. Fitness landscapes are a valuable concept in evolutionary biology, combinatorial optimization, and the physics of disordered systems. A fitness landscape is a mapping from a configuration space that is equipped with some notion of adjacency, nearness, distance or accessibility, into the real numbers. 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. This chapter gives a brief overview on recent developments in this area, focusing on "geometrical" properties of landscapes. 1 Introduction The concept of a fitness landscape originated in theoretical biology more than seventy years ago [1]. It can be thought of as a kind of "potential function" underlying the dynamics of evolutionary optimization. Implicit in this idea is both a fitness function f that assigns a fitness value to every possible genotype (or organism), and the arrangement of t...

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

Knowledge-Based potentials can be used to decide whether an amino acid sequence is likely to fold into a prescribed native protein structure. We use this idea to survey the sequencestructure relations in protein space. In particular, we test the following two propositions which were found to be important for efficient evolution: The sequences folding into a particular native fold form extensive neutral networks that percolate through sequence space. The neutral networks of any two native folds approach each other to within a few point mutations. Computer simulations using two very different potential functions, M. Sippl's PROSA pair potential and a neural network based potential, are used to verify these claims. Keywords: Knowledge-Based Potentials, Inverse Folding, Neutral Networks, Protein Evolution Babajide et al.: Knowledge Based Potentials 2 1. Introduction Mapping the sequence-structure relations of RNA, based on secondary structure predictions, has provided a theoretical bas...

The distinction between continuous and discontinuous transitions is a longstanding problem in the theory of evolution. Continuity being a topological property, we present a formalism that treats the space of phenotypes as a (finite) topological space, with a topology that is derived from the probabilities with which of one phenotype is accessible from another through changes at the genotypic level. The shape space of RNA secondary structures is used to illustrate this approach. We show that evolutionary trajectories are continuous if and only if they follow connected paths in phenotype space.

Discretized models of biopolymer structures can be used not only as approximations of the actual spatial structures but also as a computationally feasible approach to the generic features of the sequence-structure relationships. We review the combinatorics of nucleic acid secondary structures as well as lattice models of proteins, and show how properties such as the existence of extended neutral networks or shape space covering can be explained on this basis.

. RNA secondary structures provide a unique computer model for investigating the most important aspects of structural and evolutionary biology. The existence of efficient algorithms for solving the folding problem, i.e., for predicting the secondary structure given only the sequence, allows the construction of realistic computer simulations. The notion of a "landscape" 2 C. Flamm, I.L. Hofacker, P.F. Stadler underlies both the structure formation (folding) and the (in vitro) evolution of RNA. Evolutionary adaptation may be seen as hill climbing process on a fitness landscape which is determined by the phenotype of the RNA molecule (within the model this is its secondary structure) and the selection constraints acting on the molecules. We find that a substantial fraction of point mutations do not change an RNA secondary structure. On the other hand, a comparable fraction of mutations leads to very different structures. This interplay of smoothness and ruggedness (or robust...

The concept of a fitness landscape arose in theoretical biology, while that of effective fitness has its origin in evolutionary computation. Both have emerged as useful conceptual tools with which to understand the dynamics of evolutionary processes, especially in the presence of complex genotype-phenotype relations. In this contribution we attempt to provide a unified discussion of these two approaches, discussing both their advantages and disadvantages in the context of some simple models. We also discuss how fitness and effective fitness change under various transformations of the configuration space of the underlying genetic model, concentrating on coarse graining transformations and on a particular coordinate transformation that provides an appropriate basis for illuminating the structure and consequences of recombination.

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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