# The role of stochastic gene switching in determining the pharmacodynamics of certain drugs: basic mechanisms

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## Abstract

In this paper we analyze the impact of the stochastic fluctuation of genes between their ON and OFF states on the pharmacodynamics of a potentially large class of drugs. We focus on basic mechanisms underlying the onset of in vitro experimental dose-response curves, by investigating two elementary molecular circuits. Both circuits consist in the transcription of a gene and in the successive translation into the corresponding protein. Whereas in the first the activation/deactivation rates of the single gene copy are constant, in the second the protein, now a transcription factor, amplifies the deactivation rate, so introducing a negative feedback. The drug is assumed to enhance the elimination of the protein, and in both cases the success of therapy is assured by keeping the level of the given protein under a threshold for a fixed time. Our numerical simulations suggests that the gene switching plays a primary role in determining the sigmoidal shape of dose-response curves. Moreover, the simulations show interesting phenomena related to the magnitude of the average gene switching time and to the drug concentration. In particular, for slow gene switching a significant fraction of cells can respond also in the absence of drug or with drug concentrations insufficient for the response in a deterministic setting. For higher drug concentrations, the non-responding fraction exhibits a maximum at intermediate values of the gene switching rates. For fast gene switching, instead, the stochastic prediction follows the prediction of the deterministic approximation, with all the cells responding or non-responding according to the drug dose.

## Keywords

Transcriptional Network Switching Rate Stochastic Fluctuation Gene Switching Deterministic Simulation## Notes

### Acknowledgments

The authors wish to thank the anonymous referees for their useful suggestions.

### Funding

KP, Polish National Center for Science, funds granted by Decision Number DEC-2012/05/D/ST7/02072. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

## References

- 1.Mortensen SB, Jonsdottir AH, Klim S, Madsen H (2008) Introduction to PK/PD modelling with focus on PK and stochastic differential equations. IMM-Technical Report-2008-16Google Scholar
- 2.Rosenbaum SE (2011) Basic pharmacokinetics and pharmacodynamics: an integrated textbook and computer simulations. Wiley, New YorkGoogle Scholar
- 3.Donnet S, Samson A (2013) A review on estimation of stochastic differential equations for pharmacokinetic-pharmacodynamic models. Adv Drug Deliv Rev 65:929–939CrossRefPubMedGoogle Scholar
- 4.Kristensen NR, Madsen H, Ingwersen SH (2005) Using stochastic differential equations for PK-PD model development. J Pharmacokinet Pharmacodyn 32:109–141CrossRefPubMedGoogle Scholar
- 5.Macheras P, Iliadis A (2006) Modeling in biopharmaceutics, pharmacokinetics and pharmacodynamics: homogeneous and heterogeneous approaches. Springer, New YorkGoogle Scholar
- 6.Kepler TB, Elston TC (2001) Stochasticity in transcriptional regulation: origins, consequences, and mathematical representations. Biophys J 81:3116–3136CrossRefPubMedPubMedCentralGoogle Scholar
- 7.Ko MS (1991) A stochastic model for gene induction. J Theor Biol 53:181–194CrossRefGoogle Scholar
- 8.Walters MC, Fiering S, Eidemiller J, Magis W, Groudine M, Martin DIK (1995) Enhancers increase the probability but not the level of gene expression. Proc Natl Acad Sci USA 92:7125–7129CrossRefPubMedPubMedCentralGoogle Scholar
- 9.Femino AM, Fay FS, Fogary K, Singer RH (1998) Visualization of single RNA transcripts in situ. Science 280:585–590CrossRefPubMedGoogle Scholar
- 10.Rai A, Peskin CS, Tranchina D, Vargas DY, Tyagi S (2006) Stochastic mRNA synthesis in mammalian cells. PloS Biol 4:e309CrossRefGoogle Scholar
- 11.Paszek P, Lipniacki T, Brasier AR, Tian B, Nowak DE, Kimmel M (2005) Stochastic effects of multiple regulators on expression profiles in eukaryotes. J Theor Biol 233:422–433CrossRefGoogle Scholar
- 12.Lipniacki T, Paszek P, Marciniak-Czochra A, Brasier AR, Kimmel M (2006) Transcriptional stochasticity in gene expression. J Theor Biol 238:348–367CrossRefPubMedGoogle Scholar
- 13.Jaruszewicz J, Zuk PJ, Lipniacki T (2013) Type of noise defines global attractors in bistable molecular regulatory systems. J Theor Biol 317:140–151CrossRefPubMedGoogle Scholar
- 14.Puszynski K, Hat B, Lipniacki T (2008) Oscillations and bistability in the stochastic model of p53 regulation. J Theor Biol 254:452–465CrossRefPubMedGoogle Scholar
- 15.Kaern M, Elston TC, Blake WJ, Collins JJ (2005) Stochasticity in gene expression: from theories to phenotypes. Nature Rev Genet 6:451–464CrossRefPubMedGoogle Scholar
- 16.Danhof M, de Jongh J, De Lange ECM, Della Pasqua O, Ploeger BA, Voskuyl RA (2007) Mechanism-based pharmacokinetic-pharmacodynamic modeling: biophase distribution, receptor theory, and dynamical systems analysis. Ann Rev Pharmacol Toxicol 47:357–400CrossRefGoogle Scholar
- 17.Agoram BM, Martin SW, van der Graaf PH (2007) The role of mechanism-based pharmacokinetic-pharmacodynamic (PK-PD) modelling in translational research of biologics. Drug Discov Today 12:1018–1024CrossRefPubMedGoogle Scholar
- 18.Puszynsk K, Gandolfi A, d’Onofrio A (2014) The pharmacodynamics of the p53-Mdm2 targeting drug nutlin: the role of gene-switching noise. PLoS Comput Biol 10:e1003991CrossRefGoogle Scholar
- 19.Vogelstein B, Lane D, Levine AJ (2000) Surfing the p53 network. Nature 408:307–310CrossRefPubMedGoogle Scholar
- 20.Cicalese A, Bonizzi G, Pasi CE, Faretta M, Ronzoni S et al (2009) The tumor suppressor p53 regulates polarity of self-renewing divisions in mammary stem cells. Cell 138:1083–1095CrossRefPubMedGoogle Scholar
- 21.Maimets T, Neganova I, Armstrong L, Lako M (2008) Activation of p53 by nutlin leads to rapid differentiation of human embryonic stem cells. Oncogene 27:5277–5287CrossRefPubMedGoogle Scholar
- 22.Vassilev LT, Vu BT, Graves B, Carvajal D, Podlaski F et al (2004) In vivo activation of the p53 pathway by small-molecule antagonists of MDM2. Science 303:844–848CrossRefPubMedGoogle Scholar
- 23.Carvajal D, Tovar C, Yang H, Vu BT, Heimbrook DC et al (2005) Activation of p53 by MDM2 antagonists can protect proliferating cells from mitotic inhibitors. Cancer Res 65:1918–1924CrossRefPubMedGoogle Scholar
- 24.Coll-Mulet L, Iglesias-Serret D, Santidrian AF et al (2006) MDM2 antagonists activate p53 and synergize with genotoxic drugs in B-cell chronic lymphocytic leukemia cells. Blood 107:4109–4114CrossRefPubMedGoogle Scholar
- 25.Drakos E, Thomaides A, Medeiros LJ, Li J, Leventaki V et al (2007) Inhibition of p53-Murine double minute 2 interaction by Nutlin-3A stabilizes p53 and induces cell cycle arrest and apoptosis in Hodgkin lymphoma. Clin Cancer Res 13:3380–3387CrossRefPubMedGoogle Scholar
- 26.Villalonga-Planells R, Coll-Mulet L, Martinez-Soler F, Castano E, Acebes JJ, Gimenez-Bonafé P, Gil J, Tortosa A (2011) Activation of p53 by nutlin-3a induces apoptosis and cellular senescence in human glioblastoma multiforme. PLoS One 6:e18588CrossRefPubMedPubMedCentralGoogle Scholar
- 27.Sonnemann J, Palani CD, Wittig S, Becker S, Eichhorn F, Voigt A, Beck JF (2011) Anticancer effects of the p53 activator nutlin-3 in Ewing’s sarcoma cells. Eur J Cancer 47:1432–1441CrossRefPubMedGoogle Scholar
- 28.Stambolic V, MacPherson D, Sas D, Lin Y, Snow B et al (2001) Regulation of PTEN transcription by p53. Mol Cell 8:317–325CrossRefPubMedGoogle Scholar
- 29.Bertaux F, Stoma S, Drasdo D, Batt G (2014) Modeling dynamics of cell-to-cell variability in TRAIL-induced apoptosis explains fractional killing and predicts reversible resistance. PLoS Comput Biol 10:e1003893CrossRefPubMedPubMedCentralGoogle Scholar
- 30.Ridolfi L, D’Odorico P, Laio F (2011) Noise-induced phenomena in the environmental sciences. Cambridge University Press, New YorkCrossRefGoogle Scholar
- 31.Davis MHA (1984) Piecewise-Deterministic Markov processes: a general class of nondiffusion stochastic models. J R Stat Soc Ser B 46:353–388Google Scholar
- 32.Rudnicki R, Tyran-Kaminska M (2015) Piecewise deterministic Markov processes in biological models. In: Banasiak J et al (eds) Semigroup of operators, theory and applications. Springer Proceedings in Mathematics & Statistics, vol 113. Springer, New York, pp 235-255Google Scholar
- 33.Bengtsson M, Hemberg M, Rorsman P, Stahlberg A (2008) Quantification of mRNA in single cells and modelling of RT-qPCR induced noise. BMC Mol Biol 9:63CrossRefPubMedPubMedCentralGoogle Scholar
- 34.Wang YV, Wade M, Wong E, Li YC, Rodewald LW et al (2007) Quantitative analyses reveal the importance of regulated Hdmx degradation for p53 activation. Proc Natl Acad Sci USA 104:12365–12370CrossRefPubMedPubMedCentralGoogle Scholar
- 35.Funahashi A, Matsuoka Y, Jouraku A, Morohashi M, Kikuchi N, Kitano H (2008) Cell Designer 3.5: a versatile modeling tool for biochemical networks. IEEE Proc 96:1254–1265CrossRefGoogle Scholar
- 36.Klipp E, Wade RC, Kummer U (2010) Biochemical network-based drug-target prediction. Curr Opin Biotech 21:511–516CrossRefPubMedGoogle Scholar
- 37.Somogyi ET, Bouteiller JM, Glazier JA, Koenig M, Kyle Medley J, Swat MH, Sauro HM (2015) libRoadRunner: a high performance SBML simulation and analysis library. Bioinformatics 31:3315–3321CrossRefPubMedGoogle Scholar
- 38.Yamashita F, Sasa Y, Yoshida S, Hisaka A, Asai Y, Kitano H, Hashida M, Suzuki H (2013) Modeling of Rifampicin-Induced CYP3A4 activation dynamics for the prediction of clinical drug–drug interactions from in vitro data. PLoS One 8:e70330CrossRefPubMedPubMedCentralGoogle Scholar
- 39.Shi Y, Varghese SM, Huang S, White J, Pervaiz S, Tucker-Kellogg L (2009) Computational modeling of pathway dynamics for detecting drug effects: paradoxical effects of LYS303511 on TRAIL-induced apoptosis. Proc LSS Comput Syst Bioinform Conf 8:213–224Google Scholar
- 40.Kell DB, Goodacre R (2014) Metabolomics and systems pharmacology: why and how to model the human metabolic network for drug discovery. Drug Discov Today 19:171–182CrossRefPubMedPubMedCentralGoogle Scholar
- 41.Innominato PF, Levi FA, Bjarnason GA (2010) Chronotherapy and the molecular clock: clinical implications in oncology. Adv Drug Deliv Rev 62:979–1001CrossRefPubMedGoogle Scholar
- 42.Caravagna G, Mauri G, d’Onofrio A (2013) The interplay of intrinsic and extrinsic bounded noises in biomolecular networks. PLoS One 8:e51174CrossRefPubMedPubMedCentralGoogle Scholar
- 43.d’Onofrio A (ed) (2013) Bounded noises in physics biology and engineering. Birkhauser, New YorkGoogle Scholar
- 44.Puszynski K, Jaksik R, Swierniak A (2012) Regulation of p53 by siRNA in radiation treated cells: simulation studies. Int J Appl Math Comput Sci 22:1011–1018CrossRefGoogle Scholar
- 45.Gillespie DT (1976) A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J Comput Phys 22:403–434CrossRefGoogle Scholar
- 46.Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81:2340–2361CrossRefGoogle Scholar