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Computational microepidemiologic modelling can facilitate the understanding of complex biomedical systems. It provides novel methods for quantitatively studying population health dynamics from the micro level of genomes and molecules to the higher macro levels such as HIV/AIDS in humans. Untangling the dynamics between the human immunodeficiency virus-1 (HIV-1) and CD4+ lymphocyte populations and intracellular molecular kinetics of interactions in an integrative systems dynamics approach can help to understand the effective points of interventions in the HIV life cycle. With that in mind, we have developed a stochastic systems dynamics model that includes intracellular molecular level interactions. A sequence of events, molecular interactions and cytochemical kinetics are triggered when the HIV infects a CD4+ lymphocyte. The full sequence of molecular level dynamics includes: attachment and fusion; reverse transcription; integration; transcription; translation; and budding or release of new virus. The newly released virus circulates back and infects a new CD4+ lymphocyte and the cycle continues repeatedly. Mathematical models that account for these processes were developed. The model developed provides insights into how an intracellular/molecular level model can be incorporated within a macro-epidemiologic integrative systems dynamics model for examining a variety of computational experimentations. Such experimentations can help in evaluating scientific questions related to effective strategies in HIV drug therapy interventions.
In the past two decades, there has been a surge in molecular and cellular population studies of the Human Immunodeficiency Virus (HIV) and the Acquired Immunodeficiency Syndrome (AIDS) that continues to cause high morbidity and mortality of humans across the globe. Understanding the dynamic interplay of HIV within its cellular host provides the microepidemiologic basis for controlling the epidemic. Mathematical and epidemiologic models of HIV/AIDS provide important insights in population dynamics through studies at the molecular and cellular levels as well as at the human population level (, , ). The specific objectives of our study therefore were to develop a stochastic systems dynamics model that includes intracellular molecular level interactions and molecular level dynamics of HIV and use the model to study optimal chemotherapies (Fusion inhibitor, nucleoside and non-nucleoside reverse transcriptase inhibitors and protease inhibitors) for reducing the HIV viral load; and to use the model to study the pattern of mutant viral populations and resistance to drug therapies.
The epidemiologic systems dynamics model of host/agent/environment interaction for CD4+ lymphocyte populations and HIV helps to define the state transitions that occur in this disease complex. The host populations are CD4+ lymphocytes, the agent is HIV viral population and the environment is the cellular and intracellular/molecular ecosystem.
A systems analysis diagram of the cellular/molecular modelling tasks is presented in Figure 1. Virion Production and Clearance Rates: HIV production in vivo occurs continuously at high rates (, ). Virion clearance could be the result of binding and entry into cells, immune elimination, or nonspecific removal by the reticuloendothelial system .
Once the epidemiologic systems diagram was completed, Figure 1, parameter estimations and development of ordinary/partial differential equations that represent the systems dynamics followed. Each of the rates and auxiliary variables and other appropriate parameters were derived from preexisting data.
The mathematical equation for the sequence of intracellular level dynamics is given below.
Elementary probability states that, given Cu(t) uninfected CD4+ cells at time t, and M bound phage per cell, the probability that a given cell will be infected is
Trimeric gp120 on the surface of the virion binds CD4 on the surface of the target cell, inducing a conformational change in the envelope proteins that in turn allows binding of the virion to a specific subset of chemokine receptors on the cell surface.
Where kB = PBinding, ki and KD are: the binding rate, the virus entry rate and dissociation rate of the virus respectively. [VC] = the number of free virus that contacted CD4 cells. [VB] = the number of virus that bounded to CD4 cells.
The interactions between gp120, CD4, and chemokine receptors (CCR5 or CXCR4) lead to gp41-mediated fusion.
Where [VF] = the number of virus that fused to CD4 cells. kF, kRT and kD are: the fusion rate, the rate of reverse transcription and dissociation rate of the virus respectively.
Once inside the cell, the virion undergoes uncoating, likely while still associated with the plasma membrane.
After the virion is uncoated, the viral reverse transcription complex is released from the plasma membrane . This interaction, mediated by the phosphorylated matrix, is required for efficient viral DNA synthesis. The reverse transcription can be given assuming Michaeli's Menten kinetics ,
where [RNAcor] is the concentration of genomic RNA present in the viral core and [dNTP] is the concentration of the dNTP pool of the host cell. Km(RNAcor) and Km(dNTP) are the Michaelis constants for reverse transcriptase with the substrates RNAcor (= 2* [VF]) and the dNTP (Deoxyribonucleoside triphosphate) pool, respectively. kRNAcor is the degradation rate constant of the genomic RNA. Reverse transcription yields the HIV preintegration complex (PIC), composed of double-stranded viral cDNA, integrase, matrix, Vpr, reverse transcriptase, and the high mobility group DNA-binding cellular protein HMGI(Y) 
where [DNAcor] is the concentration of genomic DNA present in the cytoplasm synthesized by reverse transcription, kDNAcor is the rate constant for degradation of the proviral DNA in the viral core, and kDNA t is the rate constant for transport of the DNA from the cytoplasm to the nucleus. Integration of double-stranded viral DNA into the host chromosome is mediated by integrase, which binds the ends of the viral DNA . The viral DNA in the nucleus is circularized, integrated in the host genome, or degraded so its rate of change is given as follows:
where [DNAnuc] is the concentration of nonintegrated linear DNA in the nucleus. We assume that the rate of circularization, kcirc is equal to the rate of integration knit. kDNA nuc is the degradation rate constant of the nuclear DNA.
Integration can lead to latent or transcriptionally active forms of infection . The multiple copies of provirus that are usually integrated in a given infected cell, at least one is likely to be transcriptionally active. The transcription rate of HIV-1 is assumed to be the sum of the transcription rates due to cellular factors alone and also cellular factors in conjunction with Tat:
Where Tc is the maximum of transcription induced by cellular transcription induced by cellular transcription factors alone and TTC is equivalent to the basal transcription rate, which is about 100-fold smaller than TC.
The nuclear export of this assembly (viral RNA transcript, Rev, and CRM1/exportin 1) depends critically on yet another host factor, RanGTP.
In contrast to Tat and Rev, which act directly on viral RNA structures, Nef modifies the environment of the infected cell to optimize viral replication.
New viral particles are assembled at the plasma membrane. Each virion consists of roughly 1500 molecules of Gag and 100 Gag-Pol polyproteins,  two copies of the viral RNA genome, and Vpr.
Virion budding occurs through specialized regions in the lipid bilayer, yielding virions with cholesterol-rich membranes .
In this model, at the molecular level, we distinguish three types of variables. Uninfected CD4+ cells. Two types of virus particles: wild type virus and mutant virus. Two groups of infected CD4+ cells infected by wild type virus Ci(t) and cells infected by mutant virus Cim(t), where, for i=D, defectively infected, for i=L, latently infected, for i=P, productively infected, and for i=C, chronically infected. Variables and parameters of the model used: D (t) denotes the drug concentration at time t, β and βm are rates at which the wild type virus and mutant virus contact and bind to the CD4+ cells respectively. λ (t) the rate at which the uninfected cells are produced, from a pool of precursor cells, du the natural death rate of the uninfected cells. di's denote the rates of loss of infected cell of type i, where i = L, P, C, D; μ is the probability of mutation from wild type to resistant mutant virus; D (t) and ϕ denote the drug concentration and drug inhibition factor respectively; γi 's denote the rate at which an HIV-infected CD4 + cell of type i release virus. Thus, the kinetics of the uninfected CD4+ cells, Cu, is given by:
Where λ(t) = IC (t) − [k IC (t) / (IC (1/2)+ RNAcor) ]; IC (t) is the number of immature CD4+ cells at time t, RNAcor is the concentration of genomic RNA present in the viral core k is the product of reduction constant of immature cell and proportion of RNAcor to number of virions budding. IC(1/2) is half saturation constant of immature cell. The dynamics of cell infection by wild type virus and wild type virus production rates are represented by:
The dynamics for mutant type cell infection and mutant type virus production rates are represented.
Where ai = πiβ(1−μ) for i = L, P, C, D. The parameter πi describes the probability that upon infection a cell will become type i and the parameter μ is the probability of mutation from wild type to resistant mutant virus.
is the maximum number of virus that an infected cell of type i can produce within its life time with a survival probability until age a of the cell.
In the presence of drug, for example, fusion, kD will be modified to kDF =kD- ϕF, where
and D50 is the dose required for 50% effectiveness. εF is the efficacy of Fusion with DF(t) being the concentration of the drug fusion. Reverse transcriptase inhibitors block the ability of HIV to successfully infect a cell. Hence in the equation for Cu the term βV+βmVm becomes
The equations presented as well as other appropriately defined parameters in the model were used to create the computational model. The model is based on several assumptions and some of the parameter estimates will undoubtedly be improved over time. We do see the need for and importance of using computational models to represent complex biomedical systems that can best be studied cohesively and rationally using integrative systems dynamics modeling. Even with the advent of newer drugs, no amount of medical treatment has so far been able to prevent the eventual collapse of the immune system in people with advanced HIV infection or AIDS. The explanation for such a collapse of the immune complex is not well understood. One of the main reasons for this is our inadequate knowledge of the dynamics and interaction of the CD4+ lymphocytes with HIV, especially in the presence of the different types of anti-retroviral drugs that are presently available. Computational epidemiologic models of cellular and molecular level dynamics, if successfully developed, can be used to investigate these types of questions.
This work is supported by a Research Centers in Minority Institutions (RCMI) Award, from the National Center for Research Resources, National Institutes of Health