User:Johnsy/Lipoprotein Modelling/Model Development

A Very Simple Model

Introduction

To start exploring the method which is used to formulate models, we consider a very simple model derived from the biology of LDL particles in Brown & Goldstein, 1979. We are assuming a cell with a fixed number of LDL particles already bound to the surface of the cell. The rate of internalization is only dependent upon the number of LDL particles already on the surface. Futhermore, once the LDL particle is internalized, it is degraded by the lysosome into its constituent components (ie cholesterol esters, phospholipids, apoproteins, etc.).

We only consider three variables:

1. The concentration of surface bound LDL particles
2. The concentration of internalized LDL particles
3. The concentration of the degradation products of LDL particles (directly proportional to the intracellular cholesterol levels

Developing the Model

Equation 1:

[math]\displaystyle{ \frac{d[SB]}{dt} = -k_{I}[SB] }[/math]

In the above equation, there are no inputs as we are assuming that the LDL particles are already bound to the surface of the cell. We have also only one sink which is due to the internalization of LDL particles, determined by the rate of internalization, k_I, and the concentration of LDL particles bound to the surface of the cell.

Equation 2:

[math]\displaystyle{ \frac{d[INT]}{dt} = k_{I}[SB] - k_{deg}[INT] }[/math]

In the above equation 2, we now have a source which is the LDL particles which were internalized from equation 1. We also have a sink which is the degradation of the internalized LDL particles deteremined by rate k_deg and the concentration of internalzed LDL particles.

Equation 3:

[math]\displaystyle{ \frac{d[DEG]}{dt} = k_{deg}[INT] }[/math]

In the above equation 3, the degradation products are produced from the LDL particles which were internalized. We are not considering the fate of these degradation products, but we are assuming that they stay in the cell and accumulate. The primary degradation products are the cholesterol, fatty acids, lipoproteins, and proteins contained in the LDL particle and these are further processed and recycled by the cell. The choelsterol is converted to bile acid and cholesterol derivatives such as steriod hormones, while the fatty acids and proteins are broken down or stored for use by other cellular pathways. The lipoproteins are also recycled or are degraded by the lysosome.

Parameters and Initial Conditions

From the Panovska paper discussed below, we obtain the two parameters from literature that govern the above model.

1. k_I - the rate of internalization of LDL particles = log(2)/22 s^-1
2. k_deg - the rate of degradation of LDL particles = 1/180 s^-1

The initial conditions we enter into our model are as follows:

1. [SB]₀ - the initial concentration of surface bound LDL particles we set arbitrarily at 45
2. [INT]₀ - the initial concentration of internalized LDL particles we initial set to zero (the cells are starved of cholesterol)
3. [DEG]₀ - the initial concentration of degradation products we also set to zero since we assume that the cells have not had any cholesterol in them for a lengthy time period.

Developing the Simple Model

Introduction

We can now consider the actual concentration of cholesterol in the cell so that we can model feedback loops which are related to the concentration of cholesterol in the cell. We come up with another equation relating the rate of change of cholesterol concentration over time. Here, we again assume that we are only considering the cholesterol delivered to the cell via LDL particles and that each LDL particle has χ_L amount of cholesterol contained in them.

Adding to the Simple Model

The first equation governing the rate of change of surface bound LDL particles remains the same:

[math]\displaystyle{ \frac{d[SB]}{dt} = -k_{I}[SB] }[/math]

The second equation governing the rate of change of internalized LDL particles can now also be a function of the intracellular cholesterol concentration ([IC]) where the higher the intracellular cholesterol levels, the lower the rate of internalization. As before, the sink term of this equation is governed by the conversion of LDL particles to cholesterol.

[math]\displaystyle{ \frac{d[INT]}{dt} = k_{I}\frac{[SB]}{[IC]} - k_{con}[INT] }[/math]

The thrid equation now relates to the rate of change of intracellular cholesterol levels. We assume that the only source term is the conversion of LDL particles to free cholesterol where each LDL particle has, on average, χ_L concentration of cholesterol. The sink term is slightly different in meaning from the previous model, this time only signifying the conversion of cholesterol to bile acids and/or cholesterol derivatives, with k_deg being the rate of this degradation.

[math]\displaystyle{ \frac{d[IC]}{dt} = \chi_{L}k_{con}[INT] - k_{d}[DEG] }[/math]

The last equation is the same as in the previous model with the concentration of our degradation products.

[math]\displaystyle{ \frac{d[DEG]}{dt} = k_{d}[DEG] }[/math]

Parameters and Initial Conditions

We can again use similar parameters and initial conditions as in the previous model with a few modifications.

1. k_I - the rate of internalization of LDL particles is the same, log(2)/22 s^-1 (Panovska)
2. k_con - the rate of conversion of LDL particles to cholesterol, 1/180 s^-1 (Panovksa)
3. k_d - the rate of loss of cholesterol from the cell, 1/780 s^-1 (Panovska)
4. χ_L - the amount of cholesterol found in each LDL particle, 0.45 (August & Barahona)

Initial Conditions: Assume starved cell

HMG-CoA Reductase and Cholesterol

Cholesterol Producers (Sources)

• de novo - Acetyl CoA via HMG CoA and enzyme HMG CoA reductase
• IDL particles accepted into the cell via LDL receptors
• LDL particles accepted into the cell via LDL receptors

Cholesterol Fates (Sinks)

• Bile acids via enzyme cholesterol 7-α-hydroxylase

de novo Pathway

HMG CoA reductase is the rate limiting enzyme in the biosynthesis of cholesterol. Its transcription is upregulated by the sterol regulatory element binding protein (SREBP) which binds to the streol regulatory element (SRE) to transcribe the gene for HMG CoA reductase. SREBP is usually situated on the membrane of the endoplasmic reticulum or the nuclear membrane, but when bound by cholesterol, the protein is released via proteolysis and migrates to the nucleus where it binds to the SRE to initiate transcription.

Statins limit the action of HMG CoA reductase by acting as a competitive inhibitor since it resembles the HMG CoA molecule.

To simplify the model, we can simply say that both the concentration of cholesterol and statins will negatively affect the rate of the enzyme HMG CoA reductase and will limit the production of mevalonate (another key intermediary in cholesterol biosynthesis).

Adjusting the Model Developed by Barahona, et al.

If we have a look again at the equation which governed the intracellular concentration of cholesterol, we see that we must target this equation if we want to take into account the de novo pathway for cholesterol biosynthesis.

[math]\displaystyle{ \frac{d[IC]}{dt} = (\chi_{I}d_{I}[IDL] + \chi_{L}d_{L}[LDL])\phi_{LR} + \chi_{L}d[LDL] - d_{IC}[IC] }[/math]

From the biology, we know that the 'de novo' pathway will be a source of intracellular cholesterol that is dependent upon the concentration of cholesterol there is in the cell. Thus, we obtain the straightforward addition below, where n denotes the rate constant for the biosynthesis of cholesterol determined by the maximum rate of HMG-CoA reductase, the limiting step in the biosynthesis pathway.

[math]\displaystyle{ \frac{d[IC]}{dt} = n[IC] + (\chi_{I}d_{I}[IDL] + \chi_{L}d_{L}[LDL])\phi_{LR} + \chi_{L}d[LDL] - d_{IC}[IC] }[/math]