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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  k1 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

55 m2
4
11 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