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| − | <body>
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| − | <header>
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| − | <a class="anchor" id="top"></a>
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| − | <center>
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| − | <h1><highlight class="highlight">BioSpray</highlight></h1>
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| − | <h3><highlight class="highlight">Mathematical Modelling</highlight></h3>
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| − | </center>
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| − | </header>
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| − | <a class="anchor" id="overview"></a>
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| − | <section id="overview">
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| − | <div class="">
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| − | <img src="img/igem-sponsor.png" align="right" width="50%" height="auto" class="img-mockup">
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| − | </div>
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| − | <div class="col-lg-12 feature" style="">
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| − | <div class="row">
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| − | <h3>Overview</h3>
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| − | <p><font color="white"> The models for the BioSpray all follow a similar methodology. The law of mass action allows the description of chemical schematics or reaction pathways by equations. Ordinary differential equations (ODEs) were used to describe the binding reactions between the molecules in the BioSpray with their targets in the sample. Each ODE has one independent variable and its derivatives, describing the change of the variable over time. This allows for the investigation of the concentrations of substances left after binding has occurred, allowing for the analysis of the optimum concentration required in the BioSpray.</font></p>
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| − | </section>
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| − | <a class="anchor" id="selection"></a>
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| − | <section id="about" class="row1">
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| − | <div class="col-lg-3">
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| − | <a href="#blood" class="scroll"> <span class="glyphicon glyphicon-tint"></span></a>
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| − | <h3>Blood</h3>
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| − | <p class="about-content">Consider the binding between Haptoglobin and Haemoglobin.</p>
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| − | </div>
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| − | <div class="col-lg-3">
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| − | <a href="#semen"><span class="glyphicon glyphicon-tint" type="button"></span></a>
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| − | <h3>Semen</h3>
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| − | <p class="about-content">Consider the binding between Spermidine and PotD.</p>
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| − | </div>
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| − | <div class="col-lg-3">
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| − | <a href="#saliva"><span class="glyphicon glyphicon-tint" color="red"></span></a>
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| − | <h3>Saliva</h3>
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| − | <p class="about-content">Consider the binding between Lactoferrin and Lactoferrim Binding Protein.</p>
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| − | </div>
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| − | <div class="col-lg-3">
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| − | <a href="#nasal"><span class="glyphicon glyphicon-tint"></span></a>
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| − | <h3>Nasal Mucus</h3>
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| − | <p class="about-content">Consider the folding of the Oderant Binding Protein.</p>
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| − | </div>
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| − | </div>
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| − | </section>
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| − | <a class="anchor" id="blood"></a>
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| − | <section id="blood">
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| − | <h3>Blood: Haptoglobin and Haemoglobin Binding</h3>
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| − | <font color="white"><b>Aim</b><p>The aim of a model describing the binding between haptoglobin and haemoglobin is to find the optimum concentration and binding rates that we require for visual detection of haemoglobin in the sample from the crime scene. The more complex formed the more likely it will be that the haemoglobin will be visually detected using the BioSpray.</p><b>Results</b>
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| − | <p>Haemoglobin is a tetramer, with two \(\alpha\) chains and two \(\beta\) chains. Haptoglobin binds to haemoglobin in two stages. Firstly the haptoglobin binds to the \(\alpha\) chains of the haemeoglobin only. This first reaction is reversible and the complex can dissociate. The haptoglobin then binds to the \(\beta\) chains of the haemoglobin to form an extremely strong complex, which does not dissociate. These reactions can be described by the scheme:</p>
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| − | $$
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| − |
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| − | \ce{Hp + \alpha_{H}<=>[K_{a}][K_{d}] [Hp \cdot \alpha_{H}] ->[K_{i}] [Hp\cdot\alpha_{H}\cdot\beta_{H}]}
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| − |
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| − | $$
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| − | <p>where \(Hp\) is the amount of free haptoglobin, \(\alpha\)\(_{H}\) is the amount of free haemoglobin, \([\)Hp\(\cdot\)\(\alpha\)\(_{H}\)\(]\) is the haptoglobin-haemoglobin-\(\alpha\)-chains complex and \([\) Hp\(\cdot\)\(\alpha\)\(_{H}\)\(\cdot\)\(\beta\)\(_{H}\)\(]\) is the full haptoglobin-haemoglobin complex. \(K_{a}\), \(K_{i}\) are the forward rate reactions, and \(K_{d}\) is the reverse reaction rate.</p>
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| − | <p>The initial concentration of free haemoglobin was defined to be \(\alpha\)\(_{H0}\) and two parameters of the system were defined as:</P>
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| − | $$
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| − | \begin{equation*}
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| − | \lambda=\frac{K_{a}}{K_{d}} \alpha_{H0}, \qquad \gamma=\frac{K_{i}}{K_{d}}.
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| − | \end{equation*}
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| − | $$
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| − | <p>
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| − | Sensitivity analysis was performed to find the optimum values for the two parameters, \(\gamma\) and \(\lambda\), which give the highest concentration of the final complex.</p>
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| − | <center>
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| − | <figure>
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| − | <img src='sensitivity_analysis_plot.png'>
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| − | <figcaption>Figure 1: Sensitivity Analysis for the Binding Parameters of Haemoglobin and Haptoglobin Binding.</figcaption>
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| − | </figure>
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| − | </center>
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| − | <p>
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| − | From Figure 1, it is clear that the optimum complex formation will be when \(\gamma\) and \(\lambda\) are as high as possible. However, we also notice that if \(\lambda\) is small, even if \(\gamma\) is large, no complex is formed, and vice versa. Note that both parameters will increase when \(K_{d}\) decreases, or when either \(K_{i}\) or \(K_{a}\) increase. Thus the most efficient way to optimise complex formation would be to reduce \(K_{d}\) only in the wet lab experiments, this could be done by increasing the binding affinity of the haptoglobin and haemoglobin. This information was passed to the wet lab for there future decision making.
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| − | <b>Method</b>
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| − | <p>Using the law of mass action (Guldeberg and Waage,1879) the binding reaction schematic was written as a system of ordinary differential equations (ODEs):</p>
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| − | $$
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| − | \begin{eqnarray}
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| − | \frac{dHp}{dt}&=&K_{d}[Hp \cdot \alpha_{H}] - K_{a} Hp \alpha_{H} \nonumber \\
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| − | \frac{d \alpha_{H}}{dt}&=&K_{d}[Hp \cdot \alpha_{H}] - K_{a} Hp \alpha_{H} \nonumber\\
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| − | \frac{d[Hp \cdot \alpha_{H}]}{dt}&=& K_{a} Hp \alpha_{H} - K_{d}[Hp \cdot \alpha_{H}] - K_{i}[Hp \cdot \alpha_{H}] \label{eq1} \\
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| − | \frac{d[Hp \cdot \alpha_{H} \cdot \beta_{H}]}{dt}&=&K_{i}[Hp \cdot \alpha_{H}] \nonumber
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| − | \end{eqnarray}
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| − | $$
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| − | <p>
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| − | with initial conditions:</p>
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| − | $$
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| − | \begin{eqnarray}
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| − | Hp(0)&=&4.17 \alpha_{H0} \quad \mu M \nonumber \\ \nonumber
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| − | \alpha_{H}(0)&=&\alpha_{H0} \quad \mu M\\
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| − | \lbrack Hp \cdot \alpha_{H} \rbrack (0)&=&0 \quad \mu M\\ \label{eq2}
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| − | \lbrack Hp \cdot \alpha_{H} \cdot \beta_{H} \rbrack (0)&=&0 \quad \mu M \nonumber
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| − | \end{eqnarray}
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| − | $$
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| − | <p>
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| − | The parameters were estimated by considering the steady state of the system. Setting the left hand side of \(\eqref{eq1}\) to zero gives:</p>
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| − | $$
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| − | \begin{eqnarray}
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| − | K_{d} [Hp \cdot \alpha_{H}]&=&K_{a} Hp \alpha_{H} \nonumber \\
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| − | K_{a} Hp \alpha_{H}&=&K_{d} [Hp \cdot \alpha_{H}] - K_{i} [Hp \cdot \alpha_{H}] \label{eq3}
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| − | \end{eqnarray}
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| − | $$
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| − | <p>
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| − | Rearranging \(\eqref{eq3}\) gives:</P>
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| − | $$
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| − | \begin{equation}
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| − | \frac{[Hp \cdot \alpha_{H}]}{Hp \alpha_{H}}=\frac{K_{a}}{K_{d}} \label{eq4}
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| − | \end{equation}
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| − | $$
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| − | <p>
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| − | Considering the first binding reaction, it was found that the total amount of haptoglobin, \(HpT\), will be equal to:</p>
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| − | $$
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| − | \begin{equation}
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| − | HpT=Hp+[Hp \cdot \alpha_{H}] \label{eq5}
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| − | \end{equation}
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| − | $$
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| − | <p>
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| − | Now using \(\eqref{eq4}\) and \(\eqref{eq5}\) it can be written that:</p>
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| − | $$
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| − | \begin{equation}
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| − | \frac{Hp}{HpT}=\frac{1}{\frac{K_{a}}{K_{d}} \alpha_{H} + 1}. \label{eq6}
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| − | \end{equation}
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| − | $$
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| − | <p>
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| − | It is known that 4.17 haptoglobin per 1 haemoglobin is required for binding, and that haemoglobin and haptoglobin bind at a 1:1 ratio. Therefore the ratio of free haptoglobin to total haptoglobin will be:</p>
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| − | $$
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| − | \begin{equation}
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| − | \frac{Hp}{HpT}=\frac{3.17}{4.17}. \label{eq7}
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| − | \end{equation}
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| − | $$
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| − | <p>
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| − | By substituting \(\eqref{eq7}\) into equation \(\eqref{eq6}\) the ratio between \(K_{a}\) and \(K_{d}\) can be found:</p>
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| − | $$
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| − | \begin{equation}
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| − | \frac{K_{a}}{K_{d}}=\frac{100}{317} \quad \mu M. \label{eq8}
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| − | \end{equation}
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| − | $$
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| − | <p>
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| − | For \(\eqref{eq3}\), \(\eqref{eq6}\) and \(\eqref{eq7}\) can be used to find the ratio between \(K_{i}\) and \(K_{d}\):</p>
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| − | $$
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| − | \begin{equation}
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| − | \frac{K_{i}}{K_{d}}=\frac{83}{317}. \label{eq9}
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| − | \end{equation}
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| − | $$
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| − | <p>
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| − | From literature it is known that 2.5 \(\times\) 10\(^{-5}\) g/cm\(^{3}\) haemoglobin is found in blood plasma (Weatherby and Ferguson,2004). It is also known that the molecular weight of haemoglobin is 64458 g/mol. This can be used to calculate the expected initial concentration of haemoglobin in 1ml of blood: \(\alpha_{H0}=\) 0.3878494524 \(\mu M\). Therefore, from \(\eqref{eq8}\) and \(\eqref{eq9}\) the estimated values for \(\lambda\) and \(\gamma\) are found to be:</p>
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| − | $$
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| − | \begin{equation}
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| − | \lambda=\frac{38.78494524}{317}, \qquad \gamma=\frac{83}{317}. \label{eq10}
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| − | \end{equation}
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| − | $$
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| − | <p>
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| − | By running the ode23 solver over one hundred different values for both parameters, sensitivity analysis can be performed. The range of values has the mean as the estimated values, \(\eqref{eq10}\). The results are shown in Figure 1, where the centre of the plot represents the expected concentration of complex formed, when the expected binding rates are used.</p>
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| − | <a href="#overview" class="btn btn-primary btn-lg pull-right" role="button">Back to Top</a>
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| − | <a href="#blood" class="btn btn-primary btn-lg pull-right" role="button">Back to Start of Model</a>
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| − | </section>
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| − | <a class="anchor" id="semen"></a>
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| − | <section id="semen">
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| − | <h3>Semen: PotD and Spermidine Binding</h3>
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| − | <font color="white">Lorem ipsum dolor sit amet, nostrud maiestatis quaerendum ne sed. Reque possit ne sea. Te dico labitur mediocritatem ius. Error timeam noluisse eos ad, eam ne magna meliore contentiones, nec ei volumus persecuti.
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| − | Dicit animal definitionem et mel, nonumy tacimates nec in. Vis mucius periculis at. At est vidit scripserit repudiandae, agam porro sea ne. Sea et stet tibique praesent, vim et legere aperiri. Quo doming vocibus eleifend no.
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| − | Cu vis partem graeci facilisis. Falli inciderint mei no. Assentior suscipiantur mea id. Vis quas electram prodesset cu, choro omnium conclusionemque an his. Vis latine equidem perfecto ad.</font>
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| − | </div>
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| − | <a href="#overview" class="btn btn-primary btn-lg pull-right" role="button">Back to Top</a>
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| − | <a href="#semen" class="btn btn-primary btn-lg pull-right" role="button">Back to Start of Model</a>
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| − | </div>
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| − | </div>
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| − | </section>
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| − | <a class="anchor" id="saliva"></a>
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| − | <section id="saliva">
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| − | <div class="row3">
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| − | <div class="col-lg-12 feature" style="">
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| − | <h3>Saliva: Lactoferrin and Lactoferrin Binding Protein Binding</h3>
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| − | <p><font color="white"> Lorem ipsum dolor sit amet, nostrud maiestatis quaerendum ne sed. Reque possit ne sea. Te dico labitur mediocritatem ius. Error timeam noluisse eos ad, eam ne magna meliore contentiones, nec ei volumus persecuti.
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| − | Cu vis partem graeci facilisis. Falli inciderint mei no. Assentior suscipiantur mea id. Vis quas electram prodesset cu, choro omnium conclusionemque an his. Vis latine equidem perfecto ad.</font></p>
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| − | </div>
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| − | <a href="#saliva" class="btn btn-primary btn-lg pull-right" role="button">Back to Start of Model</a>
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| − | </div>
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| − | </section>
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| − | <a class="anchor" id="nasal"></a>
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| − | <h3>Nasal Mucus: Oderant Binding Protein Folding</h3>
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| − | <font color="white"> Lorem ipsum dolor sit amet, nostrud maiestatis quaerendum ne sed. Reque possit ne sea. Te dico labitur mediocritatem ius. Error timeam noluisse eos ad, eam ne magna meliore contentiones, nec ei volumus persecuti.
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| − | Cu vis partem graeci facilisis. Falli inciderint mei no. Assentior suscipiantur mea id. Vis quas electram prodesset cu, choro omnium conclusionemque an his. Vis latine equidem perfecto ad.</font>
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