Morphogen-DAU-Model: Diffusion-Adsorption-Uptake Model for Morphogen Validation

🔬 Project Overview

This repository presents the Diffusion-Adsorption-Uptake (DAU) Model, a rigorous theoretical and computational framework designed to efficiently analyze and validate whether a target protein fulfills the core properties of a Morphogen.

The validation of a morphogen is fundamentally governed by two core hypotheses:

1. Hypothesis 1 (H1: Gradient Formation): The protein must establish a stable, spatial concentration gradient across the tissue.
2. Hypothesis 2 (H2: Concentration-Dependent Regulation): Cells must respond to this concentration gradient in a dosage-dependent manner, activating distinct gene expression profiles in different concentration zones to specify cell fate.

The DAU Model is strictly derived from first-principle physics and higher mathematics, providing a necessary, minimal coarse-graining of the dynamics. It describes the key processes of Diffusion ($D$), Adsorption/Binding ($K$), and Cellular Uptake/Degradation ($k_{\text{uptake}}$) of the morphogen within an embryonic tissue, enabling efficient quantification and prediction of its kinetic parameters from experimental data.

⚙️ Model Schematic (model.jpg)

To provide an intuitive understanding of the model's components and processes, the DAU model schematic is included below.

Figure 1. Schematic of the DAU Model. The morphogen (C) undergoes Diffusion (D) in the extracellular space, non-endocytic Adsorption/Binding (K), and cellular Uptake/Degradation (k_uptake).

📜 Core Model Derivation

1. The Governing Partial Differential Equation (PDE)

Considering a 1D spatial domain ($x \in [0, L]$), the dynamic change in morphogen concentration $C(x, t)$ is governed by the following PDE (Partial Differential Equation):

$$ (1 + K) \frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2} - k_{\text{uptake}} C $$

Where:

• $C(x, t)$: Morphogen concentration.
• $D$: The effective diffusion coefficient.
• $K$: The binding constant (dimensionless).
• $k_{\text{uptake}}$: The first-order rate constant for uptake/degradation.
• The $(1+K)$ term implies the effective diffusivity is $D_{\text{eff}} = D / (1+K)$.

2. The Steady-State Analytical Solution for H1 (The Gradient)

When the system reaches a steady state ($\partial C / \partial t = 0$), the PDE simplifies to an Ordinary Differential Equation (ODE):

$$ D \frac{d^2 C}{d x^2} - k_{\text{uptake}} C = 0 $$

Detailed Derivation & Solution:

The ODE can be written as $\frac{d^2 C}{d x^2} = \mu^2 C$, where the characteristic decay parameter $\mu$ is defined as:

$$ \mu = \sqrt{\frac{k_{\text{uptake}}}{D_{\text{eff}}}} $$

Applying the boundary conditions ($C(0) = C_0$ and $C(L) = 0$), the final steady-state analytical solution is:

$$ C_{\text{steady}}(x) = C_0 \frac{\sinh \left( \mu (L-x) \right)}{\sinh \left( \mu L \right)} $$

The Characteristic Decay Length ($\lambda$) of the morphogen gradient is $\lambda = 1/\mu$.

💻 Code and File Structure

Filename	Description	Key Functionality
src/stimulate1.2.py	H1 Validation Script	Simulates the dynamic formation of the morphogen gradient using the Finite Difference Method (FDM) and analyzes the influence of parameters on the steady state and half-life ($T_{1/2}$).
src/stimulate1.3.py	H2 Validation Script	Analyzes the impact of the steady-state concentration gradient on downstream gene expression (H2), and simulates boundary shifts under perturbations.
paper.pdf	Original Paper	Contains the full model derivation, parameter selection, and associated wet-lab experimental protocols.
requirements.txt	Dependencies	List of Python libraries required to run the simulation scripts.

🚀 How to Run the Code

1. Environment Setup

Ensure you have Python 3 and the necessary scientific computing libraries installed:

pip install -r requirements.txt