University of Pennsylvania Chemical and Biomolecular Engineering Multiscale Modeling of...

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


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


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


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


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


OverviewProtein diffusion modelsMembrane models

Model Integration

Preliminary Results

Tale of three elastic modelsRandom walker


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)


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


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


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

×


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


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


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


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


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.


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


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.


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


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.


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.


Acknowledgments

Jonathan Nukpezah

Joshua Weinstein

Radhakrishnan Lab. Members


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.


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)


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.


Targeted Drug Delivery


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.


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.


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.


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


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].


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


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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