Team:Imperial/CBD Kinetics Model

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                             <ul>
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                                 <li><a href="#introduction" class="smoothScroll">Introduction</a>
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                                 <li><a data-scroll href="#introduction" ">Introduction</a>
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                                 <li><a href="#maths" class="smoothScroll">Mathematical Background</a>
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                                 <li><a data-scroll href="#maths" >Mathematical Background</a>
                                 </li>
                                 </li>
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                                 <li><a href="#lab" class="smoothScroll">Wet-Lab Interaction</a>
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                                 <li><a data-scroll href="#lab" >Wet-Lab Interaction</a>
                                 </li>
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                                 <li><a href="#results" class="smoothScroll">Results & Conclusions</a>
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                                 <li><a data-scroll href="#results" >Results & Conclusions</a>
                                 </li>
                                 </li>
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                                <li><a href="#references" class="smoothScroll">References</a>
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                                <li><a data-scroll href="#references" >References</a>
                                 </li>
                                 </li>
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                                                <li><a href="#code" class="smoothScroll">Code</a>
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<li><a data-scroll href="#code" >Appendix: Code</a>
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                                 </li>
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                     <section id="introduction">
                     <section id="introduction">
                         <h2>Introduction</h2>
                         <h2>Introduction</h2>
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                         <p>Based on a set of ordinary differential equations describing the kinetics of cellulose-binding modules (CBM) binding to bacterial cellulose, the model is designed to predict the efficiency and affinity of the binding process. Given an initial concentration of CBM injected into the system, the model is enabled to calculate the time allowed for a certain percentage of protein binding sites on bacterial cellulose to be saturated.</p>
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                         <p>Based on a set of ordinary differential equations (ODEs) describing the kinetics of carbohydrate-binding domains (CBDs) attaching to bacterial cellulose (BC), the model is designed to predict the efficiency and affinity of the binding process. Given an initial concentration of CBD injected into the system, the model calculates the time necessary for a certain percentage of protein binding sites on BC to be saturated.</p>
 +
                        <p>With a better understanding of the interaction between CBD and BC, <a href="https://2014.igem.org/Team:Imperial/Functionalisation">wetlab experiments</a> can be informed about the functionalisation capability of BC and therefore would be able to approximate efficiency of the contaminants-capturing process.</p>
                     </section>
                     </section>
                     <section id="maths">
                     <section id="maths">
                         <h2>Mathematical Background</h2>
                         <h2>Mathematical Background</h2>
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                        <h3>Model I</h3>
 
                         <ul>
                         <ul>
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                             <li>Variables:
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                             <li>Dependent Variables:
                                 <ul>
                                 <ul>
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                                     <li>CBS(0): initial concentration of cellulose binding sites (CBS)</li>
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<li>CBS(t): concentration of unoccupied CBS at time t</li>
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                                     <li>CBS(t): concentration of unoccupied cellulose binding sites (CBSs) at time t</li>
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<li>CBM(0): initial concentration of CBM injected into the system</li>
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<li>CBM(t): concentration of unbound CBM</li>
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                                    <li>CBD(t): concentration of unbound CBD at time t</li>
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<li>CBS_CBM(t): concentration of binding complex (CBS bounded with CBM)</li>
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                                    <li>CBS_CBD(t): concentration of binding complex (CBS bounded with CBD) at time t</li>
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                                 </ul>
                                 </ul>
                             </li>
                             </li>
                             <li>Parameters:
                             <li>Parameters:
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                            <ul>
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                                <ul>
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                                <li>k<sub>on</sub>: association rate constant</li>
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                                    <li>CBS(0): initial concentration of CBS</li>
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<li>k<sub>off</sub>: dissociation rate constant</li>
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                                    <li>CBD(0): initial concentration of CBD injected into the system</li>
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                                <li>K<sub>D</sub> = k<sub>off</sub>/k<sub>on</sub>: equilibrium dissociation constant</li>
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                                    <li>k<sub>on</sub>: association rate constant</li>
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                                    <li>k<sub>off</sub>: dissociation rate constant</li>
 +
                                    <li>K<sub>D</sub> = k<sub>off</sub>/k<sub>on</sub>: equilibrium dissociation constant</li>
 +
 
 +
 
-
                               
 
                                 </ul>
                                 </ul>
                             </li>
                             </li>
                             <li>Binding Kinetics:
                             <li>Binding Kinetics:
                                 <ul>
                                 <ul>
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                                     <li><img class="content-image " src="https://static.igem.org/mediawiki/2014/8/89/IC14-modelling-kinetics-binding1.png" height=250></li>
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                                     <li>
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                                     <li>Assume CBM(t) = CBM(0) due to large concentration of CBM injected into the system compared to initial concentration of cellulose binding sites.</li>
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                                        <img class="content-image " src="https://static.igem.org/mediawiki/2014/8/89/IC14-modelling-kinetics-binding1.png" height=150>
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                                    <li>With the boundary conditions:
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                                    </li>
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                                    <ul>
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                                     <li>Assume that CBD(t) = CBD(0) since the initial concentration of CBD injected into the system is typically very large compared to the initial concentration of cellulose binding sites, i.e. CBS(0).</li>
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                                        <li>CBS_CBM(0) = 0</li>
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                            </ul>
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                                        <li>CBS_CBM(∞) = CBS(0)</li>
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                            </li>
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                                        </ul></li>
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                            <li>Initial and Boundary Conditions:
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                                    <li>The ODE can be solved as:
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                                <ul>
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                                    <ul>
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                                    <li>CBD(0) = M<sub>0</sub>, where M<sub>0</sub> is a value determined as per experiment.</li>
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                                        <li><img class="content-image " src="https://static.igem.org/mediawiki/2014/a/ac/IC14-modelling-kinetics-binding2.png" height=60></li>
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                                    <li>CBS_CBD(0) = 0</li>
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                                         <li>where k<sub>on</sub> CBM(0) &gt;&gt; k<sub>>off</sub</li>
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                                    <li>CBS_CBD(∞) = CBS(0), assume that BC binding sites can be fully saturated given enough time, as k<sub>on</sub>CBD(0) &gt;&gt; k<sub>off</sub>.</li>
 +
 
 +
 
 +
                                </ul>
 +
                            </li>
 +
                            <li>Solution:
 +
                                <ul>
 +
                                    <li>
 +
                                        <img class="content-image " src="https://static.igem.org/mediawiki/2014/a/ac/IC14-modelling-kinetics-binding2.png" height=60>
 +
                                    </li>
 +
                                         <li>where k<sub>on</sub> CBD(0) &gt;&gt; k<sub>off</sub></li>
                                         </ul></li>
                                         </ul></li>
                                  
                                  
                                  
                                  
                                 </ul>
                                 </ul>
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                             </li>
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                         </ul>
                         </ul>
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                     </section>
                     </section>
                     <section id="lab">
                     <section id="lab">
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                         <h2>Lab Interaction</h2>
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                         <h2>Interaction with wetlab experiments</h2>
                         <figure class="content-image image-right image-small">
                         <figure class="content-image image-right image-small">
                             <img class="image-full" src="https://static.igem.org/mediawiki/2014/8/8c/IC14-modelling-kinetics-synergy.png">
                             <img class="image-full" src="https://static.igem.org/mediawiki/2014/8/8c/IC14-modelling-kinetics-synergy.png">
                             <figcaption>How our model and the wetlab interact with each other</figcaption>
                             <figcaption>How our model and the wetlab interact with each other</figcaption>
                         </figure>
                         </figure>
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                         <p>Parameters in the previous expression are determined by assay experiments where k<sub>off</sub> and k<sub>d</sub> are found, from which k<sub>on</sub> is calculated: k<sub>on</sub>=k<sub>off</sub>/k<sub>d</sub>.</p>
+
                         <p>Parameters k<sub>off</sub> and k<sub>d</sub> are determined by wetlab assays (wash experiments), from which k<sub>on</sub> is calculated: k<sub>on</sub>=k<sub>off</sub>/k<sub>d</sub>.</p>
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                        <p>LAB RESULTS HERE
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-
</p>
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<p>With the parameters determined from the experiments, the model is then instantiated and used to predict the time period necessary for a certain percentage of the binding sites to be saturated. The validity of the model is <em>a posteriori</em> checked through further wetlab experiments (please refer to <a href="https://2014.igem.org/Team:Imperial/Functionalisation">CBD expression experiments</a> for further information).</p>
-
                       
+
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                        <p>With the parameters obtained from the experiments, the model is then enabled to predict an exact time period for the binding sites to be saturated, where the validity of the model is further confirmed by the experiments (refer to CBD expression experiments in the drive).
+
<p>The chart below illustrates how our model and the wetlab experiments interact with each other.</p>
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</p>
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-
                       
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-
                     
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-
                       
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                     </section>
                     </section>
                     <section id="results">
                     <section id="results">
                         <h2>Results and Conclusions</h2>
                         <h2>Results and Conclusions</h2>
-
                       
+
 
                         <figure class="content-image image-half">
                         <figure class="content-image image-half">
                             <img class="image-full" src="https://static.igem.org/mediawiki/2014/7/7c/IC14-modelling-kinetics-results1.png">
                             <img class="image-full" src="https://static.igem.org/mediawiki/2014/7/7c/IC14-modelling-kinetics-results1.png">
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                             <figcaption>Labelled points represent 70% saturation of bacterial cellulose binding sites.
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                             <figcaption>Figure 1: CBD-BC complex concentration against time
-
</figcaption>
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                            </figcaption>
                         </figure>
                         </figure>
                         <figure class="content-image image-half">
                         <figure class="content-image image-half">
                             <img class="image-full" src="https://static.igem.org/mediawiki/2014/8/81/IC14-modelling-kinetics-results2.png">
                             <img class="image-full" src="https://static.igem.org/mediawiki/2014/8/81/IC14-modelling-kinetics-results2.png">
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                             <figcaption>It could be observed from the graph that the CBM binding capacity reduces when initial CBM concentration increases beyond 15µM; therefore, an initial CBM concentration of 15 µM would be optimum to functionalise the bacterial cellulose.</figcaption>
+
                             <figcaption>Figure 2: Time for 90% saturation of BC binding sites against initial concentration of CBD</figcaption>
                         </figure>
                         </figure>
-
                          
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                         <p>It can be observed from the second graph that the time necessary for 90% CBS saturation decreases significantly when initial CBD concentration increases beyond 0.5 µM; however, a further increase in initial CBD concentration does not contribute much to effectively reduce the time necessary for 90% CBS saturation. Therefore, an initial CBD concentration between 0.5 µM and 1 µM will be considered as optimal to efficiently functionalise BC.</p>
 +
 
                     </section>
                     </section>
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                     <section id="references">
                     <section id="references">
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                         <h3>References</h3>
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                         <h2>References</h2>
                         <ol>
                         <ol>
                             <li>Zhang, Mengmeng; Wang, Bin; Xu, Bingqian Mapping Single Molecular Binding Kinetics of Carbohydrate-Binding Module with Crystalline Cellulose by Atomic Force Microscopy Recognition Imaging J. Phys. Chem. B 2014, 118, 6714−6720s
                             <li>Zhang, Mengmeng; Wang, Bin; Xu, Bingqian Mapping Single Molecular Binding Kinetics of Carbohydrate-Binding Module with Crystalline Cellulose by Atomic Force Microscopy Recognition Imaging J. Phys. Chem. B 2014, 118, 6714−6720s
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</li>
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                            </li>
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                         </ol>
                         </ol>
                     </section>
                     </section>
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<section id="code">
-
                    <section id="code">
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<h2>Appendix: Code</h2>
-
                        <h3>Code</h3>
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-
                   
+
<p>The code for this model can be found here: <a href="https://static.igem.org/mediawiki/2014/1/16/CBD_kinetics_model_imperial_iGEM_2014.m">CBD_kinetics_model_imperial_iGEM_2014.m</a></p>
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                    </section>
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</section>
 +
 
                 </div>
                 </div>

Latest revision as of 00:13, 18 October 2014

Imperial iGEM 2014

Aqualose

CBD Kinetics Model

Introduction

Based on a set of ordinary differential equations (ODEs) describing the kinetics of carbohydrate-binding domains (CBDs) attaching to bacterial cellulose (BC), the model is designed to predict the efficiency and affinity of the binding process. Given an initial concentration of CBD injected into the system, the model calculates the time necessary for a certain percentage of protein binding sites on BC to be saturated.

With a better understanding of the interaction between CBD and BC, wetlab experiments can be informed about the functionalisation capability of BC and therefore would be able to approximate efficiency of the contaminants-capturing process.

Mathematical Background

  • Dependent Variables:
    • CBS(t): concentration of unoccupied cellulose binding sites (CBSs) at time t
    • CBD(t): concentration of unbound CBD at time t
    • CBS_CBD(t): concentration of binding complex (CBS bounded with CBD) at time t
  • Parameters:
    • CBS(0): initial concentration of CBS
    • CBD(0): initial concentration of CBD injected into the system
    • kon: association rate constant
    • koff: dissociation rate constant
    • KD = koff/kon: equilibrium dissociation constant
  • Binding Kinetics:
    • Assume that CBD(t) = CBD(0) since the initial concentration of CBD injected into the system is typically very large compared to the initial concentration of cellulose binding sites, i.e. CBS(0).
  • Initial and Boundary Conditions:
    • CBD(0) = M0, where M0 is a value determined as per experiment.
    • CBS_CBD(0) = 0
    • CBS_CBD(∞) = CBS(0), assume that BC binding sites can be fully saturated given enough time, as konCBD(0) >> koff.
  • Solution:
    • where kon CBD(0) >> koff

Interaction with wetlab experiments

How our model and the wetlab interact with each other

Parameters koff and kd are determined by wetlab assays (wash experiments), from which kon is calculated: kon=koff/kd.

With the parameters determined from the experiments, the model is then instantiated and used to predict the time period necessary for a certain percentage of the binding sites to be saturated. The validity of the model is a posteriori checked through further wetlab experiments (please refer to CBD expression experiments for further information).

The chart below illustrates how our model and the wetlab experiments interact with each other.

Results and Conclusions

Figure 1: CBD-BC complex concentration against time
Figure 2: Time for 90% saturation of BC binding sites against initial concentration of CBD

It can be observed from the second graph that the time necessary for 90% CBS saturation decreases significantly when initial CBD concentration increases beyond 0.5 µM; however, a further increase in initial CBD concentration does not contribute much to effectively reduce the time necessary for 90% CBS saturation. Therefore, an initial CBD concentration between 0.5 µM and 1 µM will be considered as optimal to efficiently functionalise BC.

References

  1. Zhang, Mengmeng; Wang, Bin; Xu, Bingqian Mapping Single Molecular Binding Kinetics of Carbohydrate-Binding Module with Crystalline Cellulose by Atomic Force Microscopy Recognition Imaging J. Phys. Chem. B 2014, 118, 6714−6720s

Appendix: Code

The code for this model can be found here: CBD_kinetics_model_imperial_iGEM_2014.m

Retrieved from "http://2014.igem.org/Team:Imperial/CBD_Kinetics_Model"