Statistical physics methods provide the exact solution to a long-standing problem of genetics
Areejit Samal and Olivier Martin
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Submission date: 24. Jul. 2014 (revised version: June 2015)
published in: Physical review letters, 114 (2014) 23, art-no. 238101
DOI number (of the published article): 10.1103/PhysRevLett.114.238101
PACS-Numbers: 02.50.Cw, 05.40.-a, 02.50.Sk
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Analytic and computational methods developed within statistical physics have found applications in numerous disciplines. In this letter, we use such methods to solve a long-standing problem in statistical genetics. The problem, posed by Haldane and Waddington [J.B.S. Haldane and C.H. Waddington, Genetics 16, 357-374 (1931)], concerns so-called recombinant inbred lines (RILs) produced by repeated inbreeding. Haldane and Waddington derived the probabilities of RILs when considering 2 and 3 genes but the case of 4 or more genes has remained elusive. Our solution uses two probabilistic frameworks relatively unknown outside of physics: Glauber's formula and self-consistent equations of the Schwinger-Dyson type. Surprisingly, this combination of statistical formalisms unveils the exact probabilities of RILs for any number of genes. Extensions of the framework may have applications in population genetics and beyond.