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Neeraj AgrawalUniversity of Pennsylvania
1
Modeling of Targeted Drug Delivery and Endocytosis
Neeraj Agrawal
Epsin
Clathrin
MembraneAp180Epsin
Clathrin
MembraneAp180
Clathrin
Neeraj AgrawalUniversity of Pennsylvania
2
Targeted Drug Delivery
Drug Carriers injected near the diseased cells Mostly drug carriers are in µm to nm scale Carriers functionalized with molecules specific to the receptors
expressed on diseased cells Leads to very high specificity and low drug toxicity
Neeraj AgrawalUniversity of Pennsylvania
3
Motivation for Modeling Targeted Drug Delivery
Predict conditions of nanocarrier arrest on cell – binding mechanics, receptor/ligand diffusion, membrane deformation, and post-attachment convection-diffusion transport interactions
Determine optimal parameters for microcarrier design – nanocarrier size, ligand/receptor concentration, receptor-ligand interaction, lateral diffusion of ligands on microcarrier membrane and membrane stiffness
Neeraj AgrawalUniversity of Pennsylvania
4
Glycocalyx Morphology and Length Scales
100 nm1,2,3Glycocalyx
10 nmAntibody
100 nmBead
20 nmAntigen
10-20 μmCell
Length Scales
1 Pries, A.R. et. al. Pflügers Arch-Eur J Physiol. 440:653-666, (2000).
2 Squire, J.M., et. al. J. of structural biology, 136, 239-255, (2001).
3 Vink, H. et. al., Am. J. Physiol. Heart Circ. Physiol. 278: H285-289, (2000).
Neeraj AgrawalUniversity of Pennsylvania
5
Effect of Glycocalyx (Experimental Data)
Mulivor, A.W.; Lipowsky, H.H. Am J Physiol Heart Circ Physiol 283: H1282-1291, 2002
Binding of carriers increases about 4 fold upon infusion of heparinase.
Glycocalyx may shield beads from binding to ICAMs
Increased binding with increasing temperature can not be explained in an exothermic reaction
0
2000
4000
6000
8000
10000
12000
4 deg C 37 deg C
nu
mb
er
of n
an
ob
ead
s b
ou
nd
/cell
In vitro experimental data from Dr. Muzykantov
Neeraj AgrawalUniversity of Pennsylvania
6
Proposed Model for Glycocalyx Resistance
21presence of glycocalyx absence of glycoca lyx
2G G kS
S
S=penetration depth
The glycocalyx resistance is a combination of
•osmotic pressure (desolvation or squeezing out of water shells),
•electrostatic repulsion
•steric repulsion between the microcarrier and glycoprotein chains of the glycocalyx
•entropic (restoring) forces due to confining or restricting the glycoprotein chains from accessing many conformations.
Neeraj AgrawalUniversity of Pennsylvania
7
Parameter for Glycocalyx Resistance
Mulivor, A.W.; Lipowsky, H.H. Am J Physiol Heart Circ Physiol 283: H1282-1291, 2002
For a nanocarrier, k = 3.9*109 J/m4
Neeraj AgrawalUniversity of Pennsylvania
8
Simulation Protocol for Nanocarrier Binding
Equilibrium binding simulated using Metropolis Monte Carlo.
New conformations are generated from old ones by-- Translation and Rotation of nanocarriers-- Translation of Antigens on endothelial cell surface
Bond formation is considered as a probabilistic event Bell model is used to describe bond deformation
Periodic boundary conditions along the cell and impenetrable boundaries perpendicular to cell are enforced
21( ) ( )2
G L G k L
System size 110.5 μm
Nanocarrier size 100 nm
Number of antibodies per nanocarrier 212
Equilibrium bond energy -7.98 × 10-20 J/molecule
Bond spring constant 1000 dyne/cm
Antigen Flexural Rigidity 700 pN-nm2
=equilibrium bond lengthL=bond length
Neeraj AgrawalUniversity of Pennsylvania
9
Select an antibody on this nanocarrier at random. Check if it’s within bond-formation distance.
Select an antigen at random. Check if it’s within bond-formation distance.
For the selected antigen, antibody; bond formation move is accepted with a probability
If selected antigen, antibody are bonded with each other, then bond breakage move accepted with a probability
Monte-Carlo moves for bond-formation
min 1,exp BG k T
min 1,exp BG k T
Glycocalyx
ICAM-1
Nanocarrier
R6.5
L σ
Endothelial cell
H
ICAM-1 flexure
ZcGlycocalyx
ICAM-1
Nanocarrier
R6.5
L σ
Endothelial cell
H
ICAM-1 flexure
Zc
Select a nanocarrier at random. Check if it’s within bond-formation distance
Neeraj AgrawalUniversity of Pennsylvania
10
Binding Mechanics
Multivalency: Number of antigens (or antibody) bound per nanocarrier
Radial distribution function of antigens: Indicates clustering of antigens in the vicinity of bound nanobeads
Energy of binding: Characterizes equilibrium constant of the reaction in terms of nanobeads
These properties are calculated by averaging four different in silico experiments.
Neeraj AgrawalUniversity of Pennsylvania
11
Effect of Antigen DiffusionIn silico experiments
0
0.5
1
1.5
2
2.5
3
3.5
640 2000antigens/m2
mul
tival
ency
Non-diffusing ICAM-1
Diffusing ICAM-1
Increasing antigen concentration diminishes the effect of antigen diffusion.
-30
-25
-20
-15
-10
-5
0
640 2000antigens/m2
Bin
ding
ene
rgy
(kca
l/mol
)
Non-diffusing ICAM-1
Diffusing ICAM-1
Neeraj AgrawalUniversity of Pennsylvania
12
Effect of Antigen FlexureIn silico experiments
Allowing antigens to flex leads to higher multivalency.
Neeraj AgrawalUniversity of Pennsylvania
13
Spatial Modulation of Antigens
Diffusion of antigens leads to clustering of antigens near bound nanocarriers
500 nanocarriers (i.e. 813 nM) on a cell with antigen density of 2000/μm2
Nanobead length scale
Neeraj AgrawalUniversity of Pennsylvania
14
Effect of GlycocalyxIn silico experiments
Presence of glycocalyx affects temperature dependence of equilibrium constant.
Based on Glycocalyx spring constant = 1.6*10-7 N/m
Neeraj AgrawalUniversity of Pennsylvania
15
Conclusions
Antigen diffusion leads to higher nanocarrier binding affinity Diffusing antigens tend to cluster near the bound nanocarriers Glycocalyx represents a physical barrier to the binding of
nanocarriers Presence of Glycocalyx not only reduces binding, but may also
reverse the temperature dependence of binding
Neeraj AgrawalUniversity of Pennsylvania
16
Multiscale Modeling of Protein-Mediated Membrane Dynamics:
Integrating Cell Signaling with Trafficking
Neeraj Agrawal
Epsin
Clathrin
MembraneAp180Epsin
Clathrin
MembraneAp180
Clathrin
Neeraj AgrawalUniversity of Pennsylvania
17
Endocytosis: The Internalization Machinery in Cells
Detailed molecular and physical mechanism of the process still evading.
Endocytosis is a highly orchestrated process involving a variety of proteins.
Attenuation of endocytosis leads to impaired deactivation of EGFR – linked to cancer
Membrane deformation and dynamics linked to nanocarrier adhesion to cells
Short-term
Quantitative dynamic models for membrane invagination: Development of a multiscale approach to describe protein-membrane interaction at the mesoscale (m)
Long-term
Integrating with signal transduction
Minimal model for protein-membrane interaction in endocytosis on the mesoscale
Neeraj AgrawalUniversity of Pennsylvania
18
Endocytosis of EGFR
A member of Receptor Tyrosine Kinase (RTK) family Transmembrane protein Modulates cellular signaling pathways – proliferation,
differentiation, migration, altered metabolism
Multiple possible pathways of EGFR endocytosis – depends on ambient conditions– Clathrin Dependent Endocytosis– Clathrin Independent Endocytosis
Neeraj AgrawalUniversity of Pennsylvania
19
Clathrin Dependent Endocytosis
One of the most common internalization pathway
Kirchhausen lab.Kirchhausen lab.
AP-2
epsin
epsi
n
AP-2
clathrin
clathrin
clathrin
AP-2
epsin
epsin
AP-2
clat
hrin
clathrin
clathrin
AP-2
ep
sin
clathrin
.
EGF
Membrane
Common theme:– Cargo Recognition – AP2– Membrane bending proteins – Clathrin, epsin
AP2
Clathrin polymerization
Neeraj AgrawalUniversity of Pennsylvania
20
Wiley, H.S., Trends in Cell biology, vol 13, 2003.
Trafficking Mechanism of EGFR
Neeraj AgrawalUniversity of Pennsylvania
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OverviewProtein diffusion modelsMembrane models
Model Integration
Preliminary Results
Tale of three elastic modelsRandom walker
Neeraj AgrawalUniversity of Pennsylvania
22
Linearized Elastic Model For Membrane: Monge-TDGL
Helfrich membrane energy accounts for membrane bending and membrane area extension.
Force acting normal to the membrane surface (or in z-direction) drives membrane deformation
2 2 4 20 0, 0, 0 02
2z x x y y
EF H z H z H H z z H
z
2 22 2 20 02 4 2 xx yy xyA
E z H H z z z z dxdy
0H Spontaneous curvature Bending modulus
Frame tension Splay modulus
Consider only those deformations for which membrane topology remains same.
z(x,y)
The Monge gauge approximation makes the elastic model amenable to Cartesian coordinate system
2
0 02
bend areaE E E
AE C H A A
In Monge notation, for small deformations, the membrane energy is
0
( ) ( )lim
E E z E z
z
Neeraj AgrawalUniversity of Pennsylvania
23
Curvature-Inducing Protein Epsin Diffusion on the Membrane
Each epsin molecule induces a curvature field in the membrane
0 ix Membrane in turn exerts a force on epsin
Epsin performs a random walk on membrane surface with a membrane mediated force field, whose solution is propagated in time using the
kinetic Monte Carlo algorithm
2 20 0
220
i i
i
x x y y
Ri
i
H C e
0 iy Bound epsin position
2 2
0 02
2
2 020 02
0 2
i i
i
x x y y
RiiA
i i
H zCEF e z H x x dxdy
x R
Extracellular
Intracellular
Membrane
x
z
yProtein proteins
KMC-move
0
2 20
4, exp
1 x
FaDrate a
kTa Z
Metric
epsin(a) epsin(a+a0)
where a0 is the lattice size, F is the force acting on epsin0 ixE
Neeraj AgrawalUniversity of Pennsylvania
24
Hybrid Multiscale Integration Regime 1: Deborah number De<<1
or (a02/D)/(z2/M) << 1
Regime 2: Deborah number De~1 or (a2/D)/(z2/M) ~ 1
KMC TDGL#=1/De #=/t
R R
( ( ) ( )) ( )P R P R P R
( ) { ( ) }BP R exp E R k T
Surface hopping switching probability
Relationship Between Lattice & Continuum Scales
Lattice continuum: Epsin diffusion changes C0(x,y)Continuum lattice: Membrane curvature introduces an energy
landscape for epsin diffusionR
Extracellular
Intracellular
Membrane
x
z Protein
Extracellular
Intracellular
Membrane
x
z Protein
Neeraj AgrawalUniversity of Pennsylvania
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Applications
Monge TDGL (linearized model)– Radial distribution function– Orientational correlation function
Surface Evolution validation, computational advantage. Local TDGL vesicle formation. Integration with signaling
– Clathrin Dependent EndocytosisInteraction of Clathrin, AP2 and epsin with membrane
– Clathrin Independent Endocytosis– Targeted Drug Delivery
Interaction of Nanocarriers with fluctuating cell membrane.
Neeraj AgrawalUniversity of Pennsylvania
26
Local-TDGL(No Hydrodynamics)
A new formalism to minimize Helfrich energy.
No linearizing assumptions made.
Applicable even when membrane has overhangs
0 200 400 600 800 10000
10
20
30
40
50
60
70
x (or y) [nm]
z [n
m]
Monge TDGL
local TDGL
exact
Exact solution for infinite boundary conditions
TDGL solutions for 1×1 µm2 fixed membrane
At each time step, local coordinate system is calculated for each grid point.
Monge-TDGL for each grid point w.r.to its local coordinates.
Rotate back each grid point to get overall membrane shape.
Neeraj AgrawalUniversity of Pennsylvania
27
Potential of Mean Force
0 50 100 150-1
0
1
2
3
4
5
6
7x 10
-15
x0 [nm]
Ene
rgy
[J]
1010 m2
55 m2
11 m2
PMF is dictated by both energetic and entropic components
Epsin experience repulsion due to energetic component when brought close.
2 22 2 20 0
2A
E H dxdy
Second variation of Monge Energy (~ spring constant).
Non-zero H0 increases the stiffness of membrane lower thermal fluctuations
Test function
Bound epsin experience entropic attraction.
2 2 4 20 0, 0, 0 02 0
2x x y yH z H z H H z z H
x0
Neeraj AgrawalUniversity of Pennsylvania
28
Glycocalyx Morphology and Length Scales
100 nm1,2,3Glycocalyx
10 nmAntibody
100 nmBead
20 nmAntigen
10-20 μmCell
Length Scales
1 Pries, A.R. et. al. Pflügers Arch-Eur J Physiol. 440:653-666, (2000).
2 Squire, J.M., et. al. J. of structural biology, 136, 239-255, (2001).
3 Vink, H. et. al., Am. J. Physiol. Heart Circ. Physiol. 278: H285-289, (2000).