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:

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:

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.

Kolmogorov-Petrovsky-Piskounov partial differential equation for our system, as derived in the reaction-diffusion model for acetaldehyde.

Dead Zone Concept

Analytical Solution

References