Team:Pasteur Paris/TrainingThomas2
General introduction
This year we did a simulation of the growth of neurons towards ou biofilm using a mathematical model. We created a determin-istic model to help the wetlab establish the optimal concen-tration gradients of Nerve Growth Factor (NGF) needed for the regrowth of the nerves. Nerve growth factor (NGF) is one of a group of small protein like molecules called neurotrophins that are responsible for the development of new neurons, and for the health and maintenance of mature ones. The concentra-tion and concentration gradient of NGF are key parameters af-fecting the growth rate and direction of neurites and axons. Neurite growth have shown to be NGF dose-dependent. At a too low concentration the NGF does not have enough effect on the growth of the neurites whereas if the concentration is too high the risks could exceed the benefits, causing diseases. In order to visualize the results of the model on a microfluidic chip we used MATLAB, App Designer, Python, Gmsh and FreeFem. This is an important part of our project because it creates the link between the wetlab and drylab.
In order to achieve our goal, we divided our project in three parts:
- Production of NGF by the E.coli genetically modified
- Simulation of the diffusion of NGF in a given environment
- Neurons growth in the presence of NGF
The context of our model
Our project aims to create a biofilm composed of genetically modified E.coli able to release a neurotrophic factor: NGF. This will help to accelerate the connection between the neurons and the implant of the prothesis; hence aiming to connect directly the prothesis to the neurons of the amputee. This will enable the patient to have more natural control of his prothesis device. The nerves will be guided toward a conductive membrane surrounding our biofilm. The membrane will then pass the electric signal of the regenerated nerves toward the electronic chip of the implant through wires. It will allow the patient to have a more instinctive and natural control than any other actual prosthesis, and a reduced reeducation time.
We will test the biofilm on a microfluidic chip as a proof of concept. The chip will be composed of two compartments: one composed of the E.Coli genetically modified to produce NGF and the other one of neurons. Both compartments will be kept alive via an input and output of nutriments. Micro canals link the two compartments in the middle of the chip, allowing the diffusion of NGF and the growth of the neurites. Our model will hence be established on a micro-fluidic chip in order to share our results with the wetlab and indicate them the optimal concentration of NGF needed according to their model.
We introduce different parameters in order to create our model :
| g | Length of the neurite outgrowth |
| Neurite outgrowth rate | |
| u(x,t) | Concentration of NGF at the position x and time t |
| NGF concentration gradient at the position x and time t | |
| Diffusion coefficient of NGF | |
| K | Gradient factor (growth rate of the neurite under the stimulation of the NGF concentration gradient) |
| Baseline growth rate (neurite growth rate in absence of NGF concentration gradient) | |
| L | Length of the conduit |
Simulation of the diffusion of NGF in a given environment
We are looking to understand the way the NGF spreads inside the conduit once it is produced. This will help us get to know the concentration of the NGF according to the distance from the production site of NGF.
Fick’s diffusion law
To simulate the diffusion of NGF in the microfluidic chip we consider a unidimensional conduit of axe x and a constant concentration of NGF introduced at one end of the canals. In this part, diffusion is assumed to be the only mechanism producing the gradient decay in the nerve conduit. We can model the diffusion characteristics of NGF with Fick’s second law of diffusion :
Cdiff is assumed to be constant inside the conduit and depends on the material used.
We also have two boundary conditions:
Indeed, in the same material, the rate transfer of the diffusing NGF through the cross section of the conduit is proportional to the concentration gradient normal to the cross section. It is assumed that the leakage of NGF at both ends of the conduit is negligible because there should be little NGF at the ends the conduits compared to the total amount of NGF and second because the diffusion rate of NGF. The equation (1) can be solved with Euler’s method and we find the NGF concentration gradient at the position x and time t.The MatLab code is the following: