Stochastic Approach to First-Order Chemical Reaction Kinetics

Darvey, Ivan G.; Staff, P. J.

doi:10.1063/1.1726855

Cited by 49 publications

(28 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is interesting that an exact solution for this general case can be found given that the model breaks detailed balance and includes a bimolecular reaction step, features not typical of the exact solutions reported in the literature. [4][5][6][7][8][9]11 In particular, to the best of our knowledge, our exact solution is a first for a gene regulatory network with a feedback loop. As we have shown, the previous exact solution claimed by Hornos et al 19 for the special case in which bound and free protein degradation are equal is incorrect because it is based on master equations which possess no coherent physical interpretation.…”

Section: Discussionmentioning

confidence: 99%

“…Exact solutions have been obtained for reaction networks obeying detailed balance [4][5][6] and for those composed of first-order (unimolecular) reactions. [7][8][9][10][11] However, these restrictions are not typical of biochemical processes inside living cells. Detailed balance conditions are characteristic of closed systems of reversible chemical reactions in thermal equilibrium conditions; they only hold for open systems in special cases.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Steady-state fluctuations of a genetic feedback loop: An exact solution

Grima

Schmidt

Newman³

2012

The Journal of Chemical Physics

132

161

View full text Add to dashboard Cite

Genetic feedback loops in cells break detailed balance and involve bimolecular reactions; hence, exact solutions revealing the nature of the stochastic fluctuations in these loops are lacking. We here consider the master equation for a gene regulatory feedback loop: a gene produces protein which then binds to the promoter of the same gene and regulates its expression. The protein degrades in its free and bound forms. This network breaks detailed balance and involves a single bimolecular reaction step. We provide an exact solution of the steady-state master equation for arbitrary values of the parameters, and present simplified solutions for a number of special cases. The full parametric dependence of the analytical non-equilibrium steady-state probability distribution is verified by direct numerical solution of the master equations. For the case where the degradation rate of bound and free protein is the same, our solution is at variance with a previous claim of an exact solution [J. E. M. Hornos, D. Schultz, G. C. P. Innocentini, J. Wang, A. M. Walczak, J. N. Onuchic, and P. G. Wolynes, Phys. Rev. E 72, 051907 (2005), and subsequent studies]. We show explicitly that this is due to an unphysical formulation of the underlying master equation in those studies.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Steady-state fluctuations of a genetic feedback loop: An exact solution

Grima

Schmidt

Newman³

2012

The Journal of Chemical Physics

132

161

View full text Add to dashboard Cite

show abstract

“…Analytical solutions of the CME are only known for special reaction systems with particular initial conditions; these include, e.g., closed (mass conserving) linear reaction systems with multinomial initial distributions [6,8], open linear reaction systems with Poisson initial distribution [9], or a two-component, linear open system with deterministic initial conditions [20,Chap. 8.4].…”

Section: Introductionmentioning

confidence: 99%

Solving the chemical master equation for monomolecular reaction systems analytically

2006

View full text Add to dashboard Cite

Solving the chemical master equation for monomolecular reaction systems analytically the date of receipt and acceptance should be inserted later Abstract The stochastic dynamics of a well-stirred mixture of molecular species interacting through different biochemical reactions can be accurately modelled by the chemical master equation (CME). Research in the biology and scientific computing community has concentrated mostly on the development of numerical techniques to approximate the solution of the CME via many realizations of the associated Markov jump process. The domain of exact and/or efficient methods for directly solving the CME is still widely open, which is due to its large dimension that grows exponentially with the number of molecular species involved. In this article, we present an exact solution formula of the CME for arbitrary initial conditions in the case where the underlying system is governed by monomolecular reactions. The solution can be expressed in terms of the convolution of multinomial and product Poisson distributions with time-dependent parameters evolving according to the traditional reaction-rate equations. This very structured representation allows to deduce any property of the solution. The model class includes many interesting examples and may also be used as the starting point for the design of new numerical methods for the CME of more complex reaction systems.

show abstract

“…The CDS solution of this problem was the subject of several earlier articles [16][17][18][19][20][21][22][23][24][25][26][27][28][29] and will only be covered here to the extent that understanding of the stochastic maps requires. First, it is necessary to identify all possible states of the system.…”

Section: The Stochastic Description Of the Reaction Networkmentioning

confidence: 99%

“…15 The stochastic description of chemically first order (or monomolecular) reaction networks has been the subject of a surprisingly large number of studies at various levels of sophistication. [16][17][18][19][20][21][22][23][24][25][26][27][28][29] From a theoretical point of view, these reaction schemes offer conceptual clarity and simplicity in mathematical formalism, which often facilitates the recognition of known probability distributions in the results. Practical interest is also increasing in these processes due to improvements in single molecule detection.…”

Section: Introductionmentioning

confidence: 99%

Stochastic mapping of first order reaction networks: A systematic comparison of the stochastic and deterministic kinetic approaches

Lente

2012

The Journal of Chemical Physics

View full text Add to dashboard Cite

Stochastic maps are developed and used for first order reaction networks to decide whether the deterministic kinetic approach is appropriate for a certain evaluation problem or the use of the computationally more demanding stochastic approach is inevitable. On these maps, the decision between the two approaches is based on the standard deviation of the expectation of detected variables: when the relative standard deviation is larger than 1%, the use of the stochastic method is necessary. Four different systems are considered as examples: the irreversible first order reaction, the reversible first order reaction, two consecutive irreversible first order reactions, and the unidirectional triangle reaction. Experimental examples are used to illustrate the practical use of the theoretical results. It is shown that the maps do not only depend on particle numbers, but the influence of parameters such as time, rate constants, and the identity of the detected target variable is also an important factor.

show abstract

Stochastic Approach to First-Order Chemical Reaction Kinetics

Abstract: Articles you may be interested inStochastic mapping of first order reaction networks: A systematic comparison of the stochastic and deterministic kinetic approaches

Cited by 49 publications

References 6 publications

Steady-state fluctuations of a genetic feedback loop: An exact solution

Steady-state fluctuations of a genetic feedback loop: An exact solution

Solving the chemical master equation for monomolecular reaction systems analytically

Stochastic mapping of first order reaction networks: A systematic comparison of the stochastic and deterministic kinetic approaches

Contact Info

Product

Resources

About