Degenerate Diffusions in Population Degenerate Diffusions in Genetics In Memory of Gennadi Population Genetics Henkin Charles L. In Memory of Gennadi Henkin Epstein MIPT Dolgoprudny The Wright-Fisher Model Degenerate Diffusions Charles L. Epstein The Semigroup on ❈ 0 ( P ) Departments of Mathematics Heat Kernel Applied Math and Computational Science Estimates University of Pennsylvania The Null Space Connections to September 13, 2016 Probability Bibliography
Acknowledgment Degenerate Diffusions in Population Genetics In Memory of Gennadi Henkin Charles L. Epstein I want to thank the organizers for inviting me to speak. The Wright-Fisher The work I will describe was done jointly with Rafe Mazzeo and Model Camelia Pop. Degenerate Diffusions The Semigroup on ❈ 0 ( P ) Heat Kernel Estimates The Null Space Connections to Probability Bibliography
Outline Degenerate Diffusions in Population 1 The Wright-Fisher Model Genetics In Memory of Gennadi Henkin 2 Degenerate Diffusions Charles L. Epstein 3 The Semigroup on ❈ 0 ( P ) The Wright-Fisher Model 4 Heat Kernel Estimates Degenerate Diffusions 5 The Null Space The Semigroup on ❈ 0 ( P ) Heat Kernel 6 Connections to Probability Estimates The Null Space 7 Bibliography Connections to Probability Bibliography
Population Genetics Degenerate Diffusions in Population Genetics In Memory of Gennadi Population genetics is the study of how the distribution of variants Henkin in a reproducing population evolves over time. There are typically Charles L. Epstein four important effects: The 1 The randomness in the number of offspring a given Wright-Fisher Model individual, or pair, has in a given generation. Degenerate 2 Mutation from one type to another type. Diffusions The Semigroup 3 Differences in “fitness” among the different types. on ❈ 0 ( P ) Heat Kernel 4 Migration in and out of the given environment. Estimates The Null Space Connections to Probability Bibliography
Population Genetics Degenerate Diffusions in Population Genetics In Memory of Gennadi Population genetics is the study of how the distribution of variants Henkin in a reproducing population evolves over time. There are typically Charles L. Epstein four important effects: The 1 The randomness in the number of offspring a given Wright-Fisher Model individual, or pair, has in a given generation. Degenerate 2 Mutation from one type to another type. Diffusions The Semigroup 3 Differences in “fitness” among the different types. on ❈ 0 ( P ) Heat Kernel 4 Migration in and out of the given environment. Estimates The Null Space Connections to Probability Bibliography
Population Genetics Degenerate Diffusions in Population Genetics In Memory of Gennadi Population genetics is the study of how the distribution of variants Henkin in a reproducing population evolves over time. There are typically Charles L. Epstein four important effects: The 1 The randomness in the number of offspring a given Wright-Fisher Model individual, or pair, has in a given generation. Degenerate 2 Mutation from one type to another type. Diffusions The Semigroup 3 Differences in “fitness” among the different types. on ❈ 0 ( P ) Heat Kernel 4 Migration in and out of the given environment. Estimates The Null Space Connections to Probability Bibliography
Population Genetics Degenerate Diffusions in Population Genetics In Memory of Gennadi Population genetics is the study of how the distribution of variants Henkin in a reproducing population evolves over time. There are typically Charles L. Epstein four important effects: The 1 The randomness in the number of offspring a given Wright-Fisher Model individual, or pair, has in a given generation. Degenerate 2 Mutation from one type to another type. Diffusions The Semigroup 3 Differences in “fitness” among the different types. on ❈ 0 ( P ) Heat Kernel 4 Migration in and out of the given environment. Estimates The Null Space Connections to Probability Bibliography
Natural Selection Degenerate Diffusions in Population Genetics In Memory of Gennadi Henkin The Darwinian concept of “Natural Selection” is the idea that Charles L. variants of greater fitness will survive longer and therefore have a Epstein larger number of offspring. The Wright-Fisher Model As a random process one cannot expect to make exact predictions Degenerate for the time evolution of a single population, but only for the Diffusions distribution of types, in an ensemble of populations, given a The Semigroup on ❈ 0 ( P ) starting distribution. Heat Kernel Estimates The Null Space Connections to Probability Bibliography
Finite populations Degenerate Diffusions in Population Genetics The earliest models assumed a fixed population size N , and a In Memory of Gennadi finite collection of possible types, { 1 , . . . , m } . The state of the Henkin Charles L. population at each time is described by an m -tuple: ( n 1 , . . . , n m ), Epstein giving the number of individuals of each type. Here we assume The that n 1 + · · · + n m = N , so the state space consists of the integer Wright-Fisher Model points in an ( m − 1 ) -simplex. Degenerate Diffusions The evolution of the population is then a Markov process specified The Semigroup by the transition probability: on ❈ 0 ( P ) Heat Kernel Estimates Prob (( k 1 , . . . , k m ) | ( n 1 , . . . , n m )). The Null Space Connections to Probability Bibliography
2-Alleles Degenerate Diffusions in Population Genetics In Memory of The simplest haploid case is when there are two types (called Gennadi Henkin alleles) and both types have the same fitness and there is also no Charles L. Epstein mutation. We use A and a to denote the types, and let X ( t ) be the number of type a at generation t ∈ � . Since n a + n A = N , in this The Wright-Fisher case the standard Wright-Fisher model is given by the binomial Model sampling formula: Degenerate Diffusions � N � � i � j � � N − j The Semigroup 1 − i on ❈ 0 ( P ) Prob ( X ( t + 1 ) = j | X ( t ) = i ) = (1) j N N Heat Kernel Estimates The Null Space Connections to Probability Bibliography
Mutation and Selection Degenerate Diffusions in Population Genetics In Memory of To incorporate mutation and selection, we change the odds. If a Gennadi Henkin and A have relative fitness ( 1 + s ) : 1 , and the rate at which Charles L. a → A is µ 1 and the rate at which A → a is µ 2 , then we let: Epstein p i = i ( 1 + s )( 1 − µ 1 ) ( N − i )µ 2 The i ( 1 + s ) + N − i + i ( 1 + s ) + N − i . Wright-Fisher Model Degenerate Diffusions We alter the transition matrix to The Semigroup � N � on ❈ 0 ( P ) p j i ( 1 − p i ) N − j Prob ( X ( t + 1 ) = j | X ( t ) = i ) = (2) Heat Kernel j Estimates The Null Space Connections to Probability Bibliography
The Infinite Population Limit Degenerate Diffusions in The main topic of this talk concerns limits of these sorts of Population Genetics processes as the population N tends to infinity. In the 1d-case, the In Memory of Gennadi rescaled processes Henkin 1 N X ( N ) ( [ t N ] ), Charles L. (3) Epstein The converge, under suitable hypotheses, to a continuous time Wright-Fisher stochastic process parametrized by the interval [ 0 , 1 ] . The Model Degenerate backward Kolmogorov operator is the second order differential Diffusions operator: The Semigroup on ❈ 0 ( P ) L ∗ f ( x ) = x ( 1 − x ) Heat Kernel ∂ 2 x f + σ x ( 1 − x )∂ x f + m 2 ( 1 − x )∂ x f − m 1 x ∂ x f . Estimates 2 The Null Space (4) Connections to Where σ = Ns , m 1 = N µ 1 and m 2 = N µ 2 are assumed fixed, as Probability N → ∞ . Bibliography
The Infinite Population Limit Degenerate Diffusions in In this limit the second order term, Population Genetics In Memory of x ( 1 − x ) Gennadi ∂ 2 x f , Henkin 2 Charles L. Epstein is related to the randomness in the number of offspring; whereas mutation and selection become deterministic forces, represented The Wright-Fisher by the vector field, Model Degenerate Diffusions σ x ( 1 − x )∂ x f + m 2 ( 1 − x )∂ x f − m 1 x ∂ x f . The Semigroup on ❈ 0 ( P ) Heat Kernel Estimates There are many different Markov chains that have the same The Null Space infinite population limit, for example the Moran Model. This limit Connections to is largely determined by the first and second moments of the Probability “offspring distribution.” Bibliography
The Classical Higher Dimensional Case Degenerate Diffusions in This Markov-chain framework can be used to model populations Population Genetics with a genome of arbitrary length. If there are M + 1 different In Memory of Gennadi types, then the infinite population limit is a Markov process on an Henkin M -simplex, � M , where the coordinates give the frequency of each Charles L. Epstein type. We can represent the simplex by � The Wright-Fisher � M = { ( x 1 , . . . , x M ) : 0 ≤ x i and x i ≤ 1 } . (5) Model i Degenerate Diffusions The generator for the infinite population limit is then an operator The Semigroup on ❈ 0 ( P ) of the form � Heat Kernel x i (δ i j − x j )∂ x i ∂ x j + V . L f = (6) Estimates i , j The Null Space Connections to V is an inward pointing vector field. We let L Kim denote the Probability second order part in (6). Bibliography
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