Evolution equation of phenotype distribution: general formulation and application to error catastrophe

Gene-influx-driven evolution

Here we analyze the evolutionary process in the presence of continuous influx of genotypes with submaximum fitness from the outside to the given habitat with finite resources. We show that strong influx from the outside allows the low-fitness genotype to win the competition with the higher fitness genotype, and in a finite population, drive the latter to extinction. We analyze a mathematical model of this phenomenon and obtain the conditions for the transition from the high-fitness to the low-fitness genotype caused by the influx of the latter. We calculate the time to extinction of the high-fitness genotype in a finite population with two alleles and find the exact analytical dynamics of extinction for the case of many genes with epistasis. We solve a related quasispecies model for a single peak (random) fitness landscape as well as for a symmetric fitness landscape. In the symmetric landscape, a nonperturbative effect is observed such that even an extremely low influx of the low-fitness genotype drastically changes the steady state fitness distribution. A similar nonperturbative phenomenon is observed for the allele fixation time as well. The identified regime of influx-driven evolution appears to be relevant for a broad class of biological systems and could be central to the evolution of prokaryotes and viruses.

Redundancy-selection trade-off in phenotype-structured populations

Realistic fitness landscapes generally display a redundancy-fitness trade-off: highly fit trait configurations are inevitably rare, while less fit trait configurations are expected to be more redundant. The resulting sub-optimal patterns in the fitness distribution are typically described by means of effective formulations, where redundancy provided by the presence of neutral contributions is modelled implicitly, e.g. with a bias of the mutation process. However, the extent to which effective formulations are compatible with explicitly redundant landscapes is yet to be understood, as well as the consequences of a potential miss-match. Here we investigate the effects of such trade-off on the evolution of phenotype-structured populations, characterised by continuous quantitative traits. We consider a typical replication-mutation dynamics, and we model redundancy by means of two dimensional landscapes displaying both selective and neutral traits. We show that asymmetries of the landscapes will generate neutral contributions to the marginalised fitness-level description, that cannot be described by effective formulations, nor disentangled by the full trait distribution. Rather, they appear as effective sources, whose magnitude depends on the geometry of the landscape. Our results highlight new important aspects on the nature of sub-optimality. We discuss practical implications for rapidly mutant populations such as pathogens and cancer cells, where the qualitative knowledge of their trait and fitness distributions can drive disease management and intervention policies.

Phenotypic-dependent variability and the emergence of tolerance in bacterial populations

Ecological and evolutionary dynamics have been historically regarded as unfolding at broadly separated timescales. However, these two types of processes are nowadays well-documented to intersperse much more tightly than traditionally assumed, especially in communities of microorganisms. Advancing the development of mathematical and computational approaches to shed novel light onto eco-evolutionary problems is a challenge of utmost relevance. With this motivation in mind, here we scrutinize recent experimental results showing evidence of rapid evolution of tolerance by lag in bacterial populations that are periodically exposed to antibiotic stress in laboratory conditions. In particular, the distribution of single-cell lag times-i.e., the times that individual bacteria from the community remain in a dormant state to cope with stress-evolves its average value to approximately fit the antibiotic-exposure time. Moreover, the distribution develops right-skewed heavy tails, revealing the presence of individuals with anomalously large lag times. Here, we develop a parsimonious individual-based model mimicking the actual demographic processes of the experimental setup. Individuals are characterized by a single phenotypic trait: their intrinsic lag time, which is transmitted with variation to the progeny. The model-in a version in which the amplitude of phenotypic variations grows with the parent's lag time-is able to reproduce quite well the key empirical observations. Furthermore, we develop a general mathematical framework allowing us to describe with good accuracy the properties of the stochastic model by means of a macroscopic equation, which generalizes the Crow-Kimura equation in population genetics. Even if the model does not account for all the biological mechanisms (e.g., genetic changes) in a detailed way-i.e., it is a phenomenological one-it sheds light onto the eco-evolutionary dynamics of the problem and can be helpful to design strategies to hinder the emergence of tolerance in bacterial communities. From a broader perspective, this work represents a benchmark for the mathematical framework designed to tackle much more general eco-evolutionary problems, thus paving the road to further research avenues.

Solution of the Crow-Kimura model with a periodically changing (two-season) fitness function

Since the origin of life, both evolutionary dynamics and rhythms have played a key role in the functioning of living systems. The Crow-Kimura model of periodically changing fitness function has been solved exactly, using integral equation with time-ordered exponent. We also found a simple approximate solution for the two-season case. The evolutionary dynamics accompanied by the rhythms provide important insights into the properties of certain biological systems and processes.

Key role of recombination in evolutionary processes with migration between two habitats

Recombination is one of the leading forces of evolutionary dynamics. Although the importance of both recombination and migration in evolution is well recognized, there is currently no exact theory of evolutionary dynamics for large genome models that incorporates recombination, mutation, selection (quasispecies model with recombination), and spatial dynamics. To address this problem, we analyze the simplest spatial evolutionary process, namely, evolution of haploid populations with mutation, selection, recombination, and unidirectional migration, in its exact analytical form. This model is based on the quasispecies theory with recombination, but with replicators migrating from one habitat to another. In standard evolutionary models involving one habitat, the evolutionary processes depend on the ratios of fitness for different sequences. In the case of migration, we consider the absolute fitness values because there is no competition for resources between the population of different habitats. In the standard model without epistasis, recombination does not affect the mean fitness of the population. When migration is introduced, the situation changes drastically such that recombination can affect the mean fitness as strongly as mutation, as has been observed by Li and Nei for a few loci model without mutations. We have solved our model in the limit of large genome size for the fitness landscapes having different peaks in the first and second habitats and obtained the total population sizes for both habitats as well as the proportion of the population around two peak sequences in the second habitat. We identify four phases in the model and present the exact solutions for three of them.

Approximate perturbative solutions of quasispecies model with recombination

Despite the major roles played by genetic recombination in ecoevolutionary processes, limited progress has been made in analyzing realistic recombination models to date, due largely to the complexity of the associated mechanisms and the strongly nonlinear nature of the dynamical differential systems. In this paper, we consider a many-loci genomic model with fitness dependent on the Hamming distance from a reference genome, and adopt a Hamilton-Jacobi formulation to derive perturbative solutions for general linear fitness landscapes. The horizontal gene transfer model is used to describe recombination processes. Cases of weak selection and weak recombination with simultaneous mutation and selection are examined, yielding semianalytical solutions for the distribution surplus of O(1/N) accuracy, where N is the number of nucleotides in the genome.

Cryptic selection forces and dynamic heritability in generalized phenotypic evolution

Individuals with different phenotypes can have widely-varying responses to natural selection, yet many classical approaches to evolutionary dynamics emphasize only how a population's average phenotype increases in fitness over time. However, recent experimental results have produced examples of populations that have multiple fitness peaks, or that experience frequency-dependence that affects the direction and strength of selection on certain individuals. Here, we extend classical fitness gradient formulations of natural selection in order to describe the dynamics of a phenotype distribution in terms of its moments-such as the mean, variance, and skewness. The number of governing equations in our model can be adjusted in order to capture different degrees of detail about the population. We compare our simplified model to direct Wright-Fisher simulations of evolution in several canonical fitness landscapes, and we find that our model provides a low-dimensional description of complex dynamics not typically explained by classical theory, such as cryptic selection forces due to selection on trait ranges, time-variation of the heritability, and nonlinear responses to stabilizing or disruptive selection due to asymmetric trait distributions. In addition to providing a framework for extending general understanding of common qualitative concepts in phenotypic evolution - such as fitness gradients, selection pressures, and heritability - our approach has practical importance for studying evolution in contexts in which genetic analysis is infeasible.

Quasispecies model of evolution with migration

We consider the model of asexual evolution with migration, which was proposed by Waclaw et al. [Phys. Rev. Lett. 105, 268101 (2010)PRLTAO0031-900710.1103/PhysRevLett.105.268101]. This model setting is based on the standard mutation scheme from the quasispecies theory but with replicators moving from one habitat to another. The primary goal is to solve exactly the infinite population-genome length version of the model for the independent random distribution of fitnesses considered in the original paper. Moreover, we propose the analytical solution for the single peak and the symmetric fitness landscape. Our analytical solution is exact at the limit of large N. We found two phases-the correlated phase with the identical distributions of mutations in both habitats and the uncorrelated phase where the second habitat is choosing another peak of distribution in sequence space compared to the peak in the first habitat.

The rich phase structure of a mutator model

We propose a modification of the Crow-Kimura and Eigen models of biological molecular evolution to include a mutator gene that causes both an increase in the mutation rate and a change in the fitness landscape. This mutator effect relates to a wide range of biomedical problems. There are three possible phases: mutator phase, mixed phase and non-selective phase. We calculate the phase structure, the mean fitness and the fraction of the mutator allele in the population, which can be applied to describe cancer development and RNA viruses. We find that depending on the genome length, either the normal or the mutator allele dominates in the mixed phase. We analytically solve the model for a general fitness function. We conclude that the random fitness landscape is an appropriate choice for describing the observed mutator phenomenon in the case of a small fraction of mutators. It is shown that the increase in the mutation rates in the regular and the mutator parts of the genome should be set independently; only some combinations of these increases can push the complex biomedical system to the non-selective phase, potentially related to the eradication of tumors.

Solution of classical evolutionary models in the limit when the diffusion approximation breaks down

The discrete time mathematical models of evolution (the discrete time Eigen model, the Moran model, and the Wright-Fisher model) have many applications in complex biological systems. The discrete time Eigen model rather realistically describes the serial passage experiments in biology. Nevertheless, the dynamics of the discrete time Eigen model is solved in this paper. The 90% of results in population genetics are connected with the diffusion approximation of the Wright-Fisher and Moran models. We considered the discrete time Eigen model of asexual virus evolution and the Wright-Fisher model from population genetics. We look at the logarithm of probabilities and apply the Hamilton-Jacobi equation for the models. We define exact dynamics for the population distribution for the discrete time Eigen model. For the Wright-Fisher model, we express the exact steady state solution and fixation probability via the solution of some nonlocal equation then give the series expansion of the solution via degrees of selection and mutation rates. The diffusion theories result in the zeroth order approximation in our approach. The numeric confirms that our method works in the case of strong selection, whereas the diffusion method fails there. Although the diffusion method is exact for the mean first arrival time, it provides incorrect approximation for the dynamics of the tail of distribution.

Mathematical Models of Quasi-Species Theory and Exact Results for the Dynamics

We formulate the Crow-Kimura, discrete-time Eigen model, and continuous-time Eigen model. These models are interrelated and we established an exact mapping between them. We consider the evolutionary dynamics for the single-peak fitness and symmetric smooth fitness. We applied the quantum mechanical methods to find the exact dynamics of the evolution model with a single-peak fitness. For the smooth symmetric fitness landscape, we map exactly the evolution equations into Hamilton-Jacobi equation (HJE). We apply the method to the Crow-Kimura (parallel) and Eigen models. We get simple formulas to calculate the dynamics of the maximum of distribution and the variance. We review the existing mathematical tools of quasi-species theory.

Punctuated equilibrium and shock waves in molecular models of biological evolution

We consider the dynamics in infinite population evolution models with a general symmetric fitness landscape. We find shock waves, i.e., discontinuous transitions in the mean fitness, in evolution dynamics even with smooth fitness landscapes, which means that the search for the optimal evolution trajectory is more complicated. These shock waves appear in the case of positive epistasis and can be used to represent punctuated equilibria in biological evolution during long geological time scales. We find exact analytical solutions for discontinuous dynamics at the large-genome-length limit and derive optimal mutation rates for a fixed fitness landscape to send the population from the initial configuration to some final configuration in the fastest way.

Evolutionary advantage via common action of recombination and neutrality

We investigate evolution models with recombination and neutrality. We consider the Crow-Kimura (parallel) mutation-selection model with the neutral fitness landscape, in which there is a central peak with high fitness A, and some of 1-point mutants have the same high fitness A, while the fitness of other sequences is 0. We find that the effect of recombination and neutrality depends on the concrete version of both neutrality and recombination. We consider three versions of neutrality: (a) all the nearest neighbor sequences of the peak sequence have the same high fitness A; (b) all the l-point mutations in a piece of genome of length l≥1 are neutral; (c) the neutral sequences are randomly distributed among the nearest neighbors of the peak sequences. We also consider three versions of recombination: (I) the simple horizontal gene transfer (HGT) of one nucleotide; (II) the exchange of a piece of genome of length l, HGT-l; (III) two-point crossover recombination (2CR). For the case of (a), the 2CR gives a rather strong contribution to the mean fitness, much stronger than that of HGT for a large genome length L. For the random distribution of neutral sequences there is a critical degree of neutrality ν(c), and for μ<μ(c) and (μ(c)-μ) is not large, the 2CR suppresses the mean fitness while HGT increases it; for ν much larger than ν(c), the 2CR and HGT-l increase the mean fitness larger than that of the HGT. We also consider the recombination in the case of smooth fitness landscapes. The recombination gives some advantage in the evolutionary dynamics, where recombination distinguishes clearly the mean-field-like evolutionary factors from the fluctuation-like ones. By contrast, mutations affect the mean-field-like and fluctuation-like factors similarly. Consequently, recombination can accelerate the non-mean-field (fluctuation) type dynamics without considerably affecting the mean-field-like factors.

Baldwin effect under multipeaked fitness landscapes: phenotypic fluctuation accelerates evolutionary rate

Phenotypic fluctuations and plasticity can generally affect the course of evolution, a process known as the Baldwin effect. Several studies have recast this effect and claimed that phenotypic plasticity accelerates evolutionary rate (the Baldwin expediting effect); however, the validity of this claim is still controversial. In this study, we investigate the evolutionary population dynamics of a quantitative genetic model under a multipeaked fitness landscape, in order to evaluate the validity of the effect. We provide analytical expressions for the evolutionary rate and average population fitness. Our results indicate that under a multipeaked fitness landscape, phenotypic fluctuation always accelerates evolutionary rate, but it decreases the average fitness. As an extreme case of the trade-off between the rate of evolution and average fitness, phenotypic fluctuation is shown to accelerate the error catastrophe, in which a population fails to sustain a high-fitness peak. In the context of our findings, we discuss the role of phenotypic plasticity in adaptive evolution.

Biological evolution in a multidimensional fitness landscape

We considered a multiblock molecular model of biological evolution, in which fitness is a function of the mean types of alleles located at different parts (blocks) of the genome. We formulated an infinite population model with selection and mutation, and calculated the mean fitness. For the case of recombination, we formulated a model with a multidimensional fitness landscape (the dimension of the space is equal to the number of blocks) and derived a theorem about the dynamics of initially narrow distribution. We also considered the case of lethal mutations. We also formulated the finite population version of the model in the case of lethal mutations. Our models, derived for the virus evolution, are interesting also for the statistical mechanics and the Hamilton-Jacobi equation as well.

Dynamics of the chemical master equation, a strip of chains of equations in d-dimensional space

We investigate the multichain version of the chemical master equation, when there are transitions between different states inside the long chains, as well as transitions between (a few) different chains. In the discrete version, such a model can describe the connected diffusion processes with jumps between different types. We apply the Hamilton-Jacobi equation to solve some aspects of the model. We derive exact (in the limit of infinite number of particles) results for the dynamic of the maximum of the distribution and the variance of distribution.

Phenotypic plasticity and robustness: evolutionary stability theory, gene expression dynamics model, and laboratory experiments

Plasticity and robustness, which are two basic concepts in the evolution of developmental dynamics, are characterized in terms of the variance of phenotype distribution. Plasticity concerns the response of a phenotype against environmental and genetic changes, whereas robustness is the degree of insensitivity against such changes. Note that the sensitivity increases with the variance, and the inverse of the variance works as a measure of the robustness. First, it is found that the response ratio is proportional to the phenotype variance, as described by extending the fluctuation-response relationship in statistical physics. Next, it is shown that through the course of robust evolution, the phenotype variance caused by genetic change decreases in proportion to that by noise during the developmental process. This evolution, resulting in increased robustness, is achieved only when the noise in the developmental process is sufficiently large; in other words, robustness to noise leads to robustness to mutation. For a system that achieves robustness in the phenotype, it is also found that the proportionality between the two variances also holds across different phenotypic traits. These general relationships for plasticity and robustness in terms of fluctuations are demonstrated using macroscopic phenomenological theory, simulations of gene-expression dynamics models with regulation networks, and laboratory selection experiments. It is also shown that an optimal noise level compatibility between robustness and plasticity is achieved to cope with a fluctuating environment.

The prebiotic evolutionary advantage of transferring genetic information from RNA to DNA

In the early 'RNA world' stage of life, RNA stored genetic information and catalyzed chemical reactions. However, the RNA world eventually gave rise to the DNA-RNA-protein world, and this transition included the 'genetic takeover' of information storage by DNA. We investigated evolutionary advantages for using DNA as the genetic material. The error rate of replication imposes a fundamental limit on the amount of information that can be stored in the genome, as mutations degrade information. We compared misincorporation rates of RNA and DNA in experimental non-enzymatic polymerization and calculated the lowest possible error rates from a thermodynamic model. Both analyses found that RNA replication was intrinsically error-prone compared to DNA, suggesting that total genomic information could increase after the transition to DNA. Analysis of the transitional RNA/DNA hybrid duplexes showed that copying RNA into DNA had similar fidelity to RNA replication, so information could be maintained during the genetic takeover. However, copying DNA into RNA was very error-prone, suggesting that attempts to return to the RNA world would result in a considerable loss of information. Therefore, the genetic takeover may have been driven by a combination of increased chemical stability, increased genome size and irreversibility.

Exact results in the large system size limit for the dynamics of the chemical master equation, a one dimensional chain of equations

We apply the Hamilton-Jacobi equation (HJE) formalism to solve the dynamics of the chemical master equation (CME). We found exact analytical expressions (in large system-size limit) for the probability distribution, including explicit expression for the dynamics of variance of distribution. We also give the solution for some simple cases of the model with time-dependent rates. We derived the results of the Van Kampen method from the HJE approach using a special ansatz. Using the Van Kampen method, we give a system of ordinary differential equations (ODEs) to define the variance in a two-dimensional case. We performed numerics for the CME with stationary noise. We give analytical criteria for the disappearance of bistability in the case of stationary noise in one-dimensional CMEs.

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.

Different fitnesses for in vivo and in vitro evolutions due to the finite generation-time effect

We consider the finite generation-time effect in virus evolution models, introducing differential equations with delay. The suggested approach more adequately describes the evolution in case of growing populations than the popular models of population genetics, especially for the viruses with large number of offspring during one life cycle. Now the mean fitness, as a coefficient for exponential population growth, could not be defined via instant characteristics of the model. For the constant population size the finite generation-time does not affect mean fitness in the steady state. The growing virus population is characterized by different fitness than the population with a constant size.

Recombination in one- and two-dimensional fitness landscapes

We consider many-site mutation-recombination models of molecular evolution, where fitness is a function of a Hamming distance from one (one-dimensional case) or two (two-dimensional case) sequences. For the one-dimensional case, we calculate the population distribution dynamics for a model with zero fitness and an arbitrary symmetric initial distribution and find an error threshold transition point in the single-peak fitness model for a given initial symmetric distribution. We calculate the recombination period in the case of a single-peak fitness function, when the original population is located at one sequence, at some Hamming distance from the peak configuration. Steady-state fitness is calculated with finite genome length corrections. We derive analytical equations for the two-dimensional mutation-recombination model.

Robustness and epistasis in mutation-selection models

We investigate the fitness advantage associated with the robustness of a phenotype against deleterious mutations using deterministic mutation-selection models of a quasispecies type equipped with a mesa-shaped fitness landscape. We obtain analytic results for the robustness effect which become exact in the limit of infinite sequence length. Thereby, we are able to clarify a seeming contradiction between recent rigorous work and an earlier heuristic treatment based on mapping to a Schrödinger equation. We exploit the quantum mechanical analogy to calculate a correction term for finite sequence lengths and verify our analytic results by numerical studies. In addition, we investigate the occurrence of an error threshold for a general class of epistatic landscapes and show that diminishing epistasis is a necessary but not sufficient condition for error threshold behaviour.

Phase diagram for the Eigen quasispecies theory with a truncated fitness landscape

Using methods of statistical physics, we present rigorous theoretical calculations of Eigen's quasispecies theory with the truncated fitness landscape which dramatically limits the available sequence space of information carriers. As the mutation rate is increased from small values to large values, one can observe three phases: the first (I) selective (also known as ferromagnetic) phase, the second (II) intermediate phase with some residual order, and the third (III) completely randomized (also known as paramagnetic) phase. We calculate the phase diagram for these phases and the concentration of information carriers in the master sequence (also known as peak configuration) x0 and other classes of information carriers. As the phase point moves across the boundary between phase I and phase II, x0 changes continuously; as the phase point moves across the boundary between phase II and phase III, x0 has a large change. Our results are applicable for the general case of a fitness landscape.

Exploration dynamics in evolutionary games

Evolutionary game theory describes systems where individual success is based on the interaction with others. We consider a system in which players unconditionally imitate more successful strategies but sometimes also explore the available strategies at random. Most research has focused on how strategies spread via genetic reproduction or cultural imitation, but random exploration of the available set of strategies has received less attention so far. In genetic settings, the latter corresponds to mutations in the DNA, whereas in cultural evolution, it describes individuals experimenting with new behaviors. Genetic mutations typically occur with very small probabilities, but random exploration of available strategies in behavioral experiments is common. We term this phenomenon "exploration dynamics" to contrast it with the traditional focus on imitation. As an illustrative example of the emerging evolutionary dynamics, we consider a public goods game with cooperators and defectors and add punishers and the option to abstain from the enterprise in further scenarios. For small mutation rates, cooperation (and punishment) is possible only if interactions are voluntary, whereas moderate mutation rates can lead to high levels of cooperation even in compulsory public goods games. This phenomenon is investigated through numerical simulations and analytical approximations.

Dynamics of the Eigen and the Crow-Kimura models for molecular evolution

We introduce an alternative way to study molecular evolution within well-established Hamilton-Jacobi formalism, showing that for a broad class of fitness landscapes it is possible to derive dynamics analytically within the 1N accuracy, where N is the genome length. For a smooth and monotonic fitness function this approach gives two dynamical phases: smooth dynamics and discontinuous dynamics. The latter phase arises naturally with no explicite singular fitness function, counterintuitively. The Hamilton-Jacobi method yields straightforward analytical results for the models that utilize fitness as a function of Hamming distance from a reference genome sequence. We also show the way in which this method gives dynamical phase structure for multipeak fitness.