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Coalescence, Recombinations and Mutations

Salamat, Majid | 2011

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 41820 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Pardoux, Etienne; Zohori Zageneh, Bijan; Zamani, Shiva
  7. Abstract:
  8. This thesis is concentrated on some subjects on population genetics. In the rst part we give formulae including the expectation and variance of the height and the length of the ancestral recombination graph (ARG) and the expectation and variance of the number of recombination events and we show that the expectation of the length of the ARG is a linear combination of the expectation of the length of Kingman's coalescent and the expectation of the height of the ARG. Also we show give a relation between the expectation of the ARG and the expectation of the number of recombination events. At the end of this part we show that the ARG comes down from innity in the sense that we can dene it with X0 = 0, while Xt < 1 for all t and we nd the speed that the ARG comes down from innity. In the second part we nd a generalization of the the Ewens sampling formula (GESF) in the presence of recombination for sample of sizes n = 2 and n = 3. In the third part of the thesis we study the ARG along the genome and we we nd the distribution of the number of mutations when we have one recombination event in the genealogy of the sample
  9. Keywords:
  10. Genetic Mutation ; Recombination ; Kingman Coalescence Theory ; Ancestral Recombination Graph ; Ewens Sampling Formula

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  • List of notations
  • Introduction
  • Preliminaries
    • Background in Genetics
      • Genes, DNA, Chromosomes
      • Mutation
      • Recombination
    • Mathematical models of alleles
      • The infinitely-many-alleles model
      • The infinitely-many-sites model
    • Wright-Fisher Model
      • The simplest model
      • Wright-Fisher with mutation
      • Wright-Fisher model with selection
    • Coalescent
    • Ewens Sampling formula (without recombination)
    • Ancestral recombination graph
  • On the height and the length of the Ancestral recombination Graph
    • Introduction and Preliminaries
    • Expectation of the height of ARG
    • Variance of the height of the ARG
    • Expectation of the length of the ARG
    • Variance of the length of the ARG
    • Expectation of the number of recombinations
    • Variance of the number of recombinations
    • The speed at which the ARG comes down from infinity
      • Proof of Proposition 2.8.3
      • Proof of Proposition 2.8.2
      • Proof of Theorem 2.8.1
    • Appendix
      • Proof of (2.2.3)
      • Proof of Theorem 2.3.2
      • Proof of Theorem 2.4.1
      • Proof of Theorem 2.5.1
      • Proof of Theorem 2.6.1
      • Proof of Theorem 2.7.1
  • Generalized Ewens Sampling Formula (GESF)
    • Introduction and Preliminaries
    • The Model
    • A sample of size n=2
      • First method
      • The second method
    • A sample of size n=3
      • The probability that all the individuals in the sample are of the same type at both loci
      • GESF for a sample of size 3.
      • The joint law of T1L and T1R
      • The conditional law of (T2L, T2R), given (T1L, T1R)
      • Computation of the expectations given in 3.4.2 to complete the GESF
    • General case
  • Recombinations and the infinitely-many-sites-model
    • The conditional law of L(1), given L(0)
      • Calculation of h0.
      • Calculation of h+.
      • Calculation of h-.
    • On the number of segregating sites
    • Appendix
      • Proof of (4.1.2)
      • E((L(0))2(L(1)-L(0))
  • Bibliographie
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