Error catastrophe for viruses infecting cells: analysis of the phase transition in terms of error classes

Phase transitions in evolutionary dynamics

Sharp changes in state, such as transitions from survival to extinction, are hallmarks of evolutionary dynamics in biological systems. These transitions can be explored using the techniques of statistical physics and the physics of nonlinear and complex systems. For example, a survival-to-extinction transition can be characterized as a non-equilibrium phase transition to an absorbing state. Here, we review the literature on phase transitions in evolutionary dynamics. We discuss directed percolation transitions in cellular automata and evolutionary models, and models that diverge from the directed percolation universality class. We explore in detail an example of an absorbing phase transition in an agent-based model of evolutionary dynamics, including previously unpublished data demonstrating similarity to, but also divergence from, directed percolation, as well as evidence for phase transition behavior at multiple levels of the model system's evolutionary structure. We discuss phase transition models of the error catastrophe in RNA virus dynamics and phase transition models for transition from chemistry to biochemistry, i.e., the origin of life. We conclude with a review of phase transition dynamics in models of natural selection, discuss the possible role of phase transitions in unraveling fundamental unresolved questions regarding multilevel selection and the major evolutionary transitions, and assess the future outlook for phase transitions in the investigation of evolutionary dynamics.

Phase transitions in virology

Viruses have established relationships with almost every other living organism on Earth and at all levels of biological organization: from other viruses up to entire ecosystems. In most cases, they peacefully coexist with their hosts, but in most relevant cases, they parasitize them and induce diseases and pandemics, such as the AIDS and the most recent avian influenza and COVID-19 pandemic events, causing a huge impact on health, society, and economy. Viruses play an essential role in shaping the eco-evolutionary dynamics of their hosts, and have been also involved in some of the major evolutionary innovations either by working as vectors of genetic information or by being themselves coopted by the host into their genomes. Viruses can be studied at different levels of biological organization, from the molecular mechanisms of genome replication, gene expression and encapsidation, to global pandemics. All these levels are different and yet connected through the presence of threshold conditions allowing for the formation of a capsid, the loss of genetic information or epidemic spreading. These thresholds, as occurs with temperature separating phases in a liquid, define sharp qualitative types of behaviour. Thesephase transitionsare very well known in physics. They have been studied by means of simple, but powerful models able to capture their essential properties, allowing us to better understand them. Can the physics of phase transitions be an inspiration for our understanding of viral dynamics at different scales? Here we review well-known mathematical models of transition phenomena in virology. We suggest that the advantages of abstract, simplified pictures used in physics are also the key to properly understanding the origins and evolution of complexity in viruses. By means of several examples, we explore this multilevel landscape and how minimal models provide deep insights into a diverse array of problems. The relevance of these transitions in connecting dynamical patterns across scales and their evolutionary and clinical implications are outlined.

Statistical Mechanics and Thermodynamics of Viral Evolution

This paper uses methods drawn from physics to study the life cycle of viruses. The paper analyzes a model of viral infection and evolution using the "grand canonical ensemble" and formalisms from statistical mechanics and thermodynamics. Using this approach we enumerate all possible genetic states of a model virus and host as a function of two independent pressures-immune response and system temperature. We prove the system has a real thermodynamic temperature, and discover a new phase transition between a positive temperature regime of normal replication and a negative temperature "disordered" phase of the virus. We distinguish this from previous observations of a phase transition that arises as a function of mutation rate. From an evolutionary biology point of view, at steady state the viruses naturally evolve to distinct quasispecies. This paper also reveals a universal relationship that relates the order parameter (as a measure of mutational robustness) to evolvability in agreement with recent experimental and theoretical work. Given that real viruses have finite length RNA segments that encode proteins which determine virus fitness, the approach used here could be refined to apply to real biological systems, perhaps providing insight into immune escape, the emergence of novel pathogens and other results of viral evolution.

Scaling law characterizing the dynamics of the transition of HIV-1 to error catastrophe

Increasing the mutation rate, μ of viruses above a threshold, μ(c) has been predicted to trigger a catastrophic loss of viral genetic information and is being explored as a novel intervention strategy. Here, we examine the dynamics of this transition using stochastic simulations mimicking within-host HIV-1 evolution. We find a scaling law governing the characteristic time of the transition: τ ≈ 0.6/(μ - μ(c)). The law is robust to variations in underlying evolutionary forces and presents guidelines for treatment of HIV-1 infection with mutagens. We estimate that many years of treatment would be required before HIV-1 can suffer an error catastrophe.

Error catastrophe and phase transition in the empirical fitness landscape of HIV

We have translated clinical sequence databases of the p6 HIV protein into an empirical fitness landscape quantifying viral replicative capacity as a function of the amino acid sequence. We show that the viral population resides close to a phase transition in sequence space corresponding to an "error catastrophe" beyond which there is lethal accumulation of mutations. Our model predicts that the phase transition may be induced by drug therapies that elevate the mutation rate, or by forcing mutations at particular amino acids. Applying immune pressure to any combination of killer T-cell targets cannot induce the transition, providing a rationale for why the viral protein can exist close to the error catastrophe without sustaining fatal fitness penalties due to adaptive immunity.

Comparative mutational analyses of influenza A viruses

The error-prone RNA-dependent RNA polymerase (RdRP) and external selective pressures are the driving forces for RNA viral diversity. When confounded by selective pressures, it is difficult to assess if influenza A viruses (IAV) that have a wide host range possess comparable or distinct spontaneous mutational frequency in their RdRPs. We used in-depth bioinformatics analyses to assess the spontaneous mutational frequencies of two RdRPs derived from human seasonal (A/Wuhan/359/95; Wuhan) and H5N1 (A/Vietnam/1203/04; VN1203) viruses using the mini-genome system with a common firefly luciferase reporter serving as the template. High-fidelity reverse transcriptase was applied to generate high-quality mutational spectra which allowed us to assess and compare the mutational frequencies and mutable motifs along a target sequence of the two RdRPs of two different subtypes. We observed correlated mutational spectra (τ correlation P < 0.0001), comparable mutational frequencies (H3N2:5.8 ± 0.9; H5N1:6.0 ± 0.5), and discovered a highly mutable motif "(A)AAG" for both Wuhan and VN1203 RdRPs. Results were then confirmed with two recombinant A/Puerto Rico/8/34 (PR8) viruses that possess RdRP derived from Wuhan or VN1203 (RG-PR8×Wuhan(PB2, PB1, PA, NP) and RG-PR8×VN1203(PB2, PB1, PA, NP)). Applying novel bioinformatics analysis on influenza mutational spectra, we provide a platform for a comprehensive analysis of the spontaneous mutation spectra for an RNA virus.

Complex dynamics of life at different scales: from genomic to global environmental issues

This introduction to the Theme Issue, Complex dynamics of life at different scales: from genomic to global environmental issues, gives a short overview on why the ideas and concepts in complexity and nonlinearity are relevant to the understanding of life in its different manifestations. Also, it discusses how life phenomena can be thought of as composing different scales of organization. Finally, the articles in this thematic publication are briefly commented on in terms of their relevance in helping to understand the complexity of life systems.