User:Johnsy/Lipoprotein Modelling/Current Lipoprotein Models

Mathematical Models of Hepatic Lipoprotein Metabolism

 * Paper for the 5th Mathematics in Medicine Study Group, Oxford University
 * Jasmina Panovska, Laura Pickersgill, Marcus Tindall, Johnathan Wattis, and Helen Byrne
 * 10 August 2006
 * Full PDF Text

Introduction
Two models are described in the paper
 * 1) Description of a discrete model of LDL partice uptake allowing for differences in the number of free, bound and internalized LDL particles to the elucidated (in the absence of VLDL particles)
 * 2) Description of a continuous model of LDL and VLDL uptake and the competition between particles for free LDL receptors

Mechanism of LDL uptake by the liver
 * 1) Particle Binding - LDL particles bind to the hepatic LDL receptors (LDLR) in coated pits (specialized regions of the liver plasma membrane)
 * 2) Particle Interaction - Interaction of LDL particles to LDLR mediated by apolipoprotein B (Apo B)
 * 3) Internalization - LDL particles are internalized upon binding to the LDLR
 * 4) Degradation of LDL particle - Fusion of endosomes with lysozymes in the cell degrade the LDL particle into its constitutent parts (cholesterol, fatty acids, and amino acids)
 * 5) Receptor fate - LDL receptors are recycled to the cell surface or degraded

Special Notations
 * Rate of LDL particle uptake - the rate at which LDL particles bind to the receptors are influenced by the amount of triglyceride-rich lipoprotein in the blood stream (ie VLDL and chylomicron particles). VLDL and CM compete with the LDL for the LDL receptors by binding to the LDLR with the Apo E proteins associated with them.

Model 1: LDL endocytosis in the absence of VLDL
Panovska, et al. first defined the biochemistry of LDL endocytosis in hepatocytes. They defined the parameters:
 * Le - the number of LDL particles bound to receptors in a pit on the surface of a hepatocyte
 * Lb - the number of bound LDL particles also internalized by the cell
 * Li - the number of LDL particles internalized by the cell
 * rC - the number of cholesterol molecules in a typical LDL particle (r) multiplied by the number of LDL particles degraded to cholesterol (C)
 * {} - represents the metabolic fate of cholesterol, either conversion to cholesterol esters to be stored, incorporated into the cell membrane, or metabolized through oxidation
 * a - the rate at which LDL particles are bound to the surface of the cell
 * b - the rate at which LDL particles are internalized
 * kid - the rate at which LDL particles are broken down into cholesterol
 * l - the rate at which cholesterol is converted into either cholesterol esters, incorporated into the cell membrane, or metabolized through oxidation.

The flow diagram for the biochemsitry of cholesterol metabolism can be summarized as: $$ L_{e} \xrightarrow{a} L_{b} \xrightarrow{b} L_{i} \xrightarrow{k_{id}} rC \xrightarrow{l} \lbrace \rbrace $$

Further developing the model with the following variables, parameters, and constants:
 * Np(t) - the number of pits with p LDL particles bound at time t
 * N0(t) - the maximum number of pits available (p = 0)
 * pm - the maximum number of LDL particles that can bind to a coated pit
 * k0 - rate at which empty pits are produced (assumed to be constant, however in reality is subject to the recycling and de novo production of LDL receptors.

We can model the attachment of LDL particles to the coated pits with the following flow diagram utilizing the parameters defined above. In this model, they crucially assume that multiple LDL particles can bind to each coated pit. $$ N_{p-1} + L_{e} \xrightarrow{a} N_{p} + L_{b} \quad p = 1, 2, ..., p_{m} $$

Development of the differential equations governing the internalization of LDL particles and the production and degradation of the coated pits. First, the number of free pits available (N0) is dependent upon the rate at which the pits are produced (k0) as well as the rate of binding of free LDL particles (Le) to the coated pits and the internalization of empty pits with the rate b0 (assumed to be less than the rate of internalization of bound LDL receptors, b. This can be expressed in the following differential equation: $$ \frac{dN_{0}}{dt} = k_{0} - aL_{e}N_{0} - b_{0}N_{0} $$

Second, the paper develops a model to account for the rate of change of the number of coated pits (Np) filled with p number of lipoproteins. This is dependent upon the binding of LDL particles to a pit with (p - 1) LDL particles as well the binding of LDL particles and the internalization of LDL particles at a rate of b when the coated pit becomes "saturated". $$ \frac{dN_{p}}{dt} = aL_{e}N_{p-1} - bN_{p} - aL_{e}N_{p} $$

Third, the paper develops a model for the rate of change of the number of pits that are filled with the maximum number of lipoproteins. The fist source term is derived from an LDL particle attaching to a pit that is filled with one less than the maximum number of lipoproteins and the drain term is derived from the rate of internalization. $$ \frac{dN_{p_{m}}}{dt} = aL_{e}N_{p_{m-1}} - bN_{p_{m}} $$

A Dynamical Model of Lipoprotein Metabolism

 * E August, KH Parker, and M Barahona
 * Bulletin of Mathematical Biology, May 2007
 * Full Text PDF

Introduction
This model presents a set of equations regarding the regulation of the different types of lipoproteins and accounts for the fluctuations in their plasma concentration. Furthermore, the model also links in the percentage of LDL receptors occupied at any given moment in time as well as the intercellular cholesterol concentration.

The biology is similar to the above model with the assumptions listed below.
 * 1) The only input to the system is VLDL secretion from the liver
 * 2) VLDL is converted to IDL via the action of Lipoprotein Lipase (LPL)
 * 3) IDL is converted to LDL also via the action of LPL
 * 4) Both IDL and LDL can attach to LDL receptors on the surface of the cell and become internalized
 * 5) LDL can also attach non-specificially to the cell and become internalized
 * 6) The recycling of receptors is dependent upon the intercellular concentration of cholesterol
 * 7) IDL and LDL particles have a certain amount of cholesterol contained in them that are released into the cell
 * 8) Intracellular cholesterol is degraded into bile salts which are lost to the outside environment, there is no accounting for the recycling of the cholesterol from reabsorption of the bile salts

Development of the Model
Five equations governing the concentration of lipoproteins, the percentage of LDL receptors, and the concentration of cholesterol were developed.

First, we consider the concentration of VLDL. We assume that it is being produced in the liver and released at a constant level uv and is converted to IDL with rate kv which is also dependent upon the concentration of VLDL in the plasma. $$ \frac{d[VLDL]}{dt} = -k_{v}[VLDL] + u_{v} $$

Next, we consider the concentration of IDL. IDL is produced from VLDL via the enzyme Lipoprotein Lipase (LPL) and is also converted to LDL with rate kI. IDL can also be internalized by the LDL receptors, which is dependent on how many LDL receptors there are and the concentration of IDL available in the plasma to be internalized. We are assuming that the IDL particles are internalized once they bind to the receptor. $$ \frac{d[IDL]}{dt} = k_{v}[VLDL] - k_{I}[IDL] - d_{I}[IDL]\phi_{LR} $$

Next, the concentration of LDL is considered. The source of LDL is the conversion of IDL to LDL particles by the enzyme Lipoprotein Lipase (LPL) as already established. LDL is also internalized both via LDL receptors and non-specifically. Another assumption to the model is that IDL is not internalized non-specifically. The specific binding and internalization is governed by the percentage of LDL receptors available and rate dL. The non-specific binding and internalization is governed by the rate d. $$ \frac{d[LDL]}{dt} = k_{I}[IDL] - d_{L}[LDL] - d[LDL] $$

The not consider the concentration of LDL receptors, but rather the percentage of free receptors on the surface of the cell. The receptors can be occupied by LDL or IDL particles, thereby reducing the fraction of free receptors. The receptors are then recycled and replaced to the cell membrane depending upon the intracellular concentration of cholesterol. With higher cholesterol levels, there is less recycling, and hence a inverse relationship. $$ \frac{d\phi_{LR}}{dt} = -b(d_{I}[IDL] + d_{L}[LDL])\phi_{LR} + c\frac{1-\phi_{LR}}{[IC]} $$

Finally, the paper considers the intracellular concentration of cholesterol. The source of this cholesterol is only the LDL and IDL particles and each has it's own amount of cholesterol it releases into the cell. As stated before, the although the IDL must be attached to LDL receptors to be internalized, LDL particles can be non-specifically internalized, and hence the third term in the equation below. Further, the only sink in this equation is the "degradation" of the cholesterol, which represents the production of cholesterol derivates such as hormones and bile acids that reduce the intracellular concentration of cholesterol. $$ \frac{d[IC]}{dt} = (\chi_{I}d_{I}[IDL] + \chi_{L}d_{L}[LDL])\phi_{LR} + \chi_{L}d[LDL] - d_{IC}[IC] $$

To model the effects of statin with our new model, we must somehow change the last equation above such that the intracellular concentration of cholesterol model takes into account the de novo cholesterol biosynthesis pathway.

Derivation of the d&Phi;LR Term
The &Phi;LR term, or the change in the fraction of bound LDL receptors can be dervied from considering the tree stages of the receptor: when the receptor is complexed with LDL/IDL particles, when it is recycled, and the nuclear (de novo) synthesis of the receptors. Below are the 4 equations governing these phenomena.

$$ \begin{alignat}{2} \frac{d[LR_{comp}]}{dt} & = \beta \alpha (d_{I}[IDL] + d_{L}[LDL])\phi_{LR} - \epsilon[LR_{comp}] \\ \frac{d[LR_{rec}]}{dt} & = \epsilon(1-\gamma)[LR_{comp}] - \delta[LR_{rec}](1-\phi_{LR}) \\ \frac{d[LR_{nuc}]}{dt} & = \frac{c_1}{[IC]} - k[LR_{nuc}] - \delta[LR_{nuc}](1-\phi_{LR}) \\ \frac{d\phi_{LR}}{dt} & = -\alpha(d_{I}[IDL] + d_{L}[LDL])\phi_{LR} + \frac{\delta}{\beta}([LR_{rec}] + [LR_{nuc}])(1-\phi_{LR})

\end{alignat} $$

Steady state conditions are assumed such that $$\frac{d[LR_{comp}]}{dt} = \frac{d[LR_{rec}]}{dt} = \frac{d[LR_{nuc}]}{dt} = 0$$ and the equations are rearranged to yield the equation below from the model.

$$ \frac{d\phi_{LR}}{dt} = -b(d_{I}[IDL] + d_{L}[LDL])\phi_{LR} + c\frac{1-\phi_{LR}}{[IC]} $$

Here, they have substituted $$b = \alpha \gamma$$ and $$c = \frac{\delta c_1}{\beta k}$$. Furthermore, the paper makes the assumption that $$k \gg \delta$$, meaning that the turnover of nucleus-synthesized LR (k) is large compared to the probability of its insertion into the membrane (&delta;).

Let us now go about interpreting the model by considering each equation in turn and deciphering the biological meaning. For the first equation relating the complexed receptors, we see that the concentration of complexed receptors grows when LDL or IDL particles are bound to the surface and internalized. Internalization is a necessary prerequisite for the the change in complexed receptor concentration. Furthermore, we take into account degradation of some receptors in this equation as the sink term $$\epsilon[LR_{comp}]$$.

Questions Regarding the Model
Some assumptions that the model makes does not necessarily correlate with the research done previously. First, in the derviation of the equation governing the rate of change of the receptor concentration, they assume that the there is more receptor degradation than receptors that make it to the cell membrane. Although in the literature (Brown 1977) it says that number of receptors that are displayed on the membrane is less than 10% of the rate of synthesis of receptors, the number of receptors produced is actually determined by the intracellular cholesterol levels and is not always much less than the maximum number of receptors. At times of starvation, the LDL receptor probability of insertion (&delta; in the model) will increase to above 10%, making the assumption no longer valid.

The another assumption the model assumes is that there is receptor-independent internalization of the LDL particles. However, this phenomenon has only been observed in rats and hamsters (Dietschy 1993). When a value was obtained in the same paper for human receptor-independent internalization, their model assumed that both receptor-dependent and receptor-independent internalization both occur, even though there was no clinical evidence to show that this occurs in humans. Furthermore, the paper only utilized a few data points (as data values for humans are difficult to obtain) to determine their findings. So is this assumption valid and should we be incorporating this into the model?

The model also makes the assumption that the cells only get cholesterol from LDL particles. Of course there is cholesterol production in the cell through the de novo pathway where the cell utilizes acetyl-CoA to produce cholesterol via the mevalonate pathway. This pathway is where we begin to add to the model in the next section of this report.