David B Saakian

Yerevan Physics Institute · Theoretical department

We study the statistical properties of a single two-level system (qubit) subject to repetitive ancilla-based measurements. This setup is a fundamental minimal model for exploring the intricate interplay between the unitary dynamics of the system and the nonunitary stochasticity introduced by quantum measurements, which is central to the phenomenon...

In this paper, we study the phase structure of the product of D * D order matrices. In each round, we randomly choose a matrix from a finite set of d matrices and multiply it with the product from the previous round. Initially, we derived a functional equation for the case of matrices with real eigenvalues and correlated choice of matrices, which l...

We consider the Crow-Kimura model in case of random transitions between different fitness landscapes. The epochs (system has constant in time fitness landscape) length is given by an exponential distribution. To solve the model exactly we need a large system of functional equations. We solve the model approximately at the limits of slow or fast trans...

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 paramete...

We consider the Crow Kimura model, modified via stochastic resetting. There are two principally different situations: First, when due to resetting the system jumps to the low fitness state, everything is rather simple in this case, we have a solution which is a slight modification of the standard Crow-Kimura model case. When there is resetting to t...

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 lat...

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 simul...

We consider the product of a large number of two 2 × 2 matrices chosen randomly (with some correlation): at any round there are transition probabilities for the matrix type, depending on the choice at previous round. Previously, a functional equation has been derived to calculate such a random product of matrices. Here, we identify the phase struct...

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...

The model of a simple self-reproducing system has been investigated. The current model has been developed in the framework of syser systems. The term syser is the abbreviation of the words “SYstem of SElf-Reproduction”. The syser model can be considered as a reasonable model of the prebiological macromolecular self-reproducing systems. The syser in...

We intend to uncover generative principles for complex, biological systems, looking the reflections as well as the analogs of decision making property in quantum physics: measurement, self-interaction of the electron, Berry phase and quantum anomalies. We assume that classical analogs of the mentioned phenomena could be related to the evolvability,...

Several models of prebiological systems are described and analyzed. The following models are characterized: a quasispecies model, a hypercycle model, a syser model (the term "syser" is an abbreviation of SYstem of SElf-Reproduction), a stochastic corrector model, a model of the origin of a primordial genome through spontaneous symmetry breaking. Th...

We study the dynamics of a finite number of replicators with different strategies in evolutionary games, using the moment closure approximation for the master equation and the Hamilton–Jacobi equation approach. These methods give finite population corrections to the results of the replicator equation. The model under investigation has two strategie...

We investigate an evolution model in which the fitness depends on the steepness of population distribution via Hamming classes (the absolute value of the logarithm of the probability ratios for the neighbor Hamming classes). The model has a rather rich phase structure with observed oscillations of the mean fitness in the dynamics. Specifically, the...

We investigate the mutator model for the asymmetric transition rates between the wild-type and mutator type. When the mutator gene changes its type, both mutation rate of genome and fitness landscape are changed. We look at smooth symmetric fitness landscapes both for the wild type (normal allele of special gene) and mutator type (mutator allele)....

We investigate the evolutionary model with recombination and random switches in the fitness function due to change in a special gene. The dynamical behaviour of the fitness landscape induced by the specific mutations is closely related to the mutator phenomenon, which, together with recombination, plays an important role in modern evolutionary stud...

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 spatia...

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 evolutiona...

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 m...

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...

We consider the product of a large number of matrices chosen randomly (with some correlation) as A or B. We introduce an exact functional equation, whose solution gives the limiting distribution and the Lyapunov exponent for the product of large number of matrices. Thus we suggest new semianalytical method to solve the random product problem of cor...

We consider the two-habitat quasispecies model, which describes evolutionary process with migration on the basis of the Eigen model. In the first habitat there is only one genotype, and here is an influx of the replicators from the first habitat to the second one with the rate h. We solve exactly the case of a single-peak fitness landscape in both...

The current work develops the previous model of interaction between learning and evolution (Red'ko, 2017). The previous model investigated this interaction by means of computer simulation. The mechanisms of the main properties of the interaction between learning and evolution (the genetic assimilation, the hiding effect, the influence of the learni...

In this study, we considered the model by Beckman and Loeb [Proc. Natl. Acad. Sci. U.S.A. 103 (2006) 14140] for the mutator phenomena. We construct an infinite population Crow-Kimura model with a mutator gene, directed mutations, a linear fitness function, and a finite genome length. We solved analytically the dynamics of the model using the genera...

We consider the model of asexual evolution with migration, which was proposed by Waclaw et al. [Phys. Rev. Lett. 105, 268101 (2010)]. 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 le...

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 ana...

The current work describes the model of sysers. Syser is the abbreviation of SYstem of SElf-Reproduction. Syser is the model of rather general and universal system of self-reproduction. The model of sysers was initially proposed as macromolecular self-reproducing systems. However, this model is close to self-reproducing system of biological cells,...

In this paper, the impact of lethal mutations on evolutionary dynamics of asexual populations is analyzed. We suggest distinguishing different definitions of lethality, which lead to different mathematical formalizations of the microscopic model. Most of the studies focus on polyphasic lethality, meaning that individuals carrying lethal mutations h...

We establish specific correspondences between notions of economics and statistical mechanics. There are several situations wherein a rather accurate correspondence has already been established, for instance in utility theory for exchange economy with quasilinear utility function, which has been mapped to analogous thermodynamics. We discuss how sta...

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 cal...

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 switch...

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...

In the present study, we have investigated the Allison mixture, a variant of the Parrondo's games where random mixing of two random sequences creates autocorrelation. We have obtained the autocorrelation function and mutual entropy of two elements. Our analysis shows that the mutual information is nonzero even if two distributions have identical av...

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 expressi...

Recently it has been found that the collective decision-making in the group is efficient only when the confidences (a version of metacognition) of the members are similar, and it has been assumed that the metacognition (self-reference) in general is crucial for the human cooperation. Our goal is to map the decision making by the cells to decision m...

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 Setayes...

Cancer is related to clonal evolution with a strongly nonlinear, collective behavior. Here we investigate a slightly advanced version of the popular Crow–Kimura evolution model, suggested recently, by simply assuming a conditional mutation rate. We investigated the steady-state solution and found a highly intriguing plateau in the distribution. The...

To describe virus evolution, it is necessary to define a fitness landscape. In this article, we consider the microscopic models with the advanced version of neutral network fitness landscapes. In this problem setting, we suppose a fitness difference between one-point mutation neighbors to be small. We construct a modification of the Wright–Fisher m...

We considered the infinite population version of the mutator phenomenon in evolutionary dynamics, looking at the uni-directional mutations in the mutator-specific genes and linear selection. We solved exactly the model for the finite genome length case, looking at the quasispecies version of the phenomenon. We calculated the mutator probability bot...

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 lo...

Lethal mutations are very common in asexual evolution, both in RNA viruses and in the clonal evolution of cancer cells. In a special case of lethal mutations (truncated selection), after a critical total number of mutations the replicator (the virus or the cell) has no offspring. We consider the Eigen and Crow-Kimura models with truncated fitness l...

We consider an evolution model, in which the mutation rates depend on the structure of population: the mutation rates from lower populated sequences to higher populated sequences are reduced. We have applied the Hamilton–Jacobi equation method to solve the model and calculate the mean fitness. We have found that the modulated mutation rates, direct...

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...

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-sel...

We consider discrete time Brownian ratchet models: Parrondo's games. Using the Fourier transform, we calculate the exact probability distribution functions for both the capital dependent and history dependent Parrondo's games. We find that in some cases there are oscillations near the maximum of the probability distribution, and after many rounds t...

We briefly review the recently developed Markov process-based isothermal chemical thermodynamics for nonlinear driven mesoscopic kinetic systems. Both the instantaneous Shannon entropy $S[\,p_{\alpha }(t)]$ and relative entropy $F[\,p_{\alpha }(t)]$ , defined based on probability distribution $\{\,p_{\alpha }(t);\alpha \in \boldsymbol {\mathsc...

We consider the general case of Parrondo's games, when there is a finite probability to stay in the current state as well as multi-step jumps. We introduce a modification of the model: the transition probabilities between different games depend on the choice of the game in the previous round. We calculate the rate of capital growth as well as the v...

Eigen model of molecular evolution is popular in studying complex biological and biomedical systems. Using the Hamilton-Jacobi equation method, we have calculated analytic equations for the steady state distribution of the Eigen model with a relative accuracy of O(1/N), where N is the length of genome. Our results can be applied for the case of sma...

We consider the decision making by mammalian cells, looking them as dynamic systems with rhythms. We calculate the effective dimension of the cell division model of the healthy mammalian cells consistent with the data: it is described via a four dimensional dynamic system. We assume that the cell's decision making property is strongly affected by t...

Crow-Kimura model is one of the famous models of population genetics. Last decade models with low-dimensional fitness landscape have been investigated. We consider the Crow-Kimura model of evolutionary dynamics on multi-dimensional fitness landscape with a single peak. We deduce exact solution for the dynamics, confirmed well by the numerics.

Complex systems in biology have attracted much attention in recent decades. We investigate the dynamics of a molecular evolution model related to the mutator gene phenomenon in biology. Here mutation in one gene drastically changes the properties of the whole genome. We investigated the Crow-Kimura version of the model, which can be mapped into a H...

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 evolutio...

We consider the evolutionary dynamics of replicators in a space for the case when the fitness is space independent, by looking at a simple generalization of the Crow-Kimura model. The dynamics of evolution in d-dimensional space is studied within the Hamilton-Jacobi formalism. We analytically derive the exact dynamics for the smooth fitness landsca...

Evolutionary games are used in various fields stretching from economics to biology. In most of these games a constant payoff matrix is assumed, although some works also consider dynamic payoff matrices. In this article we assume a possibility of switching the system between two regimes with different sets of payoff matrices. Potentially such a mode...

We calculated the mean first arrival time of the new double mutant in the Wright-Fisher and Moran models with selection where N is the population size, the mutation probability scales as 1/N and selection coefficient as . We mapped the mean first arrival time problem into Kummer equation. Our results have a relative accuracy. Our analytic result is...

We consider the dynamics of the particle influences by driving force, Gaussian and Poisson noises, which is a modification of the model by Friedman et al (2006 Phys. Rev. Lett. 97 168302). We study the relative integro-differential Fokker–Planck equation (IDFPE) for the probability density distribution and obtain the exact steady state solution of...

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 appe...

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 r...

We investigate the catalytic reactions model used in cell modeling. The reaction kinetic is defined through the energies of different species of molecules following random independent distribution. The related statistical physics model has three phases and these three phases emerged in the dynamics: fast dynamics phase, slow dynamic phase and ultra...

We consider the hierarchic tree Random Energy Model with continuous branching and calculate the moments of the corresponding partition function. We establish the multifractal properties of those moments. We derive formulas for the normal distribution of random variables, as well as for the general case. We compare our results for the moments of par...

We investigate the directed random walk on hierarchic trees. Two cases are investigated: random variables on deterministic trees with a continuous branching, and random variables on the trees constructed trough the random branching process. We derive renormalization group (partial differential) equations for the branching models with binomial, Pois...

We investigate the multi-chain 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-Ja...

We map the Markov Switching Multi-fractal model (MSM) onto the Random Energy Model (REM). The MSM is, like the REM, an exactly solvable model in 1-d space with non-trivial correlation functions. According to our results, four different statistical physics phases are possible in random walks with multi-fractal behavior. We also introduce the continu...

We consider the Eigen model with non-Poisson distribution of mutation number. We apply the Hamilton--Jacobi equation method to solve the model and calculate the mean fitness. We find that the error threshold depends on the correlation, and the suggested mechanism may give a simple solution to the error catastrophe paradox in the origin of life.

We considered a {multi-block} 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 mul...

We consider finite population size effects for Crow-Kimura and Eigen quasispecies models with single peak fitness landscape. We formulate accurately the iteration procedure for the finite population models, then derive Hamilton-Jacobi equation (HJE) to describe the dynamic of the probability distribution. The steady state solution of HJE gives the...

We map the Markov-switching multifractal model (MSM) onto the random energy model (REM). The MSM is, like the REM, an exactly solvable model in one-dimensional space with nontrivial correlation functions. According to our results, four different statistical physics phases are possible in random walks with multifractal behavior. We also introduce th...

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 mod...

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...

We investigate the multifractal random walk (MRW) model, popular in the modelling of stock fluctuations in the financial market. The exact probability distribution function (PDF) is derived by employing methods proposed in the derivation of correlation functions in string theory, including the analytical extension of Selberg integrals. We show that...

Using the Hamilton–Jacobi equation approach to study genomes of length L, we obtain 1/L corrections for the steady state population distributions and mean fitness functions for horizontal gene transfer model, as well as for the diploid evolution model with general fitness landscapes. Our numerical solutions confirm the obtained analytic equations....

We give several criteria of complexity and define different universality classes. According to our classification, at the lowest class of complexity are random graph, Markov Models and Hidden Markov Models. At the next level is Sherrington-Kirkpatrick spin glass, connected with neuron-network models. On a higher level are critical theories, spin gl...

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 pe...

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 a...

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...

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 distri...

String correlation functions in 2D gravity resemble similar objects in the statistical mechanics of directed polymer models on disordered trees. This analogy can be used to provide an approximate mapping between the two problems. Using such a mapping we derive renormalization group equations and calculate the ensuing scaling of the string correlati...

The parallel mutation-selection evolutionary dynamics, in which mutation and replication are independent events, is solved exactly in the case that the Malthusian fitnesses associated to the genomes are described by the random energy model (REM) and by a ferromagnetic version of the REM. The solution method uses the mapping of the evolutionary dyna...

We consider the optimal dynamics in the infinite population evolution models with general symmetric fitness landscape. The search of optimal evolution trajectories are complicated due to sharp transitions (like shock waves) in evolution dynamics with smooth fitness landscapes, which exist even in case of popular quadratic fitness. We found exact an...

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 a reproducing quasispecies. Depending on the mutation rates, we observe three phases, a selective one, an intermediate one with some residual...

The evolution model with parallel mutation-selection scheme is solved for the case when selection is accompanied by base substitutions, insertions, and deletions. The fitness is assumed to be either a single-peak function (i.e., having one finite discontinuity) or a smooth function of the Hamming distance from the reference sequence. The mean fitne...

We introduce a new 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 $1/N$-accuracy, where $N$ is genome length. For smooth and monotonic fitness function this approach gives two dynamical phases: smooth d...

We solve a random energy model with complex replica number and complex temperature values, and discuss the ensuing phase structure. A connection with string models and their phase structure is analyzed from the REM point of view. The REM analysis yields a few integer dimensions as special points of the REM phase diagram. For N = 1 superstrings, th...

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) select...

Mapping the economy to the some statistical physics models we get strong indications that, in contrary to the pure stock market, the stock market with derivatives could not self-regulate.

The European Physical Society (EPS) is a not for profit association whose members include 41 National Physical Societies in Europe, individuals from all fields of physics, and European research institutions. As a learned society, the EPS engages in activities that strengthen ties among the physicists in Europe. As a federation of National Physical...

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...

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 equ...

We investigate well-known models of biological evolution and address the open problem of how construct a correct continuous analog of mutations in discrete sequence space. We deal with models where the fitness is a function of a Hamming distance from the reference sequence. The mutation-selection master equation in the discrete sequence space is re...

We consider transmission of classical information through a quantum channel from one sender to several receivers. In contrast to quantum information, classical information carried by two non-orthogonal states of a quantum system can be cloned exactly. This restricts a naive extrapolation of the standard no-cloning theorem. The optimal cloning induc...

We study feedback control of classical Hamiltonian systems with the controlling parameter varying slowly in time. The control aims to change system's energy. We show that the control problems can be solved with help of an adiabatic invariant that generalizes the conservation of the phase-space volume to control situations. New mechanisms of control...

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...

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....