- Email: [email protected]
A model of two-dimensional biofilm morphology
~
Pergamon
War. Sci. Tech. Vol. 37, No. 4-5, pp. 219-222, 1998. iCI 1998 IAWQ. Published by Elsevier Science Ltd
PH: 50273-1223(98)00 110-3
Printed in Great Britain. 0273-1223/98 $19'00 + 0·00
A MODEL OF TWO-DIMENSIONAL
BIOFILM MORPHOLOGY Slawomir W. Hermanowicz
Department of Civil and Environmental Engineering, University ofCalifornia. Berkeley, CA 94720-17/0, USA
ABSTRACf A mathematical model of biofilm development was proposed. The model, which can be classified as a cellular automaton, is based on simple local rules for growth and detachment of individual cells. Results of several numerical simulations suggest that the thickness of concentration and hydrodynamic boundary layers may have an imponant effect on the developing blOfilm structure. When external mass transfer limitations are significant, model biofilms develop an open structure. When the concentration boundary layer is reduced and external mass transfer is enhanced, a dense layer of a biofilm develops. Biofilm strength to withstand erosion has a smaller but also significant effect on its structure. © 1998 IAWQ. Published by Elsevier Science Ltd
KEYWORDS Biofilm modeling; cellular automaton; biofilm structure. BIOFILM STRUcrURE Biofilms are found in many natural and engineered environments where, commonly, they are a dominant form of microbial communities. A wide variety of biofilm structure and resulting morphologies have been reported in the literature (e.g., Siegrist and Gujer, 1985; Lawrence et al., 1991; Kugaprasatharn et aI., 1992; Massoldeya et al., 1995, DeBeer et al., I994a). Biofilms range from compact and smooth to open, patchy and nuffy, often forming complicated shapes called "mushrooms," "stacks" or "tulips." At present, factors affecting biofilm structure are only surmised and even a methodology adequate for characterization of such varied morphology has not yet been formulated. However, important insights can be gained from modeling of biofilm development. At this time modeling can be useful in identification of general trends in biofilm development such as growth and (hopefully) structure but its predictive capacities are severely limited by the lack of understanding of microbial metabolism and other processes occurring inside the complex ecosystem of a biofilm. Thus, while useful for pointing directions of further studies, modeling results must be ultimately tested experimentally. MODEL OF BIOFILM DEVELOPMENT Classical approaches to modeling assumed that a biofilm could be represented by a homogeneous structure that is a site for biochemical processes (such as substrate utilization) and itself undergoes transformations (e.g., growth or detachment). Such approaches led to formulation of mass balances with phenomenological description of transport and transformation processes. However, it is possible to approach biofilm modeling 219
220
S. W. HERMANOWICZ
from a different perspective. For modeling purposes, a biofilm is represented as a two- or three-dimensional array of "cells". Each cell may be occupied by biomass or be filled with water. The size of each model cell depends on the modeling scale and may correspond to a single microbial cell or an aggregate of microbial cells. The state of each model cell (i.e., whether it is occupied by biomass or not) is a dynamic variable that changes according to prescribed rules. Thus, any biofilm structure that emerges in such a model is a result of local interactions rather than an a priori description imposed on the model. The objective of this work is to show how these general features can form a framework leading to the development of variable biofilm morphology. In particular. the implementation of simple development rules on a small scale (such as a single cell or small aggregate) results in the formation of self-organized structures. In this work. a simple two-dimensional model was constructed. At every time step, each model cell occupied by biomass could grow or detach according to a set of rules. A simulation started with a thin layer of occupied cells placed on a flat substratum and exposed to water with the bulk dimensionless substrate concentration Ca' The thickness of a boundary layer 5 a adjacent to the surface of the occupied cells was specified in units of cell size. It was assumed that outside the boundary layer, substrate concentration remained constant at Ca while inside the boundary layer and the biomass substrate was transported by diffusion. This picture of mass transport agrees qualitatively with the results of DeBeer et al. (l994b) who showed that convective mass transport is possible outside biomass clusters. DeBeer et al. (1996) measured concentrations of oxygen in a boundary layer around a biofilm and showed that its thickness depended on bulk flow velocity. As a result of this mass transport mechanism each cell in the model was exposed to a local dimensionless substrate concentration C S Ca. The probability that an occupied cell would divide and grow was given by an expression CI(I+C). If Monod growth kinetics is assumed then the dimensionless concentration C is equal to the actual concentration divided by the half-saturation constant. If a cell divides then an adjacent cell becomes occupied thus resulting in biomass growth. The growth occurred in the direction of least resistance given by the smallest distance from the growth site to the biofilm surface. All occupied cells lying in this direction were pushed by the growing cell. Finally, occupied cells on the surface of the biofilm (i.e., exposed to water) could detach under the influence of hydrodynamic shear stress. Probability that a cell on a surface would detach was given by the following expression: 1/(1 + a/t) where t is the shear stress and a denotes biofilm strength. Shear stress was assumed to be inversely proportional to 5a . Since only a dimensionless value of a/t was used in the simulations, no actual values of t were required in numerical simulations. NUMERICAL SIMULATIONS The results of three numerical simulations are shown in Figs I - 3. Each figure presents the results after 18 growth cycles. The structure of the biofilm is shown in the lower left comer with biomass in black and water in white. The other three parts of each figure show the cell growth rate. substrate con- centration and cell age with a white color representing the highest values. All of these simulations were carried out with the same values of intrinsic microbial kinetic coefficients. substrate diffusivity and bulk liquid concentration. Only two parameters were varied: boundary layer thickness and biofilm strength. In the first case (Fig. I) the thickness of the boundary layer was small (58 I) and strength of biofilm moderate (a/t 0.6). In this case the biofilm developed a dense. homogeneous structure with a relatively smooth surface.
=
=
=
In the second simulation, (Fig. 2) the thickness of the boundary layer was increased to 58 5 and other parameters were unchanged. This resulted in an extremely tenuous biofilm structure with dominant thin stacks of cells. Despite its open structure, mass transport within the biofilm occurred mostly through diffusion because the imposition of a thick boundary layer on the biomass/water interface effectively "blocked" convective transport. Whether such flow limitations occur in a real biofilm is debatable.
Two-dimensional biolilm morphology
221
concentration
high
atructure
cellilge
low
Figure I. Simulation of a dense biofilm with a Ihin boundary layer. conc:entratlon
high
atructure
cell age
low
Figure 2. Simulation with a thick boundary layer and moderate blOliIm strength.
The third simulation (Fig. 3) was carried out with 58 = 5 but the "strength" of biofilm wa~ reduced to (J/t=O.2. As a result of an increased detachment, an open biofilm structure wa~ enhanced. Bioma~s growth concentrated in somewhat bigger structures resembling "mushrooms" described by DeBeer el al. (1994a) or "tulips" characterized as fractals by Hermanowicz et al. (1995a, 1995b). The more open structure of the biofilm allowed increased substrate penetration despite a boundary layer with a large imposed thickness.
222
S. W. HERMANOWICZ
CONCLUSIONS Simulations using the developed model led to a hypothesis that external mass transfer (and especially the thickness of the boundary layer) may be an extremely imponant parameter that controls (at least to a significant ell;tent) the structure of the biofilms. The developed model is intended as a tool for funher theoretical investigations leadmg to better experimental studies. concentration high
structure
cell age
low
REFERENCES DeBeer. D. Sloooley. P.. Roe. F. and Lewandowski. Z. (1994a). Effecls of biofilm structures on oxygen dlslnhullOn and mass tran~pon BlOluh. BltHnxnx. 43.1131-1138. DeBeer. D. Sloodley. P and Lewandowski. Z. (l994b). LiqUid now 10 heterogeneous blofilms. BiOluh. Bioenxnx. 44.636-641. De Beer. D. Sloodley. P and LewandowskI. Z. (1996). LiqUid now and mass Iranspon 10 helerogeneous hlOfllms Watu Re,. 30. 2761-276~
Hermanow,c7. S W.. SchlOdler. U. and W,lderer. P. (l995a) Fraclal structure of hlofilms new tool~ for IOvesllgatlOn of morpho!l'l/Y Wal SCI. Tuh.. 3Ull). 99-105. Hermanowlc/. S W.• SchlOdler. U and Wllderer. P. (1995hl AOIsolropic morphology and fractal dimenSions of tllofilms Wat. Re.I.. 30. 753-755. Kugaprasatham. S.. Nagaoka. H. and Ohgakl. S. (1992). Effects of turbulence on nllnfylOg biofilms at non-!lmlllng suhslrate cnndlllllns Wal Res. 26. 1629-1638. Lawrence. J. R. Korher. D. R. Hoyle. B. D. Coslenon. J. W. and Caldwell. D. E. (1991). Optical secllOnlOg 01 mlcrohlal hlOhlm, J Raclu",l. 173. 65511-6567 Ma"oldcya. A A. Whallon J . Hickey R F and TledJe. J M. (1995). Channel structures 10 aerohlc hlOfilm, of fixed-film rcactor, trcatlOg conlamlOated groundwater Appl. Env. M.emblOl.. 61. 769-777 S,cgml. Hand (;ulcr. W (I'IH5l Ma" Iran,fer mechaOlsm 10 a heterotrophiC hlofilm. Wal Rrs. 19, 136'1-I17H
Pergamon
War. Sci. Tech. Vol. 37, No. 4-5, pp. 219-222, 1998. iCI 1998 IAWQ. Published by Elsevier Science Ltd
PH: 50273-1223(98)00 110-3
Printed in Great Britain. 0273-1223/98 $19'00 + 0·00
A MODEL OF TWO-DIMENSIONAL
BIOFILM MORPHOLOGY Slawomir W. Hermanowicz
Department of Civil and Environmental Engineering, University ofCalifornia. Berkeley, CA 94720-17/0, USA
ABSTRACf A mathematical model of biofilm development was proposed. The model, which can be classified as a cellular automaton, is based on simple local rules for growth and detachment of individual cells. Results of several numerical simulations suggest that the thickness of concentration and hydrodynamic boundary layers may have an imponant effect on the developing blOfilm structure. When external mass transfer limitations are significant, model biofilms develop an open structure. When the concentration boundary layer is reduced and external mass transfer is enhanced, a dense layer of a biofilm develops. Biofilm strength to withstand erosion has a smaller but also significant effect on its structure. © 1998 IAWQ. Published by Elsevier Science Ltd
KEYWORDS Biofilm modeling; cellular automaton; biofilm structure. BIOFILM STRUcrURE Biofilms are found in many natural and engineered environments where, commonly, they are a dominant form of microbial communities. A wide variety of biofilm structure and resulting morphologies have been reported in the literature (e.g., Siegrist and Gujer, 1985; Lawrence et al., 1991; Kugaprasatharn et aI., 1992; Massoldeya et al., 1995, DeBeer et al., I994a). Biofilms range from compact and smooth to open, patchy and nuffy, often forming complicated shapes called "mushrooms," "stacks" or "tulips." At present, factors affecting biofilm structure are only surmised and even a methodology adequate for characterization of such varied morphology has not yet been formulated. However, important insights can be gained from modeling of biofilm development. At this time modeling can be useful in identification of general trends in biofilm development such as growth and (hopefully) structure but its predictive capacities are severely limited by the lack of understanding of microbial metabolism and other processes occurring inside the complex ecosystem of a biofilm. Thus, while useful for pointing directions of further studies, modeling results must be ultimately tested experimentally. MODEL OF BIOFILM DEVELOPMENT Classical approaches to modeling assumed that a biofilm could be represented by a homogeneous structure that is a site for biochemical processes (such as substrate utilization) and itself undergoes transformations (e.g., growth or detachment). Such approaches led to formulation of mass balances with phenomenological description of transport and transformation processes. However, it is possible to approach biofilm modeling 219
220
S. W. HERMANOWICZ
from a different perspective. For modeling purposes, a biofilm is represented as a two- or three-dimensional array of "cells". Each cell may be occupied by biomass or be filled with water. The size of each model cell depends on the modeling scale and may correspond to a single microbial cell or an aggregate of microbial cells. The state of each model cell (i.e., whether it is occupied by biomass or not) is a dynamic variable that changes according to prescribed rules. Thus, any biofilm structure that emerges in such a model is a result of local interactions rather than an a priori description imposed on the model. The objective of this work is to show how these general features can form a framework leading to the development of variable biofilm morphology. In particular. the implementation of simple development rules on a small scale (such as a single cell or small aggregate) results in the formation of self-organized structures. In this work. a simple two-dimensional model was constructed. At every time step, each model cell occupied by biomass could grow or detach according to a set of rules. A simulation started with a thin layer of occupied cells placed on a flat substratum and exposed to water with the bulk dimensionless substrate concentration Ca' The thickness of a boundary layer 5 a adjacent to the surface of the occupied cells was specified in units of cell size. It was assumed that outside the boundary layer, substrate concentration remained constant at Ca while inside the boundary layer and the biomass substrate was transported by diffusion. This picture of mass transport agrees qualitatively with the results of DeBeer et al. (l994b) who showed that convective mass transport is possible outside biomass clusters. DeBeer et al. (1996) measured concentrations of oxygen in a boundary layer around a biofilm and showed that its thickness depended on bulk flow velocity. As a result of this mass transport mechanism each cell in the model was exposed to a local dimensionless substrate concentration C S Ca. The probability that an occupied cell would divide and grow was given by an expression CI(I+C). If Monod growth kinetics is assumed then the dimensionless concentration C is equal to the actual concentration divided by the half-saturation constant. If a cell divides then an adjacent cell becomes occupied thus resulting in biomass growth. The growth occurred in the direction of least resistance given by the smallest distance from the growth site to the biofilm surface. All occupied cells lying in this direction were pushed by the growing cell. Finally, occupied cells on the surface of the biofilm (i.e., exposed to water) could detach under the influence of hydrodynamic shear stress. Probability that a cell on a surface would detach was given by the following expression: 1/(1 + a/t) where t is the shear stress and a denotes biofilm strength. Shear stress was assumed to be inversely proportional to 5a . Since only a dimensionless value of a/t was used in the simulations, no actual values of t were required in numerical simulations. NUMERICAL SIMULATIONS The results of three numerical simulations are shown in Figs I - 3. Each figure presents the results after 18 growth cycles. The structure of the biofilm is shown in the lower left comer with biomass in black and water in white. The other three parts of each figure show the cell growth rate. substrate con- centration and cell age with a white color representing the highest values. All of these simulations were carried out with the same values of intrinsic microbial kinetic coefficients. substrate diffusivity and bulk liquid concentration. Only two parameters were varied: boundary layer thickness and biofilm strength. In the first case (Fig. I) the thickness of the boundary layer was small (58 I) and strength of biofilm moderate (a/t 0.6). In this case the biofilm developed a dense. homogeneous structure with a relatively smooth surface.
=
=
=
In the second simulation, (Fig. 2) the thickness of the boundary layer was increased to 58 5 and other parameters were unchanged. This resulted in an extremely tenuous biofilm structure with dominant thin stacks of cells. Despite its open structure, mass transport within the biofilm occurred mostly through diffusion because the imposition of a thick boundary layer on the biomass/water interface effectively "blocked" convective transport. Whether such flow limitations occur in a real biofilm is debatable.
Two-dimensional biolilm morphology
221
concentration
high
atructure
cellilge
low
Figure I. Simulation of a dense biofilm with a Ihin boundary layer. conc:entratlon
high
atructure
cell age
low
Figure 2. Simulation with a thick boundary layer and moderate blOliIm strength.
The third simulation (Fig. 3) was carried out with 58 = 5 but the "strength" of biofilm wa~ reduced to (J/t=O.2. As a result of an increased detachment, an open biofilm structure wa~ enhanced. Bioma~s growth concentrated in somewhat bigger structures resembling "mushrooms" described by DeBeer el al. (1994a) or "tulips" characterized as fractals by Hermanowicz et al. (1995a, 1995b). The more open structure of the biofilm allowed increased substrate penetration despite a boundary layer with a large imposed thickness.
222
S. W. HERMANOWICZ
CONCLUSIONS Simulations using the developed model led to a hypothesis that external mass transfer (and especially the thickness of the boundary layer) may be an extremely imponant parameter that controls (at least to a significant ell;tent) the structure of the biofilms. The developed model is intended as a tool for funher theoretical investigations leadmg to better experimental studies. concentration high
structure
cell age
low
REFERENCES DeBeer. D. Sloooley. P.. Roe. F. and Lewandowski. Z. (1994a). Effecls of biofilm structures on oxygen dlslnhullOn and mass tran~pon BlOluh. BltHnxnx. 43.1131-1138. DeBeer. D. Sloodley. P and Lewandowski. Z. (l994b). LiqUid now 10 heterogeneous blofilms. BiOluh. Bioenxnx. 44.636-641. De Beer. D. Sloodley. P and LewandowskI. Z. (1996). LiqUid now and mass Iranspon 10 helerogeneous hlOfllms Watu Re,. 30. 2761-276~
Hermanow,c7. S W.. SchlOdler. U. and W,lderer. P. (l995a) Fraclal structure of hlofilms new tool~ for IOvesllgatlOn of morpho!l'l/Y Wal SCI. Tuh.. 3Ull). 99-105. Hermanowlc/. S W.• SchlOdler. U and Wllderer. P. (1995hl AOIsolropic morphology and fractal dimenSions of tllofilms Wat. Re.I.. 30. 753-755. Kugaprasatham. S.. Nagaoka. H. and Ohgakl. S. (1992). Effects of turbulence on nllnfylOg biofilms at non-!lmlllng suhslrate cnndlllllns Wal Res. 26. 1629-1638. Lawrence. J. R. Korher. D. R. Hoyle. B. D. Coslenon. J. W. and Caldwell. D. E. (1991). Optical secllOnlOg 01 mlcrohlal hlOhlm, J Raclu",l. 173. 65511-6567 Ma"oldcya. A A. Whallon J . Hickey R F and TledJe. J M. (1995). Channel structures 10 aerohlc hlOfilm, of fixed-film rcactor, trcatlOg conlamlOated groundwater Appl. Env. M.emblOl.. 61. 769-777 S,cgml. Hand (;ulcr. W (I'IH5l Ma" Iran,fer mechaOlsm 10 a heterotrophiC hlofilm. Wal Rrs. 19, 136'1-I17H