Team:IISER-Pune-India/Model/Blue-Light

Mutatis Mutandis

Blue Light Inducible system

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. Light activation in this system occurs similar to the model of Phytochrome B dimerization^[1].

System with Chemical Species Involved:

Species	Symbol	Description	Initial Value(in μM)
YF1 mRNA	m_YF1	Amount of YF1 mRNA which is produced under the constitutive promoter in the system. Since, it is a foreign gene for E.coli, its initial concentration is considered to be 0.	0
Protein YF1	YF1	A fusion protein which was created by combining domains from YtvA(from B. subtilis) and FixL(from B. japonicum) which displayed kinase activity in the presence of blue light after dimerization.	0
Dimeric form of YF1 with both domains in dark(active) state	YF1₂(DD)	The dimeric form of YF1 has two domains, both of which can act as kinase sites(i.e get phosphorylated). These sites are active under dark conditions and become inactive when exposed to blue light. This is the dimeric form of YF1, with both of its domains active. The active form is represented by D and the inactive form is represented by L.	0
Dimeric form of YF1 with one domain in dark(active) state	YF1₂(DL/LD)	Look at the description of YF1₂(DD). This is the dimeric form of YF1, with one of its domains active and the other inactive.	0
Dimeric form of YF1 with both domains in light(inactive) state(LL)	YF1₂(LL)	Look at the description of YF1₂(DD). This is the dimeric form of YF1, with both of its domains inactive.	0
FixJ mRNA	m_FixJ	Amount of FixJ mRNA which is produced under the constitutive promoter in the system. Since, it is a foreign gene for E.coli, its initial concentration is considered to be 0.	0
Protein FixJ	FixJ	The effector protein of FixL and hence YF1, which gets phosphorylated in the presence of the active domains of YF1₂ and ATP.	0
ATP required for phosphorylation	ATP	This provides the P required for phosphorylation of FixJ, by the kinase activity of YF1.	1
Phosphorylated(active) form of FixJ	FixJ_P	This is the active form of FixJ, which acts as a positive inducer for the promoter FixK₂ by attaching to its operator region.	0
Operator of FixK₂	O_FixK₂	The operator of FixK₂ which is a constitutively repressed promoter which gets activated when FixJ_P gets attached to it.	0
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.	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.	0
Mature RFP protein	RFP_mature	The RFP protein present which has matured, and is showing fluoroscence.	0

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, $$m_{YF1} \xrightarrow{k_1} YF1 $$ $$m_{FixJ} \xrightarrow{k_2} FixJ $$ $$m_{RFP} \xrightarrow{k_3} RFP_{immature} $$

Dimerization of YF1 and cross-conversion between the different dimers:

$$2YF1 \underset{k_{dis}}{\stackrel{k_{dim}}{\rightleftharpoons}} YF1_2(DD) $$ $$YF1_2(DD) \underset{k_{rel}}{\stackrel{k_{con}}{\rightleftharpoons}} YF1_2(LD/DL) $$ $$YF1_2(LD/DL) \underset{k_{rel}}{\stackrel{k_{con}}{\rightleftharpoons}} YF1_2(LL)$$ YF1 converts into a dimer and the dimeric form has two domains. Each domain can exist in two forms, dark and light. This dimer can exist in any combination of these domains. When blue light is shined upon it, the dark form gets converted into a light form. The light form is unstable and steadily relaxes to give back the dark form of the domains.

Phosphorylation of FixJ:

$$FixJ + ATP \underset{k_{dep}}{\stackrel{k_{cat}}{\rightleftharpoons}} FixJ_P $$ with YF1₂(DD) acting as a catalyst.
YF1₂(DD) is a kinase which initially gets auto-phosphorylated and then phosphorylates FixJ to form FixJ_P

Binding of FixJ_P to O_FixK₂:

$$FixJ_P + O_{FixK_2} \xrightarrow{} O_{FixK_2} + m_{RFP}$$ FixJ_P binds to the operator region of FixK₂, after which the promoter allows transcription of the RFP gene 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:

$$m_{YF1} \xrightarrow{\lambda_{m_1}} \phi $$ $$m_{FixJ} \xrightarrow{\lambda_{m_2}} \phi $$ $$m_{RFP} \xrightarrow{\lambda_{m_3}} \phi $$ $$RFP_{immature/mature} \xrightarrow{\lambda_{p_1}} \phi$$

The Equations:

\begin{equation} \frac{d[m_{YF1}]}{dt} = a_{pNish}-{\lambda_{m_1}}[m_{YF1}] \\ \label{eq1} \end{equation} The above equation describes the production of YF1 mRNA, where m_YF1 is the amount of YF1 mRNA, a_pNish is the transcription rate of the constitutive promoter, λ_m₁ is the YF1 mRNA degradation rate constant.

\begin{equation} \frac{d[YF1]}{dt} = k_1[m_{YF1}]-2k_{dim}[YF1]+2k_{dis}[{YF1_2}(DD)] \label{eq2} \end{equation} The above equation describes the production of YF1, where k₁ is the translation rate of m_YF1, k_dim is the dimerization rate of YF1, YF1₂(DD) is the dimeric form of YF1 with both of its domains in the active state, k_dis is the dissociation rate of YF1₂(DD) to form YF1.

\begin{equation} \begin{gathered} \frac{d[{YF1_2}(DD)]}{dt} = k_{dim}[YF1]^2-k_{dis}[{YF1_2}(DD)]+2k_{rel}[{YF1_2}(LD/DL)]-2.I.k_{con}[{YF1_2}(DD)] \label{eq3} \end{gathered} \end{equation} The above equation describes the production of YF1₂(DD), where k_rel is the relaxation rate of YF1₂(LD/DL) and YF1₂(LL). Relaxation refers to the conversion of the domains from inactive state to active state. YF1₂(LD/DL) and YF1₂(LL) refers to the dimeric forms of YF1 with one and both of its domains present in inactive state, respectively. k_con is the conversion cross-section of light intensity activated production rate, which refers to the rate in which the domains of the dimeric form of YF1 gets inactivated in the presence of fixed intensity of blue light falling in a unit cross-sectional area. The assumption that the rate of cross-conversion is directly dependent on intensity comes from the assumption that light activation/inactivation occurs similarly as in phytochrome B(described in Klose et al.), where a similar system with 2 domains is present.

\begin{equation} \begin{gathered} \frac{d[{YF1_2}(LD/DL)]}{dt} = -2k_{rel}[{YF1_2}(LD/DL)]+2.I.k_{con}[{YF1_2}(DD)]+2k_{rel}[{YF1_2}(LL)]-2.I.k_{con}[{YF1_2}(LD/DL)] \label{eq4} \end{gathered} \end{equation} The above equation describes the production of YF1₂(LD/DL).

\begin{equation}\frac{d[{YF1_2}(LL)]}{dt} = -2k_{rel}[{YF1_2}(LL)]+2.I.k_{con}[{YF1_2}(LD/DL)] \end{equation} The above equation describes the production of YF1₂(LL).

\begin{equation}\frac{d[m_{FixJ}]}{dt} = a_{pNish}-{\lambda_{m_2}}[m_{FixJ}]\label{eq6} \end{equation} The above equation describes the production of FixJ mRNA, where m_FixJ is the amount of FixJ mRNA and λ_m₂ is the FixJ mRNA degradation rate constant.

\begin{equation} \begin{gathered} \frac{d[FixJ]}{dt} = k_2[m_{FixJ}]-\frac{k_{cat}[{YF1_2}(DD)][FixJ][ATP]}{K_M(ATP)[FixJ]+K_M(FixJ)[ATP]+[FixJ][ATP]}\\+k_{dep}[FixJ_P] \end{gathered} \end{equation} The above equation describes the production of FixJ, where k₂ is the translation rate of m_FixJ. FixJ is phosphorylated by ATP with YF1₂(DD) acting as a catalyst. This is described by Michaelis-Menten kinetics. k_cat refers to the catalysis rate of YF1₂(DD), ATP refers to the amount of ATP present and K_M(ATP) and K_M(FixJ) refers to the Michaelis Menten constants for ATP and FixJ respectively in the reactions. FixJ_P refers to the amount of phosphorylated FixJ present and k_dep is the dephosphorylation rate of FixJ_P.

\begin{equation}\frac{d[ATP]}{dt} = a_{ATP}-\frac{k_{cat}[{YF1_2}(DD)][FixJ][ATP]}{K_M(ATP)[FixJ]+K_M(FixJ)[ATP]+[FixJ][ATP]}+k_{dep}[FixJ_P]\label{eq8} \end{equation} The above equation describes the rate of change of ATP in the cell with a_ATP referring to the production rate of ATP or the amount of ATP made available to the system per unit time.

\begin{equation}\frac{d[FixJ_P]}{dt} = \frac{k_{cat}[{YF1_2}(DD)][FixJ][ATP]}{K_M(ATP)[FixJ]+K_M(FixJ)[ATP]+[FixJ][ATP]}-k_{dep}[FixJ_P]\label{eq9} \end{equation} The above equation describes the production rate of FixJ_P in our system.

\begin{equation}\frac{d[m_{RFP}]}{dt} = \frac{a_{FixK_2}[FixJ_P]^n}{k_A^n+[FixJ_P]^n}-{\lambda_{m_3}}[m_{RFP}]\label{eq10} \end{equation} The above equation describes the production of RFP mRNA. Here, the binding of the phosphorylated FixJ to the operator region of FixK2 is assumed to happen according to the Hill equation, which describes the binding of ligands to macromolecules. Here a_FixK₂ refers to the maximal promoter activity of the FixK2 promoter, n is the Hill coefficient, K_A refers to the concentration of FixJ_P at which half of the binding sites are occupied. λ_m₃ is the RFP mRNA degradation rate constant.

\begin{equation}\frac{d[RFP_{immature}]}{dt} = k_3[m_{RFP}]-k_{mat}[RFP_{immature}]-{\lambda_{p_1}}[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_p₁ 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_1}}[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_YF1	0.0076 min^-1	[2]
k_dim	Dimerization rate of YF1	10 μM^-1 min^-1	[3]
k_dis	Dissociation rate of YF1₂	0.5 μM^-1 min^-1	[3]
k₂	Translation rate of m_FixJ	0.0076 min^-1	[2]
k₃	Translation rate of m_RFP	0.0076 min^-1	[2]
λ_m₁	YF1 mRNA degradation rate constant	0.08 min^-1	[4]
λ_m₂	FixJ mRNA degradation rate constant	0.08 min^-1	[4]
λ_m₃	RFP mRNA degradation rate constant	0.08 min^-1	[4]
a_pNish	Transcription rate of constitutive promoter	10 μM min^-1	[5]
K_M(ATP)	Michelis Menten constant for ATP in the kinase reaction	33 μM	[3] supplementary material
K_M(FixJ)	Michelis Menten constant for FixJ in the kinase reaction	1.4 μM	[3] supplementary material
k_cat	Catalysis rate in the kinase reaction	0.0348 μM^-1 min^-1	[3] supplementary material
a_ATP	Production rate of ATP	1 min^-1	Look at 1 below
a_FixK₂	Maximal transcription rate of the FixK₂ promoter	0.048 μM min^-1	[6]
n	Hill coefficient	1	[6]
k_A	Ligand concentration producing half occupation	0.713 μM	[6]
k_dep	Dephosphorylation rate of FixJ_P	0.036 min^-1	[3] supplementary material
k_rel	Relaxation rate of YF1₂(DD) and YF1₂(LD/DL)	0.0000133 min^-1	[3] look at point 2 below
k_con	Conversion cross-section of light intensity activated production rate of YF1₂(LL) and YF1₂(LD/DL)	0.4219 m² μM^-2	[3] look at point 3 below
k_mat	RNA maturation rate constant	0.167 min^-1	[7]
λ_p₁	RFP degradation rate constant	0.00484 min^-1	[7]

1. The production rate of ATP in the ceell is taken to be 1 per minute, which means that 1 ATP is formed in the cell per minute. But, changing this parameter doesn't lead to much changes in the results.
2. Look at Moglich et al.³, Figure 3(b).
2. Look at Moglich et al.³, Figure 3(a) and (c).

Results of the propposed model:

1. Graph of RFP(in μM) versus Time(in minutes) for different intensities of light. Increasing blue light intensity (represented by thicker lines) reduces the saturation level of RFP. For the wavelength of light, λ = 435.8nm.

If we convert the units of the RFP fluorescence from μM of fluorescent molecules to the number of fluorescent molecules per cell(which can be easily related to the A.U.s measured in the experiments. For more information, take a look at the measurement page.), we get the following graph.

As we can see from the graph, the RFP production changed with the change in the intensity of light used. Hence, this shows that blue light can be used as the physical inducer needed for our system.

2. Graph of RFP(in no. of fluorescent molecules) versus Intensity(in Wm^-2) after saturation of RFP levels after 1200 minutes.

We can compare this curve with the results obtained in Ohlendorf, Möglich et al., where they created this system and called it pDawn.

The same qualitative nature of these two curves verify the accuracy of some aspects of the model. Now, as we can see from the curve, the range of intensity giving the maximum change in protein production values is 0-10 Wm^-2. So, the blue-light box, which will be used to carry out our blue light-induced/repressible experiments has been designed to operate around this range.

3. However, the dependence between RFP/protein production and Intensity in this system is an inverse one. It would be favourable for this dependence to be converted into a direct one, which is easily possible with the inclusion of a NOT gate somewhere in the system. This is where the regulatory system comes in.

References:

[1] Klose, Cornelia et al. 2015. Systematic Analysis of How Phytochrome B Dimerization Determines Its Specificity. Nature Plants 1(July):15090.
[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] Andreas Möglich, Rebecca A. Ayers, and Keith Moffat. Design and Signaling Mechanism of Light-Regulated Histidine Kinases. J Mol Biol. 2009 Feb 6; 385(5): 1433–1444. DOI: 10.1016/j.jmb.2008.12.017.
[4] 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.
[5] BBa_J23100, Warsaw 2010 iGEM
[6] Robert Ohlendorf, Roee R. Vidavski, Avigdor Eldar, Keith Moffat, Andreas Möglich Corrigendum to “From Dusk Till Dawn: One-Plasmid Systems for Light-Regulated Gene Expression” Journal of Molecular Biology, Volume 426, Issue 2, 23 January 2014, Pages 500
[7] BBa_E1010

We are people of culture too!

Team:IISER-Pune-India/Model/Blue-Light

Blue Light Inducible system

Assumptions:

System with Chemical Species Involved:

The Reactions Involved:

Translation of mRNA:

Dimerization of YF1 and cross-conversion between the different dimers:

Phosphorylation of FixJ:

Binding of FixJ_P to O_FixK₂:

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 propposed model:

References:

We are people of culture too!

Address

Contact

Sponsors

Team:IISER-Pune-India/Model/Blue-Light

Assumptions:

System with Chemical Species Involved:

The Reactions Involved:

Translation of mRNA:

Dimerization of YF1 and cross-conversion between the different dimers:

Phosphorylation of FixJ:

Binding of FixJP to OFixK2:

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 propposed model:

References:

We are people of culture too!

Address

Contact

Sponsors

Binding of FixJ_P to O_FixK₂: