This expression simplifies considerably. The value of S for a large number of particles should be zero, using the same reasoning as when determining average particle position, since an equal number are moving left and right (S2 is not zero because the squared change in position will always be positive). The middle term in equation 3.23 then drops out since it is multiplied by zero. This leaves the variance in particle position as o2(Xi(t=i)) = x2t=0) + S2 (3.24)

of step length that particles take between time points. If a group of particles all started out at position zero (meaning x2(t=1) = 0), then the variance in particle position increases by S2 every time interval. If t is the number of time steps that have elapsed for particles that started out at position zero, the variance in particle position is tS2. As intuition suggests after watching things like ink diffuse in water, the variance in particle position is not zero and increases with time.

Now we are ready to return to the diffusion coefficient. The diffusion coefficient (D) is defined as half the rate at which the variance in particle position changes as time advances. In symbols this is

1 do2

2 dt

The source of the factor of 1/2 can be seen in Fig. 3.12. Only half of the particles near the point x (within S or one step of the plane) will be headed away and increasing their dispersion while the other half will be headed toward x and not dispersing. Half the vari-

ance in particle position, —, is therefore the diffusion coefficient for physical molecules. The diffusion coefficient tells us how fast particles spread out around some point due to random movement.

Allele frequency in an ensemble population has an analog of the diffusion coefficient. Allele frequency "diffusion" is the spreading out and flattening of the allelic state distributions over time as seen in Markov chain models (see Fig. 3.11). Recall from equation 3.6

| pq that the standard error of the allele frequency is , which can also be thought of as the standard deviation of the mean allele frequency. The variance of the mean allele frequency is then the square of the pq standard deviation or-. This latter quantity is the

0 0

Post a comment