Probability Models for DNA Sequence Evolution

Durrett, Richard

doi:10.1007/978-1-4757-6285-3

Cited by 147 publications

(121 citation statements)

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“…The effect of logistic growth (relative viability) at a selected locus (i.e., the frequency of the beneficial mutation increases logistically approaching the population size, which is constant) on the level of neutral polymorphism has been studied by Stephan et al (1992). Of particular interest is the estimate of the fixation time 2 ln(2N) s (Campbell, 1999;Durrett, 2002) under logistic growth. Qualitatively, logistic growth begins like exponential growth, but slows down at the end, hence will increase both the coalescent time and the cumulative size of the coalescent.…”

Section: Logistic Increase Of Favored Allelementioning

confidence: 99%

Coalescent Size Versus Coalescent Time with Strong Selection

Campbell

2007

Bull. Math. Biol.

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This work studies the coalescent (ancestral pedigree, genealogy) of the entire population. The coalescent structure (topology) is robust, but selection changes the rate of coalescence (the time between branching events). The change in the rate of coalescence is not uniform, rather the reduction in the time between branching events is greatest when the coalescent is small (immediately after the common ancestor is the only member of the coalescent) with little change when the coalescent is large (immediately preceding when that common ancestor becomes fixed and the size of the coalescent is N). This provides that the reduction in the coalescent time due to selection is much greater than the reduction in the cumulative size of the coalescent (total number of ancestors of the present population after and including the most recent common ancestor) due to selection. If Ns>>1, the coalescent and fixation times are approximately equal to [Formula: see text] , which is much less than the value N which would result from neutral drift (N rather than the canonical haploid neutral fixation time 2N is the appropriate comparison for the model considered here), in particular, it is 70% less for Ns=10 and 95% less for Ns=100. However, for those values of Ns, and N ranging between 10(3) and 10(6), the reduction in the cumulative size of the coalescent of the entire population compared to the neutral case ranges from 17% to 65% (depending on the values of N and s). The coalescent time for two individuals for Ns of 10 and 100 is reduced by approximately 70% and 94%, respectively, compared with the neutral case. Because heterozygosity is proportional to the coalescent time for two individuals and the number of segregating alleles is proportional to the cumulative size of the coalescent, selection reduces heterozygosity more than it reduces the number of segregating alleles.

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Section: Logistic Increase Of Favored Allelementioning

confidence: 99%

Coalescent Size Versus Coalescent Time with Strong Selection

Campbell

2007

Bull. Math. Biol.

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“…To do this, we assume every death event is associated with a birth event, i.e., a Moran model (Durrett 2002). More precisely, infected cells die at rate 1 irrespective of CTL attack, and at rates λ 1 , λ 2 due to CTL attack at the epitopes e1 and e2, respectively.…”

Section: Definitionsmentioning

confidence: 99%

Sampling HIV Intrahost Genealogies Based on a Model of Acute Stage CTL Response

Leviyang

2011

Bull Math Biol

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Cytotoxic T lymphocytes (CTLs) play an important role in the immune response to HIV during the acute stage of infection, but the effect of CTLs on HIV intrahost genetic diversity is poorly understood. We introduce a model of CTL attack during the acute stage. Assuming this model, we develop a method to sample HIV intrahost genealogies. Using our sampling approach, we characterize the evolutionary forces that shape HIV genealogies. In particular, we show that early mutation events can have significant impact on HIV genealogies and that certain types of CTL attack are best at controlling HIV genetic diversity. Our sampler represents a first step toward using HIV genetic data to infer properties of CTL attack.

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“…One of the subjects of study in biology is the fixation time of a population, a random time point when the population starting from several groups of heterogeneous particles (which may be all different) first becomes homogeneous, i.e., all particles become single-type [3]. Here we deal with finite Markov chains containing only inessential and absorbing states.…”

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confidence: 99%

“…In principle, all this corresponds to the branching condition: each particle generates its own independent branching process but with the strong condition of fixed total number of offsprings. However, application of the traditional methods of the theory of branching processes is rather difficult here.We study the last population model: each of the N particles, living over the time unit, dies and produces a random amount of offsprings of the same type as itself; their distributions are exchangeable and the sum is N .One of the subjects of study in biology is the fixation time of a population, a random time point when the population starting from several groups of heterogeneous particles (which may be all different) first becomes homogeneous, i.e., all particles become single-type [3]. Here we deal with finite Markov chains containing only inessential and absorbing states.…”

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Mean fixation time estimates in constant size populations

Klokov

Топчий

2006

Sib Math J

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We consider a population consisting of N particles each of which some type is ascribed to. All particles die at the integer time moments and produce a random amount of particles of the same type as the parent. Moreover, the population retains its size N and the random vectors defining the number of offsprings of each particle have exchangeable distributions. We obtain several upper bounds for the expectation of the variable equal to the number of the generation when all particles in the population become single-type or almost single-type. Here we fix an arbitrary initial configuration of particles according to types.Keywords: Markov chain, exchangeable distribution, evolution of populations, nearest common ancestor, fixation time, simulation modeling § 1. Motivation of the Choice of the Model.The Problems Under ConsiderationMarkov models are used widely in population dynamics and population evolution analysis based on changes of the DNA. One of the popular classes of models of this kind is given by the haploid models of fixed size [1-4] without mutations. These are models with discrete time of the following form. At the beginning, there are some particles. Then each of them dies producing a random amount of offsprings or only some of the particles die and the rest survive. Anyway, the birth and death distribution is such that the total population size does not change. The term "haploid" means that each offspring has at most one parent. The absence of mutations consists in the fact that the particles produce only particles of the same type.The most popular models of the given class were introduced by Wright and Fisher, Moran, Karlin, and McGregor. In the Wright-Fisher model, the numbers of particles of different types in a generation are the parameters of a polynomial distribution of the birth of offsprings. In the Moran model, only one particle dies replacing one or several others with its offsprings. In the Karlin and McGregor model, the process branches and the distribution is redefined by the condition of fixed total number of offsprings.Some generalization of the above models is proposed in [1, 2]: Each ancestor contributes to offspring production additively and the resulting exchangeable distribution of the parent particles. In principle, all this corresponds to the branching condition: each particle generates its own independent branching process but with the strong condition of fixed total number of offsprings. However, application of the traditional methods of the theory of branching processes is rather difficult here.We study the last population model: each of the N particles, living over the time unit, dies and produces a random amount of offsprings of the same type as itself; their distributions are exchangeable and the sum is N .One of the subjects of study in biology is the fixation time of a population, a random time point when the population starting from several groups of heterogeneous particles (which may be all different) first becomes homogeneous, i.e., all particles become single-type [3]. Her...

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Probability Models for DNA Sequence Evolution

Cited by 147 publications

References 368 publications

Coalescent Size Versus Coalescent Time with Strong Selection

Coalescent Size Versus Coalescent Time with Strong Selection

Sampling HIV Intrahost Genealogies Based on a Model of Acute Stage CTL Response

Mean fixation time estimates in constant size populations

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