Team:IISER-Pune-India/Model/Bioremediation

Mutatis Mutandis

<!DOCTYPE html> Bioremediation

The Bioremediation System

The bioremediation system is an extension of the biosensor system. The transporter protein (pbrT) added downstream of pbrR allows Lead influx into the cell. The pbrD added downstream of the GFP sequesters the free Lead in the cell. Due to pbrD being downstream of the operator (pbrAP), it is produced only in the presence of Lead. Assumptions of the biosensor are followed here as well, also a few more relations are assumed.
In the bioremediation system we will follow the alternate model described in the biosensor part.

Assumptions:

1. Influx of Lead due to pbrT is assumed to follow Michaelis Menten Kinetics.

2. Most values concerned with pbrT and pbrD are assumed as no literature could be found.

3. Parameter values of the constants from the biosensor system are assumed to be same

Symbol	Species	Initial Value(nM)
$$m_T$$	transporter protein mRNA	$$0$$
$$T$$	Transporter protein (pbrT)	$$0$$
$$m_D$$	Sequestration protein mRNA	$$0$$
$$D$$	Sequestration protein (pbrD)	$$0$$
$$DA_n$$	Lead-pbrD Complex. 'n', since number of Lead bound to one pbrD is not known. It is assumed to be one in following calculations.	$$0$$

Reactions:

All reactions from the biosensor system take place here too. The additional ones are shown below.

Translation of pbrT and pbrD.

$$m_T \xrightarrow{k_{TL4}} T$$

$$m_D \xrightarrow{k_{TL3}} D$$

Lead Influx due to pbrT, here pbrT increases the rate of influx.

$$A_{out} \underset{k_{-in}}{\stackrel{T*k_{in}}{\rightleftharpoons}} A$$

Lead sequestration by pbrD

$$D + nA \underset{k_{-seq}}{\stackrel{k_{seq}}{\rightleftharpoons}} DA_n$$

$$m_D \xrightarrow{\lambda_{m_3}} \phi$$

$$m_T \xrightarrow{\lambda_{m_4}} \phi$$

The total Lead concentration $T_A$ changes due to addition of the new species $DA_n$.

$$T_O=O+{R_2}O$$

$$T_A=A+A_{out}+nDA_n+2{R_2}{A_2}$$

Equations:

The addition of the two genes pbrT and pbrD affect almost all equations. Therefore all equations used to model the system are listed below

$$\frac{d[m_R]}{dt} = v_{TX2}\frac{[G_R]^2}{K_{TX2}^2 + [G_R]^2}-{\lambda_{m_R}}[m_R]$$

Like the alternate model, this model also takes into consideration the resource competition. Here we add the effect of pbrT and pbrD on resource competition as well.

$$\frac{d[R]}{dt} = k_{TL2}[TLR]\frac{[m_R]^5}{{K_{TL2}^5}+{[m_R]^5}+{[m_F]^5}+{[m_D]^5}+{[m_T]^5}} - 2k_{2R}[R]^2 + 2k_{-2R}[R_2]$$

In thee equation below $\frac{(T_A-[A_{out}]-[A]-n[DA_n])}{2}$ is a substitute for ${R_2}{A_2}$.

$$\frac{d[R_2]}{dt} = k_{2R}[R]^2 - k_{-2R}[R_2] - k_r{[R_2]}[O] + k_{-r}(T_O - [O]) - k_{dr1}[A]^2{[R_2]} + k_{-dr1}\frac{(T_A-[A_{out}]-[A]-n[DA_n])}{2}$$

$T_{mol}$ is the number of protein molecules of pbrT. It can be calculated by multiplying the concentration of pbrT with Avogadro number and dividing by cell volume. Cell volume is assumed to be 1fL ($10^{-15}L$) [1].

$$\frac{d[A_{out}]}{dt} = -P([{A_{out}]-[A]}) - T_{mol}v_{in}\frac{[A_{out}]}{K_{in} + [A_{out}]}$$

We are assuming that Lead transport through the transporter protein (pbrT) follows Michaelis Menten kineics. Lead influx through pbrT is also multiplied by $T_{mol}$, since each pbrT protein will aid in Lead influx.
Number of Lead ions that can be sequestered by pbrD is not known hence we are denoting it by 'n'. In the calculation we have assumed n to be one.

$$\frac{d[A]}{dt} = T_{mol}v_{in}\frac{[A_{out}]}{K_{in} + [A_{out}]} + P([{A_{out}]-[A]}) - nk_{seq}[D][A]^n + nk_{-seq}[DA_n] - 2k_{dr1}[A]^2{[R_2]} + 2k_{-dr1}\frac{(T_A-[A_{out}]-[A]-n[DA_n])}{2} -2k_{dr2}[A]^2(T_O - [O]) + 2k_{-dr2}[O]\frac{(T_A-[A_{out}]-[A]-n[DA_n])}{2}$$

$$\frac{d[O]}{dt} = - k_r{[R_2]}[O] + k_{-r}(T_O - [O]) -k_{dr2}[A]^2(T_O - [O]) + k_{-dr2}[O]\frac{(T_A-[A_{out}]-[A]-n[DA_n])}{2}$$

$$\frac{d[m_F]}{dt} = v_{TX1}\frac{[O]^2}{K_{TX1}^2 + [O]^2}-{\lambda_{m_F}}[m_F] + k_{leak}(T_O - [O])$$

$$\frac{d[F_{in}]}{dt} = k_{TL1}[TLR]\frac{[m_F]^5}{{K_{TL1}^5}+{[m_R]^5}+{[m_F]^5}+{[m_D]^5}+{[m_T]^5}} - k_{mat}[F_{in}]$$

$$\frac{d[F]}{dt} = k_{mat}[F_{in}] - kgd[F]$$

pbrD is also downstream of the operator (pbrAP), hence the free operator concentration can be used find level of pbrD expression. $\lambda_{m_D}$ is the forst order decay rate of the pbrD protein. Leaky expression is same as before.

$$\frac{d[m_D]}{dt} = v_{TX3}\frac{[O]^2}{K_{TX3}^2 + [O]^2}-{\lambda_{m_D}[m_D]}+ k_{leak}(T_O - [O])$$

pbrD production is modelled using the resource competition model. $k_{seq}$ is the equilibrium constant for Lead sequestration and $k_{-seq}$ is the constant for the reverse reaction.

$$\frac{d[D]}{dt} = k_{TL3}[TLR]\frac{[m_D]^5}{{K_{TL3}^5}+{[m_R]^5}+{[m_F]^5}+{[m_D]^5}+{[m_T]^5}} - k_{seq}[D][A]^n + k_{-seq}[DA_n]$$

$$\frac{d[DA_n]}{dt} = k_{seq}[D][A]^n - k_{-seq}[DA_n]$$

pbrT is also downstream of the pbrR gene ($G_R$) , hence the $G_R$ concentration can be used find level of pbrT expression. $\lambda_{m_T}$ is the forst order decay rate of the pbrT protein.

$$\frac{d[m_T]}{dt} = v_{TX4}\frac{[G_R]^2}{K_{TX4}^2 + [G_R]^2}-{\lambda_{m_T}[m_T]}$$

$$\frac{d[T]}{dt} = k_{TL4}[TLR]\frac{[m_T]^5}{{K_{TL4}^5}+{[m_R]^5}+{[m_F]^5}+{[m_D]^5}+{[m_T]^5}}$$

$$\frac{d[TLR]}{dt} = -v_{\lambda TLR}\frac{[TLR]}{{K_{\lambda TLR}}+[TLR]}$$

References:

[1] Kubitschek HE, Friske JA. Determination of bacterial cell volume with the Coulter Counter. J Bacteriol. 1986 Dec168(3):1466-7. p.1466 table 1PubMed ID3536882

Parameters and Estimation

The parameters that are repeated from the biosensor have the same values, the additional parameter values are given below:

Parameter	Description	Value	Reference
$$v_{TX3}$$	pbrD transcription rate constant	18.2 nM min^-1	(Stögbauer et al. 2012)
$$v_{TX4}$$	pbrT transcription rate constant	18.2 nM min^-1	(Stögbauer et al. 2012)
$$K_{TX3}$$	Michaelis-Menten constant for pbrD transcription	8.5 nM	(Stögbauer et al. 2012)
$$K_{TX4}$$	Michaelis-Menten constant for pbrT transcription	8.5 nM	(Stögbauer et al. 2012)
$$λ_{m3}$$	pbrD mRNA degradation rate constant	0.08 min^-1	(Karzbrun et al. 2011)
$$λ_{m4}$$	pbrT mRNA degradation rate constant	0.08 min^-1	(Karzbrun et al. 2011)
$$k_{TL3}$$	pbrD translation rate constant	0.082 min^-1	(Stögbauer et al. 2012)
$$k_{TL4}$$	pbrT translation rate constant	0.082 min^-1	(Stögbauer et al. 2012)
$$K_{TL3}$$	Michaelis-Menten constant for translation of pbrD	29.9 nM	2015 iGEM Bielefeld
$$K_{TL4}$$	Michaelis-Menten constant for translation of pbrT	29.9 nM	2015 iGEM Bielefeld
$$v_{in}$$	Maximum rate of Lead influx through pbrT protein	1 nM min^-1	Assumed
$$K_{in}$$	Michaelis-Menten constant for influx of Lead through pbrT	100 nM	Assumed
$$k_{seq}$$	Rate constant for association of Lead-pbrD complex	1000 nM^-2 min^-1	Assumed
$$k_{-seq}$$	Rate constant for dissociation of Lead-pbrD complex	10^-15 min^-1	Assumed

References

[1] 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.
[2] Stamatakis, Michail; Mantzaris, Nikos V. (2009): Comparison of deterministic and stochastic models of the lac operon genetic network. In Biophysical journal 96 (3), pp. 887–906. DOI: 10.1016/j.bpj.2008.10.028.
[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] Huang, Lifang; Yuan, Zhanjiang; Liu, Peijiang; Zhou, Tianshou (2015): Effects of promoter leakage on dynamics of gene expression. In BMC systems biology 9, p. 16. DOI: 10.1186/s12918-015-0157-z.
[5] Cortés, Antoni; Cascante, Marta; Cárdenas, María Luz; Cornish-Bowden, Athel (2001): Relationships between inhibition constants, inhibitor concentrations for 50% inhibition and types of inhibition: new ways of analysing data. In Biochemical Journal (357), pp. 263–268.
[6] Pédelacq J-D, Cabantous S, Tran T, Terwilliger Tc, Waldo Gs (2005): Engineering and characterization of a superfolder green fluorescent protein. In Nature Biotechnology, 24(1) , 79-88. doi: 10.1038/nbt1172.