Difference between revisions of "Team:Oxford/Modeling"

Line 29: Line 29:
 
                                 We have decided to use an arabinose induced promoter for the expression of a number of our proteins. This promoter can be modelled as the following chemical system:
 
                                 We have decided to use an arabinose induced promoter for the expression of a number of our proteins. This promoter can be modelled as the following chemical system:
 
                             </p>
 
                             </p>
−
                            \[\overset{\alpha_{1}}{\rightarrow}Arab\quad\overset{\alpha_{5}}{\rightarrow}AraC\]
 
  
−
                             \[Arab+AraC\mathrel{\mathop{\rightleftharpoons}^{\mathrm{\alpha_{2}}}_{\mathrm{\alpha_{3}}}}Arab:AraC\]
+
                             \[[Arab:AraC]\overset{K}{\rightarrow}mRNA\overset{\alpha{\rightarrow}P\]
−
 
+
                             \[mRNA\overset{\gamma_{1}}{\rightarrow}\phi\quad P\overset{\gamma_{2}}{\rightarrow}\phi\]
−
                            \[\overset{K}{\rightarrow}mRNA\overset{\alpha_{4}}{\rightarrow}P\]
+
−
 
+
−
                             \[Arab\overset{\gamma_{1}}{\rightarrow}\phi\quad AraC\overset{\gamma_{2}}{\rightarrow}\phi\quad mRNA\overset{\gamma_{3}}{\rightarrow}\phi\quad P\overset{\gamma_{4}}{\rightarrow}\phi\]
+
 
                             <p>
 
                             <p>
−
                                 You can find out more information about how this promoter works here.
+
                                 You can find out more information about how this promoter works here. For this system we will assume that AraC is always in large concentration and that it's binding to arabinose happens on a faster time scale to transcription. Therefore, we do not need to consider the individual concentrations of arabinose and AraC, instead we just need to include the concentration of the complex [Arab:AraC].
 
                             </p>
 
                             </p>
 
                             <p>
 
                             <p>
−
                                 From this system of chemical reactions we can derive the following set of ODEs[<a href="#Ref4">4</a>]:
+
                                 Using this approximation, we arrive at the equations:
 
                             </p>
 
                             </p>
−
                                \[\dfrac{d\left[Arab\right]}{dt}=\alpha_{1}+\alpha_{2}\left[Arab\right]\left[AraC\right]-\alpha_{3}\left[Arab:AraC\right]-\gamma_{1}\left[Arab\right]\]
 
  
−
                                 \[\dfrac{d\left[Arab:AraC\right]}{dt}=\alpha_{3}\left[Arab:AraC\right]-\alpha_{2}\left[Arab\right]\left[AraC\right]\]
+
                                 \[\dfrac{d[mRNA]}{dt}=K_{max}\dfrac{[Arab:AraC]^{n}}{K_{half}^{n}+[Arab:AraC]^{n}}-\gamma_{1}[mRNA]\]
  
−
                                \[\dfrac{d\left[AraC\right]}{dt}=\alpha_{5}-\alpha_{2}\left[Arab\right]\left[AraC\right]+\alpha_{3}\left[Arab:AraC\right]-\gamma_{2}\left[AraC\right]\]
+
                                 \[\dfrac{d\left[P\right]}{dt}=\alpha\left[mRNA\right]-\gamma_{2}\left[P\right]\]
−
 
+
−
                                \[\dfrac{d[mRNA]}{dt}=K_{max}\dfrac{[Arab:AraC]^{n}}{K_{half}^{n}+[Arab:AraC]^{n}}-\gamma_{3}[mRNA]\]
+
−
 
+
−
                                 \[\dfrac{d\left[P\right]}{dt}=\alpha_{4}\left[mRNA\right]-\gamma_{4}\left[P\right]\]
+
 
                             <p>
 
                             <p>
 
                                 Where we define the symbols as:
 
                                 Where we define the symbols as:
Line 60: Line 51:
 
                                         <th>Fitted</th>
 
                                         <th>Fitted</th>
 
                                     </thead>
 
                                     </thead>
−
                                    <tr>
 
−
                                        <td>\([Arab]\)</td>
 
−
                                        <td>The concentration of Arabinose</td>
 
−
                                        <td>\(1\times10^{-5}M\)</td>
 
−
                                        <td>-</td>
 
−
                                    </tr>
 
−
                                    <tr>
 
−
                                        <td>\([AraC]\)</td>
 
−
                                        <td>The concentration of AraC</td>
 
−
                                        <td>\(1\times10^{-5}M\)</td>
 
−
                                        <td>-</td>
 
−
                                    </tr>
 
 
                                     <tr>
 
                                     <tr>
 
                                         <td>\([Arab:AraC]\)</td>
 
                                         <td>\([Arab:AraC]\)</td>
Line 91: Line 70:
 
                                     </tr>
 
                                     </tr>
 
                                     <tr>
 
                                     <tr>
−
                                         <td>\(\alpha_{1}\)</td>
+
                                         <td>\(\alpha\)</td>
−
                                        <td>Basal production of Arabinose</td>
+
−
                                        <td>???</td>
+
−
                                        <td>?</td>
+
−
                                    </tr>
+
−
                                    <tr>
+
−
                                        <td>\(\alpha_{2}\)</td>
+
−
                                        <td>Association constant</td>
+
−
                                        <td>\(2.8\times10^{7}s^{-1}\) [<a href="#Ref7">7</a>]</td>
+
−
                                        <td>?</td>
+
−
                                    </tr>
+
−
                                    <tr>
+
−
                                        <td>\(\alpha_{3}\)</td>
+
−
                                        <td>Dissociation constant</td>
+
−
                                        <td>\(0.022s^{-1}\) [<a href="#Ref7">7</a>]</td>
+
−
                                        <td>?</td>
+
−
                                    </tr>
+
−
                                    <tr>
+
−
                                        <td>\(\alpha_{4}\)</td>
+
 
                                         <td>Translation rate</td>
 
                                         <td>Translation rate</td>
 
                                         <td>\(15ntd\: s^{-1}\)/length of sequence [<a href="#Ref6">6</a>]</td>
 
                                         <td>\(15ntd\: s^{-1}\)/length of sequence [<a href="#Ref6">6</a>]</td>
−
                                        <td>?</td>
 
−
                                    </tr>
 
−
                                    <tr>
 
−
                                        <td>\(\alpha_{5}\)</td>
 
−
                                        <td>Basal production of AraC</td>
 
−
                                        <td>???</td>
 
 
                                         <td>?</td>
 
                                         <td>?</td>
 
                                     </tr>
 
                                     </tr>
 
                                     <tr>
 
                                     <tr>
 
                                         <td>\(\gamma_{1}\)</td>
 
                                         <td>\(\gamma_{1}\)</td>
−
                                        <td> Degradation rate of Arabinose </td>
 
−
                                        <td>\(5.13\times10^{-4}s^{-1}\) [<a href="#Ref5">5</a>]</td>
 
−
                                        <td>?</td>
 
−
                                    </tr>
 
−
                                    <tr>
 
−
                                        <td>\(\gamma_{2}\)</td>
 
−
                                        <td>Degradation rate of AraC</td>
 
−
                                        <td>\(5.13\times10^{-4}s^{-1}\) [<a href="#Ref5">5</a>]</td>
 
−
                                        <td>?</td>
 
−
                                    </tr>
 
−
                                    <tr>
 
−
                                        <td>\(\gamma_{3}\)</td>
 
 
                                         <td>Degradation rate of mRNA</td>
 
                                         <td>Degradation rate of mRNA</td>
 
                                         <td>\(5.13\times10^{-4}s^{-1}\) [<a href="#Ref5">5</a>]</td>
 
                                         <td>\(5.13\times10^{-4}s^{-1}\) [<a href="#Ref5">5</a>]</td>
Line 139: Line 82:
 
                                     </tr>
 
                                     </tr>
 
                                     <tr>
 
                                     <tr>
−
                                         <td>\(\gamma_{4}\)</td>
+
                                         <td>\(\gamma_{2}\)</td>
 
                                         <td>Degradation rate of product</td>
 
                                         <td>Degradation rate of product</td>
 
                                         <td>\(5.13\times10^{-4}s^{-1}\) [<a href="#Ref5">5</a>]</td>
 
                                         <td>\(5.13\times10^{-4}s^{-1}\) [<a href="#Ref5">5</a>]</td>
Line 271: Line 214:
 
                             </table>
 
                             </table>
 
                             <p>
 
                             <p>
−
                                 We now can run our model of the system by solving the set of equations using the MATLAB equation solver ode15s. Below is a plot of the concentration of product against time for each protein expressed with this inducer-promoter pair:
+
                                 We now can run our model of the system by solving the set of equations using the MATLAB equation solver ode15s. Below is a plot of the concentration of product against time for each protein expressed with this inducer-promoter pair where the expression is induced by a step function:
 
                             </p>
 
                             </p>
 
                             <div class="image image-full">
 
                             <div class="image image-full">
Line 279: Line 222:
 
                     </div>
 
                     </div>
 
                     <div class="section-spacer"></div>
 
                     <div class="section-spacer"></div>
−
                     <div class="section" id="interaction-with-environment">
+
                     <div class="section" id="delivery">
−
                         <h2>Interaction with the Environment</h2>
+
                         <h2>Delivery</h2>
 
                         <p>
 
                         <p>
 
                             With the information about the rates of production and concentrations of our products we can look at how the products behave once they leave the cell. This involves modelling the diffusion of the products in different topologies, each associated with a potential physical design of the catheter. With this information we can provide a better estimate of the time scale that our project is working on and assess any need for optimisation.
 
                             With the information about the rates of production and concentrations of our products we can look at how the products behave once they leave the cell. This involves modelling the diffusion of the products in different topologies, each associated with a potential physical design of the catheter. With this information we can provide a better estimate of the time scale that our project is working on and assess any need for optimisation.
 
                         </p>
 
                         </p>
−
                         <p>
+
                         <div id="delivery-dispersin">
−
                            In addition, one of our systems relies on the detection of the biofilm to cause lysis. We will look at the how the quorum sensing signal moves to the cells.
+
                            <h3>Dispersin B</h3>
−
                        </p>
+
                            <p>
−
                    </div>
+
                                Dispersin B is one of the anti-biofilm agents we are using in our project and will be the focus of this delivery section. As such we will assume that conclusions reached apply to all of our enzymes.
−
                    <div class="section-spacer"></div>
+
                            </p>
−
                    <div class="section" id="interaction-with-biofilm">
+
                            <p>
−
                        <h2>Interaction with the Biofilm</h2>
+
                                A concentration of Dispersin B of 60μg/ml is required to destroy a biofilm that has already formed on a surface. This equates to a concentration of 1.50μM. This is higher than the steady-state gene expression concentration we can expect from our cells, meaning that our system cannot rely solely on diffusion to transport our enzymes to the biofilm. We will therefore model these diffusion systems assuming that our cells are expressing at a 2μM concentration and later we will look at optimising the gene expression to this level.
−
                        <p>
+
                            </p>
−
                            Once the antibiofilm agents have arrived at the biofilm, what impact do they have? We will look into the required concentrations and rates of the antibiofilm agents at the biofilm to overcome its rate of growth.
+
−
                        </p>
+
−
                    </div>
+
−
                    <div class="section-spacer"></div>
+
−
                    <div class="section" id="overall">
+
−
                        <h2>Combined: What do we know about the system as a whole?</h2>
+
−
                        <p>
+
−
                            What we write here depends on the results of the previous section.
+
−
                        </p>
+
−
                        <div id="overall-time">
+
−
                            <h3>How quickly will the system run?</h3>
+
 
                         </div>
 
                         </div>
−
                         <div id="overall-quantity">
+
                         <div id="delivery-beads">
−
                             <h3>How many engineered bacteria do we need?</h3>
+
                             <h3>Beads</h3>
−
                        </div>
+
                            <div id="delivery-beads-diffusion">
−
                        <div id="overall-concentration">
+
                                <h4>Diffusion</h4>
−
                            <h3>What is the minimum steady state concentration we need to reach to overcome the growth rate of the biofilm?</h3>
+
                                <p>
 +
                                    The bead delivery system consists of our cells being contained in alginate spheres. Water is passed through the container filled with the beads allowing our enzymes to diffuse from the alginate to the required concentration. More details about the design of the system can be found here.
 +
                                </p>
 +
                                <p>
 +
                                    To determine the convection mass transfer coefficient of Dispersin B from our gel spheres we looked at the diffusion data obtained from <a href="https://2015.igem.org/Team:Oxford/Beads#03-09-2015">this experiment</a> involving the diffusion of crystal violet from our beads. By analysing the system we can produce a theoretical form for the concentration of crystal violet in the bulk water as a function of time:
 +
                                </p>
 +
 
 +
                                \[c_{f}=\dfrac{c_{bo}}{1+\frac{V_{f}}{V_{b}}}\left(1-\exp\left(\dfrac{-K_{m}A_{b}\left(1+\frac{V_{f}}{V_{b}}\right)t}{V_{f}}\right)\right)\]
 +
                                <table class="table table-striped">
 +
                                    <thead>
 +
                                        <th>Symbol</th>
 +
                                        <th>Definition</th>
 +
                                        <th>Value</th>
 +
                                        <th>Units</th>
 +
                                    </thead>
 +
                                    <tr>
 +
                                        <td>\(A_{b}\)</td>
 +
                                        <td>Total surface area of the beads</td>
 +
                                        <td>\(0.0238\)</td>
 +
                                        <td>\(m^{2}\)</td>
 +
                                    </tr>
 +
                                    <tr>
 +
                                        <td>\(V_{b}\)</td>
 +
                                        <td>Total volume of beads</td>
 +
                                        <td>\(1.3463\times10^{-5}\)</td>
 +
                                        <td>\(m^{3}\)</td>
 +
                                    </tr>
 +
                                    <tr>
 +
                                        <td>\(c_{bo}\)</td>
 +
                                        <td>Initial concentration in beads</td>
 +
                                        <td>\(0.02451107\)</td>
 +
                                        <td>\(M\)</td>
 +
                                    </tr>
 +
                                    <tr>
 +
                                        <td>\(V_{f}\)</td>
 +
                                        <td>Volume of fluid surrounding the beads</td>
 +
                                        <td>\(V_{f}=V_{fo}-\dfrac{1\times10^{-6}}{10}t\)</td>
 +
                                        <td>\(m^{3}\)</td>
 +
                                    </tr>
 +
                                    <tr>
 +
                                        <td>\(V_{fo}\)</td>
 +
                                        <td>Initial volume of fluid surrounding the beads</td>
 +
                                        <td>\(1\times10^{-4}\)</td>
 +
                                        <td>\(m^{3}\)</td>
 +
                                    </tr>
 +
                                    <tr>
 +
                                        <td>\(t\)</td>
 +
                                        <td>Time</td>
 +
                                        <td>\(-\)</td>
 +
                                        <td>\(min\)</td>
 +
                                    </tr>
 +
                                    <tr>
 +
                                        <td>\(c_{f}\)</td>
 +
                                        <td>Concentration of fluid surrounding beads</td>
 +
                                        <td>\(-\)</td>
 +
                                        <td>\(M\)</td>
 +
                                    </tr>
 +
                                    <tr>
 +
                                        <td>\(K_{m}\)</td>
 +
                                        <td>Convection mass diffusion coefficient</td>
 +
                                        <td>To be fitted</td>
 +
                                        <td>\(mmin^{-1}\)</td>
 +
                                    </tr>
 +
                                </table>
 +
                                <p>
 +
                                    The volume of fluid is also a function of time in order to account for the removal of 1ml of water every 10 minutes. The area and volume of the beads is that of 660 spheres with diameter 3.39mm.
 +
                                </p>
 +
                                <p>
 +
                                    However, the number of beads is an estimate. Because of this, in order to fit the curve to the experimental data we must scale the experimental data by an unknown factor. Therefore we preface our equation with an arbitrary scaling factor which, along with the convection diffusion coefficient - \(Km\), is determined by our fitting function.
 +
                                </p>
 +
                                <p>
 +
                                    Our fitting script, detailed here, returned the value of \(K_{m} = 1.7265\times 10^{-5} mmin^{-1}\).
 +
                                </p>
 +
                                <div class="image image-full">
 +
                                    <img src="https://static.igem.org/mediawiki/2015/b/b7/Ox_fitteddiffusion.png"/>
 +
                                    <p>
 +
                                        Our fitted curve plotted with the experimental data
 +
                                    </p>
 +
                                </div>
 +
                                <p>
 +
                                    Dispersin B is a significantly larger molecule than crystal violet so this diffusion coefficient will not be close to that for Dispersin B. To correct this we need to make use of similarity. More specifically we take the Sherwood Numbers of the systems to be equal therefore:
 +
                                </p>
 +
 
 +
                                \[\left(\dfrac{K_{m}R}{D}\right)_{crystal violet} = \left(\dfrac{K_{m}R}{D}\right)_{Dispersin B}\]
 +
 
 +
                                <table class="table table-striped">
 +
                                    <thead>
 +
                                        <th>Symbol</th>
 +
                                        <th>Definition</th>
 +
                                        <th>Value</th>
 +
                                        <th>Units</th>
 +
                                    </thead>
 +
                                    <tr>
 +
                                        <td>
 +
                                            \(D_{crystal violet}\)
 +
                                        </td>
 +
                                        <td>
 +
                                            Mass diffusivity of crystal violet in water
 +
                                        </td>
 +
                                        <td>
 +
                                            \(2.8652\times10^{9}\)
 +
                                        </td>
 +
                                        <td>
 +
                                            \(\mu m^{2}s^{-1}\)
 +
                                        </td>
 +
                                    </tr>
 +
                                    <tr>
 +
                                        <td>
 +
                                            \(D_{Dispersin B}\)
 +
                                        </td>
 +
                                        <td>
 +
                                            Mass diffusivity of Dispersin B in water
 +
                                        </td>
 +
                                        <td>
 +
                                            \(100\)
 +
                                        </td>
 +
                                        <td>
 +
                                            \(\mu m^{2}s^{-1}\)
 +
                                        </td>
 +
                                    </tr>
 +
                                    <tr>
 +
                                        <td>
 +
                                            \(R\)
 +
                                        </td>
 +
                                        <td>
 +
                                            Radius of bead
 +
                                        </td>
 +
                                        <td>
 +
                                            \(1.695\)
 +
                                        </td>
 +
                                        <td>
 +
                                            \(mm\)
 +
                                        </td>
 +
                                    </tr>
 +
                                </table>
 +
                                <p>
 +
                                    By rearranging this we arrive at \(\left(K_{m}\right)_{DispersinB} = 6.03\times10^{-13} mmin^{-1}\)
 +
                                </p>
 +
                            </div>
 +
                            <div id="delivery-beads-mass-exchange">
 +
                                <h4>Mass Exchange</h4>
 +
                                <p>
 +
                                    This result allows us to theorise a mass exchange system. As a first estimate we will assume that the flow through the beads is sufficiently slow to use the convection diffusion coefficient found above. It is also assumed that the gene expression happens on a faster time scale than the diffusion from the beads to the water, enabling us to assume the concentration of enzyme in the beads remains constant. This is supported by our gene expression models. We can now visualize how the concentrations of the fluid will vary with distance along the mass exchanger:
 +
                                </p>
 +
                                <p>
 +
                                    The overall system can now be described with the equation:
 +
                                </p>
 +
 
 +
                                \[J = K_mA\dfrac{c_{fo}-c_{fi}}{\ln\left(\dfrac{c_{B}-c_{fi}}{c_{B}-c_{fo}}\right)}\]
 +
                                <p>
 +
                                    Therefore
 +
                                </p>
 +
 
 +
                                \[A = J\dfrac{\ln\left(\dfrac{c_{B}-c_{fi}}{c_{B}-c_{fo}}\right)}{K_{m}\left(c_{fo}-c_{fi}\right)}\]
 +
                                <p>
 +
                                    Where \(J=Q\left(c_{fo}-c_{fi}\right)\) and \(Q\) is the volume flow rate of water. We have chosen a flow rate range of 10-100ml/min as this is accepted as a safe artificial bladder fill rate. This range results in the following number of beads required to reach the desired concentration:
 +
                                </p>
 +
                                <div class="image image-full">
 +
                                    <img src="https://static.igem.org/mediawiki/2015/1/18/Ox_numbervsflow.png"/>
 +
                                    <p>
 +
                                        The number of beads required against the flow rate of water through the mass exchange section
 +
                                    </p>
 +
                                </div>
 +
                                <p>
 +
                                    Therefore a volume of between \(20.3-203m^3\) of beads is required, assuming a packing efficiency of 64%.
 +
                                </p>
 +
                                <p>
 +
                                    However, as stated earlier this estimation relies on the fluid flowing around the beads is slow enough to be approximated as stationary, meaning that mass transfer occurs as natural convection. Although there may be a very large volume of beads and a slow fluid flow rate, the area through which the fluid can flow is likely small enough that the velocity of the fluid is non-negligable.
 +
                                </p>
 +
                            </div>
 
                         </div>
 
                         </div>
 
                     </div>
 
                     </div>
Line 330: Line 431:
 
                     </li>
 
                     </li>
 
                     <li>
 
                     <li>
−
                         <a href="#interaction-with-environment">Interaction with the Environment</a>
+
                         <a href="#delivery">Delivery</a>
−
                    </li>
+
−
                    <li>
+
−
                        <a href="#interaction-with-biofilm">Interaction with the Biofilm</a>
+
−
                    </li>
+
−
                    <li>
+
−
                        <a href="#overall">Combined: What do we know about the system as a whole?</a>
+
 
                         <ul class="nav nav-stacked">
 
                         <ul class="nav nav-stacked">
−
                             <li><a href="#overall-time">How quickly will the system run?</a></li>
+
                             <li><a href="#delivery-dispersin">Dispersin B</a></li>
−
                             <li><a href="#overall-quantity">How many engineered bacteria do we need?</a></li>
+
                             <li>
−
                            <li><a href="#overall-concentration">What is the minimum steady state concentration we need to reach to overcome the growth rate of the biofilm?</a></li>
+
                                <a href="#delivery-beads">Beads</a>
 +
                                <ul class="nav nav-stacked">
 +
                                    <li><a href="#delivery-beads-diffusion">Diffusion</a></li>
 +
                                    <li><a href="#delivery-beads-mass-exchange">Mass Exchanger</a></li>
 +
                                </ul>
 +
                            </li>
 
                         </ul>
 
                         </ul>
 
                     </li>
 
                     </li>

Revision as of 10:43, 15 September 2015

Modelling

Introduction

Mathematical modelling plays a crucial role in Synthetic Biology by acting as a link between the conception and the physical realisation of a biological circuit. Our modelling team has focussed on building a better picture of the project to evaluate the effectiveness of initial designs, as well as to provide insight into how the system can (or must) be improved. To do this we have split our modelling efforts into three main sections: Characterising our Cells, Interaction with the Environment, and Interaction with the Biofilm. By combining the information gathered in each of these section we hope to ultimately answer the question: Is our system feasible? If not, where should the design be altered?

To help readers of all kinds and specialisations understand this page we have produced guides for all the modelling techniques used in this section which are available on the Modelling Tutorial page and will be linked to when relevant on this page.

Characterising Our Cells

In this section we look at our cells in isolation in order to assess their functionality and answer important questions such as “How long does it take to produce a certain concentration of product?”, and “What are the main limiting rates/concentrations?”. The first will help us assess the feasibility of our project, ie are our cells too slow? The second will aid us in further optimising our design.

Arabinose Induced

We have decided to use an arabinose induced promoter for the expression of a number of our proteins. This promoter can be modelled as the following chemical system:

\[[Arab:AraC]\overset{K}{\rightarrow}mRNA\overset{\alpha{\rightarrow}P\] \[mRNA\overset{\gamma_{1}}{\rightarrow}\phi\quad P\overset{\gamma_{2}}{\rightarrow}\phi\]

You can find out more information about how this promoter works here. For this system we will assume that AraC is always in large concentration and that it's binding to arabinose happens on a faster time scale to transcription. Therefore, we do not need to consider the individual concentrations of arabinose and AraC, instead we just need to include the concentration of the complex [Arab:AraC].

Using this approximation, we arrive at the equations:

\[\dfrac{d[mRNA]}{dt}=K_{max}\dfrac{[Arab:AraC]^{n}}{K_{half}^{n}+[Arab:AraC]^{n}}-\gamma_{1}[mRNA]\] \[\dfrac{d\left[P\right]}{dt}=\alpha\left[mRNA\right]-\gamma_{2}\left[P\right]\]

Where we define the symbols as:

Symbol Definition Initial Value/Literature Value Fitted
\([Arab:AraC]\) The concentration of associated Arabinose and AraC \(0\) -
\([mRNA]\) The concentration of mRNA \(0\) -
\([P]\) The concentration of our product \(0\) -
\(\alpha\) Translation rate \(15ntd\: s^{-1}\)/length of sequence [6] ?
\(\gamma_{1}\) Degradation rate of mRNA \(5.13\times10^{-4}s^{-1}\) [5] ?
\(\gamma_{2}\) Degradation rate of product \(5.13\times10^{-4}s^{-1}\) [5] ?
\(K_{max}\) Maximal transcription rate \(50ntd\: s^{-1}\)/length of sequence [6] ?
\(K_{half}\) Half-maximal transcription rate \(160\mu M\) [8] ?
\(n\) Hill coefficient \(2.65\) [3] ?

This table contains literature values for the parameters, found from a number of sources. Later we will fit the parameters to the experimental data found by the wet lab team. For now though we can plot the expression graphs using the literature values. This will provide an estimate to the time scales involved.

There are mutliple products being expressed using this inducer-promoter pair, each of different sequence lenghts. Here is a table showing the relevant proteins and sequence lengths:

Product Sequence Length (/bp)
pBAD HisB DNase DsbA 621
pBAD HisB DspB YebF
pBAD HisB DspB
pBAD HisB MccS 414
pBAD HisB DspB Fla
pBAD HisB Art-175 DsbA 987
pBAD HisB Art-175 YebF 1284
pBAD HisB Art-E 632
pBAD HisB Art-175 Fla 1095
pBAD HisB Art-175 936
pBAD HisB DNase 570

We now can run our model of the system by solving the set of equations using the MATLAB equation solver ode15s. Below is a plot of the concentration of product against time for each protein expressed with this inducer-promoter pair where the expression is induced by a step function:

Delivery

With the information about the rates of production and concentrations of our products we can look at how the products behave once they leave the cell. This involves modelling the diffusion of the products in different topologies, each associated with a potential physical design of the catheter. With this information we can provide a better estimate of the time scale that our project is working on and assess any need for optimisation.

Dispersin B

Dispersin B is one of the anti-biofilm agents we are using in our project and will be the focus of this delivery section. As such we will assume that conclusions reached apply to all of our enzymes.

A concentration of Dispersin B of 60μg/ml is required to destroy a biofilm that has already formed on a surface. This equates to a concentration of 1.50μM. This is higher than the steady-state gene expression concentration we can expect from our cells, meaning that our system cannot rely solely on diffusion to transport our enzymes to the biofilm. We will therefore model these diffusion systems assuming that our cells are expressing at a 2μM concentration and later we will look at optimising the gene expression to this level.

Beads

Diffusion

The bead delivery system consists of our cells being contained in alginate spheres. Water is passed through the container filled with the beads allowing our enzymes to diffuse from the alginate to the required concentration. More details about the design of the system can be found here.

To determine the convection mass transfer coefficient of Dispersin B from our gel spheres we looked at the diffusion data obtained from this experiment involving the diffusion of crystal violet from our beads. By analysing the system we can produce a theoretical form for the concentration of crystal violet in the bulk water as a function of time:

\[c_{f}=\dfrac{c_{bo}}{1+\frac{V_{f}}{V_{b}}}\left(1-\exp\left(\dfrac{-K_{m}A_{b}\left(1+\frac{V_{f}}{V_{b}}\right)t}{V_{f}}\right)\right)\]
Symbol Definition Value Units
\(A_{b}\) Total surface area of the beads \(0.0238\) \(m^{2}\)
\(V_{b}\) Total volume of beads \(1.3463\times10^{-5}\) \(m^{3}\)
\(c_{bo}\) Initial concentration in beads \(0.02451107\) \(M\)
\(V_{f}\) Volume of fluid surrounding the beads \(V_{f}=V_{fo}-\dfrac{1\times10^{-6}}{10}t\) \(m^{3}\)
\(V_{fo}\) Initial volume of fluid surrounding the beads \(1\times10^{-4}\) \(m^{3}\)
\(t\) Time \(-\) \(min\)
\(c_{f}\) Concentration of fluid surrounding beads \(-\) \(M\)
\(K_{m}\) Convection mass diffusion coefficient To be fitted \(mmin^{-1}\)

The volume of fluid is also a function of time in order to account for the removal of 1ml of water every 10 minutes. The area and volume of the beads is that of 660 spheres with diameter 3.39mm.

However, the number of beads is an estimate. Because of this, in order to fit the curve to the experimental data we must scale the experimental data by an unknown factor. Therefore we preface our equation with an arbitrary scaling factor which, along with the convection diffusion coefficient - \(Km\), is determined by our fitting function.

Our fitting script, detailed here, returned the value of \(K_{m} = 1.7265\times 10^{-5} mmin^{-1}\).

Our fitted curve plotted with the experimental data

Dispersin B is a significantly larger molecule than crystal violet so this diffusion coefficient will not be close to that for Dispersin B. To correct this we need to make use of similarity. More specifically we take the Sherwood Numbers of the systems to be equal therefore:

\[\left(\dfrac{K_{m}R}{D}\right)_{crystal violet} = \left(\dfrac{K_{m}R}{D}\right)_{Dispersin B}\]
Symbol Definition Value Units
\(D_{crystal violet}\) Mass diffusivity of crystal violet in water \(2.8652\times10^{9}\) \(\mu m^{2}s^{-1}\)
\(D_{Dispersin B}\) Mass diffusivity of Dispersin B in water \(100\) \(\mu m^{2}s^{-1}\)
\(R\) Radius of bead \(1.695\) \(mm\)

By rearranging this we arrive at \(\left(K_{m}\right)_{DispersinB} = 6.03\times10^{-13} mmin^{-1}\)

Mass Exchange

This result allows us to theorise a mass exchange system. As a first estimate we will assume that the flow through the beads is sufficiently slow to use the convection diffusion coefficient found above. It is also assumed that the gene expression happens on a faster time scale than the diffusion from the beads to the water, enabling us to assume the concentration of enzyme in the beads remains constant. This is supported by our gene expression models. We can now visualize how the concentrations of the fluid will vary with distance along the mass exchanger:

The overall system can now be described with the equation:

\[J = K_mA\dfrac{c_{fo}-c_{fi}}{\ln\left(\dfrac{c_{B}-c_{fi}}{c_{B}-c_{fo}}\right)}\]

Therefore

\[A = J\dfrac{\ln\left(\dfrac{c_{B}-c_{fi}}{c_{B}-c_{fo}}\right)}{K_{m}\left(c_{fo}-c_{fi}\right)}\]

Where \(J=Q\left(c_{fo}-c_{fi}\right)\) and \(Q\) is the volume flow rate of water. We have chosen a flow rate range of 10-100ml/min as this is accepted as a safe artificial bladder fill rate. This range results in the following number of beads required to reach the desired concentration:

The number of beads required against the flow rate of water through the mass exchange section

Therefore a volume of between \(20.3-203m^3\) of beads is required, assuming a packing efficiency of 64%.

However, as stated earlier this estimation relies on the fluid flowing around the beads is slow enough to be approximated as stationary, meaning that mass transfer occurs as natural convection. Although there may be a very large volume of beads and a slow fluid flow rate, the area through which the fluid can flow is likely small enough that the velocity of the fluid is non-negligable.

References