Volume 11 Supplement 1
Selected articles from the Eighth Asia-Pacific Bioinformatics Conference (APBC 2010)
Formal reasoning on qualitative models of coinfection of HIV and Tuberculosis and HAART therapy
- Anil Sorathiya^{1}Email author,
- Andrea Bracciali^{2} and
- Pietro Liò^{1}
DOI: 10.1186/1471-2105-11-S1-S67
© Sorathiya et al; licensee BioMed Central Ltd. 2010
Published: 18 January 2010
Abstract
Background
Several diseases, many of which nowadays pandemic, consist of multifactorial pathologies. Paradigmatic examples come from the immune response to pathogens, in which cases the effects of different infections combine together, yielding complex mutual feedback, often a positive one that boosts infection progression in a scenario that can easily become lethal. HIV is one such infection, which weakens the immune system favouring the insurgence of opportunistic infections, amongst which Tuberculosis (TB). The treatment with antiretroviral therapies has shown effective in reducing mortality.
An in-depth understanding of complex systems, like the one consisting of HIV, TB and related therapies, is an open great challenge, on the boundaries of bioinformatics, computational and systems biology.
Results
We present a simplified formalisation of the highly dynamic system consisting of HIV, TB and related therapies, at the cellular level. The progression of the disease (AIDS) depends hence on interactions between viruses, cells, chemokines, the high mutation rate of viruses, the immune response of individuals and the interaction between drugs and infection dynamics.
We first discuss a deterministic model of dual infection (HIV and TB) which is able to capture the long-term dynamics of CD4 T cells, viruses and Tumour Necrosis Factor (TNF). We contrast this model with a stochastic approach which captures intrinsic fluctuations of the biological processes. Furthermore, we also integrate automated reasoning techniques, i.e. probabilistic model checking, in our formal analysis. Beyond numerical simulations, model checking allows general properties (effectiveness of anti-HIV therapies) to be verified against the models by means of an automated procedure. Our work stresses the growing importance and flexibility of model checking techniques in bioinformatics.
In this paper we i) describe HIV as a complex case of infectious diseases; ii) provide a number of different formal descriptions that suitably account for aspects of interests; iii) suggest that the integration of different models together with automated reasoning techniques can improve the understanding of infections and therapies through formal analysis methodologies.
Conclusion
We argue that the described methodology suitably supports the study of viral infections in a formal, automated and expressive manner. We envisage a long-term contribution of this kind of approaches to clinical Bioinformatics and Translational Medicine.
Background
Human diseases result from abnormalities in an extremely complex system of molecular processes that are often caused by viral or bacterial infections. In these pathological processes, virtually no molecular entity acts in isolation and complexity is caused by the vast amount of dependencies between molecular and phenotype features. The key player of the human survival is the immune system which is a complex system that can be described as a large network of dynamical agents (cells and signalling molecules). A great challenge for contemporary molecular medicine is the modelling, description and ultimately the comprehension of the multistep and multiscale nature of the immune response to pathogens. An even greater challenge is when viral and bacterial infections occur together in the same patient. TB, which is caused by Mycobacterium tuberculosis, is the most frequent co-infection in patients infected by the human immunodeficiency virus type 1 (HIV-1). TB makes more complicated the development of effective therapies.
Efforts for gaining further insights into the pathological mechanisms and novel therapeutic targets benefit from the integration of genomic, proteomic, metabolomic and environmental information. In this work we follow an integrated approach that relies on several sources, such as phylogeny information, traditionally explored within Bioinformatics, differential equations and stochastic modelling, both largely used in Systems Biology, and a formal reasoning technique, viz. Model Checking, developed in Theoretical Computer Science to assess properties of complex computational systems. Furthermore, the Bioinformatics approach in the study of viral dynamics has also focused on identifying variable regions in genomes and pathogenic islands in bacteria [1]. These integrated frameworks appear to be effective for the research in the biomedical problems.
Mathematical and computer science approaches have shown to be more effective in dissecting the network connectivity of cellular circuits and the corresponding dynamical characteristics. The mathematical description of the variation of biomolecular concentrations as a set of ODEs offers quantitative basis for predicting the behaviour and evolution of the system and for testing non-linearities. An alternative approach is to use stochastic simulation via the Gillespie algorithm which provides an exact algorithmic solution of a set of reactions and a meaningful way to consider the noise [2]. The use of mathematical models in immunology has been very successful and has represented an insightful and essential complement to in vivo and in vitro experimental design and interpretation. Kinetic Model suggests that HIV-1 in vivo is continuous and highly productive and that leads CD4 T cells count low [3]. Nowak and May has proposed a simple model on HIV dynamics by considering population of non-infected T cells, infected T cells and viral population [4]. Perelson has proposed mathematical models on interaction of HIV virus and T cells dynamic by considering one more variable (un-infectious viruses) [5]. The HIV quasi-species models have been inspired from the molecular quasi-species model in chemistry [6, 7]. Indeed mathematical models of HIV dynamics have proven valuable in understanding the mechanisms of many of the observed features of the progression of the HIV infection, see for example [3, 6–13]. In this paper we model coinfection of HIV and TB and the effect of HAART therapy on the model dynamics.
Recently, the observation that biological systems often exhibit interactive and concurrent behavior, similarly to computational concurrent systems, has led to the adoption of formal methods originally developed for the description and analysis of complex software systems in computer science. This abstraction "cell as computation", similar to the "DNA as string" and "protein as labeled graph" abstractions which have originated bioinformatics, has inspired the adoption of model checking methodologies to validate biological complex systems [14]. The growing success of model checking (see for instance [15–18]) relies in the specification of a biological property of interest which is expressed in a formal language, typically a formula of a suitable logic, and its verification is carried out by a fully automated procedure that returns either a positive response or a counter example [19].
The aim of this paper is to pipeline bioinformatics and quantitative models of infectious processes and anti-HIV therapies, and then show how model checking techniques can contribute to the interpretation of the in-silico results obtained from quantitative models of infectious diseases dynamics. Given that viral infections and the functioning of therapies often present stochastic aspects, we introduce a succinct but descriptive stochastic model of HIV infection associated with TB opportunistic infection. Then we extend it by modelling (the effects of) HAART anti-HIV therapy. These models have then been implemented in PRISM, a state of the art probabilistic model checker supporting a logical language [20]. We illustrate, by means of two simple properties, the flavour of the verification made possible by these techniques and how this can contribute to a precise assessment of the information conveyed by the mathematical models.
The complexity of HIV infection and anti-HIV therapies
Human immunodeficiency virus type 1 (HIV-1) infection is characterised by the progressive loss of CD4 T cells. Anti-HIV therapies act to eradicate or lower the concentration of the virus from the body and replenish the CD4 T cells reservoir. Infection by most strains of HIV requires interaction with CD4 T cells and a chemokine receptor, either CXCR4 or CCR5. Viral strains often use CCR5 during early stages of HIV-1 and then switches to CXCR4 to enter into the cells. This switch emerges in more than 50% of patients [21, 22] and it has been linked with progression to AIDS because of an increased virulence through the formation of cell syncytia and the stimulation of the cellular factor called Tumour Necrosis Factor (TNF) which inhibits the replication of R5 HIV strains while has no effect on X4 HIV [23, 24].
It takes on an average 10 years to get infected by AIDS after having HIV infection. Some patients died within 2 years after getting infected by HIV, while others remained free of AIDS for more than 15 years. The within-patient evolutionary process of viral sequence mutations during HIV infection has suggested improvements in anti-HIV therapies. Anti-HIV drugs are most effective when taken in a combination of three or more at the same time. This is called combination therapy or HAART (Highly Active Antiretroviral Therapy). Physicians recommend starting the therapies if you are ill because of HIV, or if your CD4 T cells count if low (below 200 cells per microL). HAART combinations usually include two drugs that are nucleoside analogues, and one protease inhibitor. The nucleoside analogues drugs result in targeting the viral reverse transcriptase which codes the viral RNA into the DNA that can be integrated into human cells, so transforming the cell into a factory for building blocks of the virus. The protease inhibitor acts as preventing an infected cell from producing new infectious virus particles.
Bioinformatic links between HIV and TB
The disruption caused by the dual infection (HIV and TB) focuses on RANTES which blocks CCR5 and whose expression is upregulated by TNF. Therefore we incorporated TNF in our deterministic model to predict the potential effect of HAART on coinfection of HIV and TB. That evidence would explain the increase in CXCR4 usage with the increase of TNF concentration [31]. The relationship between HIV and TB seems to be very profound since quasispecies of both HIV and TB influence each other differently [32, 33]. Our phylogenetic results suggest that both diseases act synergically and not simply additively to alter the chemokine network.
Results and Discussion
Although differential equation models have long been used for the immune system and viral infection modelling, they focus on the average behavior of large populations of perfectly mixed, identical individuals. An improved realism is perhaps provided by stochastic simulations, which however are computationally intensive. Given that differential equations and stochastic descriptions have important pros and cons and a good degree of complementarity, a framework based on the implementation of both approaches, although time and resources expensive, appears advisable. Stochastic models, then, support the well developed probabilistic model checking analysis. We have extended a model firstly presented in [7]. We have incorporated the Cytotoxic T Lymphocytes (CTLs) response and the dynamics of TB opportunistic infection so as to analyse the co-infection of HIV strains and another disease associated with it.
Furthermore, we have introduced an abstract representation of the HAART therapy treatment by altering the model's parameters that rule the dynamics of our model according to the known effects of the treatment. We are presenting results of the analysis of the HIV and the opportunistic TB infection dynamics and an associated therapy through differential equations, stochastic modelling and formal reasoning techniques.
HIV and the opportunistic TB infection
Modelling HAART therapy against HIV and TB
A stochastic description of HAART, HIV and TB dynamics
Many natural and biological phenomena are intrinsically stochastic and discrete, and they can not always be properly described by means of a deterministic and continuous description. For instance, systemic emergent properties can be sensitive to the local presence of minimal (integer) quantities of molecules [34]. Roughly speaking, a stochastic model associates a probability to each state transition of the modelled system, expressed as a rate associated to the transition. Associated probability distributions are typically memoryless, and hence the overall system behaviour can be interpreted as a Markov Chain. Often, models are used to simulate possible evolutions of the system from initial conditions, [35].
Starting from the deterministic model of HIV, TB and HAART, we have determined a corresponding stochastic model. This has been done via standard transformation from deterministic rates into stochastic ones, according to a fixed reference volume of the model (see, e.g., [36]).
Assessing HAART therapy against HIV and TB infection
Observing the quantitative results of the stochastic simulations in Figure 4, one can have an idea about the fact that the HAART therapy is somehow effective. However, we discuss now an example on how it is possible to move from a correct and informative, but somehow empirical, interpretation of data, as the idea above, to a more precise and formal approach by specifying properties of interest as logical formulae. This has the advantage of requiring a definition of all the aspects of interest, of referring to a formal and unambiguous semantics, and of having an automated procedure for verifying formulae (as the one we adopted [20]). Differently, observations rely on the capability of properly interpreting the graphs, which sometimes may appear obscure and difficult to interpret. More details about the methodology can be found in the Methods section.
Therapy effects
We focus on two properties that can give a measure of the infection progression.
where R{"i_to_v"} represents the number of occurrences of an event labelled with i_to_v, i.e. the transition from an infected CD4 T cell to virus replication. The question mark says we are interested in determining the number of such possible occurrences in the model dynamics. C < 5.0 stands for the time interval;
Experimental results
Quantitative results from the automated verification of the effects of HAART therapy. While the number of viral replications due to CD4 T cell infection is comparable with and without HAART treatment for the time interval considered, a much stronger probability of a failure of the immune system, close to twice as much as, is observed without the HAART treatment.
HIV+TB | HIV+TB+HAART | |
---|---|---|
R{"i_to_v"} = ? [C < = 9.0] | 250 | 269 |
P = ? [F[5.0, 8.0]tc < = 20] | 0.885 | 0.429 |
Automated verification yields quantitative measures of the investigated properties. While the number of viral replications due to CD4 T cell infection is comparable (but without HAART many other infected cells contribute to virus replication), the probability of a failure of the immune system is much stronger without HAART treatment, closely to twice as much as.
Conclusion
We have illustrated the potential benefits of formal methods and quantitative models when applied to the study of viral infections and therapy assessment within a computational bioinformatics approach. We have done this by presenting experiments on a proof-of-concept scenario regarding HIV infection and the relative TB opportunistic infection. We have adopted an integrated approach, combining deterministic and stochastic techniques and illustrating how properties of interests for the study of viral infections can be formalised in a general purpose logic, as the one supported by PRISM. Our work stresses the growing importance and flexibility of model checking techniques in bioinformatics. Noteworthily, the verification of these properties can precisely characterise the numerical results of simulations, and this can be helpful in comparing and assessing different antiviral therapies. In conclusion, the modelling of HIV infection has two important linked benefits. i) it has matured to the stage of allowing us to combine bioinformatics, computational modelling and formal reasoning techniques, as model checking, i.e. it provides a solid bridge between biological systems and the computational objects used to describe them; ii) these approaches together capture information that can be valuable in therapy validation suggesting the possibility of moving in the future to realistic cases, i.e. translational medicine and clinical bioinformatics.
Methods
In this section we provide further details about the construction of the used models and analysis methodologies.
A deterministic model of HIV strains and TB time evolution
Our work is based on a deterministic model of HIV-1 dynamics, firstly appeared in [7], which takes into account the models developed by Perelson and his followers [3, 10, 13, 37]. These models are well presented and take specific biological reality into account. Our initial model has been extended here by adding two more variables, Cytotoxic T Lymphocytes (CTLs) cells and TB, to capture dynamics of co-infection of HIV and TB.
The stochastic model
The stochastic model has been derived from the deterministic one. As a simplification we consider only two viral strains: R5 viral strain (V 5) and X4 viral strain (V 4), and correspondingly only two kinds of infected cells (I4, I5 and I* for the union of the two). Analogously, quasispecies dynamics revert to V 5 strains that transform into V 4 strains (at rate β_{k 54}). These two strains are the ones we focus on when observing the time course of infections. As for the deterministic model, we have included sufficient details of interaction among the species to express the properties of interest.
As mentioned, we needed to translate some parameters from deterministic to stochastic ones, accordingly to the reference volume of interest. Other parameters, particularly due to lack of existing data in literature, have been approximated by tuning the model on known macroscopic behaviour. We considered a volume of reaction large enough to contain a statistical reliable number of agents (viruses and cells) and values of extensive quantities are scaled according with the reaction volume considered.
(01) |
| N_{ U }= 100 |
---|---|---|
(02) |
| δ^{ U }= 0.1 |
(03) |
| = 0.00001 |
(04) |
| β_{k 5}= 0.0025 |
(05) |
| β_{k 4}= 0.0025 |
(06) |
| β_{k 54}= 0.025 |
(07) |
| δ_{ T }= 0.1 |
(08) |
| δ_{ I }= 0.8 |
(09) |
| δ_{ I }= 0.8 |
(10) |
| k_{ Z }= 0.01 |
(11) |
| = 0.004 |
(12) |
| π = 2.5 |
(13) |
| π = 2.5 |
(14) |
| c_{5} = 2.5 |
(15) |
| c = 2.5 |
(16) |
| K_{ F }= 0.1 |
(17) |
| α = 2.5 if T < 20 otherwise 0 |
(18) |
| K_{ B }= 10 |
(19) |
| = 0.00001 if T < 20 otherwise 0 |
(20) |
| δ^{ F }= 0.001 |
(21) |
| b = 0.2 |
where I* stands for I 4 and I 5 - two rules. |
As far as (the effects of) HAART therapy is concerned, it has been modelled by means of a stochastic triggering event that activates and deactivates the treatment. Activation consists in the modification of interactions (12) and (13), whose rate is downgraded to 1, representing reduced morbidity of the virus, which is one of the main effects of HAART.
Probabilistic Model Checking
Being an introduction to probabilistic model checking beyond the scope of this paper, we refer the interested reader to the cited literature and references therein. Other recent works are [15, 16, 38]. We introduce informally the main concepts by examples.
The information encoded in a stochastic model describes, roughly speaking, the states in which the modelled system can find itself and the associated probability of being in a state at a certain time. For many natural phenomena that follow a memory-less probability distribution this amounts to a (Continuous Time) Markov Chain (CTMC). Furthermore, given a stochastic model, several algorithms to simulate the possible quantitative evolutions of the system have been defined, noticeably the Gillespie's algorithm [2]. State semantics, such as Transition Systems, have been extensively studied and used to precisely describe the behaviour of dynamical systems, like distributed and concurrent computational systems. State semantics may often become hardly manageable due to the exponential explosion of the number of states. However, several efficient and automated analysis techniques have been defined. Logical formulas can represent "homogeneous" sets of states. A formula like
which, informally speaking, represents all the traces for which ϕ holds with probability bigger than p. Other examples are F >= t x > 2, the traces in which eventually x > 2 after t time units, and G x < 10, globally, for all the traces, the value of x is less than 10.
PRISM
Declarations
Acknowledgements
The authors wish to thank the anonymous referees for their helpful comments on a previous version of this paper. This project is partially supported by EC IST SOCIALNETS and WADA projects.
This article has been published as part of BMC Bioinformatics Volume 11 Supplement 1, 2010: Selected articles from the Eighth Asia-Pacific Bioinformatics Conference (APBC 2010). The full contents of the supplement are available online at http://www.biomedcentral.com/1471-2105/11?issue=S1.
Authors’ Affiliations
References
- Lió P, Vannucci M: Finding pathogenicity islands and gene transfer events in genome data. Bioinformatics 2000, 16: 932–940. 10.1093/bioinformatics/16.10.932View ArticlePubMedGoogle Scholar
- Gillespie DT: Exact Stochastic Simulation of Coupled Chemical Reactions. The Journal of Physical Chemistry 1977, 81(25):2340–2361. 10.1021/j100540a008View ArticleGoogle Scholar
- Ho D, Neumann AU, Perelson AS, Chen W, Leonard JM, M M: Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection. Nature 1995, 373: 123–126. 10.1038/373123a0View ArticlePubMedGoogle Scholar
- Nowak MA, May RM: Virus dynamics. mathematical principles of immunology and virology. Oxford University Press; 2000.Google Scholar
- Perelson AS: Modelling viral and immune system dynamics. Nature Review Immunology 2002, 2(1):28–36. 10.1038/nri700View ArticleGoogle Scholar
- Eigen M, McCaskill J, Schuster P: The Molecular Quasi-Species. Advances in Chemical Physics 1989, 75: 149–263. full_textGoogle Scholar
- Sguanci L, Bagnoli F, Li P: Modeling HIV quasispecies evolutionary dynamics. BMC Evolutionary Biology 2007, 7(2):S5. 10.1186/1471-2148-7-S2-S5PubMed CentralView ArticlePubMedGoogle Scholar
- Ronga L, Perelson AS: Modeling HIV persistence, the latent reservoir, and viral blips. Journal of Theoretical Biology 2009.Google Scholar
- Bagnoli F, Lio' P, Sguanci L: Modeling viral coevolution. HIV multi-clonal persistence and competition dynamics. Physica A 2005, 366: 333–346. 10.1016/j.physa.2005.10.055View ArticleGoogle Scholar
- Chao L, Davenport MP, Forrest S, Perelson AS: A stochastic model of cytotoxic T cell responses. Journal Theoretical Biology 2004, 228: 227–240. 10.1016/j.jtbi.2003.12.011View ArticleGoogle Scholar
- Celada F, Seiden PE: Affinity maturation and hypermutation in a simulation of the humoral immune response. Journal of Theoretical Biology 1996, 26: 1350–1358.Google Scholar
- De Boer RJ, Perelson AS: Towards a general function describing T cell proliferation. European Journal of Immunology 1995, 175(4):567–576.Google Scholar
- Wiegel FW, Perelson AS: Some scaling principles for the immune system. Immunology and Cell Biology 2004, 82: 127–131. 10.1046/j.0818-9641.2004.01229.xView ArticlePubMedGoogle Scholar
- Regev A, Shapiro E: Cells as computation. Nature 2002, 419: 343. 10.1038/419343aView ArticlePubMedGoogle Scholar
- Mateus D, Gallois JP, Comet ea: Symbolic modeling of genetic regulatory networks. Journal of Bioinformatics and Computational Biology 2007, 5(2B):627–640. 10.1142/S0219720007002850View ArticlePubMedGoogle Scholar
- Fromentin J, Comet JP, Le Gall P, Roux O: Analysing gene regulatory networks by both constraint programming and model-checking. Conference proceedings, IEEE Engineering in Medicine and Biology 2007, 4595–4598.Google Scholar
- Batt G, Yordanov B, Weiss R, Belta C: Robustness analysis and tuning of synthetic gene networks. Bioinformatics 2007, 23: 2415–2422. 10.1093/bioinformatics/btm362View ArticlePubMedGoogle Scholar
- Batt G, Ropers D, De Jong H, Geiselmann J, Mateescu R, Page M, Schneider D: Validation of qualitative models of genetic regulatory networks by model checking. Analysis of the nutritional stress response in E. coli. Bioinformatics 2005, 21(Suppl 1):i19-i28. 10.1093/bioinformatics/bti1048View ArticlePubMedGoogle Scholar
- Clarke EM, Grumberg O, Peled DA: Model Checking. The MIT Press; 1999.Google Scholar
- The PRISM model checker[http://www.prismmodelchecker.org]
- Karlsson I, Antonsson L, Shi Y, Oberg M, Karlsson A, Albert J, Olde B, Owman C, M J, Fenyo EM: Coevolution of RANTES sensitivity and mode of CCR5 receptor use by human immunodeficiency virus type 1 of the R5 phenotype. Journal of Virology 2004, 78: 11807–11815. 10.1128/JVI.78.21.11807-11815.2004PubMed CentralView ArticlePubMedGoogle Scholar
- Gorry PR, Churchill M, Crowe SM, Cunningham AL, Gabuzda D: Pathogenesis of macrophage tropic HIV. Current HIV Research 2005, 3: 53–60. 10.2174/1570162052772951View ArticlePubMedGoogle Scholar
- Gray L, Sterjovski J, Churchill M, Ellery P, Nasr N, Lewin SR, Crowe SM, Wesselingh SL, Cunningham AL, Gorry PR: Uncoupling coreceptor usage of human immunodeficiency virus type 1 (HIV-1) from macrophage tropism reveals biological properties of CCR5-restricted HIV-1 isolates from patients with acquired immunodeficiency syndrome. Virology 2005, 337: 384–398. 10.1016/j.virol.2005.04.034View ArticlePubMedGoogle Scholar
- Pavlakis GN, Valentin A, Morrow M, Yarchoan R: Differential effects of TNF on HIV-1 expression. R5 inhibition and implications for viral evolution. Internation Conference AIDS 2004, 15: 11–16.Google Scholar
- Parczewski M, Leszczyszyn-Pynka M, M K, Adler G, Binczak-Kuleta A, Loniewska B, Boron-Kaczmarska A, Ciechanowicz A: Sequence variants of chemokine receptor genes and susceptibility to HIV-1 infection. Journal of Applied Genetics 2009, 50(2):159–166.View ArticlePubMedGoogle Scholar
- Hatse S, Huskens D, Princen K, Vermeire K, Bridger GJ, De Clercq E, Rosenkilde MM, Schwartz TW, D S: Modest human immunodeficiency virus coreceptor function of CXCR3 is strongly enhanced by mimicking the CXCR4 ligand binding pocket in the CXCR3 receptor. Journal of Virology 2007, 81(7):3632–3639. 10.1128/JVI.01941-06PubMed CentralView ArticlePubMedGoogle Scholar
- Wen W, Chen S, Cao Y, Zhu Y, Yamamoto Y: HIV-1 infection initiates changes in the expression of a wide array of genes in U937 promonocytes and HUT78 T cells. Virus Research 2005, 113(1):26–35. 10.1016/j.virusres.2005.04.002View ArticlePubMedGoogle Scholar
- Mayanja-Kizza H, Wu M, Aung H, Liu S, Luzze H, Hirsch C, Z T: The interaction of monocyte chemoattractant protein-1 and tumour necrosis factor-alpha in Mycobacterium tuberculosis-induced HIV-1 replication at sites of active tuberculosis. Scandinavian Journal of Immunology 2009, 69(6):516–520. 10.1111/j.1365-3083.2009.02246.xView ArticlePubMedGoogle Scholar
- Maddocks S, Scandurra GM, Nourse C, Bye C, Williams RB, Slobedman B, Cunningham AL, J BW: Gene expression in HIV-1/Mycobacterium tuberculosis co-infected macrophages is dominated by M. tuberculosis. Tuberculosis (Edinb) 2009, 89(4):285–293. 10.1016/j.tube.2009.05.003View ArticleGoogle Scholar
- Volpe E, Cappelli G, Grassi M, Martino A, Serafino A, Colizzi V, Sanarico N, Mariani F: Gene expression profiling of human macrophages at late time of infection with Mycobacterium tuberculosis. Immunology 2006, 118(4):449–460.PubMed CentralPubMedGoogle Scholar
- Hirano F, Komura K, Fukawa E, Makino I: Tumor necrosis factor alpha (TNF-alpha)-induced RANTES chemokine expression via activation of NF-kappaB and p38 MAP kinase. roles of TNF-alpha in alcoholic liver diseases. Journal of Hepatology 2003, 38(4):483–489. 10.1016/S0168-8278(02)00456-7View ArticlePubMedGoogle Scholar
- Ranjbar S, Boshoff HI, Mulder A, Siddiqi N, Rubin EJ, Goldfeld AE: HIV-1 replication is differentially regulated by distinct clinical strains of Mycobacterium tuberculosis. PLoS One 2009, 4(7):e6116. 10.1371/journal.pone.0006116PubMed CentralView ArticlePubMedGoogle Scholar
- Collins KR, Quinones-Mateu ME, Wu M, Luzze H, Johnson JL, Hirsch C, Toossi Z, Arts EJ: Human immunodeficiency virus type 1 (HIV-1) quasispecies at the sites of Mycobacterium tuberculosis infection contribute to systemic HIV-1 heterogeneity. Journal of Virology 2002, 76(4):1697–1706. 10.1128/JVI.76.4.1697-1706.2002PubMed CentralView ArticlePubMedGoogle Scholar
- Wilkinson D: Stochastic Modelling for Systems Biology. CRC press; 2006.Google Scholar
- Van-Kampen NG: Stochastic processes in physics and in chemistry. Elsevier, Amsterdam; 1992.Google Scholar
- Kierzek AM: Stocks. Stochastic kinetic simulations of biochemical system with Gillespie algorithm. Bioinformatics 2002, 18: 470–481. 10.1093/bioinformatics/18.3.470View ArticlePubMedGoogle Scholar
- Perelson AS, Neumann AU, Markowitz M, Leonard JM, Ho DD: HIV-1 Dynamics in Vivo. Virion Clearance Rate, Infected Cell Life-Span, and Viral Generation Time. Science 1996, 271: 1582–1586. 10.1126/science.271.5255.1582View ArticlePubMedGoogle Scholar
- Fisher J, Piterman N, Hajnal A, Henzinger TA: Predictive modeling of signaling crosstalk during C. elegans vulval development. PLoS Computational Biology 2007, 3(5):e92. 10.1371/journal.pcbi.0030092PubMed CentralView ArticlePubMedGoogle Scholar
- Aziz A, Sanwal K, Singhal V, Brayton R: Model checking continuous time Markov chains. ACM Transactions on Computational Logic 2000, 1(1):162–170. 10.1145/343369.343402View ArticleGoogle Scholar
- Baier C, Haverkort B, Hermanns H, Katoen JP: Model-checking algorithms for continuous-time Markov chains. IEEE Transactions on Software Engineering 2003, 29(6):524–541. 10.1109/TSE.2003.1205180View ArticleGoogle Scholar
- Kwiatkowska M, Norman G, Parker D: Quantitative Verification Techniques for Biological Processes. Edited by: Condon A, Harel D, Kok J, Salomaa A, Winfree E. Algorithmic Bioprocesses, Springer; 2009.View ArticleGoogle Scholar
- Bruch CL, Chao L: Evolvability of an RNA virus determined by its mutational neighborhood. Nature 2000, 406: 625–628. 10.1038/35020564View ArticleGoogle Scholar
- Asquith B, Edwards CTT, Lipsitch M, McLean AR: Inefficient Cytotoxic T Lymphocyte Mediated Killing of HIV-1 Infected Cells In Vivo. PLOS Biology 2006, 4(4):e90. 10.1371/journal.pbio.0040090PubMed CentralView ArticlePubMedGoogle Scholar
- Althaus CL, Be Boer RJ: Dynamics of Immune escape during HIV/SIV Infection. PLOS Computational Biology 2008, 4(7):e1000103. 10.1371/journal.pcbi.1000103PubMed CentralView ArticlePubMedGoogle Scholar
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