Post on 15-Jan-2016
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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
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).
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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
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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.
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Parameter for Glycocalyx Resistance
Mulivor, A.W.; Lipowsky, H.H. Am J Physiol Heart Circ Physiol 283: H1282-1291, 2002
For a nanocarrier, k = 1.6*10-6 N/m
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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
1. Muro, et. al. J. Pharma. And expt. Therap. 2006 317, 1161.2. Eniola, A.O. Biophysical Journal, 89 (5): 3577-3588
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 [1]
Bond spring constant 100 dyne/cm [2]
=equilibrium bond lengthL=bond length
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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
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
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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
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.
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Effect of Antigen DiffusionIn silico experiments
For nanocarrier concentration of 800 nM, binding of nanocarriers is not competitive for antigen concentration of 2000 antigens/ μm2
0
5
10
15
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25
200 antigens 2000 antigens
Mu
ltiv
ale
nc
y
Antigens can diffuse
Antigens can't diffuse
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5
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5 beads 50 beads
Mu
ltiv
ale
nc
y
Antigens can diffuse
Antigens can't diffuse
Carriers: 80 nM Antigen: 2000 / μm2
/ μm2 / μm2
80 nM 800 nM
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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
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Effect of GlycocalyxIn silico experiments
0
5
10
15
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25
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35
4 deg C 37 deg C
Mu
ltiv
ale
nc
y
No glycocalyx
with glycocalyx
0
500
1000
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2500
3000
4 deg C 37 deg Cln
K
No glycocalyx
with glycocalyx
Presence of glycocalyx affects temperature dependence of equilibrium constant though multivalency remains unaffected
Based on Glycocalyx spring constant = 1.6*10-7 N/m
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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.
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Acknowledgments
Vladimir MuzykantovWeining Qiu
David EckmannAndres Calderon
Portonovo Ayyaswamy
University of Pennsylvania
Calculation of Glycocalyx spring constant
forwardk=glycocalyx
forwardk
5001
500lnTBkG=glycocalyxG
Forward rate (association) modeled as second order reactionBackward 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=glycocalyxK5001
500lnresistance glycocalyx TBk
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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.
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 waterLee, G.M.; JCB 120: 25-35 (1993).
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Bell Model
Bell (Science, 1978) 0 expr rB
fk f k
k T
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
Muro, et. al. J. Pharma. And expt. Therap. 2006 317, 1161.
( ) lnB d
G k T K
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Glycocalyx morphology
Weinbaum, S. et. al. PNAS 2003, 100, 7988.
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Fitting to Lipowsky data
B C B is constant in a flow experiment
1 max 2
dCk B B C k C
dt
1max 1 2
1 2
1 expk B
C t B k B k tk B k