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University of Pennsylvania Department of Bioengineering
Hybrid Mesoscale Models For Protein-Membrane Interactions
Neeraj Agrawal, Jonathan Nukpezah, Joshua Weinstein, Ravi Radhakrishnan
Targeted Therapeutics
Bridging Intracellular
Signaling with Trafficking
Endocytosis: the internalization machinery in cells
University of Pennsylvania Department of Bioengineering
… “In spite of the above, we felt that it would be redundant to elaborate on his career as a scientist, for that aspect of him is rather well-known. Moreover, we wanted to celebrate one of the perhaps less awarded, but not less important, aspects of Keith: his human nature. Anyone who has worked with him will recognize this in several ways.” Coray Colina, Biography of Keith E. Gubbins,
J. Phys. Chem. C, 111 (43), 15479 -15480, 2007
University of Pennsylvania Department of Bioengineering
1997: Nagarhole Tiger Sanctuary
Karnataka, India
Keith M.S. Ananth: Keith’s 1st PhD student
Madhu, Guide
Keith seen with his high-tech luxury Yacht
University of Pennsylvania Department of Bioengineering
Hierarchical Multiscale Modeling
Weinan E, Bjorn Engquist, Notices of the ACM, 2003
K E Gubbins et al., J. Phys.: Cond. Matter, 2006
Minimal model for protein-membrane interaction in endocytosis is focused on the mesoscale
University of Pennsylvania Department of Bioengineering
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 Department of Bioengineering
C0 :Intrinsic curvaturek: Bending Modulusk: Gaussian Curvature Modulus
Helfrich Free Energy
Cartesian (Monge) notation: h(x,y)
1
2
R1>0, R2>0H>0, K>0
R1>0, R2<0H=0, K<0
H1/2[1/R1+1/R2]K1/R11/R2
Plane: H=0, K=0
Nelson, Piran, Weinberg, 1987
Mesoscale linearized elastic model for membraneMesoscale linearized elastic model for membrane
University of Pennsylvania Department of Bioengineering
Linearized Elastic Model For Membrane: Monge Gauge
Helfrich membrane energy accounts for membrane bending and membrane area extension.
In Monge notation, for small deformations, the membrane energy is
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.
White noise
z(x,y)
The Monge gauge approximation makes the elastic model amenable to Cartesian coordinate system
University of Pennsylvania Department of Bioengineering
Hydrodynamics of the Linearized Elastic Membrane
Non-inertial Navier-Stokes equation
Dynamic viscosity
2
0
p v F
v
------------- -
Solution of the above PDEs yields the Oseen tensor, (Generalized Mobility).
( ') ( ') 'v r r F r dr ------------- -
Oseen tensor 1
8I rr
r
Fluid velocity is same as membrane velocity at the membrane boundary no slip condition given by:
; where, z E
M Mt z
This results in the Time-Dependent Ginzburg Landau (TDGL) Equation
z(x,y)
xy
Extracellular
Intracellular
Membrane
x
z
yProtein
University of Pennsylvania Department of Bioengineering
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 zCEe z H x x dxdy
x R
0
2 20
4, exp
1 x
FaDrate a
kTa Z
For 2 D
Metricepsin(a) epsin(a+a0)
where a0 is the lattice size, F is the force acting on epsin
0i
E
x
Extracellular
Intracellular
Membrane
x
z
yProtein proteins
KMC-move
University of Pennsylvania Department of Bioengineering
KMC-TDGL 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
Weinstein, Radhakrishnan, 2006
Constant Temperature Protein-Mediated Membrane
DynamicsC
0/µ
m-1
R/nm
R, Range
C0, Intrinsic Curvature
*, Surface Density
University of Pennsylvania Department of Bioengineering
Membrane-Mediated Potential of Mean Force between Epsins
PMF is dictated by both energetic and entropic components
Energy: Epsin experience repulsion due to energetic component when brought close.
2 22 2 20 0
2A
E H dxdy
Entropy:
Regions of non-zero H0 assume increased stiffness and hence reduced membrane fluctuations
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
Therefore, the system can lower its free energy by localizing epsins on the membrane leading to membrane-mediated epsin-epsin attraction
2E~ spring constant;=test function
University of Pennsylvania Department of Bioengineering
Membrane Dynamics, R= 40nmMembrane-Mediated Protein-Protein Spatial Correlations *=0.004, C0=20 *=0.03, C0=20
Localization
F/kTC0*=20R*=40 nm
Threshold
No effective membrane-mediated attraction; no nucleation below threshold curvature and range
University of Pennsylvania Department of Bioengineering
Membrane Dynamics, R=60nmMembrane-Mediated Protein-Protein Spatial Correlations
*=0.008, C0=10 *=0.008, C0=40 *=0.008, C0=60
Localization
F/kT C0**: 30-50
F(r)kBTln g(r)
Nucleation limited only by diffusional timescale of association (NVA)
University of Pennsylvania Department of Bioengineering
Membrane Dynamics, R=100nmMembrane-Mediated Protein-Protein Spatial Correlations *=0.016, C0=30
Localization
F/kTThreshold
C0*: 10-30
Nucleation occurs following spatial localization of epsin
University of Pennsylvania Department of Bioengineering
xy
1st Shell 2nd Shell
Epsin arrangement
xy θj
Sustained orientational correlations beyond nearest-neighbors drives nucleation
Nucleation via Hexatic Orientational Ordering: NVOO
University of Pennsylvania Department of Bioengineering
Membrane Dynamics, R=80nmMembrane Temporal Correlations*=0.02, C0=5
2 22 2 20 0
2A
E H dxdy
Regions of non-zero H0 assume increased stiffness and hence reduced membrane fluctuations
High protein-surface density drives the membrane phase into a glass-like dynamical behavior
University of Pennsylvania Department of Bioengineering
Global Phase Diagram*
C0 R
NVLRO
NVA
No N
GT
GT: Glass-like transition; No N: No nucleationNVOO: Nucleation via orientational orderingNVA: Nucleation via diffusional association
C0 / µm-1
R /
nm
20 40 600
2040
60
80100
NVANVOO
No NGT
1 2 3
0.20.40.6
0.8
1.0
g(r=r0)
6(
r=r 0
)
University of Pennsylvania Department of Bioengineering
Conclusions
The KMC-TDGL approach is successful in describing the dynamic processes associated with the interaction of proteins and membranes at a coarse-grained level
Membrane-mediated protein-protein repulsion effects short- and long-ranged ordering of epsins
Two modes of nucleation observed
-- Nucleation via Association : Effected by large C0
-- Nucleation via Orientational Ordering: Effected by persistence of orientational correlations
In the regime of large surface density, a glass-like transition is observed
A global phase diagram is proposed
University of Pennsylvania Department of Bioengineering
Integrating Signaling and Trafficking
Extracellular
Intracellular
(MAP Kinases)
Ras
Raf
MEK
ERK
Cbl Clathrin, AP2
Endph epsin
Proliferation
Nucleus
Um
bre
lla
Sam
pli
ng
KM
C+
TD
GL
Hypothesis for receptor internalization
Clathrin Coat
University of Pennsylvania Department of Bioengineering
Acknowledgments
Graduate StudentsGraduate StudentsAndrew Shih (PhD, BE)Yingting Liu (PhD, BE)Jeremy Purvis (PhD, GCB)Shannon Telesco (PhD, BE)Jonathan Nukpezah (PhD, BE)Neeraj Agrawal (PhD, CBE)
Undergraduate StudentsJoshua Weinstein (Senior, PHYS)
CollaboratorsMark Lemmon, (Penn) Sung-Hee Choi, (Penn)Boris Kholodenko, (TJU)
FundingFundingNSF; Whitaker Foundation; NIH training grant; NPACI supercomputing allocations; Greater Philadelphia Bioinformatics Alliance
Co-Authors: Neeraj Agrawal, Jonathan Nukpezah, Joshua Weinstein
University of Pennsylvania Department of Bioengineering
Paradigms of Membrane CurvatureMcMahon, Gallop, Nature reviews, 2005
University of Pennsylvania Department of Bioengineering
Epsin
Clathrin
Membrane
Ap180
Imaging Endocytosis
Ford et al., Nature, 2002
University of Pennsylvania Department of Bioengineering
Ap180+Clathrin Epsin+Clathrin Ap180+Epsin+Clathrin
Ford et al., Nature, 2002
Epsin Clathrin Ap180
Endocytosis Machinery
Receptor Inactivation to Neurotransmitters
University of Pennsylvania Department of Bioengineering
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