F fuS MRT 1 F fwS MRT

Not surprisingly, in the absence of horizontal transmission p = p = 0, a parasitic plasmid cannot satisfy the persistence condition since F = F+ in that case. In the interesting case that p,p > 0, we cannot assert that F = F+ since F+ depends on p and p. However, it is reasonable to speculate that F - F+ is small. Putting F = F+, (24) for plasmid persistence becomes:

MRT • [F • pu + (1 - F) • pw] > 1 - (1 - c)(1 - q) = c + q - cq (26)

The left hand side of the inequality gives the number of plasmid transfers made by a single plasmid-bearing cell before being washed out of the chemo-stat. It must exceed a positive threshold which depends on the cost of carriage c and the probability of miss-segregation q for the plasmid to survive.

Equation (26) indicates that the conjugation terms must exceed a threshold to maintain a parasitic plasmid. In order to see this more clearly, we fix a = a+ = 0.1 and ¡3 = ¡3+ = 0.4, with all other parameters as in Fig. 6.4, except that p = p x 10-3. We then vary p and plot the resulting stable steady state value of u+ + Sw+ in Fig. 6.5. This was done by integrating the differential equations, starting with a tiny inoculum of plasmid-bearing cells, to steady state. If u+ + Sw+ = 0, as it does for small p, that means the stable steady state is the plasmid-free state; if u+ + Sw+ > 0, then we are plotting the coexistence steady state value of u+ + Sw+. This bifurcation diagram shows that the critical value pc « 6 x 104 at which the coexistence steady state appears is much larger than our order of magnitude estimate of a biologically reasonable value of p « 1.

We are particularly interested in the case that the plasmid-free organism cannot form a macroscopically significant biofilm, for example, this may mean that it can form only a monolayer (see e.g. Pratt and Kolter (1998) and O'Toole and Kolter (1998)), while the plasmid-bearing organism can form a healthy biofilm. In this case, it is reasonable to assume that the plasmid-bearing organisms sloughing rate does not exceed that of the plasmid-free organism and that its adhesion rate constant is not less than that of the plasmid-free organism:

Strict inequality is assumed to hold in at least one of these. In this case, as noted in the appendix, the plasmid-bearing organism has a residence time

Fig. 6.5. Bifurcation of coexistence steady state from the plasmid-free steady state at a critical value of ^. Vertical axis is the coexistence steady state value of total plasmid-bearing biomass u+ + Sw+. Horizontal axis is the conjugation rate n

advantage over its plasmid-free rival:

and it should spend more of this time on the wall than the plasmid-free organism so we conjecture that:

where, presumably, bacterial densities are higher than in the fluid state and contact rates between organisms are higher:

(24) says that these advantages must outweigh the cost of carriage and seg-regational loss.

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