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University of Pennsylvania Chemical and Biomolecular Engineering
Multiscale Modeling of Protein-Mediated Membrane Dynamics:Integrating Cell Signaling with Trafficking
Neeraj Agrawal
Epsin
Clathrin
MembraneAp180Epsin
Clathrin
MembraneAp180
Clathrin
Advisor: Ravi Radhakrishnan
Thesis Project Proposal
University of Pennsylvania Chemical and Biomolecular Engineering
Previous WorkMonte-Carlo Simulations
Agrawal, N.J. Radhakrishnan, R.; Purohit, P. Biophys J. submitted
Agrawal, N.J. Radhakrishnan, R.; J. Phys. Chem. C. 2007, 111,
15848.
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.Kirchhausen lab.
AP-2
epsin
epsi
n
AP-2
clathrin
clathrin
clathrin
AP-2
epsi
n
epsin
AP-2
clat
hrin
clathrin
clathrin
AP-2
epsi
n
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
OverviewProtein diffusion modelsMembrane models
Model Integration
Preliminary Results
Tale of three elastic modelsRandom walker
University of Pennsylvania Chemical and Biomolecular Engineering
Multiscale Modeling of Membranes
Length scale
Tim
e sc
ale
nm
ns
µm
s
Fully-atomistic MD
Coarse-grained MD
Generalized elastic model
Bilayer slippage
Monolayer viscous dissipation Viscoelastic model
2
0
2
2
0
flat
A
ij ij i j j i
zz
E H H dA A A
u
u P
T P u u
ET
z
2
0
2
2~
2
0
0
flat
A
ij ij i j j i
zz
x xz
E H H dA A A
u
u P
T P u u
ET
z
F T v b v v
2
2( )
rm F U r
t
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
2 2 4 20 0, 0, 0 02
2z x x y y
EF H z H z H H z z H
z
2 22 2 20 02 4 2 xx yy xyA
E z H H z z z z dxdy
0H Spontaneous curvature Bending 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
20 02
bend areaE E E
AE C H A A
In Monge notation, for small deformations, the membrane energy is
0
( ) ( )lim
E E z E z
z
University of Pennsylvania Chemical and Biomolecular Engineering
Hydrodynamics of the Monge-TDGL
z E
t z
Non inertial Navier Stoke equation
Dynamic viscosity of surrounding fluid
2
0
p u F
u
5555555555555 5
Solution of the above PDEs results in Oseen tensor, (Generalized Mobility).
( ') ( ') 'u r r F r dr 5555555555555 5
Oseen tensor 1
8I rr
r
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)
xy
Extracellular
Intracellular
Membrane
x
z
yProtein
Hydrodynamic coupling
White noise2 1
' , '
( ) 0
( ) ( ') 2 ( ')
k
k k B k k k
t
t t k TL t t
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 =
2 2
32 2 2
1 1 2
1
x yy y xx x y xy
x y
z z z z z z z
z z
xx yyz zLinearization
xx yyz zLocal-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
u
Non-inertial Navier Stoke equation
Dynamic viscosity of surrounding fluid
2
2~
0
0
ij ij i j j i
zz
x xz
p u F
u
T P u u
ET
z
F T v
5555555555555 5
Fluid velocity is same as membrane
velocity at the membrane boundary
zFxF
Surface viscosity of bilayer
v
Surrounding fluid velocity
Membrane velocity
University of Pennsylvania Chemical and Biomolecular Engineering
Surface Evolution
For axisymmetric membrane deformation
' 0 0s
Exact minimization of Helfrich energy possible for any (axisymmetric) membrane deformation
Membrane parameterized by arc length, s and angle φ.
3 22
2 2 3 22
2 2
2 22
23 2
2
' sin 2sin cos 3sin'''sin '' '
2 2
( )sin ( )sin 2 ( )cos sin( ) ( )sin
2
3cos sin 2 ( )sin1 cos sin 2( )cos sin
2 sin ( )sin( ) cos 2
2 2
R R
R R RR R
R R R
R
R RRR R
RR
'
S=0
S=L
' cosR
0s L
0 0s
00R s R
0R s L
University of Pennsylvania Chemical and Biomolecular Engineering
Solution Protocol for Monge-TDGL
Divergence removed by neglecting mode k=0 (rigid body translation)
' '
1
8i j ij i j ij
dz E
dt zr r
The harmonic series is a diverging series for a periodic system. We sum in Fourier space (k1, k2)
2 1/ 2 1 /( ) 1
4jk n ik n
i j ij
dz k Ee e k
dt k z
Periodic boundary conditions for membrane. Numerical solution using discrete version of membrane dynamics
equation
‘n’ is number of grid points
Explicit Euler scheme with h4 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
0 ix 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
2 20 0
220
i i
i
x x y y
Ri
i
H C e
0 iy Bound epsin position
2 2
0 02
2
2 020 02
0 2
i i
i
x x y y
RiiA
i i
H zCEF e z H x x dxdy
x R
Extracellular
Intracellular
Membrane
x
z
yProtein proteins
KMC-move
0
2 20
4, exp
1 x
FaDrate a
kTa Z
Metric
epsin(a) epsin(a+a0)
where a0 is the lattice size, F is the force acting on epsin0 ixE
University of Pennsylvania Chemical and Biomolecular Engineering
Hybrid Multiscale Integration Regime 1: Deborah number De<<1
or (a2/D)/(z2/M) << 1
Regime 2: Deborah number De~1 or (a2/D)/(z2/M) ~ 1
KMC TDGL#=1/De #=/t
R R
( ( ) ( )) ( )P R P R P R
( ) { ( ) }BP R exp E R k T
Surface hopping switching probability
Relationship Between Lattice & Continuum Scales
Lattice continuum: Epsin diffusion changes C0(x,y)Continuum lattice: Membrane curvature introduces an energy
landscape for epsin diffusion
R
Extracellular
Intracellular
Membrane
x
z Protein
Extracellular
Intracellular
Membrane
x
z Protein
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
0 200 400 600 800 10000
10
20
30
40
50
60
70
x (or y) [nm]
z [n
m]
Monge TDGL
local TDGL
exact
Exact solution for infinite boundary conditions
TDGL solutions for 1×1 µm2 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
0 50 100 150-1
0
1
2
3
4
5
6
7x 10
-15
x0 [nm]
Ene
rgy
[J]
1010 m2
55 m2
11 m2
PMF is dictated by both energetic and entropic components
Epsin experience repulsion due to energetic component when brought close.
2 22 2 20 0
2A
E H dxdy
Second variation of Monge Energy (~ spring constant).
Non-zero H0 increases the stiffness of membrane lower thermal fluctuations
Test function
Bound epsin experience entropic attraction.
2 2 4 20 0, 0, 0 02 0
2x x y yH z H z H H z z H
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 ~ 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
Fluctuation spectrum of bilayer bending rigidity and tension
Intrinsic curvature
24 2B
k
k TAh
k k
02
xx yy
H z dzz
Blood, P. D.; Voth, G. A., PNAS 2006, 103, (41), 15068-15072.
Marsh, D., Biophys. J. 2001, 81, 2154.
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 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.
University of Pennsylvania Chemical and Biomolecular Engineering
Correlations
Radial Distribution function
Measures hexagonal ordering
Orientational Correlation function *6 6(0) ( )r 6 ( )
6 ( ) ji r
j
r e
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 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
( )
( , ) ( , )i ijj
kk
i ijj
P t dt P t dt
P P
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.
1
( , ) expM
j jj
P h c h c
where hµ = number of distinct combinations for reaction Rµ to happen
cµ = mean rate of reaction Rµ.
11
ln 1/T
ra
T i ii
a h c1
21 1
i i T i ii i
h c r a h c
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 EM
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.
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