Modeling of the HIV-1 Life Cycle in Productively Infected Cells to Predict Novel Therapeutic Targets.

There are many studies that model the within-host population dynamics of Human Immunodeficiency Virus Type 1 (HIV-1) infection. However, the within-infected-cell replication of HIV-1 remains to be not comprehensively addressed. There exist rather few quantitative models describing the regulation of the HIV-1 life cycle at the intracellular level. In treatment of HIV-1 infection, there remain issues related to side-effects and drug-resistance that require further search "...for new and better drugs, ideally targeting multiple independent steps in the HIV-1 replication cycle" (as highlighted recently by Teldury et al., The Future of HIV-1 Therapeutics, 2015). High-resolution mathematical models of HIV-1 growth in infected cells provide an additional analytical tool in identifying novel drug targets. We formulate a high-dimensional model describing the biochemical reactions underlying the replication of HIV-1 in target cells. The model considers a nonlinear regulation of the transcription of HIV-1 mediated by Tat and the Rev-dependent transport of fully spliced and singly spliced transcripts from the nucleus to the cytoplasm. The model is calibrated using available information on the kinetics of various stages of HIV-1 replication. The sensitivity analysis of the model is performed to rank the biochemical processes of HIV-1 replication with respect to their impact on the net production of virions by one actively infected cell. The ranking of the sensitivity factors provides a quantitative basis for identifying novel targets for antiviral therapy. Our analysis suggests that HIV-1 assembly depending on Gag and Tat-Rev regulation of transcription and mRNA distribution present two most critical stages in HIV-1 replication that can be targeted to effectively control virus production. These processes are not covered by current antiretroviral treatments.


Introduction
The Human Immunodeficiency Virus Type 1 (HIV-1) continues to threaten global health and represents a significant burden for health-care systems. Intensive research on HIV and its induced acquired immunodeficiency syndrome (AIDS) for more than 35 years has uncovered unprecedented details of virus and virus-host biology. However, it remains unclear how to cure individuals that are of the stages in a non-mechanistic way, i.e., using gamma distribution functions for the net rates of the replication markers. The parameters of the functions were calibrated using original in vitro data.
The primary aim of our study here was to formulate a high-resolution deterministic model of the HIV-1 replication cycle and to calibrate the model using available data on the kinetics and process parameters. The second purpose of the work was to use the model for predicting novel targets of intervention of HIV-1 production via sensitivity analysis of the model. To this end, we applied the variational approach based on adjoint equations.

Mathematical Model
In this section, we formulate the equations of the mathematical model describing the sequence of biochemical reactions underlying the intracellular replication of HIV-1 in productively infected cells. The schematic representation is shown in Figure 1 modified from [3]. We consider the following replication stages: entry, reverse transcription, integration, transcription, translation, assembly, budding and maturation.

Entry
The entry stage is split into three steps represented in Figure 2: 1. binding of virion to CD4 receptors (glycoprotein gp120 binds to CD4 receptors on the T cell surface), 2.
binding to the co-receptor (CCR5 or CXCR4), 3. fusion, i.e., the nucleocapsid is uncoated and the viral RNA is injected into the cell. The binding of the virion to CD4 is described by equations describing the rates of changes of freeand receptors-bound virion: where [V f ree ] is the number of free virions outside the cell, [V bound ] is the number of virions bound to CD4 and the co-receptor. The respective parameters of the model are described in Table 1.

Reverse Transcription
The fusion of bound virions with the host cell membrane results in the release of the content of the virion into the cell cytoplasm, starting the phase of reverse transcription [11]. During reverse transcription, a double-stranded DNA is created from two single-stranded RNA genomes. According to the scheme presented in Figure 3, the reverse transcription is modeled as three sequential processes: 1.
synthesis of minus-strand DNA from viral RNA, 2.
double-strand DNA formation. The rates of change of the amount of RNA and proviral DNA species are described by equations: where [RN A cor ] is the number of genomic RNA molecules in cytoplasm, [DN A cor ] is the number of proviral DNA molecules synthesized by reverse transcription. The respective parameters of the above equations are described in Table 1.

Integration
After the proviral DNA is synthesized, it associates with virus-encoded integrase (IN) and other proteins as a high-molecular-weight nucleoprotein complex (pre-integration complex, PIC) that is transported into the nucleus for subsequent integration [12]. Integration is the process of viral DNA insertion into chromosomal DNA of the host cell. The viral DNA can also undergo several circularization reactions thus losing the capability to support subsequent replication [12]. The change in the number of viral DNA in the nucleus and the number of integrated DNA are modeled with the following equations: where [DN A nuc ] is the number of DNA molecules in the nucleus, [DN A int ] stands for the number of integrated DNA.

Transcription
HIV transcription starts when the host cell receives activation signals. It is a process of messenger RNA (mRNA) synthesis. There are three types of mRNA species: full-length (around 9 kb), singly spliced (around 4 kb), doubly spliced (around 2 kb) [3]. After transcription, mRNAs are transported to the cell cytoplasm. There is a temporal regulation of transcription and mRNA distribution by viral Tat and Rev proteins. To describe these stages, we used the scheme in Figure 4 and the parameterization of the feedback regulation similar to [5,6], as specified below: and the Michaelis-Menten type equations are used for with θ Tat = (K Tat ) −1 , and θ Rev = (K Rev ) −1 (see [4,5]), respectively. The parameters of the above equations are described in Table 1.

Translation
The viral mRNAs are decoded by ribosomes to produce specific proteins. The proteins then fold into active proteins. The full-length mRNA codes for Gag and Gag-Pol proteins. The singly spliced mRNAs code for gp160, Vif, Vpu and Vpr proteins. The doubly spliced mRNAs code for Nef, Tat and Rev. In our model we account for the kinetics of Gag-Pol, Gag, gp160, Tat and Rev proteins. Their turnover is described by the following set of equations: where [P j ] is the number of protein molecules j, j ∈ {Gag-Pol, Gag, gp160, Tat, Rev} and f i,j are the fractions of mRN A i coding [P j ], i ∈ {g, ss, ds}. The respective parameters of the model are described in Table 1.

Assembly, Budding and Maturation
The late phase of the HIV-1 replication cycle includes the trafficking of the regulatory and accessory proteins and viral glycoproteins to the plasma membrane, the assembly of Gag and Gag-Pol proteins at the plasma membrane with a subsequent encapsidation of the viral RNA genomes, budding of the new virions and their maturation [13]. The viral proteins Gag-Pol, Gag, gp160 undergo several post-translational modifications such as folding, oligomerization, glycosylation, and phosphorylation [4]. After that, the proteins and the full-length mRNA molecules are transported to the membrane where they associate (pre-virion complex) and combine to generate a new virion. The scheme of this late phase of virus replication is shown in Figure 5. The transport processes to the membrane are modeled by the following kinetic equations (∆ ≡ {Gag-Pol, Gag, gp160}): The membrane thus encloses viral RNAs with the proteins Gag, Gag-Pol, Vif, Nef, and Vpr while gp160 is anchored in the membrane. The assembly of the pre-virion complex then leads to virion budding. The initially immature budded virions subsequently mature and become infectious for other cells. These last stages are described by equations: where [V pre-virion ] is the number of virions on the membrane, [V bud ] is the number of free viruses after budding from the cell and [V mat ] is the number of mature virions outside the cell.

Model Parameters
The model was calibrated using available information on the kinetics and process parameters presented in [4][5][6]9,[14][15][16][17][18][19][20][21]. The estimated values and admissible ranges of the model parameters are summarized in Table 1. The variation of threshold parameter θ Tat results in a temporal shift of the overall kinetics (increase of θ Tat increases the delay before virion release), while the value of θ Rev positively influences the rate of virion release. The combination of θ Tat and θ Rev parameters influences the overall dynamics in nonlinear way, therefore, they were tuned manually to achieve the expected temporal kinetics of replication cycle stages [10] and physiological levels of transcripts, proteins and mature virions. The initial values of low number of infectious virions (V f ree (0) ≤ 5 virions) result in the integration of V int ≤ 2 proviral DNA that resembles in vivo scenarios [22] rather than experimental setups with highly susceptible cell lines and high multiplicity of infection. The overall dynamics of the individual components of the HIV-1 replication cycle in activated CD4+ T cells infected with V f ree (0) = 4 infectious virions through their life span of about 36 h [23] is presented in Figure 6. We note from the solution of the calibrated model that the number of membrane anchored Gag molecules is a limiting factor for the assembly of pre-virion complexes since all Gag molecules are being incorporated into the complexes while the abundant levels of other proteins and genomic RNA remain at the membrane.   Table 1.

Sensitivity Analysis
The calibrated model provides a tool for predicting the dependence of the HIV-1 production by an infected cell to variations of the rates of underlying biochemical processes. The prediction is based on the sensitivity analysis of the model solutions or some functions depending on them [45]. Let us characterize the net outcome of the virus replication cycle by the total number of released virions, Here y ≡ y(t, p) ∈ R N + is the reference (unperturbed) solution to the following initial value problem: . . , p L ] ∈ R L + is a parameter vector. Let the model parameters be changed by small δp = p − p. The local sensitivity of the functional J(y) with respect to the model parameters can be computed using the following system of equations The function w(t) ≡ w(t, p) is the solution to the adjoint problem  Figure 7. The left one ranks the sensitivity coefficients for the model parameters which negatively impact the net HIV-1 production when their values are increased. The right bar plot ranks the sensitivity coefficients for the parameters which positively impact the net virus production with their increasing values. The summary of the most critical control parameters characterized by the normalized sensitivity values larger than 1000 and larger than 100, are summarized in Table 2. By comparing the sensitivity analysis data with the existing targets of antiretroviral therapy [3] specified in Figure 8, we identify additional processes which could be considered for ART. θTat keRNA ss dp,Rev dp,Gag dDNA nuc dRNA ss dp,Tat dbound dDNA int dmem,Gag kcombNGag−Pol dmem,gp160 dp,gp160 kcombNgp160 Thus, the ranked processes which display the strongest impact on HIV-1 replication are: 1. maximal achievable level of transcription rate that can be induced by Tat; 2.
HIV-1 assembly, specifically sensitive to availability and translation rate of Gag molecules; 3. full-length RNA transport to membrane and degradation; 4.
transport of pre-integration complex into nucleus and DNA integration; 5.
Rev-mediated regulation of splicing rates and export of full-length RNA; 6.
binding and fusion of free virions.

Discussion
We have formulated mathematical model (system of ODEs) which describes the biochemical reactions underlying the replication life cycle of HIV-1 in activated target cells. The model considers nonlinear regulation of the transcription of HIV-1 mediated by Tat and the Rev-dependent transport of fully spliced and singly spliced transcripts from nucleus to the cytoplasm. The model has been calibrated using available information on the kinetics of various stages of HIV-1 replication.
We have performed the sensitivity analysis of the model to rank the biochemical processes of HIV-1 replication with respect to their impact on the net production of virions by one actively infected cell. The ranking of the sensitivity factors provides a quantitative basis for identifying novel targets for antiviral therapy. Our analysis suggests that (1) HIV-1 assembly depending on Gag and (2) Tat-Rev regulation present two critical stages in HIV-1 replication that can be targeted to effectively control virus production. These processes while being considered to be interesting antiviral targets (i.e., [46,47]) are not targeted by the current antiretroviral treatments. The vital role of the Gag polyprotein precursor and the mature Gag proteins has been discussed recently in [13]. However, the efforts in developing inhibitors of Gag function have not yet resulted in efficacious drugs. The biochemistry of Gag-Pol processing for the HIV-1 assembly deserved the attention from the modeling point of view previously [8]. Our study puts the HIV-1 assembly stage into the context of the whole HIV-1 life cycle and identifies it as the most crucial (in terms of sensitivity) process.
The formulated model and its reduced versions can be used as building blocks for multiscale hybrid models of HIV-1 infection [48,49]. The current version of our model has the limitation that is does not consider the discrete nature and stochasticity of the biochemistry of the infection. This is inherent to the model set up as we considered a deterministic description for it. From the biological point of view, the model lacks a description of intracellular antiviral defense mechanisms and co-factors that facilitate virus propagation. Likewise, we did not incorporate error-prone virus replication that would lead to quasi-species distributions. All these factors can be incorporated to address specific issues that may arise including optimization of antiviral drug combinations and immune-based therapies. Further extensions of the model will require the analysis of stochastic effects to understand the link between the variability across the biochemical parameter values and the heterogeneity of virus production including the response to therapeutic perturbations.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.