1Computational GenomicsComputational GenomicsPopulation Genetics:Population Genetics:meiosis and recombinationmeiosis and recombinationEric XingEric XingLecture 2, January 18, 2007Reading:Source:http://www.accessexcellence.orgMeiosis is a process which starts with•a diploid cell having one set of maternal and one of paternal chromosomes,• and ends up with four haploidcells, each of which has a single set of chromosomes• these being mosaics of the parental ones 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.Meiosis2The problemsRecombination is a major source of genome diversityz How to study recombination?z How frequently they occur?z What phenotypical changes they can cause?z How do they imply on the evolutionary history of the genome? sisterchromatidssisterchromatids4-strand bundle (bivalent)Two exchanges4 meiotic products2 parental chromosomes12The simplified model: four-strand bundle and exchanges3z Number of exchanges along the 4-strand bundlez Positions of the exchanges z Strands involved in the exchangesz Spindle-centromere attachment at the 1st meiotic divisionz Spindle-centromere attachment at the 2nd meiotic divisionz Sampling of meiotic productsDeviations from randomness are called interference.Chance aspects of meiosisA stochastic model for meiosisz A point process X for exchanges along the 4-strand bundlez A model for determining strand involvement in exchangesz A model for determining the outcomes of spindle-centromereattachments at both meiotic divisionsz A sampling model for meiotic productsRandom at all stages defines the no-interference or Poissonmodel.4A model for strand involvementz The standard assumption here isNo Chromatid Interference (NCI):each non-sisterpair 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 crossoversz 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.z Mather's formular:Under NCI we find that if n>0, pr(R(J) | X(J) = n ) = 1/2, sopr(R(J)) = 1/2 ×pr( X(J) > 0 )………(*) (Proof?)5123r12r23r13r13≠ r12+ r23The 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)).Recombination and mappingz Map distance: d12= E{C(1--2)} = av # COs in 1--2z Unit: Morgan, or centiMorgan.z The expectation says nothing definitive about the relationship physicaldistance and genetic distancez Genetic mapping or applied meiosis: a BIG businessz Placing genes and other markers along chromosomes;z Ordering them in relation to one another;z Assigning map distances to pairs, and then globally.123d12d23d13d13= d12+ d23(how to prove this?)Map distance and mapping6Haldane's model:These crossovers occur as a Poisson process of rate d(per Morgan). Then the probability of observing a recombination:ρ(d) is an increasing function of d, ρ(d) → 1/2 as d →∞, and ρ(d) ≈ d as d → 0. )).2-exp(-1(21!)-(!21!)(0-odd -dkdkdekdedkkkdkkd=⎟⎟⎠⎞⎜⎜⎝⎛−==∑∑∞=ρGenetic linkageThe program from now onz With these preliminaries, we turn now to the data and models in the literature which throw light on the chance aspects of meiosis.z 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.7SegregationIn 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.Random spindle-centromereattachment at 1st meiotic division8Tetradsz 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.z In some cases - e.g. N. crassa, the red bread mold - there has been one further mitotic division, but the resulting octads are ordered.Meiosis in N.crassa9Using ordered tetrads to study meiosisz Data from ordered tetrads tell us a lot about meiosis. For example, we can see clear evidence of 1st and 2nd division segregation. z 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.z Finally, aberrant segregations can occasionally be observed in octads.First-division segregation patternsSeparating the chromosomesSeparating the chromotids10Second-division segregation patternsUnder random spindle-centromere attachment, all four patterns should be equally frequent.Different 2nd division segregation patterns11Lindegren’s 1932 N. crassa dataThere is a nice connection 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 FDS12k ex'sFk+1Sk+1: ? (homework)An inductive calculationz 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 findF0 =1 and So= 0, while for k>0, Fk+1 = 2Sk, and Sk+1 = 4Fk+ 2Sk.z Assuming NCI, the proportion Skof second-division segregants among meioses having kexchanges between our locus and the centromere isz If the probabilities of the # of exchanges is (xk), then the frequency of SDSs isz If the distribution is Poisson (2d) then we findz This is a map-function: between the unobservable map