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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 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 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 55 m2 4 11 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