Reddit reviews Population Genetics: A Concise Guide

Yes absolutely. Evolution is a highly mathematic science with a long history of mathematical theory describing how populations change over time. RA Fisher and Sewell Wright were some of the first and many many have followed.

For future reference, only populations can evolve, not individual organisms. Unfortunately most introductory material on evolution doesn't get into the math. I'd recommend this book as a good introduction to the math of evolution, but it assumes some pretty basic knowledge of the concepts of evolution already.

I know you said you aren't interested in debating, but I have a question: how can you be against something for theological reasons if you admit you don't fully understand what that thing is?

But to also provide a book recommendation: https://www.amazon.com/Population-Genetics-John-H-Gillespie/dp/0801880092 it wouldn't be a good starting point for someone without much background in math and science, but I think that you have enough background to be able to follow it.

Have you considered becoming a population geneticist? All of those questions are things that evolutionary/population geneticists are very interested in.

Let's break this into pieces. The first piece:

What is the eventual fate of a new mutation, and how does it depend on a) it's selection coefficient (a measure of how beneficial/deleterious it is), and b) the population size.

The selection coefficient (which we'll denote by s) is a measure of the "per generation percent fitness advantage" enjoyed by an individual who carries a particular mutation, relative to those who do not carry the mutation. To a first order approximation, the probability that a beneficial mutation that has just arisen (and thus resides only in a single individual) escapes loss from the population and eventually becomes "fixed" (i.e. present in every individual) does not depend on the population size, and is equal to about 2s, or two times the selection coefficient. In other words, if a particular mutation causes its carriers to leave approximately 1% more offspring to the next generation, relatively to non-carriers, then it has about a 2% chance of not being lost from the population. If it's not lost during those early generations, then it will eventually rise in frequency and become fixed.

Now, this is a rough approximations, and with a better approximation, we find that the population size does matter somewhat. This is because when the population size gets small, the chance events of genetic drift become more impactful, and it becomes harder for selection to overcome them. This is basically exactly the example you gave, but in reverse. Basically, even if a mutation has a fitness advantage, if it's present in only 10 out of 100 individuals in a population, it can happen to be lost by chance if it has a couple bad years in a row. In a population of 1 million, however, a mutation that's at 10% frequency would take a lot of bad years in a row, in order to be lost, which is very unlikely, so natural selection will eventually win out and push the mutation to fixation.

However, it is true that mutations change frequency faster in populations of smaller size (pretty much for the reason you surmise). If we condition on (i.e. assume that) the mutation eventually becomes fixed, then it is more likely to have done so quickly if the population size was small than if it was larger. The time it takes for a beneficial mutation to become fixed, assuming it does become fixed, is proportional to log(N), the logarithm of the population size. So if you increase the population size by a factor of 10, it takes twice as long for a beneficial mutation to transit through the population. By a factor of 100: three times as long.

However, there's one last factor we should consider, which is how the population size interacts with the mutation rate. Consider a population that exists in some environment in which it has an "adaptive need". In other words, the environmental conditions are such that if a certain mutation (or class of mutations, if we consider that mutations at multiple different base pairs might be able to solve the same problem) would be beneficial, were it to arise, then we can ask how long until we expect the population to adapt. If we say that the per individual rate at which beneficial alleles are created is given by µ, and there are N individuals in the population, then to a first order approximation in each generation there is a 2Nµ probability that a beneficial mutation arises somewhere in the population (there's a 2 because we're thinking about diploids). Then, using the simple 2s approximation from above (which is good enough for this point), the probability that a mutation both arises somewhere in the population and manages to escape being lost in those early generations is 4Nµs.

Using the properties of the geometric distribution, this means that we expect it to take about (4Nµs)^(-1) generations until the mutation that will eventually come to dominate the population arises. Then, it will take of order log(N) generations for the mutation to sweep through the population.

So when this effect is factored in, an increase in population size of 10-fold means you wait roughly one tenth as long for a beneficial mutation to arise, but only twice as long for it to fix. A 100-fold increase in population size means you wait roughly 1/100 as long for the mutation to arise, but then only 3 times as long for it to sweep through the population, meaning that in general, larger populations should adapt faster than smaller populations. However, if we're thinking about populations that are already so large that beneficial mutations occur somewhere in the population almost every generations (like bacteria, for example), then a different set of mathematics takes over, and this is still an active area of research (see here for a recent review).

These calculations all rely on what are pretty much "standard" results in population genetics, so any good population genetics text book should work as a decent reference. If you're interested in these kind of questions, a good place to start might be John Gillespie's "Population Genetics: A Concise Guide"

You're very welcome. And, yes, you do have to do the double sum, over all possible n_B, good catch!

Why Poisson? There are some biological reasons that it's reasonable (that are currently eluding me) but also because an individual can't have 1.532 offspring. A discrete outcome needs a discrete probability distribution. The poisson happens to be discrete and unbounded, so it fits the bill. A negative binomial or geometric could also work, if you just want to plug in a distribution. It is not, however, inordinately hard to simulate a Poisson RV given the ability to simulate a uniform(0,1) RV.

I wouldn't say that the example is unfair by merit of using the same survival probabilities. There are two ways for an allele/genotype to have a higher fitness than other alleles/genotypes: higher survival and/or higher fecundity. By merit of the way you set up the problem A is already fitter than B. If you want to assign s_A and s_B you can do so as well, the binomial distributions used to calculate the probabilities of n_A and n_B simply change. In general (or at least in a lot of classical population genetics), people abstract away from survival vs fecundity effects and simply talk in terms of relative or absolute fitnesses (the product of survival and fecundity).

Last note: if you're interested in population genetics, it has a very rich theoretical foundation, and you should do some reading on the subject if you're curious. I think you'll find that most problems under the sun have been discussed somewhere at some point. As starting points, Felsenstein has a free and surprisingly comprehensive book available online. Gillespie has a not free and surprisingly concise book. Both are excellent.

Most mutations are indeed deleterious/bad, but there are also beneficial/good mutations. A good example is the ability to metabolize lactose as adults - since we've domesticated cows/goats/etc. and milk is a good source of nutrition, there is positive selection on this trait being maintained on a population. That's why this mutation, or trait, is very common among European and African populations.

It's helpful to think of selection as a "force" that pushes a trait to become more or less common in a population. If selection against a trait is strong enough, it will die out (go to 0% incidence in a population), and if it is strong enough, it will "fix" (go to 100%).

In addition to selection, there is genetic "drift." Because of how people pair off to mate and how traits get passed on, there is a degree of randomness that causes the percentage incidence of a trait in a population to fluctuate - like a random walk in physics.

Just from these factors alone, with the introduction of new traits (mostly bad, but some good), there is always going to be diversity within a population. But because we have two copies of each chromosome and each gene, one from mom and one from dad, you can also have interesting situations where having one mutation has a very different outcome than none or two. In some of these cases, you can get a stable percentage of a trait in a population at a value between 0% and 100%. A good example is the mutation that causes sickle cell anemia if you have two copies, but may protect against malaria if you have one copy.

If you want to learn more, Population Genetics by Gillespe is an accessible (and cheap) book on this subject. I think a little bit of calculus helps for the math.

http://www.amazon.com/Population-Genetics-A-Concise-Guide/dp/0801880092/ref=sr_1_1?ie=UTF8&qid=1408234938&sr=8-1&keywords=population+genetics