Team:Dundee/Modeling/Biospray

FluID

Mathematical Modelling

Overview

Models were created to investigate the binding of the proteins in FluID with their targets in the sample fluids. These models allow us to calculate the optimum binding rates and the optimal concentration of proteins to use in FluID. Calculating the optimal conditions is required to achieve maximum fluorescence from the nanobeads, making detection more efficient. Ordinary Differential Equations (ODEs) were formed using the law of mass action, to describe the change in concentration of each of substances over time. Initial concentrations and binding rate parameters were estimated using information from literature, then sensitivity analysis was used to find the optimal value for these parameters. The ODE’s were solved using MATLAB’s ode23 solver which uses a Rosenbrock formula, the simulations were then run for a range of values for the parameters to perform sensitivity analysis. To see the results of these methods and the specific method for each of the four substances in FluID, click below. We also considered the FluID as a whole and the effect that both the nanobeads and the other proteins would have within the spray.

Blood Model

Consider the binding between haptoglobin and haemoglobin.

Semen Model

Consider the binding between spermidine and potD.

Saliva Model

Consider the binding between lactoferrin and lactoferrin binding protein.

Nasal Mucus Model

Consider the folding of the odorant binding protein.

FluID Concentration

Consider FluID as a whole.

Blood: Haptoglobin and Haemoglobin Binding Model

Aim

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 FluID.

Results

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 schematic:

$$ \ce{Hp + \alpha_{H}<=>[K_{a}][K_{d}] [Hp \cdot \alpha_{H}] ->[K_{i}] [Hp\cdot\alpha_{H}\cdot\beta_{H}]} $$

where $Hp$ is the concentration of free haptoglobin, $\alpha$$_{H}$ is the concentration of free haemoglobin, $[$Hp$\cdot$$\alpha$$_{H}$$]$ is the concentration of haptoglobin-haemoglobin-$\alpha$-chains complex and $[$ Hp$\cdot$$\alpha$$_{H}$$\cdot$$\beta$$_{H}$$]$ is the concentration of the full haptoglobin-haemoglobin complex. $K_{a}$, $K_{i}$ are the forward reaction rates, 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:

$$ \large{ \begin{equation*} \lambda=\frac{K_{a}}{K_{d}} \alpha_{H0}, \qquad \gamma=\frac{K_{i}}{K_{d}}. \end{equation*}} $$

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.

Figure 1: Sensitivity analysis for the binding parameters of haemoglobin and haptoglobin binding showed that optimal complex formation was when both parameters were as high as possible. Note that no complex is formed if either parameter is too small. The higher the complex formation the more fluorescence will be visualised.

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. Note that the effect of concentrations of the four components together within FluID was considered as a whole in the FluID concentration section.

Method

Using the law of mass action (1) the binding reaction schematic was written as a system of ordinary differential equations (ODEs):

$$ \large{ \begin{eqnarray} &\frac{dHp}{dt}=K_{d}[Hp \cdot \alpha_{H}] - K_{a} Hp \alpha_{H}, \nonumber \\ &\frac{d \alpha_{H}}{dt}=K_{d}[Hp \cdot \alpha_{H}] - K_{a} Hp \alpha_{H}, \nonumber\\ &\frac{d[Hp \cdot \alpha_{H}]}{dt}=K_{a} Hp \alpha_{H} - K_{d}[Hp \cdot \alpha_{H}] - K_{i}[Hp \cdot \alpha_{H}],\tag{Eq 1}\\ &\frac{d[Hp \cdot \alpha_{H} \cdot \beta_{H}]}{dt}=K_{i}[Hp \cdot \alpha_{H}], \nonumber \end{eqnarray} } $$

with initial conditions:

$$ \large{ \begin{eqnarray} Hp(0)&=&4.17 \alpha_{H0} \quad \mu M, \nonumber \\ \nonumber \alpha_{H}(0)&=&\alpha_{H0} \quad \mu M,\\ \lbrack Hp \cdot \alpha_{H} \rbrack (0)&=&0 \quad \mu M, \tag{Eq 2}\\ \lbrack Hp \cdot \alpha_{H} \cdot \beta_{H} \rbrack (0)&=&0 \quad \mu M, \nonumber \end{eqnarray} } $$

The parameters were estimated by considering the steady state of the system. Setting the left hand side of (Eq 1) to zero gives:

$$ \large{ \begin{eqnarray} K_{d} [Hp \cdot \alpha_{H}]&=&K_{a} Hp \alpha_{H}, \nonumber \\ K_{a} Hp \alpha_{H}&=&K_{d} [Hp \cdot \alpha_{H}] - K_{i} [Hp \cdot \alpha_{H}]\tag{Eq 3} . \end{eqnarray} } $$

Rearranging (Eq 3) gives:

$$ \large{ \begin{equation} \frac{[Hp \cdot \alpha_{H}]}{Hp \alpha_{H}}=\frac{K_{a}}{K_{d}} \tag{Eq 4}. \end{equation} } $$

Considering the first binding reaction, it was found that the total concentration of haptoglobin, $HpT$, will be equal to:

$$ \large{ \begin{equation} HpT=Hp+[Hp \cdot \alpha_{H}].\tag{Eq 5} \end{equation} } $$

Now using (Eq 4) and (Eq 5) it can be written that:

$$ \large{ \begin{equation} \frac{Hp}{HpT}=\frac{1}{\frac{K_{a}}{K_{d}} \alpha_{H} + 1}. \tag{Eq 6} \end{equation} } $$

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:

$$ \large{ \begin{equation} \frac{Hp}{HpT}=\frac{3.17}{4.17}. \tag{Eq 7} \end{equation} } $$

By substituting (Eq 7) into equation (Eq 6) the ratio between $K_{a}$ and $K_{d}$ can be found:

$$ \large{ \begin{equation} \frac{K_{a}}{K_{d}}=\frac{100}{317} \quad \mu M. \tag{Eq 8} \end{equation} } $$

For (Eq 3), (Eq 6) and (Eq 7) can be used to find the ratio between $K_{i}$ and $K_{d}$:

$$ \large{ \begin{equation} \frac{K_{i}}{K_{d}}=\frac{83}{317}. \tag{Eq 9} \end{equation} } $$

From literature it is known that 2.5 $\times$ 10$^{-5}$ g/cm$^{3}$ haemoglobin is found in blood plasma (2) . It is also known that the molecular weight of haemoglobin is 64458 g/mol. This can be used to calculate the initial concentration of haemoglobin in 1ml of blood: $\alpha_{H0}=$ 0.39 $\mu M$. Therefore, from (Eq 8) and (Eq 9) the estimated values for $\lambda$ and $\gamma$ are found to be:

$$ \large{ \begin{equation} \lambda=\frac{38.78}{317}, \qquad \gamma=\frac{83}{317}. \tag{Eq 10} \end{equation} } $$

By running the ode23 solver (3) over one hundred different values for both parameters, sensitivity analysis can be performed. The range of values has the mean as the estimated values, (Eq 10). 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

Guldberg CM, Waage P. Concerning chemical affinity. Erdmanns Journal fr Practische Chemie 1879; 127: 69-114.
Weatherby D, Ferguson S. Blood Chemistry and CBC Analysis, 4 ed. : Weatherby and Associates; 2004.
Bogacki P, Shampine LF. A 3 (2) pair of Runge-Kutta formulas. Applied Mathematics Letters 1989; 2(4): 321-325.

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Semen: PotD and Spermidine Binding Model

Aim

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 FluID.

Results

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 for finding traces of semen at a crime scene. The binding reaction can be described by the schematic:

$$ \ce{P + S <=>[k_{on}][k_{off}] C } $$

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 initial concentrations of 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:

$$ \large{ \begin{equation*} R_{0}=\frac{P_{0}}{S_{0}}. \end{equation*} } $$

A non-dimensional binding rate parameter was defined as:

$$ \large{ \begin{equation*} \kappa=\frac{k_{on}\cdot S_{0}}{k_{off}}. \end{equation*} } $$

Sensitivity analysis was performed to find the optimum values of both $\kappa$ and $R_{0}$ which give the optimal complex formation.

Figure 2: Sensitivity analysis for the binding efficiency and ratio of initial concentrations, $R_{0}$ for potD and spermidine binding. We see that there are optimal values for both the binding efficiency and the level of potD we add, where adding more makes no significant difference to complex produced. The more complex formed the more fluorescence there will be allowing for visual detection of spermidine.

The sensitivity analysis indicates that there are optimal values for both $\kappa$ and $R_{0}$, where increasing the value has very little effect on the complex formation. This was further investigated, firstly by setting $\kappa$ as the expected value from (Eq 16), and varying $R_{0}$ as above

Figure 3: Complex formation with increasing $R_{0}$, that is the level of potD. The more complexed formed the more fluorescence there will be, allowing for visual detection. From the plot we can see that there is an optimal value of the ratio $R_{0}$ and thus calculate the level of potD required in the biospray.

This suggests that as long as $R_{0}=1.4$, that is there is at least $578.31 \mu M$ of PotD in FluID, there will be enough complex formed to visualise via the nanobeads. The effect of increasing the parameter $\kappa$ was also investigated by setting $R_{0}=1.4$, and varying $\kappa$.

Figure 4: Complex formation with increasing $\kappa$, a parameter that defines the binding efficiency. The more complexed formed the more fluorescence there will be, allowing for visual detection. From the plot we can see that there is an optimal value of $\kappa$ and from this we can calculate the minimum binding efficiency for optimal results.

Varying $\kappa$ highlights that a slightly higher binding rate will allow for a higher level of complex formation. Therefore it is suggested that an optimal value for $\kappa$ is:

$$ \large{ \begin{equation*} \kappa=150.0, \end{equation*} } $$

rather than the value as stated in (Eq 16). Thus to increase the amount of complex formed, the initial concentration of potD and/or the binding affinity needs to be increased in FluID. The information provided by this model was passed to the lab to aid in experimental decision making. Note that the effect of concentrations of the four components together within FluID was considered as a whole in the FluID concentration section.

Method

Using the law of mass action, the reaction scheme can be described by a system of ordinary differential equations (ODEs) (1):

$$ \large{ \begin{eqnarray} \frac{dP}{dt}&=&k_{off}C-k_{on}PS, \nonumber\\ \frac{dS}{dt}&=&k_{off}C-k_{on}PS, \tag{Eq 11} \\ \frac{dC}{dt}&=&k_{on}PS-k_{off}C. \nonumber \end{eqnarray} } $$

where each equation describes the change over time of the three substances in the binding reaction, with initial conditions:

$$ \large{ \begin{eqnarray} P(0)&=&P_{0} , \nonumber\\ S(0)&=&S_{0}, \tag{Eq 12}\\ C(0)&=&0 .\nonumber \end{eqnarray} } $$

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 (2), 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 (3) it was found that a suggested concentration ratio of potD to spermidine is 1:2. Using this it is assumed that the expected initial concentration of potD is:

$$ \large{ \begin{equation} P_{0} = 206.54 \mu M. \tag{Eq 13} \end{equation} } $$

Therefore from (Eq 13) an expected value for $R_{0}$ is:

$$ \large{ \begin{equation} R_{0}=\frac{1}{2} .\tag{Eq 14} \end{equation} } $$

It is also known that one molecule of spermidine binds to one molecule of potD. In Kashiwagi's paper (3) it was also stated that the dissociation equilibrium constant for potD and spermidine binding is:

$$ \large{ \begin{equation} \frac{k_{off}}{k_{on}}=3.2 \mu M. \tag{Eq 15} \end{equation} } $$

Therefore from (Eq 15) an expected value for $\kappa$ is:

$$ \large{ \begin{equation} \kappa = 129.09 .\tag{Eq 16} \end{equation} } $$

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, (Eq 14) and (Eq 16) as half of the maximum value and a quarter of the maximum value, respectively. The results are shown in Figure 2.

References

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

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Saliva: Lactoferrin and Lactoferrin Binding Protein Binding Model

Aim

The aim of modelling of the binding between lactoferrin and lactoferrin binding protein (LPB) is to understand the optimum concentration and binding rates that are required for visual detection of saliva in the sample from the crime scene. The more complex formed the more likely that a visual detection of saliva in the sample will be obtained using FluID.

Results

Lactoferrin is a protein from the transferrin family that is present on saliva and other bodily fluids. Lactoferrin is involved in many biological processes including the transportation of iron. Lactoferrin binding protein A and B form a complex, lactoferrin binding protein (LBP), which binds to lactoferrin to allow iron transportation. LBP will be used as a detector of lactoferrin within the FluID to allow for the detection of saliva in a sample. The binding reaction of lactoferrin and LBP can be described by the schematic:

$$ \ce{Lf + LBP <=>[k_{f}][k_{r}] [Lf\cdot LBP] } $$

where $Lf$ is the concentration of lactoferrin, $LBP$ the concentration of LBP and $[Lf\cdot LBP]$ is the concentration of the lactoferrin-LBP complex. The reaction rate constants are $ k_{f}$ for the forward reaction and $ k_{r}$ for the reverse reaction. The initial concentrations of lactoferrin and LBP are denoted, $Lf_{0}$ and $LBP_{0}$ respectively. The ratio of the initial concentration of LBP to the initial concentration of lactoferrin was defined as:

$$ \large{ \begin{equation*} v_{0}=\frac{LBP_{0}}{Lf_{0}}. \end{equation*}} $$

A non-dimensional binding rate parameter was defined as:

$$ \large{ \begin{equation*} \theta=\frac{k_{f}\cdot Lf_{0}}{k_{r}}. \end{equation*}} $$

Sensitivity analysis was performed to find the optimum values of both $\theta$ and $v_{0}$ which give the optimal complex formation.

Figure 5: Sensitivity analysis for the binding efficiency defined by, binding parameter $\theta$ and ratio of initial concentrations, $v_{0}$, which represents the level of lactoferrin binding protein, LBP. We see that there are relatively low optimal values for both the binding efficiency and the level of LBP we add, where adding more makes no significant difference to complex produced. The more complex formed the more fluorescence there will be allowing for visual detection of lactoferrin.

The sensitivity analysis indicates that there are optimal values for $\theta$, where increasing the value of itself or $v_{0}$ has very little effect on the complex formation. This was further investigated, firstly by setting $\theta$ as the expected value from (Eq 20), and varying $v_{0}$ as above.

Figure 6: Complex formation with increasing levels of lactoferrin binding protein determined by $v_{0}$. The more complexed formed the more fluorescence there will be, allowing for visual detection. From the plot we can see that there is a relatively high optimal value of the ratio $v_{0}$ and thus calculate the level of lactoferrin binding protein required in the biospray to detect lactoferrin.

This suggests that if we increase the ratio of $LBP_{0}$ to $Lf_{0}$ so that it is larger than the expected, more complex will be formed. Since we know the expected value of $Lf_{0}$, it can be suggested that there should be a concentration of:

$$ \large{ \begin{equation*} LBP_{0}=59.22 \quad \mu M, \end{equation*} } $$

contained within the detector to get the optimal result. The effect of increasing the parameter $\theta$ was also investigated by setting $v_{0}=1$, and varying $\theta$.

Figure 7: Complex formation with increasing $\theta$, a parameter that defines the binding efficiency. The more complexed formed the more fluorescence there will be, allowing for visual detection. From the plot we can see that there is an optimal value of $\theta$ and from this we can calculate the minimum binding efficiency for optimal results.

Varying $\theta$ highlights that a higher binding rate will allow for a higher level of complex formation. Since we know the expected value of $Lf_{0}$, it can be suggested that:

$$ \large{ \begin{equation*} \frac{k_{f}}{k_{r}}=14025.97 \mu M^{-1}. \end{equation*}} $$

The information provided by this model was passed to the lab to aid in experimental decision making. Thus to increase the amount of complex formed the initial concentration of $LBP$ and/or the binding affinity needs to be increased in the detector. The information provided by this model was passed to the lab to aid in experimental decision making. Note that the effect of concentrations of the four components together within FluID was considered as a whole in the FluID concentration section.

Method

Using the law of mass action (1), the reaction scheme can be described by a system of ordinary differential equations (ODEs) (Guldberg,1879):

$$ \large{ \begin{eqnarray} &\frac{dLf}{dt}=k_{r}[Lf\cdot LBP]-k_{f}LfLBP, \nonumber\\ &\frac{dLBP}{dt}=k_{r}[Lf\cdot LBP]-k_{f}LfLBP,\tag{Eq 17} \\ &\frac{d[Lf\cdot LBP]}{dt}=k_{f}LfLBP-k_{r}[Lf\cdot LBP]. \nonumber \end{eqnarray} } $$

where each equation describes the change over time of the three substances in the binding reaction, with initial conditions:

$$ \large{ \begin{eqnarray} Lf(0)&=&Lf_{0} , \nonumber\\ LBP(0)&=&LBP_{0},\tag{Eq 18} \\ [Lf\cdot LBP](0)&=&0 .\nonumber \end{eqnarray} } $$

It is assumed that there will be no complex at the start of the reaction, and that there will be some concentration of lactoferrin and LBP. Viejo-Diaz (2), states that there is approximately $1.2 mg \ ml^{-1}$ of lactoferrin in human saliva. This can be used to find that there is an expected concentration of $15.58 \mu M$ in human saliva. Therefore it is assumed that the expected initial concentration of lactoferrin is $Lf_{0}=15.58 \mu M$.

In Perkins-Balding's paper (3) it was also stated that the average dissociation equilibrium constant for lactoferrin and LBP binding is:

$$ \large{ \begin{equation} \frac{k_{r}}{k_{f}}=0.03 \mu M. \tag{Eq 19} \end{equation} } $$

Therefore from (Eq 19) an expected value for $\theta$ is:

$$ \large{ \begin{equation} \theta=623.42. \tag{Eq 20} \end{equation} } $$

Due to a lack of information in literature it is assumed that $v_{0}=1$. Sensitivity analysis was then performed on the non-dimensionalised system by using MATLAB's ode23 solver. A range of values for both $\theta$ and $v_{0}$ were chosen with the estimated values, (Eq 20) and $v_{0}=1$ as half of the maximum value and a quarter of the maximum value, respectively. The results are shown in Figure 5.

References

Guldberg CM, Waage P. Concerning chemical affinity. Erdmanns Journal fr Practische Chemie 1879; 127: 69-114.
Viejo-Díaz M, Andrés MT, & Fierro JF. Modulation of in vitro fungicidal activity of human lactoferrin against Candida albicans by extracellular cation concentration and target cell metabolic activity. Antimicrobial agents and chemotherapy 2004; 48(4): 1242-1248.
Perkins-Balding D, Ratliff-Griffin M, & Stojiljkovic I. Iron transport systems in Neisseria meningitidis. Microbiology and molecular biology reviews 2004; 68(1): 154-171.

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Nasal Mucus: Odorant Binding Protein Folding Model

Aim

The aim of modelling of the folding of odorant binding protein (OBP) is to understand the optimum concentration and binding rates that are required for visual detection of nasal mucus 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 FluID.

Results

Odorant binding protein 2A (OBPIIa) is found in human nasal mucus and are involved in odorant 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 odorant detection by the OBP, this is called folding (1). For detection of nasal mucus at the scene of a crime, modified $\beta$-barrels will be within FluID. 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:

$$ \large{ \begin{equation*} D_{0}=\frac{\beta_{0}}{\alpha_{0}}. \end{equation*}} $$

A non-dimensionalised parameter describing the rate of folding was also defined as:

$$ \large{ \begin{equation*} \psi=\frac{k_{1}\cdot\alpha_{0}}{k_{-1}}. \end{equation*}} $$

Sensitivity analysis was performed to find the optimum values of both $\psi$ and $D_{0}$, which give the optimal level of complex formation.

Figure 8: Sensitivity analysis for the binding efficiency, defined by parameter $\psi$ and ratio of initial concentrations, $D_{0}$ for potD and spermidine binding. We see that there are optimal values for both the binding efficiency and the level of OBP $\beta$-chains we add, where adding more makes no significant difference to complex produced. The more complex formed the more fluorescence there will be allowing for visual detection of odorant binding proteins.

The sensitivity analysis indicates that there are optimal values for both $\psi$ and $D_{0}$, where increasing the value has very little effect on the complex formation. This was further investigated, firstly by setting $\psi$ as the expected value from (Eq 28), and varying $D_{0}$ as above.

Figure 9: Complex formation over time with increasing $D_{0}$. Complex formation with increasing levels of odorant binding protein $\beta$-chains determined by $D_{0}$. The more complexed formed the more fluorescence there will be, allowing for visual detection. From the plot we can see that there is a relatively high optimal value of the ratio $D_{0}$ and thus calculate the level of $\beta$-chains required in the biospray to detect odorant binding protein.

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:

$$ \large{ \begin{equation*} \beta_{0}=39.68 \quad \mu M, \end{equation*} } $$

contained within the FluID to get the optimal result. The effect of increasing the parameter $\psi$ was also investigated by setting $D_{0}=1.5$, and varying $\psi$.

Figure 10: Complex formation with increasing $\psi$, a parameter that defines the binding efficiency. The more complexed formed the more fluorescence there will be, allowing for visual detection. From the plot we can see that there is an optimal value of $\psi$ and from this we can calculate the minimum binding efficiency for optimal results.

Varying $\psi$ highlights that a slightly higher binding rate will allow for a higher level of OBP formation. Since we know the expected value of $\alpha_{0}$, it can be suggested that:

$$ \large{ \begin{equation*} \frac{k_{1}}{k_{-1}}=226.8, \end{equation*}} $$

rather than $200$ as stated in (Eq 28). The information provided by this model was passed to the lab to aid in experimental decision making. Thus to increase the amount of complex formed the initial concentration of $\beta$-barrels and/or the binding affinity needs to be increased in FluID. The information provided by this model was passed to the lab to aid in experimental decision making. Note that the effect of concentrations of the four components together within FluID was considered as a whole in the FluID concentration section.

Method

Using the law of mass action, the scheme describing the folding can be written as a system of ordinary differential equations (ODEs) (2):

$$ \large{ \begin{eqnarray} \frac{d\alpha}{dt}&=&k_{-1}OBP-k_{1}\alpha\beta, \nonumber\\ \frac{d\beta}{dt}&=&k_{-1}OBP-k_{1}\alpha\beta, \tag{Eq 21}\\ \frac{dOBP}{dt}&=&k_{1}\alpha\beta-k_{-1}OBP.\nonumber \end{eqnarray}} $$

where each equation describes the change over time of the three substances in the folding reaction, with initial concentrations:

$$ \large{ \begin{eqnarray} \alpha(0)&=&\alpha_{0} \quad \mu \text{M}, \nonumber\\ \beta(0)&=&\beta_{0}\quad \mu \text{M}, \tag{Eq 22}\\ OBP(0)&=&0 \quad \mu \text{M}.\nonumber \end{eqnarray}} $$

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 odorant binding protein, $\alpha_{0}$ and $\beta_{0}$, can be calculated from values provided from literature. From work by Schiefner (1) and Briand (3), the expected initial concentration of $\alpha$ was calculated as:

$$ \large{ \begin{equation} \alpha_{0}=26.46 \quad \mu \text{M}.\tag{Eq 23} \end{equation}} $$

From literature (1) (3) it was also estimated that the expected ratio between $\alpha$ and $\beta$ was 1:1, therefore the expected value of $D_{0}$ is:

$$ \large{ \begin{equation} D_{0}=1. \tag{Eq 24} \end{equation} } $$

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, (4), to be:

$$ \large{ \begin{equation} K_{d}=\frac{k_{-1}}{k_{1}}=10^{-8} \rightarrow 10^{2} \mu \text{M}. \tag{Eq 25} \end{equation} } $$

Using (Eq 23) and the mean value of (Eq 25), an estimated expected value for $\psi$, can be determined as:

$$ \large{ \begin{equation} \psi=1322751323. \tag{Eq 26} \end{equation} } $$

However, this value was too large to compute using MATLAB so a smaller value was chosen, where:

$$ \large{ \begin{equation} \frac{k_{1}}{k_{-1}}=200 \quad \mu M^{-1}. \tag{Eq 27} \end{equation} } $$

From (Eq 23) and (Eq 27) a more suitable value for $\psi$ was chosen to be:

$$ \large{ \begin{equation} \psi=5291.01. \tag{Eq 28} \end{equation} } $$

The system of non-dimensionalised ODE's derived from (Eq 21) were solved using MATLAB's ode23 solver (5). Sensitivity analysis was performed by solving the system with a range of values for $\psi$ and $D_{0}$, where the expected values, (Eq 23) and (Eq 28), were a quarter of the maximum value and a half of the maximum value respectively. The results are shown in Figure 8.

References

Schiefner A, Freier R, Eichinger A, Skerra A. Crystal structure of the human odorant binding protein, OBPIIa. Proteins: Structure, Function, and Bioinformatics 2015; 83(6): 1180-1184.
Guldberg CM, Waage P. Concerning chemical affinity. Erdmanns Journal fr Practische Chemie 1879; 127: 69-114.
Briand L, Eloit C, Nespoulous C, Bezirard V, Huet JC, Henry C, Pernollet JC. Evidence of an odorant-binding protein in the human olfactory mucus: location, structural characterization, and odorant-binding properties. Biochemistry 2002; 41(23): 7241-7252.
Archakov AI, Govorun VM, Dubanov AV, Ivanov YD, Veselovsky AV, Lewi P, Janssen P. Protein protein interactions as a target for drugs in proteomics. Proteomics 2003; 3(4): 380-391.
Bogacki P, Shampine LF. A 3 (2) pair of Runge-Kutta formulas. Applied Mathematics Letters 1989; 2(4): 321-325.

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FluID Concentration

Aim

This section considers the effect of the nanobeads and the effect of the proteins on each other.

Nanobeads

Nanobeads will be attached to the proteins within FluID to allow for visual detection. We wanted to consider whether or not the nanobeads would affect the binding affinities of the proteins in the spray with their targets in the fluid sample. After some research it was found that since biotin is smaller than most labels it is unlikely to affect the protein function or protein-protein interactions (1). Therefore we do not need to consider this further, we then wanted to investigate the affect of the proteins on each other within the spray.

Interactions within FluID

From the four binding models suggested concentrations for each of the proteins within the spray were given. However these concentrations will only be for when it is only the one protein in the solution. We want to calculate the concentrations required for FluID to contain all four proteins at once.

Firstly recall the suggested concentrations from each of the binding models:

$$ \large{ \begin{eqnarray*} Hp_{0}=1.62 \mu M, \\ P_{0}=578.31 \mu M, \\ LBP_{0}=59.22 \mu M, \\ \beta_{0}=39.68\mu M. \end{eqnarray*} } $$

Where $Hp, P, LBP, \beta$ represent haptoglobin, potD, lactoferrin binding protein and OBP $\beta$-chains respectively. Using the equation:

$$ \large{ \begin{equation*} Concentration(M)=\frac{moles}{Volume(L)}, \end{equation*}} $$

the number of moles of each protein can be calculated.

$$ \large{ \begin{eqnarray*} Hp_{0}&=&1.62 \times 10^{-8} moles, \\ P_{0}&=&5.78\times 10^{-6} moles, \\ LBP_{0}&=&5.92\times 10^{-7} moles, \\ \beta_{0}&=&3.97\times 10^{-7} moles. \end{eqnarray*} } $$

Then by multiplying all by Avogadro's constant ($6.022 \times 10^{23}$), the number of molecules can be found.

$$ \large{ \begin{eqnarray*} Hp_{0}&=&9.74 \times 10^{15} molecules, \\ P_{0}&=&3.48 \times 10^{18} molecules, \\ LBP_{0}&=&3.57 \times 10^{17} molecules, \\ \beta_{0}&=&2.39 \times 10^{17} molecules. \end{eqnarray*} } $$

Since we require all four proteins within the one spray, we divide the number of molecules required by four.

$$ \large{ \begin{eqnarray*} Hp_{0}&=&2.43 \times 10^{15} molecules, \\ P_{0}&=&8.71 \times 10^{17} molecules, \\ LBP_{0}&=&8.92 \times 10^{16} molecules, \\ \beta_{0}&=&5.97 \times 10^{16} molecules. \end{eqnarray*}} $$

Then using the equation

$$ \large{ \begin{equation*} mass=\frac{MW \times molecules}{6.022 \times 10^{23}}, \end{equation*}} $$

along with the molecular weights, found from literature (2,3,4,5):

$$ \large{ \begin{eqnarray*} Hp_{0}&=&0.04 Da, \\ P_{0}&=&0.04 Da, \\ LBP_{0}&=&0.18 Da, \\ \beta_{0}&=&0.02 Da, \end{eqnarray*}} $$

the mass (grams) can be calculated:

$$ \large{ \begin{eqnarray*} Hp_{0}&=&1.78 \times 10^{-10} g, \\ P_{0}&=&5.93 \times 10^{-8} g, \\ LBP_{0}&=&2.69 \times 10^{-8} g, \\ \beta_{0}&=&2.80 \times 10^{-9} g. \end{eqnarray*}} $$

Finally the equation:

$$ \large{ \begin{equation*} Concentration=\frac{Mass}{Volume \times MW}, \end{equation*}} $$

can be used to calculate the new concentrations required in FluID for each of the proteins:

$$ \large{ \begin{eqnarray*} Hp_{0}=0.40 \mu M, \\ P_{0}=144.58 \mu M, \\ LBP_{0}=14.81 \mu M, \\ \beta_{0}=14.81\mu M. \end{eqnarray*}} $$

This information was passed onto the lab for future decision making.

References

ThermoFisher Scientific. 2015. Overview of Protein Labeling. [ONLINE] Available at: https://www.thermofisher.com/uk/en/home/life-science/protein-biology/protein-biology-learning-center/protein-biology-resource-library/pierce-protein-methods/overview-protein-labeling.html#/legacy=www.piercenet.com. [Accessed 26 August 15].
Segawa T, Amatsuji H, Suzuki K, Suzuki M, Yanagisawa M, Itou T, Nakanishi T. Molecular characterization and validation of commercially available methods for haptoglobin measurement in bottlenose dolphin. Results in Immunology 2003; 3: 57-63.
Shah P, Swiatlo E. Immunization with polyamine transport protein PotD protects mice against systemic infection with Streptococcus pneumoniae. Infection and Immunity 2006; 74(10): 5888-5892.
Perkins-Balding D, Ratliff-Griffin M, Stojiljkovic I. Iron transport systems in Neisseria meningitidis. Microbiology and Molecular Biology Reviews 2004; 68(1): 154-171.
Schiefner A, Freier R, Eichinger A, Skerra A. Crystal structure of the human odorant binding protein, OBPIIa. Proteins: Structure, Function, and Bioinformatics 2015; 83(6): 1180-1184

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