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.
Semen: PotD and Spermidine Binding
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Nasal Mucus: Oderant Binding Protein Folding
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