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Thesis Project Proposal Multiscale Modeling of Protein-Mediated Membrane Dynamics: Integrating Cell Signaling with Trafficking Neeraj Agrawal Epsin Ap180 Membrane Clathrin Clathrin Advisor: Ravi Radhakrishnan University of Pennsylvania Chemical and Biomolecular Engineering Previous Work Monte-Carlo Simulations Protein-Mediated DNA Looping Role of Glycocalyx in mediating nanocarriercell adhesion Agrawal, N.J. Radhakrishnan, R.; Purohit, P. Biophys J. submitted Agrawal, N.J. Radhakrishnan, R.; J. Phys. Chem. C. 2007, 111, 15848. 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 Minimal model for protein-membrane interaction multiscale approach to describe protein-membrane interaction at the in mesoscale endocytosis (m) is focused on the mesoscale Long-term Integrating with signal transduction 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 Dependent Endocytosis – Clathrin Independent Endocytosis University of Pennsylvania Chemical and Biomolecular Engineering Clathrin Dependent Endocytosis One of the most common internalization pathway Membrane . EGF clathrin AP2 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. Clathrin polymerization Kirchhausen lab. University of Pennsylvania Chemical and Biomolecular Engineering Overview Membrane models Protein diffusion models Model Integration Random walker Tale of three elastic models Preliminary Results University of Pennsylvania Chemical and Biomolecular Engineering Multiscale Modeling of Membranes s E A 2 H H0 2 dA A A flat A u 0 2 H H0 2 dA A A flat 2u P Tij P ij i u j j ui Tij P ij i u j j ui E Tzz z ~ 2 Fx Txz v b v v 0 E Tzz z Generalized elastic model Monolayer viscous dissipation Bilayer slippage Coarse-grained MD r F U (r ) 2 t 2 ns u 0 2u P Time scale E m Molecular Dynamics (MD) Length scale Fully-atomistic MD nm University of Pennsylvania Viscoelastic model µm Chemical and Biomolecular Engineering Linearized Elastic Model For Membrane: Monge-TDGL Helfrich membrane energy accounts for membrane bending and membrane area extension. In notation, for small deformations, the membrane energy is E Monge Ebend E area A 22 2 A2 2 E E 2C z H 0 AH z z z z 0 0 xx yy xy dxdy A 2 2 2 4 Bending modulus H 0 Spontaneous curvature Frame tension Splay modulus z(x,y) E onlyEthose ( z ) E ( z ) for which membrane topology Consider deformations lim remains 0 z same. Force acting normal to the membrane surface (or in z-direction) drives membrane deformation Fz E 2 H 0 z x H 0, x z y H 0, y H 02 2 z 4 z 2 H 0 z 2 The Monge gauge approximation makes the elastic model amenable to Cartesian coordinate system University of Pennsylvania Chemical and Biomolecular Engineering Hydrodynamics of the Monge-TDGL Non inertial Navier Stoke equation z(x,y) p 2 u F u 0 Dynamic viscosity of surrounding fluid Solution of the above PDEs results in Oseen tensor, (Generalized Mobility). u (r r ') F (r ')dr ' Oseen tensor I rr 8 r 1 Hydrodynamic coupling Extracellular Fluid velocity is same as membrane velocity at the membrane boundary no slip condition given by: z E t z k (t ) 0 x y Membrane z White noise k (t ) k ' (t ') 2k BTL2 k1 k , k ' (t t ') Protein Intracellular y x This results in the Time-Dependent Ginzburg Landau (TDGL) Equation University of Pennsylvania Chemical and Biomolecular Engineering 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. Monge-TDGL, mean curvature = 1 z z 1 z z 1 z z 2 x 2 yy y xx 2 z x z y z xy Local Monge Gauges z xx z yy A new formalism to minimize x y Helfrich energy. No linearizing assumptions made. Local-TDGL, mean curvature = z xx z yy Applicable even when membrane has overhangs 2 2 3 Linearization 2 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 p 2 u F u 0 Tij P ij i u j j ui E Tzz z ~ 2 Fx Txz v 0 Dynamic viscosity of surrounding fluid Surface viscosity of bilayer u Surrounding fluid velocity v Membrane velocity Fz Fx Fluid velocity is same as membrane velocity at the membrane boundary University of Pennsylvania Chemical and Biomolecular Engineering Solution Protocol for Monge-TDGL Periodic boundary conditions for membrane. Numerical solution using discrete version of membrane dynamics equation dz 1 E dt ij i j 8 rij ri ' j ' z Explicit Euler scheme with h4 spatial accuracy The harmonic series is a diverging series for a periodic system. ‘n’ is number of grid points We sum in Fourier space (k1, k2) dz (k ) 1 E 2 jk e dt 4 k i j z ij 1 / n 2 ik e 1 / n k Divergence removed by neglecting mode k=0 (rigid body translation) 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 Extracellular Membrane z Protein Intracellular y H 0 Ci e proteins Membrane in turn exerts a force on epsin E Ci F 2 x0i Ri A x x0 i 2 y y0 i 2 e 2 Ri2 epsin(a) epsin(a+a0) 2 Ri2 i KMC-move x x x0 i 2 y y0 i 2 x 0i y 0i Bound epsin position 2 2 H 0 z z H0 x x0i dxdy 2 4D Fa0 rate, a 2 exp 2 kT a0 1 Z x where a0 is the lattice size, F is the force acting on epsin x E 0i 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 University of Pennsylvania Chemical and Biomolecular Engineering Hybrid Multiscale Integration Regime 1: Deborah number De<<1 or (a02/D)/(z2/M) << 1 KMC #=1/De TDGL #=/t Regime 2: Deborah number De~1 or (a2/D)/(z2/M) ~ 1 Surface hopping switching probability R R P( R) exp{E( R)kBT } ( P( R) P( R)) P( R) Extracellular Membrane Relationship Between Lattice & Continuum Scales z Protein Intracellular x Lattice continuum: Epsin diffusion changes C0(x,y) Continuum lattice: Membrane curvature introduces an energy landscape for epsin diffusion University of Pennsylvania R Chemical and Biomolecular Engineering 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. University of Pennsylvania Chemical and Biomolecular Engineering Local-TDGL (No Hydrodynamics) 70 Monge TDGL 60 local TDGL 50 z [nm] A new formalism to minimize Helfrich energy. No linearizing assumptions made. Applicable even when membrane has overhangs exact 40 30 20 10 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. 0 0 200 400 600 800 1000 x (or y) [nm] Exact solution for infinite boundary conditions TDGL solutions for 1×1 µm2 fixed membrane 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. x0 -15 7 x 10 6 Second variation of Monge Energy (~ spring constant). 10 10 m2 5 Energy [J] 2 H 0 z x H 0, x z y H 0, y H 02 2 z 4 z 2 H 0 0 2 55 m2 4 11 m2 3 2 1 0 E 2 2 A 2 2 H 02 dxdy 0 2 -1 0 50 100 150 x0 [nm] Test function Non-zero H0 increases the stiffness of membrane lower thermal fluctuations Bound epsin experience entropic attraction. 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. Parameter sensitivity analysis. TIMETABLE 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 SurfaceEvolution. 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 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. '''sin ' sin 3 2 2sin cos 3sin 2 '' ' R 2R 2 ( R) sin 2 ( R) sin 3 2 S=L 2 ( R) cos 2 sin ( R) ( R) sin 2 2R R R 3cos 2 sin 2 ( R) sin 2 2 2 2 1 cos sin 2 R R 2 ' ( R) cos sin 3 2 2R sin ( R) sin ( R ) cos 2 2R2 2 s L 0 s 0 0 ' s 0 0 R s 0 R0 R s L 0 R ' cos 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 ~ 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) University of Pennsylvania Chemical and Biomolecular Engineering Molecular Dynamics MD on bilayer and epsin incorporated bilayer k TA Fluctuation spectrum of bilayer bending rigidity hk2 4 B k k2 and tension xx yy Intrinsic curvature 2 H 0 z dz Marsh, D., Biophys. J. 2001, 81, 2154. z Blood, P. D.; Voth, G. A., PNAS 2006, 103, (41), 15068-15072. 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 PoissonBoltzmann 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 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). Ehrlich, M. et. al. Cell 2004, 118, 8719. – Rate of growth ~ one triskelion/(2 s) – Rate of dissociation inferred from mean life time of clathrin cluster 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. University of Pennsylvania Chemical and Biomolecular Engineering Correlations Radial Distribution function Probability of two particles being at distance ‘r’ compared to that of uniformly distribution. Orientational Correlation function Measures hexagonal ordering 6* (0) 6 (r ) 6 (r ) e i 6 j ( r ) j 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 University of Pennsylvania Chemical and Biomolecular Engineering 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 Pi (t , dt ) Pij (t , dt ) j k Pi (k ) Pij j 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 Rµ reaction. M P( , ) h c exp h j c j j 1 where hµ = number of distinct combinations for reaction Rµ to happen cµ = mean rate of reaction Rµ. aT hi ci i 1 ln 1/ r1 aT 1 hc i 1 i i r2 aT hi ci i 1 where both r1 and r2 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]. z E M t z 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. University of Pennsylvania Chemical and Biomolecular Engineering