Team:IISER-Pune-India/Model/Biosensor

Mutatis Mutandis

Biosensor

The Biosensor

The biosensor gene circuit has two promoters, one constitutively expressed (BBa_K1758333) and the other can be repressed (pbrAP). The gene downstream of the constitutive promoter expresses the pbrR protein. The pbrR protein dimer binds pbrAP, thus repressing expression of the gene downstream of pbrAP. Lead ions bind to pbrR dimer, preventing it from repressing pbrAP. GFP is placed downstream of pbrAP, this allows us to relate GFP fluorescence intensity to Lead concentration.

The biosensor model uses mass action kinetics for processes such as repression, derepression and dimerisation. Hill functions and Michaelis Menten kinetics are used for describing dynamics of mRNA and protein production. Proper approximations are chosen in places where the parameters are unknown.
In the biosensor and in the bioremediation system, the GFP used is superfolder GFP (sf-GFP). This is the case for all instances of GFP mentioned in the biosensor and bioremediation system modelling.

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. Ribosome Binding Site (RBS) efficiency is directly proportional to the translation rate of mRNA.

System:

Species and symbols:

Symbol	Species	Initial Value(nM)
$$G_R$$	Repressor Gene	$$5$$
$$m_R$$	Repressor mRNA	$$0$$
$$R$$	Repressor Monomer (pbrR)	$$0$$
$$R_2$$	Repressor Dimer	$$0$$
$$O$$	Free Operator Gene (pbrAP)	$$35$$
$${R_2}O$$	Repressor-Operator Complex (Responsible for repression)	$$0$$
$$A_{out}$$	Free Analyte(Lead) outside the cell	$$0\; to \;10^5$$
$$A$$	Free Lead inside the cell	$$0$$
$${R_2}{A_2}$$	Repressor-Lead Complex (Responsible for derepression)	$$0$$
$$m_F$$	Reporter (GFP) mRNA	$$0$$
$$F_{in}$$	Inactive Reporter (GFP)	$$0$$
$$F$$	Matured/Active Reporter (GFP)	$$0$$
$$TLR$$	Translation Resources	$$1520$$

Reporter DNA concentration same as Free Operator concentration since GFP is downstream of pbrAP.

Reactions

Transcription of DNA

These represent the transcription of genes

$$G_R \xrightarrow{} m_R$$

$$O \xrightarrow{} O+ m_F$$

Translation of mRNAs

These represent the translation of mRNA

$$m_R \xrightarrow{k_{TL2}} R$$

$$m_F \xrightarrow{k_{TL1}} F_{in}$$

Repression of the operator(pbrAP)

We have assumed here that only the dimer of pbrR can repress the operator. First, the pbrR protein froms a dimer. Then the dimer binds pbrAP

$$2R \underset{k_{-2R}}{\stackrel{k_{2R}}{\rightleftharpoons}} R_2$$

$$R_2+O \underset{k_-r}{\stackrel{k_r}{\rightleftharpoons}} {R_2}O$$

Derepression of operator(pbrAP)

Lead influx into the cell by diffusion. Here we have assumed that Lead can only bind to the pbrR dimer. The lead can either directly bind to the pbrR protein dimer or the repressor-operator complex. This increases the free operator concentration.

$$A_{out} \underset{-P}{\stackrel{P}{\rightleftharpoons}} A$$

$$2A + R_2 \underset{k_{-dr1}}{\stackrel{k_{dr1}}{\rightleftharpoons}} {R_2}{A_2}$$

$$2A +{R_2}O \underset{k_{-dr2}}{\stackrel{k_{dr2}}{\rightleftharpoons}} {R_2}{A_2}+O$$

Leaky expression of the operator

$${R_2}O \xrightarrow{k_{leak}} {R_2}O + m_F$$

GFP Maturation

$$F_{in} \underset{-}{\stackrel{k_{mat}}{\rightleftharpoons}} F$$

mRNA and protein Degradation

We have assumed pbrR to be a stable protein with very slow degradation. mRNA and GFP have a degradation rate in the time-scale in which we want to observe the model's behaviour.

$$m_F \xrightarrow{\lambda_{m_1}} \phi$$

$$m_R \xrightarrow{\lambda_{m_2}} \phi$$

$$F \xrightarrow{\lambda_{m_F}} \phi$$

The total concentration of $O$, i.e., bound and free operator concentration does not change over time. This is because there is not influx or degradation of the operator. Hence we can write

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

Here $O$ and ${R_2}O$ are intital concentration of the operator and the complex respectively.

Similarly total concentration of Lead (inside the cell + outside + bound to repressor) does not change over time.

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

The values here are also the initial concentration values as in the case above.

Equations

The equation below describes the transcription of the repressor (pbrR) mRNA. The hill function was taken from 2015 iGEM Bielefeld Team. It is modelled as a hill function with hill coefficient equal to 2.$v_{TX2}$ is the maximum transcription rate of $m_R$. Since all degradation terms in the biosensor are assumed to be first order reactions, $\lambda_{m_R}$ is the denaturation/degradation constant.

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

The equation describes the pbrR protein dynamics. pbrR production is modelled using a hill function, with hill coefficient equal to 1. The remaining terms deal with the mass action kinetics of dimerisation of pbrR. $k_{2R}$ is the equilibrium constant of dimerisation, $k_{-2R}$ is the constant of the reverse reaction.

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

$R_2$ is a key ingredient of almost all reactions in the system. $k_r$ is the repression reaction constant, \k_{-r}\) is the constant of the reverse reaction. $k_{dr1}$ is the equilibrium constant of the first derepression pathway. The first derepression path is $2A + R_2 \underset{k_{-dr1}}{\stackrel{k_{dr1}}{\rightleftharpoons}} {R_2}{A_2}$. Here we see the terms $(T_O - [O])$ and $\frac{(T_A - [A] - [A_{out}])}{2}$, these are substitutes for ${R_2}O$ and ${R_2}{A_2}$ respectively.

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

The foloowing equation is of Fick's law of diffusion. The permeability value of $Pb^{2+}$ is not known. It is taken to be close to that of a known cation.

$$\frac{d[A_{out}]}{dt} = -P([A_{out}]-[A])$$

Free Lead inside the cell is involved in all the derepression reactions. $k_{dr2}$ is the equilibrium constant of the second derepression pathway. It is given by the reaction $2A +{R_2}O \underset{k_{-dr2}}{\stackrel{k_{dr2}}{\rightleftharpoons}} {R_2}{A_2}+O$

$$\frac{d[A]}{dt} = P([A_{out}]-[A]) - 2k_{dr1}[A]^2{[R_2]} + 2k_{-dr1}\frac{(T_A - [A] - [A_{out}])}{2} -2k_{dr2}[A]^2(T_O - [O]) + 2k_{-dr2}[O]\frac{(T_A - [A] - [A_{out}])}{2}$$

the free operator (pbrAP) concentration is taken to be the same as the free GFP gene concentration, since GFP is placed downstream of the operator.

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

Here we again have a hill function ,of hill coefficient 2, representing transcription and we also have a first order decay term. $k_{leak}$ here represents the leaky transcription in the system, i.e., expression of GFP in the absence of Lead.

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

We see an extra $TLR$ term in the hill function in the case of translation (Even in $\frac{d[R]}{dt}$) . This term is added to mimic the limited number of resources in the system. Without it, even when Lead is added to the system above a certain threshold, the protein production increases linearly without any bound. This is because once Lead saturates all the pbrR dimers, it can continually derepress the system (Lead cannot degrade and once the diffusion reaches equilibrium, Lead concentration in the cell will be constant), and show infinite expression.

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

GFP takes time to mature, this is represented by the $k_{mat}$ term. First order decay term of GFP is also taken.

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

A constantly reducing supply of translation resources is given by $TLR$. the decay is represented by a negative hill function.

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

${R_2}O$ and ${R_2}{A_2}$ are not being solved for in the above equations. Since these values can be obtained from the equations of $ T_O$ and $T_A$ respectively.

Parameter values are shown in the 'Parameters' tab.

Table representing the different parameters and their values:

Parameter	Description	Value	Reference
$$v_{TX1}$$	reporter transcription rate constant	18.2 nM min^-1	(Stögbauer et al. 2012)
$$v_{TX2}$$	repressor transcription rate constant	18.2 nM min^-1	(Stögbauer et al. 2012)
$$K_{TX1}$$	Michaelis-Menten constant for reporter transcription	8.5 nM	(Stögbauer et al. 2012)
$$K_{TX2}$$	Michaelis-Menten constant for repressor transcription	8.5 nM	(Stögbauer et al. 2012)
$$λ_{m1}$$	reporter mRNA degradation rate constant	0.08 min^-1	(Karzbrun et al. 2011)
$$λ_{m2}$$	repressor mRNA degradation rate constant	0.08 min^-1	(Karzbrun et al. 2011)
$$k_{TL1}$$	reporter translation rate constant	0.082 min^-1	(Stögbauer et al. 2012)
$$k_{TL2}$$	repressor translation rate constant	0.082 min^-1	(Stögbauer et al. 2012)
$$K_{TL1}$$	Michaelis-Menten constant for translation of reporter	29.9 nM	2015 iGEM Bielefeld
$$K_{TL2}$$	Michaelis-Menten constant for translation of repressor	29.9 nM	2015 iGEM Bielefeld
$$v_{λTLR}$$	Translation resources degradation rate constant	13.5 nM min^-1	2015 iGEM Bielefeld
$$K_{λTLR}$$	Michaelis-Menten constant for degradation of translation resources	53.2 nM min^-1	2015 iGEM Bielefeld
$$k_{mat}$$	reporter maturation rate constant	0.1155 min^-1	(Pédelacq JD et al. 2005)
$$k_gd$$	first order decay constant of reporter protein	0.001155 min^-1	Assumed (Half life of sf GFP is assumed to be 10hrs)
$$k_{2R}$$	repressor dimerization rate constant	50 nM^-1 min^-1	(Stamatakis, Mantzaris 2009)
$$k_{2R}$$	repressor dimer dissociation rate constant	0.001 nM^-1 min^-1	(Stamatakis, Mantzaris 2009)
$$k_r$$	association rate constant for repression	960 nM^-1 min^-1	(Stamatakis, Mantzaris 2009)
$$k_{-r}$$	dissociation rate constant for repression	2.4 min^-1	(Stamatakis, Mantzaris 2009)
$$k_{dr1}$$	association rate constant for first derepression mechanism	3 * 10⁷ nM^-2 min^-1	(Stamatakis, Mantzaris 2009)
$$k_{-dr1}$$	dissociation rate constant for first derepression mechanism	12 nM^-2 min^-1	(Stamatakis, Mantzaris 2009)
$$k_{dr2}$$	association rate constant for second derepression mechanism	3 * 10⁷ nM^-2 min^-1	(Stamatakis, Mantzaris 2009)
$$k_{-dr2}$$	dissociation rate constant for second derepression mechanism	4800 nM^-2 min^-1	(Stamatakis, Mantzaris 2009)
$$k_{leak}$$	leak reporter transcription rate constant	0.0033 min^-1	(Stamatakis, Mantzaris 2009)
$$P$$	permeability of ion through cell membrane	1.178 * 10^-3 nM min^-1	("Physical Biology of the Cell", Rob Phillips, Jane Kondev and Julie Theriot (2009))

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.

Time series

We first have the time series plot of the GFP fluorescence intensity

Fig1. Shows the number of molecules of GFP in the cell through time (mins). This is when the cell is placed in an environment with Lead concentration of 5 μM

In the above graph we see that the number of GFP molecules reaches its maximum value in about one hour. The GFP stays constant after this till 100 minutes and then a decrease in GFP is observed. This is the behaviour of the system in 5 μM of Lead.
The constant value of GFP is observed because of a balance in the derepression and repression mechanisms. This would be the state of the system if not for the degradation of the GFP. The longer half-life of the sf-GFP used makes taking measurements easier. For faster degrading GFPs, fluorescence would have to be measured at the exact time for all samples to compare GFP readouts.

Modification of RBS (Ribosome Binding Site)

It was observed that changing the rate of transcription of the proteins was changing the GFP readout for various Lead concentrations. Seen in the graph below

Fig.2 Shows the GFP readout at 60 minutes at various Lead concentrations with different rates of transcription. GFP readout is taken at 60 minutes so that it is maximum for all systems.

From the graph above we see that changing the rate of transcription can change the range of detection of the biosensor. This is because the range linear GFP increase is different for different transcription rates. Transcription rate is true for the model though, in the real system the transcription rate would mean the RBS binding affinity. To change RBS binding affinity we change the RBS. We wanted a biosensor that could show good contrast of GFP readout in the range of 0μM to 10μM of Lead. Various RBSs were chosen from the RBS/Cat page. This page lists RBSs with their descriptions and binding efficiencies (if known).

Comparing experiment data with the model

Experiments were performed to measure the GFP readout of the cell at various concentrations of Lead. The values obtained are compared to those obtained by the model in the graph below.

Fig.3 Shows the GFP readout shown by the experiment and by the model.

Most points lie on the line predicted by the model. The error bars plotted show the standard error of the experiment plot points. This graph shows that the model predictions match the experiment results within limits of error. Hence these parameters can be used for further predictions to improve biosensor.

Improving the Biosensor

Fig.4 Shows the model predictions of GFP readout. The cyan line shows the predicted GFP output with the original RBS. The red line shows the predicted GFP output with the RBS BBa_B0032. This RBS has an efficiency of 0.3 comapred to the original RBS.

Fig.2 and Fig.4 show that biosensor systems with lower rate of translation can be used to measure lower concentrations of Lead and vice-versa. This implies that lower rates of translation correspond to more sensitivity of the biosensor. However this sensitivity is only at low Lead concentrations because as seen in Fig.2, less efficient RBS systems show saturation of GFP at higher Lead concentrations. Fig.4 shows that using a weaker RBS (less efficient RBS) gives us more contrast in the range of 0μM to 10μM. This is the range of detection that we want to work in, since the local river has Lead concentration in the range of 4μM to 7μM.

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