Formation of chiasmata.
Each is in resulting gamete with probability 1/2 (or 1/3)
Hence Mather's formula for recombination frequency
Poisson probabilities
Suppose number of chiasmata is Poisson (Haldane's model)
What is distribution of number of crossovers?
What is the recombination frequency?
Exponential distributions
What is the distribution of the interval between chiasmata?
What is the distribution of the interval between crossovers?
Uniform distributions
Binomial, geometric, and negative binomial probabilities
Given n chiasmata, how many crossovers?
How many chiasmata to the first that is a crossover. Multinomial data , in recombination outcomes
WHAT IS MEIOSIS?
Mendelian segregation, chromosomes,
chromatids, chiasmata
crossovers, gametes, recombinations
Genetic distance, scales the crossover process to be at rate 1
WHY A STOCHASTIC MODEL?
Continuous vs discrete outcomes:
The biophysical processes of meiosis could (in principal) be solved,
but the outcome, at a locus, is either the maternal or paternal gene.
Sensitivity to initial conditions: equations of motion of a coin flipped with given initial vertical (u) and angular (w) velocities (ignoring air resistance, coin irregularities etc etc.). For usual values of initial angular velocities (w > 200 rad/sec), high sensitivity to u/g.
Continuous vs discrete initial conditions:
Why can computers beat most of us at chess but not at dice?
The simplest model; Poisson process
Chiasmata occur independently and uniformly at random (rate 2)
So crossovers on a gamete occur independently and uniformly at random (rate 1).
Number of crossovers in distance d is Poisson, with mean d:
P(X = n) = dn exp(-d)/ n!
So P(recombination) = P(X odd) = (1/2) (1- exp(-2d)). (see section 6.1)
Note it is an increasing function of d.
INTERFERENCE; count, position, chomatid
COUNT: For well-regulated meiosis, must have at least one
chiasma in every chromasome pair (or chromatid tetrad);
Does NOT imply a recombination in every gamete.
Does imply every chromasome is at least 50 cM.
POSITION: form usually modelled -- presence of chiasma inhibits nearby chiasmata.
CHROMATID: May or may not exist. The chromatids involved in different chiasmata are not independent of each other. Suppose, for example, second chiasma always involved other two chromatids, but no other interference; so, the number of chiasmata C(d) is Poisson with mean 2d
Then, at distance d,
r(d) = sum_{n=4k+2} P(C(d)=n) + (1/2) sum_{n odd} P(C(d)=n)
If d = 1, so C(d) has mean 2, r(d) > 1/2.
Fisher (1948), with comments by Weinstein, and Lederberg.
Mather's formula
Suppose chromatid interference does not exist.
P(chiasma is crossover in resulting gamete) = 1/2,
independently for each chiasma.
Back to coin tossing:
P( odd number of heads in n tosses (n>0)) = 1/2
C(d) is number of chaismata in tetrad, in distance d
Z(d) is number of crossovers in gamete, in distance d
r(d) = P (Z(d) odd) = (1/2) P( C(d) > 0)
= (1/2) ( 1 - P( C(d) =0))