Eigen's Quasispecies Model Research Articles

Expansion of error thresholds for the Moran model

We study a finite population of individuals evolving through mutation and selection. We generalize the Eigen quasispecies model to a finite population with the Moran model. This model also presents an asymptotic phase transition, and a proper definition of the critical parameter is discussed. We retrieve the same expression for the error threshold appearing in the Eigen model along with a correction term due to the finiteness of the population. To achieve this, we estimate the average lifetime of master sequences and find it grows like an exponential in the size of the population. Our technique consists in bounding from above and below the number of master sequences in the Moran model by two simpler birth and death chains. The expectation of this lifetime is then computed with the help of explicit formulas which are in turn expanded with Laplace method.

The effect analysis of the diapause duration on the hatching parameters of the artificial semi-annual population of Lymantria dispar caterpillars

The paper investigates the existence of a relationship between the diapause duration and the temperature threshold of hatching, the variances in temperature and hatching time of the artificial six-month population of Lymantria dispar caterpillars. A half-year population gives two generations a year under artificial conditions. A comparative analysis of these parameters with similar parameters of the hybrid population of L. dispar, as well as studies by other authors, has been carried out. The Eigen quasispecies model was used for the mathematical formalization of the evolution process. The results of this work can be used to study the evolution of L. dispar.

Intermittency as metastability: A predictive approach to evolution in rugged landscapes

Many systems across the sciences evolve through the interaction of multiplicative growth and diffusive transport. In the presence of disorder, these opposing forces can generate localized structures and bursty dynamics, a phenomenon known as “intermittency” in non-equilibrium physics and as “punctuated equilibrium” in evolutionary theory. This behaviour is difficult to forecast; in particular there is no general principle to locate the regions where the system will settle, how long it will stay there, or where it will jump next. Here I introduce a Markovian representation of growth-transport dynamics that closes these gaps. I show that any autonomous positive linear system can be mapped onto a generalization of the “maximal entropy random walk”, a Markov process with non-local transition rates. From this transformation follow a characterization of localization islands as minima of an effective potential, a general scheme to coarse-grain the state space along the basins of attraction of these minima, and an entropic Lyapunov function. These results unify the concepts of linear intermittency and metastability, and provide a generally applicable method to reduce, and predict, the dynamics of disordered linear systems. Applications range from Zeld’dovich's parabolic Anderson model to Eigen's quasispecies model of molecular evolution.

Asymptotic Behavior of Eigen's Quasispecies Model.

We study Eigen's quasispecies model in the asymptotic regime where the length of the genotypes goes to [Formula: see text] and the mutation probability goes to 0. A limiting infinite system of differential equations is obtained. We prove convergence of trajectories, as well as convergence of the equilibrium solutions. We give analogous results for a discrete-time version of Eigen's model, which coincides with a model proposed by Moran.

Convergence of a Moran model to Eigen's quasispecies model

We prove that a Moran model converges in probability to Eigen's quasispecies model in the infinite population limit. We show further that the invariant probability measure of the Moran model converges to the unique stationary solution of Eigen's quasispecies model.

Quasispecies on Class-Dependent Fitness Landscapes.

We study Eigen's quasispecies model in the asymptotic regime where the length of the genotypes goes to [Formula: see text] and the mutation probability goes to 0. We give several explicit formulas for the stationary solutions of the limiting system of differential equations.

On Eigen's Quasispecies Model, Two-Valued Fitness Landscapes, and Isometry Groups Acting on Finite Metric Spaces.

A two-valued fitness landscape is introduced for the classical Eigen's quasispecies model. This fitness landscape can be considered as a direct generalization of the so-called single- or sharply peaked landscape. A general, non-permutation invariant quasispecies model is studied, and therefore the dimension of the problem is [Formula: see text], where N is the sequence length. It is shown that if the fitness function is equal to [Formula: see text] on a G-orbit A and is equal to w elsewhere, then the mean population fitness can be found as the largest root of an algebraic equation of degree at most [Formula: see text]. Here G is an arbitrary isometry group acting on the metric space of sequences of zeroes and ones of the length N with the Hamming distance. An explicit form of this exact algebraic equation is given in terms of the spherical growth function of the G-orbit A. Motivated by the analysis of the two-valued fitness landscapes, an abstract generalization of Eigen's model is introduced such that the sequences are identified with the points of a finite metric space X together with a group of isometries acting transitively on X. In particular, a simplicial analog of the original quasispecies model is discussed, which can be considered as a mathematical model of the switching of the antigenic variants for some bacteria.

On the role of the degradation rates for the dominance of the fittest in Eigen's quasispecies model

The existence of an error threshold of the mutation rate in Eigen's quasispecies model has been computationally demonstrated exclusively in the case when the degradation rates of all model genotypes are equal. Here we explore the case with different degradation rates and demonstrate examples for which the type that has highest fitness in the absence of mutation can preserve its dominance independently of the value of the mutation rate. The examples are formulated based on analysis of the equilibria at the two extreme mutation rate values and suggest absence of an error threshold in a number of cases, most prominently when the degradation rate of the wild type is much smaller that the degradation rates of the mutants.

Effcient Solution of Evolution Models for Virus Populations

The computation of the quasispecies in Eigen's quasispecies model requires the solution of a very large scale eigenvalue problem. Since the problem dimension is of an exponential growing nature the well known methods for dealing with such a problem run out of resources already far away from practically relevant cases. We propose the use of an implicit matrix vector product using the special problem structure as building block for eigenvalue solvers to partially overcome the exponential growth, which let us reach unexpected large problem sizes. As we will show our implicit matrix vector product is a prime example for an algorithm perfectly matching the requirements of GPU computing since it has low space and high parallel computation requirements. Therefore we will also present an GPU implementation delivering a speedup factor of about 100 compared to a standard implementation.

Detailed analysis of an Eigen quasispecies model in a periodically moving sharp-peak landscape

The Eigen quasispecies model in a periodically moving sharp-peak landscape considered in previous seminal works [M. Nilsson and N. Snoad, Phys. Rev. Lett. 84, 191 (2000)] and [C. Ronnewinkel, in, edited by L. Kallel, B. Naudts, and A. Rogers (Springer-Verlag, Heidelberg, 2001)] is analyzed in greater detail. We show here, through a more rigorous analysis, that results in those papers are qualitatively correct. In particular, we obtain a phase diagram for the existence of a quasispecies with the same shape as in the above cited paper by C. Ronnewinkel, with upper and lower thresholds for the mutation rate between which a quasispecies may survive. A difference is that the upper value is larger and the lower value is smaller than the previously reported ones, so that the range for quasispecies existence is always larger than thought before. The quantitative information provided might also be important in understanding genetic variability in virus populations and has possible applications in antiviral therapies. The results in the quoted papers were obtained by studying the populations only at some few genomes. As we will show, this amounts to diagonalizing a 3×3 matrix. Our work is based instead in a different division of the population allowing a finer control of the populations at various relevant genetic sequences. The existence of a quasispecies will be related to Perron-Frobenius eigenvalues. Although huge matrices of sizes 2(ℓ), where ℓ is the genome length, may seem necessary at a first look, we show that such large sizes are not necessary and easily obtain numerical and analytical results for their eigenvalues.

Optimal mutation rates in dynamic environments: The Eigen model

We consider the Eigen quasispecies model with a dynamic environment. For an environment with sharp-peak fitness in which the most-fit sequence moves by k spin-flips each period T we find an asymptotic stationary state in which the quasispecies population changes regularly according to the regular environmental change. From this stationary state we estimate the maximum and the minimum mutation rates for a quasispecies to survive under the changing environment and calculate the optimum mutation rate that maximizes the population growth. Interestingly we find that the optimum mutation rate in the Eigen model is lower than that in the Crow-Kimura model, and at their optimum mutation rates the corresponding mean fitness in the eigenmodel is lower than that in the Crow-Kimura model, suggesting that the mutation process which occurs in parallel to the replication process as in the Crow-Kimura model gives an adaptive advantage under changing environment.

Breaking records in the evolutionary race

We explore some aspects of the relationship between biological evolution processes and the mathematical theory of records. For Eigen's quasispecies model with an uncorrelated fitness landscape, we show that the evolutionary trajectories traced out by a population initially localized at a randomly chosen point in sequence space can be described in close analogy to record dynamics, with two complications. First, the increasing number of genotypes that become available with increasing distance from the starting point implies that fitness records are more frequent than for the standard case of independent, identically distributed random variables. Second, fitness records can be bypassed, which strongly reduces the number of genotypes that take part in an evolutionary trajectory. For exponential and Gaussian fitness distributions, this number scales with sequence length N as N , and it is of order unity for distributions with a power law tail. This is in strong contrast to the number of records, which is of order N for any fitness distribution.

Punctuated evolution for the quasispecies model

Biological evolution in a sequence space with random fitnesses is studied within Eigen's quasispecies model. A strong selection limit is employed, in which the population resides at a single sequence at all times. Evolutionary trajectories start at a randomly chosen sequence and proceed to the global fitness maximum through a small number of intermittent jumps. The distribution of the total evolution time displays a universal power law tail with exponent −2. Simulations show that the evolutionary dynamics is very well represented by a simplified shell model, in which the sub-populations at local fitness maxima grow independently. The shell model allows for highly efficient simulations, and provides a simple geometric picture of the evolutionary trajectories.

A theory of pragmatic information and its application to the quasi-species model of biological evolution

‘Standard’ information theory says nothing about the semantic content of information. Nevertheless, applications such as evolutionary theory demand consideration of precisely this aspect of information, a need that has motivated a largely unsuccessful search for a suitable measure of an ‘amount of meaning’. This paper represents an attempt to move beyond this impasse, based on the observation that the meaning of a message can only be understood relative to its receiver. Positing that the semantic value of information is its usefulness in making an informed decision, we define pragmatic information as the information gain in the probability distributions of the receiver's actions, both before and after receipt of a message in some pre-defined ensemble. We then prove rigorously that our definition is the only one that satisfies obvious desiderata, such as the additivity of information from logically independent messages. This definition, when applied to the information ‘learned’ by the time evolution of a process, defies the intuitions of the few previous researchers thinking along these lines by being monotonic in the uncertainty that remains after receipt of the message, but non-monotonic in the Shannon entropy of the input ensemble. It also follows that the pragmatic information of the genetic ‘messages’ in an evolving population is a global Lyapunov function for Eigen's quasi-species model of biological evolution. A concluding section argues that a theory such as ours must explicitly acknowledge purposeful action, or ‘agency’, in such diverse fields as evolutionary theory and finance.

Isomorphism of Quasispecies and Percolation Models

Population dynamics of tRNA-like macromolecules and viruses have been interpreted by Eigen (1971,Naturwissenschaften58 , 465–526) on the basis of the “quasispecies” model. The present paper contains a qualitative analysis of the similarities between Eigen's quasispecies model and percolation models. In fact, different phenomena characterized by an analogous inner structure can conceivably be described by quite similar mathematical formalisms. The occurrence of a threshold in specific processes predicted by the models is considered first. Secondly, Ising's model of ferromagnetism is taken into account in the last section. An interpretation of the above-mentioned biological theory in terms of percolation, implying a zeroth-order approximation to the real situation, might be a point of departure to a deeper insight obtainable with more refined approaches. A better comprehension of biological phenomena might in any case arise from a percolative approach, even if the description of the systems is simplified. An overview of some quasispecies results and some plausible applications are presented.

Exact solution of the quasispecies model in a sharply peaked fitness landscape

We reconsider Eigen's quasispecies model for competing self-reproductive macromolecules in populations characterized by a single-peaked fitness landscape. The use of ideas and tools borrowed from polymer theory and statistical mechanics allows us to exactly solve the model for generic DNA lengths $d$. The mathematical shape of the quasispecies confined around the master sequence is perturbatively found in powers of $1/d$ at large $d$. We rigorously prove the existence of the error-threshold phenomena and study the quasispecies formation in the general context of critical phase transitions in physics. No sharp transitions exist at any finite $d$, and at $d\ensuremath{\rightarrow}\ensuremath{\infty}$ the transition is of first order. The typical rms amplitude of a quasispecies around the master sequence is found to diverge algebraically with exponent ${\ensuremath{\nu}}_{\ensuremath{\perp}}=1$ at the transition to the delocalized phase in the limit $d\ensuremath{\rightarrow}\ensuremath{\infty}$.

An evolutionary version of the random energy model

We consider Eigen's quasispecies model (1971) of molecular evolution in the case in which the reproduction rates of different molecular species are quenched independent random variables. We show, by analogy with the random energy model, the existence of two phases: a 'neutral' phase, where no adaptation effects are exhibited, and a (pathological) 'adapted' phase, where the population is made of identical, optimally adapted molecules, and does not evolve.

Spatial stability analysis of Eigen's quasispecies model and the less than five membered hypercycle under global population regulation

Spatial stability analysis of Eigen's quasispecies model and the less than five membered hypercycle under global population regulation

Spatial stability analysis of Eigen's quasispecies model and the less than five membered hypercycle under global population regulation

A generalization of the “constant overall organization” constraint of Eigen's quasispecies and hypercycle models, called herein “global population regulation”, is shown to lead to mathematically tractable spatial generalizations of these two models. The spatially uniform steady state of Eigen's quasispecies model is shown to be stable and globally attracting for all possible values of the mutation and replication rates. In contrast, the spatially and temporally uniform solutions to the hypercycle with fewer than five members, the only ones insensitive to stochastic perturbations, are shown to be unstable, and a lower bound to the spatial inhomogeneities is obtained. The prospect that the spatially localized hypercycle might be immune to various instabilities cited in the literature is then briefly considered. Although spatial localization makes possible a much richer dynamical repertoire than previously considered, it is also more difficult to understand how Darwinian selection of hypercycles could result in a unique genetic code.

Optimization by stimulating molecular evolution.

Based on the analogy between mathematical optimization and molecular evolution and on Eigen's quasi-species model of molecular evolution, an evolutionary algorithm for combinatorial optimization has been developed. This algorithm consists of a versatile variation scheme and an innovative decision rule, the essence of which lies in a radical revision of the conventional philosophy of optimization: A number of configurations of variables with better values, instead of only a single best configuration, are selected as starting points for the next iteration. As a result the search proceeds in parallel along a number of routes and is unlikely to get trapped in local optima. An important innovation of the algorithm is introduction of a constraint to let the starting points always keep a certain distance from each other so that the search is able to cover a larger region of space effectively. The main advantage of the algorithm is that it has more chances to find the global optimum and as many local optima as possible in a single run. This has been demonstrated in preliminary computational experiments.