recombination increases human immunodeficiency virus fitness, but not necessarily diversity

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Recombination increases human immunodeficiencyvirus fitness, but not necessarily diversity

N. N. V. Vijay,1 Vasantika,1 Rahul Ajmani,2 Alan S. Perelson2

and Narendra M. Dixit1

Correspondence

Narendra M. Dixit

[email protected]

1Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India

2Theoretical Biology and Biophysics, Theoretical Division, Los Alamos National Laboratory,Los Alamos, NM 87545, USA

Received 19 December 2007

Accepted 21 February 2008

Recombination can facilitate the accumulation of mutations and accelerate the emergence of

resistance to current antiretroviral therapies for human immunodeficiency virus (HIV) infection. Yet,

since recombination can also dissociate favourable combinations of mutations, the benefit of

recombination to HIV remains in question. The confounding effects of mutation, multiple infections

of cells, random genetic drift and fitness selection that underlie HIV evolution render the influence

of recombination difficult to unravel. We developed computer simulations that mimic the

genomic diversification of HIV within an infected individual and elucidate the influence of

recombination. We find, interestingly, that when the effective population size of HIV is small,

recombination increases both the diversity and the mean fitness of the viral population. When the

effective population size is large, recombination increases viral fitness but decreases diversity. In

effect, recombination enhances (lowers) the likelihood of the existence of multi-drug resistant

strains of HIV in infected individuals prior to the onset of therapy when the effective population

size is small (large). Our simulations are consistent with several recent experimental observations,

including the evolution of HIV diversity and divergence in vivo. The intriguing dependencies on the

effective population size appear due to the subtle interplay of drift, selection and epistasis,

which we discuss in the light of modern population genetics theories. Current estimates of the

effective population size of HIV have large discrepancies. Our simulations present an avenue

for accurate determination of the effective population size of HIV in vivo and facilitate

establishment of the benefit of recombination to HIV.

INTRODUCTION

The identification of improved treatment protocols and thedevelopment of new, more potent drugs and vaccines forhuman immunodeficiency virus (HIV) infection hinge onour understanding of the mechanism(s) that underlie theability of HIV to diversify its genomic content and developmulti-drug resistance during current antiretroviral ther-apies and/or escape vaccine-induced cell-mediatedimmune responses (Barouch et al., 2002; Clavel & Hance,2004; Simon et al., 2006; Thomson et al., 2002). Severalcompeting factors influence HIV diversification in infectedindividuals: the high mutation rate of HIV, approximately361025 mutations per nucleotide per replication (Mansky& Temin, 1995), and the enormous production rate,approximately 1010 virions per day in a chronically infectedindividual (Perelson et al., 1996), enable rapid sampling ofbroad spectra of genomic variations by HIV. Consequently,viral genomes with drug resistance mutations may exist inindividuals prior to the onset of therapy (Lech et al., 1996;

Ribeiro & Bonhoeffer, 2000). On the other hand, randomgenetic drift, significant in the suggested small effective cellpopulations in vivo (Shriner et al., 2004b), and fitnessselection (Kils-Hutten et al., 2001) may impede HIVdiversification. Experiments reveal complex patterns of thegenomic evolution of HIV in patients and suggest that thepatterns are correlated with disease progression (Herbecket al., 2006; Markham et al., 1998; Shankarappa et al.,1999). Several recent studies present insights into thecomplex interplay of mutation, drift and fitness selectionthat underlies HIV diversification in vivo (Frost et al., 2000,2001; Rouzine et al., 2001; Williamson et al., 2005; Yuste etal., 2002). The role of recombination, however, remainspoorly understood (Bocharov et al., 2005; Bretscher et al.,2004; Fraser, 2005; Otto & Lenormand, 2002).

HIV has a high recombination rate, several times largerthan its point mutation rate (Jetzt et al., 2000; Levy et al.,2004; Shriner et al., 2004a; Suryavanshi & Dixit, 2007).Further, recent studies suggest that infected cells harbourmultiple HIV proviruses (Chen et al., 2005; Dang et al.,2004; Jung et al., 2002; Levy et al., 2004), which enables theSupplementary material is available with the online version of this paper.

Journal of General Virology (2008), 89, 1467–1477 DOI 10.1099/vir.0.83668-0

0008-3668 Printed in Great Britain 1467


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formation of heterozygous virions and presents thenecessary substrate for recombination to induce genomicdiversification (Rhodes et al., 2003). Recombination maythus significantly accelerate viral diversification (Bocharovet al., 2005; Charpentier et al., 2006) and facilitate theemergence of multi-drug resistance in infected individuals(Althaus & Bonhoeffer, 2005; Blackard et al., 2002;Christiansen et al., 1998; Kellam & Larder, 1995;Moutouh et al., 1996).

Conversely, just as recombination may induce the accu-mulation of favourable mutations, it may also drivebeneficial combinations of mutations apart and thereforenot necessarily benefit HIV (Bonhoeffer et al., 2004; Fraser,2005). Mathematical models (Bretscher et al., 2004; Fraser,2005) suggest that the effect of recombination dependssensitively on epistasis (E), which is a measure of theinfluence of interactions between mutations on viral fitness(in a two-locus-two-allele model, where haplotype ab hasfitness fab, E5fabfAB2faBfAb). In particular, when drug–sensitive loci interact antagonistically in reducing viralfitness, i.e. when E.0, recombination may decelerate theemergence of multi-drug resistance in a large population ofcells (Bretscher et al., 2004). Recent experiments suggest thatthe HIV-1 fitness landscape is characterized by mean E.0,raising doubts on the benefits of recombination to HIV-1(Bonhoeffer et al., 2004). With moderate values of theeffective population size, Ne, recombination may acceleratethe emergence of drug resistance significantly even with E.0(Althaus & Bonhoeffer, 2005). When Ne is smaller than acritical value, however, the viral population in an individualmay converge to a clone in the absence of mutation, leavinglittle opportunity for recombination to induce diversifica-

tion (Rouzine & Coffin, 2005). Current estimates of Ne invivo have large discrepancies (Kouyos et al., 2006a) and leaveunclear the benefit of recombination to HIV.

Our aim is to develop a fundamental understanding of theeffect of recombination on HIV diversification in aninfected individual. Mathematical modelling of HIVdiversification requires integration of the effects ofmutation, multiple infections of cells, random geneticdrift, fitness selection, and recombination, which remainsdifficult especially when fitness interactions betweenmultiple loci are important. Conversely, experimentalapproaches suffer from the difficulty involved in thedeconvolution of the effects of recombination from thoseof mutation, drift and selection. Here, we developcomputer simulations that elucidate the influence ofrecombination on the genomic diversification of HIV.

METHODS

We performed bit-string simulations of the within-host genomic

evolution of HIV (Fig. 1). We commenced simulations with a pool of

identical homozygous virions that synchronously infect a pool of

uninfected cells. Following infection, viral RNAs in cells are reverse-

transcribed into proviral DNA. During reverse transcription,

mutation and recombination introduce genomic variations. The

proviral DNAs are transcribed into viral RNAs, which are assorted

randomly into pairs, copackaged and released as new virions. The

progeny virions form a new viral pool from which virions are chosen

according to their fitness to infect a new pool of uninfected cells, and

the replication cycle is repeated. In each generation, we determine the

average diversity and fitness of the proviral DNA and their average

divergence from the proviral DNAs at the start of infection. Below, we

present details of the simulation procedure.

Fig. 1. A schematic representation of thesimulation procedure.

N. N. V. Vijay and others

1468 Journal of General Virology 89


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Initialization of the viral pool. We consider a pool of Nv virions.Each virion is a string of 2L nucleotides (A, G, C and U); the first L

nucleotides represent one RNA strand and the next L nucleotides the

other RNA strand of the virion. At the start of infection, a sequence,

F, of length L is generated by randomly selecting a nucleotide (fromA, G, C and U) at each position. The sequence is assumed to be the

fittest and is assigned the relative fitness 1. We mutate F with

probability j per nucleotide (see below) and let the resulting sequence

be the starting or founder sequence. The founder sequence is copiedto the two substrings of length L in each of the Nv virions to form the

initial viral pool.

Infection of cells. We let M virions infect each cell in a pool of Ne cells.

Virions are chosen for infection from the viral pool according to theirrelative fitness (see below) and their genomes are transferred to cells.

Reverse transcription. Following infection, reverse transcription

converts the RNA strands (length 2L) to proviral DNA (length L).Reverse transcription involves recombination, which we simulate

first, and mutation (Bocharov et al., 2005).

Recombination. The average recombination rate, r, is approximately

861024 per site per reverse transcription (Levy et al., 2004;Suryavanshi & Dixit, 2007) so that for the genome lengths we

consider (L540–100), more than one crossover per reverse

transcription is unlikely. Thus, during each reverse transcription,

we allow a single crossover with probability rL. The crossover occursat position lc, which we choose randomly from a uniform distribution

on 1 to L. We begin reverse transcription from the first nucleotide of

one of the two substrings of the viral RNA chosen randomly with

probability 0.5. We copy the nucleotide sequence of the startingsubstring from the first to the lcth position to the first lc positions of

the resulting proviral DNA. The template is then switched and the

sequence from the (lc+1)th position to the last position of the other

substring is copied to the corresponding positions in the proviralDNA. This process is repeated for all infecting virions.

Mutation. Mutations in HIV are dominated by substitutions of which

approximately 90 % are transitions (A«G and C«T) (Mansky &Temin, 1995). For simplicity, we therefore consider transitions alone.

Thus, starting from the first position of a proviral DNA, ‘A’ is

substituted with ‘G’ and ‘C’ with ‘T’, and vice versa, at all positions, each

substitution occurring with probability equal to the mutation rate, m.

Calculation of diversity, divergence and fitness. The normalized

Hamming distance between two proviral DNAs, i and j, is

dij~1

L

XL

k~1

½1{d(xki {xk

j )� ð1Þ

where xki is the nucleotide in the kth position of provirus i, and

d(xki {xk

j )~1 if xki ~xk

j and d(xki {xk

j )~0 if xki =xk

j (2). dij is thus thenumber of differences per position between sequences i and j; dij50 if

the two strings are identical, whereas dij51 if the two strings are

different at every position.

We determine the average diversity of the Q (5MNe) proviruses in

any generation as

dG~

PQ{1

i~1

PQ

j~1z1

dij

12

Q(Q{1)ð3Þ

where the Hamming distance between every possible pair ofproviruses is averaged. The denominator, (1/2)Q(Q21), is the

number of possible pairs among Q proviruses.

Similarly, the average divergence of the proviruses in any generationfrom those at the onset of infection, is

dS~1Q

PQ

i~1

di0 ð4Þ

where di0 is the Hamming distance of provirus i in a given generation

from the founder sequence (subscript 0).

The fitness landscape for the protease and reverse transcriptaseregions of HIV-1 has recently been determined (Bonhoeffer et al.,2004). On this landscape, the relative fitness, fi, of genome i dependson the Hamming distance, diF, between the amino acid sequence ofgenome i and that of the fittest genome, F. As an approximation, weassume that unit Hamming distance between two nucleotidesequences is equal to unit Hamming distance between thecorresponding amino acid sequences. We find then that theexperimental fitness landscape is captured by (Supplementary Fig.S1, available with the online version of this paper),

fi~1{(1{fmin)(diF )n

(diF )nz(d50)n ð5Þ

where fmin50.269 is the minimum fitness of sequences attained at

arbitrarily large absolute Hamming distances from F; d50L520.4 is

that Hamming distance from F at which fi50.5(1+fmin), i.e. the

average of the fittest and the least fit sequences; and n53 is analogous

to the Hill coefficient. The average fitness of proviruses in any

generation is

favg~1

Q

XQ

i~1

fi ð6Þ

Translation, assortment and viral production. Following reversetranscription, each infected cell produces P virions. For each progenyvirion arising from a cell, two of the M proviral DNAs from the cellare chosen at random and their sequences are copied to the two L-bitsubstrings to form the RNA genome of the virion. When M51, thesame proviral sequence is copied to all the new virions arising fromthat cell. The new virions thus formed constitute the viral pool forinfecting the next generation of target cells.

Fitness selection. The relative fitness of a virion is determined byequation 5 with diF the average Hamming distance of its two RNA strandsfrom the fittest sequence. The virion is chosen for infection randomlywith a probability equal to its relative fitness. The process is repeated forother virions in the pool until every cell is infected with M virions.

The above infection and replication process is repeated and theevolution of viral diversity, divergence and fitness with the number ofreplication cycles determined. Several realizations are averaged todetermine the expected evolution. The simulations are implementedusing a computer program written in C++.

RESULTS

We begin simulations with the minimal processes essentialfor viral diversification and gradually build complexityuntil the influence of recombination is elucidated. We thenperform simulations with parameter values representativeof conditions in vivo and draw links with patient data.

Mutation

We examine first the effect of mutation. We consider aviral pool of Nv540 virions, each virion with a genome of

Influence of recombination on HIV diversification

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L540 nucleotides. We assume individual cells to beinfected by single virions (M51) and let each infected cellproduce a single progeny virion (P51). The cell poolcontains Ne540 cells. We ignore recombination (r50) andassume a flat fitness landscape (fi51).

In Fig. 2, we present the evolution of diversity, dG, anddivergence, dS (Methods), with the number of replicationcycles or generations, g, for different values of the mutationrate, m. As expected, dG5dS50 when g50, as all the virions inthe initial viral pool are identical. As g increases, both dG anddS increase due to mutation. The rate of increase is higher forlarger values of m. For large g, both dG and dS approachequilibrium values, d‘

G and d‘S, respectively, in an exponential

manner; we find that d‘G5d‘

S 50.5 independent of m.

We capture these effects of mutation in a mathematicalmodel (Supplementary Information, available with theonline version of this paper). Model predictions are inexcellent agreement with our simulations (Fig. 2) andprovide a quantitative understanding of the influence of

mutation on HIV diversification as well as a successful testof our simulation procedure.

Recombination (r.0) does not alter the genomic evolu-tion in Fig. 2 because, in the absence of multiple infections,all progeny virions are homozygous and no substrate existsfor recombination to induce diversification (Supple-mentary Fig. S2). We therefore introduce multiple infec-tions of cells next.

Multiple infections of cells

With Ne540 cells, we expand the viral pool to Nv5NeM sothat every cell is infected by M virions, and let P5M, sothat all progeny virions infect cells. We choose m50.001substitutions per nucleotide per replication, r50 and fi51.

In Fig. 3(a), we present the evolution of dG for differentvalues of M5P. For M51, the evolution is identical to thatin Fig. 2. For M52, surprisingly, dG decreases significantlyfrom that for M51. The decrease in dG despite multipleinfections is attributed to enhanced viral production fromcells. When P52, each cell produces two progeny virions,which can be identical. The production of identical virionslowers the diversity of the proviruses in the followinggeneration, and, in consequence, dG. This effect is

Fig. 2. The evolution of HIV diversity (a) and divergence (b) atdifferent mutation rates, viz., 0.05 (�), 0.01 (+), 0.005 (e), 0.001(g), 0.0005 (h), and 0.0001 (#) substitutions per nucleotide perreplication. The grey bands represent deviations to the mean(±SD) due to stochastic variations in approximately 1000realizations. Solid lines are model predictions (SupplementaryInformation).

Fig. 3. The evolution of HIV diversity at different frequencies ofmultiple infections of cells (M) and numbers of progeny virions percell (P). (a) M5P51 (g), 2 (#), 3 (h), 4 (e); (b) M51; P52 (#),3 (h), 4 (e); (c) P53; M51 (#), 2 (h), 3 (e).




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emphasized in Fig. 3(b), where we present dG for M51 andP52, 3 and 4. As P increases, the probability that identicalvirions infect target cells increases and causes a decrease indG. We note, however, that when P.M, the decrease in dG

is the combined effect of the production of identical virionsand random genetic drift, as only a fraction of the progenyvirions infects cells. The effect of drift is minimized whenP5M.

As M(5P) increases above 2, dG increases (Fig. 3a).Enhanced multiple infections increase the likelihood ofdiverse proviruses forming progeny virions and cause theobserved increase in dG. This phenomenon is illustratedfurther in Fig. 3(c), where we present the evolution of dG

for M51, 2 and 3, but for fixed P53. As M increases, dG

increases.

Multiple infections (M.1) and drift (P.M) do notinfluence HIV divergence; in the absence of fitnessselection and recombination, the evolution of individualsequences is governed by mutation. dS therefore remainsunaltered by variations in M and P (Supplementary Fig.S3).

Even with multiple infections, recombination (r.0) doesnot alter HIV diversification (Supplementary Fig. S4).Whereas recombination can bring mutations together, itcan also drive mutations apart. Thus, when no fitnessbenefit exists for accumulating (or separating) mutations,i.e. when no non-random association of mutations isfavoured, HIV diversification remains independent ofrecombination. We examine next the role of fitnessselection.

Fitness selection

To delineate the effect of fitness selection, we ignoremultiple infections (M51) and select virions for infectionaccording to equation 5 from a pool of size Nv5NeP withP.1. Without loss of generality, we let the foundersequence be the fittest (j50) (Methods). The fitness ofindividual genomes is thus a function of their divergence.

In Fig. 4, we present the evolution of dG, dS, and theaverage fitness, favg, for P53 and different values of m

(when r50). We find that fitness selection influences bothdG and dS. d‘

G and d‘S decrease compared with their values

with a flat fitness landscape (Figs 2, 3 and 4). The relativefitness of genomes decreases as their divergence increases(equation 5). Thus, fitness selection lowers d‘

S (Fig. 4b).The fewer variations allowed due to fitness penalties alsolower d‘

G correspondingly (Fig. 4a). A higher m, however,forces the sampling of larger genomic variations and causesd‘

G and d‘S to increase (Fig. 4). Correspondingly, the

equilibrium fitness, f ‘avg, decreases as m increases (Fig. 4c),

in accordance with the classical mutation–selection balance(Hartl & Clark, 2007).

The evolution of dG is more complex (Fig. 4a). For smalland large values of m, dG increases monotonically to d ‘

G.

For intermediate values of m, the evolution is non-monotonic: dG first rises and then declines to d ‘

G (Fig. 4ainset). To understand this non-monotonic evolution, werecognize that, as dS increases with g (Fig. 4b), the averagenumber of mutations per genome increases. Initially(g~0), when all genomes are nearly identical, mutationstend to occur at distinct positions on different strands,causing dG to rise. As the number of mutations accruedincreases, the likelihood that new mutations occur atnovel positions decreases. Eventually, new mutationsoccur increasingly at positions where other strands carry

Fig. 4. The evolution of HIV diversity (a), divergence (b) andaverage fitness (c) with fitness selection according to equation 5 atdifferent mutation rates, viz., 0.1 (e), 0.01 (h), 0.005 (#), 0.001(g) and 0.0001 (�) substitutions per nucleotide per replication.Each cell is infected with M51 virion and produces P53 progenyvirions. The inset in (a) shows two datasets on a semi-log plot.




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mutations, causing dG to decline until the mutation–selection balance is attained. When m is small, themaximum number of mutations accrued is so small thatno reduction in dG occurs, whereas when m is large,forceful diversification outweighs fitness selection andresults in the monotonic evolution of dG, similar to thatin the absence of fitness selection (Fig. 2).

Recombination does not alter the HIV diversificationobserved in Fig. 4 as expected from the lack of multipleinfections of cells and the consequent absence of hetero-zygous virions (Supplementary Fig. S5).

Recombination

With multiple infections of cells and fitness selection,recombination influences HIV diversification. In Fig.5(a–c), we present the evolution of dG, dS and favg withM52 and fitness selection according to equation 5 fordifferent recombination rates, r. Remarkably, we findthat recombination does not alter the initial rate ofevolution of dG, dS or favg, but alters d ‘

G, d ‘S and f ‘

avg. As r

increases, d ‘G and f ‘

avg increase and d ‘S decreases (Fig. 5a–

c). These qualitative effects of recombination are robustto variations in the mutation rate and quantitativevariations in the fitness landscape (i.e. changes in d50 orfmin in equation 5) (not shown), but are sensitive to theeffective population size.

Effective population size

Upon increasing the cell population, Ne, at any value of r, d‘G

increases because a larger number of genomic variants islikely to exist in larger populations. At the same time, f ‘

avg

increases, and hence d‘S decreases, because fitness selection

operates more effectively in larger populations (Fig. 5).Remarkably, however, as Ne increases, d‘

G first increasesupon increasing r (Fig. 5a and d) and then decreases(Fig. 5g). The difference in d‘

G, which we denote Dd‘G, upon

increasing r from 0 to 0.02 crossovers per nucleotide perreplication cycle, decreases from approximately 0.015 atNe540 to approximately 20.028 at Ne55000 (Fig. 6a).Thus, recombination increases HIV diversity for small Ne

but decreases diversity for large Ne. The correspondingdifference in the fitness (divergence), Df ‘

avg (Dd‘S ), is positive

(negative) for all values of Ne considered (Fig. 6a), indicatingthat recombination consistently increases (lowers) the meanfitness (divergence) of the viral population. The extent of theincrease in fitness, however, decreases upon increasing Ne.We discuss these intriguing results in the light of modernpopulation genetics theories below.

In vivo parameter estimates

To determine whether the above effects of recombinationmay hold in vivo, we perform simulations with parametervalues that mimic conditions in vivo. We let m5361025

Fig. 5. The evolution of HIV diversity (top),divergence (middle), and average fitness (bot-tom panels) at different recombination rates,viz., 0 (red), 0.0002 (orange), 0.002 (blue) and0.02 (green) crossovers per nucleotide perreplication, and at different sizes of the cellpool, viz., 40 (a–c), 200 (d–f) and 1000 (g–i)cells. Mutations occur at 0.001 substitutionsper nucleotide per replication and fitnessselection follows equation 5. Each cell isinfected by M52 virions and produces P53progeny virions.




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substitutions per position per replication (Mansky &Temin, 1995) and M53 (Jung et al., 2002). The viral burstsize in vivo is estimated to be in the range 102–104 (Haaseet al., 1996; Hockett et al., 1999). Most virions produced,however, may be non-infectious (Dimitrov & Martin, 1995;Dimitrov et al., 1993; Piatak et al., 1993) and individualcells may yield as few as 2–3 infectious virions (Dimitrov &Martin, 1995). Here, we therefore choose P55. We assumefitness selection to follow equation 5. We vary r from 0 to961023 crossovers per position per replication, approxi-mately tenfold higher than the experimental recombinationrate (Levy et al., 2004), to emphasize the influence ofrecombination. The best current estimates of the effectivepopulation size are of the order of 103 (Shriner et al.,2004b). We therefore vary Ne from 40 to 104. We alsoconsider a larger genome length, L5100, corresponding tothe variable portion of the V1V2 region of env (Bocharovet al., 2005). With L5100, we span the experimental fitnesslandscape, which extends to a Hamming distance ofapproximately 60 (Supplementary Fig. S1).

In Fig. 6(b), we present the resulting effect of recombina-tion and Ne on d‘

G , d‘S and f ‘

avg. In agreement with ourobservations above (Fig. 6a), we find that, for small valuesof Ne, recombination increases both the mean fitness andthe diversity of the viral genomes, whereas, for large valuesof Ne, recombination increases the mean fitness but lowersviral diversity (Fig. 6b).

Links with patient data

We demonstrate finally that the predicted evolution ofdiversity and divergence are consistent with correspondingobservations in HIV patients. In Fig. 7, we compare oursimulations with the evolution following seroconversion ofthe diversity and the divergence of the C2–V5 region of theHIV-1 env gene in one (Patient 2) of nine patients observedexperimentally (Shankarappa et al., 1999). The patient dataare reported as the mean pairwise distance between viralDNA sequences determined using either a two-parameterKimura model or a general time-reversible model with site-to-site variation in substitution rates, both methodsyielding similar results (Shankarappa et al., 1999). Whenthe distances are small, as is the case in the experimentaldata, the Kimura two-parameter distance reduces to theHamming distance (Kimura, 1980), allowing us tocompare our simulation results directly with the data.

We choose the above in vivo parameter estimates, fixr5961024 crossovers per position per replication (Levyet al., 2004), and let the viral generation time be 1 day (Dixitet al., 2004; Markowitz et al., 2003; Perelson et al., 1996;Rodrigo et al., 1999). j, which determines the fitness of thefounder sequence, and Ne for the patient remain unknown.We vary these parameters and find that good comparisonsbetween our simulations and the patient data are obtainedfor Ne51500, j50.05–0.08 and Ne55000, j50.05–0.06 (Fig.7). In the patient, viral diversity rises following seroconver-sion, appears to attain a maximum and then declines slightlyto an equilibrium value. With j50.05 and Ne51500 or5000, our simulations capture accurately the initial rise andthe equilibrium diversity (Fig. 7a and b). The evolution ofdiversity, however, is monotonic. Given the large uncer-tainties associated with the experimental data, whether themaximum observed is significant remains to be ascertained.

Our simulations with the same parameter values alsocapture the evolution of viral divergence in the patient(Fig. 7c and d). We find here that the experimental data areencompassed by our predictions with j50.05 and 0.08 forNe51500 and j50.05 and 0.06 for Ne55000. We assume aunique founder sequence in our simulations. In thepatient, however, even if infected with a unique founder,a mixture of genomes, perhaps with fitness values in therange determined by the above ranges of j, may exist atseroconversion. Note that variation of j has little impacton the evolution of diversity (Fig. 7a and b).

The reasonable agreement between the data and oursimulations (Fig. 7) suggests that our simulations are

Fig. 6. The change in the equilibrium diversity, Dd ‘G (#),

divergence, Dd ‘S (e) and fitness, Df ‘

avg (h), due to recombinationat different effective population sizes. Parameter values (see text):(a) L540 nucleotides, m50.001 substitutions per nucleotide perreplication, M52, P53, and r is varied from 0 to 0.02 crossoversper nucleotide per replication. (b) L5100 nucleotides, m53�10”5

substitutions per nucleotide per replication, M53, P55, and r isvaried from 0 to 9�10”3 crossovers per nucleotide per replication.Fitness selection follows equation 5.




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representative of the scenario in vivo. The agreement alsosuggests that Ne in the patient considered is in theapproximate range of 1500–5000 cells. [Simulations withNe5500 underpredict and Ne510000 overpredict thepatient data (Fig. 7). Note that j does not influence theequilibrium diversity or fitness but alters equilibriumdivergence (Supplementary Information). Also note thatthe error bars in Fig. 7 are extreme values and not standarderrors of the mean.] With Ne~1500–5000, our simulationspredict that recombination enhances mean viral fitness butlowers viral diversity in the patient (Fig. 6b). Whetherrecombination exerts a similar influence in other patientsremains to be ascertained.

DISCUSSION

Our simulations are in agreement with several recentstudies of the effects of recombination on HIV evolution.Resistance mutations in HIV-1 appear to exhibit meanpositive epistasis (E.0) (Bonhoeffer et al., 2004). WithE.0, the reduction due to recombination of the mean timefor the establishment of double mutants is predicted toincrease, reach a maximum, and then decrease as theeffective population size, Ne, increases (Althaus &Bonhoeffer, 2005). In our simulations, the equilibriumdiversity, d‘

G, provides a qualitative measure of the waitingtime for the establishment of drug resistance. The larger thevalue of d‘

G, the greater is the likelihood of the existence ofdiverse genomes in the viral pool, and hence the shorter isthe waiting time for the establishment of desired mutants.

Our simulations employ the fitness landscape identifiedexperimentally (Bonhoeffer et al., 2004). Indeed, we findthat recombination increases d‘

G when Ne is small anddecreases d‘

G when Ne is large (Fig. 6).

Recent computer simulations suggest that recombinationand multiple infections act synergistically in the evolutionof HIV diversity (Bocharov et al., 2005). With multipleinfections, the mean diversity and fitness of HIV increase inthe presence of recombination. Our simulations are inagreement with these predictions and provide furtherinsights. We note first that recombination enhances bothviral fitness and diversity when Ne is small. Second, anincrease in the frequency of multiple infections (M)increases diversity when viral production per cell (P) isrelatively unaffected. If P increases with M, which mayhappen when viral production is limited by a viral ratherthan a host-cellular factor, then HIV diversity may firstdecrease and then increase as the frequency of multipleinfections increases (Fig. 3).

In vivo, recombination allows the conservation of genomicregions affected by evolutionary bottlenecks and increasesdiversity in other regions (Charpentier et al., 2006). Thus,where selection dominates, as with large Ne, recombinationfacilitates selection of fitter genomes, and where selection isweak, recombination increases diversity, as observed in oursimulations (Figs 5 and 6).

The patterns of change of viral diversity, dG, anddivergence, dS, predicted by our simulations have beenobserved in experiments (Herbeck et al., 2006; Markham et

Fig. 7. Comparisons of the evolution of HIVdiversity (top) and divergence (bottom panels)predicted by our simulations (lines) andobserved in a patient (symbols) (Shankarappaet al., 1999) with time post-seroconversion.Simulation parameters (see text): (a) and (c)Ne51500, j50.05 (thick line); Ne51500,j50.08 (thin line); Ne5500, j50.05 (dottedline), and (b) and (d) Ne55000, j50.05 (thickline); Ne55000, j50.06 (thin line);Ne510000, j50 (dotted line). r59�10”4

crossovers per nucleotide per replication.Other parameter values are those in Fig. 6(b).The error bars represent extreme valuesobserved at individual time points (see Fig. 1in Shankarappa et al., 1999).




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al., 1998; Shankarappa et al., 1999). For instance, dG inpatients gradually increased following infection, reached amaximum and then either stabilized or declined, whereasdS increased monotonically to a plateau (Shankarappa etal., 1999). In our simulations, these patterns arise due tothe interplay of mutation and fitness selection (Fig. 4). Theobserved patterns may also arise due to HIV-mediatedcollapse of the immune system, i.e. immune relaxation(Williamson et al., 2005), which we ignore. We haveapplied our simulations to describe the time-evolution ofdG and dS in one HIV patient for a period of approximately10 years following seroconversion. Our simulations capturethe data well (Fig. 7), suggesting that our simulations arerepresentative of the scenario in vivo.

Current population genetics theories provide insights intothe influence of recombination elucidated by our simula-tions (Ewens, 2004; Hartl & Clark, 2007; Kouyos et al., 2007;Otto & Lenormand, 2002; Rice, 2002). According to thesetheories, recombination acts to reduce the magnitude oflinkage disequilibrium (LD). In a two-locus-two-allelemodel, where gab is the frequency of haplotype ab,LD5gabgAB2gaBgAb measures the extent to which thefrequency of double-mutants is different from that expectedfrom the frequencies of single-mutants. In the absence ofselection and with large Ne, LD vanishes. Recombinationthen will not influence viral diversification (SupplementaryFigs S3–S5). Random genetic drift generates negative LD –which implies an overrepresentation of single-mutants –according to the well-known Hill–Robertson effect (Hill &Robertson, 1966). Epistasis also introduces LD; then LD hasthe same sign as E (Eshel & Feldman, 1970). When Ne issmall, drift may dominate epistasis and create net negativeLD. When recombination lowers negative LD, the frequencyof double-mutants and hence viral diversity increases (Hartl& Clark, 2007; Kouyos et al., 2007; Otto & Lenormand,2002). When Ne is large, drift is reduced. LD is thendetermined by epistasis. In our simulations, we haveassumed a fitness landscape that extends over multiple lociand has mean E.0 (Bonhoeffer et al., 2004). Although LDdepends in a complicated manner on the distribution ofepistasis and not the mean epistasis alone (Kouyos et al.,2006b), the assumed landscape may be expected to generatepositive LD. When recombination lowers positive LD, viraldiversity is expected to decrease (Hartl & Clark, 2007;Kouyos et al., 2007; Otto & Lenormand, 2002). Thus, acompetition between negative LD introduced by drift andpositive LD due to epistasis appears to underlie the effect ofrecombination and Ne on viral diversity observed in oursimulations.

Our simulations also suggest that for small Ne (,103), theenhancement in viral diversity due to recombinationincreases with increasing Ne (Fig. 6b). For very small Ne,the Hill–Robertson effect may not apply (Otto & Barton,2001). Thus, the gradual emergence of the Hill–Robertsoneffect with increasing, yet small, Ne may underlie theobserved enhanced effect of recombination. We observefinally that in all our simulations, recombination increases

mean viral fitness (Fig. 6) and hence decreases themutational load, an effect suggested as one of the plausiblecauses of the evolutionary origins of recombination (Hartl& Clark, 2007; Kouyos et al., 2007; Otto & Lenormand,2002).

Prediction of the influence of recombination on viraldiversification and consequently the emergence of drugresistance has been precluded by the lack of robustestimates of Ne in vivo (Kouyos et al., 2006a). Modelsbased on neutral evolution estimate Ne to be approximately103 (Achaz et al., 2004; Brown, 1997; Nijhuis et al., 1998;Rodrigo et al., 1999; Seo et al., 2002; Shriner et al., 2004b).HIV evolution in vivo, however, is expected to be driven byselection. A model that assumes viral evolution withselection estimates Ne to be .105–106 (Rouzine & Coffin,1999). The model, however, is restricted to fitnessinteractions between pairs of loci and ignores recombina-tion, which may overestimate Ne (Shriner et al., 2004b). Inour simulations, we consider fitness interactions betweenmultiple loci and predict the evolution of viral diversityand divergence. The evolution of viral diversity anddivergence are sensitive to Ne. Comparison of ourpredictions with data of the evolution of viral diversityand divergence from patients therefore presents an avenuefor obtaining more accurate estimates of Ne in vivo. Wedemonstrate the applicability of this methodology byanalysing data from one patient and estimate Ne to beapproximately in the range 1500–5000. Analysis of datafrom a larger set of patients – a promising potentialapplication of our simulations – would provide robustestimates of Ne in vivo and establish the benefit ofrecombination to HIV.

ACKNOWLEDGEMENTS

Portions of this work were done under the auspices of the US

Department of Energy under contract DE-AC52-06NA25396. This

work was supported by the National Institutes of Health grants

AI065334 (NMD), AI28433 and RR06555 (A. S. P.), and the NIH-

funded Center for HIV/AIDS Vaccine Immunology.

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recombination increases human immunodeficiency virus fitness, but not necessarily diversity

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