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

Viral quasispecies

Viral quasispecies refers to a population structure that consists of extremely large numbers of variant genomes, termed mutant spectra, mutant swarms or mutant clouds. Fueled by high mutation rates, mutants arise continually, and they change in relative frequency as viral replication proceeds. The term quasispecies was adopted from a theory of the origin of life in which primitive replicons) consisted of mutant distributions, as found experimentally with present day RNA viruses. The theory provided a new definition of wild type, and a conceptual framework for the interpretation of the adaptive potential of RNA viruses that contrasted with classical studies based on consensus sequences. Standard clonal analyses and deep sequencing methodologies have confirmed the presence of myriads of mutant genomes in viral populations, and their participation in adaptive processes. The quasispecies concept applies to any biological entity, but its impact is more evident when the genome size is limited and the mutation rate is high. This is the case of the RNA viruses, ubiquitous in our biosphere, and that comprise many important pathogens. In virology, quasispecies are defined as complex distributions of closely related variant genomes subjected to genetic variation, competition and selection, and that may act as a unit of selection. Despite being an integral part of their replication, high mutation rates have an upper limit compatible with inheritable information. Crossing such a limit leads to RNA virus extinction, a transition that is the basis of an antiviral design termed lethal mutagenesis.

Internal Disequilibria and Phenotypic Diversification during Replication of Hepatitis C Virus in a Noncoevolving Cellular Environment

Viral quasispecies evolution upon long-term virus replication in a noncoevolving cellular environment raises relevant general issues, such as the attainment of population equilibrium, compliance with the molecular-clock hypothesis, or stability of the phenotypic profile. Here, we evaluate the adaptation, mutant spectrum dynamics, and phenotypic diversification of hepatitis C virus (HCV) in the course of 200 passages in human hepatoma cells in an experimental design that precluded coevolution of the cells with the virus. Adaptation to the cells was evidenced by increase in progeny production. The rate of accumulation of mutations in the genomic consensus sequence deviated slightly from linearity, and mutant spectrum analyses revealed a complex dynamic of mutational waves, which was sustained beyond passage 100. The virus underwent several phenotypic changes, some of which impacted the virus-host relationship, such as enhanced cell killing, a shift toward higher virion density, and increased shutoff of host cell protein synthesis. Fluctuations in progeny production and failure to reach population equilibrium at the genomic level suggest internal instabilities that anticipate an unpredictable HCV evolution in the complex liver environment.IMPORTANCE Long-term virus evolution in an unperturbed cellular environment can reveal features of virus evolution that cannot be explained by comparing natural viral isolates. In the present study, we investigate genetic and phenotypic changes that occur upon prolonged passage of hepatitis C virus (HCV) in human hepatoma cells in an experimental design in which host cell evolutionary change is prevented. Despite replication in a noncoevolving cellular environment, the virus exhibited internal population disequilibria that did not decline with increased adaptation to the host cells. The diversification of phenotypic traits suggests that disequilibria inherent to viral populations may provide a selective advantage to viruses that can be fully exploited in changing environments.

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.