User:Johnsy/Lipoprotein Modelling/Model Analysis

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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: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \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. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle [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, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 64 \times 10^{-9} M} (Theivagt)
  2. Km1 - The michaelis-menten constant for HMG-CoA reductase, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 20 \times 10^{-6} M} (Theivagt)
  3. Ki1 - Dissociation constant for average statin, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 1.9 \times 10^{-9} M} (Flambers)
  4. dic - Degradation rate of cholesterol, estimated Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 2 \times 10^{-4} min^{-1}}
  5. H0 - Average amount of HMG-CoA in the cell, assumed constant, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 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.

Figure 1. Statin vs. Steady State Concentration of Cholesterol