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Selection in a time-periodic environment is modeled via the continuous-time two-player replicator dynamics, which for symmetric payoffs reduces to the Fisher equation of mathematical genetics. For a sufficiently rapid and cyclic (fine-grained) environment, the time-averaged population frequencies are shown to obey a replicator dynamics with a nonlinear fitness that is induced by environmental changes. The nonlinear terms in the fitness emerge due to populations tracking their time-dependent environment. These terms can induce a stable polymorphism, though they do not spoil the polymorphism that exists already without them. In this sense polymorphic populations are more robust with respect to their time-dependent environments. The overall fitness of the problem is still given by its time-averaged value, but the emergence of polymorphism during genetic selection can be accompanied by decreasing mean fitness of the population. The impact of the uncovered polymorphism scenario on the models of diversity is exemplified via the rock-paper-scissors dynamics, and also via the prisoner's dilemma in a time-periodic environment.

We consider the Wright-Fisher model of the finite population evolution on a fitness landscape defined in the sequence space by a path of nearly neutral mutations. We study a specific structure of the fitness landscape: One of the intermediate mutations on the mutation path results in either a large fitness value (climbing up a fitness hill) or a low fitness value (crossing a fitness canyon), the rest of the mutations besides the last one are neutral, and the last sequence has much higher fitness than any intermediate sequence. We derive analytical formulas for the first arrival time of the mutant with two point mutations. For the first arrival problem for the further mutants in the case of canyon crossing, we analytically deduce how the mean first arrival time scales with the population size and fitness difference. The location of the canyon on the path of sequences has a crucial role. If the canyon is at the beginning of the path, then it significantly prolongs the first arrival time; otherwise it just slightly changes it. Furthermore, the fitness hill at the beginning of the path strongly prolongs the arrival time period; however, the hill located near the end of the path shortens it. We optimize the first arrival time by applying a nonzero selection to the intermediate sequences. We extend our results and provide a scaling for the valley crossing time via the depth of the canyon and population size in the case of a fitness canyon at the first position. Our approach is useful for understanding some complex evolution systems, e.g., the evolution of cancer.

A major challenge to understanding the genetic basis of complex behavioral evolution is the quantification of complex behaviors themselves. Deer mice of the genus Peromyscus vary in their burrowing behavior, which leaves behind a physical trace that is easily preserved and measured. Moreover, natural burrowing behaviors are recapitulated in the lab, and there is a strong heritable component. Here we discuss potential mechanisms driving variation in burrows with an emphasis on two sister species: P. maniculatus, which digs a simple, short burrow, and P. polionotus, which digs a long burrow with a complex architecture. A forward-genetic cross between these two species identified several genomic regions associated with burrow traits, suggesting this complex behavior has evolved in a modular fashion. Because burrow differences are most likely due to differences in behavioral circuits, Peromyscus burrowing offers an exciting opportunity to link genetic variation between natural populations to evolutionary changes in neural circuits., (Copyright © 2016 Elsevier Ltd. All rights reserved.)

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.

Lifespan is a remarkably diverse trait ranging from a few days to several hundred years in nature, but the mechanisms underlying the evolution of lifespan differences remain elusive. Here we de novo assemble a reference genome for the naturally short-lived African turquoise killifish, providing a unique resource for comparative and experimental genomics. The identification of genes under positive selection in this fish reveals potential candidates to explain its compressed lifespan. Several aging genes are under positive selection in this short-lived fish and long-lived species, raising the intriguing possibility that the same gene could underlie evolution of both compressed and extended lifespans. Comparative genomics and linkage analysis identify candidate genes associated with lifespan differences between various turquoise killifish strains. Remarkably, these genes are clustered on the sex chromosome, suggesting that short lifespan might have co-evolved with sex determination. Our study provides insights into the evolutionary forces that shape lifespan in nature., (Copyright © 2015 Elsevier Inc. All rights reserved.)

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.

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 100-1000 nucleotides, while for the most primitive genome the length is around 7000-20,000. 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.

We calculate the mean fitness for evolution models, when the fitness is a function of the Hamming distance from a reference sequence, and there is a probability that this fitness is nullified (Eigen model case) or tends to the negative infinity (Crow-Kimura model case). We calculate the mean fitness of these models. The mean fitness is calculated also for the random fitnesses with logarithmic-normal distribution, reasonably describing sometimes the situation with RNA viruses.

We investigate the phenomenon of selection via flatness. In the static case, the finiteness of the population does not seriously influence the increase of mean fitness of population due to flatness around a peak. The effect is proportional to 1/square root(L), where L is the genome length. We investigated the two peak model (high peak and a flat peak). We find that the selection of flatness for long genome lengths occurs as a dynamic phenomenon in the case of evolution with small populations. We found that two factors are crucial to define the role of flatness: special initial distribution (the population is located at centers of peaks) allows flat peak to attract more population, and the large value of mutations per population per virus life cycle sometimes also increases the role of flatness. We suggested simple criteria to identify the phenomenon of dynamical arresting of population around flat peak by experiment. We infer that selection via robustness is possible in evolution as a nonequilibrium phenomenon.

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.

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.

We use a path integral representation to solve the Eigen and Crow-Kimura molecular evolution models for the case of multiple fitness peaks with arbitrary fitness and degradation functions. In the general case, we find that the solution to these molecular evolution models can be written as the optimum of a fitness function, with constraints enforced by Lagrange multipliers and with a term accounting for the entropy of the spreading population in sequence space. The results for the Eigen model are applied to consider virus or cancer proliferation under the control of drugs or the immune system.

We present an exact solution of Eigen's quasispecies model with a general degradation rate and fitness functions, including a square root decrease of fitness with increasing Hamming distance from the wild type. The found behavior of the model with a degradation rate is analogous to a viral quasispecies under attack by the immune system of the host. Our exact solutions also revise the known results of neutral networks in quasispecies theory. To explain the existence of mutants with large Hamming distances from the wild type, we propose three different modifications of the Eigen model: mutation landscape, multiple adjacent mutations, and frequency-dependent fitness in which the steady-state solution shows a multicenter behavior.

In a recent paper [Phys. Rev. E 69, 046121 (2004)]], we used the Suzuki-Trottere formalism to study a quasispecies biological evolution model in a parallel mutation-selection scheme with a single-peak fitness function and a point mutation. In the present paper, we extend such a study to evolution models with more general fitness functions or multiple mutations in the parallel mutation-selection scheme. We give some analytical equations to define the error thresholds for some general cases of mean-field-like or symmetric mutation schemes and fitness functions. We derive some equations for the dynamics in the case of a point mutation and polynomial fitness functions. We derive exact dynamics for two-point mutations, asymmetric mutations, and the four-value spin model with a single-peak fitness function. The same method is applied for the model with a royal road fitness function. We derive the steady-state distribution for the single-peak fitness function.