Interpreting the effect of vaccination on steady state infection in animals challenged with Simian immunodeficiency virus.

A representative vaccinated macaque challenged with SIVmac251 establishes a persistent infection with a lower virus load, higher CTL frequencies, and much higher helper cell frequencies, than a representative control animal. The reasons for the difference are not fully understood. Here we interpret this effect using a mathematical model we developed recently to explain results of various experiments on virus and CTL dynamics in SIV-infected macaques and HIV-infected humans. The model includes two types of cytotoxic lymphocytes (CTLs) regulated by antigen-activated helper cells and directly by infected cells, respectively, and predicts the existence of two steady states with different viremia, helper cell and CTL levels. Depending on the initial level of CTL memory cells and helper cells, a representative animal ends up in either the high-virus state or the low-virus state, which accounts for the observed differences between the two animal groups. Viremia in the low-virus state is proportional to the antigen sensitivity threshold of helper cells. Estimating the infectivity ratio of activated and resting CD4 T cells at 200-300, the correct range for the critical memory cell percentage and the viremia peak suppression is predicted. However, the model does not explain why viremia in the "low-virus state" is surprisingly high , relative to vaccinated animals infected with SHIV, and broadly distributed among challenged animals. We conclude that the model needs an update explaining extremely low sensitivity of uninfected helper cells to antigen in vaccinated animals.

Model with two types of CTL regulation and experiments on CTL dynamics.

Recently, we developed a mathematical model of interaction between the HIV and the immune system to match various dynamic experiments carried out in HIV-infected humans and SIV-infected macaques. The model includes helper cell-dependent and helper cell-independent cytotoxic lymphocytes (CTLs) and predicts two stable steady states, a state with a high virus load and few helper cells, and another state with a low virus load and many helper cells. Here we upgrade the model to take into account recent reports on the link between the activation status of infected cells and their ability to produce virus, the effect of helper cells at the time of priming on CTL differentiation, and virus dynamics in unvaccinated macaques with a broad genetic background acutely infected with SIVmac251. We also discuss in detail the experimental justification of the CTL block and the robustness of model predictions with respect to the hypothesis of two CTL subtypes.

Highly fit ancestors of a partly sexual haploid population.

Earlier, using the semi-deterministic solitary wave approach, we have investigated accumulation of pre-existing beneficial alleles in genomes consisting of a large number of simultaneously evolving sites in the presence of selection and infrequent recombination with small rate r per genome. Our previous results for the dynamics of the fitness distribution of genomes are now interpreted in terms of the life cycle of recombinant clones. We show that, at sufficiently small r, the clones dominating fitness classes, at the moment of their birth, are nearly the best fit in a population. New progeny clones are mostly generated by parental genomes whose fitness falls within a narrow interval in the middle of the high-fitness tail of fitness distribution. We also derive the fitness distribution for the distant ancestors of sites of a randomly sampled genome and show that its form is controlled by a single composite model parameter proportional to r. The ancestral fitness distribution differs dramatically from the fitness distribution of the entire ancient population: it is much broader and localized in the high-fitness tail of the ancient population. We generalize these results to the case of moderately small r to conclude that, regardless of fitness of an individual, all its distant ancestors are exceptionally well fit.

Increasing sequence correlation limits the efficiency of recombination in a multisite evolution model.

The accumulation of preexisting beneficial alleles in a haploid population, under selection and infrequent recombination, and in the absence of new mutation events is studied numerically by means of detailed Monte Carlo simulations. On the one hand, we confirm our previous work, in that the accumulation rate follows modified single-site kinetics, with a timescale set by an effective selection coefficient s(eff) as shown in a previous work, and we confirm the qualitative features of the dependence of s(eff) on the population size and the recombination rate reported therein. In particular, we confirm the existence of a threshold population size below which evolution stops before the emergence of best-fit individuals. On the other hand, our simulations reveal that the population dynamics is essentially shaped by the steady accumulation of pairwise sequence correlation, causing sequence congruence in excess of what one would expect from a uniformly random distribution of alleles. By sequence congruence, we understand here the opposite of genetic distance, that is, the fraction of monomorphic sites of specified allele type in a pair of genomes (individual sequences). The effective selection coefficient changes more rapidly with the recombination rate and has a higher threshold in this parameter than found in the previous work, which neglected correlation effects. We examine this phenomenon by monitoring the time dependence of sequence correlation based on a set of sequence congruence measures and verify that it is not associated with the development of linkage disequilibrium. We also discuss applications to HIV evolution in infected individuals and potential implications for drug therapy.

Two types of cytotoxic lymphocyte regulation explain kinetics of immune response to human immunodeficiency virus.

The organization of the cytotoxic T lymphocyte (CTL) response at organismal level is poorly understood. We propose a mathematical model describing the interaction between HIV and its host that explains 20 quantitative observations made in HIV-infected individuals and simian immunodeficiency virus-infected monkeys, including acute infection and response to various antiretroviral therapy regimens. The model is built on two modes of CTL activation: direct activation by infected cells and indirect activation by CD4 helper cells activated by small amounts of virus. Effective infection of helper cells by virus leads to a stable chronic infection at high virus load. We assume that CTLs control virus by killing infected cells. We explain the lack of correlation between the CTL number and the virus decay rate in therapy and predict that individuals with a high virus load can be switched to a low-viremia state that will maintain stability after therapy, but the switch requires fine adjustment of therapy regimen based on the model and individual parameters.

Evolution of human immunodeficiency virus under selection and weak recombination.

To predict emergence of drug resistance in patients undergoing antiretroviral therapy, we study accumulation of preexisting beneficial alleles in a haploid population of N genomes. The factors included in the model are selection with the coefficient s and recombination with the small rate per genome r (r << s sqrt of k, where k is the average number of less-fit loci per genome). Mutation events are neglected. To describe evolution at a large number of linked loci, we generalize the analytic method we developed recently for an asexual population. We show that the distribution of genomes over the deleterious allele number moves in time as a "solitary wave" that is quasi-deterministic in the middle (on the average) but has stochastic edges. We arrive at a single-locus expression for the average accumulation rate, in which the effects of linkage, recombination, and random drift are all accounted for by the effective selection coefficient s lnNr/lnNs(2)k/r. At large N, the effective selection coefficient approaches the single-locus value s. Below the critical size N(c) approximately 1/r, a population eventually becomes a clone, recombination cannot produce new sequences, and virus evolution stops. Taking into account finite mutation rate predicts a small, finite rate of evolution at N < N(c). We verify the accuracy of the results analytically and by Monte Carlo simulation. On the basis of our findings, we predict that partial depletion of the HIV population by combined anti-retroviral therapy can suppress emergence of drug-resistant strains.

Transition between stochastic evolution and deterministic evolution in the presence of selection: general theory and application to virology.

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.

Search for the mechanism of genetic variation in the pro gene of human immunodeficiency virus.

To study the mechanism of evolution of the human immunodeficiency virus (HIV) protease gene (pro), we analyzed a database of 213 pro sequences isolated from 11 HIV type 1-infected patients who had not been treated with protease inhibitors. Variation in pro is restricted to rare variable bases which are highly diverse and differ in location among individuals; an average variable base appears in about 16% of individuals. The average intrapatient distance per individual variable site, 27%, is similar for synonymous and nonsynonymous sites, although synonymous sites are twice as abundant. The latter observation excludes selection for diversity as an important, permanently acting factor in the evolution of pro and leaves purifying selection as the only kind of selection. Based on this, we developed a model of evolution, both within individuals and along the transmission chain, which explains variable sites as slightly deleterious mutants slowly reverting to the better-fit variant during individual infection. In the case of a single-source transmission, genetic bottlenecks at the moment of transmission effectively suppress selection, allowing mutants to accumulate along the transmission chain to high levels. However, even very rare coinfections from independent sources are, as we show, able to counteract the bottleneck effect. Therefore, there are two possible explanations for the high mutant frequency. First, the frequency of coinfection in the natural host population may be quite low. Alternatively, a strong variation of the best-adapted sequence between individuals could be caused by a combination of an immune response present in early infection and coselection.

Linkage disequilibrium test implies a large effective population number for HIV in vivo.

The effective size of the HIV population in vivo, although critically important for the prediction of appearance of drug-resistant variants, is currently unknown. To address this issue, we have developed a simple virus population model, within which the relative importance of stochastic factors and purifying selection for genetic evolution differs over, at least, three broad intervals of the effective population size, with approximate boundaries given by the inverse selection coefficient and the inverse mutation rate per base per cycle. Random drift and selection dominate the smallest (stochastic) and largest (deterministic) population intervals, respectively. In the intermediate (selection-drift) interval, random drift controls weakly diverse populations, whereas strongly diverse populations are controlled by selection. To estimate the effective size of the HIV population in vivo, we tested 200 pro sequences isolated from 11 HIV-infected patients for the presence of a linkage disequilibrium effect which must exist only in small populations. This analysis demonstrated a steady-state virus population of 10(5) infected cells or more, which is either in or at the border of the deterministic regime with respect to evolution of separate bases.