15— Some Elementary Attempts at Numerical Modeling of Problems Concerning Rates of Evolutionary Processes: With R. Schrandt (LA-4573-MS, December 1970)

In order to study, on a computer, the rate at which a population can acquire a sequence of mutations we needed a set of more amenable parameters, which, it is hoped, could eventually be scaled down to the set given above. For the first problem, called ADAM, we used the following set: N = 100, K = 100, a = .02, and 7 = .1.

The method was as follows: In any one generation, each member of the population N had a probability a of acquiring one new independent mutation. Each individual then had one child, with a probability of extra children given by y(Kg - Ko), where Kg = the total number of different mutations possessed by this individual and, Ko = the minimum number of mutations possessed by any individual in the population. (If n/-y < Kg - K <(n + l)/7y, n = 1, 2, 3, ..., the individual had n extra children, and a probability of y(Kg - Ko- n/y) for an (n+ l)th child.)

The children were then given the number of mutations possessed by their parent. The parent population then was assumed to have died, and the children formed the new population. The numbers of

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mutations in the population were recorded in categories by counts ni, i = 0, 1, 2, 3, ..., where ni = the number of individuals having a total of i mutations, and Eni =N, the size of the new population. In the next generation, each member of this new population had the same probability a of acquiring another new mutation, and had children according to the above recipe. These children with their number of mutations recorded then became the population for the succeeding generation, etc.

It was necessary to renorm the population periodically, since the number of children increased in each generation. This was done by reducing the count ni in each category by 1/2, to the nearest smaller integer, when the population reached 200, which is double the initial population.

The weighted average number p of mutations possessed by the population was computed for each generation from the categories ni. This average was then plotted as ordinate against the generation time as abscissa. The slope of this curve is then the rate at which the population can acquire a sequence of mutations, as a function of the parameters a and y.

This rate of acquiring mutations turned out indeed to be linear. For the parameters a = .02 and y = .1, the slope was about .1; or a majority of the population acquired an additional mutation every 10 generations. Several problems were run with smaller values of y, that is, y' = f -y, where f = a fraction. The graphs were all linear, with decreasing slopes, which decreased more closely with Vf than with f itself. There was no appreciable change in the slope by doubling the initial population to 200. Figure 1 shows a plot of 3 cases: N = 100, a = .01, and y = .1, .05 and .01 respectively.

A second version of ADAM was run with the Ko in the probability recipe defined as the average number of mutations in the population, instead of the minimum number. Those individuals having fewer than the average number of mutations (Kg < Ko) had a probability of 7(Ko - Kg) of no children, (if Ko- Kg> 1/7, they had no children deterministically), and a probability of 1 - 7(Ko - Kg) of one child. The individuals with more than the average number of mutations had the same probability for extra children as the one defined above. This version required fewer renormings of the population and it led to a somewhat greater slope than the Ko = min. recipe. The graph was still linear. Figure 2 illustrates the two versions with the same parameters N = 100, a = .02, and - =.1. The recipe with Ko = average was used in all subsequent problems.

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NUMBER OF GENERATIONS Fig. 1.

15— Some Elementary Attempts at Numerical Modeling of Problems Concerning Rates of Evolutionary Processes: With R. Schrandt (LA-4573-MS, December 1970)