Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay

Anti-viral drug treatment of human immunodeficiency virus type I (HIV-1) and hepatitis B virus (HBV) infections causes rapid reduction in plasma virus load. Viral decline occurs in several phases and provides information on important kinetic constants of virus replication in vivo and pharmacodynamical properties. We develop a mathematical model that takes into account the intracellular phase of the viral life-cycle, defined as the time between infection of a cell and production of new virus particles. We derive analytic solutions for the dynamics following treatment with reverse transcriptase inhibitors, protease inhibitors, or a combination of both. For HIV-1, our results show that the phase of rapid decay in plasma virus (days 2-7) allows precise estimates for the turnover rate of productively infected cells. The initial quasi-stationary phase (days 0-1) and the transition phase (days 1-2) are explained by the combined effects of pharmacological and intracellular delays, the clearance of free virus particles, and the decay of infected cells. Reliable estimates of the first three quantities are not possible from data on virus load only; such estimates require additional measurements. In contrast with HIV-1, for HBV our model predicts that frequent early sampling of plasma virus will lead to reliable estimates of the free virus half-life and the pharmacological properties of the administered drug. On the other hand, for HBV the half-life of infected cells cannot be estimated from plasma virus decay.

The curse of the pharaoh: the evolution of virulence in pathogens with long living propagules

We used a mathematical model to evaluate the hypothesis that parasites and pathogens with long living propagules should evolve high levels of virulence, i.e. high rates of pathogen-induced host mortality. Our model shows that the optimal level of virulence is independent of the longevity of the propagules either: (i) if the density or the prevalence of infected hosts is in (or fluctuates around) equilibrium; or (ii) if the death rate of the infected host population is high compared with that of the propagules. The level of virulence that maximizes the parasite's fitness (Multhusian parameter) increases with increasing longevity of its propagules only if the host-parasite system has not reached equilibrium and the death rate of the propagules is high relative to that of the infected hosts. Therefore, for parasites that have recently invaded a susceptible host population, greater propagule longevity may initially favour higher virulence; but once the equilibrium is reached the optimal virulence is independent of propagule longevity.

Viral dynamics in hepatitis B virus infection

Treatment of chronic hepatitis B virus (HBV) infections with the reverse transcriptase inhibitor lamivudine leads to a rapid decline in plasma viremia and provides estimates for crucial kinetic constants of HBV replication. We find that in persistently infected patients, HBV particles are cleared from the plasma with a half-life of approximately 1.0 day, which implies a 50% daily turnover of the free virus population. Total viral release into the periphery is approximately 10(11) virus particles per day. Although we have no direct measurement of the infected cell mass, we can estimate the turnover rate of these cells in two ways: (i) by comparing the rate of viral production before and after therapy or (ii) from the decline of hepatitis B antigen during treatment. These two independent methods give equivalent results: we find a wide distribution of half-lives for virus-producing cells, ranging from 10 to 100 days in different patients, which may reflect differences in rates of lysis of infected cells by immune responses. Our analysis provides a quantitative understanding of HBV replication dynamics in vivo and has implications for the optimal timing of drug treatment and immunotherapy in chronic HBV infection. This study also represents a comparison for recent findings on the dynamics of human immunodeficiency virus (HIV) infection. The total daily production of plasma virus is, on average, higher in chronic HBV carriers than in HIV-infected patients, but the half-life of virus-producing cells is much shorter in HIV. Most strikingly, there is no indication of drug resistance in HBV-infected patients treated for up to 24 weeks.

Can live attenuated virus work as post-exposure treatment?

Simple mathematical models for the competition between different virus variants in the presence of a crossreactive immune response show that, contrary to expectation, selection can favour variants that induce low viral loads. Here, Sebastian Bonhoeffer and Martin Nowak suggest that such 'competitively superior', but 'less pathogenic', mutants may be a possibility for post-exposure treatment of persistent virus infections.

Viral dynamics in human immunodeficiency virus type 1 infection

The dynamics of HIV-1 replication in vivo are largely unknown yet they are critical to our understanding of disease pathogenesis. Experimental drugs that are potent inhibitors of viral replication can be used to show that the composite lifespan of plasma virus and virus-producing cells is remarkably short (half-life approximately 2 days). Almost complete replacement of wild-type virus in plasma by drug-resistant variants occurs after fourteen days, indicating that HIV-1 viraemia is sustained primarily by a dynamic process involving continuous rounds of de novo virus infection and replication and rapid cell turnover.

Intra-host versus inter-host selection: viral strategies of immune function impairment

We investigate the evolution of viral strategies to counteract immunological attack. These strategies can be divided into two classes: those that impair the immune response inside or at the surface of a virus-infected cell and those that impair the immune response outside an infected cell. The former strategies confer a "selfish" individual selective advantage for intra-host competition among viruses. The latter strategies confer an "unselfish" selective advantage to the virus population as a group. A mutant, defective in the gene coding for the extracellular immune function-impairment strategy, may be protected from immune attack because the wild-type virus in the same host successfully impairs the host's immune function. Such "unselfish" defense strategies are neutral with respect to intra-host competition. We present simple models of viral intra-host and combined inter- and intra-host evolution. We show that selfish strategies can evolve by intra-host evolution. Unselfish strategies may evolve if inter-host selection pressures outweigh intra-host selection, suggesting that such strategies can only evolve in viruses with low mutation rates.

Spatial games and the maintenance of cooperation

The Prisoner's Dilemma (PD) is a widely employed metaphor for problems associated with the evolution of cooperative behavior. We have recently proposed an alternative approach to the PD, by exploring "spatial games" in which players--who are either pure cooperators, C, or pure defectors, D--interact with neighbors in some spatial array; in each generation, players add up the scores from all encounters, and in the next generation a given cell is retained by its previous owner or taken over by a neighbor, depending on who has the largest score. Over a wide range of the relevant parameters, we find that C and D persist together indefinitely (without any of the complex strategies that remember past encounters, and anticipate future ones, which characterize essentially all previous work on the iterated PD). Our earlier work, however, dealt with symmetric spatial lattices in two dimensions, deterministic winning and discrete time. We show here that the essential results remain valid in more realistic situations where the spatial distributions of cells are random in two or three dimensions, and where winning is partly probabilistic (rather than being determined by the largest local total). The essential results also remain valid (pace Huberman and Glance [Huberman, B. A. & Glance, N. S. (1993) Proc. Natl. Acad. Sci. USA 90, 7716-7718]) when interactions occur in continuous rather than discrete time.

RNA multi-structure landscapes. A study based on temperature dependent partition functions

Statistical properties of RNA folding landscapes obtained by the partition function algorithm (McCaskill 1990) are investigated in detail. The pair correlation of free energies as a function of the Hamming distance is used as a measure for the ruggedness of the landscape. The calculation of the partition function contains information about the entire ensemble of secondary structures as a function of temperature and opens the door to all quantities of thermodynamic interest, in contrast with the conventional minimal free energy approach. A metric distance of structure ensembles is introduced and pair correlations at the level of the structures themselves are computed. Just as with landscapes based on most stable secondary structure prediction, the landscapes defined on the full biophysical GCAU alphabet are much smoother than the landscapes restricted to pure GC sequences and the correlation lengths are almost constant fractions of the chain lengths. Correlation functions for multi-structure landscape exhibit an increased correlation length, especially near the melting temperature. However, the main effect on evolution is rather an effective increase in sampling for finite populations where each sequence explores multiple structures.