Chance and Selection

Chapter 5 The force that joins populations

Gene flow

Migrants carry genes between populations, and a few per generation are enough to hold populations together against drift.

8 min read · 4 interactive figures, 1 question

An island of thirty beetles where orange starts rare; each generation a fraction m of its gene copies come from a mainland where orange is at 80 %. The island is pulled toward the dotted mainland line while drift jostles it. Set m to zero and it drifts on its own.
On this page
  1. An island and a continent
  2. Many islands: drift against migration
  3. Migration against selection: Lake Erie’s water snakes
  4. Why gene flow matters

Gene flow is the movement of genes from one population into another, when migrants arrive and breed. Its effect on allele frequencies is simple: it pulls the receiving population toward the frequencies of the migrants. Where drift makes populations differ and local selection adapts each to its own conditions, gene flow makes them alike. Most of what is interesting about it comes from that tug-of-war, and the surprising part is how little migration it takes to win against drift.

This also answers a common exam question, how gene flow differs from genetic drift. Drift is a random change within a population and has no direction; gene flow is a change caused by an outside source and has one, toward the source. Drift makes isolated populations diverge; gene flow holds connected populations together.

An island and a continent

The simplest model has one large mainland population with allele frequency pm and an island population. Each generation a fraction m of the island’s genes come from mainland migrants. Then the island’s frequency is a weighted average[1]:

p′ = (1 − m) p + m pm,     pt − pm = (1 − m)t (p0 − pm)(5.1)
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The gap between island and mainland shrinks by a factor 1 − m every generation. With m = 0.01, one gene in a hundred from migrants, the gap halves in about 69 generations. Nothing else is needed to explain why human populations that have been in contact for centuries have converged in their allele frequencies, or why island populations of birds and lizards so often resemble the nearest mainland.

Figure 1 Thirty islands of N individuals start with A at 10 %; each generation a fraction m of their genes arrive from a mainland where A is at 70 %. The dashed curve is equation (5.1) without drift. Set m to 0 to see pure drift, then raise it. Notice that migration pulls every island toward the mainland and stops any of them staying fixed, while drift keeps them scattered around the curve, more so when N is small.#

The islands in Figure 1 do not sit on the dashed curve. Each one drifts, and migration only pulls the average. With m = 0.02 and 25 individuals, Nm = 0.5 migrants arrive per generation, and the islands stay noticeably different from one another even after they have caught up with the mainland on average. How much they differ is the next question.

Many islands: drift against migration

Now take many islands, each of N individuals, exchanging migrants with one another. Drift pushes each island toward its own random fate; migration pulls them all toward the archipelago’s average. Sewall Wright worked out where the two balance[2]. The standard measure of how different the islands are is FST, the share of the total genetic variance that lies between islands: 0 if all islands have the same frequency, 1 if each is fixed for one allele or the other. In Wright’s island model with many islands, its expected value follows an exact recursion[1]

FST′ = (1 − m)² [ 12N + (1 − 12N) FST ](5.2)
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The term 1/2N is drift creating differences, as in equation (3.3); the factor (1 − m)² is migration erasing them (both gene copies compared must be non-migrants for their identity to survive). At equilibrium, for small m,

FST ≈ 11 + 4N m(5.3)
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Question 1 Prediction before the next figure.#
Figure 2 Twenty-four islands, each a tube filled to its share of A, all starting at 50 %. Choose the island size and the number of migrants each island receives per generation (Nm). Middle: each island over time, with the archipelago average in black. Bottom: FST from the simulation against the exact recursion (dashed) and Wright’s equilibrium. Set Nm to 0, then to 1, then change N at fixed Nm.#

Equation (5.3) has a remarkable consequence. N and m enter only as their product Nm, the number of migrants per island per generation. Big islands drift slowly but need many migrants to make up a given fraction m; small islands drift fast but are swamped by a few. One migrant per generation gives FST ≈ 0.2 whatever the island size: the islands differ, but none is likely to lose an allele the others keep. With Nm = 0.1, FST ≈ 0.7 and the islands go their own way; with Nm = 10, FST ≈ 0.02 and they behave as one population. established

That result became a rule of thumb in conservation: connect fragmented populations with at least one migrant per generation. Mills and Allendorf examined it in 1996 and concluded that one migrant per generation is a desirable minimum but may be inadequate for many natural populations, suggesting between 1 and 10 migrants per generation as a general rule for genetic purposes[3]. The reasons are the ones that run through this site: the migrants that matter are those that breed, so effective numbers are lower than head counts, and real islands are not equal-sized and evenly connected.

Migration against selection: Lake Erie’s water snakes

Gene flow does not only fight drift. When selection on an island favours one allele and migrants keep bringing in the other, the island settles where the two forces balance, and the balance depends on their relative strength. The island keeps its own character only while the migration rate stays below a threshold set by the selection coefficient, and dominance moves that threshold a long way. In the model below the banded allele is selected against on the island, so in this site’s convention (banded homozygotes have fitness 1 + s) its s is negative. An island is swamped by the mainland once m passes about |s|/2 if the alleles are additive, but already at about |s|/4 if banded is dominant: then the favoured unbanded allele is recessive, and once migrants make it rare it hides in banded-looking carriers[4].

The water snakes of the Lake Erie islands are the textbook case. On the Ohio and Ontario mainland, northern water snakes (Nerodia sipedon) are banded. On the small limestone islands in the western basin of the lake, many snakes are unbanded or only faintly banded, and the explanation that has been studied for decades is that selection, thought to act through camouflage on the bare island shorelines, favours unbanded snakes on the islands while banded migrants keep arriving from the mainland. Richard King and Robin Lawson measured the gene flow directly with genetic markers in 1995. It is high: about 25.5 migrants per generation among island sites, 9.2 between the Ontario mainland and the islands and 3.6 between the Ohio mainland and the islands, corresponding to migration rates m between island and mainland of 0.0008 to 0.01. They concluded that the colour-pattern variation on the islands reflects a balance between gene flow and natural selection[5]. documented

Figure 3 An island receiving banded migrants at rate m from a mainland where every snake is banded, while selection acts against the banded allele on the island (s below zero: banded homozygotes have fitness 1 + s). The first panel follows the island from an all-unbanded start; the second (beside it on a wide screen, below it on a phone) gives the equilibrium for every migration rate, with the range King and Lawson estimated shaded and the swamping threshold m* dashed. Notice that with banded dominant the equilibrium does not rise gently to the mainland: it jumps there at m* ≈ |s|/4 (0.005 for s = −0.02, right at the default). Switch banded to additive and the threshold moves out to about |s|/2, approached smoothly.#

Within the shaded range, the outcome depends on s, which is harder to measure than m. With s = −0.02 on a dominant banded allele (banded snakes 2 % less fit) and m = 0.005, just below the threshold of about 0.0051, the island settles with the banded allele at about 45 %, and seven snakes in ten look banded. Halve the migration and the banded allele falls to about 15 % (a quarter of the snakes banded). Double it to 0.01 and the island is swamped completely: unbanded is recessive, so once it becomes rare it hides in banded-looking carriers and selection can no longer find it to protect it. The point King and Lawson drew is the general one: gene flow at these rates is too high for drift to explain the island populations, and too high for the islands to differ at all unless selection on colour pattern is substantial. reconstructed

Why gene flow matters

Gene flow is the reason species stay species across large ranges: a few migrants per generation are enough to stop drift from splitting populations apart at neutral genes. It is also why local adaptation is fragile. A population can adapt to its own conditions only where selection is stronger than the stream of genes from elsewhere. And it is the only fast cure for a population that has lost variation through a bottleneck: mutation would take millions of years to rebuild what a handful of immigrants can bring in a generation. Conservation genetic rescue, deliberately moving individuals into small inbred populations, is equation (5.1) applied on purpose.

The four forces have now each had a chapter. The last one sets the two that matter most for adaptation, selection and drift, against each other.