Backwards in Time

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It’s hard to have a recessive lethal hang around for a long time without some kind of heterozygote advantage: selection reduces its frequency. If the population is even moderately large, more than a few thousand, changes in allele frequency over time are very predictable: deterministic.

That also means that one can calculate past frequencies, as long as as these assumptions hold (i.e. as long as there was no tight bottleneck & selection coefficients were the same).

Going forward in time , the frequency of a recessive lethal with no het advantage declines more and more slowly, since the ratio of homozygotes to heterozygotes declines as the allele frequency declines. But if you go backward in time, the frequency grows, and it grows more and more rapidly as you go further and further back in time. This doesn’t continue indefinitely: the frequency can’t go above 100%. Project the frequency of such a recessive lethal back in time and you hit a singularity.

Today, lethal cystic fibrosis alleles have a frequency of 2% in northern Europeans. Unless I’m wrong, it takes 50 generations for a recessive lethal to go from almost 100% to 2%, and another 50 to go from 2% to 1%, assuming no reproductive compensation. ‘Reproductive compensation’ means that parents have another kid when one dies young and thus end up with the same number of children raised to adulthood. This effect weakens, but does not eliminate, selection against lethal recessives. With full reproductive compensation, it takes 80 generations for a recessive lethal to go from 99.5% to 2%, and another 75 to go from 2% to 1%.

If the frequency of lethal CF alleles is 2% today, there must have either been a selective advantage in heterogygotes over most the of the past two thousand years, or the population of northern Europe must have crashed down to a few hundred or less sometime during that period.

There was no such crash, which would have been worse than a nuclear war. Indeed, there was no bottleneck of any kind in that time period: we know this from the historical record. Events like the Black Death do not a bottleneck make: you need to get the population down into the low thousands or less. The Black Death left tens of millions.

So lethal CF mutants had some kind of selective advantage, or were closely linked to some allele that did.

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15 Comments

  1. Here are some theories.

  2. It sounds like a recessive allele would most likely go extinct due to drift in the long run.

  3. Eventually, if nothing else interfered, the gene would be in mutation-selection equilibrium; with new deleterious alleles being generated by mutation at the same rate that old deleterious alleles were being eliminating by selection. Mutation rates are low enough that you wouldn’t see any lethals at 2%.

  4. Greg, this is off-topic, but – assuming the poulation is a constant size – what is the minimum population needed for selection to work? Obviously it is a function of the mutation rate, but what is the function? And what is it in real life, say for humans? 
     
    Thanks in advance.

  5. Could we pick a huge sample of CF-allele-carrying people, and then find out which diseases (if any) are under-represented in this sample? Even better: compare siblings with and without the allele. 
     
    Of course, there is the additional difficulty that modern hygiene and medicine in the West have strongly reduced the incidence of many diseases, which in turn must have strongly reduced any such effect – perhaps even to undetectability. But on the other hand, the CF allele has the “advantage” (*cough*) of being relatively frequent, so there’s a lot of available data. 
     
    The “hitch-hiking” hypothesis (to which gc seems to be alluding in the last sentence of the post) should be more easily tested, by observing whether adjacent regions of the genome show the effects of selection. Who knows, perhaps the data is already there?

  6. David, 
     
    If the product of the population size and the selection coefficient (favored type leaves 101 offspring for every 100 left by disfavored type -> selection coefficient = 0.01) is much larger than one, then the allele frequency will change more or less deterministically. If the product is much smaller than one, then random drift predominates. If the product is between these extremes … then it is more complicated.

  7. a selective advantage in heterogygotes over most the of the past two thousand years 
     
    How about a huge advantage for some of the past two thousand years, as during relatively brief but devastating famines or plagues – would that also work?

  8. How about a huge advantage for some of the past two thousand years, as during relatively brief but devastating famines or plagues – would that also work? 
     
    a selection coefficient of 0.10 is REAL big. the black death resulted in a reduction in population of around 1/3. i don’t think you get the numbers to work.

  9. Thanks, Darth. So how does that translate to real life? Can a population of 1000 be stable long-term? How about 100?

  10. Thanks, Darth. So how does that translate to real life? Can a population of 1000 be stable long-term? How about 100? 
     
    define “long term.” last i checked conservation biologists thought 1,000 was a real small number to avoid exogenous stochastic shocks. a population of 100 is probably susceptible to mutational meltdown. more empirically i think a census size of 1000 is probably not the malthusian limit for most lineages that last for a while. but the long term effec. pop size is the harmonic mean, so the crashes have a disproportionate effect.

  11. define “long term.” 
     
    Forever.  
     
    last i checked conservation biologists thought 1,000 was a real small number to avoid exogenous stochastic shocks. a population of 100 is probably susceptible to mutational meltdown 
     
    I’m aware of that. What I would like to know is if this is confirmed by the theory. I know that mutation rates vary for different parts of the genome, but can’t we account for that, too, at least approximately? 
     
    As a specific example, I understand that the Tasmanian population was very small for ~10,000 years. Also, the Andaman Islanders seem to have some adaptation to their environment, even though their population is small. Of course, neither population is as low as 100. But maybe it got that low at some points?

  12. David, with regard to your questions here are some thoughts that may be relevant: 
     
    By “selection to work” I assume you mean that genome information is maintained. I.e., selection converts noise into information as fast as mutation + drift + changing_environment converts information into noise. 
     
    Mating patterns and number of offspring affect the amount of selection a population experiences. When only the best males mate then many slightly harmful mutations will be removed at each generation. 
     
    Due to crossover during meiosis, good alleles can accumulate on chromosome segments. Such segments will have a significant fitness advantage over competing segments. In this way good alleles can “cooperate” to eliminate harmful alleles. With larger populations there is higher probability of fortuitous crossovers producing high fitness chromosome segments. It is also more likely that linkage between a very beneficial allele and a harmful allele will be broken before the beneficial allele sweeps the population. I.e., harmful mutations aren’t spread as far by draft. Thus large populations are more efficient at removing moderately harmful mutations. 
     
    A changing environment can reduce genomic information as good adaptations become bad. 
     
    Species with large populations and many offspring can maintain high amounts of genomic information. However, the importance of the genomic information differs across the genetic elements (a power law?). As harmful mutations accumulate the specie becomes less adapted to its environment until a new balance is reached and the total genomic information is again maintained by selection in the population. Only the more important genomic information is preserved by selection. If a population was a very successful competitor in an environmental niche, a bottleneck might significantly lower specie adaptation without causing extinction. However, if the specie were barely surviving then a bottleneck event might be the end. 
     
    Depending on the specie biology, the environment, and the specie competitors, a reduction in genome information could lead to extinction.

  13. Thanks, Fly. 
     
    I’m finding it odd that evidently we don’t have good models for what happens to small populations.

  14. More thoughts: 
     
    Total population genetic diversity depends upon population size. Diversity in the specie population genome provides the raw genetic material by which selection produces adaptation to new environments. This could be important if the environment is rapidly changing. Also genetic diversity provides protection against pathogens. Small populations that survived for thousands of years could be wiped out when contact with other people introduced new pathogens too frequently. 
     
    To model the affect of small population size on a given trait you would also need to know what DNA affects that trait. Traits that depend upon only a few short DNA segments will behave differently compared to traits that depend on thousands of DNA segments of varying size spread across the genome. In a small population, selection might be sufficient to maintain the trait quality in the first case but not in the second. (E.g., consider the consequence of mutational meltdown in the trait, fertility.)

  15. Here’s an example of a tiny population that survived thousands of years: 
     
    http://en.wikipedia.org/wiki/Devil%27s_Hole_pupfish

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