Eigen Model Research Articles

Eigen model as a quantum spin chain: exact dynamics.

We map the Eigen model of biological evolution [Naturwissenschaften 58, 465 (1971)] into a quantum spin model with non-Hermitian Hamiltonian. Based on such a connection, we derive exact relaxation periods for the Eigen model to approach static energy landscape from various initial conditions. We also study a simple case of dynamic fitness function.

An introduction to physical theory of molecular evolution

Abstract This work is a tutorial in Molecular Evolution from the point of view of Physics. We discuss Eigen's model, a link between evolutionary theory and physics. We will begin by assuming the existence of (marco) molecules or replicators with the template property, that is, the capacity to self-replicate. According to this assumption, information will be randomly generated and destroyed by mutations in the code (i.e., errors in the copying process) and new bits of information will be fixed (made stable) by the existence of an external pressure on the system (i.e., selection), and the ability of the molecules to replicate themselves. Our aim is to build a model in order to describe molecular evolution from as general a standpoint as possible. As we will see, even very simple models from the theoretical point of view will have surprisingly deep consequences.

Evolution and ageing

The idea of this review is to connect the different models of evolution to those of biological ageing through Darwin's theory. We start with the Eigen model of quasispecies for microevolution, then introduce the Bak–Sneppen model for macroevolution and, finally, present the Penna model for biological ageing and some of its most important results. We also explore the concept of coevolution using this model.

About a three states molecular evolution model.

The Eigen model of molecular evolution has recently been reconsidered by taking into account the existence of non-reproducing sequences. New results obtained with this model are presented here: several time-scales characterize the population dynamics and the location of the error threshold, shifted towards larger values of the probability of mutation, depends on both the size of the population and time. Thus this model can be used to analyse extinction events. The distribution of individuals in sequence space beyond the threshold is also discussed.

Dynamics in Eigen Quasispecies Model

In this work the Eigen model of quasispecies is reconsidered. Our main aim is to find to what extent mutational meltdown changes previously presented results, when the finite population can decrease or increase during evolution.

Hamiltonian dynamics of Darwin systems

We present a Hamiltonian approach for the well-known Eigen model of the Darwin selection dynamics. Hamiltonization is carried out by means of the embedding of the population variable space, describing behavior of the system, into the space of double dimension by introducing auxiliary dynamic variables. Besides the study of the formalism, we try to interpret its basic elements (phase space, Hamiltonian, geometry of solutions) in terms of theoretical biology. A geometric treatment is given for the considered system dynamics in terms of the geodesic flows in the Euclidean space where the population variables serve as curvilinear coordinates. Time evolution of the distribution function is found for arbitrary distributed initial values of the population variables.

Stochastic model of evolving populations

A general formalism is described which allows the study of a wide range of stochastic problems related to evolution. We apply it to the treatment of the evolution of a finite population in the presence of mutations and selective advantage. The results are close to those of Eigen and Schuster's error threshold model, in the presence of randomly drawn sterile sequences. The theoretical results are in excellent agreement with numerical simulations.

Speciation as Pattern Formation by Competition in a Smooth Fitness Landscape

We investigate the problem of speciation and coexistence in simple ecosystems when the competition among individuals is included in the Eigen model for quasispecies. By suggesting an analogy between the competition among strains and the diffusion of a chemical inhibitor in a reaction-diffusion system, the speciation phenomenon is considered the analogous of chemical pattern formation in genetic space. In the limit of vanishing mutation rate we obtain analytically the conditions for speciation. Using different forms of the competition interaction we show that the speciation is absent for the genetic equivalent of a normal diffusing inhibitor, and is present for shorter-range interactions.

Reduced-Order Aeroservoelastic Model with an Unsteady Aerodynamic Eigen Formulation

The purpose of this study is to construct a new aeroservoelastic model using a two dimensional, incompressible aerodynamic eigenformulation in the continuous time domain. The work is based on the discrete time domain eigen model by Hall. To obtain a reduced order model with real coefficients, a modal decomposition technique that transforms the aerodynamic matrix into a canonical modal form is introduced. A simple typical section model with two strain actuated control inputs is used for an aeroservoelastic design

The Origin of Synergistic Symbiosis

A dominant theme in the history of life has been the evolutionary innovations of cooperative symbioses: the first genomes near the origin of life, integrated prokaryotic cells, the complex symbiotic communities that evolved into modern eukaryotic cells, lichens, mycorrhizae, and so on. In this paper, a mode l of cooperative symbiosis that shows a threshold condition for the evolution of cooperation is analyzed. The threshold is not easily passed, but cooperative evolution proceeds rapidly once a symbiosis overcomes the threshold. In the model presented here, each species has genetic variability for a symbiotic trait. The trait imposes a reproductive cost on its bearer but enhances the reproduction of its partner species. For example, in the origin of genetic systems, the trait may cause biochemical synergism for the rate of replication of primitive RNA strands as in Eigen and Schuster's hypercycle model. Models of growth are contrasted with synergism, which are most appropriate for the evolution of genetic systems and for mutualisms such as lichens, with the strategic and psychological applications of the Prisoner's Dilemma model.

Counterion effects in intramolecular charge transfer in radical anions

The influence of the presence of an electrolyte in a low-polarity solvent (tetrahydrofuran) on the rate of a weakly exoergic ([approx]50 meV) nonadiabatic intramolecular charge shift reaction and on the position of the charge transfer absorption band in a degenerate system has been studied by pulse radiolysis. The investigation focused in particular on the dependence of the effect on the size of the cation of the chemically inert electrolyte. Electrolytes caused dramatic rate reductions of 2-3 orders of magnitude. Unexpectedly, the rate reduction was found to increase with an increasing size of the cation. The recorded spectral blue shifts were as large as 0.5 eV and exhibited the expected tendency; the smaller counterions produced larger spectral shifts, due to a stronger Coulombic interaction. In both cases the effects exceeded the limits predicted by the continuum approach and pointed to the ion-pairing mechanism. The discrepancy between the dependence of the thermal electron transfer and the optically assisted electron transfer on the radius of the counterion could not be easily explained on purely energetic grounds, and the dynamics of the counterion had to be considered. Unfortunately, no agreement between the experimental results and the Eigen model of dissociative diffusion of ion pairsmore » in the zero ionic strength limit could be found. However, a qualitative correlation between the measured rates and the dissociation constants of the used salts, as well as the counterion mobilities was observed. 34 refs., 8 figs., 3 tabs.« less

Error thresholds for molecular quasispecies as phase transitions: From simple landscapes to spin-glass models.

The correspondence between Eigen's model [Naturwissenschaften 58, 465 (1971)] for molecular quasispecies and the equilibrium properties of a lattice system proposed by Leuthausser [J. Chem. Phys. 84, 1884 (1986); J. Stat. Phys. 48, 343 (1987)] is used to characterize the error thresholds for the existence of quasispecies as phase transitions. For simple replication landscapes the error threshold is related to a first-order phase transition smoothed by the complete wetting of the time surface

Evolutionary search with soft selection

A new method of global random search in R n devised from a model of evolution is proposed. The model can be regarded as a randomization of Eigen's model of macromolecular evolution. The novelty of the method is in a random choice of modification basepoints with probabilities depending on results of the last group of trials -- soft selection. In other aspects, the proposed search and the widely used evolutionary search are identical, and the performance of the former can be conveniently assessed against the latter. The method appears especially efficient in crossing gaps between regions of attraction of local optima. This effect, crucial for global optimization, comes from an ability of soft selection to shift distributions of trials toward gaps. Numerical estimates show significant advantages of soft selection which grow with the gap size and the problem dimension.

An extremal principle of macromolecular evolution

This paper shows that the logarithm of the mean asymptotic productivity of the Eigen model of polynucleotide replication satisfies an extremal principle which is formally identical to the minimization of the free energy in statistical mechanics. This extremal principle implies that: (a) Fluctuations in the asymptotic mean productivity due to variations in the replication and mutation rates of the polynucleotides are determined by a macroscopic parameter which is the analogue of the mean energy in statistical mechanics: (b) The logarithm of the asymptotic mean productivity is a convex function of the sequence length of the polynucleotides. This convexity property implies evolutionary stability, the analogue of thermodynamic stability in Ising models.The macromolecular replication model represents another class of dynamical systems in evolutionary biology which are thermodynamically stable, in the sense that the dynamical system possesses an asymptotic limit in which analogues of the laws of equilibrium thermodynamics hold.

Statistical mechanics of Eigen's evolution model

The correspondence between Eigen's model of macromolecular evolution and the equilibrium statistical mechanics of an inhomogeneous Ising system is developed. The free energy landscape of random Ising systems with the Hopfield Hamiltonian as a special example is applied to the replication rate coefficient landscape. The coupling constants are scaled with 1/l, since the maxima of any landscape must not increase with the length of the macromolecules. The calculated error threshold relation then agrees with Eigen's expression, which was derived in a different way. It gives an explicit expression for the superiority parameter in terms of the parameters of the landscape. The dynamics of selection and evolution is discussed.

Selection under random mutations in stochastic Eigen model

The concept of natural selection is examined, which is one of basic principles for the Darwinian interpretation of evolution. In this model selection is defined as a solution of the deterministic Eigen equation. Next, the random effect is introduced through the mutation term. However, the probability of finding the solution expressing the selection is shown to be smallest. The validity of the model and its applicability to polynucleotide replication are discussed.

Selection under random mutations in stochastic eigen model

Selection under random mutations in stochastic eigen model

Non-equilibrium thermodynamics of the selection of biological macromolecules

The Eigen's model for the Darwinian selection of self-reproductive macromolecules is viewed by non-equilibrium thermodynamics of chemical reactions. An excess entropy suitable for the purpose is introduced. It, as the case may be, becomes a Liapunov function to investigate the stability of the selected state characterized by an extremum principle. The selection under the condition for the total concentration of self-reproductive macromolecular species to remain constant in time is discussed on the three cases; the selection described by the simplest Eigen's equation that the effect of mutation is neglected, the selection of quasi-species, and the selective behavior of elementary hypercycle.

Multi-step kinetics of aquo aluminum(III)-ferron (8-hydroxy-7-iodo-5-quinoline) complexation reactions

The interaction of excess hydrolyzed aluminum(III) cation in mildly acid aqueous media with ferron (7-iodo-8-hydroxyquinoline-5-sulfonic acid) has been studied unsing stopped-flow. Three distict transients are observed. The shortest-lived, which is largest in amplitude, has been assigned to mono-complex formation. The other two transients, which are much smaller in amplitude, are interpreted as reactions of dimeric and other polymeric aluminum(III) species. The hydrogen ion dependences of the observed rate constants for mono-complex formation indicate participation of Al(OH) + 2 in the dominant pathway. Analysis of these results, using the Eigen model of complexation, leads to a water exchange rate constant of(4.0 ± 0.4) × 10 7 sec −1 for Al(OH) − 2. The slower transients could not be adequately explained by formation of bis and tris-complexes. Assignment of these transients to reactions of μ-hydroxy-bridged aluminum dimer with ferron, coupled to mono-complex formation leads to an adequate data fit. The polyphasic nature of the aluminum-ferron interaction over the pH range 2–5 can be understood as showing different groups of reactions, rahter than differently reacting metal-containing species.

Stability and selection in self-reproducing macromolecular systems

In this paper we develop stability criteria applicable to Eigen's model for the selection and evolution of biological macromolecules. We show that it is possible to characterize the time evolution of the macromolecular system by Lyapounov-like functionals. The Lyapounov method is used to discuss the general stability of the system and the metastable nature of the selected states are discussed.