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Related Concept Videos

Concentration and Rate Law03:03

Concentration and Rate Law

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The rate of a reaction is affected by the concentrations of reactants. Rate laws (differential rate laws) or rate equations are mathematical expressions describing the relationship between the rate of a chemical reaction and the concentration of its reactants.
For example, in a generic reaction aA + bB ⟶ products, where a and b are stoichiometric coefficients, the rate law can be written as:
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Le Chatelier's Principle: Changing Concentration02:27

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A system at equilibrium is in a state of dynamic balance, with forward and reverse reactions taking place at equal rates. If an equilibrium system is subjected to a change in conditions that affects these reaction rates differently (a stress), then the rates are no longer equal and the system is not at equilibrium. The system will subsequently experience a net reaction in the direction of a greater rate (a shift) that will re-establish the equilibrium. This phenomenon is summarized by Le...
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Dynamic Equilibrium02:20

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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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The relative amount of a given solution component is known as its concentration. Often, though not always, a solution contains one component with a concentration that is significantly greater than that of all other components. This component is called the solvent and may be viewed as the medium in which the other components are dispersed or dissolved. Solutions in which water is the solvent are, of course, very common on our planet. A solution in which water is the solvent is called an aqueous...
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Diffusion01:12

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Diffusion is the passive movement of substances down their concentration gradients—requiring no expenditure of cellular energy. Substances, such as molecules or ions, diffuse from an area of high concentration to an area of low concentration in the cytosol or across membranes. Eventually, the concentration will even out, with the substance moving randomly but causing no net change in concentration. Such a state is called dynamic equilibrium, which is essential for maintaining overall...
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Proteins show rotational as well as lateral diffusion across the membrane. The lateral diffusion of proteins was confirmed through the cell fusion experiment where mouse and human cells were fused, resulting in hybrid cells. When the human and mouse cells fused, the specific membrane proteins on human and mouse cells were marked with the red and green-fluorescent markers, respectively. Initially, the red and green fluorescence was located on the respective hemisphere of the cell. As time...
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Spot Variation Fluorescence Correlation Spectroscopy for Analysis of Molecular Diffusion at the Plasma Membrane of Living Cells
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Concentration-Dependent Domain Evolution in Reaction-Diffusion Systems.

Andrew L Krause1, Eamonn A Gaffney2, Benjamin J Walker3

  • 1Mathematical Sciences Department, Durham University, Upper Mountjoy Campus, Stockton Rd, Durham, DH1 3LE, UK. andrew.krause@durham.ac.uk.

Bulletin of Mathematical Biology
|January 13, 2023
PubMed
Summary
This summary is machine-generated.

This study explores pattern formation in reaction-diffusion systems with concentration-dependent evolving domains. We found that domain evolution significantly impacts pattern dynamics, especially in higher dimensions.

Keywords:
Evolving domainsLinear instability analysisPattern formation

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Area of Science:

  • Mathematical Biology
  • Pattern Formation
  • Reaction-Diffusion Systems

Background:

  • Pattern formation is crucial in developmental biology and is often studied in evolving domains.
  • Existing models typically use prescribed domain evolution, not concentration-dependent dynamics, which are biologically relevant.
  • Understanding concentration-dependent domain evolution is key for realistic biological modeling.

Purpose of the Study:

  • To investigate concentration-dependent domain evolution in reaction-diffusion systems.
  • To elucidate fundamental aspects of these complex models in biological contexts.
  • To explore pattern formation dynamics in both one-dimensional (1D) and N-dimensional settings.

Main Methods:

  • Developed a general form of one-dimensional domain evolution and extended it to N-dimensional manifolds.
  • Employed linear stability analysis for 1D systems around homogeneous equilibria.
  • Utilized numerical simulations to demonstrate dynamical behaviors in 1D and 2D planar geometries.

Main Results:

  • Demonstrated various dynamical behaviors and new phenomena, such as peak-splitting instabilities, particularly near critical bifurcation boundaries.
  • Showcased the significant qualitative and quantitative impact of concentration-dependence on nonlinear dynamics under fast growth/contraction.
  • Highlighted crucial differences between 1D and higher-dimensional models, revealing limitations of linear analysis and the importance of constitutive choices.

Conclusions:

  • Concentration-dependent domain evolution introduces complex dynamics and phenomena not observed in simpler models.
  • Higher-dimensional models present significant analytical challenges compared to 1D, emphasizing the need for careful model formulation.
  • This work opens avenues for further mathematical and biological modeling research, particularly in understanding pattern robustness in developmental systems.