User:Johnsy/Lipoprotein Modelling/De Novo Cholesterol

Background

We first start off with a diagram of our pathway showing all of the components that we wish to model in the de novo cholesterol biosynthesis.

The Model

The equation for the production of HMG-CoA reductase taking into account the genetic inhibition from the intercellular cholesterol pools.

[math]\displaystyle{ \frac{d[HMGR]}{dt} = \frac{k_1}{b_1 + [IC]} - d_1[HMGR] }[/math]

The equation for the production of cholesterol taking into account the action of statins (as allosteric inhibitor of HMGR).

[math]\displaystyle{ \frac{d[IC]}{dt} = \frac{k_2[HMGR][HMG-CoA]}{k_{m1} + [HMG-CoA] + \frac{k_{m1}}{k_i}[statin]} - d_{ic}[IC] }[/math]

Parameters

• k₁ – transcription rate
• b₁ – attenuation factor for the regulation of HMGR by cholesterol
• d₁ – degradation rate of HMGR

• k₂ – rate constant for conversion of HMG-CoA to cholesterol mediated by HMGR
• k_m1 – Michaelis-Menten constant for HMGR
• k_i – inhibition constant for the action of statins
• d_ic – degradation rate of cholesterol to cholesterol derivatives

Assumptions

HMGR Equation

An increase in cholesterol will decrease the activity of the enzyme by reducing the gene expression via the action of SREBP (not shown). This is a gross simplification of the actual mechanism for controlling gene expression of sterol regulatory enzymes but has been taken to reduce the complexity of the model and to obtain a general understanding of the dynamics involved.

We assume that the enzyme is constantly expressed with a rate constant of k₁, but this may not be true in vivo. The most probable regulation would be an inhibitory mechanism, and our regulation by cholesterol strives to achieve this inhibition with an inverse relationship. A more precise equation would include Hill cooperativity. Again, this would only serve to complicate the model and increase the number of parameters that need to be analyzed. The lack of parameters, especially for such genetic components in the literature has prevented a more thorough model from being developed.

Cholesterol equation

We have discarded the rest of the mevalonate pathway and have focused on the rate limiting step of cholesterol synthesis, the HMGR catalyzed step from HMG-CoA to mevalonate. The subsequent steps to form cholesterol from mevalonate are relatively fast and so are the steps from the cholesterol precursor acetyl-CoA to HMG-CoA. We have thus eliminated them from the model to simplify not only the number of parameters, but the species involved. In effect, all the rate constants that are involved in the complete pathway can be incorporated into k₂.

We assume that the action of statins is an allosteric inhibitor as shown with the equation. An increase in the amount of statins will tend to decrease the level of intracellular cholesterol by blocking the HMGR enzyme from producing more cholesterol (hence the overall inverse relationship).

For now, we have just put in a degradation term, but this is only temporary until we are able to complete the entire model. The degradation term represents the conversion of cholesterol to steroid hormones and bile acids (in the liver) and other sources of loss such as from sloughing off of the skin cells or the intestinal epithelium.

We have also assumed that there is sufficient HMG-CoA concentration for cholesterol production to take place and that the presence of HMGR will be the limiting factor in this enzymatic step.

Quasi Steady State Approximation

We have decided to assume that the level of enzyme reaches steady-state quickly, even though the actual genetic regulation of gene expression takes on the order of 40 minutes to act. With this approximation, we can eliminate one of the variables as follows:

[math]\displaystyle{ \frac{d[HMGR]}{dt} = 0 }[/math]

Solving for the steady state value of [HMGR]:

[math]\displaystyle{ [HMGR]^* = \frac{k_1}{d_1(b_1 + [IC])} }[/math]

Substituting into the equation governing cholesterol production, we are left with a one dimensional equation for the de novo synthesis of cholesterol.

[math]\displaystyle{ \frac{d[IC]}{dt} = \frac{k_1}{d_1(b_1 + [IC])}\frac{k_2[HMG-CoA]}{k_{m1} + [HMG-CoA] + \frac{k_{m1}}{k_i}[statin]} - d_{ic}[IC] }[/math]

Justifying the Genetic Regulation Term for the Production of HMGR

We use the following network to justify the inverse regulatory component from intercellular cholesterol that we use for the production of HMGR:

We can come up with the following differential equations to model the genetic system of transcription and translation.

[math]\displaystyle{ \frac{d[mRNA]}{dt} = \frac{g_1g_m^n}{k_m^n + [IC]^n} - w_1[mRNA] }[/math]

[math]\displaystyle{ \frac{d[HMGR]}{dt} = g_2[mRNA] - w_2[HMGR] }[/math]

The parameters of the this model are defined below:

• g₁ - transcription rate
• g₂ - translation rate
• g_k - DNA binding constant
• n - Hill cooperativity
• w₁ - mRNA degradation rate
• w₂ - HMGR degradation rate

We again assume that the mRNA will be at steady state:

[math]\displaystyle{ \frac{d[mRNA]}{dt} = 0 }[/math]

So that we can come up with the fixed point and substitute into the HMGR equation:

[math]\displaystyle{ \frac{d[HMGR]}{dt} = \frac{g_1 g_2 g_m^n}{w_1(g_m^n + [IC]^n)} - w_2[HMGR] }[/math]

Comparing this with the above equation for HMGR, we can see some equivalents to the equation which will help us to determine the value of the parameters that we are using. For the equations above, we assume that the Hill cooperativity constant n = 1, as we assume that only one molecule of cholesterol (actually cholesterol equivalent through the action of SREBP) will be sufficient to begin transcription of the genes required to express HMGR.

[math]\displaystyle{ k_1 = \frac{g_1 g_2 g_m}{w_1} }[/math]

Also,

[math]\displaystyle{ b_1 = g_m }[/math]

and

[math]\displaystyle{ d_1 = w_2 }[/math]