scholarly journals Moment-based approximations for the Wright-Fisher model of population dynamics under natural selection at two linked loci

2021 ◽  
Author(s):  
Zhangyi He ◽  
Wenyang Lyu ◽  
Mark Beaumont ◽  
Feng Yu
Keyword(s):  
Population Dynamics ◽  
Natural Selection ◽  
Probability Distribution ◽  
Diffusion Approximation ◽  
Normal Approximation ◽  
Genetic Recombination ◽  
Transition Probability ◽  
Computational Cost ◽  
Fisher Model ◽  
Support Issue

AbstractProperly modelling genetic recombination and local linkage has been shown to bring a significant improvement to the inference of natural selection from time series genetic data under a Wright-Fisher model. Existing approaches that can take genetic recombination effect and local linkage information into account are built upon either the diffusion approximation or the moment-based approximation of the Wright-Fisher model. However, such approximations are either limited to the increased computational cost like the diffusion approximation or suffer from the distribution support issue like the normal approximation, which can seriously affect computational efficiency and accuracy. In this work, we propose two novel moment-based approximations for the Wright-Fisher model of population dynamics subject to natural selection at a pair of linked loci. Our key innovation is that we extend existing approaches to calculate the mean and (co)variance of the two-locus Wright-Fisher model with selection and suggest a logistic normal distribution or a hierarchical beta distribution as a parametric continuous probability distribution to approximate the Wright-Fisher model by matching its first two moments to those of the Wright-Fisher model. Compared with the diffusion approximation, our approximations enable the computation of the transition probability distribution of the Wright-Fisher model at a far smaller computational cost and also allow us to avoid the distribution support issue found in the normal approximation.

scholarly journals Numerical simulation of the two-locus Wright-Fisher stochastic differential equation with application to approximating transition probability densities

2020 ◽  
Author(s):  
Zhangyi He ◽  
Mark Beaumont ◽  
Feng Yu
Keyword(s):  
Differential Equation ◽  
Natural Selection ◽  
Probability Density Function ◽  
Stochastic Differential Equation ◽  
Probability Density ◽  
Density Function ◽  
Genetic Recombination ◽  
Transition Probability ◽  
Superior Performance ◽  
Transition Probability Density

AbstractOver the past decade there has been an increasing focus on the application of the Wright-Fisher diffusion to the inference of natural selection from genetic time series. A key ingredient for modelling the trajectory of gene frequencies through the Wright-Fisher diffusion is its transition probability density function. Recent advances in DNA sequencing techniques have made it possible to monitor genomes in great detail over time, which presents opportunities for investigating natural selection while accounting for genetic recombination and local linkage. However, most existing methods for computing the transition probability density function of the Wright-Fisher diffusion are only applicable to one-locus problems. To address two-locus problems, in this work we propose a novel numerical scheme for the Wright-Fisher stochastic differential equation of population dynamics under natural selection at two linked loci. Our key innovation is that we reformulate the stochastic differential equation in a closed form that is amenable to simulation, which enables us to avoid boundary issues and reduce computational costs. We also propose an adaptive importance sampling approach based on the proposal introduced by Fearnhead (2008) for computing the transition probability density of the Wright-Fisher diffusion between any two observed states. We show through extensive simulation studies that our approach can achieve comparable performance to the method of Fearnhead (2008) but can avoid manually tuning the parameter ρ to deliver superior performance for different observed states.

scholarly journals Effects of the Ordering of Natural Selection and Population Regulation Mechanisms on Wright-Fisher Models

2016 ◽  
Author(s):  
Zhangyi He ◽  
Mark Beaumont ◽  
Feng Fu
Keyword(s):  
Population Dynamics ◽  
Life Cycle ◽  
Natural Selection ◽  
Population Regulation ◽  
Genetic Research ◽  
Natural Populations ◽  
Gene Migration ◽  
Local Selection ◽  
Fisher Model ◽  
Regulation Mechanisms

AbstractThe Wright-Fisher model and its extensions are of central importance in population genetics, and so far, they have formed the basis of most theoretical and applied population genetic research. In the present work, we explore the effect that the ordering of natural selection and population regulation in the life cycle has on the resulting population dynamics under the Wright-Fisher model, especially for the evolution of one- and two-locus systems. With weak natural selection, the details of how to order natural selection and population regulation in the life cycle do not matter in the Wright-Fisher model and its diffusion approximation. By contrast, we show that when there is strong natural selection and the population is in linkage disequilibrium, there can be appreciable differences in the resulting population dynamics under the Wright-Fisher model, depending on whether natural selection occurs before or after population regulation in the life cycle. We argue that this effect may be of significance in natural populations subject to gene migration and local selection.F.Y. supported in part by EPSRC Grant EP/I028498/1.

Celebrating Darwin’s Errors

2017 ◽  
Author(s):  
Douglas Allchin
Keyword(s):  
Functional Analysis ◽  
Population Dynamics ◽  
Natural Selection ◽  
General Principle ◽  
Charles Darwin ◽  
Genetic Recombination ◽  
Comparative Anatomy ◽  
Common View ◽  
The Common

Charles Darwin was truly amazing. In 1859 he introduced a robust understanding of descent with modification by means of natural selection. His concepts would help unify taxonomy, biogeography, comparative anatomy, heredity, functional analysis of form, embryology, paleontology, population dynamics, and ecology, and even human moral behavior. Darwin showed how to explain organic “design” as well as the limitations of contingent history, adaptive structures as well as vestigial ones. Every lesson in biology, properly framed, expresses and celebrates Darwin’s achievement. How, then, might one mark so august an occasion as his two hundredth birthday (also the sesquicentennial year of his premier work, the Origin of Species)? Many will no doubt parade Darwin’s many triumphs. But allow me to take exception to the common view (another sacred bovine?) that science is best reflected only by its successful theories. If science is fundamentally about discovery, then its “failures” or errors along the way may be just as important as the ultimately reliable insights. I wish to celebrate science as a process. Here, then, I acknowledge Darwin’s mistakes and show how understanding them gives us a deeper understanding both of Darwin and of science more generally. My tribute is to forgo the mythologized legend and appreciate so remarkable a scientist as Darwin in familiarly human terms. First, one may note that Darwin’s errors generate interest largely because of his many achievements. His credentials are unimpeachable. If he made mistakes, it was not for want of scientific ability. One cannot rudely dismiss his errors as due to ineptitude. Indeed, Darwin’s contributions are wider and their theoretical coherence deeper than popularly known. He produced four volumes on the taxonomy of barnacles, demonstrating his skills in detailed observation and analysis of evolutionary classification. In his first work after the Origin, he showed the importance of orchid form in promoting outcrossing through pollination, thereby contributing to an understanding of the role of sex and genetic recombination in evolution. Later, he explained heterostyly—the occurrence of flowers with styles of different lengths—as illustrating the same general principle (see essay 16).

scholarly journals Effects of the Ordering of Natural Selection and Population Regulation Mechanisms on Wright-Fisher Models

2017 ◽  
Vol 7 (7) ◽  
pp. 2095-2106 ◽  
Author(s):  
Zhangyi He ◽  
Mark Beaumont ◽  
Feng Yu
Keyword(s):  
Linkage Disequilibrium ◽  
Natural Selection ◽  
Population Regulation ◽  
Genetic Recombination ◽  
Gene Migration ◽  
Fisher Model ◽  
Fecundity Selection ◽  
Regulation Mechanisms ◽  
The Difference ◽  
Significant Linkage

Abstract We explore the effect of different mechanisms of natural selection on the evolution of populations for one- and two-locus systems. We compare the effect of viability and fecundity selection in the context of the Wright-Fisher model with selection under the assumption of multiplicative fitness. We show that these two modes of natural selection correspond to different orderings of the processes of population regulation and natural selection in the Wright-Fisher model. We find that under the Wright-Fisher model these two different orderings can affect the distribution of trajectories of haplotype frequencies evolving with genetic recombination. However, the difference in the distribution of trajectories is only appreciable when the population is in significant linkage disequilibrium. We find that as linkage disequilibrium decays the trajectories for the two different models rapidly become indistinguishable. We discuss the significance of these findings in terms of biological examples of viability and fecundity selection, and speculate that the effect may be significant when factors such as gene migration maintain a degree of linkage disequilibrium.

scholarly journals Dependence of Cytoplasmic temperatures on the Vibrational energies of molecules in a eukaryotic cell Nucleus

2019 ◽  
Author(s):  
Muhammad Ali ◽  
Muhammad Haider ◽  
Naeem Akhtar
Keyword(s):  
Natural Selection ◽  
Probability Distribution ◽  
Cell Nucleus ◽  
Canonical Ensemble ◽  
Molecular Level ◽  
Quadratic Function ◽  
Eukaryotic Cell ◽  
Effect Of Temperature ◽  
Vibrational Energies ◽  
Vibrational Kinetic

This paper presents a statistical model to show the dependence of cytoplasmic temperatures on the vibrational kinetic energies of molecules in a eukaryotic cell nucleus. The probability distribution 𝑃(𝐸_𝑁) of energy states of cell nucleus 𝐸_𝑁 is derived using canonical ensemble framework and the vibrational energies of molecules are quadratic function of Temperature. It has been postulated that vibrational energies changes the reaction potentials of processes making certain reactions favorable. These favorable reactions explains the evolutionary processes such as mutation at molecular level. Natural Selection is simply just favorable reactions of molecules affected by the surrounding Temperature. The effect of temperature on vibrational energies of molecules can be effectively used to study cancerous mutations and astrobiology.

scholarly journals Ecological and evolutionary dynamics of interconnectedness and modularity

2018 ◽  
Vol 115 (4) ◽  
pp. 750-755 ◽  
Author(s):  
Jan M. Nordbotten ◽  
Simon A. Levin ◽  
Eörs Szathmáry ◽  
Nils C. Stenseth
Keyword(s):  
Population Dynamics ◽  
Natural Selection ◽  
Evolution Equation ◽  
Evolutionary Dynamics ◽  
Population Distribution ◽  
Theoretical Framework ◽  
Ecological System ◽  
Macro Level ◽  
Population Distributions ◽  
Trait Space

In this contribution, we develop a theoretical framework for linking microprocesses (i.e., population dynamics and evolution through natural selection) with macrophenomena (such as interconnectedness and modularity within an ecological system). This is achieved by developing a measure of interconnectedness for population distributions defined on a trait space (generalizing the notion of modularity on graphs), in combination with an evolution equation for the population distribution. With this contribution, we provide a platform for understanding under what environmental, ecological, and evolutionary conditions ecosystems evolve toward being more or less modular. A major contribution of this work is that we are able to decompose the overall driver of changes at the macro level (such as interconnectedness) into three components: (i) ecologically driven change, (ii) evolutionarily driven change, and (iii) environmentally driven change.

Analysis of transient behaviour of certain processes with return to a central state

1983 ◽  
Vol 20 (01) ◽  
pp. 61-70
Author(s):  
Peter G. Buckholtz ◽  
L. Lorne Campbell ◽  
Ross D. Milbourne ◽  
M. T. Wasan
Keyword(s):  
Probability Distribution ◽  
Diffusion Equation ◽  
Diffusion Approximation ◽  
Cash Management ◽  
Central State ◽  
Transient Behaviour ◽  
Management Problems ◽  
Transient Probability ◽  
Birth Death

In economics, cash management problems may be modelled by birth-death processes which reset to central states when a boundary is reached. The nature of the transient behaviour of the probability distribution of such processes symmetric about a central state is investigated. A diffusion approximation of such processes is given and the transient probability behaviour derived from the diffusion equation.

scholarly journals The Strassen invariance principle for certain non-stationary Markov–Feller chains

2020 ◽  
Vol 121 (1) ◽  
pp. 1-34 ◽  
Author(s):  
Dawid Czapla ◽  
Katarzyna Horbacz ◽  
Hanna Wojewódka-Ściążko
Keyword(s):  
Gene Expression ◽  
Mathematical Model ◽  
Population Dynamics ◽  
Invariance Principle ◽  
Probability Function ◽  
Stochastic Dynamics ◽  
Transition Probability ◽  
Type Condition ◽  
Iterated Logarithm ◽  
Transition Probability Function

We propose certain conditions implying the functional law of the iterated logarithm (the Strassen invariance principle) for some general class of non-stationary Markov–Feller chains. This class may be briefly specified by the following two properties: firstly, the transition operator of the chain under consideration enjoys a non-linear Lyapunov-type condition, and secondly, there exists an appropriate Markovian coupling whose transition probability function can be decomposed into two parts, one of which is contractive and dominant in some sense. Our criterion may serve as a useful tool in verifying the functional law of the iterated logarithm for certain random dynamical systems, developed e.g. in biology and population dynamics. In the final part of the paper we present an example application of our main theorem to a mathematical model describing stochastic dynamics of gene expression.

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