Efficient analysis of stochastic gene dynamics in the non-adiabatic regime using piecewise deterministic Markov processes

Lin, Yen Ting; Buchler, Nicolas E.

doi:10.1098/rsif.2017.0804

Efficient analysis of stochastic gene dynamics in the non-adiabatic regime using piecewise deterministic Markov processes

Journal Article · Wed Jan 31 00:00:00 EST 2018 · Journal of the Royal Society Interface

DOI:https://doi.org/10.1098/rsif.2017.0804· OSTI ID:1419760

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Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Univ. of Manchester, Manchester (United Kingdom)
Duke Univ., Durham, NC (United States); Center for Genomic and Computational Biology, Durham, NC (United States)

Single-cell experiments show that gene expression is stochastic and bursty, a feature that can emerge from slow switching between promoter states with different activities. In addition to slow chromatin and/or DNA looping dynamics, one source of long-lived promoter states is the slow binding and unbinding kinetics of transcription factors to promoters, i.e. the non-adiabatic binding regime. Here, we introduce a simple analytical framework, known as a piecewise deterministic Markov process (PDMP), that accurately describes the stochastic dynamics of gene expression in the non-adiabatic regime. We illustrate the utility of the PDMP on a non-trivial dynamical system by analysing the properties of a titration-based oscillator in the non-adiabatic limit. We first show how to transform the underlying chemical master equation into a PDMP where the slow transitions between promoter states are stochastic, but whose rates depend upon the faster deterministic dynamics of the transcription factors regulated by these promoters. We show that the PDMP accurately describes the observed periods of stochastic cycles in activator and repressor-based titration oscillators. We then generalize our PDMP analysis to more complicated versions of titration-based oscillators to explain how multiple binding sites lengthen the period and improve coherence. Finally, we show how noise-induced oscillation previously observed in a titration-based oscillator arises from non-adiabatic and discrete binding events at the promoter site.

View Accepted Manuscript (DOE)

Research Organization:: Los Alamos National Laboratory (LANL)

Sponsoring Organization:: LDRD; USDOE

Grant/Contract Number:: AC52-06NA25396

OSTI ID:: 1419760

Report Number(s):: LA-UR-17-29798

Journal Information:: Journal of the Royal Society Interface, Journal Name: Journal of the Royal Society Interface Journal Issue: 138 Vol. 15; ISSN 1742-5689

Publisher:: The Royal SocietyCopyright Statement

Country of Publication:: United States

Language:: English

References (63)

Gene expression noise is affected differentially by feedback in burst frequency and burst size Bokes, Pavol; Singh, Abhyudai Journal of Mathematical Biology, Vol. 74, Issue 6 https://doi.org/10.1007/s00285-016-1059-4	journal	September 2016
Non-equilibrium Thermodynamics of Piecewise Deterministic Markov Processes Faggionato, A.; Gabrielli, D.; Ribezzi Crivellari, M. Journal of Statistical Physics, Vol. 137, Issue 2 https://doi.org/10.1007/s10955-009-9850-x	journal	October 2009
A general method for numerically simulating the stochastic time evolution of coupled chemical reactions Gillespie, Daniel T. Journal of Computational Physics, Vol. 22, Issue 4 https://doi.org/10.1016/0021-9991(76)90041-3	journal	December 1976
Transcriptional stochasticity in gene expression Lipniacki, Tomasz; Paszek, Pawel; Marciniak-Czochra, Anna Journal of Theoretical Biology, Vol. 238, Issue 2 https://doi.org/10.1016/j.jtbi.2005.05.032	journal	January 2006
Emergence of coherent oscillations in stochastic models for circadian rhythms Gonze, Didier; Halloy, José; Goldbeter, Albert Physica A: Statistical Mechanics and its Applications, Vol. 342, Issue 1-2 https://doi.org/10.1016/j.physa.2004.04.082	journal	October 2004
Solutions of ordinary differential equations as limits of pure jump markov processes Kurtz, Thomas G. Journal of Applied Probability, Vol. 7, Issue 01 https://doi.org/10.1017/s0021900200026929	journal	April 1970
Direct measurement of transcription factor dissociation excludes a simple operator occupancy model for gene regulation Hammar, Petter; Walldén, Mats; Fange, David Nature Genetics, Vol. 46, Issue 4 https://doi.org/10.1038/ng.2905	journal	February 2014
Directed cell migration towards softer environments Isomursu, Aleksi; Park, Keun-Young; Hou, Jay Nature Materials, Vol. 21, Issue 9 https://doi.org/10.1038/s41563-022-01294-2	journal	July 2022
Determination of binding constant of transcription factor AP-1 and DNA Kwon, Haeyoung; Park, Seyeon; Lee, Sangkyou European Journal of Biochemistry, Vol. 268, Issue 3 https://doi.org/10.1046/j.1432-1327.2001.01897.x	journal	February 2001
Mechanisms of noise-resistance in genetic oscillators Vilar, J. M. G.; Kueh, H. Y.; Barkai, N. Proceedings of the National Academy of Sciences, Vol. 99, Issue 9 https://doi.org/10.1073/pnas.092133899	journal	April 2002
On schemes of combinatorial transcription logic Buchler, N. E.; Gerland, U.; Hwa, T. Proceedings of the National Academy of Sciences, Vol. 100, Issue 9 https://doi.org/10.1073/pnas.0930314100	journal	April 2003
Massively parallel measurements of molecular interaction kinetics on a microfluidic platform Geertz, M.; Shore, D.; Maerkl, S. J. Proceedings of the National Academy of Sciences, Vol. 109, Issue 41 https://doi.org/10.1073/pnas.1206011109	journal	September 2012
On the dephasing of genetic oscillators Potoyan, Davit A.; Wolynes, Peter G. Proceedings of the National Academy of Sciences, Vol. 111, Issue 6 https://doi.org/10.1073/pnas.1323433111	journal	January 2014
Phenotypic switching in gene regulatory networks Thomas, P.; Popovi , N.; Grima, R. Proceedings of the National Academy of Sciences, Vol. 111, Issue 19 https://doi.org/10.1073/pnas.1400049111	journal	April 2014
Quantitative model for gene regulation by lambda phage repressor. Ackers, G. K.; Johnson, A. D.; Shea, M. A. Proceedings of the National Academy of Sciences, Vol. 79, Issue 4 https://doi.org/10.1073/pnas.79.4.1129	journal	February 1982
Stochastic hybrid model of spontaneous dendritic NMDA spikes Bressloff, Paul C.; Newby, Jay M. Physical Biology, Vol. 11, Issue 1 https://doi.org/10.1088/1478-3975/11/1/016006	journal	January 2014
Mean first passage times for piecewise deterministic Markov processes and the effects of critical points Bressloff, Paul C.; Lawley, Sean D. Journal of Statistical Mechanics: Theory and Experiment, Vol. 2017, Issue 6 https://doi.org/10.1088/1742-5468/aa71df	journal	June 2017
Globally coupled stochastic two-state oscillators: synchronization of infinite and finite arrays Rosas, Alexandre; Escaff, Daniel; Pinto, Italo’Ivo Lima Dias Journal of Physics A: Mathematical and Theoretical, Vol. 49, Issue 9 https://doi.org/10.1088/1751-8113/49/9/095001	journal	January 2016
Stochastic switching in biology: from genotype to phenotype Bressloff, Paul C. Journal of Physics A: Mathematical and Theoretical, Vol. 50, Issue 13 https://doi.org/10.1088/1751-8121/aa5db4	journal	February 2017
Effective phase dynamics of noise-induced oscillations in excitable systems Schwabedal, Justus T. C.; Pikovsky, Arkady Physical Review E, Vol. 81, Issue 4 https://doi.org/10.1103/physreve.81.046218	journal	April 2010
Gene expression dynamics with stochastic bursts: Construction and exact results for a coarse-grained model Lin, Yen Ting; Doering, Charles R. Physical Review E, Vol. 93, Issue 2 https://doi.org/10.1103/physreve.93.022409	journal	February 2016
Intrinsic noise in systems with switching environments Hufton, Peter G.; Lin, Yen Ting; Galla, Tobias Physical Review E, Vol. 93, Issue 5 https://doi.org/10.1103/physreve.93.052119	journal	May 2016
Lévy-noise-induced transport in a rough triple-well potential Li, Yongge; Xu, Yong; Kurths, Jürgen Physical Review E, Vol. 94, Issue 4 https://doi.org/10.1103/physreve.94.042222	journal	October 2016
Emergent Lévy behavior in single-cell stochastic gene expression Jia, Chen; Zhang, Michael Q.; Qian, Hong Physical Review E, Vol. 96, Issue 4 https://doi.org/10.1103/physreve.96.040402	journal	October 2017
Linking Stochastic Dynamics to Population Distribution: An Analytical Framework of Gene Expression Friedman, Nir; Cai, Long; Xie, X. Sunney Physical Review Letters, Vol. 97, Issue 16 https://doi.org/10.1103/physrevlett.97.168302	journal	October 2006
Evaluation of MafG interaction with Maf recognition element arrays by surface plasmon resonance imaging technique Kyo, Motoki; Yamamoto, Tae; Motohashi, Hozumi Genes to Cells, Vol. 9, Issue 2, p. 153-164 https://doi.org/10.1111/j.1356-9597.2004.00711.x	journal	February 2004
Path-Integral Methods for Analyzing the Effects of Fluctuations in Stochastic Hybrid Neural Networks Bressloff, Paul C. The Journal of Mathematical Neuroscience, Vol. 5, Issue 1 https://doi.org/10.1186/s13408-014-0016-z	journal	February 2015
Effective phase description of noise-perturbed and noise-induced oscillations Schwabedal, Justus T. C.; Pikovsky, Arkady arXiv https://doi.org/10.48550/arxiv.1006.3173	text	January 2010
Phase Description of Stochastic Oscillations Schwabedal, Justus T. C.; Pikovsky, Arkady arXiv https://doi.org/10.48550/arxiv.1301.6852	text	January 2013
Breakdown of fast-slow analysis in an excitable system with channel noise Newby, Jay M.; Bressloff, Paul C.; Keener, James P. arXiv https://doi.org/10.48550/arxiv.1304.6952	text	January 2013
Effects of moderate noise on a limit cycle oscillator: Counter rotation and bistability Newby, Jay M.; Schwemmer, Michael A. arXiv https://doi.org/10.48550/arxiv.1310.2227	text	January 2013
On the Hamiltonian structure of large deviations in stochastic hybrid systems Bressloff, Paul; Faugeras, Olivier arXiv https://doi.org/10.48550/arxiv.1410.2152	preprint	January 2014
A stochastic and dynamical view of pluripotency in mouse embryonic stem cells Lin, Yen Ting; Hufton, Peter G.; Lee, Esther J. arXiv https://doi.org/10.48550/arxiv.1710.08542	text	January 2017
Transcriptional Bursting Diversifies the Behaviour of a Toggle Switch: Hybrid Simulation of Stochastic Gene Expression Bokes, Pavol; King, John R.; Wood, Andrew T. A. Bulletin of Mathematical Biology, Vol. 75, Issue 2 https://doi.org/10.1007/s11538-013-9811-z	journal	January 2013
Landscape and Global Stability of Nonadiabatic and Adiabatic Oscillations in a Gene Network Feng, Haidong; Han, Bo; Wang, Jin Biophysical Journal, Vol. 102, Issue 5 https://doi.org/10.1016/j.bpj.2012.02.002	journal	March 2012
Exact stochastic simulation of coupled chemical reactions Gillespie, Daniel T. The Journal of Physical Chemistry, Vol. 81, Issue 25 https://doi.org/10.1021/j100540a008	journal	December 1977
On the Hamiltonian structure of large deviations in stochastic hybrid systems Bressloff, Paul C.; Faugeras, Olivier Journal of Statistical Mechanics: Theory and Experiment, Vol. 2017, Issue 3 https://doi.org/10.1088/1742-5468/aa64f3	journal	March 2017
Bursting noise in gene expression dynamics: linking microscopic and mesoscopic models Lin, Yen Ting; Galla, Tobias Journal of The Royal Society Interface, Vol. 13, Issue 114 https://doi.org/10.1098/rsif.2015.0772	journal	January 2016
Molecular stripping, targets and decoys as modulators of oscillations in the NF-κB/IκBα/DNA genetic network Wang, Zhipeng; Potoyan, Davit A.; Wolynes, Peter G. Journal of The Royal Society Interface, Vol. 13, Issue 122 https://doi.org/10.1098/rsif.2016.0606	journal	September 2016
Core genetic module: The mixed feedback loop François, Paul; Hakim, Vincent Physical Review E, Vol. 72, Issue 3 https://doi.org/10.1103/PhysRevE.72.031908	journal	September 2005
Self-regulating gene: An exact solution Hornos, J. E. M.; Schultz, D.; Innocentini, G. C. P. Physical Review E, Vol. 72, Issue 5 https://doi.org/10.1103/PhysRevE.72.051907	journal	November 2005
Noise-induced dynamics in the mixed-feedback-loop network motif Li, Difei; Li, Chunguang Physical Review E, Vol. 77, Issue 1 https://doi.org/10.1103/PhysRevE.77.011903	journal	January 2008
Effective phase dynamics of noise-induced oscillations in excitable systems Schwabedal, Justus T. C.; Pikovsky, Arkady Physical Review E, Vol. 81, Issue 4 https://doi.org/10.1103/PhysRevE.81.046218	journal	April 2010
Globally coupled stochastic two-state oscillators: Fluctuations due to finite numbers Pinto, Italo'Ivo Lima Dias; Escaff, Daniel; Harbola, Upendra Physical Review E, Vol. 89, Issue 5 https://doi.org/10.1103/PhysRevE.89.052143	journal	May 2014
Arrays of stochastic oscillators: Nonlocal coupling, clustering, and wave formation Escaff, Daniel; Pinto, Italo'Ivo Lima Dias; Lindenberg, Katja Physical Review E, Vol. 90, Issue 5 https://doi.org/10.1103/PhysRevE.90.052111	journal	November 2014
Role of DNA binding sites and slow unbinding kinetics in titration-based oscillators Karapetyan, Sargis; Buchler, Nicolas E. Physical Review E, Vol. 92, Issue 6 https://doi.org/10.1103/PhysRevE.92.062712	journal	December 2015
Gene expression dynamics with stochastic bursts: Construction and exact results for a coarse-grained model Lin, Yen Ting; Doering, Charles R. Physical Review E, Vol. 93, Issue 2 https://doi.org/10.1103/PhysRevE.93.022409	journal	February 2016
Intrinsic noise in systems with switching environments Hufton, Peter G.; Lin, Yen Ting; Galla, Tobias Physical Review E, Vol. 93, Issue 5 https://doi.org/10.1103/PhysRevE.93.052119	journal	May 2016
Lévy-noise-induced transport in a rough triple-well potential Li, Yongge; Xu, Yong; Kurths, Jürgen Physical Review E, Vol. 94, Issue 4 https://doi.org/10.1103/PhysRevE.94.042222	journal	October 2016
Feynman-Kac formula for stochastic hybrid systems Bressloff, Paul C. Physical Review E, Vol. 95, Issue 1 https://doi.org/10.1103/PhysRevE.95.012138	journal	January 2017
Simplification of Markov chains with infinite state space and the mathematical theory of random gene expression bursts Jia, Chen Physical Review E, Vol. 96, Issue 3 https://doi.org/10.1103/PhysRevE.96.032402	journal	September 2017
Emergent Lévy behavior in single-cell stochastic gene expression Jia, Chen; Zhang, Michael Q.; Qian, Hong Physical Review E, Vol. 96, Issue 4 https://doi.org/10.1103/PhysRevE.96.040402	journal	October 2017
Delay-Induced Degrade-and-Fire Oscillations in Small Genetic Circuits Mather, William; Bennett, Matthew R.; Hasty, Jeff Physical Review Letters, Vol. 102, Issue 6 https://doi.org/10.1103/PhysRevLett.102.068105	journal	February 2009
Phase Description of Stochastic Oscillations Schwabedal, Justus T. C.; Pikovsky, Arkady Physical Review Letters, Vol. 110, Issue 20 https://doi.org/10.1103/PhysRevLett.110.204102	journal	May 2013
Breakdown of Fast-Slow Analysis in an Excitable System with Channel Noise Newby, Jay M.; Bressloff, Paul C.; Keener, James P. Physical Review Letters, Vol. 111, Issue 12 https://doi.org/10.1103/PhysRevLett.111.128101	journal	September 2013
Effects of Moderate Noise on a Limit Cycle Oscillator: Counterrotation and Bistability Newby, Jay M.; Schwemmer, Michael A. Physical Review Letters, Vol. 112, Issue 11 https://doi.org/10.1103/PhysRevLett.112.114101	journal	March 2014
Synchronization of Degrade-and-Fire Oscillations via a Common Activator Mather, William; Hasty, Jeff; Tsimring, Lev S. Physical Review Letters, Vol. 113, Issue 12 https://doi.org/10.1103/PhysRevLett.113.128102	journal	September 2014
Stochastic Phenotype Transition of a Single Cell in an Intermediate Region of Gene State Switching Ge, Hao; Qian, Hong; Xie, X. Sunney Physical Review Letters, Vol. 114, Issue 7 https://doi.org/10.1103/PhysRevLett.114.078101	journal	February 2015
Linking Stochastic Dynamics to Population Distribution: An Analytical Framework of Gene Expression Friedman, Nir; Cai, Long; Xie, X. Sunney Physical Review Letters, Vol. 97, Issue 16 https://doi.org/10.1103/PhysRevLett.97.168302	journal	October 2006
Error Bound for Piecewise Deterministic Processes Modeling Stochastic Reaction Systems Jahnke, Tobias; Kreim, Michael Multiscale Modeling & Simulation, Vol. 10, Issue 4 https://doi.org/10.1137/120871894	journal	January 2012
Effective phase description of noise-perturbed and noise-induced oscillations Schwabedal, J. T. C.; Pikovsky, A. The European Physical Journal Special Topics, Vol. 187, Issue 1 https://doi.org/10.1140/epjst/e2010-01271-6	journal	September 2010
Dichotomous Markov Noise: Exact Results for Out-Of-Equilibrium Systems Bena, Ioana International Journal of Modern Physics B, Vol. 20, Issue 20 https://doi.org/10.1142/S0217979206034881	journal	August 2006
Solutions of ordinary differential equations as limits of pure jump markov processes Kurtz, Thomas G. Journal of Applied Probability, Vol. 7, Issue 1 https://doi.org/10.2307/3212147	journal	April 1970

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59 BASIC BIOLOGICAL SCIENCES
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stochastic cycles

Efficient analysis of stochastic gene dynamics in the non-adiabatic regime using piecewise deterministic Markov processes

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