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Revision as of 14:55, 12 August 2015
Fingerprint Aging
Analysis and Modeling
Principal Component Analysis
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Squalene epoxide and Lanosterol Synthase Binding Model
AimThe aim of a model describing the binding between squalene epoxide and lanosterol synthase is to find the optimum concentration and binding rates that we require for visual detection of squalene epoxide in the fingermark sample from the crime scene. The more squalene epoxide and lanosterol synthase binding the more likely it will be that squalene epoxide will be visually detected.
ResultsThese reactions can be described by the schematic:
$$ \ce{LS + SE<=>[K_{1}][K_{2}] PC ->[K_{3}] La}. $$Where \(LS\) is the concentration of lanosterol synthase, \(SE\) is the concentration of squalene epoxide, \(PC\) is the concentration of the 1st intermdeiate, protosterol cation, and \(La\) is the concentration of lanosterol, the full complex. \(K_{1}\), \(K_{3}\) are the forward reaction rates, and \(K_{2}\) is the reverse reaction rate.
The initial concentration of SE was defined to be \(SE_{0}\) and two parameters of the system were defined as:
$$ \begin{equation*} \lambda=\frac{K_{1}}{K_{2}} SE_{0}, \qquad \gamma=\frac{K_{3}}{K_{2}}. \end{equation*} $$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, lanosterol.
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):
$$ \begin{eqnarray} \frac{dLS}{dt}&=&K_{2}PC - K_{1} LS \cdot SE \nonumber \\ \frac{dSE{dt}&=&K_{2}PC - K_{1} LS \cdot SE \nonumber\\ \frac{dPC}{dt}&=& K_{1} LS \cdot SE - K_{2} PC- K_{3}PC \\ \frac{dLa}{dt}&=&K_{3}PC \nonumber \end{eqnarray} $$with initial conditions:
$$ \begin{eqnarray} LS(0)&=&LS_{0} \quad \mu M \nonumber \\ \nonumber SE(0)&=&SE_{0} \quad \mu M\\ PC(0)&=&0 \quad \mu M\\ La(0)&=&0 \quad \mu M \nonumber \end{eqnarray} $$The parameters were estimated by considering the steady state of the system. Setting the left hand side of (1) to zero gives:
$$ \begin{eqnarray} K_{2} PC&=&K_{1} LS \cdot SE \nonumber \\ K_{1} LS \cdot SE&=&K_{2} PC - K_{3} PC. \end{eqnarray} $$Rearranging (3) gives:
$$ \begin{equation} \frac{PC}{LS \cdot SE=\frac{K_{1}}{K_{2}} \end{equation} $$Considering the first binding reaction, it was found that the total concentration of lanosterol synthase, \(LS\), will be equal to:
$$ \begin{equation} LST=LS+PC. \end{equation} $$Now using (4) and (5) it can be written that:
$$ \begin{equation} \frac{LS}{LST}=\frac{1}{\frac{K_{1}}{K_{2}} SE_{0} + 1}. \end{equation} $$It is known that the ratio between lanosterol synthase and squalene epoxide is 2.561:1 is required for binding, and that they bind at a 1:1 ratio (Boutaud, 1992). Therefore the ratio of free lanosterol synthase to total lanosterol synthase will be:
$$ \begin{equation} \frac{LS}{LST}=\frac{1.561}{2.561}. \end{equation} $$By substituting (7) into equation (6) the ratio between \(K_{1}\) and \(K_{2}\) can be found:
$$ \begin{equation} \frac{K_{1}}{K_{2}}= 0.7810323048\quad \mu M^{-1}. \end{equation} $$For (3), (6) and (7) can be used to find the ratio between \(K_{3}\) and \(K_{2}\):
$$ \begin{equation} \frac{K_{3}}{K_{2}}=1.000223733. \end{equation} $$From Goodman's 1964 paper, it can be calculated that the expected initial concentration of squalene is: \(SE_{0}=\) 0.8202157402 \(\mu M\). It is then assumed that this will be a reasonable estimate for the initial concentration of squalene epoxide. Therefore, from (8) and (9) the estimated values for \(\lambda\) and \(\gamma\) are found to be:
$$ \begin{equation} \lambda=0.64061499, \qquad \gamma=1.000223733. \end{equation} $$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, (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- Boutaud, O., Dolis, D., & Schuber, F. (1992). Preferential cyclization of 2, 3 (S): 22 (S), 23-dioxidosqualene by mammalian 2, 3-oxidosqualene-lanosterol cyclase. Biochemical and biophysical research communications, 188(2), 898-904.
- Goodman, D. S. (1964). Squalene in human and rat blood plasma. Journal of Clinical Investigation, 43(7), 1480.
