Correlation of Automorphism Group Size and Topological Properties with Program-size Complexity Evaluations of Graphs and Complex Networks

We show that numerical approximations of Kolmogorov complexity (K) applied to graph adjacency matrices capture some group-theoretic and topological properties of graphs and empirical networks ranging from metabolic to social networks. That K and the size of the group of automorphisms of a graph are correlated opens up interesting connections to problems in computational geometry, and thus connects several measures and concepts from complexity science. We show that approximations of K characterise synthetic and natural networks by their generating mechanisms, assigning lower algorithmic randomness to complex network models (Watts-Strogatz and Barabasi-Albert networks) and high Kolmogorov complexity to (random) Erdos-Renyi graphs. We derive these results via two different Kolmogorov complexity approximation methods applied to the adjacency matrices of the graphs and networks. The methods used are the traditional lossless compression approach to Kolmogorov complexity, and a normalised version of a Block Decomposition Method (BDM) measure, based on algorithmic probability theory.Comment: 15 2-column pages, 20 figures. Forthcoming in Physica A: Statistical Mechanics and its Application

A computational study of genotype-phenotype mutation patterns

Understanding properties of genotype-phenotype maps is important for understanding biology and evolution. In this project we make a computational study of the statistical effects of genetic mutations, in particular computing the probabilities of each phenotype transitioning to any other phenotype. We also investigate the importance of the local phenotypic environment of a single genotype, and its role in determining mutation transition probabilities. We use HP protein folding, RNA structure, and a simplified GRN matrix model to study these questions

Phenotype bias determines how natural RNA structures occupy the morphospace of all possible shapes

The relative prominence of developmental bias versus natural selection is a long standing controversy in evolutionary biology. Here we demonstrate quantitatively that developmental bias is the primary explanation for the occupation of the morphospace of RNA secondary structure (SS) shapes. By using the RNAshapes method to define coarse-grained SS classes, we can measure the frequencies that non-coding RNA SS shapes appear in nature. Our main findings are firstly that only the most frequent structures appear in nature; the vast majority of possible structures in the morphospace have not yet been explored. Secondly, and perhaps more surprisingly, these frequencies are accurately predicted by the likelihood that structures appear upon uniform random sampling of sequences. The ultimate cause of these patterns is not natural selection, but rather strong phenotype bias in the RNA genotype-phenotype (GP) map, a type of developmental bias which tightly constrains evolutionary dynamics to only act within a reduced subset of structures that are easy to “find”. Preprint on: https://www.biorxiv.org/content/10.1101/2020.12.03.410605v

An ab initio study of Fe-doped Nickel Oxide

Abstract In the following pages we investigate the magnetic properties of the dilute magnetic semiconductor Fe-doped NiO from an ab initio approach; the incentive being the need to discover how the observation of ferromagnetism in the doped crystal mentioned has come about. Hartree-Fock and B3LYP Hamiltonians are employed to find the natural state of pure and doped NiO in various arrangements of the magnetic ions' spin, and finally geometry optimisations are performed of the most stable configuration

Probabilistic Bias in Genotype-Phenotype Maps

Among the most fundamental features shared by all organisms is the mapping of information encoded in genotypes (genetic material) to generate phenotypes (biological structures, functions, and traits). Hence, elucidating the structure of genotype-phenotype (GP) maps is important for understanding evolution and biology. While it is known that often GP maps are highly degenerate with many different genotypes adopting the same phenotype, the distribution of genotypes over phenotypes is less well studied. In this thesis we investigate the question of the distribution of genotypes over phenotypes, or put differently the distribution of neutral set sizes (NSS), where a neutral set is the collection of all genotypes in a GP map which map to the same phenotype. We focus on examining phenotypic bias in GP maps, where some phenotypes have disproportionally large NSS as compared to others. We find phenotypic bias to be ubiquitous in the broad range of GP maps that we analyse, from the genetic code up to molecular RNA to a model of neuronal connections, and hence we hypothesise bias to be a common property of GP maps. Further, we also consider the implications that this bias has for evolutionary outcomes, and we argue that bias is a significant influencing factor in determining evolutionary outcomes. Finally, we propose a method to predict a phenotype's NNS via estimating the phenotype's structural complexity, without using detailed knowledge about the specifics of the relevant GP map. We achieve this via a novel application of algorithmic information theory and especially Levin's coding theorem.This thesis is not currently available on ORA

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 the increasing phenotype descriptional complexity. This means that phenotypes with 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 [Formula: see text], where [Formula: see text] 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 facilitate the prediction of transition probabilities directly from examining phenotype themselves, without utilizing detailed knowledge of the GP map

Maximum mutational robustness in genotype-phenotype maps follows a self-similar blancmange-like curve.

Peer reviewed: TrueFunder: Engineering and Physical Sciences Research Council; doi: http://dx.doi.org/10.13039/501100000266Funder: Royal Society; doi: http://dx.doi.org/10.13039/501100000288Funder: Gatsby Charitable Foundation; doi: http://dx.doi.org/10.13039/501100000324Phenotype robustness, defined as the average mutational robustness of all the genotypes that map to a given phenotype, plays a key role in facilitating neutral exploration of novel phenotypic variation by an evolving population. By applying results from coding theory, we prove that the maximum phenotype robustness occurs when genotypes are organized as bricklayer's graphs, so-called because they resemble the way in which a bricklayer would fill in a Hamming graph. The value of the maximal robustness is given by a fractal continuous everywhere but differentiable nowhere sums-of-digits function from number theory. Interestingly, genotype-phenotype maps for RNA secondary structure and the hydrophobic-polar (HP) model for protein folding can exhibit phenotype robustness that exactly attains this upper bound. By exploiting properties of the sums-of-digits function, we prove a lower bound on the deviation of the maximum robustness of phenotypes with multiple neutral components from the bricklayer's graph bound, and show that RNA secondary structure phenotypes obey this bound. Finally, we show how robustness changes when phenotypes are coarse-grained and derive a formula and associated bounds for the transition probabilities between such phenotypes