defective deletion mutant amplification

J. theor. Biol. (2000) 206, 449}464doi:10.1006/jtbi.2000.2128, available online at http://www.idealibrary.com on

Defective Deletion Mutant Ampli5cation

BARRY MCLEOD AND NIGEL BURROUGHS*

¹he Mathematics Institute, ;niversity of=arwick, Coventry, C<4 7A¸, ;K

(Received on 21 May 1999, Accepted in revised form on 19 June 2000)

Defective deletion mutants can be replicated in superinfected cells by parasitism of the intactvirus' replication machinery, and through replication with the host cell. We show by analysisof a mathematical model that dynamic stability of superinfected cell growth is crucial indetermining the frequency of deletion mutant infected cells, i.e. there is a critical infectivitythreshold o

scbelow which the density of proliferative virus is signi"cantly reduced by the

presence of defective deletion mutants. Above osc, proliferative virus principally occurs as

superinfected cells (wild type with defective deletion mutant). The threshold osc, and the

interference e!ects of the deletion mutant, increase with deletion mutant parasitism of thewild-type replication machinery in superinfected cells. The interaction of virally infected cellswith host homeostasis determines whether immune escape by deletion mutant infected cells isnecessary for the interference window to exist. Only when the deletion mutant has a detrimen-tal e!ect on infected host cell replication did we observe periodic behaviour.

( 2000 Academic Press

1. Introduction

Deletion mutants are viral strains that have lostsections of their genome. They have been detec-ted in a number of host}virus systems includinghumans, HIV-1 (Li et al., 1991; Sanchez et al.,1997), HTLV 1 (Hiramatsu & Yoshikura, 1986;Shuh et al., 1999), hepatitis A (NuK esch et al.,1989), other mammals, caprine arthritis encepha-litis virus (Gazit et al., 1992), Moloney murineleukemia virus (Shields et al., 1978), visna virus(Molineaux & Clements, 1983) and plants, citrustristeza virus (Ayllon et al., 1999), tomato spottedwilt virus (Inoue-Nagata et al., 1998). Deletionsare believed to arise because of slip-page/skip-ping of the polymerase along the viral genomeduring replication (Huang, 1977; Lazzarini et al.,1981; Perrault, 1981; Roux et al., 1991), anidea supported by the recent analysis of deletion

*Author to whom correspondence should be addressed.

0022}5193/00/200449#16 $35.00/0

mutants in HIV where deletions in central partsof the genome were more frequent (Sanchez et al.,1997). Most deletion mutants are likely to bedefective because of deletion/destruction ofessential genes, especially as the deleted regioncan be large (Hiramatsu & Yoshikura, 1986; Liet al., 1991; Perrault, 1981; Sanchez et al., 1997).However, deletion mutants can interfere withwild-type virus replication and confer protection(Barrett & Dimmock, 1986; Roux et al., 1991), orcause more pathogenic forms of disease than thewild-type virus (Chattopadhyay et al., 1989;Overbaugh et al., 1988) possibly through expres-sion of novel (partially deleted) viral proteins.Deletion mutants are potentially problematic invirus quanti"cation by PCR techniques sincePCR will amplify sequences on genomes thatmay not be complete and therefore overestimateviable virus, or primer sequences may lie in adeleted region failing to detect the presence ofthe virus.

( 2000 Academic Press

450 B. MCLEOD AND N. BURROUGHS

The traditional explanation for the highfrequency and prevalence of non-replicative dele-tion mutants is an escape from immune surveil-lance (e.g. the deleted region containing theimmunodominant epitopes). Consideration ofproduction and removal processes gives a rela-tive frequency of defective infected cells to wild-type infected cells of

(1!Q)Q

dwt

ddm

(1)

where Q is the "delity of the polymerase, i.e.1!Q is the probability of producing a deletionmutant, and d

wt, d

dmare the death rates of the

cells infected with the wild-type virus and dele-tion mutant, respectively. Thus, deletion mutantsbecome more frequent as the immune responseagainst the wild-type strengthens, i.e. increasingdwt

with respect to ddm

. For instance, the half-lifeof an HIV-infected lymphocyte is about 2 days(Ho et al., 1995; Wei et al., 1995). Thus, therewould be a (10}100)-fold increase in defectivevirus infected cell numbers by immune escapesince T-cells have a lifespan of 1 week to 1 year(Freitas & Rocha, 1993).

If the probability 1!Q of producing a defec-tive mutant is of the order 0.1}10% then thissimple model is su$cient to explain the frequencyat which defective mutants have been observed(Sanchez et al., 1997). However, there is as yet nodirect estimate of the probability of producinga deletion in a replication event, a problem that isfurther complicated by the fact that deletion mu-tants are more prevalent in in vivo samples thanin vitro cultures (Sanchez et al., 1997). This indi-cates that in vivo dynamics are important in theirreplication. The discrete bands under gel elec-trophoresis observed in Sanchez et al. (1997)suggest that arbitrary length deletions are notpresent in the population but only a few distinctdeletion mutants. Thus either slippage of thepolymerase occurs frequently at a limited num-ber of speci"c sites, e.g. constrained by sequencesimilarity restrictions for recombination, or theprobability of producing a deletion mutant issmall, and once produced they are ampli"edthrough selection to levels capable of prod-ucing the strong bands observed in this study.

Mechanisms capable of increasing deletion mu-tant numbers include positive selection underimmune evasion, co-replication of the proviruswith the host cell, or through concurrent replica-tion with the wild type under superinfection.Only immune escape is considered in the simplemodel (1), and therefore the models validity is inqu estion. In this paper, we examine the e!ect ofsuperinfection and host cell replication in am-plifying defective deletion mutants and demon-strate that defective deletion mutant infected cellscan numerically dominate even in the absence ofimmune escape. Further, we develop an idealizedmodel that can be analysed explicitly leading to anunderlying structure that survives perturbation ofthis system and thus applies to more general virusmodels. Although our model is primarily aimed atHIV, it applies to more general viruses when theassumptions hold. In Section 2 the basic idealizedmodel and the underlying assumptions are pre-sented. In Sections 3 and 4, the dynamics of thesystem are investigated, showing that the outcomeof the system is either loss of proliferating virus orloss of uninfected host cells. The model is modi"edin Section 5 by generic perturbations where weprove that the basic structure of the unperturbedsystem survives. We show two examples. In Sec-tion 6 a further modi"cation is examined whichshows periodic behaviour.

Defective viral particles have been modelledpreviously: (engineered) defective interfering par-ticles as inhibitors of wild-type growth (Nelson& Perelson, 1995), the interference e!ects of defec-tive interfering particles (Bangham & Kirkwood,1990), and as escape defective virus in HIV wheninterference on the immune system (CD4` cells)is retained (Berry & Nowak, 1994).

2. A Superinfection Model

Defective deletion mutants are likely to havetheir life cycle interrupted at the stage of genetranscription of the genome, i.e. the defectivegenome is successfully packaged in the parenthost cell and infects a new host as e$ciently asthe wild-type virions. This is because the originalwild-type proteins are involved in this part of thelife cycle and there is only a weak dependence onthe viral genome sequence at this stage. When thegenome regions necessary for packaging, proviral

DEFECTIVE DELETION MUTANT AMPLIFICATION 451

insertion or e$cient replication are in the deletedregion, then the life cycle is likely to be interrup-ted at an earlier stage causing loss of the deletionmutant (Li et al., 1991). In HIV, these essentialsequences are in the long terminal repeats(Frankel & Young, 1998), possibly explaining thecentral deletion region genotype of the mutants(Sanchez et al., 1997; Konishi et al., 1984). How-ever, end deletions would be di$cult to detect bylong-distance PCR since it requires primers atthe extreme ends of the genome.

There should be strong selection pressureagainst deletion mutants since they cannot repli-cate unaided. However, the frequent observationof mutants (estimated to be 25}40% of allHTLV-1 genomes; Shuh et al., 1999), includingmutants with large deletions, means that theyprobably undergo replication through a parasiticmechanism either on the host or wild-type virus.Evolution of deletion mutants to enhanced inter-ference (e.g. increased deletion mutant produc-tion in superinfected cells) may explain thisprevalence; however, without an initial parasiticmechanism deletion mutants would not be insu$cient frequency to enable evolution on a rea-sonable time-scale. We develop a mathematicalmodel to examine parasitic growth of deletionmutants that have not yet evolved strong interfer-ence characteristics. This is therefore a model ofthe initial stages of their appearance.

Our analysis is motivated by an idealizedmodel that displays a basic dynamic structure,that underlies more general deletion mutant sys-tems. We assume, in the idealized model, thatthe defective virus genome is not transcribedalthough it is inserted successfully into the hostgenome, i.e. the necessary signalling sequences forgenome replication, packaging and proviral in-sertion are present. This can be justi"ed by anabsence of transactivating proteins, e.g. Tat, Tax,Rex and early regulatory gene products (Flint& Shenk, 1997; Shuh et al., 1999), while retaining,for example, the LTR in HIV. The lack of viraltranscription, and thus viral proteins means thatfor host cells infected with the defective virus(1) superinfection is possible since suprinfectioninhibition mechanisms (e.g. CD4 down modula-tion in HIV: Mangasarian & Trono 1997; Steven-son et al., 1988) are silent, (2) immune responsesare ine!ective against these cells in the absence of

virus proteins, i.e. viral epitopes are not expressedand (3) the replication processes of the host cellare not in#uenced by the presence of the defectivevirus, and the viral genome is replicated whenthe host cell doubles since it is incorporatedinto the host genome. In superinfected cells, thedeletion mutant genome is replicated providedsequences to initiate replication are intact. Someexpression of deletion mutant genes may occur insuperinfected cells which may alter the balance ofgenome replication and packaging although weignore this e!ect (but it could be examinedthrough the analysis of Section 5). In HTLV,deletion, mutants lacking tax are transcription-ally silent, and although transcribed in the pres-ence of a helper virus, virus production was notsigni"cantly a!ected by the presence of mutantvirus proteins (Shuh et al., 1999). The systemdescribed above is an idealization of the interac-tion of deletion mutants with the host cell and thewild-type virus; however, it is important in that itclari"es the basic mechanisms in the dynamics,and exhibits a mathematical structure thatunderlies more general virus models.

The basic model consists of virus infected cells:cells that are either infected only by the wild typeat density v

1, only by a defective virus at density

v2, or by both, v

3(Fig. 1). Host cells at density

¹ are assumed to replicate under logistic growthwith a capacity K, i.e. they replicate in responseto removal of cells. Virions are not modelledexplicitly; virion half-life is assumed to be muchshorter than the infected cell half-life and there-fore can be removed by a pseudo-steady-stateassumption. Even for viruses such as HIV wherethe half-life of infected cells is short, this is likelyto be a good approximation in that the reducedsystem and the original one display similar dy-namics. Since our analysis concerns "xed points,the only concern is whether stability has changed,which is unlikely. Host cells infected with thedefective virus (v

2) are assumed to replicate nor-

mally under host cell replication signals. Cellsinfected with proliferating virus (v

1, v

3), are

assumed not to replicate or interact with host cellhomeostasis mechanisms. This can be justi"edfor viruses with rapid disruption of host cell func-tion. For viruses where this is questionable, anidealized model with modi"ed homeostasis dy-namics can be used (see below), i.e. none of our

FIG. 1. Model schematic of interacting infected cell populations viand host cells ¹ of the idealized system. DDM refers to

defective deletion mutants. Infected cells produce virions that infect host cells, or currently infected cells (super-infection),DDM virions being produced by mutation from wild-type infected cells and from superinfected cells by partition ofreplication time between the wild-type and DDM genomes. All cells die at rate d, or if there is an immune response, cellpopulations v

1and v

3are cleared with rate sd, s*1. Parameter 0)b)1 parameterizes the superinfectability of wild-type

infected cells relative to uninfected and DDM infected cells. Virions are not modelled explicitly since they are proportional tothe respective infected cell populations under the assumption of fast virion kinetics which is likely to be the case: wild-typevirion density proportional to Qv

1#aQv

3, DDM virion density proportional to (1!Q)v

1#(1!aQ)v

3, where a is the time

competition factor. Cells infected with DDM alone do not produce virions or express viral proteins, because of, for example,missing tat, tax, rex gene products.


results depend on the exact form of the homeo-stasis dynamics. The model is as follows:

vR1"

oK

¹Q (v1#av

3)!sdv

1

!boK

v1((1!Q)v

1#(1!aQ)v

3), (2)

vR2"wA1!

¹#v2

K B v2#

oK

¹ ((1!Q)v1

#(1!aQ)v3)!dv

2!

oK

Qv2(v

1#av

3)

(3)

vR3"

oK

(v1b ((1!Q)v

1#(1!aQ)v

3)

#Qv2(v

1#av

3))!sdv

3, (4)

¹Q "wA1!¹#v

2K B¹!d¹

!

oK

(v1#v

3)¹. (5)

Here Q is the "delity of the viral polymerase, o isthe e!ective virion infection rate of cells (incor-porating virion production and infection of a newhost cell) and w is the replication rate of hostcells. Throughout, we ignore mutations otherthan deletions. Cells have a basic death rate d,which is increased by a factor s for productivelyinfected cells through cellular cytopathicity ora cellular immune response against those cells.Superinfection gives rise to the quadratic terms ineqns (2) and (3), i.e. the term v

2(v

1#av

3) in eqn

(3) arises from superinfection of v2

host cells bywild-type virions produced from the v

1or v

3populations. No contribution to host cell infec-tion by v

2is included because defective infected

cells do not produce virions. The parameter bis the e$ciency of superinfection of wild-typeinfected cells (if b"1 there is no inhibition ofsuperinfection). We also assume that a super-infected cell (v

3) produces the same quantity of

virions as a wild-type infected cell but genomereplication and packaging is partitioned betweenthat of the wild-type provirus and that of thedefective provirus in the proportion a : 1!a, i.e.


wild type to deletion mutant virions are inthe ratio aQ : 1!aQ since the viral polymerasecan introduce further deletions. Throughout, weonly consider the case a"1

2for simplicity (no

competition advantage of wild type or deletionmutant) but our results generalize to arbitrarya(1.

Our assumption that multiple provirus infec-ted cells have the same productivity as singleprovirus infected cells means that cells infectedwith multiple copies of the wild type virus (forexample) do not need to be modelled. This allowsus to reduce the system to cells infected only withwild type v

1, only with the defective v

2, and with

both v3

(Fig. 1). Our models can be extended ifthis is not the case (Sekellick & Marcus, 1980).

This model has three kinetic parameters o, d,w (one of which determines the time-scale) andfour phenotypic parameters characterizing thedi!erences between wild-type and defective infec-ted cells Q, a, b, s. The quantity K determines thescale of measurement for cells, and setting K"1normalizes cell numbers relative to the size of thehost population in the absence of infection. Thus,we will treat v

iand ¹ as dimensionless in all

simulations. By rescaling w, K and o as

w@"w#(s!1)d,

K@"A1#(s!1)d

w B K,

o@"oK@K

, (6)

the death terms in eqns (3) and (5) can be madeequal to sd as in eqns (2) and (4), which impliesthat the parameter s is redundant in determiningbehaviour. Throughout, we use s"1 to charac-terize the dynamics, and discuss the changes un-der s'1 that can be deduced from this rescaling.A time rescaling can be used to remove d, w oro which reduces the parameters to "ve, Thus,there is an approximate symmetry between o andd~1, i.e. high infectivity has a similar e!ect to lowdeath rate. In the following, we consider time asmeasured in days for illustration, and mainlyconsider variation with o since this is a parameterintrinsic to the virus, whereas d and w relate to

the host dynamics. We use w"2 day~1, i.e.a T-cell replicates on a scale of 8 hr, andd"0.05 day~1, i.e. a lifespan of about 20 dayswhich is the middle of the range of estimates(Freitas & Rocha, 1993), although there is largevariation in both of these time-scales whichdepend on cell type and environment. Since thedynamics is determined by the ratios w/d and o/d,exact values are not important. Dependence onthese parameter combinations is clari"ed by theanalysis.

For HIV, where the host cells are predomi-nantly CD4/helper T-cells, the three mainassumptions of the model are satis"ed: (1) super-infection is observed (Ott et al., 1995); however,CD4 downregulation reduces superinfection ofT-cells infected with pro"cient virus (Mangasar-ian & Trono, 1997; Stevenson et al., 1988) andtherefore b(1. (2) Viral gene transcription re-quires the transcriptional regulator Tat, an earlygene product that is not packaged in the virion(Fields et al., 1996). Thus, deletion mutants witha defective tat will have low levels of viral tran-scription, a defect that can be corrected by thepresence of wild-type (pro"cient) provirus. Theassumption of no immune response against de-fective virus infected cells is therefore justi"ed.The lack of viral transcription also justi"es (3),defectively infected cells will replicate normallyunless the provirus has disrupted an essentialhost gene. The HIV virion carries two copies ofthe HIV genome that are recombined into asingle provirus (DeStephano et al., 1994; Katz& Skalka, 1990). Therefore, Q is the probabilityof not producing a deletion mutant from theconsecutive events of replication and recombina-tion of two genomes. This also implies that HIVinfected host cells harbour a single provirus un-less multiple infection through superinfectionoccurs (Ott et al., 1995), which is important sinceotherwise deletion mutants would not requiresuperinfection for replication. HIV-infected cellsalso produce virions rapidly once activated anddisrupt cell function, suggesting that v

1and v

3can be considered as non-replicating and notlikely to express cytokines a!ecting ¹ cellhomeostasis. Thus, we obtain the homeostasismechanism of eqns (2)}(5). Our model ignorestwo aspects of HIV that are important in under-standing host infections: HIV transcription is


dependent on the activation status of the T-cell(Bassuk et al., 1997; Frankel & Young, 1998), andthus our model will not have the correct long-term dynamics which will depend on subsequentpathogen infections. However, average behav-iour will be captured. Secondly, the homo-geneous mixing assumption implicit in theordinary di!erential equation formulation doesnot capture the di!erences between di!erent en-vironments in the body, e.g. lymph nodes wherethere is possibly a higher frequency of infectedcells, and di!erent chemokine receptor T-cellsubsets (Horuk, 1999; Moore et al., 1997).The models can be further compartmentalized(Nelson & Perelson, 1995) to incorporate thesee!ects. We expect our results to extend to thesecases.

For HTLV, 25}50% of virus in infectedlymphocytes is defective, and a signi"cantfraction of these have substantial deletions(Hiramatsu & Yoshikura, 1986; Konishi et al.,1984). In addition, mutants lacking tax are tran-scriptionally silent, while replication occurs in thepresence of replication competent provirus (Shuhet al., 1999). Cell}cell contact is the principalmeans of host cell infection, and thus eqns (2)}(5)follow immediately without the virion pseudo-equilibrium approximation. However, for virusessuch as HTLV with a long-term associationwith their host cells, a more suitable homeo-stasis dynamics is a logistic rate of growth(1!(¹#v

1#v

2#v

3)/K) per cell in each of

FIG. 2. Virus densities as functions of time showing the cresource limitation. The "gure shows log

10v1, log

10v2, log

10b"0.001. If b'0, v3

always dominates, but for small b thereobserved for b"0 (not shown). The model can be non-dimenconditions are consistent with a small initial exposure to thek"1 cell~1, s"1. Line key: ( ) v

1; ( ) v

2; ( ) v

3

eqns (3)}(5). This system has one more parameterthan the earlier case since the symmetry (6)cannot be used to remove s. This modi"cation isdiscussed in Section 4. Interference of growth andcytokine production (Li & Gaynor, 1999) byHTLV suggests that the v

1and v

3compartments

may have enhanced e!ects on homeostasis orenhanced growth. The results also hold underthese additional generalisations.

3. Virus Growth in Absence of ResourceLimitation

The model has three processes that determinethe dynamics: superinfection, resource (host cell)limitation and immune escape of the defectivemutant. In order to understand the contributionsof these individual processes we "rst considera simpler case consisting of virus with unlimitedresources/hosts, i.e. the host has no dynamics.Replication terms in the host cell and v

2compart-

ments are removed, e!ectively "xing host cells ata constant density, ¹"K, to give

vR1"oQ (v

1#av

3)!sdv

1

!bkov1((1!Q)v

1#(1!aQ)v

3), (7)

vR2"o ((1!Q)v

1#(1!aQ)v

3)

!dv2!koQv

2(v

1#av

3), (8)

hange in the numerically dominant strain when there is nov3

and two values for the superinfectivity b. (a) b"1.00; (b)is an interval of v

1and v

3exponential growth similar to that

sionalized so that o and k can be chosen to be 1. The initialvirus. Parameters Q"0.7, a"0.5, d/o"0.05, o"1 day~1,.


vR3"ko (bv

1((1!Q)v

1#(1!aQ)v

3)

#Qv2(v

1#av

3))!sdv

3, (9)

where k"K~1.This system has very simple dynamics which

di!ers signi"cantly between the cases b"0, nosuperinfection of wild-type infected cells, andb'0 where superinfection occurs. Typical tra-jectories are shown in Fig. 2. Provided thegrowth rate (infection rate) is su$ciently large(Qo'sd) the virus grows exponentially, initiallywith v

1dominating at a growth rate oQ!sd. The

"nal state consists of exponential growth of v3

alone if b'0 (growth rate o!sd), or v1

and v3

growing exponentially in the ratio v3/v

1"kv

2, v

2being constant, if b"0. This di!erence in behav-iour is caused by the loss of v

1through superin-

fection [quadratic loss in v1

in eqn (7)] whenb'0. For small b, an intermediate phase of v

1,

v3

exponential growth similar to the b"0 caseoccurs, turning into an exponential growth phasesimilar to that of b"1 at higher viral densities(Fig. 2). The two exponential phases can bededuced from eqns (7)}(9) by searching for ex-ponential growth solutions, in which case thequadratic terms only have the correct exponen-tial growth rate if certain populations areconstant.

4. Idealized Defective Deletion Mutant Dynamics

The basic criterion for virus growth is that thehost density is su$cient to balance replicationwith loss, i.e. in epidemiological terms the basicreproductive rate R

0'1 (Anderson & May,

1992). Thus, for example, if we consider the earlystages of an infection (negligible v

2, v

3), the

virus will grow if o¹Q'sdK from eqn (2), i.e.(1!d/w)oQ'sd since ¹"(1!d/w)K initially.This is an invasion criterion, i.e. a condition on theparameters that determines if a minority popula-tion grows. The dynamics of this system is totallydetermined by two invasion criteria. The "rst isthe growth of the virus at the early stages ofinfection as given above. The second follows sincethe density of host cells decreases as the virusgrows, reducing the growth rate of v

1. For the

system (2)}(5) the population of wild-type infec-ted cells v

1is eventually lost since the number of

host cells falls below, and remains below, thatrequired for growth. This can be proved using theLyapunov function ¸"v

2/¹ for which 0̧ *0

follows from the equations of motion. Equality isobtained if v

1"v

3"0. Thus v

2always increases

its frequency relative to ¹. This also implies thatthere are no physical coexistence states for allfour variables. The "nal state is determined bywhether doubly infected cells can grow on v

2hosts, the second invasion criterion. The para-meter values determine which of the followingtwo "nal states (attractors) occur:

I. v1"v

3"0, v

2#¹"(1!d/w)K. This

corresponds to complete loss of proliferat-ing virus. The ratio v

2/¹ is determined by

the initial conditions.II. ¹"v

1"0, v

2"sdK/oaQ, v

3"K (w(1!

v2/K)!d)/oaQ. The defective deletion

mutant infects all of the host cells, pro-liferating virus surviving only as super-infected cells (deletion mutant togetherwith wild type).

E!ectively, the deletion mutant has a selectionadvantage over the uninfected cell population inthe presence of proliferating virus; the mutant isproduced from replication of the wild-type virus,whereas in the absence of proliferating virus thereis no selection against deletion mutant infectedcells relative to the uninfected host cells. If dele-tion mutant infected cells replicate slower, wipe-out does not occur and a coexistence state will beestablished (Section 5).

For attractor I to be attained from an initialinoculum it is necessary that the virus can growin an uninfected host, i.e. we have the invasioncriterion

Qo'sdw

(w!d). (10)

Similarly, to attain attractor II it is necessary thatsuperinfected cells v

3can grow on v

2hosts, i.e.

that

aQo'sdw

(w!d). (11)


When considered as conditions on the infectionrate o, we can de"ne the critical infectivity o

cby

eqn (10), and the superinfection critical infectivityosc'o

cby eqn (11). Thus, if o(o

cthe infected

cell population v1

decays immediately. In theinfectivity window o

c(o(o

sc, v

1initially in-

creases, but will ultimately die out as host celldensity ¹ declines. Viral population v

1declines

when v2'v*

2, where

v*2"

11!a A1!

dw!

sdoQB . (12)

This is the condition of stability of attractor I. Ifo'o

sc, there are no values of v

2)(1!d/w)K

FIG. 4. Virus densities as functions of time showing the lossbetween critical infectivities; (b) o"0.11 day~1 above bothinfectivity lies between the two critical infectivities o

c"0.0523

the infectivity is above osc, since v

3grows on DDM infected hos

virus alone or superinfected. Parameters Q"0.98, a"0.5, d( ) v

1; ( ) v

2; ( ) v

3; ( ) ¹.

FIG. 3. Schematic characterization of attractor stabilityand "nal state under variation of the infectibility o. Thresh-old v*

2is given in eqn (12).

for which manifold I is stable. Hence, the "nalstate has no uninfected hosts, 0̧ '0 for all timeand ¹P0. Since o'o

scsuperinfected cells can-

not die out and the system approaches attractorII. These conditions are summarized in Fig. 3,and typical trajectories are shown in Fig. 4. Therelative densities of the v

2and v

3steady-state

populations as functions of o are shown in Fig. 5.These conclusions are independent of whethersuperinfection of wild-type infected cells occurs,i.e. independent of b.

We de"ne an idealized defective deletionmutant system as one where ¸"v

2/¹ is a

Lyapunov function. All idealized DDM systemswill have a dynamic structure similar to theabove, in particular uninfected hosts ¹ will belost unless proliferating virus is cleared. Forinstance, the analogous full model eqns (2)}(5)incorporating virion dynamics is ideal. The sys-tem remains ideal under a variety of modi"ca-tions. For example, the homeostatic mechanismcan be modi"ed to allow all cell populations toreplicate, i.e. logistic growth with a replicationrate per host cell

wA1!(¹#v

1#v

2#v

3)

K B , (13)

in eqns (2)}(5). This system has identical behav-iour to the above, i.e. there are analogousattractors I and II and ¸"v

2/¹ is a Lyapunov

function. The only di!erences are that s isreplaced by s!1 in the critical susceptibility

of infected cell type v1

after initial growth. (a) o"0.07 day~1critical infectivities. Proliferating virus dies out when theday~1 and o

sc"0.1046 day~1. Proliferating virus survives if

ts v2. In this case ¹P0, i.e. all host are infected with defective

"0.05 day~1, K"1, s"1, w"2 day~1, b"1. Line key:

FIG. 5. Proportions of the various virally infected celltypes, p

i"v

i/+

jvjfor the idealized DDM model. Cells infec-

ted with wild-type alone always die out, i.e. p1"0. Only

above the superinfection critical infectivity osc"0.1047 does

proliferating virus survive (as superinfected cells). Betweenthe critical infectivities o

c, o

scthe ratio v

2:¹ is not "xed,

except that v2'v*

2since attractor I must be stable for the

proliferating virus to die out. Parameters: Q"0.98, a"0.5,d"0.05 day~1, K"1, s"1, w"2 day~1, b"1. Line key:( ) p

2; ( ) p

3.


expressions (10) and (11), and in the attractorI stability threshold v*

2[eqn (12)]. Attractor II

is given by v2#v

3"(s!1)dK/oaQ, v

3"

K (w (1!(s!1)d/oaQ)!d )oaQ. In this case, animmune response is essential for the existence ofthe window o

c(o(o

sc.

The utility of these idealized systems is inidentifying the critical infectivities as the keyto understanding the dynamics. However, theseidealized cases are unrealistic for a number ofreasons:

f The host population is unlikely to be closed.For instance, naive T-cells continuously arrivefrom the thymus. Alternatively, infection ofa host cell may depend on its status, i.e. theremay be an uninfectable precursor subset lack-ing required viral receptors. Input from anexternal (uninfected) source dilutes the defec-tive deletion mutant infected cell popula-tion v

2.

f Host cells infected with deletion mutants mayhave an increased death rate relative to un-infected cells. This could be caused by MHCupregulation, a cytotoxic immune response, orapoptosis. Deletion mutant infection may alsointerfere with replication signals, reducing hostcell proliferation. These e!ects may be a result

of damage to the host cell function caused bythe insertion of the virus in the host genome,a low transcription rate of the virus or becauseof the detection of double-stranded RNA andtriggering of anti-viral activity.

f The DDM genome may be lost under hostreplication. This is more applicable for virusesthat are not inserted in the host genome butoccur separately in the nucleus or cytoplasm.

In all these three cases the Mv2, ¹N manifold is

no longer neutral and v2P0 in the absence of

proliferating virus, i.e. ¸ is no longer Lyapunov.In the next section, we demonstrate that theunderlying structure determined by the criticalinfectivities remains when any of these e!ectsare incorporated, and the population densitiesessentially follow that of the idealized system.However, the behaviour is signi"cantly di!erentin that proliferating virus is not removed wheno'o

cand a coexistence state of the four cell

populations exists.

5. Host Kinetics with a Small UninfectedExternal Source

In this section, we consider a host populationwith a small input from an external (uninfectable)source. Our equations are identical to eqns (2)}(5)except that the host dynamics changes to

¹Q "eF#wA1!¹#v

2K B¹!d¹

!

oK

(v1#v

3)¹, (14)

where e is a small dimensionless perturbationparameter. Possible choices for the perturbationF include F"wK for a thymic input to the T-cellpool and F"wK (1!(¹#v

2)/K) for a subset

of precursor host cells. Our analysis applies togeneric F although all illustrations use the thymicinput form. More general perturbations can alsobe included, for instance if deletion mutants pro-liferate slower than uninfected host cells thena term !ew (1!(¹#v

2)/K)v

2can be added to

eqn (3). The analysis of these more general sys-tems is identical to that of eqn (14) and the results


are similar. Analogous analysis and results canalso be obtained for systems with the alternativehomeostatic mechanism (13) and their associatedattractors. Thus, our results are robust againstmodel assumptions.

The variation of the steady state with infectiv-ity o is shown in Figs 6}8. The trajectories dis-play a generic structure that is determined by theunderlying attractors I and II of the idealized(unperturbed) system. In particular, for o

c(o(

osc

although population v1

is initially dominantnear o"o

cits proportion rapidly declines as

v2/¹ increases with o, i.e. the solution ap-

proaches the attractor I as o increases. When o isabove the superinfection critical infectivity o

scthe

solution follows that of attractor II, compare

FIG. 7. Infected cell type proportions pi"v

i/+

jvj

and invariation of the infection rate of the host cells o for the case o(a) detail for small o; (b) detail for large o. Asymptotically v

1P

thymic input. Perturbation e"0.01 corresponds to Fig. 6(a). Oday~1, K"1, s"1. Line key: ( ) p

1; ( ) p

2; ( ) p

3

FIG. 6. Infected cell type fractions pi"v

i/+

jvj

and infevariation of the infection rate of the host cells o for the model wdi!ering values for the thymic input. (a) e"0.01; (b) e"0.001.day~1, K"1, s"1, b"1. Line key: ( ) p

1; ( ) p

2; (

Fig. 5 with Figs 6 and 7. As oPR the solutionsdeviate from that of the unperturbed system sincehost replication becomes negligible compared toexternal input. The behaviour has a dependenceon b reminiscent of the unlimited resourcecase dealt with earlier. The infectivity windowoc(o(o

scis where maximal interference of the

deletion mutant occurs, i.e. proliferative viralload is reduced compared to the system that doesnot produce mutants. The width of this windowis determined by the competition factor a forreplication and packaging of the deletion mutant.This structure is now considered in more detail.

Input from an external source continuouslydilutes population v

2from the host population.

Therefore, if o(octhe only physical steady state

fected host population fraction f"+jvj/(¹#+

jvj) with

f no superinfection of wild-type infected cells v1, i.e. b"0.

constant similar to the system without DDM and a smallther parameters: Q"0.98, a"0.5, d"0.05 day~1, w"2.0; ( ) f.

cted host population fraction f"+jvj/(¹#+

jvj) with

ith a small thymic input ewK. The two graphs correspond toOther parameters: Q"0.98, a"0.5, d"0.05 day~1, w"2

) p3; ( ) f.

FIG. 8. Viral load and host density for the thymic inputmodel of Fig. 6(a), e"0.01, b"1. Between the criticalinfectivities o

c"0.0523 and o

sc"0.1047 proliferative virus

density is signi"cantly reduced compared to the virus systemthat does not produce deletion mutants (solid line). Uninfec-ted host density is reduced relative to this system because ofthe generation of hosts infected with defective deletion mu-tant virus. Other parameters: Q"0.98, a"0.5, d"0.05day~1, w"2.0 day~1, K"1, s"1, b"1. Line key: ( )v; ( ) v

1; ( ) v

2; ( ) v

3; ( ) ¹.


is v1"v

2"v

3"0 and ¹"1!sd/w, i.e. the de-

letion mutant is lost from the host population.The deletion mutant only survives when prolif-erating virus is present, a conclusion that appliesfor all values of o. Above the critical infectivity o

cwe have v

2/v

1, v

3/v

1P0 as oPo

c. Approximate

densities near ocare given by

v1&

Ko A

eF¹

#wA1!¹

KB!dB,

v2&

o (1!Q)eFK

¹2v1

(15)

with ¹&sdK/oQ and v3"O((1!Q)v2

1). The

defective virus equilibrium is determined by thebalance of generation from mutation, the termo (1!Q )¹v

1, with the dilution rate eF/¹,

emphasizing the need for proliferating virus tomaintain the deletion mutant population. Thesingular dependence as eP0 arises becausethe solution approaches attractor I where v

2is

undetermined.As o increases away from the critical suscepti-

bility octhe density of v

2rapidly increases at the

expense of the uninfected host population, and thesteady state approaches attractor I. Perturbing

around attractor I we have leading order expres-sions v

2"v*

2(12), the earlier stability boundary

for the unperturbed system (Fig. 3), and¹#v

2&(1!d/w)K, i.e. ¹ and v

2are on attrac-

tor I to this order. The remaining populations aregiven by

ov1&eF

v2

v2#¹

K(1!Q)¹#(1!aQ)v

2

(16)

while v3&(v

2/¹ )v

1. This solution remains

valid until ¹&eFo~1 near o"osc

where theapproximation above is singular, ¹P0 asoPo

scon attractor I. This singular behaviour

re#ects the increase in v3

in the unperturbedsystem. A perturbation expansion about attrac-tor II gives

eF¹

&

1!aQaQ AwA1!

v2

KB!dB (17)

v1&(¹/v

2) v

3as previously, and v

2, v

3on

attractor II to leading order. This analysis dem-onstrates that provided v

1remains a minority

population (speci"cally o is away from the criticalinfectivity o

c) the steady state is well approxi-

mated by the unperturbed system as indicated bycomparison of Fig. 5 with Figs 6 and 7. The formof the perturbation is not important, although ase decreases the steady states are closer to those ofthe idealized systems.

As oPR the external source determinesthe dynamics, i.e. the dynamics changes froma perturbation of the resource limitation solutionto a solution independent of the resource limita-tion process. This is because ¹P0 as o increases[eqn (18)] and therefore replication within thehost population becomes negligible relative tothe external input. The asymptotic states (oPR)are

v3"

eFsd

, (o independent), ¹"

sdKo

,

v1"

aQ1!aQ

¹, v2"

1!aQaQ

¹ (18)


in the case b"1 and

v1"

eFsd

(o independent), ¹"

sdKoQ

,

v2"

1!QQ

¹, v3"

1!QQ

v1

(19)

in the case b"0 (Fig. 7). These solutions areanalogous to the two exponentially growingstates for b'0 and b"0 under no resourcelimitation. This is because they derive fromthe same balances for equilibrium in the threepopulations, except for the constraint on thetotal virus population. For b"0 the change indominance between v

1and v

3occurs when

oQ/sd&w/eF. Matching of these expansionswould give an approximation to the steady stateover the range o*o

c.

The parameter a determines the windowbetween the two critical infectivities where thedefective deletion mutants have maximal e!ect,i.e. the proliferating virus load is signi"cantlyreduced compared to the virus system withoutdeletion mutants (Fig. 8). This e!ect is reducedas the probability of deletion 1!Q falls sincethe boundary layer around the critical infectivityoc

increases in width [eqn (15)]. For a giveninfectivity o and mutation probability Q deletionmutant interference of the wild type increases asa falls, i.e. as the replication advantage of deletionmutants in superinfected cells increases thedensity of the proliferative virus decreases (Rouxet al., 1991). Reduced total virion output ofsuperinfected cells has a similar e!ect. Theappearance of successive DDM that decreasea or reduce the virion output of superinfectedcells would lead to the evolution of defectiveinterfering particles (widening the window ofinterference o

c(o(o

sc). Evolution of DIP and

counter evolution of helper viruses have beenobserved (Roux et al., 1991). As a decreases,superinfection inhibition (parametrized by b)may become important within the infectivityrange o

c(o(o

sc, where the deletion mutants

have a signi"cant e!ect on proliferating virusload, i.e. interchange of the dominant strain mayoccur for o(o

sc.

In systems far from the idealized system,e.g. large e, the general structure of the idealized

system is retained. This is because the criticalinfectivities are independent of the nature ofthe host replication system and only depend onthe viral replicative properties. Consider thecase where the virus does not infect proliferatinghost cells, proliferating cells remaining at a "xeddensity. Then we obtain a host dynamics of theform

¹Q "uKA1!¹#v

2K B!d¹

!

oK

(v1#v

3)¹, (20)

where uK is the rate of the production of non-replicating host cells, the host density beingcontrolled at the level of total numbers. We illus-trate this example with respect to d to emphasizethe inverse relationship of d and o, althoughincreasing the infectivity o and decreasing thedeath rate d are not identical since the latter isequivalent to increasing o and u concurrently (orw in the previous model). This has the e!ect thatat low d the virus load remains high, since e!ec-tively high host replication continues to supplyhost cells for infection. In Fig. 9 as d increasesfrom zero we move through the asymptotic solu-tions (18), (19) with their dependence on b. Thenthe behaviour is similar to the idealized systemwith the proportion of v

2increasing to a max-

imum between the critical infectivities/criticaldeath rates and decreasing to zero as the deathrate approaches sd"oQ (1!d/w).

6. Loss of Virus at Host Replication

In this section, a modi"cation to the carriage ofthe virus by the host is introduced. We havepreviously assumed that the virus is replicatedwith the host since it is inserted in the genome.However, cytoplasmic viruses may be lost underreplication. The model is as follows:

vR2"

oK

¹ ((1!Q)v1#(1!aQ)v

3)!dv

2

!

oK

Qv2(v

1#av

3), (21)

FIG. 9. Infected cell type proportions pi"v

i/+

jvjwith variation of the death rate of the host cells d for the model of

uninfectable precursor cells at "xed density. Cases (a) with superinfection (b"1) and (b) without superinfection of wild-typeinfected cells (b"0). Other parameters Q"0.98, K"1, a"0.5, o"1 day~1, u"1.4 day~1, s"1. Line key: ( ) p

1;

( ) p2; ( ) p

3; ( ) f .

FIG. 10. Virus dynamics for the model where the presence of deletion mutants prevents the host cell from replicating p"0.Depending on the host replication rate the "nal state can be (a) a "xed point w"1 day~1; (b) with a limit cyclew"0.3 day~1. The "gure shows log v

1, log v

2, log v

3, log ¹ as functions of time. Parameters: Q"0.98, K"1, a"0.5,

d"0.04 day~1, o"1 day~1, s"1. Line key: ( ) v1; ( ) v

3.


¹Q "wA1!¹#pv

2K B (¹#pv

2)!d¹

!

oK

(v1#v

3)¹, (22)

with v1

and v3

dynamics identical to eqns (2)and (4). In this model the deletion mutant isnot replicated with the host but remains in oneof the daughter cells. However, the presence ofthe virus in the host reduces the replication rateof that cell by a factor p. There are two extremes,p"1 when hosts carrying defective virus repli-cate normally and p"0 when hosts carrying thedefective virus do not replicate. The reasons forintroducing p are that these two extremes behave

signi"cantly di!erently. Rede"nition of w andK cannot be used to remove s as in the originalmodel.

A typical trajectory for the case when v2

infec-ted host cells do not replicate (p"0) is givenin Fig. 10. The "xed point can be unstable andthe "nal state periodic. Such cycling has beenobserved in in vitro experiments with DIP(Bangham & Kirkwood, 1990; Huang, 1973). The"xed point becomes stable as the death rate dincreases, but has a more complex dependence onthe host replication rate (Fig. 11). Stability of the"xed point is also improved through an immuneresponse, increase in s (not shown). The limitcycle is not robust since a relatively low contribu-tion of v

2to replication (p'0) destroys the limit

cycle (Fig. 12). We suspect that cycling behaviour

FIG. 11. Bifurcation diagram in d and w showing a limitcycle (region enclosed) and a stable "xed point when hoststhat carry the deletion mutant fail to replicate. Figure 10(a)and (b) correspond to parameter values outside and insidethe enclosed region. Other parameters Q"0.98, K"1,a"0.5, s"1, o"1 day~1, p"0.

FIG. 12. Bifurcation diagram in d and p showing limitcycle (region enclosed) and stable "xed point for the modelwhere the deletion mutant is lost on host replication, andhosts carrying the deletion mutant have replicative e$ciencyp. Limit cycle is lost as the replicative e$ciency ofcells infected with defective virus (v

2) increases. Other

parameters: Q"0.98, K"1, a"0.5, w"0.4 day~1,o"1 day~1, s"1, b"1.


is more robust as interference e!ects of themutants increase, e.g. aP0.

7. Conclusions

In this paper, we have examined the frequencyand replication of defective deletion mutantsin a number of virus dynamics models. Threeprocesses are signi"cant in determining DDM

frequency: replication of the defective virus underhost replication, superinfection of wild-typeinfected cells and an immune response againstcells infected with proliferating virus. We foundthat these systems display a robust structure de-termined by two thresholds which we consideredin terms of the viral infectivity: the critical infec-tivity o

cfor growth of the virus in a naive host,

and the superinfection critical infectivity osc

forthe virus to grow in host cells harbouring defec-tive virus. For an infectivity between these criticalvalues, o

c(o(o

sc, the proliferating virus is sup-

pressed by the presence of the deletion mutantthrough a dynamic instability of proliferativevirus, and host cells infected with the deletionmutant alone can numerically dominate. The de-crease in the proliferative, and total, viral loadscaused by the presence of the deletion mutantresults from a reduced output of wild-type virionsfrom superinfected cells compared to cells infec-ted with wild type alone. This was motivated bycompetition for replication and packaging fromDDM genomes, the total virion output remain-ing unchanged. In particular, we found that adi!erential immune response is not required toincrease the relative proportion of deletion mu-tants, although this was not a robust result sinceit was dependent on the nature of the interactionof the virus with the homeostatic mechanisms.Above o

scsuper-infected cells account for the

majority of the virally infected cells, and only aso increases through this critical infectivity doesthe frequency of uninfected cells fall rapidly. Onlyat high infectivities was any dependence on thesuperinfection e$ciency of wild-type infectedcells observed. Thus, deletion mutants are pre-dicted to be ampli"ed, and therefore observable,either as a signi"cant population of deletion mu-tant infected cells (o

c(o(o

sc), or as superinfec-

ted cells (o'osc), except in the case of strong

superinfection inhibition of wild-type infectedcells and high infectivity o when DDM frequencycan be low (Fig. 7). The dependence on the criti-cal infectivities was underpinned by analysis ofidealized systems, the basic dynamic attractorsbeing robust against perturbation and thusextending to general systems. For systemsclose to ideal, approximate expressions were de-rived for the steady states by perturbationanalysis and demonstrated that the nature of the


perturbation is not critical. An idealized defectivedeletion mutant system was de"ned as a deletionmutant that was replicated when the host celldoubled, did not a!ect the host replication ratewhen infecting that host alone and there was noexternal source feeding into the host population.Mathematically, an idealized DDM system wasone where v

2/¹ was a Lyapunov function.

The window of interference oc(o(o

scis de-

termined in our model by the degree of deletionmutant parasitism on wild-type virion produc-tion in a superinfected cell. We argue thatalthough total virion production is likely to beunchanged (for unselected DDM), the deletionmutant genome will be copied and packaged ina fraction of the time a, provided packagingsignals (in the LTR in HIV) are present. InAIDS, uninfected CD4 cells numerically domin-ate which suggests that the infectivity lies be-tween the critical infectivities o

cand o

sc. In this

parameter range, deletion mutants can play a sig-ni"cant role in reducing proliferative virus, whichmay apply to AIDS where signi"cant quantitiesof non-viable virus are produced. For HTLV,the predominant association of deletion mutantswith intact provirus in the same cell (Konishiet al., 1984) suggests that HTLV lies in the regimeof superinfected cell dominance, o'o

sc. Our

models assume a constant immune response, andboth thresholds depend linearly on its strength.An interesting question is the e!ects of a dynamicimmune response on DDM frequency and DDMinterference, especially as removal of the virusrequires the immune response to move the infec-tivity window through the infectivity o. Theimmune response may receive insu$cient stimu-lation when the infectivity lies in the infectivitywindow to eliminate the virus because of interfer-ence e!ects, i.e. defective virus may be a contribut-ing cause to viral chronicity.

Defective virus mutants also arise from frameshifts or point mutations that introduce STOPcodons into the genome. Their frequency can behigh (46% estimated for HIV in Li et al., 1991,10}15% for a defective tat alone in Meyerhanset al., 1989). The models presented here for dele-tion mutants also apply to these defective strains,although because of partial genome transcriptionthe system is likely to be far from idealization.This means that perturbation theory away from

idealization may be inappropriate; however, thesame general dependence on the critical infectivi-ties is expected to survive (Fig. 9).

Our analysis has demonstrated the importanceof considering host replication as a means fordefective provirus replication. However, sinceour analysis grouped all defective deletion mu-tants together we are unable to conclude that theband patterns observed in Sanchez et al. (1997)is consistent with the replication processes forDDM analysed here. It cannot be ruled out thatthe deletion mutants observed have not beenselected on an interference trait such as preferen-tial replication of the deletion mutant in superin-fected cells (small a) (Roux et al., 1991). Ourmodels can be extended to examine these ques-tions; the frequencies of speci"c genotypes wouldneed to be traced and interaction e!ects betweendi!erent strains included. As emphasized here,neutral or weak selection probably operates onDDM relative to uninfected cells, and thereforestochasticity should possibly be included. Sucha model would lead to an examination of theevolution of DDM and DIPS.

We are grateful for useful discussions with NigelDimmock (Univ. Warwick) and David Robertson(CNRS, Marseille), and to Hugo van den Berg (Univ.Warwick) and the anonymous referees for commentson the paper. NJB was supported by an EPSRCAdvanced Fellowship, B/94/AF/1822. BMcL wassupported by a Mathematics in Medicine InitiativeVisitors grant from the University of Warwick. Bifur-cation diagrams were generated with CONTENT.

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defective deletion mutant amplification

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