Stochastic chemical kinetics A review of the modelling and simulation approaches
---
paper_title: Stochastic simulation of cell signaling pathways

paper_content:

1324127 Making tyre inner tubes RUBBERFABRIEK VREDESTEIN NV 11 Oct 1971 [14 Oct 1970] 47153/71 Heading B7C Apparatus for making tyre inner tubes comprises a tubing conveyer 1, means 3, e.g. a supply roll or an extrusion apparatus, for feeding a continuous length of tubing 4, e.g. of natural or synthetic rubber, on to the conveyer 1, and a carriage 5 reciprocably movable parallel to the conveyer 1, the carriage 5 carrying means 9 for forming holes for air valves in the tubing, means 8 for positioning and fitting air valves, and a cutter 10 for periodically cutting the tubing into lengths. The carriage 5 may be formed of two parts 5A, 5B interconnected by a piston and cylinder device 7 whereby, in use, the means 9 may be operated as the carriage moves along the at same speed as the conveyer 1, and then the part 6B of the carriage may be accelerated with respect to the conveyer by actuating the device 7, the means 8 thus being brought over the holes formed during operation of the means 9. Alternatively the means 8 and 9 may be combined such that a hole forming punch is provided which acts through an air valve as it is fitted. The means 8 and 9 may be arranged to operate before or after the cutter 10. A conveyer belt 14 is provided to supply air valves to the means 8, and the valves may have rubber bases to which adhesive is applied by a device 13. A conveyer 12 receives the cut lengths of tubing 4 with air valves fitted, and the ends of each length are then joined and the tube vulcanized.
---

Title: Stochastic Chemical Kinetics: A Review of the Modelling and Simulation Approaches

Section 1: Introduction
Description 1: This section introduces the concept of stochastic chemical kinetics, highlights its significance, and provides an overview of the paper's structure.

Section 2: The Master Equation
Description 2: Discusses the master equation starting from Markov processes and explains how the master equation is derived and used in stochastic processes.

Section 3: The Chemical Master Equation
Description 3: Explains the chemical master equation specifically for chemical reactions and outlines its application and limitations.

Section 4: Molecular Approach to Chemical Kinetics
Description 4: Describes the difference between the deterministic and stochastic approaches in modeling chemical kinetics and reviews the conversion of rate constants.

Section 5: Fundamental Hypothesis of Stochastic Chemical Kinetics
Description 5: Generalizes the concepts discussed earlier, formalizes the stochastic reaction constant, and introduces the reaction propensity per unit time.

Section 6: The Reaction Probability Density Function
Description 6: Introduces the reaction probability density function necessary for understanding Gillespie's stochastic simulation algorithm.

Section 7: The Stochastic Simulation Algorithms
Description 7: Reviews various stochastic simulation algorithms including the Direct Method, First Reaction Method, and Next Reaction Method.

Section 8: Time-dependent Extension of the First Reaction Method
Description 8: Explores an extension of the First Reaction Method to account for time-dependent changes in rate constants due to variations in volume and temperature.

Section 9: Approximate Stochastic Simulation Algorithms
Description 9: Discusses methods for approximating stochastic simulations to reduce computational effort, including the Poisson timestep method and the τ-leap method.

Section 10: StochSim Algorithm
Description 10: Describes the StochSim algorithm, its differences from Gillespie's approach, and its application in biological systems.

Section 11: Advantages and Drawbacks of Gillespie Algorithm
Description 11: Discusses the benefits and limitations of Gillespie's algorithm, particularly in terms of efficiency and precision versus complexity.

Section 12: Spatio-temporal Algorithms
Description 12: Covers algorithms that incorporate spatial dimensions into stochastic simulations and their importance in modeling realistic biological systems.

Section 13: The Langevin Equation
Description 13: Examines the Langevin equation's role in including external fluctuations in the modeling of chemical systems and discusses its uses and limitations.

Section 14: Conclusions
Description 14: Summarizes the review, highlights the importance of choosing the right mathematical model, and suggests future directions including spatio-temporal models and hybrid methods.