An informational transition in conditioned Markov chains: Applied to genetics and evolution.
In this work we assume that we have some knowledge about the state of a population at two known times, when the dynamics is governed by a Markov chain such as a Wright-Fisher model. Such knowledge could be obtained, for example, from observations made on ancient and contemporary DNA, or during laboratory experiments involving long term evolution. A natural assumption is that the behaviour of the population, between observations, is related to (or constrained by) what was actually observed. The present work shows that this assumption has limited validity. When the time interval between observations is larger than a characteristic value, which is a property of the population under consideration, there is a range of intermediate times where the behaviour of the population has reduced or no dependence on what was observed and an equilibrium-like distribution applies. Thus, for example, if the frequency of an allele is observed at two different times, then for a large enough time interval between observations, the population has reduced or no dependence on the two observed frequencies for a range of intermediate times. Given observations of a population at two times, we provide a general theoretical analysis of the behaviour of the population at all intermediate times, and determine an expression for the characteristic time interval, beyond which the observations do not constrain the population's behaviour over a range of intermediate times. The findings of this work relate to what can be meaningfully inferred about a population at intermediate times, given knowledge of terminal states.
Centre for Computational Systems Biology, Fudan University, 220 Handan Road, Shanghai 200433, PR China.
Lei
L
Zhao
Centre for Computational Systems Biology, Fudan University, 220 Handan Road, Shanghai 200433, PR China; Evolutionary Biology Center, Department of Ecology and Genetics, Uppsala University, Uppsala 75236, Sweden.
Martin
M
Lascoux
Centre for Computational Systems Biology, Fudan University, 220 Handan Road, Shanghai 200433, PR China. Electronic address: davidwaxman@fudan.edu.cn.
David
D
Waxman
D000483:Alleles; D005075:Biological Evolution; D003198:Computer Simulation; D008390:Markov Chains; D008957:Models, Genetic; D013997:Time Factors
Ancient DNA; Conditional distribution; Frequency trajectories; Population genetics theory; Random genetic drift