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Squeezing Stationary Distributions of Stochastic Chemical Reaction Systems

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Abstract

Stochastic modeling of chemical reaction systems based on master equations has been an indispensable tool in physical sciences. In the long-time limit, the properties of these systems are characterized by stationary distributions of chemical master equations. In this paper, we describe a novel method for computing stationary distributions analytically, based on a parallel formalism between stochastic chemical reaction systems and second quantization. Anderson, Craciun, and Kurtz showed that, when the rate equation for a reaction network admits a complex-balanced steady-state solution, the corresponding stochastic reaction system has a stationary distribution of a product form of Poisson distributions. In a formulation of stochastic reaction systems using the language of second quantization initiated by Doi, product-form Poisson distributions correspond to coherent states. Pursuing this analogy further, we study the counterpart of squeezed states in stochastic reaction systems. Under the action of a squeeze operator, the time-evolution operator of the chemical master equation is transformed, and the resulting system describes a different reaction network, which does not admit a complex-balanced steady state. A squeezed coherent state gives the stationary distribution of the transformed network, for which analytic expression is obtained.

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Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Notes

  1. For monomolecular reaction networks, time-dependent solutions of master equations can be obtained [4], which are parametrized by the solution of rate equations.

  2. Here, a complex is represented by a formal linear combination with integer coefficients. For example, the reaction of ethanol combustion can be expressed as

    figure a

    where \(\{ \mathrm{C_2 H_5 OH}, \mathrm{O_2}, \mathrm{CO_2},\mathrm{H_2 O} \}\) are the set of chemical species and coefficients in front of them in Eq. (1) correspond to \(s_{iA}\) and \(t_{iA}\).

  3. The Kronecker delta for \(n,m \in {\mathbb {N}}^V\) is defined by \(\delta _{n,m} = \prod _i \delta _{n_i,m_i}\).

  4. figure b

  5. Note that this is a property of deterministic reaction systems, and not a property of stochastic ones.

  6. The Kronecker delta of two complexes \(s,s' \in {\mathbb {N}}^V\) should be understood as \(\delta _{s, s'} :=\prod _i \delta _{s_{i}, s'_{i}} \).

  7. For an alternative approach to constrain the steady-state properties of deterministic chemical reaction systems with generic kinetics base on a different topological index, see Refs. [21,22,23].

  8. A tricky point here is that the rate equation is parametrized by \(k_A\), which are the parameters of the stochastic reaction systems, and not those of the rate equation obtained by the deterministic limit (7) of the stochastic reaction system under consideration. In fact, the theorem holds even for non-mass-action stochastic kinetics [5, 11], whose deterministic limit does not have reaction rates with mass-action kinetics.

  9. Note that this is the abbreviation of the following expression,

    figure c

  10. The following derivation is a slightly simplified version of the one given in Ref. [20]. See also Ref. [24].

  11. The operator \(S(\xi )\) is a unitary operator and \(S^{-1}(\xi ) = S^\dag (\xi ) =S(-\xi )\).

  12. Using the Rodrigues’s formula for \(H_n(x)\), the function \(h_n(x)\) can be expressed as

    figure f

    Since the derivatives acts only on \(e^{x^2}\) and its derivatives, all the terms of \(h_n(x)\) are of positive coefficients, and thus \(h_n(x)>0\) for \(x>0\).

  13. Using the Rodrigues’ formula for generalized Laguerre polynomials, we have

    figure i

    Since the derivatives act on \(x^{n+p}\) and its derivatives, all the coefficients are positive, and \(L^{(p)}_n (-x)>0 \) for \(x>0\).

  14. Similarly, if there is a reaction containing one \(v_j\) as a source, two additional reactions appear from the reaction via the two-mode squeezing.

  15. A duality relation for stochastic processes has been discussed [29] based on the Doi–Peliti formalism.

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Acknowledgements

The authors are grateful to Hyukpyo Hong and Bryan Hernandez for helpful discussions. Y. H. and R. H. are supported by an appointment of the JRG Program at the APCTP, which is funded through the Science and Technology Promotion Fund and Lottery Fund of the Korean Government, and is also supported by the Korean Local Governments of Gyeongsangbuk-do Province and Pohang City. Y. H. is also supported by the National Research Foundation (NRF) of Korea (Grant No. 2020R1F1A1076267) funded by the Korean Government (MSIT).

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Authors and Affiliations

  1. Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk, 37673, Korea

    Yuji Hirono & Ryo Hanai

  2. Department of Physics, Pohang University of Science and Technology, Pohang, Gyeongbuk, 37673, Korea

    Yuji Hirono & Ryo Hanai

  3. RIKEN iTHEMS, RIKEN, Wako, 351-0198, Japan

    Yuji Hirono

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Correspondence to Yuji Hirono.

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Communicated by Shin-ichi Sasa.

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Hirono, Y., Hanai, R. Squeezing Stationary Distributions of Stochastic Chemical Reaction Systems. J Stat Phys 190, 86 (2023). https://doi.org/10.1007/s10955-023-03096-5

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