RNA based evolutionary optimization

The notion of an RNA world has been introduced for a prebiotic scenario that is dominated by RNA molecules and their properties, in particular their capabilities to act as templates for reproduction and as catalysts for several cleavage and ligation reactions of polynucleotides and polypeptides. This notion is used here also for simple experimental assays which are well suited to study evolution in the test tube. In molecular evolution experiments fitness is determined in essence by the molecular structures of RNA molecules. Evidence is presented for adaptation to environment in cell-free media. RNA based molecular evolution experiments have led to interesting spin-offs in biotechnology, commonly called applied molecular evolution, which make use of Darwinian trial-and-error strategies in order to synthesize new pharmacological compounds and other advanced materials on a biological basis.Error-propagation in RNA replication leads to formation of mutant spectra called quasispecies. An increase in the error rate broadens the mutant spectrum. There exists a sharply defined threshold beyond which heredity breaks down and evolutionary adaptation becomes impossible. Almost all RNA viruses studied so far operate at conditions close to this error threshold. Quasispecies and error thresholds are important for an understanding of RNA virus evolution, and they may help to develop novel antiviral strategies.Evolution of RNA molecules can be studied and interpreted by considering secondary structures. The notion of sequence space introduces a distance between pairs of RNA sequences which is tantamount to counting the minimal number of point mutations required to convert the sequences into each other. The mean sensitivity of RNA secondary structures to mutation depends strongly on the base pairing alphabet: structures from sequences which contain only one base pair (GC or AU are much less stable against mutation than those derived from the natural (AUGC) sequences. Evolutionary optimization of two-letter sequences in thus more difficult than optimization in the world of natural RNA sequences with four bases. This fact might explain the usage of four bases in the genetic language of nature.Finally we study the mapping from RNA sequences into secondary structures and explore the topology of RNA shape space. We find that neutral paths connecting neighbouring sequences with identical structures go very frequently through entire sequence space. Sequences folding into common structures are found everywhere in sequence space. Hence, evolution can migrate to almost every part of sequence space without hill climbing and only small fractions of the entire number of sequences have to be searched in order to find suitable structures.     

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     

Determinants of simulated RNA evolution

Models of RNA secondary structure folding are widely used to study evolution in theory and simulation. However, systematic studies of the parameters involved are rare. In this paper, we study by simulation how RNA evolution is influenced by three different factors, namely the mutation rate, scaling of the fitness function, and distance measure. We found that for low mutation rates the qualitative evolutionary behavior is robust with respect to the scaling of the fitness function. For efficient mutation rates, which are close to the error threshold, scaling and distance measure have a strong influence on the evolutionary behavior. A global distance measure that takes sequence information additively into account lowers the error threshold. When using a local sequence-structure alignment for the distance, we observed a smoother evolution of the fitness over time. Finally, in addition to the well known error threshold, we identify another threshold of the mutation rate, called divergence threshold, where the qualitative transient behavior changes from a localized to an exploratory search.     

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.     

Interactions between evolutionary processes at high mutation rates

Evolution at high mutation rates is minimally affected by six processes: mutation-selection balance, error catastrophes, Muller's Ratchet, robustness and compensatory evolution, and clonal interference. Including all of these processes in a tractable, analytical model is difficult, but they can be captured in simulations that utilize realistic genotype-phenotype-fitness maps, as done here by modeling RNA folding. Subjecting finite, asexual populations to a range of mutation rates revealed simple criteria that predict when particular evolutionary processes are important. Populations were initiated with a genotype encoding the most fit phenotype. When purifying selection was strong relative to mutation, the initial genotype was replaced by one more mutationally robust, and the maximally fit phenotype was maintained in a mutation-selection balance where the deleterious mutation rate determined mean fitness. With weaker purifying selection, the most fit genotypes were lost. Although loss of the best genotype was ongoing and might have led to a progressive fitness decline, continual compensatory evolution led to an approximate fitness equilibration. Per total genomic mutation rate, mean fitness was similar for strong and weak purifying selection. These results represent a first step at separating interactions between evolutionary processes at high mutation rate, but additional theory is needed to interpret some outcomes.     

A computer model of evolutionary optimization

Molecular evolution is viewed as a typical combinatorial optimization problem. We analyse a chemical reaction model which considers RNA replication including correct copying and point mutations together with hydrolytic degradation and the dilution flux of a flow reactor. The corresponding stochastic reaction network is implemented on a computer in order to investigate some basic features of evolutionary optimization dynamics. Characteristic features of real molecular systems are mimicked by folding binary sequences into unknotted two-dimensional structures. Selective values are derived from these molecular 'phenotypes' by an evaluation procedure which assigns numerical values to different elements of the secondary structure. The fitness function obtained thereby contains nontrivial long-range interactions which are typical for real systems. The fitness landscape also reveals quite involved and bizarre local topologies which we consider also representative of polynucleotide replication in actually occurring systems. Optimization operates on an ensemble of sequences via mutation and natural selection. The strategy observed in the simulation experiments is fairly general and resembles closely a heuristic widely applied in operations research areas. Despite the relative smallness of the system--we study 2000 molecules of chain length v = 70 in a typical simulation experiment--features typical for the evolution of real populations are observed as there are error thresholds for replication, evolutionary steps and quasistationary sequence distributions. The relative importance of selectively neutral or almost neutral variants is discussed quantitatively. Four characteristic ensemble properties, entropy of the distribution, ensemble correlation, mean Hamming distance and diversity of the population, are computed and checked for their sensitivity in recording major optimization events during the simulation.     

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.     

In silico identification of functional divergence between the multiple groEL gene paralogs in Chlamydiae

Background

RNA molecules, through their dual appearance as sequence and structure, represent a suitable model to study evolutionary properties of quasispecies. The essential ingredient in this model is the differentiation between genotype (molecular sequences which are affected by mutation) and phenotype (molecular structure, affected by selection). This framework allows a quantitative analysis of organizational properties of quasispecies as they adapt to different environments, such as their robustness, the effect of the degeneration of the sequence space, or the adaptation under different mutation rates and the error threshold associated.

Results

We describe and analyze the structural properties of molecular quasispecies adapting to different environments both during the transient time before adaptation takes place and in the asymptotic state, once optimization has occurred. We observe a minimum in the adaptation time at values of the mutation rate relatively far from the phenotypic error threshold. Through the definition of a consensus structure, it is shown that the quasispecies retains relevant structural information in a distributed fashion even above the error threshold. This structural robustness depends on the precise shape of the secondary structure used as target of selection. Experimental results available for natural RNA populations are in qualitative agreement with our observations.

Conclusion

Adaptation time of molecular quasispecies to a given environment is optimized at values of the mutation rate well below the phenotypic error threshold. The optimal value results from a trade-off between diversity generation and fixation of advantageous mutants. The critical value of the mutation rate is a function not only of the sequence length, but also of the specific properties of the environment, in this case the selection pressure and the shape of the secondary structure used as target phenotype. Certain functional motifs of RNA secondary structure that withstand high mutation rates (as the ubiquitous hairpin motif) might appear early in evolution and be actually frozen evolutionary accidents.     

Critical Mutation Rate Has an Exponential Dependence on Population Size in Haploid and Diploid Populations

Understanding the effect of population size on the key parameters of evolution is particularly important for populations nearing extinction. There are evolutionary pressures to evolve sequences that are both fit and robust. At high mutation rates, individuals with greater mutational robustness can outcompete those with higher fitness. This is survival-of-the-flattest, and has been observed in digital organisms, theoretically, in simulated RNA evolution, and in RNA viruses. We introduce an algorithmic method capable of determining the relationship between population size, the critical mutation rate at which individuals with greater robustness to mutation are favoured over individuals with greater fitness, and the error threshold. Verification for this method is provided against analytical models for the error threshold. We show that the critical mutation rate for increasing haploid population sizes can be approximated by an exponential function, with much lower mutation rates tolerated by small populations. This is in contrast to previous studies which identified that critical mutation rate was independent of population size. The algorithm is extended to diploid populations in a system modelled on the biological process of meiosis. The results confirm that the relationship remains exponential, but show that both the critical mutation rate and error threshold are lower for diploids, rather than higher as might have been expected. Analyzing the transition from critical mutation rate to error threshold provides an improved definition of critical mutation rate. Natural populations with their numbers in decline can be expected to lose genetic material in line with the exponential model, accelerating and potentially irreversibly advancing their decline, and this could potentially affect extinction, recovery and population management strategy. The effect of population size is particularly strong in small populations with 100 individuals or less; the exponential model has significant potential in aiding population management to prevent local (and global) extinction events.     

Selection for mutational robustness in finite populations

We investigate the evolutionary dynamics of a finite population of RNA sequences replicating on a neutral network. Despite the lack of differential fitness between viable sequences, we observe typical properties of adaptive evolution, such as increase of mean fitness over time and punctuated-equilibrium transitions, after initial mutation-selection balance has been reached. We find that a product of population size and mutation rate of approximately 30 or larger is sufficient to generate selection pressure for mutational robustness, even if the population size is orders of magnitude smaller than the neutral network on which the population resides. Our results show that quasispecies effects and neutral drift can occur concurrently, and that the relative importance of each is determined by the product of population size and mutation rate.     

Thermodynamical interpretation of evolutionary dynamics on a fitness landscape in an evolution reactor,II

In our previous report [Aita, T., Morinaga, S., Hosimi, Y., 2004. Thermodynamical interpretation of evolutionary dynamics on a fitness landscape in an evolution reactor I. Bull. Math. Biol. 66, 1371–1403], an analogy between thermodynamics and adaptive walks on a Mt. Fuji-type fitness landscape in an artificial selection system was presented. Introducing the ‘free fitness’ as the sum of a fitness term and an entropy term and ‘evolutionary force’ as the gradient of free fitness on a fitness coordinate, we demonstrated that the adaptive walk (=evolution) is driven by the evolutionary force in the direction in which free fitness increases. In this report, we examine the effect of various modifications of the original model on the properties of the adaptive walk. The modifications were as follows: first, mutation distance d was distributed obeying binomial distribution; second, the selection process obeyed the natural selection protocol; third, ruggedness was introduced to the landscape according to the NK model; fourth, a noise was included in the fitness measurement. The effect of each modification was described in the same theoretical framework as the original model by introducing ‘effective’ quantities such as the effective mutation distance or the effective screening size.     

The ascent of the abundant: how mutational networks constrain evolution

Evolution by natural selection is fundamentally shaped by the fitness landscapes in which it occurs. Yet fitness landscapes are vast and complex, and thus we know relatively little about the long-range constraints they impose on evolutionary dynamics. Here, we exhaustively survey the structural landscapes of RNA molecules of lengths 12 to 18 nucleotides, and develop a network model to describe the relationship between sequence and structure. We find that phenotype abundance—the number of genotypes producing a particular phenotype—varies in a predictable manner and critically influences evolutionary dynamics. A study of naturally occurring functional RNA molecules using a new structural statistic suggests that these molecules are biased toward abundant phenotypes. This supports an “ascent of the abundant” hypothesis, in which evolution yields abundant phenotypes even when they are not the most fit.     

Genotypes with phenotypes: adventures in an RNA toy world

Evolution has created the complexity of the animate world and deciphering the language of evolution is the key towards understanding nature. The dynamics of evolution is simplified by considering it as a superposition of three less sophisticated processes: population dynamics, population support dynamics, and genotype-phenotype mapping. Evolution of molecules in laboratory assays provides a sufficiently simple system for the quantitative analysis of the three phenomena. Coarse-grained notions of structures like RNA secondary structures are used as model phenotypes. They provide an excellent tool for a comprehensive analysis of the entire complex of molecular evolution. The mapping from RNA genotypes into secondary structures is highly redundant. In order to find at least one sequence for every common structures one need only search a (relatively) small part of sequence space. The existence of selectively neutral phenotypes plays an important role for the the success and the efficiency of evolutionary optimization. Molecular evolution found a highly promising technological application in the design of biomolecules with predefined properties.     

Local Neutral Networks Help Maintain Inaccurately Replicating Ribozymes

The error threshold of replication limits the selectively maintainable genome size against recurrent deleterious mutations for most fitness landscapes. In the context of RNA replication a distinction between the genotypic and the phenotypic error threshold has been made; where the latter concerns the maintenance of secondary structure rather than sequence. RNA secondary structure is treated as a proxy for function. The phenotypic error threshold allows higher per digit mutation rates than its genotypic counterpart, and is known to increase with the frequency of neutral mutations in sequence space. Here we show that the degree of neutrality, i.e. the frequency of nearest-neighbour (one-step) neutral mutants is a remarkably accurate proxy for the overall frequency of such mutants in an experimentally verifiable formula for the phenotypic error threshold; this we achieve by the full numerical solution for the concentration of all sequences in mutation-selection balance up to length 16. We reinforce our previous result that currently known ribozymes could be selectively maintained by the accuracy known from the best available polymerase ribozymes. Furthermore, we show that in silico stabilizing selection can increase the mutational robustness of ribozymes due to the fact that they were produced by artificial directional selection in the first place. Our finding offers a better understanding of the error threshold and provides further insight into the plausibility of an ancient RNA world.     

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     

Evolutionary biotechnology – theory,facts and perspectives

Molecular evolution has recently been applied in biotechnology which consist of the development of evolutionary strategies in the design of biopolymers with predefined properties and functions. At the heart of this new technology are the in vitro replication and random synthesis of RNA or DNA molecules, producing large libraries of genotypes that are subjected to selection techniques following DARWIN's principle. By means of these evolutionary methods, RNA molecules were derived which specifically bind to predefined target molecules. Ribozymes with new catalytic functions were obtained as well as RNA molecules that are resistant to cleavage by specific RNases. In addition, the catalytic specificities of group I introns, a special class of ribozymes, were modified by variation and selection. Efficient applications of molecular evolution to problems in biotechnology require a fundamental and detailed understanding of the evolutionary process. Two basic questions are of primary importance: (i) How can evolutionary methods be successful as the numbers of possible genotypes are so large that the chance of obtaining a particular sequence by random processes is practically zero, and (ii) how can populations avoid being caught in evolutionary traps corresponding to local fitness optima? This review is therefore concerned with an abridged account of the theory of molecular evolution, as well as its application to biotechnology. We add a brief discussion of new techniques for the massively parallel handling and screening of very small probes as is required for the spatial separation and selection of genotypes. Finally, some imminent prospects concerning the evolutionary design of biopolymers are presented.     

Quasispecies made simple

Quasispecies are clouds of genotypes that appear in a population at mutation–selection balance. This concept has recently attracted the attention of virologists, because many RNA viruses appear to generate high levels of genetic variation that may enhance the evolution of drug resistance and immune escape. The literature on these important evolutionary processes is, however, quite challenging. Here we use simple models to link mutation–selection balance theory to the most novel property of quasispecies: the error threshold—a mutation rate below which populations equilibrate in a traditional mutation–selection balance and above which the population experiences an error catastrophe, that is, the loss of the favored genotype through frequent deleterious mutations. These models show that a single fitness landscape may contain multiple, hierarchically organized error thresholds and that an error threshold is affected by the extent of back mutation and redundancy in the genotype-to-phenotype map. Importantly, an error threshold is distinct from an extinction threshold, which is the complete loss of the population through lethal mutations. Based on this framework, we argue that the lethal mutagenesis of a viral infection by mutation-inducing drugs is not a true error catastophe, but is an extinction catastrophe.     

Widespread compensatory evolution conserves DNA-encoded nucleosome organization in yeast

Evolution maintains organismal fitness by preserving genomic information. This is widely assumed to involve conservation of specific genomic loci among species. Many genomic encodings are now recognized to integrate small contributions from multiple genomic positions into quantitative dispersed codes, but the evolutionary dynamics of such codes are still poorly understood. Here we show that in yeast, sequences that quantitatively affect nucleosome occupancy evolve under compensatory dynamics that maintain heterogeneous levels of A+T content through spatially coupled A/T-losing and A/T-gaining substitutions. Evolutionary modeling combined with data on yeast polymorphisms supports the idea that these substitution dynamics are a consequence of weak selection. This shows that compensatory evolution, so far believed to affect specific groups of epistatically linked loci like paired RNA bases, is a widespread phenomenon in the yeast genome, affecting the majority of intergenic sequences in it. The model thus derived suggests that compensation is inevitable when evolution conserves quantitative and dispersed genomic functions.     

Error thresholds and the constraints to RNA virus evolution

RNA viruses are often thought of as possessing almost limitless adaptability as a result of their extreme mutation rates. However, high mutation rates also put a cap on the size of the viral genome by establishing an error threshold, beyond which lethal numbers of deleterious mutations accumulate. Herein, I argue that a lack of genomic space means that RNA viruses will be subject to important evolutionary constraints because specific sequences are required to encode multiple and often conflicting functions. Empirical evidence for these constraints, and how they limit viral adaptability, is now beginning to accumulate. Documenting the constraints to RNA virus evolution has important implications for predicting the emergence of new viruses and for improving therapeutic procedures.