xoi (version 0.72)

first.given.two: Location of first crossover given there are two

Description

Calculates the density of the location of the first crossover on a random meiotic product, given that there are precisely two crossovers, for the gamma model.

Usage

first.given.two(
  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.

Warning

We sometimes have difficulty with the numerical integrals. You may need to use large min.subd (e.g. 25) to get accurate results.

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 first crossover in a product with two crossovers by calculating the joint distribution of the location of the two crossovers, given that they both occur before L and the third occurs after L, and then integrating out the location of the second crossover.

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

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

Examples

f1 <- first.given.two(1, L=200, n=101)
plot(f1, type="l", lwd=2, las=1,
     ylim=c(0,0.011), yaxs="i", xaxs="i", xlim=c(0,200))

f2 <- first.given.two(2.6, L=200, n=101)
lines(f2, col="blue", lwd=2)

if (FALSE) {
f3 <- first.given.two(4.3, L=200, n=101)
lines(f3, col="red", lwd=2)

f4 <- first.given.two(7.6, L=200, n=101)
lines(f4, col="green", lwd=2)
}