Team:Pasteur Paris/Model

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General introduction

The aim of our mathematical model is to simulate the growth of neurons towards our biofilm in response to the presence of Nerve Growth Factor (NGF). Nerve growth factor is one of a group of small proteins called neurotrophins that are responsible for the development of new neurons, and for the health and maintenance of mature ones. We created a deterministic model to help the wetlab establish the optimal concentration 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 neurites. Neurites growth has shown to be NGF dosedependent: if NGF concentration is 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, Spaceclaim 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:

  1. Production of NGF by the E. coli genetically modified
  2. Simulation of the diffusion of NGF in a given environment
  3. 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 the prothesis and the amputee's neurons directly. This will enable the patient to have a more instinctive control of his prosthetic 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: the genetically modifed E. coli that produces NGF and the other one of neurons. Micro channels 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 our 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
Cdiff Diffusion coefficient of NGF
K Gradient factor (growth rate of the neurite under the stimulation of the NGF concentration gradient)
Gθ Baseline growth rate (neurite growth rate in absence of NGF concentration gradient)
L Length of the conduit

NGF Production by genetically modified E. coli

As we want to obtain the best fitted NGF concentration, we first simulate the production and secretion of our recombinant NGF by transformed E. coli, in order to help the wetlab to optimize the induction and obtain the desired concentration, and to check whether we can theoretically obtain the optimal concentration for neurite growth.

Model Description

In this model, we include transcription, translation, translocation through E. coli membrane, protein folding and mRNA and protein degradation in cytoplasm and medium. NGF synthesis is placed under Plac promoter, so we also modelled the IPTG induction. Finally, NGF is secreted to the medium through Type I secretion system in which the export signal peptide is not cleaved during translocation. Our Biobrick is design to synthetize and export TEV protease in order to cleave signal peptide and thus produce functional NGF.

The molecular mechanism included in our model appears schematically in:

Our model includes the following variables:

Name Meaning
Iex IPTG outside the cell
Iin IPTG in the cytoplasm
Po Plac promoter occupied by repressor, prevent transcription
Pf Plac promoter with free lacO site
m mRNA for TEV and NGF
m-r Ribosome-bound mRNA
NGFc NGF in cytoplasm
TEVc TEV protease in cytoplasm
(N-T)c NGF-TEV complex in cytoplasm
NGFcc Cleaved NGF in cytoplasm, cannot be exported
NGFt NGF bound to transporter channel
TEVt TEV bound to transporter channel
t Transmembrane transporter
NGFum Unfolded NGF in medium with export peptide
NGFm Folded NGF in medium with export peptide
N-Tm Complex between NGF with export peptide and functional TEV
TEVm TEV in medium with export peptide
NGFf Functional NGF in the medium

1. NGF and TEV synthesis in the cytoplasm

The synthesis of NGF and TEV is placed under the control of Plac promoter. The promoter can be in two different states: occupied (Po) by the repressor lacI, preventing RNA polymerase from binding and thus preventing transcription, or free (Pf) thanks to IPTG binding to the repressor. We assume that one IPTG molecule binds with one repressor molecule, freeing the promoter and restoring RNA polymerase binding capacity. The real mechanism of promoter Plac is more complex, as described in [1], but this simplification is sufficient for our model.

The transport of IPTG from outside the cell to cytoplasm is considered to be only due to free diffusion through the membrane by two first order reaction with the same kinetic constant.

IPTG is not considered to be degraded neither in the cytoplasm nor in the medium.

For the TEV and NGF transcription, we use a first-order reaction where the rate of mRNA production (m) depends on the concentration of the free promoter (Pf).

For the TEV and NGF translation, we first consider binding of ribosomes to ribosome binding site (the same association constant is used since the r.b.s. are the same), and then translation rate is proportional to the protein length. Since TEV and NGF have approximately the length, we consider only one translation rate β.

Even though it still has an export peptide, TEV is assumed to be functional in the cytoplasm (although less functional than if it had no export peptide). Since NGF has TEV cleaving site between the coding sequence and the export peptide, a fraction of NGF is cleaved inside the cytoplasm and thus cannot be secreted. We use a simple model to simulate TEV kinetics: TEV recognizes the signal sequence ENLYFQ, bind to its substrate and then cleave the export peptide. This process can thus be modeled by the following equations:

K1, k-1 and k2 are taken lower than constants found in literature, in order to model the fact that TEV still has its signal peptide and is consequently less functional than usually.

2. NGF and TEV secretion to the medium

The transport of NGF and TEV with their export signal peptide from inside the cell to the medium is assumed to follow Michaelis-Menten enzymatic kinetics in which the transporter channel (composed of HlyB in the inner membrane, bound to HlyD and recruiting TolC in the outer membrane) plays the role of the enzyme and intracellular protein the role of the substrate.

Each protein (NGF and TEV) via its export signal peptide HlyA can bind to the HlyB-HlyD complex pore, forming a protein-transporter complex (NGFt or TEVt). Translocation correspond to the dissociation of this complex, resulting in restoring a free transporter and secreting NGF or TEV in the medium (NGFum and TEVm), which stand for the products.

3. Including growth rate

This model is valid for one bacterial cell, but for our model to fit with our proof of concept system, which is a microfluidic chip chamber containing 100 μL of bacterial culture, we need to integrate the number of bacteria contained in the chamber. Therefore, our model helps to determine which is the most accurate bacteria amount we need to put in our chip to produce the appropriate NGF concentration.

4. NGF folding and export peptide cleavage by TEV

Once in the medium, both NGF and TEV are still bound to the export signal peptide HlyA. We assume there is a very small amount of functional TEV, that is sufficient to cleave TEV signal peptide, producing more functional TEV.

As for the transporter, we use a simple model in which TEV recognizes the signal sequence ENLYFQ, bind to its substrate (which can be either NGF with its export peptide or TEV with its export peptide) and then cleave the export peptide. This process can thus be modeled by the following equations:

5. mRNA and protein degradation

Finally, in cytoplasm and in the medium, mRNA and protein are degraded and all degradations are assumed to follow first-order kinetic reactions.

MODEL PARAMETRISATION

From these equations, we obtained a system of differential equations mostly based on mass action kinetics (GET IT HERE). We numerically solveD the ordinary differential equations system using Euler method implemented in Python. The constants we used were mainly determined from literature AND are given in table …

NAME DESCRIPTION VALUE UNIT SOURCE
kt IPTG diffusion rate across the membrane 0.92 min-1
ki Association rate for derepression mechanism by IPTG 3 x 10-5 nM-1min-1
k-i Dissociation rate for derepression mechanism 4.8 x 103 min-1
Transcription rate 2 mRNA.min-1nM-1
kr Association rate of ribosome with r.b.s 1 min-1mRNA-1
k-r Dissociation rate of ribosome with r.b.s 1 min-1
Translation rate 4 nM.min-1mRNA-1
k1 Association rate of TEV with its substrate in the cytoplasm 7.8 x 10-7
k-1 Dissociation rate of TEV with its substrate in the cytoplasm 6 x 10-4
k2 Cleaving rate by TEV in cytoplasm 1.38 x 10-2
k3 Association rate of NGF and TEV with transmembrane transporter 6 x 10-4
k-3 Dissociation rate of NGF and TEV with transporter 2.34
k4 Translocation rate within the transporter 2.1
kf NGF folding rate in the medium 0.28
k5 Association rate of TEV with its substrate in the medium 7.8 x 10-5
k-5 Dissociation rate of TEV with its substrate in the medium 0.06
k6 Cleaving rate by TEV in the medium 1.38
mRNA degradation rate 0.462 min-1
Protein degradation rate in cytoplasm 0.2 min-1
Protein degradation rate in extracelular medium 0.1 min-1

MODEL RESULTS

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 = Cdiff 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, Cdiff) to verify if the program corresponds to a diffusion phenomenon described in Fick’s second law of diffusion.

Observations:

  1. 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.
  2. 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:

  1. NGF concentration at x=0.1 cm is 675 000 ng.ml-1 for a diffusion coefficient Cdiff = 15*10-7 cm2.s-1
  2. 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 ≤ Tlag :     dg / dt = 0
else, if: t ≥ Tlag :     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 :    

gn+1 - gn / dt |(g(t),t) = Gθ + k ∂u / ∂x |(g(t),t)

We can therefore have an expression of gn+1:    

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

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

References

  • Defining the concentration gradient of nerve growth factor for guided neurite outgrowth, XCao M.SShoichet, March 2001

  • Immobilized Concentration Gradients of Neurotrophic Factors Guide Neurite Outgrowth of Primary Neurons in Macroporous Scaffolds, Moore K, MacSween M, Shoichet M, feb 2006

  • Mathematical Modeling of Guided Neurite Extension in an Engineered Conduit with Multiple Concentration Gradients of Nerve Growth Factor (NGF), Tse TH, Chan BP, Chan CM, Lam J, sep 2007

  • Mathematical modelling of multispecies biofilms for wastewater treatment, Maria Rosaria Mattei, november 2005