1. University of Pennsylvania Department of Bioengineering Hybrid Mesoscale Models For Protein- Membrane Interactions Neeraj Agrawal, Jonathan Nukpezah, Joshua Weinstein, Ravi Radhakrishnan Targeted Therapeutics Bridging Intracellular Signaling with Trafficking Endocytosis: the internalization machinery in cells

2. University of Pennsylvania Department of Bioengineering Hierarchical Multiscale Modeling Weinan E, Bjorn Engquist, Notices of the ACM, 2003 K E Gubbins et al., J. Phys.: Cond. Matter, 2006 Minimal model for protein-membrane interaction in endocytosis is focused on the mesoscale

3. University of Pennsylvania Department of Bioengineering Multiscale Modeling of Membranes Length scale Time scale nm ns µmµm s Fully-atomistic MD Coarse-grained MD Generalized elastic model Bilayer slippage Monolayer viscous dissipation Viscoelastic model Molecular Dynamics (MD)

4. University of Pennsylvania Department of Bioengineering C 0 :Intrinsic curvature k: Bending Modulus k: Gaussian Curvature Modulus Helfrich Free Energy Cartesian (Monge) notation: h(x,y) 1 2 R 1 >0, R 2 >0 H>0, K>0 R 1 >0, R 2 <0 H=0, K<0 H  1/2[1/R 1 +1/R 2 ] K  1/R 1  1/R 2 Plane: H=0, K=0 Nelson, Piran, Weinberg, 1987 Mesoscale linearized elastic model for membrane

5. University of Pennsylvania Department of Bioengineering Linearized Elastic Model For Membrane: Monge Gauge Helfrich membrane energy accounts for membrane bending and membrane area extension. In Monge notation, for small deformations, the membrane energy is Force acting normal to the membrane surface (or in z-direction) drives membrane deformation Spontaneous curvatureBending modulus Frame tension Splay modulus Consider only those deformations for which membrane topology remains same. White noise z(x,y) The Monge gauge approximation makes the elastic model amenable to Cartesian coordinate system

6. University of Pennsylvania Department of Bioengineering Hydrodynamics of the Linearized Elastic Membrane Non inertial Navier Stoke equation  Dynamic viscosity Solution of the above PDEs results in Oseen tensor, (Generalized Mobility). Oseen tensor Fluid velocity is same as membrane velocity at the membrane boundary  no slip condition given by: This results in the Time-Dependent Ginzburg Landau (TDGL) Equation z(x,y) x y

7. University of Pennsylvania Department of Bioengineering Curvature-Inducing Protein Epsin Diffusion on the Membrane Each epsin molecule induces a curvature field in the membrane 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 Bound epsin position For 2 D Metric epsin(a)  epsin(a+a 0 ) where a 0 is the lattice size, F is the force acting on epsin KMC-move

8. University of Pennsylvania Department of Bioengineering KMC-TDGL Hybrid Multiscale Integration Regime 1: Deborah number De<<1 or (a 2 /D)/(z 2 /M) << 1 Regime 2: Deborah number De~1 or (a 2 /D)/(z 2 /M) ~ 1 KMC TDGL #=1/De #=  /  t Surface hopping switching probability Weinstein, Radhakrishnan, 2006 Constant Temperature Protein- Mediated Membrane Dynamics C 0 /µm -1 R/nm R, Range C 0, Intrinsic Curvature  *, Surface Density

9. University of Pennsylvania Department of Bioengineering Membrane-Mediated Potential of Mean Force between Epsins PMF is dictated by both energetic and entropic components Energy: Epsin experience repulsion due to energetic component when brought close. Entropy: Regions of non-zero H 0 assume increased stiffness and hence reduced membrane fluctuations Therefore, the system can lower its free energy by localizing epsins on the membrane leading to membrane-mediated epsin- epsin attraction  2 E~ spring constant;  =test function

10. University of Pennsylvania Department of Bioengineering Membrane Dynamics, R= 40nm Membrane-Mediated Protein-Protein Spatial Correlations  *=0.004, C 0 =20  *=0.03, C 0 =20 Localization F/kT C 0 *=20 R*=40 nm Threshold No effective membrane-mediated attraction; no nucleation below threshold curvature and range

11. University of Pennsylvania Department of Bioengineering Membrane Dynamics, R=60nm Membrane-Mediated Protein-Protein Spatial Correlations  *=0.008, C 0 =10  *=0.008, C 0 =40  *=0.008, C 0 =60 Localization F/kT C 0 **: 30-50 F(r)  k B Tln g(r) Nucleation limited only by diffusional timescale of association (NVA)

12. University of Pennsylvania Department of Bioengineering Membrane Dynamics, R=100nm Membrane-Mediated Protein-Protein Spatial Correlations  *=0.016, C 0 =30 Localization F/kT Threshold C 0 *: 10-30 Nucleation occurs following spatial localization of epsin

13. University of Pennsylvania Department of Bioengineering x y 1 st Shell 2 nd Shell Epsin arrangement x y θjθj Sustained orientational correlations beyond nearest- neighbors drives nucleation Nucleation via Hexatic Orientational Ordering: NVOO

14. University of Pennsylvania Department of Bioengineering Membrane Dynamics, R=80nm Membrane Temporal Correlations  *=0.02, C 0 =5 Regions of non-zero H 0 assume increased stiffness and hence reduced membrane fluctuations High protein-surface density drives the membrane phase into a glass-like dynamical behavior

15. University of Pennsylvania Department of Bioengineering Global Phase Diagram ** C0C0 R NVLRO NVA No N GT GT: Glass-like transition; No N: No nucleation NVOO: Nucleation via orientational ordering NVA: Nucleation via diffusional association C 0 / µm -1 R / nm 20 40600 20 40 60 80 100 NVA NVOO No N GT 123 0.2 0.4 0.6 0.8 1.0 g(r=r 0 )  6 (r=r 0 )

16. University of Pennsylvania Department of Bioengineering Conclusions The KMC-TDGL approach is successful in describing the dynamic processes associated with the interaction of proteins and membranes at a coarse-grained level Membrane-mediated protein-protein repulsion effects short- and long-ranged ordering of epsins Two modes of nucleation observed -- Nucleation via Association : Effected by large C 0 -- Nucleation via Orientational Ordering: Effected by persistence of orientational correlations In the regime of large surface density, a glass transition is observed A global phase diagram is proposed

17. University of Pennsylvania Department of Bioengineering Integrating Signaling and Trafficking Extracellular Intracellular (MAP Kinases) Ras Raf MEK ERK CblClathrin, AP2 Endphepsin Proliferation Nucleus Umbrella Sampling KMC+TDGL Mechanism for receptor internalization

18. University of Pennsylvania Department of Bioengineering Acknowledgments Graduate Students Andrew Shih (PhD, BE) Yingting Liu (PhD, BE) Jeremy Purvis (PhD, GCB) Shannon Telesco (PhD, BE) Jonathan Nukpezah (PhD, BE) Undergraduate Students Joshua Weinstein (Senior, PHYS) Collaborators Mark Lemmon, (Penn) Sung-Hee Choi, (Penn) Boris Kholodenko, (TJU) Funding NSF; Whitaker Foundation; NIH training grant; NPACI supercomputing allocations; Greater Philadelphia Bioinformatics Alliance Co-Authors: Neeraj Agrawal, Jonathan Nukpezah, Joshua Weinstein

19. University of Pennsylvania Department of Bioengineering

20. University of Pennsylvania Department of Bioengineering Local TDGL Formulation for Extreme Deformations A new formalism to minimize Helfrich energy. No linearizing assumptions made. Applicable even when membrane has overhangs Exact solution for infinite boundary conditions TDGL solutions for 1×1 µm 2 fixed membrane Surface represented in terms of local coordinate system. Monge TDGL valid for each local coordinate system. Overall membrane shape evolution – combination of local Monge-TDGL. Monge-TDGL, mean curvature = Linearization Local-TDGL, mean curvature = Local Monge Gauges

21. University of Pennsylvania Department of Bioengineering Surface Evolution For axisymmetric membrane deformation Exact minimization of Helfrich energy possible for any (axisymmetric) membrane deformation Membrane parameterized by arc length, s and angle φ.

22. University of Pennsylvania Department of Bioengineering

23. University of Pennsylvania Department of Bioengineering Paradigms of Membrane Curvature McMahon, Gallop, Nature reviews, 2005

24. University of Pennsylvania Department of Bioengineering Epsin Clathrin Membrane Ap180 Imaging Endocytosis Ford et al., Nature, 2002

25. University of Pennsylvania Department of Bioengineering Ap180+ClathrinEpsin+ClathrinAp180+Epsin+Clathrin Ford et al., Nature, 2002 EpsinClathrinAp180 Endocytosis Machinery Receptor Inactivation to Neurotransmitters

26. University of Pennsylvania Department of Bioengineering 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