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.     

Chaotic dynamics may determine the effect of inter-patch migration on metapopulation survival

A simple, strategic model of a system of habitat fragments connected by conservation corridors is presented. The intrinsic dynamics of the population on each fragment are stochastic. In addition, at each generation there is a probability of a catastrophic event occurring which affects all the habitat fragments by greatly reducing the size of the population on each. Global extinction is considered to occur when all the populations simultaneously fall below a threshold value. If the intrinsic dynamics on each fragment are simple cycles or a stable equilibrium, then the addition of conservation corridors does not reduce the frequency of global extinction. This is because migration between fragments induces their populations to have values which are similar to each other. However, if the intrinsic population dynamics are chaotic then the probability of global extinction is greatly reduced by the introduction of conservation corridors. Although local extinction is likely, the chaos acts to oppose the synchronising effect of migration. Often a subset of the populations survive a catastrophe and can recolonize the other patches.     

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.     

Error threshold in RNA quasispecies models with complementation

A general assumption of quasispecies models of replicons dynamics is that the fitness of a genotype is entirely determined by its sequence. However, a more biologically plausible situation is that fitness depends on the proteins that catalyze metabolic reactions, including replication. In a stirred population of replicons, such as viruses replicating and accumulating within the same cell, the association between a given genome and the proteins it encodes is not tight as it can be replicated by proteins translated from other genomes. We have investigated how this complementation phenomenon affects the error threshold in simple quasispecies mean field models. We first studied a model in which the master and the mutant genomes code for wild-type and mutant replicases, respectively. We assume that the mutant replicase has a reduced activity and that the wild-type replicase does not have increased affinity for the master genome. The whole pool of replicases can bind and replicate both genomes. We then analyze a different model considering a more extreme case of mutant genomes, the defective interfering particles (DIPs) described in many cases of viral infection. DIPs, with a higher replication rate owed to their shorter genomes, do not code for replicase, but they are able of using the replicase translated from the master genome. Our models allow to study how the probability of interaction between the genomes and the whole pool of replicases affects the error threshold. In both systems we characterize the scenario of coexistence between master and mutant genomes, providing the critical values of mutation rate, μ_c, and the critical interaction rate between master genomes and replicases, γ_c, at which the quasispecies enters into error catastrophe, a situation in which the mutant genomes dominate the population. In both cases, we showed that the error-threshold transition is given by transcritical-like bifurcations, suggesting a continuous phase transition. We have also found that the region in the parameter space (μ,γ) in which the master sequence survives is reduced when DIPs are introduced into the system.     

On the Spectrum of Prebiotic Chemical Systems

We reexamine Eigen’s paradox using the asymptotic limit theorems of information theory. Applying the homology between information source uncertainty and free energy density, under rate distortion constraints, the error catastrophe emerges as the lowest energy state for simple prebiotic systems without error correction. Invoking the usual compartmentalization – i.e., ‘vesicles’ – and using a Red Queen argument, suggests that information crosstalk between two or more properly interacting structures can initiate a coevolutionary dynamic having at least two quasi-stable states. The first is a low energy realm near the error threshold, and, depending on available energy, the second can approach zero error as a limit. A large deviations argument produces jet-like global transitions which, over sufficient time, may enable shifts between the many quasi-stable modes available to more complicated structures, ‘locking in’ to some subset of the various possible low error rate chemical systems, which become subject to development by selection and chance extinction. Energy availability, according to the model, is thus a powerful necessary condition for low error rate replication, suggesting that some fundamental prebiotic ecosystem transformation entrained reproductive fidelity. This work, then, supports speculation that our RNA/DNA world may indeed be only the chance result of a very broad prebiotic evolutionary phenomenon. Processes in vitro, or ex planeta, might have other outcomes.     

Pathways to extinction: beyond the error threshold

Since the introduction of the quasispecies and the error catastrophe concepts for molecular evolution by Eigen and their subsequent application to viral populations, increased mutagenesis has become a common strategy to cause the extinction of viral infectivity. Nevertheless, the high complexity of virus populations has shown that viral extinction can occur through several other pathways apart from crossing an error threshold. Increases in the mutation rate enhance the appearance of defective forms and promote the selection of mechanisms that are able to counteract the accelerated appearance of mutations. Current models of viral evolution take into account more realistic scenarios that consider compensatory and lethal mutations, a highly redundant genotype-to-phenotype map, rough fitness landscapes relating phenotype and fitness, and where phenotype is described as a set of interdependent traits. Further, viral populations cannot be understood without specifying the characteristics of the environment where they evolve and adapt. Altogether, it turns out that the pathways through which viral quasispecies go extinct are multiple and diverse.     

Predicting patterns of long‐term adaptation and extinction with population genetics

Population genetics struggles to model extinction; standard models track the relative rather than absolute fitness of genotypes, while the exceptions describe only the short‐term transition from imminent doom to evolutionary rescue. But extinction can result from failure to adapt not only to catastrophes, but also to a backlog of environmental challenges. We model long‐term adaptation to long series of small challenges, where fitter populations reach higher population sizes. The population's long‐term fitness dynamic is well approximated by a simple stochastic Markov chain model. Long‐term persistence occurs when the rate of adaptation exceeds the rate of environmental deterioration for some genotypes. Long‐term persistence times are consistent with typical fossil species persistence times of several million years. Immediately preceding extinction, fitness declines rapidly, appearing as though a catastrophe disrupted a stably established population, even though gradual evolutionary processes are responsible. New populations go through an establishment phase where, despite being demographically viable, their extinction risk is elevated. Should the population survive long enough, extinction risk later becomes constant over time.     

Mutation induced extinction in finite populations: lethal mutagenesis and lethal isolation

Reproduction is inherently risky, in part because genomic replication can introduce new mutations that are usually deleterious toward fitness. This risk is especially severe for organisms whose genomes replicate "semi-conservatively," e.g. viruses and bacteria, where no master copy of the genome is preserved. Lethal mutagenesis refers to extinction of populations due to an unbearably high mutation rate (U), and is important both theoretically and clinically, where drugs can extinguish pathogens by increasing their mutation rate. Previous theoretical models of lethal mutagenesis assume infinite population size (N). However, in addition to high U, small N can accelerate extinction by strengthening genetic drift and relaxing selection. Here, we examine how the time until extinction depends jointly on N and U. We first analytically compute the mean time until extinction (τ) in a simplistic model where all mutations are either lethal or neutral. The solution motivates the definition of two distinct regimes: a survival phase and an extinction phase, which differ dramatically in both how τ scales with N and in the coefficient of variation in time until extinction. Next, we perform stochastic population-genetics simulations on a realistic fitness landscape that both (i) features an epistatic distribution of fitness effects that agrees with experimental data on viruses and (ii) is based on the biophysics of protein folding. More specifically, we assume that mutations inflict fitness penalties proportional to the extent that they unfold proteins. We find that decreasing N can cause phase transition-like behavior from survival to extinction, which motivates the concept of "lethal isolation." Furthermore, we find that lethal mutagenesis and lethal isolation interact synergistically, which may have clinical implications for treating infections. Broadly, we conclude that stably folded proteins are only possible in ecological settings that support sufficiently large populations.     

Lethal mutants and truncated selection together solve a paradox of the origin of life

Background

Many attempts have been made to describe the origin of life, one of which is Eigen''s cycle of autocatalytic reactions [Eigen M (1971) Naturwissenschaften 58, 465–523], in which primordial life molecules are replicated with limited accuracy through autocatalytic reactions. For successful evolution, the information carrier (either RNA or DNA or their precursor) must be transmitted to the next generation with a minimal number of misprints. In Eigen''s theory, the maximum chain length that could be maintained is restricted to nucleotides, while for the most primitive genome the length is around . This is the famous error catastrophe paradox. How to solve this puzzle is an interesting and important problem in the theory of the origin of life.

Methodology/Principal Findings

We use methods of statistical physics to solve this paradox by carefully analyzing the implications of neutral and lethal mutants, and truncated selection (i.e., when fitness is zero after a certain Hamming distance from the master sequence) for the critical chain length. While neutral mutants play an important role in evolution, they do not provide a solution to the paradox. We have found that lethal mutants and truncated selection together can solve the error catastrophe paradox. There is a principal difference between prebiotic molecule self-replication and proto-cell self-replication stages in the origin of life.

Conclusions/Significance

We have applied methods of statistical physics to make an important breakthrough in the molecular theory of the origin of life. Our results will inspire further studies on the molecular theory of the origin of life and biological evolution.     

Does red noise increase or decrease extinction risk? Single extreme events versus series of unfavorable conditions

Recent theoretical studies have shown contrasting effects of temporal correlation of environmental fluctuations (red noise) on the risk of population extinction. It is still debated whether and under which conditions red noise increases or decreases extinction risk compared with uncorrelated (white) noise. Here, we explain the opposing effects by introducing two features of red noise time series. On the one hand, positive autocorrelation increases the probability of series of poor environmental conditions, implying increasing extinction risk. On the other hand, for a given time period, the probability of at least one extremely bad year ("catastrophe") is reduced compared with white noise, implying decreasing extinction risk. Which of these two features determines extinction risk depends on the strength of environmental fluctuations and the sensitivity of population dynamics to these fluctuations. If extreme (catastrophic) events can occur (strong noise) or sensitivity is high (overcompensatory density dependence), then temporal correlation decreases extinction risk; otherwise, it increases it. Thus, our results provide a simple explanation for the contrasting previous findings and are a crucial step toward a general understanding of the effect of noise color on extinction risk.     

Genetic instability and the quasispecies model

Genetic instability is a defining characteristic of cancers. Microsatellite instability (MIN) leads to by elevated point mutation rates, whereas chromosomal instability (CIN) refers to increased rates of losing or gaining whole chromosomes or parts of chromosomes during cell division. CIN and MIN are, in general, mutually exclusive. The quasispecies model is a very successful theoretical framework for the study of evolution at high mutation rates. It predicts the existence of an experimentally verified error catastrophe. This catastrophe occurs when the mutation rates exceed a threshold value, the error threshold, above which replicative infidelity is incompatible with cell survival. We analyse the semiconservative quasispecies model of both MIN and CIN tumors. We consider the role of post-methylation DNA repair in tumor cells and demonstrate that DNA repair is fundamental to the nature of the error catastrophe in both types of tumors. We find that CIN introduces a plateau in the maximum viable mutation rate for a repair-free model, which does not exist in the case of MIN. This provides a plausible explanation for the mutual exclusivity of CIN and MIN.     

A Quasistationary Analysis of a Stochastic Chemical Reaction: Keizer’s Paradox

For a system of biochemical reactions, it is known from the work of T.G. Kurtz [J. Appl. Prob. 8, 344 (1971)] that the chemical master equation model based on a stochastic formulation approaches the deterministic model based on the Law of Mass Action in the infinite system-size limit in finite time. The two models, however, often show distinctly different steady-state behavior. To further investigate this “paradox,” a comparative study of the deterministic and stochastic models of a simple autocatalytic biochemical reaction, taken from a text by the late J. Keizer, is carried out. We compute the expected time to extinction, the true stochastic steady state, and a quasistationary probability distribution in the stochastic model. We show that the stochastic model predicts the deterministic behavior on a reasonable time scale, which can be consistently obtained from both models. The transition time to the extinction, however, grows exponentially with the system size. Mathematically, we identify that exchanging the limits of infinite system size and infinite time is problematic. The appropriate system size that can be considered sufficiently large, an important parameter in numerical computation, is also discussed.     

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.     

How patch configuration affects the impact of disturbances on metapopulation persistence

Disturbances affect metapopulations directly through reductions in population size and indirectly through habitat modification. We consider how metapopulation persistence is affected by different disturbance regimes and the way in which disturbances spread, when metapopulations are compact or elongated, using a stochastic spatially explicit model which includes metapopulation and habitat dynamics. We discover that the risk of population extinction is larger for spatially aggregated disturbances than for spatially random disturbances. By changing the spatial configuration of the patches in the system--leading to different proportions of edge and interior patches--we demonstrate that the probability of metapopulation extinction is smaller when the metapopulation is more compact. Both of these results become more pronounced when colonization connectivity decreases. Our results have important management implication as edge patches, which are invariably considered to be less important, may play an important role as disturbance refugia.     

Error thresholds for quasispecies on single peak Gaussian-distributed fitness landscapes

Based on the Eigen and Crow-Kimura models with a single peak fitness landscape, we propose that the fitness values of all molecules be Gaussian distributed random variables to incorporate the fluctuation effects of the fitness landscapes (noise of environments). And we investigate the quasispecies distribution and error threshold using ensemble average method within this theoretical framework. Numerical results show that a small fluctuation of the fitness landscape causes only a slight change in the concentration distribution and error threshold, which implies that the error threshold is stable against small perturbations. However, for a sizable fluctuation, quite different from the previous deterministic models, our statistical results reveal that the transition from quasi-species to error catastrophe is no longer so sharp, indicating the error threshold is located within a certain range and shifts toward a larger value.     

Error catastrophe in and the evolution of the protein synthesizing machinery

The theory of the propagation of errors in the system of enzymes translating genetic information proposed earlier is developed further. It is shown that if initially for some of these enzymes the normal molecules are more than, for some less than, and for some equal to the erroneous ones, the evolution of the system can lead to many asymptotic conditions including error catastrophe and complete recovery from errors.     

Population oscillations in a discrete model of neural networks of the brain

A discrete model of biological neural networks is used to find out how synchronized firing of neurons emerges in a randomly connected neural population. The objective is to understand the mechanisms underlying brain waves and to find and characterize conditions which support spontaneous switching from disordered to rhythmic population activity as in case of an epileptic seizure. The model is kept as simple as possible to achieve on one hand a fast performance of computer simulations of networks with up to 10,000 neurons and to keep on the other hand an overview of parameter dependences. Dynamics of the model can be classified into different regimes: random fluctuations, rhythmic oscillations and silence. When the ratio of the inhibitory/excitatory connectivity is raised the system crosses from the fluctuating regime through the rhythmic oscillating region to the silence regime. Close to the boundary between the fluctuating and the oscillating regimes the network shows spontaneous bursting of high amplitude rhythmic oscillations, which is characteristic of epileptiform behavior. The simulation results are in agreement with recent theories saying that focal epilepsy after injury of the brain could result from axonal sprouting of GABAergic neurons in the injured region.     

Hybrid lethal systems in theDrosophila melanogaster species complex

Lethal phases of the hybrids betweenDrosophila melanogaster and its sibling species,D. simulans are classified into three types: (1) embryonic lethality in hybrids carryingD. simulans cytoplasm andD. melanogaster X chromosome, (2) larval lethality in hybrids not carryingD. simulans X, and (3) temperature-sensitive pupal lethality in hybrids carryingD. simulans X. The same lethal phases are also observed when either of the two other sibling species,D. mauritiana orD. sechellia, is employed for hybridization withD. melanogaster. Here, we describe genetic analyses of each hybrid lethality, and demonstrate that these three types of lethality are independent phenomena. We then propose two models to interpret the mechanisms of each hybrid lethality. The first model is a modification of the conventional X/autosome imbalance hypothesis assuming a lethal gene and a suppressor gene are involved in the larval lethality, while the second model is for embryonic lethality assuming an interaction between a maternal-effect lethal gene and a suppressor gene.     

Fire regimes and shifting community patterns: a case study and method for species with complex fire requirements

Anthropogenic alteration of fire regimes is implicated in the extinction or decline of species across the globe. Active management of fire regimes using specified guidelines can help sustain diversity and counter this loss. However, some species occur across multiple communities with differing fire regime requirements or may exploit the dynamics of associated ecotones. These complexities are not easily understood via a simple consideration of life history traits or unravelled by a simple experimental framework. Population modelling, however, may provide valuable insights in these instances, but to date this remains unexplored. A population model based on detailed demographic, habitat and fire regime data was developed for the species Prostanthera askania. The species occurs across rainforest/eucalypt forest ecotones subject to long-term alteration, fragmentation and invasion by exotic species. Modelling revealed that current recommended fire guidelines will not sustain this species under these novel conditions. Fire regimes that minimised extinction risk in some scenarios had double or five times the fire frequencies of recommended regimes. Managing fire was also more important in specific habitat (gully habitat rather than on slopes or ridges). Currently applied management actions for Prostanthera askania do not include any fire management and the regimes that apply to most habitat in which the species occurs (no fire or fire at up to 50-year intervals in gully habitat) is not optimal for the species. Management will be substantially more effective if specific fire treatments revealed by this study are employed (e.g. fire intervals of 20–30 years in gully habitat). The study demonstrates that a conceptualisation and consideration of communities as both temporally and spatially dynamic can substantially contribute to better fire management outcomes. The approach used herein can be readily adapted and applied to a plethora of species via a range of software packages and codes.