# Final Model Development

## Basic Key Aspects

1. Lipoprotein production, conversion, and metabolism
2. LDL receptor production and display
3. De novo cholesterol synthesis via the mevalonate pathway
4. Bile acid production and cholesterol recycling in the liver

### Lipoprotein production, conversion, and metabolism

• We assume a fixed rate of production and excretion of VLDL from the cells (especially hepatocytes)
• $\frac{d[VLDL]}{dt} = u_v$
• The conversion of VLDL to IDL depends on the concentration of VLDL present in the blood plamsa
• $\frac{d[VLDL]}{dt} = -k_v[VLDL]$
• $\frac{d[IDL]}{dt} = k_v[VLDL]$
• The conversion of IDL to LDL depends on the concentration of IDL present in the blood plasma
• $\frac{d[IDL]}{dt} = -k_i[IDL]$
• $\frac{d[LDL]}{dt} = k_i[IDL]$
• Binding and internaliztion of IDL and LDL occur via the LDL receptor
• $\frac{d[IDL]}{dt} = -d_i[IDL]\phi_{LR}$
• $\frac{d[LDL]}{dt} = -d_l[LDL]\phi_{LR}$
• $\frac{d\phi_{LR}}{dt} = -b(d_i[IDL]+d_l[LDL])\phi_{LR}$
• Once lipoproteins are internatlized, they are broken down to their constitutive components such as proteins and fatty acids).
• $\frac{d[IC]}{dt} = (\chi_id_i[IDL]+\chi_ld_l[LDL])\phi_{LR}$

### LDL receptor production and display

• Production of LDL receptors occurs via transcription of genes that are itself dependent upon binding of SREBP to SRE's regulating transcription of gene sites.
• Receptors are either degraded or recycled back to the cell surface once they have been internalized.
• $\frac{d\phi_{LR}}{dt} = c\frac{1-\phi_{LR}}{[IC]}$ --> recycle term depending upon intracellular cholesterol concentration taking into account nuclear synthesized receptors

### De Novo Cholesterol synthesis

• Cholesterol can be derived from the metabolite acetyl-CoA and via a multi-stage process partially known as the mevalonate pathway.
• The critical rate determining step of the conversion is the enzymatic reaction of HMG-CoA to mevalonate utilizing the enzyme HMG-CoA reductase.
• We have taken the simple assumption that the concentration of the enzyme (HMG-CoA reductase) remains the same throughout time. Although this assumption is valid in the short term, this is no longer valid in the long term. The enzyme is produced by transcription and control of this transcription is thought also to be via SRE's.
• The effect of statins (HMG-CoA reductase inhibitors) is to competitively inhibit the active site of the enzyme as well as cause a structural change to render the enzyme unable to preform its functions.
• Cholesterol levels in the cell attenuate the action of HMG-CoA reductase such that higher cholesterol levels decrease enzymatic activity
• The Reaction Mechanism for the de novo synthesis of cholesterol is outlined below.
• Let S = HMG-CoA, E = HMG-CoA Reductase, P = Cholesterol (assume that the concentration of cholesterol is equivalent to the concentration of mevalonate since the reaction sequences are fast compared to this rate determining step), T = Statin

$E + S \begin{matrix} k_1 \\ \longrightarrow \\ \longleftarrow \\ k_{-1} \end{matrix} ES \xrightarrow{k_2} E + P$
AND
$E + T \begin{matrix} k_3 \\ \longrightarrow \\ \longleftarrow \\ k_{-3} \end{matrix} ET$

Individual equations were drawn up for each reaction

\begin{alignat}{2} \frac{d[E]}{dt} & = -k_1[E][S] + k_{-1}[ES] + k_2[ES] - k_3[E][T] + k_{-3}[ET] \\ \frac{d[S]}{dt} & = -k_1[E][S] + k_{-1}[ES] \\ \frac{d[ES]}{dt} & = k_1[E][S] - k_{-1}[ES] - k_2[ES] \\ \frac{d[ET]}{dt} & = k_3[E][T] - k_{-3}[ET] \\ \frac{d[T]}{dt} & = - k_3[E][T] + k_{-3}[ET] \\ \frac{d[P]}{dt} & = k_2[ES] \end{alignat}
We also include the synthesis of the enzyme that is controlled by the level of intracellular cholesterol as well as a degradation term for the enzyme shown below. We assume a very simple inverse relationship between the cholesterol level and the production of enzyme such that a decrease in cholesterol will produce an increase in enzyme level and vice versa. $\frac{d[E]}{dt} = \frac{j}{[P]+w} - d_g[E]$
Under the assumption that $\frac{d[ES]}{dt} = 0$ (steady state approximation), we can derive the michaelis-menten constant, $k_m = \frac{k_{-1} + k_2}{k_1}$ and the relationship $k_m = \frac{[E][S]}{[ES]}$. Under the assumption that $\frac{d[ET]}{dt} = 0$, we can also derive the dissociation constant, $k_i = \frac{k_{-3}}{k_3}$ and the relationship $k_i = \frac{[E][T]}{[ET]}$.
The complete differential equation describing the enzyme concentration levels is shown below.
$\frac{d[E]}{dt} = \frac{j}{[P]+w} - d_g[E] -k_1[E][S] + k_{-1}[ES] + k_2[ES] - k_3[E][T] + k_{-3}[ET]$
Under the steady state approximation, we again get to the relation that $\frac{d[E]}{dt} = \frac{j}{[P]+w} - d_g[E]$, which is slightly unreasonable since it does not incorporate the level of statins into the model clearly meaning that one of our assumptions is incorrect.

### Bile acid production and cholesterol recycling in the liver

• Intracellular cholesterol is converted to bile acids as one of the metabolites of cholesterol. This can be modeled with the following chemical reaction:
$IC \xrightarrow{d_{BA}} BA$

And the following mathematical differential equations describing the changes to the intracelllular cholesterol levels and the bile acid levels:
\begin{alignat}{2} \frac{d[BA]}{dt} & = d_{BA}[IC] \\ \frac{d[IC]}{dt} & = - d_{BA}[IC] \end{alignat}

• The bile acids are excreted from the liver and stored in the gall bladder before excretion into the gut as a fat emulsifier
• Most of the bile acids that are excreted are taken up by the gut and are returned to the liver
• Only about 3% of the bile acids are not absorbed representing the only exit for cholesterol. This can be modeled as a degradation of the bile acids with a rate constant Bd and the following chemical reaction, where η is the fraction of bile acids that is returned to the liver:
$BA \xrightarrow{B_d(1-\eta)} \phi$

The following mathematical differential equation can describe this process:
$\frac{d[BA]}{dt} = -B_d(1-\eta)[BA]$

• Bile acid binding resins prevent the reabsorption of bile acids and will hence increase the outflow of cholesterol from the body. This can be modeled using an inverse relationship such that an increase in the amount of resins will decrease the rate at which the bile acids are recycled back to the liver. In the equations below, BR is the rate at which the bile acids are recycled, taking into account the reabsorption of the bile acids as well as the processing of the bile acids back to cholesterol. We are assuming that this can be modelled as a constant process. The following mathematical differential equations can be used to describe this process:

\begin{alignat}{2} \frac{d[BA]}{dt} & = - \frac{B_R\eta}{[R]} [BA] \\ \frac{d[IC]}{dt} & = \frac{B_R\eta}{[R]} [BA] \end{alignat}

Taking everything into consideration, we can combine the equations to get two general equations for the modelling of the bile acid recycling (or enteropatic circulation) of cholesterol.
\begin{alignat}{2} \frac{d[BA]}{dt} & = d_{BA}[IC] - B_d(1-\eta)[BA] - \frac{B_R\eta}{[R]} [BA] \\ \frac{d[IC]}{dt} & = - d_{BA}[IC] + \frac{B_R\eta}{[R]} [BA] \end{alignat}

## Further Considerations

1. HDL synthesis and metabolism
2. Internalization of lipoproteins via clathrin coated pits
3. The effect of cholesterol on SREBP with its implications on transcription of important genes

### HDL synthesis and metabolism

• HDL is produced as nascent particles by the liver
• Uses the enzyme CETP to transfer cholesterol esters from VLDL, IDL and LDL particles
• Has the effect of reducing the cholesterol levels within VLDL, IDL and LDL
• HDL is taken up by the liver via the receptor SRB1 and the cholesterol returned to the intracellular cholesterol pool.
• In peripheral tissues, HDL attaches to HDL receptors which transfer excess cholesterol to the HDL particle to be transported back to the liver conferring anti-atherosclerotic properties to HDL

### Internalization of lipoproteins via clathrin coated pits

• Although LDL receptors are implanted into the plasma membrane at random, they are required to be within clathrin coated pits before internalization/endocytosis can occur
• It has not been observed in humans that lipoproteins are internalized without being bound to receptors, most probably due to the size of the lipoprotein itself

### Effect of cholesterol on SREBP with its implications on transcription of important genes

• SREBP regulates the transcription of genes necessary for the synthesis of fatty acids and cholesterol
• SREBP is a membrane bound protein possessing two cleavage sites. Cleavage at site 1, the required initial cleavage, separates two membrane bound segments of premature SREBP.
• Cleavage at site 2 is the activating cleavage, splicing mature SREBP from the membrane bound protein allowing it to enter the nucleus.
• SREBP itself is regulated through the action of SCAP (SREBP cleavage activating protein), which has a sterol-sensing domain.
• In the absence of sterols, SCAP activates cleavage of the SREBP membrane bound protein resulting in an increase in enzyme levels for cholesterol biosynthesis.

## Parameter Values

### Lipoprotein production, conversion, and metabolism

• $k_v = 0.3 h^{-1}$, source Packard et al, 2000
• $k_i = 0.05 h^{-1}$, source Packard et al, 2000
• $u_v = 0.3 g(lh)^{-1}$, source White et al, 1984
• $b = 0.1 lg^{-1}$ (approximately 10% of receptors recycled to the cell surface
• $\chi_i = 0.35$, source August 2007 (35% of the IDL by mass is cholesterol)
• $\chi_l = 0.45$, source August 2007 (45% of the LDL by mass is cholesterol)
• $d_i = 1 h^{-1}$, source August 2007 (solve using steady state values of IDL and LDL and equilibrium concentrations of lipoproteins)
• $d_l = 0.01 h^{-1}$, source August 2007 (solve using steady state values of IDL and LDL and equilibrium concentrations of lipoproteins)

### LDL receptor production and display

• $c = 0.05 g(lh)^{-1}$, source Goldstein & Brown, 1977