Two-dimensional cellular automaton model for mixed-culture biofilm

G.E. Pizarro*, C. Garca*, R. Moreno** and M.E. Seplveda**

* Department of Hydraulic and Environmental Engineering, Pontificia Universidad Catlica de Chile, Casilla306 Correo 22, Santiago, Chile (E-mail: gpizarro@ing.puc.cl; cpgarcia@puc.cl)

** Department of Computer Science, Pontificia Universidad Catlica de Chile, Casilla 306 Correo 22,Santiago, Chile (E-mail: ramoreno@puc.cl; marcos@ing.puc.cl)

Abstract Structural and microbial heterogeneity occurs in almost any type of biofilm system. Generalapproaches for the design of biofilm systems consider biofilms as homogeneous and of constant thickness.In order to improve the design of biofilms systems, models need to incorporate structural heterogeneity andthe effect of inert microbial mass. We have improved a 2D biofilm model based on cellular automata (CA)and used it to simulate multidimensional biofilms with active and inert biomass including a self-organizingdevelopment. Results indicate that the presence of inert biomass within biofilm structures does not changeconsiderably the substrate flux into the biofilm because the active biomass is located at the surface of thebiofilm. Long-term simulations revealed that although the biofilm system is highly heterogeneous and themicrostructure is continuously changing, the biofilm reaches a dynamic steady-state with prediction ofbiofilm thickness and substrate flux stabilizing on a delimited range.Keywords Biofilm modeling; cellular automata; dynamic steady-state; multidimensional model; self-organization

IntroductionIt has been demonstrated that heterogeneity (physical and microbial) occurs in almost anytype of biofilm system (Hermanowicz, 2002; Massol-Dey et al., 1995; Stoodley et al.,1999). General approaches for the design of biofilm systems consider biofilms as homoge-neous and of constant thickness (Sez and Rittmann, 1992; Williamson and McCarty,1976). More developed models incorporate microbial heterogeneity but still assume a one-dimensional (1D) biofilm (Rittmann and Manem, 1992; Wanner and Gujer, 1986; Wannerand Reichert, 1996). In order to improve the design of biofilms systems, models need toalso incorporate structural heterogeneity. An important factor for a structurally heteroge-neous biofilm model is the detachment process. For the 1D model, detachment is assumedas a first order term added to the maintenance decay coefficient. This way of representingdetachment does not take into account the location of specific cells within the biofilm. A structurally heterogeneous biofilm model can represent detachment as a naturalconsequence of the decay of bacteria that form the biofilm and does not need a detachmentcoefficient (Pizarro, 2001).

One of the important features of the multi-species models is the incorporation of the fateof decayed biomass in the form of inert biomass. For the 1D models, this process results inthe accumulation of inert biomass closer to the substratum while active biomass remains atthe surface (Rittmann and Manem, 1992; Wanner and Gujer, 1986; Wanner and Reichert,1996). Since the detachment coefficient applies to all cells within the biofilm, steady-statebiofilms with maximum thickness can be obtained. For the structurally heterogeneousbiofilm model, where detachment is modeled only as the loss of connectivity of the cells ofthe biofilm to the substratum, when inert biomass is incorporated the result is a biofilm thatincreases its thickness continually because there is no mechanism to remove the inactive

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cells that remain close to the substratum. Using experimental techniques, researchers haveelucidated some implications of the death/lysis process: decreasing the biofilm density indeep biofilm layers (Bishop et al., 1995) and biofilm shrinking and cavity formation(Ohashi and Harada, 1996). This suggests that inert biomass has to lose its mechanicalproperties and at some point leave the deep layers of the biofilm open and allow sloughingto occur. Thus, the main modeling advances presented here are to incorporate the formationof inert biomass within a structurally heterogeneous biofilm, to incorporate mechanismsfor the decay (in terms of its mechanical properties) of the inert biomass and to include aself-organizing development of the biofilm structure. This is attained with a 2D biofilmmodel based on cellular automata (CA) (Pizarro et al., 2001).

Mathematical modelThe CA biofilm model used here describes substrate and biomass as discrete particles exist-ing and interacting in a specified 2D physical domain. Substrate particles move by randomwalks, simulating molecular diffusion. The concentration of substrate at a given location isrepresented by the density of substrate particles in the neighborhood of that location.Active microorganisms in the biofilm can grow attached to a surface or to other microbialparticles, consume substrate particles, and duplicate if a sufficient amount of substrate isconsumed. Inert microbial particles do not consume substrate and can serve as support foractive microbial particles. The dynamics of the system are simulated using stochasticprocesses that represent the occurrence of specific events, such as substrate diffusion, sub-strate utilization, biofilm growth, and biofilm decay (active and inert). Detachment (orsloughing) is a natural consequence of the dynamics of the simulation and is determined byevaluating the connectivity of the cells to the substratum. Diffusion and reaction events areseparated from growth and decay events using the concept of characteristic times (Kissel etal., 1984).

Model Assumptions: The model considers only one limiting substrate. The biofilm is composed of two types of bacteria, active and inert. When an active biomass particle decays, it has a probability to change into an inert parti-

cle or to disappear from the physical domain. Inert biomass particles have a probability to disappear from the physical domain. The probability of a substrate particle being consumed depends on the concentration of

substrate particles in the neighborhood. A boundary layer is assumed to be parallel to the substratum, and to have a defined min-

imum thickness, which is measured from the bulk liquid to the maximum height of thesimulated biofilm.

Sloughing occurs only when biofilm cells lose connectivity to the substratum and areassumed to leave the domain.Detailed description of the CA biofilm model, the definition of the boundary conditions

and the relationships with the kinetic and physical parameters can be found elsewhere(Noguera et al., in press; Pizarro, 2001).

Model resultsSimulations were run using typical parameters for heterotrophic biofilms (Sez andRittmann, 1992), shown in Table 1. The formation of inert biomass considered the fractionof active biomass not biodegradable (1-fd) (Laspidou and Rittmann, 2002). Part of thedecayed biomass will form inert biomass at a rate rinert_f (Eq. (1)), where b is the decay coef-ficient and Xf is the active biomass density. To incorporate the loss of structural resistanceof inert biomass we introduced a decay rate of inert biomass (binac), which considers that

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part of this inert biomass will disappear from the system at a rate rinert_d (Eq. (2)), where Xi is the inert biomass density. The net inert biomass accumulation is represented by Eq. (3).

(1)(2)

(3)

Three kinds of simulations were performed with values for the formation of inert biomassand the decay of inert biomass. The values of the parameters used are presented in Table 2for the three cases analyzed. Case A includes only endogenous decay and thus active bio-mass disappears from the domain and there is no inert biomass formation. Case B considersthat part of the decayed biomass will form inert biomass (at a rate rinert_f) and part of thisinert biomass will disappear from the system (at a rate rinert_d). Finally, Case C also consid-ers the formation of biomass but inert biomass will accumulate since there is no disappear-ance from the system.

Results show that the steady state substrate flux predictions of the three model simula-tions A, B, and C are very similar. This similarity is due to the heterogeneous structures andthe distribution of active biomass on the external part of the biofilm. However, the patternfor biofilm sloughing changes. Figure 1 shows snapshots of the three cases for two bulksubstrate concentrations, 2 and 40 mg/L at 50 days of simulation.

It was observed that the distribution of the biomass with biofilm depth changes as thetime increases. Figure 1 also shows the normalized density of active and inert biomass withbiofilm depth at 50 days of simulation.

As for previous results (Pizarro et al., 2001), long-term simulations revealed thatalthough the biofilm system is highly heterogeneous and the microstructure is continuouslychanging, the biofilm reaches a dynamic steady-state with prediction of biofilm thick-ness and substrate flux stabilizing on a delimited range (Figure 2 after 50 days of simula-tion). Note that this is not the case for simulation C, where binac is zero, even though the substrate flux into the biofilm remains fairly constant, biofilm thickness increasescontinuously.

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Table 1 Parameters used for the simulation of astructurally heterogeneous biofilm

Parameter Value Units

q 8.0 mg/mg-dayKs 10.0 mg/LY 0.5 mg/mgb 0.1 day1

Dl 0.8 cm2/day

Df 0.64 cm2/day

r b f Xr b XdXdt

r r

inert f d f

inert d inac i

iinert f inert d

_

_

_ _

= ( )=

=

1

Table 2 Values for parameters fd and binac for the three simulations analyzed

Parameter Case A Case B Case C

fd 1 0.8 0.8binac (d

1) 0 0.1 0

Discussion and conclusionsThe CA model presented confirms the concept of dynamic steady-state and the importanceof self-organization in biofilm structure formation. The results of the simulations demon-strated that the presence of inert biomass within biofilm structures does not change consid-erably the substrate flux into the biofilm because the active biomass is located mainly at thesurface of the biofilm. Table 3 presents the values for the density of biomass per unit area(Mact and Minert) and mean biofilm thickness (Lf) for active and inert biomass for cases A,B, and C for two bulk substrate concentrations. It can be noted that the mass of active cellsin the three cases is very similar, what changes is the amount of inert biomass.

For the structurally heterogeneous biofilm model, the way of obtaining a finite biofilm

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Figure 1 Snapshots of biofilm structure after 50 days of simulation for a bulk substrate concentration (Sb)of 2 and 40 mg/L with A) no inert biomass production; B) inert biomass production and loss and C) inertbiomass production and accumulation. Normalized density of biomass at different depths of the biofilm isshown at the right side of each panel. Light color represents active cells while dark represents inert cells

thickness is allowing detachment to occur. Given the assumptions of the 2D CA model pre-sented here, detachment was obtained by adding a new first order parameter, binac, to repre-sent the loss of structural resistance of inert biomass, allowing sloughing to occur. In thisway, biofilms can reach a dynamic steady-state with substrate fluxes similar to simulationswithout inactive biomass formation and with finite biofilm thickness.

Biofilm density varies with the biofilm depth according to the observations made byBishop and colleagues (Bishop et al., 1995) even though it changes continuously as slough-ing events take place.

AcknowledgementsThis research was funded by a grant from FONDECYT (National Fund for theDevelopment of Science and Technology, project 1020736) and by a grant for youngresearchers from Fundacin Andes (C-13760).

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Figure 2 Average biofilm thickness and substrate flux over time for Sb = 40 mg/L. Cases A) with no inertbiomass production; B) with inert biomass production and loss, and C) with inert biomass production andaccumulation

Table 3 Steady-state values for parameters in simulations of cases A, B, and C for Sb = 2 and 40 mg/L

Substrate Sb= 2 mg/L Sb= 40 mg/L

Parameter A B C A B C

Jb (mg/cm2d) 0.07 0.06 0.07 0.42 0.37 0.27

Lf (m) 74 84 266 312 334 863Mact (mg/cm

2) 0.25 0.24 0.29 0.93 0.96 0.97Minert (mg/cm

2) 0.00 0.04 0.46 0.00 0.12 0.71

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