Computer codes for computation and comparison of RNA secondary structures, the Vienna RNA package, are presented, that are based on dynamic programming algorithms and aim at predictions of structures with minimum free energies as well as at computations of the equilibrium partition functions and base pairing probabilities. An efficient heuristic for the inverse folding problem of RNA is introduced. In addition we present compact and efficient programs for the comparison of RNA secondary structures based on tree editing and alignment. All computer codes are written in ANSI C. They include implementations of modified algorithms on parallel computers with distributed memory. Performance analysis carried out on an Intel Hypercube shows that parallel computing becomes gradually more and more efficient the longer the sequences are.

Random graph theory is used to model relationships between sequences and secondary structures of RNA molecules. Sequences folding into identical structures form neutral networks which percolate sequence space if the fraction of neutral nearest neighbors exceeds a threshold value. The networks of any two different structures almost touch each other, and sequences folding into almost all "common" structures can be found in a small ball of an arbitrary location in sequence space. The results from random graph theory are compared with data obtained by folding large samples of RNA sequences. Differences are explained in terms of RNA molecular structures. 1.

Fitness landscapes are an important concept in molecular evolution. Many important examples of landscapes in physics and combinatorial optimation, which are widely used as model landscapes in simulations of molecular evolution and adaptation, are "elementary", i.e., they are (up to an additive constant) eigenfuctions of a graph Laplacian. It is shown that elementary landscapes are characterized by their correlation functions. The correlation functions are in turn uniquely determined by the geometry of the underlying configuration space and the nearest neighbor correlation of the elementary landscape. Two types of correlation functions are investigated here: the correlation of a time series sampled along a random walk on the landscape and the correlation function with respect to a partition of the set of all vertex pairs.

Global relations between RNA sequences and secondary structues are understood as mappings from sequence space into shape space. These mappings are investigated by exhaustive folding of all GC and AU sequences with chain lengths up to 30. The technique od tries is used for economic data storage and fast retrieval of information. The computed structural data are evaluated through exhaustive enumeration and used as an exact reference for testing analytical results derived from mathematical models and sampling based of statistical methods. Several new concepts of RNA sequence to secondary structure mappings are investigated, among them the structure of neutral networks (being sets of sequences folding into the same structure), percolation of sequence space by neutral networks, and the principle of shape space covering . The data of exhaustive enumeration are compared to the analytical results of a random graph model that reveals the generic properties of sequence to structure mappings based on some base pairing logic. The differences between the numerical and the analytical results are interpreted in terms of specific biophysical properties of RNA molecules.

In this paper we view the folding of polynucleotide (RNA) sequences as a map that assigns to each sequence a minimum free energy pattern of base pairings, known as secondary structure. Considering only the free energy leads to an energy landscape over the sequence space. Taking into account structure generates a less visualizable non-scalar "landscape", where a sequence space is mapped into a space of discrete "shapes". We investigate the statistical features of both types of landscapes by computing autocorrelation functions, as well as distributions of energy and structure distances, as a function of distance in sequence space. RNA folding is characterized by very short structure correlation lengths compared to the diameter of the sequence space. The correlation lengths depend strongly on the size and the pairing rules of the underlying nucleotide alphabet. Our data suggest that almost every minimum free energy structure is found within a small neighborhood of any random sequence. The...

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 present and study the behavior of a simple kinetic model for the melting of RNA secondary structures, given that those structures are known. The model is then used as a map that assigns structure dependent overall rate constants of melting (or refolding) to a sequence. This induces a "landscape" of reaction rates, or activation energies, over the space of sequences with fixed length. We study the distribution and the correlation structure of these activation energies. 1. Introduction Single stranded RNA sequences fold into complex three-dimensional structures. A tractable, yet reasonable, model for the map from sequences to structures considers a more coarse grained level of resolution known as the secondary structure. The secondary structure is a list of base pairs such that no pairings occur between bases located in different loop regions. Algorithms based on empirical energy data have been developed to compute the minimum free energy secondary structure of an RNA sequence (Zuker...

The evolution of RNA molecules in replication assays, viroids and RNA viruses can be viewed as an adaptation process on a 'fitness' landscape. The dynamics of evolution is hence tightly linked to the structure of the underlying landscape. Global features of landscapes can be described by statistical measures like number of optima, lengths of walks, and correlation functions. The evolution of a quasispecies on such landscapes exhibits three dynamical regimes depending on the replication fidelity: Above the "localization threshold" the population is centered around a (local) optimum. Between localization and "dispersion threshold" the population is still centered around a consensus sequence, which, however, changes in time. For very large mutation rates the population spreads in sequence space like a gas. The critical mutation rates separating the three domains depend strongly on characteristics properties of the fitness landscapes. Statistical characteristics of RNA landscapes are acces...

A new mathematical representation is proposed for the configuration space structure induced by recombination which we called "P-structure". It consists of a mapping of pairs of objects to the power set of all objects in the search space. The mapping assigns to each pair of parental "genotypes" the set of all recombinant genotypes obtainable from the parental ones. It is shown that this construction allows a Fourier-decomposition of fitness landscapes into a superposition of "elementary landscapes". This decomposition is analogous to the Fourier decomposition of fitness landscapes on mutation spaces. The elementary landscapes are obtained as eigenfunctions of a Laplacian operator defined for P-structures. For binary string recombination the elementary landscapes are exactly the p-spin functions (Walsh functions), i.e. the same as the elementary landscapes of the string point mutation spaces (i.e. the hypercube). This supports the notion of a strong homomorphisms between string mutation ...

We view the folding of RNA-sequences as a map that assigns a pattern of base pairings to each sequence, known as secondary structure. These preimages can be constructed as random graphs (i.e. the neutral networks associated to the structure s). By interpreting the secondary structure as biological information we can formulate the so called Error Threshold of Shapes as an extension of Eigen's et al. concept of an error threshold in the single peak landscape [5]. Analogue to the approach of Derrida & Peliti [3] for a flat landscape we investigate the spatial distribution of the population on the neutral network. On the one hand this model of a single shape landscape allows the derivation of analytical results, on the other hand the concept gives rise to study various scenarios by means of simulations, e.g. the interaction of two different networks [29]. It turns out that the intersection of two sets of compatible sequences (with respect to the pair of secondary structures) plays a key role in the search for "fitter" secondary structures.