A model-based Bayesian estimation of the rate of evolution of VNTR loci in Mycobacterium tuberculosis

Abstract

Variable numbers of tandem repeats (VNTR) typing is widely used for studying the bacterial cause of tuberculosis. Knowledge of the rate of mutation of VNTR loci facilitates the study of the evolution and epidemiology of Mycobacterium tuberculosis. Previous studies have applied population genetic models to estimate the mutation rate, leading to estimates varying widely from around 10⁻⁵ to 10⁻² per locus per year. Resolving this issue using more detailed models and statistical methods would lead to improved inference in the molecular epidemiology of tuberculosis. Here, we use a model-based approach that incorporates two alternative forms of a stepwise mutation process for VNTR evolution within an epidemiological model of disease transmission. Using this model in a Bayesian framework we estimate the mutation rate of VNTR in M. tuberculosis from four published data sets of VNTR profiles from Albania, Iran, Morocco and Venezuela. In the first variant, the mutation rate increases linearly with respect to repeat numbers (linear model); in the second, the mutation rate is constant across repeat numbers (constant model). We find that under the constant model, the mean mutation rate per locus is 10⁻²·⁰⁶ (95% CI: 10⁻²·⁶¹,10⁻¹·⁵⁸)and under the linear model, the mean mutation rate per locus per repeat unit is 10⁻²·⁴⁵ (95% CI: 10⁻³·⁰⁷,10⁻¹·⁹⁴). These new estimates represent a high rate of mutation at VNTR loci compared to previous estimates. To compare the two models we use posterior predictive checks to ascertain which of the two models is better able to reproduce the observed data. From this procedure we find that the linear model performs better than the constant model. The general framework we use allows the possibility of extending the analysis to more complex models in the future.

Figure 1. Marginal posterior distributions for and using simulated data.

Plots show the marginal posterior distribution of (left) and (right) using four simulated data sets generated from the constant (left) and linear (right) VNTR models. The known values of and used to generate the data, and , are indicated by vertical lines.

Figure 2. Genetic diversity of VNTR loci for each published dataset.

Left plots: Empirical cumulative distribution function of gene diversity across loci. The gene diversity is computed at each locus as where is the number of isolates with repeat size at locus . Right plots: Heat-map diversity, following Aminian et al (2009), illustrating the proportion of tandem repeats for each locus (ordered according to the original study).

Figure 3. Marginal posterior estimates for , and .

Here is the per-locus mutation rate for a locus with a single repeat under the linear model; is the same quantity scaled by the mean number of repeats observed in the sample; is the per-locus mutation rate for any repeat number under the constant model.

Figure 4. Posterior predictive model checks.

Scatterplots of the posterior predictive distributions of (the difference between maximum and minimum range of repeat numbers over loci), versus (the same quantity substituting variance for range). Columns represent constant (left) and linear (right) models. Rows represent the Albanian dataset (top), artificially generated data from the constant model (middle) and artificially generated data from the linear model (bottom). The indicates the statistics derived from the observed dataset.

Figure 5. Further posterior predictive model checks.

Scatterplots of the posterior predictive distributions of (the maximum range of repeat numbers over loci) versus (the intercept at one repeat) under the linear model, for each observed dataset. The indicates the statistics derived from the observed dataset.