Revision as of 16:09, 30 October 2017

MathJax AsciiMath Test Page

Model

Abstract

In order to combine our sensing device and synthetic biotechnology. First of all, we have to build a theoretical model (for our promoter PyeaR) to describe the whole reaction and mechanism. Then, we use simbiology to simulate the whole reaction. Next, to find out each substance varied with time. Afterward, using approximate method to choose a good model fitting GFP fluorescence variation curve. Furthermore, to help our sensing detection, we use nonlinear regression and calibration to build a statistical model. Last but not least, creating new method to analysis sensing data by our empirical model (constructed by more than 150 experiments’ data) , which can distinguish 5 interval of nitrate concentration.

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 activity is controlled two gene, which represent two binding sites. One is NarL, the other is NsrR. NarL can senses nitrate and nitrite, promoting PyeaR produce GFP further. NsrR has the ability to repress whole reaction except when nitric oxide come to the biding site. Repression become weak, and the block to GFP generation is gone.

Equations of Our Sensing Pathway Model

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

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

$\frac{d {[{N O}_{3}}^{-}]}{d t} = - \frac{{V m a x}_{(N a p)} \times {[{N O}_{3}}^{-}]}{{K m}_{(N a p)} + {[{N O}_{3}}^{-}]} - {k f}_{({N O}_{3}^{-})} \times {[{N O}_{3}}^{-}]$ ------(1)

NO2- can be produced by Nap enzyme , and will be consumed by NarL and NirK enzyme to become mRNA of GFP and NO.

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

$\frac{d {[{N O}_{2}}^{-}]}{d t} = \frac{{V m a x}_{(N a p)} \times {[{N O}_{3}}^{-}]}{{K m}_{(N a p)} + {[{N O}_{3}}^{-}]} - {k f}_{({N O}_{2}^{-})} \times {[{N O}_{2}}^{-}] - \frac{{V m a x}_{(N i r K)} \times {[{N O}_{2}}^{-}]}{{K m}_{(N i r K)} + {[{N O}_{2}}^{-}]}$ ------(2)

NO can be produced by NirK enzyme.

The rate of [NO] can be expressed by:

$\frac{d [N O]}{d t} = \frac{{V m a x}_{(N i r K)} \times {[{N O}_{2}}^{-}]}{{K m}_{(N i r K)} + {[{N O}_{2}}^{-}]}$ ------(3)

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

The rate of [mGFP] can be expressed by:

$\frac{d [m G F P]}{d t} = {k f}_{({N O}_{3}^{-})} \times {[{N O}_{3}}^{-}] + {k f}_{({N O}_{2}^{-})} \times {[{N O}_{2}}^{-}] + \frac{k t r a n s c r i p t i o n \times {[P y e a R]}_{a c t i v i t y}}{1 + \{\frac{[N s r R]}{1 + {\{\frac{[N O]}{K_{N O}}\}}^{2}}\} \times {k d}_{N s r R}} - r_{m G F P} \times [m G F P] - k t r a n s l a t i o n \times [m G F P]$ ------(4)

hyperlink

Finally , mRNA of GFP will be translated into GFP.

The rate of [GFP] can be expressed by:

$\frac{d [G F P]}{d t} = k t r a n s l a t i o n \times [m G F P] - r_{G F P} \times [G F P]$ ------(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
[P_yeaR]_activity	Concentration of P_yeaR	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
$r_{m G F P}$	mGFP degradation rate	5x10^-5	1/s
$r_{G F P}$	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
k_NO	Dissociation constant of NO	1.4x10^-4	mol/m³
[NsrR]	Concentration of NsrR	10^-6	mol/m³

Simulation

We use (Matlab) simbiology to simulate the model:

Get a Function to Describe GFP varied with time(t)

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

In order to know each term how to influence GFP , we divide GFP(t) into 4 parts.

(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}

Obviously , we can easily know A and B terms’ influence are much essential than C’s and D’s . (in 2hr)
So , we can use simple equation, which neglact C and D terms to fit our data. (our device and with powder)
By using general model Exp2:

$f (x) = d \times e^{g x} + h \times e^{j x}$

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

μvolt $(t) = 3382 \times e^{0.04229 t} - 70.48 \times e^{- 1.119 t}$ (for 60 ppm)

Statistical Model

We randomly set 15 ppm of concentration of nitrate as the separating level of clean water and polluted one. Hence, we dichotomize the concentration of nitrate into binary data (the outcomes of Bernoulli trials). Then, fitting a general linear model using the concentration of nitrate and time as explain variables, the electrical signals collected from our device as response variables. The regression equation is:

$Y_{i j} = \frac{248.8 T_{i j}}{781.9 + T_{i j}} + 0.1374 T_{i j} ． X_{i} + ε_{i j}$

Y_ij	Electrical signals collected from i ^th time series data with j^th second.
T_ij	Time of i^th time series data with j^th second.
X_i = 1	When the concentration of nitrate above 15 ppm.
X_i = 0	When the concentration of nitrate below 15 ppm.
ε_ij	Random error term of i_th time series data with j_th second.

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

Empirical Model

In order to build a empirical model for our sensing boat, we did more than 150 times experiments to set up our database.
We choose 5 interval:

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.

@@ Line 253: / Line 253: @@
            <tr>
              <td>
+              <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>r</mi></mrow><mrow><mi>m</mi><mi>G</mi><mi>F</mi><mi>P</mi></mrow></msub></math>
              </td>
              <td>
@@ Line 267: / Line 267: @@
            <tr>
              <td>
+              <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>r</mi></mrow><mrow><mi>G</mi><mi>F</mi><mi>P</mi></mrow></msub></math>
              </td>
              <td>

Difference between revisions of "Team:NCKU Tainan/test formula"