Thesis Project Proposal Multiscale Modeling of Protein-Mediated Membrane Dynamics: Integrating Cell Signaling with Trafficking Neeraj Agrawal Clathrin Advisor: Ravi Radhakrishnan

Previous Work Monte-Carlo Simulations Role of Glycocalyx in mediating nanocarrier-cell adhesion Protein-Mediated DNA Looping Agrawal, N.J. Radhakrishnan, R.; Purohit, P. Biophys J. submitted Agrawal, N.J. Radhakrishnan, R.; J. Phys. Chem. C. 2007, 111, 15848. biological process of DNA transcription, replication, recombination and repair involve binding of proteins at multiple sites on same DNA molecules. These proteins may interact with each other leading to formation of DNA loop. DNA to membrane  1 D to 2D Implications of DNA elasticity under applied force will be seen later in membrane elasticity TDD: inclusion of membrane curvature Collaboraters DNA elasticity under applied force

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 Click after Long-term Minimal model for protein-membrane interaction in endocytosis on the mesoscale

Endocytosis of EGFR A member of Receptor Tyrosine Kinase (RTK) family Transmembrane protein Modulates cellular signaling pathways – proliferation, differentiation, migration, altered metabolism At low EGF, exlusively by CDE. At higher EGF, both CDE and CIE. Multiple possible pathways of EGFR endocytosis – depends on ambient conditions Clathrin Dependent Endocytosis Clathrin Independent Endocytosis

Clathrin Dependent Endocytosis One of the most common internalization pathway AP - 2 epsin clathrin AP-2 . EGF Membrane AP2 Common theme: Cargo Recognition – AP2 Membrane bending proteins – Clathrin, epsin Clathrin polymerizes in vitro under non-physiological condn. Or in presence of AP2 Alpha subunit of AP2 interact with PtdIns(4,5)P2 lipid with 5-10 microM. AP2 bilayer int. is weak  coat is dynamic Mu2 of AP2 interact with FYRALM of EGFR. Both AP2 and clathrin can bind to epsin increasing its conc in coat. Clathrin energetics by mean size of clathrin cluster FYRALM = Phenylalanina-Tyr-Arginine-Alanine-Leucine-Methionine Hypothesis: Clathrin+AP2 assembly alone is not enough for vesicle formation, accessory curvature inducing proteins required. Clathrin polymerization Kirchhausen lab.

Overview Membrane models Protein diffusion models Model Integration Random walker Tale of three elastic models Preliminary Results

Multiscale Modeling of Membranes Generalized elastic model Time scale Monolayer viscous dissipation Viscoelastic model Bilayer slippage Fully atomistic MD simulations provide a great deal of information about membrane dynamics but only for short time and length scale. One tech. for performing simulations at larger time scale is to define coarse grained model lipids which are then studeid by MD. However, in order to model micron scale membrane surfaces, one one need to coarse-grain membrane further. Molecular Dynamics (MD) Coarse-grained MD Length scale ns Fully-atomistic MD nm µm

Linearized 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 z(x,y) Spontaneous curvature Bending modulus Frame tension Splay modulus Consider only those deformations for which membrane topology remains same. Force acting normal to the membrane surface (or in z-direction) drives membrane deformation By plugging in the expressions for mean curvature and area of the surface and linearizing the terms, we get the energy as a functional. Force is negative gradient of potential. In our case, it is the negative variational derivative of energy functional. H0 is the parameter that will depend on the protein-membrane interaction. Monge parameterization says that we can represent a surface by (x,y,h(x,y)) The Monge gauge approximation makes the elastic model amenable to Cartesian coordinate system

Hydrodynamics of the Monge-TDGL Non inertial Navier Stoke equation z(x,y) Dynamic viscosity of surrounding fluid Solution of the above PDEs results in Oseen tensor, (Generalized Mobility). y x Oseen tensor Hydrodynamic coupling Fluid velocity is same as membrane velocity at the membrane boundary  no slip condition given by: What’s ginzburg-landau dynamics ? Check hydrodynamics ? Random term is delta correlated – obtained using fluctuation dissipation theorem Xi (zai) White noise This results in the Time-Dependent Ginzburg Landau (TDGL) Equation

Local-TDGL Formulation for Extreme Deformations × 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.   Local Monge Gauges Monge-TDGL, mean curvature = Linearization A new formalism to minimize Helfrich energy. No linearizing assumptions made. Applicable even when membrane has overhangs Local-TDGL, mean curvature = Membrane elastic forces act in x, y and z directions

Hydrodynamics of the Local-TDGL Non-inertial Navier Stoke equation Dynamic viscosity of surrounding fluid Surface viscosity of bilayer Surrounding fluid velocity Membrane velocity Fluid velocity is same as membrane velocity at the membrane boundary What’s ginzburg-landau dynamics ?

Solution Protocol for Monge-TDGL Periodic boundary conditions for membrane. Numerical solution using discrete version of membrane dynamics equation Explicit Euler scheme with h4 spatial accuracy The harmonic series is a diverging series for a periodic system. We sum in Fourier space (k1, k2) ‘n’ is number of grid points Harmonic series – diverges but slowely Random term drawn from gaussian distribution. Divergence removed by neglecting mode k=0 (rigid body translation)

Curvature-Inducing Protein Epsin Diffusion on the Membrane Each epsin molecule induces a curvature field in the membrane KMC-move Membrane in turn exerts a force on epsin Bound epsin position epsin(a)  epsin(a+a0) Diffusion in force field on a manifold – no analytical solution available. where a0 is the lattice size, F is the force acting on epsin 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

Hybrid Multiscale Integration Regime 1: Deborah number De<<1 or (a02/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 Add lattice size reduction <C> function in adiabatic limit Non-adiabatic: integrating non-adiabatic electronic transitions with Born-Oppenheimer dynamics Write about transition probability. Lattice  continuum: Epsin diffusion changes C0(x,y) Continuum  lattice: Membrane curvature introduces an energy landscape for epsin diffusion R

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.

Local-TDGL (No Hydrodynamics) A new formalism to minimize Helfrich energy. No linearizing assumptions made. Applicable even when membrane has overhangs 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. Exact solution for infinite boundary conditions TDGL solutions for 1×1 µm2 fixed membrane Define the problem: Membrane profile for a given H0 function. Mismatch because of different boundary conditions

Potential of Mean Force PMF is dictated by both energetic and entropic components x0 Epsin experience repulsion due to energetic component when brought close. Second variation of Monge Energy (~ spring constant). Weak attraction since as epsin moves farther from boundaries, energy decreases. Enthaplic attraction without membrane fluctuation. Test function Non-zero H0 increases the stiffness of membrane  lower thermal fluctuations Bound epsin experience entropic attraction.

Research Plan Extension to Methods Include protein-dynamics in Local-TDGL. Non-adiabatic formalism Numerical solver for Surface Evolution approach to validate Local-TDGL. Extension to Applications Inclusion of relevant information about Clathrin and AP2 in the model. Parameter sensitivity analysis. Phase diagram: effect of protein concentrations, effect of curvature functoin, and other system parameters.

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.

Radhakrishnan Lab. Members Acknowledgments Jonathan Nukpezah Joshua Weinstein Radhakrishnan Lab. Members

TIME-TABLE Method development 1-1.25 year Study of CDE 0.5-1 year

Surface Evolution For axisymmetric membrane deformation Exact minimization of Helfrich energy possible for any (axisymmetric) membrane deformation S=0 Membrane parameterized by arc length, s and angle φ. S=L Inputs from Talid Total curve length, L not known a priori

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.

Parameters Bending Rigidity ~ 4kBT = 1.6*10-13 erg Tension ~ 3 µm Diffusion coeff. in cell membrane ~ 0.01 µm2/s Cytoplasm viscosity ~ 0.006 Pa.s a0 = 3*3 nm (ENTH domain size)

Molecular Dynamics MD on bilayer and epsin incorporated bilayer Fluctuation spectrum of bilayer  bending rigidity and tension Intrinsic curvature Marsh, D., Biophys. J. 2001, 81, 2154. Blood, P. D.; Voth, G. A., PNAS 2006, 103, (41), 15068-15072.

Targeted Drug Delivery

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.

Clathrin and AP2 models Clathrin AP2 H0 = H0(r,t,t0,r0) t0 and r0: time and position of nucleation H0 grows in position as a function of time. Rate of appearance ~ 3 events/(100 µm2-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, 8719. AP2 α-subunit of AP2 interact with PtdIns(4,5)P2 lipid with 5-10 µM. AP2 interacts with FYRALM motif on EGFR  Docking studies to find KD. Add AP2 dynamics

Correlations Radial Distribution function Probability of two particles being at distance ‘r’ compared to that of uniformly distribution. Orientational Correlation function Measures hexagonal ordering Potential of Mean Force PMF of a system with N molecules is the potential that gives the average force over all the configurations of all the n+1...N molecules acting on a particle ‘j’ at any fixed configuration keeping fixed a set of molecules 1...n

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

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 Rµ reaction. where hµ = number of distinct combinations for reaction Rµ to happen cµ = mean rate of reaction Rµ. where both r1 and r2 are uniform random number in [0,1].

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]. More like an inertial motion, i.e. high Re number limit.

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. More like an inertial motion, i.e. high Re number limit.