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 |
Iex |
IPTG outside the cell |
Iin |
IPTG in the cytoplasm |
Po |
Plac promoter occupied by repressor, prevent transcription |
Pf |
Plac promoter with free lacO site |
m |
mRNA for TEV and NGF |
m-r |
Ribosome-bound mRNA |
NGFc |
NGF in cytoplasm |
TEVc |
TEV protease in cytoplasm |
(N-T)c |
NGF-TEV complex in cytoplasm |
NGFcc |
Cleaved NGF in cytoplasm, cannot be exported |
NGFt |
NGF bound to transporter channel |
TEVt |
TEV bound to transporter channel |
t |
Transmembrane transporter |
NGFum |
Unfolded NGF in medium with export peptide |
NGFm |
Folded NGF in medium with export peptide |
N-Tm |
Complex between NGF with export peptide and functional TEV |
TEVm |
TEV in medium with export peptide |
NGFf |
Functional NGF in the medium |
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 |
kt |
IPTG diffusion rate across the membrane |
0.92 |
min-1 |
[1] |
ki |
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] |
kr |
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] |
kf |
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] |
δm |
mRNA degradation rate |
0.462 |
min-1 |
[1] |
δpc |
Protein degradation rate in cytoplasm |
0.2 |
min-1 |
[1] |
δpm |
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
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.):
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