RNA shape space topology

Connected pretopology in recombination space

Topological features evolved from genetic operators in the recombination space play a crucial role in the course of evolution. Different crossover models generated in a recombination space can be structured by pretopological space. This paper deals with several types of connectedness properties of recombination space in three different cases of unequal crossover of recombination space structured as pretopology. It is shown that unrestricted unequal crossover is one-sidedly connected and consequently hyper-connected, apo-connected and connected as well. Restricted unequal crossover is hyper-connected and connected. Unequal sister chromatid exchange is connected.

Topological model for recombination of DNA and RNA

The aim of this paper is to use topological concepts in the construction of flexible mathematical models in the field of biological mathematics. Also, we will build new topographic types to study recombination of deoxyribonucleic acid (DNA) and ribonucleic acid (RNA). Finally, we study the topographical properties of constructed operators and the associated topological spaces of DNA and RNA.

In the Beginning was a Mutualism - On the Origin of Translation

The origin of translation is critical for understanding the evolution of life, including the origins of life. The canonical genetic code is one of the most dominant aspects of life on this planet, while the origin of heredity is one of the key evolutionary transitions in living world. Why the translation apparatus evolved is one of the enduring mysteries of molecular biology. Assuming the hypothesis, that during the emergence of life evolution had to first involve autocatalytic systems which only subsequently acquired the capacity of genetic heredity, we propose and discuss possible mechanisms, basic aspects of the emergence and subsequent molecular evolution of translation and ribosomes, as well as enzymes as we know them today. It is possible, in this sense, to view the ribosome as a digital-to-analogue information converter. The proposed mechanism is based on the abilities and tendencies of short RNA and polypeptides to fold and to catalyse biochemical reactions. The proposed mechanism is in concordance with the hypothesis of a possible chemical co-evolution of RNA and proteins in the origin of the genetic code or even more generally at the early evolution of life on Earth. The possible abundance and availability of monomers at prebiotic conditions are considered in the mechanism. The hypothesis that early polypeptides were folding on the RNA scaffold is also considered and mutualism in molecular evolutionary development of RNA and peptides is favoured.

A topological approach to chemical organizations

Large chemical reaction networks often exhibit distinctive features that can be interpreted as higher-level structures. Prime examples are metabolic pathways in a biochemical context. We review mathematical approaches that exploit the stoichiometric structure, which can be seen as a particular directed hypergraph, to derive an algebraic picture of chemical organizations. We then give an alternative interpretation in terms of set-valued set functions that encapsulate the production rules of the individual reactions. From the mathematical point of view, these functions define generalized topological spaces on the set of chemical species. We show that organization-theoretic concepts also appear in a natural way in the topological language. This abstract representation in turn suggests the exploration of the chemical meaning of well-established topological concepts. As an example, we consider connectedness in some detail.

Quasi-independence, homology and the unity of type: a topological theory of characters

In this paper Lewontin's notion of "quasi-independence" of characters is formalized as the assumption that a region of the phenotype space can be represented by a product space of orthogonal factors. In this picture each character corresponds to a factor of a region of the phenotype space. We consider any region of the phenotype space that has a given factorization as a "type", i.e. as a set of phenotypes that share the same set of phenotypic characters. Using the notion of local factorizations we develop a theory of character identity based on the continuation of common factors among different regions of the phenotype space. We also consider the topological constraints on evolutionary transitions among regions with different regional factorizations, i.e. for the evolution of new types or body plans. It is shown that direct transition between different "types" is only possible if the transitional forms have all the characters that the ancestral and the derived types have and are thus compatible with the factorization of both types. Transitional forms thus have to go over a "complexity hump" where they have more quasi-independent characters than either the ancestral as well as the derived type. The only logical, but biologically unlikely, alternative is a "hopeful monster" that transforms in a single step from the ancestral type to the derived type. Topological considerations also suggest a new factor that may contribute to the evolutionary stability of "types". It is shown that if the type is decomposable into factors which are vertex irregular (i.e. have states that are more or less preferred in a random walk), the region of phenotypes representing the type contains islands of strongly preferred states. In other words types have a statistical tendency of retaining evolutionary trajectories within their interior and thus add to the evolutionary persistence of types.

Modelling 'evo-devo' with RNA

The folding of RNA sequences into secondary structures is a simple yet biophysically grounded model of a genotype-phenotype map. Its computational and mathematical analysis has uncovered a surprisingly rich statistical structure characterized by shape space covering, neutral networks and plastogenetic congruence. I review these concepts and discuss their evolutionary implications.

Generalized topological spaces in evolutionary theory and combinatorial chemistry

The search spaces in combinatorial chemistry as well as the sequence spaces underlying (molecular) evolution are conventionally thought of as graphs. Recombination, however, implies a nongraphical structure of the combinatorial search spaces. These structures, and their implications for search process itself, are heretofore not well understood in general. In this contribution we review a very general formalism from point set topology and discuss its application to combinatorial search spaces, fitness landscapes, evolutionary trajectories, and artificial chemistries.

Recombination Spaces, Metrics, and Pretopologies

The topological features of genotype spaces given a genetic operator have a substantial impact on the course of evolution. We explore the structure of the recombination spaces arising from five different unequal crossover models in the context of pretopological spaces. We show that four of the models are incompatible with metric distance measures due to a lack of symmetry.

The topology of the possible: formal spaces underlying patterns of evolutionary change

The current implementation of the Neo-Darwinian model of evolution typically assumes that the set of possible phenotypes is organized into a highly symmetric and regular space equipped with a notion of distance, for example, a Euclidean vector space. Recent computational work on a biophysical genotype-phenotype model based on the folding of RNA sequences into secondary structures suggests a rather different picture. If phenotypes are organized according to genetic accessibility, the resulting space lacks a metric and is formalized by an unfamiliar structure, known as a pre-topology. Patterns of phenotypic evolution-such as punctuation, irreversibility, modularity--result naturally from the properties of this space. The classical framework, however, addresses these patterns by exclusively invoking natural selection on suitably imposed fitness landscapes. We propose to extend the explanatory level for phenotypic evolution from fitness considerations alone to include the topological structure of phenotype space as induced by the genotype-phenotype map. We introduce the mathematical concepts and tools necessary to formalize the notion of accessibility pre-topology relative to which we can speak of continuity in the genotype-phenotype map and in evolutionary trajectories. We connect the factorization of a pre-topology into a product space with the notion of phenotypic character and derive a condition for factorization. Based on anecdotal evidence from the RNA model, we conjecture that this condition is not globally fulfilled, but rather confined to regions where the genotype-phenotype map is continuous. Equivalently, local regions of genotype space on which the map is discontinuous are associated with the loss of character autonomy. This is consistent with the importance of these regions for phenotypic innovation. The intention of the present paper is to offer a perspective, a framework to implement this perspective, and a few results illustrating how this framework can be put to work. The RNA case is used as an example throughout the text.