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.     

Quasispecies spatial models for RNA viruses with different replication modes and infection strategies

Empirical observations and theoretical studies suggest that viruses may use different replication strategies to amplify their genomes, which impact the dynamics of mutation accumulation in viral populations and therefore, their fitness and virulence. Similarly, during natural infections, viruses replicate and infect cells that are rarely in suspension but spatially organized. Surprisingly, most quasispecies models of virus replication have ignored these two phenomena. In order to study these two viral characteristics, we have developed stochastic cellular automata models that simulate two different modes of replication (geometric vs stamping machine) for quasispecies replicating and spreading on a two-dimensional space. Furthermore, we explored these two replication models considering epistatic fitness landscapes (antagonistic vs synergistic) and different scenarios for cell-to-cell spread, one with free superinfection and another with superinfection inhibition. We found that the master sequences for populations replicating geometrically and with antagonistic fitness effects vanished at low critical mutation rates. By contrast, the highest critical mutation rate was observed for populations replicating geometrically but with a synergistic fitness landscape. Our simulations also showed that for stamping machine replication and antagonistic epistasis, a combination that appears to be common among plant viruses, populations further increased their robustness by inhibiting superinfection. We have also shown that the mode of replication strongly influenced the linkage between viral loci, which rapidly reached linkage equilibrium at increasing mutations for geometric replication. We also found that the strategy that minimized the time required to spread over the whole space was the stamping machine with antagonistic epistasis among mutations. Finally, our simulations revealed that the multiplicity of infection fluctuated but generically increased along time.     

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.     

Probability of fixation of an advantageous mutant in a viral quasispecies

The probability that an advantageous mutant rises to fixation in a viral quasispecies is investigated in the framework of multitype branching processes. Whether fixation is possible depends on the overall growth rate of the quasispecies that will form if invasion is successful rather than on the individual fitness of the invading mutant. The exact fixation probability can be calculated only if the fitnesses of all potential members of the invading quasispecies are known. Quasispecies fixation has two important characteristics: First, a sequence with negative selection coefficient has a positive fixation probability as long as it has the potential to grow into a quasispecies with an overall growth rate that exceeds that of the established quasispecies. Second, the fixation probabilities of sequences with identical fitnesses can nevertheless vary over many orders of magnitudes. Two approximations for the probability of fixation are introduced. Both approximations require only partial knowledge about the potential members of the invading quasispecies. The performance of these two approximations is compared to the exact fixation probability on a network of RNA sequences with identical secondary structure.     

Simple quasispecies models for the survival-of-the-flattest effect: The role of space

The survival-of-the-flattest effect postulates that under high mutation rates natural selection does not necessarily favor the faster replicators. Under such conditions, genotypes which are robust against deleterious mutational effects may be favored instead, even at the cost of a slower replication. This tantalizing hypothesis has been recently proved using digital organisms, subviral RNA plant pathogens (viroids), and an animal RNA virus. In this work we study a simple theoretical system composed by two competing quasispecies which are located at two widely different fitness landscapes that represent, respectively, a fit and a flat quasispecies. The fit quasispecies is characterized by high replication rate and low mutational robustness, whereas the flat quasispecies is characterized by low replication rate but high mutational robustness. By using a mean field model, in silico simulations with digital replicons and a two-dimensional spatial model given by a stochastic cellular automata (CA), we predict the presence of an absorbing first-order phase transition with critical slowing down between selection for replication speed and selection for mutational robustness, where the surpassing of a critical mutation rate involves the outcompetition of the fit quasispecies by the flat one. Furthermore, it is shown that space, which involves a lower critical mutation rate, broadens the conditions at which the survival-of-the-flattest may occur.     

The replication of kinetoplast DNA networks in Crithidia fasciculata

Kinetoplast DNA from the mitochondria of Crithidia is in the form of a two-dimensional network of thousands of minicircles each containing about 2.5 kb, and a small number of maxicircles each containing about 40 kb. Fractionation of kinetoplast DNA by equilibrium centrifugation in a CsCl-propidium dilodide gradient resolves it into three types of networks. Form I networks band at high density and contain minicircles which are covalently closed; form II networks band at low density and contain minicircles which are nicked or gapped; and replicating networks band at intermediate density and contain some minicircles of each type. Form I networks contain about 5000 minicircles; form II networks contain about 11,000; and replicating networks contain an intermediate number. When cells are pulse-labeled with ³H-thymidine, radioactivity in mitochondrial DNA is preferentially incorporated into replicating networks, but after a chase it appears first in form II networks and finally in form I. Examination of replicating networks by electron microscopy in the presence of ethidium bromide reveals that minicircles in the central region of the network are twisted and therefore covalently closed, whereas those in the peripheral region are not twisted and therefore must be nicked or gapped. The pulse-label is incorporated into the nicked or gapped minicircles of the replicating networks. These results indicate that replication of form I networks begins in peripheral minicircles and that progeny minicircles remain nicked or gapped. As replication proceeds, the size of the network increases, and the peripheral zone of nicked or gapped minicircles enlarges. Finally, when all minicircles have replicated, the network, now form II, is double the size of form I and contains only nicked or gapped minicircles. The final step in replication presumably includes both the cleavage of the network into two form I species and the covalent closure of all the minicircles.     

Late replicating domains are highly recombining in females but have low male recombination rates: implications for isochore evolution

In mammals sequences that are either late replicating or highly recombining have high rates of evolution at putatively neutral sites. As early replicating domains and highly recombining domains both tend to be GC rich we a priori expect these two variables to covary. If so, the relative contribution of either of these variables to the local neutral substitution rate might have been wrongly estimated owing to covariance with the other. Against our expectations, we find that sex-averaged recombination rates show little or no correlation with replication timing, suggesting that they are independent determinants of substitution rates. However, this result masks significant sex-specific complexity: late replicating domains tend to have high recombination rates in females but low recombination rates in males. That these trends are antagonistic explains why sex-averaged recombination is not correlated with replication timing. This unexpected result has several important implications. First, although both male and female recombination rates covary significantly with intronic substitution rates, the magnitude of this correlation is moderately underestimated for male recombination and slightly overestimated for female recombination, owing to covariance with replicating timing. Second, the result could explain why male recombination is strongly correlated with GC content but female recombination is not. If to explain the correlation between GC content and replication timing we suppose that late replication forces reduced GC content, then GC promotion by biased gene conversion during female recombination is partly countered by the antagonistic effect of later replicating sequence tending increase AT content. Indeed, the strength of the correlation between female recombination rate and local GC content is more than doubled by control for replication timing. Our results underpin the need to consider sex-specific recombination rates and potential covariates in analysis of GC content and rates of evolution.     

Mammalian DNA replication: mutation biases and the mutation rate

Experimental studies have shown that the fidelity of DNA replication can be affected by the concentrations of free deoxyribonucleotides present in the cell. Replication of mammalian chromosomes is achieved using pools of newly-synthesized deoxyribonucleotides which fluctuate during the cell cycle. Since regions of mammalian chromosomes are replicated sequentially, there is the potential for differences among mammalian loci in both the relative and absolute frequencies of the various transitional and transversional mutations which may occur. Where these mutations are effectively neutral, at silent sites in genes and in non-coding sequences, this may result in different rates of evolution and in different base compositions, as have been observed in data from mammalian genes. A simple model of the DNA replication process is developed to describe how the mutation rate could be affected by the G + C contents of the deoxyribonucleotide pools and of the replicating DNA. Mutation rates are predicted to vary from locus to locus; only in the particular case of identical G + C contents in the DNA locus and the deoxyribonucleotide pools, and no proofreading, will the mutation rate be uniform over all loci.     

Evidence for the non-quasispecies evolution of RNA viruses [corrected

The quasispecies model of RNA virus evolution differs from those formulated in conventional population genetics in that neutral mutations do not lead to genetic drift of the population, and natural selection acts on the mutant distribution as a whole rather than on individual variants. By computer simulation, we show that this model could be inappropriate for many RNA viruses because the neutral sequence space may be too large to allow the formation of a quasispecies distribution. This view is supported by our analysis of gene sequences from vesicular stomatitis virus, which is considered a prototype RNA virus quasispecies. Our results are relevant to the evolution of RNA systems in general.     

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.     

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.     

Two optimal mutation rates in obligate pathogens subject to deleterious mutation

Pathogen species with high mutation rates are likely to accumulate deleterious mutations that reduce their reproductive potential within the host. By altering the within-host growth rate of the pathogen, the deleterious mutation load has the potential to affect epidemiological properties such as prevalence, mean pathogen load, and the mean duration of infections. Here, I examine an epidemiological model that allows for multiple segregating mutations that affect within-host replication efficiency. The model demonstrates a complex range of outcomes depending on pathogen mutation rate, including two distinct, widely separated mutation rates associated with high pathogen prevalence. The low mutation rate prevalence peak is associated with small amounts of genetic diversity within the pathogen population, relatively stable prevalence and infection dynamics, and genetic variation partitioned between hosts. The high mutation rate peak is characterized by considerable genetic diversity both within and between hosts, relatively frequent invasions by more virulent types, and is qualitatively similar to an RNA virus quasispecies. The two prevalence peaks are separated by a valley where natural selection favors evolution toward the optimal within-host state, which is associated with high virulence and relatively rapid host mortality. Both chronic and acute infections are examined using stochastic forward simulations.     

Statistical tests for detecting gene conversion

Statistical tests for detecting gene conversion are described for a sample of homologous DNA sequences. The tests are based on imbalances in the distribution of segments on which some pair of sequences agrees. The methods automatically control for variable mutation rates along the genome and do not depend on a priori choices of potentially monophyletic subsets of the sample. The tests show strong evidence for multiple intragenic conversion events at two loci in Escherichia coli. The gnd locus in E. coli shows a highly significant excess of maximal segments of length 70-200 bp, which suggests conversion events of that size. The data also indicate that the rate of these short conversion events might be of the order of neutral mutation rate. There is also evidence for correlated mutation in adjacent codon positions. The same tests applied to a locus in an RNA virus were negative.      

The relationship between the error catastrophe, survival of the flattest, and natural selection

Background  

The quasispecies model is a general model of evolution that is generally applicable to replication up to high mutation rates. It predicts that at a sufficiently high mutation rate, quasispecies with higher mutational robustness can displace quasispecies with higher replicative capacity, a phenomenon called "survival of the flattest". In some fitness landscapes it also predicts the existence of a maximum mutation rate, called the error threshold, beyond which the quasispecies enters into error catastrophe, losing its genetic information. The aim of this paper is to study the relationship between survival of the flattest and the transition to error catastrophe, as well as the connection between these concepts and natural selection.     

Quasispecies and recombination

Recombination is introduced into Eigen's theory of quasispecies evolution. Comparing numerical simulations of the rate equations in the non-recombining and recombining cases show that recombination has a strong effect on the error threshold and, for a wide range of mutation rates, gives rise to two stable fixed points in the dynamics. This bi-stability results in the existence of two error thresholds. However, we prove that, for low mutation rates the bi-stability breaks down and the unique equilibrium distribution is concentrated around the sequence with highest fitness.     

GENETIC VARIATION AND DNA REPLICATION TIMING,OR WHY IS THERE LATE REPLICATING DNA?

Mutation rates vary significantly within the genome and across species. Recent studies revealed a long suspected replication-timing effect on mutation rate, but the mechanisms that regulate the increase in mutation rate as the genome is replicated remain unclear. Evidence is emerging, however, that DNA repair systems, in general, are less efficient in late replicating heterochromatic regions compared to early replicating euchromatic regions of the genome. At the same time, mutation rates in both vertebrates and invertebrates have been shown to vary with generation time (GT). GT is correlated with genome size, which suggests a possible nucleotypic effect on species-specific mutation rates. These and other observations all converge on a role for DNA replication checkpoints in modulating generation times and mutation rates during the DNA synthetic phase (S phase) of the cell cycle. The following will examine the potential role of the intra-S checkpoint in regulating cell cycle times (GT) and mutation rates in eukaryotes. This article was published online on August 5, 2011. An error was subsequently identified. This notice is included in the online and print versions to indicate that both have been corrected October 4, 2011.     

Epistasis can lead to fragmented neutral spaces and contingency in evolution

In evolution, the effects of a single deleterious mutation can sometimes be compensated for by a second mutation which recovers the original phenotype. Such epistatic interactions have implications for the structure of genome space--namely, that networks of genomes encoding the same phenotype may not be connected by single mutational moves. We use the folding of RNA sequences into secondary structures as a model genotype-phenotype map and explore the neutral spaces corresponding to networks of genotypes with the same phenotype. In most of these networks, we find that it is not possible to connect all genotypes to one another by single point mutations. Instead, a network for a phenotypic structure with n bonds typically fragments into at least 2(n) neutral components, often of similar size. While components of the same network generate the same phenotype, they show important variations in their properties, most strikingly in their evolvability and mutational robustness. This heterogeneity implies contingency in the evolutionary process.     

Longitudinal Evaluation of the Structure of Replicating and Circulating Hepatitis C Virus Quasispecies in Nonprogressive Chronic Hepatitis C Patients

In previous cross-sectional studies, we demonstrated that, in most patients with chronic hepatitis C, the composition and complexity of the circulating hepatitis C virus (HCV) population do not coincide with those of the virus replicating in the liver. In the subgroup of patients with similar complexities in both compartments, the ratio of quasispecies complexity in the liver to that in serum (liver/serum complexity ratio) of paired samples correlated with disease stage. In the present study we investigated the dynamic behavior of viral population parameters in consecutive paired liver and serum samples, obtained 3 to 6 years apart, from four chronic hepatitis C patients with persistently normal transaminases and stable liver histology. We sequenced 359 clones of a genomic fragment encompassing the E2(p7)-NS2 junction, in two consecutive liver-serum sample pairs from the four patients and in four intermediate serum samples from one of the patients. The results show that the liver/serum complexity ratio is not stable but rather fluctuates widely over time. Hence, the liver/serum complexity ratio does not identify a particular group of patients but a particular state of the infecting quasispecies. Phylogenetic analysis and signature mutation patterns showed that virtually all circulating sequences originated from sequences present in the liver specimens. The overall behavior of the circulating viral quasispecies appears to originate from changes in the relative replication kinetics of the large mutant spectrum present in the infected liver.     

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.     

Stochastic Simulations Suggest that HIV-1 Survives Close to Its Error Threshold

The use of mutagenic drugs to drive HIV-1 past its error threshold presents a novel intervention strategy, as suggested by the quasispecies theory, that may be less susceptible to failure via viral mutation-induced emergence of drug resistance than current strategies. The error threshold of HIV-1, , however, is not known. Application of the quasispecies theory to determine poses significant challenges: Whereas the quasispecies theory considers the asexual reproduction of an infinitely large population of haploid individuals, HIV-1 is diploid, undergoes recombination, and is estimated to have a small effective population size in vivo. We performed population genetics-based stochastic simulations of the within-host evolution of HIV-1 and estimated the structure of the HIV-1 quasispecies and . We found that with small mutation rates, the quasispecies was dominated by genomes with few mutations. Upon increasing the mutation rate, a sharp error catastrophe occurred where the quasispecies became delocalized in sequence space. Using parameter values that quantitatively captured data of viral diversification in HIV-1 patients, we estimated to be substitutions/site/replication, ∼2–6 fold higher than the natural mutation rate of HIV-1, suggesting that HIV-1 survives close to its error threshold and may be readily susceptible to mutagenic drugs. The latter estimate was weakly dependent on the within-host effective population size of HIV-1. With large population sizes and in the absence of recombination, our simulations converged to the quasispecies theory, bridging the gap between quasispecies theory and population genetics-based approaches to describing HIV-1 evolution. Further, increased with the recombination rate, rendering HIV-1 less susceptible to error catastrophe, thus elucidating an added benefit of recombination to HIV-1. Our estimate of may serve as a quantitative guideline for the use of mutagenic drugs against HIV-1.