Gene Expression

You only go extinct once….

Assume that you have a new mutation, totally novel. What’s its probability of going extinct in one generation? That is, it doesn’t get passed on….

Consider, you have a population of N individuals. Fix the population size across nonoverlapping generations. So, in generation t you have N individuals and in t + 1 you have N individuals. In the first generation of the mutation the proportion in the population is 1/N, that is, there is one mutant amongst N individuals (ergo, N – 1 other copies). The probability that the mutant is never “drawn” (copied) to the next generation in this fixed population is (1 – 1/N)N. 1 – 1/N represents the non-mutants, and there are N draws since the population across generations is fixed. For example, if there are 100 individuals (haploid) and 99 are non-mutants, and the next generation will also have 100 individuals, there are 100 opportunities for the 99 out of 100 instead of the 1 out of 100 to be drawn, i.e., (1 – 0.01)100.

This equation converges upon ~ 0.37 as N approaches ∞. Here are some values generated for a given N:

10 → 0.34867844
100 → 0.366032341
1000 → 0.367695425
10000 → 0.367861046

In other words, for a neutral allele the probability of extinction in one generation is relatively insensitive to population size. Also, 0.367861046… is equivalent to 1/e. An assumption of the above model is that the mean number of offspring is 1, and that the variance is Poisson distributed, in other words the variance and mean are the same. What about selection, which would alter the mean? (i.e., shift up the expectation of offspring for an individual about or below the mean)

The model then becomes approximately 1/e X ( 1 – s), where s is the selection coefficient (e.g., s of 0.10 ~ 10% increased fitness vs. population mean, in this case, 1.1 instead of 1 as expectation). Plugging in….

0.001 → 0.37
0.01 → 0.36
0.10 → 0.33
0.20 → 0.29
0.30 → 0.26
0.40 → 0.22
0.50 → 0.18

As you can see, a fitness advantage for a new mutant reduces its probability of extinction in one generation, but it is still non-trivial even if it increases fitness half-again! This is important because empirically we know that a fitness increase of 0.10 is extremely powerful. I incremented up to 0.50 simply for illustrative purposes, biologically 0.01 to 0.10 is probably the norm. And, as you can see the risk of extinction does not decrease much with such a selective value. It’s a hard world out there for a new mutant trying to make a go of it, and the fates aren’t kind. The risk of extinction is high, and as you know the probability of fixation for a new allele in a diploid organism is 2s. That is, if it confers a 0.10 fitness advantage its chance of sweeping through the population is only 20%, ergo, its chance of extinction is 80%!

Note: Adapted from chapter 5 of Evolutionary Genetics: Concepts & Case Studies.

Comments

  1. #1 Fly
    October 29, 2006

    “The probability that the mutant is never “drawn” (copied) to the next generation in this fixed population is (1 – 1/N)N. 1 – 1/N represents the non-mutants, and there are N draws since the population across generations is fixed.”

    Typical family size also affects the calculation.

    Consider the extreme limit case in which each parent has exactly two children. Then the mutant would have 0.25 probability of not passing on the mutation and this is independent of population size.

    In a real population some people will have no children while others have many and so there is some chance that the mutant will have no children. So the probability of not passing on the mutation will be more than 0.25. But one woman can’t pump out an unlimited number of offspring so the (1 – 1/N) estimate for each draw overstates the probability of a non-mutant in all but the first draw. It is somewhat like saying that the chance of not drawing the ace of spades from a deck of cards with 52 tries is (1 – 1/52)**52 instead of (1-1/52) (1-1/51)…(1 – 1/1) = 0.

    In the case of flies, one mating couple could have very many offspring so using (1 – 1/N) for each of the N draws is a good approximation. For humans that isn’t so clear.

  2. #2 razib
    October 29, 2006

    In the case of flies, one mating couple could have very many offspring so using (1 – 1/N) for each of the N draws is a good approximation. For humans that isn’t so clear.

    well, the above assumes a poisson and non-changing pop. reality obviously isn’t like that…and in fact, it seems in most pops variance of offspring # exceeds mean, so the probability of extinction is higher than that # by a bit.

  3. #3 Agnostic
    October 29, 2006

    there is one mutant amongst N individuals (ergo, N other copies)

    Priggish correction: should be “ergo, N-1 other copies.” (lets teacher continue the lecture…)

  4. #4 razib
    October 29, 2006

    mathematical exposition demands priggishness, so no offense taken.

  5. #5 Agnostic
    October 29, 2006

    I only point out stuff like that since you don’t know who’s readig this; a person might be thrown by something they should be able to mentally fix themselves. I remember a press conference on estimates of autism — the journalists kept saying, “OK, 1.3% — what is that in terms of 1 out of X?” over and over. You could tell the researcher thought it was easy and pointless (something they could do afterwards with a pocket calculator), and once a journalist broke down and cried “All right, we know, but some of us — most of us — are innumerate!” xD

  6. #6 Fly
    October 29, 2006

    “well, the above assumes a poisson and non-changing pop. reality obviously isn’t like that…and in fact, it seems in most pops variance of offspring # exceeds mean, so the probability of extinction is higher than that # by a bit.”

    Yes.

    I want readers to understand the limits of the mathematical models and to consider the limit cases. I’ve seen people apply formulas from a book without considering the implicit assumptions.

  7. #7 Darth Quixote
    October 29, 2006

    Here’s an interesting conceptual HW question for newbies. Genetic drift increases in potency as population size decreases. Yet the probability that a novel mutant is lost in one generation increases with population size. How can this seeming paradox be explained?

  8. #8 yeti
    October 30, 2006

    by mutation, I’m assuming you mean of the ninja turtle variety

  9. #9 David B
    October 30, 2006

    I needed a jolt of strong coffee before I could follow Razib’s model at all, but having taken my shot I find it ingenious! But the assumption of strictly fixed population size, even for very small populations (say, N less than 10), is unrealistic, and leads to the kind of paradox that Darth mentions. If you relax the assumption of strictly fixed population size, and simply assume mean offspring number of 1 per gene, as in Fisher’s model, the probability of extinction of a single mutant gene is not affected by population size.

  10. #10 Darth Quixote
    October 30, 2006

    David, that’s not the answer that I was looking for. Shame on you! :)

    The resolution of the paradox is thus. The effect of decreasing the population size should be thought of as “freeing” the allele frequency to take larger jumps. In the case of a novel mutant, this effect is asymmetrical; any downward jump can take the allele frequency no further than zero, but an upward jump boosts the frequency sufficiently in excess of 1/(2N) to decrease the probability of loss (when this occurrence is sufficiently weighted by its probability in the summation over the outcome space) by some marginal amount. As the population size increases, however, the total number of alleles becomes more or less restricted to moving by one.

    The point of this exercise is to make sure that people understand what genetic drift really is. Instead of a blind acceptance of “genetic drift = increased probability of fixation,” they should be thinking “genetic drift = enabling of stochastic jumps in allele frequency.”

  11. #11 razib
    October 30, 2006

    phew :) that was my argument….

  12. #12 David B
    October 30, 2006

    Been out, come back… This morning I was only half-awake, now I’m half-asleep again. So I may have missed Darth’s point. But he may also have missed my point. In standard models of population genetics (Fisher/Wright/Haldane) any gene has a certain probability of having 0, 1, 2… ‘offspring’ in the next generation. The usual assumption for neutral genes is a Poisson distribution with mean 1, so that the average expected size of the population does not change, though there will be stochastic fluctuation up and down. Or we could take a simpler algebraic model with, say, 1/4 probability of 0 offspring, 1/2 probability of 1 offspring, and 1/4 probability of 2 offspring. For each individual gene there is a non-zero probability of having no offspring (extinction). This probability is the same regardless of population size. And since the probability distribution for each gene is independent of all the other genes, there is a non-zero probability that *all* of the genes will have no offspring. For a large population this probability is negligible, but for a very small one (e.g. N < 10) it is not negligible. But in Razib’s model total extinction, or indeed any fluctuation in population size, is ruled out. The probability of extinction for each gene is no longer independent of other genes: if the first 9 out of 10 genes ‘drawn’ for a population of 10 all have no offspring, then the remaining gene *must* have 10 offspring. So the fact that in Razib’s model the probability of extinction for a new mutation in one generation is lower in a very small population than in a large one is just an artifact of his unusual assumptions. Since total extinction is ruled out, the probability of extinction for each gene must also be reduced (whether new mutation or otherwise). To take the most extreme case, if N = 1, and the 1 happens to be a mutant, it has a zero probability of extinction.

  13. #13 David B
    October 30, 2006

    (continued) (e.g. N less than 10) it is not negligible. But in Razib’s model total extinction, or indeed any fluctuation in population size, is ruled out. The probability of extinction for each gene is no longer independent of other genes: if the first 9 out of 10 genes ‘drawn’ from a population of 10 all have no offspring, then the remaining gene *must* have 10 offspring. So the fact that in Razib’s model the probability of extinction for a new mutation in one generation is lower in a very small population than in a large one is just an artifact of his unusual assumptions. Since total extinction is ruled out, the probability of extinction for each gene must also be reduced (whether it is a new mutation or otherwise). To take the most extreme case, if N = 1, and the 1 happens to be a mutant, it has a zero probability of extinction.

  14. #14 David B
    October 30, 2006

    Sorry, the system seems to have stopped taking long comments. I might (or might not) try again later.

  15. #15 razib
    October 30, 2006

    no, it gets messed up by ‘greater than’ and ‘less than’ symbols. i changed it.

  16. #16 Darth Quixote
    October 30, 2006

    I’ll have to think about this more. Maybe I’ll weigh in again tomorrow …

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