Lecture 3 of the Molecular Genomics series. Physical and genetic distance, the Morgan, and what crossing over does to marker associations.
We have markers and we know their genotypes. To do anything with them we also need to know where they are, and what happens to their arrangement between generations. Both are the subject of this lecture, and both are prerequisites for linkage disequilibrium, which is where the course turns genuinely quantitative.
There are two answers to the question "how far apart are these two loci", and they measure different things.
They are correlated but not proportional, and the difference between them carries biological information.
Physical distance is counted in base pairs (bp), kilobase pairs (kb, one thousand) and megabase pairs (Mb, one million). A mammalian genome is about
The chicken genome is roughly \(1.2\times 10^{9}\) bp, about 40% of the mammalian figure, which is a reminder that genome size is not a measure of biological complexity.
Physical position is the coordinate system used by essentially all modern software, because it is what a reference assembly provides. A marker is identified by a chromosome number and a base-pair position, and every result in this course is ultimately reported in those terms.
The genetic map measures the same interval by how often it is broken.
A chromosome segment has a genetic length of one Morgan (M) if, on average, one crossover occurs within it per meiosis. One Morgan equals 100 centiMorgan (cM). Named for T.H. Morgan, 1919.
Over short distances, one centiMorgan corresponds to a recombination fraction of about one percent, written \(\theta = 0.01\): two loci one cM apart are separated in about one gamete in a hundred.
The approximation is short-range only. Recombination fraction cannot exceed 0.5, because loci on different chromosomes, or very far apart on the same one, assort independently and are separated in half of all gametes. Genetic distance in Morgans, however, is additive and unbounded: a chromosome can be three Morgans long. The two agree only when the interval is small enough that double crossovers are negligible, since a second crossover restores the original arrangement and is invisible.
For mammals the two scales are related by a convenient approximation.
Physical genome length about 3,000 Mb, genetic map length about 30 M = 3,000 cM. Therefore, roughly,
1 cM ≈ 1 Mb ≈ 1% recombination
Equivalently, about one crossover per 100 Mb of chromosome.
The arithmetic is worth doing once. If a chromosome is 350 Mb long, then it is about 350 cM, which is 3.5 M, so we expect about 3.5 crossovers on that chromosome per meiosis.
Bovine chromosome 1 is approximately 158 Mb. Applying the rule, its genetic length is about 158 cM = 1.58 M, so we expect between one and two crossovers on it per meiosis.
Two loci 5 Mb apart on this chromosome are about 5 cM apart, so they are separated in roughly 5% of gametes and co-inherited in the other 95%. That is the sense in which nearby markers carry information about each other, and it is the seed of linkage disequilibrium.
Treat the rule as an order-of-magnitude guide. It is an average over a genome in which the local rate varies by more than an order of magnitude, and it does not transfer across taxa: in chicken the genetic map is far longer relative to the physical map than in mammals.
During meiosis the maternal and paternal copies of a chromosome pair along their length. The arms overlap, the molecules break and rejoin, and material is exchanged between the two. The event is a crossover; the outcome, a chromosome carrying alleles that came from both parental copies, is a recombinant.
Two consequences follow, and they pull in opposite directions.
Recombination creates variation. Without it, a chromosome would pass through the generations as an indivisible block and the number of distinct genotypes a population could produce would be tiny. Crossing over lets alleles be recombined into new arrangements, so offspring can carry allele combinations that neither parent had. This is why recombination is best understood not as an accident of meiosis but as a mechanism that generates and maintains variability.
Recombination destroys association. The same process that creates new combinations breaks up existing ones. A marker and a nearby causal variant travel together only for as long as no crossover falls between them. Over many generations, association decays at a rate set by the recombination fraction. Everything in Lecture 5 follows from this tension.
A precise statement of when two loci recombine is more useful than an intuitive one.
Two loci are recombinant if an odd number of crossovers occurs between them. An even number, including zero, restores the original arrangement.
This matters because crossovers are not directly observable. What we observe is the gamete, and a gamete produced by two crossovers between the same pair of loci is indistinguishable from one produced by none. Recombination fraction is therefore always an underestimate of the true number of crossover events, and the underestimate grows with distance.
This is precisely why recombination fraction saturates at 0.5 while genetic distance does not. Over a long interval, crossovers are numerous and the parity of their count is effectively a coin toss, so half of gametes are recombinant no matter how long the interval becomes. Genetic map distances are constructed by summing short intervals, where double crossovers are rare, and are additive for that reason.
The rule of thumb averages over substantial and systematic variation.
Between the sexes. Recombination is usually more frequent in the homogametic sex, which is the female in mammals and the male in birds, so that map is longer. In humans the female to male map length ratio is roughly two to one. A genetic map is therefore sex-specific unless it is explicitly a sex-averaged map.
Along the genome. Recombination clusters into hot spots separated by cold regions where it is rare. This is not noise: it is a property of the local sequence and chromatin. A consequence is that linkage disequilibrium is patchy, with blocks of strongly associated markers bounded by hot spots, which is what makes haplotype blocks a useful idea in Lecture 4.
Between individuals and populations. Recombination rate is itself a heritable trait, with detectable family differences and a demonstrated response to selection. Domesticated animals generally recombine more than their wild relatives, which is one of the more intriguing correlates of domestication.
On short chromosomes. Very short chromosomes, such as the microchromosomes of chicken or the pseudoautosomal region of the sex chromosomes, often require at least one crossover per meiosis for correct segregation, the so-called obligatory chiasma. Their genetic length is therefore much greater than their physical length would suggest, and the 1 cM per Mb rule fails badly.
Applying the rule. A chromosome is 120 Mb long. Estimate its genetic length and the expected number of crossovers per meiosis. Two markers on it are 250 kb apart; what recombination fraction do you expect between them, and how often will they be co-inherited?
Using 1 cM per Mb, the chromosome is about 120 cM = 1.2 M, so we expect about 1.2 crossovers per meiosis.
250 kb is 0.25 Mb, so the markers are about 0.25 cM apart, giving \(\theta \approx 0.0025\). They are separated in about one gamete in 400 and co-inherited in about 99.75%.
Note how strong that is. At the marker spacing of a 50K bovine chip, roughly 50 kb, adjacent markers are co-inherited more than 99.9% of the time. This is why dense marker panels track chromosome segments rather than independent points, and it is the reason genomic prediction works at all.
The parity argument. Two loci are 60 cM apart. Naively this suggests a recombination fraction of 0.60. Explain why that is impossible and what the true recombination fraction is closer to.
Recombination fraction is a probability that a gamete is recombinant, and it cannot exceed 0.5. The maximum is reached when the loci assort independently, which is the situation for loci on different chromosomes.
The reason is the odd-number rule. Over an interval of 0.6 M the expected number of crossovers is 0.6, and gametes are recombinant only when that number is odd. Modelling crossovers as a Poisson process with mean \(m\) and ignoring interference gives Haldane's mapping function,
With \(m = 0.6\), \(\theta = \tfrac12(1-e^{-1.2}) = \tfrac12(1-0.301) = 0.35\). So the true recombination fraction is about 0.35, not 0.60, and as \(m\) grows \(\theta\) approaches but never reaches 0.5.
This is the formal reason genetic distance is additive and recombination fraction is not.
Hot spots and mapping resolution. Two research groups fine-map the same trait in the same species. One finds an association spanning 2 Mb, the other, using the same marker density, resolves it to 150 kb. Give a genomic explanation that does not involve either group making a mistake.
The regions differ in local recombination rate. Mapping resolution is set by how much recombination has occurred between the marker and the causal variant in the ancestry of the sample, not by marker density alone. In a recombination cold region, long stretches travel together and association extends far, so the signal is broad and the causal variant cannot be localised. Near a hot spot, association decays within a short distance and the signal is correspondingly narrow.
Two secondary explanations are also legitimate. The populations may differ in effective population size or history, since a population that passed through a recent bottleneck has long-range linkage disequilibrium everywhere. And the number of generations since the causal mutation arose matters: an old variant has had more meioses in which to be separated from its original haplotype.
The practical implication is that resolution is a property of the region and the population, and is not something the analyst controls by adding markers.
Physical versus genetic maps. You are told that a 10 Mb region in chicken has a genetic length of 60 cM. What does this tell you about the region, and what would go wrong if you assumed the mammalian rule of thumb?
The observed ratio is 6 cM per Mb, six times the mammalian rule. Recombination in this region is unusually frequent, which is characteristic of chicken microchromosomes: they are short enough that an obligatory crossover is needed for correct segregation, so the genetic length is inflated relative to physical length.
Assuming 1 cM per Mb would cause two errors, in opposite directions but both damaging. You would substantially underestimate how quickly linkage disequilibrium decays, and therefore expect a given marker density to provide far better coverage of the region than it does. And in fine-mapping you would infer that an association signal implicates a wide interval, when in fact the high recombination rate means the signal is more localised than the physical distance suggests.
The general lesson is that the mammalian rule of thumb is a mammalian rule of thumb.
