Abstract

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In order to combine the sensing device with synthetic biotechnology, first of all, we built a theoretical model for our promoter P_yeaR to describe the whole mechanism and reaction, which was simulated by simbiology.
Then we found out the concentration of each substance varied with time by Matlab, and approximate method was applied for choosing a proper model fitting with GFP fluorescence variation curve.
Furthermore, to improve our sensing detection, we built a statistical model by nonlinear regression and calibration, and created analysis method for sensing data by our empirical model constructed with data from more than 150 experiments; this model was able to distinguish 5 interval of nitrate concentration.

Motivation of Improve P_yeaR Model

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After taking a look to the research result of other pervious teams, we found out that it’s not enough describing the sensing reaction of P_yeaR just with NsrR; so to realize the mechanism, the paper, Activation of yeaR-yoaG Operon Transcription by the Nitrate-Responsive Regulator NarL Is Independent of Oxygen-Responsive Regulator Fnr in Escherichia coli K-12, was referred. From figure 1. in the paper, no matter with or without O2, the influence of NsrR and NarL to NO₃^- had not much difference.
Instead, from Figure 2. the influence of NsrR and NarL to P_yeaR promotor was significant, which was the reason for us to consider the effects of NsrR and NarL while building a more complete model.

P_yeaR Mechanism

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There are Nap and NirK enzymes that can catalyze NO₃^- to NO₂^- and NO₂^- to NO separately in our E. coli system. After paper searching, we found that the promotor’s reaction was controlled by two gene representing two binding sites, one of which was NarL, and the other was NsrR. NarL is able to sense both nitrate and nitrite, promoting P_yeaR to produce GFP further. NsrR has the ability to repress the whole reaction except for the situation of nitric oxide on the biding site with the repression becoming weak and the block to GFP generation gone.

Equations of Our Sensing Pathway Model

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NO₃^- will be consumed in two ways, one of which is turning into NO₂^- by Nap enzyme , and the other becomes mRNA of GFP by NarL.

The rate of [NO₃^-] can be expressed by:

d[NO3-]dt=-VmaxNap×[NO3-]KmNap+[NO3-]-kfNO3-×[NO3-]------(1)

NO₂^- can be produced by Nap enzyme, be consumed by NarL and NirK enzyme, and become mRNA of GFP along with NO.

The rate of [NO₂^-] can be expressed by:

d[NO2-]dt=VmaxNap×[NO3-]KmNap+[NO3-]-kfNO2-×[NO2-]-VmaxNirK×[NO2-]KmNirK+[NO2-]------(2)

NO can be produced by NirK enzyme.

The rate of [NO] can be expressed by:

d[NO]dt=Vmax(NirK)×[NO2-]Km(NirK)+[NO2-]------(3)

There are 3 sources cause mRNA of GFP production; one is from NO₃^-, another is from NO₂^-, and the other is from NO.
Also, owing to translating into GFP and gradually being degraded, mGFP will decreased.

The rate of [mGFP] can be expressed by:

Wrap(How can we derive this term)

Finally , mRNA of GFP will be translated into GFP.

The rate of [GFP] can be expressed by:

d[GFP]dt=ktranslation×mGFP-rGFP×GFP------(5)

Parameter Table

	Description	Value	Unit(SI)
[NO3-] (100ppm)	Nitrate initial value	1.6x10-6	mol/m3
Km (Nap)	the NO3- at which the reaction rate is at half-maximum	8x10-3	mol/m3
Vmax(Nap)	Maximum velocity of Nap	4.7x10-1	mol/s x m3
Km(NirK)	the NO2- at which the reaction rate is at half-maximum	2.5x10-1	mol/m3
Vmax(NirK)	Maximum velocity of NirK	5x10-3	mol/s x m3
[PyeaR]activity	Concentration of PyeaR	10-10	mol/m3
ktranscription	Rate of mGFP synthesis	1.8x10-5	1/s
kfno3	Related constant of NO3- and NarL	3x10-4	1/s
kfno2	Related constant of NO2- and NarL	6x10-5	1/s
rmGFP	mGFP degradation rate	5x10-5	1/s
rGFP	GFP degradation rate	2.5x10-6	1/s
ktranslation	Rate of GFP synthesis	4x10-4	1/s
kd(NsrR)	Dissociation constant of NsrR	3.5x10-6	m3/mol
kNO	Dissociation constant of NO	1.4x10-4	mol/m3
[NsrR]	Concentration of NsrR	10-6	mol/m3

Simulation

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Simbiology of Matlab is used to simulate the model:

Get a Function to Describe GFP varied with time

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By taking the parameters into the (1)~(5) equations, the [GFP] can be described by:

GFP(t)= (2.9x10^-8)e^-2.5x10-6t - (3.08x10^-8)e^-4.485x10-4t + (6.06x10^-10)e^-2x10-2t - (1.48x10^-9)e^-1.74x10-2t

(This equation is for initial concentration of nitrate 100ppm)

The Fitting Results

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As to know each term how to influence GFP, we divide GFP(t) into 4 parts:

1. (2.9x10^-8)e^-2.5x10-6t
2. (3.08x10^-8)e^-4.485x10-4t
3. (6.06x10^-10)e^-2x10-2t
4. (1.48x10^-9)e^-1.74x10-2t

Obviously, we easily know the influence within 2 hours of terms A and B are more essential than that of terms C and D. Consequently, we modify equation to be simpler, which neglact C and D terms, to fit our data getting from experiments with our device and with powder.
By using general model Exp2:

fx=d×egx+h×ejx

Coefficients (with 95% confidence bounds):

Coefficients Table

	value	min	max
d	3382	3364	3399
g	0.04229	0.03939	0.4518
h	-70.48	-86.6	-54.36
j	-1.119	-1.22	-1.018

Goodness of fit:
SSE: 3.151e+04
R-square: 0.9981
Adjusted R-square: 0.998
RMSE: 11.53

μvoltt=3382×e0.04229t-70.48×e-1.119t (for 60 ppm)

Statistical Model

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We randomly set 15 ppm of concentration of nitrate as the separating level of clean and polluted water. Hence, we dichotomized the concentration of nitrate into binary data (the outcomes of Bernoulli trials). Then, in order to fit a general linear model we use the concentration of nitrate, time as explain variables, and the electrical signals collected from our device as response variables.
Here is the regression equation:

Yij=248.8Tij781.9+Tij+0.1374Tij．Xi+εij

Yij	Electrical signals collected from i th time series data with jth second.
Tij	Time of ith time series data with jth second.
Xi = 1	When the concentration of nitrate above 15 ppm.
Xi = 0	When the concentration of nitrate below 15 ppm.
εij	Random error term of ith time series data with jth second.

After that, we use calibration to forecast the concentration of nitrate within the time series data with our model by minimizing the sum of square of time series data from 1^stsecond to 〖1800〗^th second. The training sensitivity and specificity are shown in the table below.

Empirical Model

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In order to build an empirical model for our sensing boat, we had done more than 150 times experiments setting up our database.
We chose 5 intervals:

0 – 4 ppm
4 – 10 ppm
10 – 20 ppm
20 – 60 ppm
Over 60 ppm

And our sensing device only need to detect the Optical signal on 5 min, 10 min, 15 min, 20 min. By using this method, we can easily, quickly and precisely distinguish the concentration into this 5 intervals.

Conclusion

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Not only to build a more complete model, according to species and application in the reality, we can also set different intervals by empirical method and any separating level by statistical method. With this advance, we are able to get the results within 20 to 30 minutes, which is quicker and more precise as well.

References

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