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Figure 15.2 A Iransmission electron micrograph of a Pseudomonas aeruginosa biofilm consisting almost entirely of a base film. (From P. A. Wilderer and W. G. Characklis, Structure and function of biofilms. In Structure and Function of Biofilms, W. G. Characklis and P. A. Wilderer, eds. Wiley, New York, pp. 5-17, 1989. Reprinted by permission of John Wiley & Sons, Inc.)

between the bulk fluid and the biofilm, on the other hand, is dominated by advection and turbulent diffusion.Is These concepts dominate all mathematical models today.

Due to the development of new tools for the study of biofilms, a different picture of the internal structure of the base film is emerging.'""12 Figure 15.3 is an artist's conceptualization of the architecture of a biofilm based on the observations of several researchers.1' Biofilms now appear to be nonuniform structures consisting of discrete cell clusters attached to each other and to the solid support with extracellular polymeric material.1'The spaces between clusters form vertical and horizontal voids, with the vertical voids acting as pores and the horizontal voids acting as channels. As a result, biomass distribution within a biofilm is not uniform,"nor are physical factors such as porosity and density."' The cell clusters are microbial aggregates cemented with extracellular polymeric material, whereas the voids are open structures relatively free of it. The significance of the voids is that liquid can flow through them.1" This has a profound effect on mass transfer in the biofilm because it suggests that it can occur by both diffusion and advection, with diffusion dominating in the cell clusters. However, because advection brings materials to the clusters, diffusion can occur from almost any direction into a cluster, rather than just from the liquid-biofilm interface. In addition, it also appears that the cell clusters have small conduits through them, adding another level of complexity to the biofilm." Finally, many factors, such as the texture of the substratum, the nature of the flow past the biofilm, and the geometry of the bioreactor, influence the heterogeneity of the biofilm that develops.1" These observations suggest that the commonly accepted use of a single transport parameter, such as an effective diffusion coefficient,

Figure 15.3 Artist's conceptualization of the architecture of a biofilm. (From J. W. Cos-terton, Z. Lewandowski, D. F. Caldwell, D. R. Korber, and H. M. Lappin-Scott, Microbial bioftlms. Annual Review of Microbiology 49:711-745, 1995. Copyright © Annual Reviews, Inc.; reprinted with permission.)

for describing the transport of substrate, electron acceptor, etc. within a biofilm is inadequate.11 12 In fact, various researchers have shown that effective diffusion coefficients vary with biofilm depth,"'"'"1' which is consistent with changes in the structure of the biofilm. Nevertheless, because most current mathematical models for biofilm reactors assume that transport within a biofilm is by diffusion alone with a constant diffusion coefficient, that is the approach we will take herein. However, the reader should be aware of the limitations of such an approach. Recent mathematical models attempt to consider the variability in the diffusion coefficient," suggesting that different approaches to modeling transport within the biofilm will be used in the future.

The conceptual models presented above are for a simple heterotrophic biofilm in which the bacteria are using a single electron donor with a single electron acceptor. However, just as heterotrophic and autotrophic bacteria can grow together in suspended growth bioreactors, they can also grow together in attached growth reactors. In this instance they have different electron donors (organic matter and ammonia-N), but compete for the same electron acceptor (oxygen). They also must compete for space in the biolilm. The assumed spatial arrangements of the competing species within the biofilm can take several forms in mathematical models.1'' However, the most realistic approach assumes that all types of bacteria are available for growth at any point within a biofilm, but that their ultimate distribution is determined by their competition for shared nutrients and space,lv2:24 ' which is consistent with observation.'"" Although the mathematical models for this competition were developed before the advent of the conceptual model in the preceding paragraph, we will consider them because of the importance of the interactions between heterotrophs and autotrophs in attached growth reactors. Multispecies models have also been developed for methanogenic cultures containing three trophic levels. However, because of space constraints, they will not be covered here.

The importance of competition for space in determining the ultimate distribution of competing species within a biofilm can be visualized by considering the traditional conceptualization of a base biofilm. Consider first a single species biofilm. Because substrate can only move into the biofilm by diffusion, a substrate concentration gradient will exist through the biofilm as illustrated by Figure 15.4. This means that bacteria near the liquid-biofilm interface are growing faster than those in the interior. However, as bacteria in the interior grow, they occupy more space, pushing those that are closer to the liquid-biofilm interface further away from the solid support. In addition, all of the bacteria are subject to decay, regardless of their position in the biofilm, resulting in the accumulation of biomass debris. The net effect of both processes is to cause a migration of particlcs from the interior of the film to the exterior where surface shear forces remove them, allowing a biofilm of constant thickness to develop. Even for a single species biofilm, however, the distribution of active organisms will not be the same throughout the depth of the biofilm.4" Rather, active biomass will predominate in the outer regions of the film and biomass debris in the inner regions, as shown by the simulation results in Figure 15.5.w

If we have two species that do not compete for any nutrient, but only for space, their ultimate distribution will depend upon their relative specific growth rates at any point within the biofilm. Consider two species, A and B, growing on different substrates, but sharing oxygen as the clcctron acceptor. Oxygen is assumed to be present in excess, so as not to limit either species. Species A has a higher maximum specific growth rate coefficient on its substrate than species B does on its substrate. Species A will dominate the outer regions and species B will dominate the inner regions, as shown by the simulation results in Figure 15.6."' Species B is confined to the inner regions because there the substrate concentration for species A will have been diminished sufficiently to allow species B to grow as fast as, or faster than, species A. When the two species compete for a resource, such as oxygen, the distribution of organisms can become even more complex, depending upon the relative Ks values for the shared resource, as well as the growth kinetics of each species on its individual substrate.

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