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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...
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Integration of Synaptic Events

Synaptic integration mainly includes the summation of graded potentials. Graded potentials, regardless of their type, cause subtle alterations in membrane voltage, resulting in either depolarization or hyperpolarization. These incremental changes, when combined or summed, can propel the neuron toward its threshold. Consider, for example, a membrane experiencing a +15 mV shift, causing it to depolarize from -70 mV to -55 mV. In this scenario, graded potentials govern the membrane's ability to...
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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...
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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...
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...
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The Use of Chemostats in Microbial Systems Biology
13:19

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Published on: October 14, 2013

Sensitivity summation theorems for stochastic biochemical reaction systems.

Kyung Hyuk Kim1, Herbert M Sauro

  • 1Department of Bioengineering, University of Washington, William H. Foege Building, Box 355061, Seattle, WA 98195-5061, USA. kkim@u.washington.edu

Mathematical Biosciences
|May 8, 2010
PubMed
Summary

We extended metabolic control analysis (MCA) to stochastic processes, revealing new summation theorems for reaction flux variances. These findings link time-window dependencies to power-law scaling in complex network fluctuations.

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Area of Science:

  • Biochemical reaction dynamics
  • Systems biology
  • Non-equilibrium statistical mechanics

Background:

  • Stochasticity is crucial in biological systems.
  • Deterministic Metabolic Control Analysis (MCA) is limited for stochastic processes.
  • Understanding external perturbations' effects on reaction dynamics is essential.

Purpose of the Study:

  • Extend deterministic MCA to the stochastic regime.
  • Introduce and analyze stochastic sensitivities for mean and covariance.
  • Investigate the time-window dependency of flux variance summation theorems.

Main Methods:

  • Developed stochastic sensitivities for reactant concentrations and reaction fluxes.
  • Formulated MCA-like summation theorems for stochastic sensitivities.
  • Introduced a time-scale separation measure to quantify time-window dependency.

Main Results:

  • Established MCA-like summation theorems for stochastic sensitivities.
  • Demonstrated that flux variance summation theorems depend on the measurement time window (τ).
  • Quantified the significant τ-dependency in multi-time-scale systems, linking it to power-law scaling.

Conclusions:

  • Stochastic MCA provides new insights into reaction dynamics under perturbations.
  • The time-window dependency of flux variances is a key feature of stochastic processes.
  • This framework connects theoretical sensitivities to observed power-law scaling in complex networks.