A Characterization of Crossover Models That Possess Map Functions

This paper concerns genetic map functions and a particular probability model for crossovers. S. Karlin and U. Liberman (1979, Adv. Appi. Prob. 11, 479-501) and N. Risch and K. Lange (1979, Biometrics39, 949-963) independently introduced what the former called the count-location (point) process model for crossovers, which leads to a probability model for multilocus recombination under the assumption of no chromatid interference. U. Liberman and S. Karlin (1984, Theor. Popul. Biol. 25, 331-346) later explored the constraints on genetic map functions resulting from the requirement that they be realisable in terms of a probability model for multilocus recombination. In this note we prove that under the assumption of no chromatid interference, the class of probability models for multilocus recombination that possess map functions is precisely the class of count-location processes. As a consequence, we give a complete analytic characterisation of the functions that can arise as map functions for some probability model of multilocus recombination under the assumption of no chromatid interference.