Linkage, recombination and gene mapping

Mendel's law of independent assortment holds for genes on different chromosomes. But every chromosome carries hundreds or thousands of genes, and genes on the same chromosome tend to be inherited together. Such genes are linked. Linkage is only partial, because crossing over in meiosis can separate linked alleles, and the further apart two genes are, the more often a crossover falls between them. This simple idea, worked out by Alfred Sturtevant in 1913 when he was still an undergraduate, lets geneticists build maps of chromosomes from breeding data alone. Genetic maps guided the search for disease genes for most of the twentieth century and remain essential in plant and animal breeding today.

Discovering linkage

Bateson and Punnett noticed in 1905 that two sweet pea traits, flower color and pollen shape, did not show a 9 : 3 : 3 : 1 ratio; the parental combinations were far too common. They called it coupling, but could not explain it. Morgan explained it in 1911 using Drosophila. He crossed flies and found that certain traits tended to stay together in the combinations they had in the parents, and proposed that the genes lay on the same chromosome. The occasional new combinations, he suggested, arose from crossing over, which the cytologist Frans Janssens had recently proposed as an explanation for the chiasmata seen in meiosis.

Parental and recombinant gametes

Consider a fly heterozygous for two linked genes, with the dominant alleles A and B on one homologue and the recessive alleles a and b on the other. This arrangement, with dominant alleles together, is called coupling (or cis); the opposite arrangement, A b on one homologue and a B on the other, is repulsion (or trans). If no crossover occurs between the genes, the fly makes only parental gametes, AB and ab. A crossover between the genes produces recombinant gametes, Ab and aB.

A single crossover involves only two of the four chromatids, so even when a crossover happens between two genes in every meiosis, at most half the gametes are recombinant. This is why the recombination frequency between two genes can never exceed 50 percent. Genes very far apart on the same chromosome recombine so often that they behave as if they were unlinked, giving 50 percent recombinants, just like genes on different chromosomes.

Measuring recombination frequency

The easiest way to count recombinant gametes is a testcross: cross the double heterozygote to a double homozygous recessive (ab/ab). The tester contributes only ab gametes, so the phenotype of each offspring reveals the gamete it received from the heterozygous parent. The recombination frequency (RF) is the proportion of offspring that are recombinant:

Sturtevant's insight was that RF can serve as a measure of distance. He defined one map unit, now called a centimorgan (cM) in Morgan's honor, as the distance giving 1 percent recombinant offspring.

The three-point testcross

Two-point crosses give distances but can miss double crossovers, which restore the parental combination for the outer genes and so make distant genes look closer than they are. A three-point testcross, following three linked genes at once, solves this and also reveals the gene order. The procedure is the same every time, and it is worth learning as a recipe.

Interference

If crossovers in two neighboring intervals happened independently, the expected frequency of double crossovers would be the product of the two RFs. In practice there are usually fewer double crossovers than this, because a crossover tends to inhibit another nearby. The coefficient of coincidence (c) and interference (I) measure this:

In the corn example, the expected number of double crossovers is 0.393 × 0.113 × 1,000 ≈ 44, but only 11 were seen, so c = 11/44 ≈ 0.25 and I ≈ 0.75. Three-quarters of the expected double crossovers were suppressed. Interference is usually strong over short distances, often near 1 in intervals of a few centimorgans.

from itertools import combinations

# Three-point testcross data (corn): genotype of the gamete from the heterozygote -> count
data = {("v", "lg", "gl"): 270, ("+", "+", "+"): 235, ("v", "+", "gl"): 62, ("+", "lg", "+"): 40,
        ("+", "+", "gl"): 4, ("v", "lg", "+"): 7, ("v", "+", "+"): 165, ("+", "lg", "gl"): 217}
genes = ["v", "lg", "gl"]
total = sum(data.values())
ranked = sorted(data.items(), key=lambda kv: -kv[1])
parental = ranked[0][0]
double = ranked[-1][0]
# middle gene: the double crossover differs from one of the two parental classes at exactly one gene
parentals = [ranked[0][0], ranked[1][0]]
for par in parentals:
    diff = [g for i, g in enumerate(genes) if par[i] != double[i]]
    if len(diff) == 1:
        middle = diff
print("parental:", parental, "double crossover:", double, "middle gene:", middle[0])

def rf(i, j):
    """Recombinant fraction between genes i and j: alleles separated relative to parental."""
    rec = 0
    for cls, n in data.items():
        same_i = (cls[i] == parental[i]); same_j = (cls[j] == parental[j])
        if same_i != same_j:
            rec += n
    return rec / total

dist = {}
for i, j in combinations(range(3), 2):
    dist[(genes[i], genes[j])] = rf(i, j)
    print(f"{genes[i]}-{genes[j]}: RF = {rf(i, j):.3f}")
m = genes.index(middle[0])
outer = [k for k in range(3) if k != m]
r1, r2 = rf(outer[0], m), rf(m, outer[1])
dco_obs = ranked[-1][1] + ranked[-2][1]
expected = r1 * r2 * total
print(f"map: {genes[outer[0]]} --{100*r1:.1f} cM-- {middle[0]} --{100*r2:.1f} cM-- {genes[outer[1]]}")
print(f"double crossovers observed {dco_obs}, expected {expected:.1f}; "
      f"coincidence = {dco_obs/expected:.2f}, interference = {1 - dco_obs/expected:.2f}")

parental: ('v', 'lg', 'gl') double crossover: ('+', '+', 'gl') middle gene: gl
v-lg: RF = 0.484
v-gl: RF = 0.393
lg-gl: RF = 0.113
map: v --39.3 cM-- gl --11.3 cM-- lg
double crossovers observed 11, expected 44.4; coincidence = 0.25, interference = 0.75

The direct two-point estimate for the outer genes, v and lg, ignores the double crossovers and comes out shorter than the sum of the intervals. This is the main reason for building maps from short intervals.

Map distance and physical distance

Genetic maps measure recombination, not physical length. Crossovers are not evenly distributed: they are rare near centromeres, often concentrated in hotspots a few thousand base pairs long, and more frequent in female than in male meiosis in humans (lesson 6). Across the human genome as a whole, 1 cM corresponds on average to roughly a million base pairs, but the local ratio can vary tenfold or more. The total human genetic map is about 3,500 cM averaged over the sexes, longer in females than in males.

Because double and multiple crossovers go undetected, RF underestimates map distance for genes far apart. Mapping functions correct for this. The simplest, Haldane's mapping function, assumes crossovers occur at random with no interference and converts an observed RF (r) into map distance (d, in morgans):

For small r, \(d \approx r\), so the correction matters only for larger distances. An observed RF of 0.30 corresponds to a Haldane distance of about 0.46 morgans, or 46 cM.

Linkage in human families and LOD scores

In humans, geneticists cannot design testcrosses, so they test for linkage by following a disease and a genetic marker through families. Because family sizes are small, evidence from many families must be combined. The standard measure is the LOD score (logarithm of the odds), introduced by Newton Morton in 1955. It compares the probability of the observed family data if the disease gene and the marker are linked at a given recombination fraction θ with the probability if they are unlinked (θ = 0.5):

By convention, a LOD score of 3 or more (odds of 1,000 to 1 in favor of linkage) is accepted as evidence of linkage, and a score of −2 or less as evidence against. LOD scores from different families can simply be added, which is why they suited the gradual collection of family data.

import math

def lod(recombinants, nonrecombinants, theta):
    """LOD score for phase-known meioses: R recombinants and N non-recombinants."""
    n = recombinants + nonrecombinants
    linked = (theta ** recombinants) * ((1 - theta) ** nonrecombinants)
    unlinked = 0.5 ** n
    return math.log10(linked / unlinked)

# Combined data from several families: 2 recombinants among 20 informative meioses (invented)
R, N = 2, 18
best = max((lod(R, N, t / 100), t / 100) for t in range(1, 50))
for theta in (0.01, 0.05, 0.10, 0.20, 0.30, 0.40):
    print(f"theta = {theta:.2f}: LOD = {lod(R, N, theta):5.2f}")
print(f"maximum LOD = {best[0]:.2f} at theta = {best[1]:.2f}")

theta = 0.01: LOD =  1.94
theta = 0.05: LOD =  3.02
theta = 0.10: LOD =  3.20
theta = 0.20: LOD =  2.88
theta = 0.30: LOD =  2.19
theta = 0.40: LOD =  1.23
maximum LOD = 3.20 at theta = 0.10

Linkage analysis with LOD scores mapped the genes for Huntington disease (to chromosome 4 in 1983), cystic fibrosis (chromosome 7, 1985) and many others before they were identified. The cystic fibrosis gene was then found in 1989 by a team at the Hospital for Sick Children in Toronto and the University of Michigan, one of the first human disease genes found purely from its map position. Today most new disease genes are found by sequencing, but linkage still helps narrow the search in large families.

Uses of genetic maps

Genetic maps are not just of historical interest. Plant and animal breeders use maps of molecular markers to follow valuable genes through crosses, a technique called marker-assisted selection: a seedling can be tested for markers linked to a disease-resistance gene long before the plant is old enough to show resistance. Maps also underpin the study of quantitative traits (lesson 11), the forensic and ancestry use of DNA markers, and the assembly of genome sequences, which must be anchored to chromosomes.