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An allele is one of the alternative inherited states at a genetic locus, often distinguished by a difference in DNA sequence. A population can contain many alleles, although a two-copy diploid individual carries at most two at that genetic locus, one inherited from each parent. See NHGRI: allele.

Count copies, not just individuals, when computing allele frequency: in 100 individuals with genotype counts $49AA$, $42Aa$, $9aa$, there are $2(49)+42=140$ copies of $A$ among 200, so $p_A=0.70$. The heterozygotes contribute to both allele counts. This converts observed genotypes into the population quantity needed by the Hardy-Weinberg principle.

The allele frequency is the proportion of gene copies at a specified genetic locus carrying an allele. Two unrelated copies can have the same observed allele without being identical by descent; under independent sampling their matching probability is the sum of squared allele frequencies. Pedigree autozygosity instead traces copies to the same ancestral copy.

A chromosomal crossover is a reciprocal exchange of segments between nonsister chromatids of homologous chromosomes during meiosis. It can change the haplotypes inherited at selected genetic loci; whether it does so depends on the transmitted chromatid and interval.

A chromosome carries genes and other genetic material. A haplotype records the alleles carried on one chromosome at specified genetic loci.

Diploid biological cells have two sets of chromosomes. In the ordinary one-locus inheritance model, a diploid genotype therefore contains two allele copies at each genetic locus.

Experimental tractability is the practical ability to cultivate a system, manipulate the variables of interest, and obtain an interpretable readout with appropriate scientific controls. In Saccharomyces cerevisiae, inexpensive cultivation, accessible haploid and diploid states, and tools for introducing DNA make controlled changes of genotype feasible. A haploid biological strain exposes a changed allele without a second copy at that genetic locus, while crosses and meiosis permit tests of inheritance. ^[ref]

An accessible readout is not necessarily a direct measurement of the mechanism. Growth of a biological colony integrates many cellular functions. Failure to grow could reflect the intended gene deletion, unsuitable conditions, failed genetic transformation, or another mutation. Matching the genetic background and using scientific controls narrows these alternatives. Ease of doing an experiment and strength of its inference are distinct advantages.

A gamete is a reproductive biological cell that can fuse with another gamete during fertilization. In the ordinary diploid inheritance model, a haploid gamete carries one copy at each genetic locus.

A gene is an inherited functional unit of genetic material. In cellular organisms, a gene is a region of DNA whose information produces a functional RNA or contributes to a protein. This connects inheritance to cellular chemistry: a change in a gene can alter the amount or activity of a protein, and thereby a phenotype. See NHGRI: gene.

Alternative inherited versions at a genetic locus are alleles. The distinction matters: two parents can carry the same gene but different alleles, and a genetic locus can have variation outside a gene. Neither the word gene nor a change in DNA alone specifies the resulting phenotype.

// Target: biology.bigb

A genetic genetic carrier has one disease-associated recessive allele and one normal allele at a genetic locus, and is unaffected under a fully penetrant recessive model. Two genetic carriers have affected-offspring probability $1/4$ under Mendelian segregation; a genetic carrier and an individual with no disease allele have affected-offspring probability zero under that same mutation-free model.

Why can two individually fair allele transmissions fail to be independent? Genetic loci on the same chromosome can travel together through meiosis. This tendency is genetic linkage; in an ordinary two-locus model, detectable genetic linkage means a recombination fraction below $1/2$. A genetic marker can therefore help track a neighboring genetic locus through a pedigree. See Clark, Douglas and Choi, Biology 2e: chromosomal theory and genetic linkage.

Take a diploid individual heterozygous at two genetic loci, with phase-known genotype $AB/ab$: one homologous chromosome carries alleles $A,B$, the other $a,b$. The parental haplotypes are $AB,ab$; the recombinant haplotypes are $Ab,aB$. Assume fair Mendelian segregation, equal probabilities within each reciprocal pair of gametes, and no allele-dependent differences in gamete recovery. If the recombination fraction is $r=0.2$, the gamete probabilities are $0.4,0.4,0.1,0.1$ in that order. Each genetic locus still transmits its two alleles equally, but $P(AB)=0.4$ exceeds $P(A)P(B)=0.25$: fairness at each genetic locus does not imply independence between genetic loci.

At $r=0$, only the parental haplotypes occur, each with probability $0.5$: this is complete genetic linkage. At $r=0.5$, all four haplotypes have probability $0.25$, giving independent assortment. A recombination fraction near $0.5$ alone cannot establish whether the genetic loci lie on different chromosomes or far apart on one chromosome.

Genetic linkage concerns joint transmission through meiosis. Linkage disequilibrium concerns haplotype frequencies in a population, while genetic association compares allele or genotype frequencies with a phenotype. These questions require different observations; pedigrees directly reveal co-transmission.

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.