Statistics 246 Spring 2006 Meiosis and Recombination Week 3 Lecture 1 1 the process which starts with a diploid cell having one set of maternal and one of paternal chromosomes and ends up with four haploid cells each of which has a single set of chromosomes these being mosaics of the parental ones Source http www accessexcellence org 2 The action of interest to us happens around here Chromosomes replicate but stay joined at their centromeres Bivalents form Chiasmata appear Bivalents separate by attachment of centromeres to spindles Source http www accessexcellence org 3 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 4 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 5 A stochastic model for meiosis A stochastic point process X for exchanges along the 4 strand bundle A stochastic 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 6 Point process for exchanges We ll come to this in a moment going first to strand involvement in exchanges to derive an important result that is independent of the model for exchanges Usually we can t observe exchanges they occur inside meioses and we usually observe only single meiotic products but on suitably marked meiotic products chromosomes we can track crossovers see next slide These give us indirect information about the exchange process We can make inferences about exchanges when all the products of a single meiosis can be recovered together and we ll see cases of this 7 From exchanges to crossovers Changes of parental origin along meiotic products are called crossovers They form the crossover point process C along the single chromosomes A meiotic product is called recombinant across an interval J we ll denote the event by R J if the parental origins of its endpoints differ i e if an odd number of crossovers have occurred along J and parental otherwise Assays exist for determining whether this is so usually simply genotyping segregating markers at the endpoints of J 8 A model for strand involvement The standard 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 pretty well but of course not perfectly There are broader models though not in general use It is interesting to make up models with different strand involvement rules and explore their consequences Formally NCI says that the crossover process C is a Bernoulli thinning of the exchange process X with p 0 5 that is we copy an exchange point to a crossover point on a meiotic product with probability 1 2 independently for all exchanges 9 The relation between exchanges and crossovers Key to making inferences about unobserved exchanges from observable crossovers are the assumptions connecting them NCI is our principal one but before we look carefully at that let s do draw some pictures and get an idea of the combinatorics involved We ll begin by defining an interval J by segregating markers at its endpoints and looking at possible meiotic products Our interest is in whether a meiotic product is recombinant or parental across the interval and how often this occurs 10 Two exchanges 4 16 cases 11 Exercise With Emerson Rhodes The American Naturalist Vol 67 No 711 Jul Aug 1933 374 377 extend this table to exchanges with 3 4 or more exchanges in the interval and conclude that in all cases apart from 0 exchanges exactly one half of the meiotic products are parental and one half recombinant Develop a proof of their conclusion that under NCI for n 0 pr R J X J n 1 2 and so pr R J 1 2 pr X J 0 12 Recombination fractions The recombination fraction across an interval J is defined to be r pr R J Under NCI tells us that 0 r 1 2 In a sense it gives an indication of the chromosomal length of the interval J Exercise Prove that under NCI r r J is monotone in the length of J However the use of r as a metric must be limited as it is bounded by 1 2 13 The Poisson no interference model for exchanges Suppose that the exchange process X is a Poisson point process with mean measure That is for a chromosomal interval J the distribution of X J is Poisson with mean J and exchanges across disjoint intervals are mutually independent The crossover process C the thinned version of X is thus also a Poisson process with mean measure d 2 Further under NCI we have the formula 1 1 J 2d J r J 1 e 1 e 2 2 14 Recombination fractions and mapping 1 r12 2 r23 3 r13 r13 r12 r23 The recombination fraction does not define a metric However Sturtevant 1913 used recombination fractions to order genes Exercise Can we determine the order of three loci on a chromosome from their 3 pairwise recombination fractions Can you extend this to any number of loci along a chromosome 15 More on the Poisson model for exchanges Exercise Prove that for the Poisson model we have the following formula for loci in order 1 2 3 r13 r12 r23 2r12r23 Extend to 4 loci 1 2d r 1 e Relations such as 2 between observable recombination fractions and parameters d of unobservable exchange processes are called map functions This one is known as Haldane s after JBS Haldane who introduced it in 1919 16 Multilocus recombination probabilities Let s forget the Poisson model for a moment Suppose that we have 3 loci 1 2 3 along a chromosome Write Eij for the event of at least one exchange between loci i and j during meiosis and Rij for the event of recombination between loci i and j on a meiotic product Now define probs p and q as follows where denotes the complementary event q00 pr E12 E 23 p00 pr R12 R23 q01 pr E12 E 23 p01 pr R12 R23 q10 pr E12 E 23 q11 pr E12 E 23 p10 pr R12 R23 p11 pr R12 R23 17 Exercise Prove the following formulae under NCI and extend to n 3 loci 1 p11 q11 4 1 1 p10 q11 q10 4 2 1 1 p01 q11 q01 4 2 1 1 1 p00 q11 q10 q01 q00 4 2 2 Infer that we can order any number of loci under NCI from recombination
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