User:Johnsy/Lipoprotein Modelling/Model Analysis

=Model Analysis=

De Novo Synthesis Pathway and Degradation
Let us start by considering a simple system taking into account only cholesterol synthesis from HMG-CoA and it's degradation to either bile acids, steroid hormones, or other cholesterol derivatives. We can also model the action of statins as a competitive inhibitor of the enzyme HMG-CoA reductase, the main limiting enzyme of cholesterol biosynthesis. One of the key assumptions that is made is that the level of enzyme is constant (quasi-steady state approximation). Although this does not hold due to the genetic component, we will investigate the use of delay differential equations when considering a further extension to the model.

The equation we first consider is: $$\frac{d[IC]}{dt} = \frac{V_1[H_0]}{K_{m1}+[H_0]+\frac{k_{m1}}{k_{i1}}[Statin]} - d_{ic}[IC]$$

Solving for the fixed point of the equation is straightforward and we are left with the following steady state transfer function. $$[IC]* = \frac{V_1[H_0]}{d_{ic}(K_{m1}+[H_0]+\frac{k_{m1}}{k_{i1}}[Statin])}$$

The parameters in the equation are shown below with their approximate values and references.
 * 1) V1 - The Vmax rate for HMG-CoA reductase, $$64 \times 10^{-9} M$$ (Theivagt)
 * 2) Km1 - The michaelis-menten constant for HMG-CoA reductase, $$20 \times 10^{-6} M$$ (Theivagt)
 * 3) Ki1 - Dissociation constant for average statin, $$1.9 \times 10^{-9} M$$ (Flambers)
 * 4) dic - Degradation rate of cholesterol, estimated $$2 \times 10^{-4} min^{-1}$$
 * 5) H0 - Average amount of HMG-CoA in the cell, assumed constant, $$30 \times 10^{-6} M$$ (Corsini)

The graph in Figure 1 shows the effect of an increase in statin levels versus the steady state concentration of cholesterol.