Meiosis Stat 246 Lecture I Jan 22 2002 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 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 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 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 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 spindle centromere attachments at both meiotic divisions A sampling model for meiotic products Random at all stages defines the no interference or Poisson model 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 pretty well but there are broader models Changes of 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 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 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 Under NCI and so we find that if n 0 pr R J X J n 1 2 pr R J 1 2 pr X J 0 Proof Recombination and mapping The recombination fraction pr R J gives an indication of the chromosomal length of the interval J under NCI it is monotone in J Sturtevant 1913 first used recombination fractions to order i e map genes How Problem the recombination fraction does not define a metric Put r ij pr R i j 1 r12 2 r23 r13 r13 r12 r23 3 Map distance and mapping Map distance d12 E C 1 2 av COs in 1 2 Unit Morgan or centiMorgan 1 d 12 d 23 2 d d 13 d 12 3 13 d 23 Genetic mapping or applied meiosis a BIG business Placing genes and other markers along chromosomes Ordering them in relation to one another Assigning map distances to pairs and then globally 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 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 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 Using ordered tetrads to study meiosis Data from ordered tetrads tell us a lot about meiosis For example we can see clear evidence of 1st and 2nd division segregation We first learned definitively that normal exchanges occur at the 4 stand stage using data from N crassa and we can also see that random spindle centromere attachment is the case for this organism Finally aberrant segregations can occasionally be observed in octads Meiosis in N crassa First division segregation patterns Second division segregation patterns Different 2nd division segregation patterns Under random spindle centromere attachment all four patterns should be equally frequent Lindegren s 1932 N crassa data 2 strand double exchanges lead to FDS There is a nice connexion between the frequencies of multiple exchanges between a locus and its centromere and the frequency of 2nd division segregations at that locus A simple calculation and result Let F k resp S k denote the number of strand choice configurations for k exchanges leading to first resp second division segregation at a segregating locus By simple counting we find F0 1 and So 0 while for k 0 Fk 1 2Sk and S k 1 4Fk 2Sk Assuming NCI the proportion s k of second division segregants among meioses having k exchanges between our locus and the centromere is s k 2 3 1 1 2 k k 0 If the distribution of the of exchanges is x then the frequency of SDSs is s x 1 1 2 x 2 3 4 x 3 k If the distribution is Poisson 2d then we find s 2 3 1 e 3 d This is a map function between the unobservable map distance d and the observable SDS frequency s Interference the state of play Total number of exchanges on an arm rarely Poisson Positions of exchanges rarely Poisson in map distance i e crossover interference is the norm Strand involvement generally random i e chromatid interference is rare Spindle centromere attachment generally random non random attachments are quite rare The biological basis for crossover interference is only slowly becoming revealed stay tuned Testing and generalizing NCI NCI implies inequality constraints on multilocus recombination probabilities which can be tested against statistical alternatives We also have biological alternatives models for strand choice going beyond NCI The best known is due to Weinstein 1938 which postulates a Markov model for the pairs of nonsister chromatids being involved in successive exchanges the cost is just two extra parameters There is not much evidence that it is needed The Poisson model implies independence of recombination across disjoint intervals 1 pr R 1 2
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