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Adv Syst Sci Appl. Author manuscript; available in PMC 2010 May 5.

Published in final edited form as:

Adv Syst Sci Appl. 2008; 8(1): 40–45.

PMCID: PMC2864528

NIHMSID: NIHMS129552

Center for Computational Epidemiology, Bioinformatics & Risk Analysis (CCEBRA), College of Veterinary Medicine, Nursing & Allied Health, Tuskegee University, Tuskegee, AL 36088, http://compepid.tuskegee.edu

Berhanu Tameru: ude.eegeksut@uremat

See other articles in PMC that cite the published article.

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 ([5], [11], [13]). 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 ([5], [13]). Virion clearance could be the result of binding and entry into cells, immune elimination, or nonspecific removal by the reticuloendothelial system [10].

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 *C _{u}(t)* uninfected CD4+ cells at time t, and

$$\begin{array}{c}{P}_{\mathit{\text{Binding}}}\mathit{(}{C}_{u}\mathit{(}t\mathit{)},M\mathit{)}={\mathit{[}\mathit{1}-\mathit{(}\mathit{1}/{C}_{u}\mathit{(}t\mathit{)}\mathit{)}\mathit{]}}^{M\phantom{\rule{0.2em}{0ex}}\mathit{\text{Cu}}\phantom{\rule{0.2em}{0ex}}\mathit{(}t\mathit{)}}\\ {P}_{\mathit{\text{Binding}}}\mathit{(}{C}_{u}\mathit{(}t\mathit{)},M\mathit{)}\to {e}^{-M}\phantom{\rule{0.2em}{0ex}}\text{as}\phantom{\rule{0.2em}{0ex}}{C}_{u}\mathit{(}t\mathit{)}\to \infty \left[3\right].\end{array}$$

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.

$$\frac{d\left[{V}_{B}\right]}{dt}={k}_{B}\left[{V}_{C}\right]-\left({k}_{i}+{k}_{D}\right)\left[{V}_{B}\right]$$

Where *k _{B}* =

The interactions between gp120, CD4, and chemokine receptors (CCR5 or CXCR4) lead to gp41-mediated fusion.

$$\frac{d\left[{V}_{F}\right]}{dt}={k}_{F}\left[{V}_{B}\right]-{k}_{RT}\left[{V}_{F}\right]-{k}_{D}\left[{V}_{F}\right]$$

Where [*V _{F}*] = the number of virus that fused to CD4 cells.

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 [7]. 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 [12],

$$\frac{d\left[{\mathit{\text{RNA}}}_{\mathit{\text{cor}}}\right]}{dt}=-\frac{{V}_{m}\left[{\mathit{\text{RNA}}}_{\mathit{\text{cor}}}\right]}{{K}_{m\left({\mathit{\text{RNA}}}_{\mathit{\text{cor}}}\right)}+\left[{\mathit{\text{RNA}}}_{\mathit{\text{cor}}}\right]}\cdot \frac{\left[\mathit{\text{dNTP}}\right]}{{K}_{m\left(\mathit{\text{dNTP}}\right)}+\left[\mathit{\text{dNTP}}\right]}-{k}_{{\mathit{\text{RNA}}}_{\mathit{\text{cor}}}}\left[{\mathit{\text{RNA}}}_{\mathit{\text{cor}}}\right]$$

where [RNA* _{cor}*] is the concentration of genomic RNA present in the viral core and [dNTP] is the concentration of the dNTP pool of the host cell.

$$\frac{d\left[{\mathit{\text{DNA}}}_{\mathit{\text{cor}}}\right]}{dt}=-\frac{{V}_{m}\left[{\mathit{\text{RNA}}}_{\mathit{\text{cor}}}\right]}{{K}_{m\left({\mathit{\text{RNA}}}_{\mathit{\text{cor}}}\right)}+\left[{\mathit{\text{RNA}}}_{\mathit{\text{cor}}}\right]}\cdot \frac{\left[\mathit{\text{dNTP}}\right]}{{K}_{m\left(\mathit{\text{dNTP}}\right)}+\left[\mathit{\text{dNTP}}\right]}-{k}_{{\mathit{\text{DNA}}}_{\mathit{\text{cor}}}}\left[{\mathit{\text{DNA}}}_{\mathit{\text{cor}}}\right]-{k}_{{\mathit{\text{DNA}}}_{t}}\left[{\mathit{\text{DNA}}}_{\mathit{\text{cor}}}\right]$$

where [DNA* _{cor}*] is the concentration of genomic DNA present in the cytoplasm synthesized by reverse transcription,

$$\frac{d\left[{\mathit{\text{DNA}}}_{\mathit{\text{nuc}}}\right]}{dt}={k}_{{\mathit{\text{DNA}}}_{t}}\left[{\mathit{\text{DNA}}}_{\mathit{\text{cor}}}\right]-{k}_{\mathit{\text{cir}}}\left[{\mathit{\text{DNA}}}_{\mathit{\text{nuc}}}\right]-{k}_{\text{int}}\left[{\mathit{\text{DNA}}}_{\mathit{\text{nuc}}}\right]-{k}_{{\mathit{\text{DNA}}}_{\mathit{\text{nuc}}}}\left[{\mathit{\text{DNA}}}_{\mathit{\text{nuc}}}\right]$$

where [DNA* _{nuc}*] is the concentration of nonintegrated linear DNA in the nucleus. We assume that the rate of circularization,

Integration can lead to latent or transcriptionally active forms of infection [1]. 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:

$$\mathit{\text{Transcripts per time}}=\mathit{\text{TR}}={T}_{C}{F}_{C}+{T}_{TC}{F}_{TC}$$

Where *Tc* is the maximum of transcription induced by cellular transcription induced by cellular transcription factors alone and *T _{TC}* is equivalent to the basal transcription rate, which is about 100-fold smaller than

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, [14] 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 [9].

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 C _{i}(t)* and

$$d{C}_{u}\mathit{(}t\mathit{)}/\mathit{\text{dt}}=\lambda \mathit{(}t\mathit{)}-\mathit{[}{d}_{u}+\beta V\mathit{(}t\mathit{)}+{\beta}_{m}{V}_{m}\mathit{(}t\mathit{)}\mathit{]}{C}_{u}\mathit{(}t\mathit{)}$$

Where *λ(t)* = *IC (t)* − *[k IC (t) / (IC _{(1/2)}*+

$$\begin{array}{l}\frac{\partial {C}_{i}\left(t,a\right)}{\partial t}+\frac{\partial {C}_{i}\left(t,a\right)}{\partial a}=-{d}_{i}{C}_{i}\left(t,a\right)\phantom{\rule{0.2em}{0ex}}\mathit{\text{with}}\phantom{\rule{0.5em}{0ex}}{C}_{i}\left(t,0\right)=\beta {C}_{i}\left(t\right)V\left(t\right)\phantom{\rule{0.5em}{0ex}}\mathit{\text{for}}\phantom{\rule{0.5em}{0ex}}i\in \left\{L,P,C,D\right\}\\ \frac{dV\left(t\right)}{dt}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}\sum _{i\in \left\{L,P,C\right\}}\underset{{a}_{iP}}{\overset{{a}_{\text{max}}}{\int}}{\gamma}_{i}\left(a\right){C}_{i}\left(t,a\right)da-uV\left(t\right)\end{array}$$

The dynamics for mutant type cell infection and mutant type virus production rates are represented.

$$\begin{array}{l}\frac{\partial {C}_{im}\left(t,a\right)}{\partial t}+\frac{\partial {C}_{im}\left(t,a\right)}{\partial a}=-{d}_{im}{C}_{im}\left(t,a\right)\phantom{\rule{0.5em}{0ex}}\mathit{\text{with}}\phantom{\rule{0.5em}{0ex}}{C}_{im}\left(t,0\right)={\pi}_{im}\beta {C}_{i}\left(t,0\right)V\left(t\right)\phantom{\rule{0.5em}{0ex}}\mathit{\text{for}}\phantom{\rule{0.5em}{0ex}}i\in \left\{L,P,C,D\right\}\\ \frac{d{V}_{m}}{dt}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}\sum _{i\in \left\{L,P,C\right\}}\underset{{a}_{iP}^{m}}{\overset{{a}_{\text{max}}}{\int}}{\gamma}_{im}\left(a\right){C}_{im}\left(t,a\right)da-{u}_{m}{V}_{m}\left(t\right)\end{array}$$

Where *a _{i}* =

$${\gamma}_{i}(a)={\gamma}_{\text{max}}(1-{e}^{-{\gamma}_{i}(a-{a}_{ip})})\phantom{\rule{0.8em}{0ex}}\mathit{\text{for}}\phantom{\rule{0.8em}{0ex}}a>{a}_{ip}\phantom{\rule{0.8em}{0ex}}\mathit{\text{and}}\phantom{\rule{0.8em}{0ex}}{\gamma}_{i}(a)=0\phantom{\rule{0.8em}{0ex}}\mathit{\text{for}}\phantom{\rule{0.8em}{0ex}}a<{a}_{ip}\phantom{\rule{0.8em}{0ex}}\mathit{\text{where}}\phantom{\rule{0.8em}{0ex}}{\gamma}_{\text{max}}$$

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, *k _{D}* will be modified to

$${\mathit{\varphi}}_{F}=\frac{1}{{\text{D}}_{\text{F}}(\text{t})}-\frac{{\epsilon}_{\text{F}}}{({\text{D}}_{\text{F}}(\text{t})+{\text{D}}_{50})}$$

and *D _{50}* is the dose required for 50% effectiveness.

$$\mathit{(}\mathit{1}-{\epsilon}_{\phantom{\rule{0.2em}{0ex}}RT}\mathit{)}\beta \phantom{\rule{0.2em}{0ex}}V\mathit{+}\mathit{(}\mathit{1}-{\epsilon}_{\phantom{\rule{0.2em}{0ex}}\mathit{\text{RTm}}}\mathit{)}{\beta}_{m}{V}_{m}$$

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

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