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Model reduction, coherence, and information transfer in stochastic biochemical systems

Juan David Marmolejo-Lozano1, Nikola Popović2,3, and Ramon Grima1,*

  • *Contact author: ramon.grima@ed.ac.uk
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Phys. Rev. E 113, 044416 – Published 29 April, 2026

DOI: https://doi.org/10.1103/9h19-b6ht

Abstract

Simplified stochastic models are widely used in the study of frequency-resolved noise propagation in biochemical reaction networks, a common measure being the coherence between random fluctuations in molecule number trajectories. Such models have also found widespread application in the quantification of how information is transmitted in reaction networks via the mutual information (MI) rate. A common assumption is that under timescale separation, estimates for the coherence and MI rate obtained from simplified (reduced) models closely approximate those in the underlying full models. Here, we challenge that assumption by showing that while reduced models can faithfully reproduce low-order statistics of molecular counts, they frequently incur substantial discrepancies in the coherence spectrum, especially at intermediate and high frequencies. These errors, in turn, lead to significant inaccuracies in the resulting estimates for the MI rates. We show that the observed discrepancies are due to the interplay between the structure of the underlying reaction networks, the specific model reduction method that is applied, and the asymptotic limits relating the full and reduced models. We illustrate our results in canonical models of enzyme catalysis and gene expression, highlighting practical implications for quantifying information flow in cells.

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References (54)

  1. P. Thomas, R. Grima, and A. V. Straube, Rigorous elimination of fast stochastic variables from the linear noise approximation using projection operators, Phys. Rev. E 86, 041110 (2012).
  2. J. K. Kim, K. Josić, and M. R. Bennett, The validity of quasi-steady-state approximations in discrete stochastic simulations, Biophys. J. 107, 783 (2014).
  3. Y. M. Song, H. Hong, and J. K. Kim, Universally valid reduction of multiscale stochastic biochemical systems using simple nonelementary propensities, PLoS Comput. Biol. 17, e1008952 (2021).
  4. N. Popović, C. Marr, and P. S. Swain, A geometric analysis of fast-slow models for stochastic gene expression, J. Math. Biol. 72, 87 (2016).
  5. H.-W. Kang, W. R. KhudaBukhsh, H. Koeppl, and G. A. Rempała, Quasi-steady-state approximations derived from the stochastic model of enzyme kinetics, Bull. Math. Biol. 81, 1303 (2019).
  6. P. Thomas, N. Popović, and R. Grima, Phenotypic switching in gene regulatory networks, Proc. Natl. Acad. Sci. USA 111, 6994 (2014).
  7. T. Jahnke, On reduced models for the chemical master equation, Multiscale Model. Simul. 9, 1646 (2011).
  8. S. Smith, C. Cianci, and R. Grima, Model reduction for stochastic chemical systems with abundant species, J. Chem. Phys. 143, 214105 (2015).
  9. A. Hellander and P. Lötstedt, Hybrid method for the chemical master equation, J. Comput. Phys. 227, 100 (2007).
  10. Z. Cao and R. Grima, Linear mapping approximation of gene regulatory networks with stochastic dynamics, Nat. Commun. 9, 3305 (2018).
  11. C. Jia and R. Grima, Holimap: An accurate and efficient method for solving stochastic gene network dynamics, Nat. Commun. 15, 6557 (2024).
  12. M. Barrio, A. Leier, and T. T. Marquez-Lago, Reduction of chemical reaction networks through delay distributions, J. Chem. Phys. 138, 104114 (2013).
  13. A. Leier, M. Barrio, and T. T. Marquez-Lago, Exact model reduction with delays: Closed-form distributions and extensions to fully bi-directional monomolecular reactions, J. R. Soc. Interface 11, 20140108 (2014).
  14. J. Zhang and T. Zhou, Markovian approaches to modeling intracellular reaction processes with molecular memory, Proc. Natl. Acad. Sci. USA 116, 23542 (2019).
  15. J. Szavits-Nossan and R. Grima, Solving stochastic gene-expression models using queueing theory: A tutorial review, Biophys. J. 123, 1034 (2024).
  16. K. Öcal, G. Sanguinetti, and R. Grima, Model reduction for the chemical master equation: An information-theoretic approach, J. Chem. Phys. 158, 114113 (2023).
  17. Y. Cao, D. T. Gillespie, and L. R. Petzold, The slow-scale stochastic simulation algorithm, J. Chem. Phys. 122, 014116 (2005).
  18. J. Goutsias, Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems, J. Chem. Phys. 122, 184102 (2005).
  19. C. V. Rao and A. P. Arkin, Stochastic chemical kinetics and the quasi-steady-state assumption: Application to the Gillespie algorithm, J. Chem. Phys. 118, 4999 (2003).
  20. D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem. 81, 2340 (1977).
  21. P. Thomas, A. V. Straube, and R. Grima, The slow-scale linear noise approximation: An accurate, reduced stochastic description of biochemical networks under timescale separation conditions, BMC Syst. Biol. 6, 39 (2012).
  22. M. Ma, J. Szavits-Nossan, A. Singh, and R. Grima, Analysis of a detailed multistage model of stochastic gene expression using queueing theory and model reduction, Math. Biosci. 373, 109204 (2024).
  23. J. Elf and M. Ehrenberg, Fast evaluation of fluctuations in biochemical networks with the linear noise approximation, Genome Res. 13, 2475 (2003).
  24. E. Ziv, I. Nemenman, and C. H. Wiggins, Optimal signal processing in small stochastic biochemical networks, PLoS ONE 2, e1077 (2007).
  25. M. Rodríguez Martínez, J. Soriano, T. Tlusty, Y. Pilpel, and I. Furman, Messenger RNA fluctuations and regulatory RNAs shape the dynamics of a negative feedback loop, Phys. Rev. E 81, 031924 (2010).
  26. J. Holehouse and R. Grima, Revisiting the reduction of stochastic models of genetic feedback loops with fast promoter switching, Biophys. J. 117, 1311 (2019).
  27. F. Tostevin and P. R. ten Wolde, Mutual information between input and output trajectories of biochemical networks, Phys. Rev. Lett. 102, 218101 (2009).
  28. F. Tostevin and P. R. ten Wolde, Mutual information in time-varying biochemical systems, Phys. Rev. E 81, 061917 (2010).
  29. M. Meijers, S. Ito, and P. R. ten Wolde, Behavior of information flow near criticality, Phys. Rev. E 103, L010102 (2021).
  30. R. Fan and A. Hilfinger, Characterizing the nonmonotonic behavior of mutual information along biochemical reaction cascades, Phys. Rev. E 110, 034309 (2024).
  31. T. Munakata and M. Kamiyabu, Stochastic resonance in the Fitzhugh-Nagumo model from a dynamic mutual information point of view, Eur. Phys. J. B 53, 239 (2006).
  32. A. Komaee, Mutual information rate between stationary Gaussian processes, Results Appl. Math. 7, 100107 (2020).
  33. P. P. Mitra and J. B. Stark, Nonlinear limits to the information capacity of optical fibre communications, Nature (London) 411, 1027 (2001).
  34. N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier, Amsterdam, 1992), Vol. 1.
  35. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 2012).
  36. D. Schnoerr, G. Sanguinetti, and R. Grima, Approximation and inference methods for stochastic biochemical kinetics—A tutorial review, J. Phys. A: Math. Theor. 50, 093001 (2017).
  37. C. W. Gardiner et al., Handbook of Stochastic Methods (Springer-Verlag, Berlin, 1985), Vol. 3.
  38. M. Reinhardt, G. Tkačik, and P. R. ten Wolde, Path weight sampling: Exact Monte Carlo computation of the mutual information between stochastic trajectories, Phys. Rev. X 13, 041017 (2023).
  39. A.-L. Moor and C. Zechner, Dynamic information transfer in stochastic biochemical networks, Phys. Rev. Res. 5, 013032 (2023).
  40. M. Reinhardt, A. J. Tjalma, A.-L. Moor, C. Zechner, and P. R. Ten Wolde, Mutual information rate–linear noise approximation and exact computation, arXiv:2508.21220.
  41. P. Thomas, A. V. Straube, and R. Grima, Communication: limitations of the stochastic quasi-steady-state approximation in open biochemical reaction networks, J. Chem. Phys. 135, 181103 (2011).
  42. K. R. Sanft, D. T. Gillespie, and L. R. Petzold, Legitimacy of the stochastic Michaelis-Menten approximation, IET Syst. Biol. 5, 58 (2011).
  43. J. Janssen, The elimination of fast variables in complex chemical reactions. III. Mesoscopic level (irreducible case), J. Stat. Phys. 57, 187 (1989).
  44. P. D. Welch, The use of fast fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms, IEEE Trans. Audio Electroacoust. 15, 70 (1967).
  45. Gillespie's SSA samples paths from the CME of the reduced system, given by Eq. (29), which has propensities k1Ω for the reaction ⌀→k1S, k′nS=k4ϕTnSΩKM+nS for the reaction S→k′P, and k5np for the reaction P→k5⌀; here, Ω is the volume of the system and ni denotes the number of molecules of species i.
  46. T. Maier, M. Güell, and L. Serrano, Correlation of mRNA and protein in complex biological samples, FEBS Lett. 583, 3966 (2009).
  47. S. H. Payne, The utility of protein and mRNA correlation, Trends Biochem. Sci. 40, 1 (2015).
  48. C. Buccitelli and M. Selbach, mRNAs, proteins and the emerging principles of gene expression control, Nat. Rev. Genet. 21, 630 (2020).
  49. D. Popovic, B. Koch, M. Kueblbeck, J. Ellenberg, and L. Pelkmans, Multivariate control of transcript to protein variability in single mammalian cells, Cell Syst. 7, 398 (2018).
  50. C. Jia, Simplification of Markov chains with infinite state space and the mathematical theory of random gene expression bursts, Phys. Rev. E 96, 032402 (2017).
  51. I. Golding, J. Paulsson, S. M. Zawilski, and E. C. Cox, Real-time kinetics of gene activity in individual bacteria, Cell 123, 1025 (2005).
  52. Z. Cao and R. Grima, Analytical distributions for detailed models of stochastic gene expression in eukaryotic cells, Proc. Natl. Acad. Sci. USA 117, 4682 (2020).
  53. A.-L. Moor, A. Tjalma, M. Reinhardt, P. R. Ten Wolde, and C. Zechner, State-versus reaction-based information processing in biochemical networks, arXiv:2505.13373.
  54. R. Ramaswamy, N. González-Segredo, I. F. Sbalzarini, and R. Grima, Discreteness-induced concentration inversion in mesoscopic chemical systems, Nat. Commun. 3, 779 (2012).

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Phys. Rev. E 113, 044416– Published 29 April, 2026
  • Received 7 November 2025
  • Accepted 6 April 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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