Effective Single-Step Posttranscriptional Dynamics Allowing for a Direct Maximum Likelihood Estimation of Transcriptional Activity and the Quantification of Sources of Gene Expression Variability with an Illustration for the Hypoxia and TNFα Regulated Inflammatory Pathway
Data analysis methods for estimating promoter activity from gene reporter data frequently involve the reconstruction of the dynamics of unobserved species and numerical search algorithms for determining optimal model parameters. In contrast, we argue that posttranscriptional dynamics effectively behave like a singlestep stochastic process when gene expression variability is relatively low and, half-lives of the unobserved species are relatively small compared to characteristic observation time scales. In this case, by means of maximum likelihood estimators, for which analytical expressions exist, transcriptional activity of gene promoters can be estimated directly from observed gene reporter data without the need for numerical search algorithms and the reconstruction of unobserved variables. In addition, the model-based data analysis approach yields a single variable that measures the effective strength of the sources that give rise to gene expression variability. The approach is applied to conduct a model-based analysis of the inflammatory pathway under hypoxia condition and stimulation with tumor necrosis factor alpha in HEK293 cells. 1. Introduction A problem in the field of computational biology is how to model and determine quantitatively promoter activity from observed reporter data. Deterministic approaches suggest that when the activity of a promoter is constant over a period of time, then reporter data should be linearly increasing (see Section 2.1). The steepness of the increase is proportional to the activity of the promoter. Linear regression analysis may be applied to determine the rate of increase [1]. This deterministic perspective is limited in its scope, which becomes clear when considering stochastic approaches as alternatives. For example, deterministic approaches regard gene expression fluctuations as errors. In contrast, according to stochastic approaches gene expression fluctuations indicate that the transcriptional machinery functions properly because the machinery is based on biochemical reactions that are stochastic in the very nature. Stochastic accounts, for example, based on chemical Langevin equations, provide a mathematical framework to address both the deterministic and stochastic components of promoter activity [2]. In the literature, modeling of gene transcription often starts at the promoter level with the transcription event [3–6]. Accordingly, mRNA is produced at a certain rate . Subsequently, mRNA is translated into proteins at a rate . The proteins are finally exported out of the cell with an export rate .
References
[1]
U. Bruning, S. F. Fitzpatrick, M. Birtwistle, T. Frank, C. T. Taylor, and A. Cheong, “NFκB and HIF display synergistic behaviour during hypoxic inflammation,” Cellular and Molecular Life Sciences, vol. 69, no. 8, pp. 1319–1329, 2012.
[2]
D. J. Wilkinson, Stochastic Modelling for Systems Biology, Chapman & Hall/CRC, London, UK, 2006.
[3]
B. Finkenst?dt, E. A. Heron, M. Komorowski et al., “Reconstruction of transcriptional dynamics from gene reporter data using differential equations,” Bioinformatics, vol. 24, no. 24, pp. 2901–2907, 2008.
[4]
T. D. Frank, A. Cheong, M. Okada-Hatakeyama, and B. N. Kholodenko, “Catching transcriptional regulation by thermostatistical modeling,” Physical Biology, vol. 9, no. 4, Article ID 045007, 2012.
[5]
C. V. Harper, B. Finkenst?dt, D. J. Woodcock et al., “Dynamic analysis of stochastic transcription cycles,” PLoS Biology, vol. 9, no. 4, Article ID e1000607, 2011.
[6]
P. J. Ingram, M. P. H. Stumpf, and J. Stark, “Network motifs: structure does not determine function,” BMC Genomics, vol. 7, article 108, 2006.
[7]
J. Alam and J. L. Cook, “Reporter genes: application to the study of mammalian gene transcription,” Analytical Biochemistry, vol. 188, no. 2, pp. 245–254, 1990.
[8]
A. J. Millar, S. R. Short, N.-H. Chua, and S. A. Kay, “A novel circadian phenotype based on firefly luciferase expression in transgenic plants,” Plant Cell, vol. 4, no. 9, pp. 1075–1087, 1992.
[9]
J. F. Schmedtje Jr., Y.-S. Ji, W.-L. Liu, R. N. DuBois, and M. S. Runge, “Hypoxia induces cyclooxygenase-2 via the NF-κB p65 transcription factor in human vascular endothelial cells,” Journal of Biological Chemistry, vol. 272, no. 1, pp. 601–608, 1997.
[10]
A. J. Millar, I. A. Carre, C. A. Strayer, N.-H. Chua, and S. A. Kay, “Circadian clock mutants in Arabidopsis identified by luciferase imaging,” Science, vol. 267, no. 5201, pp. 1161–1163, 1995.
[11]
T. Nakakuki, M. R. Birtwistle, Y. Saeki et al., “Ligand-specific c-fos expression emerges from the spatiotemporal control of ErbB network dynamics,” Cell, vol. 141, no. 5, pp. 884–896, 2010.
[12]
S. Kuttykrishnan, J. Sabina, L. L. Langton, M. Johnston, and M. R. Brent, “A quantitative model of glucose signaling in yeast reveals an incoherent feed forward loop leading to a specific, transient pulse of transcription,” Proceedings of the National Academy of Sciences of the United States of America, vol. 107, no. 38, pp. 16743–16748, 2010.
[13]
E. Maltepe and O. D. Saugstad, “Oxygen in health and disease: regulation of oxygen homeostasis-clinical implications,” Pediatric Research, vol. 65, no. 3, pp. 261–268, 2009.
[14]
G. L. Semenza, “Mechanisms of disease: oxygen sensing, homeostasis, and disease,” The New England Journal of Medicine, vol. 365, no. 6, pp. 537–547, 2011.
[15]
R. H. Wenger, “Mammalian oxygen sensing, signalling and gene regulation,” Journal of Experimental Biology, vol. 203, no. 8, pp. 1253–1263, 2000.
[16]
G. L. Semenza, “Hypoxia-inducible factors in physiology and medicine,” Cell, vol. 148, no. 3, pp. 399–408, 2012.
[17]
J. Pouysségur, F. Dayan, and N. M. Mazure, “Hypoxia signalling in cancer and approaches to enforce tumour regression,” Nature, vol. 441, no. 7092, pp. 437–443, 2006.
[18]
R. Deulofeut, A. Critz, I. Adams-Chapman, and A. Sola, “Avoiding hyperoxia in infants ≤1250g is associated with improved short- and long-term outcomes,” Journal of Perinatology, vol. 26, no. 11, pp. 700–705, 2006.
[19]
O. D. Saugstad, “Optimal oxygenation at birth and in the neonatal period,” Neonatology, vol. 91, no. 4, pp. 319–322, 2007.
[20]
H. K. Eltzschig and P. Carmeliet, “Hypoxia and inflammation,” The New England Journal of Medicine, vol. 364, no. 7, pp. 656–665, 2011.
[21]
L. K. Nguyen, M. A. S. Cavadas, C. C. Scholz et al., “A dynamic model of hypoxia-induced factor 1-alpha (HIF-1alpha) network,” Journal of Cell Sciences, vol. 126, pp. 1454–1463, 2013.
[22]
C. Culver, A. Sundqvist, S. Mudie, A. Melvin, D. Xirodimas, and S. Rocha, “Mechanism of hypoxia-induced NF-kappaB,” Molecular and Cellular Biology, vol. 30, no. 20, pp. 4901–4921, 2010.
[23]
E. P. Cummins, E. Berra, K. M. Comerford et al., “Prolyl hydroxylase-1 negatively regulates IκB kinase-β, giving insight into hypoxia-induced NFκB activity,” Proceedings of the National Academy of Sciences of the United States of America, vol. 103, no. 48, pp. 18154–18159, 2006.
[24]
S. F. Fitzpatrick, M. M. Tambuwala, U. Bruning et al., “An intact canonical NF-κB pathway is required for inflammatory gene expression in response to hypoxia,” Journal of Immunology, vol. 186, no. 2, pp. 1091–1096, 2011.
[25]
A. C. Koong, E. Y. Chen, and A. J. Giaccia, “Hypoxia causes the activation of nuclear factor κB through the phosphorylation of IκBα on tyrosine residues,” Cancer Research, vol. 54, no. 6, pp. 1425–1430, 1994.
[26]
H. Li, J. Alyce Bradbury, R. T. Dackor et al., “Cyclooxygenase-2 regulates Th17 cell differentiation during allergic lung inflammation,” American Journal of Respiratory and Critical Care Medicine, vol. 184, no. 1, pp. 37–49, 2011.
[27]
K. A. Ryall, D. O. Holland, K. A. Delaney, M. J. Kraeutler, A. J. Parker, and J. J. Saucerman, “Network reconstruction and systems analysis of cardiac myocyte hypertrophy signaling,” Journal of Biological Chemistry, vol. 287, pp. 42259–42268, 2012.
[28]
H. Risken, The Fokker-Planck Equation: Methods of Solutions and Applications, Springer, Berlin, Germany, 1989.
[29]
H. Haken, Advanced Synergetics, Springer, Berlin, Germany, 1987.
[30]
G. Sch?ner and H. Haken, “The slaving principle for stratonovich stochastic differential equations,” Zeitschrift für Physik B, vol. 63, no. 4, pp. 493–504, 1986.
[31]
B. C. Eu, Kinetic Theory and Irreversible Thermodynamics, John Wiley & Sons, New York, NY, USA, 1992.
[32]
T. D. Frank, A. Daffertshofer, P. J. Beek, and H. Haken, “Impacts of noise on a field theoretical model of the human brain,” Physica D, vol. 127, no. 3-4, pp. 233–249, 1999.
[33]
T. D. Frank, “A limit cycle oscillator model for cycling mood variations of bipolar disorder patients from cellular biochemical reaction equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, pp. 2107–2119, 2013.
[34]
T. D. Frank, Nonlinear Fokker-Planck Equations: Fundamentals and Applications, Springer, Berlin, Germany, 2005.
[35]
C. W. Gardiner, Handbook of Stochastic Methods For Physics, Chemistry and the Natural Sciences, Springer, Berlin, Germany, 1985.
[36]
T. D. Frank, A. M. Carmody, and B. N. Kholodenko, “Versatility of cooperative transcriptional activation: a thermodynamical modeling analysis for greater-than-additive and less-than-additive effects,” PLoS One, vol. 7, no. 4, Article ID e34439, 2012.
[37]
B. K. Slinker, “The statistics of synergism,” Journal of Molecular and Cellular Cardiology, vol. 30, no. 4, pp. 723–731, 1998.
[38]
P. J. Diggle, Time Series: A Biostatistical Introduction, Clarendon Press, Oxford, UK, 1990.