Does a better gene always win?

Every new genetic variant starts the same way: as a single copy, in a single individual, created by a copying error in DNA. Suppose this variant is genuinely useful. Its carriers have, on average, 1% more offspring than everyone else. Over thousands of generations, surely it takes over?

The common picture of evolution says yes. The mathematics says: probably not. A variant with a 1% advantage, starting as one copy in a large population, ends up in every member of the population only about 2% of the time. The other 98% of the time it vanishes, usually within a few generations of appearing. To see why, start with what happens when no variant is better than any other.

Chance alone

Imagine a population that stays the same size, N individuals, generation after generation. To make the next generation, we pick a parent at random from the current one, copy it, and repeat N times. Some individuals are picked twice or three times; others not at all. This is the Wright–Fisher model, named after Sewall Wright and Ronald Fisher, who used it around 1930 to found the mathematics of evolution. It describes organisms with one copy of each gene (haploid), like bacteria.

Below, the population starts half gold and half green. Neither colour is better. Watch the share of gold wander up and down, in eight separate populations at once, until it hits 0% or 100%. That wandering is called genetic drift. When a variant reaches 100% it is fixed: there is nothing left for it to compete with, and only a new mutation can change things. Try a small population (N = 10) and a large one (N = 500).

Two things to notice. First, drift always ends. Sooner or later one colour is lost for good, because once it is gone, no parent can pass it on. Second, small populations get there much faster. Starting from half and half, the average time until one colour wins is about 1.4 × N generations: about 14 generations when N = 10, about 700 when N = 500.

There is a neat consequence. With no advantage on either side, every individual alive today has the same chance to be the ancestor of the whole future population. So a single new neutral variant, one copy among N, has a chance of exactly 1 in N of ending up fixed.

A better gene appears

Now give one variant an edge. Its carriers are a little more likely to be picked as parents: if the variant is rare, each carrier has on average 1 + s offspring, while everyone else has 1. The number s is the selection coefficient; s = 0.01 means a 1% advantage. That is a large advantage by the standards of real mutations.

The trouble is the start. One carrier doesn’t have exactly 1.01 offspring. It has 0, 1, 2 or 3, at random. In the Wright–Fisher model the chance of having none at all is about e⁻¹, or 37%, and a 1% advantage lowers that only to 36%. A small advantage barely changes the odds while there are only a handful of copies. Selection is a statement about averages, and averages need numbers.

The figure runs 1,000 separate populations of 1,000 individuals each. In every one, a single copy of the better variant appears. Each line follows the number of copies in one population, on scales that stretch out the early generations. Press Run 1,000 populations and watch most of the lines drop out almost at once. Then change s and run again; every run adds a point to the chart of the fraction that won.

The fraction that wins is small, and it grows in step with s: roughly twice the advantage. This result, that a new beneficial variant survives with probability about 2s, was worked out by J. B. S. Haldane in 1927. A 1% advantage wins about 2% of the time; a 5% advantage about 9%. Even a large advantage loses most of the time. Once a variant has more than roughly 1/s copies (50 copies when s = 2%), though, its fate is nearly sealed: drift can no longer pull it back down, and selection carries it the rest of the way.

Notice too that the better variant still does far better than a neutral one. At s = 2% it wins about 40 times as often as the neutral 1 in 1,000. Selection works; it just works on odds, not on certainties.

Chance against selection

Motoo Kimura found a formula, in 1962, that covers any population size and any s, including harmful variants (s below zero). For one new copy in a haploid population of size N, the chance that it ends up fixed is close to

P ≈ (1 − e^−2s) / (1 − e^−2Ns)

In words: the top is about 2s, Haldane’s answer. The bottom is close to 1 when N·s is large, and then the population size doesn’t matter. When N·s is small, the bottom is about 2Ns, the s cancels out, and P is about 1/N, just as if the variant were neutral.

So what matters is the product N·s. If it is much bigger than 1, selection decides. If it is much smaller than 1 (either sign), drift decides, and the population behaves as if the variant made no difference at all. Try N = 100 and s = −0.1%: the variant is harmful, yet it takes over almost as often as a neutral one. Then make the population 100 times larger.

This is why population size matters so much. In a population of a million, a variant with a 0.01% advantage has N·s = 100, and selection sees it clearly. In a population of a few hundred, the same variant is invisible to selection; harmful variants with small effects can drift to fixation, which is one reason small, isolated populations accumulate damaging mutations.

Kimura took the idea further. In 1968 he proposed the neutral theory of molecular evolution: that most of the differences in DNA between species were fixed by drift, not by selection, because most of them have |N·s| well below 1. How much of evolution that covers is still debated. That drift is a force alongside selection is not.