Meiosis recombination fractions and genetic distance Statistics 246 Spring 2004 Lecture 2A January 22 Initially pages 1 11 Later pages 12 18 1 the process which starts with a diploid cell having one set of maternal and one of paternal The action of interest to us chromosomes and ends up happens around here with four haploid cells each Chromosomes replicate but of which hasata their single set of stay joined centromeres chromosomes Bivalents form these being mosaics ofappear the parental ones Chiasmata Bivalents separate by attachment of centromeres to spindles Source http www accessexcellence org 2 Four strand bundle and exchanges one chromosome arm depicted sister chromatids sister chromatids 2 parental chromosomes Two exchanges 4 strand bundle bivalent 4 meiotic products 3 Chance aspects of meiosis Number of exchanges along the 4 strand bundle Positions of the exchanges Strands involved in the exchanges Spindle centromere attachment at the 1st meiotic division Spindle centromere attachment at the 2nd meiotic division Sampling of meiotic products Deviations from randomness called interference 4 A stochastic model for meiosis A point process X for exchanges along the 4 strand bundle A model for determining strand involvement in exchanges A model for determining the outcomes of spindlecentromere attachments at both meiotic divisions A sampling model for meiotic products Random at all stages defines the no interference or Poisson model 5 6 A model for strand involvement The standard random assumption here is No Chromatid Interference NCI each non sister pair of chromatids is equally likely to be involved in each exchange independently of the strands involved in other exchanges NCI fits the available data pretty well but there are broader models 7 The crossover process on meiotic products 1 change 2 changes 1 change no change Changes of grand parental origin along meiotic products are called crossovers They form the crossover point process C along the single chromosomes Under NCI C is a Bernoulli thinning of X with p 0 5 that is each exchange has a probability of 1 2 of involving a given chromatid independently of the involvement of other 8 exchanges From exchanges to crossovers Usually we can t observe exchanges but on suitably marked chromosomes we can track crossovers Call a meiotic product recombinant across an interval J and write R J if the grand parental origins of its endpoints differ i e if an odd number of crossovers have occurred along J Assays exist for determining whether this is so We usually write pr R J r and call r the recombination fraction Recombination across the interval No recombination Recombination No recombination 9 Counting recombinants R and non recombinants NR across the interval AB 4 NR 2R 2NR 4NR 2R 2NR 2R 2NR 4R 10 Mather s formula Under NCI if n 0 pr R J X J n 1 2 Proof Suppose that n 0 Consider a particular chromatid It has a probability of 1 2 of being involved in any given exchange and its involvement in any of the n separate exchanges are independent events Thus the chance that it is involved in an odd number of exchanges is the sum over all odd k of the binomial probabilities b k n 1 2 which equals 1 2 check Corollary Mather pr R J 1 2 pr X J 0 It follows that under NCI the recombination fraction r pr R J 11 is monotone increasing in the size of J and 1 2 The Poisson model Suppose that the exchange process X is a Poisson process i e that the numbers of exchanges in any pairwise disjoint set of intervals are mutually independent Poisson random variables Denoting the mean number of exchanges in interval J by J we can make a monotone change of the chromosome length scale to convert this mean to J where J is the length of J This foreshadows the important notion of genetic or map distance where rate length Exercise Prove that if X is a Poisson process so is the crossover process C 12 Recombination and mapping Sturtevant 1913 first used recombination fractions to order i e map genes Problem the recombination fraction does not define a metric Let s consider 3 loci denoted by 1 2 and 3 and put rij pr R i j 1 r12 2 r23 3 r13 In general r13 r12 r23 13 Triangle inequality We will prove that under NCI r13 r12 r23 To see this define p00 pr R 1 2 R 2 3 p01 pr R 1 2 R 2 3 p10 pr R 1 2 R 2 3 p11 pr R 1 2 R 2 3 where the denotes the complement negation of the event Now notice that R 1 2 R 2 3 R 1 2 R 2 3 R 1 2 R 1 2 R 2 3 R 1 2 R 2 3 R 2 3 and R 1 2 R 2 3 R 1 2 R 2 3 R 1 3 think about this one Thus we have p10 p11 r12 p01 p11 r23 and p00 p11 1 r13 Adding the three equations and using the fact that the pij sum to 1 gives r12 r23 r13 2p11 0 In general this inequality is strict Under the Poisson model p11 r12r23 14 Map distance and mapping Map distance d12 E C 1 2 av COs in 1 2 Unit Morgan or centiMorgan 1 d12 2 d23 3 d13 d13 d12 d23 Genetic mapping or applied meiosis a BIG business Placing genes and other markers along chromosomes Ordering them in relation to one another 15 Assigning map distances to pairs and then globally Haldane s map function Suppose that X is a Poisson process and that the map length of an interval J is d Then the mean number J of exchanges across J is 2d and by Mather the recombination fraction across J is 1 2d r 1 e 2 More generally map functions relate recombination 16 fraction to genetic distance r d for r small The program from now on With these preliminaries we turn now to the data and models in the literature which throw light on the chance aspects of meiosis Mendel s law of segregation a result of random sampling of meiotic products with allele variant pairs generally segregating in precisely equal numbers As usual in biology there are exceptions 17 18 Random spindle centromere attachment at 1st meiotic division x larger smaller In 300 meioses in an grasshopper heterozygous for an inequality in the size of one of its chromosomes the smaller of the two chromosomes moved with the single X 146 times while the larger did so 154 times Carothers 1913 19 Tetrads In some organisms fungi molds yeasts all four products of an individual meiosis can be recovered together in what is known as an ascus These are called tetrads The four ascospores can be typed individually In some cases e g N crassa the red bread mold there has been one further mitotic division but the resulting octads are ordered 20 21 Using ordered tetrads to study meiosis Data from ordered tetrads tell us a lot about meiosis For example we can …
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