The chemical birth-death process with additive noise

Vastola, John J.

doi:10.48550/arxiv.1910.09117

Cited by 5 publications

(8 citation statements)

References 35 publications

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“…There are probably many more results (in both stochastic gene regulation and studies of active matter dynamics) that can be derived by exploiting useful parallels between quantum mechanics and stochastic chemical reaction dynamics, as several pieces of past work have suggested [50,51,[57][58][59].…”

Section: Discussionmentioning

confidence: 99%

“…Parenthetically, we may note that the use of these ladder operators to understand the birth-death process evokes striking parallels with the quantum harmonic oscillator; the analogy is even more striking in the continuous limit, where Hermite polynomials replace the Charlier polynomials (see Sec. VI and [50]). In some sense, a single chemical species being randomly produced and degraded behaves like a free boson.…”

Section: Hermitian Conjugatesmentioning

confidence: 92%

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Analytic solution of chemical master equations involving gene switching. I: Representation theory and diagrammatic approach to exact solution

Vastola¹,

Gorin²,

Pachter³

et al. 2021

Preprint

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The chemical master equation (CME), which describes the discrete and stochastic molecule number dynamics associated with biological processes like transcription, is difficult to solve analytically.It is particularly hard to solve for models involving bursting/gene switching, a biological feature that tends to produce heavy-tailed single cell RNA counts distributions. In this paper, we present a novel method for computing exact and analytic solutions to the CME in such cases, and use these results to explore approximate solutions valid in different parameter regimes, and to compute observables of interest. Our method leverages tools inspired by quantum mechanics, including ladder operators and Feynman-like diagrams, and establishes close formal parallels between the dynamics of bursty transcription, and the dynamics of bosons interacting with a single fermion. We focus on two problems: (i) the chemical birth-death process coupled to a switching gene/the telegraph model, and (ii) a model of transcription and multistep splicing involving a switching gene and an arbitrary number of downstream splicing steps. We work out many special cases, and exhaustively explore the special functionology associated with these problems. This is Part I in a two-part series of papers; in Part II, we explore an alternative solution approach that is more useful for numerically solving these problems, and apply it to parameter inference on simulated RNA counts data.

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Section: Discussionmentioning

confidence: 99%

Section: Hermitian Conjugatesmentioning

confidence: 92%

Analytic solution of chemical master equations involving gene switching. I: Representation theory and diagrammatic approach to exact solution

Vastola¹,

Gorin²,

Pachter³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

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“…The Onsager-Machlup [78,79,80,81] and Martin-Siggia-Rose-Janssen-De Dominicis [82,83,84,85,81] path integrals are two other examples, which offer an alternative to the Fokker-Planck equation in the same way the Doi-Peliti path integral is an alternative to the CME. While exact computations of these path integrals are also tedious, they are just as mechanicalone can 'turn the crank' and generate answer, without relying on (for example) a priori knowledge of special functions to solve differential equations [86,75].…”

Section: Discussionmentioning

confidence: 99%

Solving the chemical master equation for monomolecular reaction systems analytically: a Doi-Peliti path integral view

Vastola

2019

Preprint

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The chemical master equation (CME) is a fundamental description of interacting molecules commonly used to model chemical kinetics and noisy gene regulatory networks. Exact time-dependent solutions of the CME-which typically consists of infinitely many coupled differential equations-are rare, and are valuable for numerical benchmarking and getting intuition for the behavior of more complicated systems. Jahnke and Huisinga's landmark calculation of the exact time-dependent solution of the CME for monomolecular reaction systems is one of the most general analytic results known; however, it is hard to generalize, because it relies crucially on properties of monomolecular reactions. In this paper, we rederive Jahnke and Huisinga's result on the time-dependent probability distribution and moments of monomolecular reaction systems using the Doi-Peliti path integral approach, which reduces solving the CME to evaluating many integrals. While the Doi-Peliti approach is less intuitive, it is also more mechanical, and hence easier to generalize. To illustrate how the Doi-Peliti approach can go beyond the method of Jahnke and Huisinga, we also find the exact time-dependent solution to a problem involving an autocatalytic reaction that Jahnke and Huisinga identified as not solvable using their method. Problem statementWe will consider a system with n species S 1 , ..., S n , whose reaction list readsi.e. all possible monomolecular reactions (birth, death, and conversion) are allowed. Note that the rates are allowed to have arbitrary time-dependence as long as c jk (t) ≥ 0 for all j, k and all times t. The corresponding CME reads

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“…1, we will call this system the chemical birth-death process with Gillespie noise. It is closely related to an even simpler approximation (which is only valid when the system is sufficiently close to its steady state value x ss = k/γ) of the same system, the chemical birth-death process with additive noise [40].…”

Section: Introductionmentioning

confidence: 99%

The chemical birth-death process with Gillespie noise

Vastola

2019

Preprint

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A nontrivial technical issue has long plagued the literature on stochastic path integrals: it is not clear which definition is correct in the case of multiplicative/statedependent noise. One reason for this is the unavailability of exactly solvable toy problems with state-dependent noise, that could in principle be used to compare the correctness of different approaches. In this paper, we provide an exact path integral calculation of the transition probability corresponding to a one-dimensional system with state-dependent noise. In particular, we solve the chemical birth-death process with Gillespie noise (the canonical continuous approximation to the discrete birth-death process often used as a toy model in chemical kinetics) using a Martin-Siggia-Rose-Janssen-De Dominicis (MSRJD) path integral. We verify that our result is correct by solving the Fokker-Planck equation via eigenfunction expansion.1 See Sec. 6.1 of Wang [34], for example, where an optimal path missing a saddle point is attributed to a Jacobian-derived term in the action. This term is not present in some other path integral actions [27,28].

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The chemical birth-death process with additive noise

Cited by 5 publications

References 35 publications

Analytic solution of chemical master equations involving gene switching. I: Representation theory and diagrammatic approach to exact solution

Analytic solution of chemical master equations involving gene switching. I: Representation theory and diagrammatic approach to exact solution

Solving the chemical master equation for monomolecular reaction systems analytically: a Doi-Peliti path integral view

The chemical birth-death process with Gillespie noise

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