MeiosisPowerPoint PresentationFour-strand bundle and exchanges (one chromosome arm depicted)Chance aspects of meiosisA stochastic model for meiosisA model for strand involvementSlide 7From exchanges to crossoversRecombination and mappingMap distance and mappingThe program from now onSlide 12Slide 13TetradsSlide 15Using ordered tetrads to study meiosisSlide 17Slide 18Slide 19Slide 20Slide 21Slide 22A simple calculation and resultSlide 24Interference: the state of playTesting and generalizing NCIThe Poisson model implies independence of recombination across disjoint intervalsMorgan’s D. melanogaster data (1935)A measure of crossover interferenceAn observation concerning crossover interferenceStochastic models for exchangesSlide 32Slide 33Evidence in support of the chi-squared model, ISlide 35Evidence in support of the chi-squared model, IISlide 37Slide 38Biological interpretation of the chi-squared or Cx(Co)m modelSlide 40Failure of the Cx(Co)m model with yeastVery brief summary of some current research on recombinationChallenges in the statistical study of meiosisAcknowledgementsReferencesWhere we have got to?MeiosisStat 246, Lecture I, Jan 22, 2002Source:http://www.accessexcellence.orgThe 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.- 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 onesFour-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 exchanges Strands involved in the exchangesSpindle-centromere attachment at the 1st meiotic divisionSpindle-centromere attachment at the 2nd meiotic divisionSampling of meiotic products Deviations from randomness called interference.A stochastic model for meiosisA point process X for exchanges along the 4-strand bundleA model for determining strand involvement in exchangesA 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-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 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 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 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 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 divisionIn 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.xsmallerlargerTetrads 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.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 connexion between the frequencies of multiple exchanges between a locus and its centromere and the frequency of 2nd division segregations 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 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 Sk+1 = 4Fk + 2Sk . Assuming NCI, the proportion sk of second-division segregants
View Full Document
Unlocking...