Computational Modelling of Intracellular Viral Kinetics and CD4+ Cellular Population Dynamics of...

Computational Modelling of Intracellular Viral Kinetics and CD4+Cellular Population Dynamics of HIV/AIDS

Berhanu Tameru, Tsegaye Habtemariam, David Nganwa, Lekan Ayanwale, GemechuBeyene, Vinaida Robnett, and Wanda WilsonCenter for Computational Epidemiology, Bioinformatics & Risk Analysis (CCEBRA), College ofVeterinary Medicine, Nursing & Allied Health, Tuskegee University, Tuskegee, AL 36088,http://compepid.tuskegee.eduBerhanu Tameru: [email protected]

AbstractComputational microepidemiologic modelling can facilitate the understanding of complexbiomedical systems. It provides novel methods for quantitatively studying population healthdynamics 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 anintegrative systems dynamics approach can help to understand the effective points of interventionsin the HIV life cycle. With that in mind, we have developed a stochastic systems dynamics modelthat includes intracellular molecular level interactions. A sequence of events, molecular interactionsand cytochemical kinetics are triggered when the HIV infects a CD4+ lymphocyte. The full sequenceof molecular level dynamics includes: attachment and fusion; reverse transcription; integration;transcription; translation; and budding or release of new virus. The newly released virus circulatesback and infects a new CD4+ lymphocyte and the cycle continues repeatedly. Mathematical modelsthat account for these processes were developed. The model developed provides insights into howan intracellular/molecular level model can be incorporated within a macro-epidemiologic integrativesystems dynamics model for examining a variety of computational experimentations. Suchexperimentations can help in evaluating scientific questions related to effective strategies in HIVdrug therapy interventions.

Keywordsintracellular/molecular modelling; computational microepidemiologic modelling; HIV/AIDS

1 IntroductionIn the past two decades, there has been a surge in molecular and cellular population studies ofthe 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 themicroepidemiologic basis for controlling the epidemic. Mathematical and epidemiologicmodels of HIV/AIDS provide important insights in population dynamics through studies at themolecular and cellular levels as well as at the human population level ([5], [11], [13]). Thespecific objectives of our study therefore were to develop a stochastic systems dynamics modelthat includes intracellular molecular level interactions and molecular level dynamics of HIVand use the model to study optimal chemotherapies (Fusion inhibitor, nucleoside and non-nucleoside reverse transcriptase inhibitors and protease inhibitors) for reducing the HIV viral

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Published in final edited form as:Adv Syst Sci Appl. 2008 ; 8(1): 40–45.

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load; and to use the model to study the pattern of mutant viral populations and resistance todrug therapies.

2 Materials and Methods2.1. Epidemiologic Modelling

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 diseasecomplex. The host populations are CD4+ lymphocytes, the agent is HIV viral population andthe environment is the cellular and intracellular/molecular ecosystem.

Cellular and Intracellular level dynamics—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 bethe result of binding and entry into cells, immune elimination, or nonspecific removal by thereticuloendothelial system [10].

2.2 The Mathematical ModelOnce the epidemiologic systems diagram was completed, Figure 1, parameter estimations anddevelopment of ordinary/partial differential equations that represent the systems dynamicsfollowed. Each of the rates and auxiliary variables and other appropriate parameters werederived from preexisting data.

2.2.1 Intracellular kinetics—The mathematical equation for the sequence of intracellularlevel dynamics is given below.

Probability of binding: Elementary probability states that, given Cu(t) uninfected CD4+ cellsat time t, and M bound phage per cell, the probability that a given cell will be infected is

Kinetics of Binding: Trimeric gp120 on the surface of the virion binds CD4 on the surface ofthe target cell, inducing a conformational change in the envelope proteins that in turn allowsbinding 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 rateof the virus respectively. [VC] = the number of free virus that contacted CD4 cells. [VB] = thenumber of virus that bounded to CD4 cells.

Kinetics of Fusion: The interactions between gp120, CD4, and chemokine receptors (CCR5or CXCR4) lead to gp41-mediated fusion.

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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.

Uncoating: Once inside the cell, the virion undergoes uncoating, likely while still associatedwith the plasma membrane.

Reverse transcription: After the virion is uncoated, the viral reverse transcription complexis released from the plasma membrane [7]. This interaction, mediated by the phosphorylatedmatrix, is required for efficient viral DNA synthesis. The reverse transcription can be givenassuming Michaeli's Menten kinetics [12],

where [RNAcor] is the concentration of genomic RNA present in the viral core and [dNTP] isthe concentration of the dNTP pool of the host cell. Km(RNAcor) and Km(dNTP) are the Michaelisconstants for reverse transcriptase with the substrates RNAcor (= 2* [VF]) and the dNTP(Deoxyribonucleoside triphosphate) pool, respectively. kRNAcor is the degradation rate constantof the genomic RNA. Reverse transcription yields the HIV preintegration complex (PIC),composed of double-stranded viral cDNA, integrase, matrix, Vpr, reverse transcriptase, andthe high mobility group DNA-binding cellular protein HMGI(Y) [8]

where [DNAcor] is the concentration of genomic DNA present in the cytoplasm synthesizedby reverse transcription, kDNAcor is the rate constant for degradation of the proviral DNA inthe viral core, and kDNA t is the rate constant for transport of the DNA from the cytoplasm tothe nucleus. Integration of double-stranded viral DNA into the host chromosome is mediatedby integrase, which binds the ends of the viral DNA [8]. The viral DNA in the nucleus iscircularized, 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 assumethat the rate of circularization, kcirc is equal to the rate of integration knit. kDNA nuc is thedegradation rate constant of the nuclear DNA.

Transcription: 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 oneis likely to be transcriptionally active. The transcription rate of HIV-1 is assumed to be the sum

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of the transcription rates due to cellular factors alone and also cellular factors in conjunctionwith Tat:

Where Tc is the maximum of transcription induced by cellular transcription induced by cellulartranscription factors alone and TTC is equivalent to the basal transcription rate, which is about100-fold smaller than TC.

Transportation: The nuclear export of this assembly (viral RNA transcript, Rev, and CRM1/exportin 1) depends critically on yet another host factor, RanGTP.

HIV Replication: In contrast to Tat and Rev, which act directly on viral RNA structures, Nefmodifies the environment of the infected cell to optimize viral replication.

Assembly: New viral particles are assembled at the plasma membrane. Each virion consistsof roughly 1500 molecules of Gag and 100 Gag-Pol polyproteins, [14] two copies of the viralRNA genome, and Vpr.

Budding: Virion budding occurs through specialized regions in the lipid bilayer, yieldingvirions with cholesterol-rich membranes [9].

2.2.2 Cellular level Equations—In this model, at the molecular level, we distinguish threetypes of variables. Uninfected CD4+ cells. Two types of virus particles: wild type virus andmutant virus. Two groups of infected CD4+ cells infected by wild type virus Ci(t) and cellsinfected 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 ofthe model used: D (t) denotes the drug concentration at time t, β and βm are rates at which thewild type virus and mutant virus contact and bind to the CD4+ cells respectively. λ (t) the rateat which the uninfected cells are produced, from a pool of precursor cells, du the natural deathrate 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 ϕdenote the drug concentration and drug inhibition factor respectively; γi 's denote the rate atwhich an HIV-infected CD4 + cell of type i release virus. Thus, the kinetics of the uninfectedCD4+ 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 theproduct of reduction constant of immature cell and proportion of RNAcor to number of virionsbudding. IC(1/2) is half saturation constant of immature cell. The dynamics of cell infection bywild type virus and wild type virus production rates are represented by:

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

Where ai = πiβ(1−μ) for i = L, P, C, D. The parameter πi describes the probability that uponinfection a cell will become type i and the parameter μ is the probability of mutation from wildtype to resistant mutant virus.

is the maximum number of virus that an infected cell of type i can produce within its life timewith a survival probability until age a of the cell.

2.2.3 Effects of drugs—In the presence of drug, for example, fusion, kD will be modifiedto 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 ofHIV to successfully infect a cell. Hence in the equation for Cu the term βV+βmVm becomes

3 Simulation Results and DiscussionThe equations presented as well as other appropriately defined parameters in the model wereused to create the computational model. The model is based on several assumptions and someof the parameter estimates will undoubtedly be improved over time. We do see the need forand importance of using computational models to represent complex biomedical systems thatcan 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 toprevent the eventual collapse of the immune system in people with advanced HIV infection orAIDS. The explanation for such a collapse of the immune complex is not well understood. Oneof the main reasons for this is our inadequate knowledge of the dynamics and interaction ofthe CD4+ lymphocytes with HIV, especially in the presence of the different types of anti-retroviral drugs that are presently available. Computational epidemiologic models of cellularand molecular level dynamics, if successfully developed, can be used to investigate these typesof questions.

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AcknowledgmentsThis work is supported by a Research Centers in Minority Institutions (RCMI) Award, from the National Center forResearch Resources, National Institutes of Health

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Figure 1. An epidemiologic systems model of Intracellular HIV and CD4+ Cellular Populationdynamics

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