Abstract
The dynamic pattern of viral load in a patient’s body critically depends on the host’s genes. For this reason, the identification of those genes responsible for virus dynamics, although difficult, is of fundamental importance to design an optimal drug therapy based on patients’ genetic makeup. Here, we present a differential equation (DE) model for characterizing specific genes or quantitative trait loci (QTLs) that affect viral load trajectories within the framework of a dynamic system. The model is formulated with the principle of functional mapping, originally derived to map dynamic QTLs, and implemented with a Markov chain process. The DE-integrated model enhances the mathematical robustness of functional mapping, its quantitative prediction about the temporal pattern of genetic expression, and therefore its practical utilization and effectiveness for gene discovery in clinical settings. The model was used to analyze simulated data for viral dynamics, aimed to investigate its statistical properties and validate its usefulness. With an increasing availability of genetic polymorphic data, the model will have great implications for probing the molecular genetic mechanism of virus dynamics and disease progression.
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References
Bertsekas DP (2003) Nonlinear programming, 2nd edn. Athena Scientific, Belmont
Bonhoeffer S, Coffin JM, Nowak MA (1997) Human immuodeficiency virus drug therapy and virus load. J Virol 71: 6971–6976
Bonhoeffer S, May RM, Shaw GM, Nowak MA (1999) Virus dynamics and drug therapy. Proc Natl Acad Sci USA 94: 6971–6976
Bremaud P (1999) Markov chains: Gibbs fields, Monte Carlo simulation and queues. Springer, New York
Churchill GA, Doerge RW (1994) Empirical threshold values for quantitative trait mapping. Genetics 138: 963–971
Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via EM algorithm. J R Stat Soc B 39: 1–38
Ho DD, Neumann AU, Perelson AS, Chen W, Leonard JM, Markowitz M (1999) Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection. Nature 373: 123–126
Lander ES, Bostein D (1989) Mapping Mendelian factors underlying quantitative traits using RFLP linkage maps. Genetics 121: 185–199
Little RJA, Rubin DB (2002) Statistical analysis with missing data, 2nd edn. Wiley, New York
Li Q, Wu RL (2009) A multilocus odel for constructing a linkage disequilibrium map in human populations. Stat Appl Mol Genet Biol 8:Article 18
Ma CX, Casella G, Wu RL (2002) Functional mapping of quantitative trait loci underlying the character process: a theoretical framework. Genetics 161: 1751–1762
Nowak MA, May RM (2000) Virus dynamics. Oxford University, New York
Perelson AS, Neumann AU, Markowitz M, Leonard JM, Ho DD (1996) HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science 271: 1582–1586
Perelson AS, Essunger P, Cao Y, Vesanen M, Hurley A, Saksela K, Markowitz M, Ho DD (1997) Decay characteristics of HIV-1-infected compartments during combination therapy. Nature 387: 188–191
Ribeiro RM, Bonhoeffer S (2000) Production of resistant HIV mutants during antiretroviral therapy. Proc Natl Acad Sci USA 97: 7681–7686
Sedaghat AR, Dinoso JB, Shen L, Wilke CO, Siliciano RF (2008) Decay dynamics of HIV-1 depend on the inhibited stages of the viral life cycle. Proc Natl Acad Sci 105: 4832–4837
The International HapMap Consortium (2003) The International HapMap Project. Nature 426:789–794
Wang ZH, Wu RL (2004) A statistical model for high-resolution mapping of quantitative trait loci determining HIV dynamics. Stat Med 23: 3033–3051
Wang ZH, Hou W, Wu RL (2005) A statistical model to analyze quantitative trait locus interactions for HIV dynamics from the virus and human genomes. Stat Med 25: 495–511
Wei X, Ghosh SK, Taylor ME, Johnson VA, Emini EA, Deutsch P, Lifson JD, Bonhoeffer S, Nowak MA, Hahn BH, Saag MS, Shaw GM (2004) Viral dynamics in human immunodeficiency virus type 1 infection. Nature 23: 3033–3051
Wodarz D, Nowak MA (2000) HIV therapy: managing resistance. Proc Natl Acad Sci USA 97: 8193–8195
Wu RL, Lin M (2006) Functional mapping—how to study the genetic architecture of dynamic complex traits. Nat Rev Genet 7: 229–237
Wu RL, Ma C-X, Casella G (2002) Joint linkage and linkage disequilibrium mapping of quantitative trait loci in natural populations. Genetics 160: 779–792
Wu RL, Ma CX, Lin M, Casella G (2004a) A general framework for analyzing the genetic architecture of developmental characteristics. Genetics 166: 1541–1551
Wu RL, Ma CX, Lin M, Wang ZH, Casella G (2004b) Functional mapping of growth quantitative trait loci using a transform-both-sides logistic model. Biometrics 60: 729–738
Wu RL, Wang ZH, Zhao W, Cheverud JM (2004c) A mechanistic model for genetic machinery of ontogenetic growth. Genetics 168: 2383–2394
Wu S, Yang J, Wu RL (2006) Multilocus linkage disequilibrium mapping of quantitative trait loci that affect HIV dynamics: a simulation approach. Stat Med 25: 3826–3849
Zhao W, Hou W, Littell RC, Wu RL (2005) Structured antedependence models for functional mapping of multivariate longitudinal quantitative traits. Stat Appl Mol Genet Biol 4:Article 33
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Luo, J., Hager, W.W. & Wu, R. A differential equation model for functional mapping of a virus-cell dynamic system. J. Math. Biol. 61, 1–15 (2010). https://doi.org/10.1007/s00285-009-0288-1
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DOI: https://doi.org/10.1007/s00285-009-0288-1