I threw together some simulations a while ago for a lecture on selection. As usual they led to thinking about some of the implications of selection that I knew but that I had never really internalized.
Here, for example is a simulation of 100 histories, i.e. parallel worlds, in which a single mutation causing lactase persistence appears in an adequately mixed population of 50,000 reproducing adults who are dairying people. The mutation is dominant, so both carriers and homozygotes enjoy a 5% fitness advantage due to the ability to digest lactose and hence obtain 40% more calories from the same diet as people without the mutation: see here for the details. We don’t imagine this advantage to be important every year, but in times of stress lactase persistents (LPs) would have been more likely to survive and, perhaps, avoid slaughtering their herds to eat them.
In twelve (the expected number is ten) of the one hundred worlds the mutant persisted long enough that deterministic forces took over and it increased in frequency just as theory predicts. In the other 88 worlds the mutation may have hung around a while at low frequency but was eventually lost to drift. Since the time until determinism takes over is random, even where the mutant eventually won the time is highly variable. In the green world it took about four thousand years to reach a frequency of one half while in the tardy red and dark blue worlds it took about six thousand years.
This may provide some quantitative understanding of history. A figure tossed about for the age of the west Eurasian mutation is eight thousand years. With a five percent advantage it only reaches a frequency of seventy to eighty percent in that time while the frequency in northern Europe today approaches fixation. The advantage must have been greater than five percent.
On the other hand there are grounds for sorry pessimism here. I have the naive human idea that history makes sense while this exercise shows that it is worse than a crapshoot. An extremely rare random accident, the mutation, caused one of the greatest population blooms we know about, if our theory of the Indo-European tie in with LP is correct.
When we teach population genetics we study gene frequency change, usually with the assumption that population size is held constant. In this case we can’t defend such an assumption since the 5% advantage of LP is huge. In this Malthusian dairying environment the advantage would lead immediately to explosive population growth, which is almost certainly what did happen.
This figure is from the same simulation program as the figure above but the outcome is slightly different (randomness). Only five of the one hundred mutations persisted and the simulation only goes to five thousand rather than ten thousand years (drawing a million binomial samples of size two slows my poor desktop to a crawl so I stop things early).
In the five rare cases where the new mutant persists it leads to population growth, but how much? In one history, the right red one, the population has risen to about seventy thousand people, while at the other extreme it has risen to nearly eight hundred thousand people because it happened to become common faster. With the more realistic advantage of ten percent these number would be much higher. The full figure is not shown because my poor desktop begged me to give it a rest.
Are these simulated population numbers implausible? I think not, recalling that the Indo-European expansion is like the explosion of a huge inkball in western Eurasia from the perspective of history. Toward the end of the five thousand year interval people at the center must have been moving away as fast as they could to avoid being trampled.