## Abstract

At meiosis, each member of a pair of homologous chromosomes replicates to form two **sister** chromosomes known as **chromatids**. The maternally and paternally derived sister pairs then perfectly align to form a bundle of four chromatids. Crossing-over occurs at points along the bundle known as **chiasmata**. At each **chiasma**, one sister chromatid from each pair is randomly selected and cut at the crossover point. The cell then rejoins the partial paternal chromatid above the cut to the partial maternal chromatid below the cut, and vice versa, to form two hybrid maternal-paternal chromatids. The preponderance of evidence suggests that the two chromatids participating in a chiasma are chosen nearly independently from chiasma to chiasma [30]. This independence property is termed lack of **chromatid interference**. After crossing-over has occurred, the recombined chromatids of a bundle are coordinately separated by two cell divisions so that each of the four resulting gametes receives exactly one chromatid.

## Keywords

Renewal Process Recombination Fraction Renewal Function Positive Interference Chiasma Interference## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Bailey NTJ (1961)
*Introduction to the Mathematical Theory of Genetic Linkage*. Oxford University Press, LondonMATHGoogle Scholar - [2]Baum L (1972) An inequality and associated maximization technique in statistical estimation for probabilistic functions of Markov processes.
*Inequalities*3:1–8Google Scholar - [3]Carter TC, Falconer DS (1951) Stocks for detecting linkage in the mouse and the theory of their design.
*J Genet*50:307–323CrossRefGoogle Scholar - [4]Carter TC, Robertson A (1952) A mathematical treatment of genetical recombination using a four-strand model.
*ProcRSocLondB*139:410–426.Google Scholar - [5]Cox DR, Isham V (1980)
*Point Processes*, Chapman and Hall, New YorkMATHGoogle Scholar - [6]Devijver PA (1985) Baum’s forward-backward algorithm revisited.
*Pattern Recognition Letters*3:369–373MATHCrossRefGoogle Scholar - [7]Feller W (1971)
*An Introduction to Probability Theory and its Applications, Vol*2,2nd ed. Wiley, New YorkGoogle Scholar - [8]Felsenstein J (1979) A mathematically tractable family of genetic mapping functions with different amounts of interference.
*Genetics*91:769–775MathSciNetGoogle Scholar - [9]Fisher, RA, Lyon MF, Owen ARG (1947) The sex chromosome in the house mouse.
*Heredity*1:335–365.Google Scholar - [10]Haidane JBS (1919) The combination of linkage values, and the calculation of distance between the loci of linked factors.
*J Genet*8:299–309Google Scholar - [11]Karlin S (1984) Theoretical aspects of genetic map functions in recombination processes. In
*Human Population Genetics: The Pittsburgh Symposium*, Chakravarti A, editor, Van Nostrand Reinhold, New York, pp 209–228Google Scholar - [12]Karlin S, Liberman U (1979) A natural class of multilocus recombination processes and related measures of crossover interference.
*Adv Appl Prob*11:479–501MATHCrossRefMathSciNetGoogle Scholar - [13]Karlin S, Liberman U (1983) Measuring interference in the chiasma renewal formation process.
*Adv Appl Prob*15:471–487MATHCrossRefGoogle Scholar - [14]Karlin S, Taylor HM (1975)
*A First Course in Stochastic Processes*, 2nd ed. Academic Press, New YorkMATHGoogle Scholar - [15]Kosambi DD (1944) The estimation of map distance from recombination values.
*Ann Eugen*12:172–175Google Scholar - [16]Lange K, Risch N (1977) Comments on lack of interference in the four-strand model of crossingover.
*J Math Biol*5:55–59CrossRefMathSciNetGoogle Scholar - [17]Lange K, Zhao H, Speed TP (1997) The Poisson-skip model of crossingover.
*Ann Appl Prob*(in press)Google Scholar - [18]Mather K (1938) Crossing-over.
*Biol Reviews Camb Phil Soc*13:252–292CrossRefGoogle Scholar - [19]Morgan TH, Bridges CB, Schultz J (1935) Constitution of the germinal material in relation to heredity.
*Carnegie Inst Washington Yearbook*34:284–291Google Scholar - [20]Owen ARG (1950) The theory of genetical recombination.
*Adv Genet*3:117–157CrossRefGoogle Scholar - [21]Payne LC (1956) The theory of genetical recombination: A general formulation for a certain class of intercept length distributions appropriate to the discussion of multiple linkage.
*Proc Roy Soc B*144:528–544.CrossRefGoogle Scholar - [22]Risch N, Lange K (1979) An alternative model of recombination and interference.
*Ann Hum Genet*43:61–70MATHCrossRefGoogle Scholar - [23]Risch N, Lange K (1983) Statistical analysis of multilocus recombination.
*Biometrics*39:949–963MATHCrossRefMathSciNetGoogle Scholar - [24]Ross SM (1983)
*Stochastic Processes*. Wiley, New YorkMATHGoogle Scholar - [25]Schnell FW (1961) Some general formulations of linkage effects in inbreeding.
*Genetics*46:947–957Google Scholar - [26]Speed TP (1996) What is a genetic map function? In
*Genetic Mapping and DNA Sequencing, IMA Vol 81 In Mathematics and its Applications*. Speed TP, Waterman MS, editors, Springer-Verlag, New York, pp 65–88CrossRefGoogle Scholar - [27]Stahl FW (1979)
*Genetic Recombination: Thinking about it in Phage and Fungi*. WH Freeman, San FranciscoGoogle Scholar - [28]Sturt E (1976) A mapping function for human chromosomes.
*Ann Hum Genet*40:147–163CrossRefGoogle Scholar - [29]Whitehouse HLK (1982)
*Genetic Recombination: Understanding the Mechanisms*. St. Martin’s Press, New YorkGoogle Scholar - [30]Zhao H, McPeek MS, Speed TP (1995) Statistical analysis of chromatid interference.
*Genetics*139:1057–1065Google Scholar - [31]Zhao H, Speed TP (1996) On genetic map functions.
*Genetics*142:1369–1377Google Scholar - [32]Zhao H, Speed TP, McPeek MS (1995) Statistical analysis of crossover interference using the chi-square model.
*Genetics*139:1045–1056.Google Scholar