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.

Quantifying the stochasticity of policy parameters in reinforcement learning problems.

The stochastic dynamics of reinforcement learning is studied using a master equation formalism. We consider two different problems-Q learning for a two-agent game and the multiarmed bandit problem with policy gradient as the learning method. The master equation is constructed by introducing a probability distribution over continuous policy parameters or over both continuous policy parameters and discrete state variables (a more advanced case). We use a version of the moment closure approximation to solve for the stochastic dynamics of the models. Our method gives accurate estimates for the mean and the (co)variance of policy variables. For the case of the two-agent game, we find that the variance terms are finite at steady state and derive a system of algebraic equations for computing them directly.

A solution of the Crow-Kimura evolution model on fluctuating fitness landscape.

The article discusses the Crow-Kimura model in the context of random transitions between different fitness landscapes. The duration of epochs, during which the fitness landscape is constant over time, is modeled by an exponential distribution. To obtain an exact solution, a system of functional equations is required. However, to approximate the model, we consider the cases of slow or fast transitions and calculate the first-order corrections using either the transition rate or its inverse. Specifically, we focus on the case of slow transitions and find that the average fitness is equal to the average fitness for evolution on static fitness landscapes, but with the addition of a load term. We also investigate the model for a small number of genes and identify the exact transition points to the transient phase.

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.

A simple statistical physics model for the epidemic with incubation period.

Based on the classical SIR model, we derive a simple modification for the dynamics of epidemics with a known incubation period of infection. The model is described by a system of integro-differential equations. Parameters of our model are directly related to epidemiological data. We derive some analytical results, as well as perform numerical simulations. We use the proposed model to analyze COVID-19 epidemic data in Armenia.

Weak mixed phase in the mutator model.

We consider the mutator model with unidirected transitions from the wild type to the mutator type, with different fitness functions for the wild types and mutator types. We calculate both the fraction of mutator types in the population and the surpluses, i.e., the mean number of mutations in the regular part of genomes for the wild type and mutator type, which have never been derived exactly. We identify the phase structure. Beside the mixed (ordinary evolution phase with finite fraction of wild types at large genome length) and the mutator phase (the absolute majority is mutators), we find another new phase as well-it has the mean fitness of the mixed phase but an exponentially small (in genome length) fraction of wild types. We identify the phase transition point and discuss its implications.

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.

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.

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.

Allele fixation probability in a Moran model with fluctuating fitness landscapes.

Evolution on changing fitness landscapes (seascapes) is an important problem in evolutionary biology. We consider the Moran model of finite population evolution with selection in a randomly changing, dynamic environment. In the model, each individual has one of the two alleles, wild type or mutant. We calculate the fixation probability by making a proper ansatz for the logarithm of fixation probabilities. This method has been used previously to solve the analogous problem for the Wright-Fisher model. The fixation probability is related to the solution of a third-order algebraic equation (for the logarithm of fixation probability). We consider the strong interference of landscape fluctuations, sampling, and selection when the fixation process cannot be described by the mean fitness. Such an effect appears if the mutant allele has a higher fitness in one landscape and a lower fitness in another, compared with the wild type, and the product of effective population size and fitness is large. We provide a generalization of the Kimura formula for the fixation probability that applies to these cases. When the mutant allele has a fitness (dis-)advantage in both landscapes, the fixation probability is described by the mean fitness.

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.

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.

Diploid biological evolution models with general smooth fitness landscapes and recombination.

Using a Hamilton-Jacobi equation approach, we obtain analytic equations for steady-state population distributions and mean fitness functions for Crow-Kimura and Eigen-type diploid biological evolution models with general smooth hypergeometric fitness landscapes. Our numerical solutions of diploid biological evolution models confirm the analytic equations obtained. We also study the parallel diploid model for the simple case of recombination and calculate the variance of distribution, which is consistent with numerical results.

Looking for the optimal rate of recombination for evolutionary dynamics.

We consider many-site mutation-recombination models of evolution with selection. We are looking for situations where the recombination increases the mean fitness of the population, and there is an optimal recombination rate. We found two fitness landscapes supporting such nonmonotonic behavior of the mean fitness versus the recombination rate. The first case is related to the evolution near the error threshold on a neutral-network-like fitness landscape, for moderate genome lengths and large population. The more realistic case is the second one, in which we consider the evolutionary dynamics of a finite population on a rugged fitness landscape (the smooth fitness landscape plus some random contributions to the fitness). We also give the solution to the horizontal gene transfer model in the case of asymmetric mutations. To obtain nonmonotonic behavior for both mutation and recombination, we need a specially designed (ideal) fitness landscape.

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.

Exact solution of a ratchet with switching sawtooth potential.

We consider the flashing potential ratchet model with general asymmetric potential. Using Bloch functions, we derive equations which allow for the calculation of both the ratchet's flux and higher moments of distribution for rather general potentials. We indicate how to derive the optimal transition rates for maximal velocity of the ratchet. We calculate explicitly the exact velocity of a ratchet with simple sawtooth potential from the solution of a system of 8 linear algebraic equations. Using Bloch functions, we derive the equations for the ratchet with potentials changing periodically with time. We also consider the case of the ratchet with evolution with two different potentials acting for some random periods of time.

Accurate analytic solution of chemical master equations for gene regulation networks in a single cell.

Studying gene regulation networks in a single cell is an important, interesting, and hot research topic of molecular biology. Such process can be described by chemical master equations (CMEs). We propose a Hamilton-Jacobi equation method with finite-size corrections to solve such CMEs accurately at the intermediate region of switching, where switching rate is comparable to fast protein production rate. We applied this approach to a model of self-regulating proteins [H. Ge et al., Phys. Rev. Lett. 114, 078101 (2015)PRLTAO0031-900710.1103/PhysRevLett.114.078101] and found that as a parameter related to inducer concentration increases the probability of protein production changes from unimodal to bimodal, then to unimodal, consistent with phenotype switching observed in a single cell.

Solution of the Crow-Kimura model with changing population size and Allee effect.

The Crow-Kimura model is commonly used in the modeling of genetic evolution in the presence of mutations and associated selection pressures. We consider a modified version of the Crow-Kimura model, in which population sizes are not fixed and Allee saturation effects are present. We demonstrate the evolutionary dynamics in this system through an analytical approach, examining both symmetric and single-peak fitness landscape cases. Especially interesting are the dynamics of the populations near extinction. A special version of the model with saturation and degradation on the single-peak fitness landscape is investigated as a candidate of the Allee effect in evolution, revealing reduction tendencies of excessively large populations, and extinction tendencies for small populations. The analytical solutions for these dynamics are presented with accuracy O(1/N), where N is the number of nucleotides in the genome.

Kinetics of biochemical sensing by single cells and populations of cells.

We investigate the collective stationary sensing using N communicative cells, which involves surface receptors, diffusive signaling molecules, and cell-cell communication messengers. We restrict the scenarios to the signal-to-noise ratios (SNRs) for both strong communication and extrinsic noise only. We modified a previous model [Bialek and Setayeshgar, Proc. Natl. Acad. Sci. USA 102, 10040 (2005)PNASA60027-842410.1073/pnas.0504321102] to eliminate the singularities in the fluctuation correlations by considering a uniform receptor distribution over the surface of each cell with a finite radius a. The modified model enables a simple and rigorous mathematical treatment of the collective sensing phenomenon. We then derive the scaling of the SNR for both juxtacrine and autocrine cases in all dimensions. For the optimal locations of the cells in the autocrine case, we find identical scaling for both two and three dimensions.

Exact solution of the hidden Markov processes.

We write a master equation for the distributions related to hidden Markov processes (HMPs) and solve it using a functional equation. Thus the solution of HMPs is mapped exactly to the solution of the functional equation. For a general case the latter can be solved only numerically. We derive an exact expression for the entropy of HMPs. Our expression for the entropy is an alternative to the ones given before by the solution of integral equations. The exact solution is possible because actually the model can be considered as a generalized random walk on a one-dimensional strip. While we give the solution for the two second-order matrices, our solution can be easily generalized for the L values of the Markov process and M values of observables: We should be able to solve a system of L functional equations in the space of dimension M-1.