Neeraj Agrawal University of Pennsylvania 1 Modeling of Targeted Drug Delivery and Endocytosis...

Neeraj AgrawalUniversity of Pennsylvania

1

Modeling of Targeted Drug Delivery and Endocytosis

Neeraj Agrawal

Epsin

Clathrin

MembraneAp180Epsin

Clathrin

MembraneAp180

Clathrin


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


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


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).


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


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.


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


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


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


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.


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


12

Effect of Antigen FlexureIn silico experiments

Allowing antigens to flex leads to higher multivalency.


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


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


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


16

Multiscale Modeling of Protein-Mediated Membrane Dynamics:

Integrating Cell Signaling with Trafficking

Neeraj Agrawal

Epsin

Clathrin

MembraneAp180Epsin

Clathrin

MembraneAp180

Clathrin


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


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


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


20

Wiley, H.S., Trends in Cell biology, vol 13, 2003.

Trafficking Mechanism of EGFR


21

OverviewProtein diffusion modelsMembrane models

Model Integration

Preliminary Results

Tale of three elastic modelsRandom walker


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


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


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


25

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.


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.


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


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).

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Neeraj Agrawal University of Pennsylvania 1 Modeling of Targeted Drug Delivery and Endocytosis...

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