Models for DNA substitution

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

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

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

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

Other Deletions, insertions Insertions in reverse order

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

Other assumptions Stationarity Reversibility

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

Models – discrete time

Jukes – Cantor model All substitutions are equally probable

Stationary distribution

Spectral decomposition of Pn

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.

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

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

Stationary distribution

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

Stationary distribution

Spectral decomposition

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

Stationary distribution where Remark: this model is not reversible

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

Stationary distribution

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

Eigenvalues of P

Left (row) eigenvectors

Right (column) eigenvectors

General 12 parameter model Tavare, 1986

Stationary distribution

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

Continuous – time models

Matrix of transition probabilites Q – intensity matrix

Jukes – Cantor model

Spectral decomposition of P(t)

Kimura model

Spectral decomposition of P(t)

Parameter estimation

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

Probability that the nucleotides are different in two descendants

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

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