Simulating Haldane's Dilemma

In my previous post, I calculated how long it would take for multiple beneficial mutations to substitute into a population, and showed that it happened much faster than substitution one at a time. What happens when you try this in a simulation?

Remine's web-page at http://saintpaulscience.com/Haldane.htm notes that "Cost arguments can apply to any evolutionary scenario, for any species, in any environment, and even including computer simulations." So results of a computer simulation (given the right assumptions) should be a valid test.

The simulation I set up worked like this: each individual has two sets of chromosomes. Each chromosome can either contain or not contain a beneficial mutation. At reproduction time, two individuals contribute one of each of their pairs of chromosomes (not necessarily all from the same set) to each child they have. The species is monogamous, so each parent mates with only one other parent, having a total of four children (2 per individual). The population starts with one of each mutation.

That would double the next generation's population, so some must be eliminated. First we kill off a randomly chosen set of individuals, to bring the numbers back to 1.1 per parent. This simulates accidents, harmful mutations, random predation, disease, and all the other things that interfere with selection. Now we take the best of the remaining individuals and make that the next generation.

Selection can be done in two ways. We can count the number of beneficial mutations each individual possesses, and sort by that. This assumes that each is about equal, and the more you have the better. Or we can sort "alphabetically" - that is, the first mutation rules unless both individuals have it; try the second if the first was even; then try the third, etc. This assumes that some mutations are better than others - the mutations do not interact especially favorably, but are still always advantageous.

The results of the simulation are naturally going to be variable; particularly in the early stages of a substitution, it's easy for a beneficial mutation to drift out when its numbers are small. The simulation assumes that beneficial mutations are available; if any vanish, they are replaced with a new single instance.

The result? I had a population size of 1024, so Remine and I would both agree that selecting one mutation takes about 80 generations (log2048 / log1.1 - it's 2048 because there are two copies of each gene). Where we differ is in how long it takes to substitute more. He would claim 80 generations for each extra substitution; I would claim only 11, and some of that to take place during the initial 80 generations.

With 1 mutation to substitute, 100 trials gave results anywhere between 53 and 192 generations, at an average of 86. That matches the prediction reasonably well (because of the random heritability, a mutation may leave anywhere from 0-4 copies, so sometimes a mutation will grow faster than expected or even shrink, particularly early in the run). With 5 mutations, Remine would predict 80 generations each - a total of 400. I would predict 80 for the first one, and less than 11 each for the rest (total of 124), as some of the substitution would be done in parallel.
Results of 100 trials: 92-243, at an average of 130.

With 20 traits, I predict 80 + 19 * 11 = 289; Remine predicts 1600.Results of 100 trials: 238-366, at an average of 279. Note that as larger numbers are involved, the results are more predictable.

Above I mentioned two different selection types; what happens if these are tried? The alphabetical approach means the first mutation always substitutes first, and so on in order. The counting approach means all the mutations end up substituting about the same time. But there is no significant difference in the overall approach (20 traits over 100 trials, 236-361, average 277).

Remine complains that low selection coefficients are generally assumed, which lowers the cost. So let's try 2 as the growth rate instead of 1.1:
1 Trait - both of us predict about 11 generations.
100 trials: 9-14, at an average of 11.
5 Traits - I predict about 11 + 4*2 = 19, Remine predicts 5*11 = 55
100 trials: 18-24, at an average of 20
20 Traits - I predict about 11 + 19*2 = 49, Remine predicts 20*11 = 220
100 trials: 54-65, at an average of 57

As you can see, Remine's expectations are not met.

Remine does make some valid points about the assumptions made; only some mutations will substitute this fast. But if you look at a genome, it will typically be dominated by the fast-substituting mutations, because more of them can substitute in a given time. Nor is it much of a surprise that the first species that's capable of figuring this out would come at the end of a burst of faster-than-average mutation; any new capability such as provided by human intelligence opens up a wide field of potentially beneficial mutation.

But the computer simulation confirms what the mathematics shows - the 300 generation speed-limit that Remine talks about is far slower than what is possible in reality. His argument fails to stand up, at least here.

This assumes truncation selection, which Remine doesn't like. So my next effort was to try individual selection, to see if that made a difference. Once again, this post is long enough already, so I'll leave that for another.

(Incidentally, if you're curious to see what the simulation looks like, I've added that in a separate post).