Line 251: | Line 251: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
โ | <th scope="row"> | + | <th scope="row"> [๐๐๐
๐๐ดโ๐
๐ต๐โ๐๐
๐๐ด<sub>sfGFP</sub>]</th> |
<td align="center">A</td> | <td align="center">A</td> | ||
<td align="center">15</td> | <td align="center">15</td> | ||
โ | <td align="center"> | + | <td align="center">mol/L</td> |
</tr> | </tr> | ||
<tr> | <tr> | ||
โ | <th scope="row"> | + | <th scope="row">[๐๐
๐๐ด<sub>Csy4</sub>]</th> |
<td align="center">B</td> | <td align="center">B</td> | ||
<td align="center">0</td> | <td align="center">0</td> | ||
โ | <td align="center"> | + | <td align="center">mol/L</td> |
</tr> | </tr> | ||
<tr> | <tr> | ||
โ | + | <th scope="row">[Protein<sub>๐ถ๐ ๐ฆ4</sub>]</th> | |
<td align="center">C</td> | <td align="center">C</td> | ||
<td align="center">0</td> | <td align="center">0</td> | ||
โ | <td align="center"> | + | <td align="center">mol/L</td> |
</tr> | </tr> | ||
<tr> | <tr> | ||
โ | + | <th scope="row">[Protein<sub>๐ถ๐ ๐ฆ4</sub>-crRNA-RBS-mRNA<sub>sfGFP</sub>]</th> | |
<td align="center">D</td> | <td align="center">D</td> | ||
<td align="center">0</td> | <td align="center">0</td> | ||
โ | <td align="center"> | + | <td align="center">mol/L</td> |
</tr> | </tr> | ||
<tr> | <tr> | ||
โ | + | <th scope="row">[๐๐
๐๐ด<sub>sfGFP</sub>]</th> | |
<td align="center">E</td> | <td align="center">E</td> | ||
<td align="center">0</td> | <td align="center">0</td> | ||
โ | <td align="center"> | + | <td align="center">mol/L</td> |
</tr> | </tr> | ||
<tr> | <tr> | ||
โ | + | <th scope="row">[Protein<sub>๐ถ๐ ๐ฆ4</sub>-๐๐๐
๐๐ด]</th> | |
<td align="center">F</td> | <td align="center">F</td> | ||
<td align="center">0</td> | <td align="center">0</td> | ||
โ | <td align="center"> | + | <td align="center">mol/L</td> |
</tr> | </tr> | ||
<tr> | <tr> | ||
โ | <th scope="row"> | + | <th scope="row">[Protein<sub>sfGFP</sub>]</th> |
<td align="center">G</td> | <td align="center">G</td> | ||
<td align="center">0</td> | <td align="center">0</td> | ||
โ | <td align="center"> | + | <td align="center">mol/L</td> |
</tr> | </tr> | ||
</tbody> | </tbody> | ||
Line 306: | Line 306: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
โ | + | <th scope="row">๐<sub>1</sub></th> | |
โ | <td align="center">The transcription rate of | + | <td align="center">The transcription rate of [๐๐๐
๐๐ดโ๐
๐ต๐โ๐๐
๐๐ด<sub>sfGFP</sub>]</td> |
โ | + | <td align="center">h<sub>-1</sub></td> | |
<td align="center">1.955</td> | <td align="center">1.955</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
โ | + | <th scope="row">๐<sub>d1</sub></th> | |
โ | <td align="center">The degradation rate of | + | <td align="center">The degradation rate of [๐๐๐
๐๐ดโ๐
๐ต๐โ๐๐
๐๐ด<sub>sfGFP</sub>]</td> |
โ | <td align="center"> | + | <td align="center">h<sub>-1</sub></td> |
<td align="center">0.002</td> | <td align="center">0.002</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
โ | + | <th scope="row">๐<sub>2</sub></th> | |
โ | <td align="center">The transcription rate of | + | <td align="center">The transcription rate of [๐๐
๐๐ด<sub>Csy4</sub>] </td> |
โ | <td align="center"> | + | <td align="center">h<sub>-1</sub></td> |
<td align="center">1.116</td> | <td align="center">1.116</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
โ | + | <th scope="row">๐<sub>d2</sub></th> | |
โ | <td align="center">The degradation rate of | + | <td align="center">The degradation rate of [๐๐
๐๐ด<sub>Csy4</sub>] </td> |
โ | <td align="center"> | + | <td align="center">h<sub>-1</sub></td> |
<td align="center">0.241</td> | <td align="center">0.241</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
โ | + | <th scope="row">๐<sub>p2</sub></th> | |
โ | <td align="center">The translation rate of | + | <td align="center">The translation rate of [Protein<sub>๐ถ๐ ๐ฆ4</sub>] </td> |
โ | <td align="center"> | + | <td align="center">h<sub>-1</sub></td> |
<td align="center">1.134</td> | <td align="center">1.134</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
โ | + | <th scope="row">๐<sub>dp2</sub></th> | |
โ | <td align="center">The degradation rate of | + | <td align="center">The degradation rate of [Protein<sub>๐ถ๐ ๐ฆ4</sub>] </td> |
โ | <td align="center"> | + | <td align="center">h<sub>-1</sub></td> |
<td align="center">6.547</td> | <td align="center">6.547</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
โ | + | <th scope="row">๐<sub>on</sub></th> | |
โ | <td align="center">The binding constant of | + | <td align="center">The binding constant of [Protein<sub>๐ถ๐ ๐ฆ4</sub>] and [๐๐๐
๐๐ดโ๐
๐ต๐โ๐๐
๐๐ด<sub>sfGFP</sub>]</td> |
โ | <td align="center"> | + | <td align="center">h<sub>-1</sub></td> |
<td align="center">23995.469</td> | <td align="center">23995.469</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
โ | + | <th scope="row">๐<sub>off</sub></th> | |
โ | <td align="center">The dissociation constant of | + | <td align="center">The dissociation constant of [Protein<sub>๐ถ๐ ๐ฆ4</sub>] and [๐๐๐
๐๐ดโ๐
๐ต๐โ๐๐
๐๐ด<sub>sfGFP</sub>]</td> |
โ | <td align="center"> | + | <td align="center">h<sub>-1</sub></td> |
<td align="center">2.703</td> | <td align="center">2.703</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
โ | + | <th scope="row">๐<sub>dc1</sub></th> | |
โ | <td align="center">The degradation rate of | + | <td align="center">The degradation rate of [Protein<sub>๐ถ๐ ๐ฆ4</sub>-crRNA-RBS-mRNA<sub>sfGFP</sub>] </td> |
โ | <td align="center"> | + | <td align="center">h<sub>-1</sub></td> |
<td align="center">0.024</td> | <td align="center">0.024</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
โ | + | <th scope="row">๐<sub>obs</sub></th> | |
โ | <td align="center">The cleavage rate of | + | <td align="center">The cleavage rate of [Protein<sub>๐ถ๐ ๐ฆ4</sub>] </td> |
โ | <td align="center"> | + | <td align="center">h<sub>-1</sub></td> |
<td align="center">0.327</td> | <td align="center">0.327</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
โ | + | <th scope="row">๐<sub>d3</sub></th> | |
โ | <td align="center">The degradation rate of | + | <td align="center">The degradation rate of [๐๐
๐๐ด<sub>sfGFP</sub>] </td> |
โ | <td align="center"> | + | <td align="center">h<sub>-1</sub></td> |
<td align="center">0.472</td> | <td align="center">0.472</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
โ | + | <th scope="row">๐<sub>dc2</sub></th> | |
โ | <td align="center">The degradation rate of | + | <td align="center">The degradation rate of [Protein<sub>๐ถ๐ ๐ฆ4</sub>-๐๐๐
๐๐ด] </td> |
โ | <td align="center"> | + | <td align="center">h<sub>-1</sub></td> |
<td align="center">0.024</td> | <td align="center">0.024</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
โ | + | <th scope="row">๐<sub>p1</sub></th> | |
<td align="center">The translation rate of \[Protein_{sfGFP}\]</td> | <td align="center">The translation rate of \[Protein_{sfGFP}\]</td> | ||
โ | <td align="center"> | + | <td align="center">h<sub>-1</sub></td> |
<td align="center">1.711</td> | <td align="center">1.711</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
โ | + | <th scope="row"><sub>๐dp1</sub></th> | |
โ | <td align="center">The degradation rate of | + | <td align="center">The degradation rate of [Protein<sub>sfGFP</sub>] </td> |
โ | <td align="center"> | + | <td align="center">h<sub>-1</sub></td> |
<td align="center">0.479</td> | <td align="center">0.479</td> | ||
</tr> | </tr> |
Revision as of 14:35, 16 October 2018
miniToe
Reactions
We can describe our miniToe system to be followings:
\[\rightarrow [๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{sfGFP}]\]
\[\rightarrow [๐๐
๐๐ด_{Csy4}]\]
\[[๐๐
๐๐ด_{๐ถ๐ ๐ฆ4}]\rightarrow [๐๐
๐๐ด_{๐ถ๐ ๐ฆ4}] + [๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}]\]
\[[Protein_{๐ถ๐ ๐ฆ4}]+[crRNA-RBS-mRNA_{sfGFP}]\leftrightarrow [๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-crRNA-RBS-๐๐
๐๐ด_{sfGFP}]\]
\[[Protein_{๐ถ๐ ๐ฆ4}-crRNA-RBS-mRNA_{sfGFP}]\rightarrow [๐๐
๐๐ด_{sfGFP}] + [๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-๐๐๐
๐๐ด]\]
\[[๐๐
๐๐ด_{sfGFP}]\rightarrow [๐๐
๐๐ด_{sfGFP}] + [๐๐๐๐ก๐๐๐_{sfGFP}]\]
\[[crRNA-RBS-mRNA_{sfGFP}]\rightarrow \emptyset\]
\[[๐๐
๐๐ด_{๐ถ๐ ๐ฆ4}]\rightarrow \emptyset\]
\[[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}]\rightarrow \emptyset\]
\[[Protein_{๐ถ๐ ๐ฆ4}-crRNA-RBS-mRNA_{sfGFP}]\rightarrow \emptyset\]
\[[๐๐
๐๐ด_{sfGFP}]\rightarrow \emptyset\]
\[[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-๐๐๐
๐๐ด]\rightarrow \emptyset\]
\[[๐๐๐๐ก๐๐๐_{sfGFP}]\rightarrow \emptyset\]
Two equations, describing the functional binding and cleavage of Csy4 protein in biology, and three parameters: kon, koff, kobs, describing the same things in mathematics, are the core of our model.
ODEs
To simulate the dynamics of sfGFP, we use ordinary differential equations to model the reactions above. And ODEs are given as follows:
\[\frac{d[๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{sfGFP}]}{dt}=๐_{1}-๐_{d1}[๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{sfGFP}]\]
\[-๐_{on}[Protein_{๐ถ๐ ๐ฆ4}][๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{sfGFP}]\]
\[+๐_{off}[Protein_{๐ถ๐ ๐ฆ4}-๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{sfGFP}]\]
\[\frac{d[๐๐
๐๐ด_{๐ถ๐ ๐ฆ4}]}{dt}=๐_{2}-๐_{d2}[๐๐
๐๐ด_{๐ถ๐ ๐ฆ4}]\]
\[\frac{d[Protein_{๐ถ๐ ๐ฆ4}]}{dt}=๐_{p2}[๐๐
๐๐ด_{๐ถ๐ ๐ฆ4}]-๐_{dp2}[Protein_{๐ถ๐ ๐ฆ4}]\]
\[-๐_{on}[Protein_{๐ถ๐ ๐ฆ4}][๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{sfGFP}]\]
\[+๐_{off}[Protein_{๐ถ๐ ๐ฆ4}-๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{sfGFP}]\]
\[\frac{d[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-crRNA-RBS-๐๐
๐๐ด_{sfGFP}]}{dt}=๐_{on}[Protein_{๐ถ๐ ๐ฆ4}][๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{sfGFP}]\]
\[-๐_{of}[Protein_{๐ถ๐ ๐ฆ4}][Protein_{๐ถ๐ ๐ฆ4}-crRNA-RBS-mRNA_{sfGFP}]\]
\[-๐_{d1}[Protein_{๐ถ๐ ๐ฆ4}][Protein_{๐ถ๐ ๐ฆ4}-crRNA-RBS-mRNA_{sfGFP}]\]
\[-๐_{obs}[Protein_{๐ถ๐ ๐ฆ4}][Protein_{๐ถ๐ ๐ฆ4}-crRNA-RBS-mRNA_{sfGFP}]\]
\[\frac{d[๐๐
๐๐ด_{sfGFP}]}{dt}=๐_{obs}[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-crRNA-RBS-๐๐
๐๐ด_{sfGFP}]-๐_{d3}[๐๐
๐๐ด_{sfGFP}]\]
\[\frac{d[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-crRNA]}{dt}=๐_{obs}[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-crRNA-RBS-๐๐
๐๐ด_{sfGFP}]-๐_{dc2}[๐๐๐๐ก๐๐๐_{๐ถ๐ ๐ฆ4}-๐๐๐
๐๐ด]\]
\[-๐_{on}[Protein_{๐ถ๐ ๐ฆ4}][๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{sfGFP}]\]
\[+๐_{off}[Protein_{๐ถ๐ ๐ฆ4}-๐๐๐
๐๐ด โ ๐
๐ต๐ โ ๐๐
๐๐ด_{sfGFP}]\]
\[\frac{d[Protein_{sfGFP}]}{dt}=๐_{p1}[๐๐
๐๐ด_{sfGFP}]-๐_{dp1}[Protein_{sfGFP}]\]
For the readability, the complex symbol is simplified as:
\[\frac{d[A]}{dt}=๐_{1}-๐_{d1}[A]-๐_{on}[C][A]+๐_{off}[D]\]
\[\frac{d[B]}{dt}=๐_{2}-๐_{d2}[B]\]
\[\frac{d[C]}{dt}=๐_{p2}[B]-๐_{dp2}[C]-๐_{on}[C][A]+๐_{off}[D]\]
\[\frac{d[D]}{dt}=๐_{on}[C][A]-๐_{dp2}[C]-๐_{off}[D]-๐_{dc1}[D]-๐_{obs}[D]\]
\[\frac{d[E]}{dt}=๐_{obs}[D]-๐_{d3}[E]\]
\[\frac{d[F]}{dt}=๐_{obs}[D]-๐_{d3}[F]\]
\[\frac{d[G]}{dt}=๐_{p1}[E]-๐_{dp1}[G]\]
Data Processing
The leak in the experiment is a big problem in estimating parameters in our ODEs model, so we processing the data by following formula๏ผ \[Data(without leak)=Data(+IPTG)-Data(-IPTG)\]
By doing this, we can reduce some factor which may be influenced estimation, not just the leak, but also some background noise. So we can get more precise parameters of the Csy4.
Species, symbols and parameters
Species | Symbol | Initial value | Units |
---|---|---|---|
[๐๐๐ ๐๐ดโ๐ ๐ต๐โ๐๐ ๐๐ดsfGFP] | A | 15 | mol/L |
[๐๐ ๐๐ดCsy4] | B | 0 | mol/L |
[Protein๐ถ๐ ๐ฆ4] | C | 0 | mol/L |
[Protein๐ถ๐ ๐ฆ4-crRNA-RBS-mRNAsfGFP] | D | 0 | mol/L |
[๐๐ ๐๐ดsfGFP] | E | 0 | mol/L |
[Protein๐ถ๐ ๐ฆ4-๐๐๐ ๐๐ด] | F | 0 | mol/L |
[ProteinsfGFP] | G | 0 | mol/L |
Because the [๐๐๐ ๐๐ด โ ๐ ๐ต๐ โ ๐๐ ๐๐ดsfGFP] is under controlled by a constitutive promoter, so we set the initial concentration to 15mol/L .
The other parament we used in the ODEs is listed in the following table:
Parameters | Definition | Units | Value |
---|---|---|---|
๐1 | The transcription rate of [๐๐๐ ๐๐ดโ๐ ๐ต๐โ๐๐ ๐๐ดsfGFP] | h-1 | 1.955 |
๐d1 | The degradation rate of [๐๐๐ ๐๐ดโ๐ ๐ต๐โ๐๐ ๐๐ดsfGFP] | h-1 | 0.002 |
๐2 | The transcription rate of [๐๐ ๐๐ดCsy4] | h-1 | 1.116 |
๐d2 | The degradation rate of [๐๐ ๐๐ดCsy4] | h-1 | 0.241 |
๐p2 | The translation rate of [Protein๐ถ๐ ๐ฆ4] | h-1 | 1.134 |
๐dp2 | The degradation rate of [Protein๐ถ๐ ๐ฆ4] | h-1 | 6.547 |
๐on | The binding constant of [Protein๐ถ๐ ๐ฆ4] and [๐๐๐ ๐๐ดโ๐ ๐ต๐โ๐๐ ๐๐ดsfGFP] | h-1 | 23995.469 |
๐off | The dissociation constant of [Protein๐ถ๐ ๐ฆ4] and [๐๐๐ ๐๐ดโ๐ ๐ต๐โ๐๐ ๐๐ดsfGFP] | h-1 | 2.703 |
๐dc1 | The degradation rate of [Protein๐ถ๐ ๐ฆ4-crRNA-RBS-mRNAsfGFP] | h-1 | 0.024 |
๐obs | The cleavage rate of [Protein๐ถ๐ ๐ฆ4] | h-1 | 0.327 |
๐d3 | The degradation rate of [๐๐ ๐๐ดsfGFP] | h-1 | 0.472 |
๐dc2 | The degradation rate of [Protein๐ถ๐ ๐ฆ4-๐๐๐ ๐๐ด] | h-1 | 0.024 |
๐p1 | The translation rate of \[Protein_{sfGFP}\] | h-1 | 1.711 |
๐dp1 | The degradation rate of [ProteinsfGFP] | h-1 | 0.479 |
Simulation
With the help of Simbiology toolbox in Matlab๏ผwe simulate our miniToe system for 30h, the result can be seen in the Fig.1.
Fig.1 The dynamics of sfGFP by model prediction
Fig.2 The comparison between experimental data and simulation data
Discussion
Combining the biology and math, we now discuss why the dynamics of sfGFP is like the curve in the Fig.1. In order to explain in detail, we plot the dynamics of all species in the miniToe system in Fig.3.
Fig.3 The dynamics of all species in the miniToe system
Sensitivities Analysis
After building the ODE model, we try to do something more deeply to our miniToe system by analyzing the sensitivity of parameters. Fig.4 shows the numerical integration of sensitivities of parameters in 30 hours.
Fig.4 The numerical integration of sensitivities of parameters in 30h
Enlighten by the sensitivity analysis, we give a prediction curve that shows what will happen in the sfGFP expression curve if we change the Csy4, and it can be seen in the Fig.5.
Fig.5 The curve of sfGFP with the changing cleavage rate
Fig.6 The relationship between the stable expression level of sfGFP and the claevage rate
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