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Phys. Biol. 12 (2015) 054001
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Scaling law characterizing the dynamics of the transition of HIV-1 to
error catastrophe
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Vipul Gupta and Narendra M Dixit
Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India
E-mail: [email protected]
Keywords: error catastrophe, quasispecies, HIV, mutagen, scaling law
Supplementary material for this article is available online
Abstract
Increasing the mutation rate, μ, of viruses above a threshold, μc , 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 − μc ). 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.
Introduction
The molecular quasispecies theory, developed by
Eigen and coworkers in an attempt to understand the
origin of life (Eigen 1971, Eigen et al 1989), has yielded
key insights into viral evolution and suggested a novel
intervention strategy for treating viral infections
(Domingo et al 2012). The theory predicts that when
the viral mutation rate, μ, is below a threshold, μc , the
viral population in an infected individual is dominated
by the fittest, or master, sequence. When μ ⩾ μc ,
however, the distribution of sequences delocalizes in
sequence space, obliterating the domination of the
master sequence and resulting in a catastrophic loss of
genetic information. Intervention to increase μ
beyond μc could thus prove lethal to the virus
(Eigen 2002, Domingo et al 2012). Supporting
evidence comes from in vitro experiments with the
poliovirus, where 4-fold increase in μ lowered viral
infectivity by 70% (Crotty et al 2001), prompting
efforts to develop drugs called mutagens that can
increase μ. For treating HIV-1 infection, for instance,
where current antiretroviral drugs are susceptible to
failure by viral mutation-driven development of
resistance (Wensing et al 2014), this novel strategy of
increasing μ may be particularly promising. Indeed,
several HIV-1 mutagens are in the pipeline (Loeb
et al 1999, Harris et al 2005, Smith et al 2005, Dapp
© 2015 IOP Publishing Ltd
et al 2009, Li et al 2014), including one that entered
clinical trials (Mullins et al 2011).
The success of treatment with mutagens relies on
accurate estimates of μc as well as the timescale of the
transition to error catastrophe, τ . μc determines the
minimum desired level of exposure to a mutagen:
Typically, μ increases with mutagen dosage (e.g., see
Dapp et al 2009) and the dosage employed must
ensure that μ > μc . τ , referred to also as the relaxation
time or the characteristic time of the transition, determines the requisite duration of treatment given
μ > μc . In a recent clinical trial involving an HIV-1
mutagen, for instance, over the 124 days of treatment,
no significant viral load change was observed in
patients, although the viral mutational pattern was
altered (Mullins et al 2011). This absence of antiviral
activity was attributed to the poor knowledge of the
dosage of the mutagen and the duration of treatment
necessary to induce an error catastrophe in HIV-1
(Mullins et al 2011).
Application of the quasispecies theory to estimate
μc and τ is precluded by the assumptions the theory
makes that often do not hold for specific viruses. For
instance, the quasispecies theory considers the asexual
evolution of an infinitely large population of haploid
organisms on an isolated peak fitness landscape. HIV1, in contrast, is diploid, undergoes recombination
(Onafuwa-Nuga and Telesnitsky 2009), has a finite
population in infected individuals (Kouyos et al 2006),
Phys. Biol. 12 (2015) 054001
V Gupta and N M Dixit
and experiences a much broader fitness landscape
(Bonhoeffer et al 2004, Hinkley et al 2011, Ferguson
et al 2013). Although attempts to relax several of the
assumptions of the quasispecies theory have been successful (e.g., see Nilsson and Snoad 2000, Saakian and
Hu 2006, Park and Deem 2007, Nagar and Jain 2009,
Alonso and Fort 2010), incorporating all of the above
features of HIV-1 in the theory has not been possible.
Recently, as an alternative, we developed population genetics-based stochastic simulations that incorporate all the above features of HIV-1 and closely
mimic patient data of viral genomic evolution over
extended durations following infection (Vijay
et al 2008, Balagam et al 2011). With these simulations,
we found that HIV-1 experiences a sharp transition to
error catastrophe and estimated μc to be 2–6 fold
higher than its natural mutation rate (Tripathi
et al 2012). A guideline for the desired level of exposure to HIV-1 mutagens thus emerges, and has been
applied to the drug KP1212 (Li et al 2014). The
required duration of treatment, however, remains
unknown.
In the present study, we employ the stochastic
simulations above to estimate τ as a function of
μ > μc . We find that τ obeys a scaling law:
τ ≈ 0.6/ ( μ − μc ). We examine the dependence of this
scaling on simulation parameters, viz., population
size, genome length, cellular superinfection frequency,
recombination rate, and the fitness landscape, and
find that the scaling is robust to variations in these
parameters although μc is not. With this scaling law,
we estimate that treatment with mutagens would have
to last several years before the transition of HIV-1 to
error catastrophe can be realized.
( )
Methods
Simulation procedure
Our simulations proceed as follows. We represent viral
genomes as bit-strings of length L. Each virus contains
two such genomic strings. We initiate infection with a
founder sequence, F, which we assume without loss of
generality to be the fittest sequence (see figure S1). We
construct F by randomly choosing a sequence of
nucleotides, A, G, C, and U. We let all the V virions in
the initial viral pool contain F. We infect C uninfected
cells synchronously with virions drawn from the viral
pool. Infection involves transferring the chosen viral
genomes to the cells. Each cell is infected by M virions,
where M is drawn randomly from a distribution that
mimics experiments (stacks.iop.org/PB/12/054001/
mmedia). Following infection, the two genomes from
each infecting virion undergo reverse transcription,
which involves mutation and recombination. We
commence reverse transcription from one end of one
of the genomes chosen with a probability ½. The
nucleotide sequence of that string is copied bit by bit
until a switch to the other genome is implemented,
2
which then acts as the template. At each position, the
switch occurs with a probability ρ , the recombination
rate. The resulting string is thus a mosaic of the two
parent genomes. This recombined string is mutated
with a probability μ per position. The mutated string
yields the proviral DNA produced from reverse
transcription of the viral genomes. We repeat this
procedure for all the viral genomes infecting each cell.
Infected cells then produce progeny virions. The
proviral DNA within a cell are randomly chosen in
pairs with repetition and copied as the genomes of
progeny virions. Each cell is assumed to produce P
progeny virions. The progeny virions are subjected to
fitness selection, where virions are chosen with a
probability equal to their relative fitness to yield a new
pool of virions for the infection of a fresh set of C
uninfected cells. The fitness of a virion is assumed to
be the average of the fitness of its genomes. Genomic
fitness is determined by an experimentally identified
fitness landscape for HIV-1 (see text S1).
In each generation, we compute the distribution of
genomes in different Hamming classes. Genomes carrying i mutations relative to F belong to Hamming
class i. We also compute the Shannon entropy asso1 2L
ciated with the distribution, H = − L ∑ j = 1v j log 2 v j ,
where v j is the frequency (or relative prevalence) of
genome j. The time when the mean Shannon entropy
reaches 0.9 is taken as τ , the time when the quasispecies has practically completely delocalized in
sequence space, indicating error catastrophe. We estimate τ as a function of μ > μc and other simulation
parameters. We set parameters to values representative of HIV-1 infection in vivo (Tripathi et al 2012,
Thangavelu et al 2014) and also examine implications
of their variations (text S1). For each parameter setting, we run simulations for up to 10000 generations
and average over up to 10 realizations (figure S2). We
perform the simulations using a code written in C++.
( )
Results
Dynamics of the transition to error catastrophe
We start with a viral population comprising the
founder sequence and let the mutation rate, μ, exceed
the error threshold, μc . Initially, the distribution of
genomic sequences is localized at Hamming class 0.
With time, mutations accumulate and the distribution
shifts to higher Hamming classes. When μ ≈ 60μc , for
instance, the peak of the distribution is at a Hamming
class ∼2000 after 50 generations and reaches Hamming class ∼5000 in ∼500 generations, where it
subsequently stays (figure 1(a)). Note that for the full
length genomes considered here, i.e., L = 10000
nucleotides, the peak at L/2=5000 nucleotides marks
complete delocalization of the quasispecies in
sequence space, indicating completion of the transition to error catastrophe. (When μ < μc , the peak
stabilizes at a Hamming class < L/2 (Tripathi
Phys. Biol. 12 (2015) 054001
V Gupta and N M Dixit
Figure 1. Dynamics and scaling of the transition to error catastrophe. (a) The distribution of genomes in different Hamming classes at
various time points in the transition when the mutation rate μ = 5 × 10−3 substitutions/site/replication. Grey bars represent
standard deviations across runs (see figure S4). (b) The evolution of the mean Shannon entropy, H, with time for different values of μ
(substitutions/site/replication). The dashed line indicates H = 0.9. (c) The transition time, τ , for the data in (b) plotted as a function
of μ − μc (symbols) and the best-fit (line) yielding the scaling. For the parameter values employed (see text S1), μc = 8.4 × 10−5
substitutions/site/replication (Tripathi et al 2012).
et al 2012).) The corresponding evolution of the
Shannon entropy, H, shows a rise from zero at the
start of the simulations to an asymptotic plateau to
unity at the end (figure 1(b)). H here serves as an
order parameter. When all genomes are identical,
indicating complete localization, H = 0, whereas
when all possible genomes are equally represented,
indicating complete delocalization in sequence space,
H = 1. In our previous simulations to determine μc
(Tripathi et al 2012), we assumed that when H crossed
0.9, the quasispecies was sufficiently delocalized to be
treated as having suffered an error catastrophe. In
accordance, here we define τ , the time required for the
transition to error catastrophe to be practically complete, as the time when H = 0.9. (Our key conclusions
are not sensitive to the latter value of H (figure S3).)
Thus, when μ ≈ 60μc , τ ≈ 100 generations.
Scaling law
We examine next the effect of altering μ on τ .
Decreasing μ 10-fold to ≈6μc , increases τ 10-fold to
≈1000 generations (figure 1(b)). This suggests an
inverse relationship between μ and τ . Indeed, we find
upon performing simulations with several values of μ
and analyzing the data that τ scales as ~1/ ( μ − μc )
(figure 1(c)). This scaling is consistent with the
expectation that as μ approaches μc from above, τ
diverges, and with the observation that when μ ≫ μc ,
an n-fold increase in μ results in an n-fold decrease in
τ . We examine next whether the scaling is robust to
parameter variations.
Robustness of the scaling law
We vary the population size of cells, C, the genome
length, L, the frequency of multiple infections of cells,
3
M, the recombination rate, ρ , and the fitness landscape. For many parameter combinations, estimates
of μc are available from our previous study (Tripathi
et al 2012); where unavailable, we estimate μc here
(table S1). We find that μc varies with the parameters
(figure S5). For instance, consistent with our previous
findings (Tripathi et al 2012), μc scales as 1/ C . Yet,
intriguingly, τ depends only on μ − μc , regardless of
the parameters. The dependence of τ on parameters is
thus subsumed in μc . Further, not only is the scaling
applicable to all parameter variations, data from the
various parameter sweeps overlap (figure 2). Thus, the
scaling appears remarkably robust to parameter variations. Using a least squares fit to all our simulation
data, we obtain the scaling law: τ ≈ 0.6/ ( μ − μc )
(figure 2).
Estimate of the minimum treatment duration with
mutagens
We apply the scaling law to estimate the duration of
treatment with an HIV-1 mutagen for ensuring that
the transition to error catastrophe is practically
complete. We estimated μc for HIV-1 in vivo previously to be 7 × 10−5 − 1 × 10−4 substitutions/site/
replication (Tripathi et al 2012). Using this estimate in
the scaling law, we predict that if a mutagen drives μ
5-fold above μc , then τ ≈ 0.6/4μc ≈ 1500−2000 generations, which assuming a mean generation time of
∼2 d for HIV-1 in vivo (Dixit et al 2004), would
correspond to a duration of ≈8−12 years. If the
mutagen were to drive μ 10-fold above μc , the
corresponding duration would be reduced to ≈3.5−5
years. Thus, the transition appears slow and treatment
over several years appears necessary to drive HIV-1 to
an error catastrophe.
Phys. Biol. 12 (2015) 054001
V Gupta and N M Dixit
Figure 2. Robustness of the scaling law. τ as a function of μ − μc obtained from our simulations (symbols) for a wide range of values of
the parameters listed. Data for each parameter sweep is colour coded. For each parameter setting, variations in μ are depicted using a
symbol type. The details of the parameter values employed are in table S1. The corresponding variations of μc are in figure S5. The
black line represents the best-fit to all of the data and the grey lines 95% confidence limits.
Discussion
The notion inspired by the quasispecies theory that
increasing the viral mutation rate can drive viruses
past an error threshold is being explored as a novel
treatment strategy for viral infections and holds
particular promise against infections such as HIV-1
because it may be less susceptible to failure via viral
mutation-driven development of resistance than current antiretroviral treatments (Domingo et al 2012).
Indeed, several mutagens are in development for HIV1 infection. In a previous study, we estimated the error
threshold, μc , of HIV-1 to be 2-6-fold higher than its
natural mutation rate, suggesting that modest
increases in the mutation rate could drive HIV-1 past
its error threshold (Tripathi et al 2012). A recent study,
using an empirical fitness landscape of the HIV-1
protein p6, has also argued that HIV-1 survives close
to its error threshold (Hart and Ferguson 2015). Here,
we identify a scaling law, τ ≈ 0.6/ ( μ − μc ), that relates
the timescale, τ , of the transition to error catastrophe
to the mutation rate, μ, presenting guidelines for the
duration of treatment with HIV-1 mutagens. We
estimate that such treatments would have to last
several years before HIV-1 suffers an error
catastrophe.
Our findings may explain why the short span of
124 days might not have resulted in a significant viral
load decline in patients in a recent clinical trial with an
HIV-1 mutagen (Mullins et al 2011). According to the
scaling law we identify, >50-fold increase in the mutation rate would be required to realize an error catastrophe in such a short span, which is unlikely to be
4
achieved with the dosage employed. Exposure to the
mutagen KP1212 has been observed to increase mutational frequencies by 40–90% (i.e., <2-fold) in the
HIV-1 envelope gene in vitro (Harris et al 2005). Its
pro-drug, KP1461, employed in the trial, appeared to
yield a much smaller, ∼20%, increase in the so-called
private mutational frequency in vivo (Mullins
et al 2011). Clearly, much longer treatment durations
appear to be necessary with the dosages employed.
We find the scaling law to be robust to parameter
variations. Parameter variations alter the strengths of
underlying evolutionary forces. For instance, increasing the frequency of multiple infections of cells increases the contribution of recombination to viral
genomic diversification. Similarly, increasing the
population size lowers stochastic effects and hence
random genetic drift. The scaling law is thus robust to
variations in the underlying evolutionary forces. Parameter variations, however, do affect μc . Thus, the
effects of parameter variations are all subsumed in μc
leaving the scaling law unaffected. The origin of this
intriguing scaling behaviour remains difficult to unravel. Campos and Fontanari (1999) found using a Markov processes approach that the time for the master
sequence to vanish from the population depended on
an unknown function of (Q − 1/a) C , where Q is
the probability of correct copying of an infinitely long
sequence, and a is the selective advantage of the master sequence in an isolated peak fitness landscape.
Here, in closer agreement with the definition of error
catastrophe, we estimate τ as the time for the quasispecies to delocalize in sequence space and not for the
master sequence to vanish. While a Markov chain that
Phys. Biol. 12 (2015) 054001
V Gupta and N M Dixit
mimics our simulations in the absence of recombination can be set up, analyzing its properties and in particular estimating its so-called mixing time, which
would yield τ , poses an outstanding challenge (Dixit
et al 2012, Vishnoi 2015). A further complication is
introduced by the lack of reliable estimates of the fitness of strains carrying multiple mutations leaving the
global fitness landscape inadequately characterized
(Hinkley et al 2011, Ferguson et al 2013).
We recognize several limitations of our study.
First, our simulations ignore lethal mutations that
could be introduced by a mutagen. Lethal mutations
lower viral infectivity and/or productivity, which may
decrease the viral population size continuously leading
to an extinction rather than an error catastrophe
(Tejero et al 2010), as also predicted with APOBEC3Ginduced hypermutations (Thangavelu et al 2014). The
required treatment duration may then be shorter than
estimated by the present scaling law. Using mutagens
in combination with other drugs may similarly
shorten the required treatment duration. Further, we
employ estimates of the viral generation time (∼2 d),
obtained by continuum viral dynamics models (Dixit
et al 2004), that are larger than those obtained by fits of
our simulations, which assume discrete generations
with synchronous infections, to patient data (∼1 d)
(Balagam et al 2011). Lower generation times would
again imply shorter required treatment durations. Our
estimate of the required treatment duration is thus a
conservative one. Second, our simulations do not
incorporate latent reservoirs of HIV-1 (Siliciano and
Greene 2011) because the reservoirs do not contribute
significantly to active viral diversification and would
remain unaffected by mutagens. Eliminating the reservoirs remains a key challenge to HIV-1 eradication
(Deng et al 2015). Finally, our simulations also do not
explicitly consider the host immune response (Perreau
et al 2013). The immune response, to a first approximation, may be viewed as resulting in an altered fitness landscape (Shekhar et al 2013), which we expect
not to affect the scaling law we identify (see above).
Besides, with the effective population size we employ,
our simulations are in agreement with patient data of
viral diversification over extended durations (Vijay
et al 2008, Balagam et al 2011) so that the effects of the
immune response on viral diversification are implicitly accounted for. Nonetheless, the full scope of the
interactions between HIV-1 and the host immune system is yet to be unravelled (Perreau et al 2013), leaving
its implications for treatment with mutagens the subject of future study.
In summary, we unravel a new scaling law characterizing the dynamics of the transition of HIV-1 to
error catastrophe and estimate that treatment of HIV1 infection with mutagens may have to last several
years for the transition to be realized.
5
Acknowledgments
This work was supported by a Wellcome Trust/
DBT India Alliance Senior Fellowship (IA/S/14/1/
501307).
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