Genetic Maps and Recombination Mapping Functions

Why Convert Recombination Fractions to Genetic Distance?

Two-point linkage analysis describes how often a meiosis is recombinant between a pair of loci. A recombination fraction summarizes that evidence, but a genetic map is usually expressed as a distance along a chromosome. Mapping functions provide the conversion between these quantities. They matter because the same recombination fraction can imply different map distances under different assumptions about crossovers.

The recombination fraction is estimated from inheritance evidence, such as informative meioses in a pedigree. The mapping function does not change that evidence or the estimate; it supplies a model-dependent interpretation of the estimate as genetic distance. In particular, a recombination fraction is not itself a distance in centimorgans.

Definition: The recombination fraction, denoted $r$, is the probability that a gamete is recombinant between two loci in a meiosis. It lies between 0 and $0.5$. Genetic distance, denoted $d$, is a map interval measured in Morgans; one Morgan corresponds to 100 centimorgans (cM). A mapping function specifies a model-based relationship between $r$ and $d$.

The upper bound $r=0.5$ reflects the fact that, for two loci, recombinant and nonrecombinant outcomes cannot be distinguished more often than a fair split under the usual linkage model. As loci become far apart, multiple crossovers can restore the parental arrangement, so the observed recombination fraction does not keep increasing in proportion to distance. Mapping functions account for this hidden multiple-crossing possibility in different ways.

The Haldane Mapping Function

Haldane’s function assumes that crossovers occur along the interval according to a Poisson process with a constant rate and no crossover interference. No interference means that a crossover does not alter the probability of another crossover nearby. For a genetic interval of $d$ Morgans, the number of crossovers in a meiosis is modeled as $N\sim\operatorname{Poisson}(d)$. Each crossover switches the transmitted chromosomal segment between the two homologous chromosomes. Thus, the gamete is recombinant at the endpoints if and only if the interval contains an odd number of crossovers.

Haldane mapping function: Under the Poisson, no-interference model, $$ r=\frac{1-e^{-2d}}{2}, \qquad d=-\frac{1}{2}\ln(1-2r). $$ Here $d$ is in Morgans and $r<0.5$. To report distance in centimorgans, multiply $d$ by 100.

The forward formula gives the recombination fraction implied by a distance; the inverse formula gives the distance implied by a recombination fraction. The inverse is defined for $r<0.5$. At $r=0.5$, the inferred distance tends to infinity, expressing saturation rather than providing a useful finite estimate.

Worked Example: Haldane Distance From a Recombination Fraction

Suppose a hypothetical two-point analysis gives an estimated recombination fraction of $r=0.12$. Find the Haldane distance.

The Haldane inverse function is $d=-\frac{1}{2}\ln(1-2r)$. Substituting the estimate, $$ d=-\frac{1}{2}\ln(1-2(0.12)) =-\frac{1}{2}\ln(0.76) =0.1372\ \text{M}. $$

Since one Morgan is 100 cM, the distance is $0.1372(100)=13.72$ cM. This is the distance under the no-interference Poisson model; it is not a model-free restatement of the observed 12% recombination fraction.

Why the Haldane Formula Counts Odd Crossovers

For $N\sim\operatorname{Poisson}(d)$, the probability of an odd number of crossovers can be obtained by separating the even and odd probabilities. The probability generating function of a Poisson random variable gives $E[(-1)^N]=e^{-2d}$. Since $(-1)^N$ equals 1 for even $N$ and $-1$ for odd $N$, this expectation is the probability of an even count minus the probability of an odd count. If those probabilities are $P_{\mathrm{even}}$ and $P_{\mathrm{odd}}$, then

P_{\mathrm{even}}+P_{\mathrm{odd}}=1, \qquad P_{\mathrm{even}}-P_{\mathrm{odd}}=e^{-2d}.

Subtracting the second equality from the first gives $2P_{\mathrm{odd}}=1-e^{-2d}$, and hence $P_{\mathrm{odd}}=(1-e^{-2d})/2$. Because endpoint recombination occurs exactly when the crossover count is odd under this model, this is the Haldane mapping function.

The Kosambi Mapping Function

Kosambi’s mapping function gives a different relationship between distance and recombination fraction. It is commonly used as a model that allows for positive crossover interference: crossovers tend to be spaced farther apart than they would be under Haldane’s independent Poisson model. The function is a model for converting between the quantities, not a universal description of crossover biology. Its assumptions may be more or less suitable depending on the organism, chromosome region, and analysis.

One mathematical characterization of the Kosambi relationship is the differential mapping relation $dr/dd=1-4r^2$, with $r=0$ at $d=0$. Integrating this relation gives the distance as a function of recombination fraction:

d=\int_0^r \frac{du}{1-4u^2} =\frac{1}{4}\ln\left(\frac{1+2r}{1-2r}\right).

Solving this expression for $r$ gives the forward form. Exponentiating yields $e^{4d}=(1+2r)/(1-2r)$; rearranging gives $r=(e^{4d}-1)/(2(e^{4d}+1))$, which is equivalent to the hyperbolic-tangent expression below.

Kosambi mapping function: $$ r=\frac{1}{2}\tanh(2d), \qquad d=\frac{1}{4}\ln\left(\frac{1+2r}{1-2r}\right). $$ As with Haldane, $d$ is in Morgans, the distance in centimorgans is $100d$, and the inverse requires $r<0.5$.

Worked Example: Kosambi Distance From the Same Estimate

Use the hypothetical estimate $r=0.12$ again, this time under the Kosambi model.

Substitute into the inverse function: $$ d=\frac{1}{4}\ln\left(\frac{1+2(0.12)}{1-2(0.12)}\right) =\frac{1}{4}\ln\left(\frac{1.24}{0.76}\right) =\frac{1}{4}\ln(1.6316) =0.1224\ \text{M}. $$

The corresponding distance is $0.1224(100)=12.24$ cM. For this recombination fraction, the Kosambi distance is shorter than the Haldane distance of 13.72 cM. The difference follows from the distinct mapping assumptions, not from any change in the hypothetical linkage data.

Converting Distance to Recombination Fraction

The same functions can be used in the forward direction. For a specified map interval, Haldane predicts a recombination fraction by accounting for the parity of Poisson crossovers. Kosambi predicts a recombination fraction using its interference-based mapping relationship. Keep the units straight: the formulas take distance in Morgans, so a distance reported in centimorgans must first be divided by 100.

Worked Example: Comparing Predictions for a 15 cM Interval

Consider a hypothetical map interval of 15 cM, or $d=0.15$ M. Calculate the recombination fraction under each function.

Under Haldane, $$ r=\frac{1-e^{-2(0.15)}}{2} =\frac{1-e^{-0.30}}{2} =\frac{1-0.7408}{2} =0.1296. $$

Under Kosambi, $$ r=\frac{1}{2}\tanh(2(0.15)) =\frac{1}{2}\tanh(0.30) =\frac{1}{2}(0.2913) =0.1457. $$

Thus, for a 15 cM interval, Haldane predicts about 13.0% recombinant gametes and Kosambi about 14.6%. These are model predictions, not observed proportions. The interval distance is the same in both calculations, but the assumed crossover patterns differ.

What the Functions Imply—and Where They Differ

For short intervals, both functions give approximately $r=d$ when $d$ is measured in Morgans. For example, a distance of $0.01$ M corresponds to approximately a 1% recombination fraction under either function. The agreement is close because multiple crossovers are relatively unlikely in a short interval. As intervals grow, the functions diverge: multiple crossovers become more consequential, and the assumptions used to model them matter more.

For a given $d>0$, the Kosambi forward function gives a larger recombination fraction than Haldane’s function. Conversely, for a given $r$ between 0 and $0.5$, Kosambi gives a smaller distance. This comparison describes the formulas, not a universal biological rule that one model must always be correct.

Feature	Haldane	Kosambi
Core crossover assumption	Poisson crossovers with no interference	Mapping relationship commonly used to allow positive interference
Distance from $r$	$-\frac{1}{2}\ln(1-2r)$	$\frac{1}{4}\ln\left(\frac{1+2r}{1-2r}\right)$
Recombination fraction from $d$	$\frac{1-e^{-2d}}{2}$	$\frac{1}{2}\tanh(2d)$
Behavior near $r=0.5$	Implied distance grows without bound	Implied distance grows without bound

Map distances are coordinates along a genetic map, whereas recombination fractions are probabilities for endpoints. Do not add recombination fractions to obtain a longer interval: an odd number of crossovers in one subinterval can be offset by an odd number in another. Distances along a map are additive by coordinate, but the recombination fraction for the full interval must be interpreted under an appropriate mapping model. In particular, the simple independent-interval reasoning associated with Haldane should not be silently imposed when using a model that represents interference.

Worked Example: Choosing a Conversion for a Short Interval

Suppose a hypothetical pedigree analysis estimates $r=0.02$ between two closely spaced markers. Calculate both distances and explain how a model choice affects the report.

For Haldane, $$ d=-\frac{1}{2}\ln(1-2(0.02)) =-\frac{1}{2}\ln(0.96) =0.02041\ \text{M} =2.041\ \text{cM}. $$

For Kosambi, $$ d=\frac{1}{4}\ln\left(\frac{1+2(0.02)}{1-2(0.02)}\right) =\frac{1}{4}\ln\left(\frac{1.04}{0.96}\right) =0.02001\ \text{M} =2.001\ \text{cM}. $$

The distances are close for this small recombination fraction. If the analysis specifies a no-interference Poisson model, report the Haldane conversion; if it specifies the Kosambi mapping relationship, report the Kosambi conversion. The small difference does not justify switching models without stating the change in assumption.

Assumptions and Interpretation in Linkage Analysis

A mapping function is most useful when its assumptions are made explicit. Haldane is appropriate when the intended model treats crossovers as occurring independently along the interval at a constant rate. Kosambi is often selected when positive crossover interference is considered a reasonable approximation. Neither conversion removes uncertainty in the estimated recombination fraction, and neither makes map distance directly observed. A calculated distance inherits uncertainty from the data and dependence on the chosen mapping model.

Both functions approach $r=0.5$ as $d$ becomes large. Consequently, a two-point recombination fraction near $0.5$ contains little information for distinguishing a very long linked interval from unlinked loci. The inverse formulas become increasingly sensitive as $r$ approaches $0.5$, because their denominators $1-2r$ approach zero. A finite estimate close to that boundary can therefore imply a very large, unstable distance. It should not be read as a precise map location.

Key takeaway: Use the Haldane function when a Poisson crossover model with no interference is intended, and the Kosambi function when its interference-based mapping relationship is intended. State the mapping function and the distance units whenever converting a recombination fraction into map distance.

Common Mistakes

Treating $r$ as map distance. An estimate such as $r=0.12$ is a probability, not 12 cM. Apply a specified mapping function before reporting distance.
Using centimorgans as Morgans in the formula. Convert 15 cM to $0.15$ M before substituting for $d$; otherwise the calculation is off by a factor of 100.
Assuming the conversion is model-free. Haldane and Kosambi make different assumptions, so a converted distance should be associated with the function used.
Adding recombination fractions across intervals. Endpoint recombination depends on crossover parity, not the sum of interval probabilities. Use a map distance and a mapping model for the full interval instead.
Interpreting $r$ near 0.5 as a precise long distance. Recombination fractions saturate at 0.5, so two-point evidence cannot precisely resolve increasingly large distances near this limit.

Check Your Understanding

Work through these questions using the formulas and assumptions above.

What does the Haldane no-interference assumption mean for the occurrence of crossovers along a chromosome interval?
Under Haldane, what event makes a gamete recombinant at the two endpoints: an even or an odd number of crossovers?
For a hypothetical estimate $r=0.08$, write the expression for the Haldane distance in Morgans before evaluating it.
A map interval is 8 cM. What unit conversion must be made before using either forward mapping function?
Why does a recombination fraction close to 0.5 provide little precision about a large genetic distance?
For the same $r$ below 0.5, which function gives the shorter distance, Haldane or Kosambi, and why must the model still be stated?