Team:Ecuador/Model
Index
- List of variables and constants
- IPTG-inducible model
- Entry of IPTG into the cell
- CBD cipA-BMP2 expression
- Kinetic constants
- Results of the IPTG-inducible model
- Calculation of entry of IPTG into the cell
- IPTG-LacI repressor interaction
- Calculation and validation of CBD cipA-BMP2 expression
- Conclusions
INTRODUCTION
This is an IPTG-inducible model for the expression of the fusion protein CBD cipA-BMP2 under the control of Plac promoter and cloned in psb1c3. The vector does not contain the ORF of LacI and thus the LacI repressor protein concentration is that in E. coli strains (10nM per cell).1
The model consists of systems of differential equations that were solved in Python using Spyder 3.6 (Anaconda).
The model consists of systems of differential equations that were solved in Python using Spyder 3.6 (Anaconda).
LIST OF VARIABLES AND CONSTANTS
Variable/Constant |
Definition |
Unit/Value |
[IPTG] TOTAL |
Total IPTG concentration in the medium |
nM |
[IPTG]ext |
IPTG concentration outside E. coli |
nM |
[IPTG]int |
IPTG concentration inside E. coli |
nM |
[IPTG]TOTAL |
Total LacI concentration in E. coli |
nM |
[LacI]act |
Active LacI concentration in E. coli |
nM |
[CB]mRNA |
CBD-BMP2 mRNA concentration in E. coli |
nM |
[CB] |
CBD-BMP2 protein concentration in E. coli |
nM |
δ[CB] |
Degradation rate of CBD-BMP2 mRNA |
min-1 |
β[CB] |
Translational rate of CBD-BMP2 protein |
min-1 |
σ[CB] |
Degradation rate of CBD-BMP2 protein |
min-1 |
t |
time |
min |
α[CB] |
Transcriptional leakage of Plac promoter |
% |
kPlac |
Maximum transcription rate of Plac promoter |
0.5 nM min-1 |
kIPTGupt |
Rate constant of IPTG uptake |
0.92 min-1 |
kIPTGout |
Rate constant of IPTG output |
0.05 min-1 |
KIPTG |
Michaelis constant of IPTG-LacI binding |
600 nM |
hIPTG |
Hill coefficient of IPTG-LacI binding |
2 |
HLacI |
Hill coefficient of LacI-Plac promoter binding |
2 |
IPTG-INDUCIBLE MODEL FOR CBD CIPA-BMP2 PROTEIN EXPRESSION IN E. coli
Entry of IPTG into E. coli
Mimicking lactose, IPTG is a chemical widely used in scientific research because it cannot be metabolized. IPGT induces the expression of genes under the control of Plac promoter.
Mimicking lactose, IPTG is a chemical widely used in scientific research because it cannot be metabolized. IPGT induces the expression of genes under the control of Plac promoter.
IPTG enters into E. coli cells in a concentration dependent manner. This uptake is mediated by lactose permease and passive diffusion at low and high concentrations respectively.2 The IPTG uptake through passive diffusion can be described with the law of mass action expressing the balance between IPTG concentration outside (IPTGext) and inside (IPTGint) of E. coli :
Eq 1.
The first term on the right side of Eq 1 represents the forward rate where kIPTGupt is the rate constant of IPTG uptake (0.92 min-1)3 while the second term has negative sign because it represents the backward rate where kIPTGout is the rate constant of IPTG output (0.05 min-1).3 Because of [IPTG]ext = [IPTG]TOTAL - [IPTG]int, it can be replaced in equation 1 in order to have IPTGint as the unique independent variable:
Eq 2.
The LacI-IPTG interaction is assumed to be so fast and when the IPTG concentration largely exceeds that of lacI. The LacI active ([LacI]act), i.e that unbounded to IPTG, can be described with the Hill repression function.4,2
Eq 3.
Equation 3 (Eq 3) describes the amount of unbounded LacI that actively repress the expression of a protein driven by the Plac promoter; the expression of CBD cipA-BMP2 in the present study. [LacI]TOTAL is the concentration of LacI inside E. coli, KIPTG is the Michaelis constant of IPTG-LacI binding (6000 M) 5 and hIPTG is the associated Hill coefficient (2).3,6
Eq 1.
Eq 2.
Eq 3.
CBD cipA-BMP2 EXPRESSION
CBD cipA-BMP2 mRNA
The CBD cipA-BMP2 mRNA concentration ([CB]mRNA) at time t can be described as follows:
Eq 4.
The first term on the right side of the equation 2.6 represents the synthesis of mRNA. kPlac is the maximum transcriptional rate corresponding to the Plac promoter (0.5 nM min-1).3 αCEB+(1-αCEB) is the leakiness factor where αCEB is the transcriptional leakage of Plac promoter. is the repressing Hill function of Plac promoter where KLacI is the Michaelis constant of LacI-Plac promoter binding (6000 M), 3 and hLacI is the associated Hill coefficient(2).3 The second term represents the degradation of the [CB]mRNA where δ[CB] is the degradation rate.3,6,7
Because there is not constitutive expression of LacI protein from the plasmid cloned in E. coli carrying the CBD cipA-BMP2 construct and based on our preliminary measurements of total protein concentration using BCA method, the transcriptional leakage can be taken to be 0.8 and thus the balance for [CB]mRNA becomes:
Eq 5.
Figure 1. Rate balance of IPTG diffusion through E. coli membrane. Forward and backward rates are plotted versus normalized [IPTG]int
Because there is not constitutive expression of LacI protein from the plasmid cloned in E. coli carrying the CBD cipA-BMP2 construct and based on our preliminary measurements of total protein concentration using BCA method, the transcriptional leakage can be taken to be 0.8 and thus the balance for [CB]mRNA becomes:
Eq 5.
CBD cipA-BMP2 protein
The concentration of CBD cipA-BMP2 protein ([CB]) can be described as follows:
Eq 6.
The first term on the right side of the equation 6 represents the synthesis of [CB] where β[CB] is the translational rate of [CB]. The second term represents the degradation of [CB] where σ[CB] is the degradation rate.
The concentration of CBD cipA-BMP2 protein ([CB]) can be described as follows:
Eq 6.
KINETIC CONSTANTS
The time cell division of E. coli is function of factors such as pressure, temperature and culture medium and ranging from 20-30 min under optimal conditions. Culturing in universal common medium such as Nutrient Broth and Luria-Bertani at 35 ℃, the time cell division can be assumed to be 30 min.8,9,10
The half life of mRNAs in E. coli has been well reported by Bernstein and coworkers who determined that it depends of culture medium. In, LB 99% of mRNAs had a half-life time between 1-15 min with a mean of 5.2 min.11 Therefore, 5.2 min can be assumed as the half-life time of CBD cipA -BMP2 mRNA. Consequently, δ[CB]= 0.226 min-1.
In E. coli , the translation takes place at 12-21 amino acids per second (aa s-1).12 We choose 19 aa s-1 because the codon optimization. Because CBD cipA-BMP2 contains 292 aa, β[CB] can be taken to be 3.904 min-1.The half life of CBD cipA-BMP2 protein can be assumed to be 60 min-1, 13 consequently, σ[CB]= 0.05min-1:
The time cell division of E. coli is function of factors such as pressure, temperature and culture medium and ranging from 20-30 min under optimal conditions. Culturing in universal common medium such as Nutrient Broth and Luria-Bertani at 35 ℃, the time cell division can be assumed to be 30 min.8,9,10
The half life of mRNAs in E. coli has been well reported by Bernstein and coworkers who determined that it depends of culture medium. In, LB 99% of mRNAs had a half-life time between 1-15 min with a mean of 5.2 min.11 Therefore, 5.2 min can be assumed as the half-life time of CBD cipA -BMP2 mRNA. Consequently, δ[CB]= 0.226 min-1.
In E. coli , the translation takes place at 12-21 amino acids per second (aa s-1).12 We choose 19 aa s-1 because the codon optimization. Because CBD cipA-BMP2 contains 292 aa, β[CB] can be taken to be 3.904 min-1.The half life of CBD cipA-BMP2 protein can be assumed to be 60 min-1, 13 consequently, σ[CB]= 0.05min-1:
RESULTS OF THE IPTG-INDUCIBLE MODEL FOR CBD cipA-BMP2 PROTEIN EXPRESSION IN E. coli
Calculation of entry of IPTG into E. coli
Figure 1 is a rate balance plot derived from Equation 1 whith both forward and backward rates as functions of the normalized [IPTG]int concentration whose value in the equilibrium (steady state) can be obtained where the two functions intersect. Analitically, this point is [IPTG]int =0.9485[IPTG]TOTAL
Figure 1 is a rate balance plot derived from Equation 1 whith both forward and backward rates as functions of the normalized [IPTG]int concentration whose value in the equilibrium (steady state) can be obtained where the two functions intersect. Analitically, this point is [IPTG]int =0.9485[IPTG]TOTAL
rate
Figure 1. Rate balance of IPTG diffusion through E. coli membrane. Forward and backward rates are plotted versus normalized [IPTG]int
IPTG-LacI repressor interaction
Because of IPTG is the inducer, it is important to evaluate its interaction with LacI repressor at different initial concentrations added to the system ([IPTG]TOTAL). For these values the [LacI]act can be calculated with Equation 3 (Table 1). From table 1 and figure 4, it can be deduced that a concentration of IPTG = 0.1 mM is saturating because the solutions derived from higher IPTG values converge.
Because of IPTG is the inducer, it is important to evaluate its interaction with LacI repressor at different initial concentrations added to the system ([IPTG]TOTAL). For these values the [LacI]act can be calculated with Equation 3 (Table 1). From table 1 and figure 4, it can be deduced that a concentration of IPTG = 0.1 mM is saturating because the solutions derived from higher IPTG values converge.
Table 1. [LacI]act for different initial IPTG concentrations.
[IPTG]TOTAL (mM) |
[LacI]act (nM) |
0,001 |
9.7562 |
0,02 |
0.9095 |
0,03 |
0.4257 |
0.1 |
0.0399 |
1.0 |
0.0004 |
Calculation and validation of CBD cipA-BMP2 expression
The mathematical model for the expression of CBD cipA-BMP2 is:
Eq 7.
Eq 7.
The stability of the system can be analyzed with the Jacobian matrix associated to the system of differential equations:
Because the two igenvalues (λ1= -0.226 y λ2 = -0.05) are negative, the equilibrium point is a stable node.
JACOBIAN
Time step selection and phase plane
A reasonable error less than 0.01 for both mRNA and protein concentrations was obtained with a time step Dt=0.1/8. Therefore, this Dt was considered to evaluate a phase plane plot which indicates the equilibrium points of the mathematical model. To determine the maximum concentrations of CBD cipA-BMP2 mRNA and CBD cipA-BMP2 protein, a saturating concentration of IPTG (IPTG=0.1 mM) was evaluated. This is a monostable system where the equilibrium point at the steady state (Figure 2) is 2.21 nM and 172.69nM for CBD cipA-BMP2 mRNA and CBD cipA-BMP2 protein respectively.
A reasonable error less than 0.01 for both mRNA and protein concentrations was obtained with a time step Dt=0.1/8. Therefore, this Dt was considered to evaluate a phase plane plot which indicates the equilibrium points of the mathematical model. To determine the maximum concentrations of CBD cipA-BMP2 mRNA and CBD cipA-BMP2 protein, a saturating concentration of IPTG (IPTG=0.1 mM) was evaluated. This is a monostable system where the equilibrium point at the steady state (Figure 2) is 2.21 nM and 172.69nM for CBD cipA-BMP2 mRNA and CBD cipA-BMP2 protein respectively.
Plano
Figure 2. Phase plane plot of the molecular species of CBD cipA-BMP2 with IPTG=0.1 mM as saturating concentration. The stable equilibrium point is at 2.21 nM and 172.69 nM for [CB]mRNA and [CB], respectively.
Expression at different IPTG inducer concentrations
Different values of [LacI]act cause a different effect in the mathematical model for expression of [CB]mRNA and [CB]. Figure 3 and Figure 4 show the concentrations of [CB]mRNA and [CB], respectively, as functions of time with a time step size Dt=0.1/8. It is worth to note that the higher IPTG dosage the higher [CB]mRNA and [CB] until the saturating level of IPTG = 0.1 mM.
Different values of [LacI]act cause a different effect in the mathematical model for expression of [CB]mRNA and [CB]. Figure 3 and Figure 4 show the concentrations of [CB]mRNA and [CB], respectively, as functions of time with a time step size Dt=0.1/8. It is worth to note that the higher IPTG dosage the higher [CB]mRNA and [CB] until the saturating level of IPTG = 0.1 mM.
RNA
Figure 3. CBD cipA-BMP2 mRNA concentration as function of time. The concentration is dependent of levels of IPTG inducer until the saturating concentration (IPTG=0.1 mM).
CB protein
Figure 4. CBD cipA-BMP2 protein concentration as function of time. The concentration is dependent of the IPTG inducer until the saturating level (IPTG=0.1 mM).
CONCLUSIONS
The present IPTG-inducible mathematical model, where a fusion protein CBD cipA-BMP2 was cloned in psb1c3, can be analyzed in two stages. First, the concentration of the IPTG inducer added to the culture medium interacts with the repressor protein LacI following the Hill equation. Then, the expression of CBD cipA-BMP2 becomes relieved, and at saturating concentrations of IPTG ≥ 0.1 mM it reaches the highest value (CBD cipA-BMP2 protein = 172.69nM, CBD cipA-BMP2 mRNA=2.21 nM).
The IPTG concentration inside E. coli is 94,845% of the total IPTG added to the culture medium. Moreover, the phase line analysis of the ODE representing the IPTG concentration inside E. coli as function of time showed that the equilibrium point is stable.
The linearization of the mathematical model revealed an stable equilibrium point for the expression of CBD cipA-BMP2 because two negative igenvalues,
The IPTG concentration inside E. coli is 94,845% of the total IPTG added to the culture medium. Moreover, the phase line analysis of the ODE representing the IPTG concentration inside E. coli as function of time showed that the equilibrium point is stable.
The linearization of the mathematical model revealed an stable equilibrium point for the expression of CBD cipA-BMP2 because two negative igenvalues,
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The main goal of our work is to develop a functionalized cellulose-based biomaterial for bone and cartilage regeneration in order to allow a faster injuries recovery time and overcome remaining challenges in repairing these connective tissues. In biomedicine, emergent strategies have focused the use biomaterials loaded with drugs.1-3 We aimed to use bacterial cellulose, a biocompatible biomaterial, as scaffold for the delivery of the human bone morphogenetic protein 2 (BMP2). It is worth to note that the diffusion of a drug far of the action site can become ectopic tissue formation, representing a serious problem. Therefore, we focused to attach the BMP2 by the use of a cellulose binding domain (CBD). We choose the CBD of the scaffolding protein of C. thermocellum (CBD cipA) because the higher affinity compared to other CBDs, reported by the Team Imperial_2014. Moreover, in an effort for a longer broad application of a functionalized bacterial cellulose, such as wound healing, we also fused the CBD cipA with an elastin like polypeptide (ELP_C5) derived from the ELP- [V-150].