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

2. 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,
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

3. 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

4. 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

5. 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.

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

7. 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

8. 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

9. 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

10. 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

11. 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 ?

12. 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)

13. 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

14. 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

15. 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.

16. 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

17. 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.

18. 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.

19. 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.

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

21. TIME-TABLE
Method development
year
Study of CDE
0.5-1 year

22. 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

23. 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.

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

25. 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),

26. Targeted Drug Delivery

27. 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.

28. 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

29. 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

30. 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

31. 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].

32. 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.

33. 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.