Quasispecies Model Research Articles

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.

Uncertainty of the second order

We study the stochastic dynamics of the quasispecies model with inverse Bayesian inference under environmental uncertainty. Inverse Bayesian inference is introduced through the correspondence between Bayesian inference and the replicator equation. We consider environmental uncertainty that is not modeled as the stochastic fitness called uncertainty of the second order. This is in contrast to uncertainty of the first order that can be subsumed by the stochastic fitness. The difference between these two kinds of uncertainty is discussed in the framework of categorical Bayesian probability theory. We analytically show that if the time scale of inverse Bayesian inference is sufficiently larger than that of Bayesian inference, then the quasispecies model exhibits a noise-induced transition. The theoretical result is verified by a numerical simulation.

Eigen model version of evolution with directed migration

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 habitats, when in the first habitat there are no mutations. The Eigen model version of the model is more adequately describes the real biological experiments than the Crow-Kimura model, as can be related to the serial transfer experiments in chemical reactor.

Generalized quasispecies model on finite metric spaces: isometry groups and spectral properties of evolutionary matrices.

The quasispecies model introduced by Eigen in 1971 has close connections with the isometry group of the space of binary sequences relative to the Hamming distance metric. Generalizing this observation we introduce an abstract quasispecies model on a finite metric space X together with a group of isometries [Formula: see text] acting transitively on X. We show that if the domain of the fitness function has a natural decomposition into the union of tG-orbits, G being a subgroup of [Formula: see text], then the dominant eigenvalue of the evolutionary matrix satisfies an algebraic equation of degree at most [Formula: see text], where R is the orbital ring that is defined in the text. The general theory is illustrated by three detailed examples. In the first two of them the space X is taken to be the metric space of vertices of a regular polytope with the natural "edge" metric, these are the cases of a regular m-gon and of a hyperoctahedron; the final example takes as X the quotient rings [Formula: see text] with p-adic metric.

Quasispecies model of evolution with migration.

We consider the model of asexual evolution with migration, which was proposed by Waclaw etal. [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.

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.

Trans-heteroclinic bifurcation: a novel type of catastrophic shift.

Global and local bifurcations are extremely important since they govern the transitions between different qualitative regimes in dynamical systems. These transitions or tipping points, which are ubiquitous in nature, can be smooth or catastrophic. Smooth transitions involve a continuous change in the steady state of the system until the bifurcation value is crossed, giving place to a second-order phase transition. Catastrophic transitions involve a discontinuity of the steady state at the bifurcation value, giving place to first-order phase transitions. Examples of catastrophic shifts can be found in ecosystems, climate, economic or social systems. Here we report a new type of global bifurcation responsible for a catastrophic shift. This bifurcation, identified in a family of quasi-species equations and named as trans-heteroclinic bifurcation, involves an exchange of stability between two distant and heteroclinically connected fixed points. Since the two fixed points interchange the stability without colliding, a catastrophic shift takes place. We provide an exhaustive description of this new bifurcation, also detailing the structure of the replication–mutation matrix of the quasi-species equation giving place to this bifurcation. A perturbation analysis is provided around the bifurcation value. At this value the heteroclinic connection is replaced by a line of fixed points in the quasi-species model. But it is shown that, if the replication–mutation matrix satisfies suitable conditions, then, under a small perturbation, the exchange of heteroclinic connections is preserved, except on a tiny range around the bifurcation value whose size is of the order of magnitude of the perturbation. The results presented here can help to understand better novel mechanisms behind catastrophic shifts and contribute to a finer identification of such transitions in theoretical models in evolutionary biology and other dynamical systems.

Mechanisms of interaction between learning and evolution

Abstract The main properties of the interaction between learning and evolutionary optimization are quantitatively analyzed. These properties are the following: (1) the genetic assimilation, (2) the hiding effect, (3) the influence of the learning load on the interaction between learning and evolution. The genetic assimilation means that the acquired features of organisms obtained by means of individual learning can become inherited during many generations of Darwinian evolution. The hiding effect means that strong learning can inhibit the evolutionary search for the optimal genotype. The learning load means that the fitness of an organism can be reduced due to learning because the learning process needs some energy and time. Our analysis is based on the quasispecies model of evolution (Eigen, 1971; Eigen & Schuster, 1979). Basing on this model, it is sufficient to consider only single significant variable, the distance from the optimum. This ensures the clear quantitative analysis of mechanisms of the interaction between learning and evolution. The interaction between learning and evolution is analyzed by means of computer simulation.

Comparative analysis of cherry virus A genome sequences assembled from deep sequencing data.

Cherry virus A (CVA) is a ubiquitous graft-transmissible virus that mainly infects Prunus spp. Next-generation sequencing was applied to 39 tree fruit specimens infected with CVA, and 75 full and 16 partial-length CVA genome sequences were assembled. Phylogenetic analysis of these and 11 previously sequenced CVA genomes resulted in six major clusters with no observable relationship between the host and the assembled genome sequences. Recombination analysis detected four recombinants. Consistent single-nucleotide polymorphism (SNP) patterns were observed between the 75 full-length genomes and their sequence clouds, which supports a quasispecies model for CVA evolution.

A new bio-inspired, population-level approach to the socioeconomic evolution of dynamic spectrum access services

Evolutionary species and quasispecies models provide the universal and flexible basis for a large-scale description of the dynamics of evolutionary systems, which can be built conceived as a constraint satisfaction dynamics. It represents a general framework to design and study many novel, technologically contemporary models and their variants. Here, we apply the classical quasispecies concept to model the emerging dynamic spectrum access (DSA) markets. The theory describes the mechanisms of mimetic transfer, competitive interactions between socioeconomic strata of the end-users, their perception of the utility and inter-operator switching in the variable technological environments of the operators offering the wireless spectrum services. The algorithmization and numerical modeling demonstrate the long-term evolutionary socioeconomic changes which reflect the end-user preferences and results of the majorization of their irrational decisions in the same manner as the prevailing tendencies which are embodied in the efficient market hypothesis.

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.

Analysis of transcription-factor binding-site evolution by using the Hamilton-Jacobi equations

We investigate a quasi-species mutation-selection model of transcription-factor binding-site evolution. By considering the mesa and the crater fitness landscapes designed to describe these binding sites and point mutations, we derive an evolution equation for the population distribution of binding sequences. In the long-length limit, the evolution equation is replaced by a Hamilton-Jacobi equation which we solve for the stationary state solution. From the stationary solution, we derive the population distributions and find that an error threshold, separating populations in which the binding site does or does not evolve, only exists for certain values of the fitness parameters. A phase diagram in this parameter space is derived and shows a critical line below which no error threshold exists. We also investigate the evolution of multiple binding sites for the same transcription factor. For two binding sites, we perform an analysis similar to that for a single site and determine a phase diagram showing different phases with both, one, or no binding sites selected. In the phase diagram, the phase boundary between the one-or-two selected site phases is qualitatively different for the mesa and the crater fitness landscapes. As fitness benefits for a second bound transcription factor tend to zero, the minimum mutation rate at which the two-site phase occurs diverges in the mesa landscape whereas the mutation rate at the phase boundary tends to a finite value for the crater landscape.

Increase in Complexity and Information through Molecular Evolution

Biological evolution progresses by essentially three different mechanisms: (I) optimization of properties through natural selection in a population of competitors; (II) development of new capabilities through cooperation of competitors caused by catalyzed reproduction; and (III) variation of genetic information through mutation or recombination. Simplified evolutionary processes combine two out of the three mechanisms: Darwinian evolution combines competition (I) and variation (III) and is represented by the quasispecies model, major transitions involve cooperation (II) of competitors (I), and the third combination, cooperation (II) and variation (III) provides new insights in the role of mutations in evolution. A minimal kinetic model based on simple molecular mechanisms for reproduction, catalyzed reproduction and mutation is introduced, cast into ordinary differential equations (ODEs), and analyzed mathematically in form of its implementation in a flow reactor. Stochastic aspects are investigated through computer simulation of trajectories of the corresponding chemical master equations. The competition-cooperation model, mechanisms (I) and (II), gives rise to selection at low levels of resources and leads to symbiontic cooperation in case the material required is abundant. Accordingly, it provides a kind of minimal system that can undergo a (major) transition. Stochastic effects leading to extinction of the population through self-enhancing oscillations destabilize symbioses of four or more partners. Mutations (III) are not only the basis of change in phenotypic properties but can also prevent extinction provided the mutation rates are sufficiently large. Threshold phenomena are observed for all three combinations: The quasispecies model leads to an error threshold, the competition-cooperation model allows for an identification of a resource-triggered bifurcation with the transition, and for the cooperation-mutation model a kind of stochastic threshold for survival through sufficiently high mutation rates is observed. The evolutionary processes in the model are accompanied by gains in information on the environment of the evolving populations. In order to provide a useful basis for comparison, two forms of information, syntactic or Shannon information and semantic information are introduced here. Both forms of information are defined for simple evolving systems at the molecular level. Selection leads primarily to an increase in semantic information in the sense that higher fitness allows for more efficient exploitation of the environment and provides the basis for more progeny whereas understanding transitions involves characteristic contributions from both Shannon information and semantic information.

Abrupt transitions to tumor extinction: a phenotypic quasispecies model.

The dynamics of heterogeneous tumor cell populations competing with healthy cells is an important topic in cancer research with deep implications in biomedicine. Multitude of theoretical and computational models have addressed this issue, especially focusing on the nature of the transitions governing tumor clearance as some relevant model parameters are tuned. In this contribution, we analyze a mathematical model of unstable tumor progression using the quasispecies framework. Our aim is to define a minimal model incorporating the dynamics of competition between healthy cells and a heterogeneous population of cancer cell phenotypes involving changes in replication-related genes (i.e., proto-oncogenes and tumor suppressor genes), in genes responsible for genomic stability, and in house-keeping genes. Such mutations or loss of genes result into different phenotypes with increased proliferation rates and/or increased genomic instabilities. Despite bifurcations in the classical deterministic quasispecies model are typically given by smooth, continuous shifts (i.e., transcritical bifurcations), we here identify a novel type of bifurcation causing an abrupt transition to tumor extinction. Such a bifurcation, named as trans-heteroclinic, is characterized by the exchange of stability between two distant fixed points (that do not collide) involving tumor persistence and tumor clearance. The increase of mutation and/or the decrease of the replication rate of tumor cells involves this catastrophic shift of tumor cell populations. The transient times near bifurcation thresholds are also characterized, showing a power law dependence of exponent [Formula: see text] of the transients as mutation is changed near the bifurcation value. These results are discussed in the context of targeted cancer therapy as a possible therapeutic strategy to force a catastrophic shift by simultaneously delivering mutagenic and cytotoxic drugs inside tumor cells.

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.

Enhancement of robustness by mutation in the host-parasite coevolution

We investigate the host-parasite coevolution with mutation in a single-locus system. We use the single-locus quasi-species model with a frequency-dependent fitness landscape and find the phase diagrams for given sets of evolution parameters by means of a linear stability analysis and numerical solutions of the coevolutionary dynamics equations. For a large mutation rate, there is only one internal fixed point. However, for a small mutation rate, within the linear order approximation, we find four steady equilibrium points and one oscillatory equilibrium point. All fixed points show the polymorphism in the host and the parasite populations in the presence of mutation whereas only one oscillatory equilibrium point shows the polymorphism in the absence of mutation. By starting from random initial host-parasite populations, we determine the separatrix and the basins of attraction in evolution parameter space. In comparison with results without mutation, we show that even a very small mutation rate enhances the robustness of the oscillatory equilibrium point; therefore, mutation is significant to maintain diversity in the evolution of finite populations. We also present the corresponding stochastic model for the host-parasite coevolution, and by computer simulation, we find that the stochastic simulation results are consistent with those of the quasi-species model.

When Competing Viruses Unify: Evolution, Conservation, and Plasticity of Genetic Identities.

In the early 1970s, Manfred Eigen and colleagues developed the quasispecies model (qs) for the population-based origin of RNAs representing the early genetic code. The Eigen idea is basically that a halo of mutants is generated by error-prone replication around the master fittest type which will behave similarly as a biological population. But almost from the start, very interesting and unexpected observations were made regarding competition versus co-operation which suggested more complex interactions. It thus became increasingly clear that although viruses functioned similar to biological species, their behavior was much more complex than the original theory could explain, especially adaptation without changing the consensus involving minority populations. With respect to the origin of natural codes, meaning, and code-use in interactions (communication), it also became clear that individual fittest type-based mechanisms were likewise unable to explain the origin of natural codes such as the genetic code with their context- and consortia-dependence (pragmatic nature). This, instead, required the participation of groups of agents competent in the code and able to edit code because natural codes do not code themselves. Three lines of inquiry, experimental virology, quasispecies theory, and the study of natural codes converged to indicate that consortia of co-operative RNA agents such as viruses must be involved in the fitness of RNA and its involvement in communication, i.e., code-competent interactions. We called this co-operative form quasispecies consortia (qs-c). They are the essential agents that constitute the possibility of evolution of biological group identity. Finally, the basic interactional motifs for the emergence of group identity, communication, and co-operation-together with its opposing functions-are explained by the "Gangen" hypothesis.

The Solution of the Spatial Quasispecies Model

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 landscapes. Our main interest is to find the proper scaling for the large genome size limit for our solutions.

Dynamics and bifurcations in a simple quasispecies model of tumorigenesis

Cancer is a complex disease and thus is complicated to model. However, simple models that describe the main processes involved in tumoral dynamics, e.g., competition and mutation, can give us clues about cancer behavior, at least qualitatively, also allowing us to make predictions. Here, we analyze a simplified quasispecies mathematical model given by differential equations describing the time behavior of tumor cell populations with different levels of genomic instability. We find the equilibrium points, also characterizing their stability and bifurcations focusing on replication and mutation rates. We identify a transcritical bifurcation at increasing mutation rates of the tumor cells. Such a bifurcation involves a scenario with dominance of healthy cells and impairment of tumor populations. Finally, we characterize the transient times for this scenario, showing that a slight increase beyond the critical mutation rate may be enough to have a fast response towards the desired state (i.e., low tumor populations) by applying directed mutagenic therapies.