Stochastic effects are important in intrahost HIV evolution
even when viral loads are high
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Citation
Read, E. L., A. A. Tovo-Dwyer, and A. K. Chakraborty.
“Stochastic Effects Are Important in Intrahost HIV Evolution Even
When Viral Loads Are High.” Proceedings of the National
Academy of Sciences 109.48 (2012): 19727–19732.
As Published
http://dx.doi.org/10.1073/pnas.1206940109
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National Academy of Sciences (U.S.)
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Thu May 26 20:25:21 EDT 2016
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http://hdl.handle.net/1721.1/78837
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Detailed Terms
Stochastic effects are important in intrahost HIV
evolution even when viral loads are high
Elizabeth L. Reada,b, Allison A. Tovo-Dwyerb,c, and Arup K. Chakrabortya,b,c,d,e,1
Departments of aChemical Engineering, cChemistry, dBiological Engineering, and ePhysics, Massachusetts Institute of Technology, Cambridge, MA 02139;
and bRagon Institute of Massachusetts General Hospital, Massachusetts Institute of Technology, and Harvard, Boston, MA 02114
rare events
| viral dynamics
D
iscerning why some individuals maintain low levels of virus
and remain AIDS-free indefinitely, whereas others harbor
high viral loads and develop AIDS rapidly on HIV infection will
provide important clues for the development of a vaccine and
improved therapies. HIV replicates at an enormous rate during
the asymptomatic phase of infection (1). The virus’ fast and errorprone replication allows it to accumulate mutations that can
evade immune pressure (2), and the immune system responds by
shifting its points of attack (3, 4). Thus, HIV infection is characterized by complex dynamics of host–virus interaction, which
remain poorly understood.
Elite controllers and controllers are individuals who are able to
control HIV infection sustainably without therapy. Many lines of
evidence suggest that the genetic determinants of controlling HIV
are protective HLA-I alleles (e.g., B57) (5), which encode the
molecules that present antigenic peptides to cytotoxic T lymphocytes (CTLs). Various factors have been identified as contributing to the efficacy of HLA-I–associated immune responses
in controllers, including preferential targeting of conserved Gag
epitopes (6, 7), cross-reactivity of the T-cell repertoire (8), and
targeting of coordinately conserved regions (9). However, most
people with protective alleles do not control HIV (10), and
a significant fraction of controllers lack these alleles (11).
CTLs play a key role in suppressing both HIV and simian immunodeficiency virus infection (12–15), consistent with the finding
of HLA-I involvement. Many HLA-I–associated viral escape
mutations have been identified that interfere with the CTL response while incurring a cost of reduced viral replicative fitness at
the same time (2, 16, 17). Such sequence polymorphisms driven by
CTL-imposed selection pressure often appear in a coordinated
manner with secondary mutations, suggesting that fitness defects
caused by primary escape mutations can be compensated for by
additional changes at structurally or functionally linked sites within
the protein (2, 18, 19). Thus, the interplay between immune evasion
www.pnas.org/cgi/doi/10.1073/pnas.1206940109
at points in the HIV proteome under CTL attack and maintenance
of replicative fitness gives rise to complex viral evolutionary dynamics, which can affect clinical outcomes. For example, emergence of escape mutations has been observed to precipitate
progression to AIDS (13, 20), and distinct mutation patterns distinguish B57+ controllers from B57+ progressors (21, 22).
Despite the rapidity of HIV evolution and the clear benefit to
the virus of making particular HLA-I–associated mutations, the
waiting time to appearance of such mutations varies greatly (13, 19,
22, 23). These observations raise questions. Why are the time required by the virus to mutate to an escape variant and the path
taken through sequence space not predictable? Is this heterogeneity related to the unpredictability of clinical outcomes? In particular, why is elite control highly associated with protective alleles
(e.g., B57) yet still rare in individuals harboring these alleles?
It would appear that stochastic effects may contribute to
unpredictability of disease outcomes in HIV infection. A
number of studies have considered stochastic effects in HIV
evolution, particularly in the context of antiretroviral therapy
(ART) resistance mutations (24–26). These have largely proceeded within the framework of evolutionary population dynamics, in which stochastic effects manifest as genetic drift,
significant only when the effective size of the mutating population is small or the selective pressures favoring particular
mutations are weak (27). Within this framework, the effective
size of the virus population within an infected individual, and
thus the importance of stochastic evolutionary processes, has
remained controversial (28–31).
However, these studies neglected the dynamics of CTLs, which
impose critical selection pressure in natural infection, and a realistic description of the viral fitness landscape (e.g., compensatory mutations). In a hybrid model of compensatory mutation
under CTL pressure, in which CTL–virus interaction dynamics
were assumed to be deterministic and virus mutations arose stochastically, stochastic effects were found to play a minor role (32).
We studied a fully stochastic computational model of host–virus
interaction that incorporates the interplay between the evolving
virus and epitope-specific CTL responses. We show that stochastic effects in viral evolution cannot be neglected, even at very
large HIV population sizes, and can produce qualitatively different disease outcomes in identical individuals. These stochastic
effects are associated with the virus traversing a fitness barrier to
escape, rather than resulting from standard genetic drift. The
results highlight the necessity of considering additional types of
stochastic processes beyond genetic drift in evolutionary models
of HIV. The results also give insight into the high person-toperson variability observed in HIV and demonstrate how genetic
factors influence the probability with which HIV can be controlled
without therapy.
Author contributions: E.L.R. and A.K.C. designed research; E.L.R. and A.A.T.-D. performed
research; E.L.R. and A.A.T.-D. analyzed data; and E.L.R. and A.K.C. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
1
To whom correspondence should be addressed. E-mail: arupc@mit.edu.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.
1073/pnas.1206940109/-/DCSupplemental.
PNAS | November 27, 2012 | vol. 109 | no. 48 | 19727–19732
IMMUNOLOGY
Blood plasma viral loads and the time to progress to AIDS differ
widely among untreated HIV-infected humans. Although people
with certain HLA (HLA-I) alleles are more likely to control HIV infections without therapy, the majority of such untreated individuals
exhibit high viral loads and progress to AIDS. Stochastic effects are
considered unimportant for evolutionary dynamics in HIV-infected
people when viral load is high or when selective forces strongly drive
mutation. We describe a computational study of host–pathogen interaction demonstrating that stochastic effects can have a profound
influence on disease dynamics, even in cases of high viral load and
strong selective pressure. These stochastic effects are pronounced
when the virus must traverse a fitness “barrier” in sequence space
to escape the host’s cytotoxic T-lymphocyte (CTL) response, as often
occurs when a fitness defect imposed by a CTL-driven mutation must
be compensated for by other mutations. These “barrier-crossing”
events are infrequent and stochastic, resulting in divergent disease
outcomes in genetically identical individuals infected by the same
viral strain. Our results reveal how genetic determinants of the CTL
response control the probability with which an individual is able to
control HIV infection indefinitely, and thus provide clues for
vaccine design.
CHEMISTRY
Edited by Michael L. Klein, Temple University, Philadelphia, PA, and approved September 28, 2012 (received for review April 25, 2012)
Model
Our host–pathogen interaction model has the core features of
a mutating virus and epitope-specific CTL activation and killing of
infected cells. The infecting virus is composed of a small number
of mutable sites, some of which represent CTL epitopes. Mutations leading to sequence changes within CTL epitopes may diminish CTL susceptibility, according to the specificities encoded
in the CTL repertoire. Mutation at any site can result in altered
viral fitness (according to a specified fitness landscape) and is
reflected by a change in the viral replication rate. Parameters of
the model are quantitatively in line with experimental measurements, where available. Fitness landscapes and CTL repertoires
reflect qualitative experimental findings. Infections in individual
hosts were simulated according to either fully stochastic dynamics
(master equations) or deterministic [ordinary differential equation (ODE)] descriptions (Methods).
We first studied a small model system exhibiting CTL escape
and compensatory mutation. The model is a two-allele, two-locus
virus, with one locus subject to CTL pressure (Fig. 1A). Infection
begins with a single founding strain (33), which can evolve via point
mutations or simultaneous double mutations; the latter are rarer,
and so arise at a slower rate. Each possible viral strain (indexed by
k ∈ f00; 01; 10; 11g, where 0 denotes the WT allele at a locus and
1 indicates mutation) is assigned a replication rate, sk. The “intrinsic” replicative fitness, fk, is the replication rate minus the
clearance rate (u), which is assumed to be identical for all strains
(fk = sk − u). Mutation at either site incurs a fitness penalty, corresponding to a reduction of fk by a factor of δk (δk <1). If there are
mutations at both loci, the replicative fitness loss imposed by a
single mutation may be completely or partially restored (if δ11 >
δ01, δ10). Mutation in the epitope reduces the CTL pressure [by
abrogating HLA binding or T-cell receptor (TCR) recognition] by
a factor of α (α < 1). Although fk denotes intrinsic fitness, a measure of effective in-host fitness is given by Fk = fk=χ , or the ratio
k
of intrinsic fitness to CTL susceptibility, χ k. This definition arises
because the difference in Fk serves as a bifurcation parameter
determining which viral strain has the competitive advantage when
multiple strains are present in the host (SI Appendix, Fig. S1). The
effective fitness landscape is “compensatory” if F11 > F00 > F01,
F10. We have also studied more complex variants of this simple
model, which differ in the character of the virus’ fitness landscape
and nature of the CTL response.
A
B
Results
Setpoint Viral Load Exhibits Stochastically Driven Bimodal Outcomes.
We studied various viral fitness landscapes within the two-locus
model of Fig. 1A. The immunodominant B27-KK10 epitope is an
example where a CTL escape mutation completely destroys virus
viability unless it is preceded by another mutation (2). Accordingly, we studied a case where a single mutant at the site subject
to CTL pressure is completely unfit, the other single mutant is
mildly impaired relative to the infecting strain, and the double
mutant has intermediate (intrinsic) fitness. Qualitative results were
robust and did not depend on a particular choice of parameters
(SI Appendix, Figs. S2–S4, S8, and S9).
Deterministic simulation of infection for this model (using
ODEs) shows rapid escape to the double mutant (Fig. 1 C and
D). Viral load peaks in I00 (cells infected by the founding strain)
shortly after infection, when the activated CTL response begins
to exert control on the virus. The single mutants (I01 and I10)
appear rapidly, followed by the double mutant (I11) within ∼25 d,
followed by setpoint with the double mutant (I11) dominant.
Although the double mutant has reduced intrinsic replicative
fitness relative to the founding strain, it is effectively the fittest
strain because it is subject to reduced CTL pressure.
Simulations that explicitly account for stochastic effects using
the Gillespie method were carried out for precisely the same
model with identical parameters and initial conditions. Two
types of behavior emerge: immediate appearance and outgrowth
of I11 during acute infection, similar to the ODE dynamics (Fig. 1
C and D, Middle), or attainment of setpoint with I00 remaining
dominant (Fig. 1 C and D, Bottom). In the latter case, I11 often
appears transiently but fails to expand. Statistics from many
stochastic simulations (Fig. 2A) show that total viral load
“measured” at 500 d after infection exhibits a bimodal pattern,
with the peaks at low and high viral loads corresponding to
unescaped and escaped trajectories, respectively. We observed
similar stochastic bimodality of viral loads in systems with various
compensatory fitness landscapes (SI Appendix, Fig. S2–S4).
Analysis of the corresponding ODE systems reveals a single
stable state representing viral escape in all cases (SI Appendix).
Thus, purely stochastic effects cause individuals with identical
immune repertoires and infecting viruses to exhibit different
disease phenotypes.
In contrast, if the virus can escape CTL pressure by making
a single mutation that does not incur too large a defect in replicative capacity (F01 > F00), stochastic effects are irrelevant and
C
D
Fig. 1. (A) Schematic for model virus under CTL pressure. The virus has two loci: one representing an epitope site under CTL pressure (yellow) and the other
representing a functionally linked site not directly targeted by CTLs. Arrows represent evolution by single (m(1)) and double (m(2)) mutations. Mutation at the
CTL-targeted site results in reduction of CTL recognition, according to the parameter α. Mutation at either site can result in reduced intrinsic viral fitness fk
(units, day−1). (B) Choice of parameters δk and α determines the effective in-host replicative fitness, Fk, of each strain. (C) Simulated dynamics of the four viral
strains after infection of the host by strain 00 for a compensatory fitness landscape (Fk/F00 = {1, 0, .93, 9.5}, where k ∈ {00, 01, 10, 11}, respectively). All
parameters given in SI Appendix. Colors of the time traces correspond to the colors representing the different viral strains in A. Deterministic (ODE) dynamics
show rapid escape to double mutant (Top); stochastic Gillespie simulations show two types of behavior, rapid escape similar to ODE distribution (Middle) or
prevention of escape beyond setpoint (Bottom), giving a 20-fold lower setpoint viral load (dotted black line at y = 5 × 104 shown to guide the eye). The
increase in viral load on escape is largely determined by the parameter α. (D) Same as C but with α = 0.01, giving 100-fold higher setpoint viral loads in escape
trajectories. We assume virus particles of strain k are in quasi-steady state with respect to cells infected by strain k, Ik, and therefore present viral loads in terms
of total-body infected cells. Stochastic simulations were performed for volumes consistent with realistic estimates of total-body infected cells (SI Appendix).
19728 | www.pnas.org/cgi/doi/10.1073/pnas.1206940109
Read et al.
3
3
Occurrence (/10 )
10
0
200
400
Time (Days)
2
1
0
4.5 5 5.5 6 6.5
Viral Load at 500 Days
5
10
3
10
0
200
400
Time (Days)
1
0.5
0
4.5 5 5.5 6 6.5
Viral Load at 500 Days
Fig. 2. Statistics from Gillespie simulations of HIV infection show bimodal
distribution of setpoint viral loads when the fitness landscape is compensatory. (A) Simulations for model virus, where mutation of the epitope
imposes a severe cost to replicative fitness, which is partially compensated
on mutation at the other site (same as Fig. 1C). (Upper) Stochastic trajectories (gray) show two behaviors: early escape as in deterministic (ODE) dynamics (red) or escape later or not at all within the simulation time. (Lower)
Distribution of viral loads is bimodal, with 10% preventing escape beyond
500 d. Escaped viral loads center around the ODE result (red line), whereas
unescaped trajectories center around the ODE result with reduced topology
of the viral mutational network (blue line; see main text pertaining to Fig.
4). (B) As in A, but where cost on epitope mutation is lower, such that
the single mutant is effectively more fit than the infecting strain (Fk/F00 = {1,
5, .93, 9.5}, all parameters in SI Appendix).
escape occurs during primary infection as in the ODE simulations (Fig. 2B). Also, if the double mutant is characterized by
synergistic fitness costs that make it very unfit, escape occurs
neither deterministically or stochastically (SI Appendix, Fig. S3).
Thus, the purely stochastic bistability is a general feature of
compensatory systems, in which point mutants have decreased
(and multiple mutants have increased) effective fitness relative
to the infecting virus strain.
Barrier in Effective Fitness Landscape Results in a Stochastically
Driven Wide Distribution of Escape Times. Our results show that
stochastic effects are important when the virus has to traverse
a fitness barrier to escape the immune response. The distribution
of times to escape [tescape , where “escape” is defined as dominance of the strain with highest effective fitness (I11 ≥ 50% of
total viral load)], shows that the virus escapes in most individuals
during primary infection, as in deterministic calculations, for
a compensatory fitness landscape (Fig. 3A, Upper). However,
there is an additional long tail in tescape , corresponding to individuals who durably control the virus (lower peak of the bimodal
viral load distribution at 500 d postinfection). Some of these
individuals continue to prevent escape for decades or longer.
The virus may be able to acquire multiple mutations simultaneously within one replication cycle, either due to errors during
reverse transcription at key escape sites or by recombination of
two different templates harboring the single-site mutations in a
superinfected cell. When recombination is neglected [such that
the double-mutation probability is the square of the singlemutation probability, m(2) = (m(1))2], the dominant pathway
through sequence space to the compensatory double mutant is
through the single mutant (Fig. 3A, Lower), and a long tail in the
distribution of tescape is seen. Increasing the double-mutation rate
to allow recombination (Fig. 3B and SI Appendix) increases the
contribution of escape occurring directly from the infecting
strain, akin to “tunneling” through the barrier. Here again, tescape
can exhibit a long-tailed distribution, leading to a bimodal viral
load (Fig. 3B).
A process that occurs by crossing a significant barrier is a rare
event. Such rare events are strongly influenced by stochastic
effects (34). The long tail in the distribution of tescape arises from
the importance of stochastic effects in the rare crossing of a fitness barrier (an unfit intermediate strain) or the rarity of occurRead et al.
Virus Population Size Influences the Relative Importance of Stochastic
Effects. Our results show that stochastic effects in viral evolution
can cause divergent disease outcomes if the virus has to cross
a significant fitness barrier to escape the immune response. The
minimum requirements for such a landscape are that the escape
strain be different from the founding strain in at least two sites;
thus, for escape to occur, either a rare double mutation must
arise and/or the virus must traverse through an intervening strain
with a deleterious single mutation. The simplest model that
includes these features is a three-strain model (infecting, intervening, and escape strains), with the effective fitnesses of the
strains satisfying the condition Fintervening < Finfecting < Fescape.
The size of the viral population within a host (relative to the
inverse mutation rate) is central to discussions of stochastic
effects in viral evolution (28–30). Therefore, we studied the effect of population size on tescape , by comparing three “cohorts”:
elite controllers, controllers, and progressors, defined by low,
intermediate, and high setpoint viral loads, respectively. We
approximated the system by a 1D time-homogeneous birth-death
Markov process over the number of copies of Iescape in the system, and calculated the density and mean first passage times for
the corresponding master equation according to Gillespie’s
A
B
C
Fig. 3. Distributions of tescape and pathways of escape through sequence
space. (A) Simulation results for the compensatory model virus of Fig. 2A.
Distribution of tescape from Gillespie simulations (Top, gray bars) has an early
peak, corresponding to individuals in whom the virus escaped during acute
infection, and an extended tail, corresponding to individuals who durably
control the virus for years. (Top) Deterministic (ODE) dynamics give only early
escape (red line). (Middle) Colored bars show occurrence of mutation events
to the double-mutant from the other three strains. Blue: strain 00; green:
strain 10; magenta: strain 01. (Bottom) Qualitative fitness landscape predicted
by Fk; arrows indicate dominant escape pathway through strain 10. (B) Simulation results for the parameter regime in which direct mutation from the
infecting strain (“tunneling” through the fitness barrier) is the dominant
pathway to escape. Although the virus can escape without occupying one of
the less fit, single-mutant strains, the rarity of double mutation gives an extended tail in tescape (Fk/F00 = {1, 0, .48, 9.5}, all parameters in SI Appendix). (C)
Noncompensatory, as in Fig. 2B, the virus increases effective fitness by mutating to strain 01. With no barrier to escape, all stochastic trajectories escape
immediately, showing a narrow distribution of tescape centered around the
ODE result. The dominant pathway to escape is through strain 01.
PNAS | November 27, 2012 | vol. 109 | no. 48 | 19729
CHEMISTRY
10
7
10
rence of simultaneous double mutations that allow escape. The
barrier to escape due to the initial decrease in Fk on point mutation is easier to overcome when viral loads are high during
primary infection, resulting in the early peak in tescape resembling
the deterministic response. After setpoint, with lower viremia,
the barrier is more difficult to overcome, stochastic effects are
more important, and long and widely distributed escape times
result. When there is no barrier to escape from the immune
pressure, as when Fk increases with each successive point mutation (F00 < F01 < F11), tescape is short and narrowly distributed,
and stochastic effects are unimportant (Fig. 3C, Upper).
IMMUNOLOGY
5
3
7
10
Total Viral Load
B
Occurrence (/10 )
Total Viral Load
A
eigenvalue approach (35) (SI Appendix). This approach, which
circumvents the computational challenge of full simulations
when the viral load is high (i.e., progressors), required choosing
a threshold “absorbing” state commensurate with escape. We
chose this state to be 50 copies of the escape strain (Iescape = 50
copies), and calculated t0→50, the time required to reach this
state. The calculated t0→50 slightly underestimates tescape but is
otherwise identically distributed (SI Appendix).
Stochastic waiting times to escape (proxied by <t0→50>) show
strong dependence on relative effective fitnesses (Fintervening,
Fescape) and size of virus population (Fig. 4). Elite controllers
prevent escape for an average of 9 y to 560 y beyond setpoint.
Over the same range of parameters, progressors prevent escape
for an average of only 35 d to 2.7 y. Notably, the ODE dynamics
underestimate tescape even in progressors, with 107 infected cells
[corresponding to m(1)N = 300, where m(1) is the rate of (singlepoint) mutation and N is the population size of total infected cells,
N = Iinfecting + Iintervening + Iescape], demonstrating that even for
large population sizes (i.e., m(1)N >> 1), these stochastic effects
cannot be neglected.
The dependence of tescape on Fintervening is pronounced for stochastic dynamics, but less so in the deterministic case (Fig. 4 and SI
Appendix, Fig. S8). Deterministically, the virus continuously
mutates to all possible strains (albeit at a slow rate): When an
infinitesimally small amount of Iescape (<<1) has been made, which
occurs immediately after infection, the subsequent waiting time is
dependent only on the time required for outgrowth of Iescape,
rendering deterministic dynamics more sensitive to Fescape than
Fintervening. In contrast, stochastically, the system is not “aware” of
the existence of fitter strains until a discrete mutation event to that
strain has occurred. This is evident in Fig. 2A, where the lower viral
load peak corresponds to the deterministic (ODE) result predicted
for a “reduced topology” mutational network [i.e., with mutation
to the double-mutant strain blocked (mk11 = 0)]. Even when the
total setpoint virus population size is large (m(1)N >> 1), that of
the intervening strain is small (m(1)Iintervening << 1) and dependent
on Fintervening. A low copy number of Iintervening means that mutation events producing Iescape are infrequent, and thus sensitive to
stochastic fluctuations.
Fig. 4. Mean time to escape after setpoint <t0→50> in three representative
cohorts as a function of relative effective fitness of intermediate strain
(Fintermediate) and escape strain (Fescape) in the three-strain model with a compensatory fitness landscape. Stochastic <t0→50> (Top) and deterministic t0→50
from ODE (Bottom) are shown. Elite controllers (Left) have setpoint viral
loads of 50 RNA/mL, giving N = 5 × 104 (parameters in SI Appendix). Controllers have setpoint viral loads of 2,000 RNA/mL (N = 2 × 106) (Middle),
whereas progressors (Right) have setpoint viral loads of 104 RNA/mL (N = 107).
The times required for the virus to escape are indicated by the colors (hotter
colors represent shorter waiting times; bar code). Although N is greater than
the inverse mutation rate in every cohort (m(1)N = 1.5, 60, and 300 from left
to right), deterministic t0→50 values are much shorter than stochastic waiting
times for most parameters. Even in progressors, stochastic predicted escape
times are, on average, 2.3 times longer than corresponding ODE predictions
over the range of parameters shown.
19730 | www.pnas.org/cgi/doi/10.1073/pnas.1206940109
Probability of Control Is Determined by CTL Repertoire and Targeted
Epitopes. Cohorts of individuals who control HIV are statistically
correlated with the possession of certain HLA genes (5). Recently,
it has been shown that these individuals disproportionately target
collectively evolving groups of amino acids within Gag, wherein,
compared with other such groups, a greater proportion of combinations of mutations are deleterious for the virus (9). Targeting
multiple sites within such a group of amino acids is likely to trap the
virus between the CTL pressure and multiply mutated escape
strains that are unfit. Motivated by these findings, we asked whether
targeting such a group of amino acids, as opposed to one where
escape via multiple mutations is easier, can bias the probability with
which an individual controls HIV. We constructed a model virus
with two independent groups, each containing three epitope sites
targeted by one of six CTL clones and characterized by different
numbers of positive and negative correlations (i.e., the groups differed in vulnerability to multiple mutations as described above;
details are provided in SI Appendix). We simulated infections in
individuals who differed in terms of their CTL targeting (presenting
and targeting epitopes from either the more or less vulnerable
group). In all cases, significant mutation occurred solely within the
targeted epitopes; individuals targeting the more vulnerable group
preferentially drove mutation to less fit variants, resulting in decreased average viral loads relative to individuals targeting the
other group (Fig. 5). The probability of controlling the virus was
also higher for those individuals who target the vulnerable group of
amino acids.
It has also been proposed that individuals with certain HLA
genes associated with control have a higher likelihood of having
cross-reactive CTLs (8, 36, 37). Fig. 5 shows that more cross-reactive CTL repertoires (likelihood that CTLs continued to recognize mutated epitopes) further enhances the probability of control.
Discussion
Identifying the causes of person-to-person variability in natural
HIV infection can inform understanding of disease pathogenesis
and the design of prophylactic and therapeutic vaccines. To this
end, we report fully stochastic computer simulations of the CTL
response to natural infection using simplified but realistic models
of the fitness landscape of HIV. We find that purely stochastic
effects can be an important contributing factor underlying person-to-person variability, giving rise to qualitatively distinct disease outcomes (durable control vs. high viremia) in infected
people with the same genotype. These distinct outcomes reflect a
striking range of times (days to decades) required for viral escape
from the CTL response.
Our findings are surprising because our simulations are carried
out under conditions wherein standard criteria would suggest that
stochastic effects are negligible. Under standard assumptions,
stochastic effects are significant only when selective forces are
weak and/or the effective population size is small, such that
m(1)N << 1 where m(1) is the (per base pair) mutation rate and N
is the effective population size (30). The qualitatively distinct
stochastic outcomes we report arise under conditions of strong
CTL-imposed selection pressure and high viral loads (in the
simulations, m(1)N ranges from >1 to >>1, depending on the stage
of infection and strength of the CTL response). Therefore, our
results challenge the common criteria for determining the importance of stochastic effects during viral evolution.
We provide a mechanism underlying this phenomenon. We
show that when the virus must traverse a significant fitness barrier to escape immune responses, as is the case in compensatory
mutation, stochastic effects are important for the rare “barriercrossing events.” We point out that the correct way of discerning
whether stochastic effects are important in such cases is not the
standard criterion (m(1)N << 1), but rather that m(1)Iintervening <<
1, where Iintervening is the population size of an intervening, relatively unfit strain that must emerge en route to a multiple mutant
escape strain. The stochastic bistability in viral loads arises because escape depends on the presence of discrete copies of these
intervening single-mutant strains, which are scarce after setpoint
Read et al.
More Vulnerable
Sector Targeted
Less Vulnerable
Sector Targeted
100
0
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5
5.5
6
Log Viral Load at 500 Days
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Log Viral Load at 500 Days
Fig. 5. Effects of CTL targeting and cross-reactivity. Distribution of viral
loads after 500 d in individuals targeting the more vulnerable coevolving
group of amino acids (as discussed in main text) of the viral proteome (blue)
or the less vulnerable group (red), with CTLs unable to recognize mutated
sites (Left, less cross-reactive) or able to recognize mutated sites with a
probability of 0.2 (Right, more cross-reactive). The probability of controlling the
virus is higher for individuals who target the more vulnerable coevolving
group of amino acids, and viral loads are lower. Increased cross-reactivity aids
the probability of controlling the virus (all parameters in SI Appendix).
due to their impaired fitness. Of note, the stochastic bistability
occurs even though there is no underlying deterministic bistability, similar to a finding in cell signaling (38).
We also report on how aspects of the CTL response that correlate with HLA alleles associated with control, including targeting of regions in which multiple mutations limit viability (9)
and cross-reactivity to mutants (8, 37) can bias the probability
of controlling HIV successfully. This elaborates why elite controller cohorts are enriched with people with certain genotypes,
and also why many people with these genes do not control HIV.
This finding can inform the design of the CTL arm of a vaccine. In
our simulations, a necessary but not sufficient condition for durable control is strong CTL recognition of key viral epitopes at
early time points postinfection. These findings suggest that a
general strategy for combating HIV should combine a rapid
overall reduction in viral load with directed targeting to regions of
the virus where coordinated mutations limit viability.
Using realistic numbers for the infected cell population within
a host (5 × 104 in elite controllers to >107 in progressors), we
showed striking differences between stochastic and deterministic
estimates of escape times, particularly in individuals with low
setpoint viral loads. Within the smallest, two-loci model, the
majority of individuals exhibit rapid (deterministic-like) escape
within tens of days after infection, whereas a small subset prevents
escape for years to decades, or longer. This suggests that the virus’
best opportunity to cross a barrier in sequence space is during
primary infection, when viremia is high, suggesting a contributing
factor underlying the finding that characteristics of primary infection appear to determine long-term disease outcomes.
Studies of infected individuals with shared HLA alleles show
disparate dynamics of escape. Mutation of the B27-restricted Gag
KK10 has been observed to occur in both acute and chronic
infections (13, 39). B57-restricted Gag TW10 is one of the fastest
escaping epitopes on average, generally showing variation within
months of infection (40); however, the dominant T242N mutation
still fails to appear in some B57+ individuals. Furthermore, B57+
individuals show different patterns of mutations associated with
T242N escape, and the presence or absence of certain T242Nassociated compensatory mutations correlates with HIV progression or control, respectively (21). Consistent with these findings,
our model predicts that emergence of compensatory mutations is
unpredictable, even among individuals exerting identical immune
pressures, and that these mutations can markedly increase viral
load. Therefore, our model suggests a potential stochastic contribution to elite control of HIV, whereby the virus is prevented
from making compensatory mutations in a subset of individuals
who consequently exhibit durable virus control.
The influence of particular HLA-I alleles may be confounded
by the combined effects of different haplotypes. However, longitudinal studies of HIV-infected identical twins report similar
immunodominance patterns and concordant evolution, but with
Read et al.
clear instances of sequence divergence in a subset of CTL epitopes (41, 42). This divergence may be attributed to differences in
TCR repertoires based on random gene rearrangement (43).
Our results suggest that dynamic stochasticity may be an additional cause of evolutionary divergence.
All model variants that we have studied show the same qualitative results, suggesting that the behavior we observe is general
and not specific to these models. We have assumed a well-mixed
system, but spatial inhomogeneity may exist due to slow equilibration between blood and lymph nodes (44). Inhomogeneities
would result in a smaller effective population size, and would
thus enhance the stochastic effects we report.
The potential importance of stochastic effects in intrahost
HIV evolutionary dynamics has been considered in the context
of resistance to ART, where variability was observed in the timing
and types of drug-resistant mutations that arose in patients under
identical ART programs (26, 45). Although viral loads are generally much lower during ART than in natural infection, making
it possible that m(1)N << 1, the effective population size has been
controversial (28, 29, 31, 45). It has remained unclear whether
genetic drift can account for observed variability. Although we
considered a separate problem (HIV escape from CTL pressure
during untreated infection), ART selection also gives rise to
compensatory fitness landscapes (46), suggesting that stochastic
phenomena similar to those we describe could contribute to
variability in ART resistance evolution. Thus, observed stochasticity in ART resistance may not necessarily imply a small effective population size of total virus (i.e., m(1)N << 1) as has
generally been assumed, but rather that the population of a less fit
strain on the path to escape is small.
Methods
The model describing viral evolution and host–pathogen interaction comprises infected cells and CTLs of naive, memory, or effector phenotype. Cells
are assumed to be infected by a viral strain, indexed by k, which may comprise multiple peptide epitopes, indexed by j. A naive (Ni) or memory (Mi)
CTL of clonotype i may be activated into the cytotoxic effector pool (Ei) on
interaction with an infected cell, Ik, if it bears a recognized viral epitope.
Recognition depends on the affinity of the TCR of clonotype i for epitopes
from viral strain k according to χ ik .
The host–pathogen interaction network is given by (ODE equations in SI
Appendix):
sk’ mk’k
Ik’ ! Ik + Ik’
u
Ik → ∅
w
∅ → Ni
b
Ni → 2Ni
c
Ni → ∅
αχ ik
Ni + Ik → Ei1 + Ik
r
Ein → 2Ein+1
n = 1 . . . ND − 1
d
Ein → ∅ n = 1 . . . ND
g
Ein → Mi n = 1 . . . ND
pχ ik
Ik + Ein ! Ein n = 1 . . . ND
α’χ ik
Mi + Ik ! Ei1 + Ik
h
Mi → ∅
Infected cells produce “offspring” at a rate sk, which is the replication rate
of a particular viral strain. The virus (and thus its targeted epitopes) may
accumulate mutations during replication, with the probability of the virus
making a particular mutation from strain k′ to strain k during a replication
event given by mk′k [e.g., if strains k and k′ differ at one site, mk’k = mð1Þ ]
(Table 1). When k = k′, mk′k denotes the probability that the virus is repliP
mk’k = 1 (47). Infected cells die at a constant rate
cated with fidelity; thus,
u, in addition to death byk CTL killing with rate pχ ik. Naive CD8+ T cells are
produced from the thymus at rate w, proliferate at rate b, and die at rate e.
Activation of naive cells occurs at rate aχ ik; reactivation of memory cells
occurs at a faster rate, a′χ ik. On activation into the effector pool, T cells
PNAS | November 27, 2012 | vol. 109 | no. 48 | 19731
CHEMISTRY
Occurrence
200
More Vulnerable
Sector Targeted
Less Vulnerable
Sector Targeted
IMMUNOLOGY
20% Cross−reactive
0% Cross−reactive
300
undergo a proliferation program (dividing 7–10 times at rate r) (48, 49).
Commitment by cells to a memory vs. effector lineage occurs early after
activation (50), transforming into memory cells at rate g. Effector and
memory cells die at rates d and h, respectively.
We studied several related models that differed, for example, in the
details of T-cell proliferation and differentiation. The networks all contained the core features of viral mutation and nonlinear activation/killing
by T cells (i.e., predator–prey-like dynamics) and gave qualitatively similar
results [stochastic bimodality for a wide range of fitness parameters (SI
Appendix)]. We performed numerical simulations in the deterministic limit
using MATLAB (MathWorks) ODE solver ode15s. Stochastic simulations were
performed assuming Gillespie dynamics (51) in a well-mixed system using
Stochastic Simulation Compiler (SSC), a fast and convenient implementation
of the Gillespie algorithm (52). Where possible, parameter values are based
on experimental results. Further details of the model, parameters, and master
equation approximation are provided in SI Appendix.
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19732 | www.pnas.org/cgi/doi/10.1073/pnas.1206940109
ACKNOWLEDGMENTS. We thank Dr. Bruce D. Walker for fruitful discussions.
This study was supported by the Jane Coffins Childs Foundation (E.L.R.), Ragon
Institute of Massachusetts General Hospital, Massachusetts Institute of Technology, and Harvard (A.K.C.), and a National Institutes of Health Director’s
Pioneer award (to A.K.C.).
Read et al.