Conditions for invasion of synapse-forming HIV variants.

Cesar Augusto Vargas Garcia1, Ryan Zurakowski2, and Abhyudai Singh3,∗

Abstract— Infection by Human Immunodeficiency Virus is awidespread cause of progressive immunodeficiency and death.The mechanisms of virus transmission and infection are well-documented from many experimental studies. These showthat the infection of CD4+ T Cells by HIV happens by twodistinct mechanisms: transmission by free viruses, and cell-celltransmission in which viral particles are transmitted directlyacross a tight junction or synapse between an infected andan uninfected cell. In this paper, a mathematical model ofHIV transmission including both the free virus and cell-celltransmission pathways is introduced. A variation of this modelis considered including two populations of virus. The first infectscells only by the free virus pathway, and the second infects cellsby either the free virus or the cell-cell pathway. Steady-stateand bifurcation analyses are performed on this model. A simpleformula is presented describing the bifurcation point for localstability of a steady-state solution consisting entirely of the firstviral subtype. This is equivalent to the conditions for invasion bya synapse-forming HIV variant. Synapse-forming HIV is shownto provide an evolutionary advantage relative to non synapse-forming virus when the average number of virus transmittedacross a synapse is a sufficiently small fraction of the burstsize. The exact bifurcation point depends on the fitness ofthe non synapse-forming virus and the likelihood of successfulinfection as a function of multiplicity of infection. These resultsare important for understanding synaptic transmission in HIV,which has been identified as a possible cause of continuedreplication during antiviral therapy.

I. INTRODUCTION

Human Immunodeficiency Virus (HIV) is a human retro-virus that infects certain immune cells, primarily the CD4+helper T cells and the macrophages. Untreated infection canlead to the eventual collapse of the immune system, resultingin severe immunodeficiency and death due to opportunisticinfections [1]. Over 34 million people are infected, with asmany as 2.5 million new infections each year [2].

The lifecycle of the HIV virus has been extensivelystudied. The virus particle consists of two positive single-stranded RNA copies of the viral genome, together with thefunctional HIV enzymes Vif, Vpr, Nef, Protease, Integrase,and Tat, are enclosed in a protein capsid. This capsid is inturn encased within a lipid-bilayer envelope studded with theviral glycoproteins gp41 and gp120.

1 C.A. Vargas Garcia is a graduate student in the Department of Electricaland Computer Engineering, University of Delaware, Newark, DE 19716cavargar@udel.edu

2 R. Zurakowski is an Assistant Professor in the Department of Electricaland Computer Engineering, University of Delaware, Newark, DE 19716ryanz@udel.edu

3 A. Singh is an Assistant Professor in the Department of Electricaland Computer Engineering, University of Delaware, Newark, DE 19716a2singh@udel.edu

AS is supported by the National Science Foundation Grant DMS-1312926, University of Delaware Research Foundation (UDRF) and OakRidge Associated Universities (ORAU)

The envelope protein gp120 has a high affinity for thecell marker CD4, which is a characteristic marker of thehelper-T cells and the macrophages [3], and to a co-receptor,either the CCR5 or CXCR4 transmembrane proteins on thetarget cells [4], [5]. Binding of gp120 to these two receptorsresults in a conformational change which exposes an activesite of the viral protein gp41. The exposed gp41 mediatesfusion of the viral membrane with the host cell membrane,which decapsulates the virus [6]. The viral membrane andall integrated proteins, including gp41 and gp120, becomespart of the host cell’s membrane. Attachment inhibitors suchas maraviroc may interfere with this step [1], [7].

Once the virus has entered the host cell, viral reverse-transcriptase creates a DNA copy of the viral RNA, a stepthat may be interrupted by the presence of reverse transcrip-tase inhibitors [1]. The viral DNA is transported to the cellnucleus, and integrated into the host-cell chromosomes bythe viral integrase enzyme. This step can be interrupted bythe presence of integrase inhibitors such as raltegravir orelvitegravir [8], [9]. The integrated HIV genome is thenexpressed by the normal cellular RNA transcription ma-chinery, although a hairpin structure in the evolving HIVRNA transcript can result in aborted transcription unlessthe viral protein Tat is present. The viral RNA transcriptis transported to the cytosol, where ribosomes transcribeseveral non-functional super-proteins, which must be cleavedby the viral protease enzyme into their functional forms. Thevirus products aggregate on the cell surface and form viralparticles.

These viral particles bud off of the surface of the hostcell and are released into the surrounding fluid, able toinfect other cells. However, prior to the formation of mature,budding HIV particles, there is an accumulation of gp41and gp120 complexes on the surface of the infected cell. Inthe same manner in which these viral proteins mediate thebinding of and fusion of viral envelopes to the target cell,they are also capable of allowing the infected cell to bindto an uninfected target cell and partially fuse membranes,resulting in the formation of a viral synapse. Formation of thesynapse allows for the direct transmission of viral particlesbetween the infected and the uninfected cell [10]–[17]. Thishas been observed experimentally, and it has been shown thatas many as several hundred virus particles can be transmittedacross a single synapse [18].

It has been suggested that the cell-cell transmission path-way provides an evolutionary advantage, either by allow-ing the virus to evade the host immune response [15], tooverwhelm antiviral drug activity [19], [20], or simply by amore efficient mode of infection. Previous modeling work

considering the potential benefits of cell-cell transmissionhas shown that the optimal number of viruses transmittedby a synapse is small [21], [22]. This previous work didnot explicitly consider the competition between synapse-forming and non-synapse forming viruses, as we do here.The previous work was also based on nominal parameters,where we use parameter values identified from experimentalpatient data [23].

We present a novel model of HIV virus dynamics thataccounts for transmission by both the free virus and cell-cellpathways. We analyze the stability of the stationary pointsof this model, and derive bifurcation conditions. We showthat the steady-state viral loads in a synaptic transmissionmodel increase as a function of the probability of successfulinfection given a cell-entry event. We further show that thesteady-state virus load increases with synaptic multiplicityof infection when the fraction of total burst size is small, butbegins to decrease with increasing multiplicity of infectiononce a relatively small fraction of the burst size is reached.We show that a synapse forming virus variant will success-fully invade against an established non synapse-forming viruswhen the multiplicity of infection as a fraction of total burstsize is less than a critical value determined primarily by thefitness of the non synapse-forming virus.

The paper is organized as follows. Section II introducesthe basic model of HIV dynamics as developed by Perel-son et al. [24]–[26]. Section III introduces the model ofsynapse-forming virus, develops stability conditions on thestationary points of the model, and explores the effect ofvarying multiplicity of infection and probability of success-ful infection on the steady-state virus loads. Section IVintroduces a competition model between synapse-formingand non synapse-forming virus, and develops a bifurcationcondition for the stability of the stationary point with nosynapse-forming virus. Section V summarizes the results,and discusses implications for HIV treatment and futurework.

II. HIV MODEL

The free virus transmission mechanism is described usingthe extensively studied model introduced by [24]. In thismodel the behavior of uninfected, infected cells and HIVvirus is given by

T = λ︸︷︷︸T-cell

Production

− dT T︸︷︷︸T-cellDeath

− βf T Vf︸ ︷︷ ︸Free VirusInfection

(1a)

If = βf T Vf︸ ︷︷ ︸Free VirusInfection

− dI If︸ ︷︷ ︸Infected Cell

Death

(1b)

Vf = k If︸︷︷︸Free VirusProduction

− dV Vf︸ ︷︷ ︸Free Virus

Death

(1c)

Here T (t), If (t) and Vf (t) represent uninfected, infectedcells and virus population at time t, respectively. The rateof production of uninfected cells is represented by λ. Deathrate of uninfected, infected cells and virus are dT , dI and

dV . k represents the number of free virus particles producedper infected cell per time unit. The infection rate is given byβf . Table I shows experimental values for these parameterscalculated from experimental patient data in [23].

If we assume that virus copies decay at a rate much higherthan the infected cells’ death rate (dV >> dI ), then the viruspopulation can be assumed to always be in a quasi-steadystate, with Vf = k

dVIf . Thus Equation (1) reduces to

T = λ− dT T − k

dVβf T If (2a)

If =k

dVβf T If − dI If (2b)

Steady state analysis of Equation (2) reveals the conditionsfor a successful establishment of infection. Equation (2) hastwo steady state solutions:

T =λ

dT, If = 0, (3)

which means there is no infection in the host, and

T =dT dVβfk

, If =βfkλ− dIdT dV

dIβfk, (4)

which corresponds to a successful virus infection. Deter-mining if local stability take place at point represented byEquation (3) is equivalent to determining if infection can takeplace by means of the free virus transmission mechanism.The eigenvalues at Equation (3) are −dT which is alwaysnegative, and −dI + λkβf

dV dTwhich can be either positive or

negative. Let

R0f =λkβfdV dIdT

(5)

be the basic reproductive ratio of HIV infection by meansof the free virus pathway. If R0 > 1 then Equation (3) isunstable and the infection will take place.

III. MODELING SYNAPTIC VIRUS

Equation (1) describes HIV infection by the free viruspathway. However that is not the only method of HIVtransmission. Infection may also happen by direct interactionbetween cells, as described previously.

We modify Equation (1) to model synaptic transmission.Let Vs (t) and Is (t) be populations of HIV virus capableof forming synapses and cells infected with this synapticvirus at time t, respectively. Also let s be the multiplicityof infection, which is the average number of viruses sentthrough one synapse. These viruses may cause new infectionsby the free virus pathway at a rate βf T Vs, or by the cell-cell pathway at a rate p (s)βsTIs. Here βs is the rate ofinteraction between infected and uninfected cells, and p (s)is the probability that an uninfected cell will be infected byreceiving s virus particles through synapses, given there isinteraction between infected and uninfected cells. Probabilityp (s) is defined as

p (s) = f (s) σ (s) , (6)

where σ (s) is the probability that uninfected and infectedcell form synapses given there is interaction. f (s) is the

(a) (b) (c)

Fig. 1. Synaptic virus mechanism. A synaptic virus has the capability to infect cells by means of free pathway (a) and also through synapse formation(b). In the free pathway (a), infected cells produce RNA strings (red lines) using virus information stored in its genome (blue and red line), encapsulatesthem (blue and red concentric circles) and send this capsids outside the cell. Uninfected cells absorb them releasing RNA virus strings (opened blue circle)which integrates with cell’s DNA (blue line). Synaptic interactions may occur between infected and uninfected (b) or infected-infected cells (c). The viruscopies in (b) sent through synapses are not used in the infection of other cells.

probability that sending s viruses through given synapsesleads to an infection; this can be any monotonically increas-ing function on s. If we treat each virus transmitted by thesynaptic pathway to have equal probability of successfulinfection r, then the probability that s viruses transmittedwill result in at least one successful infection of the cellfollows the equation:

f (s) = (1− (1− r)s) , (7)

Synapses can form both between an infected and anuninfected cell and between two infected cells. The formerleads to an infection with probability p (s). This results ina loss of s σ (s)βsT Is virus copies that cannot be used infurther infections. The other scenario arises because there isno discrimination mechanism that leads infected cells to formsynapses with uninfected cells only, thus infected-infectedinteractions also should occur. Infected-infected synapseslead to a waste of s σ (s)βsI2s virus copies that does notproduce any additional infection, because both cells arealready infected. Figure 1 shows all three possible synapticvirus pathways: free virus transmission, infected-uninfectedand infected-infected virus transmission through synapses.

Using the synaptic mechanism illustrated above and in-cluding it in Equation (1) leads to

T = λ− dT T − βf T Vs − p (s)βs T Is︸ ︷︷ ︸SynapticInfection

(8a)

Is = βf T Vs − dI Is + p (s)βs T Is︸ ︷︷ ︸SynapticInfection

(8b)

Vs = k Is − s σ (s)βs (T + Is) Is︸ ︷︷ ︸Reduction of

Virus Production

−dV Vs. (8c)

By again assuming that dV >> dI and im-posing the quasi steady-state approximation Vs =

0.00 0.05 0.10 0.15 0.20 0.25 0.30r11 000

11 500

12 000

12 500

13 000

13 500

Virus Copies

Fig. 2. Steady state behavior of virus load with different fractions ofsynaptic size s

B, with probability r of successful infection. Increasing virus

load when sB< 0.1 is observed. Fast growth is observed for r close to 1.

Burst size used on this simulations is B = 2× 103. Probability of formingsynapses used is σ (s) = 1

(1− s

k σ (s)βs (T + Is))kdVIs), Equation (8) reduces to

T = λ− dT T (9a)

−(1− s

kσ (s)βs (T + Is)

) k

dVβfT Is

− p (s)βs T Is

Is =(1− s

kσ (s)βs (T + Is)

) k

dVβfT Is − dI Is (9b)

+ p (s)βs T Is,

which has two stationary points, one of them being theuninfected state

T =λ

dT, Is = 0. (10)

Infection will occur (this point is unstable) if

R0s =

(1− s

kσ(s)βs

λ

dT

)R0f + p (s)βs

λ

dIdT> 1. (11)

The other stability point is not difficult to calculate, howeveris not included here due space limits.

Figure 2 illustrates the steady-state virus load as a functionof the infection success probability r for the condition whenthe number of viruses sent through a synapse is a smallfraction of the total virus produced by the infected cell. HereB = k

dIrepresents the burst size of the cell, which is the

number of virus particles that an infected cell produces overits lifespan. s

B is the synaptic size as a fraction of this burstsize. For small values ( sB < 0.1) steady state virus loadincreases with increasing s

B , for a maximum increase ofmore than 20%. Increasing the probability of infection ofa single virus r increases speed of infection growth. Figure3, however, shows that for values of s

B larger than 0.1, viralload decreases with increased s

B , eventually dropping belowthe level of a virus that does not form synapses as s is closeto burst size. The parameter values used in this simulationare shown in Table I.

0.02 0.04 0.06 0.08 0.10

s

B

11 000

11 500

12 000

12 500

13 000

13 500

Virus Copies

Fig. 3. Steady state behavior of virus load for different synaptic sizefractions s

B, with probability r = 0.1 of successful infection. Steady-state

virus load decreases with sB

when sB> 0.1, and drops below the non

synapse-forming equilibrium for sB

∼ 1. Burst size used on this simulationsis B = 2× 103. Probability of forming synapses is σ (s) = 1

IV. COMPETITION MODEL

In order to determine if a synapse-forming virus could suc-cessfully invade against an established non synapse-formingvirus, we introduce a mathematical model of competitionbetween synaptic and non-synaptic virus. Merging (1) and(8) yields the form:

T = λ− dT T − βf T (Vs + Vf )− p (s)βs T Is (12a)

If = βf T Vf − dI If (12b)

Vf = k If − dV Vf (12c)

Is = βf T Vs − dI Is + p (s)βs T Is (12d)

Vs = k Is − s σ (s)βs (T + Is + If ) Is − dV Vs. (12e)

Note that in Equation (12a) βf T (Vs + Vf ) represents thetotal infection rate by the free virus transmission for both

non-synaptic and synaptic virus. In this model, cells infectedby synaptic virus (Is) are able to establish synaptic interac-tion with uninfected (T ), cells infected by free (If ) and cellsinfected by synaptic virus (Is).

In order to study stability of the competition model weassume dV >> dI , thus Equation (12) reduces to

T = λ− dT T (13a)

− k

dVβf T

((1− s

kσ (s)βs (T + Is + If )

)Is + If

)− p (s)βs T Is

If =k

dVβf T If − dI If (13b)

Is =(1− s

kσ (s)βs (T + Is + If )

) k

dVβfTIs − dI Is

+ p (s)βs T Is.

To determine whether a synapse-forming virus would suc-cessfully invade against an established non synapse-formingvirus, we evaluate the input-to-state stability of the steadystate where no synapse-forming virus is present, adding εcopies of synapse-forming virus. Assuming we start at thefree virus equilibrium (Equation (4)), the stability analysisof Equation (13) reduces to determining whether

∂Is∂Is

=βsσ (s)

βfk2(dV dIkf (s)− dV d

2Is

+ dV dIdT s− 2k sIsβf )

− βsσ (s)

kλs (14)

is positive (unstable) for Is = 0. This is satisfied if

s

B<

f (s)dTdIR0f − dT

dI+ 1

. (15)

Figure 4 shows a bifurcation diagram for the success ofsynapse-forming virus invasion against the values of s

B andR0f . It is worth pointing out that the bifurcation describedby Equation (15) does not depend on the reproductive ratioof synaptic virus R0s; any value R0s > 1 is sufficient for thesynapse-forming virus to invade and drive the non synapse-forming virus extinct, so long as s

B is below the upper limitimposed by the value of R0f . Figure 5 shows competitionsimulations between synaptic and non-synaptic virus.

V. CONCLUSIONS

In this paper, we have introduced a novel model of HIVdynamics accounting for both the free virus and cell-cellmechanisms of viral transmission. We explicitly consider theincrease in likelihood of infection due to multiple cell entryevents, the reduced production of free virus by cells formingsynapses, and the loss of virus due to synapse formationbetween infected cells. We derived the stationary points ofthese models, and evaluated local stability under a variety ofparametric conditions.

When realistic parameter values identified from clinicaldata were used, our model showed that steady-state viralload of a synapse forming virus increased monotonically with

Parameter Value Units Biological meaning Parameter Value Units Biological meaningλ 7× 102 cells

µL×day T cell birth rate k 2× 103 copiescell×day Production rate of free virus per cell

dT 0.1 1day

T cell death rate βf 2× 10−6 mLcopies×day Rate of uninfected-virus infection

dI 1 1day

Infected cell death rate βs 10−5 µLcells×day Rate of infected-uninfected interaction

dV 23 1day

Virus decay rate

TABLE IPARAMETER VALUES FOR SIMULATIONS ON THIS PAPER. ALL PARAMETERS VALUES EXCEPT βs WERE TAKEN FROM [23]. IT IS ASSUMED THAT THE

RATIO OF INFECTIONS BY THE FREE VIRUS PATHWAY TO INFECTIONS BY SYNAPTIC PATHWAY IS 20.

Free Virus

outperforms

Synaptic Virus

outperforms

1 2 3 4 5 6 7 8 9 10R0 f

0.4

0.5

0.6

0.7

0.8

0.9

1.0

s

B

Fig. 4. Bifurcation diagram showing regions where synaptic virus outper-forms free virus. The larger basic reproductive ratio of the free virus R0f

is, the smaller the region where the synaptic virus outperforms gets.

the probability of successful infection r. Steady-state viralload also increased with the number of viruses transmittedper synapse s until this reached a threshold measured asa fraction of total burst size B of approximated 1-2% forthe parameters used in this study. This is consistent withthe results reported in [21], and reflects the fact that oncethe increased probability of successful infection begins tosaturate, additional viruses transmitted via the cell-cell path-way are “wasted”, in that they reduce the number of virusestransmitted by the free virus pathway without significantlyincreasing the probability of success of the cell-cell pathway.However, by using realistic parameter values, our modelmakes it clear that 1-2% of the burst size is approximately20-40 viruses, which is on the same order of magnitude asthe observed transmission numbers for synapses in vitro [18].Figure 3 illustrates the fact that a much higher penalty ispaid (measured in viral load at equilibrium) for values of s

Bthat are smaller than optimal rather than for values largerthan optimal. It is feasible that the synaptic multiplicity ofinfection s has evolved to a value larger than the optimumdue to decreased sensitivity of the viral fitness around thislevel.

When the synapse-forming HIV variant was considered incompetition with a non synapse-forming variant, we wereable to derive conditions for the invasion of a synapse-forming HIV variant against an established non synapse-forming variant. This condition was best expressed as an

upper-bound on the number of viruses transmitted via asynapse as a fraction of total burst size s

B . The upperbound was most sensitive to the fitness of the non synapse-forming virus R0f , and was as high as 90% for R0f ∼ 2,which is consistent with values measured during chronicinfection [23], [27], to as low as 50% for R0f ∼ 10, whichis consistent with values measured during acute infection[28]. It is important to remember that these are not theoptimal muliplicities of infection; as discussed previously,the optimal multiplicity of infection as a fraction of burstsize is on the order of 1%.

We have developed a model that shows an evolutionary ad-vantage for synapse forming virus, and which predicts mul-tiplicities of infection for the cell-cell transmission pathwaythat are consistent with those observed in experiment. Thismodel will serve as the basis for future work investigating theimpact of cell-cell transmission on viral persistence duringsuppressive therapy.

Future Work

The results obtained from the model developed in thiswork suggest several avenues of future research. We havetreated multiplicity of infection as multiple independenttrials in this work; future work will consider more generalformulations for the function f(s), including synergistic andantagonistic behavior between the multiple virions invadingthe cell. We have considered all cells to have the sameviral production rate and the same death rate regardless ofmultiplicity of infection. The authors of [21] suggest thatvirus production rate may scale with multiplicity of infection;future work will include this possibility as well. Viral fitnessis known to decrease dramatically from acute to chronicphase infection, and our model shows that this significantlychanges the optimal multiplicity of infection. Furthermore,the decrease in viral fitness is almost certainly due to anincreased immune response [28], and several authors havesuggested that cell-cell transmission evades certain immunesystem mechanisms [15]. The implications of this for theevolution of synapse formation rates over the course ofinfection will be investigated in future versions of the model.This may be related to the emergence of syncytium-inducingvariants of the virus, which has been associated with diseaseprogression in the final stages of immune collapse into AIDS[29].

300 400 500 600 700Days0

5000

10 000

15 000

20 000

25 000

30 000

Virus Copies

(a) R0f = 2, sB< 0.91

1000 1200 1400 1600Days0

10 000

20 000

30 000

40 000

Virus Copies

(b) R0f = 5, sB< 0.71

2000 2500 3000 3500Days0

10 000

20 000

30 000

40 000

50 000

Virus Copies

(c) R0f = 10, sB< 0.52

Fig. 5. Population of free virus and infected cells with free virus are begin in steady state (Equation (4)) for all three cases. There is no infected cellswith synaptic virus at time 0. One particle of synaptic virus is present at t = 0. The fraction of synaptic size is s/B = 0.2. In all three cases synapticvirus outperforms free virus regardless increasing value of R0f .

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