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Modeling of Targeted Drug
Delivery
Neeraj Agrawal
University of Pennsylvania
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
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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
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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).
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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
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8000
6000
4000
2000
0
4 deg C
37 deg C
In vitro experimental data
from Dr. Muzykantov
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.
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S
S=penetration depth
Parameter for Glycocalyx Resistance
For a nanocarrier,
k = 1.6*10-6 N/m
Mulivor, A.W.; Lipowsky, H.H. Am J Physiol Heart Circ Physiol 283: H1282-1291, 2002
University of Pennsylvania
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 [1]
Bond spring constant
100 dyne/cm [2]
1. Muro, et. al. J. Pharma. And expt. Therap. 2006 317, 1161.
2. Eniola, A.O. Biophysical Journal, 89 (5): 3577-3588
University of Pennsylvania
2
Monte-Carlo moves for bond-formation
Select a nanocarrier at random. Check if it’s within bond-formation
distance
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  
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Computational Details

Program developed in-house.

Mersenne Twister random number generator (period of 219937-1)

Implemented using Intel C++ and MPICH used for parallelization

System reach steady state within 200 million monte-carlo moves.

Relatively low computational time required (about 4 hours on multiple
processors)
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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.
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Effect of Antigen Diffusion
In silico experiments
Antigen: 2000 / μm2
Carriers: 80 nM
25
Antigens can diffuse
30
Antigens can't diffuse
Antigens can't diffuse
25
Multivalency
20
Multivalency
Antigens can diffuse
15
10
5
20
15
80 nM
800 nM
10
5
0
0
200 antigens /
μm2
2000 antigens /
μm2
5 beads
50 beads
For nanocarrier concentration of 800 nM, binding of nanocarriers
is not competitive for antigen concentration of 2000 antigens/ μm2
University of Pennsylvania
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
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Effect of Glycocalyx
In silico experiments
Based on Glycocalyx spring constant = 1.6*10-7 N/m
35
No glycocalyx
with glycocalyx
30
No glycocalyx
with glycocalyx
3000
2500
2000
20
ln K
Multivalency
25
1500
15
10
1000
5
500
0
0
4 deg C
37 deg C
4 deg C
37 deg C
Presence of glycocalyx affects temperature dependence of
equilibrium constant though multivalency remains unaffected
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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
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Work in Progress

For larger glycocalyx
resistance, importance
sampling does not give
accurate picture

Implementation of umbrella
sampling protocol
Near Future Work
To include membrane deformation using Time-dependent Ginzburg-Landau
equation.
University of Pennsylvania
Acknowledgments
Vladimir Muzykantov
Weining Qiu
David Eckmann
Andres Calderon
Portonovo Ayyaswamy
University of Pennsylvania
Calculation of Glycocalyx spring constant
Forward rate (association) modeled as second order reaction
Backward rate (dissociation) modeled as first order reaction
Rate constants derived by fitting Lipowsky data to rate
equation.
Presence of glycocalyx effects only forward rate contant.
k glycocalyx= 1 k
forward
500 forward
K glycocalyx= 1 K
500
G glycocalyx= G  k T ln500
B
glycocalyx resistance  k T ln 500
B
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Glycocalyx thickness
Squrie et. al.
50 – 100 nm
Vink et. al.
300 – 500 nm
Viscosity of glycocalyx phase ~ 50-90 times higher than
that of water
Lee, G.M.; JCB 120: 25-35 (1993).
Review chapters on glycocalyx
•
Robert, P.; Limozin, L.; Benoliel, A.-M.; Pierres, A.; Bongrand, P. Glycocalyx regulation
of cell adhesion. In Principles of Cellular engineering (M.R. King, Ed.), pp. 143-169,
Elsevier, 2006.
•
Pierres, A.; Benoliel, A.-M.; Bongrand, P. Cell-cell interactions. In Physical chemistry of
biological interfaces (A. Baszkin and W. Nord, Eds.), pp. 459-522, Marcel Dekker, 2000.
University of Pennsylvania
Bell Model

Bell (Science, 1978)
f 
kr  f   kr  0  exp 

k
T
 B 
we can loosely associate  with  L   
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Umbrella Sampling

A biasing potential added to the system along the desired coordinate
to make overall potential flatter

Probability distribution along the bottleneck-coordinate calculated

New biasing potential = -ln (P)

For efficient sampling, system divided into smaller windows.

WHAM (weighted histogram analysis method) used to remove the
artificial biasing potential at the end of the simulation to get free
energy profile along the coordinate.
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Additional Simulation Parameters
ICAM size
19 nm × 3 nm
R 6.5 size
15 nm
Chemical cut-off
1.3 nm
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Determination of reaction free energy change
G( )  kBT ln Kd
Muro, et. al. J. Pharma. And expt. Therap. 2006 317, 1161.
University of Pennsylvania
Glycocalyx morphology
Weinbaum, S. et. al. PNAS 2003, 100, 7988.
University of Pennsylvania
Fitting to Lipowsky data
B   C
B is constant in a flow
experiment
dC
 k1 B  Bmax  C   k2C
dt


k1 B
C t  
Bmax 1  exp    k1 B  k2  t 
k1 B  k2
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