1. Models for DNA substitution

2. http://www.stat.rice.edu/ ~mathbio/Polanski/stat655/

3. Plan Basics Models in discrete time Model is continuous time
Parameter estimation

4. Nucleotides Adenine ( A ) or ( a ) Guanine ( G ) or ( g ) purines
Cytosine ( C ) or ( c )
Thymine ( T ) or ( t )
purines
pyrimidines

5. Substitution Purine Purine Transitions Pyrimidine Pyrimidine Purine
AG, G A, C T, T C
Purine
Pyrimidine
Pyrimidine
Purine
Transversions
AT, T A, A C, C A
GT, T G, G C, C G

6. Other
Deletions, insertions
Insertions in reverse order

7. Hypothesis
Substitution of nucleotides in the evolution of DNA sequences can be modeled by a Markov chain or Markov process

8. Other assumptions
Stationarity
Reversibility

9. Transition matrix P = a g c t paa pag pac pat a g pga pgg pgc pgt c
pca
pcg
pcc
pct t
pta
ptg
ptc
ptt

10. Models – discrete time

11. Jukes – Cantor model
All substitutions are equally probable

12. Stationary distribution

13. Spectral decomposition of Pn

14. Remark
When learning and researching Markov models for nucleotide substitution, it greatly helps to use a software for symbolic computation, like Mathematica, Maple, Scientific Workplace.

15. Kimura models  - probability of a transition
 - probability of a specific transversion

16. Kimura 3ST model
 - probability of : AG, C T
 - probability of : AC, G T
 - probability of : AT, C G

17. Stationary distribution

18. Generalizations of Kimura models
By Ewens:
 - probability of : AG, C T
 - probability of : AC, A  T, G C, G T
 - probability of : CA, T  A, C G, T G

19. Stationary distribution

20. Spectral decomposition

21. By Blaisdell:
 - probability of : AG, CT
 - probability of : GA, TC
 - probability of : AC, A  T, G C, G T
 - probability of : CA, T  A, C G, T G

22. Stationary distribution
where
Remark: this model is not reversible

23. Felsenstein model
Probability of substitution of any nucleotide by another
is proportional to the stationary probability of the substituting
nucleotide

24. Stationary distribution

25. HKY model Hasegawa, Kishino, Yano
Different rates for transitions and transversions

26. Eigenvalues of P

27. Left (row) eigenvectors

28. Right (column) eigenvectors

29. General 12 parameter model
Tavare, 1986

30. Stationary distribution

31. Reversibility A=D, B=G, C=J, E=H, F=K, I=L
Conclusion – the most general reversible model has
12 – 6 = 6 free parameters

32. Continuous – time models

33. Matrix of transition probabilites
Q – intensity matrix

34. Jukes – Cantor model

35. Spectral decomposition of P(t)

36. Kimura model

37. Spectral decomposition of P(t)

38. Parameter estimation

39. Jukes – Cantor model Three things are equivalent due to reversibility:
Ancestor (A) D2 A D1 D1 A D2 D1 D2

40. Probability that the nucleotides are different in two descendants

41. Estimating p We have two DNA sequences of length N
D1: ACAATACAGGGCAGATAGATACAGATAGACACAGACAGAGCAGAGACAG
D2: ACAATACAGGACAGTTAGATACAGATAGACACAGACAGAGCAGAGACAG
Number of differences p = N

42. Kimura model p – probability of two different purines or pyrimidines
q – probability of purine and pyrimidine