Center for Biofilm Engineering, Montana State University, Bozeman, MT 59717-3980, USA
Correspondence
Stephen M. Hunt
steve_h{at}erc.montana.edu
![]() |
ABSTRACT |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
![]() |
INTRODUCTION |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
It is known that biofilms are heterogeneous structures from which both single cells and multicellular aggregates slough (Stewart et al., 1994). It has been suggested that freely diffusible chemical signals play an important role in biofilm development and maintenance (Davies et al., 1998
; Kolter et al., 1998
). Although the mechanisms have not been established, there is some experimental evidence that a subset of these chemical signals plays an active role in biofilm detachment (Allison et al., 1999
; Boyd & Chakrabarty, 1994
). Boyd & Chakrabarty (1994)
state that when expressed from a regulated promoter, alginate lyase can induce enhanced sloughing of cells due to degradation of the alginate.
Mathematical models have been used for the last three decades to synthesize and integrate our knowledge about the behaviour of microbial biofilms. Early models represented biofilms as homogeneous steady-state films containing a single species (Rittmann & McCarty, 1980). They later evolved to dynamic multisubstratemultispecies biofilm computer models (Rittmann & Manem, 1992
; Wanner & Gujer, 1986
; Wanner & Reichert, 1996
). Although these models were advanced descriptions, they were governed exclusively by one-dimensional mass transport and biochemical interactions and the models could not account for the experimentally observed three-dimensional heterogeneity resulting from bacterial attachment, growth and detachment. The morphology was essentially predetermined by the modeller. Detachment was represented by an arbitrary uniform removal rate or velocity. These models are generally suitable for representing aggregate biofilm activity on many square millimetres of surface area.
The subsequent generation of biofilm models focused on a smaller scale. Most utilized discrete methods, such as cellular automata, to simulate the rules that govern the lives of microbial cells. These models produced realistic, structurally heterogeneous biofilms (Barker & Grimson, 1993; Colasanti, 1992
; Eberl et al., 2000
; Ermentrout & Edelstein-Keshet, 1993
; Hermanowicz, 1998
, 1999
; Kreft et al., 1998
; Noguera et al., 1999
; Picioreanu et al., 1998
; Wimpenny & Colasanti, 1997
). They allowed the artificial biofilm structure to evolve as a self-organization process, emulating how bacterial cells organize themselves into biofilms. Some models ignored detachment, while others viewed detachment as a process completely dependent on the shear-stress induced by the flowing bulk liquid. No published models consider other potential detachment mechanisms.
The aim of this article is to evaluate the implications of a chemically mediated detachment mechanism through computer experimentation. We describe a three-dimensional biofilm model that combines conventional diffusion-reaction equations for chemicals to model solute transport and a cellular automata algorithm to simulate the bacterial growth, movement and detachment. Three different plausible cellular automata detachment rules are examined by conducting 20 replicate simulations for each detachment rule. Detachment via a hypothetical bacterially produced chemical detachment factor produces structures compatible with the known morphology and dynamic behaviour of biofilms. We conclude that the simulation results are both qualitatively and quantitatively similar to those for laboratory biofilms. The conjecture of a chemically mediated detachment mechanism is not invalidated.
![]() |
MODEL DESCRIPTION |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
The domain.
The spatial domain, , of the model consists of a box, typically 900 µm per side, containing two overlapping computational grids. The first grid is a fixed lattice used to represent any soluble components (e.g. substrate, detachment factor) in the model. The second grid partitions the model space into many small cubes, each cube being a volume element large enough (here assumed 3 µm per side) to include a bacterial cell and its associated extracellular polymeric substance (Characklis, 1989
). Coordinates of each cube and lattice point are then uniquely given by the set of vectors {(x,y,z) such that x=0...Nx-1, y=0...Ny-1, z=0...Nz-1}. There are Nx·Ny·Nz total lattice points and cubes. In our simulations Nx=Ny=Nz=300. Here z=0 indicates an element on the substratum, and z=Nz-1 indicates an element the furthest from the substratum. There is a one-to-one correspondence between lattice points and cubes.
The temporal domain is discrete with equally spaced time points, typically 1 h apart, chosen to be small with respect to biofilm development.
State variables.
At any time point, three primary arrays are used to represent the state of the system: S={CS(x,y,z)}, F={CF(x,y,z)} and B={B(x,y,z)}, where CS(x,y,z) denotes the concentration of the limiting substrate, CF(x,y,z) denotes the concentration of the detachment factor and B(x,y,z) denotes the occupation state at location (x,y,z).
Each element within the substrate and detachment arrays, CS(x,y,z) and CF(x,y,z), contains a positive real value corresponding to that solute's concentration at that node location. The occupation state of a cube, B(x,y,z), is represented by an identity pointer to a vector, Ibacterium, containing all relevant information (e.g. bacterial species, kinetic parameters, etc.) about an individual bacterium. If the cube is unoccupied by a bacterium, it is represented by a null identity pointer. When computing the state of the system at the next time point, S and F are updated using conventional differential equations, while B is updated using cellular automata rules.
Differential equations for solutes.
In the aqueous environment being modelled, the bulk liquid is well-mixed, but imposes no shear-stress on the biofilm. The solutes are transported solely by diffusion in the biofilm. The concentrations CS(x,y,z) and CF(x,y,z) are a result of molecular diffusion and reaction (consumption or production) with the bacteria. The diffusional time constant is approximately 100 orders of magnitude smaller than that for bacterial cell division (Picioreanu, 1999). Thus, molecular diffusion can be assumed to be at steady-state with respect to the bacterial growth. Suppressing the location indices (x,y,z), let Ci denote the concentration of solute i, where i is either the limiting substrate, S, or the chemical detachment factor, F. Let the parameter Di denote the diffusivity coefficient of solute i. The diffusivity in the biofilm is calculated by multiplying the diffusivity of the solute in the aqueous or bulk phase, Di,aq, by the relative effective diffusivity, Di,e/Di,aq. The variable X denotes the biomass density (calculated as average cell mass per cube volume for occupied cubes and 0 otherwise), and ri(CS,X) denotes the reaction term (to be defined below) corresponding to substrate i's consumption or production by the bacteria. A negative ri value indicates substrate conversion into biomass and a positive ri value indicates that the bacteria are producing the solute, as is the case for the chemical detachment factor. Equation (1) is the three-dimensional representation for diffusive transport and reaction.
![]() |
![]() |
Let the parameter k denote the detachment factor production coefficient. Equation (3) is an assumed first order kinetic expression in CS for the detachment factor production. This first order expression in CS attempts to correlate the detachment factor production with cellular activity. It is therefore assumed that, when a cell is in a starved state, energy is conserved and extra cellular chemicals are not actively produced.
![]() |
The model's substrate uptake from the surrounding environment is dictated by the set of boundary conditions used in the simulation. The substratum is modelled as an impermeable surface by specifying a no flux boundary condition at z=0.
![]() |
To eliminate edge effects, the model utilizes a periodic boundary in the x- and y-direction. For example, the periodic boundary condition implemented in the y-direction means that the node (x,Ny-1,z) is a nearest neighbour to the node at (x,0,z). Therefore, a particle going past the boundary on one side of the box results in the particle being wrapped back to the corresponding opposite side.
Cellular automata rules for bacterial behaviour.
Cellular automata rules are used to update the occupation array, B, at each time step. The rules are locally applied to each bacterium to determine its new state as a function of local environment and the previous state of that bacterium. The rules specify whether each bacterium divides, moves or detaches.
Let the parameter mavg denote the average mass of an individual bacterium. For a bacterium to divide it must consume enough substrate to create a new daughter cell (mavg/YXS). Therefore, each bacterium, when created, is assigned a random division threshold denoted by mn, which is the cumulative mass of substrate needed for the bacterium to divide. The mn value is drawn at random from the uniform distribution on the interval [0·9x(mavg/YXS), 1·1x(mavg/YXS)] (Evans et al., 1993
). Both mn and the cumulative amount of substrate consumed by a bacterium to form biomass are stored in the I vector for that particular cell. Substrate consumption for each time increment is determined by multiplying (2) by the time step,
t, and node volume, 27 µm3. The cumulative amount of substrate consumed by the bacterium increases until it exceeds mn, in which case the cell divides. Any excess substrate consumed (above mn) is kept with the parent bacterium as part of a new mn for future division. (After completing these simulation experiments, we revised this rule to split the excess substrate equally between the parent and daughter cells. Direct comparison indicates that due to the relatively small excess of substrate left after cellular division there is no difference in population growth between the two rules).
The location (cube) of the newly created daughter cell is chosen at random from the 26 locations bordering the location of the mother cell (17 if the mother cell is on the substratum). If the selected cube is occupied, the daughter cell will displace cells in that direction until an empty cube is encountered. If the substratum or the original location (due to the periodic boundary conditions) is reached before encountering an empty volume element, a different daughter cell location is chosen at random.
In BacLAB, detachment is entirely governed by the local concentration of the detachment factor. Let the parameter CF,max denote a predetermined threshold concentration. Detachment occurs if and only if CF>CF,max at the cell's location or if a cell is no longer anchored to the substratum due to other cells detaching. Three different detachment rules were investigated in a computer experiment. The first presumes removal of any bacterium at each node point that has reached the detachment factor threshold, CF,max. This rule is referred to as local detachment. The second rule is similar to the first, but additionally removes any cell within a specified radius of detachment, Rd, of the node point where CF exceeds CF,max. This rule attempts to account for degradation of the extracellular polymeric substance as described by Boyd & Chakrabarty (1994). A hollowing of the biofilm structure is commonly observed with this detachment method, and it is therefore referred to as the hollow method. The third removal rule is similar to the second, with the exception that it also initiates the detachment of the plug or cylinder of biomass directly above the hollowed region. These detached particles typically resemble cylinders and, therefore, this method is referred to as the cylinder method.
Computational steps.
Fig. 1 shows the algorithm that defines BacLAB. The numbered steps in Fig. 1
correspond to the following sequence of operations. 1, Initialize the carrier surface with Nc randomly placed spherical colonies of radius Rc. Each cell within the colonies is itself inoculated with a random amount of substrate relative to division denoted by M, where M is chosen from a uniform (0,mn) distribution. 2, Generate the substrate distribution for the current time step, t, by finding the steady-state solution to (1). 3, Generate the detachment factor distribution for the current time step, t, by finding the steady-state solution to (1). 4, For each cube in
, determine if it is occupied by a bacterium. If the cube is unoccupied, nothing further is done with that volume element at the current time step. If the cube is occupied, further calculations are performed. 5, Each bacterium consumes substrate based on (2) and the local concentration. The cumulative amount of substrate consumed for each cell, since its last division, is then updated. 6, Determine if CF in the cube is above the detachment factor threshold, CF,max. 7, Remove the bacterium in the current cube and any additional bacteria in other cubes according to the detachment rule specified. Additionally, identify and remove any floating clusters of bacteria. 8, Check if the bacterium has consumed enough substrate to divide. 9, Create a new bacterium neighbouring the parent and leave excess substrate (not required for the creation of a daughter cell) with the parent bacterium according to the rules specified. 10, Move forward in time by
t based on the events that occurred in steps 49.
|
![]() |
COMPUTER EXPERIMENT |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
|
|
Computational resources.
BacLAB was written in C++ and compiled using the GCC compiler collection available from the Free Software Foundation. The differential equation solver, pois3d, was obtained as Fortran source code which was compiled and linked to BacLAB, again with the GCC compiler collection.
All experiments were performed on a dual 800 MHz Pentium III Mandrake Linux workstation with 1·5 gigabytes of PC133 SDRAM. A single experiment required approximately 700 megabytes of RAM. A simulation of 500 biofilm hours typically took about 8 computer hours. Although all experiments were conducted in a Linux/Unix environment, BacLAB has successfully been ported to a Microsoft Windows platform.
Analysis.
The three-dimensional location data of each cell provide the ability to plot the biofilm in three-space and display the structural heterogeneity attained by BacLAB. Furthermore, by showing these images sequentially we are able animate the structural development of the biofilm.
To visualize important quantitative characteristics for each simulation, we plot the biofilm thickness and cell areal density as functions of time. The geometric mean cell areal density and mean biofilm thickness of the 20 simulations were plotted at each time step for all detachment rules. For each detachment rule, an 80 % envelope was constructed by sorting the 20 simulation values at each time step and plotting the third smallest and third largest values. This envelope provides an upper and a lower bound between which 80 % of our data lie.
To quantitatively compare the different detachment rules, the mean log10(cell areal density) and mean biofilm thickness for each simulation were calculated in the transition and stationary phases, defined as 101300 and 301500 h, respectively. These times were chosen as oscillatory semi-steady values were observed after 300 h. Scatter plots of the means, log10(cell areal density) versus thickness, were created to see whether the data were clustered according to the detachment rule. Paired sample t-tests were then conducted to compare means between detachment rules.
![]() |
RESULTS |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Fig. 3(ac) present cell areal density data representing the 20 individual simulations for the local, hollow and cylinder rules, respectively, while Fig. 3(df) display the smoothed geometric mean and 80 % envelope. Fig. 4 displays the biofilm thickness data in a similar manner. Because each simulation describes the biofilm characteristics on a 0·0081 cm2 area of the surface, the cell areal density averaged over the 20 simulations corresponds to the areal density for a field of view of 0·162 cm2. The variation among the 20 simulations shows that different areas of the biofilm are not synchronized, but can grow and detach at different times.
|
|
|
|
![]() |
DISCUSSION |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Modern biofilm research suggests that extracellular products play an important role in the development of biofilms. The BacLAB simulations assumed a chemical detachment factor accumulating locally leads to cellular detachment. Three rules were used to simulate the behaviour of this chemically mediated detachment. While these detachment rules proved statistically different from one another, the differences are of little practical significance. The maximum difference in cell areal density between any two computer experiments in Fig. 5 is less than a log. Furthermore, the difference in biofilm thickness is difficult to interpret because it is a measurement of the most extreme point and, as such, highly variable (see Fig. 4
). What we believe to be of practical significance is the observed trend (regardless of detachment rule) of a biofilm life cycle comparable to that observed in the laboratory. This life cycle begins when a carrier surface is inoculated with bacteria which undergo exponential growth until the biofilm reaches such a point that growth and detachment begin to offset one another. At this point the biofilm enters an oscillatory steady-state where the biofilm is maintained as a heterogeneous entity by constant growth and periodic detachment. If the detachment rule is removed from BacLAB, the simulated biofilm grows as an ever-thickening slab and no steady-state is reached.
We have examined one reasonable, growth-related theoretical explanation to the process of biofilm development by assuming a detachment mechanism in systems that impose no shear-stress on the biofilm. Conceivably, a combination of shear and normal forces induced by flowing bulk liquid also contributes to biofilm detachment. It is perhaps more plausible that chemical mediation and bulk fluid force acting in concert are responsible for the detachment of biofilms. The development of mushroom-like biofilm clusters and interstitial voids and channels is typically thought to be a result of positive processes, like cell attachment, cell division and polymer production, and generally thought to result from some type of cellular organization. By refocusing on negative processes, such as cell detachment, we have shown that structures observed in real biofilms can be reproduced.
![]() |
ACKNOWLEDGEMENTS |
---|
![]() |
REFERENCES |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Barker, G. C. & Grimson, M. J. (1993). A cellular automaton model of microbial growth. Binary 5, 132137.
Boyd, A. & Chakrabarty, A. M. (1994). Role of alginate lyase in cell detachment of Pseudomonas aeruginosa. Appl Environ Microbiol 60, 23552359.[Abstract]
Characklis, W. G. (1989). Biofilms, pp. 5589, p. 114. Edited by W. G. Characklis & K. C. Marshall. New York: Wiley.
Colasanti, R. L. (1992). Cellular automata models of microbial colonies. Binary 4, 191193.
Davies, D. G., Parsek, M. R., Pearson, J. P., Iglewski, B. H., Costerton, J. W. & Greenberg, E. P. (1998). The involvement of cell-to-cell signals in the development of a bacterial biofilm. Science 280, 295298.
Eberl, H. J., Picioreanu, C., Heijnen, J. J. & van Loosdrecht, M. C. M. (2000). A three-dimensional numerical study on the correlation of spatial structure, hydrodynamic conditions, and mass transfer and conversion in biofilms. Chem Eng Sci 55, 62096222.[CrossRef]
Ermentrout, G. B. & Edelstein-Keshet, L. (1993). Cellular automata approaches to biological modeling. J Theor Biol 160, 97133.[CrossRef][Medline]
Evans, M., Hastings, N. & Peacock, B. (1993). Rectangular (uniform) continuous distribution. In Statistical Distributions, pp. 137140. New York: Wiley.
Hermanowicz, S. W. (1998). Model of two-dimensional biofilm morphology. Water Sci Technol 37, 219222.
Hermanowicz, S. W. (1999). Two-dimensional simulations of biofilm development: effects of external environmental conditions. Water Sci Technol 39, 107114.
Jackson, G., Beyenal, H., Rees, W. M. & Lewandowski, Z. (2001). Growing reproducible biofilms with respect to structure and viable cell counts. J Microbiol Methods 37, 110.
Kolter, R. & Losick, R. (1998). One for all and all for one. Science 280, 226227.
Kreft, J. U., Booth, G. & Wimpenny, J. W. T. (1998). BacSim, a simulator for individual-based modelling of bacterial colony growth. Microbiology 144, 32753287.[Abstract]
Monod, J. (1949). The growth of bacterial cultures. Annu Rev Microbiol 3, 371394.[CrossRef]
Noguera, D. R., Pizarro, G., Stahl, D. A. & Rittmann, B. E. (1999). Simulation of multispecies biofilm development in three dimensions. Water Sci Technol 39, 123130.
Picioreanu, C., van Loosdrecht, M. C. M. & Heijnen, J. J. (1998). A new combined differential-discrete cellular automaton approach for biofilm modeling: application for growth in gel beads. Biotechnol Bioeng 57, 718731.[CrossRef][Medline]
Picioreanu, C., van Loosdrecht, M. C. M. & Heijnen, J. J. (1999). Discrete-differential modeling of biofilm structure. Water Sci Technol 39, 115122.[CrossRef]
Potera, C. (1999). Forging a link between biofilms and disease. Science 283, 18371839.
Rittmann, B. E. & Manem, J. A. (1992). Development and experimental evaluation of a steady-state, multispecies biofilm model. Biotechnol Bioeng 39, 914922.[CrossRef]
Rittmann, B. E. & McCarty, P. L. (1980). Model of steady-state-biofilm kinetics. Biotechnol Bioeng 22, 23432357.
Sauer, K., Camper, A. K., Ehrlich, G. D., Costerton, J. W. & Davies, D. G. (2002). Pseudomonas aeruginosa displays multiple phenotypes during development as a biofilm. J Bacteriol 184, 11401154.
Stewart, P. S., Peyton, B. M., Drury, W. J. & Murga, R. (1994). Quantitative observations of heterogeneities in Pseudomonas aeruginosa biofilms. Appl Environ Microbiol 59, 327329.
Stoodley, P., Wilson, S., Hall-Stoodley, L., Boyle, J. D., Lappin-Scott, H. M. & Costerton, J. W. (2001). Growth and detachment of cell clusters from mature mixed species biofilms. Appl Environ Microbiol 67, 56085613.
Wanner, O. & Gujer, W. (1986). Multispecies biofilm model. Biotechnol Bioeng 28, 314328.
Wanner, O. & Reichert, P. (1996). Mathematical modeling of mixed-culture biofilms. Biotechnol Bioeng 49, 172184.[CrossRef]
Watnick, P. & Kolter, R. (2000). Biofilm, city of microbes. J Bacteriol 182, 26722679.
Wimpenny, J. W. T. & Colasanti, R. (1997). A unifying hypothesis for the structure of microbial biofilms based on cellular automaton models. FEMS Microb Ecol 22, 116.[CrossRef]
Received 19 November 2002;
revised 22 January 2003;
accepted 23 January 2003.