The genetic diffusion coefficient depends on both the allele frequency and the size of the population. Diffusion of allele frequency is greatest when p = q = 1/2 and declines to zero as p approaches zero or one. Populations (as with particles) tend to diffuse to areas where the diffusion coefficient is lowest and then get stuck there since the rate of spread of particles (the variance in position per time step) is reduced. The rate of diffusion also depends on the size of the population, decreasing as N increases. For particles, this is due to more frequent collision that reduces the ability to move as the concentration of particles increases (think of trying to walk in a straight line while in a large crowd of people). In biological populations, the diffusion coefficient depends on N since the population size determines the amount of sampling error from generation to generation. It is satisfying that both of these features of allele frequency diffusion agree with our previous generalizations about genetic drift obtained with distinct approaches to the problem.

Next, we would like to keep track of the chance that a particle is at a given location along the axis of diffusion. The resulting probability distribution shows how many particles out of a large number should be at each point along the axis, just as Markov chain models show the expected number of populations at each allelic state. Making such a probability distribution requires that we know the flux or the net number of particles moving through a defined area per time interval. Let's define the area, call it A, where we will determine the flux through the plane at the point x (Fig. 3.12). The particles that will move through plane A in one time step must be within plus or minus 8 of x because a particle travels the distance 8 in one time step. The net number of particles moving to the right through A is the same thing as the difference in the number of particles moving from the left and from the right:

where N represents the number of particles moving left (L) or right (R) through A, and the 1/2 is because only half of the particles on each side of x will move

The flux is defined per area per time, so we need to divide by area (A) and time (t):

where Jx represents the flux at point x. We can multiply equation 3.31 by 82/82 (or 1) and then rearrange it to get

Now notice that the number of particles moving left and moving right are both divided by an area times a distance along the axis of diffusion (A8). As you can see in Fig. 3.12, this defines a three-dimensional volume. The number of particles per volume is equivalent to a concentration, so we can use C to represent the concentration of particles. Also notice 82

■ is the diffusion coefficient for one time step that

0 0

Post a comment