Team:ETH Zurich/Modeling/Analytical Approximation
From 2011.igem.org
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In mathematics, the problem posed by general reaction-diffusion systems form is known as the '''Kolmogorov-Petrovsky-Piskounov Equation'''. In the case of acetaldehyde as diffusing and reacting molecule, the equation has the following general structure: | In mathematics, the problem posed by general reaction-diffusion systems form is known as the '''Kolmogorov-Petrovsky-Piskounov Equation'''. In the case of acetaldehyde as diffusing and reacting molecule, the equation has the following general structure: | ||
[[File:ETH-AcAl-Reaction-Diffusion.png|317px|center|thumb|'''General Kolmogorov-Petrovsky-Piskounov partial differential equation''' for an acetaldehyde-based reaction-diffusion system. '''D(AcAl(x,z),z)''' is the diffusive term, '''R(AcAl(x,z))''' is the uniform (independent of the spatial z coordinate) reaction term.]] | [[File:ETH-AcAl-Reaction-Diffusion.png|317px|center|thumb|'''General Kolmogorov-Petrovsky-Piskounov partial differential equation''' for an acetaldehyde-based reaction-diffusion system. '''D(AcAl(x,z),z)''' is the diffusive term, '''R(AcAl(x,z))''' is the uniform (independent of the spatial z coordinate) reaction term.]] | ||
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+ | In our case, it has the following, less general form: [[File:ETHZ-AcAl-Diffusion-Degradation.png|573px|center|thumb|'''Kolmogorov-Petrovsky-Piskounov partial differential equation''' for our system, as derived in [[Team:ETH_Zurich/Modeling/Microfluidics#Model|the reaction-diffusion model for acetaldehyde]].]] | ||
== Dead Zone Concept == | == Dead Zone Concept == |
Revision as of 18:31, 27 October 2011
Analytical Approximation
GFP Band: Dimensionless Model
1. Dimensionless Species
We nondimensionalized all species occurring in the equations for the GFP band circuit.
2. Dimensionless Equations
The dimensionless equations for the dynamics of the band-generating system then read
3. Steady State
4. Dimensionless Groupings
The equation system can be simplified by introducing the following dimensionless groupings:
This yields the final equations for steady state:
Gradient Approximation
We derived the gradient formation dynamics analytically already in the reaction-diffusion model. However, there we only solved the resulting partial differential equation numerically. Here, we are going to venture forth and give a suitable analytical approximation!
Kolmogorov-Petrovsky-Piskounov Equation
In mathematics, the problem posed by general reaction-diffusion systems form is known as the Kolmogorov-Petrovsky-Piskounov Equation. In the case of acetaldehyde as diffusing and reacting molecule, the equation has the following general structure:
Dead Zone Concept
Analytical Solution
References