03-511/711, 15-495/856 Course Notes - Sep. 22, 2009

An introduction to finite state, discrete Markov chains

An short introduction to this material can be found in Chapter 4 of Ewens and Grant.

The Jukes-Cantor model of sequence evolution

We that observe that a pairwise alignment of two sequences r and s of length n has m mismatches (no gaps).

In the regime where neutral change dominates, we suspect that this is an underestimate of the number of positions at which change occurred. For example, if there is an A at the same position in both sequences, it could be because the ancestral state was also A and no change occured (left hand figure) or because parallel changes occured in both sequences (right hand figure).

Given the observed number of mismatches, we wish to estimate the number of substitutions that actually occurred. We do this via the following steps:

p= m/n is the observed frequency of mismatches.
p is an estimator for the true probability of a mismatch, p=P_mismatch.

• all substitutions are equally likely
• constant rate of change, λ, in both lineages
• site independence

1. Markov model of sequence evolution
2. Use model to derive an expression for P_mismatch in terms of λt; p = f(λt)
3. Solve for λt = f ^-1(p)
4. E [substitutions/site] ≈ 2f ^-1(p)

First, we define a Markov model of instantaneous change. The simplest such model for DNA is the Jukes-Cantor model, which assumes

• that all point mutations (A->C, A->G, A->T, C->A...) are equally likely and occur at a rate a.
• that the substitution probabilities at each site are independent

              To:    A        G        C        T
                   ---------------------------------
               A  | 1-3a      a         a       a
       From:   G  |  a       1-3a       a       a
               C  |  a        a        1-3a     a
               T  |  a        a         a      1-3a

• the rate of substitution is λ=3 a,
• the expected number of substitutions per site is 2λt=6 a t, where t is the time since the two sites diverged from their common ancestor,
• given sufficient time, all nucleotides will appear equally frequently in the sequence. In other words, the stationary distribution of this Markov chain is (0.25,0.25,0.25,0.25).

Second, using this Markov chain we derive an expression describing how changes accumulate over a period of time t. The probability that residue X is in site k after time t is

                                         - 4 a t
               pXX(t)   =     1/4 + 3/4 e     

                                         - 4 a t
               pYX(t)   =     1/4 - 1/4 e     

Third, using the above formulae, we derive an expression for P_mismatch, the expected number of observable differences, between the two sequences:

                Pmismatch   =    3/4 ( 1   -  e - 8 a t).

We solve the above equation for to obtain an expression for a t in terms of P_mismatch:

                a t  =    -1/8 ln ( 1   -  4/3 p  ).

Note that the expected number of substitutions per site (whether they are observable or not) is 6 a t (see above). Multiplying both sides of the equation by 6, we obtain an expression for the expected fraction of sites at which at least one substitution occurred in terms of the number of differences observed:

                Psub  =   - 3/4 ln  ( 1  -  4/3 Pmismatch ).

                Psub   =   - 3/4 ln  ( 1  -  4/3 m/n ).

For more details, see Mona Singh's Phylogeny notes or Durbin, et al: 8.1, 8.2.