Team 5 – Dr. Ross

Chuan XueDavid Tello

Skyler SpeakmanRina Santos

Shilpa Das Gupta

Mathematical Models of the

Delipidation Cascade of

Low Density Lipoproteins (LDL)

PURPOSE:

Why? Small, dense LDL has been linked to coronary artery disease (CAD). We’d like to evaluate what measure to use: LDL itself, Apo-B/Apo-A1 Ratio, MR, C-Reactive Protein…and how prescribed drugs play a part in the process.

What? Devise a mathematical model of LDL size distribution that is characterized by its components and processes.

List of Chemical Players for the Model

LDL Components

Cholesterol Ester (CE)Triglyceride (TG)

Protein – ApoB-100

CETP (Cholesterol Ester Transfer Protein)

Note: Component list is not all-inclusive

HDL Components

Cholesterol Ester (CE)Triglyceride (TG)Protein – ApoA-I

HL (Hepatic Lipase)

Liver and artery wall (Source and Sink)

Components of LDL

ApoB-100 CE TG

Components of HDL

ApoA-I CE TG

Team 5’s Initial Model involving ODE’s

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AfTArdt

dT

jArArAfAfTdt

dA

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dA

Assume we have reaction:

Which can be described by the following system:

j1-j ATA jif ,1

jir ,

Team 5’s Initial Model involving ODE’s

0

11

1

111

00

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Steady states of the system

Team 5 – 2nd Model of LDL Delipidation

Packard’s DiagramChemical reactions incorporated:

1. CETP facilitates the exchange of CE for TG from VLDL to LDL

2. CETP also facilitates the process taking CE from HDL to LDL.

3. HL removes TG from LDL transforming LDL/IDL to small, dense LDL

These three reactions are directly responsible for LDL delipidation.

Team 5 – 2nd Model of LDL Delipidation

i : # of CE wrapped in LDL particle

j : # of TG wrapped in LDL particle

denotes LDL particle

Blue reactions push LDL distribution to small dense direction

Green reaction pushes LDL distribution back to large buoyant direction.

Elevated VLDL concentration in blood enhance blue reaction and have a impact making LDL smaller denser.

ji,B

Team 5 – 2nd Model of LDL Delipidation

FINAL MODEL

The Final Model

Previous models required 3 objects to interact simultaneously. However, the probability for 3 particles to interact is much smaller than 2 particles.

This lead us to disband the “swapping” ability of CETP for a more realistic series of reactions.

Pool of CETP

We introduced a pool (finite number) of CETP. CETP exists in 3 states: Free Carrying CE Carrying TG

CETP serves as a transport for CE and TG between the lipoproteins (both HDL and LDL). There is no direct interaction between lipoproteins.

Actions of CETP

The aforementioned series of reactions can be broken down into 4 sub-reactions: CETP may pick up a CE or TG CETP may drop off a CE or TG

Hepatic Lipase

CETP accounts for all of the interactions involving CE and TG except Hepatic Lipase.

Hepatic Lipase is a enzyme that takes TG from LDL. This process contributes to the formation of small dense LDL.

Chemical Reactions:

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AHA

PACA

PATA

CAPA

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PBCB

PBTB

CBPB

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System of Differential Equations:

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BTBCBPdt

dB

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dA

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,,,,,,,,

,1,1,,1,11,1,

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,,,,,,,,

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System of Differential Equations:

HHAHBdt

dH

TAPATBPBdt

dT

CAPACBPBdt

dC

PTACAPAPA

TBCBPBPBdt

dP

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Numerical Approximations using MATLAB

An example of a source distribution for LDL

The final distribution of LDL corresponding to the above source distribution

Increased CETP This allows for more exchanges between LDL. This is represented by the smoother graph on

the right.

Increased Hepatic Lipase This removes TG from LDL. The distribution has shifted towards less TG in

LDL.

Increased HDL source HDL removes CE from the system. The distribution has shifted towards fewer CE and an

abundance of TG in the LDL. Note the decrease in sdLDL and increase in VLDL.

Summary

Through studying biological papers and speaking with a cardiologist, Dr. Duprez, we developed a model that resembled the processes involved in the LDL delipidation cascade.

We formed a system of ODE’s to describe our model and simulated the system in Matlab.

We explored the numerical model and interpreted the results.

Future Work

Incorporate real source terms and reaction rates to improve our ODE model, and add in drugs to test the model.

Continue steady-state analysis of our ODE model.

Try to interpret the macroscopic flow rate in the 1st order hyperbolic PDE model we get in terms of microscopic chemical reaction rates.

Continue adding features to the Matlab code.

Acknowledgements

IMADr. RossProfessor Braun Professor SantosaDr. Duprez M.D.Valjean Eleander“John” from Rice University

Thank you!

CETP deposits CE on HDL instead of on LDL. HDL then leaves the system. Resulting in higher TG and lower CE on VLDL / IDL

CETP

CETP (P)

CE CETP with CE (C)

TG CETP with TG (T)

HDL

CE CE CE CE

TG TG TG TG

ApoA-I

CE i

TG j

ApoA-I with i cholesterol ester(CE) and j triglyceride (TG)

ji,A

LDL

CE CE CE CE

TG TG TG TG

ApoB-100

CE i

TG j

ApoB-100 with i cholesterol (CE)

and j triglyceride (TG)ji,B

Size, Data and Composition Table from Camilla Anderson’s PhD Thesis, 2003

Table 1 Size, density and composition of the major plasma lipoproteins

VLDL IDL LDL HDLDensity (g/mL) .95-1.006 1.006-1.019 1.019-1.063 1.063-1.210Diameter (nm) 30-80 25-35 18-25 5-15

Composition (% dry wt)Protein 10 18 25 33Triglyceride (TG) 50 31 9 8Cholesterol (CE) 22 29 45 30Apolipoproteins (Apo- ) B-100 B-100 B-100 A-1,A-2