Nucleotide substitution models

Transcription

1 Nucleotide substitution models Alexander Churbanov University of Wyoming, Laramie Nucleotide substitution models p. 1/23

2 Jukes and Cantor s model [1] The simples symmetrical model of DNA evolution All sites change independently All sites have the same stochastic process working at them Nucleotide substitution models p. 2/23

3 Probability for nucleotide (1) Let us assume nucleotide residing at certain cite in DNA sequence is A, Consider probability that p A(t) that the site will be occupied by A at time t, Since we start with A, p A(0) = 1, At a time 1 probability of still having A is p A(1) = 1 3α. Nucleotide substitution models p. 3/23

4 Probability for nucleotide (2) At a time 2 p A(2) = (1 3α)p A(1) + α(1 p A(1) ) 1. The nucleotide has remained unchanged with probability 1 3α 2. The nucleotide did change to T, C, G, but subsequently reverted to A with probability α The following recurrence holds p A(t+1) = (1 3α)p A(t) + α(1 p A(t) ), p A(t+1) p A(t) = 3αp A(t) + α(1 p A(t) ), p A(t) = 3αp A(t) + α(1 p A(t) ) = 4αp A(t) + α. Nucleotide substitution models p. 4/23

5 Probability for nucleotide (2) At a time 2 p A(2) = (1 3α)p A(1) + α(1 p A(1) ) 1. The nucleotide has remained unchanged with probability 1 3α 2. The nucleotide did change to T, C, G, but subsequently reverted to A with probability α The following recurrence holds p A(t+1) = (1 3α)p A(t) + α(1 p A(t) ), p A(t+1) p A(t) = 3αp A(t) + α(1 p A(t) ), p A(t) = 3αp A(t) + α(1 p A(t) ) = 4αp A(t) + α. Nucleotide substitution models p. 5/23

6 Continuous time dp A(t) dt = 4αp A(t) + α. This first-order linear differential equation has solution p A(t) = 1 ( 4 + p A(0) 1 ) e 4αt 4 Initial probability is p A(0) = 1, thererefore p A(t) = e 4αt Nucleotide substitution models p. 6/23

7 Probabilities If the initial nucleotide is not A, then p A(0) = 0 and p A(t) = e 4αt generalizing for nucleotides i and j, where i j p ii(t) = e 4αt p ij(t) = e 4αt Nucleotide substitution models p. 7/23

8 Graphical interpretation Nucleotide substitution models p. 8/23

9 Sequence similarity (1) Nucleotide substitution models p. 9/23

10 Sequence similarity (2) A common measure for sequence similarity is the proportion of identical nucleotides between the two sequences under study. The expected value of this proportion is equal to the probability I (t) that the nucleotide at a given site at a time t is the same in both sequences. Cases include nucleotide conservation p 2 ii(t) and parallel substitutions p 2 ij(t). I (t) = p 2 AA(t) + p2 AT(t) + p2 AC(t) + p2 AG(t), I (t) = e 8αt. Nucleotide substitution models p. 10/23

11 Estimating substitutions (1) The probability that the two sequences are different at a site at time t is p = 1 I (t) p = 3 4 ( 1 e 8αt ), 8αt = ln (1 43 p ). Nucleotide substitution models p. 11/23

12 Estimating substitutions (2) We estimate K, the actual number of substitutions per site since the divergence between the two sequences. In the one parameter model, K = 2(3αt), where 3αt is the expected number of substitutions per site in one lineage. ( ) 3 K = ln (1 43 ) 4 p Nucleotide substitution models p. 12/23

13 Kimura model [2] The method has the merit of incorporating the possibility that sometimes transition type substitutions (with rate α) may occur more frequently than transversion type substitutions (with rate β). Nucleotide substitution models p. 13/23

14 Kimura model [2] Same UU CC AA GG Total (Frequency) (R 1 ) (R 2 ) (R 3 ) (R 4 ) (R) Different, Type I UC CU AG GA Total (Frequency) (P 1 ) (P 1 ) (P 2 ) (P 2 ) (P ) Different, UA AU UG GU TypeII (Q 1 ) (Q 1 ) (Q 2 ) (Q 2 ) Total CA AC CG GC (Q) (Frequency) (Q 3 ) (Q 3 ) (Q 4 ) (Q 4 ) Nucleotide substitution models p. 14/23

15 Kimura model (1) Total rate of substitutions per site per year is k = α + 2β P is the probability of homologous sites showing a type I difference Q is the probability of homologous sites showing a type II difference R is the probability of homologous sites to be the same We denote probability of identity at homologous sites at time T as R(T) = 1 P(T) Q(T) Nucleotide substitution models p. 15/23

16 Kimura model (2) We can derive the equation for P and Q at time T + T in terms of P, Q and R at time T We can distinguish three ways by which UC (U at homologous position of organism 1 corresponding to C in organism 2) at time T + T is derived from various base pairs at time T. 1. Pair UC is derived from UC. Since probability of substitution in short time interval is T is (α + 2β) T. Thus the probability of no change occurring in both homologous sites is [1 (α + 2β) T] 2, so this case contribution is [1 (α + 2β) T] 2 P 1 (T) Nucleotide substitution models p. 16/23

17 Kimura model (3) 2. Pair UC is derived either from UU or from CC with probability α T [R 1 (T) + R 2 (T)] 3. Pair UC could be derived from UA, UG, AC and GC with probability β T[Q 1 (T) + Q 2 (T) + Q 3 (T) + Q 4 (T)] = β T Q(T) 2 Nucleotide substitution models p. 17/23

18 Kimura model (4) Combining contributions coming from different classes resulting in UC and disregarding terms with ( T) 2, we get (1) P 1 (T + T) = [1 (2α + 4β) T] P 1 (T) + α T [R 1 (T) + R 2 (T)] + β T Q(T) 2 Similarly, for the base pair AG we get (2) P 2 (T + T) = [1 (2α + 4β) T] P 2 (T) + α T [R 3 (T) + R 4 (T)] + β T Q(T) 2 Nucleotide substitution models p. 18/23

19 Kimura model (5) Summing equations (1) and (2), and noting P(T) = 2P 1 (T) + 2P 2 (T), we get (3) P(T) T = 2α 4(α + β)p(t) 2(α β)q(t) Q(T) (4) = 4β 8βQ(T) T Converting (3) and (4) to continuous case, we get dp(t) dt = 2α 4(α + β)p(t) 2(α β)q(t) dq(t) dt = 4β 8βQ(T) Nucleotide substitution models p. 19/23

20 Kimura model (5) The solution for these equations satisfy the initial condition P(0) = Q(0) = 0, i.e. no base difference exists at T = 0 P(T) = e 4(α+β)T e 8βT Q(T) = e 8βT Nucleotide substitution models p. 20/23

21 Kimura model (6) It follows that (5) and (6) so that (7) 4(α + β)t = ln(1 2P(T) Q(T)) 8βT = ln(1 2Q(T)) 4αT = ln(1 2P(T) Q(T)) ln(1 2Q(T)) Nucleotide substitution models p. 21/23

22 Kimura model (7) Since evolutionary rate is k = α + 2β, the total number of substitutions per two diverged sequences is K = 2Tk = 2αT + 4βT By omitting index T and following equations (6) and (7) we obtain K = 1 {(1 2 ln 2P Q) } 1 2Q Nucleotide substitution models p. 22/23

23 References [1] T.H. Jukes and C.R. Cantor, Evolution of protein molecules, Mammalian protein metabolism (H.N. Munro, ed.), Academic Press, New York, 1969, pp [2] M. Kimura, A simple method for estimating evolutionary rate of base substitutions through comparative studies of nucleotide sequences, Journal of Molecular Evolution 16 (1980), Nucleotide substitution models p. 23/23