Maximum population size and speciation time: Here is an idea you may try on for size.  It is not critical, and I have no data to support it, so don’t put your heart into it.

While puzzling over what a maximum population size might be, I reasoned thus.  Assume, and I think it has been demonstrated, that crossing over, or as it is now called recombination, does not occur at the same rate throughout the genome.  So imagine the genome with the largest areas of reduced recombination flagged.  Perhaps they are near the centromeres. 

Next assume, and it seems reasonable, that mutation rates are not the same throughout the genome and in particular the mutations that produce detuning of one gene against another are not uniformly distributed throughout.  Choose from among the flagged regions the one with the highest mutation rate of this sort.  Imagine tying a ribbon around it, just one ribbon for the entire population that is randomly mating.  Now in each generation you will tie a ribbon around any similar region that is the direct descendant of the original region with the ribbon.

Now choose a randomly mating population in which these ribbons will be found.  There are of course homologous regions throughout the population, two per individual unless you have chosen a region that is on the sex chromosome, in which case choose a new one.  As you follow these regions from generation to generation, some will go extinct and some will increase in number simply from stochastic variation.  Some will just be lucky.  Assume you have chosen one that neither goes extinct nor increases dramatically in number. 

Now as you watch, from time to time one of the regions with a ribbon will be pared with another region with a ribbon in some individual within the population.  At this time, the region is tested for compatibility with its counterpart.  If too many mutations have occurred in one of them, there are no offspring and it is eliminated.  And mutations accumulate over time.

The number of generations that it takes for one region with a ribbon to be matched with another region with a ribbon is equal to the number of such regions, with or without ribbon, divided by the number of ribbons.  We have assumed the number of ribbons is small, so let us just say it is two.  So the average time between tests is the number of generations that equals the size of the population.  If the population size is excessive, then the number of mutations that have accumulated on average will be excessive and the population will become unstable and numbers will fall. 

This, of course, is what we demonstrated in the first place.  Then we were looking at population size.  Now we are looking at the number of generations between specific regions being tested against each other, but the numbers are related, and we are suggesting they are about the same.

The region with the ribbon will behave no differently whether it is in the same population or a different one.  Assume we divide the population in two by some geographic event.  A glacier moves in splitting the environment.  Individuals east of the glacier cannot mate with individuals west of the glacier, but otherwise all is as before.  The region with a ribbon behaves the same on both sides of the glacier, maintaining the low recombination rate and high mutation rate and being tested periodically by being in the same individual as another region with a ribbon.

The glacier melts.  The two populations merge again.  A number of generations have passed.

What I propose is this.  If the number of generations of separation is much smaller than the size of a stable population, then when two regions with ribbons are tested against each other, their separation time and their new mutational burden is not much greater than usual and the two populations become a single surviving population, always assuming that the population size remains within limits and the environment remains supportive.

On the other hand, if the glacier has lasted much longer than the maximum number of members in a stable population, then the number of accumulated mutations is much greater, and is effectively the same as if the population size was enormous.  When the regions with ribbons are tested against each other, and when all other homologous regions are tested against each other, there are no offspring.  There will be hybrid infertility, and we can declare that the two are now different species. 

So the proposition can now be stated thus.  The number of generations it takes for two recently isolated populations to undergo speciation is approximately the same as the size of the maximum stable population.  At one time, this seemed important, because speciation is a well established phenomenon, while maximum stable population size remains an obscure idea. 

The concept could be tested with the same computer program that I have used to develop the predictions on the Main Page of this site.  There are some difficulties with this.  It is not clear from the graph exactly what the maximum stable population size might be, but that could be addressed by doing enough runs to get clean statistics and then choosing the population size where the growth rate falls to zero.  It is not clear exactly what to use as a definition of speciation, but taking the onset of hybrid vigor with hybrid infertility should do even if the cut off is not perfectly clean.

The third problem is that it would be very laborious, but that should not be an excuse.

The fourth and most serious problem is that there are no real world data to compare the results with.  No one yet has pointed to a human or animal population and said, “So big it can grow, and no bigger.”  To be sure, you could look at the maximum effective gene pool size in large urban areas, but they are already too big.  In rich countries they are already very infertile, far larger than the maximum stable population size.

Sadder is the fact that there are no estimates I have seen as to just how long speciation takes.  It is forgivable that there is no other mathematical model of speciation, but it would seem that anybody with an interest in it would be running around looking at animals and trying to decide whether they are the same species and how long they have been separated.  From the calculations of population size already given, maximum stable population sizes are probably in the low hundreds for people and in the low thousands for mollusks.  Surely the mutational clock is able to pick up genetic distances over that range, or if not, at least the limits of what can be picked up should be well known and much discussed.  If there is discussion, I have not heard it. 

The best data I have discovered to date has to do with camels.  Camels come in 2 forms, the two humped Bactrian and the one humped dromedary.  They can be crossed, and the result, as I remember, is a one humped camel with a dent in the top of the hump.  I believe the reference is Adrian M. Lister, Remedies for Windy Camels, NATURE vol 390, December 18/25, 1997 p 658.  The two lines were separated when camels were brought from Asia to Egypt about 4,500 years ago.  Now when you follow these hybrids, the hump-and-a-half camels, they can have offspring but the line eventually becomes infertile.  This is the well established phenomenon of “hybrid breakdown,” and it stands to reason that it is an early stage in speciation, occurring before hybrid sterility. 

I happened to meet Dr. Kamal Khazanehdari of the Central Veterinary Research Laboratory, Dubai, United Arab Emirates at a conference.  We chatted only briefly, but I pressed him on what the generation time of camels was.  He pointed out the enormous difficulties of making an estimate but thought it might be about 10 years.  There were a lot of distractions, so it would not be fair to hold him to that number, but let us take it as a possibility.  Wikipedia says a camel lives 40 or 50 years, so if fertility declines little with age, that might mean a generation time of 20 years. 

Between the two estimates, and assuming that speciation with hybrid breakdown has occurred quite recently, that would mean 225 to 450 generations to speciation and suggest that the maximum stable gene pool size for camels and perhaps other mammals is indeed in the low hundreds.  That is consistent with everything else we know, but I should not take it to the bank.  There need to be a lot more estimates of time to speciation before it can be taken at all seriously.

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