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In [[genetics]], '''mapping functions''' are used to model the relationship between [[Gene mapping|map]] distancedistances (measured in map units or [[Centimorgan|centimorganscentimorgan]]s) between [[Genetic marker|markers]] and [[Genetic recombination|recombination]] frequencyfrequencies, particularly as these measurements relate to regions encompassed between markers[[genetic marker]]s. One utility of this approach is that it allows valuesone to beobtain obtainedvalues for genetic distances, whichin isgenetic typicallymapping notunits estimable,directly from recombination fractions, whichas map distances cannot typically arebe obtained from empirical experiments.<ref>{{Cite book |last1=Broman |first1=Karl W. |url=https://www.worldcat.org/title/669122118 |title=A guide to QTL mapping with R/qtl |last2=Sen |first2=Saunak |date=2009 |publisher=Springer |isbn=978-0-387-92124-2 |series=Statistics for biology and health |___location=Dordrecht |pages=14 |oclc=669122118}}</ref>
The simplest mapping function wasis the '''Morgan Mapping Function''', eponymously devised by [[Thomas Hunt Morgan]]. Other well-known mapping functions include the '''Haldane Mapping Function''' introduced by [[J. B. S. Haldane]] in 1919,<ref>{{Cite journal |last=Haldane |first=J.B.S. |date=1919 |title=The combination of linkage values, and the calculation of distances between the loci of linked factors |url=https://www.ias.ac.in/article/fulltext/jgen/008/04/0299-0309 |journal=Journal of Genetics |volume=8 |issue=29 |pages=299–309}}</ref> and the '''Kosambi Mapping Function''' introduced by [[Damodar Dharmananda Kosambi]] in 1944.<ref>{{Cite journal |last=Kosambi |first=D. D. |date=1943 |title=The Estimation of Map Distances from Recombination Values |url=https://onlinelibrary.wiley.com/doi/10.1111/j.1469-1809.1943.tb02321.x |journal=Annals of Eugenics |language=en |volume=12 |issue=1 |pages=172–175 |doi=10.1111/j.1469-1809.1943.tb02321.x |issn=2050-1420|url-access=subscription }}</ref><ref name=":0">{{Cite book |last1=Wu |first1=Rongling |url=https://books.google.com/books?id=-NlGKOEQuEsC&dq=haldane%20mapping%20function&pg=PA65 |title=Statistical genetics of quantitative traits: linkage, maps, and QTL |last2=Ma |first2=Chang-Xing |last3=Casella |first3=George |date=2007 |publisher=Springer |isbn=978-0-387-20334-8 |___location=New York |pages=65 |oclc=141385359}}</ref> Few mapping functions are used in practice other than Haldane and Kosambi.<ref name=":1" /> The main difference between them is in how [[crossover interference]] is incorporated.<ref name=":3">{{Cite journal |last1=Peñalba |first1=Joshua V. |last2=Wolf |first2=Jochen B. W. |date=2020 |title=From molecules to populations: appreciating and estimating recombination rate variation |url=https://www.nature.com/articles/s41576-020-0240-1 |journal=Nature Reviews Genetics |language=en |volume=21 |issue=8 |pages=476–492 |doi=10.1038/s41576-020-0240-1 |issn=1471-0064|url-access=subscription }}</ref>
== Morgan Mapping Function ==
=== Overview ===
Two properties of the Haldane Mapping Function is that it limits recombination frequency up to, but not beyond 50%, and that it represents a linear relationship between the frequency of recombination and map distance up to recombination frequencies of 10%.<ref>{{Cite web |title=mapping function |url=https://www.oxfordreference.com/display/10.1093/oi/authority.20110803100132641 |access-date=2024-04-29 |website=Oxford Reference |language=en |doi=10.1093/oi/authority.20110803100132641?rskey=srzx3w&result=6|doi-broken-date=2024-04-30 }}</ref> It also assumes that crossovers occur at random positions and that they do so independent of one another. This assumption therefore also assumes no [[crossover interference]] takes place;<ref name=":1">{{Cite book |title=Mammalian genomics |date=2005 |publisher=CABI Pub |isbn=978-0-85199-910-4 |editor-last=Ruvinsky |editor-first=Anatoly |___location=Wallingford, Oxfordshire, UK ; Cambridge, MA, USA |pages=15 |editor-last2=Graves |editor-first2=Jennifer A. Marshall}}</ref>; but using this assumption allows Haldane to model the mapping function using a [[Poisson distribution]].<ref name=":0" />
=== Definitions ===
=== Formula ===
Based on the definitions, [[Mather's formula]] can be used to derive the Haldane mapping function as:
<math>\ r = \frac{1}{2} (1-e^{-2d})</math>
=== Inverse ===
<math>\ d = -\frac{1}{2} \ln (1-2r)</math>
== Kosambi Mapping Function ==
=== Formula ===
<math>\ r = \frac{1}{2} \tanh (2d) = \frac{1}{2}\frac{e^{4d}-1}{e^{4d}+1}</math>
=== Inverse ===
<math>\ d = \frac{1}{2} \tanh^{-1} (2r) = \frac{1}{4} \ln(\frac{1+2r}{1-2r})</math>
== Comparison and application ==
Below 10% recombination frequency, there is little mathematical difference between different mapping functions and the relationship between map distance and recombination frequency is linear (that is, 1 map unit = 1% recombination frequency).<ref name=":2" /> WhileWhen many mapping functions now exist,<ref>{{Cite journal |last=Crow |first=J F |date=1990 |title=Mapping functions. |url=https://academic.oup.com/genetics/article/125/4/669/6000769 |journal=Genetics |language=en |volume=125 |issue=4 |pages=669–671 |doi=10.1093/genetics/125.4.669 |issn=1943genome-2631wide |pmc=1204092SNP |pmid=2204577}}</ref> in practice functions other than Haldanesampling and Kosambimapping aredata rarelyis used.<ref name=":1" /> More specificallypresent, Thethe Haldane function is preferred when distancedifference between markers is relatively smaller, whereas the Kosambi functionfunctions is preferrednegligible whenoutside distancesof betweenregions markersof ishigh largerrecombination, andsuch crossoversas needrecombination tohotspots beor accountedends forof chromosomes.<ref>{{Cite book |urlname=https"://www.google.ca/books/edition/Handbook_of_Computational_Molecular_Biol/3Ss-ws2Zm6IC?hl=en&gbpv=1&pg=SA17-PA11&printsec=frontcover3" |title=Handbook of computational molecular biology |date=2006 |publisher=CRC Press |isbn=978-1-58488-406-4 |editor-last=Aluru |editor-first=Srinivas |series= |___location= |pages=17-10–17-11 |oclc=}}</ref>
While many mapping functions now exist,<ref>{{Cite journal |last=Crow |first=J F |date=1990 |title=Mapping functions. |url=https://academic.oup.com/genetics/article/125/4/669/6000769 |journal=Genetics |language=en |volume=125 |issue=4 |pages=669–671 |doi=10.1093/genetics/125.4.669 |issn=1943-2631 |pmc=1204092 |pmid=2204577}}</ref><ref>{{Cite journal |last=Felsenstein |first=Joseph |date=1979 |title=A Mathematically Tractable Family of Genetic Mapping Functions with Different Amounts of Interference |url=https://academic.oup.com/genetics/article/91/4/769/5993247 |journal=Genetics |language=en |volume=91 |issue=4 |pages=769–775 |doi=10.1093/genetics/91.4.769 |issn= |pmc=1216865 |pmid=17248911}}</ref><ref>{{Cite journal |last1=Pascoe |first1=L. |last2=Morton |first2=N.E. |date=1987 |title=The use of map functions in multipoint mapping |journal=American Journal of Human Genetics |volume=40 |issue=2 |pages=174–183|pmid=3565379 |pmc=1684067 }}</ref> in practice functions other than Haldane and Kosambi are rarely used.<ref name=":1" /> More specifically, the Haldane function is preferred when distance between markers is relatively small, whereas the Kosambi function is preferred when distances between markers is larger and crossovers need to be accounted for.<ref>{{Cite book |url=https://books.google.com/books?id=3Ss-ws2Zm6IC&pg=SA17-PA11 |title=Handbook of computational molecular biology |date=2006 |publisher=CRC Press |isbn=978-1-58488-406-4 |editor-last=Aluru |editor-first=Srinivas |series= |___location= |pages=17-10–17-11 |oclc=}}</ref>
== References ==
{{Reflist}}
== Further reading ==
* Bailey, N.T.J., 1961 ''Introduction to the Mathematical Theory of Genetic Linkage''. Clarendon Press, Oxford.
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