University of Pennsylvania Department of Bioengineering Hybrid Mesoscale Models For Protein- Membrane Interactions Neeraj Agrawal, Jonathan Nukpezah, Joshua

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


… “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


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


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


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)


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


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


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


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


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


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


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


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)


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


xy

1st Shell 2nd Shell

Epsin arrangement

xy θj

Sustained orientational correlations beyond nearest-neighbors drives nucleation

Nucleation via Hexatic Orientational Ordering: NVOO


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


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

)


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


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


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




Paradigms of Membrane CurvatureMcMahon, Gallop, Nature reviews, 2005


Epsin

Clathrin

Membrane

Ap180

Imaging Endocytosis

Ford et al., Nature, 2002


Ap180+Clathrin Epsin+Clathrin Ap180+Epsin+Clathrin

Ford et al., Nature, 2002

Epsin Clathrin Ap180

Endocytosis Machinery

Receptor Inactivation to Neurotransmitters


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

Documents

University of Pennsylvania Department of Bioengineering Hybrid Mesoscale Models For Protein- Membrane Interactions Neeraj Agrawal, Jonathan Nukpezah, Joshua