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Modeling of Targeted Drug
Delivery and Endocytosis
Neeraj Agrawal
Epsin Ap180
Membrane
Clathrin
Clathrin
Neeraj Agrawal
University of Pennsylvania
1
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 Agrawal
University of Pennsylvania
2
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 Agrawal
University of Pennsylvania
3
Glycocalyx Morphology and Length Scales
Length Scales
Cell
10-20 μm
Antigen
20 nm
Bead
100 nm
Antibody
10 nm
Glycocalyx
100 nm1,2,3
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 Agrawal
University of Pennsylvania
4
Effect of Glycocalyx (Experimental Data)
Binding of carriers increases about
4 fold upon infusion of heparinase.
Glycocalyx may shield beads from
binding to ICAMs
number of nanobeads bound/cell
12000
10000
Mulivor, A.W.; Lipowsky, H.H. Am J Physiol Heart Circ Physiol 283: H1282-1291, 2002
Increased binding with increasing temperature can
not be explained in an exothermic reaction
Neeraj Agrawal
University of Pennsylvania
8000
6000
4000
2000
0
4 deg C
37 deg C
In vitro experimental data
from Dr. Muzykantov
5
Proposed Model for Glycocalyx Resistance
1
G  presence of glycocalyx   G absence of glycoca lyx   kS 2
2
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 Agrawal
University of Pennsylvania
S
S=penetration depth
6
Parameter for Glycocalyx Resistance
For a nanocarrier,
k = 3.9*109 J/m4
Mulivor, A.W.; Lipowsky, H.H. Am J Physiol Heart Circ Physiol 283: H1282-1291, 2002
Neeraj Agrawal
University of Pennsylvania
7
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
G( L)  G( )  1 k  L   
2
Bond formation is considered as a probabilistic event
Bell model is used to describe bond deformation
=equilibrium bond length
L=bond length
Periodic boundary conditions along the cell and impenetrable boundaries
perpendicular to cell are enforced
System size
110.5 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
Neeraj Agrawal
University of Pennsylvania
8
2
Monte-Carlo moves for bond-formation
R6.5
Nanocarrier
ICAM-1 flexure

H
Glycocalyx

L
σ
Zc
ICAM-1
Select a nanocarrier at random.
Check if it’s within bond-formation
distance
Endothelial cell
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 min 1,exp   G kBT  
If selected antigen, antibody are bonded with each other, then bond
breakage move accepted with a probability min 1,exp  G kBT  
Neeraj Agrawal
University of Pennsylvania
9
Binding Mechanics
Multivalency: Number of
antigens (or antibody) bound per
nanocarrier
Energy of binding:
Characterizes equilibrium
constant of the reaction in
terms of nanobeads
Radial distribution function of
antigens: Indicates clustering of
antigens in the vicinity of bound
nanobeads
These properties are
calculated by averaging four
different in silico experiments.
Neeraj Agrawal
University of Pennsylvania
10
Effect of Antigen Diffusion
In silico experiments
640
3.5
Binding energy (kcal/mol)
Diffusing ICAM-1
2.5
multivalency
2000
0
Non-diffusing ICAM-1
3
antigens/m2
2
1.5
1
0.5
-5
-10
-15
-20
-25
0
640
antigens/m2
2000
Non-diffusing ICAM-1
Diffusing ICAM-1
-30
Increasing antigen concentration diminishes the effect of antigen
diffusion.
Neeraj Agrawal
University of Pennsylvania
11
Effect of Antigen Flexure
In silico experiments
Allowing antigens to flex leads to higher multivalency.
Neeraj Agrawal
University of Pennsylvania
12
Spatial Modulation of Antigens
500 nanocarriers (i.e. 813 nM)
on a cell with antigen density
of 2000/μm2
Nanobead length
scale
Diffusion of antigens leads to clustering of
antigens near bound nanocarriers
Neeraj Agrawal
University of Pennsylvania
13
Effect of Glycocalyx
In silico experiments
Based on Glycocalyx spring constant = 1.6*10-7 N/m
Presence of glycocalyx affects temperature dependence of
equilibrium constant.
Neeraj Agrawal
University of Pennsylvania
14
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 Agrawal
University of Pennsylvania
15
Multiscale Modeling of Protein-Mediated Membrane
Dynamics:
Integrating Cell Signaling with Trafficking
Epsin Ap180
Membrane
Clathrin
Clathrin
Neeraj Agrawal
Neeraj Agrawal
University of Pennsylvania
16
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
Minimal
model
for protein-membrane
interaction
a multiscale
approach
to describe
protein-membrane interaction
at thein
mesoscale (m) endocytosis on the mesoscale

Long-term
Integrating with signal transduction
Neeraj Agrawal
University of Pennsylvania
17
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 Agrawal
University of Pennsylvania
18
Clathrin Dependent Endocytosis

One of the most common internalization pathway
Membrane
.
EGF
clathrin
AP2
 Common theme:
– Cargo Recognition – AP2
– Membrane bending proteins – Clathrin, epsin
Clathrin polymerization
Kirchhausen lab.
Neeraj Agrawal
University of Pennsylvania
19
Trafficking Mechanism of EGFR
Wiley, H.S., Trends in Cell biology, vol 13, 2003.
Neeraj Agrawal
University of Pennsylvania
20
Overview
Membrane models
Protein diffusion models
Model Integration
Random walker
Tale of three elastic models
Preliminary Results
Neeraj Agrawal
University of Pennsylvania
21
Linearized Elastic Model For Membrane: Monge-TDGL
 Helfrich membrane energy accounts for membrane bending and
membrane area extension.
 In
notation,
for small deformations, the membrane energy is
E Monge
 Ebend  E
area
 A 2
2
   A2  
2
E  E    C
z  H 0   AH


z


z
z

z



0
0
xx yy
xy  dxdy

A 2 2
2
4
 Bending modulus
H 0 Spontaneous curvature

22
Frame tension

z(x,y)
Splay modulus
 E only E
( z  
)  E ( z ) for which membrane topology
 Consider
those
deformations
same.
lim
remains
 z  0

 Force acting normal to the membrane surface (or in z-direction)
drives membrane deformation
E
 2
 2
Fz  
 2 H 0  z x H 0, x  z y H 0, y    H 0      z    4 z   2 H 0
z
2

The Monge gauge approximation makes the elastic model amenable to Cartesian
coordinate system
Neeraj Agrawal
University of Pennsylvania
22
Curvature-Inducing Protein Epsin Diffusion on the Membrane
 Each epsin molecule induces a
curvature field in the membrane
Extracellular
Membrane
z
Protein
Intracellular
y
H 0   Ci e
proteins
 Membrane in turn exerts a force on epsin
E  Ci
F 
 2
x0i
Ri

A

e
 x  x0 i 2  y  y0 i 2
2 Ri2
epsin(a)  epsin(a+a0)
2 Ri2
i
KMC-move
x

 x  x0 i 2  y  y0 i 2
x 0 i y 0i Bound epsin position
2
 2
H 0  z  
  z  H0 
  x  x0i  dxdy


2


4D
 Fa0 
rate, a  2
exp  

2
kT
a0 1  Z x 


where a0 is the lattice size, F is the force acting on epsin  x E
0i
Metric
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
Neeraj Agrawal
University of Pennsylvania
23
Hybrid Multiscale Integration
 Regime 1: Deborah number De<<1
or (a02/D)/(z2/M) << 1
KMC
#=1/De
TDGL
#=/t
 Regime 2: Deborah number De~1 or (a2/D)/(z2/M) ~ 1
Surface hopping switching
probability
R  R
P( R)  exp{E( R)kBT }
  ( P( R)  P( R)) P( R)
Extracellular
Membrane
Relationship Between Lattice & Continuum Scales
z
Protein
Intracellular
x
Lattice  continuum: Epsin diffusion changes C0(x,y)
Continuum  lattice: Membrane curvature introduces an energy
landscape for epsin diffusion
Neeraj Agrawal
University of Pennsylvania
R
24
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 Endocytosis
Interaction of Clathrin, AP2 and epsin with membrane
– Clathrin Independent Endocytosis
– Targeted Drug Delivery
Interaction of Nanocarriers with fluctuating cell membrane.
Neeraj Agrawal
University of Pennsylvania
25
Local-TDGL
(No Hydrodynamics)
70
Monge TDGL
60
local TDGL
50
z [nm]
 A new formalism to minimize Helfrich
energy.
 No linearizing assumptions made.
 Applicable even when membrane has
overhangs
exact
40
30
20
10
 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.
0
0
200
400
600
800
1000
x (or y) [nm]
Exact solution for infinite boundary conditions
TDGL solutions for 1×1 µm2 fixed membrane
 Rotate back each grid point to get
overall membrane shape.
Neeraj Agrawal
University of Pennsylvania
26
Potential of Mean Force
 PMF is dictated by both energetic and entropic
components
 Epsin experience repulsion due to
energetic component when brought close.
x0
-15
7
x 10
6
 Second variation of Monge Energy (~ spring
constant).
10 10 m2
5
Energy [J]


2 H 0  z x H 0, x  z y H 0, y    H 02      2 z    4 z   2 H 0  0
2

55 m2
4
11 m2
3
2
1
0
 E          
2
2
A
2
2


  H 02       dxdy  0
2

-1
0
50
100
150
x0 [nm]

Test function
 Non-zero H0 increases the stiffness of membrane  lower thermal
fluctuations
Bound epsin experience entropic attraction.
Neeraj Agrawal
University of Pennsylvania
27
Glycocalyx Morphology and Length Scales
Length Scales
Cell
10-20 μm
Antigen
20 nm
Bead
100 nm
Antibody
10 nm
Glycocalyx
100 nm1,2,3
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 Agrawal
University of Pennsylvania
28