The Simple Ratio

pg. 349 Statistical Genomics: Linkage, mapping and QTL Analysis. B.H. Liu.

"Marker coverage is the simple ratio between genomic map length and the total genome length. Map density is the average or the maximum map distance between adjacent markers."

The total genome length is the sum of the lengths of each chromosome in the genome. Suppose we have three non-homologous chromosomes with lengths of 100cM each. Then the total genome length is:
L₁ + L₂ + L₃ = 100 +100 + 100 = L = 300cM.

Suppose we want every QTL to be within a distance 2d from a marker. Let d = 10cM. If markers are evenly spaced on the genome then:

L/2d = number of markers = n

In our example,  L  = 300 = 15 = n.
In our example, 2d     20

We only need 15 markers to have a marker located within 20cM of each QTL.

However, markers are not evenly distributed across the genome, but markers are randomly distributed.

Suppose we want to know the probability that when markers are randomly distributed, at least one marker will be located within d cM with a certain probability of coverage of the genome.

Let c = marker coverage

Let c = genomic map length
Let c =   total map length

n = number of markers to cover genome within d cM
d = at least one marker on a segment of length d cM
L = total map length

The probability that the genome is not covered by the markers within d cM is:

1 - c = (1 - 2d)ⁿ

The marker coverage can be estimated using

c = 1 - e^2dn/L

n = -L ln (1-c)
n =     2d

Example:

L = 2000 cM or 20 M for soybean.
Let d = 10 cM, c = 95% coverage.

n = -2000ln(0.05) = 300
           20

We would need 300 markers to cover 95% of the length of the soybean genome such that a marker was within 10cM of each QTL.