Some Consequences of Demographic Stochasticity in Population Genetics

By Parsons, Todd L.; Quince, Christopher et al. | Genetics, August 2010 | Go to article overview

Some Consequences of Demographic Stochasticity in Population Genetics


Parsons, Todd L., Quince, Christopher, Plotkin, Joshua B., Genetics


ABSTRACT

Much of population genetics is based on the diffusion limit of the Wright-Fisher model, which assumes a fixed population size. This assumption is violated in most natural populations, particularly for microbes. Here we study a more realistic model that decouples birth and death events and allows for a stochastically varying population size. Under this model, classical quantities such as the probability of and time before fixation of a mutant allele can differ dramatically from their Wright-Fisher expectations. Moreover, inferences about natural selection based on Wright-Fisher assumptions can yield erroneous and even contradictory conclusions: at small population densities one allele will appear superior, whereas at large densities the other allele will dominate. Consequently, competition assays in laboratory conditions may not reflect the outcome of long-term evolution in the field. These results highlight the importance of incorporating demographic stochasticity into basic models of population genetics.

(ProQuest: ... denotes formulae omitted.)

MATHEMATICAL descriptions of allele frequencies are typically built upon the Wright-Fisher model (Wright 1931; Fisher 1958) or, more accurately, its diffusion limit (Kimura 1962; Ewens 2004). This model forms the basis of Kimura's work on fixation probabilities (Kimura 1955), Ewens' sampling formula (Ewens 1972; Lessard 2007), Kingman's coalescent (Kingman 1982), tests of neutrality (Hudson et al. 1987; Tajima 1989; McDonald and Kreitman 1991; Fu and Li 1992; Fay and Wu 2000), and techniques for inferring mutation rates and selection pressures (Sawyer and Hartl 1992; Yang and Bielawski 2000; Bustamante et al. 2001).

The Wright-Fisher model has been generalized to account for a variety of complications, such as nonrandom mating, migration, and multiple loci, among others (Ewens 2004; Durrett 2009). The standard diffusion approximation of Kimura (Kimura 1962; Ewens 2004) and its coalescent (Kingman 1982) are remarkably robust to violations of the model's underlying assumptions. For example, when the population is stratified, or when the population size varies rapidly and independently of the genetic composition of the population, allele frequency dynamics are still accurately approximated by the Wright-Fisher diffusion, under a suitable change of timescale or choice of effective population size (Ewens 1967; Otto and Whitlock 1997; Wakeley 2005, 2009). In fact, most population-genetic models-including the Moran process (Moran 1958), Karlin's conditional branching process (Karlin and McGregor 1964), and some Cannings processes (Cannings 1974; Ewens 2004)- share the same diffusion limit as the Wright-Fisher model (Mühle 2001). As a result, Kimura's diffusion approximation has had an enormous impact on the development of theoretical and applied population genetics.

Despite its robustness, the Wright-Fisher diffusion is not appropriate in all circumstances. Many natural populations experience substantial stochastic variation. However, with few exceptions (e.g., Kaj and Krone 2003; Lambert 2005, 2006; Champagnat and Lambert 2007), models typically assume fixed or deterministically varying population number (e.g., Ewens 1967; Kimura and Ohta 1974; Donnelly 1986; Griffiths and Tavaré 1994; Otto and Whitlock 1997).

Here, we consider an alternative approach, inspired by Moran'smodel (Moran 1958) and the Gause-Lotka- Volterra model (Lotka 1925; Volterra 1926; Gause 1934): individuals give birth and die with rates that vary with the total population number. Populations are kept finite by density-dependent factors (e.g., resource limitation) such as those that have been empirically verified in microbial populations (Gause 1934; Vandermeer 1969; Pascual and Kareiva 1996). This approach produces stochastic variation in the population size because birth events are not immediately followed by death events. In this study, we formulate the simplest population process that incorporates such demographic stochasticity. …

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