Multiscale Modeling of Protein-Mediated Membrane Processes Joshua Weinstein*, Mark Goulian*, & Ravi Radhakrishnan¶ Departments of Physics* and Bioengineering¶, University of Pennsylvania

Endocytosis: The Internalization Machinery in Cells

Paradigms of Membrane Curvature McMahon, Gallop, Nature reviews, 2005

Imaging Endocytosis Ap180 Ford et al., Nature, 2002 Epsin Membrane Clathrin

Endocytosis Machinery Ford et al., Nature, 2002 Epsin Clathrin Ap180 Ap180+Clathrin Epsin+Clathrin Ap180+Epsin+Clathrin Receptor Inactivation to Neurotransmitters

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

Hierarchical Multiscale Modeling K E Gubbins et al., J. Phys.: Cond. Matter, 2006 Weinan E, Bjorn Engquist, Notices of the ACM, 2003

Hierarchical Multiscale Modeling Spatial Multiscale Methods (Traditional) Boundary layer Analysis, Perturbation Theory Multigrid Method Domain Decomposition, Adaptive Mesh Refinement Temporal Multiscale Methods Separation of timescales, discrete frequency spectrum Multiple-timestep Molecular Dynamics, -leap KMC Free energy methods, Density of States Methods Path-based approaches: Transition path sampling, Nudged elastic band, Stochastic String Fast Evaluation Methods Effective treatment of long-range interactions Fast Multipole Method Ewald-summation Linear Scaling Density Functional Theory Based on a single Lagrangian, Hamiltonian, energy function, or equation of motion

Hierarchical Multiscale Modeling Coarse Graining Hamiltonians Field-theoretic expansion based on order parameters Frequency filtering Effective potential for Force-matching Effective potential for Property matching Dynamical Coarse Graining Time-stepping with up and down shifting Hybrid Lagrangian, Hamiltonians Carr-Parrinello MD, Born-Oppenheimer MD Mixed Quantum Mechanics Molecular Mechanics Hybrid Integrators Mixed deterministic and Stochastic Integrators in kinetic modeling of networks Renormalization of the Hamiltonian, Energy Function Same effective phenomenology

Hierarchical Multiscale Modeling Hybrid Representations (Heterogeneous Multiscale Modeling) Decoupled Approach Transition State Dynamics Coupled Approaches Integration of Disparate Representations Non-adiabatic surface hopping dynamics MD+Navier-Stokes (flux matching) KMC+Diffusion field equation (flux matching) Mixed phenomenology Deterministic+Stochastic Domain Decomposotion The hybrid multiscale approach is well suited for modeling the complexities biological systems because it enables us to leverage the strengths of pre-validated phenomenological theories in different fields and combine them together

Model Components for Endocytosis Ap180 Epsin Vesicle membrane motion Hohenberg and Halperin, 1977 Nelson, Piran, Weinberg, 1987 Membrane z(x,y,t) membrane coordinates;  interfacial tension;  bending rigidity; M membrane mobility,  Langevin noise; F elastic free energy; C(x,y) is the intrinsic mean curvature of the membrane Epsin diffusion Gillespie, 1977 Clathrin Kinetic Monte Carlo: diffusion on a lattice

Epsin Localization Causes Membrane Invagination Time-dependent Ginzburg Landau Equation Dissipative Dynamics C(x,y)

Epsin-Membrane Interaction Parameters Range (R) r*, Surface Density Hardsphere exclusion C0 (intrinsic curvature) Measurable quantities: C0, D, ,  Micropipette, FRAP, Microscopy C(x,y) is the mean intrinsic curvature of the membrane determined by epsins adsorbed on the membrane. C(x,y) is dynamically varying because of lateral diffusion of epsins

Hybrid Multiscale Integration Regime 1: Deborah number De<<1 or (a2/D)/(z2/M) << 1 Regime 2: Deborah number De~1 or (a2/D)/(z2/M) ~ 1 KMC TDGL #=1/De #=/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 diffusion R R F C0 r C0/t=-(1/M)F/C0=-(C0/tDiffusion,Free)F/C0 tDiffusion,Membrane/tDiffusion,Free=1/(F/C0)

Membrane Dynamics (Constant Temperature) Epsin Localization Process Spontaneous? Activated? F/kT Determinants of Nucleation

Membrane Dynamics, R= 40nm Membrane-Mediated Protein-Protein Spatial & Orientational Correlations Membrane Dynamics, R= 40nm r*=0.004, C0=20 r*=0.03, C0=20

No Nucleation Below Threshold Curvature and Range Process Spontaneous? Activated? F/kT Threshold C0*=20 R*=40 nm Localization

sequence Membrane Dynamics, R=60nm Membrane-Mediated Protein-Protein Spatial & Orientational Correlations Membrane Dynamics, R=60nm r*=0.008, C0=10 sequence r*=0.008, C0=40 r*=0.008, C0=60

Membrane Dynamics, R=80nm Membrane-Mediated Protein-Protein Spatial & Orientational Correlations r*=0.008, C0=30 r*=0.008, C0=50

Nucleation Limited by Diffusional Timescale of Association Process Spontaneous? Activated? F(r)kBT ln g(r) F/kT C0**: 30-50 C0**> 50 Membrane Rupture? Localization

Membrane-Mediated Protein-Protein Spatial & Orientational Correlations Membrane Dynamics, R=80nm r*=0.008, C0=30 r*=0.016, C0=30

Membrane-Mediated Protein-Protein Spatial & Orientational Correlations Membrane Dynamics, R=100nm r*=0.016, C0=30

Membrane-Mediated Protein-Protein Spatial & Orientational Correlations Membrane Dynamics, R=80nm r*=0.02, C0=5 r*=0.02, C0=10 r*=0.03, C0=20

Nucleation occurs following the emergence of long-ranged spatial and orientational correlations Process Spontaneous? Activated? F/kT Requirements: Under conditions where orientational correlations persist Under conditions that preclude a glass transition C0*: 10-30 Localization

Global Phase Diagram r* C0 R GT No N NVLRO NVA GT: Glass transition No N: No nucleation NVLRO: Nucleation via long range order NVA: Nucleation via association

Conclusions The hybrid multiscale 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 and attraction effects short- and long-ranged ordering Two modes of nucleation observed -- NVA: Effected by large C0 -- NVLRO: Effected by persistence of orientational correlations In the regime of small intrinsic curvature and large density, a glass transition is observed A global phase diagram is proposed

Activation of Endocytosis as a Multiscale Problem Extracellular Molecular Dynamics Intracellular (MAP Kinases) Ras Mixed Quantum Mechanics Molecular Mechanics PLC IP3 DAG Raf Ca++ PKC MEK Nucleus ERK Proliferation KMC+TDGL

Acknowledgments Mr. Josh Weinstein, UG, (Co-Author) Prof. Mark Goulian (Co-Author) Mr. Neeraj Agrawal, PhD student, CBE Ms. Yingting Liu, PhD Student, BE Mr. Jeremy Weiss, UG student, BMB

Helfrich Free Energy Plane: H=0, K=0 1 R1>0, R2<0 H=0, K<0 C0 :Intrinsic curvature k: Bending Modulus k: Gaussian Curvature Modulus 2 R1>0, R2>0 H>0, K>0 Cartesian (Monge) notation: h(x,y) H1/2[1/R1+1/R2] K1/R11/R2 Nelson, Piran, Weinberg, 1987

The Variational Problem

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r* C0 R Global State Diagram GT No N NVLRO NVA GT: Glass transition No N: No nucleation NVLRO: Nucleation via long range order NVA: Nucleation via association

Multiscale modeling of Protein-mediated membrane dynamics Develop a New Heterogeneous Multiscale Approach Aim 1(a): Generalize and extend the multiscale approach Aim 2: Determine parameters from atomistic simulations Aim 3: Biological Application to endocytosis Aim 1(b): Develop a global state diagram Dielectric experiments of Dr. Bartkowiak, Poznan, Poland Membrane biophysical experiments of Dr. Tobias Baumgart, Penn. Cell biology experiments of Dr. Mark Lemmon, Penn Compare and validate Apply Long-term objective of integrating with flow, cytoskeleton motion, biochemical signal transduction Long-term Multiscale modeling of Protein-mediated membrane dynamics

Clathrin

Epsin Clathrin