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  1. Article ; Online: Coalescence and sampling distributions for Feller diffusions.

    Burden, Conrad J / Griffiths, Robert C

    Theoretical population biology

    2023  Volume 155, Page(s) 67–76

    Abstract: Consider the diffusion process defined by the forward equation ... ...

    Abstract Consider the diffusion process defined by the forward equation u
    Language English
    Publishing date 2023-12-12
    Publishing country United States
    Document type Journal Article
    ZDB-ID 3948-2
    ISSN 1096-0325 ; 0040-5809
    ISSN (online) 1096-0325
    ISSN 0040-5809
    DOI 10.1016/j.tpb.2023.12.001
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  2. Article ; Online: An estimator for the recombination rate from a continuously observed diffusion of haplotype frequencies.

    Griffiths, Robert C / Jenkins, Paul A

    Journal of mathematical biology

    2023  Volume 86, Issue 6, Page(s) 98

    Abstract: Recombination is a fundamental evolutionary force, but it is difficult to quantify because the effect of a recombination event on patterns of variation in a sample of genetic data can be hard to discern. Estimators for the recombination rate, which are ... ...

    Abstract Recombination is a fundamental evolutionary force, but it is difficult to quantify because the effect of a recombination event on patterns of variation in a sample of genetic data can be hard to discern. Estimators for the recombination rate, which are usually based on the idea of integrating over the unobserved possible evolutionary histories of a sample, can therefore be noisy. Here we consider a related question: how would an estimator behave if the evolutionary history actually was observed? This would offer an upper bound on the performance of estimators used in practice. In this paper we derive an expression for the maximum likelihood estimator for the recombination rate based on a continuously observed, multi-locus, Wright-Fisher diffusion of haplotype frequencies, complementing existing work for an estimator of selection. We show that, contrary to selection, the estimator has unusual properties because the observed information matrix can explode in finite time whereupon the recombination parameter is learned without error. We also show that the recombination estimator is robust to the presence of selection in the sense that incorporating selection into the model leaves the estimator unchanged. We study the properties of the estimator by simulation and show that its distribution can be quite sensitive to the underlying mutation rates.
    MeSH term(s) Haplotypes ; Computer Simulation ; Biological Evolution ; Recombination, Genetic ; Models, Genetic
    Language English
    Publishing date 2023-05-26
    Publishing country Germany
    Document type Journal Article ; Research Support, Non-U.S. Gov't
    ZDB-ID 187101-8
    ISSN 1432-1416 ; 0303-6812
    ISSN (online) 1432-1416
    ISSN 0303-6812
    DOI 10.1007/s00285-023-01931-7
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  3. Book ; Online: An estimator for the recombination rate from a continuously observed diffusion of haplotype frequencies

    Griffiths, Robert C. / Jenkins, Paul A.

    2022  

    Abstract: Recombination is a fundamental evolutionary force, but it is difficult to quantify because the effect of a recombination event on patterns of variation in a sample of genetic data can be hard to discern. Estimators for the recombination rate, which are ... ...

    Abstract Recombination is a fundamental evolutionary force, but it is difficult to quantify because the effect of a recombination event on patterns of variation in a sample of genetic data can be hard to discern. Estimators for the recombination rate, which are usually based on the idea of integrating over the unobserved possible evolutionary histories of a sample, can therefore be noisy. Here we consider a related question: how would an estimator behave if the evolutionary history actually was observed? This would offer an upper bound on the performance of estimators used in practice. In this paper we derive an expression for the maximum likelihood estimator for the recombination rate based on a continuously observed, multi-locus, Wright--Fisher diffusion of haplotype frequencies, complementing existing work for an estimator of selection. We show that, contrary to selection, the estimator has unusual properties because the observed information matrix can explode in finite time whereupon the recombination parameter is learned without error. We also show that the recombination estimator is robust to the presence of selection in the sense that incorporating selection into the model leaves the estimator unchanged. We study the properties of the estimator by simulation and show that its distribution can be quite sensitive to the underlying mutation rates.

    Comment: 28 pages, 3 figures
    Keywords Quantitative Biology - Populations and Evolution ; Mathematics - Probability ; Mathematics - Statistics Theory ; 92D15 (Primary) 62M05 (Secondary)
    Subject code 519
    Publishing date 2022-12-15
    Publishing country us
    Document type Book ; Online
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  4. Book ; Online: Coalescence and sampling distributions for Feller diffusions

    Burden, Conrad J. / Griffiths, Robert C.

    2022  

    Abstract: Consider the diffusion process defined by the forward equation $u_t(t, x) = \tfrac{1}{2}\{x u(t, x)\}_{xx} - \alpha \{x u(t, x)\}_{x}$ for $t, x \ge 0$ and $-\infty < \alpha < \infty$, with an initial condition $u(0, x) = \delta(x - x_0)$. This equation ... ...

    Abstract Consider the diffusion process defined by the forward equation $u_t(t, x) = \tfrac{1}{2}\{x u(t, x)\}_{xx} - \alpha \{x u(t, x)\}_{x}$ for $t, x \ge 0$ and $-\infty < \alpha < \infty$, with an initial condition $u(0, x) = \delta(x - x_0)$. This equation was introduced and solved by Feller to model the growth of a population of independently reproducing individuals. We explore important coalescent processes related to Feller's solution. For any $\alpha$ and $x_0 > 0$ we calculate the distribution of the random variable $A_n(s; t)$, defined as the finite number of ancestors at a time $s$ in the past of a sample of size $n$ taken from the infinite population of a Feller diffusion at a time $t$ since since its initiation. In a subcritical diffusion we find the distribution of population and sample coalescent trees from time $t$ back, conditional on non-extinction as $t \to \infty$. In a supercritical diffusion we construct a coalescent tree which has a single founder and derive the distribution of coalescent times.

    Comment: 32 pages, 6 figures. New diagrams have been added and some rewording to sections 1 to 6. Section 7 of the original manuscript contained an error which has necessitated rewriting the original sections 7 to 9 as a new and more straightforward Section 7 which contains new results as a theorem and a corollary
    Keywords Mathematics - Probability ; Quantitative Biology - Populations and Evolution
    Subject code 519
    Publishing date 2022-10-23
    Publishing country us
    Document type Book ; Online
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  5. Article ; Online: The transition distribution of a sample from a Wright-Fisher diffusion with general small mutation rates.

    Burden, Conrad J / Griffiths, Robert C

    Journal of mathematical biology

    2019  Volume 79, Issue 6-7, Page(s) 2315–2342

    Abstract: The transition distribution of a sample taken from a Wright-Fisher diffusion with general small mutation rates is found using a coalescent approach. The approximation is equivalent to having at most one mutation in the coalescent tree of the sample up to ...

    Abstract The transition distribution of a sample taken from a Wright-Fisher diffusion with general small mutation rates is found using a coalescent approach. The approximation is equivalent to having at most one mutation in the coalescent tree of the sample up to the most recent common ancestor with additional mutations occurring on the lineage from the most recent common ancestor to the time origin if complete coalescence occurs before the origin. The sampling distribution leads to an approximation for the transition density in the diffusion with small mutation rates. This new solution has interest because the transition density in a Wright-Fisher diffusion with general mutation rates is not known.
    MeSH term(s) Gene Frequency ; Genetic Drift ; Genetics, Population/methods ; Models, Genetic ; Mutation Rate
    Language English
    Publishing date 2019-09-17
    Publishing country Germany
    Document type Journal Article
    ZDB-ID 187101-8
    ISSN 1432-1416 ; 0303-6812
    ISSN (online) 1432-1416
    ISSN 0303-6812
    DOI 10.1007/s00285-019-01430-8
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  6. Book ; Online: The stationary and quasi-stationary properties of neutral multi-type branching process diffusions

    Burden, Conrad J. / Griffiths, Robert C.

    2021  

    Abstract: The stationary asymptotic properties of the diffusion limit of a multi-type branching process with neutral mutations are studied. For the critical and subcritical processes the interesting limits are those of quasi-stationary distributions conditioned on ...

    Abstract The stationary asymptotic properties of the diffusion limit of a multi-type branching process with neutral mutations are studied. For the critical and subcritical processes the interesting limits are those of quasi-stationary distributions conditioned on non-extinction. Pedagogical derivations are given for known results that the limiting distributions for supercritical and critical processes are found to collapse onto rays aligned with stationary eigenvectors of the mutation rate matrix, in agreement with discrete multi-type branching processes. For the sub-critical process the previously unsolved quasi-stationary distribution is obtained to first order in the overall mutation rate, which is assumed to be small. The sampling distribution over allele types for a sample of given finite size is found to agree to first order in mutation rates with the analogous sampling distribution for a Wright-Fisher diffusion with constant population size.

    Comment: 37 pages, 2 figures. Propositions 1 and 2 and Lemma 1 reworked, and clarification of various points
    Keywords Mathematics - Probability ; Quantitative Biology - Populations and Evolution
    Subject code 519
    Publishing date 2021-07-28
    Publishing country us
    Document type Book ; Online
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  7. Article ; Online: The stationary distribution of a sample from the Wright-Fisher diffusion model with general small mutation rates.

    Burden, Conrad J / Griffiths, Robert C

    Journal of mathematical biology

    2018  Volume 78, Issue 4, Page(s) 1211–1224

    Abstract: The stationary distribution of a sample taken from a Wright-Fisher diffusion with general small mutation rates is found using a coalescent approach. The approximation is equivalent to having at most one mutation in the coalescent tree to the first order ... ...

    Abstract The stationary distribution of a sample taken from a Wright-Fisher diffusion with general small mutation rates is found using a coalescent approach. The approximation is equivalent to having at most one mutation in the coalescent tree to the first order in the rates. The sample probabilities characterize an approximation for the stationary distribution from the Wright-Fisher diffusion. The approach is different from Burden and Tang (Theor Popul Biol 112:22-32, 2016; Theor Popul Biol 113:23-33, 2017) who use a probability flux argument to obtain the same results from a forward diffusion generator equation. The solution has interest because the solution is not known when rates are not small. An analogous solution is found for the configuration of alleles in a general exchangeable binary coalescent tree. In particular an explicit solution is found for a pure birth process tree when individuals reproduce at rate [Formula: see text].
    MeSH term(s) Alleles ; Animals ; Computational Biology ; Gene Frequency ; Genetics, Population ; Markov Chains ; Mathematical Concepts ; Models, Genetic ; Mutation Rate ; Probability
    Language English
    Publishing date 2018-11-13
    Publishing country Germany
    Document type Journal Article ; Research Support, Non-U.S. Gov't
    ZDB-ID 187101-8
    ISSN 1432-1416 ; 0303-6812
    ISSN (online) 1432-1416
    ISSN 0303-6812
    DOI 10.1007/s00285-018-1306-y
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  8. Article ; Online: Stationary distribution of a 2-island 2-allele Wright-Fisher diffusion model with slow mutation and migration rates.

    Burden, Conrad J / Griffiths, Robert C

    Theoretical population biology

    2018  Volume 124, Page(s) 70–80

    Abstract: The stationary distribution of the diffusion limit of the 2-island, 2-allele Wright-Fisher with small but otherwise arbitrary mutation and migration rates is investigated. Following a method developed by Burden and Tang (2016, 2017) for approximating the ...

    Abstract The stationary distribution of the diffusion limit of the 2-island, 2-allele Wright-Fisher with small but otherwise arbitrary mutation and migration rates is investigated. Following a method developed by Burden and Tang (2016, 2017) for approximating the forward Kolmogorov equation, the stationary distribution is obtained to leading order as a set of line densities on the edges of the sample space, corresponding to states for which one island is bi-allelic and the other island is non-segregating, and a set of point masses at the corners of the sample space, corresponding to states for which both islands are simultaneously non-segregating. Analytic results for the corner probabilities and line densities are verified independently using the backward generator and for the corner probabilities using the coalescent.
    MeSH term(s) Alleles ; Computer Simulation ; Gene Frequency ; Genetic Drift ; Genetics, Population ; Models, Genetic ; Mutation ; Probability
    Language English
    Publishing date 2018-10-09
    Publishing country United States
    Document type Journal Article ; Research Support, Non-U.S. Gov't
    ZDB-ID 3948-2
    ISSN 1096-0325 ; 0040-5809
    ISSN (online) 1096-0325
    ISSN 0040-5809
    DOI 10.1016/j.tpb.2018.09.004
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  9. Article ; Online: Ancestral inference from haplotypes and mutations.

    Griffiths, Robert C / Tavaré, Simon

    Theoretical population biology

    2018  Volume 122, Page(s) 12–21

    Abstract: We consider inference about the history of a sample of DNA sequences, conditional upon the haplotype counts and the number of segregating sites observed at the present time. After deriving some theoretical results in the coalescent setting, we implement ... ...

    Abstract We consider inference about the history of a sample of DNA sequences, conditional upon the haplotype counts and the number of segregating sites observed at the present time. After deriving some theoretical results in the coalescent setting, we implement rejection sampling and importance sampling schemes to perform the inference. The importance sampling scheme addresses an extension of the Ewens Sampling Formula for a configuration of haplotypes and the number of segregating sites in the sample. The implementations include both constant and variable population size models. The methods are illustrated by two human Y chromosome datasets.
    MeSH term(s) Algorithms ; Computer Simulation ; Databases, Genetic ; Evolution, Molecular ; Genealogy and Heraldry ; Genetics, Population ; Haplotypes ; Humans ; Models, Genetic ; Mutation ; Probability
    Language English
    Publishing date 2018-04-25
    Publishing country United States
    Document type Journal Article
    ZDB-ID 3948-2
    ISSN 1096-0325 ; 0040-5809
    ISSN (online) 1096-0325
    ISSN 0040-5809
    DOI 10.1016/j.tpb.2018.04.006
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  10. Article ; Online: Corrigendum to "A coalescent dual process for a Wright-Fisher diffusion with recombination and its application to haplotype partitioning" [Theor. Popul. Biol. 112 (2016) 126-138].

    Griffiths, Robert C / Jenkins, Paul A / Lessard, Sabin

    Theoretical population biology

    2019  Volume 130, Page(s) 203

    Language English
    Publishing date 2019-08-05
    Publishing country United States
    Document type Published Erratum
    ZDB-ID 3948-2
    ISSN 1096-0325 ; 0040-5809
    ISSN (online) 1096-0325
    ISSN 0040-5809
    DOI 10.1016/j.tpb.2019.08.001
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