xoi (version 0.72)

location.given.one: Location of crossover given there is one

Description

Calculates the density of the location of the crossover on a random meiotic product, given that there is precisely one crossover, for the gamma model.

Usage

location.given.one(
  nu,
  L = 103,
  x = NULL,
  n = 400,
  max.conv = 25,
  integr.tol = 0.00000001,
  max.subd = 1000,
  min.subd = 10
)

Value

A data frame with two columns: x is the location (between 0 and L, in cM) at which the density was calculated and f is the density.

Arguments

nu

The interference parameter in the gamma model.

L

The length of the chromsome in cM.

x

If specified, points at which to calculate the density.

n

Number of points at which to calculate the density. The points will be evenly distributed between 0 and L. Ignored if x is specified.

max.conv

Maximum limit for summation in the convolutions to get inter-crossover distance distribution from the inter-chiasma distance distributions. This should be greater than the maximum number of chiasmata on the 4-strand bundle.

integr.tol

Tolerance for convergence of numerical integration.

max.subd

Maximum number of subdivisions in numerical integration.

min.subd

Minimum number of subdivisions in numerical integration.

Author

Karl W Broman,

Details

Let \(f(x;\nu)\) denote the density of a gamma random variable with parameters shape=\(\nu\) and rate=\(2\nu\), and let \(f_k(x;\nu)\) denote the density of a gamma random variable with parameters shape=\(k \nu\) and rate=\(2\nu\).

The distribution of the distance from one crossover to the next is \(f^*(x;\nu) = \sum_{k=1}^{\infty} f_k(x;\nu)/2^k\).

The distribution of the distance from the start of the chromosome to the first crossover is \(g^*(x;\nu) = 1 - F^*(x;\nu)\) where \(F^*\) is the cdf of \(f^*\).

We calculate the distribution of the location of the crossover on a product with a single crossover as the convolution of \(g^*\) with itself, and then rescaled to integrate to 1 on the interval (0,L).

References

Broman, K. W. and Weber, J. L. (2000) Characterization of human crossover interference. Am. J. Hum. Genet. 66, 1911--1926.

Broman, K. W., Rowe, L. B., Churchill, G. A. and Paigen, K. (2002) Crossover interference in the mouse. Genetics 160, 1123--1131.

McPeek, M. S. and Speed, T. P. (1995) Modeling interference in genetic recombination. Genetics 139, 1031--1044.

See Also

first.given.two(), distance.given.two(), joint.given.two(), ioden(), firstden(), xoprob(), gammacoi()

Examples

f1 <- location.given.one(1, L=200, n=201)
plot(f1, type="l", lwd=2, las=1,
     ylim=c(0,0.006), yaxs="i", xaxs="i", xlim=c(0,200))

f2 <- location.given.one(2.6, L=200, n=201)
lines(f2, col="blue", lwd=2)

f3 <- location.given.one(4.3, L=200, n=201)
lines(f3, col="red", lwd=2)

f4 <- location.given.one(7.6, L=200, n=201)
lines(f4, col="green", lwd=2)