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| - | <td width="750" bgcolor="#414141" valign="top">
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| - | <br>
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| - | <div class="button">
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| - | <img src="http://i1001.photobucket.com/albums/af132/igemcalgary/Mo.gif" align="left">
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| - | </div>
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| - | <div class="heading">
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| - | DIFFERENTIAL EQUATIONS MODELLING METHODS
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| - | </div>
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| - | <div class="desc">
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| - | The simbiology interface from Matlab was used to simulate the differential equations model. Chemical Kinetic equations were used to build the model for simulation.
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| - | <br><br>
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| - | <center> <img src="https://static.igem.org/mediawiki/2009/9/9b/ReactA.JPG"> </center> <br>
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| - | <center> Fig : The Reaction of Species A with B to produce C and D </center><br>
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| - | <br>
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| - | <center><img src="https://static.igem.org/mediawiki/2009/archive/b/bd/20091019225744%21Rate.JPG">
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| - | </center> <br>
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| - | <center> Fig : The Chemical Kinetic Rate Equation </center>
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| - | <br><br>
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| - | k is the kinetic rate constant. The size of k will determine the speed of the reaction. A smaller value of k will produce a slow reaction rate while a larger value of k will produce a fast reaction rate. <br><br>
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| - | [A] is the amount of reactant A present. <br><br>
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| - | The simulations were run for 50000 seconds . It was considered to be enough time for the system to reach equilibrium after disturbance.<i>
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| - | <br><br>
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| - | <b>Sundial Solver</b>
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| - | The sundial solver (SUNDIALS) was developed so that robust time integrators and non-linear solvers can be easily combined with already existing simulation codes. Minimal information from user is required and this solver allow users to easily supply their own data structures. The Sundials solvers are part of a third-party package developed at Lawrence Livermore National Laboratory. Built-in ordinary differential equation (ODE) solvers (ode45 and ode15s) are also part of the interface.
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| - | <br><br>
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| - | When sundials solver is selected, the program selects one of teh two sundials solvers that suits your model: CVODE or IDA. CVODE is used for systems of ODEs (stiff or nonstiff) and this type of solver is usually used for a model that has no algebraic rules. IDA is a differential-algebraic equation (DAE) solver and it is usually used when there is one more algebraic rules. Since our model incorporates an event (the addition of autoinducer-II (AI-2)), this type of solver was used in our model. More information can be found here: https://computation.llnl.gov/casc/sundials/description/description.html <i>
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| - | </div>
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| - | <br>
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| - | <div class="heading">The Reactions</div>
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| - | <div class="desc">
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| - | The system was represented by the following reactions. The reactions with double headed arrows have two rate constants(forward/ reverse rate constant). All reactions were assumed to be elementary reactions. </div>
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| - | <br>
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| - | <center><img src ="https://static.igem.org/mediawiki/2009/5/5f/Reactions1.tif"></center>
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| - | <br>
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| - | <div class="heading">Parameter Rationale</div>
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| - | <div class="desc">
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| - | </div>
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| - | <center><b>Table: Initial Values of the Species in the System</b> </center>
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| - | <br>
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| - | <table width="700" border="1" bgcolor="#414141" align = "center">
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| - | <tr>
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| - | <td>Species</td>
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| - | <td> Initial Value </td>
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| - | <td>Rationale</td>
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| - | </tr>
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| - | <tr>
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| - | <td>AI-2</td>
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| - | <td>0</td>
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| - | <td><align = "left">Initially the amount of AI-2 is constant at 0. After an equilibruim is established variable amounts of AI-2 are added at different simulations. <br><br> </td>
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| - | </tr>
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| - | <tr>
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| - | <td>LuxPQ</td>
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| - | <td>10</td>
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| - | <td>The amount of LuxPQ varies depending on the simulation run.<br><br> </td>
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| - | </tr>
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| - | <tr>
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| - | <td>AI-2:LuxPQ</td>
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| - | <td>0</td>
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| - | <td>This value is kept at time = 0 because the initial concentration of AI-2 is 0. <br><br></td>
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| - | </tr>
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| - | <tr>
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| - | <td>LuxU:p</td>
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| - | <td>2</td>
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| - | <td>----</td>
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| - | </tr>
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| - | <tr>
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| - | <td>LuxU</td>
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| - | <td>1000</td>
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| - | <td>There is a lot of this species present in the cell in nature. To signify plenty a value of 1000 is assigned.<br><br></td>
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| - | </tr>
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| - | <tr>
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| - | <td>LuxO:p</td>
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| - | <td>2</td>
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| - | <td>Equal amounts of LuxO:p and LuxU:p was considered in the model because LuxU:p phosphorylates LuxO . The phosphorylation reaction is considered to be a fast reaction therefore there are equal amounts of the two proteins present.<br><br></td>
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| - | </tr>
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| - | <tr>
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| - | <td>LuxO</td>
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| - | <td>1000</td>
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| - | <td>There are equal amounts of LuxO and LuxU present since they are encoded within the same operon in the cell.<br><br></td>
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| - | </tr>
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| - | <tr>
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| - | <td>p</td>
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| - | <td>10.0658</td>
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| - | <td>We assume that there is enough p is the environment that it doesn’t become a limiting factor. For that reason we assign p as a constant value in simbiology. (It doesn’t really matter that the initial amount is presented as a comparatively small number in this case. ) <br><br></td>
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| - | </tr>
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| - | <tr>
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| - | <td>sigma54</td>
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| - | <td>0.14183</td>
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| - | <td>This amount is kept at a constant value to ensure that this value does not become a limiting factor.<br><br></td>
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| - | </tr>
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| - | <tr>
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| - | <td>sigma54:LuxO:p:Pqrr4</td>
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| - | <td>0.63</td>
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| - | <td> There is only 1 copy of Pqrr4 present in each cell . since in the reaction equations Pqrr4 is shared between 2 other equations we decided to break the concentration of Pqrr4 between 3 species: sigma54:LuxO:p:Pqrr4 , Pqrr4 , sigma54:Pqrr4 . The initial values of the three species add up to one. The fractions of the Pqrr4 combination species are weighted differently . Since the Pqrr4 promotor stays on most of the time we decided the sigma54:LuxO:p:Pqrr4 complex should recieve the most weight. Pqrr4 is assumed to stay unbound from any complex for the least amount of time therefore Pqrr4 initial amount is the smallest. <br><br></td>
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| - | </tr>
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| - | <tr>
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| - | <td>Sigma54:Pqrr4</td>
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| - | <td>0.345</td>
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| - | </tr>
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| - | <tr>
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| - | <td>Pqrr4</td>
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| - | <td>0.025</td>
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| - | </tr>
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| - | <tr>
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| - | <td>GFP</td>
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| - | <td> 0</td>
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| - | <td> The model assumes that initially we have no GFP present . The simulation is allowed to run till the protein reaches equilibrium . The AI-2 is added after equilibrium conditions and a drop in GFP levels is observed.<br><br></td>
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| - | </tr>
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| - | <tr>
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| - | <td>mRNA</td>
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| - | <td>0</td>
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| - | <td>The justification for the initial value of mRNA is the same as GFP<br><br></td>
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| - | </tr>
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| - | </table>
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| - | <br>
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| - | <center><b> Table: The Kinetic Rate Constant Values</b> </center>
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| - | <br>
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| - | <table width="200" border="1" bgcolor="#414141" align = "center">
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| - | <tr>
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| - | <td>Rate Constants</td>
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| - | <td>Constant Value</td>
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| - | <td>Rationale</td>
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| - | </tr>
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| - | <tr>
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| - | <td> kPhosU</td>
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| - | <td> </td>
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| - | <td> </td>
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| - | </tr>
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| - | <tr>
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| - | <td>kPhosO</td>
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| - | <td> </td>
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| - | <td> </td>
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| - | </tr>
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| - | <tr>
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| - | <td>kTranscription</td>
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| - | <td> </td>
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| - | <td> </td>
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| - | </tr>
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| - |
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| - | <tr>
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| - | <td>kTranslation</td>
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| - | <td> </td>
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| - | <td> </td>
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| - | </tr>
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| - | <tr>
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| - | <td>kProtDegrad</td>
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| - | <td> </td>
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| - | <td> </td>
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| - | </tr>
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| - | <tr>
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| - | <td> kAI2bind </td>
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| - | <td> </td>
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| - | <td> </td>
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| - | </tr>
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| - | <tr>
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| - | <td>kAI2unbind</td>
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| - | <td> </td>
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| - | <td> </td>
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| - | </tr>
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| - | <tr>
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| - | <td>kPQphosphatase</td>
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| - | <td> </td>
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| - | <td> </td>
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| - | </tr>
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| - | <tr>
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| - | <td>kNSPU</td>
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| - | <td> </td>
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| - | <td> </td>
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| - | </tr>
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| - | <tr>
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| - | <td>kNSPO</td>
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| - | <td> </td>
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| - | <td> </td>
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| - | </tr>
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| - | <tr>
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| - | <td>kPqrr4Sig54unbind</td>
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| - | <td> </td>
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| - | <td> </td>
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| - | </tr>
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| - | <tr>
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| - | <td>kPqrr4Sig54bind</td>
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| - | <td> </td>
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| - | <td> </td>
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| - | </tr>
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| - | <tr>
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| - | <td>kOPqrr4Unbind</td>
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| - | <td> </td>
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| - | <td> </td>
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| - | </tr>
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| - | <tr>
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| - | <td>kOPqrr4bind</td>
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| - | <td> </td>
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| - | <td> </td>
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| - | </tr>
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| - | <tr>
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| - | <td>kRNAdegrad</td>
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| - | <td> </td>
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| - | <td> </td>
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| - | </tr>
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| - | </table>
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| | <br> | | <br> |
"
