Mathematical Model of Gene Transfer in a Biofilm

Mudassar Imran and Hal L. Smith

Summary. Based on our previous work, a model of plasmid transfer between micro-organisms in a heterogeneous environment consisting of a biofilm immersed in a fluid medium is constructed. A review of previous modeling of gene transfer is provided in order to place our work in context. The key question is whether the plasmid can persist in the bacterial population. We answer this question by constructing a basic reproductive number which takes into account the advantages conferred by the plasmid and its costs to the bacterial host.

6.1 Introduction

Plasmids, small circular strands of DNA separate from the main genome of the organism, are common in natural bacterial populations such as soils, lakes and stream and in the gut of mammals. They often carry genes for such beneficial factors as resistence to antibiotics and heavy metals, the ability to ferment sugars, or to produce toxins. Some carry genes for pili production and mating pair formation that allow the infectious transfer of the plasmid to other bacteria - a process called conjugation. However, many plasmids have no known function in bacteria and may simply be parasitic. Vertical transmission of plasmids occurs during cell division when the plasmids in the cell are duplicated and partitioned among the daughter cells; rarely, however, one daughter cell may end up without plasmid while the other daughter cell receives multiple copies. This loss of plasmid is referred to a segregative loss. Furthermore, there is some cost to an organism carrying plasmids since the cell may produce plasmid gene products and must duplicate it during cell division which leads to a reduced reproductive rate. See Simonsen (1991) for a readable review, particularly of modeling aspects, and Summers (1996) for a general review. Because the benefits of plasmid carriage, if any, depend on ever-changing environmental conditions while the costs are always present, a longstanding focus of theoretical studies has been to determine conditions under which plasmids can be maintained in bacterial populations. See Stewart and Levin (1977), Levin and Rice (1980) and Bergstrom, Lipsitch, and Levin (2000) for modeling results related to this problem. According to Angles and Goodman (2000):

Biofilms are environments of high microbial density where cell-cell contact is likely. Such conditions create a favorable niche for the spread of self-transmissible as well as mobilisable plasmids among members of the bacterial communities. Studies have demonstrated plasmid transfer among bacteria in a wide range of biofilm habitats, including the surface of stones in a river, the air-water interface, surfaces in soil and water microcosms, plant surfaces and insect as well as animal intestinal surfaces.

In a recent paper Ghigo (Ghigo 2001) established that several natural con-jugative plasmids express factors that induce some bacteria to form biofilms. Experimental studies showed that a strain of E. Coli bearing a certain plas-mid formed a thick biofilm within one day while those not carrying the plas-mid produced no macroscopically observable biofilm. Interestingly, Ghigo's results suggest that the pili responsible for the horizontal transfer of the plasmid, may also act as an adhesion factor for cell-to-surface contact. See also Pratt and Kolter (1998) and O'Toole and Kolter (1998). Ghigo points out the many beneficial aspects for bacteria in biofilms relative to the fluid environment and speculates that such factors "may provide a rationale for the unexplained vertical maintenance of the numerous uninfectious cryptic plasmids found in natural populations". He also observes that by inducing bacteria to form the denser communities characteristic of biofilms the plas-mid increases the likelihood of its own horizontal transfer via conjugation.

In this chapter, we explore the suggested link between plasmid maintenance and biofilms by modifying slightly the mathematical model proposed by us in (Imran et al. preprint) of a bacterial population consisting of plasmid-bearing and plasmid-free organisms in a continuous culture with a surface on which a biofilm may form. The question we address is under what circumstances can the plasmid be maintained in a population. Heuristically, the advantageous genes carried by the plasmid together with the ability of the plasmid-bearing organism to pass the plasmid to other organisms must compensate for the energetic cost of bearing the plasmid and the occasional segregative loss of the plasmid during cell division. We seek to quantify this trade-off. Our model builds on the plasmid model of Stephanopoulus and Lapidus (1988) and Ryder and DiBiasio (1984), includes conjugation terms used by Stewart and Levin (1977), and models the biofilm following the model of Pilyugin and Waltman (1999). Consequently, we briefly review these models in order that the foundation of our model is made more clear.

Two cases are considered: (1) the plasmid is parasitic, conferring no advantage on its host, and (2) the plasmid codes for enhanced biofilm forming ability in its bacterial host which in its absence can form only a macro-scopically unobservable biofilm. In the first case, the question is under what circumstances can a parasitic plasmid can be maintained. In the second case, the question is under what circumstances can the ability to form a robust biofilm community in which conjugative transfer of the plasmid may occur be sufficiently advantageous for the plasmid-bearing organism to compensate for the energetic cost of bearing the plasmid and the segregative loss of the plasmid. In each case, we provide a quantitative expression of a potential mechanism which may be significant in plasmid maintenance.

Our work corroborates the conjectures of Ghigo. The availability of colon-alizable surfaces that provide a selective advantage for an organism carrying a plasmid containing a biofilm-enhancing gene may contribute to the maintenance of such plasmids in natural bacterial populations.

The same models developed in this paper could also be used to study the important phenomena of horizontal spread of antibiotic resistance in the gut. Rather than assuming the plasmid codes for enhanced biofilm forming ability one would assume that it codes for antibiotic resistance. Selection for the resistant strain could, of course, be arranged by adding antibiotic. Ingestion of bacteria containing plasmid coding for antibiotic resistance could lead to the spread of resistance to the gut microflora. This phenomena may play a significant role in the proliferation of antibiotic resistant pathogens (Summers 1996).

6.2 A model of plasmid transfer with wall growth

We consider a population of bacteria in a continuous culture which colonize both the fluid environment and a portion of a surface immersed in the

Table 6.1. Model parameters for the chemostat: t =time, m = mass, l =length

Symbol

Description

Dimension

u, u+

biomass concentration of planktonic bacteria.

ml-3

w, w+

areal biomass density of adherent bacteria.

ml-2

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