university of pennsylvania modeling of targeted drug delivery neeraj agrawal

University of Pennsylvania

Modeling of Targeted Drug Delivery

Neeraj Agrawal


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


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


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).


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


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.


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


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


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


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)


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.


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

20

25

200 antigens 2000 antigens

Mu

ltiv

ale

nc

y

Antigens can diffuse

Antigens can't diffuse

0

5

10

15

20

25

30

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


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


Effect of GlycocalyxIn silico experiments

0

5

10

15

20

25

30

35

4 deg C 37 deg C

Mu

ltiv

ale

nc

y

No glycocalyx

with glycocalyx

0

500

1000

1500

2000

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


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


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.


Acknowledgments

Vladimir MuzykantovWeining Qiu

David EckmannAndres Calderon

Portonovo Ayyaswamy


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


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).


Bell Model

Bell (Science, 1978) 0 expr rB

fk f k

k T

we can loosely associate with L


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.


Additional Simulation Parameters

ICAM size 19 nm × 3 nm

R 6.5 size 15 nm

Chemical cut-off 1.3 nm


Determination of reaction free energy change

Muro, et. al. J. Pharma. And expt. Therap. 2006 317, 1161.

( ) lnB d

G k T K


Glycocalyx morphology

Weinbaum, S. et. al. PNAS 2003, 100, 7988.


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

university of pennsylvania modeling of targeted drug delivery neeraj agrawal

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