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General introduction

The aim of our mathematical model is to simulate the growth of neurons towards our biofilm in response of the presence of Nerve Growth Factor (NGF). Nerve growth factor is one of a group of small proteins called neurotrophins that are re-sponsible for the development of new neurons, and for the health and maintenance of mature ones. We created a determin-istic model to help the wetlab establish the optimal concen-tration gradients of NGF needed for the regrowth of the nerves. NGF concentration and concentration gradient are key parameters affecting the growth rate and direction of neu-rites. Neurites growth have shown to be NGF dose-dependent: if NGF concentration si too low or too high, the growth rate is attenuated. 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 since it creates the link between the wetlab and drylab.

We divided our model 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

Context of our model

Our project aims at creating a biofilm composed of genetically modified E. coli able to release a neurotrophic factor: NGF. It helps to accelerate the connection between the neurons and the implant of the prothesis; hence aiming at connecting directly the prothesis amputee’s neurons. This will enable the patient to have a more instinctive control of his prothesis device. The nerves will be guided towards a conductive membrane surrounding our genetically modified biofilm. This membrane will then pass the neural signal of the regenerated nerves towards the electronic chip of the implant through wires. It will allow the patient to have a more instinctive and natural control than any other current prosthesis, and a reduced reeducation time.

The aim of the wetllab is to test the biofilm on a microfluidic chip as a proof of concept. The chip is composed of two compartments: one made of the E. coli genetically modified to produce NGF and the other one of neurons. 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 shape 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
dg/dt	Neurite outgrowth rate
u(x,t)	Concentration of NGF at the position x and time t
du/dt	NGF concentration gradient at the position x and time t
C_diff	Diffusion coefficient of NGF
K	Gradient factor (growth rate of the neurite under the stimulation of the NGF concentration gradient)
G_θ	Baseline growth rate (neurite growth rate in absence of NGF concentration gradient)
L	Length of the conduit

NGF Production by genetically modified E. coli

NGF diffusion diffusion 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 to determine the NGF concentration u(x,t) (ng.mL^-1) as a function of the distance x (cm) from the production site of NGF.

Fick’s diffusion law

To simulate NGF diffusion 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 micro canals. We can model the diffusion characteristics of NGF with Fick’s second law of diffusion:
du / dt = C_diff d²u / dx² (1)

Cdiff is assumed to be constant inside the conduit and depends on the material used.
There are also two boundary conditions:
at x=0: du / dx |_(0,t) (2)
at x=L: du / dx |_(L,t) (3)

Indeed, in the same material, the rate transfer of the diffusing NGF through the cross section of the micro canal is proportional to the concentration gradient normal to the cross section. It is assumed that the leakage of NGF at both ends of the micro canal is negligible because there should be little NGF at the ends the micro canals compared to the total amount of NGF and second because of a low NGF diffusion rate. 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:

We displayed our results showing a decrease of the concentration of NGF (u(x,t)) depending on the distance of the conduit x.

We used the following parameters for the model:

Length of the conduit: L	0.1 cm
Diffusion coefficient of NGF : Cdiff	7,8*10^-7 cm².s^-1
Time length of the experiment: t_final	3 600 s

We obtain the following graphs:

Optimisation of the NGF gradient

To optimize the accuracy of the NGF gradient we interpolate the curve u(x)=f(x). Consequently, we obtain the f polynomial function easier to derive and a polynomial function of the gradient with a better accuracy than with the first method. The program is the following:

With the same parameters as with the previous model we obtain the following graphs:

Analysis of the model

To validate the model, we vary the three parameters (L, t_final, C_diff) to verify if the program corresponds to a diffusion phenomenon described in Fick’s second law of diffusion.

Observations:

When the length of the conduit increases but the duration of the experiment is fixed the NGF doesn’t have the time to diffuse in the entire conduit.
For instance, with a t_final= 3 600s the NGF molecules can’t diffuse further than x=0.2cm.

The higher the diffusion coefficient, the faster the molecules will diffuse in the conduit. Indeed, we observe in the model that with a fixed t_final:

NGF concentration at x=0.1 cm is 675 000 ng.ml^-1 for a diffusion coefficient C_diff = 15*10^-7 cm².s^-1
For a diffusion coefficient two times lower, the NGF concentration is 380 ng.ml¹

The results confirm the prediction of the Fick’s law model.

When the time length of the experiment lasts from 1 hour to 2 hours, the concentration of NGF is almost homogeneous in the entire conduit. At the end of the conduit, for x= 0.1 cm, the concentration of NGF equals to 910 ng.ml-1 when t_final= 7 200s whereas the concentration is 3 900 ng.ml^-1 when t_final=3 600s.

It is interesting to observe that when the duration of the experiment increases, the stationary regime is established: the NGF concentration in the conduit becomes independent of the position and time. Indeed, the concentation gradient of NGF in the conduit moves toward 0 for any position.

Neurons growth in the presence of NGF

In this part our goal is to determine the length of the neurite outgrowth (g(t)) in response to the gradient concentration of NGF.

Explanation of the model

Baseline growth rate:

In our mathematical model, neurites grow at a constant growth rate defined as the baseline growth rate G0 when the concentration is below the threshold (assumed to be 995 ng.mL^-1). Neurites stop growing when the NGF concentration is higher than the threshold concentration. The value for the baseline growth rate G0 has been fixed at 20 μm.h^-1 for this model.

Concentration Gradient:

The extent of directional guidance is gradient steepness-dependent provided that the concentration gradient reaches the threshold value. The gradient factor k is a gradient steepness-dependent positive effect on the neurite growth rate.

In this model we assume that the baseline growth rate and the growth rate in the presence of concentration gradient follow an additive rule. This can be explained by the fact that both the NGF concentration and the its gradient can both individually contribute to neurite extension. The equation governing neurite outgrowth thus becomes:

dg / dt = G₀ + k ∂u / ∂x |_(g(t),t) (4)

We can introduce a time parameter Tlag because the time taken to transmit the NGF signal is finite. The experiments show that the time lag for the cells to respond to NGF is approximately 1 day. The experiments show:
if t ≤ T_lag : dg / dt = 0
else, if: t ≥ T_lag : dg / dt = G₀ + k ∂u / ∂x |_(g(t),t)

Solving the model

To solve the equation (4) we are using Euler’s method forward because the gradient concentration of NGF depends on the length of the neurite (since neurites consume NGF).

The Equation (4):
dg / dt = G₀ + k ∂u / ∂x |_(g(t),t)

Can be written as:

g' = G₀ + k*f(g,t)

Which can be written as :

g_n+1 - g_n / dt |_(g(t),t) = G_θ + k ∂u / ∂x |_(g(t),t)

We can therefore have an expression of g_n+1:

g_n+1 = g_n + dt*[G₀ + k ∂u / ∂x |_(g(t),t)]

With initial values of g_θ, t_θ and ∂u / ∂x |_(g(t),t) we can find all the values of the g

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