1 Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia
2 Department of Mathematics, University of Chester, Chester, UK
3 Department of Virology, University of the Saarland, Homburg, Germany
4 Unité de Rétrovirologie Moléculaire, Institut Pasteur, Paris, France
Correspondence
Andreas Meyerhans
Andreas.Meyerhans{at}uniklinik-saarland.de
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ABSTRACT |
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Supplementary material is available in JGV Online.
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INTRODUCTION |
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As the virion is diploid, recombination can also occur. Retroviral recombinants are generated via a copy-choice mechanism involving switching between RNA viral templates during reverse transcription (Coffin, 1979). For HIV-1, the number of crossovers between the two RNA templates has been estimated to range from three to nine per genome per round of replication (Jetzt et al., 2000
; Levy et al., 2004
). As the recombination rate exceeds the mutation rate by a factor of more than 10, once mutations are generated, recombination should be the prime driving force in HIV-1 evolution. This conclusion presupposes that HIV-1-infected cells are both multiply infected in vivo and infected by divergent virus genomes.
The paradigm for viral infections in general assumes that the m.o.i., which is the number of virions needed to set up a productively infected cell, is equal to 1. For HIV, this means that an infected cell harbours a single provirus. De facto this reduces the impact of recombination to zero because transcription of a single provirus would lead to the co-packaging of two identical RNA copies. Hence, even if recombination occurred at a rate of about three to nine per genome per replication round, the effects of recombination on evolution would be trivial. Thus, the obvious chimeras between different HIV-1 clades were understood as arising from rare cases of multi-infected cells (Carr et al., 1998; Hoelscher et al., 2001
; Peeters et al., 1999
; Takehisa et al., 1999
). Recently, however, it has been shown that the majority of HIV-1-infected cells in vivo can contain multiple proviruses (Jung et al., 2002
). Indeed, for the two cases studied in detail, the number of proviruses ranged from one to eight copies per infected splenocyte, with a mean around three. Furthermore, there was extensive sequence variation among HIV genomes from the same nucleus up to 29 % at the amino acid level among hypervariable regions of the HIV-1 envelope protein gp120. This set the stage for recombination as a major player in the intrapatient evolution of HIV.
Understanding intrapatient HIV evolution is compounded by the complexity of the virus dynamics in vivo. For example, there is the meta-population structure of virus replication, bottlenecking inherent in the chronic phase of the infection and clearance by the innate and acquired immune system (Cheynier et al., 1994; Frost et al., 2001
; Gratton et al., 2000
; Grossman et al., 1998
; Wain-Hobson, 1993
). Many groups have analysed viral kinetics following highly active antiretroviral therapy and made inferences about HIV dynamics (Ho et al., 1995
; Perelson et al., 1996
; Wei et al., 1995
). By contrast, only a few have attempted to simulate HIV sequence evolution (Ribeiro & Bonhoeffer, 2000
) and examine recombination effects (Boerlijst et al., 1996
), although in a very restricted manner covering a handful of sites. In addition, no attempt has been made to incorporate the discrete steps inherent to HIV replication.
To appreciate the impact of a high frequency of multi-infected cells, with attendant recombination, on viral evolution we have developed an in silico stochastic model to explore the effects of major microscopic parameters (e.g. the point-mutation and recombination rates, the proviral copy number per cell, etc.) on the dynamics of macroscopic characteristics such as the Hamming distance and the abundance of n-point mutants. It is shown that (i) the effect of recombination depends on the nature of the distribution of mutants in the population, (ii) multi-infection increases the effective mutation rate, and (iii) stochastic events have to be considered as an important factor for the variability in the emergence of mutants under selection pressure.
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METHODS |
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(i) The total viral population Vtotal is constant.
(ii) Only four types of mutants are considered.
(iii) The law of mass action is applied to describe the kinetics of the four types of mutants generated by mutation and recombination.
(iv) The mutation rate is uniform across the whole HIV genome.
(v) All four mutants have identical fitness.
(vi) The frequency of two sequential mutations per replication cycle is so small that it is neglected.
Details of the model are given in Supplementary material S1 available in JGV Online.
A stochastic approach to model intrahost HIV evolution
Representation of viral genomes and sequence diversity.
We represent the HIV-1 genome (nucleotide sequence) as a bit-string of length L (=100). Each position can take two different values (0 or 1). The population (P) of N virions at any given time tn is a set of the bit-strings,
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The Hamming distance between two genomes yi, yj is defined by the formula
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Accordingly, the mean population Hamming distance (the normalized pairwise differences between strings) is computed using the expression
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Modelling HIV replication cycles.
Although the evolution of HIV is a result of many localized bursts of HIV infection (Grossman et al., 1998), we reduced it in the present stochastic model to a sequence of synchronous replication cycles of the whole population of bit-strings, P.
The general structure of the model is given by the following pseudo-code:
t:=0:initialize P(0)
While t<tmax do
(i) P*(t):=recombine/mutate [P(t)]
(ii) P**(t):=expand/co-pack [P*(t)]
(iii) P(t):=select [P**(t)]
(iv) t:=t+1
The fundamental elements of the model and the respective parameters are described in the Results and in Supplementary material available in JGV Online.
Computer implementation.
The genetic-algorithm source code was written in Fortran 90. The supercomputing facilities at the Joint Supercomputer Center of the Russian Academy of Sciences (Moscow) were used. To estimate means of the Hamming distance, we ran 50 single simulations in parallel. Message passing interface (MPI) programming was used to control the data flow.
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RESULTS |
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Qualitative insights into recombination effects
Depending on the circumstances, recombination may either increase or decrease the frequency of a particular mutant within a virus population. As has been shown previously, recombination determines the kinetics of convergence to a linkage equilibrium (Baake, 2001). Thus, the effect of recombination should be easily visible in a situation when a virus population is in a state of linkage disequilibrium. In practical terms, such disequilibrium conditions exist shortly after antiviral treatment.
Let us consider the simple chemical kinetics framework for the relative roles of mutation and recombination in the generation of the basic case of two-point mutants carrying mutations at positions p1 and p2. The point-mutation rate µ for HIV-1 per base and replication round is 0·25x104 (Mansky & Temin, 1995). The position-specific mutation rate is 1·95x105 with a genome length of about 104 bases (see parameters in Supplementary material S1 available in JGV Online). The recombination of the HIV-1 genome was estimated to occur at the rate of
{39}x104 per base per cycle (Jetzt et al., 2000
; Levy et al., 2004
). We now examine the recombination of virus Vp1 with virus Vp2, which carry mutations at positions p1 and p2, respectively. In the absence of hot or cold spots, the distance between p1 and p2 obviously affects the probability that a recombination will produce a two-point mutant. The frequency of such a recombination can vary by as much as four orders of magnitude, given a genome size of 104 bases.
With such a range of recombination rates between two markers, we analysed the population dynamics of single- and two-point mutants with a law of mass action-type model under a set of simplifying assumptions (see Supplementary material S1 available in JGV Online). The rate of change of the density of a two-point mutant Vp1p2 is a function of the recombination-rate times, the linkage disequilibrium and forward and backward mutations. In situations distant from linkage equilibrium e.g. when the virus population is dominated by single-point mutants, then recombination would greatly accelerate the accumulation of Vp1p2 (Fig. 1a). However, when Vp1 or Vp2 would be underrepresented, then recombination would decrease the density of Vp1p2 accordingly (Fig. 1b
).
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(ii) Variation operators.
A mutation process was modelled in two stages: first, the 2L-strings are selected randomly from the whole virus population PHIV with a uniform probability µ, which equals the experimentally determined mutation rate corrected for the reduced length of the model genome (see Supplementary Table available in JGV Online). Then the point-mutation operator is applied, which randomly selects one position of a string and inverts it from zero to one or one to zero. A uniform probability distribution for selection and mutation is used. A recombination operator is implemented as a two-point crossover allowing zero, one or two crossover events per segment. It produces single L-strings referring to proviral DNA from the 2L-strings representing the viral genomes as follows: two random positions along a string are selected (uniform probability distribution) and the sections appearing before the first selected position and after the second selected position are spliced with the segment from the second string that is in between the selected positions. For simplicity, recombination is incorporated as a first variation operator, while mutation is incorporated second. This is a valid option as the mutation and the recombination operators are commutative (Baake, 2001).
(iii) Production of the next generation of virus genomes.
Biologically this involves expansion of viral genomes through transcription of proviral DNA and co-packaging of viral RNA into virions. For the ease of algorithmic implementation, co-packaging precedes the expansion step. These operations are commutative as the number and structure of offspring are independent of the order of the two steps. Co-packaging: from the set PDNA of mutated and recombined L-strings (standing for the population of proviral DNA) we select randomly Ninf-cell groups consisting of m strings. Assuming random co-packaging (linking) of 2L-strings, we generate as many as
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In the second scenario one may assume that the multiplication factor is directly proportional to the proviral copy number per cell, therefore
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(iv) Selection principle.
The HIV population produced at the end of every replication cycle exceeds the number of infected cells by several orders of magnitude (Cavert et al., 1997). Each productively infected CD4+ T cell may produce several thousand virions but only a small fraction takes part in the next round of productive infection. Therefore a similar scale of reduction in the HIV population size has to be modelled to represent the combined effect of bottlenecking (random sampling) and fitness-based selection. In simulations presented below, we considered the random selection of NRNA 2L-strings of the total set of either NRNAx
or NRNAx
xm offspring virions. The bottlenecking factor is taken as being inversely proportional to the proviral copy number, i.e. either as
or
depending on the expansion scheme. Overall, we implemented the so-called (M,
)-evolution strategy (Baeck et al., 1997
) such that M parent genomes create
offspring HIV genomes (
>M), of which M are randomly selected using a uniform probability distribution or a fitness-related probability function.
The development of sequence diversity
An increase in the multiplicity of integrated proviruses per cell accelerates the development of sequence diversity and the appearance of n-point mutants in the virus population (Fig. 3). To characterize an evolving HIV population within a host, we used the mean Hamming distance as a measure of the genetic diversity between individual viral genomes and the appearance of n-point mutants as a measure of the rate of evolution. We considered an initial homogeneous population of 200 infected cells, harbouring one, three, five and eight identical proviruses, a range observed experimentally in vivo (Jung et al., 2002
). As expected from theoretical considerations of stochastic, finite population models, the mean nucleotide diversity increases in a manner directly proportional to the provirus copy number (Fig. 3a
). The maximal divergence (
) was in the range of 14 % within 1200 replication rounds (Fig. 3a
). Such values are within the bounds of a 56-year-old HIV infection. Concerning the rate of evolution, a higher m.o.i. can reduce by up to 60 % the time needed for the emergence of mutants that differ in two to six positions from the founder genome (Fig. 3b
).
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DISCUSSION |
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The network structure of HIV sequence sets from infected individuals suggests that rampant HIV recombination occurs in vivo but becomes visible only above a certain sequence diversity (Cheynier et al., 2001; Kils-Hutten et al., 2001
; Wain-Hobson et al., 2003
). Indeed, novel polymorphisms at a particular viral gene locus can only be introduced by mutations. Recombination may then shuffle such mutated loci between individual genomes and enhance their genetic variation. However, when considering a whole virus population exposed to recombination, then mutant frequencies have to be taken into account. As known from classical evolutionary genetics, recombination will force the mutant distribution to change to an equilibrium state called linkage equilibrium (Maynard Smith, 1989
). Thus, as shown in Fig. 1
, recombination may either favour or reduce the presence of specific mutants, dependent on the frequency distribution of the mutant spectrum. Only for a highly simplified framework, i.e. a two-locus two-allele model, is this easy to demonstrate (Bretscher et al., 2004
; Christiansen et al., 1998
; Maynard Smith, 1989
; Rouzine & Coffin, 1999
; Rouzine et al., 2001
). Yet such a model is of limited value if one seeks to quantify the effects of multi-infection and recombination on genetic evolution during an HIV infection in vivo.
In the present work, we have used a genetic evolutionary algorithm approach to model HIV evolution. It considers in finer detail the biology of HIV replication and mirrors the highly localized nature of the secondary immune system as well as the intense restriction of virus replication by immune responses. Accordingly, extensive bottlenecking is to be expected, reflecting a situation where the majority of progeny die before entering productive infection. The present model shows that the time to build up n-point mutants is enhanced by multi-infection compared with the previous view of a single provirus per cell (Fig. 3b). In a situation where no selection is occurring, mutants can be temporarily fixed over numerous rounds of replication before becoming extinct (Fig. 4
). This is consistent with experimental data, suggesting that the majority of mutations observed in cross-sectional analyses do not arise from strong selection (Kils-Hutten et al., 2001
; Plikat et al., 1997
). However, whenever a strong selection pressure is applied to a few sites, as occurring under antiviral treatment, there is rapid emergence of variants encoding the selected traits (Figs 5 and 6
). Obviously, the extinction of lineages due to bottlenecking and the fixation of mutations in the absence of selection both imply that mutants are far from being in linkage equilibrium. Thus, under the conditions of an initial homogeneous infection, the selection of n-point mutants is generally accelerated by multi-infection and recombination, even though there was great variation in the kinetics of fixation (Figs 5 and 6
).
The in silico simulations serve to highlight the situation if the m.o.i. in vivo would be highly constrained. For example, under such a scenario, the accumulation of resistance mutations would be slower (Fig. 3). It is tempting to suggest that this might be the case under conditions of highly active antiretroviral therapy, when the plasma viral load is reduced frequently by two to three orders of magnitude. Unfortunately this is not axiomatic because the proviral distribution per cell for the two patients studied previously was very similar despite a 25-fold difference in plasma viraemia (Jung et al., 2002
).
Other experimental unknowns suggest that it is probably imprudent to rule out alternative assumptions. For example, what is the limiting factor in virus expansion? Does multi-infection result in more virus production per cell or is production limited by some cellular cofactors? Resolving these questions experimentally would advance our understanding of the dynamics of HIV evolution.
Mathematical approaches to investigate many of the various aspects of HIV infection are still at an early stage. Except for the non-linear regression analysis of virus decay after antiviral treatment, many more elaborated models are based on simplifying assumptions that do not even approach the true complexity and causality of the phenomenon under investigation. For example, for a viral lineage going back 15 years with a low estimate of the intrinsic recombination rate of approximately three crossover events per genome per cycle and no hot or cold spots, and assuming approximately 200 continuous rounds of replication per year (Perelson et al., 1996), then it is possible that something of the order of 9000 crossovers (approx. 15x200x3) are embedded in the lineage. Given the sequence complexity within an individual, up to 1020 % amino acid variation within the hypervariable regions of the envelope glycoprotein, the pertinence of experimentally trying to define a fitness value for what is a precise yet highly ephemeral genome, captured by cloning, or to deduce the true sequence history merits discussion.
In conclusion, a stochastic model for the analysis of HIV evolution was developed that reflects in some detail both the biology of HIV replication and the infection process within a host. With this model we could segregate the contribution of the inherently linked processes of multi-infection and recombination and demonstrate a substantial variation in the mutant dynamics. The model provides a versatile platform for predicting the response of HIV towards therapeutic interventions. In particular, with fitness values for drug resistant mutants, one can examine whether resistance mutations can arise de novo or need to pre-exist at the time of treatment.
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ACKNOWLEDGEMENTS |
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Received 26 April 2005;
accepted 25 July 2005.
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