Jukes-Cantor (3)

From last time, we have two equations for sequences changing according to Jukes-Cantor:

PXX(t) = 1/4 + 3/4*e-4*α*t PXY(t) = 1/4 - 1/4*e-4*α*t

We can look at this as the probability that a single site will change in this way (from X to Y) over time, but we can also look at it as the fraction of a collection of sites that will change. Since Y can be any one of three nucleotides, the total fraction of sites that differ between the ancestral sequence and a present-day descendant sequence is three times P_XY(t) or:

PXN(t) = 3/4 - 3/4*e-4*α*t

Two present-day homologs (common ancestor) have effectively evolved for twice the time because there are two stretches of evolution of time t. The proportion of sites that differ is:

p = 3/4 - 3/4*e-8*α*t

The above equation is what we observe when we look at the sequences. However, our estimate of the true distance, or actual number of substitutions per site:

d = 2*t * 3*α = 6*α*t

We usually do not know either α or t individually, but we can say that:

α*t = d/6 p = 3/4 - 3/4*e-8*α*t 4/3*p = 1 - e-8/6*d d = -3/4 * ln(1 - 4/3*p)

This is what we've been after. These equations relate the actual evolutionary distance to the observed changes and vice-versa.

p = proportion or fraction of sites that are observed to be different
d = distance or actual number of substitutions per site

Their relationship is plotted at the top.

Plot code:

from math import e import numpy as np import matplotlib.pyplot as plt def d(at): return 6*at def p(at): return -0.75*e**(-8*at) + 0.75 at = A = np.arange(0,2.001,0.001) plt.plot(at,d(at),lw=3,color='r') plt.plot(at,p(at),lw=3,color='b') ax = plt.axes() ax.set_xlim(0, 1.0) ax.set_ylim(0, 2.0) ax.grid(True) plt.savefig('example.png')