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A First Course in Stochastic Processes

This chapter explores Markov Chains and their properties, using nucleotide evolution as an example. Topics include types of point mutations, Kimura's 2-parameter model, and Markov Chain properties.

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A First Course in Stochastic Processes

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  1. A First Course in Stochastic Processes Chapter Two: Markov Chains

  2. X2= X2=2 X3=1 X4=3 X1=1

  3. X1 X2 X3 X4 X5 etc

  4. P =

  5. G C T Example Two: Nucleotide evolution A

  6. α β β β β α Types of point mutation A G Purine Transitions Transversions T C Pyramidine Transitions

  7. A G C T A G P = C T Kimura’s 2 parameter model (K2P)

  8. G C C G A C T G A T G C G C T G T A C A G T A T C A T

  9. G C C G A C T G A T G C A G T A T C A T The Markov Property

  10. The Markov Property

  11. Markov Chain properties accessible aperiodic communicate recurrent irreducible transient

  12. A G C T A G P = C T Accessible 0

  13. A G C T A G P = C T Accessible A (and G) are no longer accessible from C (or T). 0 0 0 0

  14. A G C T A G P = C T Accessible But C (and T) are still accessible from A (or G). 0 0 0 0

  15. A G C T A G P = C T Communicate Reciprocal accessibility

  16. A G C T A G P = C T Irreducible All elements communicate

  17. A G C T A P1 0 0 0 G P = = 0 P2 0 0 C 0 0 T 0 0 A G C T P1 = P2 = A C G T Non-irreducible

  18. Repercussions of communication • Reflexivity • Symmetry • Transitivity

  19. P = Periodicity

  20. Periodicity • The period d(i) of an element i is defined as the greatest common divisor of the numbers of the generations in which the element is visited. • Most Markov Chains that we deal with do not exhibit periodicity. • A Markov Chain is aperiodicif d(i) = 1 for all i.

  21. recurrent transient Recurrence

  22. More on Recurrence • and i is recurrent then j is recurrent • In a one-dimensional symmetric random walk the origin is recurrent • In a two-dimensional symmetric random walk the origin is recurrent • In a three-dimensional symmetric random walk the origin is transient

  23. Markov Chain properties accessible aperiodic communicate recurrent irreducible transient

  24. Markov Chains Examples

  25. X1=1 X1 X2 X3 X4 X5 etc

  26. P =

  27. Diffusion across a permeable membrane (1D random walk)

  28. Brownian motion (2D random walk)

  29. Wright-Fisher allele frequency model X1=1

  30. Haldane (1927) branching process model of fixation probability 2 3 4 4 4 4 2

  31. Haldane (1927) branching process model of fixation probability

  32. Haldane (1927) branching process model of fixation probability Pi,j = coefficient of sj in the above generating function

  33. Haldane (1927) branching process model of fixation probability Probability of fixation = 2s

  34. Markov Chain properties accessible aperiodic communicate recurrent irreducible transient

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