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A genetic map turns observations of inherited haplotypes into an ordered account of genetic loci along a chromosome. Its distances describe genetic recombination, which makes the map useful for locating a genetic locus relative to known genetic markers. A genetic map measures inheritance separation; separation measured in base pairs describes positions along DNA. Recombination hotspots help explain why equal numbers of base pairs need not give equal genetic map distances. See T. A. Brown, Genomes, 2nd edition (2002), chapter 5: Mapping Genomes.

Write $d$ for the expected value of the number of chromosomal crossovers encountered along one transmitted chromatid between the chosen genetic loci, measured in morgans; $100d$ is the distance in centimorgans. This counts exchanges along that transmitted chromatid, rather than all exchanges in the four-chromatid meiotic bundle. For a short interval with very few multiple exchanges, the recombination fraction satisfies $r\approx d$: about $1\%$ recombinant gametes then corresponds to about one centimorgan.

For a longer interval, hidden even numbers of chromosomal crossovers make that approximation inaccurate. Under the Haldane mapping function, $d=0.2$ morgans gives $r\approx 0.164840$; conversely $r=0.2$ gives $d\approx 0.255413$ morgans, or $25.5413$ centimorgans. Applying the inverse formula to an already specified genetic map distance would confuse a count with a probability.

Genetic map distances add along ordered adjacent intervals because the total exchange count is the sum of its interval counts. For independent random variables describing counts in disjoint intervals of the Haldane mapping function model, the endpoint recombination fraction instead obeysExactly one interval must change parental origin. Thus $r_1=r_2=0.2$ gives $r_{12}=0.32$, although the two genetic map distances add to approximately $0.510826$ morgans. The simple composition formula requires the stated independence; crossover interference can invalidate it.

Genetic recombination erodes linkage disequilibrium by reshuffling alleles between haplotypes, while preserving marginal allele frequencies in the idealized model. Let $f_{AB,t}$ denote the haplotype frequency in the gamete pool of generation $t$, let $p,q$ denote its marginal allele frequencies, and let $r\in [0,1/2]$ be the constant recombination fraction between two autosomal genetic loci. Assume random mating within one population, nonoverlapping generations, fair Mendelian segregation, equal reciprocal gamete production, the same $r$ in both sexes, and a deterministic population large enough to neglect genetic drift. Exclude natural selection, mutation and gene flow. ThenHere $p$ and $q$ remain constant: recombining changes which alleles occur together, not their total numbers. The other haplotype frequencies change by $+r{D}_t$ for $Ab,aB$ and $-r{D}_t$ for $ab$. Consequently the joint probability distribution stays within its allowed bounds and approaches the product of its marginal distributions when $r>0$.

For the original positive example $p=0.60,q=0.40,D_0=0.12$, a recombination fraction $r=0.10$ gives $D_1=0.108$ and $D_{10}\approx 0.04184$. The negative example $D_0=-0.12$ has exactly the opposite sequence: genetic recombination reduces the magnitude without reversing the sign. With $r=0$, the initial linkage disequilibrium persists in this model; even independent assortment at $r=1/2$ only halves $D$ each generation. A population association does not vanish after one generation merely because the genetic loci are unlinked.

The geometric sequence gives a generation scale: for $r>0$, the continuous interpolation reaches $|D_0|/2$ at $t_{1/2}=\log (1/2)/\log (1-r)$. Actual generations are integers; the first at or below half is $\lceil t_{1/2}\rceil$. For small $r$, $t_{1/2}\approx (\log 2)/r$. Because the denominator of allelic correlation stays fixed, $r_{\mathrm{LD},t}^2=(1-r{)}^{2t}{r}_{\mathrm{LD},0}^2$; its decay is faster than the magnitude of $D_t$.

A short derivation uses random mating to assign frequency $2f_{AB}{f}_{ab}$ to the double heterozygote $AB/ab$ and $2f_{Ab}{f}_{aB}$ to $Ab/aB$. Relative to $r=0$, each loses or gains an $AB$ gamete fraction $r/2$, respectively. Their net contribution is $r(f_{Ab}{f}_{aB}-f_{AB}{f}_{ab})=-rD$. The other genotypes produce no net change in $f_{AB}$, proving the recurrence and, by iteration, its solution.

This temporal model does not identify elapsed generations from physical separation alone. The recombination fraction depends on genetic recombination, including recombination hotspots, while population stratification, natural selection, gene flow or genetic drift can create or maintain linkage disequilibrium. Repeated sampling across time would also require accounting for sample size and haplotype estimation. A cross-sectional plot of association against distance between genetic loci is therefore a different observation from the idealized trajectory plotted here.

Visible divergence and reproductive incompatibility measure different outcomes of evolution. A few changes in a gene regulatory network can transform a conspicuous phenotype, whereas differences in the machinery of meiosis or biological development can prevent reproduction without greatly changing adult appearance. Neither membership of the same genus nor morphological similarity specifies divergence time or the extent of reproductive isolation.

First separate barriers. premating isolation can arise through differences in courtship, habitat or breeding season even when a forced cross produces fertile offspring. postzygotic isolation concerns the survival or fertility of the offspring themselves. Thus fertile captive hybrids do not refute substantial reproductive isolation in nature: they may never encounter suitable mates, may have intermediate ecological requirements, or may be unattractive to both parental biological populations. Under the biological species concept, the relevant issue is effective gene flow among natural biological populations, not the success of one laboratory mating. Conversely, a laboratory failure can reflect husbandry rather than an intrinsic incompatibility.

Rapid visible divergence is especially plausible when sexual selection changes plumage or other signals, or natural selection changes a feeding structure. Such changes can involve a small part of the genome, leaving most interactions required for fertility compatible. In Darwin's finches, genome comparisons associated an ALX1 region with beak shape and also revealed interspecific gene flow. This is an example of conspicuous adaptive differentiation coexisting with exchange of alleles, rather than evidence that all avian hybrids are fertile.^[ref] Continued gene flow can itself help maintain compatibility through much of the genome while divergent natural selection preserves differences at selected genetic loci.

For cryptic incompatibility, consider a Dobzhansky-Muller incompatibility. Two lineages start with compatible alleles $a,b$. One evolves $A$ in the presence of $b$, while the other evolves $B$ in the presence of $a$. Each change is tolerable in its own genetic background, but the combination $A,B$ disrupts function in hybrids. This requires neither lineage to pass through a low-fertility intermediate. Such epistasis can affect spermatogenesis, chromosome pairing, or interactions between nuclear and mitochondrial genomes, while leaving the adult body plan nearly unchanged. The relevant changes can arise through natural selection, genetic drift, or conflict over inheritance; adaptive divergence in external form is not required.

A particularly informative mammalian experiment concerns PRDM9, which helps specify recombination hotspots. In a particular house-mouse cross, changing its DNA-binding region to the corresponding human sequence restored chromosome pairing and male fertility. The experiment implicates compatibility between a genetic recombination regulator and its binding sites, rather than visible morphological divergence. Crucially, this was a rescue in a specified cross and genetic background, not a demonstration that PRDM9 explains all mammalian speciation.^[ref]

Comparative work on birds lacking PRDM9 found relatively stable recombination hotspots associated with functional genomic features.^[ref] This suggests a possible difference in the routes to incompatibility: turnover of PRDM9 and its targets is unavailable in those lineages. The inference is mechanistic, however; hotspot stability by itself does not demonstrate that avian hybrids will be fertile.

Differences in karyotypes provide another mechanism. A balanced chromosomal rearrangement can have little visible effect in either parent yet cause faulty pairing or unbalanced gametes in a heterozygous hybrid. However, a different chromosome count alone does not prove sterility: the arrangement and segregation behavior matter. Differences in genomic imprinting may also disturb placental development or embryonic growth in mammalian crosses, sometimes producing direction-dependent outcomes. Parent-of-origin growth abnormalities and disrupted genomic imprinting in particular rodent hybrids provide evidence that this mechanism can matter.^[ref] These are hypotheses to test in the relevant lineages, not a universal explanation for the bird–mammal comparison.

The sex of the offspring matters. Haldane's rule predicts that, when one sex of first-generation hybrids is absent, rare or sterile, it tends to be the heterogametic sex: typically XY males in mammals and ZW females in birds.^[ref] Recessive incompatibilities on a sex chromosome can be exposed when there is no corresponding second copy; rapid evolution of reproductive functions can contribute too. Therefore a report of fertile avian males is insufficient without testing females. The rule predicts a sex bias in a given cross, not that birds as a class have weaker reproductive isolation.

To discriminate these mechanisms, compare multiple independently diverged pairs, match their estimated divergence times using phylogenetics, and avoid treating taxonomic rank as a clock. Perform reciprocal crosses with parental within-lineage controls; measure fertilization, embryonic survival, both sexes' fertility, and second-generation and backcross success. A fertile first generation can conceal hybrid breakdown after genetic recombination disrupts coadapted combinations. Pair karyotype analysis and observations of meiosis with genome mapping and carefully controlled rescue experiments. Field measurements of mating and ancestry then determine whether compatibility translates into gene flow. The contrast is explained by different rates and genetic targets of visible divergence and reproductive divergence, with sex, cross direction, ancestry and assay conditions determining the observed outcome.

Across a human chromosome, neighboring single-nucleotide polymorphisms often show substantial linkage disequilibrium, while average allelic correlation tends to decline with increasing genetic distance. A new mutation starts on one ancestral haplotype; repeated genetic recombination separates it from progressively more distant markers. In an ideal randomly mating two-locus population without further mutation, migration or selection, gamete formation gives $p_{11}^{\prime }=(1-c)p_{11}+c{p}_A{p}_B$, where $c$ is the recombination fraction. The marginal allele frequencies remain fixed, soHere the prime means the next-generation value, not a separately standardized LD statistic. This simple decay explains the distance trend, rather than asserting an exact deterministic law for real finite populations.

The local pattern is heterogeneous. Many regions have haplotype blocks with strong LD and a small number of common haplotypes, separated by intervals showing more historical recombination, often recombination hotspots. A hotspot can therefore cause a rapid drop in LD between markers that are close in physical distance. Block lengths and boundaries are not uniform, perfectly sharp or independent of the population and the statistical block definition. These empirical features are documented in the HapMap data.

Population history also matters. Population bottlenecks can reduce the number of ancestral haplotypes and increase the extent of LD; population stratification or admixture can create longer-range associations, including between distant loci. Different populations can consequently have different haplotype frequencies and LD patterns at the same genomic interval. Raw $D$ also varies with allele frequency, so comparing LD across markers requires attention to the chosen measure. The typical pattern is locally strong, block-like and highly variable LD, with a broad decline over genetic distance rather than a uniform chromosome-wide decay.

Five consequences of the ancestral recombination graph are:

Shared ancestry explains correlation; recombination makes ancestry a mosaic; mutation on the resulting branches converts genealogy into observed variation. These explanations use the neutral ARG, with a variable recombination map where stated. Selection, population subdivision or changing population size can also be studied genealogically, but require extending the event model rather than being consequences of the constant-size panmictic neutral process alone.

A regulator of the positions of meiotic recombination hotspots in relevant mammals. An engineered DNA-binding region rescued fertility in a particular mouse hybrid cross.^[ref]