User:Johnsy/Lipoprotein Modelling/Model Development

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: $$ \frac{d[SB]}{dt} = -k_{I}[SB] $$

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, kI, and the concentration of LDL particles bound to the surface of the cell.

Equation 2: $$ \frac{d[INT]}{dt} = k_{I}[SB] - k_{deg}[INT] $$

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 kdeg and the concentration of internalzed LDL particles.

Equation 3: $$ \frac{d[DEG]}{dt} = k_{deg}[INT] $$

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) kI - the rate of internalization of LDL particles = log(2)/22 s-1
 * 2) kdeg - 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.

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 &chi;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 now takes into account the inverse relation between the rate of internalization as well as the intracellular cholesterol level: $$ \frac{d[SB]}{dt} = -k_{I}\frac{[SB]}{[IC]} $$

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. $$ \frac{d[INT]}{dt} = k_{I}\frac{[SB]}{[IC]} - k_{con}[INT] $$

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, &chi;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 kdeg being the rate of this degradation and is related to the concentration of cholesterol already in the cell. $$ \frac{d[IC]}{dt} = \chi_{L}k_{con}[INT] - k_{d}[IC] $$

The last equation is the same as in the previous model with the concentration of our degradation products. $$ \frac{d[DEG]}{dt} = k_{d}[IC] $$

Parameters and Initial Conditions
We can again use similar parameters and initial conditions as in the previous model with a few modifications.
 * 1) kI - the rate of internalization of LDL particles is the same, log(2)/22 s-1 (Panovska)
 * 2) kcon - the rate of conversion of LDL particles to cholesterol, 1/180 s-1 (Panovksa)
 * 3) kd - the rate of loss of cholesterol from the cell, 1/780 s-1 (Panovska)
 * 4) &chi;L - the amount of cholesterol found in each LDL particle, 0.45 (August & Barahona)

Initial Conditions: Assume starved cell
 * 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) [IC]0 - the initial intracellular cholesterol concentration also set to 0.4 (arbitrarily small value to prevent error)
 * 4) [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 and thus have no cholesterol derivatives and degradation products yet

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).

HMG-CoA Reductase as an Enzyme
HMG-CoA Reductase is the limiting enzymatic step in the biosynthesis of cholesterol and we can assume that the conversion of mevalonate to cholesterol is fast as well as the conversion from acetyl-CoA to HMG-CoA. We can use Michaelis-Menten kinetics to describe the action of HMG-CoA reductase and incorporate it into our equation governing the rate of change of intracellular cholesterol concentration. We can recall that the Michaelis-Menten Equation is the following: $$ \frac{d[P]}{dt} = V_{max}\frac{[S]}{K_{m}+[S]} $$

Parameters of the Michaelis-Menten Equation:
 * 1) [P] - Concentration of the Product, in our case we can assume that it is cholesterol
 * 2) [S] - Concentration of the Substrate, it is immediately HMG-CoA, but we can also use the concentration of acteyl-CoA as our substrate in replacement since the conversion from acteyl-CoA to HMG-CoA is fast
 * 3) Km - The Michaelis-Menten constant describing the activity of the enzyme.
 * 4) Vmax - The maximum rate of turnover of the enzyme

Key Assumptions to the Michaelis-Menten Equation
 * We are assuming that the concentration of enzyme is fixed at a certain concentration [E]0.
 * Quasi-steady state approximation where the rate of change of the enzyme-substrate complex is 0.
 * We have taken a four dimensional problem and reduced it down to a one-dimensional problem, so we have effectively neglected any change in the enzyme concentration. With the action of statins inhibiting the enzyme concentration, we might have to take this into account when building more complex models to accurately reflect the biochemistry involved in LDL metabolism.

Expanding the Previous Model
The Michaelis-Menten term governing the HMG-CoA reductase enzymatic reaction would be incorporated into the rate equation for intracellular cholesterol concentration as a source term as shown below. Furthermore, the action of the enzyme is regulated by the concentration of cholesterol within the cell at any given time. The more cholesterol in the cell, the lower the activity of HMG-CoA reductase, so again, we have an inverse relationship between the activity of the enzyme and the concentration of intracellular cholesterol that has to be taken into account of. $$ \frac{d[IC]}{dt} = \chi_{L}k_{con}[INT] + V_{HCR}\frac{[AC]}{[IC](K_{HCR}+[AC])} - k_{d}[IC] $$
 * [AC] - the concentration of Actyl-CoA at any time in the cell. Actually, the model should strictly reflect the concentration of HMG-CoA; however we assume that the concentrations of intermediates up to HMG-CoA are in equilibrium and will be in effect constant (see below)
 * VHCR - Vmax for HMG-CoA reductase enzyme
 * KHCR - the Michaelis-Menten constant, Km for the HMG-CoA reductase enzyme

The other equations governing the LDL concentration of internalized or surface bound LDL remain the same. $$ \begin{alignat}{2} \frac{d[SB]}{dt} & = -k_{I}\frac{[SB]}{[IC]} \\ \frac{d[INT]}{dt} & = k_{I}\frac{[SB]}{[IC]} - k_{con}[INT] \\ \frac{d[DEG]}{dt} & = k_{d}[IC] \end{alignat} $$

For the model to work, we also must incorporate an equation dealing with the concentration of Acetyl-CoA. We assume that the mechanisms within the cell will keep the concentration of Acetyl-CoA constant, since Acetyl-CoA is an essential metabolite of the cell and is used in several cellular networks (e.g. Citric Acid Cycle). $$ \frac{d[AC]}{dt} = 0 $$

Since we have made the above quasi-steady state approximation, we can alter the equation dealing with intracellular concentration once more before moving on to extending the model to incorporate other lipoprotein molecules such as VLDL and IDL as explored in August, 2005. We have assumed that the concentration of HMG-CoA will be constant in the cell as there should be sufficient Acetyl-CoA to put HMG-CoA in equilibrium at a value H.

$$ \frac{d[IC]}{dt} = \chi_{L}k_{con}[INT] + V_{HCR}\frac{H}{[IC](K_{HCR}+H)} - k_{d}[IC] $$

Considering the model above, we can see that the activity of statins, which affect the enzyme HMG-CoA reductease by competitive inhibition, will affect the KHCR value of the enzyme. We can consider this effect with the constant &alpha;, a modulator for the enzyme which is always greater than one (one signifying no inhibition of the enzyme). And so our complete intracellular concentration equation for now is as follows.

$$ \frac{d[IC]}{dt} = \chi_{L}k_{con}[INT] + V_{HCR}\frac{H}{[IC](\alpha K_{HCR}+H)} - k_{d}[IC] $$

Parameters of the Model Incorporating HMG-CoA Reductase
We can again use similar parameters and initial conditions as in the previous model with a few modifications.
 * 1) kI - the rate of internalization of LDL particles is the same, log(2)/22 s-1 (Panovska)
 * 2) kcon - the rate of conversion of LDL particles to cholesterol, 1/180 s-1 (Panovksa)
 * 3) kd - the rate of loss of cholesterol from the cell, 1/780 s-1 (Panovska)
 * 4) &chi;L - the amount of cholesterol found in each LDL particle, 0.45 (August & Barahona)
 * 5) VHCR - the Vmax value for HMG-CoA Reductase, 0.2 nmol/min/mg (Smythe 2002)
 * 6) KHCR - the Km value for HMG-CoA Reductease, 4.2 &mu;M (Smythe 2002)
 * 7) &alpha; - modulation coefficient for the action of a competitive inhibitor for HMG-CoA reductase such as statins, initially set to 1 (no inhibition) (see biochemical models for details)

Initial Conditions: Assume starved cell
 * 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) [IC]0 - the initial intracellular cholesterol concentration also set to 0.4 (arbitrarily small value to prevent error)
 * 4) [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 and thus have no cholesterol derivatives and degradation products yet
 * 5) [AC]0 - the initial and steady concentration of Acetyl-CoA within the cell, arbitrarily given the value 5 (pending value)

Investigation Into the Effects of HMG-CoA Reductease on the Model
We want to analyze the portion of the equation dealing with the intracellular cholesterol concentrations to get a better understanding to how HMG-CoA reductase affects the kinetics of the system. We start by only considering the equation with the cholesterol source given by the Michaelis-Menten term and a degradation term as shown below.

$$ \frac{d[IC]}{dt} = V_{HCR}\frac{H}{[IC](\alpha K_{HCR} + H)} - k_d[IC] $$

We solve simply for the fixed points of the system by setting $$\frac{d[IC]}{dt} = 0 $$ to obtain the fixed point below.

$$ [IC]* = \sqrt{\frac{V_{HCR}H}{k_d(\alpha K_{HCR} + H)}} $$

So from the above equation, we can see that the action of statins will lower the fixed point through the variable &alpha;. We can also decreased the fixed point (ie lowering the intracellular levels) by changing the value of kd (e.g. through altering the rate of reaction of 7-&alpha;-hydroxylase) or by altering the HMG-CoA reductase enzyme Km value. Furthermore, this fixed point is a stable fixed point there is a negative gradient at that point, so all initial conditions will tend to that value.

The equation above also gives us a method by which we can obtain values for &alpha;. Included in &alpha; are several parameters that we are currently neglecting, such as degradation of the statins (half-life in the body) and the potency of the drug on HMG-CoA reductase. Pharmaceutical companies obtained dosages by a trial-and-error processace, something we hope to reduce the need of through modelling. From Schachter 2004, values of the average reduction in cholesterol for given dosages of statins are given. Each statin is different, and will thus have a different value for &alpha;. For example, If we consider lovastatin, a 40 mg daily dosage saw a decrease in LDL choelsterol of 34%. We assume that a decrease in LDL cholesterol is the product of a proportional decrease in intracellular cholesterol, an assumption that we should take more time to consider)

Let us first assume that the value of &alpha; in the equation above is directly proportional to the dosage , and that without the effect of statins, the value of &alpha; = 1. This allows the full turnover rate (KHCR) to occur. The value of &alpha; is shown below and is dependent upon the percent reduction of cholesterol.

$$ \alpha = \frac{1}{\sqrt{red}}(1+\frac{H(1-\sqrt{red})}{K_{HCR}}) $$

In the above equation, it can be seen that an increasing dosage (increasing &alpha;) does not lead to a uniform reduction (red) in intracellular cholesterol (nor LDL cholesterol via our assumptions). This agrees with Schachter's 2004 findings of this non-linear behavior.

Other Lipoproteins (VLDL and IDL)
Now that we have gained a better understanding of the biochemistry within the cellular level, we can take a step back to look at the other plasma lipoproteins such as VLDL and IDL. In this section, we attempt to incorporate the mathematical modelling developed in August & Barahona, 2005. We begin by taking our existing set of 5 equations and slowly moving out. First, the intracellular cholesterol concentration does not only come from LDL, but can also come from IDL particles which bind to the cell. The internalized particles include both LDL and IDL, so we can make the substitution $$ [INT] = [LDL] + [IDL]$$ However, we must now incorporate different parameters for the internalization rate of IDL, since it is different from LDL. $$ \frac{d[IC]}{dt} = \chi_{L}d_{L}[LDL]+\chi_{I}d_{I}[IDL] + V_{HCR}\frac{H}{[IC](\alpha K_{HCR}+H)} - k_{d}[IC] $$

In the above equation, &chi;I now is the amount of cholesterol contained within each IDL molecule and dI is the rate at which IDL particles are internalized into the cell. In fact, we can adapt the model described in August, 2005 as discussed in a previous section to derive the equations found below. We notice that instead of just being concerned with the overall concentration of internalized lipoproteins, the model proposed focuses on the fraction of LDL receptors available to bind to LDL or IDL particles. Furthermore, the concentration of surface bound particles have been incorporated in the change in fraction of free LDL receptors, but we now take into account the amounts of LDL and IDL in the circulation. Another assumption that is made is that the concentration everywhere in the blood stream is constant and that the concentration gradient between the center of the vessel and the vessel wall is zero. Due to the non-laminar nature of the blood flow and the relativly fast velocity of the flow with respect to the vessel diameter (aka Reynolds number), we can assume that the boundary layer is very small and neglible.

Another note to make is the presence of the internalization term of LDL without being attached to the LDL receptors. Although the majority of LDL particles are internalized via the LDL receptor (aka receptor dependent endocytosis), there is a very small amount of LDL particles that are internalized via receptor independent methods. Experimental evidence from patients with familial hypercholesterolemia n published papers have shown that LDL internaliation does not occur when not bound to the receptor; however, one paper argues that since this phenomenon is seen in other animals such as the rat or the rabbit, receptor independent internalization must also occur in humans. Rates have been extracted from experimental data on humans on the assumption that receptor-independent internalization occurs, which might be misleading. The rate at which this occurs is almost neglible (less than 0.5% of all LDL particles are internalized via receptor-independent methods), but keeping with accuracy of the model, this phenomenon is included in the complete model.

$$ \begin{alignat}{2} \frac{d[VLDL]}{dt} & = -k_V[VLDL] + u_V \\ \frac{d[IDL]}{dt} & = k_V[VLDL] - k_I[IDL] -d_I[IDL]\phi_{LR} \\ \frac{d[LDL]}{dt} & = k_I[IDL] - d_L[LDL]\phi_{LR} - d[LDL] \\ \frac{d\phi_{LR}}{dt} & = -b(d_I[IDL] + d_L[LDL])\phi_{LR} + c\frac{1-\phi_{LR}}{[IC]} \\ \frac{d[IC]}{dt} = & (\chi_{L}d_{L}[LDL]+\chi_{I}d_{I}[IDL])\phi_{LR}+\chi_{I}d_{I}[IDL] +V_{HCR}\frac{H}{[IC](\alpha K_{HCR}+H)} - k_{d}[IC]

\end{alignat} $$

Parameters of the Model Incorporating other Lipoproteins
Variables of the model:
 * 1) [VLDL] - concentration of VLDL particles in the blood plamsa
 * 2) [IDL] - concentration of IDL particles in the blood plamsa
 * 3) [LDL] - concentration of LDL particles in the blood plamsa
 * 4) &phi;LR - fraction of receptors occupied on the cell surace
 * 5) [IC] - concentration of intracellular cholesterol

We can again use similar parameters and initial conditions as in the previous model with a few modifications.
 * 1) kI - the rate of internalization of LDL particles is the same, log(2)/22 s-1 (Panovska)
 * 2) kcon - the rate of conversion of LDL particles to cholesterol, 1/180 s-1 (Panovksa)
 * 3) kd - the rate of loss of cholesterol from the cell, 1/780 s-1 (Panovska)
 * 4) &chi;L - the amount of cholesterol found in each LDL particle, 0.45 (August & Barahona)
 * 5) VHCR - the Vmax value for HMG-CoA Reductase, 0.2 nmol/min/mg (Smythe 2002)
 * 6) KHCR - the Km value for HMG-CoA Reductease, 4.2 &mu;M (Smythe 2002)
 * 7) &alpha; - modulation coefficient for the action of a competitive inhibitor for HMG-CoA reductase such as statins, initially set to 1 (no inhibition) (see biochemical models for details)