Local LD as a Clue to Recombination History
Linkage disequilibrium (LD) describes non-independent association of alleles at different loci in a population. Across a genomic region, LD is not generally uniform: some neighboring variants may occur together on the same haplotypes much more often than expected from their allele frequencies, while other pairs show weaker association. Such local patterns can provide evidence about recombination history. They do not, by themselves, reveal a unique history or identify the process that produced the pattern.
A useful interpretation keeps three things separate: the observed pattern of haplotypes and LD, a historical explanation that could produce it, and the strength of the evidence for that explanation. A sharp local reduction in LD can be consistent with frequent recombination in the intervening region. But the same present-day pattern can also be shaped by allele frequencies, population history, selection, mutation, gene conversion, and sampling.
The values \(D\) and \(r^2\) describe association, not recombination rate. In particular, \(r^2\) is a standardized measure useful for comparing associations across pairs with different allele frequencies, while \(D\) retains the direction of association. A local LD profile typically displays a measure such as \(r^2\) for many pairs of variants, often against their genomic positions. A pattern in that profile is a summary of current population data, not a direct record of every historical crossover.
How Recombination Can Shape a Local Pattern
For two loci, recombination can put alleles onto new haplotype backgrounds. If \(c\) is the recombination fraction between the loci, then \(0\leq c\leq 1/2\). Under the standard two-locus model, with random mating and no other forces changing haplotype frequencies, the disequilibrium after one generation is \(D_{\text{next}}=(1-c)D\). Thus, for the same initial \(D\), a larger \(c\) leaves less LD after that generation.
The relationship follows directly from the haplotype-frequency update. With \(p_A\) and \(p_B\) unchanged by recombination, a fraction \(1-c\) of the \(AB\) frequency is retained in expectation, and the recombination contribution is \(c p_Ap_B\). Subtracting the product of the allele frequencies gives:
If the same assumptions hold for \(t\) generations and \(c\) remains constant, repeated application gives \(D_t=(1-c)^tD_0\). This is a model-based expectation, not a general rule for every real population: mutation, selection, migration, drift, changing population size, or nonrandom mating can alter the trajectory.
In a region with multiple markers, a recombination hotspot is a short interval with an elevated rate of recombination relative to surrounding intervals. Recombination there can weaken associations between variants on opposite sides of the interval, while variants within either side may remain more strongly associated. A local decline in LD around an interval can therefore be consistent with a hotspot. It does not establish that a hotspot caused the decline, nor does it imply that every pair spanning the interval has low LD. Allele frequencies, the ages and origins of alleles, and the number of historical generations all matter.
Worked Example: Interpreting a Local Decline in Pairwise LD
Suppose a hypothetical sample has phased chromosomes for four neighboring variants. Estimated pairwise \(r^2\) values for selected pairs are as follows. The entries are illustrative estimates, not measurements from a real study.
| Variant pair | Relative position | Estimated \(r^2\) |
|---|---|---|
| V1–V2 | Left side | 0.64 |
| V2–V3 | Across a short middle interval | 0.13 |
| V3–V4 | Right side | 0.58 |
| V1–V4 | Across the region | 0.09 |
The immediate pattern is that the two within-side pairs have higher estimated \(r^2\) than the pair spanning the middle interval. This is compatible with a history in which recombination in that interval has more often separated alleles that were previously correlated. The V1–V4 value is also low, which is compatible with weaker association across a longer span, but physical distance alone does not determine \(r^2\).
The interpretation should stop short of claiming that the middle interval is a proven hotspot. These four estimates do not establish the recombination rate, and the values can depend on the allele frequencies and histories of the variants. A small sample could also produce uncertain estimates. The pattern motivates a recombination hypothesis; additional evidence would be needed to distinguish that explanation from alternatives.
Reading Haplotype Frequencies Alongside LD
Pairwise LD compresses information about a pair of loci. Haplotype frequencies provide a more direct view of which allele combinations are common. For two biallelic loci, the four haplotype frequencies are \(p_{AB}\), \(p_{Ab}\), \(p_{aB}\), and \(p_{ab}\); they sum to one. Their marginal allele frequencies are \(p_A=p_{AB}+p_{Ab}\) and \(p_B=p_{AB}+p_{aB}\). These relationships help check that calculations and interpretations agree.
Worked Example: Calculating LD for a Pair Spanning an Interval
Imagine 1,000 phased chromosomes from a hypothetical sample. For two variants on opposite sides of a candidate interval, the observed haplotype counts are \(AB=420\), \(Ab=100\), \(aB=100\), and \(ab=380\). These counts sum to 1,000, so the estimated frequencies are 0.420, 0.100, 0.100, and 0.380.
The allele frequencies are \(p_A=0.420+0.100=0.520\) and \(p_B=0.420+0.100=0.520\). Under independence, the expected \(AB\) frequency is \(p_Ap_B=(0.520)(0.520)=0.2704\). Therefore:
To calculate \(r^2\), use the allele-frequency variances in the denominator:
The positive \(D\) indicates that \(AB\) occurs more often than expected under independence; \(r^2\) is about 0.359. These are measures of association in this sample. They do not show that recombination occurred in the interval, and \(r^2\) should not be read as a proportion of recombination events. The numerical estimate alone also gives no account of sampling uncertainty.
Why the Same LD Can Have Different Histories
A present-day haplotype distribution is an outcome of population processes. Different sequences of events can lead to the same frequencies, so LD cannot usually be inverted to recover a single history. Recombination is one possible contributor, but it is not the only process that can generate or reduce association.
Worked Example: Two Histories With the Same Present-Day LD
Consider a hypothetical two-locus population with current haplotype frequencies \(AB=0.34\), \(Ab=0.16\), \(aB=0.16\), and \(ab=0.34\). The allele frequencies are \(p_A=p_B=0.50\), giving:
One possible history is a single generation of recombination. Suppose the preceding frequencies were \(AB=0.35\), \(Ab=0.15\), \(aB=0.15\), and \(ab=0.35\), so \(p_A=p_B=0.50\) and \(D_0=0.35-0.25=0.10\). If \(c=0.10\), then \(D_1=(1-0.10)(0.10)=0.09\). With the allele frequencies unchanged, the resulting haplotype frequencies are \(AB=0.25+0.09=0.34\), \(Ab=0.50-0.34=0.16\), \(aB=0.16\), and \(ab=0.34\).
A different history is mixture of two populations that each had independent alleles at these loci. In population 1, let \(p_A=p_B=0.80\), giving haplotype frequencies \(0.64, 0.16, 0.16, 0.04\) in the order \(AB, Ab, aB, ab\). In population 2, let both allele frequencies be 0.20, giving frequencies \(0.04, 0.16, 0.16, 0.64\). An equal mixture of their chromosomes has frequencies \(0.34, 0.16, 0.16, 0.34\), exactly the same current distribution as above. The mixture produces positive LD because allele frequencies differed together across the source populations.
Thus, the same \(D\), \(r^2\), and haplotype frequencies are compatible with a recombination history or with mixture of populations that had independent alleles within each source. The observed LD alone cannot tell which history occurred.
What Local Patterns Can and Cannot Support
A careful reading of local LD begins with description, then considers explanations. A profile with relatively high LD on either side of a short interval and lower LD for pairs spanning it is consistent with recombination having separated correlated alleles across that interval. If this pattern is localized and supported by many variants, it can make a hotspot hypothesis plausible. But the evidence remains indirect unless recombination is assessed using additional information.
Several factors complicate this inference:
- Allele frequencies: \(r^2\) depends on the frequencies at both loci. Two pairs with similar \(D\) can have different \(r^2\), and rare alleles can limit the attainable values.
- Population history: bottlenecks, drift, migration, and admixture can change haplotype frequencies and produce local differences in LD.
- Other molecular processes: mutation introduces alleles, while gene conversion can transfer short DNA segments; neither pattern is summarized by a simple crossover-only account.
- Selection: selection on one or more variants can change the frequencies of haplotypes carrying them.
- Sampling and estimation: estimated haplotype frequencies and LD measures vary with sample size and phase uncertainty. A visible drop in an estimate need not reflect a true population difference.
The scale matters as well. Recombination acts between loci, while an LD profile displays pairwise associations across positions. A single interval can affect several pairs that span it, but the exact pattern depends on marker placement and allele frequencies. Do not infer a precise recombination boundary from only one pair, and do not treat a smooth decline with distance as a universal law. The later study of haplotype blocks considers how regions of correlated variants and their boundaries are summarized; here, the central task is to interpret local patterns as evidence rather than as a unique reconstruction.
Common Mistake
A second error is to equate a high \(r^2\) value with “little recombination” as if it measured a current rate. \(r^2\) is a measure of present association between alleles. It can reflect recombination accumulated over generations, but also allele age, demography, and other processes. A strong interpretation distinguishes an LD statistic from a recombination-rate estimate.
Check Your Understanding
Answer each question using the definitions and distinctions in this tutorial.
- If LD is high among variants on either side of a short interval but low for pairs spanning it, what recombination explanation is consistent with the pattern, and what can the pattern not establish by itself?
- For \(p_A=0.40\), \(p_B=0.50\), and \(p_{AB}=0.25\), calculate \(D\). Is the association positive or negative?
- Under the two-locus model with no other forces, what fraction of the initial \(D\) remains after one generation when \(c=0.20\)?
- How can mixture of populations with different allele frequencies create LD even if each source population had independent alleles at the two loci?
- Why is it incorrect to describe \(r^2\) as a direct estimate of the recombination rate?