IGEM:IMPERIAL/2008/New/Genetic Circuit
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+  __NOTOC__  
{{Imperial/StartPage2}}  {{Imperial/StartPage2}}  
  
  +  {{Imperial/Box2The Genetic Circuit  
  +  An accurate mathematical description of the genetic circuit is essential for projects involving synthetic biology. Such descriptions are an integral component of part submission to the Registry, as exemplified by the canonical characterisation of part F2620 <cite>#1</cite>. The ability to capture part behaviour as a mathematical relationship between input and output is useful for future reuse of the part and modification of integration into novel genetic circuits.}}  
  +  {{Imperial/Box1Modelling Constitutive Gene Expression  
+  A simple [[Imperial_College/Courses/Spring2008/Synthetic_Biology/Computer_Modelling_Practicals/Practical_2  synthesisdegradation model]] is assumed for the modelling of the expression of a protein under the control of a constitutive promoter, with the same model assumed for all four [[IGEM:IMPERIAL/2008/Prototype/Wetlab/test_constructs  promoterRBS constructs]]. The synthesisdegradation model assumes a steady state level of mRNA.  
  +  <math>\frac{d[protein]}{dt} = k_{1}  d_{1}[protein]</math>  
  
  
  
  
  
In this case, <math>[protein]</math> represents the concentration of GFP, <math>k_{1}</math> represents the rate of sythesis and <math>d_{1}</math> represents the degradation rate.  In this case, <math>[protein]</math> represents the concentration of GFP, <math>k_{1}</math> represents the rate of sythesis and <math>d_{1}</math> represents the degradation rate.  
We can easily simulate this synthesisdegradation model using matlab:<br>  We can easily simulate this synthesisdegradation model using matlab:<br>  
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[[Media:SimulationGfp.mSimulation File]]  [[Media:SimulationGfp.mSimulation File]]  
<br>  <br>  
  +  
  +  We can also solve this ODE analytically.  
  +  
  +  <math>[protein]=\frac{k_1}{d_1}(1{e^{d_1t }})</math>  
  +  
  +  
+  Consider the steadystate behaviour of <math>[protein]</math>.  
+  
+  <math>\frac{d[protein]}{dt}=0 \Rightarrow [protein]_{steadystate}=\frac{k_1}{d_1}</math>  
+  
+  This relationship can be seen in the parameter scan graphs on the right.  
+  
From the wetlab experiments it is likely that we will obtain steadystate data for each of the four promoterRBS constructs. If we assume the same rate of degradation of GFP in each case, we can have some measure of the relative rate of transcription through each promoter which will help us with the selection of the most appropriate promoter to use for Phase 2. In order to obtain an absolute measure of transcription (as opposed to a relative measure of transcriptional strength) we require constitutive expression in terms of molecules per cell (as opposed to fluorescene in arbitrary units).  From the wetlab experiments it is likely that we will obtain steadystate data for each of the four promoterRBS constructs. If we assume the same rate of degradation of GFP in each case, we can have some measure of the relative rate of transcription through each promoter which will help us with the selection of the most appropriate promoter to use for Phase 2. In order to obtain an absolute measure of transcription (as opposed to a relative measure of transcriptional strength) we require constitutive expression in terms of molecules per cell (as opposed to fluorescene in arbitrary units).  
<br>  <br>  
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*In the case where <math>k_1 = 0</math>, no GFP is sythesised.  *In the case where <math>k_1 = 0</math>, no GFP is sythesised.  
*In the case where <math>d_1 = 0</math>, the concentration of protein does not reach a steady state.  *In the case where <math>d_1 = 0</math>, the concentration of protein does not reach a steady state.  
+  [[Image:Phase 1.PNGthumb300pxConstitutive expression of antibiotic resistance (AB) and GFP. GFP brick is part E0040, GFPmut3b. Terminator is part B0015, the doublestop.]]  
  ==Modelling Inducible Gene Expression  +  <gallery caption="Parameter Scan" align="right"> 
+  Image:ParamterScanGFPk1.jpgParameter scan on <math>k_1</math>  
+  Image:ParamterScanGFPd1.jpgParameter scan on <math>d_1</math>  
+  </gallery>}}  
+  
+  {{Imperial/Box1Modelling Inducible Gene Expression  
+  [[Image:Phase 2linduced.PNGthumb300px]]  
The repressor is constitutively expressed. Hence we can assume the constitutive expression model from the previous characterisation step.  The repressor is constitutively expressed. Hence we can assume the constitutive expression model from the previous characterisation step.  
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When the inducer is added it binds reversibly to the repressor.  When the inducer is added it binds reversibly to the repressor.  
  <math> R + I \  +  <math> R + I \rightleftharpoons RI </math> 
  +  Repressor only binds to the promoter when it is in its unbound form, hence transcription will be a function of free repressor concentration.  
<math> Transcription = \frac{\beta.{[R]}^n}{{K_m}^n+[R]^n}</math><br>  <math> Transcription = \frac{\beta.{[R]}^n}{{K_m}^n+[R]^n}</math><br>  
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<math>\frac{d[protein]}{dt} = Transcription  d_2[protein]</math>  <math>\frac{d[protein]}{dt} = Transcription  d_2[protein]</math>  
+  The [[media:ConcentrationsODE.mODEs]] and [[media:SimulationconcentrationsODE.msimulation]] may be found in the Appendices section of the Dry Lab hub.  
+  
+  <biblio>  
+  #1 pmid=18612302  
+  </biblio>  
+  }}  
+  
+  {{Imperial/Box1Equations for iGEM wiki  
+  
+  <math>2LacI + IPTG \rightleftharpoons IPTGLacI_2</math>  
+  
+  <math>k_{on} = k_2</math>  
+  
+  <math>k_{off} = k_3</math>  
+  
+  <math>k_{\alpha} = \frac{k_2}{k_3}</math>  
+  
+  <math>2LacI + P \rightleftharpoons PLacI_2</math>  
+  
+  <math>IPTG + PLacI_2 \rightleftharpoons P + IPTGLacI_2</math>  
  {{Imperial/EndPage}}  +  }} 
+  {{Imperial/EndPageGrowth_CurveMotility}} 
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