Speaker
Description
We consider the system of differential equations describing genetic recombination, which is well known to be equivalent to the law of mass action of a chemical reaction network \cite{Mueller_Hofbauer_15,Alberti_21}. It is equally well known that this nonlinear system can be solved via its dual process backward in time (see \cite{Baake_Baake_21} for a review). This dual process is a (linear) Markov chain in continuous time, which describes how an individual's genes are partitioned across its ancestors when looking back into the past.
We find an explicit representation of the semigroup of the partitioning process, and thus an explicit solution of the recombination equation, both for the simple case of single crossovers and for general recombination distributions. The latter works by representing the realisations of the partitioning process as trees and establishing an inclusion-exclusion principle for the decomposition of these trees into subtrees.
Based on the semigroup, we attack the embedding problem for recombination in discrete time, where the partitioning is described by a discrete-time Markov chain.
The embedding problem of Markov transition matrices into Markov semigroups is a classic problem that goes back to Kingman \cite{Kingman_62}; the question is whether a discrete-time Markov chain may be represented as the semigroup of a continuous-time Markov process. We solve the problem for short gene sequences and give an outlook to the general case.
Bibliography
@Article{Mueller_Hofbauer_15,
author = {M\"uller, S. and Hofbauer, J.},
title = {Genetic recombination as a chemical reaction network},
journal = {Math. Model. Nat. Phenom.},
year = {2015},
volume = {10},
pages = {84--99}
}
@Article{Alberti_21,
author = {Alberti, F.},
title = {Genetic recombination as a generalised gradient flow},
journal = {Monatsh. Math.},
year = {2021},
volume = {196},
pages = {645--663}
}
@InCollection{Baake_Baake_21,
author = {Baake, E. and Baake, M.},
title = {Ancestral lines under recombination},
booktitle = {Probabilistic Structures in Evolution},
publisher = {EMS Press},
year = {2021},
editor = {Baake, E. and Wakolbinger, A.},
pages = {365--382},
address = {Berlin}
}
@Article{Kingman_62,
author = {Kingman,J.~F.~C.},
title = {The imbedding problem for finite {M}arkov chains},
journal = {Z. Wahrscheinlichkeitsth. verw. Geb.},
year = {1962},
volume = {1},
pages = {14--24}
}
