# How does rate of evolution/innovation scale with population size?

I understand that there will be many other factors that affect rate of evolution/innovation. However, other things being equal, how will the rate of evolution vary between two populations of different sizes? Does doubling the population size double the rate of introduction of new traits?

Is this answerable for evolving life in general, or do different forms of life have different scaling laws? For example, is it different for sexual and asexual life?

• I'm no evolutionary biologist but surely the rate (meaning the frequency) for a given mutation won't change if two populations are of different size, but the likelihood of observing it at a given time probably will. Obviously I'm assuming any mating won't change/affect the rates. But I could be totally wrong! – Behzad Rowshanravan Nov 16 '14 at 15:18
• @Bez you mean the rate for an individual will be the same, but its rate of occurrence in the population will be proportional to the population size? – trichoplax Nov 16 '14 at 15:20
• Yes! that sounds about right! – Behzad Rowshanravan Nov 16 '14 at 15:22
• @Bez I can see intuitively that the rate of appearance of a given mutuation (good/bad/neutral) will be proportional to the population size, but it isn't clear to me whether the same will apply to new traits being adopted. For example, if a trait requires 3 separate mutations to occur in a specific order, will its probability of occurring still scale linearly? – trichoplax Nov 16 '14 at 15:23
• It still should since you can calculate the probability of a sequential mutation just as you can calculate what are the chances of getting a 4, 5 and 6 in that order after throwing the dice exactly 3 times! The higher the number of throws, the higher its probability of occurrence but the overall chances of getting that particular order won't change! Obviously I'm discarding the concept of selection pressure etc! – Behzad Rowshanravan Nov 16 '14 at 15:27

Interesting question (+1) but not that easy to answer. I'll give a try!

First, the rate of evolution and rate of adaptation are two different things.

There are two main thing processes that quantitatively differ in populations of different sizes. 1) population-wide mutation rate and 2) genetic drift.

1. Population-wide mutation rate

Large populations create more mutations at each generation and therefore if the rate of evolution or the rate of adaptation is limited by the genetic variance, then one can expect large population to have higher rate of evolution and higher rate of adaptation.

--> Large populations may have higher rate of evolution and higher rate of adaptation

2. Genetic drift

However, small population have higher drift, resulting in a higher probability of fixation for new neutral (or slightly deleterious) mutations. It results that a small population will fix (fixed: frequency = 1) a higher proportions of the new mutations through time than large populations.nActually the probability of fixation of a neutral mutation is \frac{1}{2N}. Therefore, the rate at which neutral alleles fix is $\mu\cdot 2N \frac{1}{2N} = \mu$, where $\mu$ is the mutation rate. So, the rate of fixation of neutral alleles is independent of the population size.

--> Small populations have more drift but this doesn't necessarily yield to higher rate of evolution

Shifting balance theory

If adaptation is limited by the genetic architecture, that is, only mutation of very big effect on phenotype can be advantageous, then small population have the advantage to be able to accumulate slightly deleterious mutations, so that they can get off a local peak on the adaptive landscape and eventually manage to evolve toward a new higher fitness peak.

--> Small population may have higher rate of adaptation

Population-wide mutation rate vs genetic drift

Weighting the relative importance of rate of production of neutral (or quasi-neutral) mutations (See population-wide mutation rate above and the probability of fixation of neutral (or quasi-neutral) mutations (See Genetic drift above) depends also on the probability distribution of mutational effects (=probability distribution of the effect on fitness of a given new mutation).

Ceteris Paribus

The comparison between large and small populations are dependent on the state of other parameters. So even when saying all else being equal (ceteris paribus), we need to know what these other parameters are equal to, in order to answer the question because there are interactions between populations size and other parameters in explaining the rate of adaptation and rate of evolution.

Other impact of population size on genetic features

Population size has many impacts on various genetic features, such as pattern of genetic diversity, distribution of per locus Fst between subpopulation, and other things. This paper is likely of interest

• There is probably other processes I forget about. Also, there is probably some sources that offer some quantitative measure of the general effect of population size on adaptation. – Remi.b Nov 17 '14 at 3:42
• On point 2: for strictly neutral mutations, the increase in fixation probability ($1/N$) in small populations is exactly canceled by the reduction in the mutation flux ($N\mu$), so the rate of evolution is independent of population size. – Daniel Weissman Nov 17 '14 at 5:09
• @DanielWeissman I'd be interested to see an answer expanding on this. – trichoplax Nov 17 '14 at 11:54
• @Remi.b I'm not sure I understand your comment. Drake's Rule refers to the product of the per-base mutation rate and the genome size, not the population size. – Daniel Weissman Nov 17 '14 at 18:12
• Oops. You are totally right! Temporary brain malfunctioning! To defend myself more than I should, I would note that I would expect a rough tendency for species with small genome size to have large population size. I deleted my comment above. Thanks! – Remi.b Nov 17 '14 at 18:47