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            <h1><highlight class="highlight">BioSpray</highlight></h1>
            <h3><highlight class="highlight">Mathematical Modelling</highlight></h3>
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                <h3>Overview</h3>
             <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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          <h3>Blood</h3>
          <p class="about-content">Consider the binding between Haptoglobin and Haemoglobin.</p>
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          <a href="#semen"><span class="glyphicon glyphicon-tint" type="button"></span></a>
          <h3>Semen</h3>
          <p class="about-content">Consider the binding between Spermidine and PotD.</p>
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          <a href="#saliva"><span class="glyphicon glyphicon-tint" color="red"></span></a> 
          <h3>Saliva</h3>
          <p class="about-content">Consider the binding between Lactoferrin and Lactoferrim Binding Protein.</p>
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          <a href="#nasal"><span class="glyphicon glyphicon-tint"></span></a> 
          <h3>Nasal Mucus</h3>
          <p class="about-content">Consider the folding of the Oderant Binding Protein.</p>
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                <h3>Blood: Haptoglobin and Haemoglobin Binding</h3>
             <font color="white"> <b>Objective</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>Model Formation</b>
<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. This reaction is not reversible. These reactions can be described by the scheme:

<!--Double dollar signs surround any math notation copied straight from latex, for example, equations, this will automatically be centered-->
$$
\ce{Hp + \alpha_{H}<=>[K_{a}][K_{d}] [Hp \cdot \alpha_{H}] ->[K_{i}] [Hp\cdot\alpha_{H}\cdot\beta_{H}]}
$$
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>


<p>Using the law of mass action (Guldeberg and Waage,1879) the scheme can be written as  a system of ordinary differential equations (ODEs):

$$
\begin{eqnarray}
\frac{dHp}{dt}&=&K_{d}[Hp \cdot \alpha_{H}] - K_{a} Hp \alpha_{H} \nonumber\\
\frac{d \alpha_{H}}{dt}&=&K_{d}[Hp \cdot \alpha_{H}] - K_{a} Hp \alpha_{H} \nonumber\\
\frac{d[Hp \cdot \alpha_{H}]}{dt}&=& K_{a} Hp \alpha_{H} - K_{d}[Hp \cdot \alpha_{H}] -  K_{i}[Hp \cdot \alpha_{H}] \label{eq1}\\
\frac{d[Hp \cdot \alpha_{H} \cdot \beta_{H}]}{dt}&=&K_{i}[Hp \cdot \alpha_{H}] \nonumber
\end{eqnarray}
$$
with initial conditions:
$$
\begin{eqnarray}
Hp(0)&=&4.17 \alpha_{H0} \nonumber\\
\alpha_{H}(0)&=&\alpha_{H0} \nonumber\\
\lbrack Hp \cdot \alpha_{H} \rbrack (0)&=&0 \label{eq2}\\
\lbrack Hp \cdot \alpha_{H} \cdot \beta_{H} \rbrack (0)&=&0 \nonumber
\end{eqnarray}
$$
</p>

<b>Parameter Finding</b>
<p>The parameters were estimated by considering the steady state of the system.Setting the left hand side of equations \(\eqref{eq1}\) to zero, then the following equations are given:
$$
\begin{eqnarray}
K_{d} [Hp \cdot \alpha_{H}]&=&K_{a} Hp \alpha_{H} \nonumber \\
K_{a} Hp \alpha_{H}&=&K_{d} [Hp \cdot \alpha_{H}] - K_{i} [Hp \cdot \alpha_{H}] \label{eq3}
\end{eqnarray}
$$
Rearranging \(\eqref{eq3}\)gives:
$$
\begin{equation}
\frac{[Hp \cdot \alpha_{H}]}{Hp \alpha_{H}}=\frac{K_{a}}{K_{d}} \label{eq4}
\end{equation}
$$
Considering the first binding reaction, it is found that the total amount of haptoglobin, \(HpT\), will be equal to:
$$
\begin{equation}
HpT=Hp+[Hp \cdot \alpha_{H}] \label{eq5}
\end{equation}
$$
Now using \(\eqref{eq4}\) and \(\eqref{eq5}\) it can be written that:
$$
\begin{equation}
\frac{Hp}{HpT}=\frac{1}{\frac{K_{a}}{K_{d}} \alpha_{H} + 1} \label{eq6}
\end{equation}
$$
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:
$$
\begin{equation}
\frac{Hp}{HpT}=\frac{3.17}{4.17} \label{eq7}
\end{equation}
$$
By substituting \(\eqref{eq7}\) into \(\eqref{eq6}\) the ratio between \(K_{a}\) and \(K_{d}\) can be found:
$$
\begin{equation}
\frac{K_{a}}{K_{d}}=\frac{100}{317} \label{eq8}
\end{equation}
$$
For \(\eqref{eq3}\) we can use equations \(\eqref{eq6}\)and \(\eqref{eq7}\) to find the ratio between \(K_{i}\) and \(K_{d}\):
$$
\begin{equation}
\frac{K_{i}}{K_{d}}=\frac{83}{317} \label{eq9}
\end{equation}
$$ </p>



<b>Non-Dimensionalisation</b>
<p>To implement system \(\eqref{eq1}\) in a model of the haptoglobin and haemoglobin binding, non-dimensionalisation is used to simplify the equations to the following system:</p>

$$
\begin{eqnarray}
\frac{du}{dt}&=&w - \lambda  uv \nonumber\\
\frac{dv}{dt}&=&w - \lambda uv \nonumber\\
\frac{dw}{dt}&=&\lambda uv - w - \gamma w \label{eq10}\\
\frac{dx}{dt}&=&\gamma w \nonumber
\end{eqnarray}
$$
with new initial conditions:
$$
\begin{eqnarray}
u(0)&=&4.17 \nonumber \\
v(0)&=&1 \nonumber \\
w(0)&=&0 \lable{eq11}\\
x(0)&=&0 \nonumber
\end{eqnarray}
$$
<p>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.
From this the initial concentration of hemoglobin in 1ml of blood is \(\alpha_{H0}=\) 0.3878494524 \(\mu M\)
  From \(\eqref{eq8}\) and \(\eqref{eq9}\) the values for both \(\lambda\) and \(\gamma\) are known.
 </p>


<b>Initial Results</b>
 <p> Numerical simulations of the non-dimensionalised system of ODEs were run using MATLAB's ode23 solver (Bogacki, 1989).</p>
<!--The following center tag is the inclusion of a png file, from MATLAB, into our document and is centred in the page-->
 <center><figure><img src="https://static.igem.org/mediawiki/2015/8/82/TeamDundee-Hapto_edit_1.png" style="border:4px solid black">


 <figcaption>Amount of haemoglobin, haptoglobin, the intermediate complex and the full complex over time, with \(u(0)=4.17\), \(v(0)=1\) and \(\alpha_{H0}=0.3878494524\).</figcaption>
 </figure>
</center>



<b>Steady State Analysis</b>
<p>The steady state of the system was found to be when:</p>


$$
\begin{equation*}
(u,v,w,x)=(3.17,0,0,1).
\end{equation*}
$$



<b>Sensitivity Analysis</b>


<p>The parameters \(\gamma\) and \(\lambda\) are estimated above, however we can assess the optimal value of these. Notice that the value of both \(\gamma\) and \(\lambda\) are governed by the binding rates of the system. By running the ode23 solver over one hundred different values for both parameters. The range of values has the mean as the estimated values from above. That is the max values, A and B, are twice the estimated values for \(\lambda\) and \(\gamma\) respectively. All other conditions were kept the same as previous analysis. </p>
 <center>
 <figure>
 <img src="https://static.igem.org/mediawiki/2015/b/b3/TeamDundee-Haptosens1.png" style="border:4px solid black">
 <figcaption>Surface showing the effect of different values for \(\gamma\) and \(\lambda\) on complex formation.</figcaption>
 </figure>
</center>
<p>where the colour bar represents the concentration of the final complex and:</p>
$$
\begin{eqnarray*}
A&=&2 \times \frac{83}{317}, \\
B&=&2 \times \frac{100}{317} \times  0.3878494524.
\end{eqnarray*}
$$
<p>Conclusions that can drawn from Figure 2 will be discussed in the conclusions section.</p>



<b>Conclusions</b>
<p>From the model of haemoglobin and haptoglobin binding several conclusions are inferred. The binding reaction should be relatively quick, meaning that a visual change should be apparent after a short period of time. This is demonstrated in Figure 1. The optimal binding rates were investigated via sensitivity analysis, and it was discovered that \(\gamma\) and \(\lambda\) should be as high as possible for the optimal visualistion of the traces within the sample. The previously estimated values of both parameters is the equivalent to the centre of the figure 2. From the figure it can be seen that the optimal values for both parameters is as large as possible. This suggests that the association rates should be much greater than the dissociation rates, to ensure optimal binding. This could be done by modifying haptoglobin to become more sticky, and thus more likely to bind to haemoglobin and less like to dissociate from the complex. The estimated values for \(\gamma\) and \(\lambda\) represent the natural binding reaction, if these are increased then more efficient reactions will occur.
</p>

<b>References</b>
<ul>
  <li>Bogacki, P.,  Shampine, L. F. (1989). A 3 (2) pair of Runge-Kutta formulas. Applied Mathematics Letters, 2(4), 321-325.</li>
  <li>Guldberg, C. M.,  Waage, P. (1879). Concerning chemical affinity. Erdmanns Journal fr Practische Chemie, 127, 69-114.</li>
  <li>Weatherby, D.,  Ferguson, S. (2004). Blood Chemistry and CBC Analysis (Vol. 4). Weatherby and Associates, LLC.</li>
</ul></font>
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                <h3>Semen: PotD and Spermidine Binding</h3>
             <font color="white"><b>Objective</b>
<p>The aim of modelling of the binding between spermidine and PotD is to understand the optimum concentration and binding rates that are required for visual detection of spermidine in the sample from the crime scene. The more complex formed the more likely that a visual detection of spermidine in the sample will be obtained using the BioSpray.</p>




<b>Model Formation</b>
<p> PotD is a polyamine substrate-binding protein found in E.Coli. PotD binds to spermidine, allowing it to then bind to PotA, PotB and PotC, which allows for movement of the spermidine. For the project only the initial binding of PotD to spermidine is important, as the aim is to used PotD as a detector fo finding traces of semen at a crime scene.
The binding reaction can be described by the scheme:
<!--Double dollar signs surround any math notation copied straight from latex, for example, equations, this will automatically be centered-->
$$  \ce{P + S
<=>[k_{on}][k_{off}] C
}  $$
<!--The notation \(....\) is used where the four dots are a math notation that we want in a text line-->
where \(P\) is the concentration of PotD, \(S\) the concentration of spermidine and \(C\) is the concentration of the PotD-spermidine complex. The reaction rate constants are \(k_{on}\) for the association reaction and \(k_{off}\) for the dissociation reaction.</p>

<p>Using the law of mass action, the binding reactions can be described by a system of ordinary differential equations (ODEs) (Guldberg,1879), the system was then non-dimensionalised to obtain:
$$
\begin{eqnarray}
\frac{dP}{d\tau }&=&\kappa C-PS, \nonumber \\
\frac{dS}{d\tau }&=&\kappa C-PS,\\
\frac{dC}{d\tau }&=&PS-\kappa C. \nonumber
\end{eqnarray}
$$
where each equation describes the change over time of the three substances in the binding reaction, with initial conditions:
$$
\begin{eqnarray}
P(0)&=&\frac{1}{2}, \nonumber\\
S(0)&=&1,\\
C(0)&=&0. \nonumber
\end{eqnarray}
$$


It is assumed that there will be no complex at the start of the reaction, and that there will be some concentration of spermidine and PotD.</p>

<b>Parameter Finding</b>
<p>Vanella (1978), states that there is \(60 \mu g \ ml^{-1}\) of spermidine in seminal fluid of humans. This can be used to find that there is a concentration of \(413.08 \mu M\) in \(1ml\) of seminal fluid of humans. Therefore it is assumed that the intial concentration of  spermidine is \(S_{0}=413.08\mu M\).</p>

<p>In a paper by Kashiwagi (1993) it was found that an optimum concentration ratio of PotD to spermidine is 1:2. Using this it is assumed that the initial concentration of PotD is:
$$
\begin{equation*}
 P_{0} = \frac{1}{2} S_{0} = 206.54 \mu M.
 \end{equation*}
$$
It is also known that one molecule of spermidine binds to one molecule of PotD. In Kashiwagi's, 1993, paper it is also stated that the dissociation equilibrium constant for PotD and spermidine binding is, \(K_{dis}=3.2\mu M\), where:</p>
$$

\begin{equation*}
K_{dis}=\frac{k_{off}}{k_{on}}.
\end{equation*}
$$


<p>As numerical values for \(K_{dis}\) and \(S_{0}\) are known, \(\kappa\) can be calculated as:
$$
\begin{equation*}
\kappa = \frac{3.2}{413.08}
\end{equation*}
$$</p>

<b>Initial Results</b>
<p> 
Numerical simulations of the non-dimensionalised system of ODEs were run using MATLAB's ode23 solver (Bogacki, 1989). The binding reaction can be considered and the concentration of each substance over time can be visualised:

</p>
<!--The following center tag is the inclusion of a png file, from MATLAB, into our document and is centred in the page-->
<center>
<figure>
<img src='https://static.igem.org/mediawiki/2015/5/58/TeamDundee-Sperm_edit_1.png' style="border:4px solid black" align="center">
<figcaption>Figure 1: PotD, Spermidine and Complex concentrations over time with \(\kappa=\frac{3.2}{413.08}\), \(P_{0}=\frac{1}{2}\), \(S_{0}=1\).</figcaption>
</figure>
</center>


<b>Steady State Analysis</b>
<p>The steady state of the system was found to be when:</p>

$$
\begin{equation*}
(P,S,C)=(0.007517200116,0.5075172001,0.4924827999).
\end{equation*}
$$
These values are consistent with those found by viewing the data statistics found in MATLAB of Figure 1.</p>



<b>Sensitivity Analysis</b>
<p>The parameter \(\kappa\) is found from literature, however the optimal value of \(\kappa\) can be assessed. The MATLAB ode23 solver was run over one hundred different values for \(\kappa\) . The range of values chosen is from 1:A, where A is twice the value of \(\kappa\) found from literature. The numerical simulation run previously was re-run with the range of values for \(\kappa\) and the results for the complex formation plotted:</p>
<center>
<figure>
<img src='https://static.igem.org/mediawiki/2015/3/36/TeamDundee-Spermsen1.png' style="border:4px solid black" align="center">
<figcaption>Figure 2: Complex formation over time with a range of values for \(\kappa\).</figcaption>
</figure>
</center>
<p>Figure 2 shows that for varying values of \(\kappa\) the concentration of complex formed stays roughly the same. This analysis can be used to compare the concentration of complex formed with increasing \(\kappa\) to find the optimum value:</p>
<center>
<figure>
<img src='https://static.igem.org/mediawiki/2015/d/d0/TeamDundee-Spermsen2.png' style="border:4px solid black" align="center">
<figcaption>Figure 3: Complex formation against the value of \(\kappa\).</figcaption>
</figure>
</center>

<p>Figure 3 demonstrates that the lower \(\kappa\) is the more complex will be formed. Recall that:</p>
$$
\begin{equation*}
\kappa = \frac{k_{off}}{k_{on} S_{0}}.
\end{equation*}
$$
<p>So for optimal complex formation, \(k_{on}\) should be much larger than \(k_{off}\). </p>



<b>Conclusions</b>
<p>
From the model of spermidine and PotD binding several conclussions can be inferred. Figure 1, shows that the binding reaction occurs very quickly, within 15 seconds, therefore we can expect a visual response to the spray reasonably quickly if spermidine is present at the scene of the crime.  The sensitivity analysis suggests that for optimal complex formation, and thus optimal viualisation, the association rate should be much larger than the dissociation rate. This could potentially be done by modifying PotD to be more likely to bind to spermidine by making it more sticky</p>



<b>References</b>
<ul>
<li>Guldberg, C. M., Waage, P. (1879). Concerning chemical affinity. Erdmannas Journal fr Practische Chemie, 127, 69-114.</li>
<li>Kashiwagi, K., Miyamoto, S., Nukui, E., Kobayashi, H., Igarashi, K. (1993). Functions of potA and potD proteins in spermidine-preferential uptake system in Escherichia coli. Journal of Biological Chemistry, 268(26), 19358-19363.</li>
<li>Vanella, A., Pinturo, R., Vasta, M., Piazza, G., Rapisarda, A., Savoca, S., Panella, M. (1978). Polyamine levels in human semen of unfertile patients: effect of S-adenosylmethionine. Acta Europaea Fertilitatis, 9(2), 99-103.</li>

</ul></font>
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                <h3>Saliva: Lactoferrin and Lactoferrin Binding Protein Binding</h3>
             <p><font color="white"> BlahHhhhhhh</font></p>
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                <h3>Nasal Mucus: Oderant Binding Protein Folding</h3>
             <font color="white"> <b>Objective</b>
<p>The aim of modelling of the folding of oderant binding protein (OBP) is to understand the optimum concentration and binding rates that are required for visual detection of oderants in the sample from the crime scene. The more folded OBP formed the more likely that a visual detection of nasal mucus in the sample will be obtained using the BioSpray.</p>

<b>Model Formation</b>

<p>Oderant binding protein 2A (OBPIIa) is found in human nasal mucus and are involved in oderant detection. Within the protein there is an eight sranded \(\beta\)-barrel and an \(\alpha\) helix. The \(\beta\)-barrel and the $\alpha$ helix bind via a disulphide bridge, to allow for oderant detection by the OBP, this is called folding (Schiefner, 2015). For detection of nasal mucus at the scene of a crime, modified \(\beta\)-barrels will be within a BioSpray. The \(\beta\)-barrels will fluoresce when bound to an \(\alpha\) helix within an OBP in the sample. The reaction that has to be considered is the natural binding between the \(\alpha\) helix and the \(\beta\)-barrel, so that the modified \(\beta\)-barrel can be designed to be more likely to bind than its natural counterpart. The folding process can be described by the scheme:
$$  \ce{\alpha + \beta
<=>[k_{on}][k_{off}] OBP
}  $$
where \(\alpha\) represents the concentration of \(\alpha\) helices, \(\beta\) represents the concentration of the \(\beta\)-barrels and \(OBP\) represents the folded protein concentration. The kinetic association and dissociation rates are \(k_{on}\) and \(k_{off}\) respectively.</p>

<p>Using the law of mass action, the binding reactions can be described by a system of ordinary differential equations (ODEs) (Guldberg,1879), the system was then non-dimensionalised:
$$
\begin{eqnarray}
\frac{d\alpha}{d\tau }&=& OBP-\Psi \alpha \beta, \nonumber\\
\frac{d\beta}{d\tau }&=&OBP-\Psi \alpha \beta,\\
\frac{dOBP}{d\tau }&=&\Psi \alpha \beta-OBP. \nonumber
\end{eqnarray}
$$
where each equation describes the change over time of the three substances in the folding reaction, with initial concentrations in relation to the initial concentration of unfolded OBP:</p>
$$
\begin{eqnarray}
\alpha(0)&=&1,\nonumber \\
\beta(0)&=&\frac{\beta_{0}}{\alpha_{0}}=1,\\
OBP(0)&=&0. \nonumber
\end{eqnarray}
$$

<b>Parameter Finding</b>
<p>
It will be assumed that there is one \(\alpha\) helix and one \(\beta\)-barrel in each protein molecule. The initial concentration of un-folded oderant binding protein, \(\alpha_{0}\) and \(\beta_{0}\), can be calculated from values provided from literature. From work by Schiefner (2015) and Briand (2002), the exepcted initial concentration of unfolded OBP in nasal mucus could be calculated as:

$$
\begin{equation}
\alpha_{0}=\beta_{0}=26.45502646 \quad \mu \text{M}.
\end{equation}
$$
It will also be assumed that one \(\alpha\) helix binds to one \(\beta\)-barrel.</p>

<p>Due to a lack of experimental data and previous research a numerical value for both kinetic rates, \(k_{on}\) and \(k_{off}\), cannot be stated. However a ratio may be easier to estimate or consider, so the following parameter is defined:
$$
\begin{equation*}
K=\frac{k_{on}}{k_{off}}.
\end{equation*}
$$
A standard range for a dissociation constant, \(K_{d}\), was found by Archakov, (2003), to be:
$$
\begin{equation*}
K_{d}=\frac{k_{off}}{k_{on}}=10^{-8} \rightarrow 10^{2} \mu \text{M}.
\end{equation*}
$$
Since \(K_{d}=\frac{1}{K}\) then a range for \(K\) is assumed to be:
$$
\begin{equation}
K=10^{-2} \rightarrow 10^{8} \mu \text{M}.
\end{equation}
$$
Sensitivity analysis was used to investigate the reaction with this estimated range of \(K\).</p>


<b>Sensitivity Analysis</b>
<p>As the value of \(K\) is unknown, the system of non-dimensionalised ODEs are solved with the range of values defined in equation (8). MATLAB's ode23 solver was used to solve the equations and perform sensitivity analysis (Bogacki 1989). The solver would was too slow with the larger end values for the range of \(K\). One hundred data points were used within the range for \(K\):
$$
\begin{equation*}
 K=10^{-2} \rightarrow 10^{3} \mu \text{M}.
\end{equation*}
$$
The value of \(\alpha_{0}\) was set as that defined in equation (7) and was kept constant throughout the model.</p>

<center>
<figure>
<img src='https://static.igem.org/mediawiki/2015/a/a4/TeamDundee-OBP_large_K2.png' style="border:4px solid black" align="center">
<figcaption>Figure 1: The concentration of folded OBP formed over time, max\(K=1000\).</figcaption>
</figure>
</center>

<p>Where each line represents a different value for \(K\). From Figure 1, it is hard to tell what is the optimum value for \(K\). By considering the concentration of folded OBP formed with increasing values of \(K\), trends are more easily seen. </p>
<center>
<figure>
<img src='https://static.igem.org/mediawiki/2015/5/5e/TeamDundee-OBP_large_K.png' style="border:4px solid black" align="center">
<figcaption>Figure 2: Concentration of folded OBP formed with increasing values for \(K\), max\(K=1000\).</figcaption>
</figure>
</center>
<p>Figure 2, shows that as \(K\) increases the concentration of folded OBP formed increases. However after \(K=200\) the concentration formed does not seem to increase greatly. </p>


<p>To investigate this further the analysis was repeated with the maximum \(K=200\).</p>

<center>
<figure>
<img src='https://static.igem.org/mediawiki/2015/4/44/TeamDundee-OBP_200_K2.png' style="border:4px solid black" align="center">
<figcaption>Figure 3: The concentration of folded OBP formed over time, max\(K=200\).</figcaption>
</figure>
</center>
<p>Where each line represents a different value for \(K\). Figure 3, shows a greater difference in concentration formed than with Figure 1. Again by considering the concentration of folded OBP formed with increasing values of \(K\), trends are more easily seen. </p>
<center>
<figure>
<img src='https://static.igem.org/mediawiki/2015/f/ff/OBP_200_K.png' style="border:4px solid black" align="center">
<figcaption>Figure 4: Concentration of folded OBP formed with increasing values for \(K\), max\(K=200\).</figcaption>
</figure>
</center>
<p>From Figure 4 it can be seen that for most values of \(K\) at least 90\(\%\) of the \(\alpha\) helices and \(\beta\)-barrels will bind to form the folded OBP. However the greater the value of \(K\) the more complex will be formed.</p>

<p> From the other models for the BioSpray part of the project the equivalent values of \(K\) given by literature were roughly, \(K\)=3.2 \(\mu\)M, where the substrates binding are of similar size to those considered here. To investigate this further the analysis was repeated with the maximum \(K=10\).</p>
<center>
<figure>
<img src='https://static.igem.org/mediawiki/2015/2/25/TeamDundee-OBP_10_K2.png' style="border:4px solid black" align="center">
<figcaption>Figure 5: The concentration of folded OBP formed over time, max\(K=10\).</figcaption>
</figure>
</center>
<p>Where each line represents a different value for \(K\). Figure 5, shows a greater difference in concentration formed than with Figure 1 and 3. Again by considering the concentration of folded OBP formed with increasing values of \(K\), trends are more easily seen. </p>

<center>
<figure>
<img src='https://static.igem.org/mediawiki/2015/6/63/TeamDundee-OBP_10_K.png'  style="border:4px solid black"align="center">
<figcaption>Figure 6: Concentration of folded OBP formed with increasing values for \(K\), max\(K=10\).</figcaption>
</figure>
</center>
<p>Figure 6 demonstrates that for values of \(K\) greater then 5, around 90\(\%\) of the \(\alpha\) helices and \(\beta\)-barrels will bind to form the folded OBP. So as long as \(k_{on}\) is 5 times greater than \(k_{off}\) the binding should allow for visual detection of folded OBPs. Numerical simulations can now be run with set values of $K$ at \(K=200\) and \(K=3.2\) to understand the folding process in greater detail.</p>

<b>Numerical Simulations</b>
<p>The non-dimensionalised system of ODEs can be solved with set values for the parameter \(K\), chosen from the sensitivity analysis, with the given initial conditions. Again the value of \(\alpha_{0}\) was set as that defined in equation (7) and was kept constant throughout this part of the model. By considering the change of concentration over time of the \(\alpha\) helices, \(\beta\)-barrels and folded OBP, the binding reactions can be further understood. Firstly the system was solved using ode23 from MATLAB with \(K=200\) (Bogacki 1989).</p>
 
<center>
<figure>
<img src='https://static.igem.org/mediawiki/2015/c/c3/TeamDundee-OBP_setK_200.png' style="border:4px solid black" align='center'>
<figcaption>Figure 7: Concentrations over time with given initial conditions, \(K=200\).</figcaption>
</figure>
</center>
<p>
From Figure 7 it can be seen that the folding reaction occurs very quickly, within 0.01 seconds, when \(K=200\). The lower value of \(K=3.2\) is also considered:</p>
<center>
<figure>
<img src='https://static.igem.org/mediawiki/2015/6/62/TeamDundee-OBP_setK_3_2.png' style="border:4px solid black" align='center'>
<figcaption>Figure 8: Concentrations over time with given initial conditions, \(K=3.2\).</figcaption>
</figure>
</center>
<p>
For the lower value of \(K\), as expected, less complex is formed. However the reaction is still reasonably fast and a visual detection should be obtained within seconds of applying the spray.</p>
<p>
 As the ratio between \(\alpha\) helices and \(\beta\)-barrels is assumed in the parameter finding section, the effects of varying this value can be investigated. This is done by increasing the value of \(\beta_{0}\).
The non-dimensionalised system was solved again with a range of values for \(\beta_{0}\) with either \(K=200\) or \(K=3.2\) with all other variables the same as previous numerical simulations. Five values were chosen for \(\beta_{0}\) between 1:5. Firstly the simulation was run with \(K=200\).</p>
<center>
<figure>
<img src='https://static.igem.org/mediawiki/2015/d/df/TeamDundee-OBP_v0_Kis200.png' style="border:4px solid black" align='center'>
<figcaption>Figure 9: Concentrations over time with varying initial conditions for \(\beta_{0}\),  \(K=200\).</figcaption>
</figure>
</center>
<p>
Figure 9 demonstrates that there is a significant difference in the concentration of folded OBP formed between \(\beta_{0}=1\) and \(\beta_{0}=2\). That is, the concentration of folded OBP formed will increase if the initial concentration of \(\beta\)-barrels is twice that of the \(\alpha\) helices. The speed of reaction is also significantly increased between \(\beta_{0}=1\) and \(\beta_{0}=2\), thus the latter gives a better result. There is less significant change in concentrations for greater values of \(\beta_{0}\). This analysis can be repeated for \(K=3.2\).</p>
<center>
<figure>
<img src='https://static.igem.org/mediawiki/2015/8/8e/TeamDundee-OBP_v0_Kis3_2.png' style="border:4px solid black" align='center'>
<figcaption>Figure 10: Concentrations over time with varying initial conditions for \(\beta_{0}\),  \(K=200\).</figcaption>
</figure>
</center>
<p>
Again a significant difference is seen between the first two values for \(\beta_{0}\) in both concentration of folded OBP formed and speed of reaction. Both Figures 9 and 10 demonstrate that the optimal ratio between the \(\alpha\) helices and \(\beta\)-barrels is 1:2. Although the exact concentration of \(\alpha\) helices in the sample cannot be known, the expected value was estimated in equation (7). According to this value the optimal concentration of \(\beta\)-barrels to have in the BioSpray will be at least:
$$
\begin{equation*}
\beta_{0}=52.91005292 \mu \text{M}.
\end{equation*}
$$
The steady state of the system can be investigated to find the exact concentrations of \(\alpha\) helices, \(\beta\)-barrels and folded OBP that can be expected after the folding process has occured.</p>


<b>Steady State Analysis</b>
<p>To investigate the steady state of the system, to consider all that has been discussed previously, four cases were investigated:
<ul>
<li>Case 1: \(K=3.2\) and \(v_{0}=2\),</li>
<li>Case 2: \(K=200\) and \(v_{0}=2\),</li>
<li>Case 3: \(K=3.2\) and \(v_{0}=1\),</li>
<li>Case 4: \(K=200\) and \(v_{0}=1\).</li>
</ul>


This equation can then be solved for each case defined previously and substituted into the equations for \(v\) and \(w\) to find the steady state:
<ul>
<li>Case 1: \((u,v,w)=(0.01154291058,1.011542911,0.9884570894)\),</li>
<li>Case 2: \((u,v,w)=(0.00188928598,1.000188929,0.9998110714)\),</li>
<li>Case 3: \((u,v,w)=(0.102939438,0.102939438,0.897060562)\),</li>
<li>Case 4: \((u,v,w)=(0.01365355187,0.01365355187,0.9863464481)\).
</li></ul>
Where each steady state is in the non-dimensionalised state, so is a fraction of the initial \(\alpha\) helix concentration. From this analysis it can be seen that the optimal situation is Case 3 where \(K=200\) and \(\beta_{0}=2\), as the highest concentration of folded OBP will be formed. This supports the conclusions drawn from the sensitivity analysis and numerical simulations run previously.</p>

<b>Conclusions</b>
<p>The conclusions drawn from the model of the folding process of oderant binding proteins have been declared previously. To recap, the main points inferred by the model are that:</p>
<ul>
<li> The folding process will occur quickly, which means that if folded OBPs are present a visual detection should be expected within seconds.</li>
<li>The optimal ratio between \(\alpha\) helices in the sample and \(\beta\)-barrels in the BioSpray is 1:2 respectively.</li>
<li>The optimum ratio between \(k_{on}\) and \(k_{off}\) is 200:1. With this value almost all of the \(\alpha\) helices and \(\beta\)-barrels will react to form folded OBPs. However with lower ratios a visual change should still occur.</li></ul>


<b>References</b>
<ul>

<li>Archakov, A. I., Govorun, V. M., Dubanov, A. V., Ivanov, Y. D., Veselovsky, A. V., Lewi, P., Janssen, P. (2003). Protein protein interactions as a target for drugs in proteomics. Proteomics, 3(4), 380-391.</li>
<li>Bogacki, P., Shampine, L. F. (1989). A 3 (2) pair of Runge-Kutta formulas. Applied Mathematics Letters, 2(4), 321-325.
</li>
<li>Briand, L., Eloit, C., Nespoulous, C., Bezirard, V., Huet, J. C., Henry, C., Pernollet, J. C. (2002). Evidence of an odorant-binding protein in the human olfactory mucus: location, structural characterization, and odorant-binding properties. Biochemistry, 41(23), 7241-7252.
</li>
<li>Schiefner, A., Freier, R., Eichinger, A., Skerra, A. (2015). Crystal structure of the human odorant binding protein, OBPIIa. Proteins: Structure, Function, and Bioinformatics, 83(6), 1180-1184.</li>

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