Predicting phenotype transition probabilities via conditional algorithmic probability approximations

Unravelling the structure of genotype-phenotype (GP) maps is an important problem in biology. Recently, arguments inspired by algorithmic information theory (AIT) and Kolmogorov complexity have been invoked to uncover simplicity bias in GP maps, an exponentially decaying upper bound in phenotype probability with increasing phenotype descriptional complexity. This means that phenotypes with very many genotypes assigned via the GP map must be simple, while complex phenotypes must have few genotypes assigned. Here we use similar arguments to bound the probability P(x → y) that phenotype x, upon random genetic mutation, transitions to phenotype y. The bound is P(x → y)≲ 2 −aK˜(y|x)−b , where K˜(y|x) is the estimated conditional complexity of y given x, quantifying how much extra information is required to make y given access to x. This upper bound is related to the conditional form of algorithmic probability from AIT. We demonstrate the practical applicability of our derived bound by predicting phenotype transition probabilities (and other related quantities) in simulations of RNA and protein secondary structures. Our work contributes to a general mathematical understanding of GP maps, and may also facilitate the prediction of transition probabilities directly from examining phenotype themselves, without utilising detailed knowledge of the GP map.