The typical length scales of such fingers, which align in the radial direction and later undergo further branching, are controlled by the size parameters of the problem, whereas the emergence of branching is favoured if the diffusion is dominant on the chemotaxis. The model is able to predict the experimental morphologies, confirming that compact (resp. branched) patterns arise for fast (resp. slow) expanding colonies. Such results, while providing new insights into pattern selection in bacterial colonies, may finally have important applications for designing controlled patterns.

Mathematical modeling of dormant cell formation in growing biofilm

Understanding the dynamics of dormant cells in microbial biofilms, in which the bacteria are embedded in extracellular matrix, is important for developing successful antibiotic therapies against pathogenic bacteria. Although some of the molecular mechanisms leading to bacterial persistence have been speculated in planktonic bacterial cell, how dormant cells emerge in the biofilms of pathogenic bacteria such as Pseudomonas aeruginosa remains unclear. The present study proposes four hypotheses of dormant cell formation; stochastic process, nutrient-dependent, oxygen-dependent, and time-dependent processes. These hypotheses were implemented into a three-dimensional individual-based model of biofilm formation. Numerical simulations of the different mechanisms yielded qualitatively different spatiotemporal distributions of dormant cells in the growing biofilm. Based on these simulation results, we discuss what kinds of experimental studies are effective for discriminating dormant cell formation mechanisms in biofilms.

Individual based modeling of Pseudomonas aeruginosa biofilm with three detachment mechanisms

Individual based simulation approach has attracted more and more interest in biofilm simulation.

Cellular automata-based modelling and simulation of biofilm structure on multi-core computers

The article presents a mathematical model of biofilm growth for aerobic biodegradation of a toxic carbonaceous substrate. Modelling of biofilm growth has fundamental significance in numerous processes of biotechnology and mathematical modelling of bioreactors. The process following double-substrate kinetics with substrate inhibition proceeding in a biofilm has not been modelled so far by means of cellular automata. Each process in the model proposed, i.e. diffusion of substrates, uptake of substrates, growth and decay of microorganisms and biofilm detachment, is simulated in a discrete manner. It was shown that for flat biofilm of constant thickness, the results of the presented model agree with those of a continuous model. The primary outcome of the study was to propose a mathematical model of biofilm growth; however a considerable amount of focus was also placed on the development of efficient algorithms for its solution. Two parallel algorithms were created, differing in the way computations are distributed. Computer programs were created using OpenMP Application Programming Interface for C++ programming language. Simulations of biofilm growth were performed on three high-performance computers. Speed-up coefficients of computer programs were compared. Both algorithms enabled a significant reduction of computation time. It is important, inter alia, in modelling and simulation of bioreactor dynamics.

A model of filamentous cyanobacteria leading to reticulate pattern formation

The filamentous cyanobacterium, Pseudanabaena, has been shown to produce reticulate patterns that are thought to be the result of its gliding motility. Similar fossilized structures found in the geological record constitute some of the earliest signs of life on Earth. It is difficult to tie these fossils, which are billions of years old, directly to the specific microorganisms that built them. Identifying the physicochemical conditions and microorganism properties that lead microbial mats to form macroscopic structures can lead to a better understanding of the conditions on Earth at the dawn of life. In this article, a cell-based model is used to simulate the formation of reticulate patterns in cultures of Pseudanabaena. A minimal system of long and flexible trichomes capable of gliding motility is shown to be sufficient to produce stable patterns consisting of a network of streams. Varying model parameters indicate that systems with little to no cohesion, high trichome density and persistent movement are conducive to reticulate pattern formation, in conformance with experimental observations.

Modeling of biofilm systems: a review

The modeling of biochemical processes in biofilms is more complex compared to those in suspended biomass due to the existence of substrate gradients. The diffusion and reaction of substrates within the biofilms were simulated in 1D models in the 1970s. The quality of these simulation results was later improved by consideration of mass transfer at the bulk/biofilm interface and detachment of biomass from the surface. Furthermore, modeling of species distribution along the axis perpendicular to the substratum helped to simulate productivity and long-term behavior in multispecies biofilms. Multidimensional models that were able to give a realistic prediction of the surface structure of biofilms were published in the 1990s. The 2D or 3D representation of the distribution of the species in a matrix of extracellular polymeric substances (EPS) helped predict the behavior of multispecies biofilm systems. The influence of shear forces on such 2D or 3D biofilm structures was included by solving the Navier-Stokes equation for the liquid phase above the biofilm. More recently, the interaction between the fluid and biofilm structures was addressed more seriously by no longer considering the biofilm structures as being rigid. The latter approach opened a new door, enabling one to describe biofilms as viscoelastic systems that are not only able to grow and produce but also be deformed or even dislodged if external forces are applied.

Role of intracellular stochasticity in biofilm growth. Insights from population balance modeling

There is increasing recognition that stochasticity involved in gene regulatory processes may help cells enhance the signal or synchronize expression for a group of genes. Thus the validity of the traditional deterministic approach to modeling the foregoing processes cannot be without exception. In this study, we identify a frequently encountered situation, i.e., the biofilm, which has in the past been persistently investigated with intracellular deterministic models in the literature. We show in this paper circumstances in which use of the intracellular deterministic model appears distinctly inappropriate. In Enterococcus faecalis, the horizontal gene transfer of plasmid spreads drug resistance. The induction of conjugation in planktonic and biofilm circumstances is examined here with stochastic as well as deterministic models. The stochastic model is formulated with the Chemical Master Equation (CME) for planktonic cells and Reaction-Diffusion Master Equation (RDME) for biofilm. The results show that although the deterministic model works well for the perfectly-mixed planktonic circumstance, it fails to predict the averaged behavior in the biofilm, a behavior that has come to be known as stochastic focusing. A notable finding from this work is that the interception of antagonistic feedback loops to signaling, accentuates stochastic focusing. Moreover, interestingly, increasing particle number of a control variable could lead to an even larger deviation. Intracellular stochasticity plays an important role in biofilm and we surmise by implications from the model, that cell populations may use it to minimize the influence from environmental fluctuation.

An improved cellular automaton method to model multispecies biofilms

Biomass-spreading rules used in previous cellular automaton methods to simulate multispecies biofilm introduced extensive mixing between different biomass species or resulted in spatially discontinuous biomass concentration and distribution; this caused results based on the cellular automaton methods to deviate from experimental results and those from the more computationally intensive continuous method. To overcome the problems, we propose new biomass-spreading rules in this work: Excess biomass spreads by pushing a line of grid cells that are on the shortest path from the source grid cell to the destination grid cell, and the fractions of different biomass species in the grid cells on the path change due to the spreading. To evaluate the new rules, three two-dimensional simulation examples are used to compare the biomass distribution computed using the continuous method and three cellular automaton methods, one based on the new rules and the other two based on rules presented in two previous studies. The relationship between the biomass species is syntrophic in one example and competitive in the other two examples. Simulation results generated using the cellular automaton method based on the new rules agree much better with the continuous method than do results using the other two cellular automaton methods. The new biomass-spreading rules are no more complex to implement than the existing rules.

An efficient nonlinear finite-difference approach in the computational modeling of the dynamics of a nonlinear diffusion-reaction equation in microbial ecology

In this manuscript, we present a computational model to approximate the solutions of a partial differential equation which describes the growth dynamics of microbial films. The numerical technique reported in this work is an explicit, nonlinear finite-difference methodology which is computationally implemented using Newton's method. Our scheme is compared numerically against an implicit, linear finite-difference discretization of the same partial differential equation, whose computer coding requires an implementation of the stabilized bi-conjugate gradient method. Our numerical results evince that the nonlinear approach results in a more efficient approximation to the solutions of the biofilm model considered, and demands less computer memory. Moreover, the positivity of initial profiles is preserved in the practice by the nonlinear scheme proposed.

On the mechanics of bacterial biofilms on non-dissolvable surgical sutures: a laser scanning confocal microscopy-based finite element study

Biofilms are bacterial communities encapsulated within a self-secreted extracellular polymeric substance and are responsible for a wide range of chronic medical device related infections. Understanding and addressing the conditions that lead to the attachment and detachment of biofilms from biomedical surfaces (orthopaedic implants, sutures, intravenous catheters, cardio-vascular stents) has the potential to identify areas of the device that might be more prone to infection and predict how and when biofilms might dislodge. In this study, an integrated software methodology was devised to create image-based microscopic finite element models of real biofilm colonies of Staphylococcus aureus attached to a fragment of surgical suture. The goal was to predict how deformation of the suture may lead to the potential detachment of biofilm colonies by solving the equations of continuum mechanics using the finite element method for various loading cases. Tension, torsion and bending of the biomaterial structure were simulated, demonstrating that small strains in the suture can produce surface shear stresses sufficient to trigger the sliding of biofilms over the suture surface. Applications of this technique to other medical devices are discussed.

Biofilm development and the dynamics of preferential flow paths in porous media

A two-dimensional pore-scale numerical model was developed to evaluate the dynamics of preferential flow paths in porous media caused by bioclogging. The liquid flow and solute transport through the pore network were coupled with a biofilm model including biomass attachment, growth, decay, lysis, and detachment. Blocking of all but one flow path was obtained under constant liquid inlet flow rate and biomass detachment caused by shear forces only. The stable flow path formed when biofilm detachment balances growth, even with biomass weakened by decay. However, shear forces combined with biomass lysis upon starvation could produce an intermittently shifting location of flow channels. Dynamic flow pathways may also occur when combined liquid shear and pressure forces act on the biofilm. In spite of repeated clogging and unclogging of interconnected pore spaces, the average permeability reached a quasi-constant value. Oscillations in the medium permeability were more pronounced for weaker biofilms.

The formation and functional consequences of heterogeneous mitochondrial distributions in skeletal muscle

Diffusion plays a prominent role in governing both rates of aerobic metabolic fluxes and mitochondrial organization in muscle fibers. However, there is no mechanism to explain how the non-homogeneous mitochondrial distributions that are prevalent in skeletal muscle arise. We propose that spatially variable degradation with dependence on O(2) concentration, and spatially uniform signals for biogenesis, can account for observed distributions of mitochondria in a diversity of skeletal muscle. We used light and transmission electron microscopy and stereology to examine fiber size, capillarity and mitochondrial distribution in fish red and white muscle, fish white muscle that undergoes extreme hypertrophic growth, and four fiber types in mouse muscle. The observed distributions were compared with those generated using a coupled reaction-diffusion/cellular automata (CA) mathematical model of mitochondrial function. Reaction-diffusion analysis of metabolites such as oxygen, ATP, ADP and PCr involved in energy metabolism and mitochondrial function were considered. Coupled to the reaction-diffusion approach was a CA approach governing mitochondrial life cycles in response to the metabolic state of the fiber. The model results were consistent with the experimental observations and showed higher mitochondrial densities near the capillaries because of the sometimes steep gradients in oxygen. The present study found that selective removal of mitochondria in the presence of low prevailing local oxygen concentrations is likely the primary factor dictating the spatial heterogeneity of mitochondria in a diversity of fibers. The model results also suggest decreased diffusional constraints corresponding to the heterogeneous mitochondrial distribution assessed using the effectiveness factor, defined as the ratio of the reaction rate in the system with finite rates of diffusion to that in the absence of any diffusion limitation. Thus, the non-uniform distribution benefits the muscle fiber by increasing the energy status and increasing sustainable metabolic rates.

Multicomponent hydrodynamic model for heterogeneous biofilms: two-dimensional numerical simulations of growth and interaction with flows

We develop a tricomponent (ternary) hydrodynamic model for multiphase flows of biomass and solvent mixtures, which we employ to simulate biofilm. In this model, the three predominant effective components in biofilms, which are the extracellular polymeric substance (EPS) network, the bacteria, and the effective solvent (consisting of the solvent and nutrient, etc.), are modeled explicitly. The tricomponent fluid mixture is assumed incompressible as a whole, while intercomponent mixing, dissipation, and conversion are allowed among the effective components. Bacterial growth and EPS production due to the growing bacterial population are modeled in the biomass transport equations. Bacterial decay due to starvation and natural causes is accounted for in the bacterial population dynamics to capture the possible bacterial population reduction due to the depletion of the nutrient. In the growth regime for biofilms, the mixture behaves like a multiphase viscous fluid, in which the molecular relaxation is negligible in the corresponding time scale. In this regime, the dynamics of biofilm growth in the solvent (water) are simulated using a two-dimensional finite difference solver that we developed, in which the distribution and evolution of the EPS and bacterial volume fractions are investigated. The hydrodynamic interaction between the biomass and the solvent flow field is also simulated in a shear cell environment, demonstrating the spatially and temporally heterogeneous distribution of the EPS and bacteria under shear. This model together with the numerical codes developed provides a predictive tool for studying biomass-flow interaction and other important biochemical interactions in the biofilm and solvent fluid mixture.