Combining mathematical models and statistical methods to understand and predict the dynamics of antibiotic-sensitive mutants in a population of resistant bacteria during experimental evolution

Abstract

Temporarily discontinuing the use of antibiotics has been proposed as a means to eliminate resistant bacteria by allowing sensitive clones to sweep through the population. In this study, we monitored a tetracycline-sensitive subpopulation that emerged during experimental evolution of E. coli K12 MG1655 carrying the multiresistance plasmid pB10 in the absence of antibiotics. The fraction of tetracycline-sensitive mutants increased slowly over 500 generations from 0.1 to 7%, and loss of resistance could be attributed to a recombination event that caused deletion of the tet operon. To help understand the population dynamics of these mutants, three mathematical models were developed that took into consideration recurrent mutations, increased host fitness (selection), or a combination of both mechanisms (full model). The data were best explained by the full model, which estimated a high mutation frequency (lambda = 3.11 x 10(-5)) and a significant but small selection coefficient (sigma = 0.007). This study emphasized the combined use of experimental data, mathematical models, and statistical methods to better understand and predict the dynamics of evolving bacterial populations, more specifically the possible consequences of discontinuing the use of antibiotics.

Figure 1.—

HindIII RFLP patterns of ancestral and evolved plasmids. Lanes 1 and 7, 1-kb extended ladder, 5 μg and 10 μg, respectively; lane 2, ancestral pB10; lanes 3 and 4, plasmid DNA from Tc^S mutants; lanes 5 and 6, plasmid DNA from evolved Tc^R clones. Arrow indicates missing band in Tc^S mutants. Colors were inverted and brightness and contrast were adjusted to enhance the quality of the image.

Figure 2.—

Plot of the joint 94.8% bootstrapped confidence region for both the mutation rate λ and the selection coefficient σ. The solid diamonds represent the joint parameter estimate of the mutation rate λ and the selection coefficient σ.

Figure 3.—

Population dynamics of the tetracycline-sensitive mutants. (A) Time course of the observed (•) and predicted (—) average fractions of mutants under each of the three mathematical models. (B) Ratios of observed vs. predicted fractions of mutants under each model. The solid lines indicate identical observed and predicted values. All replicate and average values are reported in Table 1.

Figure 4.—

Pictorial representation of the bootstrap likelihood-ratio test (LRT) outcomes for each mathematical model. The plotted density is the bootstrap sampling distribution of the LRT, −2 ln Λ. The P-value (pBoot) is the area under the curve to the right of the dashed vertical lines (corresponding to the −2 ln Λ observed value). It represents the probability that a LRT value greater than or equal to the one observed would actually occur given that the hypothesized model is true.