Carsten Kutzner, Helmut Grubmüller

Jochen J. Sieber1, Katrin I. Willig2, Claas Gerding-Reimers1, Benjamin Harke2, Gerald Donnert2, Burkhard Rammner, Christian Eggeling2, Stefan W. Hell2, Thorsten Lang1

Departments of 1Neurobiology, 2NanoBiophotonics

A brownian dynamics interpretation of membrane protein clustering

1

‣ fluid mosaic model of cell membrane (Singer, Nicolson 1972): individual proteins diffuse freely in a sea of lipids

‣ however, most membrane proteins are organized in clusters

‣ no satisfactory explanation! (subplasmalemmal fences that form compartment boundaries?)

‣ cluster formation is likely functionally important since syntaxin clusters represent sites for docking & fusion of vesicles

Extracellular Fluid

Cytoplasm

Cholesterol Integral protein(Globular protein)

Peripherial protein

Glycoprotein

Carbohydrate

Glycolipid

Phospholipid bilayer

Alpha-Helix protein(Integral protein)

Surface protein

Protein channel(transport protein)

Globular protein

Filaments of cytoskeleton

Phospholipidmolecule

Hydrophilic heads

Hydrophobic tails

Questions & motivation

‣ physical principles underlying most membrane protein clusters poorly understood

‣ explain the experimentally accessible properties of the syntaxin-1 (Sx1) clusters

SNARE protein involved in membrane

fusion

2

Experiments 1: cluster density and cluster diameter

‣ STED microscopy on plasma sheets yields density of 19.6(5.7) clusters / μm2

‣ average cell surface area is 460 μm2 ⇒ 9000 clusters per cell

‣ quantitative immunoblotting: 830 000 Sx1 per cell ⇒ max. 90 Sx1 per cluster (if all Sx1 in

clusters)

‣ STED: average cluster diameter 50-60 nm ⇒ dense package

‣ overexpression ⇒ more clusters

STEDConfocal

2 µm

500 nm

Histogram of measured Sx1 cluster size

0 20 40 60 80 100 120 140 1600

40

80

120

160

F WHM (nm)

num

bero

fclu

ster

s

100 nm

3

‣ FRAP measurements with green fluorescent protein (GFP)–labeled Sx1 overall mobility

‣ recovery half times t1/2 range from 40–60 s, much longer than expected for a freely diffusing protein with a single transmembrane region (TMR)

Experiments 2: syntaxin mobility

Sx1A-GFP

0 s 25 s 100 s 256 spre5 µm

S x1A, T MR -G FP (T MR )

S x1A-G FP

0 50 100 150 200 2500.0

0.2

0.4

0.6

0.8

1.0

norm

alized

s igna l

time (s )

TMR

60

40

20

t(s)

1/2

S x1A, T MR -G FP (T MR )

S x1A-G FP

0 50 100 150 200 2500.0

0.2

0.4

0.6

0.8

1.0

norm

alized

s igna l

time (s )

TMR

60

40

20

t(s)

1/2

0

N-term.

S NAR Emotif

TMR

linker

4

‣ study mutants and deletion constructs of Sx1 C-term TMR – SNARE – linker – N-term domain

‣

‣ SNARE motifs are responsible for reduced syntaxin mobility (whereas closed conformation plays no role)

‣ reduced mobility is caused by the assembly of syntaxin into clusters

Experiments 3: which region of Sx1 is responsible for mobility restriction?

0

20

40

60

111G F P b lo ckI 0 11G F P 0 0 1G F P0

20

40

60

111G F P b lo ckII o pen111G F P 111multG F P

S x1A, T MR -G F P (T MR )

S x1A-G F PS x1A, S NAR E motif-T MR -G F P ( S NAR E )

0 50 100 150 200 2500.0

0.2

0.4

0.6

0.8

1.0

norm

aliz

edsi

gna l

time (s )0 50 100 150 200 250

0.0

0.2

0.4

0.6

0.8

1.0

norm

aliz

edsi

gnal

time (s )

S x1AmutT MR -G F P ( mut)

S x1A-G F Popen S x1A-G F P ( open)

60

40

20t

(s)

1/2

0

open mut T MRS NAR E

60

40

20

t(s

)1/

2

0

N-term.

S NAR Emotif

T MR

linker

5

‣ FRAP recovery du to ...

➊ diffusion of entire clusters? ➋ exchange of individual molecules?

‣ syntaxin spots do not change over minutes ⇒ Clusters are immobile. ⇒ Equilibrium of free and clustered syntaxins.

Experiments 4 – molecule exchange or whole cluster diffusion?

103 s 199 s 304 s0 s1 µm

0 s 304 s overlay5 µm

membrane Sx1A-GFP

S x1A-G FP

0 50 100 150 200 2500.0

0.2

0.4

0.6

0.8

1.0

norm

alized

s igna l

time (s )

6

The BD model‣ can size and dynamics of the syntaxin clusters be explained by simple physical

principles? (rather than by elaborate layers of biological regulation)

‣ MD not feasible, since we eventually need 250 s trajectories to compare with the FRAP experiments

‣ no need to describe in detail the collisions of syntaxins with surrounding lipids, consider them as a heat bath only

‣ MD becomes BD (Newtons eq. of motion ⇒ Langevin):

‣ Position Langevin eq. (Lax 1966, Zwanzig 1969):

positions r, diffusion coefficient D, random velocity process drS/dt

‣ Recursive update of molecular positions with algorithm by Ermak (1975):

‣ lateral diffusion coefficient of TMR construct D = 0.075(25) μm2 / s determined from half time of recovery according to Ficz et al. (2005)

drdt

= DF(t)+drS

dt

ri(t +!t) = ri(t)+D!tFi(t)+"!

2D!t

random number drawn from a 2d normal distribution

with variance 1

7

The BD model‣ diffusion on a 2d plane, periodic boundary conditions

‣ interaction potential between individual molecules i, j at distance r = rij :

(1) (2) and (3)

‣ (1) mutal repulsion of strength E1 and range σ that prohibits molecules to run into each other (Pauli repulsion)

‣ (2) effective attraction of strength E2 and range 2σ between the molecules, mediated by the SNARE motifs

‣ (3) fs steric hindrance due to crowding

‣ force

‣ σ is chosen such that an approximate area per TM helix of 1.5 nm2 is obtained (Takamori et al. 2006)

V (ri j) = E1 · e!r2/(2!2)!E2 · fs · e!r2/{2(2!)2}

0.5 1 1.5 2 2.5

-2

0

2

r (nm)

pote

ntia

l(kT

)

nmax=140

nc =175140

10570

350

fs = 1! nc

nmax

!F(!ri) =!!iU(!r1, . . .!rn)

U = !i< jV (ri j)

min(V ) =!

1.5 nm

8

Global algorithm

‣ generate starting configuration

‣ calculate forces

‣ make neighbourlists (every 10th step)

‣ detect clusters (every 10th step)

‣ calculate forces for particles within cutoff radius

‣ update positions

‣ calculate fluorescence recovery

‣ output

‣ ri = (x, y)i, bleached?

‣ potential energy

‣ FRAP signal intensity

‣ cluster size distribution

0.5 1 1.5 2 2.5

-2

0

2

r (nm)

pote

ntia

l(kT

)

nmax=140

nc =175140

10570

350

gridded neighbour searching with cutoff >3.5 nm

rc

9

Cluster detection‣ needed for cluster penalty term

‣ if distance d between 2 molecules < rd =1.5 nm they belong to same cluster

‣ HOW TO AUTOMATICALLY DETECT THAT?

‣ if 2 molecules belong to same cluster, they must be in same neighbourlist, since rd < rcutoff

‣ go through all molecules

‣ (if not already assigned to a cluster), build the network that connects all molecules with a mutual distance of less than 1.5 nm;mark each of those molecules with a unique cluster-number.

fs = 1! nc

nmax

0.5 1 1.5 2 2.5

-2

0

2

r (nm)

pote

ntia

l(kT

)

nmax=140

nc =175140

10570

350

neighbourlist cutoff

distance criterion

snapshots every 250 steps

10

Time step length

0 0.5 1 1.5 2 2.5 3 3.50

100

200

300

400

500

600

700

800

900Radial distribution of syntaxins (E2 = 10.5 kT)

radius (nm)

coun

t (ar

bitra

ry u

nits

)

dt= 3.05 nsdt= 15.25 nsdt= 30.5 nsdt= 61.0 nsdt= 91.5 nsdt=122 ns

0.5 1 1.5 2 2.5

-2

0

2

r (nm)

pote

ntia

l(kT

)

nmax=140

nc =175140

10570

350

11

Parallel force calculation

#include <omp.h>

...

/*******************************************************************************//* start omp parallel region *//*******************************************************************************/#pragma omp parallel for \ default(none) \ shared(signal, dvdx, dvdy, stderr) \ firstprivate(bleachedarr, cluster_size, cluster_no, numneighbours, ind, x, y) \ private(xoffset, yoffset, signalindex, xdum, ydum, xblock, yblock, blockindex, \ j, jsize, ijsize, isize, neighbour_no, index, r2, dx, dy, fac, a, \ exp1, exp2) \ reduction(+ : etot) \ schedule(static)

/* loop over molecules */for (i = 0; i < nsyntaxins; i++){

(calculate potential & force)...

}

Benchmark on Kea:CPUs time/s speedup scaling

1 165.1 1.00 1.00 serial code 1 170.6 0.97 0.97 threaded code 2 93.4 1.77 0.88 4 54.0 3.06 0.76

‣ fastest CPUs (roc) need ≈3 days for 80M time steps (250 s)

‣ icc ./synsim.c -O3 -openmp -o synsim.x

12

Simulation protocol‣ make a parameter study:

vary E2 and nmax

‣ E2 ⇒ depth of well

‣ nmax ⇒ at which cluster size force becomes

repulsive

‣ start with 1391 random positions ri

‣ iterate until equilibrated (potential energy)

‣ if cluster density matches exp. density of 19.6(5.7) clusters / μm2

‣ bleach (simulate FRAP experiment)

‣ record 250 s of fluorescence recovery

‣ now match experimental FRAP curve

‣ extract number of molecules per cluster, fraction of free molecules

0.5 1 1.5 2 2.5

-2

0

2

r (nm)

pote

ntia

l(kT

)

nmax=140

nc =175140

10570

350

bleach size 293 nm (1:10)

13

Equilibration

0 50 100 150 200 250 300 350 400 4500

500

1000

1500

time [seconds]

num

ber o

f Sx1

mol

ecul

es

0 50 100 150 200 250 300 350 400 450−2.5

−2

−1.5

−1

−0.5

0x 104

time [seconds]

pote

ntia

l ene

rgy

[kT]

single Sx1

clusters 10+ (*10)

Sx1in clusters

14

Parameter study

experimental density of 19.6(5.7) clusters/μm2

}

11 12 13 140

10

20

30

40

50

60

E2 (kT)

clus

ters

/ (

m)2

Cluster number (min/av/max)

11 12 13 140

10

20

30

40

50

60

70

80

90

E2 (kT)

<nc>

Average cluster size

Counting only clusters nc 10

nmax = 40

60

80

100

120140

nmax = 40

60

80

100

120

140

15

Example 0-11s

‣ experiment measures intensity of white molecules in central area

‣ bleached molecules (orange) are invisible

0 5 100

0.05

0.1

0.15

0.2

time [seconds]

norm

aliz

ed s

igna

l

16

Example 0-257 s

0 100 2000

0.5

1

time [seconds]

norm

aliz

ed s

igna

l

‣ experiment measures intensity of white molecules in central area

‣ bleached molecules (orange) are invisible

17

Simulation of FRAP data

‣ each unbleached molecule in the middle area adds one count to the intensity each time step

‣ bleach central area

‣ bleach 9 areas: assign each syntaxin a number from 1-9 depending on where it is at bleach time

‣ displace by half a bleach box size in x, y, x&y directions

‣ ⇒ altogether 36 bleach areas

0 50 100 150 2000

0.5

1

1.5

time [seconds]

norm

aliz

ed s

igna

l

36 signals from 1 area (9 shown)4 signals from 9 areasaverage signal from 36 areas

18

0

20

40

60

111G F P b lo ckI 0 11G F P 0 0 1G F P0

20

40

60

111G F P b lo ckII o pen111G F P 111multG F P

S x1A, T MR -G F P (T MR )

S x1A-G F PS x1A, S NAR E motif-T MR -G F P ( S NAR E )

0 50 100 150 200 2500.0

0.2

0.4

0.6

0.8

1.0

norm

aliz

edsi

gna l

time (s )0 50 100 150 200 250

0.0

0.2

0.4

0.6

0.8

1.0

norm

aliz

edsi

gnal

time (s )

S x1AmutT MR -G F P ( mut)

S x1A-G F Popen S x1A-G F P ( open)

60

40

20

t(s

)1/

2

0

open mut T MRS NAR E

60

40

20

t(s

)1/

2

0

N-term.

S NAR Emotif

T MR

linker

Image acquisition correction

‣ FRAP experiments: bleach fraction s of molecules during each image acquisition

s = 1! (Iexpa ! Iexp

b )1/10 " 0.005

19

Image acquisition correction‣ FRAP experiments: bleach fraction s of molecules during each image acquisition

‣

‣ include this bleaching effect in the simulated curves by simulating 68 data acquisitions

s = 1! (Iexpa ! Iexp

b )1/10 " 0.005

image acquisition

time

FRAP signal

correctedoriginal

I(ti) · (1! s)

I(ti) · (1! s) } s · I(ti)

R(t) = s · I(ti) · I(t! ti) t > ti

20

Image acquisition correction

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time (s)

norm

aliz

ed s

igna

l

experimental Sx1without correctionwith correctionimage acquisition

21

Simulated fluorescence recovery

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

time (s)

norm

aliz

ed s

igna

l

free Sx1 nmax E2 100% ---- 0.055% 60 11.0

21% 120 11.518% 140 11.516% 140 11.810% 140 12.54% 140 13.01% 140 14.0

---experiment

‣ simulated curves are average of 2 trajectories each

‣ 1:10 scale ⇒ 102 x faster diffusion and FRAP recovery

for simulations that match experimentally observed cluster density

clustersize 70-80

22

Snapshots87

9 nm

293

nm

a

start positions in equilibriumin equilibrium

0 s0 s 129 s129 s

start positions

b

23

Conclusions

‣ experimental data on composition and dynamics of Sx1 clusters can be explained by simple physical principles

protein-protein attraction ↔ steric hindrance

‣ Sx1 molecules are quasi-immobile when in clusters

‣ a fraction of ≈16% of the molecules diffuse freely between the clusters which contain 70–80 Sx1

‣ Clustering via self-assembly likely applies to a variety of membrane protein clusters

‣ presented a framework with syntaxin-1 as an example

24

Acknowledgments

‣ Jochen Sieber & Thorsten Lang – experiments, experiments, ...

‣ Helmut Grubmüller – Idea & model layout

‣ 10500 – thanks to all of you!

Let‘s make a model!

25

0 0.01 0.02 0.03residence time in cluster [s]

# di

ssoc

iatio

n ev

ents

(log

sca

le)

c( ) = g( ) −1 e− /

simulationfit c( )

10−6 10−5 10−4 10−3 10−2 10−1

ampl

itude

s g(

)

dissociation time constant [s]

fast dissociation

slow dissociation

26