Dynamic fitness landscapes: Expansions for small mutation rates

We study the evolution of asexual microorganisms with small mutation rate in fluctuating environments, and develop techniques that allow us to expand the formal solution of the evolution equations to first order in the mutation rate. Our method can be applied to both discrete time and continuous time systems. While the behavior of continuous time systems is dominated by the average fitness landscape for small mutation rates, in discrete time systems it is instead the geometric mean fitness that determines the system's properties. In both cases, we find that in situations in which the arithmetic (resp. geometric) mean of the fitness landscape is degenerate, regions in which the fitness fluctuates around the mean value present a selective advantage over regions in which the fitness stays at the mean. This effect is caused by the vanishing genetic diffusion at low mutation rates. In the absence of strong diffusion, a population can stay close to a fluctuating peak when the peak's height is below average, and take advantage of the peak when its height is above average.Comment: 19 pages Latex, elsart style, 4 eps figure

Finite-size scaling of the quasiespecies model

We use finite-size scaling to investigate the critical behavior of the quasiespecies model of molecular evolution in the single-sharp-peak replication landscape. This model exhibits a sharp threshold phenomenon at Q=Q_c=1/a, where Q is the probability of exact replication of a molecule of length L and a is the selective advantage of the master string. We investigate the sharpness of the threshold and find that its characteristic persist across a range of Q of order L^(-1) about Q_c. Furthermore, using the data collapsing method we show that the normalized mean Hamming distance between the master string and the entire population, as well as the properly scaled fluctuations around this mean value, follow universal forms in the critical region.Comment: 8 pages,tex. Submitted to Physical Review

Template coexistence in prebiotic vesicle models

The coexistence of distinct templates is a common feature of the diverse proposals advanced to resolve the information crisis of prebiotic evolution. However, achieving robust template coexistence turned out to be such a difficult demand that only a class of models, the so-called package models, seems to have met it so far. Here we apply Wright's Island formulation of group selection to study the conditions for the coexistence of two distinct template types confined in packages (vesicles) of finite capacity. In particular, we show how selection acting at the level of the vesicles can neutralize the pressures towards the fixation of any one of the template types (random drift) and of the type with higher replication rate (deterministic competition). We give emphasis to the role of the distinct generation times of templates and vesicles as yet another obstacle to coexistence.Comment: 7 pages, 8 figure

Molecular Evolution in Time Dependent Environments

The quasispecies theory is studied for dynamic replication landscapes. A meaningful asymptotic quasispecies is defined for periodic time dependencies. The quasispecies' composition is constantly changing over the oscillation period. The error threshold moves towards the position of the time averaged landscape for high oscillation frequencies and follows the landscape closely for low oscillation frequencies.Comment: 5 pages, 3 figures, Latex, uses Springer documentclass llncs.cl

Error Thresholds on Dynamic Fittness-Landscapes

In this paper we investigate error-thresholds on dynamics fitness-landscapes. We show that there exists both lower and an upper threshold, representing limits to the copying fidelity of simple replicators. The lower bound can be expressed as a correction term to the error-threshold present on a static landscape. The upper error-threshold is a new limit that only exists on dynamic fitness-landscapes. We also show that for long genomes on highly dynamic fitness-landscapes there exists a lower bound on the selection pressure needed to enable effective selection of genomes with superior fitness independent of mutation rates, i.e., there are distinct limits to the evolutionary parameters in dynamic environments.Comment: 5 page

The Error and Repair Catastrophes: A Two-Dimensional Phase Diagram in the Quasispecies Model

This paper develops a two gene, single fitness peak model for determining the equilibrium distribution of genotypes in a unicellular population which is capable of genetic damage repair. The first gene, denoted by $\sigma_{via}$, yields a viable organism with first order growth rate constant $k > 1$ if it is equal to some target ``master'' sequence $\sigma_{via, 0}$. The second gene, denoted by $\sigma_{rep}$, yields an organism capable of genetic repair if it is equal to some target ``master'' sequence $\sigma_{rep, 0}$. This model is analytically solvable in the limit of infinite sequence length, and gives an equilibrium distribution which depends on \mu \equiv L\eps , the product of sequence length and per base pair replication error probability, and \eps_r , the probability of repair failure per base pair. The equilibrium distribution is shown to exist in one of three possible ``phases.'' In the first phase, the population is localized about the viability and repairing master sequences. As \eps_r exceeds the fraction of deleterious mutations, the population undergoes a ``repair'' catastrophe, in which the equilibrium distribution is still localized about the viability master sequence, but is spread ergodically over the sequence subspace defined by the repair gene. Below the repair catastrophe, the distribution undergoes the error catastrophe when $\mu$ exceeds \ln k/\eps_r , while above the repair catastrophe, the distribution undergoes the error catastrophe when $\mu$ exceeds $\ln k/f_{del}$, where $f_{del}$ denotes the fraction of deleterious mutations.Comment: 14 pages, 3 figures. Submitted to Physical Review

Chance and Necessity in Evolution: Lessons from RNA

The relationship between sequences and secondary structures or shapes in RNA exhibits robust statistical properties summarized by three notions: (1) the notion of a typical shape (that among all sequences of fixed length certain shapes are realized much more frequently than others), (2) the notion of shape space covering (that all typical shapes are realized in a small neighborhood of any random sequence), and (3) the notion of a neutral network (that sequences folding into the same typical shape form networks that percolate through sequence space). Neutral networks loosen the requirements on the mutation rate for selection to remain effective. The original (genotypic) error threshold has to be reformulated in terms of a phenotypic error threshold. With regard to adaptation, neutrality has two seemingly contradictory effects: It acts as a buffer against mutations ensuring that a phenotype is preserved. Yet it is deeply enabling, because it permits evolutionary change to occur by allowing the sequence context to vary silently until a single point mutation can become phenotypically consequential. Neutrality also influences predictability of adaptive trajectories in seemingly contradictory ways. On the one hand it increases the uncertainty of their genotypic trace. At the same time neutrality structures the access from one shape to another, thereby inducing a topology among RNA shapes which permits a distinction between continuous and discontinuous shape transformations. To the extent that adaptive trajectories must undergo such transformations, their phenotypic trace becomes more predictable.Comment: 37 pages, 14 figures; 1998 CNLS conference; high quality figures at http://www.santafe.edu/~walte

Error thresholds for self- and cross-specific enzymatic replication

The information content of a non-enzymatic self-replicator is limited by Eigen's error threshold. Presumably, enzymatic replication can maintain higher complexity, but in a competitive environment such a replicator is faced with two problems related to its twofold role as enzyme and substrate: as enzyme, it should replicate itself rather than wastefully copy non-functional substrates, and as substrate it should preferably be replicated by superior enzymes instead of less-efficient mutants. Because specific recognition can enforce these propensities, we thoroughly analyze an idealized quasispecies model for enzymatic replication, with replication rates that are either a decreasing (self-specific) or increasing (cross-specific) function of the Hamming distance between the recognition or "tag" sequences of enzyme and substrate. We find that very weak self-specificity suffices to localize a population about a master sequence and thus to preserve its information, while simultaneous localization about complementary sequences in the cross-specific case is more challenging. A surprising result is that stronger specificity constraints allow longer recognition sequences, because the populations are better localized. Extrapolating from experimental data, we obtain rough quantitative estimates for the maximal length of the recognition or tag sequence that can be used to reliably discriminate appropriate and infeasible enzymes and substrates, respectively.Comment: 23 pages, 7 figures; final version as publishe

Error threshold in finite populations

A simple analytical framework to study the molecular quasispecies evolution of finite populations is proposed, in which the population is assumed to be a random combination of the constiyuent molecules in each generation,i.e., linkage disequilibrium at the population level is neglected. In particular, for the single-sharp-peak replication landscape we investigate the dependence of the error threshold on the population size and find that the replication accuracy at threshold increases linearly with the reciprocal of the population size for sufficiently large populations. Furthermore, in the deterministic limit our formulation yields the exact steady-state of the quasispecies model, indicating then the population composition is a random combination of the molecules.Comment: 14 pages and 4 figure