Difference between revisions of "Team:Dundee/Modelling/Biospray"
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\frac{dHp}{dt}&=&K_{d}[Hp \cdot \alpha_{H}] - K_{a} Hp \alpha_{H} \nonumber \\ | \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 \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}]}{dt}&=& K_{a} Hp \alpha_{H} - K_{d}[Hp \cdot \alpha_{H}] - K_{i}[Hp \cdot \alpha_{H}] |
| + | \phantomsection | ||
| + | \label{eq1} \\ | ||
\frac{d[Hp \cdot \alpha_{H} \cdot \beta_{H}]}{dt}&=&K_{i}[Hp \cdot \alpha_{H}] \nonumber | \frac{d[Hp \cdot \alpha_{H} \cdot \beta_{H}]}{dt}&=&K_{i}[Hp \cdot \alpha_{H}] \nonumber | ||
\end{eqnarray} | \end{eqnarray} | ||
Revision as of 10:21, 5 August 2015
BioSpray
Mathematical Modelling
Overview
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.
Blood: Haptoglobin and Haemoglobin Binding
AimThe 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.
ResultsHaemoglobin 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:
$$ \ce{Hp + \alpha_{H}<=>[K_{a}][K_{d}] [Hp \cdot \alpha_{H}] ->[K_{i}] [Hp\cdot\alpha_{H}\cdot\beta_{H}]} $$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.
The initial concentration of free haemoglobin was defined to be \(\alpha\)\(_{H0}\) and two parameters of the system were defined as:
$$ \begin{equation*} \lambda=\frac{K_{a}}{K_{d}} \alpha_{H0}, \qquad \gamma=\frac{K_{i}}{K_{d}}. \end{equation*} $$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.
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. Method
Using the law of mass action (Guldeberg and Waage,1879) the binding reaction schematic was written as a system of ordinary differential equations (ODEs):
$$ \begin{eqnarray} \frac{dHp}{dt}&=&K_{d}[Hp \cdot \alpha_{H}] - K_{a} Hp \alpha_{H} \nonumber \\ \frac{d \alpha_{H}}{dt}&=&K_{d}[Hp \cdot \alpha_{H}] - K_{a} Hp \alpha_{H} \nonumber\\ \frac{d[Hp \cdot \alpha_{H}]}{dt}&=& K_{a} Hp \alpha_{H} - K_{d}[Hp \cdot \alpha_{H}] - K_{i}[Hp \cdot \alpha_{H}] \phantomsection \label{eq1} \\ \frac{d[Hp \cdot \alpha_{H} \cdot \beta_{H}]}{dt}&=&K_{i}[Hp \cdot \alpha_{H}] \nonumber \end{eqnarray} $$with initial conditions:
$$ \begin{eqnarray} Hp(0)&=&4.17 \alpha_{H0} \quad \mu M \nonumber \\ \nonumber \alpha_{H}(0)&=&\alpha_{H0} \quad \mu M\\ \lbrack Hp \cdot \alpha_{H} \rbrack (0)&=&0 \quad \mu M\\ \label{eq2} \lbrack Hp \cdot \alpha_{H} \cdot \beta_{H} \rbrack (0)&=&0 \quad \mu M \nonumber \end{eqnarray} $$The parameters were estimated by considering the steady state of the system. Setting the left hand side of \(\eqref{eq1}\) to zero gives:
$$ \begin{eqnarray} K_{d} [Hp \cdot \alpha_{H}]&=&K_{a} Hp \alpha_{H} \nonumber \\ K_{a} Hp \alpha_{H}&=&K_{d} [Hp \cdot \alpha_{H}] - K_{i} [Hp \cdot \alpha_{H}] \label{eq3} \end{eqnarray} $$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 was 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 equation \(\eqref{eq6}\) the ratio between \(K_{a}\) and \(K_{d}\) can be found:
$$ \begin{equation} \frac{K_{a}}{K_{d}}=\frac{100}{317} \quad \mu M. \label{eq8} \end{equation} $$For \(\eqref{eq3}\), \(\eqref{eq6}\) and \(\eqref{eq7}\) can be used to find the ratio between \(K_{i}\) and \(K_{d}\):
$$ \begin{equation} \frac{K_{i}}{K_{d}}=\frac{83}{317}. \label{eq9} \end{equation} $$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:
$$ \begin{equation} \lambda=\frac{38.78494524}{317}, \qquad \gamma=\frac{83}{317}. \label{eq10} \end{equation} $$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.
References- Bogacki, P., Shampine, L. F. (1989). A 3 (2) pair of Runge-Kutta formulas. Applied Mathematics Letters, 2(4), 321-325.
- Guldberg, C. M., Waage, P. (1879). Concerning chemical affinity. Erdmanns Journal fr Practische Chemie, 127, 69-114.
- Weatherby, D., Ferguson, S. (2004). Blood Chemistry and CBC Analysis (Vol. 4). Weatherby and Associates, LLC.
Semen: PotD and Spermidine Binding
AimThe 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.
ResultsPotD 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:
$$ \ce{P + S <=>[k_{on}][k_{off}] C } $$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. The inital concentrations of the PotD and Spermidine are denoted, \(P_{0}\) and \(S_{0}\) respectively. The ratio of the initial concentration of PotD to the initial concentration of Spermidine was defined as:
$$ \begin{equation*} R_{0}=\frac{P_{0}}{S_{0}}. \end{equation*} $$A non-dimensional binding rate parameter was defined as:
$$ \begin{equation*} \kappa=\frac{k_{on}\cdot S_{0}}{k_{off}}. \end{equation*} $$Sensitivity analysis was performed to find the optimum values of both \(\kappa\) and \(R_{0}\) which give the optimal complex formation.
From the sensitivity analysis it can be seen that increasing \(R_{0}\) does not seem to greatly affect the complex formation, thus the minimum ratio can be used. The minimum value for \(R_{0}\) that still gives a significant level of complex formation was investigated by plotting the complex formation against increasing \(R_{0}\).
This suggests that as long as \(R_{0}=1\), that is there is at least a concentration of \(413.08 \mu M\) of PotD in the BioSpray, there will be enough complex formed to visualise via the nanobeads. The sensitivity analysis plot also suggests that the higher the value of \(\kappa\), the more complex will be formed. This could be achieved by enhancing the binding affinity of PotD and Spermidine via wet lab experiments. The information from this model will be considered in lab based experiments.
MethodUsing the law of mass action, the reaction scheme can be described by a system of ordinary differential equations (ODEs) (Guldberg,1879):
$$ \begin{eqnarray} \frac{dP}{dt}&=&k_{off}C-k_{on}PS, \nonumber\\ \frac{dS}{dt}&=&k_{off}C-k_{on}PS, \label{eq11}\\ \frac{dC}{dt}&=&k_{on}PS-k_{off}C. \nonumber \end{eqnarray} $$where each equation describes the change over time of the three substances in the binding reaction, with initial conditions:
$$ \begin{eqnarray} P(0)&=&P_{0} , \nonumber\\ S(0)&=&S_{0}, \label{eq12}\\ C(0)&=&0 .\nonumber \end{eqnarray} $$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. 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 expected initial concentration of Spermidine is \(S_{0}=413.08 \mu M\).
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 expected initial concentration of PotD is:
$$ \begin{equation} P_{0} = 206.54 \mu M. \label{eq13} \end{equation} $$Therefore from \(\eqref{eq13}\) an expected value for \(R_{0}\) is:
$$ \begin{equation} R_{0}=\frac{1}{2} \label{eq14}. \end{equation} $$It is also known that one molecule of spermidine binds to one molecule of PotD. In Kashiwagi's, 1993, paper it was also stated that the dissociation equilibrium constant for PotD and spermidine binding is:
$$ \begin{equation} \frac{k_{off}}{k_{on}}=3.2 \mu M. \label{eq15} \end{equation} $$Therefore from \(\eqref{eq15}\) an expected value for \(\kappa\) is:
$$ \begin{equation} \kappa=129.0875 \label{eq16}. \end{equation} $$Sensitivity analysis was then performed on the non-dimensionalised system by using MATLAB's ode23 solver. A range of values for both \(\kappa\) and \(R_{0}\) were chosen with the estimated values, \(\eqref{eq14}\) and \(\eqref{eq16}\) as the mean value of the range. The results are shown in Figure 2.
References- Guldberg, C. M., Waage, P. (1879). Concerning chemical affinity. Erdmannas Journal fr Practische Chemie, 127, 69-114.
- 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.
- 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.
Saliva: Lactoferrin and Lactoferrin Binding Protein Binding
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. 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. 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.
Nasal Mucus: Oderant Binding Protein Folding
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. 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. 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.
