Evolution models with lethal mutations on symmetric or random fitness landscapes

Quasispecies model of evolution with migration

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

The rich phase structure of a mutator model

We propose a modification of the Crow-Kimura and Eigen models of biological molecular evolution to include a mutator gene that causes both an increase in the mutation rate and a change in the fitness landscape. This mutator effect relates to a wide range of biomedical problems. There are three possible phases: mutator phase, mixed phase and non-selective phase. We calculate the phase structure, the mean fitness and the fraction of the mutator allele in the population, which can be applied to describe cancer development and RNA viruses. We find that depending on the genome length, either the normal or the mutator allele dominates in the mixed phase. We analytically solve the model for a general fitness function. We conclude that the random fitness landscape is an appropriate choice for describing the observed mutator phenomenon in the case of a small fraction of mutators. It is shown that the increase in the mutation rates in the regular and the mutator parts of the genome should be set independently; only some combinations of these increases can push the complex biomedical system to the non-selective phase, potentially related to the eradication of tumors.

Mathematical Models of Quasi-Species Theory and Exact Results for the Dynamics

We formulate the Crow-Kimura, discrete-time Eigen model, and continuous-time Eigen model. These models are interrelated and we established an exact mapping between them. We consider the evolutionary dynamics for the single-peak fitness and symmetric smooth fitness. We applied the quantum mechanical methods to find the exact dynamics of the evolution model with a single-peak fitness. For the smooth symmetric fitness landscape, we map exactly the evolution equations into Hamilton-Jacobi equation (HJE). We apply the method to the Crow-Kimura (parallel) and Eigen models. We get simple formulas to calculate the dynamics of the maximum of distribution and the variance. We review the existing mathematical tools of quasi-species theory.

Evolution of functional diversification within quasispecies

According to quasispecies theory, high mutation rates limit the amount of information genomes can store (Eigen’s Paradox), whereas genomes with higher degrees of neutrality may be selected even at the expenses of higher replication rates (the “survival of the flattest” effect). Introducing a complex genotype to phenotype map, such as RNA folding, epitomizes such effect because of the existence of neutral networks and their exploitation by evolution, affecting both population structure and genome composition. We reexamine these classical results in the light of an RNA-based system that can evolve its own ecology. Contrary to expectations, we find that quasispecies evolving at high mutation rates are steep and characterized by one master sequence. Importantly, the analysis of the system and the characterization of the evolved quasispecies reveal the emergence of functionalities as phenotypes of nonreplicating genotypes, whose presence is crucial for the overall viability and stability of the system. In other words, the master sequence codes for the information of the entire ecosystem, whereas the decoding happens, stochastically, through mutations. We show that this solution quickly outcompetes strategies based on genomes with a high degree of neutrality. In conclusion, individually coded but ecosystem-based diversity evolves and persists indefinitely close to the Information Threshold.

Stochastic simulations suggest that HIV-1 survives close to its error threshold

The use of mutagenic drugs to drive HIV-1 past its error threshold presents a novel intervention strategy, as suggested by the quasispecies theory, that may be less susceptible to failure via viral mutation-induced emergence of drug resistance than current strategies. The error threshold of HIV-1, , however, is not known. Application of the quasispecies theory to determine poses significant challenges: Whereas the quasispecies theory considers the asexual reproduction of an infinitely large population of haploid individuals, HIV-1 is diploid, undergoes recombination, and is estimated to have a small effective population size in vivo. We performed population genetics-based stochastic simulations of the within-host evolution of HIV-1 and estimated the structure of the HIV-1 quasispecies and . We found that with small mutation rates, the quasispecies was dominated by genomes with few mutations. Upon increasing the mutation rate, a sharp error catastrophe occurred where the quasispecies became delocalized in sequence space. Using parameter values that quantitatively captured data of viral diversification in HIV-1 patients, we estimated to be substitutions/site/replication, ∼2–6 fold higher than the natural mutation rate of HIV-1, suggesting that HIV-1 survives close to its error threshold and may be readily susceptible to mutagenic drugs. The latter estimate was weakly dependent on the within-host effective population size of HIV-1. With large population sizes and in the absence of recombination, our simulations converged to the quasispecies theory, bridging the gap between quasispecies theory and population genetics-based approaches to describing HIV-1 evolution. Further, increased with the recombination rate, rendering HIV-1 less susceptible to error catastrophe, thus elucidating an added benefit of recombination to HIV-1. Our estimate of may serve as a quantitative guideline for the use of mutagenic drugs against HIV-1.

Currently available antiretroviral drugs curtail HIV infection but fail to eradicate the virus. A strategy of intervention radically different from that employed by current drugs has been proposed by the molecular quasispecies theory. The theory predicts that increasing the viral mutation rate beyond a critical value, called the error threshold, would cause a severe loss of genetic information, potentially leading to viral clearance. Several chemical mutagens are now being developed that can increase the mutation rate of HIV-1. Their success depends on reliable estimates of the error threshold of HIV-1, which are currently lacking. The quasispecies theory cannot be applied directly to HIV-1: the theory considers an infinitely large population of asexually reproducing haploid individuals, whereas HIV-1 is diploid, undergoes recombination, and is estimated to have a small effective population size in vivo. We employed detailed stochastic simulations that overcome the limitations of the quasispecies theory and accurately mimic HIV-1 evolution in vivo. With these simulations, we estimated the error threshold of HIV-1 to be ∼2–6-fold higher than its natural mutation rate, suggesting that HIV-1 survives close to its error threshold and may be readily susceptible to mutagenic drugs.

Biological evolution in a multidimensional fitness landscape

We considered a multiblock molecular model of biological evolution, in which fitness is a function of the mean types of alleles located at different parts (blocks) of the genome. We formulated an infinite population model with selection and mutation, and calculated the mean fitness. For the case of recombination, we formulated a model with a multidimensional fitness landscape (the dimension of the space is equal to the number of blocks) and derived a theorem about the dynamics of initially narrow distribution. We also considered the case of lethal mutations. We also formulated the finite population version of the model in the case of lethal mutations. Our models, derived for the virus evolution, are interesting also for the statistical mechanics and the Hamilton-Jacobi equation as well.

Lethal mutants and truncated selection together solve a paradox of the origin of life

Background

Many attempts have been made to describe the origin of life, one of which is Eigen's cycle of autocatalytic reactions [Eigen M (1971) Naturwissenschaften 58, 465–523], in which primordial life molecules are replicated with limited accuracy through autocatalytic reactions. For successful evolution, the information carrier (either RNA or DNA or their precursor) must be transmitted to the next generation with a minimal number of misprints. In Eigen's theory, the maximum chain length that could be maintained is restricted to nucleotides, while for the most primitive genome the length is around . This is the famous error catastrophe paradox. How to solve this puzzle is an interesting and important problem in the theory of the origin of life.

Methodology/Principal Findings

We use methods of statistical physics to solve this paradox by carefully analyzing the implications of neutral and lethal mutants, and truncated selection (i.e., when fitness is zero after a certain Hamming distance from the master sequence) for the critical chain length. While neutral mutants play an important role in evolution, they do not provide a solution to the paradox. We have found that lethal mutants and truncated selection together can solve the error catastrophe paradox. There is a principal difference between prebiotic molecule self-replication and proto-cell self-replication stages in the origin of life.

Conclusions/Significance

We have applied methods of statistical physics to make an important breakthrough in the molecular theory of the origin of life. Our results will inspire further studies on the molecular theory of the origin of life and biological evolution.