MEDICINE Acta Científica Venezolana, 55: 247-263, 2004 AN EPITOPE VARIABLE-RESERVOIR MODEL FOR HIV-1 INFECTION 1 2 Horacio Ortega and Miguel Martín-Landrove 1 Facultad de Medicina, Universidad Central de Venezuela and Cátedra de Física, UNEXPO Antonio José de Sucre, VR La Yaguara, Caracas. A.P. 47636 Los Chaguaramos, Caracas, 1041-A, Venezuela. 2 Departamento de Física, Centro de Resonancia Magnética, Facultad de Ciencias, Universidad Central de Venezuela, A.P. 47586, Caracas, 1041-A, Venezuela and Instituto de Resonancia Magnética, La Florida/San Román, Caracas, Venezuela. e-mail: firstname.lastname@example.org Recibido: 26/11/02; Revisado: 29/05/03; Aceptado: 13/07/04 ABSTRACT: A model of HIV-1 infection based on immunodominance concepts is presented. The model considers not only mutating virions and T-CD4 cells, but also viral reservoirs, such as macrophages and follicular dendritic cells. It also considers strong cytotoxic attack against reservoirs and extra cellular attack on virions, which are both coordinated by T-CD4 cells. As a first case, only one viral variant is dealt with, and approximations were used to obtain a manageable model. A stability criterion was found that marks the transition between progression and regression of the viral infection. This criterion was proven valid for the model without any approximation, i.e., two randomly mutant viral epitopes with two variants each. Data suggest that a) the role played by the reservoirs in the maintenance of the viral infection is a very important one, b) maintaining CTL attack on infected cells facilitates the control of HIV-1 infection, and c) therapy considerations should be addressed to these points. Keywords: reservoirs, virions, immune response, mutability, viral escape, HIV UN MODELO DE EPITOPES VARIABLES PARA LA INFECCIÓN DE RESERVORIOS POR VIH-1 RESUMEN: Se presenta un modelo de infección por VIH-1 basado en el concepto de inmunodominancia. Además de viriones mutantes y células T-CD4, se considera también reservorios virales, entre ellos macrófagos y células dendríticas foliculares. También se considera ataque citotóxico contra los reservorios y ataque extracelular sobre los viriones, ambos coordinados por las células T-CD4. En primer lugar, se trabaja sólo con una variable viral, y se usan ciertas aproximaciones para obtener un modelo fácilmente manipulable, para el cual se encuentra un criterio de estabilidad que rige el paso de progresión a regresión de la infección. Este criterio mantiene su validez para el modelo sin uso de aproximaciones, esto es, dos epítopes que mutan al azar, cada uno de ellos con dos variantes. Los resultados sugieren: a) que el papel jugado por los reservorios en el mantenimiento de la infección es muy importante, b) que mantener el ataque de las células citotóxicas sobre los reservorios infectados facilita la labor del sistema inmune, y c) que la terapia debería orientarse preferiblemente hacia estos objetivos. Palabras clave: Reservorios, viriones, respuesta inmune, mutabilidad, escape viral, VIH INTRODUCTION persist in a non-functional state45. Recent reports state that: a) CTL attack to viral sources causes a radical decrease in viral charge in the primary stage of In 1995 Nowak et al.23 introduced a model of HIV-1 infection25,26, b) activated CTL cells are short lived, their infection based on the concept of immunodominance in presence in the blood stream needs continuous order to explain variations of the human immune stimulation9,17,38, c) CTL cells are involved both in response against virus which continually produce mutant cytopathic and non cytopathic control of virions5,19,44, d) + epitopes. HIV-1 infects T-CD4 cells causing damage genetic manipulation and induction of apoptotic HIV-1 and shortening of their life cycle. Other cells as T-CD8 , + infected cells results in durable CTL responses14,35. A macrophages, dendritic cells are also a target for the summary of CTL role in control of HIV-1 infection can be virus. Several mechanisms have been proposed to justify found in42. A controversy currently is posed on the + existence of humoral response and its effectiveness30,41, the reduction of the T-CD4 cell life cycle due to the viral but this response exists and it is susceptible of being infection, among them cytotoxicity, viral cytopathogenicity enhanced14,22,36. A long list of candidates for the viral and apoptosis. Cytotoxicity occurs due to the attack replication site exists. Virions are produced in cellular made by CTL to infected cells, the clonal and activation compartments in lymphoid tissue from the beginning of processes of immunocompetent cells being mediated by + the infection8,29. In advanced stages of AIDS, the viral T-CD4 cells16. CTL can occasionally be present in high + load is very high even when the T-CD4 cell population, numbers, yet they are not associated to the control of which are the main accepted source of free virus, is viral replication. If there is T-CD4 deficiency, CTL can depressed20,27. Moreover, there is good agreement 248 Ortega and Martín-Landrove between the number of cells infected by virus present in several epitopes, or active regions, but the immune lymphoid tissue and the quantity of free virus present in system normally recognizes just one or two of them23. blood11. Then, it is reasonable to assume the existence of We suppose that virions exhibit only two epitopes, α additional viral sources such as the follicular dendritic with N variants 1, 2,....i,.....N , and β , with M cells (FDPC) and peripheral blood mononuclear cells (PBMC), B-cells among these21. Opportunistic variants 1, 2,..... j,....M . A particular virion will be 27 infections , and immune stimulation, for instance by designated by its density, vij , this is, we write vij for vaccination40, seem to highlight the role of macrophages as: a) site of attack by HIV-1 during the incubation period vαi β j . Any cell set which is susceptible to be invaded by of AIDS8, b) source of huge quantities of free virions, at a the virus, and then to produce viral particles will be called + time when T-CD4 density is minimal, and c) not a reservoir. An infected reservoir can exhibit viral succumbing producers of both intracellular and extra residues on its surface, and when they are recognized as cellular HIV-1, although the detailed mechanisms that alien by the immune system, specific cells deliver a lethal govern this behavior are not clearly understood. hit against them. Densities for healthy reservoirs and for Therefore, the existence of a cellular population with a infected ones are denoted by m0 , and mij , respectively long lifetime, which provides virions continuously to the + blood stream, seems to be stated beyond any doubt. The (we also write mij for mαiβj ). T-CD4 cells acting mainly use of viral inhibitor drugs causes a decrease of viral as coordinators of the immune response are called z . burden among others places, in lymphoid tissu2; this Infected cells elicit a response of CTL cells addressed therapy has lead to a decrease in deceases related to against them. Cytotoxic cells directed against infected AIDS. However, some troubles persist, because even with the use of HAART (highly active antiretroviral cells bearing variant i of epitope α are denoted by xi , therapy) some subsets of cells go on producing while those directed against cells bearing variant j of virions3,34. Also, it has been reported that after epitope β , by y j . Each equation for the present system suppression of drug control in voluntaries, viral charge will be separately discussed. We begin by introducing the and viral infection come back4, suggesting the presence following variables: of sanctuaries where virions are kept safe. This fact suggests that HAART treatment is a lifetime deal, which produces no definitive cure. mi∗ = ∑ mij m∗ j = ∑ mij In the present paper we intend to describe the role of j i the reservoirs as a continuous source of HIV-1 infective particles, and the effect of both cellular and humoral (uv )∗ = ∑ uij vij ( γij vij )∗ = ∑ γij vij , immunity on the progression from HIV-1 infection to full i, j i, j AIDS. The main goal of our work is to find under what conditions the immune system controls the viral The equation that states the dynamics of healthy population in the proposed system, and how these reservoir cell density is: conditions are modified by the presence of several epitopes and mutations. To perform the stability analysis, a simplified version of the original system of equations is dm0 = λ 0 − ( γ ij v ij ) * m 0 − ω0m 0 (1) studied by standard means (linearization and evaluation dt of eigenvalues by the Routh-Hurwitz procedure), and later on the obtained stability criteria are used to verify Reservoirs are produced in the hematopoietic organs at numerically the behavior of the system without any a rate λ0 . These reservoirs migrate to the blood stream approximation, that is, situations where two epitopes can be found, each one of them with two strains that can (or to their final places), and they take virions, at a rate mutate in a continuous way. For that case analytical γij m0 vij , as a result of a contact interaction. γij methods fail, but numerical integration of the system of represents the probability for a healthy reservoir of being equations shows that the stability criteria obtained in the invaded once it contacts a virion. Reservoirs die after a simplified case are still valid. Finally we point out that our −1 model can be extended easily to situations that deal with mean time t0 = ( ω0 ) . antiviral drugs7,24. The equation for the infected reservoir cell density is: THE MODEL d m ij dt = γ ij v ij m 0 − m ij ( px i + q y j ) − ω 1 m ij In what follows we will use the notation utilized in23. HIV-1 is a highly mutant virus; its mutation capability is (2) related to its reverse transcriptase replying-based process. We consider the existence of free virions, Infected reservoirs are produced as a result of the interaction of healthy reservoirs with viral particles, at a denoted by vαβ . An antigen can simultaneously present Model for HIV-1 infection 249 rate γij m0 vij , and as they display viral epitopes, they are Virions are liberated by infected reservoirs, at a rate gij mij , and by infected T-CD4 + cells, at a rate ξuij zvij . attacked by cytotoxic cells, with density xi ( y j ), The previous hypothesis is made considering that the addressed against epitope αi ( β j ), at a rate pxi mij + lifetime of infected T-CD4 cells is much shorter than ( qy j mij ). p and q are the dynamical coefficients ruling that of healthy ones13,31, and therefore at any moment their average concentration is at quasi-static equilibrium these interactions. This also means that CTL cells are with the present density of healthy cells (a formal specific, those addressed against epitope αi cannot deduction is shown in Appendix A). Virions are taken exert any effect on epitope β j and vice versa. Infected from blood stream by reservoirs, at a rate γij m0 vij , and reservoirs are supposed to die after a mean time they are destroyed by an extra cellular response −1 + t1 = (ω1 ) , which is not affected essentially by the mediated by T-CD4 cells44. Such response includes interaction with antibodies from B-cells22, targeting of the infection27,40. 120 gp viral envelope12, or targeting of its fragments15. The equations that describe the dynamics of cytotoxic We describe this interaction by a function cell densities are: f ( B, xi , y j , z )vij , where B is the concentration of d xi plasmatic cells. For simplicity we have assumed = η ci z m i* + ci z m i* xi − ω 2 xi (3) f ( B, xi , y j , z ) = fij z , with fij a dynamic constant of dt adjustment. Other apparently simpler alternatives lead to dy an extra equation for B. In our model we do not consider j = ηk j zm ∗ j + k j zm ∗ j y j − ω2y j (4) + any direct effect of T-CD4 cells over virions. dt Equations (1) to (6) represent the complete system. We discuss its stability by means of linearization and Cytotoxic cells are produced by activation from a pool evaluation of the resultant eigenvalues. Linearization of precursor cells, at a rate ηci zmi• ( ηk j zm• j ), or by methods lead to equations with no analytical solution, proliferation from cells previously activated, at a rate and due to this fact two very simple situations with immunological relevance are considered first, in order to ci zmi • xi ( k j zm• j y j ), both processes are regulated by obtain stability criteria which later on will be compared the presence of ( z ) mediator cells, η is the ratio of the with the numerical solution without any approximations. activation rate to proliferation rate, and ci ( k j ) is the We call these simple situations “cellular approximation” and “humoral approximation”. dynamical coefficient ruling the activation23. Note that CTL cells can recognize antigens on the surface of infected, antigen presenting cells (APC), but to develop any cytopathic activity, they must also receive an CELLULAR APPROXIMATION, NO MUTATIONS + additional signal from T-CD4 cells. Activated cytotoxic −1 In this approximation we consider that CTL cells exert cells eventually die after a mean time t2 = ( ω2 ) . the main control on infected reservoirs, and that there The equation for the mediator T cell density is: exists no extra cellular attack on virions. We also consider the existence of just one viral variant with dz epitopes α and β , vαβ . We suppose a moderate viral = λz − (u v )* z − ω z z (5) dt invasion on reservoirs, γαβ vαβ → 0 in equation (1), and Mediator T cells are produced in the thymus at a rate on T-cells, uαβ vαβ →0 in equation (5), and we neglect the λz , they are infected by any viral variant at a rate • humoral attack on virions, ( ξu αβ − f αβ ) zvαβ → 0 in (uv ) z , where uij is a dynamic interaction coefficient. equation (6). We obtain as null equilibrium point Eventually they die after a mean time of t z = ( ωz ) . −1 (absence of virions and infected reservoirs); see the appendix B for details: + Note that T-CD4 cells do not fight the virus directly, but instead they act as mediators and activators of other λ0 λ cells. m0 = , z = z , vαβ = mαβ = xα = y β = 0 (7) The equation for the viral particle density is: ω0 ωz d v ij = g ij m ij + ( ξ u ij − f ij ) z v ij − γ ij m 0 v ij (6) stable if: dt 250 Ortega and Martín-Landrove g αβ < 1. (8) λz λ g αβ γαβ (11) hM ≡ ( f αβ − ξu αβ ) > 0 ω 1 ωz ω 0 ω1 Condition (8) is easily fulfilled if the production of virions and the non null equilibrium point is always unstable. We in reservoirs is minute (then g αβ = 0 ), and this inequality interpret inequality (11) by saying that the null equilibrium means that the null point is stable in this approximation if point is stable if the fraction of virions destroyed by an infected reservoir creates less than one virion during processes mediated by T-helper cells is higher than the its lifetime, an interpretation already mentioned by some fraction of virions produced by infected reservoirs in the authors although they use a very different model39. same interval (see appendix D for details). Alternative interpretations (or no interpretation at all) of inequality (11) are also possible. We call hM “humoral HUMORAL APPROXIMATION, NO MUTATIONS λ0 g α β γ α β responsiveness”, and the factor as “viral Alternatively, it is possible to consider just the attack ω 0 ω1 performed on virions by T-CD4+ humoral mediated growth”. If condition (11) is not fulfilled, then the viral response. In this case, we neglect the decline of healthy particle density increases drastically, as shown in reservoirs due to viral invasion and also the clearing of Figure 1. Linearization criterion fails both if virions from blood stream for invasion to reservoirs, γαβ = hM = 0 , and/or g αβ = hM = 0 (see below). γ αβm0vαβ → 0 in equations (1) and (6), although this term We now consider several particular cases of interest: is conserved in equation (2) due to that a) γαβ m0vαβ g αβ → 0 and/or γαβ → 0 , but hM ≠ 0 (reservoirs do not represents the source of infected reservoirs in such send virions to the blood, and/or there is no viral invasion equation, and b) ω 1 m ij is small and comparable to to them), then inequality (11) predicts that whenever a population of mediator cells exists, no matter how small it γ α β m 0 v α β (in fact t1 ≡ lifetime of infected reservoirs is is, there will be control of the viral population. If uαβ = 0 −1 rather long, therefore ω1 ≈ ( t1 ) is small). We also (there is not viral invasion to T cells, but virions continue neglect the cytotoxic attack to infected reservoirs, entering and reproducing into reservoirs), the null ( ) mαβ pxα + qy β → 0 in (2), and therefore equations λz λ g γ population is stable if > 0 αβ αβ , i.e., it is (3) and (4) become uncoupled from the system. As a ωz ω0 ω1 f αβ consequence, the following equilibrium points are obtained (see appendix C for details): necessary that the equilibrium concentration of T-helper cells without infection is bigger than the medium rate of λ0 λz virions production, which is a more restrictive condition m0 = , z= , vαβ = mαβ = 0 (9) than the previous one. If h M = 0 and γαβ ≠ 0 (there is ω0 ωz no control on the viral population contribution coming which is called the null equilibrium point, due to the from T cells, although reservoirs continue producing absence of infection. There exists also a non null virus), then any viral population is unstable. This situation equilibrium point given by: seems to be the current status of the therapy. If hM = 0 and γ αβ = 0, perturbation procedures show that the equilibrium point is given by (9), stable Lyapunov (this λ λ0 γαβ g αβ implies that minute virion densities can remain in the m0 = 0 , z = , ω0 ω0 ω1 ( f αβ − ξu αβ ) dvαβ system). Finally if hM = 0 and g αβ = 0 , then = 0 . In (10) dt ω λz ω0 ω1 ( f αβ − ξu αβ ) vαβ = z − 1 , this case, analytic approaches cannot provide an answer, u ω z λ0 g αβ γαβ but numerical integrations show that there is a finite λ0 λ0 γαβ equilibrium population given by m0 = , vαβ = v0 , mαβ = vαβ ω0 ω1 γ αβ λz m αβ = v αβ , and z = , where v0 is ω 0 ω z + u αβ v 0 Linearization procedure shows that the null equilibrium the initial viral population. There exist reports of situations point is stable if: of viral infection not progressing to full AIDS33. Model for HIV-1 infection 251 Figure 1 shows the equilibrium populations for an occurs is compatible with inequality (11). Figure 2 shows system lacking of CTL cells and possessing just one viral the temporal evolution for the same system under variable, v11 , as a function of hM , the humoral conditions that lead to viral outgrowth (weak immune response). Notice that healthy reservoir density is responsiveness. There is only viral persistence when the constant, that mediator T cells decrease, while viral load humoral response is weak. It is clear that the point at and infected reservoirs densities increase continuously. which the transition between viral growth and viral control λ0γ 11 g11 ω0ω1 Figure 1. Computer generated plot of equilibrium values of variables in our system in absence of cytolytic cells as a function of λz hM = ( f11 − ξu11 ) , humoral responsiveness. We consider the existence of just one epitope with one viral variable, which we call v11 . ωz So we use in this and in the following figures, α→1, and β → 1 . Values of parameters used in the calculations are: λ0 = 0.25 , λ z = 4 . 00 , ω 2 = 140 , γ11 = 0 .35 , g11 = 0.65 , ω z = 7 . 0 0 ω0 = ω1 = 0 .28 , u11 = ξ = 1 . 80 , and we vary f 11 between 0 and 10. With this values the threshold for convergence, λ 0 g αβ γ αβ takes the value 0.73. Viral and infected reservoirs densities ω 0 ω1 remain low only while immune response hM is higher than the threshold. The step viral saturation → viral control occurs as predicted by equation (11). For plotting, we used the 10000 for the number of steps, and we also truncated the integrator output for v11 and m11 . in the divergent region. 252 Ortega and Martín-Landrove Figure 2. Temporal evolution of a system with weak immune response. Parameters used in the calculation λ0 were, f 1 1 = 3 . 50 , ξ = 2 . 5 0 , u11 = 3 .50 and all others as in Figure 1. With these values, γ11 g11 ≅ 0 .73 and hM ≅ − 3 .00 . ω0 ω1 For this choice of parameters there exists just the null equilibrium point (unstable). As viral creation rate is higher than virions destruction rate, viral and infected reservoirs densities grow, and slow T-helper cells decay occur. Model for HIV-1 infection 253 THE FULL SYSTEM, NO MUTATIONS two epitopes with two viral variants each one, b) there are random mutations in any of the viral variants currently present, c) each time a mutation takes place and a new When considering the complete system, that is to say, viral variant of an epitope appears, a fraction α of the with the presence of cytotoxic cells and without any mutated viral variant is set as initial condition for that approximation, it is found that there exists a null variant, and a fraction (1- α ) of the not mutated viral equilibrium population, given by equation (9), and linearization procedures similar to those performed in variant is set as initial condition for that epitope, appendices B and C show that this point is stable if : additional equations for the corresponding CTL and infected reservoirs densities are added, and the remaining of the system does not vary and d) the λz λ g ( f α β − ξu α β ) > 0 γ α β ( α β − 1 ) (12) mutation rate depends on γij , the viral infectivity to ωz ω0 ω1 reservoirs (our emphasis is on the role of reservoirs in AIDS). An algorithm based on the Monte Carlo method We observe that a) inequality (12) is a combination of was coupled to our numerical integrator for evaluating inequalities (8) and (11) and b) it is easier for the immune explicitly the effects of the virus mutability. We introduced system to fulfill condition (12) than condition (11), due to a temporal evolution for each viral variant, ruled by a the existence of the negative term λ0 γαβ driving the ( ) quantity R= exp − sγ ij n∆t , where s is an adjustment ω0 parameter which we call mutability, and which we could system to the stable region. Figure 3 shows a numerical arbitrarily vary, n is the number of elapsed time steps integration for this system. Note that the transition and ∆t , the time step interval. We compared R with Γ , instability → stability occurs at values of hM compatibles the output of a random number generator between 0 and with condition (12). Also, by comparing Figure 1 and 1, and every time that R < Γ a mutation takes place. Figure 3, we can observe that the full system (with CTL Notice that for short elapsed times, R ≈ 1 and mutations cells) eradicates the infection at smaller values of hM are unlikely, while for a sufficiently long elapsed time than a system without CTL cells, (clearing of the infection R → 0 and some of the possible mutations take place. In occurs for lesser values of the immune response when such a case, n is reset to zero for the particular variant. Results obtained by this method show that a) the viral cytolytic cells are present and moreover γαβ and gαβ outgrowth is a consequence of the mutability, because can take higher values than in absence of them without for values of the parameters that would lead to viral diverging). removal (any number of viral variants, without mutations) Treatment of the general equilibrium point, with non- divergence occurs (with identical parameters and initial null densities both of virions, vαβ and of infected conditions), as shown in Figure 4, a plot of total viral reservoirs, mαβ leads to eigenvalue equations density, ∑v ij ij vs. time for some values of mutability, s. analytically unmanageable, so that we limit ourselves in b) It is sufficient that the immune system loses control the remainder of this paper to a numerical treatment. In upon a certain viral variant (through its respective hM ) to any instance our simulations show that have viral outgrowth. c) For low values of mutability and λ the range the steps computed (up to 105), we found hM = z ( fαβ − ξuαβ ) acts as a very important parameter situations of non divergent dynamics. d) For moderate ωz values of s, there can be situations in which there is a that controls the behavior of the system. Our simulations long persistence in time of low or moderate values of the (some of them running up to 106 steps) also suggest the viral variables before the collapsing or the clearing of possibility of non-progression to full AIDS for the infected them. e) For moderate and high mutability the system organism, a fact well documented33. We stress that our becomes sensible to initial conditions, the higher the system’s stability is related not only to the immune mutability, the lower initial concentrations for viral or control of viral output from T-CD4 cells, but also to the infected reservoirs resulting in system collapse. control of virions produced in reservoirs. This fact states Nowak et al.23 classified the different behavior of a departure from currently accepted evolution of HIV-1. patients with conserved epitopes (always recognized by immune system), as non-progressors, and others in which immune response failed to recognize some viral epitope (or shifted to another epitope), as fast THE CONTINUALLY MUTANT SYSTEM progressors. This fact was confirmed by our model (see Figures 5-a and 5-b). HIV-1 is a highly mutant virus; its mutation capability is Figure 5-a shows a case with fixed epitope α , and four related to its reverse transcriptase replying-based mutant variants in epitope β ( v11 , v12 , v13 , v14 ). This process. In order to simulate the mutant system the infection is easily controlled by the immune system. following assumptions were taken: a) the system exhibit Figure 5-b shows a case with two continually mutant 254 Ortega and Martín-Landrove epitopes, each one with two variants ( v11 , v12 , v21 , v22 ). based in the lacking of CTL attack on them. Virions with conserved epitopes facilitate the immune system task Now the immune system fails to control the infection and (see Figure 5-a). Mutability also results in a critical viral outgrowth occurs. dependence on the initial concentrations of the temporal behavior of the system. Minute concentrations of virions do not imply compulsive progression to full AIDS, and DISCUSSION numerical work suggests that it is possible to find situations of otherwise healthy HIV-1 infected subjects which remain in stable immune condition for long time. Inequalities (11) and (12) suggest that blocking the viral Such situation could explain the report on non invasion to reservoirs ( γ α β → 0 ), or the capability of transmission of AIDS when the concentration of virions in reservoirs of producing free virus ( g αβ → 0 ) is the blood stream is low33. Opportunistic attack on the organism often shadows easiest way of controlling viral growth. In such cases any the actual processes occurring during viral infection. It is non-null population of T-CD4+ cells is enough for the generally expected that stable values for immune fulfillment of this goal. The relative shift of the variables should lead to clearing opportunistic agents, but convergence points given by equations (11), (no numerical work shows that there exists non diverging ω1 points for the system (1) to (6), these values being reservoirs), and (12) (with reservoirs) results to be , gαβ minute for z, T helper cells density, and m0 , healthy this shift being large if there exists absence of viral reservoirs density. If in the otherwise healthy system (HIV-1 infection is maintained under control) opportunistic production by reservoirs, gαβ → 0 , or also if the lifetime infections emerge, this breakdown and further damage of for infected reservoirs is short, ω1 → ∞ , suggesting that immune system being responsible of the posterior collapse of the whole organism. therapeutic or CTL attack on them could result in better expectances for HIV-1 control. Moreover if there is no viral production inside reservoirs, gαβ → 0 , CONCLUSIONS and as it is obtained from equation (12) λz λ ( f αβ − ξu αβ ) > − 0 γαβ , a condition that is fulfilled We have made an approximate analytical approach, ωz ω0 and an exact numerical one of the viral infection. by every non-null equilibrium T-CD4+ cell population. This They show that it is important to impair virus production fact should be considered when checking the progression inside the reservoirs and/or block the infection of these of HIV-1 infected organisms to full AIDS. cells. If these requirements are fulfilled, a relatively Figure 1 and Figure 3 were plotted after numerical moderate immune response is enough to control the viral work. By comparing them, it is possible to confront infection. The control on the T-CD4 cells alone drives the asymptotic viral trends in absence/presence of cellular system to the onset of AIDS. To annihilate the viral attack on reservoirs, such attack resulting in the already infection it is necessary to annihilate the infected mentioned shift of the immune responsiveness threshold reservoirs also. Attack to the infected reservoirs by to lower values. Recent reports state the role of cytotoxic cells shows situations that are easier to handle reservoirs in supporting viral production at late stages of by the immune system, so it seems important to develop HIV-1 infection, suggesting that the immune system fails mechanisms for continually activating CTL. to recognize infected reservoirs. Some mechanisms Viral mutability can lead to the appearance of more responsible for this failure have been reported6,17,37,42. efficient viral variants (higher values of γ ij or g ij ), to the Lifetime of cytotoxic cells43, continuous production of them, and attack CTL-CD4 mediated on infected cells1 shift of the cytotoxic response ( pi and/or q j tend to should somehow be enhanced. We also mention the zero), or simply to viral accumulation and concerted finding of non diverging dynamics (up to 105 computation attack on T cells. In any case, there is viral escape. It steps) with the full system (one viral variable, no seems clear that any future therapy should include the mutations). control of any cell susceptible of being infected by the Mutation results in higher instability of the system, HIV-1, not only of T-helper cells, the only clinical convergence points shifting toward higher values of the measure available at this moment. Also, therapy should immune response. High mutability implies viral escape aim to the enhancement of the performance and lifetime for situations that normally would produce immune of CTL. We recall that many advances have taken place control if mutability, s, was low23, (see Figure 4). in the interpretation of the illness related to the HIV-1 There is viral control only for low values of s. All infection, and it is possible to reflect these changes in the simulations displayed were performed with identical therapy, so that reports of cures soon appear. Until this parameter values, with the exception of the rate of happens, the avoidance of infection seems to be the best mutation, s , so the advantage of new viral variables is prophylaxis against AIDS. Model for HIV-1 infection 255 λ0 g λ0γ 11 g11 γ αβ ( αβ − 1) ω0 ω1 ω0ω1 λz Figure 3. Equilibrium values of variables in our system with the presence of CTL cells, as a function of hM = ( f 11 − ξ u 11 ) . ωz Parameters used in the calculation are p1 = q1 = 2 .05 ,k1 = c1 = 0 .45 ; and η = 0.02 , ω0 = ω1 = 0 .35 , and all others as in Figure 1. Resulting densities of xi , and y j are identical for this particular choice of parameters, and they overlap each other. Now there exists viral saturation for small values of hM , but the convergence point is shifted to lower values as predicted by equation (12), this shift is caused by attack of CTL on virus producing cells. Note the existence of finite non diverging values for viral and infected reservoirs densities. Cytotoxic cells exist just for appreciable density of reservoirs. 256 Ortega and Martín-Landrove Figure 4. Time evolution of total viral density ∑ ij v ij for three instances of the mutability s. For s = 0.01 , low mutability, total virions density fades quickly. For s = 0.1 , moderate mutability, viral density oscillates before slow growth. For s = 1.0 , high mutability, total virions density diverges. After mutations (marked by changes in the slope of the curves s = 0.01 , and s = 0.1 ),there are transient increases in the viral density caused by the current delay in CTL specific response. We used f11 = 5 .00 ,ξ = 1 .90 , u11 = 2 .25 , pi = q j = 2 .05 , ci = k j = 0.45 , η = 0 . 02 , and the remaining parameters as in Figure 3. Now λ0 g λ γ11 ( 11 − 1 ) ≅ 0 .21 < hM = 0.41 < 0 γ11 g 11 ≅ 0 .46 . T-helper cells concentrations (not shown) are high when viral burden is low ω01 ω11 ω0 ω1 and vice versa. Model for HIV-1 infection 257 Figure 5-a. Time evolution of the total viral density ∑ ij v ij ≡ v , and T-CD4 cells, z, for a system with a conserved epitope (epitope 1). There exist v11 , v12 , v13 , v14 variants. As the attack of CTL x1 against epitope 1 is always present, the immune system easily checks the infection. We used for both 5 and 6 figures s=0.1 and for the remainder of parameters identical values as in Figure 4. 258 Ortega and Martín-Landrove Figure 5-b. Time evolution of the total viral density ∑ ij v ij ≡ v , and T-CD4 cells, z, for a system without conserved epitopes. There exist v11 , v12 , v 21 , v 22 variants. A possible evolution of the viral variants (among others) could be v11 → v12 → v 22 . As the CTL addressed against v11 (the initial existing virion) are x1 , y1 , these mutations leave the immune system void of CTL addressed against v 22 ( x 2 , y 2 ) , a delay being necessary for the creation of these cells, and viral escape occurs. APPENDIX A already explained in the text, and we include an attack on infected T-CD4 + cells by CTL cells. Equation (6) can be The model can be expressed more generally by written more generally as: equations (1) to (6), adding the following equation: dvαβ dzαβ = gαβ mαβ + rαβ zαβ − fαβ zvαβ − γ αβ m0 vαβ − uαβ zαβ vαβ = uzvαβ − ω4 zαβ − zαβ ( pxα + qy β ) (A-1) dt dt (A-2) Where rαβ is the dynamical coefficient ruling the where zαβ represents the T cell density infected by viral production of virions by infected T cells. The lifetime of an variant αβ . Most of the terms in equation (A-1) are infected T cell is much shorter than that of a healthy Model for HIV-1 infection 259 cell38,39, for this reason we suppose that the population of The null point of equilibrium of this system is given by: infected T cells is in quasi-static equilibrium with the healthy ones. Additionally we suppose that an infected T λ0 λz cell in its short time of life has few opportunities to m0 = , z= , vαβ = mαβ = xα = y β = 0 interact with a specific cytolytic cell, which is equivalent to ω0 ωz taking zαβ ( pxα + qyβ ) → 0 in (A-1). Then equation (B-7) (A-1) becomes: Its linearization matrix is: dzαβ = uzvαβ − ω 4 zαβ (A-3) −ω 0 0 0 0 0 0 dt 0 −ω1 0 0 0 γ αβ m0 and in quasi-static equilibrium: 0 ηcα z −ω 2 0 0 0 0 ηkβ z 0 −ω 2 0 0 u 0 0 0 0 −ω z 0 zαβ = zvαβ (A-5) ω4 −γ αβ m0 0 gαβ 0 0 0 When substituting equation (A-5) into equation (A-2), (B-8) rαβ and defining ξ ≡ − 1 , we obtain equation (6). w4 with eigenvalues µ1 = −ω 0 , µ 2 = µ3 = −ω 2 , µ 4 = −ω z , and: Observe that viral invasion impairs infected T-CD4+ cells mediator role, so it is unnecessary to include an equation for them in the model. There are alternative µ 2 + µ ( ω + γ αβ m 0 ) + γ αβ m 0 ( ω 1 − g αβ ) = 0 ways to arrive to equation (6), not shown here. (B-9) When applying the Routh-Hurwitz criterion to equation APPENDIX B (B-9), we obtain as condition of stability: The system, with the approximations given in the text m0γ αβ (ω1 − gαβ ) > 0 , or becomes: gαβ dm0 <1 (B-10) = λ0 − ω 0m0 (B-1) ω1 dt d mαβ APPENDIX C dt = γ α β vα β m 0 − m α β ( px α + qyβ ) The system, with the approximations mentioned in the −ω 1m αβ text, is given by: (B-2) dm 0 = λ − ω m (C-1) d xα dt 0 0 0 = η cα z m α β + cα z m α β xα − ω 2 xα dt (B-3) d m αβ = γ αβ m 0 v αβ − ω 1m αβ (C-2) dt dy β = η k β zm αβ + k β zm αβ yβ − ω 2 yβ dt d z (B-4) = λ z − u α β v α β z − ω z z (C-3) d t dz = λz − ω z z d vαβ dt (B-5) dt = g αβ m αβ − (f αβ − ξ u αβ )zv αβ (C-4) dv αβ Its linearization matrix is: = g αβ m αβ − γ αβ m 0 v αβ (B-6) dt 260 Ortega and Martín-Landrove −ω 0 0 0 0 µ 3 + µ 2 [ ω 0 + ω z + uv αβ + z ( f − ξu )] + γ v −ω1 0 γ αβ m0 αβ αβ µ [ ω 0 ( ω z + uv αβ ) + ( ω 0 + ω z )( f − ξu ) − 0 0 −uαβ vαβ − ω z − uαβ z λ0 g αβ γ αβ )] 0 gαβ ( ξ uαβ − fαβ )vαβ ( ξ uαβ − fαβ )z ω0 (C-5) λ0 gαβ γ αβ vαβ = 0 (C-12) The system has two equilibrium points, which we call ω0 the null equilibrium point, and the general equilibrium point, respectively. The independent term in equation (C-12) is negative, The eigenvalues µ , at the null equilibrium point, given thus, according to the Routh-Hurwitz criterion one of by: system-associated eigenvalues always has a non- λ0 negative real part, and therefore the equilibrium point λz m0 = , z= , vαβ = mαβ = 0 (C-6) given by equation (C-11) is unstable. This conclusion ω0 ωz does not hold if γ αβ = hM = 0 , and/or g αβ = hM = 0 for in such case an eigenvalue is null and linearization are solution of criterion fails. µ + ω1 m 0γ αβ ( µ + ω 0 )( µ + ω z ) =0 g αβ µ − ( ξ u − f )z APPENDIX D (C-7) In the following all the discussion is referred to the we obtain µ1 = −ω 0 , µ 2 = −ω z , and system (C-1) to (C-4) and all our argumentation holds in a vicinity of its null equilibrium point. We notice that in equation (C-4) the term corresponding to the rate of viral µ 2 + µ [ ω 1 − z ( ξ uαβ − fαβ )] − g αβ γ αβ m0 destruction mediated by T-helper cells is − f αβ zv αβ , but (C-8) −ω 1 ( ξ uαβ − fαβ )z = 0 the term ξu αβ zvαβ is associated to production of virions at the same cells, so the net rate of virions variation When applying the Routh-Hurwitz criterion to (C-8) we obtain that this point is stable if: ( mediated by T-CD4+ cells is f αβ − ξu αβ v αβ z . If this ) term is positive, there occurs viral destruction and the g αβ γ αβ opposite if it is negative. Now we define the z ( f αβ − γ αβ ) > m 0 (C-9) instantaneous fraction of destroyed virions as ω1 dvαβ = f − ξu ( αβ αβ ) z 1 , (observe that Or, if we use the equilibrium values given by (C-6), vαβ dt D λz λ g γ although vαβ and d v α β approach separately to zero in a ( f αβ − ξu ) > 0 αβ αβ (C-10) dt ωz ω0 ω1 neighborhood of the equilibrium point, their ratio is a well defined quantity). The equilibrium value for The general equilibrium point is given by: T-CD4+cells is λ z = z , therefore we define λ0 λ0 γαβ g αβ ωz m0 = ,z = , λz ω0 ω0 ω1 ( f − ξu ) hM ≡ ( f αβ − ξu αβ ) , and then we arrive to ωz ωz λz ω0 ω1 ( f − ξu ) λ0 γαβ v αβ = − 1 ,m αβ = vαβ 1 dv αβ , this is, hM represents the u ωz λ0 g αβ γαβ ω1 hM ≈ v αβ dt D (C-11) fractional rate of virus population destroyed in equilibrium, by the immune system, by processes (there is infection, and only positive equilibrium points having biological meaning, we should restrain the mediated by T-helper cells. We call hM “humoral parameters values so that points obeying (C-11) fulfill this responsiveness”. Now, the term corresponding to requirement). The correspondent eigenvalue equation is: infected reservoirs creation in equation (C-2) is Model for HIV-1 infection 261 γ αβ m0 vαβ , and we define the rate of creation of hand side of (11) represents the fractional rate of virions produced in equilibrium by an infected reservoir in its infected reservoirs as dm α β . meantime, and we call “viral growth” to this factor. Then = γαβ m 0 vαβ this approximate system possesses a null and stable dt C solution (no virus) if and only if the fraction of virions Similarly, the term corresponding to virions creation destroyed by processes mediated by T-helper cells is processes in equation (C-4) is given by g αβ m αβ greater than the fraction of virions produced by infected reservoirs in their meantime, this is if the humoral and we define the rate of virions creation as responsiveness is higher than the viral growth. d vαβ . Finally, 1 ≈ τ 1 , is the = g αβ m αβ ω1 AKNOWLEDGEMENTS dt C meantime of an infected cell. Then we obtain that: The authors wish to thank to Carla Ortega DiGregorio (Brunel University, Uxbridge, London) for grammatical λ0 γ α β g α β m0 m and editing revision of this paper. = g αβ γαβ ≈ g αβ γαβ 0 ω 0 ω1 ω1 ω1 1 d vαβ 1 d m αβ = × τ1 ≈ m αφ d t C vαβ d t , 1 d vαβ 1 d m αβ × τ1 vαβ d t m αβ d t C C where the factors mαβ and vαβ have been rearranged and included in the parenthesis. We see that the right REFERENCES 1. Boaz, M., Waters, A., Murad, S., Easterbrook, P. and 6. Collins, K., Chen, B., Kalams, S., Walker, B. and Vyakarman A. Presence of HIV-1 gag-specific IFN- γ IL- + Baltimore, D. HIV-1 Nef proteine protects infected primary + 2 CD4 T cell responses is associated with nonprogression cells against killing by cytotoxic T lymphocytes. Nature in HIV-1 infection. J. Immunol. 169: 6376-6385, 2002. 391: 397-401, 1998. 2. Carvert, W., Notermans, D., Staskus, K., Wietgrefe, S., 7. De Boer, R. and Perelson, A. 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