Stochastic theory of large-scale enzyme-reaction networks: finite copy number corrections to rate equation models.

  title={Stochastic theory of large-scale enzyme-reaction networks: finite copy number corrections to rate equation models.},
  author={Philipp Thomas and Arthur V. Straube and Ramon Grima},
  journal={The Journal of chemical physics},
  volume={133 19},
Chemical reactions inside cells occur in compartment volumes in the range of atto- to femtoliters. Physiological concentrations realized in such small volumes imply low copy numbers of interacting molecules with the consequence of considerable fluctuations in the concentrations. In contrast, rate equation models are based on the implicit assumption of infinitely large numbers of interacting molecules, or equivalently, that reactions occur in infinite volumes at constant macroscopic… 

Figures from this paper

Systematic approximation methods for stochastic biochemical kinetics

An asymptotic series for the moments of the Chemical Master Equation that can be computed to arbitrary precision in the system size expansion and a diagrammatic technique based on the path-integral method that allows to compute time-correlation functions are devised.

Discreteness-induced concentration inversion in mesoscopic chemical systems.

A theoretical framework to understand the effects of discreteness on the steady state of a monostable chemical reaction network and theoretically predict the critical volumes and verify, by exact stochastic simulations, that rate equations are qualitatively incorrect in sub-critical volumes.

Model reduction for stochastic chemical systems with abundant species.

It is proved that in this limit, the fluctuations in the molecule numbers of non-abundant species are accurately described by a hybrid stochastic description consisting of a chemical master equation coupled to deterministic rate equations.

The slow-scale linear noise approximation: an accurate, reduced stochastic description of biochemical networks under timescale separation conditions

A new general method is derived and shown to correctly describe the statistics of intrinsic noise about the macroscopic concentrations under timescale separation conditions, which is expected to be of widespread utility in studying the dynamics of large noisy reaction networks.

Approximation and inference methods for stochastic biochemical kinetics—a tutorial review

An introduction to basic modelling concepts as well as an overview of state of the art methods for deterministic and stochastic methods for modelling chemical networks are introduced and several approximation methods are discussed.

Approximation and inference methods for stochastic biochemical kinetics—a tutorial review

An introduction to basic modelling concepts as well as an overview of state of the art methods for stochastic chemical kinetics, including the chemical Langevin equation and several approximation methods.

Non-linear corrections to the time-covariance function derived from a multi-state chemical master equation.

It is demonstrated that non-linear effects can reveal features of the underlying dynamics, such as reaction stoichiometry, not available in linearised theory, in models that exhibit noise-induced oscillations.

Computation of biochemical pathway fluctuations beyond the linear noise approximation using iNA

The linear noise approximation is commonly used to obtain intrinsic noise statistics for biochemical networks. These estimates are accurate for networks with large numbers of molecules. However it is

Intrinsic Noise Analyzer: A Software Package for the Exploration of Stochastic Biochemical Kinetics Using the System Size Expansion

The software package intrinsic Noise Analyzer (iNA) is introduced, which allows for systematic analysis of stochastic biochemical kinetics by means of van Kampen’s system size expansion of the Chemical Master Equation, and allows for quick approximate analysis of time-dependent mean concentrations, variances, covariances and correlations coefficients.

Queueing Theory-Based Perspective of the Kinetics of “Channeled” Enzyme Cascade Reactions

This work compares the results of a model using differential equations to describe concentration changes with a queueing model and finds that in two enzyme cascades, the queueingmodel predicts at most a 50% smaller throughput than the continuum model even if the waiting room size is smaller.



An effective rate equation approach to reaction kinetics in small volumes: theory and application to biochemical reactions in nonequilibrium steady-state conditions.

  • R. Grima
  • Physics
    The Journal of chemical physics
  • 2010
The theory of effective mesoscopic rate equations generalizes the conventional rate equations of physical chemistry to describe kinetics in systems of mesoscopic size such as biological cells and shows that these differences depend sensitively on the Fano factors and on the inherent structure and topology of the chemical network.

Investigating the robustness of the classical enzyme kinetic equations in small intracellular compartments

Novel mesoscopic rate equations are derived which describe subcellular enzyme reaction kinetics, taking into account, for the first time, the simultaneous influence of both intrinsic noise and the mode of transport.

Fast evaluation of fluctuations in biochemical networks with the linear noise approximation.

The method complements bifurcation studies of the system's parameter dependence by providing estimates of sizes, correlations, and time scales of stochastic fluctuations by suitable variable changes and elimination of fast variables.

Statistical‐mechanical theory of many‐body effects in reaction rates

Many‐body effects in reaction rates depend on the ratio e of a rate coefficient to the product of a diffusion coefficient and a radius, and on the reduced volume fraction φ0 of one or more reactants.

Stochastic chemical kinetics and the quasi-steady-state assumption: Application to the Gillespie algorithm

Using the QSSA, stochastic Michaelis–Menten rate expressions for simple enzymatic reactions are derived and it is illustrated how theQSSA is applied when modeling and simulating a simple genetic circuit.

Reduction and solution of the chemical master equation using time scale separation and finite state projection.

This work presents a model reduction method for study of stochastic chemical kinetic systems that takes advantage of multiple time scales and is implemented in a novel numerical algorithm that exploits the time scale separation to achieve model order reductions while enabling error checking and control.

Adiabatic coarse-graining and simulations of stochastic biochemical networks

We propose a universal approach for analysis and fast simulations of stiff stochastic biochemical networks, which rests on elimination of fast chemical species without a loss of information about

Stochastic simulation of chemical kinetics.

  • D. Gillespie
  • Chemistry
    Annual review of physical chemistry
  • 2007
Some recent advances in methods for using that theory to make numerical simulations include improvements to the exact stochastic simulation algorithm (SSA) and the approximate explicit tau-leaping procedure, as well as the development of two approximate strategies for simulating systems that are dynamically stiff.

Extinction times in autocatalytic systems.

Through the use of the Poisson representation, an exact analytical expression for the mean extinction time of the X population is identified and it is shown that the exact result can be neatly approximated by an Arrhenius-like expression involving an effective activation energy separating a quasi-stationary state from the extinct state.

Exact Stochastic Simulation of Coupled Chemical Reactions

There are two formalisms for mathematically describing the time behavior of a spatially homogeneous chemical system: The deterministic approach regards the time evolution as a continuous, wholly