Team:Dundee/Modelling/Biospray
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):
$$ \begin{eqnarray} \frac{dHp}{dt}&=&K_{d}[Hp \cdot \alpha_{H}] - K_{a} Hp \alpha_{H} \nonumber \\ \frac{d \alpha_{H}}{dt}&=&K_{d}[Hp \cdot \alpha_{H}] - K_{a} Hp \alpha_{H} \nonumber\\ \frac{d[Hp \cdot \alpha_{H}]}{dt}&=& K_{a} Hp \alpha_{H} - K_{d}[Hp \cdot \alpha_{H}] - K_{i}[Hp \cdot \alpha_{H}] \label{eq1} \\ \frac{d[Hp \cdot \alpha_{H} \cdot \beta_{H}]}{dt}&=&K_{i}[Hp \cdot \alpha_{H}] \nonumber \end{eqnarray} $$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
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Nasal Mucus: Oderant Binding Protein Folding
AimThe 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.
ResultsOderant binding protein 2A (OBPIIa) is found in human nasal mucus and are involved in oderant detection. Within the protein there is an eight stranded \(\beta\)-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 \(k_{1}\) and \(k_{-1}\) respectively. The initial concentrations of \(\alpha\), \(\beta\) and \(OBP\) are denoted \(\alpha_{0}\), \(\beta_{0}\) and \(OBP_{0}\), respectively, where the ratio of \(\beta_{0}\) to \(\alpha_{0}\) is denoted:
$$ \begin{equation*} D_{0}=\frac{\beta_{0}}{\alpha_{0}}. \end{equation*} $$A non-dimensionalised parameter describing the rate of folding was also defined as:
$$ \begin{equation*} \psi=\frac{k_{1}\cdot\alpha_{0}}{k_{-1}}. \end{equation*} $$Sensitivity analysis was performed to find the optimum values of both \(\psi\) and \(D_{0}\), which give the optimal level of complex formation.
The sensitivity analysis indicates that there are optimal values for both \(\psi\) and \(D_{0}\), where increasing there value has very little effect on the complex formation. This was further investigated, firstly by setting \(\psi\) as the expected value from \(\eqref{eq24}\), and varying \(D_{0}\) as above.
This suggests that if we increase the ratio of \(\beta_{0}\) to \(\alpha_{0}\) so that it is larger than the expected, more complex will be formed. Since we know the expected value of \(\alpha_{0}\), it can be suggested that there should be a concentration of:
$$ \begin{equation*} \beta_{0}=39.68253969 \quad \mu M, \end{equation*} $$contained within the BioSpray to get the optimal result. The effect of increasing the parameter \(\psi\) was also investigated by setting \(D_{0}=1.5\), and varying \(\psi\).
Varying \(\psi\) highlights that even with a reduced binding rate, the same level of complex will be formed. Since we know the expected value of \(\alpha_{0}\), it can be suggested that:
$$ \begin{equation*} \frac{k_{1}}{k_{-1}}=151.2 \quad \mu M^{-1}, \end{equation*} $$rather than \(200\) as previously stated in \(\eqref{eq23}\). Thus to increase the amount of complex formed only the initial concentration of \(\beta\)-barrels needs to be increased in the BioSpray. The information provided by this model was passed to the lab to aid in experimental decision making.
MethodUsing the law of mass action, the scheme describing the folding can be written as a system of ordinary differential equations (ODEs) (Guldberg,1879):
$$ \begin{eqnarray} \frac{d\alpha}{dt}&=&k_{-1}OBP-k_{1}\alpha\beta, \nonumber\\ \frac{d\beta}{dt}&=&k_{-1}OBP-k_{1}\alpha\beta, \label{eq17}\\ \frac{dOBP}{dt}&=&k_{1}\alpha\beta-k_{-1}OBP.\nonumber \end{eqnarray} $$where each equation describes the change over time of the three substances in the folding reaction, with initial concentrations:
$$ \begin{eqnarray} \alpha(0)&=&\alpha_{0} \quad \mu \text{M}, \nonumber\\ \beta(0)&=&\beta_{0}\quad \mu \text{M}, \label{eq18}\\ OBP(0)&=&0 \quad \mu \text{M}.\nonumber \end{eqnarray} $$It is assumed that there will be no folded OBP at the start of the reaction, and that there will be some concentration of \(\alpha\) helices and \(\beta\)-barrels.
The initial expected 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 expected initial concentration of \(\alpha\) was calculated as:
$$ \begin{equation} \alpha_{0}=26.45502646 \quad \mu \text{M}. \label{eq19} \end{equation} $$From literature (Schiefner (2015) and Briand (2002)) it was also estimated that the expected ratio between \(\alpha\) and \(\beta\) was 1:1, therefore the expected value of \(D_{0}\) is:
$$ \begin{equation} D_{0}=1. \label{eq20} \end{equation} $$Due to a lack of experimental data and previous research a specific expected value for both kinetic rates, \(k_{1}\) and \(k_{-1}\), cannot be stated. However, a standard range for a dissociation constant, \(K_{d}\), was found by Archakov, (2003), to be:
$$ \begin{equation*} K_{d}=\frac{k_{-1}}{k_{1}}=10^{-8} \rightarrow 10^{2} \mu \text{M}. \label{eq21} \end{equation*} $$Using \(\eqref{eq19}\) and the mean value of \(\eqref{eq21}\), an estimated expected value for \(\psi\), can be determined as:
$$ \begin{equation} \psi=1322751323. \label{eq22} \end{equation} $$However, this value was too large to compute using MATLAB so a smaller value was chosen, where:
$$ \begin{equation} \frac{k_{1}}{k_{-1}}=200 \quad \mu M^{-1}. \label{eq23} \end{equation} $$From \(\eqref{eq19}\) and \(\eqref{eq23}\) a more suitable value for \(\psi\) was chosen to be:
$$ \begin{equation} \psi=5291.005292. \label{eq24} \end{equation} $$The system of non-dimensionalised ODE's derived from \(\eqref{eq17}\) were solved using MATLAB's ode23 solver (Bogacki 1989). Sensitivity analysis was performed by solving the system with a range of values for \(\psi\) and \(D_{0}\), where the mean of the range was set as the expected values, \(\eqref{eq19}\) and \(\eqref{eq24}\). The results are shown in Figure 4.
References- 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.
- Bogacki, P., Shampine, L. F. (1989). A 3 (2) pair of Runge-Kutta formulas. Applied Mathematics Letters, 2(4), 321-325.
- 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.
- 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.
