Meiosis Recombination Probabilities from Haplotype Transmission

From a Parent’s Haplotypes to Gamete Probabilities

At two linked loci, an offspring receives one haplotype from each parent. A parent’s genotype at both loci does not, by itself, tell us which alleles are carried together on the same chromosome. When that arrangement is known, it is called the parent’s phase. Given phase and a recombination fraction, we can assign probabilities to the haplotypes that a gamete may carry.

Use \(A\) and \(a\) for alleles at the first locus, and \(B\) and \(b\) for alleles at the second. A parent with haplotypes \(AB\) and \(ab\) carries \(A\) with \(B\) on one chromosome and \(a\) with \(b\) on the other. The two intact haplotypes are the parental types. A gamete carrying \(Ab\) or \(aB\) has a recombinant combination of alleles.

The split into two equally probable types within each class is part of the standard model: the two homologous chromosomes are transmitted without bias, and the reciprocal recombinant types are treated symmetrically. Thus the total probability of the parental class is \(1-\theta\), and the total probability of the recombinant class is \(\theta\). The four probabilities sum to one:

The recombination fraction concerns whether the alleles inherited at the two loci form a recombinant combination. It is not, in general, the probability that exactly one physical crossover occurred: multiple crossovers can occur, and some do not change the observed two-locus combination. At \(\theta=0\), only the parental combinations are transmitted in this model. At \(\theta=0.5\), each of the four haplotypes has probability \(0.25\); at these two loci, transmission then has the same distribution as independent assortment.

Assigning the Four Haplotype Probabilities

For a parent with phase \(AB/ab\), the parental types are \(AB\) and \(ab\), while \(Ab\) and \(aB\) are recombinant. If instead the phase is \(Ab/aB\), the parental types are \(Ab\) and \(aB\), and \(AB\) and \(ab\) are recombinant. The value of \(\theta\) is unchanged by this switch; the labels “parental” and “recombinant” depend on the parent’s haplotypes.

This distinction matters when calculating a particular transmission. A gamete carrying \(Ab\) is relatively likely when \(Ab\) is one of the parent’s intact haplotypes, but less likely when \(Ab\) is recombinant. If parental haplotypes are distinct, the probability assigned to any one gamete is obtained by identifying its class and applying the corresponding formula.

Calculating an Offspring Genotype Probability

A two-locus offspring genotype results from one gamete from each parent. To calculate its probability, list the gamete pairs that produce the specified genotype, multiply the two transmission probabilities for each compatible pair, and add those products. The multiplication uses the standard assumption that the two parents’ meioses are independent, conditional on their genotypes and phases. The addition is necessary because an unphased offspring genotype can arise through more than one pair of transmitted haplotypes.

For example, the offspring genotype \(A/a\) at the first locus and \(B/b\) at the second can be produced by gametes \(AB\) and \(ab\), or by \(Ab\) and \(aB\), with either parent supplying either member of the pair. Those routes are distinct transmissions, so their probabilities must be summed rather than selecting just one.

When Different Haplotypes Produce the Same Allele Outcome

The four-haplotype formula applies directly when a parent is heterozygous at both loci and the two phased haplotypes are distinct. If the parent is homozygous at one locus, some of the four named haplotypes may be indistinguishable by their alleles. The underlying transmissions can still be described, but probabilities for the same observable gamete outcome must be added.

A Short Justification of the Transmission Rule

Under the two-locus model, the parent’s two intact haplotypes define the parental class. By definition, the total probability of transmitting a recombinant combination is \(\theta\); the probability of transmitting a parental combination is therefore \(1-\theta\). With no transmission bias and symmetry between the two homologues, each of the two parental haplotypes receives half of the parental-class probability. The same symmetry divides the recombinant-class probability equally between its two reciprocal products:

This justification depends on the model assumptions: the phase is specified, allele transmission follows Mendelian segregation, the reciprocal products are symmetric, and there is no transmission distortion or mutation. If those assumptions are not appropriate, these formulas need not describe the observed gamete probabilities. When the phase or a parent’s genotype is uncertain, a single fixed set of haplotype probabilities is not enough; that uncertainty must be handled in the analysis.

Common Mistakes and Why These Probabilities Matter

• Using \(\theta\) for each recombinant haplotype. The total probability of the recombinant class is \(\theta\); each of the two reciprocal types has probability \(\theta/2\) under the standard model.
• Forgetting to identify phase. “Parental” refers to the haplotypes actually carried by that parent. For \(Ab/aB\), \(Ab\) is parental and \(AB\) is recombinant.
• Failing to add compatible routes. A specified offspring genotype may arise through several gamete pairs. The probability is the sum of their products, not the probability of just one route.
• Counting duplicate gamete outcomes twice. If the parent is homozygous at one locus, different transmission classes can yield the same observed haplotype. Add their probabilities when they produce the same allele outcome.
• Interpreting \(\theta\) as the probability of exactly one crossover. It is the probability of a recombinant two-locus gamete under the model, not necessarily a count of physical crossover events.

These calculations provide the basic transmission probabilities needed to describe inheritance across a pair of loci. They separate the meiosis-level question—what haplotype can a parent transmit, and with what probability?—from the later task of combining transmissions across a pedigree or comparing competing linkage models.