Linkage and Genetic Mapping

Mendel's second law says that genes assort independently, but a human cell has about 20,000 genes and only 23 pairs of chromosomes, so most genes must share a chromosome with hundreds of others. Genes that travel together break Mendel's ratios in a way that turns out to be enormously useful: how often they are separated tells us how far apart they are. This chapter shows how to turn counts of offspring into maps of chromosomes, from fruit flies to the human disease genes found by following families.

Section 1Linkage and crossing over

Independent assortment works because nonhomologous chromosomes line up independently at metaphase I. Two genes on different chromosomes therefore sort independently. Two genes on the same chromosome are physically connected, and unless something breaks that connection, the alleles that entered meiosis together on one chromosome will leave together in the same gamete. Genes that tend to be inherited together because they lie near each other on a chromosome are linked, and a group of genes that are all linked to one another is a linkage group. In a well mapped species the number of linkage groups equals the haploid chromosome number: four in Drosophila melanogaster, ten in maize, and 23 in humans (counting the X; 24 if the Y is counted separately).

The first hints

The earliest sign of linkage came in 1905, when William Bateson, Edith Saunders and Reginald Punnett crossed sweet peas differing in flower color (purple, dominant, versus red) and pollen shape (long, dominant, versus round). The F2 did not give 9:3:3:1. Instead, the two parental combinations, purple long and red round, were much more common than expected, and the new combinations, purple round and red long, were rarer. Bateson and Punnett called this "coupling," but they had no chromosomal explanation. In 1911 Thomas Hunt Morgan, studying X linked genes in Drosophila such as white eyes and miniature wings, saw the same excess of parental combinations and proposed the explanation we still use: the genes lie on the same chromosome, and the new combinations arise when homologous chromosomes exchange segments during meiosis. He called the exchange crossing over, and he suggested that the farther apart two genes are, the more often a crossover will fall between them.

Parental and recombinant gametes

Consider an individual heterozygous for two linked genes, with A and B on one homolog and a and b on the other. Its gametes fall into two classes:

• Parental (nonrecombinant) gametes carry the same allele combinations that the individual received from its parents: AB and ab.
• Recombinant gametes carry new combinations: Ab and aB.

The arrangement of alleles on the two homologs is called the linkage phase. When the two dominant alleles are on one chromosome and the two recessive alleles on the other (AB/ab), the genes are in coupling, also called the cis configuration. When each chromosome carries one dominant and one recessive allele (Ab/aB), the genes are in repulsion, or the trans configuration. The phase determines which gametes are parental. For a repulsion heterozygote, Ab and aB are the parental gametes and AB and ab are the recombinants. Students often forget this, so always identify the phase before labeling classes.

The slash notation is worth learning. Writing AB/ab means one homolog carries A and B and the other carries a and b. Writing the genotype as AaBb tells you only that the individual is a double heterozygote and says nothing about phase.

Crossing over happens at the four strand stage

Crossing over takes place in prophase I of meiosis, after DNA replication, when homologous chromosomes are tightly paired by the synaptonemal complex. The process begins when the protein Spo11 makes deliberate double strand breaks in the DNA; some of these breaks are repaired using a nonsister chromatid as a template and resolved as crossovers, which reciprocally exchange the chromosome arms beyond the break. When the homologs begin to separate in diplotene, the crossover sites remain visible as chiasmata, X shaped connections that hold the homologs together until anaphase I. Each pair of homologs normally forms at least one crossover, the obligate crossover, which is necessary for the pair to orient and segregate properly. Failure of crossing over raises the risk of nondisjunction.

Three facts follow from the timing. First, crossing over involves two of the four chromatids at any one site, so one crossover between two genes yields two recombinant and two parental chromatids. Second, for that reason the frequency of recombinant gametes is half the frequency of meioses in which a crossover occurs in the interval. Third, because a crossover can also involve different pairs of chromatids at different sites, double crossovers can restore the parental combination of outside markers, a point we will return to in Section 3.

Proof that crossing over is physical exchange

In 1931 Harriet Creighton and Barbara McClintock in maize, and independently Curt Stern in Drosophila, showed that genetic recombination is accompanied by physical exchange of chromosome parts. Creighton and McClintock used a maize chromosome 9 that carried two visible landmarks, a dark knob at one end and an extra piece of another chromosome at the other, along with alleles for kernel color and starch type. Plants that were recombinant for the alleles had also exchanged the landmarks, exactly as the crossing over model predicts. Genetic maps were from then on understood as maps of real physical objects.

Complete and incomplete linkage

If two genes are so close together that a crossover essentially never occurs between them, or if the species does not cross over in one sex, linkage is complete and only parental gametes are made. Drosophila males have no meiotic crossing over at all, and neither do females of the silkworm moth. More often linkage is incomplete: parental gametes are in the majority but some recombinants appear. The best way to see the difference is a testcross to a homozygous recessive, in which each offspring's phenotype directly reveals the gamete it received from the heterozygous parent.

Section 2Recombination frequency and map units

In 1911 Alfred Sturtevant, then an undergraduate in Morgan's laboratory, realized that the frequency of recombination between linked genes could serve as a measure of the distance between them. That night, as he later recalled, he neglected his homework and produced the first genetic map, ordering five X linked genes in Drosophila. The idea rests on a simple assumption: crossovers occur more or less at random along a chromosome, so the chance that one falls between two genes is proportional to the distance separating them.

Calculating recombination frequency

The recombination frequency (RF) between two genes is the proportion of gametes, or of testcross progeny, that are recombinant:

RF = number of recombinant offspring ÷ total number of offspring

One percent recombination defines one map unit (m.u.), also called a centimorgan (cM) in honor of Morgan. So genes with 8 percent recombination are 8 map units apart. One hundred centimorgans make one Morgan, a unit used mainly in mapping functions.

Why recombination frequency cannot exceed 50 percent

As two genes get farther apart, more than one crossover can occur between them. Each crossover switches which homolog the chromatid is copying, so an odd number of crossovers on a chromatid between two genes gives a recombinant, and an even number (two, four and so on) restores the parental combination. When genes are very far apart, crossovers are so frequent that the chance of an odd number approaches the chance of an even number, and recombinants approach, but never exceed, 50 percent. That is exactly the value for genes on different chromosomes. So:

• An RF near 50 percent means the genes are either on different chromosomes or far apart on the same chromosome. Such genes are syntenic (on the same chromosome) but genetically unlinked.
• RF underestimates distance once double crossovers become common, roughly above 10 to 15 map units, because double crossovers between two markers are counted as parentals.
• Map distances are therefore best measured over short intervals and added up. Added in this way, a chromosome's total map length can be much longer than 50 cM. The Drosophila X chromosome is about 66 cM long, and a large human chromosome such as chromosome 1 is more than 250 cM long.

Testing for linkage with chi square

Data from real crosses never match expectations exactly, so we need a way to decide whether a deviation from 1:1:1:1 in a testcross is due to chance or to linkage. The chi square test compares observed numbers (O) with the numbers expected (E) under a null hypothesis; for linkage questions the null hypothesis is that the genes assort independently. The statistic is

χ² = Σ (O − E)² / E

summed over all classes. The degrees of freedom are the number of classes minus one. The calculated value is compared with a critical value from a chi square table at the chosen significance level, usually P = 0.05. If the calculated value is larger than the critical value, the deviation is too large to blame on chance and the null hypothesis is rejected.

Genetic distance versus physical distance

Map units measure crossover frequency, not DNA length, and the two do not convert at a fixed rate. Averaged over the human genome, about 1 cM corresponds to roughly 1 million base pairs (1 Mb), but in baker's yeast, which has a tiny genome and very high crossover rates, 1 cM is only about 3,000 base pairs. Even within one genome, the relationship varies widely:

• Hotspots and coldspots. In humans and mice, most crossovers cluster in short hotspots one or two thousand base pairs long, many of them marked by binding sites for the protein PRDM9, which places histone marks that attract Spo11. Regions near centromeres and in heterochromatin have very little crossing over, so genes there look closer on the genetic map than they really are.
• Sex differences. The human female map is roughly 1.6 times longer than the male map, meaning women make more crossovers per meiosis. In Drosophila the difference is absolute: males make none.
• Structural changes. In an individual heterozygous for an inversion, crossovers within the inverted region produce unbalanced gametes that are not recovered, so recombination appears suppressed (Chapter 6).

Despite these quirks, the order of genes on a genetic map matches their physical order on the chromosome. Only the spacing differs.

Section 3Three point crosses and gene order

Mapping genes two at a time works, but it is slow and it leaves the order of genes uncertain when distances are similar. If A and B are 10 map units apart and B and C are 4 apart, C could lie between A and B or outside them. A three point testcross solves both problems at once. A female (or other individual) heterozygous for three linked genes is crossed to a tester that is homozygous recessive for all three. Every offspring then reveals, gene by gene, which alleles were on the chromatid it received, and a single cross gives the order of the genes, the two distances and a direct count of double crossovers.

The eight classes

A trihybrid makes 2³ = 8 kinds of gametes, so a three point testcross gives eight phenotypic classes. They pair up into four reciprocal pairs, and the pairs come in characteristic sizes:

• Parental (noncrossover) classes: the two largest classes, carrying the allele combinations of the two chromosomes in the heterozygote.
• Single crossover classes in region I: a pair produced by a crossover between the first and middle genes.
• Single crossover classes in region II: a pair produced by a crossover between the middle and last genes.
• Double crossover (DCO) classes: the two smallest classes, produced by one crossover in each region. Since the probability of two independent events is the product of their probabilities, double crossovers are rarer than either kind of single crossover.

Each reciprocal pair should contain roughly equal numbers, because each crossover produces both reciprocal products. If the numbers in a pair are wildly different, suspect reduced viability of one phenotype.

Finding the middle gene

The key insight of three point mapping is that a double crossover swaps only the middle gene relative to the two outside genes. Compare a parental class with a double crossover class that shares the same alleles at two genes: the one gene that differs is the gene in the middle. Once the middle gene is known, rewrite every class in the correct order, and the single crossover classes sort themselves into region I and region II.

Calculating the distances

The distance between two adjacent genes is the frequency of all offspring recombinant in that interval. Double crossovers are recombinant in both intervals, so they must be added to each:

• Distance in region I = (single crossovers in region I + double crossovers) ÷ total
• Distance in region II = (single crossovers in region II + double crossovers) ÷ total

The distance between the two outside genes is the sum of the two intervals. If you instead calculate it directly by counting only offspring recombinant for the two outside genes, you miss the double crossovers, which look parental for those genes. The direct value is smaller than the sum by exactly twice the DCO frequency, the numerical proof that two point mapping over long distances underestimates the truth.

A systematic routine

1. Pair up the eight classes as reciprocals and add each pair.
2. Call the largest pair parental and the smallest pair double crossovers.
3. Find the middle gene by comparing the parentals with the DCOs.
4. Rewrite all classes in the correct gene order.
5. Assign the remaining two pairs to region I or region II.
6. Compute each interval, remembering to add the DCOs to both.
7. Check that the outer distance equals the sum of the intervals, then compute interference (Section 4).

Section 4Interference, coefficient of coincidence and mapping functions

Expected versus observed double crossovers

If crossovers in two adjacent regions occurred independently, the probability of a double crossover would equal the product of the two single interval recombination frequencies. In the three point example, that expectation is 0.12 × 0.08 = 0.0096, or 9.6 double crossovers per 1,000 offspring. Only 6 were observed. Fewer double crossovers than expected is the usual finding, and it means that one crossover reduces the chance of another nearby. This phenomenon is called crossover interference (or chiasma interference).

Interference is measured in two related ways:

• The coefficient of coincidence (c.o.c.) is the ratio of observed to expected double crossovers: c.o.c. = observed DCO ÷ expected DCO.
• Interference (I) is its complement: I = 1 − c.o.c.

Interference is strongest over short distances and fades as intervals get longer. In Drosophila and in humans, two crossovers almost never occur within about 10 cM of each other on the same chromatid pair. The mechanism is still being worked out, but the leading models propose that the formation of one crossover sends a signal, possibly mechanical stress relieved along the chromosome axis, that inhibits others nearby. Most organisms have two classes of crossovers: an interfering class, which accounts for the majority, and a smaller noninterfering class made by a different resolution pathway. Interference also helps ensure that each chromosome pair gets its obligate crossover without wasting crossovers close together.

Mapping functions

Because double crossovers hide, recombination frequency is a biased estimate of true map distance, especially above 10 to 15 cM. A mapping function is a formula that converts an observed recombination frequency (r) into an estimated map distance (d) by correcting for undetected multiple crossovers.

Haldane's mapping function assumes no interference: crossovers fall at random, so the number per chromatid in an interval follows a Poisson distribution. If d is the map distance in Morgans, the mean number of crossovers per meiosis in the interval is 2d (each crossover involves only two of the four chromatids). The probability of no crossover is e^−2d, and every meiosis with at least one crossover produces 50 percent recombinant chromatids on average. Therefore

r = ½(1 − e^−2d), and rearranged, d = −½ ln(1 − 2r)

As d becomes large, e^−2d approaches zero and r approaches 1/2, which is why recombination frequency levels off at 50 percent.

Kosambi's mapping function allows for interference that is strong over short distances and weakens with distance, which fits most real data better:

d = ¼ ln[(1 + 2r) ÷ (1 − 2r)]

For small values of r, both functions give almost exactly d = r; they diverge as r grows. Human linkage maps are usually reported in Kosambi or Haldane centimorgans, and the choice should always be stated.

Section 5Mapping in humans: LOD scores and molecular markers

Humans cannot be testcrossed, families are small, and for most of the twentieth century there were few genes whose alleles could be followed through a family. Human gene mapping became practical when two things came together: a statistical method for combining evidence from many small families, and DNA molecular markers, harmless sequence differences scattered by the hundreds of thousands across the genome.

Informative meioses and phase

Linkage can be measured only in meioses from a parent who is heterozygous at both loci: the disease gene and the marker. Such a meiosis is informative. To count recombinants we also need to know the phase, which marker allele sits on the same chromosome as the disease allele. In a phase known family, the grandparents' genotypes reveal the phase. In a phase unknown family, both phases must be considered, which weakens the evidence from each child.

The LOD score

In 1955 Newton Morton introduced the LOD score (logarithm of the odds) to weigh the evidence for linkage. For a chosen recombination fraction θ (theta, the human geneticist's symbol for RF), the LOD score compares the probability of the observed family data if the loci are linked at θ with the probability if they are unlinked (θ = 0.5):

LOD(θ) = log₁₀ [ L(θ) ÷ L(0.5) ]

For phase known meioses with R recombinants and NR nonrecombinants, L(θ) = θ^R(1 − θ)^NR and L(0.5) = 0.5^{R + NR}. LOD scores are calculated for a series of θ values, and the value of θ that gives the highest LOD is the best estimate of the recombination fraction. Because logarithms of independent likelihoods add, LOD scores from different families can simply be summed at each θ, which is the method's great strength.

The conventional thresholds are:

• LOD of 3 or more (odds of 1,000 to 1 in favor of linkage) is accepted as evidence of linkage. The bar is high because any two loci picked at random are unlikely to be linked; with that low prior probability, a LOD of 3 corresponds to roughly a 5 percent chance that the linkage is false.
• LOD of −2 or less (odds of 100 to 1 against) excludes linkage at that θ.
• Values between −2 and 3 are inconclusive; more families are needed.

For phase unknown families, the likelihood is averaged over the two possible phases, each with probability 1/2. For example, for a father whose phase is unknown and whose children show R of one kind and NR of the other, L(θ) = ½[θ^R(1 − θ)^NR + θ^NR(1 − θ)^R]. In practice, computer programs calculate LOD scores for complex pedigrees, taking account of penetrance, missing individuals and marker allele frequencies.

Molecular markers

A good marker is polymorphic (many people are heterozygous, so meioses are informative), common across the genome, easy to type and inherited codominantly, so both alleles are visible. Several generations of markers have been used:

Markers transformed medical genetics through positional cloning: find a marker linked to the disease, narrow the interval with more markers and more families, then search the interval for the gene. Huntington disease was linked to a marker on chromosome 4 in 1983, using one of the first RFLPs, and the gene itself, HTT, was identified ten years later. Cystic fibrosis was mapped to chromosome 7 and its gene, CFTR, cloned in 1989. Today whole genome sequencing often finds rare disease genes directly, but linkage still narrows the search.

Linkage disequilibrium and association studies

Linkage analysis in families follows a handful of meioses. Population studies take advantage of thousands of past meioses. When a new mutation arises, it sits on a particular chromosome with a particular set of nearby marker alleles, a haplotype. Over many generations, recombination breaks the association with distant markers, but very close markers can remain associated for thousands of years. This nonrandom association of alleles at nearby loci is called linkage disequilibrium. Genome wide association studies (GWAS) type hundreds of thousands of SNPs in thousands of cases and controls and look for SNPs whose allele frequencies differ; because of linkage disequilibrium, a SNP that is not itself causal can flag a nearby causal variant. GWAS has found thousands of common variants with small effects on traits such as height, type 2 diabetes and coronary disease, complementing family linkage studies, which are best at finding rare variants of large effect.

Words worth knowing