MeiosisStat 246, Lecture I, Jan 22, 2002Source:http://www.accessexcellence.org- the process which starts witha diploid cell having one set ofmaternal and one of paternalchromosomes, and ends upwith four haploid cells, each ofwhich has a single set ofchromosomes, these beingmosaics of the parental onesSource:http://www.accessexcellence.orgThe action of interest to ushappens around here :•Chromosomes replicate, butstay joined at their centromeres•Bivalents form•Chiasmata appear•Bivalents separate by attachmentof centromeres to spindles.Four-strand bundle and exchanges(one chromosome arm depicted)sisterchromatidssisterchromatids4-strand bundle (bivalent)Two exchanges4 meiotic products2 parental chromosomesChance aspects of meiosisNumber of exchanges along the 4-strand bundlePositions of the exchangesStrands involved in the exchangesSpindle-centromere attachment at the 1st meioticdivisionSpindle-centromere attachment at the 2nd meioticdivisionSampling of meiotic products Deviations from randomness called interference.A stochastic model for meiosisA point process X for exchanges along the 4-strandbundleA model for determining strand involvement inexchangesA model for determining the outcomes of spindle-centromere attachments at both meiotic divisionsA sampling model for meiotic products Random at all stages defines the no-interferenceor 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 tobe involved in each exchange, independently of thestrands involved in other exchanges. NCI fits pretty well, but there are broader models. Changes of parental origin along meiotic products arecalled crossovers. They form the crossover pointprocess 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 suitablymarked chromosomes we can track crossovers. Call a meiotic product recombinant across an intervalJ, and write R(J), if the parental origins of its endpointsdiffer, i.e. if an odd number of crossovers haveoccurred along J. Assays exist for determiningwhether this is so.Under NCI we find that if n>0, pr(R(J) | X(J) = n ) = 1/2,and so pr(R(J)) = 1/2 × pr( X(J) > 0 )………(*) (Proof?)1 2 3r12r23r13r13 ≠ r12 + r23 Recombination and mapping The recombination fraction pr(R(J)) gives anindication of the chromosomal length of theinterval 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 notdefine a metric. Put rij = pr(R(i--j)).Map distance: d12 = E{C(1--2)} = av # COs in 1--2 Unit: Morgan, or centiMorgan.1 2 3d12 d23 d13d13 = d12 + d23 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.Map distance and mappingThe program from now on With these preliminaries, we turn now to thedata and models in the literature which throwlight on the chance aspects of meiosis. Mendel’s law of segregation: a result ofrandom sampling of meiotic products, withallele (variant) pairs generally segregating inprecisely equal numbers. As usual in biology, there are exceptions.Random spindle-centromere attachment at 1st meiotic divisionIn 300 meioses in angrasshopper heterozygousfor an inequality in the size ofone of its chromosomes,the smaller of the twochromosomes moved withthe single X 146 times, whilethe larger did so 154 times.Carothers, 1913.xsmallerlargerTetrads In some organisms - fungi, molds, yeasts - allfour products of an individual meiosis can berecovered together in what is known as anascus. These are called tetrads. The fourascospores can be typed individually. In some cases - e.g. N. crassa, the red breadmold - there has been one further mitoticdivision, 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 and2nd division segregation. We first learned definitively that normal exchangesoccur at the 4-stand stage using data from N. crassa,and we can also see that random spindle-centromereattachment is the case for this organism. Finally, aberrant segregations can occasionally beobserved in octads.Meiosis in N.crassaFirst-division segregation patternsSecond-division segregation patternsDifferent 2nd division segregation patternsUnder random spindle-centromere attachment,all four patterns should be equally frequent.Lindegren’s 1932 N. crassa dataThere is a nice connexionbetween the frequenciesof multiple exchangesbetween a locus and itscentromere and thefrequency of 2nd divisionsegregations at that locus.2-strand double exchanges lead to FDSA simple calculation and result Let Fk (resp. Sk ) denote the number of strand-choice configurations for k exchanges leading tofirst (resp. second) division segregation at asegregating locus. By simple counting we find F0 =1 and So = 0, while for k>0, Fk+1 = 2Sk , and Sk+1 = 4Fk + 2Sk . Assuming NCI, the proportion sk of second-division segregants among meioses having kexchanges between our locus and thecentromere is€ € s ed= −−2313( ).€ € s kkk= − − >231120[ ( ) ], .€ s x x x= + + +1 2 31234..... If the distribution of the # of exchanges is (xk), then the frequency of SDSs is If the distribution is Poisson (2d) then we findThis 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 mapdistance (i.e. crossover interference is the norm)• Strand involvement generally random (i.e. chromatidinterference is rare)• Spindle-centromere attachment generally random(non-random attachments are quite rare)• The biological basis for crossover interferenceis only slowly becoming revealed; stay tuned.Testing and generalizing NCINCI implies inequality constraints on (multilocus)recombination probabilities which can be testedagainst statistical alternatives.We also have biological alternatives: models forstrand choice going beyond NCI.The best known is due to Weinstein (1938) whichpostulates a Markov
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