Stochastic effects on the genetic structure of populations

Basic problem

The genetic structure of natural populations is strongly affected by random genetic drift: random effects can destroy the genetic diversity built up by mutation, counteract the effect of selection, and build up statistical associations between different loci. Therefore, a proper understanding of most questions in evolutionary biology requires random effects to be taken into account.

General approach

Use simple population genetic models that include mutation, selection, recombination (in the advanced part) and random sampling of offspring, in order to obtain an understanding of the interplay between these different factors.

What can be learned

Concepts

• Population genetic models (how to model if your primary interest is gene frequencies)
• Simulation of stochastic models (sampling methods, random numbers)
• Potential evolutionary benefits of recombination

Methods

• Iteration of discrete generation models
• Sampling via binomially (or multinomially) distributed random numbers
• Histograms as approximations of frequency distributions

Starting point

Download reader and scripts describing the equations for the basic and the extended model. Plot different time courses of the allele frequencies; vary parameters to obtain a feeling for their importance; then try to answer the questions below (detailed version in the reader).

Interesting questions that you can investigate

• How does drift reduce the diversity that mutation builds up?
• How does drift affect the elimination of detrimental alleles through selection?
• How do bottlenecks affect the diversity at neutral and selected loci?
• What do effective population sizes tell about the magnitude of stochastic effects?

Advanced questions:

• For which parameters does drift create conditions under which recombination accelerates the fixation of beneficial mutations at several loci? (Applied version: When does recombination accelerate the emergence of double or multiple resistance mutations?)
• What is the relative importance of epistasis and drift for the effect of recombination?

Glossary

• Ne: the effective population size.  Gives the size of an ideal population (e.g. according to the Wright-Fisher model) in which the stochastic effects (on a given quantity) have the same strength as in the natural population under consideration. The concept of an effective population size is not unproblematic, mainly because Ne depends on the quantity of interest, see (Kouyos et al. 2006).
• Deterministic models: models that assume that the current state of a population fully determines the behaviour of the population in the future. Deterministic models are often appropriate when populations are large.
• Stochastic models: models that assume that the future of a population is determined in part by its current state and in part by random effects. Random effects due to sampling increase with decreasing population size and therefore stochastic models are often more appropriate than deterministic models when populations are small.
• Wright-Fisher model (WFM): The WFM describes discrete and non-overlapping generations in a population with fixed size N. Every generation, each of the N genomes undergoes mutation with probability mu. Then the N genomes for the next generation are determined by drawing every offspring genome with uniform probability from the N parental genomes. Note that the WFM assumes selective neutrality. In this module we consider an "extended" WFM that allows for selection.
• Genetic drift: Change of genotype frequencies due to random sampling.
• Binomial distribution: If N bowls are drawn randomly from an urn containing black and white bowls with fractions p and 1-p, respectively, then the binomial distribution with parameters N and p gives the distribution of the number of black bowls drawn. In general, this distribution describes the occurrences of an event in a series of N experiments that test an event which occurs with probability p in each individual experiment.
• Multinomial distribution: Analogous to drawing from an urn containing bowls of k>2 different colors (e.g. red, blue, green, etc). The multinomial distribution with parameters N, k, and p=(p1,p2,…,pk) gives the probability of drawing a given number of red, blue, green etc. bowls. In general, the distribution describes the outcome of a series of repeated experiments that have exactly k distinct possible outcomes that occur with the given probabiities in each repetition.

Literature

For stochastic effects in HIV-1:

• Kouyos, R. D., Althaus, C. L. and Bonhoeffer, S. (2006). Stochastic or deterministic: what is the effective population size of HIV-1? Trends in Microbiology 14: 507-511.
• Nijhuis, M., Boucher, C. A., Schipper, P, Leitner, T., Schuurman, R. et al. (1998). Stochastic processes strongly influence HIV-1 evolution during suboptimal protease-inhibitor therapy. Proc Natl Acad Sci U S A 95: 14441-14446.

For the interplay between stochastic effects and the evolution of recombination:

• Althaus, C. L. and Bonhoeffer, S. (2005). Stochastic interplay between mutation and recombination during the acquisition of drug resistance mutations in human immunodeficiency virus type 1. J Virol 79: 13572-13578.
• Bretscher, M. T., Althaus, C. L., Müller, V. and Bonhoeffer, S. (2004). Recombination in HIV and the evolution of drug resistance: for better or for worse? Bioessays 26: 180-188.
• Otto, S. P. and Lenormand, T, 2002. Resolving the paradox of sex and recombination. Nature Reviews Genetics 3: 252-261.

And a recent experimental demonstration of founder effects (in lizards):

• Kolbe, J. J., Leal, M., Schoener, T. W., Spiller, D. A. and Losos, J. B. (2012). Founder effects persist despite adaptive differentiation: a field experiment with lizards.