For a heterozygous genome, there are two allelic configurations with the possibility of recombination between the homologs. The cross over resultant is dependent on the original arrangement of cis or trans arrangement. (Coupling or repulsion).

**Recombination frequency and crossover rates**

The frequency of cross over of linked genes is specific to each gene pair irrespective of the position of the allele whether cis or trans. The recombination frequencies were used to construct the first genetic map in 1913 by Morgan and Sturtevant. For a genetic map of 1cM (centi Morgan) or 1 map unit, the percentage of cross over between the alleles was found to be 1 %. If the genes are farther apart from each other, the cross over frequency was higher.

The cross over rate is used to construct the genetic map. This may not be the actual physical map since the crossover rates are not uniform across the chromosome. In genetic crosses, recombination frequency is a reliable indicator of the possible progenies, especially the recombinants. Genes with a recombination frequency of 50 % is considered to be linked.

Genes may be unlinked if they are on separate chromosomes or are placed far on the same chromosome. Multiple crossovers can result if the loci are located on the same chromosome far from each other. Genome mapping using the recombination frequency gives an accurate representation of physical positions of the loci.

**Detecting genetic linkage by test cross**

A test cross is another ideal method to know whether the genes are linked or not. Any deviation from the ratio of progenies as expected by the law of independent assortment is to be verified for linkage. A test cross is one with one of the parents being homozygous recessive. All the progeny exhibits the possible combinations of traits in equal ratio if the alleles are not linked and the other parent f the original cross is heterozygous. Any significant deviation from this denotes the possibility of linkage.

Approaches to test cross can include two point test crosses for double heterozygotes and three point testcrosses for analysis with three genes. These techniques allow simultaneous analysis of two or more traits.

**Special cases of test crosses**

Autosomal recessive: Aa/Bb (heterozygous)x aa/bb (homozygous recessive)

Autosomal dominant: Aa/Bb (heterzygous) x AA/BB (homozygous dominant)

X-linked recessive: female double heterozygote Aa/Bb x hemizygous male ab/Y

X-linked dominant: female double heterozygote Aa/Bb x hemizygous AB/Y

Determining linkage through chi square analysis.

This can be used to analyze the results of test cross to determine the significance of any variation. The null hypothesis specified that the genes are unlinked. In such a case the test cross ratio should be 1:1 for parental phenotype: recombinants.

χ2 = Σ (O-E)2 / E

Where O - Observed value

And E - Expected value

The resultant chi square value and the degrees of freedom is used to determine probability with which the variations in expected value can occur by random chance.

A 'p value' more than 0.05 means that more than 5 out of 100 observations show deviation by chance and the null hypothesis is accepted.

If the 'p value' is less than 0.5, the deviation is considered statistically significant and the null hypothesis is rejected. A 'p value' less than 0.01 is said to be highly significant.

**Coefficient and Interference**

Usually double crossovers are not found to occur at the expected rates as single crossovers. Crossover reduces the chances of formation of other chiasmata nearby thereby causing interference. An interference of 1 denotes that there can be no crossovers nearby and is also called total interference.

The extent of interference is dependent on the coefficient of coincidence. They are inversely related by the equation

Interference = 1 - coefficient of coincidence.

**Mapping functions**

Recombination frequencies are used to map distances in a genetic map. However, as the distance increases the accuracy of progeny analysis decreases. The recombinants are all considered to be results of single crossover. However, a single crossover and any odd number of crossovers will produce the same number of recombinant chromosomes. For double crossovers, the parental phenotype is restored and hence it will be unnoticed in the progeny even though they are actually recombinants.

Hence, with increasing distances over 7μ, the mapping is not accurate if done directly from recombination frequencies. Mapping functions aim to correct such errors in estimation of map distance.

Haldane's mapping function is given by

c = (1 - e-2m) /2.

Or m = -(1/2) ln(1-2c)

Where m is the expected number of crossovers, and c is the observed recombination frequency

This assumes that there is no interference between different loci. For very small map distances, m = c and for large distances c = ½

There are other mapping functions like Kosambi's which takes into account the number of double crossovers also.Corrections can be introduced to account for bias and systematic errors in calculation.

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