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Related Concept Videos

Chemical Equilibria: Systematic Approach to Equilibrium Calculations01:21

Chemical Equilibria: Systematic Approach to Equilibrium Calculations

Equilibrium calculations for systems involving multiple equilibria are often complex. For example, to calculate the solubility of a sparingly soluble salt in an aqueous solution in the presence of a common ion, one must consider all the equilibria in this solution. Calculations for these systems can be complicated and tedious, so a systematic approach with a series of steps is often helpful. The process is detailed below.
The first step is to identify all the chemical reactions involved, The...
Chemical Reactions02:26

Chemical Reactions

A balanced chemical equation provides the information of chemical formulas of the reactants and products involved in the chemical change. A reaction’s stoichiometry helps predict how much of the reactant is needed to produce the desired amount of product, or in some cases, how much product will be formed from a specific amount of the reactant.
The relative amounts of reactants and products represented in a balanced chemical equation are often referred to as stoichiometric amounts. However, in...
Chemical Reactions01:19

Chemical Reactions

A chemical reaction is a process by which the bonds in the atoms of substances are rearranged to generate new substances. Matter cannot be created or destroyed in a chemical reaction—the same type and number of atoms that make up the reactants are still present in the products. Merely, the rearrangement of chemical bonds produces new compounds.
Chemical Reactions Rearrange Atoms into New Substances
A chemical reaction takes starting materials—the reactants—and changes them into different...
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...
Chemical Equations03:10

Chemical Equations

Chemical equations represent the identities and relative quantities of substances involved in a chemical reaction. The substances undergoing reaction are called reactants, and their formulas are placed on the left side of the equation. The substances generated by the reaction are called products, and their formulas are placed on the right side of the equation. Plus signs (+) separate individual reactant and product formulas, and an arrow (→) separates the reactant and product (left and right)...
Dynamic Equilibrium02:20

Dynamic Equilibrium

A reversible chemical reaction represents a chemical process that proceeds in both forward (left to right) and reverse (right to left) directions. When the rates of the forward and reverse reactions are equal, the concentrations of the reactant and product species remain constant over time and the system is at equilibrium. A special double arrow is used to emphasize the reversible nature of the reaction. The relative concentrations of reactants and products in equilibrium systems vary greatly;...

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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 closure scheme for chemical master equations.

Patrick Smadbeck1, Yiannis N Kaznessis

  • 1Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA.

Proceedings of the National Academy of Sciences of the United States of America
|August 14, 2013
PubMed
Summary
This summary is machine-generated.

We present a new closure scheme to solve the master probability equation for biomolecular networks. This method accurately simulates molecular population dynamics and enables rapid computation of steady states for stochastic reaction networks.

Keywords:
entropy maximizationinformation theorystatistical mechanicsstochastic models

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Area of Science:

  • Biochemistry
  • Theoretical Biology
  • Computational Biology

Background:

  • Probability is fundamental to biology, influencing organismal fate and evolution.
  • The master probability equation offers a complete model for biomolecular networks but remains largely unsolved for complex systems.
  • This computational intractability has limited the application of master equations in biological discovery.

Purpose of the Study:

  • To develop a practical method for solving the master probability equation in chemical and biochemical reaction networks.
  • To enable accurate simulation of molecular population dynamics and computation of system steady states.
  • To overcome the 70-year-old challenge of solving complex master equations.

Main Methods:

  • Casting the master equation into ordinary differential equations for probability distribution moments.
  • Utilizing a closure scheme based on maximizing information entropy to approximate solutions.
  • Simulating dynamic evolution and comparing results with kinetic Monte Carlo methods.

Main Results:

  • The closure scheme accurately solves the master probability equation for small reaction networks.
  • Simulations demonstrate the dynamic evolution of molecular populations.
  • A significant finding is the ability to compute steady states in a single step, bypassing time-consuming simulations.

Conclusions:

  • The developed closure scheme provides an accurate and efficient solution for the master probability equation.
  • This method facilitates the study of stochastic reaction networks and molecular population dynamics.
  • The ability to rapidly compute steady states opens new avenues for biological discovery using probabilistic models.