Academic journal article Genetics

Fixation Probabilities When Generation Times Are Variable: The Burst-Death Model

Academic journal article Genetics

Fixation Probabilities When Generation Times Are Variable: The Burst-Death Model

Article excerpt


Estimating the fixation probability of a beneficial mutation has a rich history in theoretical population genetics. Typically, to attain mathematical tractability, we assume that generation times are fixed, while the number of offspring per individual is stochastic. However, fixation probabilities are extremely sensitive to these assumptions regarding life history. In this article, we compute the fixation probability for a "burst-death" life-history model. The model assumes that generation times are exponentially distributed, but the number of offspring per individual is constant. We estimate the fixation probability for populations of constant size and for populations that grow exponentially between periodic population bottlenecks. We find that the fixation probability is, in general, substantially lower in the burst-death model than in classical models. We also note striking qualitative differences between the fates of beneficial mutations that increase burst size and mutations that increase the burst rate. In particular, once the burst size is sufficiently large relative to the wild type, the burst-death model predicts that fixation probability depends only on burst rate.

(ProQuest-CSA LLC: ... denotes formulae omitted.)

ESTIMATING the fixation probability of an initially rare beneficial mutation is fundamental to our understanding of adaptation. Such estimates are critical to studies of evolution under controlled laboratory conditions and are also essential for predicting the rate of adaptation of natural populations-for example, the rate of adaptation in response to environmental change or the rate of emergence of novel, or drug-resistant, pathogens.

To date, a considerable body of theoretical literature has been devoted to this question, beginning with classic articles such as FISHER (1922) and HALDANE (1927). To attain mathematical tractability, these approaches have necessarily imposed a number of simplifying assumptions. Inparticular, thelife-historymodelsof classicalpopulation genetics typically assume that generation times are fixed, with no death between generations. However, recent work in our group (e.g., WAHL and DEHAAN 2004) has emphasized that fixation probabilities are extremely sensitive to the underlying life-history model.

The work described here is motivated, in part, by the extensive and important body of literature regarding the experimental evolution of lytic viruses (e.g., BULL et al. 1997, 2000; BURCH and CHAO 1999, 2000; WICHMAN et al. 1999; SANJUÁN et al. 2005; MANRUBIA et al. 2005). For lytic particle "searches" for a cell to infect, and a lysis time, the time between attachment and lysis, during which phage accumulate within the infected cell. Although the lysis (or "phage-accumulation") time may be tightly regulated, attachment times are well modeled by a first-order process; i.e., they are exponentially distributed (ABEDON et al. 2001).

When attachment times are very short, the standard assumption of fixed generation times is clearly valid. In this article, however, we develop a life-history model in which generation times vary. In particular, we treat the case when "death" occurs at a constant rate, and thus generation times (or, more correctly, lifetimes) are exponentially distributed. This may be an appropriate model for estimating fixation probabilities for lytic viruses, particularly when the attachment time is long compared to the phage-accumulation time, and may also be applied to other natural and experimental populations where generation times vary widely.


We have developed the "burst-death" life-historymodel in analogy to the well-studied birth-death process. There are two ways, intuitively, to derive this model. First, we can consider an individual with a constant probability of death, MΔt, in any small interval of time, Δt. An example of such a process is the clearance of free virus or bacteria from a chemostat, or, more generally, any failure of a virion to infect a new cell. …

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