Abstract—Self-adaptation is used in all main paradigms of evolutionary computation to increase efficiency. We claim that the basis of self-adaptation is the use of neutrality. In the absence of external control neutrality allows a variation of the search distribution without the risk of fitness loss. I.

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

Theoretical concepts and experiments dealing with evolution of molecules in vitro reached a state that allows for direct applications to the design of biomolecules with predefined properties. RNA evolution in vitro represents a basis for the development of a new and comprehensive model of evolution focusing on the phenotype and its fitness relevant properties. Relations between genotypes and phenotypes are described by mappings from genotype space onto a space of phenotypes, which are many-to-one and thus give ample room for neutrality as expressed by existence of extended neutral networks in genotype space. The RNA model reduces genotype-phenotype relations to mappings from sequences into secondary structures of minimal free energies and allows for derivation of otherwise inaccessible quantitative results. Continuity and discontinuity in evolution are defined through a new notion of accessibility in phenotype space that provides a basis for straightforward interpretation of computer simulations on RNA optimization and reveal the constructive role of random genomic drift in the search for phenotypes of higher fitness. The effects of population size on the course of evolutionary optimization can be predicted quantitatively by means of a simple stochastic model based on a birth-and-death process with immigration.

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Abstract. The secondary structure for nucleic acids provides a level of description that is both abstract enough to allow for efficient algorithms and realistic enough to provide a good approximate to the thermodynamic and kinetics properties of RNA structure formation. The secondary structure model has furthermore been successful in explaining salient features of RNA evolution in nature and in the test tube. In this contribution we review the computational chemistry of RNA secondary structures using a simplified algorithmic approach for explanation.

Abstract. Neutral genotype-phenotype mappings can be observed in natural evolution and are often used in evolutionary computation. In this article, important aspects of such encodings are analyzed. First, it is shown that in the absence of external control neutrality allows a variation of the search distribution independent of phenotypic changes. In particular, neutrality is necessary for self-adaptation, which is used in a variety of algorithms from all main paradigms of evolutionary computation to increase efficiency. Second, the average number of fitness evaluations needed to find a desirable (e.g., optimally adapted) genotype depending on the number of desirable genotypes and the cardinality of the genotype space is derived. It turns out that this number increases only marginally when neutrality is added to an encoding presuming that the fraction of desirable genotypes stays constant and that the number of these genotypes is not too small.

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Representation techniques are important issues when designing successful evolutionary algorithms. Within this field the use of neutrality plays an important role. We examine the use of bit-wise neutrality introduced by Poli and López (2007) from a theoretical point of view and show that this mechanism only enhances mutation-based evolutionary algorithms if not the same number of genotypic bits for each phenotypic bit is used. Using different numbers of genotypic bits for the bits in the phenome we point out by rigorous runtime analyses that it may reduce the optimization time significantly. 1

RNA secondary structures are derived from RNA sequences, which are strings built form the natural four letter nucleotide alphabet, {AUGC}. These coarse-grained structures, in turn, are tantamount to constrained strings over a three letter alphabet. Hence, the secondary structures are discrete objects and the number of sequences always exceeds the number of structures. The sequences built from two letter alphabets form perfect structures when the nucleotides can form a base pair, as is the case with {GC} or {AU}, but the relation between the sequences and structures differs strongly from the four letter alphabet. A comprehensive theory of RNA structure is presented, which is based on the concepts of sequence space and shape space, being a space of structures. It sets the stage for modelling processes in ensembles of RNA molecules like evolutionary optimization or kinetic folding as dynamical phenomena guided by mappings between the two spaces. The number of minimum free energy (mfe) structures is always smaller than the number of sequences, even for two letter alphabets. Folding of RNA molecules into mfe energy structures constitutes a non-invertible mapping from sequence space onto shape space. The preimage of

INTRODUCTION N etworks of different kinds and sizes appear at different levels of molecular evolution. Networks have a natural representation as graphs or hypergraphs of various kinds. It is not surprising, therefore, that graph theoretical methods are of utmost importance for both analyzing data and building models. Here we present a brief overview of networks with increasing complexity of the systems to be modeled. The single biopolymer molecule is considered first. Conformational diversity of biopolymers as encapsulated by a free energy surface allows for straightforward application of tree-like networks representing the saddle hierarchies in conformation space. Replication and mutation are the basis of evolution based on asexual reproduction. The inherent limitations of such dynamical systems are overcome by another class of dynamical bimolecular processes, catalysis and inhibition representing the positive or negative expression of this interaction, respectively. The results ar