RNA structures with pseudo-knots: Graph-theoretical, combinatorial, and statistical properties

The secondary structures of nucleic acids form a particularly important class of contact structures. Many important RNA molecules, however, contain pseudo-knots, a structural feature that is excluded explicitly from the conventional definition of secondary structures. We propose here a generalization of secondary structures incorporating ‘non-nested’ pseudo-knots, which we call bi-secondary structures, and discuss measures for the complexity of more general contact structures based on their graph-theoretical properties. Bi-secondary structures are planar trivalent graphs that are characterized by special embedding properties. We derive exact upper bounds on their number (as a function of the chain length n) implying that there are fewer different structures than sequences. Computational results show that the number of bi-secondary structures grows approximately like 2.35ⁿ. Numerical studies based on kinetic folding and a simple extension of the standard energy model show that the global features of the sequence-structure map of RNA do not change when pseudo-knots are introduced into the secondary structure picture. We find a large fraction of neutral mutations and, in particular, networks of sequences that fold into the same shape. These neutral networks percolate through the entire sequence space.     

Revisiting Robustness and Evolvability: Evolution in Weighted Genotype Spaces

Robustness and evolvability are highly intertwined properties of biological systems. The relationship between these properties determines how biological systems are able to withstand mutations and show variation in response to them. Computational studies have explored the relationship between these two properties using neutral networks of RNA sequences (genotype) and their secondary structures (phenotype) as a model system. However, these studies have assumed every mutation to a sequence to be equally likely; the differences in the likelihood of the occurrence of various mutations, and the consequence of probabilistic nature of the mutations in such a system have previously been ignored. Associating probabilities to mutations essentially results in the weighting of genotype space. We here perform a comparative analysis of weighted and unweighted neutral networks of RNA sequences, and subsequently explore the relationship between robustness and evolvability. We show that assuming an equal likelihood for all mutations (as in an unweighted network), underestimates robustness and overestimates evolvability of a system. In spite of discarding this assumption, we observe that a negative correlation between sequence (genotype) robustness and sequence evolvability persists, and also that structure (phenotype) robustness promotes structure evolvability, as observed in earlier studies using unweighted networks. We also study the effects of base composition bias on robustness and evolvability. Particularly, we explore the association between robustness and evolvability in a sequence space that is AU-rich – sequences with an AU content of 80% or higher, compared to a normal (unbiased) sequence space. We find that evolvability of both sequences and structures in an AU-rich space is lesser compared to the normal space, and robustness higher. We also observe that AU-rich populations evolving on neutral networks of phenotypes, can access less phenotypic variation compared to normal populations evolving on neutral networks.     

Mutational Robustness Accelerates the Origin of Novel RNA Phenotypes through Phenotypic Plasticity

Novel phenotypes can originate either through mutations in existing genotypes or through phenotypic plasticity, the ability of one genotype to form multiple phenotypes. From molecules to organisms, plasticity is a ubiquitous feature of life, and a potential source of exaptations, adaptive traits that originated for nonadaptive reasons. Another ubiquitous feature is robustness to mutations, although it is unknown whether such robustness helps or hinders the origin of new phenotypes through plasticity. RNA is ideal to address this question, because it shows extensive plasticity in its secondary structure phenotypes, a consequence of their continual folding and unfolding, and these phenotypes have important biological functions. Moreover, RNA is to some extent robust to mutations. This robustness structures RNA genotype space into myriad connected networks of genotypes with the same phenotype, and it influences the dynamics of evolving populations on a genotype network. In this study I show that both effects help accelerate the exploration of novel phenotypes through plasticity. My observations are based on many RNA molecules sampled at random from RNA sequence space, and on 30 biological RNA molecules. They are thus not only a generic feature of RNA sequence space but are relevant for the molecular evolution of biological RNA.     

Mutational Robustness Accelerates the Origin of Novel RNA Phenotypes through Phenotypic Plasticity

Novel phenotypes can originate either through mutations in existing genotypes or through phenotypic plasticity, the ability of one genotype to form multiple phenotypes. From molecules to organisms, plasticity is a ubiquitous feature of life, and a potential source of exaptations, adaptive traits that originated for nonadaptive reasons. Another ubiquitous feature is robustness to mutations, although it is unknown whether such robustness helps or hinders the origin of new phenotypes through plasticity. RNA is ideal to address this question, because it shows extensive plasticity in its secondary structure phenotypes, a consequence of their continual folding and unfolding, and these phenotypes have important biological functions. Moreover, RNA is to some extent robust to mutations. This robustness structures RNA genotype space into myriad connected networks of genotypes with the same phenotype, and it influences the dynamics of evolving populations on a genotype network. In this study I show that both effects help accelerate the exploration of novel phenotypes through plasticity. My observations are based on many RNA molecules sampled at random from RNA sequence space, and on 30 biological RNA molecules. They are thus not only a generic feature of RNA sequence space but are relevant for the molecular evolution of biological RNA.     

Exploring protein sequence space using knowledge-based potentials

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 sequence-structure 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.     

Exploring phenotype space through neutral evolution

RNA secondary-structure folding algorithms predict the existence of connected networks of RNA sequences with identical secondary structures. Fitness landscapes that are based on the mapping between RNA sequence and RNA secondary structure hence have many neutral paths. A neutral walk on these fitness landscapes gives access to a virtually unlimited number of secondary structures that are a single point mutation from the neutral path. This shows that neutral evolution explores phenotype space and can play a role in adaptation. Received: 23 December 1995 / Accepted: 17 March 1996     

Generic properties of combinatory maps: Neutral networks of RNA secondary structures

Random graph theory is used to model and analyse the relationship between sequences and secondary structures of RNA molecules, which are understood as mappings from sequence space into shape space. These maps are non-invertible since there are always many orders of magnitude more sequences than structures. Sequences folding into identical structures formneutral networks. A neutral network is embedded in the set of sequences that arecompatible with the given structure. Networks are modeled as graphs and constructed by random choice of vertices from the space of compatible sequences. The theory characterizes neutral networks by the mean fraction of neutral neighbors (λ). The networks are connected and percolate sequence space if the fraction of neutral nearest neighbors exceeds a threshold value (λ>λ*). Below threshold (λ<λ*), the networks are partitioned into a largest “giant” component and several smaller components. Structure are classified as “common” or “rare” according to the sizes of their pre-images, i.e. according to the fractions of sequences folding into them. The neutral networks of any pair of two different common structures almost touch each other, and, as expressed by the conjecture ofshape space covering 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 to data obtained by folding large samples of RNA sequences. Differences are explained in terms of specific features of RNA molecular structures. Deicated to professor Manfred Eigen     

The effect of recombination on the neutral evolution of genetic robustness

Conventional population genetics considers the evolution of a limited number of genotypes corresponding to phenotypes with different fitness. As model phenotypes, in particular RNA secondary structure, have become computationally tractable, however, it has become apparent that the context dependent effect of mutations and the many-to-one nature inherent in these genotype-phenotype maps can have fundamental evolutionary consequences. It has previously been demonstrated that populations of genotypes evolving on the neutral networks corresponding to all genotypes with the same secondary structure only through neutral mutations can evolve mutational robustness [E. van Nimwegen, J.P. Crutchfield, M. Huynen, Neutral evolution of mutational robustness, Proc. Natl. Acad. Sci. USA 96(17), 9716-9720 (1999)], by concentrating the population on regions of high neutrality. Introducing recombination we demonstrate, through numerically calculating the stationary distribution of an infinite population on ensembles of random neutral networks that mutational robustness is significantly enhanced and further that the magnitude of this enhancement is sensitive to details of the neutral network topology. Through the simulation of finite populations of genotypes evolving on random neutral networks and a scaled down microRNA neutral network, we show that even in finite populations recombination will still act to focus the population on regions of locally high neutrality.     

An in silico exploration of the neutral network in protein sequence space

Designating amino-acid sequences that fold into a common main-chain structure as "neutral sequences" for the structure, regardless of their function or stability, we investigated the distribution of neutral sequences in protein sequence space. For four distinct target structures (alpha, beta,alpha/beta and alpha+beta types) with the same chain length of 108, we generated the respective neutral sequences by using the inverse folding technique with a knowledge-based potential function. We assumed that neutral sequences for a protein structure have Z scores higher than or equal to fixed thresholds, where thresholds are defined as the Z score for the corresponding native sequence (case 1) or much greater Z score (case 2). An exploring walk simulation suggested that the neutral sequences mapped into the sequence space were connected with each other through straight neutral paths and formed an inherent neutral network over the sequence space. Through another exploring walk simulation, we investigated contiguous regions between or among the neutral networks for the distinct protein structures and obtained the following results. The closest approach distance between the two neutral networks ranged from 5 to 29 on the Hamming distance scale, showing a linear increase against the threshold values. The sequences located at the "interchange" regions between the two neutral networks have intermediate sequence-profile-scores for both corresponding structures. Introducing a "ball" in the sequence space that contains at least one neutral sequence for each of the four structures, we found that the minimal radius of the ball that is centered at an arbitrary position ranged from 35 to 50, while the minimal radius of the ball that is centered at a certain special position ranged from 20 to 30, in the Hamming distance scale. The relatively small Hamming distances (5-30) may support an evolution mechanism by transferring from a network for a structure to another network for a more beneficial structure via the interchange regions.     

Local Connectivity of Neutral Networks

This paper studies local connectivity of neutral networks of RNA secondary and pseudoknot structures. A neutral network denotes the set of RNA sequences that fold into a particular structure. It is called locally connected, if in the limit of long sequences, the distance of any two of its sequences scales with their distance in the n-cube. One main result of this paper is that is the threshold probability for local connectivity for neutral networks, considered as random subgraphs of n-cubes. Furthermore, we analyze local connectivity for finite sequence length and different alphabets. We show that it is closely related to the existence of specific paths within the neutral network. We put our theoretical results into context with folding algorithms into minimum-free energy RNA secondary and pseudoknot structures. Finally, we relate our structural findings with dynamics by discussing the role of local connectivity in the context of neutral evolution.     

Evolution in silico and in vitro: the RNA model.

Theoretical concepts and experiments dealing with the 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 the 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 straight forward interpretation of computer simulations on RNA optimization; furthermore, it reveals 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-anddeath process with immigration.     

New structural variation in evolutionary searches of RNA neutral networks

RNA secondary structure is an important computational model to understand how genetic variation maps into phenotypic (structural) variation. Evolutionary innovation in RNA structures is facilitated by neutral networks, large connected sets of RNA sequences that fold into the same structure. Our work extends and deepens previous studies on neutral networks. First, we show that even the 1-mutant neighborhood of a given sequence (genotype) G0 with structure (phenotype) P contains many structural variants that are not close to P. This holds for biological and generic RNA sequences alike. Second, we analyze the relation between new structures in the 1-neighborhoods of genotypes Gk that are only a moderate Hamming distance k away from G0, and the structure of G0 itself, both for biological and for generic RNA structures. Third, we analyze the relation between mutational robustness of a sequence and the distances of structural variants near this sequence. Our findings underscore the role of neutral networks in evolutionary innovation, and the role that high robustness can play in diminishing the potential for such innovation.     

Robustness and evolvability: a paradox resolved

Understanding the relationship between robustness and evolvability is key to understand how living things can withstand mutations, while producing ample variation that leads to evolutionary innovations. Mutational robustness and evolvability, a system's ability to produce heritable variation, harbour a paradoxical tension. On one hand, high robustness implies low production of heritable phenotypic variation. On the other hand, both experimental and computational analyses of neutral networks indicate that robustness enhances evolvability. I here resolve this tension using RNA genotypes and their secondary structure phenotypes as a study system. To resolve the tension, one must distinguish between robustness of a genotype and a phenotype. I confirm that genotype (sequence) robustness and evolvability share an antagonistic relationship. In stark contrast, phenotype (structure) robustness promotes structure evolvability. A consequence is that finite populations of sequences with a robust phenotype can access large amounts of phenotypic variation while spreading through a neutral network. Population-level processes and phenotypes rather than individual sequences are key to understand the relationship between robustness and evolvability. My observations may apply to other genetic systems where many connected genotypes produce the same phenotypes.     

Replication and mutation on neutral networks

Folding of RNA sequences into secondary structures is viewed as a map that assigns a uniquely defined 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 pre-images 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 fitness 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 fitter than all the others. On such a single-shape landscape the superior fitness value is assigned to all sequences belonging to the master shape. All other shapes are lumped together and their fitness values are averaged in a way that is reminiscent of mean field theory. Replication and mutation on neutral networks are modeled by phenomenological rate equations as well as by a stochastic birth-and-death model. In analogy to the error threshold in sequence space the phenotypic error threshold separates two scenarios: (i) a stationary (fittest) master shape surrounded by closely related shapes and (ii) populations drifting through shape space by a diffusion-like process. The error classes of the quasispecies model are replaced by distance classes between the master shape and the other structures. Analytical results are derived for single-shape landscapes, in particular, simple expressions are obtained for the mean fraction of master shapes in a population and for phenotypic error thresholds. The analytical results are complemented by data obtained from computer simulation of the underlying stochastic processes. The predictions of the phenomenological approach on the single-shape landscape are very well reproduced by replication and mutation kinetics of tRNA(phe). Simulation of the stochastic process at a resolution of individual distance classes yields data which are in excellent agreement with the results derived from the birth-and-death model.     

Degeneracy and genetic assimilation in RNA evolution

Background

The neutral theory of Motoo Kimura stipulates that evolution is mostly driven by neutral mutations. However adaptive pressure eventually leads to changes in phenotype that involve non-neutral mutations. The relation between neutrality and adaptation has been studied in the context of RNA before and here we further study transitional mutations in the context of degenerate (plastic) RNA sequences and genetic assimilation. We propose quasineutral mutations, i.e. mutations which preserve an element of the phenotype set, as minimal mutations and study their properties. We also propose a general probabilistic interpretation of genetic assimilation and specialize it to the Boltzmann ensemble of RNA sequences.

Results

We show that degenerate sequences i.e. sequences with more than one structure at the MFE level have the highest evolvability among all sequences and are central to evolutionary innovation. Degenerate sequences also tend to cluster together in the sequence space. The selective pressure in an evolutionary simulation causes the population to move towards regions with more degenerate sequences, i.e. regions at the intersection of different neutral networks, and this causes the number of such sequences to increase well beyond the average percentage of degenerate sequences in the sequence space. We also observe that evolution by quasineutral mutations tends to conserve the number of base pairs in structures and thereby maintains structural integrity even in the presence of pressure to the contrary.

Conclusions

We conclude that degenerate RNA sequences play a major role in evolutionary adaptation.

     

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.     

Neutral evolution of model proteins: diffusion in sequence space and overdispersion.

We stimulate the evolution of model protein sequences subject to mutations. A mutation is considered neutral if it conserves (1) the structure of the ground state, (2) its thermodynamic stability and (3) its kinetic accessibility. All other mutations are considered lethal and are rejected. We adopt a lattice model, amenable to a reliable solution of the protein folding problem. We prove the existence of extended neutral networks in sequence space-sequences can evolve until their similarity with the starting point is almost the same as for random sequences. Furthermore, we find that the rate of neutral mutations has a broad distribution in sequence space. Due to this fact, the substitution process is overdispersed (the ratio between variance and mean is larger than 1). This result is in contrast with the simplest model of neutral evolution, which assumes a Poisson process for substitutions, and in qualitative agreement with the biological data.     

Adaptive evolutionary walks require neutral intermediates in RNA fitness landscapes

In RNA fitness landscapes with interconnected networks of neutral mutations, neutral precursor mutations can play an important role in facilitating the accessibility of epistatic adaptive mutant combinations. I use an exhaustively surveyed fitness landscape model based on short sequence RNA genotypes (and their secondary structure phenotypes) to calculate the minimum rate at which mutants initially appearing as neutral are incorporated into an adaptive evolutionary walk. I show first, that incorporating neutral mutations significantly increases the number of point mutations in a given evolutionary walk when compared to estimates from previous adaptive walk models. Second, that incorporating neutral mutants into such a walk significantly increases the final fitness encountered on that walk - indeed evolutionary walks including neutral steps often reach the global optimum in this model. Third, and perhaps most importantly, evolutionary paths of this kind are often extremely winding in their nature and have the potential to undergo multiple mutations at a given sequence position within a single walk; the potential of these winding paths to mislead phylogenetic reconstruction is briefly considered.     

Unified approach to partition functions of RNA secondary structures

RNA secondary structure formation is a field of considerable biological interest as well as a model system for understanding generic properties of heteropolymer folding. This system is particularly attractive because the partition function and thus all thermodynamic properties of RNA secondary structure ensembles can be calculated numerically in polynomial time for arbitrary sequences and homopolymer models admit analytical solutions. Such solutions for many different aspects of the combinatorics of RNA secondary structure formation share the property that the final solution depends on differences of statistical weights rather than on the weights alone. Here, we present a unified approach to a large class of problems in the field of RNA secondary structure formation. We prove a generic theorem for the calculation of RNA folding partition functions. Then, we show that this approach can be applied to the study of the molten-native transition, denaturation of RNA molecules, as well as to studies of the glass phase of random RNA sequences.     

Uncovering network systems within protein structures

Traditionally, proteins have been viewed as a construct based on elements of secondary structure and their arrangement in three-dimensional space. In a departure from this perspective we show that protein structures can be modelled as network systems that exhibit small-world, single-scale, and to some degree, scale-free properties. The phenomenological network concept of degrees of separation is applied to three-dimensional protein structure networks and reveals how amino acid residues can be connected to each other within six degrees of separation. This work also illuminates the unique features of protein networks in comparison to other networks currently studied. Recognising that proteins are networks provides a means of rationalising the robustness in the overall three-dimensional fold of a protein against random mutations and suggests an alternative avenue to investigate the determinants of protein structure, function and folding.