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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
01
1111
11000
:1,)(
jjj
jjj
jjjjjjjjj
AfTArdt
dT
jArArAfAfTdt
dA
ArATfdt
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
01
jjj
jjj
jj
jjjjj
Af
Ar
T
rTf
ArATfA
Ar
TfA
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:
1,,
,1,
1,,
,1,
1,,
,
,
,
,
,
jiji
jiji
jiji
jiji
jiji
AHA
PACA
PATA
CAPA
TAPA
Aji
Aji
Aji
Aji
Aji
1,,
,1,
1,,
,1,
1,,
,
,
,
,
,
jiji
jiji
jiji
jiji
jiji
BHB
PBCB
PBTB
CBPB
TBPB
Bji
Bji
Bji
Bji
Bji
System of Differential Equations:
jiBji
Bji
Bji
Bji
Bji
Bji
Bji
Bjiji
Bjiji
Bjiji
Bji
jiBjiji
Bjiji
Bji
Bji
ji
jiAji
Aji
Aji
Aji
Aji
Aji
Aji
Ajiji
Ajiji
Ajiji
Aji
jiAjiji
Ajiji
Aji
Aji
ji
BBPBPBH
BHBTBCBC
BTBCBPdt
dB
AAPAPAH
AHATACAC
ATACAPdt
dA
,1,1,,1,11,1,
,,1,1,,1,1,1,1
,,,,,,,,
,1,1,,1,11,1,
,,1,1,,1,1,1,1
,,,,,,,,
ˆ
)(
ˆ
)(
System of Differential Equations:
HHAHBdt
dH
TAPATBPBdt
dT
CAPACBPBdt
dC
PTACAPAPA
TBCBPBPBdt
dP
jiAjiji
ji
Bji
jiAjiji
Ajiji
Bjiji
ji
Bji
jiAjiji
Ajiji
Bjiji
ji
Bji
jiAjiji
Ajiji
Ajiji
Aji
jiBjiji
Bjiji
Bjiji
ji
Bji
ˆ)(
)(
)(
ˆ)
(
,,,1,
,
1,1,,,1,1,,1,
,
,1,1,,,1,1,1,
,
1,1,,1,1,,,,
1,1,,1,1,,,1,
,
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