Error Threshold Transition Research Articles

Error catastrophe in populations under similarity-essential recombination.

Organisms are often more likely to exchange genetic information with others that are similar to themselves. One of the most widely accepted mechanisms of RNA virus recombination requires substantial sequence similarity between the parental RNAs and is termed similarity-essential recombination. This mechanism may be considered analogous to assortative mating, an important form of non-random mating that can be found in animals and plants. Here we study the dynamics of haplotype frequencies in populations evolving under similarity-essential recombination. Haplotypes are represented by a genome of B biallelic loci and the Hamming distance between individuals is used as a criterion for recombination. We derive the evolution equations for the haplotype frequencies assuming that recombination does not occur if the genetic distance is larger than a critical value G and that mutation occurs at a rate μ per locus. Additionally, uniform crossover is considered. Although no fitness is directly associated to the haplotypes, we show that frequency-dependent selection emerges dynamically and governs the haplotype distribution. A critical mutation rate μc can be identified as the error threshold transition, beyond which this selective information cannot be stored. For μ<μc the distribution consists of a dominant sequence surrounded by a cloud of closely related sequences, characterizing a quasispecies. For μ>μc the distribution becomes uniform, with all haplotypes having the same frequency. In the case of extreme assortativeness, where individuals only recombine with others identical to themselves (G=0), the error threshold results μc=1/4, independently of the genome size. For weak assortativity (G=B-1)μc=2(-(B+1)) and for the case of no assortativity (G=B) μc=0. We compute the mutation threshold for 0<G<B and show that, for large B, it depends only on the ratio G/B. We discuss the consequences of these results for recombination in viruses and for speciation.

Error threshold in RNA quasispecies models with complementation

A general assumption of quasispecies models of replicons dynamics is that the fitness of a genotype is entirely determined by its sequence. However, a more biologically plausible situation is that fitness depends on the proteins that catalyze metabolic reactions, including replication. In a stirred population of replicons, such as viruses replicating and accumulating within the same cell, the association between a given genome and the proteins it encodes is not tight as it can be replicated by proteins translated from other genomes. We have investigated how this complementation phenomenon affects the error threshold in simple quasispecies mean field models. We first studied a model in which the master and the mutant genomes code for wild-type and mutant replicases, respectively. We assume that the mutant replicase has a reduced activity and that the wild-type replicase does not have increased affinity for the master genome. The whole pool of replicases can bind and replicate both genomes. We then analyze a different model considering a more extreme case of mutant genomes, the defective interfering particles (DIPs) described in many cases of viral infection. DIPs, with a higher replication rate owed to their shorter genomes, do not code for replicase, but they are able of using the replicase translated from the master genome. Our models allow to study how the probability of interaction between the genomes and the whole pool of replicases affects the error threshold. In both systems we characterize the scenario of coexistence between master and mutant genomes, providing the critical values of mutation rate, μc, and the critical interaction rate between master genomes and replicases, γc, at which the quasispecies enters into error catastrophe, a situation in which the mutant genomes dominate the population. In both cases, we showed that the error-threshold transition is given by transcritical-like bifurcations, suggesting a continuous phase transition. We have also found that the region in the parameter space (μ,γ) in which the master sequence survives is reduced when DIPs are introduced into the system.

Recombination in one- and two-dimensional fitness landscapes

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 distribution and find an error threshold transition point in the single-peak fitness model for a given initial symmetric distribution. We calculate the recombination period in the case of a single-peak fitness function, when the original population is located at one sequence, at some Hamming distance from the peak configuration. Steady-state fitness is calculated with finite genome length corrections. We derive analytical equations for the two-dimensional mutation-recombination model.

Error threshold transition in the random-energy model.

We perform a statistical analysis of the error threshold transition in quasispecies evolution on a random-energy fitness landscape. We obtain a precise description of the genealogical properties of the population through extensive numerical simulations. We find a clear phase transition and can distinguish two regimes of evolution: The first, for low mutation rates, is characterized by strong selection, and the second, for high mutation rates, is characterized by quasineutral evolution.

How the quasispecies evolution depends on the topology of the genome space

We compared the properties of the error threshold transition in quasispecies evolution for three different topologies of the genome space. They are (a) hypercube (b) rugged landscape modelled by an ultrametric space, and (c) holey landscape modelled by Bethe lattice. In all studied topologies, the phase transition exists. We calculated the critical exponents in all the cases. For the critical exponent corresponding to appropriately defined susceptibility we found super-universal value.

Optimal adaptive performance and delocalization in NK fitness landscapes

We investigate the evolutionary dynamics of a finite population of sequences adapting to NK fitness landscapes. We find that, unlike in the case of an infinite population, the average fitness in a finite population is maximized at a small but finite, rather than vanishing, mutation rate. The highest local maxima in the landscape are visited for even larger mutation rates, close to a transition point at which the population delocalizes (i.e., leaves the fitness peak at which it was localized) and starts traversing the sequence space. If the mutation rate is increased even further, the population undergoes a second transition and loses all sensitivity to fitness peaks. This second transition corresponds to the standard error threshold transition first described by Eigen. We discuss the implications of our results for biological evolution and for evolutionary optimization techniques.

Finite-size scaling of the error threshold transition in finite populations

The error threshold transition in a stochastic (i.e. finite population) version of the quasispecies model of molecular evolution is studied using finite-size scaling. For the single-sharp-peak replication landscape, the deterministic model exhibits a first-order transition at $Q=Q_c=1/a$, where $% Q$ is the probability of exact replication of a molecule of length $L \to \infty$, and $a$ is the selective advantage of the master string. For sufficiently large population size, $N$, we show that in the critical region the characteristic time for the vanishing of the master strings from the population is described very well by the scaling assumption $\tau = N^{1/2} f_a \left [ \left (Q - Q_c) N^{1/2} \right ] $, where $f_a$ is an $a$-dependent scaling function.

Eigen's Error Threshold and Mutational Meltdown in a Quasispecies Model

We introduce a toy model for interacting populations connected by mutations and limited by a shared resource. We study the presence of Eigen's error threshold and mutational meltdown. The phase diagram of the system shows that the extinction of the whole population due to mutational meltdown can occur well before an eventual error threshold transition.

Species Formation in Simple Ecosystems

In this paper we consider a microscopic model of a simple ecosystem. The basic ingredients of this model are individuals, and both the phenotypic and genotypic levels are taken in account. The model is based on a long range cellular automaton (CA); introducing simple interactions between the individuals, we get some of the complex collective behaviors observed in a real ecosystem. Since our fitness function is smooth, the model does not exhibit the error threshold transition; on the other hand the size of total population is not kept constant, and the mutational meltdown transition is present. We study the effects of competition between genetically similar individuals and how it can lead to species formation. This speciation transition does not depend on the mutation rate. We present also an analytical approximation of the model.

Population genetics approach to the quasispecies model.

A population genetics formulation of Eigen's molecular quasispecies model is proposed and several simple replication landscapes are investigated analytically. Our results show a remarcable similarity to those obtained with the original kinetics formulation of the quasispecies model. However, due to the simplicity of our approach, the space of the parameters that define the model can be explored. In particular, for the simgle-sharp-peak landscape our analysis yelds some interesting predictions such as the existence of a maximum peak height and a mini- mum molecule length for the onset of the error threshold transition.