Standard Compartment-Based Morphogen Gradient Model

Available identifiers

standard compartment-based model for morphogen gradient modelling

Signaling molecules (A) are produced locally, and morphogen molecules (B) are produced from A. Diffusion corresponds to two chains of chemical reactions (picture).

List of mathematical expressions with quantities

Product Of Poisson Distributions	$ϕ (𝐧, 𝐦) = \exp [- \sum_{i = 1}^{K} ({\bar{A}}_{i} + {\bar{B}}_{i})] \prod_{i = 1}^{K} \frac{{\bar{A}}_{i}^{n_{i}} {\bar{B}}_{i}^{m_{i}}}{n_{i}! m_{i}!}$
$\overline{A}$ symbol represents Average Number Of Molecules Of Signal
$\overline{B}$ symbol represents Average Number Of Molecules Of Morphogen
$ϕ (𝕟, 𝕞)$ symbol represents Stationary Distribution

Reaction Diffusion Master Equation	$\frac{\partial p}{\partial t} (𝐧, 𝐦, t) = (𝒟 + ℛ) p (𝐧, 𝐦, t)$
$𝒟$ symbol represents Diffusion Operator
$ℛ$ symbol represents Reaction Operator
$p (𝐧, 𝐦, t)$ symbol represents Joint Probability

Steady State Equations	$\begin{aligned} (\frac{D_{A}}{h^{2}} S - k_{2} I) \overline{𝐀} = - k_{1} 𝐞_{1} (\frac{D_{B}}{h^{2}} S - k_{3} I) \overline{𝐁} = - k_{2} \overline{𝐀} \end{aligned}$
$D_{A}$ symbol represents Diffusion Coefficient A
$D_{B}$ symbol represents Diffusion Coefficient B
$S$ symbol represents Matrix S
$\frac{D_{A}}{h^{2}}$ symbol represents Jump Rate of A
$\frac{D_{B}}{h^{2}}$ symbol represents Jump Rate of B
${\overline{A}}_{i}$ symbol represents Average Number Of Molecules Of Signal
${\overline{B}}_{i}$ symbol represents Average Number Of Molecules Of Morphogen
$h$ symbol represents Length
$k_{1}$ symbol represents Reaction Rate Constant
$k_{2}$ symbol represents Reaction Rate Constant
$k_{3}$ symbol represents Reaction Rate Constant

Stationary Reaction-Diffusion Master Equation	$0 = (𝒟 + ℛ) ϕ (𝐧, 𝐦)$
$𝒟$ symbol represents Diffusion Operator
$ℛ$ symbol represents Reaction Operator
$ϕ$ symbol represents Stationary Distribution