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Related Concept Videos

Reaction Mechanisms: The Steady-State Approximation01:26

Reaction Mechanisms: The Steady-State Approximation

The steady-state approximation, also referred to as the quasi-steady-state approximation to differentiate it from a true steady state, is a widely used method for simplifying calculations in complex reaction mechanisms. This approach is particularly useful when dealing with multi-step reactions that involve reverse reactions or several steps, which can significantly increase mathematical complexity and make the reactions nearly unsolvable analytically.The steady-state approximation operates on...
Transition State Theory01:25

Transition State Theory

Transition-state theory, also known as activated-complex theory, provides a molecular-level explanation of reaction rates in both gas-phase and solution-phase reactions. It extends earlier kinetic models by considering the formation of a short-lived, high-energy configuration during a reaction.The progress of a chemical reaction can be represented using a reaction profile, which plots potential energy against the reaction coordinate. As two reactant molecules approach one another, their...
Reaction Mechanisms: Rate-limiting Step Approximation01:29

Reaction Mechanisms: Rate-limiting Step Approximation

The rate-determining step, or RDS, in a chemical reaction is the slowest step that determines the overall reaction rate. It is identified by using the observed rate law and typically involves approximation methods like the RDS approximation or the steady-state approximation.In the RDS approximation, also known as the rate-limiting-step or equilibrium approximation, the reaction mechanism consists of one or more reversible reactions near equilibrium, followed by a slower RDS, and then one or...
Multi-Step Reactions02:31

Multi-Step Reactions

Chemical reactions often occur in a stepwise fashion involving two or more distinct reactions taking place in a sequence. A balanced equation indicates the reacting species and the product species, but it reveals no details about how the reaction occurs at the molecular level. The reaction mechanism (or reaction path) provides details regarding the precise, step-by-step process by which a reaction occurs. Each of the steps in a reaction mechanism is called an elementary reaction. These...
Consecutive Reactions01:22

Consecutive Reactions

Consecutive reactions involve a sequence where the product of a preceding reaction becomes the reactant for the subsequent one. In a simple scheme, A transforms into B, which further reacts to form C, with rate constants k1 and k2, respectively. This concept is evident in the radioactive decay series. Assuming an initial state with only A present, the conservation of matter leads to three coupled differential equations, determining the concentrations of A, B, and C over time.The rate of change...
Standard Entropy Change for a Reaction03:00

Standard Entropy Change for a Reaction

Entropy is a state function, so the standard entropy change for a chemical reaction (ΔS°rxn) can be calculated from the difference in standard entropy between the products and the reactants.

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Plasmid-derived DNA Strand Displacement Gates for Implementing Chemical Reaction Networks
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Published on: November 25, 2015

A probability generating function method for stochastic reaction networks.

Pilwon Kim1, Chang Hyeong Lee

  • 1Ulsan National Institute of Science and Technology, Ulsan Metropolitan City 689-798, South Korea. pwkim@unist.ac.kr

The Journal of Chemical Physics
|July 12, 2012
PubMed
Summary
This summary is machine-generated.

This study introduces a probability generating function (PGF) method for analyzing stochastic reaction networks. The approach accurately calculates probability distributions and moments for complex chemical systems.

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Ligand-Mediated Nucleation and Growth of Palladium Metal Nanoparticles
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Area of Science:

  • Computational biology
  • Chemical kinetics
  • Systems biology

Background:

  • Stochastic reaction networks are fundamental to understanding biological processes.
  • Analyzing these networks often involves complex mathematical models like the master equation.
  • Accurate computation of probability distributions and moments is crucial for predicting system behavior.

Purpose of the Study:

  • To develop a novel computational approach for analyzing stochastic reaction networks.
  • To provide accurate numerical schemes for probability distributions and moments.
  • To validate the proposed method using established biochemical models.

Main Methods:

  • Conversion of the master equation to a partial differential equation for the probability generating function (PGF).
  • Application of power series expansion and Padé approximation to the PGF.
  • Numerical simulation and validation using chemical reaction examples.

Main Results:

  • The probability generating function (PGF) approach effectively analyzes stochastic reaction networks.
  • Numerical schemes accurately determine probability distributions and key moments (first and second).
  • The method demonstrates high accuracy across diverse models, including ultrasensitive switches and cell cycle transitions.

Conclusions:

  • The PGF approach offers a powerful and accurate tool for stochastic reaction network analysis.
  • This method facilitates the quantitative understanding of complex biological systems.
  • The numerical schemes are robust and applicable to various biochemical reaction systems.