1. A biochemical reaction network model is developed to explore APP-C99 homodimer formation and its binding to cholesterol, which are believed to play roles in Alzheimer's disease. 2. The model consists of kinetic equations that capture the association and dissociation of APP-C99 monomers, dimers, and APP-C99 bound to cholesterol. 3. The model is parameterized using experimental data and simulations of APP-C99 and cholesterol diffusion, and solved using particle-based and stochastic simulations.

1. Modeling the Biochemical Network of APP-C99:
Dimerization and Interactions with Cholesterol
Longjiaxin Zhong and John E. Straub
Boston University, Chemistry Department, 590 Commonwealth Avenue, Boston, MA 02215
Abstract
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Introduction
Methods
Mathematical Model for Reaction Network Conclusions
Future Direction
References
Acknowledgements
Alzheimer's Disease (AD) constitutes one of the leading causes of death in the elderly.
The goal of this work is to explore the nature of homodimer formation of the amyloid
precursor protein (APP-C99), and APP-C99 binding to cholesterol, which is believed to
play a critical role in the etiology of AD.
The model takes the form of a set of non-linear ordinary differential equations
describing the conception of C99 dimerization and binding to cholesterol. The rate of
homodimerization of APP-C99 was determined by theoretical calculation, computer
simulation, and Gillespie stochastic simulation. The computational association rates were
generated using a particle-based random walk and collision model written in python.
Preliminary results suggest that this simple theoretical model can capture essential
aspects of the association of APP-C99 and cholesterol in a cellular environment. In the
future, our network model may be further generalized to include more detailed models,
including two lipid domains (liquid-ordered and liquid-disordered) and additional lipids
(sphingomyelin) that may bind cholesterol.
Amyloid Precursor Protein (APP) is a
transmembrane protein that undergoes a series of
proteolytic processing steps involving secretase
enzymes, ultimately leading to the formation of
amyloid beta (Aβ) protein. Processing of APP produces
the 99 amino acid fragment known as APP-C99. APP-
C99 contains a single transmembrane helical domain
that is subsequently cleaved by the γ-secretase enzyme
to produce the Aβ protein. Aβ protein aggregates are
believed to act as pathogenic agents in AD. Fig 1. Model of C99 subdomains
imbedded in cell membrane and the Aβ
amyloid fragment produced from
cleavage by γ-secretase. 1
Fig 3. Depiction of C99 dimer in a
POPC lipid bilayer (left) and in a
DPC micelle (right) .2
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143 (2012).
2. Dominguez, L; Foster L; Straub,J.E.; Thirumalai, D, J. Am. Chem. Soc., 136 , 9619–9626 (2014)
3. Panahi,A; Bandara, A; Pantelopulos ,G; Straub, J.E., J. Phys. Chem. Lett., 7, 3535–3541 (2016)
A complete
understanding of the
kinetics of processing of
APP-C99 by γ-secretase
must include detailed
knowledge of the
distribution of APP-C99
substrates available for
cleavage, and how that
distribution depends on
APP-C99 protein sequence
and concentration of
cholesterol.
The goal of this research is to develop the first detailed kinetic model of the
network of reactions that determine the equilibrium concentrations of APP-C99 in
its various forms - monomer, homodimer, and monomer bound to cholesterol. Our
model consists of kinetic equations that capture the association and dissociation of
the key molecular species APP-C99 and cholesterol.
Thanks to my advisor Professor John Straub for scientific
instruction, my colleagues in the lab for helping me with SCC and
programing, and Doctor Ryo Urano for coding up the Gillespie
Algorithm.
This work was supported by Undergraduate Research Opportunities
Program (UROP).
The goal of this work is to explore the nature of APP-C99 homodimer formation, and
its role in Alzheimer's Disease (AD), as well as the role of APP-C99 in sensing
cholesterol.
1. BUILDING BIOCHEMICAL REACTION NETWORK: A novel biochemical
reaction network has been developed and used to explore cholesterol's influence on APP-
C99 homodimerization
2. PARAMETERIZING THE REACTION NETWORK MODEL: The biochemical
reaction network has been parameterized using results of experiment (equilibrium
constants for APP-C99 dimerization) and simulation (rates of APP-C99 and cholesterol
diffusion). The model explores realistic concentrations of peptide and cholesterol that
are found in in vitro experiments as well as in vivo conditions.
3. SOLUTION OF REACTION NETWORK MODEL: Two approaches are used
identify solutions of the biochemical reaction network: (1) particle-based simulation
models involving diffusion and reaction and (2) stochastic simulation of the reaction
kinetics.
In our model, [A] is the concentration of C99 monomer, [A2] is the concentration of
C99 dimer, and [C] and [C2] are concentrations of cholesterol monomer and cholesterol
dimer.
The basic kinetic model is
Fig 2. Representation of C99-
cholesterol complex. Cholesterol
molecules are marked in blue, lipid
molecules are in gray, and peptides
are shown as blue cylinders.3
Expressed in differential equation form
Additional Background
In the lipids of biomembranes,
phosphatidylcholine–based
lipids with cholesterol can
adopt the two coexisting fluid
phases that are Liquid-
Disordered phase (Ld) and
Liquid-Ordered phase (Lo). Lo
phase domains, high in
cholesterol are known as “raft”.
The reaction network can be further generalized to include two lipid
domains (liquid ordered and liquid disordered) and additional lipids
(sphingomyelin) that may bind cholesterol. Following the validation of
our model, a broad range of "initial conditions" for the concentrations
of APP-C99 and cholesterol (and later sphingomyelin) might be
studied. Our ultimate goal is to develop a detailed understanding of
the predominant species expected for each range of concentration of
protein and cholesterol. This knowledge will be used to address
critical questions regarding the state of the APP-C99 protein, namely:
"Is APP-C99 predominantly a monomer or a dimer?" and "Is APP-C99
free or bound to cholesterol?" Addressing these open questions will
represent a fundamental contribution to AD research.
Fig5. Representation of ripple phase, solid-
ordered phase, liquid-disordered phase, and
liquid-ordered phase.
Eeman, M; Deleu, M, Base, 14(4), 2010(2009)
Represented of Lo and Ld domains may be included as
Fig 4. Another representation of Ld and Lo
domains.
Meer, G; Voelker,D ; Feigenson, G, Nature Reviews
Molecular Cell Biology , 9, 112-124 (2008)
Fig 6. Liquid-ordered phase is on the left of the origin and
the Liquid-disorder is on the right. There are two C99
monomers which marked in red and blue. (Left) moving
trajectory of the particles. (Right) change of number of
particles. Red line represents the monomer, and blue line
represents the dimer.
A = C99 monomer
𝐴2= C99 dimer
C = cholesterol
monomer
𝐶2 = cholesterol dimer
AC = C99 – cholesterol
heterodimer
𝑘1 = Association rate of
two C99 monomers
𝑘−1 = Dissociation rate
of C99 dimer
𝑘2 = Association rate of
C99 monomer and
cholesterol monomer
𝑘−2 = Dissociation rate
of C99 – cholesterol
heterodimer
𝑘3 = Association rate of
two cholesterol
monomers
𝑘−3 = Dissociation rate
of cholesterol dimer
We used diffusion rates, encounter radii, and dissociation constants of C99 and
cholesterol in POPC lipid bilayer.
Temperature: 29.85 degree Celsius
Time span: 50 – 200 nanoseconds
Liquid disordered phase:
Diffusion rate of C99 (POPC): 6.0 × 10−9
𝑐𝑚 2
/𝑠
Diffusion rate of Cholesterol (POPC): 3.9 × 10−8 𝑐𝑚 2/𝑠
Encounter radii of C99 homo-dimerization: 9.3 Angstrom
Encounter radii of cholesterol homo-dimerization: 6.9 Angstrom
Encounter radii of C99 - cholesterol hetero-dimerization: 9.5 Angstrom
There is no experimental study for the diffusion rate of the C99 dimer. Following
Saffman and Delbruck (1975) we assumed the diffusion rate of the homodimer is twice
as fast as the monomer.
The particle-based numerical model includes a continuous time random walk model and
Brownian motion. The continuous time random walk model approximates diffusive
motion of the protein in a lipid membrane.
1. Modeling the diffusion in liquid ordered phase and disordered
phases
By programing in python, the simplest model with two phases that
separated from the origin was built up. Particles in the model are set at
random positions at the beginning and then do random walks in certain
time steps. We can observe from the trajectory that particles move faster
in the disordered phase than in the ordered phase.
Fig 7. (Left)There are 10 C99 monomers marked in
different colors in the system. (Right) the system includes
association and dissociation.
2. Simulating the association of
C99 and Cholesterol Homo-
dimerization
We ignored the dissociation of the
dimer and assumed the concentration
of the C99 or cholesterol in the
system doesn’t change in the process
of dimerization. The association rate
of C99 and cholesterol dimerization
was then calculated. According to
the stochastic model, the association
rate for C99 is 2.8 × 1010 1
𝑐𝑚2 𝑠
and
the association rate for cholesterol is
1.41× 1018 1
𝑐𝑚2 𝑠
.
Fig 8. Inverse number of C99 versus time (seconds)
where the slope will give us the association rate in
units of 1/s. The blue line is the data generated from
the stochastic model with the standard deviation in
blue. The red line and red region are data from
Gillespie algorithm.