APGt aPfit a PGt PGt

Expanding the terms on the right side of this equation gives

which then simplifies to

which increases from zero to 1/4 as time increases. Recall that 1/4 is the probability that a site in a sequence assembled from randomly drawn nucleo-tides (that are equally frequent) matches the same site in an existing sequence. Also notice that the approach to 1/4 will be faster as the substitution rate a increases due to the -4at term in the exponent.

Let's not forget that the original goal was to correct observed divergence between sequences or p distances for multiple hit mutations. The model of sequence change we have so far is the foundation of a correction, but we need to do some more work to obtain an actual correction method. If we think of two DNA sequences originally identical by descent at every nucleotide site at time 0, at some later time t the probability that any site will possess the same nucleotide is

The model we have considered to this point treats time as discrete steps, as shown in Fig. 8.8. If we consider the rate of change at any time t, then the change in the probability that a nucleotide site appears the same with changes in time is a differential dP

equation, —GitI = a - 4aP . The solution to this dt equation is:

0 0