University of Pennsylvania Chemical and Biomolecular Engineering Multiscale Modeling of Protein-Mediated Membrane Dynamics: Integrating Cell Signaling with Trafficking Neeraj Agrawal Clathrin Advisor: Ravi Radhakrishnan Thesis Project Proposal

University of Pennsylvania Chemical and Biomolecular Engineering Previous Work Monte-Carlo Simulations Agrawal, N.J. Radhakrishnan, R.; Purohit, P. Biophys J. submitted Agrawal, N.J. Radhakrishnan, R.; J. Phys. Chem. C. 2007, 111, Protein-Mediated DNA Looping Role of Glycocalyx in mediating nanocarrier- cell adhesion DNA elasticity under applied force

University of Pennsylvania Chemical and Biomolecular Engineering 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 is focused on the mesoscale

University of Pennsylvania Chemical and Biomolecular Engineering 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 Mediated Endocytosis –Clathrin Independent Endocytosis

University of Pennsylvania Chemical and Biomolecular Engineering Clathrin Dependent Endocytosis One of the most common internalization pathway Kirchhausen lab. AP - 2 epsin AP - 2 clathrin AP-2 epsin AP - 2 clathrin AP - 2 epsin clathrin. EGF Membrane Common theme: –Cargo Recognition – AP2 –Membrane bending proteins – Clathrin, epsin Hypothesis: Clathrin+AP2 assembly alone is not enough for vesicle formation, accessory curvature inducing proteins required. AP2 Clathrin polymerization

University of Pennsylvania Chemical and Biomolecular Engineering Overview Protein diffusion models Membrane models Model Integration Preliminary Results Tale of three elastic models Random walker

University of Pennsylvania Chemical and Biomolecular Engineering 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)

University of Pennsylvania Chemical and Biomolecular Engineering 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 Spontaneous curvatureBending 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 In Monge notation, for small deformations, the membrane energy is

University of Pennsylvania Chemical and Biomolecular Engineering Hydrodynamics of the Monge-TDGL Non inertial Navier Stoke equation Dynamic viscosity of surrounding fluid 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 Hydrodynamic coupling White noise

University of Pennsylvania Chemical and Biomolecular Engineering Local-TDGL Formulation for Extreme Deformations A new formalism to minimize Helfrich energy. No linearizing assumptions made. Applicable even when membrane has overhangs 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 Membrane elastic forces act in x, y and z directions ×

University of Pennsylvania Chemical and Biomolecular Engineering Hydrodynamics of the Local-TDGL Non-inertial Navier Stoke equation Dynamic viscosity of surrounding fluid Fluid velocity is same as membrane velocity at the membrane boundary Surface viscosity of bilayer Surrounding fluid velocity Membrane velocity

University of Pennsylvania Chemical and Biomolecular Engineering 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. S=0 S=L

University of Pennsylvania Chemical and Biomolecular Engineering Solution Protocol for Monge-TDGL Divergence removed by neglecting mode k=0 (rigid body translation) The harmonic series is a diverging series for a periodic system. We sum in Fourier space (k 1, k 2 ) Periodic boundary conditions for membrane. Numerical solution using discrete version of membrane dynamics equation ‘n’ is number of grid points Explicit Euler scheme with h 4 spatial accuracy

University of Pennsylvania Chemical and Biomolecular Engineering 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 KMC-move Metric epsin(a)  epsin(a+a 0 ) where a 0 is the lattice size, F is the force acting on epsin

University of Pennsylvania Chemical and Biomolecular Engineering 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 R Other approach: Reduce protein lattice size.

University of Pennsylvania Chemical and Biomolecular Engineering Applications Monge TDGL (linearized model)  Phase transitions –Radial distribution function –Orientational correlation function Surface Evolution  validation, computational advantage. Local TDGL  vesicle formation. Integration with signaling –Clathrin Dependent Endocytosis –Clathrin Independent Endocytosis –Targeted Drug Delivery

University of Pennsylvania Chemical and Biomolecular Engineering Local-TDGL (No Hydrodynamics) 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 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.

University of Pennsylvania Chemical and Biomolecular Engineering 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. x0

University of Pennsylvania Chemical and Biomolecular Engineering Research Plan Include protein-dynamics in Local-TDGL. Non-adiabatic formalism Numerical solver for Surface Evolution approach to validate Local-TDGL. Inclusion of relevant information about Clathrin and AP2 in the model. Development of Global Phase Diagram.

University of Pennsylvania Chemical and Biomolecular Engineering Summary A Monte Carlo study to show the importance of glycocalyx and antigen flexural rigidity for nanocarrier binding to cell surface. Effect of protein size on DNA loop formation probability demonstrated using Metropolis, Gaussian sampling and Density of State Monte Carlo. Two new formalisms developed for calculating membrane shape for non-zero spontaneous curvature  Local-TDGL and Surface- Evolution. Interaction between two membrane bound epsin studied.

University of Pennsylvania Chemical and Biomolecular Engineering Acknowledgments Jonathan Nukpezah Joshua Weinstein Radhakrishnan Lab. Members

University of Pennsylvania Chemical and Biomolecular Engineering Hydrodynamics Main assumptions – validity ? –Surrounding fluid extends to infinity –Membrane is located at z=0, i.e. deformations are low. Hydrodynamics in cellular environment is much more complicated. Can be used to compare system (dynamic and equilibrium) behavior in absence and presence of hydrodynamic interactions. Can be used to validate results against in vitro experiments.

University of Pennsylvania Chemical and Biomolecular Engineering Parameters Bending Rigidity ~ 4k B T = 1.6* erg Tension ~ 3 µm Diffusion coeff. in cell membrane ~ 0.01 µm 2 /s Cytoplasm viscosity ~ Pa.s a 0 = 3*3 nm (ENTH domain size)

University of Pennsylvania Chemical and Biomolecular Engineering Molecular Dynamics MD on bilayer and epsin incorporated bilayer Fluctuation spectrum of bilayer  bending rigidity and tension Intrinsic curvature Blood, P. D.; Voth, G. A., PNAS 2006, 103, (41), Marsh, D., Biophys. J. 2001, 81, 2154.

University of Pennsylvania Chemical and Biomolecular Engineering Targeted Drug Delivery

University of Pennsylvania Chemical and Biomolecular Engineering Atomistic to Block-Model Each protein – a combination of blocks. Charge per block determined by solving non-linear Poisson- Boltzmann equation. Implicit solvent. LJ parameters – sum of LJ parameters of all atom types in a block. Electrostatics & vDW are relevant only for distances of 30 Å. Specific interaction.

University of Pennsylvania Chemical and Biomolecular Engineering Clathrin and AP2 models Clathrin H 0 = H 0 (r,t,t 0,r 0 ) t 0 and r 0 : time and position of nucleation –H 0 grows in position as a function of time. –Rate of appearance ~ 3 events/(100 µm 2 -s). –Rate of growth ~ one triskelion/(2 s) –Rate of dissociation inferred from mean life time of clathrin cluster Ehrlich, M. et. al. Cell 2004, 118, AP2 α-subunit of AP2 interact with PtdIns(4,5)P 2 lipid with 5-10 µM. AP2 interacts with FYRALM motif on EGFR  Docking studies to find K D.

University of Pennsylvania Chemical and Biomolecular Engineering Correlations Radial Distribution function Measures hexagonal ordering Orientational Correlation function Probability of two particles being at distance ‘r’ compared to that of uniformly distribution.

University of Pennsylvania Chemical and Biomolecular Engineering Non-adiabatic Monte Carlo System can hop from one adiabatic energy surface to other. Let p i (t) and p i (t’) be probability of system being in state ‘i’ at time ‘t’ and time t’ = t+dt Define P i (t,dt) = p i (t) - p i (t’) A transition from state ‘i’ to state ‘k’ is now invoked if P i (k) < ζ < P i (k+1) ζ (0≤ ζ ≤ 1) is a uniform random number

University of Pennsylvania Chemical and Biomolecular Engineering Kinetic Monte Carlo P(τ,µ)dτ = probability at time t that the next reaction will occur in time interval (t+τ, t+τ+dτ) and will be an R µ reaction. where h µ = number of distinct combinations for reaction R µ to happen c µ = mean rate of reaction R µ. where both r 1 and r 2 are uniform random number in [0,1].

University of Pennsylvania Chemical and Biomolecular Engineering Ginzburg-Landau theory Based on Landau’s theory of second-order phase transition, Ginzburg and Landau argued that the free energy, F near the transition can be expressed in terms of a complex order parameter. This type of Landau-Ginzburg equation is also referred to as potential motion [i.e. it, by itself, attempts to drive the membrane shape to an equilibrium state corresponding to the minimum in the free energy (F) of the membrane].

University of Pennsylvania Chemical and Biomolecular Engineering Bilayer Experiments Micropipette aspiration: Use Laplace law to find surface tension of membrane. Constant area experiments. Thermal fluctuation spectrum  bending rigidity Membrane tether formation: tension of a cell membrane can be measured via the force (applied by an optical trap) to pull a membrane tether.