Figure 1.
An illustration of the basic model, defining the three dimensionless model parameters (T, z0 and p0).
Two lineages compete for resources during the growth interval before catastrophe occurs at time T. The initial size of the focal lineage is x0, expressed as a proportion of the total carrying capacity. The initial size of the total population is z0 = x0+y0, where y0 is the initial size of the competitor lineage. The initial frequency of the focal lineage is p0 = x0/z0. Persister cells are represented by the shaded areas, and non-persister cells are unshaded. Upon the catastrophe occurring, all persister cells survive and all non-persister cells are destroyed.
Table 1.
A summary of model notation.
Figure 2.
The evolution of persister function, assuming fixed time until catastrophe.
Numerical solutions for the ESS persister allocation are given for a range of parameter values. The ESS allocation to persister function (π*) decreases as the growth time (T) before catastrophe increases, and the ESS increases with increasing resource competition (z0) and genetical relatedness (p0). Note that T→∞ does not imply infinite growth, but rather that the catastrophe occurs after resources are exhausted and growth has ceased. Also, some proportion of persisters is always favoured (i.e. π*>0), but the quantity predicted may be vanishingly small and hence appear to be zero in the figure.
Figure 3.
The evolution of persister function, with random waiting time until catastrophe.
Numerical results for the stochastic version of the model in which the probability of catastrophe occurring is at any time is constant through time, with average waiting time T̅. The ESS allocation to persister function (π*) decreases as the average growth time (T̅) before catastrophe increases, and the ESS increases with increasing resource competition (z0) and genetical relatedness (p0). Note that T̅→∞ does not imply infinite growth, but rather that the catastrophe (almost always) occurs after resources are exhausted and growth has ceased. Also, some proportion of persisters is always favoured (i.e. π*>0), but the quantity predicted may be vanishingly small and hence appear to be zero in the figure.
Figure 4.
The evolution of persister function, with less extreme differences in persister and nonpersister survival and growth, and differences in efficiency of resource use.
(A) ESS persister allocation (π*) is a decreasing function of the relative survival (s) of nonpersister cells. We assume: p0 = 1, T = 5, g = 0, a = 1 and a range of z0. (B) ESS persister allocation (π*) is an increasing function of the relative growth rate (g) of persister cells. We assume: p0 = 1, T = 5, s = 0, a = 1 and a range of z0. (C) ESS persister allocation (π*) may depend on the relative competitive strain on resources imposed by persister cells (a), but this is negligible when they appear only infrequently in bacterial populations (low π*). We assume: p0 = 1, T = 5, s = 0, g = 0 and a range of z0.