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Introduction

First aspect modeled : secretion, diffusion and influence of NGF

The aim of our mathematical model is to simulate the growth of neurons towards our biofilm in response to the presence of pro Nerve Growth Factor (NGF) (Figure 1). NGF is part of a family of proteins called neurotrophins. They are responsible for the development of new neurons, and for the growth and maintenance of mature ones. We created a deterministic model to help the wet lab 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 dose-dependent: 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 microchannel, we used MATLAB and Python. This is an important part of our project since it creates the link between the wet lab and dry lab.

Figure 1: Drawing of the NGF (purple) required for the growth of the neurons

We divided our model in three parts:

Production of NGF by the genetically modified Escherichia coli
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 prosthesis; hence aiming at connecting the prosthesis 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 (Figure 2). 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 re-education time.

Figure 2: Drawing of the E. coli confined in our membrane

Figure 3: Drawing of our microfluidic chip

The aim of the wet lab is to test the biofilm on a microfluidic chip as a proof of concept. The chip is composed of two compartments: one contains the genetically modified E. coli that produce NGF and the other one contains neurons (Figure 3). Microchannels 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 microfluidic chip shape in order to share our results with the wet lab and indicate them the optimal concentration of NGF needed according to our model. All the codes we used in this part are available here.

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

Table 1: Model variables

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 modeled the IPTG induction. Finally, NGF is secreted in the medium through Type I secretion system in which the export signal peptide is not cleaved during translocation. Our Biobrick is designed 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 Figure 4.

Figure 4: Secretion mechanism of TEV and NGF by our engineered bacteria

Our model includes the following variables:

Name	Meaning
I_ex	IPTG outside the cell
I_in	IPTG in the cytoplasm
P_o	Plac promoter occupied by repressor, prevent transcription
P_f	Plac promoter with free lacO site
m	mRNA for TEV and NGF
m-r	Ribosome-bound mRNA
NGF_c	NGF in cytoplasm
TEV_c	TEV protease in cytoplasm
(N-T)_c	NGF-TEV complex in cytoplasm
NGF_cc	Cleaved NGF in cytoplasm, cannot be exported
NGF_t	NGF bound to transporter channel
TEV_t	TEV bound to transporter channel
t	Transmembrane transporter
NGF_um	Unfolded NGF in medium with export peptide
NGF_m	Folded NGF in medium with export peptide
N-T_m	Complex between NGF with export peptide and functional TEV
TEV_m	TEV in medium with export peptide
NGF_f	Functional NGF in the medium

Table 2: Model parameters

1. NGF and TEV synthesis in the cytoplasm

The synthesis of NGF and TEV is placed under the control of the 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 reactions 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 same 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, binds to its substrate and then cleaves 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, bounded 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 corresponds to the dissociation of this complex, resulting in restoring a free transporter and secreting NGF or TEV in the medium (NGFum and TEVm), which are 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 bounded 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 the following table.

NAME	DESCRIPTION	VALUE	UNIT	SOURCE
k_t	IPTG diffusion rate across the membrane	0.92	min^-1	[1]
k_i	Association rate for derepression mechanism by IPTG	3 x 10^-5	nM^-1min^-1	[1]
k_-i	Dissociation rate for derepression mechanism	4.8 x 10³	min^-1	[1]
α	Transcription rate	2	mRNA.min^-1nM^-1	[3]
k_r	Association rate of ribosome with r.b.s	1	min^-1mRNA^-1	[2]
k_-r	Dissociation rate of ribosome with r.b.s	1	min^-1	[2]
β	Translation rate	4	nM.min^-1mRNA^-1	[3]
k₁	Association rate of TEV with its substrate in the cytoplasm	7.8 x 10^-7	min^-1nM^-1	Estimated from [4]
k_-1	Dissociation rate of TEV with its substrate in the cytoplasm	6 x 10^-4	min^-1	Estimated from [4]
k₂	Cleaving rate by TEV in cytoplasm	1.38 x 10^-2	min^-1	Estimated from [4]
k₃	Association rate of NGF and TEV with transmembrane transporter	6 x 10^-4	min^-1nM^-1	[5]
k_-3	Dissociation rate of NGF and TEV with transporter	2.34	min^-1	[5]
k₄	Translocation rate within the transporter	2.1	min^-1	[5]
k_f	NGF folding rate in the medium	0.28	min^-1
k₅	Association rate of TEV with its substrate in the medium	7.8 x 10^-5	min^-1nM^-1	[4]
k_-5	Dissociation rate of TEV with its substrate in the medium	0.06	min^-1nM	[4]
k₆	Cleaving rate by TEV in the medium	1.38	min^-1nM^-1	[4]
δ_m	mRNA degradation rate	0.462	min^-1	[1]
δ_pc	Protein degradation rate in cytoplasm	0.2	min^-1	[1]
δ_pm	Protein degradation rate in extracelular medium	0.1	min^-1	[1]

Table 3: Values of constants

MODEL RESULTS

We determined the temporal evolution of secreted NGF concentration in the medium, in order to get the u(0,t) term used in our following diffusion model.

Figure 5: Comparison of cytoplasmic and secreted NGF with a single-cell model (IPTG induction 1 mM)

After the initial dynamics, concentration of secreted NGF quickly reaches a steady state , which is then only driven by the bacterial population dynamics. If we consider a bacterial culture in stationary phase, we can consequently consider that the initial NGF concentration is constant. Our model predicts that the majority of recombinant protein remains cytoplasmic or is secreted but not functional (we consider as "non-functional NGF" the recombinant proteins that are not folded or still have a C-terminal HlyA signal peptide), as it appears in Figure 4.

The aim of this first model is to demonstrate that we can expect an appropriate secreted recombinant NGF concentration to observe neurite growth. However, we had to make several assumptions to parametrize the model. We scanned different parameter values for the values we assumed (such as the number of transporters or kinetic parameters for translocation) in order to check the range of NGF amount we can reasonably expect. We also studied the influence of IPTG induction and number of bacteria, since they are parameters our wet lab can control to best fit recombinant NGF secretion with what we need.

Influence of number of transporters

We co-transformed our bacteria with a plasmid expressing HlyB and HlyD, two of the components of the secretion pore. However, we did not quantify the number of pores each cell contains, and we are only able to estimate it, based on assumptions made in [5]. Consequently, we scanned a range of different values for the number of transporters in order to see the range of NGF concentration we can expect.

The following graph shows the predicted NGF concentration in the microfluidic chip chamber for a number of pores varying: no pore (A.), 10 per cell (B.), 100 per cell (C.) and 500 per cell (D.):

Figure 6: Comparison of cytoplasmic and secreted NGF when the number of transporters varies

Influence of translocation rate

Figure 7: Secreted NGF as a function of translocation rate

As expected, the more transporters the cell has, the more recombinant NGF is secreted, but the amount of functional secreted NGF (in blue) remains limited due to TEV protease cleaving efficiency.

Taking in account the number of E. coli cells and the dilution factor between intracellular and extracellular space, we obtain for 500 transporters a concentration of functional NGF of 1 nM, which corresponds to 24 ng/mL. This is still 10 times lower than what we need to observe neurite growth. Enhancing signal peptide cleavage by a more efficient enzyme should help solve the problem since we could expect 5 nM functional NGF if the totality of the secreted NGF were cleaved.

IPTG induction level

One of the parameters our wet lab team is able to adjust is IPTG induction in the microchannel chip in order to optimize the obtained NGF concentration. Consequently, we studied the dependence of secreted NGF with IPTG initial concentration.

As expected the final NGF concentration (both in the cytoplasm and in extracellular medium) is an increasing function of IPTG induction. As our wet lab did not succeed in quantifying the secreted NGF, it is hard to figure out whether or not the desired concentration was obtained, but if our assumptions are valid, it could be reached with reasonable IPTG concentrations. Production of NGF with the tag has been detected by Mass spectrometry.

Figure 8: Comparison of cytoplasmic and secreted NGF for different IPTG induction level

PERSPECTIVES

Our model is based on assumptions but it shows that within realistic parameters values, we can reasonably expect to obtain the optimal NGF concentration needed for neurite growth in the microfluidic chamber and it consequently paves the way to a functional proof of concept.

Next modeling steps:

It would be worth isolating and quantifying secreted recombinant NGF in order to confront model and experiments, and be able to determine some of the kinetics parameters values we used (such as translocation rate)

This program is designed to model the microchip proof-of-concept experiment but we will adapt it to our final biofilm device to predict its behavior

NGF diffusion simulation in a given environment

We are trying to understand the way the NGF spreads inside the conduit once it is produced. This will help us 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 (cm) and a constant concentration rate 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. According to Fick's diffusion law :
du / dt = C_diff d²u / dx² (1)

C_diff is assumed to be constant inside the conduit and depends on the material used.

The equation (1) can be solved with Euler’s method and we find the NGF concentration gradient at the position x and time t. We displayed our results showing a decrease in the concentration of NGF (u(x,t)) depending on the distance of the conduit x.

We used the following parameters for the model: ^[8]

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

Table 4: Fick's diffusion law parameters

We obtain the following graphs:

Figure 9: NGF gradient

Optimization 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.

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

Analysis of the model

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

Figure 10: Fick's second law model validation (a)

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.

Figure 11: Fick's second law model validation (b)

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 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 90 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. This step is the last one in our neurotrophin modelization. It aims at building a persistent model which should give two relevant pieces of information regarding the use of the interface NeuronArch :
-The model must be able to indicate an estimated value of the time needed for the nerves to grow of a certain distance
-The model must be of use to provide the optimized parameters to boost the nerves growth

Explanation of the model

Baseline growth rate:

In the mathematical model studied ^[8], 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 its gradient can 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)

Euler's Method

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.

Figure 12: Schematic representation of NGF diffusion

Solving the Model

We noticed that there was a mistake in the article regarding the gradient steepness-dependent factor k. Therefore, by examining the results of the article, we were able to find a new coherent value of k, different from what was originally written in the article. To see the details of the mistake and our strategy to find the right value, click here.

In the following graphs, the red curve corresponds to a nerve growth inside a unidimensional canal without any NGF while the blue one corresponds to the situation where there is an NGF gradient inside the canal.

Our strategy to compute this phenomenon is the following.

For each different time, (spaced by the value dt) the position (in cm) of an axon is put inside two unidimensional matrices, g and g_control. The matrix g holds the values of positions when there is a gradient of NGF, while there isn’t for g_control. At each time and for the corresponding position, the script we used to calculate the gradient of NGF (in part 2 ) is ran with those new parameters. As we obtain the value of the gradient of NGF at this time and at a position g_n, we can calculate the new position at the end of the axon g_n+1 by using the formula written above. Tlag is set as 10 000 s for the rest of the modeling.

The first set of parameters we use is the following :

Length of the device L	0.4 cm
Time of the experiment t_final	100 000 s
Initial concentration of NGF U1	995 ng.mL^-1

Table 5: Parameters for testing Euler's method (a)

Figure 13: Evolution of the Nerve growth as a function of time

Not only does the result clearly demonstrates the need of an NGF gradient in order to boost the nerve growth, it also indicates how fast will a neuron grow in a certain amount of time, knowing the NGF concentration at the beginning of the canal.

We then carried on the study regarding two different parameters.

Dependence of the length of the canal

We studied the influence of the length of the canal on the nerve growth. We worked with 2 sets of parameters:

	Set 1	Set 2
Length of the device L	0.4 cm	0.1 cm
Time of the experiment t_final	50 000 s	50 000 s
Initial concentration of NGF U1	995 ng.mL^-1	995 ng.mL^-1

Table 6: Parameters for testing Euler's method (b)

Figure 14: SET 1: L=0.4 cm

Figure 15: SET 2: L=0.1 cm

Figure 16: SET 1: L=0.4 cm

Figure 17: SET 2: L=0.1 cm

The more the length of the device rises, the longer it will take for the NGF concentration to be homogenous inside the canal. Therefore, to boost the growth nerve by having a gradient concentration of NGF in the media, the length of the device must be taken into account. The difference is significant since, at t=50 000 s, the magnitude of the gradient concentration of NGF in the canal worth 10³ when L = 0.4 cm while it worth 10^-5 when L= 0.1 cm.

If the length is too small, the gradient concentration won’t be high enough to boost the growth nerves significantly, as shown in the graph Evolution of the nerve growth above.

Dependence of the initial concentration

We studied the influence of the length of the canal on the nerve growth. We worked with two sets of parameters

	Set 1	Set 2
Length of the device L	0.4 cm	0.1 cm
Time of the experiment t_final	50 000 s	50 000 s
Initial concentration of NGF U1	400 ng.mL^-1	995 ng.mL^-1

Table 7:Parameters for testing Euler's method (c)

The results are shown below.

Figure 18: SET 1

Figure 19: SET 2

The results show that the more the initial concentration of NGF increases, the more its gradient reaches higher values and therefore the faster the neurons grow. It would appear that increasing the initial concentration of NGF would help to boost the nerve growth. However, if the concentration of NGF is too high, it would cause the opposite effect as neurons would start to die. Finding the right compromise, depending on the length of our final device and the coefficient diffusion of NGF in the future media, will be of paramount importance. Finding this compromise will be possible thanks to our model.

COMPARISON WITH EXPERIMENTAL WORKS

Our modeling work has permitted us to study the secretion, diffusion, and influence of NGF on the growth of the neurons. With our model, we are able to optimize two parameters: the length of the microchannels and the initial concentration. The wet lab took into consideration our results to do the experiments on the influence of the concentration of NGF on the growth of the axons. Indeed, they observed the growth of the axons of E18 cortex cells for different concentrations of NGF: 0, 50, 250, 500, 750 and 900 ng/mL. The wet lab’s results were coherent with our model.

The experiments show that until a certain concentration the growth of the neurons increases with the presence of NGF. For a concentration between 250 and 750 ng/mL, the presence of NGF increases significantly the growth of the axons. On the model (Figure 20 and 21), the higher the NGF concentration, the higher the gradient concentration of NGF so the faster the neurons will grow.

Figure 20: Experimental results of the wet lab showing the importance of the growth of the axons for different concentrations of NGF. The β-III Tubulin is a bio marker of neuron cell differentiation that indicates the growth of axons (to know more about the experimental results of the cell culture, click here)

Figure 21: The modeling results showing the nerves growth (cm) for different concentrations of NGF (purple: 0 ng/mL, blue: 250 ng/mL, red: 500 ng/mL, yellow: 750 ng/mL)

The model helped the wet lab establish the concentration limit of NGF above which the NGF doesn’t have any more influence on the growth of the neurons. The wet lab’s concentration limit is coherent with ours: their concentration limit is approximately 900 ng/mL whilst the model shows a concentration limit of 995 ng/mL^[8].

The wet lab has done the series of experiments on a 96 wells plate in order to optimize the number of samples. The next step for the wet lab is to experimentally verify the influence of the length of the microchannels in the microfluidic chip on the growth of the nerves. The model is able to provide information on the optimization of the length of the microchannels which could be of use for the wet lab. Another improvement would be to calculate the diffusion coefficient in the microfluidic chip media.

PERSPECTIVES

Our model will be used to prototype the final device to help and establish the NGF concentration needed to control nerves’ growth. The length of the nerves needed to reach the interface depends on the individual. As mentioned in the design scenario, chemical induction for bacteria regarding NGF production might be considered. Since the model manages to link induction to diffusion to nerves growth, it will enable to know how much NGF needs to be produced for each individual.

The next step consists in keep trying to get in touch with the authors of the article or contacting other experts to make our model completely fulfill its major role in NeuronArch.

Second aspect modeled : mechanical modeling

Neuronarch aims at making the prosthesis of the future and making it more comfortable and protective for the patient. For this sake and to facilitate surgical interventions we modeled the behavior of a bone under mechanical stress. We presented our tools and scripts to Dr. Laurent Sedel, an orthopedic surgeon at Hôpital Lariboisière and researcher at the Hôpital Ambroise Paré – Hôpitaux Universitaires Paris Ile-de-France Ouest, in the hopes of using our tools to improve the lifespan of prosthesis.

Representation of our calculated model of the deformation and the stress (von Mises) on a straight line inside a humerus

Download here the full PDF of the Mechanical Modeling

REFERENCES

M. Stamatakis and N. V. Mantzaris, "Comparison of deterministic and stochastic models of the lac operon genetic network," Biophys. J., vol. 96, no. 3, pp. 887-906, 2009.
A. Y. Weiße, D. A. Oyarzún, V. Danos, and P. S. Swain, "Mechanistic links between cellular trade-offs, gene expression, and growth," Proc. Natl. Acad. Sci., vol. 112, no. 9, pp. E1038-E1047, 2015.
R. Milo, "Useful fundamental BioNumbers handout.doc," pp. 1-2, 2008.
M. S. Packer, H. A. Rees, and D. R. Liu, "Phage-assisted continuous evolution of proteases with altered substrate specificity," Nat. Commun., vol. 8, no. 1, 2017.
H. Benabdelhak et al., "A specific interaction between the NBD of the ABC-transporter HlyB and a C-terminal fragment of its transport substrate haemolysin A," J. Mol. Biol., vol. 327, no. 5, pp. 1169-1179, 2003.
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 modeling of multispecies biofilms for wastewater treatment, Maria Rosaria Mattei, november 2005

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                  </div>
                  <div class="block two-third">
-                     <p>The aim of our mathematical model is to simulate the growth of neurons towards our biofilm in response to the presence of pro Nerve Growth Factor (proNGF). proNGF is part of a family of proteins called neurotrophins. They are responsible for the development of new neurons, and for the growth and maintenance of mature ones. We created a deterministic model to help the wet lab establish the optimal concentration gradients of proNGF needed for the regrowth of the nerves. proNGF concentration and concentration gradient are key parameters affecting the growth rate and direction of neurites. Neurites growth has shown to be proNGF dose-dependent: if proNGF concentration is too low or too high, the growth rate is attenuated. In order to visualize the results of the model on a micro channel, 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 wet lab and dry lab. </p>
+                     <p>The aim of our mathematical model is to simulate the growth of neurons towards our biofilm in response to the presence of pro Nerve Growth Factor (NGF) (Figure 1). NGF is part of a family of proteins called neurotrophins. They are responsible for the development of new neurons, and for the growth and maintenance of mature ones. We created a deterministic model to help the wet lab 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 dose-dependent: 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 microchannel, we used MATLAB and Python. This is an important part of our project since it creates the link between the wet lab and dry lab. </p>
                  </div>
                  <div class="block one-third">
                      <img src="https://static.igem.org/mediawiki/2018/2/23/T--Pasteur_Paris--neurone%2BNGF%2Bchip.png" style="max-width: 450px">
+                    <div class="legend"><b>Figure 1: </b>Drawing of the NGF (purple) required for the growth of the neurons</div>
                  </div>
                  <div class="block full">
                      <p style="text-align: center;">We divided our model in three parts:
                      <ol style="text-align: left;">
-                         <li>Production of proNGF by the <i>E. coli</i> genetically modified</li>
+                         <li>Production of NGF by the genetically modified <i>Escherichia coli</i></li>
-                         <li>Simulation of the diffusion of proNGF in a given environment</li>
+                         <li>Simulation of the diffusion of NGF in a given environment</li>
-                         <li>Neurons growth in the presence of proNGF</li>
+                         <li>Neurons growth in the presence of NGF</li>
                      </ol>
                      </p>
@@ Line 98: / Line 96: @@
                  </div>
                  <div class="block half">
-                     <p>Our project aims at creating a biofilm composed of genetically modified <i>E. coli</i> able to release a neurotrophic factor: proNGF. It helps to accelerate the connection between the neurons and the implant of the prosthesis; hence aiming at connecting the prosthesis 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 re-education time.</p>
+                     <p>Our project aims at creating a biofilm composed of genetically modified <i>E. coli</i> able to release a neurotrophic factor: NGF. It helps to accelerate the connection between the neurons and the implant of the prosthesis; hence aiming at connecting the prosthesis 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 (Figure 2). 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 re-education time.</p>
                  </div>
                  <div class="block half">
                      <img src="https://static.igem.org/mediawiki/2018/d/d4/T--Pasteur_Paris--membrane-plus-egal.png">
+                    <div class="legend"><b>Figure 2: </b>Drawing of the <i>E. coli</i> confined in our membrane</div>
                  </div>
                  <div class="block one-third">
                      <img src="https://static.igem.org/mediawiki/2018/b/b7/T--Pasteur_Paris--micropuce.png" style="max-width: 300px">
+                    <div class="legend"><b>Figure 3: </b>Drawing of our microfluidic chip</div>
                  </div>
                  <div class="block two-third">
-                     <p>The aim of the wet lab is to test the biofilm on a microfluidic chip as a proof of concept. The chip is composed of two compartments: one contains the genetically modified <i> E. coli </i> that produce proNGF and the other one contains neurons. Microchannels link the two compartments in the middle of the chip, allowing the diffusion of proNGF and the growth of the neurites.  Our model will hence be established on a microfluidic chip shape in order to share our results with the wet lab and indicate them the optimal concentration of proNGF needed according to our model.</p>
+                     <p>The aim of the wet lab is to test the biofilm on a microfluidic chip as a proof of concept. The chip is composed of two compartments: one contains the genetically modified <i> E. coli </i> that produce NGF and the other one contains neurons (Figure 3). Microchannels 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 microfluidic chip shape in order to share our results with the wet lab and indicate them the optimal concentration of NGF needed according to our model. All the codes we used in this part are available <a href="https://github.com/samueljaoui/iGEM-Pasteur-Paris-2018-codes-for-secretion-diffusion-and-influence-" style="font-weight: bold ; color:#85196a;"target="_blank">here.</a></p>
                  </div>
                  <div class="block two-third center">
@@ Line 126: / Line 126: @@
                          <tr>
                              <td>u(x,t)</td>
-                             <td>Concentration of proNGF at the position x and time t</td>
+                             <td>Concentration of NGF at the position x and time t</td>
                          </tr>
                          <tr>
@@ Line 134: / Line 134: @@
                                  </span>
                              </td>
-                             <td>proNGF concentration gradient at the position x and time t</td>
+                             <td>NGF concentration gradient at the position x and time t</td>
                          </tr>
                          <tr>
                              <td>C<SUB>diff</SUB></td>
-                             <td>Diffusion coefficient of proNGF</td>
+                             <td>Diffusion coefficient of NGF</td>
                          </tr>
                          <tr>
                              <td>K</td>
-                             <td>Gradient factor (growth rate of the neurite under the stimulation of the proNGF concentration gradient)</td>
+                             <td>Gradient factor (growth rate of the neurite under the stimulation of the NGF concentration gradient)</td>
                          </tr>
                          <tr>
                              <td>G<SUB><FONT face="Raleway">&theta;</FONT></SUB></td>
-                             <td>Baseline growth rate (neurite growth rate in absence of proNGF concentration gradient)</td>
+                             <td>Baseline growth rate (neurite growth rate in absence of NGF concentration gradient)</td>
                          </tr>
                          <tr>
@@ Line 153: / Line 153: @@
                          </tr>
                      </table>
+                </div>
+<div class= "block half">
+<div class="legend"><b>Table 1: </b>Model variables</div>
                  </div>
                  <div class="block separator"></div>
-             <!-- First Onglet Production of proNGF-->
+             <!-- First Onglet Production of NGF-->
                  <div class="block full bothContent">
                      <div class="block dropDown" id="Production">
-                         <h4>proNGF production by genetically modified <i>E. coli</i></h4>
+                         <h4>NGF production by genetically modified <i>E. coli</i></h4>
                      </div>
@@ Line 165: / Line 168: @@
                          <span class="closeCross"><img src="https://static.igem.org/mediawiki/2018/6/67/T--Pasteur_Paris--CloseCross.svg"></span>
                          <div class="block title">
-                             <h1 style="padding-top: 50px;">proNGF production by genetically modified <i>E. coli</i></h1>
+                             <h1 style="padding-top: 50px;">NGF production by genetically modified <i>E. coli</i></h1>
-                             <p><i>As we want to obtain the best fitted proNGF concentration, we first simulate the production and secretion of our recombinant proNGF by transformed <i> E. coli</i>, 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.</i></p>
+                             <p><i>As we want to obtain the best fitted NGF concentration, we first simulate the production and secretion of our recombinant NGF by transformed <i> E. coli</i>, 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.</i></p>
                          </div>
                          <div class="block full">
                              <h3>Model Description</h3>
-                             <p>In this model, we include transcription, translation, translocation through <i> E. coli </i> membrane, protein folding and mRNA and protein degradation in cytoplasm and medium. proNGF synthesis is placed under Plac promoter, so we also modeled the IPTG induction. Finally, proNGF is secreted in the medium through Type I secretion system in which the export signal peptide is not cleaved during translocation. Our Biobrick is designed to synthetize and export TEV protease in order to cleave signal peptide and thus produce functional proNGF.</p>
+                             <p>In this model, we include transcription, translation, translocation through <i> E. coli </i> membrane, protein folding and mRNA and protein degradation in cytoplasm and medium. NGF synthesis is placed under Plac promoter, so we also modeled the IPTG induction. Finally, NGF is secreted in the medium through Type I secretion system in which the export signal peptide is not cleaved during translocation. Our Biobrick is designed to synthetize and export TEV protease in order to cleave signal peptide and thus produce functional NGF.</p>
-                             <p>The molecular mechanism included in our model appears schematically in Figure 1.</p>
+                             <p>The molecular mechanism included in our model appears schematically in Figure 4.</p>
                          </div>
                          <div class="block two-third center">
-                             <img src="https://static.igem.org/mediawiki/2018/8/8f/T--Pasteur_Paris--schéma_model.png">
+                             <img src="https://static.igem.org/mediawiki/2018/5/5b/T--Pasteur_Paris--schemamodel.png">
-                     <div class="legend"><b>Figure 1: </b>Secretion mechanism of TEV and proNGF by our engineered bacteria</div>
+                     <div class="legend"><b>Figure 4: </b>Secretion mechanism of TEV and NGF by our engineered bacteria</div>
                          </div>
                          <div class="block two-third">
@@ Line 205: / Line 208: @@
                                  <tr>
                                      <td><b>m</b></td>
-                                     <td>mRNA for TEV and proNGF</td>
+                                     <td>mRNA for TEV and NGF</td>
                                  </tr>
                                  <tr>
@@ Line 212: / Line 215: @@
                                  </tr>
                                  <tr>
-                                     <td><b>proNGF<sub>c</sub></b></td>
+                                     <td><b>NGF<sub>c</sub></b></td>
-                                     <td>proNGF in cytoplasm</td>
+                                     <td>NGF in cytoplasm</td>
                                  </tr>
                                  <tr>
@@ Line 221: / Line 224: @@
                                  <tr>
                                      <td><b>(N-T)<sub>c</sub></b></td>
-                                     <td>proNGF-TEV complex in cytoplasm</td>
+                                     <td>NGF-TEV complex in cytoplasm</td>
                                  </tr>
                                  <tr>
-                                     <td><b>proNGF<sub>cc</sub></b></td>
+                                     <td><b>NGF<sub>cc</sub></b></td>
-                                     <td>Cleaved proNGF in cytoplasm, cannot be exported</td>
+                                     <td>Cleaved NGF in cytoplasm, cannot be exported</td>
                                  </tr>
                                  <tr>
-                                     <td><b>proNGF<sub>t</sub></b></td>
+                                     <td><b>NGF<sub>t</sub></b></td>
-                                     <td>proNGF bound to transporter channel</td>
+                                     <td>NGF bound to transporter channel</td>
                                  </tr>
                                  <tr>
@@ Line 240: / Line 243: @@
                                  </tr>
                                  <tr>
-                                     <td><b>proNGF<sub>um</sub></b></td>
+                                     <td><b>NGF<sub>um</sub></b></td>
-                                     <td>Unfolded proNGF in medium with export peptide</td>
+                                     <td>Unfolded NGF in medium with export peptide</td>
                                  </tr>
                                  <tr>
-                                     <td><b>proNGF<sub>m</sub></b></td>
+                                     <td><b>NGF<sub>m</sub></b></td>
-                                     <td>Folded proNGF in medium with export peptide</td>
+                                     <td>Folded NGF in medium with export peptide</td>
                                  </tr>
                                  <tr>
                                      <td><b>N-T<sub>m</sub></b></td>
-                                     <td>Complex between proNGF with export peptide and functional TEV</td>
+                                     <td>Complex between NGF with export peptide and functional TEV</td>
                                  </tr>
                                  <tr>
@@ Line 256: / Line 259: @@
                                  </tr>
                                  <tr>
-                                     <td><b>proNGF<sub>f</sub></b></td>
+                                     <td><b>NGF<sub>f</sub></b></td>
-                                     <td>Functional proNGF in the medium</td>
+                                     <td>Functional NGF in the medium</td>
                                  </tr>
                              </table>
                          </div>
+<div class= "block half">
+<div class="legend"><b>Table 2: </b>Model parameters</div>
+                </div>
                          <div class="block separator"></div>
                          <div class="block title">
-                             <h4 style="text-align: left;">1. proNGF and TEV synthesis in the cytoplasm</h4>
+                             <h4 style="text-align: left;">1. NGF and TEV synthesis in the cytoplasm</h4>
                          </div>
                          <div class="block full">
-                             <p>The synthesis of proNGF 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.</p>
+                             <p>The synthesis of NGF and TEV is placed under the control of the 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.</p>
                          </div>
                          <div class="block one-third center">
@@ Line 279: / Line 285: @@
                          <div class="block full">
                              <p>IPTG is not considered to be degraded neither in the cytoplasm nor in the medium.</p>
-                             <p>For the TEV and proNGF transcription, we use a first-order reaction where the rate of mRNA production (m) depends on the concentration of the free promoter (Pf).</p>
+                             <p>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).</p>
                          </div>
                          <div class="block one-third center">
@@ Line 285: / Line 291: @@
                          </div>
                          <div class="block full">
-                             <p>For the TEV and proNGF 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 proNGF have approximately the same length, we consider only one translation rate <FONT face="Raleway">&beta;</FONT>.</p>
+                             <p>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 same length, we consider only one translation rate <FONT face="Raleway">&beta;</FONT>.</p>
                          </div>
                          <div class="block one-third center">
@@ Line 291: / Line 297: @@
                          </div>
                          <div class="block full">
-                             <p>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 proNGF has TEV cleaving site between the coding sequence and the export peptide, a fraction of proNGF 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, binds to its substrate and then cleaves the export peptide. This process can thus be modeled by the following equations:</p>
+                             <p>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, binds to its substrate and then cleaves the export peptide. This process can thus be modeled by the following equations:</p>
                          </div>
                          <div class="block one-third center">
@@ Line 301: / Line 307: @@
                          <div class="block separator"></div>
                          <div class="block title">
-                             <h4 style="text-align: left;">2. proNGF and TEV secretion to the medium</h4>
+                             <h4 style="text-align: left;">2. NGF and TEV secretion to the medium</h4>
                          </div>
                          <div class="block full">
-                             <p>The transport of proNGF 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, bounded to HlyD and recruiting TolC in the outer membrane) plays the role of the enzyme and intracellular protein the role of the substrate.</p>
+                             <p>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, bounded to HlyD and recruiting TolC in the outer membrane) plays the role of the enzyme and intracellular protein the role of the substrate.</p>
                          </div>
                          <div class="block two-third">
-                             <p>Each protein (proNGF and TEV) via its export signal peptide HlyA can bind to the HlyB-HlyD complex pore, forming a protein-transporter complex (proNGFt or TEVt). Translocation corresponds to the dissociation of this complex, resulting in restoring a free transporter and secreting proNGF or TEV in the medium (proNGFum and TEVm), which are the products.</p>
+                             <p>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 corresponds to the dissociation of this complex, resulting in restoring a free transporter and secreting NGF or TEV in the medium (NGFum and TEVm), which are the products.</p>
                          </div>
                          <div class="block one-third">
@@ Line 317: / Line 323: @@
                          </div>
                          <div class="block full">
-                             <p>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 <FONT face="Raleway">&mu;</FONT>L 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 proNGF concentration.</p>
+                             <p>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 <FONT face="Raleway">&mu;</FONT>L 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.</p>
                          </div>
                          <div class="block separator"></div>
                          <div class="block title">
-                             <h4 style="text-align: left;">4. proNGF folding and export peptide cleavage by TEV</h4>
+                             <h4 style="text-align: left;">4. NGF folding and export peptide cleavage by TEV</h4>
                          </div>
                          <div class="block full">
-                             <p>Once in the medium, both proNGF and TEV are still bounded 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.</p>
+                             <p>Once in the medium, both NGF and TEV are still bounded 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.</p>
-                             <p>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 proNGF 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:</p>
+                             <p>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:</p>
                          </div>
                          <div class="block one-third center">
@@ Line 342: / Line 348: @@
                          <div class="block full">
                              <h3>MODEL PARAMETRISATION</h3>
-                             <p>From these equations, we obtained a system of differential equations mostly based on mass action kinetics (get it <a href="https://static.igem.org/mediawiki/2018/d/d4/T--Pasteur_Paris--secretion_equations.pdf" target="_blank">here</a>. 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 the following table.</p>
+                             <p>From these equations, we obtained a system of differential equations mostly based on mass action kinetics (get it <a href="https://static.igem.org/mediawiki/2018/d/d4/T--Pasteur_Paris--secretion_equations.pdf" style="font-weight: bold ; color:#85196a;" target="__blank">here</a>. 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 the following table.</p>
                              <table class="tableData" style="margin: auto;">
                                  <tr>
@@ Line 423: / Line 429: @@
                                  <tr>
                                      <td>k<sub>3</sub></td>
-                                     <td>Association rate of proNGF and TEV with transmembrane transporter</td>
+                                     <td>Association rate of NGF and TEV with transmembrane transporter</td>
                                      <td>6 x 10<sup>-4</sup></td>
                                      <td>min<sup>-1</sup>nM<sup>-1</sup></td>
@@ Line 430: / Line 436: @@
                                  <tr>
                                      <td>k<sub>-3</sub></td>
-                                     <td>Dissociation rate of proNGF and TEV with transporter</td>
+                                     <td>Dissociation rate of NGF and TEV with transporter</td>
                                      <td>2.34</td>
                                      <td>min<sup>-1</sup></td>
@@ Line 444: / Line 450: @@
                                  <tr>
                                      <td>k<sub>f</sub></td>
-                                     <td>proNGF folding rate in the medium</td>
+                                     <td>NGF folding rate in the medium</td>
                                      <td>0.28</td>
                                      <td>min<sup>-1</sup></td>
@@ Line 493: / Line 499: @@
                              </table>
                          </div>
+<div class= "block half">
+<div class="legend"><b>Table 3: </b>Values of constants</div>
+                </div>
                          <div class="block separator"></div>
                          <div>
@@ Line 498: / Line 507: @@
                          </div>
                          <div class="block full">
-                             <p>We determined the temporal evolution of secreted proNGF concentration in the medium, in order to get the u(0,t) term used in our following diffusion model.</p>
+                             <p>We determined the temporal evolution of secreted NGF concentration in the medium, in order to get the u(0,t) term used in our following diffusion model.</p>
                          </div>
                          <div class="block half">
                              <img src="https://static.igem.org/mediawiki/2018/4/43/T--Pasteur_Paris--model1.png">
-                     <div class="legend"><b>Figure 2: </b>Comparison of cytoplasmic and secreted proNGF with a single-cell model (IPTG induction 1 mM)</div>
+                     <div class="legend"><b>Figure 5: </b>Comparison of cytoplasmic and secreted NGF with a single-cell model (IPTG induction 1 mM)</div>
                          </div>
                          <div class="block half">
-                             <p> After the initial dynamics, concentration of secreted proNGF quickly reaches a <b>steady state </b>, which is then only driven by the bacterial population dynamics. If we consider a bacterial culture in stationary phase, we can consequently consider that the initial proNGF concentration is constant. Our model predicts that the majority of recombinant protein remains cytoplasmic or is secreted but not functional (we consider as "non-functional proNGF" the recombinant proteins that are not folded or still have a C-terminal HlyA signal peptide), as it appears in Fig1.</p>
+                             <p> After the initial dynamics, concentration of secreted NGF quickly reaches a <b>steady state </b>, which is then only driven by the bacterial population dynamics. If we consider a bacterial culture in stationary phase, we can consequently consider that the initial NGF concentration is constant. Our model predicts that the majority of recombinant protein remains cytoplasmic or is secreted but not functional (we consider as "non-functional NGF" the recombinant proteins that are not folded or still have a C-terminal HlyA signal peptide), as it appears in Figure 4.</p>
                          </div>
                          <div class="block full">
-                             <p>The aim of this first model is to demonstrate that we can expect an appropriate secreted recombinant proNGF concentration to observe neurite growth. However, we had to make several assumptions to parametrize the model. We scanned different parameter values for the values we assumed (such as number of transporters or kinetic parameters for translocation) in order to check the range of proNGF amount we can reasonably expect. We also studied influence of IPTG induction and number of bacteria, since they are parameters our wetlab can control to best fit recombinant proNGF secretion with what we need.</p>
+                             <p>The aim of this first model is to demonstrate that we can expect an appropriate secreted recombinant NGF concentration to observe neurite growth. However, we had to make several assumptions to parametrize the model. We scanned different parameter values for the values we assumed (such as the number of transporters or kinetic parameters for translocation) in order to check the range of NGF amount we can reasonably expect. We also studied the influence of IPTG induction and number of bacteria, since they are parameters our wet lab can control to best fit recombinant NGF secretion with what we need.</p>
                          </div>
                          <div class="block title">
@@ Line 514: / Line 523: @@
                          </div>
                          <div class="block full">
-                             <p>We co-transformed our bacteria with a plasmid expressing HlyB and HlyD, two of the components of the secretion pore. However, we did not quantify the number of pores each cell contains, and we are only able to estimate it, based on assumptions made in [5]. Consequently, we scanned a range of different values for the quantity of transporters in order to see the range of proNGF concentration we can expect.</p>
+                             <p>We co-transformed our bacteria with a plasmid expressing HlyB and HlyD, two of the components of the secretion pore. However, we did not quantify the number of pores each cell contains, and we are only able to estimate it, based on assumptions made in [5]. Consequently, we scanned a range of different values for the number of transporters in order to see the range of NGF concentration we can expect.</p>
-                             <p>The following graph shows the predicted proNGF concentration in the microfluidic chip chamber for a number of pores varying: no pore (A.), 10 per cell (B.), 100 per cell (C.) and 500 per cell (D.):</p>
+                             <p>The following graph shows the predicted NGF concentration in the microfluidic chip chamber for a number of pores varying: no pore (A.), 10 per cell (B.), 100 per cell (C.) and 500 per cell (D.):</p>
                          </div>
-                         <div class="block two-third center">
+                         <div class="block full">
                              <img src="https://static.igem.org/mediawiki/2018/d/d8/T--Pasteur_Paris--model2.png">
-                     <div class="legend"><b>Figure 3: </b>Comparison of cytoplasmic and secreted proNGF when the number of transporters varies</div>
+                     <div class="legend"><b>Figure 6: </b>Comparison of cytoplasmic and secreted NGF when the number of transporters varies</div>
-                        </div>
-                        <div class="block full">
-                            <p>We co-transformed our bacteria with a plasmid expressing HlyB and HlyD, two of the components of the secretion pore. However, we did not quantify the number of pores each cell contains, and we are only able to estimate it, based on assumptions made in [5]. Consequently, we scanned a range of different values for the quantity of transporters in order to see the range of proNGF concentration we can expect.</p>
-                            <p>The following graph shows the predicted proNGF concentration in the microfluidic chip chamber for a number of pores varying: no pore (A.), 10 per cell (B.), 100 per cell (C.) and 500 per cell (D.):</p>
                          </div>
                          <div class="block title">
                              <h4 style="text-align: left;">Influence of translocation rate</h4>
@@ Line 532: / Line 538: @@
                              <img src="https://static.igem.org/mediawiki/2018/8/8f/T--Pasteur_Paris--model3.png">
-                     <div class="legend"><b>Figure 4: </b>Secreted proNGF as a function of translocation rate</div>
+                     <div class="legend"><b>Figure 7: </b>Secreted NGF as a function of translocation rate</div>
                          </div>
                          <div class="block two-third">
-                              <p>As expected, the more transporters the cell has, the more recombinant proNGF is secreted, but the amount of functional secreted proNGF (in blue) remains limited due to TEV protease cleaving efficiency. </p>
+                              <p>As expected, the more transporters the cell has, the more recombinant NGF is secreted, but the amount of functional secreted NGF (in blue) remains limited due to TEV protease cleaving efficiency. </p>
-                              <p>Taking in account the number of <i> E. coli </i> cells and the dilution factor between intracellular and extracellular space, we obtain for 500 transporters a concentration of functional proNGF of 1 nM, which corresponds to 24 ng/mL. This is still 10 times lower than what we need to observe neurite growth.
+                              <p>Taking in account the number of <i> E. coli </i> cells and the dilution factor between intracellular and extracellular space, we obtain for 500 transporters a concentration of functional NGF of 1 nM, which corresponds to 24 ng/mL. This is still 10 times lower than what we need to observe neurite growth.
-Enhancing signal peptide cleavage by a more efficient enzyme should help solve the problem, since we could expect 5 nM functional proNGF if the totality of the secreted proNGF were cleaved.
+Enhancing signal peptide cleavage by a more efficient enzyme should help solve the problem since we could expect 5 nM functional NGF if the totality of the secreted NGF were cleaved.
 </p>
                          </div>
@@ Line 544: / Line 550: @@
                          </div>
                          <div class="block two-third">
-                              <p> One of the parameters our wetlab team is able to adjust is IPTG induction in the microchannel chip in order to optimize the obtained proNGF concentration. Consequently, we studied the dependence of secreted proNGF with IPTG initial concentration.</p>
+                              <p> One of the parameters our wet lab team is able to adjust is IPTG induction in the microchannel chip in order to optimize the obtained NGF concentration. Consequently, we studied the dependence of secreted NGF with IPTG initial concentration.</p>
-                              <p> As expected the final proNGF concentration (both in the cytoplasm and in extracellular medium) is an increasing function of IPTG induction. As our wetlab did not succeed in quantifying the secreted proNGF, it is hard to figure out whether or not the desired concentration was obtained, but if our assumptions are valid, it could be reached with reasonable IPTG concentrations. Production of proNGF with the tag has been detected by Mass spectrometry.</p>
+                              <p> As expected the final NGF concentration (both in the cytoplasm and in extracellular medium) is an increasing function of IPTG induction. As our wet lab did not succeed in quantifying the secreted NGF, it is hard to figure out whether or not the desired concentration was obtained, but if our assumptions are valid, it could be reached with reasonable IPTG concentrations. Production of NGF with the tag has been detected by Mass spectrometry.</p>
                          </div>
                          <div class="block one-third">
                               <img src="https://static.igem.org/mediawiki/2018/5/5b/T--Pasteur_Paris--model4.png">
-                     <div class="legend"><b>Figure 5: </b>Comparison of cytoplasmic and secreted proNGF for different IPTG induction level</div>
+                     <div class="legend"><b>Figure 8: </b>Comparison of cytoplasmic and secreted NGF for different IPTG induction level</div>
                          </div>
                          <div>
@@ Line 555: / Line 561: @@
                          </div>
                          <div class="block full">
-                             <p>Our model is based on assumptions but it shows that within <b>realistic parameters values</b>, we can reasonably expect to obtain the optimal proNGF concentration needed for neurite growth in the microfluidic chamber and it consequently paves the way to a functional proof of concept. </p>
+                             <p>Our model is based on assumptions but it shows that within <b>realistic parameters values</b>, we can reasonably expect to obtain the optimal NGF concentration needed for neurite growth in the microfluidic chamber and it consequently paves the way to a functional proof of concept. </p>
                  <i style="text-align: left;"><p>Next modeling steps:<br>
-                     <ul>
+                     <ul style="list-style: disc;">
-                         <li> It would be worth isolating and <b>quantifying secreted recombinant proNGF</b> in order to confront model and experiments, and be able to determine some of the kinetics parameters values we used (such as translocation rate)</li>
+                         <li> It would be worth isolating and <b>quantifying secreted recombinant NGF</b> in order to confront model and experiments, and be able to determine some of the kinetics parameters values we used (such as translocation rate)</li>
                          <li> This program is designed to model the microchip proof-of-concept experiment but we will adapt it to our final <b>biofilm</b> device to predict its behavior</li>
                      </ul><br></p>
@@ Line 569: / Line 575: @@
                  <div class="block separator"></div>
-             <!-- Second Onglet Diffusion of proNGF -->
+             <!-- Second Onglet Diffusion of NGF -->
                  <div class="block full bothContent">
                      <div class="block dropDown" id="Diffusion">
-                         <h4>proNGF diffusion simultation in a given environment</h4>
+                         <h4>NGF diffusion simulation in a given environment</h4>
                      </div>
@@ Line 578: / Line 584: @@
                          <span class="closeCross"><img src="https://static.igem.org/mediawiki/2018/6/67/T--Pasteur_Paris--CloseCross.svg"></span>
                          <div class="block title">
-                             <h1 style="padding-top: 50px;">proNGF diffusion diffusion in a given environment</h1><br>
+                             <h1 style="padding-top: 50px;">NGF diffusion simulation in a given environment</h1><br>
-                             <p><i>We are trying to understand the way the proNGF spreads inside the conduit once it is produced. This will help us determine the proNGF concentration u(x,t) (ng.mL<SUP>-1</SUP>) as a function of the distance x (cm) from the production site of proNGF.</i></p>
+                             <p><i>We are trying to understand the way the NGF spreads inside the conduit once it is produced. This will help us determine the NGF concentration u(x,t) (ng.mL<SUP>-1</SUP>) as a function of the distance x (cm) from the production site of NGF.</i></p>
                          </div>
                          <!-- Fick's diffusion law -->
                              <div class="block full">
                                  <h3>Fick’s diffusion law </h3>
-                                 <p>To simulate proNGF diffusion in the microfluidic chip we consider a unidimensional conduit of axe x (cm) and a constant concentration rate of proNGF 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. This corresponds to a unidimensional version of equation (1) with s = 0 and the following Neumann boundary conditions:<br>
+                                 <p>To simulate NGF diffusion in the microfluidic chip we consider a unidimensional conduit of axe x (cm) and a constant concentration rate 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. According to Fick's diffusion law :<br>
                                  <span style="position: relative; display: inline-block; width: 100%; text-align: center;">
                                      <span class="frac">
@@ Line 600: / Line 606: @@
                                  </span>
                                  </p>
-                                 <p>Cdiff is assumed to be constant inside the conduit and depends on the material used.<br>
+                                 <p>C<SUB>diff</SUB> is assumed to be constant inside the conduit and depends on the material used.<br></p>
-                                There are also two boundary conditions:<br>
-                                at x=0: &emsp;&emsp;
+                                 <p>The equation (1) can be solved with Euler’s method and we find the NGF concentration gradient at the position x and time t. We displayed our results showing a decrease in the concentration of NGF (u(x,t)) depending on the distance of the conduit x.</p>
-                                    <span class="frac">
-                                        <span>du</span>
-                                        <span class="symbol">/</span>
-                                        <span class="bottom">dx</span>
-                                    </span>
-                                     |<SUB>(0,t)</SUB> = u<sub>0</sub>&emsp;&emsp;&emsp;(2)<br>
-                                 at x=L: &emsp;&emsp;
-                                 <span style="text-align: left;">
-                                    <span class="frac">
-                                        <span>du</span>
-                                        <span class="symbol">/</span>
-                                        <span class="bottom">dx</span>
-                                    </span>
-                                     |<SUB>(L,t)</SUB> = 0&emsp;&emsp;&emsp;(3)<br>
-                                </span>
-                                </p>
-                                <p>Indeed, in the same material, the rate transfer of the diffusing proNGF 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 proNGF at both ends of the micro canal is negligible because there should be little proNGF at the ends the micro canals compared to the total amount of proNGF and second because of a low proNGF diffusion rate.
-                                The equation (1) can be solved with Euler’s method and we find the proNGF concentration gradient at the position x and time t. The MatLab code is the following:</p>
-                            </div>
-                            <div class="block two-third center">
-                                <img src="https://static.igem.org/mediawiki/2018/6/66/T--Pasteur_Paris--code.1.svg">
-                            </div>
-                            <div class="block two-third center">
-                                <p>We displayed our results showing a decrease of the concentration of proNGF (u(x,t)) depending on the distance of the conduit x.</p>
-                                <img src="https://static.igem.org/mediawiki/2018/f/f3/T--Pasteur_Paris--code-plot.1.svg">
                              </div>
                              <div class="block half center">
-                                 <p>We used the following parameters for the model: </p>
+                                 <p>We used the following parameters for the model: <sup>[8]</sup> </p>
                                  <table class="tableData">
                                      <tr>
@@ Line 637: / Line 620: @@
                                      </tr>
                                      <tr>
-                                         <td>Diffusion coefficient of proNGF : Cdiff</td>
+                                         <td>Diffusion coefficient of NGF : Cdiff</td>
                                          <td>7,8*10<SUP>-7</SUP> cm<SUP>2</SUP>.s<SUP>-1</SUP></td>
                                      </tr>
@@ Line 646: / Line 629: @@
                                  </table>
                              </div>
+<div class= "block full">
+<div class="legend"><b>Table 4: </b>Fick's diffusion law parameters</div>
+                </div>
                              <div class="block full">
                                  <p style="text-align: center;">We obtain the following graphs: </p>
@@ Line 656: / Line 643: @@
                                  <img src="https://static.igem.org/mediawiki/2018/1/14/T--Pasteur_Paris--gifcouleurspuce.gif" style="max-width: 500px; box-shadow: 0px 0px 8px -2px;">
                              </div>
-                         <!-- Optimisation of the gradient -->
+<div class= "block full">
+<div class="legend"><b>Figure 9: </b>NGF gradient</div>
+                </div>
+                         <!-- Optimization of the gradient -->
                              <div class="block full">
-                                 <h3>Optimisation of the proNGF gradient</h3>
+                                 <h3>Optimization of the NGF gradient</h3>
                              </div>
                              <div class="block two-third center">
-                                 <p>To optimize the accuracy of the proNGF 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:</p>
+                                 <p>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.</p>
-                                <img src="https://static.igem.org/mediawiki/2018/2/2f/T--Pasteur_Paris--interpolation-plot.1.svg">
                                  <p>With the same parameters as with the previous model we obtain the following graphs: </p>
                              </div>
@@ Line 673: / Line 665: @@
                                  <img src="https://static.igem.org/mediawiki/2018/4/41/T--Pasteur_Paris--gif-fct-L.gif" style="max-width: 500px; box-shadow: 0px 0px 8px -2px;">
                              </div>
+<div class= "block full">
+<div class="legend"><b>Figure 10: </b>Fick's second law model validation (a)</div>
+                </div>
                              <div class="block full">
                                  <p>Observations:<br>
                                      <ol style="text-align: left; list-style-type: disc;">
-                                         <li>When the length of the conduit increases but the duration of the experiment is fixed the proNGF doesn’t have the time to diffuse in the entire conduit.</li>
+                                         <li>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.</li>
-                                         <li>For instance, with a t_final= 3 600s the proNGF molecules can’t diffuse further than x=0.2cm.</li>
+                                         <li>For instance, with a t_final= 3 600s the NGF molecules can’t diffuse further than x=0.2cm.</li>
                                      </ol>
                                  </p>
@@ Line 684: / Line 681: @@
                                  <img src="https://static.igem.org/mediawiki/2018/0/07/T--Pasteur_Paris--fct-Cdiff.gif" style="max-width: 500px; box-shadow: 0px 0px 8px -2px;">
                              </div>
+<div class= "block full">
+<div class="legend"><b>Figure 11: </b>Fick's second law model validation (b)</div>
+                </div>
                              <div class="block full">
                                  <p>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:<br>
                                      <ol style="text-align: left; list-style-type: disc;">
-                                         <li>proNGF concentration at x=0.1 cm is 675 000 ng.ml<SUP>-1</SUP> for a diffusion coefficient C<SUB>diff</SUB> = 15*10<SUP>-7</SUP> cm<SUP>2</SUP>.s<SUP>-1</SUP></li>
+                                         <li>NGF concentration at x=0.1 cm is 675 ng.ml<SUP>-1</SUP> for a diffusion coefficient C<SUB>diff</SUB> = 15*10<SUP>-7</SUP> cm<SUP>2</SUP>.s<SUP>-1</SUP></li>
-                                         <li>For a diffusion coefficient two times lower, the proNGF concentration is 380 ng.ml<SUP>1</SUP></li>
+                                         <li>For a diffusion coefficient two times lower, the NGF concentration is 380 ng.ml<SUP>1</SUP></li>
                                      </ol>
                                  </p>
@@ Line 694: / Line 694: @@
                              </div>
                              <div class="block full">
-                                 <p>When the time length of the experiment lasts from 1 hour to 2 hours, the concentration of proNGF is almost homogeneous in the entire conduit. At the end of the conduit, for x= 0.1 cm, the concentration of proNGF equals to 910 ng.ml-1 when t_final= 7 200s whereas the concentration is 3 900 ng.ml<SUP>-1</SUP> when t_final=3 600s. </p>
+                                 <p>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 90 ng.ml<SUP>-1</SUP> when t_final=3 600s. </p>
-                                 <p>It is interesting to observe that when the duration of the experiment increases, the stationary regime is established: the proNGF concentration in the conduit becomes independent of the position and time. Indeed, the concentation gradient of proNGF in the conduit moves toward 0 for any position. </p>
+                                 <p>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. </p>
                              </div>
                              <div class="block two-third">
@@ Line 710: / Line 710: @@
                  <div class="block full bothContent">
                      <div class="block dropDown" id="Growth">
-                         <h4>Neurons growth in the presence of proNGF</h4>
+                         <h4>Neurons growth in the presence of NGF</h4>
                      </div>
@@ Line 716: / Line 716: @@
                          <span class="closeCross"><img src="https://static.igem.org/mediawiki/2018/6/67/T--Pasteur_Paris--CloseCross.svg"></span>
                          <div class="block title">
-                             <h1>Neurons growth in the presence of proNGF</h1><br>
+                             <h1>Neurons growth in the presence of NGF</h1><br>
-                             <p><i>In this part our goal is to determine the length of the neurite outgrowth (g(t)) in response to the gradient concentration of proNGF. This step is the last one in our neurotrophin modelisation. It aims at building a persistent model which should  give two relevant informations regarding the use of the interface NeuronArch :
+                             <p><i>In this part our goal is to determine the length of the neurite outgrowth (g(t)) in response to the gradient concentration of NGF. This step is the last one in our neurotrophin modelization. It aims at building a persistent model which should  give two relevant pieces of information regarding the use of the interface NeuronArch :
-</br>-The model must be able to indicate an estimated value of the time needed for the nerves to grow of a certain distance
+                            </br>-The model must be able to indicate an estimated value of the time needed for the nerves to grow of a certain distance
-</br>-The model must be of use to provide the optimizes parameters to boost the nerves growth
+                            </br>-The model must be of use to provide the optimized parameters to boost the nerves growth
-</i></p>
+                            </i></p>
                          </div>
                          <!-- Explanation of the model -->
@@ Line 728: / Line 728: @@
                              <div class="block full">
                                  <h4 style="text-align: left">Baseline growth rate: </h4>
-                                 <p>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<SUP>-1</SUP>). Neurites stop growing when the proNGF concentration is higher than the threshold concentration. The value for the baseline growth rate G0 has been fixed at 20 <FONT face="Raleway">&mu;</FONT>m.h<SUP>-1</SUP> for this model. </p>
+                                 <p>In the mathematical model studied <sup>[8]</sup>, 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<SUP>-1</SUP>). 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 <FONT face="Raleway">&mu;</FONT>m.h<SUP>-1</SUP> for this model. </p>
                                  <h4 style="text-align: left">Concentration Gradient:</h4>
                                  <p>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. </p>
-                                 <p>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 proNGF concentration and its gradient can both individually contribute to neurite extension. The equation governing neurite outgrowth thus becomes:<br><br>
+                                 <p>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 its gradient can individually contribute to neurite extension. The equation governing neurite outgrowth thus becomes:<br><br>
                                  <span style="position: relative; display: inline-block; text-align: center; width: 100%">
                                      <span class="frac">
@@ Line 748: / Line 748: @@
                              </div>
                              <div class="block full">
-                                 <p>We can introduce a time parameter Tlag because the time taken to transmit the proNGF signal is finite. The experiments show that the time lag for the cells to respond to proNGF is approximately 1 day. The experiments show:<br>
+                                 <p>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:<br>
                                  if t <FONT face="Raleway">&le;</FONT> T<SUB>lag</SUB> : &emsp; &emsp;
                                      <span class="frac">
@@ Line 771: / Line 771: @@
                          <!-- Solving the model -->
                              <div class="block full">
-                                 <h3>Solving the model</h3>
+                                 <h3>Euler's Method</h3>
                              </div>
                              <div class="block full">
-                                 <p>To solve the equation (4) we are using Euler’s method forward because the gradient concentration of proNGF depends on the length of the neurite (since neurites consume proNGF). <br><br>
+                                 <p>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). <br><br>
                                  The Equation (4): &emsp; &emsp; <br>
                                  <span style="position: relative; display: inline-block; width: 100%; text-align: center;">
@@ Line 825: / Line 825: @@
                                      </span>
                                       |<SUB>(g(t),t)</SUB>
-                                   we can find all the values of the g </p>
+                                   we can find all the values of the g.</br>  </p>
+                            </div>
+                            <div class="block full">
+                                <img src="https://static.igem.org/mediawiki/2018/1/10/T--Pasteur_Paris--ModelGrowthFigure1bis.png">
+                            </div>
+<div class="block full">
+<div class="legend"><b>Figure 12: </b>Schematic representation of NGF diffusion </div>
+</div>
+                            <div class="block full"><h3>Solving the Model</h3></div>
+                            <div class="block full">
+                                <p>We noticed that there was a mistake in the article regarding the gradient steepness-dependent factor k. Therefore, by examining the results of the article, we were able to find a new coherent value of k, different from what was originally written in the article. To see the details of the mistake and our strategy to find the right value, <a href="https://static.igem.org/mediawiki/2018/2/28/T--Pasteur_Paris--ModelisationMistakes.pdf"style="font-weight: bold ; color:#85196a;" target="__blank">click here.</a></p>
+                                <p>In the following graphs, the red curve corresponds to a nerve growth inside a unidimensional canal without any NGF while the blue one corresponds to the situation where there is an NGF gradient inside the canal. </p>
+                                <p>Our strategy to compute this phenomenon is the following. </p>
+                                <p>For each different time, (spaced by the value dt) the position (in cm) of an axon is put inside two unidimensional matrices, g and g<sub>control</sub>. The matrix g holds the values of positions when there is a gradient of NGF, while there isn’t for g<sub>control</sub>. At each time and for the corresponding position, the script we used to calculate the gradient of NGF (in part 2 ) is ran with those new parameters. As we obtain the value of the gradient of NGF at this time and at a position g<sub>n</sub>, we can calculate the new position at the end of the axon g<sub>n+1</sub> by using the formula written above. Tlag is set as 10 000 s for the rest of the modeling. </p>
+                                <p>The first set of parameters we use is the following : </p>
+                                <table class="tableData" style="margin: auto;">
+                                    <tr>
+                                        <td>Length of the device L </td>
+                                        <td>0.4 cm </td>
+                                    </tr>
+                                    <tr>
+                                        <td>Time of the experiment t_final</td>
+                                        <td>100 000 s </td>
+                                    </tr>
+                                    <tr>
+                                        <td>Initial concentration of NGF U1 </td>
+                                        <td>995 ng.mL<sup>-1</sup></td>
+                                    </tr>
+                                </table>
+<div class="block full">
+<div class="legend"><b>Table 5: </b>Parameters for testing Euler's method (a)</div>
+                </div>
+                            </div>
+                            <div class="block two-third center">
+                                <img src="https://static.igem.org/mediawiki/2018/c/c9/T--Pasteur_Paris--ModelGrowthFigure2.png">
+                                <div class="legend"><b>Figure 13: </b>Evolution of the Nerve growth as a function of time</div>
+                            </div>
+                            <div class="block full">
+                                <p>Not only does the result clearly demonstrates the need of an NGF gradient in order to boost the nerve growth, it also indicates how fast will a neuron grow in a certain amount of time, knowing the NGF concentration at the beginning of the canal.</p>
+                                <p>We then carried on the study regarding two different parameters.</p>
+                            </div>
+                            <div class="block title"><h4 style="text-align: left;">Dependence of the length of the canal </h4></div>
+                            <div class="block full">
+                                <p>We studied the influence of the length of the canal on the nerve growth. We worked with 2 sets of parameters:</p>
+                                <table class="tableData" style="margin: auto;">
+                                    <tr>
+                                        <td></td>
+                                        <td>Set 1</td>
+                                        <td>Set 2</td>
+                                    </tr>
+                                    <tr>
+                                        <td>Length of the device L</td>
+                                        <td>0.4 cm</td>
+                                        <td>0.1 cm </td>
+                                    </tr>
+                                    <tr>
+                                        <td>Time of the experiment t_final</td>
+                                        <td>50 000 s </td>
+                                        <td>50 000 s </td>
+                                    </tr>
+                                    <tr>
+                                        <td>Initial concentration of NGF U1</td>
+                                        <td>995 ng.mL<sup>-1</sup></td>
+                                        <td>995 ng.mL<sup>-1</sup></td>
+                                    </tr>
+                                </table>
+ <div class="block full">
+<div class ="legend"><b> Table 6: </b> Parameters for testing Euler's method (b)
+                            </div>
+                            <div class="block half">
+                              <img src="https://static.igem.org/mediawiki/2018/4/40/T--Pasteur_Paris--ModelGrowthFigure3.png">
+                                <div class="legend"><b>Figure 14: </b>SET 1: L=0.4 cm</div>
+                            </div>
+                            <div class="block half">
+                              <img src="https://static.igem.org/mediawiki/2018/a/a8/T--Pasteur_Paris--ModelGrowthFigure4.png">
+                                <div class="legend"><b>Figure 15: </b>SET 2: L=0.1 cm</div>
+                            </div>
+                            <div class="block half">
+                                <img src="https://static.igem.org/mediawiki/2018/8/82/T--Pasteur_Paris--ModelGrowthFigure5.png">
+                                <div class="legend"><b>Figure 16: </b>SET 1: L=0.4 cm</div>
+                            </div>
+                            <div class="block half">
+                                <img src="https://static.igem.org/mediawiki/2018/7/79/T--Pasteur_Paris--ModelGrowthFigure6.png">
+                                <div class="legend"><b>Figure 17: </b>SET 2: L=0.1 cm</div>
+                            </div>
+                            <div class="block full">
+                                <p>The more the length of the device rises, the longer it will take for the NGF concentration to be homogenous inside the canal. Therefore, to boost the growth nerve by having a gradient concentration of NGF in the media, the length of the device must be taken into account. The difference is significant since, at t=50 000 s, the magnitude of the gradient concentration of NGF in the canal worth 10<sup>3</sup> when L = 0.4 cm while it worth 10<sup>-5</sup> when L= 0.1 cm.</p>
+                                <p>If the length is too small, the gradient concentration won’t be high enough to boost the growth nerves significantly, as shown in the graph Evolution of the nerve growth above.</p>
+                            </div>
+                            <div class="block title"><h4 style="text-align: left;">Dependence of the initial concentration </h4></div>
+                            <div class="block full">
+                                <p>We studied the influence of the length of the canal on the nerve growth. We worked with two sets of parameters </p>
+                                <table class="tableData" style="margin: auto;">
+                                    <tr>
+                                        <td></td>
+                                        <td>Set 1</td>
+                                        <td>Set 2</td>
+                                    </tr>
+                                    <tr>
+                                        <td>Length of the device L</td>
+                                        <td>0.4 cm</td>
+                                        <td>0.1 cm </td>
+                                    </tr>
+                                    <tr>
+                                        <td>Time of the experiment t_final</td>
+                                        <td>50 000 s </td>
+                                        <td>50 000 s </td>
+                                    </tr>
+                                    <tr>
+                                        <td>Initial concentration of NGF U1</td>
+                                        <td>400 ng.mL<sup>-1</sup></td>
+                                        <td>995 ng.mL<sup>-1</sup></td>
+                                    </tr>
+                                </table>
+<div class="block full">
+<div class = "legend"><b>Table 7:</b>Parameters for testing Euler's method (c) </div>
+</div>
+                                <p>The results are shown below.</p>
+                            </div>
+                            <div class="block half">
+                              <img src="https://static.igem.org/mediawiki/2018/3/38/T--Pasteur_Paris--ModelGrowthFigure7.png">
+                                <div class="legend"><b>Figure 18: </b>SET 1</div>
+                            </div>
+                            <div class="block half">
+                              <img src="https://static.igem.org/mediawiki/2018/1/17/T--Pasteur_Paris--ModelGrowthFigure8.png">
+                                <div class="legend"><b>Figure 19: </b>SET 2</div>
+                            </div>
+                            <div class="block full">
+                                <p>The results show that the more the initial concentration of NGF increases, the more its gradient reaches higher values and therefore the faster the neurons grow. It would appear that increasing the initial concentration of NGF would help to boost the nerve growth. However, if the concentration of NGF is too high, it would cause the opposite effect as neurons would start to die. Finding the right compromise, depending on the length of our final device and the coefficient diffusion of NGF in the future media, will be of paramount importance. Finding this compromise will be possible thanks to our model. </p>
+                            </div>
+                            <div class="block title"><h3>COMPARISON WITH EXPERIMENTAL WORKS</h3></div>
+                            <div class="block full">
+                                <p>Our modeling work has permitted us to study the secretion, diffusion, and influence of NGF on the growth of the neurons.  With our model, we are able to optimize two parameters: the length of the microchannels and the initial concentration.
+The wet lab took into consideration our results to do the experiments on the influence of the concentration of NGF on the growth of the axons. Indeed, they observed the growth of the axons of E18 cortex cells for different concentrations of NGF: 0, 50, 250, 500, 750 and 900 ng/mL.  The wet lab’s results were coherent with our model.
+</p> <p>The experiments show that until a certain concentration the growth of the neurons increases with the presence of NGF. For a concentration between 250 and 750 ng/mL, the presence of NGF increases significantly the growth of the axons. On the model (Figure 20 and 21), the higher the NGF concentration, the higher the gradient concentration of NGF so the faster the neurons will grow.</p>
+<div class="block full">
+<img src="https://static.igem.org/mediawiki/2018/5/50/T--Pasteur_Paris--fig9.png">
+<div class="legend"><b>Figure 20: </b> Experimental results of the wet lab showing the importance of the growth of the axons for different concentrations of NGF. The <FONT face="Raleway">&beta;</FONT>-III Tubulin is a bio marker of neuron cell differentiation that indicates the growth of axons (to know more about the experimental results of the cell culture, click <a href="https://2018.igem.org/Team:Pasteur_Paris/Results" style="font-weight: bold;" target="_blank">here</a>)</div>
+</div>
+<br>
+<div class="block full">
+<img src="https://static.igem.org/mediawiki/2018/c/c8/T--Pasteur_Paris--fig10.png">
+<div class="legend"><b>Figure 21:</b> The modeling results showing the nerves growth (cm) for different concentrations of NGF (purple: 0 ng/mL, blue: 250 ng/mL, red: 500 ng/mL, yellow: 750 ng/mL)</div>
+</div>
+<p>The model helped the wet lab establish the concentration limit of NGF above which the NGF doesn’t have any more influence on the growth of the neurons. The wet lab’s concentration limit is coherent with ours: their concentration limit is approximately 900 ng/mL whilst the model shows a concentration limit of 995 ng/mL<sup>[8]</sup>.</p>
+<p>The wet lab has done the series of experiments on a 96 wells plate in order to optimize the number of samples. The next step for the wet lab is to experimentally verify the influence of the length of the microchannels in the microfluidic chip on the growth of the nerves. The model is able to provide information on the optimization of the length of the microchannels which could be of use for the wet lab. Another improvement would be to calculate the diffusion coefficient in the microfluidic chip media.
+</p>
+                            </div>
+                            <div class="block title"><h3>PERSPECTIVES </h3></div>
+                            <div class="block full">
+                                <p>Our model will be used to prototype the final device to help and establish the NGF concentration needed to control nerves’ growth. The length of the nerves needed to reach the interface depends on the individual. As mentioned in the <a href="https://2018.igem.org/Team:Pasteur_Paris/Scenario" style="font-weight: bold ; color:#85196a;"target="_blank">design scenario</a>, chemical induction for bacteria regarding NGF production might be considered. Since the model manages to link induction to diffusion to nerves growth, it will enable to know how much NGF needs to be produced for each individual.</p>
+<p>The next step consists in keep trying to get in touch with the authors of the article or contacting other experts to make our model completely fulfill its major role in NeuronArch.
+</p>
                              </div>
                      </div>
                  </div>
+             </div>
+           </div>
                  <div class="block separator"></div>
+        <div class="block title" id="Introduction">
+                    <h1 id="Mechanical"> Second aspect modeled : mechanical modeling </h1>
+                </div>
+        <div class="block two-full">
+                    <p>Neuronarch aims at making the prosthesis of the future and making it more comfortable and protective for the patient. For this sake and to facilitate surgical interventions we modeled the behavior of a bone under mechanical stress. We presented our tools and scripts to Dr. Laurent Sedel, an orthopedic surgeon at Hôpital Lariboisière and researcher at the Hôpital Ambroise Paré – Hôpitaux Universitaires Paris Ile-de-France Ouest, in the hopes of using our tools to improve the lifespan of prosthesis.</p></div>
+<div class="block two-third center">
+                    <img src="https://static.igem.org/mediawiki/2018/1/13/T--Pasteur_Paris--ModelMecha.png">
+                    <div class="legend">Representation of our calculated model of the deformation and the stress (von Mises) on a straight line inside a humerus</div>
+                </div>
-             <!--Fourth Onglet Mechanical Modeling-->
-                <div class="block full bothContent">
+            <div class="block title">
-                    <div class="block dropDown" id="Mechanical">
+                            <h3><a href="https://static.igem.org/mediawiki/2018/2/2c/T--Pasteur_Paris--MechanicalModeling.pdf"style="font-weight: bold ; color:#85196a;" target="__blank">Download here the full PDF of the Mechanical Modeling</a></h3>
-                        <h4>Mechanical Modeling</h4>
-                    </div>
-                    <div class="block hiddenContent">
-                        <span class="closeCross"><img src="https://static.igem.org/mediawiki/2018/6/67/T--Pasteur_Paris--CloseCross.svg"></span>
-                        <div class="block title">
-                            <h3><a href="">Download here the full pdf of the Mechanical Modeling</a></h3>
                          </div>
-                        <div class="block full">
-                        	<p><i>Neuronarch aims at making the prosthesis of the future and making it more comfortable and protective for the patient. For this sake and to facilitate surgical interventions we modeled the behavior of a bone under certain constraints. We presented our tools and scripts to Dr.Laurent Sedel , an orthopedic surgeon at Hôpital Lariboisière and researcher at the Hôpital Ambroise Paré – Hôpitaux universitaires Paris Ile-de-France Ouest, in the hopes of using our tools to improve the life span of prosthesis.</i></p>
+</div>
-                        </div>
+</div>
-                    </div>
-                </div>
@@ Line 855: / Line 1,019: @@
              </div>
              <div class="block full">
-                 <ul style="text-align: left;">
+                 <ul style="text-align: left;list-style: disc;">
                      <li style="list-style-type: decimal;">M. Stamatakis and N. V. Mantzaris, "Comparison of deterministic and stochastic models of the lac operon genetic network," Biophys. J., vol. 96, no. 3, pp. 887-906, 2009.<br><br></li>
                      <li style="list-style-type: decimal;">A. Y. Weiße, D. A. Oyarzún, V. Danos, and P. S. Swain, "Mechanistic links between cellular trade-offs, gene expression, and growth," Proc. Natl. Acad. Sci., vol. 112, no. 9, pp. E1038-E1047, 2015.<br><br></li>
@@ Line 863: / Line 1,027: @@
                      <li style="list-style-type: decimal;">Defining the concentration gradient of nerve growth factor for guided neurite outgrowth, XCao M.SShoichet, March 2001<br><br></li>
                      <li style="list-style-type: decimal;">Immobilized Concentration Gradients of Neurotrophic Factors Guide Neurite Outgrowth of Primary Neurons in Macroporous Scaffolds, Moore K, MacSween M, Shoichet M, feb 2006<br><br></li>
-                     <li style="list-style-type: decimal;">Mathematical Modeling of Guided Neurite Extension in an Engineered Conduit with Multiple Concentration Gradients of Nerve Growth Factor (proNGF), Tse TH, Chan BP, Chan CM, Lam J, sep 2007<br><br></li>
+                     <li style="list-style-type: decimal;">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<br><br></li>
                      <li style="list-style-type: decimal;">Mathematical modeling of multispecies biofilms for wastewater treatment, Maria Rosaria Mattei, november 2005<br><br></li>
                  </ul>

Difference between revisions of "Team:Pasteur Paris/Model"

Latest revision as of 14:48, 10 November 2018

First aspect modeled : secretion, diffusion and influence of NGF

Context of our model

NGF production by genetically modified E. coli

NGF production by genetically modified E. coli

Model Description

1. NGF and TEV synthesis in the cytoplasm

2. NGF and TEV secretion to the medium

3. Including growth rate

4. NGF folding and export peptide cleavage by TEV

5. mRNA and protein degradation

MODEL PARAMETRISATION

MODEL RESULTS

Influence of number of transporters

Influence of translocation rate

IPTG induction level

PERSPECTIVES

NGF diffusion simulation in a given environment

NGF diffusion simulation in a given environment

Fick’s diffusion law

Optimization of the NGF gradient

Analysis of the model

Neurons growth in the presence of NGF

Neurons growth in the presence of NGF

Explanation of the model

Baseline growth rate:

Concentration Gradient:

Euler's Method

Solving the Model

Dependence of the length of the canal

Dependence of the initial concentration

COMPARISON WITH EXPERIMENTAL WORKS

PERSPECTIVES

Second aspect modeled : mechanical modeling

Download here the full PDF of the Mechanical Modeling

REFERENCES