Team:IISER-Pune-India/Model/regulatory

Mutatis Mutandis

Regulatory system

[1] https://2015.igem.org/Team:Oxford/Modeling
[2] Stögbauer, Tobias; Windhager, Lukas; Zimmer, Ralf; Rädler, Joachim O. (2012): Experiment and mathematical modeling of gene expression dynamics in a cell-free system. In Integrative biology : quantitative biosciences from nano to macro 4 (5), pp. 494–501. DOI: 10.1039/c2ib00102k
[3] Karzbrun, Eyal; Shin, Jonghyeon; Bar-Ziv, Roy H.; Noireaux, Vincent (2011): Coarse-Grained Dynamics of Protein Synthesis in a Cell-Free System. In Phys. Rev. Lett. 106 (4). DOI: 10.1103/PhysRevLett.106.048104.
[4]BBa_R0051
[5]Nandan Kumar Jana, Siddhartha Roy, Bhabatarak Bhattacharyya, Nitai Chandra Mandal, Amino acid changes in the repressor of bacteriophage lambda due to temperature-sensitive mutations in its cI gene and the structure of a highly temperature-sensitive mutant repressor, Protein Engineering, Design and Selection, Volume 12, Issue 3, March 1999, Pages 225–233, https://doi.org/10.1093/protein/12.3.225
[6]BBa_E1010

Assumptions:

1. The rates of production, binding, and degradation of mRNAs and proteins are assumed to be linear which results in a first order differential equations system.
2. The effects of dilution/ growth are neglected. Thus the cell’s volume is assumed to be constant over time.
3. The repression due to the cI and cIts2 repressor is assumed to occur through Hill repression.
4. The diffusion of Arabinose into the system is considered to be instant.

System with Chemical Species Involved:

Species	Symbol	Description
Arabinose attached with AraC	A₁AraC	Amount of Arabinose binded to AraC. The rate of diffusion of Arabinose outside the cell to inside is assumed to be very fast and hence all the arabinose added is directly assumed to be inside the system. ALso, the binding of Arabinose to AraC is considered to be instant. These assumptions were considered by the Oxford 2014 iGEM team, when building the model for pBAD.
cI mRNA	m_cI	Amount of cI mRNA which is produced under the pBad promoter in the system. Since, it is a foreign gene for E.coli, its initial concentration is considered to be 0.
Protein cI	cI	cI is a protein which acts as a repressor for the constitutively active promoter cIlam after dimerization. However, here the dimeriation process is ignored and all the repression is considered to happen by the monomer of cI, since the effect of dimerization isn't important to the system and will only change the parametric values of repression, which has been accounted for.
RFP mRNA	m_RFP	Amount of RFP mRNA which is produced under the FixK₂ promoter in the system. Since, it is a foreign gene for E.coli, its initial concentration is considered to be 0.
Immature RFP protein	RFP_immature	The RFP protein created, which does not show fluoroscence yet. The time required for it to show fluoroscence, or for it to achieve maturity is 60 minutes.
Mature RFP protein	RFP_mature	The RFP protein present which has matured, and is showing fluoroscence.
cIts2 mRNA	m_cIts2	Amount of cIts2 mRNA which is produced under the cIlam promoter in the system. Since, it is a foreign gene for E.coli, its initial concentration is considered to be 0.
Protein cIts2	cIts2	cIts2 is a modified version of cI, which is temperature-sensitive and shows the same mode of repression as in cI, only weaker. The same reason why dimerization is not considered for cI, applies here too.

The Reactions Involved:

For a detailed explanation on all of the parameters in the reactions below, switch over to the next tab.

Translation of mRNA:

First, the translation of the different mRNA species into their respective proteins, $$m_{cI} \xrightarrow{k_1} cI$$ $$m_{cIts2} \xrightarrow{k_2} cIts2 $$ $$m_{RFP} \xrightarrow{k_3} RFP_{immature} $$

Binding of A₁AraC to the operator region of pBAD, causing transcription of cI:

$$A_1AraC + O_{pBAD} \xrightarrow{} O_{pBAD} + m_{cI}$$ The positive regulation of pBAD by Arabinose happens due to the binding of Arabinose to the repressor AraC, thus allowing activation. However, this can be modelled by considering Arabinose bounded to AraC as the positive inducer of the constitutively repressed pBAD promoter, as was done by 2015 Oxford Team.

Negative autoregulation of cIlam by cI and cIts2:

$$cI + O_{cIlam}+m_{RFP}/m_{cIts2} \xrightarrow{} O_{cIlam} $$ $$cIts2 + O_{cIlam}+m_{RFP}/m_{cIts2} \xrightarrow{} O_{cIlam} $$ cI and cIts2 bind to the operator region of cIlam, preventing the promoter from transcription of the genes downstream.

Maturation of the immature RFP:

$$RFP_{immature} \xrightarrow{k_{mat}} RFP_{mature}$$ The immature RFP(which does not express colour) matures to form the mature RFP(which does express colour) with a maturation time of 60 minutes.

Degradation of the different mRNA and proteins created:

$$m_{cI} \xrightarrow{\lambda_{m_1}} \phi $$ $$m_{cIts2} \xrightarrow{\lambda_{m_2}} \phi $$ $$m_{RFP} \xrightarrow{\lambda_{m_3}} \phi $$ $$RFP_{immature/mature} \xrightarrow{\lambda_{p_1}} \phi$$ $$cI \xrightarrow{\lambda_{p_2}} \phi$$ $$cIts2 \xrightarrow{\lambda_{p_3}} \phi$$

The Equations:

\begin{equation} \frac{d[m_{cI}]}{dt} = \frac{k_{max}[A_1AraC]^n}{k_{half}^n+[A_1AraC]^n}-{\lambda_{m_1}}[m_{cI}] \\ \label{eq1} \end{equation} The above equation describes the production of cI mRNA, where m_cI is the amount of cI mRNA. The production, as mentioned before is assumed to happen due to the activation of the pBAD operator by the binding of Arabinose attached with AraC to it. This is modelled by using Hill activation, where k_max is the maximal transcription rate of pBAD, n is the Hill coefficient, and k_half refers to the concentration of A₁AraC at which half of the binding sites are occupied, λ_m₁ is the cI mRNA degradation rate constant.

\begin{equation} \frac{d[cI]}{dt} = k_1[m_{cI}]-{\lambda_{p_1}}[cI] \label{eq2} \end{equation} The above equation describes the production of cI, where k₁ is the translation rate of m_cI, and λ_p₁ is the cI degradation rate constant.

\begin{equation} \frac{d[{m_{cIts2}}]}{dt} = a_{p_{cIlam}}-\frac{a_{p_{cIlam}}[cI]^{n2}}{k_{d_1}+[cI]^{n2}}-\frac{a_{p_{cIlam}}[cIts2]^{n3}}{k_{d_2}+[cIts2]^{n3}}-{\lambda_{m_2}}[m_{cIts2}] \end{equation} \begin{equation} \frac{d[{m_{RFP}}]}{dt} = a_{p_{cIlam}}-\frac{a_{p_{cIlam}}[cI]^{n2}}{k_{d_1}+[cI]^{n2}}-\frac{a_{p_{cIlam}}[cIts2]^{n3}}{k_{d_2}+[cIts2]^{n3}}-{\lambda_{m_3}}[m_{RFP}] \end{equation} The above equation describes the production of cIts2 and mRFP mRNA, where a_{p_cIlam} is the transcription rate of p_cIlam. The production rate is reduced due to repression by the cI and cIts2, which has been modelled by a Hill function where the ligands attach to the operator and reduce the transcription rate. k_d₁ and k_d₂ refers to the disscociation constant for cI and cIts2 with cIlam. n2 and n3 are the Hill coefficients for cI and cIts2, binding to the operator of cIlam. λ_m₂ and λ_m₃ are the cIts2 and RFP mRNA degradation rate constants respectively.

\begin{equation} \frac{d[cIts2]}{dt} = k_2[m_{cIts2}]-{\lambda_{p_2}}[cIts2] \end{equation} The above equation describes the production of cIts2, where k₂ is the translation rate of m_cIts2, and λ_p₂ is the cIts2 degradation rate constant.

\begin{equation}\frac{d[RFP_{immature}]}{dt} = k_3[m_{RFP}]-k_{mat}[RFP_{immature}]-{\lambda_{p_3}}[RFP_{immature}]\label{eq11} \end{equation} The above equation describes the production of immature RFP, where k₃ is the translation rate of m_RFP, k_mat is the maturation rate of immature RFP and λ_p₃ is the RFP degradation rate constant(which is the same for both immature and mature RFP).

\begin{equation}\frac{d[RFP_{mature}]}{dt} = k_{mat}[RFP_{immature}]-{\lambda_{p_3}}[RFP_{mature}]\label{eq12} \end{equation} The above equation describes the production of mature RFP.

Table representing the different parameters and their values:

Parameter	Description	Value	Reference
k₁	Translation rate of m_cI	1.503 min^-1	[1]
k_max	Maximal transcription rate of pBAD	5.01 μM min^-1	[1]
k_half	Ligand concentration of A₁AraC for half-occupation of binding sites	160 μM	[1]
n1	Hill coefficient for A₁AraC attaching to the operator site of pBAD	2.65	[1]
k₂	Translation rate of m_cIts2	0.0076 min^-1	[2]
k₃	Translation rate of m_RFP	0.0076 min^-1	[2]
λ_m₁	cI mRNA degradation rate constant	0.08 min^-1	[3]
λ_m₂	mRFP mRNA degradation rate constant	0.08 min^-1	[3]
λ_m₃	cIts2 mRNA degradation rate constant	0.08 min^-1	[3]
a_{p_cIlam}	Maximal transcription rate of cIlam promoter	0.0165 μM min^-1	[4]
n2	Hill parameter for cI	2	[4]
n3	Hill parameter for cIts2	x	[4]
k_d₁	Disscociation constant for cI with cIlam	.002 μM	[5]
k_d₂	Disscociation constant for cIts2 with cIlam	.008 min^-1	[5]
k_mat	RNA maturation rate constant	0.167 min^-1	[6]
λ_p₁	cI degradation rate constant	0.00484 min^-1	[5] refer 1 below
λ_p₂	cIts2 degradation rate constant	0.00484 min^-1	[5] refer 1 below
λ_p₃	RFP degradation rate constant	0.00484 min^-1	[6]

1. The degradation of cI and cIts2 follow a logarithmic fit as can be seen in Jana et al., FIg 2A. The fit achieved was -0.354+0.249*np.log(T) for cIts2 and -0.75+0.331*np.log(T) for cI(above 30^oC).

Results of the proposed model:

1. Graph of RFP(in μM) versus Time(in minutes) at T=20^oC.

As we can see from the graph,the RFP production increases with time until it reaches its saturation level at around 350 molecules of RFP(due to degradation of RFP)for 0 μM of Arabinose in the system.

2. Now, if we plot the production of molecules of RFP at saturation versus the amount of Arabinose in the system(in μM),we get(at T=20^oC.)

As we can see from the graph, pBAD has a very narrow range of Arabinose over wich it shows activity. As could be predicted, increase in Arabinose leads to increase in the production of cI, which decreases the amount of RFP being produced.

3. Also, with increase in Temperature from 20^oC to 42^oC, there is greater degradation of cI and cIts2 as described in Jana et al, as can be seen from the graph below,

Now, if we model this degradation of protein with temperature as a logarithmic fit, we get the following curve for RFP(after saturation) with increase in temperature for no Arabinose in the cell.

4. So, now we can convert the negative dependance of the production of the protein on the intensity of light using the regulatory system with the cI taking the place of RFP in the earlier blue light inducible system. This forms the complete blue light inducible system, which will be the part used in the mutator plasmid.

We are people of culture too!

Team:IISER-Pune-India/Model/regulatory

Regulatory system

Assumptions:

System with Chemical Species Involved:

The Reactions Involved:

Translation of mRNA:

Binding of A₁AraC to the operator region of pBAD, causing transcription of cI:

Negative autoregulation of cIlam by cI and cIts2:

Maturation of the immature RFP:

Degradation of the different mRNA and proteins created:

The Equations:

Table representing the different parameters and their values:

Results of the proposed model:

We are people of culture too!

Address

Contact

Sponsors

Team:IISER-Pune-India/Model/regulatory

Assumptions:

System with Chemical Species Involved:

The Reactions Involved:

Translation of mRNA:

Binding of A1AraC to the operator region of pBAD, causing transcription of cI:

Negative autoregulation of cIlam by cI and cIts2:

Maturation of the immature RFP:

Degradation of the different mRNA and proteins created:

The Equations:

Table representing the different parameters and their values:

Results of the proposed model:

We are people of culture too!

Address

Contact

Sponsors

Binding of A₁AraC to the operator region of pBAD, causing transcription of cI: