Understanding which phenotypes are accessible from which genotypes is fundamental for understanding the evolutionary process. This notion of accessibility can be used to define a relation of nearness among phenotypes, independently of their similarity. Because of neutrality, phenotypes denote equivalence classes of genotypes. The definition of neighborhood relations among phenotypes relies, therefore, on the statistics of neighborhood relations among equivalence classes of genotypes in genotype space. The folding of RNA sequences (genotypes) into secondary structures (phenotypes) is an ideal case to implement these concepts. We study the extent to which the folding of RNA sequences induces a "statistical topology" on the set of minimum free energy secondary structures. The resulting nearness relation suggests a notion of "continuous" structure transformation. We can, then, rationalize major transitions in evolutionary trajectories at the level of RNA structures by identifying those tra...

RNA folding from sequences into secondary structures is a simple yet powerful, biophysically grounded model of a genotype-phenotype map in which concepts like plasticity, evolvability, epistasis, and modularity can not only be precisely defined and statistically measured but also reveal simultaneous and profoundly non-independent effects of natural selection. Molecular plasticity is viewed here as the capacity of an RNA sequence to assume a variety of energetically favorable shapes by equilibrating among them at constant temperature. Through simulations based on experimental designs, we study the dynamics of a population of RNA molecules that evolve toward a predefined target shape in a constant environment. Each shape in the plastic repertoire of a sequence contributes to the overall fitness of the sequence in proportion to the time the sequence spends in that shape. Plasticity is costly, since the more shapes a sequence can assume, the less time it spends in any one of the...

It has come to be almost an article of faith amongst population biologists and GA researchers alike that the principal feature of a fitness landscape as regards evolutionary dynamics is "ruggedness", particularly as measured by the auto-correlation function. In this paper we demonstrate that auto-correlation alone may be inadequate as a mediator of evolutionary dynamics, specifically in the presence of large scale neutrality. We introduce the NKp family of landscapes (a variant on NK landscapes) which possess the remarkable property that varying the degree of neutrality has minimal effect on the correlation structure. It is demonstrated that NKp landscapes feature neutral networks which have a "constant innovation" property comparable with the neutral networks observed in models of RNA secondary structure folding landscapes. We show that evolutionary dynamics on NKp landscapes vary dramatically with the degree of neutrality - at high neutrality the dynamics are characterised by populat...

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 pretopology. 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 st...

that allows to choose the direction for the next step at random from all directions along which fitness does not decrease. Stationary states of populations correspond to local optima of the fitness landscape. Evolution is seen as a series of transitions between optima with increasing fitness values. Wright's metaphor saw a recent revival when sufficiently simple models of fitness landscapes became available [1, 41]. These models are based on spin glass theory [63, 66] or closely related to it like Kauffman's Nk model [42]. Evolution of RNA molecules has been studied by more realistic models that deal explicitly with molecular structures obtained from folding RNA sequences [23, 24]. Fitness values serving as input parameters for evolutionary dynamics were derived through evaluation of the structures. The complexity of RNA fitness landscapes originates from conflicting consequences of structural changes that are reminiscent of "frustration" in the theory of spin glasses [2]. Fitness in t

We introduce a new model of evolution on a fitness landscape possessing a tunable degree of neutrality. The model allows us to study the general properties of molecular species undergoing neutral evolution. We find that a number of phenomena seen in RNA sequence-structure maps are present also in our general model. Examples are the occurrence of “common ” structures which occupy a fraction of the genotype space which tends to unity as the length of the genotype increases, and the formation of percolating neutral networks which cover the genotype space in such a way that a member of such a network can be found within a small radius of any point in the space. We also describe a number of new phenomena which appear to be general properties of neutrally evolving systems. In particular, we show that the maximum fitness attained during the adaptive walk of a population evolving on such a fitness landscape increases with increasing degree of neutrality, and is directly related to the fitness of the most fit percolating network. 1

In this paper, we develop techniques based on evolvability statistics of the fitness landscape surrounding sampled solutions. Averaging the measures over a sample of equal fitness solutions allows us to build up fitness evolvability portraits of the fitness landscape, which we show can be used to compare both the ruggedness and neutrality in a set of tunably rugged and tunably neutral landscapes. We further show that the techniques can be used with solution samples collected through both random sampling of the landscapes and online sampling during optimization. Finally, we apply the techniques to two real evolutionary electronics search spaces and highlight differences between the two search spaces, comparing with the time taken to find good solutions through search.

Background: Protein space is explored by means of an inverse folding procedure that makes use of knowledge-based potentials of mean force. Results: Computer simulations indicate that amino acid sequences folding into a common shape are distributed homogeneously forming extended percolating networks that span the entire sequence space. Conclusions: The existence of very long neutral paths on all examined protein structures, indicates the existence of neutral networks percolating protein space. The same qualitative results were obtained for some, but not all, restricted amino acid alphabets. In this respect, the sequence-structure map of proteins seems to be very similar to the nucleic acid case.

Fitness landscapes have proven to be a valuable concept in evolutionary biology, combinatorial optimization, and the physics of disordered systems. A fitness landscape is a mapping from a configuration space into the real numbers. The configuration space is equipped with some notion of adjacency, nearness, distance or accessibility. Landscape theory has emerged as an attempt to devise suitable mathematical structures for describing the "static" properties of landscapes as well as their influence on the dynamics of adaptation. In this review we focus on the connections of landscape theory with algebraic combinatorics and random graph theory, where exact results are available.

Folding of RNA sequences into secondary structures is viewed as a map that assigns a uniquely de ned base pairing pattern to every sequence. The mapping is non-invertible since many sequences fold into the same minimum free energy (secondary) structure or shape. The preimages of this map, called neutral networks, are uniquely associated with the shapes and vice versa. Random graph theory is used to construct networks in sequence space which are suitable models for neutral networks. The theory of molecular quasispecies has been applied to replication and mutation on single-peak tness landscapes. This concept is extended by considering evolution on degenerate multi-peak landscapes which originate from neutral networks by assuming that one particular shape is tter than all others. On such a singleshape landscape the superior tness value is assigned to all sequences belonging