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

Electron donors supporting growth and electroactivity of Geobacter sulfurreducens anode biofilms

Geobacter bacteria efficiently oxidize acetate into electricity in bioelectrochemical systems, yet the range of fermentation products that support the growth of anode biofilms and electricity production has not been thoroughly investigated. Here, we show that Geobacter sulfurreducens oxidized formate and lactate with electrodes and Fe(III) as terminal electron acceptors, though with reduced efficiency compared to acetate. The structure of the formate and lactate biofilms increased in roughness, and the substratum coverage decreased, to alleviate the metabolic constraints derived from the assimilation of carbon from the substrates. Low levels of acetate promoted formate carbon assimilation and biofilm growth and increased the system's performance to levels comparable to those with acetate only. Lactate carbon assimilation also limited biofilm growth and led to the partial oxidization of lactate to acetate. However, lactate was fully oxidized in the presence of fumarate, which redirected carbon fluxes into the tricarboxylic acid (TCA) cycle, and by acetate-grown biofilms. These results expand the known ranges of electron donors for Geobacter-driven fuel cells and identify microbial constraints that can be targeted to develop better-performing strains and increase the performance of bioelectrochemical systems.

Dissipative-particle-dynamics model of biofilm growth

A dissipative-particle-dynamics model for the quantitative simulation of biofilm growth controlled by substrate (nutrient) consumption, advective and diffusive substrate transport, and hydrodynamic interactions with fluid flow (including fragmentation and reattachment) is described. The model was used to simulate biomass growth, decay, and spreading. It predicts how the biofilm morphology depends on flow conditions, biofilm growth kinetics, the rheomechanical properties of the biofilm, and adhesion to solid surfaces. The morphology of the model biofilm depends strongly on its rigidity and the magnitude of the body force that drives the fluid over the biofilm.

A mathematical model of quorum sensing regulated EPS production in biofilm communities

Background

Biofilms are microbial communities encased in a layer of extracellular polymeric substances (EPS). The EPS matrix provides several functional purposes for the biofilm, such as protecting bacteria from environmental stresses, and providing mechanical stability. Quorum sensing is a cell-cell communication mechanism used by several bacterial taxa to coordinate gene expression and behaviour in groups, based on population densities.

Model

We mathematically model quorum sensing and EPS production in a growing biofilm under various environmental conditions, to study how a developing biofilm impacts quorum sensing, and conversely, how a biofilm is affected by quorum sensing-regulated EPS production. We investigate circumstances when using quorum-sensing regulated EPS production is a beneficial strategy for biofilm cells.

Results

We find that biofilms that use quorum sensing to induce increased EPS production do not obtain the high cell populations of low-EPS producers, but can rapidly increase their volume to parallel high-EPS producers. Quorum sensing-induced EPS production allows a biofilm to switch behaviours, from a colonization mode (with an optimized growth rate), to a protection mode.

Conclusions

A biofilm will benefit from using quorum sensing-induced EPS production if bacteria cells have the objective of acquiring a thick, protective layer of EPS, or if they wish to clog their environment with biomass as a means of securing nutrient supply and outcompeting other colonies in the channel, of their own or a different species.

Influence of reaction and diffusion on spatial organization of mitochondria and effectiveness factors in skeletal muscle cell design

A mathematical model is developed to analyze the influence of chemical reaction and diffusion processes on the intracellular organization of mitochondria in skeletal muscle cells. The mathematical modeling approach uses a reaction-diffusion analysis of oxygen, ATP, and ADP involved in energy metabolism and mitochondrial function as governed by oxygen supply, volume fraction of mitochondria, and rates of reaction. Superimposed upon and coupled to the continuum species material balances is a cellular automata (CA) approach governing mitochondrial life cycles in response to the metabolic state of the cell. The effectiveness factor (η), defined as the ratio of reaction rate in the system with finite rates of diffusion to those in the absence of any diffusion limitation is used to assess diffusional constraints in muscle cells. The model shows the dramatic effects that the governing parameters have on the mitochondrial cycle of life and death and how these effects lead to changes in the distribution patterns of mitochondria observed experimentally. The model results showed good agreement with experimental results on mitochondrial distributions in mammalian muscle fibers. The η increases as the mitochondrial population is redistributed toward the fiber periphery in response to a decreased availability of oxygen. Modification of the CA parameters so that the mitochondrial lifecycle is more sensitive to the oxygen concentration caused larger mitochondrial shifts to the edge of the cell with smaller changes in oxygen concentration, and thus also lead to increased values of η. The present study shows that variation in oxygen supply, muscle activity and mitochondrial ATP supply influence the η and are the important parameters that can cause diffusion limitations. In order to prevent diffusion constraints, the cell resorts to shifts in their mitochondrial population towards the cell periphery, thus increasing η.

A modeling and simulation study of siderophore mediated antagonism in dual-species biofilms

BACKGROUND

Several bacterial species possess chelation mechanisms that allow them to scavenge iron from the environment under conditions of limitation. To this end they produce siderophores that bind the iron and make it available to the cells later on, while rendering it unavailable to other organisms. The phenomenon of siderophore mediated antagonism has been studied to some extent for suspended populations where it was found that the chelation ability provides a growth advantage over species that do not have this possibility. However, most bacteria live in biofilm communities. In particular Pseudomonas fluorescens and Pseudomonas putida, the species that have been used in most experimental studies of the phenomenon, are known to be prolific biofilm formers, but only very few experimental studies of iron chelation have been published to date for the biofilm setting. We address this question in the present study.

METHODS

Based on a previously introduced model of iron chelation and an existing model of biofilm growth we formulate a model for iron chelation and competition in dual species biofilms. This leads to a highly nonlinear system of partial differential equations which is studied in computer simulation experiments.

CONCLUSIONS

(i) Siderophore production can give a growth advantage also in the biofilm setting, (ii) diffusion facilitates and emphasizes this growth advantage, (iii) the magnitude of the growth advantage can also depend on the initial inoculation of the substratum, (iv) a new mass transfer boundary condition was derived that allows to a priori control the expect the expected average thickness of the biofilm in terms of the model parameters.

Microbial metropolis

Microorganisms can form tightly knit communities such as biofilms. Many others include marine snow, anaerobic digester granules, the ginger beer plant and bacterial colonies. This chapter is devoted to a survey of the main properties of these communities, with an emphasis on biofilms. We start with attachment to surfaces and the nature of adhesion. The growing community then forms within a matrix, generally of organic macromolecules. Inevitably the environment within such a matrix is different from that outside. Organisms respond by forming crowd-detection and response units; these quorum sensing systems act as switches between planktonic life and the dramatically altered conditions found inside microbial aggregates. The community then matures and changes and may even fail and disappear. Antimicrobial resistance is discussed as an example of multicellular behavior. The multicellular lifestyle has been modeled mathematically and responded to powerful molecular biological techniques. Latterly, microbial systems have been used as models for fundamental evolutionary processes, mostly because of their high rates of reproduction and the ease of genetic manipulation. The life of most microbes is a duality between the yin of the community and the yang of planktonic existence. Sadly far less research has been devoted to adaptation to free-living forms than in the opposite direction.

Front instabilities and invasiveness of simulated avascular tumors

We study the interface morphology of a 2D simulation of an avascular tumor composed of identical cells growing in an homogeneous healthy tissue matrix (TM), in order to understand the origin of the morphological changes often observed during real tumor growth. We use the Glazier-Graner-Hogeweg model, which treats tumor cells as extended, deformable objects, to study the effects of two parameters: a dimensionless diffusion-limitation parameter defined as the ratio of the tumor consumption rate to the substrate transport rate, and the tumor-TM surface tension. We model TM as a nondiffusing field, neglecting the TM pressure and haptotactic repulsion acting on a real growing tumor; thus, our model is appropriate for studying tumors with highly motile cells, e.g., gliomas. We show that the diffusion-limitation parameter determines whether the growing tumor develops a smooth (noninvasive) or fingered (invasive) interface, and that the sensitivity of tumor morphology to tumor-TM surface tension increases with the size of the dimensionless diffusion-limitation parameter. For large diffusion-limitation parameters, we find a transition (missed in previous work) between dendritic structures, produced when tumor-TM surface tension is high, and seaweed-like structures, produced when tumor-TM surface tension is low. This observation leads to a direct analogy between the mathematics and dynamics of tumors and those observed in nonbiological directional solidification. Our results are also consistent with the biological observation that hypoxia promotes invasive growth of tumor cells by inducing higher levels of receptors for scatter factors that weaken cell-cell adhesion and increase cell motility. These findings suggest that tumor morphology may have value in predicting the efficiency of antiangiogenic therapy in individual patients.

Conditions for partial nitrification in biofilm reactors and a kinetic explanation

Nitrification is a two-step process in which ammonia is incompletely oxidized by ammonia-oxidizing bacteria or archaea (AOB) to nitrite, which is then further oxidized to nitrate by nitrite-oxidizing bacteria (NOB). Literature reports show that segregation of initially coexisting ammonia and nitrite oxidizing populations co-immobilized in gel cubes and cultured in a set-up with three reactors in series (without recirculation) is attained. In those studies NOB were present and nitrite was oxidized mainly in the last reactor. We developed a mathematical model for immobilized biomass that allows for one-dimensional gradients of metabolites and changes of porosity within the gel due to growth. The model reproduced the experimentally observed compartmentalization under the conditions used by Noto et al. (Noto et al., 1998. Water Res 32(3): 769- 773), using standard kinetic parameters of nitrifying bacteria including free ammonia inhibition of AOB and NOB. The model predicted compartmentalization when the ammonium load was sufficiently high and liquid phase mixing sufficiently limited (close to plug-flow). Modeling results demonstrated that inhibition of NOB by free ammonia did not substantially contribute to the compartmentalization in biofilm reactors. Additional simulations identified the higher oxygen affinity of AOB as the key parameter leading to compartmentalization (i.e., partial nitrification) in artificial and natural biofilms since they enable the formation of oxygen gradients. As a result, a tendency for compartmentalization was found even at equal competitiveness.

Individual-based modelling: an essential tool for microbiology

Micro-organisms play a central role in every ecosystem and in the global biomass cycle. They are strongly involved in many fields of human interest, from medicine to the food industry and waste control. Nevertheless, most micro-organisms remain almost unknown, and nearly 99% of them have not yet been successfully cultured in vitro. Therefore, new approaches and new tools must be developed in order to understand the collective behaviour of microbial communities in any natural or artificial setting. In particular, theoretical and practical methodologies to deal with such systems at a mesoscopic level of description (covering the range from 100 to 10(8) cells) are required. Individual-based modelling (IBM) has become a widely used tool for describing complex systems made up of autonomous entities, such as ecosystems and social networks. Individual-based models (IBMs) provide some advantages over the traditional whole-population models: (a) they are bottom-up approaches, so they describe the behaviour of a system as a whole by establishing procedural rules for the individuals and for their interactions, and thus allow more realistic assumptions for the model of the individuals than population models do; (b) they permit the introduction of randomness and individual variability, so they can reproduce the diversity found in real systems; and (c) they can account for individual adaptive behaviour to their environmental conditions, so the evolution of the whole system arises from the dynamics that govern individuals in their pursuit of optimal fitness. However, they also present some drawbacks: they lack the clarity of continuous models and may easily become rambling, which makes them difficult to analyse and communicate. All in all, IBMs supply a holistic description of microbial systems and their emerging properties. They are specifically appropriate to deal with microbial communities in non-steady states, and spatially explicit IBMs are particularly appropriate to study laboratory and natural microbiological systems with spatial heterogeneity. In this paper, we review IBM methodology applied to microbiology. We also present some results obtained from the application of Individual Discrete Simulations, an IBM of ours, to the study of bacterial communities, yeast cultures and Plasmodium falciparum-infected erythrocytes in vitro cultures of Plasmodium falciparum-infected erythrocytes.

Analysis and IbM simulation of the stages in bacterial lag phase: basis for an updated definition

The lag phase is the initial phase of a culture that precedes exponential growth and occurs when the conditions of the culture medium differ from the pre-inoculation conditions. It is usually defined by means of cell density because the number of individuals remains approximately constant or slowly increases, and it is quantified with the lag parameter lambda. The lag phase has been studied through mathematical modelling and by means of specific experiments. In recent years, Individual-based Modelling (IbM) has provided helpful insights into lag phase studies. In this paper, the definition of lag phase is thoroughly examined. Evolution of the total biomass and the total number of bacteria during lag phase is tackled separately. The lag phase lasts until the culture reaches a maximum growth rate both in biomass and cell density. Once in the exponential phase, both rates are constant over time and equal to each other. Both evolutions are split into an initial phase and a transition phase, according to their growth rates. A population-level mathematical model is presented to describe the transitional phase in cell density. INDividual DIScrete SIMulation (INDISIM) is used to check the outcomes of this analysis. Simulations allow the separate study of the evolution of cell density and total biomass in a batch culture, they provide a depiction of different observed cases in lag evolution at the individual-cell level, and are used to test the population-level model. The results show that the geometrical lag parameter lambda is not appropriate as a universal definition for the lag phase. Moreover, the lag phase cannot be characterized by a single parameter. For the studied cases, the lag phases of both the total biomass and the population are required to fully characterize the evolution of bacterial cultures. The results presented prove once more that the lag phase is a complex process that requires a more complete definition. This will be possible only after the phenomena governing the population dynamics at an individual level of description, and occurring during the lag and exponential growth phases, are well understood.

A three-dimensional computer model analysis of three hypothetical biofilm detachment mechanisms

Three hypothetical mechanisms of detachment were incorporated into a three-dimensional computer model of biofilm development. The model integrated processes of substrate utilization, substrate diffusion, growth, cell advection, and detachment in a cellular automata framework. The purpose of this investigation was to characterize each of the mechanisms with respect to four criteria: the resulting biofilm structure, the existence of a steady state, the propensity for sloughing events, and the dynamics during starvation. The three detachment mechanisms analyzed represented various physical and biological influences hypothesized to affect biofilm detachment. The first invoked the concept of fluid shear removing biomass that protrudes far above the surface and is therefore subjected to relatively large drag forces. The second pathway linked detachment to changes in the local availability of a nutrient. The third pathway simulated an erosive process in which individual cells are lost from the surface of a biofilm cell cluster. The detachment mechanisms demonstrated diverse behaviors with respect to the four analysis criteria. The height-dependant mechanism produced flat, steady state biofilms that lacked sloughing events. Detachment based on substrate limitation produced significant sloughing events. The resulting biofilm structures included distinct, hollow clusters separated by channels. The erosion mechanism produced neither a non-zero steady state nor sloughing events. A mechanism combining all three-detachment mechanisms produced mushroom-like structures. The dynamics of biofilm decay during starvation were distinct for each detachment mechanism. These results show that detachment is a critical determinant of biofilm structure and of the dynamics of biofilm accumulation and loss.

A three-dimensional computer model of four hypothetical mechanisms protecting biofilms from antimicrobials

Four hypothetical mechanisms for protection of biofilms against antimicrobials were incorporated into a three-dimensional model of biofilm growth and development. The model integrated processes of substrate utilization, diffusion, growth, cell migration, death, and detachment in a cellular automaton framework. Compared to simulations of unprotected biofilms, each of the protective mechanisms provided some tolerance to antimicrobial action. When the mechanisms were compared to each other, the behaviors of the four protective mechanisms produced distinct shapes of killing curves, nonuniform spatial patterns of survival and cell type distribution, and anticipated susceptibility patterns for dispersed biofilm cells. The differences between the protective mechanisms predicted in these simulations could guide the design of experiments to discriminate antimicrobial tolerance mechanisms in biofilms. Each of the mechanisms could be a plausible avenue of biofilm protection.

Architecture of a nascent Sphingomonas sp. biofilm under varied hydrodynamic conditions

The architecture of a Sphingomonas biofilm was studied during early phases of its formation, using strain L138, a gfp-tagged derivative of Sphingomonas sp. strain LB126, as a model organism and flow cells and confocal laser scanning microscopy as experimental tools. Spatial and temporal distribution of cells and exopolymer secretions (EPS) within the biofilm, development of microcolonies under flow conditions representing varied Reynolds numbers, and changes in diffusion length with reference to EPS production were studied by sequential sacrificing of biofilms grown in multichannel flow cells and by time-lapse confocal imaging. The area of biofilm in terms of microscopic images required to ensure representative sampling varied by an order of magnitude when area of cell coverage (2 x 10(5) microm(2)) or microcolony size (1 x 10(6) microm(2)) was the biofilm parameter under investigation. Hence, it is necessary to establish the inherent variability of any biofilm metric one is attempting to quantify. Sphingomonas sp. strain L138 biofilm architecture consisted of microcolonies and extensive water channels. Biomass and EPS distribution were maximal at 8 to 9 mum above the substratum, with a high void fraction near the substratum. Time-lapse confocal imaging and digital image analysis showed that growth of the microcolonies was not uniform: adjacently located colonies registered significant growth or no growth at all. Microcolonies in the biofilm had the ability to move across the attachment surface as a unit, irrespective of fluid flow direction, indicating that movement of microcolonies is an inherent property of the biofilm. Width of water channels decreased as EPS production increased, resulting in increased diffusion distances in the biofilm. Changing hydrodynamic conditions (Reynolds numbers of 0.07, 52, and 87) had no discernible influence on the characteristics of microcolonies (size, shape, or orientation with respect to flow) during the first 24 h of biofilm development. Inherent factors appear to have overriding influence, vis-a-vis environmental factors, on early stages of microcolony development under these laminar flow conditions.

Porosity and tortuosity relations as revealed by a mathematical model of biofilm structure

A biofilm is assumed to be submerged in a fluid with given viscosity and low Reynolds number. The interaction between fluid and bacteria is modeled through streamlines. We use finite-difference and boundary element numerical schemes to predict streamlines within the biofilm. The results show that this approach can provide information about prior distribution and geometry of the biofilm structure. Theoretical values of porosity and tortuosity are computed and compared with published data.

Detachment and diffusive-convective transport in an evolving heterogeneous two-dimensional biofilm hybrid model

Under the hypothesis of correlation between biofilm survival and nutrient availability, by considering fluid drag forces and mortality due to nutrient depletion, a biofilm detachment/breaking condition is derived. The mechanisms leading to biofilm detachment/breaking are discussed. We construct and describe a hybrid model for a heterogeneous biofilm attached to walls in a channel where liquid is flowing. The model is called hybrid because it couples conservation equations with a cellular automaton. The biofilm layer is viewed as a porous medium with variable porosity, tortuosity, and permeability. The model is solved using asymptotic and finite differences methods. Results for porosity, nutrient distribution, and average surface location are presented. The model is capable of reproducing biofilm heterogeneity as well as the typical surface fingering (mushroomlike structure).

Modeling the development of biofilm density including active bacteria, inert biomass, and extracellular polymeric substances

We present the unified multi-component cellular automaton (UMCCA) model, which predicts quantitatively the development of the biofilm's composite density for three biofilm components: active bacteria, inert or dead biomass, and extracellular polymeric substances. The model also describes the concentrations of three soluble organic components (soluble substrate and two types of soluble microbial products) and oxygen. The UMCCA model is a hybrid discrete-differential mathematical model and introduces the novel feature of biofilm consolidation. Our hypothesis is that the fluid over the biofilm creates pressures and vibrations that cause the biofilm to consolidate, or pack itself to a higher density over time. Each biofilm compartment in the model output consolidates to a different degree that depends on the age of its biomass. The UMCCA model also adds a cellular automaton algorithm that identifies the path of least resistance and directly moves excess biomass along that path, thereby ensuring that the excess biomass is distributed efficiently. A companion paper illustrates the trends that the UMCCA model is able to represent and shows a comparison with experimental results.

Particle-based multidimensional multispecies biofilm model

In this paper we describe a spatially multidimensional (two-dimensional [2-D] and three-dimensional [3-D]) particle-based approach for modeling the dynamics of multispecies biofilms growing on multiple substrates. The model is based on diffusion-reaction mass balances for chemical species coupled with microbial growth and spreading of biomass represented by hard spherical particles. Effectively, this is a scaled-up version of a previously proposed individual-based biofilm model. Predictions of this new particle-based model were quantitatively compared with those obtained with an established one-dimensional (1-D) multispecies model for equivalent problems. A nitrifying biofilm containing aerobic ammonium and nitrite oxidizers, anaerobic ammonium oxidizers, and inert biomass was chosen as an example. The 2-D and 3-D models generally gave the same results. If only the average flux of nutrients needs to be known, 2-D and 1-D models are very similar. However, the behavior of intermediates, which are produced and consumed in different locations within the biofilm, is better described in 2-D and 3-D models because of the multidirectional concentration gradients. The predictions of 2-D or 3-D models are also different from those of 1-D models for slowly growing or minority species in the biofilm. This aspect is related to the mechanism of biomass spreading or advection implemented in the models and should receive more attention in future experimental studies.

Overcoming oxygen limitations in membrane-attached biofilms--investigation of flux and diffusivity in an anoxic biofilm

The possibility of overcoming oxygen limitations in membrane-attached biofilms has been investigated by using nitrate as an electron acceptor instead of oxygen in an extractive membrane bioreactor (EMB) degrading toluene. The effect of nitrate concentration on toluene flux, the effective diffusivity in the biofilm and the biofilm activity has been investigated. A counter-diffusion-reaction model is also presented, describing the pollutant flux versus biofilm thickness. The toluene flux decreased with increasing biofilm thickness under excess nitrate concentrations, similar to the experiment with low nitrate. Mathematical modelling indicated that this was either due to decreasing activity, and/or different diffusivities in the biofilm. The effective diffusivity was investigated by using an inert tracer molecule. It remained constant for biofilm thicknesses up to 1.8mm, with a value twice that in water. The biofilm activity was investigated by inactivating a mature biofilm using sodium azide. The toluene flux remained the same before and after the addition of sodium azide, suggesting that the activity in the biofilm is very low. We conclude that the decreasing toluene flux with increasing biofilm thickness is due to the diffusional resistance of the inactive biofilm.

Quantifying biofilm structure: facts and fiction

There is no doubt among biofilm researchers that biofilm structure is important to many biofilm processes, such as the transport of nutrients to deeper layers of the biofilm. However, biofilm structure is an elusive term understood only qualitatively, and as such it cannot be directly correlated with any measurable parameters characterizing biofilm performance. To correlate biofilm structure with the parameters characterizing biofilm performance, such as the rate of nutrient transport within the space occupied by the biofilms, biofilm structure must first be quantified and expressed numerically on an appropriate scale. The task of extracting numerical parameters quantifying biofilm structure relies on using biofilm imaging and image analysis. Although defining parameters characterizing biofilm structure is relatively straightforward, and multiple parameters have been described in the computer science literature, interpreting the results of such analyses is not trivial. Existing computer software developed by several research groups, including ours, for the sole purpose of analyzing biofilm images helps quantify parameters from biofilm images but does nothing to help interpret the results of such analyses. Although computing structural parameters from biofilm images permits correlating biofilm structure with other biofilm processes, the meaning of the results is not obvious. The first step to understanding the quantification of biofilm structure, developing image analysis, methods to quantify information from biofilm images, has been made by several research groups. The next step is to explain the meaning of these analyses. This presentation explains the meaning of several parameters commonly used to characterize biofilm structure. It also reviews the authors' research and experience in quantifying biofilm structure and their attempts to quantitatively relate biofilm structure to fundamental biofilm processes.

A three-dimensional, stochastic simulation of biofilm growth and transport-related factors that affect structure

Biofilm structural heterogeneity affects a broad range of microbially catalysed processes. Solute transport limitation and autoinhibitor production, two factors that contribute to heterogeneous biofilm development, were investigated using BacMIST, a computer simulation model. BacMIST combines a cellular automaton algorithm for biofilm growth with Brownian diffusion for solute transport. The simulation represented the growth of microbial unit cells in a three-dimensional domain modelled after a repeating section of a constant depth film fermenter. The simulation was implemented to analyse the effects of various levels of transport limitation on a growing single-species biofilm. In a system with rapid solute diffusion, cells throughout the biofilm grew at their maximum rate, and no solute gradient was formed over the biofilm thickness. In increasingly transport-limited systems, the rapidly growing fraction of the biofilm population decreased, and was found exclusively at the biofilm–liquid interface. Trans-biofilm growth substrate gradients also deepened with increasing transport limitation. Autoinhibitory biofilm growth was simulated for various rates of microbially produced inhibitor transport. Inhibitor transport rates affected both the biofilm population dynamics and the resulting biofilm structures. The formation of networks of void spaces in slow-growing regions of the biofilm and the development of columns in the fast-growing regions suggested a possible mechanism for the microscopically observed evolution of channels in biofilms.

A computer investigation of chemically mediated detachment in bacterial biofilms

A three-dimensional computer model was used to evaluate the effect of chemically mediated detachment on biofilm development in a negligible-shear environment. The model, BacLAB, combines conventional diffusion-reaction equations for chemicals with a cellular automata algorithm to simulate bacterial growth, movement and detachment. BacLAB simulates the life cycle of a bacterial biofilm from its initial colonization of a surface to the development of a mature biofilm with cell areal densities comparable to those in the laboratory. A base model founded on well established transport equations that are easily adaptable to investigate conjectures at the biological level has been created. In this study, the conjecture of a detachment mechanism involving a bacterially produced chemical detachment factor in which high local concentrations of this detachment factor cause the bacteria to detach from the biofilm was examined. The results show that the often observed 'mushroom'-shaped structure can occur if detachment events create voids so that the remaining attached cells look like mushrooms.

Simulation of growth and detachment in biofilm systems under defined hydrodynamic conditions

Detachment from biofilms was evaluated using a mixed culture biofilm grown on primary wastewater in a tube reactor. The growth of biofilms and the detachment of biomass from biofilms are strongly influenced by hydrodynamic conditions. In a long-term study, three biofilms were cultivated in a biofilm tube reactor. The conducted experiments of biofilm growth and detachment can be divided into three phases: 1) an exponential phase with a rapid increase of the biofilm thickness, 2) a quasi-steady-state with spontaneous fluctuation of the biofilm thickness between 500 and 1,200 microm in the investigated biofilm systems, and 3) a washout experiment with increased shear stress in three to four steps after several weeks of quasi-steady-state. Whereas the biofilm thickness during the homogeneous growth phase can be regarded constant throughout the reactor, it was found to be very heterogeneous during the quasi-steady-state and the washout experiments. Growth and detachment during all three phases was simulated with the same one-dimensional biofilm model. For each of the three phases, a different detachment rate model was used. During the homogeneous growth phase, detachment was modeled proportional to the biofilm growth rate. During the quasi-steady-state phase, detachment was described by random detachment events assuming a base biofilm thickness. Finally, the washout experiment was simulated with detachment being a function of the biofilm thickness before the increase of the shear stress.

INDISIM, an individual-based discrete simulation model to study bacterial cultures

An individual-based model has been developed and designed to simulate the growth and behaviour of bacterial colonies. The simulator is called INDISIM, which stands for INDividual DIScrete SIMulations. INDISIM is discrete in space and time, and controls a group of bacterial cells at each time step, using a set of random, time-dependent variables for each bacterium. These variables are used to characterize its position in space, biomass, state in the cellular reproduction cycle as well as other individual properties. The space where the bacterial colony evolves is also discrete. A physical lattice is introduced, subject to the appropriate boundary conditions. The lattice is subdivided into spatial cells, also defined by a set of random, time-dependent variables. These variables may include concentrations of different types of particles, nutrients, reaction products and residual products. Random variables are used to characterize the individual bacterium and the individual particle, as well as the updating of individual rules. Thus, the simulations are stochastic rather than deterministic. The whole set of variables, those that characterize the bacterial population and the environment where they evolve, enables the simulator to study the behaviour of each microorganism-such as its motion, uptake, metabolism, and viability-according to given rules suited for the system under study. These rules require the input of only a few parameters. Once this information is inputted, INDISIM simulates the behaviour of the system providing insights into the global properties of the system from the assumptions made on the properties of the individual bacteria. The relation between microscopic and global properties of the bacterial colony is obtained by using statistical averaging. In this work INDISIM has been used to study (a) biomass distributions, (b) the relationship between the rate of growth of a bacterial colony and the nutrient concentration and temperature, and (c) metabolic oscillations in batch bacterial colonies. The simulation results are found to be in very good qualitative agreement with available experimental data, and provide useful insights into the mechanisms involved in each case.

Individual-based modelling of biofilms

Understanding the emergence of the complex organization of biofilms from the interactions of its parts, individual cells and their environment, is the aim of the individual-based modelling (IbM) approach. This IbM is version 2 of BacSim, a model of Escherichia coli colony growth, which was developed into a two-dimensional multi-substrate, multi-species model of nitrifying biofilms. It was compared with the established biomass-based model (BbM) of Picioreanu and others. Both models assume that biofilm growth is due to the processes of diffusion, reaction and growth (including biomass growth, division and spreading). In the IbM, each bacterium was a spherical cell in continuous space and had variable growth parameters. Spreading of biomass occurred by shoving of cells to minimize overlap between cells. In the BbM, biomass was distributed in a discrete grid and each species had uniform growth parameters. Spreading of biomass occurred by cellular automata rules. In the IbM, the effect of random variation of growth parameters of individual bacteria was negligible in contrast to the E. coli colony model, because the heterogeneity of substrate concentrations in the biofilm was more important. The growth of a single cell into a clone, and therefore also the growth of the less abundant species, depended on the randomly chosen site of attachment, owing to the heterogeneity of substrate concentrations in the biofilm. The IbM agreed with the BbM regarding the overall growth of the biofilm, due to the same diffusion-reaction processes. However, the biofilm shape was different due to the different biomass spreading mechanisms. The IbM biofilm was more confluent and rounded due to the steady, deterministic and directionally unconstrained spreading of the bacteria. Since the biofilm shape is influenced by the spreading mechanism, it is partially independent of growth, which is driven by diffusion-reaction. Chance in initial attachment events modifies the biofilm shape and the growth of single cells because of the high heterogeneity of substrate concentrations in the biofilm, which again results from the interaction of diffusion-reaction with spreading. This stresses the primary importance of spreading and chance in addition to diffusion-reaction in the emergence of the complexity of the biofilm community.

Bacterial behavior at surfaces

Population level studies demonstrate that bacterial colonization of surfaces and subsequent biofilm architecture are controlled by a variety of factors that include the hydrodynamics, surface chemistry and genotype of the cell. New molecular tools now extend our ability to investigate among bacterial cells within a surface-associated population subtle phenotypic differences that do not involve changes in genotype. Such resolution has led to new discoveries in relationships between bacterial cells and their environment.

On the need for another type of predictive model in structured foods

Most of the models discussed up till now in predictive microbiology do not take into account the variability of microbial growth with respect to space. In structured (solid) foods, microbial growth can strongly depend on the position in the food and the assumption of homogeneity can thus not be accepted: space must be considered as an independent variable. Indeed, experimental evidence exists of bacteria competition on agar not showing the same behavior as the competition in a well-mixed liquid culture system. It is conjectured that this is due to the spatially structured habitat. Therefore, in the current paper, a prototype two species competition model proposed in previous work by the authors is extended to take space into account. The extended model describes two phenomena: (i) local evolution of biomass and (ii) transfer of biomass through the medium. The structure of the food product is taken into account by limiting the diffusion through the medium. The smaller mobility of the micro-organisms in solid foods allows spatial segregation which causes pattern formation. Evidence is given for the fact that taking space into account indeed has an influence on the behavior (coexistence/extinction) of the populations. Although the reported simulations are by no means to be interpreted as accurate predictions, the proposed model structure allows one to highlight (i) important characteristics of microbial growth in structured foods and (ii) future research trends in predictive microbiology.

Heterogeneity in biofilms

Biofilms, accumulations of microorganisms at interfaces, have been described for every aqueous system supporting life. The structure of these microbial communities ranges from monolayers of scattered single cells to thick, mucous structures of macroscopic dimensions (microbial mats; algal-microbial associations; trickling filter biofilms). During recent years the structure of biofilms from many different environments has been documented and evaluated by use of a broad variety of microscopic, physico-chemical and molecular biological techniques, revealing a generally complex 3D structure. Parallel to these investigations more and more complex mathematical models and simulations were developed to explain the development, structures, and interactions of biofilms. The forces determining the spatial structure of biofilms, including microcolonies, extracellular polymeric substances (EPS), and channels, are still the subject of controversy. To achieve conclusive explanations for the structures observed in biofilms the cooperation of both fields of investigation, modelling and experimental research, is necessary. The expanding field of molecular techniques not only allows more and more detailed documentation of the spatial distribution of species, but also of functional activities of single cells in their biofilm environment. These new methods will certainly reveal new insights in the mechanisms involved in the developmental processes involved in the formation and behavior of biofilms.

Quantification of biofilm structures by the novel computer program COMSTAT

The structural organization of four microbial communities was analysed by a novel computer program, COMSTAT, which comprises ten features for quantifying three-dimensional biofilm image stacks. Monospecies biofilms of each of the four bacteria, Pseudomonas: putida, P. aureofaciens, P. fluorescens and P. aeruginosa, tagged with the green fluorescent protein (GFP) were grown in flow chambers with a defined minimal medium as substrate. Analysis by the COMSTAT program of four variables describing biofilm structure - mean thickness, roughness, substratum coverage and surface to volume ratio - showed that the four Pseudomonas: strains represent different modes of biofilm growth. P. putida had a unique developmental pattern starting with single cells on the substratum growing into micro-colonies, which were eventually succeeded by long filaments and elongated cell clusters. P. aeruginosa colonized the entire substratum, and formed flat, uniform biofilms. P. aureofaciens resembled P. aeruginosa, but had a stronger tendency to form micro-colonies. Finally, the biofilm structures of P. fluorescens had a phenotype intermediate between those of P. putida and P. aureofaciens. Analysis of biofilms of P. aureofaciens growing on 0.03 mM, 0.1 mM or 0.5 mM citrate minimal media showed that mean biofilm thickness increased with increasing citrate concentration. Moreover, biofilm roughness increased with lower citrate concentrations, whereas surface to volume ratio increased with higher citrate concentrations.

Effect of diffusive and convective substrate transport on biofilm structure formation: a two-dimensional modeling study

A two-dimensional model for quantitative evaluation of the effect of convective and diffusive substrate transport on biofilm heterogeneity was developed. The model includes flow computation around the irregular biofilm surface, substrate mass transfer by convection and diffusion, biomass growth, and biomass spreading. It was found that in the absence of detachment, biofilm heterogeneity is mainly determined by internal mass transfer rate of substrates and by the initial percentage of carrier-surface colonization. Model predictions show that biofilm structures with highly irregular surface develop in the mass transfer-limited regime. As the nutrient availability increases, there is a gradual shift toward compact and smooth biofilms. A smaller fraction of colonized carrier surface leads to a patchy biofilm. Biofilm surface irregularity and deep vertical channels are, in this case, caused by the inability of the colonies to spread over the whole substratum surface. The maximum substrate flux to the biofilm was greatly influenced by both internal and external mass transfer rates, but not affected by the inoculation density. In general, results of the present model were similar to those obtained by a simple diffusion-reaction-growth model.

Quantifying biofilm structure using image analysis

We have developed and implemented methods of extracting morphological features from images of biofilms in order to quantify the characteristics of the inherent heterogeneity. This is a first step towards quantifying the relationship between biofilm heterogeneity and the underlying processes, such as mass-transport dynamics, substrate concentrations, and species variations. We have examined two categories of features, areal, which quantify the relative magnitude of the heterogeneity and textural, which quantify the microscale structure of the heterogeneous elements. The feature set is not exhaustive and has been restricted to two-dimensional images to this point. Included in this paper are the methods used to extract the structural information and the algorithms used to quantify the data. The features discussed are porosity, fractal dimension, diffusional length, angular second moment, inverse difference moment and textural entropy. We have found that some features are better predictors of biofilm behavior than others and we discuss possible future directions for research in this area.

Mathematical modeling of biofilm structure with a hybrid differential-discrete cellular automaton approach

A hybrid differential-discrete mathematical model has been used to simulate biofilm structures (surface shape, roughness, porosity) as a result of microbial growth in different environmental conditions. In this study, quantitative two- and three-dimensional models were evaluated by introducing statistical measures to characterize the complete biofilm structure, both the surface structure and volume structure. The surface enlargement, coefficient of roughness, fractal dimension of surface, biofilm compactness, and solids hold-up were found to be good measures of biofilm structure complexity. Among many possible factors affecting the biofilm structure, the influence of biomass growth in relation to the diffusive substrate transport was investigated. Porous biofilms, with many channels and voids between the "finger-like" or "mushroom" outgrowth, were obtained in a substrate-transport-limited regime. Conversely, compact and dense biofilms occurred in systems limited by the biomass growth rate and not by the substrate transfer rate. The surface complexity measures (enlargement, roughness, fractal dimension) all increased with increased transport limitation, whereas the volume measures (compactness, solid hold-up) decreased, showing the change from a compact and dense to a highly porous and open biofilm.