Estimating a Reaction-Diffusion System

Whilst the dynamics in systems can be defined neatly using differential equations, the approximation of their parameters becomes quickly intractable when dealing with PDEs. Reaction-Diffusion systems offer a potent toolset to analyse a set of non-linear PDEs. Here, we make use of a modified version of an expected maximisation (EM) algorithm using BSplines, which was proposed by Brewer et al. 2008 [1], to derive the parameters of a reaction-diffusion system from data points. For an interactive and explanatory guide for how to use the fitting method, see the interactive Jupyter Notebook. You can run it locally with the command

python3 -m jupyter notebook

Requirements

Please install the following software

• Python3
• pip3
• singularity (if you want to run the model in a virtual environment)

Installation of Python libraries

To install the necessary python requirements simply run

python3 -m pip install -r requirements.txt

Run the Simple Test Example

Run

python3 test_main.py

which shows the example of a predefined reaction diffusion system of the form

It will load the dummy data from the data directory and try to derive the correct parameters for the ODE. Note that the algorithm is heavily dependent on the degree that is used for the BSplines to approximate the time course. While a degree of k=5 or k=4 should produce an outcome like

setting the degree of the BSplines to k=3 has a large influence on the derived parameters and hence on the system dynamics

Run the General Example

To run the same example, but implemented with ODEs that were only defined in the main (instead of being given) run

python3 general_main.py

Use this as a reference if you want to estimate the parameters of an ODE with a customised layout.

References

[1] Brewer, Daniel, et al. "Fitting ordinary differential equations to short time course data." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 366.1865 (2008): 519-544.