Cardiff School of Biosciences, Cardiff University, PO Box 915, Cardiff CF10 3TL, UK1
Kluyver Institute of Biotechnology, Delft University of Technology, Julianalaan 67, 2628 BC Delft, The Netherlands2
Author for correspondence: Jan-Ulrich Kreft. Tel: +44 29 2087 5278. Fax: +44 29 2087 4305. e-mail: kreft{at}cardiff.ac.uk
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ABSTRACT |
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Keywords: biofilm structure, nitrification, spatial heterogeneity, chance, complexity
Abbreviations: 2D, 3D, two-, three-dimensional; IbM, individual-based model/modelling; BbM, biomass-based model/modelling; CV, coefficient of variation
a Present address: Abteilung Theoretische Biologie, Botanisches Institut, Universität Bonn, Kirschallee 1, D-53115 Bonn, Germany.
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INTRODUCTION |
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Biofilm models have undergone an evolution of increasing complexity (Noguera et al., 1999a ), prompted by a growing body of experimental evidence of biofilm heterogeneity. The first models, in the 1970s, described biofilms as uniform steady-state films of a single species with one-dimensional mass transport and biochemical reactions. In the 1980s, stratified dynamic models of multisubstrate/multispecies biofilms were developed. However, they could not generate the characteristic biofilm morphology but had to use a given biofilm structure as input into the model. With the aim to explain new experimental findings, and facilitated by the advances in both computational power and numerical methods, 2D and three-dimensional (3D) models were developed in the 1990s. They incorporate the whole range of transport processes as well as biofilm growth and detachment to various extents (Wimpenny & Colasanti, 1997
; Picioreanu et al., 1998a
, b
; Hermanowicz, 1999
; Noguera et al., 1999b
; Eberl et al., 2000
, 2001
). Such models will be referred to as biomass-based models (BbMs) in this paper. In these models, biofilm structure is an emergent property rather than the model input because they follow a bottom-up approach where the complex community emerges as a result of the actions and interactions of the biomass units with each other and the environment. They can also be classified as spatially structured population models.
Individual-based modelling (IbM) is also a bottom-up approach as it attempts to model a population or community by describing the actions and properties of the individuals comprising the population or community (Huston et al., 1988 ; DeAngelis & Gross, 1992
; Grimm, 1999
). In contrast to BbM, IbM allows individual variability and treats organisms, in our case bacterial cells, as the fundamental entities. This has important consequences regarding biomass spreading. When such an entity changes position, its biomass, fixed and variable properties (such as its genome, state of differentiation, etc.) and its cell number (one) are displaced together.
All bottom-up models have the potential to address questions about the relationship of microscopic and macroscopic properties. How do simple microscopic units (cells) give rise to complex macroscopic structures (biofilms)? How does chance affect these structures?
IbM is particularly suited to address questions about the effects of individual variability. Before we can take full advantage of this potential, however, we must first emphasize the prediction of normal biofilm structure.
We have applied IbM to a nitrifying biofilm with one ammonia- and one nitrite-oxidizing species as our model system, because it provides a simple example of a food chain and, moreover, nitrification is an important process in natural environments, sewage treatment plants and agriculture.
The aims of this paper were the development and validation of the first IbM of biofilms, based on an IbM of Escherichia coli colony growth (Kreft et al., 1998 ), and the comparison of results with a BbM adapted from Picioreanu et al. (1998a
, b
) by using the same kinetics, parameters and initial conditions.
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THE INDIVIDUAL-BASED MODEL BacSim |
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BacSim essentially consists of two parts: one deals with the simulation of the growth and behaviour of individual bacteria as autonomous agents; the other deals with the simulation of substrate and product diffusion and reaction (Fig. 1). Since biofilm growth is usually a much slower process than diffusion of substrate into the biofilm, the diffusion process can be simulated assuming the growth process to be frozen, and the growth process can be simulated assuming the diffusion process to be in a pseudo-steady state (Picioreanu et al., 1999
). This decoupling of processes based on differences in their time scale is standard for BbM and the main improvement from the previous version of BacSim (Kreft et al., 1998
). Although the numerous extensions of the features of BacSim make it impossible to compare execution times of the two versions of BacSim exactly, as a rough guideline, runs are now about 500 times faster.
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Simulation of bacteria. Each bacterium is individually simulated as a sphere of variable volume in a continuous, 3D space.
The single cell uptake rate for the nitrogen-containing substrate vN is described by a Monod equation with double substrate limitation for both the electron donor (denoted N for nitrogen-containing substrate, i.e. ammonia or nitrous acid, depending on the species) and acceptor (denoted O for oxygen, for both species), as well as a substrate inhibition term for the respective electron donor substrate:
![]() | (1) |
where MX is the amount of biomass (dry mass), Vmax,N is the maximum specific uptake rate for the nitrogen-containing substrate, CN and CO are the concentrations of the respective substrates, KS is the Monod saturation constant and Ki is the substrate inhibition constant. The uptake rate for oxygen, vO , is related to vN by the yield ratio:
![]() | (2) |
where YN is the growth yield for the electron donor and YO is the growth yield for the electron acceptor.
Although the nitrogen-containing compounds are ionized at neutral pH, the saturation and inhibition constants given in Table 1 are expressed in terms of the neutral species because NH3 and HNO2 are the real substrates. This makes the constants independent of temperature and pH. Given temperature and pH, these kinetic constants, K, can be converted to an expression for the ionized species:
|
![]() | (3) |
![]() | (4) |
where T is the absolute temperature in Kelvin (Wiesmann, 1994 ).
The observed growth rate of the cell is given by:
![]() | (5) |
where µ is the specific growth rate and m is the maintenance rate, expressed on the basis of rate of biomass decrease (Herbert model; see Kreft et al., 1998 ).
The size of a cell is important, since it affects the cells metabolic rates and starvation survival. To model the well known dependence of cell size on growth rate, we have used the Donachie model for cell division (Donachie & Robinson, 1987 ) in a descriptive form (for a detailed discussion see Kreft et al. (1998)
. Donachie and coworkers found that cells divide a constant time after initiation of DNA replication. Faster growing cells will gain more volume during this time interval than slower growing cells, therefore they will be larger at division. Donachie and coworkers also established an empirical relationship for the dependence of cell volume on growth rate, which we have used here. The volume at cell division Vd, is a function of the number of generations per hour, g:
![]() | (6) |
The minimal volume at division is a function of the maximal volume at division:
![]() | (7) |
The spreading of the growing biomass is simulated by maintenance of a minimum distance between neighbouring cells. For each cell, the vector sum of all positive overlap radii with neighbouring cells is calculated and then the position of the cell is shifted in the direction opposite to this vector. The overlap radius, Ro, is given by:
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![]() | (8) |
where Rc is the radius of the cell, Rn is the radius of the neighbouring cell, d is the euclidean distance between the centres of the cell and its neighbour and k is a multiplier on the cell radius that allows adjustment of the minimal spacing between cells (shove radius). The shove radius multiplier prevents a closer packing of cells than is physically possible (Kreft et al., 1998 ). If the cells surface extended into the substratum, the overlap with the substratum was added to the vector sum of overlap radii. This spreading mechanism is illustrated by a movie available at our website (http://www.cf.ac.uk/biosi/staff/kreft/biofilms.html) and also at Microbiology Online (http://mic.sgmjournals.org).
Boundary conditions. The x and y dimensions had periodic boundaries, that is, cells that become shifted beyond such a boundary plane re-enter the domain through the opposite boundary plane. In other words, the sides are wrapped around. This avoids edge effects, since there are, effectively, no edges. The z dimension was non-periodic, with bacteria allowed to spread towards the bulk liquid but not into the solid substratum (Fig. 2).
Initial conditions. If not mentioned otherwise, the system was inoculated with 10 bacteria of each species, placing them at randomly chosen locations on the surface of the substratum at time zero.
Sorting bacteria by location. A tree data structure was used to sort bacteria by location in the vertical xz plane to find neighbours efficiently. Each leaf of the tree stores a reference to one bacterium. Traversing this tree will visit each bacterium in turn and therefore all bacteria in a particular sequence that is determined by the tree structure. The tree is constructed from a list of bacteria by calculating the bounding box around all area patches occupied by the list members. This box is then subdivided into quadrants and the list members are sorted into these quadrants according to their location. This procedure is repeated recursively for each sublist of quadrant members until each list contains fewer than 11 members. Resorting the tree is done every 10 steps by reorganizing all entries in the tree into a single list in a way that inverts the order of this new list compared to the original list from which the tree was constructed. (This inversion is exact only if no list members were added due to cell divisions or removed due to deaths.) This new inverted list is then used to reconstruct the tree in the same way as above. If list members have changed their location during the last 10 steps, they may become sorted into a different quadrant, and the quadrants may also change in area and location.
Random variation of cell parameters. If not mentioned otherwise, both the maximal uptake rate and the volume-at-division of each bacterium were varied by random draws from a Gaussian distribution with a coefficient of variation (CV) of 10%. This CV is a value typical for the few cases were this has been studied (for a discussion see Kreft et al., 1998 ). Draws resulting in parameters outside their default ±2
were skipped, as were changes of sign. This is necessary as the Gaussian distribution has an interval of [-
,+
], which would lead to draws of parameter values that are physically impossible or beyond the range allowed by biological constraints.
Simulation of substrate fields
The concentrations of the dissolved substrates and products (here oxygen, ammonia, nitrite and nitrate), result from a solution of their mass balances, including transport and reaction terms. Here, we only consider diffusive transport. The continuous diffusion-reaction equation for each compound i is given by:
![]() | (9) |
where D denotes the diffusion coefficient and r the reaction rate of the given compound.
Note that the uptake rates are calculated by each bacterium, but this information has to be transferred to a grid for computing the substrate fields. Since the specific uptake rates of different bacteria of the same species were allowed to vary randomly, a rectangular lattice for storing the momentary reaction rates must be constructed by querying all cells for their uptake rates at given substrate concentrations. Each cells reaction rates are apportioned into the lattice elements covered by this cell on an area percentage basis (this procedure of accurately gridding the reaction rates is left out of equations 1013 below for the sake of simplicity). This process has to be repeated for every iteration of the substrate field relaxation method, because reaction rates depend on substrate concentrations. The reaction rates for the different compounds in grid element p with volume Vp relate to the single-cell uptake rates of all the bacteria j of species 1 and all the bacteria k of species 2 in grid element p as follows (assimilation of nitrogen not considered):
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For the substrate fields, the space is discretized with a rectangular mesh with Nx1xL elements, numbered from 0 to N-1, 0 and L-1 for the dimensions x, y and z, respectively. Note that this mesh was only one layer deep in the y direction but the depth of this single layer was variable. Unless stated otherwise, 100x1x100 volume elements were used with a length of 2 µm for all sides (lengths in x and y were always equal), giving a computational domain of 200x2x200 µm.
Boundary conditions. The zero-flux boundary condition (Ci/
z=0) is assumed at the substratum, z=0. At the top of the system, z=lz, the bulk liquid with constant substrate concentration (infinite reservoir) was located. This bulk liquid phase extended down to the mass transfer boundary layer, the top of which was approximated by a plane located at a constant distance above the top of the changing biofilm front (Fig. 2
). Here, we use a typical value for the boundary layer thickness of 40 µm (Picioreanu et al., 1998b
), thereby modelling the effect of convective flow rather than modelling convective flow directly. For the bulk liquid, a fixed-value boundary condition was used with the substrate concentration being set to the external substrate concentration throughout the bulk liquid phase. For the other dimensions, periodic boundaries were used.
Initial conditions. The substrate concentration was set to the external substrate concentration throughout.
Solution. The system of non-linear second order partial differential equations (913) was solved for the steady-state solution using the alternating direction implicit method (Peaceman & Rachford, 1955
; Ames, 1977
). The discretization used for each grid point was a five-point centred finite difference scheme also used in Picioreanu et al. (1998b
).
Comparison with the BbM
The chosen BbM was essentially the model developed by Picioreanu et al. (1998a , b
), tailored to the simulation of a two-species nitrifying biofilm in two dimensions. The two models are based on the same algorithms for the solution of substrate diffusion-reaction mass balances. Regarding the biomass, the IbM and BbM differed in three fundamental aspects: (a) the space for the biomass was discrete for the BbM but continuous for the IbM; (b) the biofilm consisted of uniform bricks of biomass for the BbM but individual cells with mutable parameters for the IbM; and (c) biomass spreading algorithms. In the BbM, when bricks of biomass divide, one of the halves is randomly shifted into the neighbourhood with a preference for an unoccupied site. In the IbM, the spherical cells shove each other every time step to minimize overlap. Both models use the same equations and parameters (Table 1
) for the growth of the biomass and the same system geometry and initial conditions. The maximal uptake rate and the size-at-division were randomly varied in the IbM (with the mean equal to the setting in the BbM). However, this hardly affected results (see below).
Biomass spreading in the BbM was as follows. If the maximum biomass density was reached in a given lattice element, the biomass density in that lattice element was halved and an equal amount of biomass was placed into a randomly chosen free grid element in the eight-connected neighbourhood. If none of the grid elements in this neighbourhood was free, a randomly chosen neighbour was displaced. The search for a free grid element was then recursively started over again with the displaced neighbour as the new starting point. Based on these cellular automata rules, three different versions of spreading were derived that treated the mixing of the two species differently. In the first scheme, called apart, different species occupied different grid elements and stayed apart during spreading. In the second scheme, called coupled, different species could jointly occupy the same grid element and were redistributed as a unit. In the third scheme, called mixed, different species could occupy the same grid element as in coupled, but upon reaching the maximum density allowed per grid element, only the biomass of the species constituting the larger fraction was redistributed.
It was essential that both models only differed in those respects that we intended to compare and be equivalent otherwise. Therefore, in addition to using the same equations and parameters for diffusion, reactions and growth, the following points had to be taken care of as well. The initial random distribution of bacteria along the substratum in the IbM was converted into a matrix of initial biomass distribution that was read into the BbM to ensure equal but random initial distribution of biomass in both models. Also, the mean biomass density of the biofilm resulting from the IbM was used to calculate the maximum biomass density for the BbM, using a previously established ratio of mean to maximum biomass density for the BbM. Since both models differed in the way the maintenance energy requirements were modelled, the maintenance energy was set to zero for the sake of comparison. The effective diffusion coefficient was assumed to be equal to the diffusion coefficient in water for the same reason.
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ANALYSIS OF MODEL OUTPUT |
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Biofilm structure. The aim was to establish biologically relevant quantitative descriptors of biofilm structure that allowed comparison of various biofilms, simulated or real. Many such descriptors have been proposed (Zhang & Bishop, 1994a , b
; Gibbs & Bishop, 1995
; Murga et al., 1995
; Hermanowicz et al., 1995
, 1996
; Kreft et al., 1998
; Kwok et al., 1998
; Picioreanu et al., 1998b
; Lewandowski et al., 1999
; Heydorn et al., 2000
; Yang et al., 2000
), but their relative merit has not yet been assessed by a comparative study. See Table 2
for a list of measures used.
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Discussion of biofilm structure analysis. All measures can be used to quantify the structures of model biofilms, since they are not too affected by fluctuations from time step to time step. The dependence of the shape measures on the resolution, 1 or 2 µm, of the analysis of the same run (IbM with 1 µm spatial resolution; data not shown) was very strong, particularly for measures of heterogeneity, although image analysis shape measures are supposed to be independent of resolution. Only surface enlargement and solids hold-up were the same at the different resolutions. This renders comparisons of biofilm structures sampled at different resolutions meaningless. Contrast and heterogeneity are both measures of some aspects of heterogeneity, with contrast showing higher sensitivity, differentiating between the mixed and the other two versions of the BbM as well as the IbM (see Fig. 8). Also, contrast was the only measure that varied within a given biofilm structure (data not shown) as would be expected of a good measure of heterogeneity. Heterogeneity, however, is highest close to the biofilm surface, and low and level within the biofilm (data not shown). Therefore, contrast seemed to be the better measure of internal heterogeneity. By this token, the IbM, compared with the BbM, appeared to have a more homogeneous internal structure and a less homogeneous structure close to the surface. We feel that the objective quantification of biofilm structure remains an important challenge, despite recent progress, since the utility of shape measures can only be assessed when they have been applied to the full diversity of biofilm shapes, whether real or simulated. This study is only a small contribution to this task.
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The following statistical method was found to be appropriate. The difference between a given and a reference dataset, divided by the reference dataset, was computed (called relative residuals later on). To test for a significant difference of the mean of these relative residuals from zero, a T-test can be used. Since a divergence of trends may only occur for a part of the trace, say the last half, we have used a moving T-test with a window of 10 values. If a stretch of significant deviations from random scatter around zero was found, and this stretch was long enough to allow an estimate of the trends for this time interval (five data points), we determined the range of the deviation of the trends in this window of significance. The trends were estimated by robust smoothing (3RSR, followed by two Hannings; see Tukey, 1977 ) for less noisy traces, or a stronger averaging procedure (moving median with a window of three data points to remove spikes, followed by a moving mean with a window of five data points to dampen) for noisier traces. (The trend in the data could be removed by one or two differentiations.) The relative difference (divergence) between these trends was plotted and a moving mean curve (five data point window) drawn. If the range of this curve was below 0·05, the divergence of trends was defined as negligible.
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TESTING (VALIDATION) OF THE MODEL |
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The extra dimension for bacteria. The extra dimension (y), the depth of the biofilm (Fig. 2), was kept small to avoid biomass gradients and for reasons of computational efficiency. To check for the absence of a biomass gradient across the total extent of this dimension, we simulated biofilms with depths of 1, 2, 3, 4, 5 and 10 µm (data not shown). No gradient spanned the extent of the extra dimension, but small-scale local gradients reflecting arrangement of cells into layers were found.
Spatial resolution of the lattices for the diffusion-reaction scheme. The size of the lattice cells of all the lattices used in the diffusion-reaction scheme was varied, using 1, 2, 4, 5 and 10 µm for the sides of the squares. Initially, all of the growth curves of the majority species deviated strongly from the reference run (1 µm resolution), but growth curves converged again later (Fig. 4). Specifically, growth curves were back to less than 5% relative difference when the biofilm reached a height of about 68 grid cells, e.g. 16 µm at 2 µm resolution and 30 µm at 5 µm resolution. In other words, the thinner the biofilm, the higher the resolution has to be for accurate growth rates. If a 5% relative difference is chosen as the acceptable maximal difference, then the resolution should be at least 1/8 of the biofilm height. Since we have used a constant resolution of usually 2 µm, growth results are not very accurate initially, differing by up to 12%.
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Accuracy of gridding biomasses, reaction rates and substrate concentrations. The procedure of doling out the biomass of each spherical cell, according to the cells position, into the square biomass grid was carried out in a simple and also in a more accurate way. With the simple procedure, all of the cells biomass was added to the biomass grid in which the cells centre was located. With the more accurate procedure, the biomass was doled out into the eight neighbouring biomass grid elements in addition to the central grid, in proportion to the area coverage of an equivalent-volume cube representing the spherical cell. The reaction rates were allocated and the substrate concentrations were averaged in the same manner. The growth of the majority species as well as biofilm shape parameters did not deviate considerably, but the growth of the minority species was clearly affected by the biomass apportioning. Therefore, the more accurate scheme was used for all runs. Apparently, the growth of the minority species is more sensitive to slight local changes in substrate concentration, since it only exists in a few different locations in a small number. It must be assumed that the growth of individual cells can be strongly affected by these slight changes, but for the growth of larger populations spread over many different locations, these changes are averaged out.
Time resolution for growth. The growth time step of the simulation was set to 1, 2, 5 or 10 min and the results compared. At a time step of 10 min, many of the parameters deviated; at a time step of 5 min, growth of both species deviated; at a time step of 2 min, results agreed well. Note that these results depend on the growth rate of the species, hence, time resolution should be higher for faster growing bacteria.
Discussion of model validation. For validation of the IbM, we checked that effects due to (a) code optimization (data not shown), (b) implementing the algorithms in two different programming languages (Objective-C and Java; data not shown) and (c) sequence of execution of the bacterial activities were negligible. Further, the biomass distribution in the third dimension was level. Sufficiently accurate partitioning of the reaction rates of the cells into a reaction rate matrix required distribution of rates into an eight-connected neighbourhood on the basis of area coverage. The spatial and time resolution of the simulation was fine enough and the convergence criterion for diffusion was stringent enough (data not shown).
The new mutual shoving scheme was better than the classic unilateral shoving scheme (Kreft et al., 1998 ), since the biomass was more spread out at the same setting for the minimum distance between cells (shove radius). Also, the mutual scheme is free of bias due to the sequence of shoving.
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RESULTS |
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Effect of random variations
Stochastic daughter cell placement upon division. There are three uses of random numbers in the simulations. One is varying parameters of the bacteria (see section on individual variability below), another one is in choosing the initial location of cells within given bounds (see section on stochastic initial cell locations below) and the last one is in choosing the directions of displacement of the daughter cells from the mothers position prior to division. To evaluate the effect of the latter on biofilm growth and shape, we made 50 repeats of a simulation starting with identical positions of the bacteria and drawing random numbers from one non-repeating series. Each run was compared with the mean of all runs (Table 2, Fig. 5a
, b
). The growth of the dominant species (and therefore the biofilm) did not deviate considerably in any of the runs (Fig. 5a
). But the growth of the less abundant species deviated considerably in two runs (Fig. 5b
). Almost all of the biofilm shape parameters agreed amongst the 50 runs. Therefore, the placement of daughter cells did not have a major effect in the system studied here.
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Individual variability. We compared a simulation where bacterial parameters were kept constant with one where both Vmax,N and volume-at-division were varied randomly. No differences between the two runs were found apart from temporary deviations for contrast.
Comparison of IbM with BbM
The IbM was compared with the BbM using two different sets of random initial locations for the bacteria. These sets of locations were generated with the IbM, converted into biomass matrices for the two species, saved to file and read into the BbM. Since the original version of the BbM, called apart, did not allow the coexistence of biomass of both species in the same grid element, the initial biomass matrices for the two species should not have overlapping entries. However, the other two versions, called coupled and mixed, allow such coexistence. Since the two species do not mix in the coupled version, they will stay apart if they were apart at the beginning, and results will be the same as with the apart version. Therefore, to compare all versions of the BbM with the IbM, one set of non-overlapping and one set of partially overlapping initial locations were used.
The biomass density of the BbM version was set to the biomass density of the IbM, since this density is not known a priori (Fig. 3). It emerges from the interaction of cell sizes and growth rates with the spreading mechanism of the IbM, which uses the shove radius parameter (Table 1
).
The growth of the majority species, and therefore the growth of the biofilm, agreed reasonably well between the IbM and all the versions of the BbM (Fig. 6a). Although the trend of some of the BbM versions diverged from the IbM considerably, the differences between some of the BbM versions were more pronounced than the difference to the IbM. This is also true for the growth of the minority species (Fig. 6b
), but here the discrepancies between some of the runs were dramatic.
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Finer textures have a higher contrast (Weszka et al., 1976 ). This would indicate a coarser texture for the IbM.
Results with the non-overlapping set of initial conditions were similar to the results with the partially overlapping initial conditions (data not shown).
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DISCUSSION |
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The effect of random variation of the two key growth parameters, Vmax,N and volume-at-division, was negligible in contrast to the colony model (Kreft et al., 1998 ), due to the influence of the very high heterogeneity of substrate concentration in the biofilm. The growth and shape of biofilms was independent of random variations in the placement of daughter cells upon division.
The interplay of the stochastic process of initial attachment with the heterogeneity of substrate concentration in the biofilm drastically altered the growth of single cells into clones (Fig. 9), and therefore of less abundant species with a small number of individuals in the population. By analogy, stochastic attachment events after formation of the biofilm would have a similar, though smaller effect. Furthermore, in the BbM, the biomass spreading mechanisms were themselves stochastic, explaining why the different versions of biomass spreading of the BbM resulted in strong differences in the growth of the minority species. It should be noted that this implies that modelling the growth of rare species requires careful consideration of the spreading mechanism appropriate for this species.
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The BbM biofilm shapes were clearly more related to one another than to the IbM (Fig. 8), since all versions use the same principle of biomass spreading. The observed differences in the biofilm shapes of the three versions of the BbM can be explained by the stochastic nature of the spreading mechanism which let different fingers grow better in different runs of the same version (data not shown) or in runs of the different versions (Fig. 7
). Since the spreading of the minority species is independent of the spread of the biofilm in the mixed version, results of this version were particularly stochastic. Also, the minority species had higher chances to be shifted towards higher oxygen concentrations, resulting in the best growth of all models (Fig. 6b
).
BbM biofilms from runs with a spatial resolution of 1 µm were more confluent and rounded than at 2 µm (data not shown) and therefore more similar to the IbM, which gave biofilms that were even more rounded than the 1 µm BbM biofilms. The spatial resolution of the biomass is fixed in the IbM, since it corresponds to the mean size of the bacteria, which were about 1 µm in diameter. Due to the stochastic nature and the resolution dependence of the spreading mechanism of the BbM, its biofilms often grew into different patterns of fingers at different resolutions and the growth of the minority species varied drastically. At the higher resolution, the species became more mixed in the mixed version; therefore, growth of the minority species was much better. But the growth of the majority species was still almost identical.
In conclusion, both the BbM and IbM results agreed in principle, due to modelling the same physical processes, but differed in details of biofilm shape and growth of minority species. Therefore, if one wants to focus on rare species or rare events, the choice of model should depend on which biomass spreading mechanism is supported by experimental results. Until enough is known about this, the choice of model is rather arbitrary and one might want to use both. The BbM is clearly better suited to model larger scale systems, which is particularly important when the system is characterized by larger scale heterogeneities.
Biofilm growth, driven by reaction-diffusion, interacts with biofilm shape, influenced by the biomass spreading mechanism. Chance events modify 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.
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ACKNOWLEDGEMENTS |
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Received 10 January 2001;
revised 9 July 2001;
accepted 20 July 2001.