Igor M. Rouzine

The solitary wave of asexual evolution

Using a previously undescribed approach, we develop an analytic model that predicts whether an asexual population accumulates advantageous or deleterious mutations over time and the rate at which either process occurs. The model considers a large number of linked identical loci, or nucleotide sites; assumes that the selection coefficient per site is much less than the mutation rate per genome; and includes back and compensating mutations. Using analysis and Monte Carlo simulations, we demonstrate the accuracy of our results over almost the entire range of population sizes. Two limiting cases of our results, when either deleterious or advantageous mutations can be neglected, correspond to the Fisher–Muller effect and Mullers ratchet, respectively. By comparing predictions of our model (no recombination) to those of simple single-locus models (strong recombination), we show that the accumulation of advantageous mutations is slowed by linkage over a broad, finite range of population size. This supports the view of Fisher and Muller, who argued in the 1930s that progressive evolution of organisms is slowed because loci at which beneficial mutations can occur are often linked together on the same chromosome. These results follow from our main finding, that distribution of sequences over the mutation number evolves as a traveling wave whose speed and width depend on population size and other parameters. The model explains a logarithmic dependence of steady-state fitness on the population size reported recently for an RNA virus.

Transition between Stochastic Evolution and Deterministic Evolution in the Presence of Selection: General Theory and Application to Virology

SUMMARY We present here a self-contained analytic review of the role of stochastic factors acting on a virus population. We develop a simple one-locus, two-allele model of a haploid population of constant size including the factors of random drift, purifying selection, and random mutation. We consider different virological experiments: accumulation and reversion of deleterious mutations, competition between mutant and wild-type viruses, gene fixation, mutation frequencies at the steady state, divergence of two populations split from one population, and genetic turnover within a single population. In the first part of the review, we present all principal results in qualitative terms and illustrate them with examples obtained by computer simulation. In the second part, we derive the results formally from a diffusion equation of the Wright-Fisher type and boundary conditions, all derived from the first principles for the virus population model. We show that the leading factors and observable behavior of evolution differ significantly in three broad intervals of population size, N. The “neutral limit” is reached when N is smaller than the inverse selection coefficient. When N is larger than the inverse mutation rate per base, selection dominates and evolution is “almost” deterministic. If the selection coefficient is much larger than the mutation rate, there exists a broad interval of population sizes, in which weakly diverse populations are almost neutral while highly diverse populations are controlled by selection pressure. We discuss in detail the application of our results to human immunodeficiency virus population in vivo, sampling effects, and limitations of the model.

The traveling-wave approach to asexual evolution: Muller's ratchet and speed of adaptation

We use traveling-wave theory to derive expressions for the rate of accumulation of deleterious mutations under Mullers ratchet and the speed of adaptation under positive selection in asexual populations. Traveling-wave theory is a semi-deterministic description of an evolving population, where the bulk of the population is modeled using deterministic equations, but the class of the highest-fitness genotypes, whose evolution over time determines loss or gain of fitness in the population, is given proper stochastic treatment. We derive improved methods to model the highest-fitness class (the stochastic edge) for both Mullers ratchet and adaptive evolution, and calculate analytic correction terms that compensate for inaccuracies which arise when treating discrete fitness classes as a continuum. We show that traveling-wave theory makes excellent predictions for the rate of mutation accumulation in the case of Mullers ratchet, and makes good predictions for the speed of adaptation in a very broad parameter range. We predict the adaptation rate to grow logarithmically in the population size until the population size is extremely large.

Distribution of fixed beneficial mutations and the rate of adaptation in asexual populations

When large asexual populations adapt, competition between simultaneously segregating mutations slows the rate of adaptation and restricts the set of mutations that eventually fix. This phenomenon of interference arises from competition between mutations of different strengths as well as competition between mutations that arise on different fitness backgrounds. Previous work has explored each of these effects in isolation, but the way they combine to influence the dynamics of adaptation remains largely unknown. Here, we describe a theoretical model to treat both aspects of interference in large populations. We calculate the rate of adaptation and the distribution of fixed mutational effects accumulated by the population. We focus particular attention on the case when the effects of beneficial mutations are exponentially distributed, as well as on a more general class of exponential-like distributions. In both cases, we show that the rate of adaptation and the influence of genetic background on the fixation of new mutants is equivalent to an effective model with a single selection coefficient and rescaled mutation rate, and we explicitly calculate these effective parameters. We find that the effective selection coefficient exactly coincides with the most common fixed mutational effect. This equivalence leads to an intuitive picture of the relative importance of different types of interference effects, which can shift dramatically as a function of the population size, mutation rate, and the underlying distribution of fitness effects.

Estimate of effective recombination rate and average selection coefficient for HIV in chronic infection

HIV adaptation to a host in chronic infection is simulated by means of a Monte-Carlo algorithm that includes the evolutionary factors of mutation, positive selection with varying strength among sites, random genetic drift, linkage, and recombination. By comparing two sensitive measures of linkage disequilibrium (LD) and the number of diverse sites measured in simulation to patient data from one-time samples of pol gene obtained by single-genome sequencing from representative untreated patients, we estimate the effective recombination rate and the average selection coefficient to be on the order of 1% per genome per generation (10−5 per base per generation) and 0.5%, respectively. The adaptation rate is twofold higher and fourfold lower than predicted in the absence of recombination and in the limit of very frequent recombination, respectively. The level of LD and the number of diverse sites observed in data also range between the values predicted in simulation for these two limiting cases. These results demonstrate the critical importance of finite population size, linkage, and recombination in HIV evolution.

A Hardwired HIV Latency Program

Biological circuits can be controlled by two general schemes: environmental sensing or autonomous programs. For viruses such as HIV, the prevailing hypothesis is that latent infection is controlled by cellular state (i.e., environment), with latency simply an epiphenomenon of infected cells transitioning from an activated to resting state. However, we find that HIV expression persists despite the activated-to-resting cellular transition. Mathematical modeling indicates that HIVs Tat positive-feedback circuitry enables this persistence and strongly controls latency. To overcome the inherent crosstalk between viral circuitry and cellular activation and to directly test this hypothesis, we synthetically decouple viral dependence on cellular environment from viral transcription. These circuits enable control of viral transcription without cellular activation and show that Tat feedback is sufficient to regulate latency independent of cellular activation. Overall, synthetic reconstruction demonstrates that a largely autonomous, viral-encoded program underlies HIV latency—potentially explaining why cell-targeted latency-reversing agents exhibit incomplete penetrance.

The Stochastic Edge in Adaptive Evolution

In a recent article, Desai and Fisher proposed that the speed of adaptation in an asexual population is determined by the dynamics of the stochastic edge of the population, that is, by the emergence and subsequent establishment of rare mutants that exceed the fitness of all sequences currently present in the population. Desai and Fisher perform an elaborate stochastic calculation of the mean time τ until a new class of mutants has been established and interpret 1/τ as the speed of adaptation. As they note, however, their calculations are valid only for moderate speeds. This limitation arises from their method to determine τ: Desai and Fisher back extrapolate the value of τ from the best-fit classs exponential growth at infinite time. This approach is not valid when the population adapts rapidly, because in this case the best-fit class grows nonexponentially during the relevant time interval. Here, we substantially extend Desai and Fishers analysis of the stochastic edge. We show that we can apply Desai and Fishers method to high speeds by either exponentially back extrapolating from finite time or using a nonexponential back extrapolation. Our results are compatible with predictions made using a different analytical approach (Rouzine et al.) and agree well with numerical simulations.

Link between immune response and parasite synchronization in malaria

Anti-malaria vaccines and drugs could be greatly improved if we knew which phases of Plasmodium falciparum development in red blood cells are major inducers and which are major targets of natural immune responses. This information should focus attention on relevant immunogens and prove useful in developing immune-based therapies. Here we explore the hypothesis that innate immune responses mediate synchronization between the replication cycles of parasites in different red blood cells which is reflected in periodic fevers. Based on a recently developed, rather general mathematical model, we find that periodicity is highly sensitive to the position of both the inducing phase interval and the target phase interval in the parasite replication cycle. In addition, the degree of variability in the length of the replication cycle also strongly affects periodicity. To produce synchronization, the inducing and the target phase intervals must be developmentally distant from each other. We developed a computer program which prompts for information based on measurements of the numbers of erythrocytes in two replication cycle intervals chosen by the researcher, tests our model, and predicts the two phase intervals most critical to the synchronizing immune response. The program can be obtained from the authors.

RNA Recombination Enhances Adaptability and Is Required for Virus Spread and Virulence

Mutation and recombination are central processes driving microbial evolution. A high mutation rate fuels adaptation but also generates deleterious mutations. Recombination between two different genomes may resolve this paradox, alleviating effects of clonal interference and purging deleterious mutations. Here we demonstrate that recombination significantly accelerates adaptation and evolution during acute virus infection. We identified a poliovirus recombination determinant within the virus polymerase, mutation of which reduces recombination rates without altering replication fidelity. By generating a panel of variants with distinct mutation rates and recombination ability, we demonstrate that recombination is essential to enrich the population in beneficial mutations and purge it from deleterious mutations. The concerted activities of mutation and recombination are key to virus spread and virulence in infected animals. These findings inform a mathematical model to demonstrate that poliovirus adapts most rapidly at an optimal mutation rate determined by the trade-off between selection and accumulation of detrimental mutations.

Multi-site adaptation in the presence of infrequent recombination

The adverse effect of co-inheritance linkage of a large number of sites on adaptation has been studied extensively for asexual populations. However, it is insufficiently understood for multi-site populations in the presence of recombination. In the present work, motivated by our studies of HIV evolution in infected patients, we consider a model of haploid populations with infrequent recombination. We assume that small quantities of beneficial alleles preexist at a large number of sites and neglect new mutation. Using a generalized form of the traveling wave method, we show that the effectiveness of recombination is impeded and the adaptation rate is decreased by inter-sequence correlations, arising due to the fact that some pairs of homologous sites have common ancestors existing after the onset of adaptation. As the recombination rate per individual becomes smaller, site pairs with common ancestors become more frequent, making recombination even less effective. In addition, an increasing number of sites become identical by descent across large samples of sequences, causing reversion of the direction of evolution and the loss of beneficial alleles at these sites. As a result, within a 10-fold range of the recombination rate, the average adaptation rate falls from 90% of the infinite-recombination value down to 10%. The entire transition from almost maximum to almost zero may occur at very small recombination rates. Interestingly, the strong effect of linkage on the adaptation rate is predicted in the absence of average linkage disequilibrium (Lewontins measure).