University of Pennsylvania Department of Bioengineering Hybrid Models For Protein-Membrane Interactions At Mesoscale: Bridge to Intracellular Signaling Neeraj Agrawal, Jonathan Nukpezah, Joshua Weinstein & Ravi Radhakrishnan University of Pennsylvania Clathrin

University of Pennsylvania Department of Bioengineering Objectives 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 flow and signal transduction Targeted Therapeutics

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

University of Pennsylvania Department of Bioengineering Length scale Time scale nm ns µmµm s Fully-atomistic MD Coarse-grained MD Generalized elastic model Bilayer slippage Monolayer viscous dissipation Viscoelastic model

University of Pennsylvania Department of Bioengineering Elastic Model For Membrane Monge TDGL 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

University of Pennsylvania Department of Bioengineering Hydrodynamics Non inertial Navier Stoke equation Dynamic viscosity Greens function of above PDE results in Oseen tensor, (Generalized Mobility matrix). Oseen tensor in infinite medium Fluid velocity is same as membrane velocity at the membrane boundary  no slip condition For Monge TDGL, viscous dissipation within bilayer is irrelevant. Hydrodynamic coupling

University of Pennsylvania Department of Bioengineering Epsin Diffusion Each epsin molecule induces a (additive) curvature field in the membrane Membrane in turn exerts a force on epsin Epsin performs a random walk on membrane surface with a bias dictated by force acting on epsin Diffusion in a force field on a 2 dimensional manifold – No general analytical solution exist Bound epsin position

University of Pennsylvania Department of Bioengineering Epsin Diffusion Solution propagated in time using kinetic monte carlo The elementary reaction characterized by rate: For 2 D Metric Where is the lattice size, F is the force acting on epsin, i.e. epsin(a)  epsin(a+a 0 )

University of Pennsylvania Department of Bioengineering 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 Relationship Between Lattice & Continuum Scales Lattice  continuum: Epsin diffusion changes C 0 (x,y) Continuum  lattice: Membrane curvature introduces an energy landscape for epsin diffusion F r R C0C0 R  C 0 /  t=-(1/M)  F/  C 0 =-(  C 0 /t Diffusion,Free )  F/  C 0 t Diffusion,Membrane /t Diffusion,Free =1/(  F/  C 0 )

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.

University of Pennsylvania Department of Bioengineering Local-TDGL 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 =

University of Pennsylvania Department of Bioengineering Applications Monge TDGL (linearized model)  Phase transitions Surface Evolution Local TDGL Integration with signaling –CDE –TDD

University of Pennsylvania Department of Bioengineering Local-TDGL 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

University of Pennsylvania Department of Bioengineering Potential of Mean Force PMF is dictated by both energetic and entropic components Epsin experience repulsion due to energetic component when brought close. Second variation of Monge Energy (~ spring constant). Non-zero H 0 increases the stiffness of membrane  lower thermal fluctuations Test function Bound epsin experience entropic attraction.

University of Pennsylvania Department of Bioengineering Acknowledgments