ANALYSES OF THE AGE OF GENES AND THE FIRST ARRIVAL TIMES IN A FINITE POPULATION

ABSTRACT The age of a mutant gene is studied using the infinite allele model in which every mutant is new and selectively neutral. Based on a time reversal theory of Markov processes, we develop a method of mathematical analysis that is considerably simpler for calculating the various statistics of the age than previous methods. Formulas for the mean and variance and for the distribution of age are presented together with some examples of relevance to cases in natural populations.—Theoretical studies of the first arrival time of an allele to a specified frequency, given an initially monomorphic condition of the locus, are presented. It is shown that, beginning with an allele that has frequency p = 1 or an allele with frequency p = 1/2N, there is an initial lag phase in which there is virtually no chance of an allele with a specified intermediate frequency appearing in the population. The distribution of the first arrival time is also presented. The distribution shows several characteristics that are not immediately obvious from a consideration of only the mean and variance of first arrival time. Especially noteworthy is the existence of a very long tail to the distribution. We have also studied the distribution of the age of an allele in the population. Again, the distribution of this measure is shown to be more informative for several questions than are the mean and variance alone.

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On the Mean First Arrival Time of Brownian Particles on Riemannian Manifolds

Journal de Mathématiques Pures et Appliquées ◽

10.1016/j.matpur.2021.04.006 ◽

2021 ◽

Author(s):

Medet Nursultanov ◽

Justin C. Tzou ◽

Leo Tzou

Keyword(s):

Riemannian Manifolds ◽

Arrival Time ◽

Brownian Particles ◽

First Arrival ◽

The Mean ◽

First Arrival Time

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A note on the arrival probability, first arrival time and age of a mutant gene in a finite population

Annals of Human Genetics ◽

10.1111/j.1469-1809.1976.tb00149.x ◽

1976 ◽

Vol 39 (4) ◽

pp. 435-439

Author(s):

WEN-HSIUNG LI

Keyword(s):

Arrival Time ◽

Finite Population ◽

Mutant Gene ◽

First Arrival ◽

First Arrival Time ◽

Arrival Probability

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THE AGE OF A NEUTRAL MUTANT PERSISTING IN A FINITE POPULATION

Genetics ◽

10.1093/genetics/75.1.199 ◽

1973 ◽

Vol 75 (1) ◽

pp. 199-212

Author(s):

Motoo Kimura ◽

Tomoko Ohta

Keyword(s):

Arrival Time ◽

Finite Population ◽

Low Frequency ◽

Effective Population ◽

Random Drift ◽

Current Frequency ◽

Very Low Frequency ◽

First Arrival ◽

The Mean ◽

First Arrival Time

ABSTRACT Formulae for the mean and the mean square age of a neutral allele which is segregating with frequency x in a population of effective size Ne have been obtained using the diffusion equation method, for the case of 4Nev<1 where v is the mutation rate. It has been shown that the average ages of neutral alleles, even if their frequencies are relatively low, are quite old. For example, a neutral mutant whose current frequency is 10% has the expected age roughly equal to the effective population size Ne and the standard deviation 1.4Ne (in generations), assuming that this mutant has increased by random drift from a very low frequency. Also, formulae for the mean "first arrival time" of a neutral mutant to a certain frequency x have been presented. In addition, a new, approximate method has been developed which enables us to obtain the condition under which frequencies of "rare" polymorphic alleles among local populations are expected to be uniform if the alleles are selectively neutral.—It was concluded that exchange of only a few individuals on the average between adjacent colonies per generation is enough to bring about such a uniformity of frequencies.

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Moth Mating: Modeling Female Pheromone Calling and Male Navigational Strategies to Optimize Reproductive Success

Applied Sciences ◽

10.3390/app10186543 ◽

2020 ◽

Vol 10 (18) ◽

pp. 6543

Author(s):

Tracy L. Stepien ◽

Cole Zmurchok ◽

James B. Hengenius ◽

Rocío Marilyn Caja Rivera ◽

Maria R. D’Orsogna ◽

...

Keyword(s):

Arrival Time ◽

Mathematical Framework ◽

Reproductive Process ◽

Mating Period ◽

Agent Based ◽

First Arrival ◽

The Mean ◽

First Arrival Time ◽

Over Time

Male and female moths communicate in complex ways to search for and to select a mate. In a process termed calling, females emit small quantities of pheromones, generating plumes that spread in the environment. Males detect the plume through their antennae and navigate toward the female. The reproductive process is marked by female choice and male–male competition, since multiple males aim to reach the female but only the first can mate with her. This provides an opportunity for female selection on male traits such as chemosensitivity to pheromone molecules and mobility. We develop a mathematical framework to investigate the overall mating likelihood, the mean first arrival time, and the quality of the first male to reach the female for four experimentally observed female calling strategies unfolding over a typical one-week mating period. We present both analytical solutions of a simplified model as well as results from agent-based numerical simulations. Our findings suggest that, by adjusting call times and the amount of released pheromone, females can optimize the mating process. In particular, shorter calling times and lower pheromone titers at onset of the mating period that gradually increase over time allow females to aim for higher-quality males while still ensuring that mating occurs by the end of the mating period.

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