protein dynamics and stability: universality vs....

Protein Dynamics and Stability:Universality vs. Specificity

Rony Granek

The Stella and Avram Goren-Goldstein Dept. of Biotechnology EngineeringBen-Gurion University of The Negev

Motivation.

Fractal nature of proteins. Fractons – the vibrational normal modes of a fractal (scalar elasticity).

Protein marginal stability – universal “equation of state”.

Dynamics: Vibrational & Random Walk MSD’s, return probabilities, etc.

Tensorial elasticity models.

Dynamic Structure factor – preliminary results.

Force induced unfolding – preliminary results.

Conclusions.

Coworkers

Aviv University-Tel

Shlomi Reuveni

Joseph Klafter

Gurion University-Ben

Marina de Leeuw

Roee Ben-Halevi

Amit Srivastava

RG

Long sequence of amino acids (20 types).

Thousands of different proteins. Differ by sequence and length. Fold in different ways to give different 3-D fold structure.

Conflicting requirements:

Specific folding – leads to a specific function (lock and key…). Large internal motion is needed to allow for biochemical function (enzymatic activity, antibody function, capturing and releasing ions, etc.).

Natural Proteins

–Problem a folded protein has less internal motion than an unfolded

.Protein

Sequence

3D Structure

Function ?

Dynamics ?

www.rcsb.org 2gko

Single molecule experiments in proteins

Photo-induced electron transfer (a few angstroms)

FRET: Fluorescence resonant energy transfer (tens of angstroms).

eqXtXtx )()(

W. Min et al., PRL (2005)

FL (fluorescein) and anti-FL complexTyr (tyrosine) – donor, FL – acceptor

Autocorrelation function )0()()( xtxtCx

stt

stttC

x

1

11~)(

2/1

2/1 const.

Small scale motion – VIBRATIONS?

MSD of dihedral angles

http://upload.wikimedia.org/wikipedia/commons/4/42/Bond_dihedral_angle.png

)The Gaussian Network Model (GNM

Scalar elasticity. Springs exist only below a cutoff distance Rc.

All springs have equal spring constant.

I. Bahar and coworkers

Factors) –Mean Square Displacement (B VibrationalThe

Relevance

2

2

3

8

iiuB

I. Bahar and coworkers

normal mode analysis–Vibrations

ll

otlutlumtlu

dt

dm

'

2

2

2

),(),'(),(

Equations of motion

Normal modes (eigenmodes, eigenstates)

tieltlu

)(),(

l

u

m

o

mass

Spring natural frequency

displacement

“name” of anAlpha-carbon

ll

olll

'

22

)()'()(

Density of states (DOS, density of normal modes):

0

)()( dgG

)(g

Cumulated DOS:

Modes of Motion – HIV Protease

Mode 1 Mode 2 Mode 3

Calculation of cumulated DOS using the GNM

Powerlaws!

Reuveni et al., PRL (2007)

190N

505N

1184N

Fractals Definition –

“a rough or fragmented geometric shape

that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole”

Benoît Mandelbrot (1975)

Fractal nature of proteins

Mass fractal dimension :

fd

rM ~

fd

The atoms enclosed in spheres of different radii (pdb: 1OCP )

190N

505N

1184N

D. M. Leitnerand coworkers

r

Fractons

ll

otlutlumtlu

dt

dm

'

2

2

2

),(),'(),(

Vibrations of the fractal

Normal modes (eigenmodes, eigenstates) – Fractons:

tieltlu

)(),(

l

u

m

o

mass

Spring natural frequency

displacement

“name” of a pointmass

Strongly localized eigenstates

Density of states

)( l

S. Alexander and R. Orbach (1982)

Yakubo and Nakayama (1989)

1~)(

s

dg

sd – Spectral dimension

Calculation of cumulated DOS using the GNM

sd

dgG

~)()(

0

190N

505N

1184N

Landau-Peierls Instability

u

– Amplitude of a normal mode )( l

Protein Stability & Unfolding

2

2 3

m

Tku

B

T

Thermal fluctuations of the displacements ( )

)1/2()2(

min

222~~)(

min

ss

o

dd

TTT

Nugduu

ssfddd

gNR

/1/

min~~

N – # of amino acids (“polymer index”)

If , increases with increasing !

Large fluctuations may assist enzymatic/biological activity.

NT

u2

2s

d

2s

d

Equipartition

But should not exceed the mean inter-amino acid distance,

otherwise protein must unfold (or not fold).

2/12

u

Marginal stability. To have large amplitude motion but remain folded:

Proteins can “live” in the “twilight” zone: Folded-Unfolded !

To keep proteins folded, should depend on :

should approach the value of 2 for large proteins.

sd N

sd

Instability threshold: Universal relation between exponents

)1/2(

2

2~

s

d

o

B

T

Nm

Tku

Landau-Peierls Instability:

for unfoldinglike criterion-Lindenman22

~c

surface

Ru

)1/2(222~p)-(1

s

d

bulksurface

Nuupu

Assume

2)1/2(

2~

c

d

o

BRpN

m

Tks

Unfolding:

Take:)/1(

~1

~~f

d

g

NRV

Sp

N

b

ddfs

ln1

12

Tk

Rmb

B

co

22

ln

Unfolding/Melting occurs from the surface inward

Cluster melting analog

Reuveni et al., PRL (2007)

N

ba

ddfs

ln

12

Best fit: CC=0.55, 4,1 53.4,90.0 thethe

baba

Fitting the data of 543 proteins to:

Nln1

fsdd

12

GNM cutoff length Rc=6AReuveni et al., PRL (2007)

Selecting Proteins

Structures in the PDB : 48,638

No ligands

Structures to analyze : 4249

No more than 95% ID

No RNA, no DNA

No Peptides (less than 100 amino acids)

Files with missing data - removed

High accuracy in determining ds and df ,

99.02R

Larger set:

GNM cutoff length Rc=6 A

4249 proteinsColored histogram (100X100 bins)

)ln(

12

N

b

a

fd

sd

)ln(

1

12

N

b

fd

sd

fit to

fit toa=0.95

cc=0.58

M. de Leeuw et al., PLOS-ONE, 2009

4249 proteinsX-axis separated into 100 bins

)ln(

12

N

b

a

fd

sd

)ln(

1

12

N

b

fd

sd

fit to

fit toa=1.065

cc=0.945

M. de Leeuw et al., PLOS-ONE, 2009

Dynamics

I) Time-autocorrelation function of the distance between two alpha-carbons

Motivation: Single molecule experiments in proteins (Xie and coworkers)

eqXtXtx )()(

Autocorrelation function )0()()( xtxtCx

autocorrelation function-Displacement difference time

eqXtXtx

)()(

Two point masses (alpha-carbons), and

Positions in space and

Separation vector

Equilibrium spacing

Displacement difference vector

l

'l

),( tlR

),'( tlR

),'(),()( tlRtlRtX

eqX

),'(),()( tlutlutx

Expansion in normal modes )()(),( ltutlu

+disorder averaging

)0()(|)'(|1

2)0()( utull

Nxtx

Granek & Klafter, PRL (2005)

)0Static fluctuations (t=

)12(

)12(

2

2~

sf

sf

dd

dd

o

Br

b

r

m

Tkx

rwith the real space separation between the two points Diverging

Longer separation distance Larger fluctuations

fractonsoverdampedStrongly

t

T

uutu

2

e)0()(2

Where is the local friction (Rouse-like model).

Propagation length is

m

rξ(t)t

rttxtx

sls

s

ddd

d

: timeslong

)(:sshort timeconst.1~)0()(

)12(

2/1

fd

sd

tt2

~)(

fldd

rlM ~~r - Real space separation between the two points

l - Topological space separation between the two points

ld - Topological space dimension

Granek & Klafter, PRL (2005)

Hydrodynamic interaction (Zimm)

Oseen hydrodynamic interaction tensor

),(),'(),''()',(),(

''''

2tltlutlullOmtlu

dt

d

lll

o

rllO

1~)',(

noise

propagation length

fssf

s

dddd

d

tt2

~)(

timeslong

sshort timeconst.1~)0()(

2

22

2

2

fss

sls

fss

s

ddd

ddd

ddd

d

t

txtx

Granek, Phys. Rev. E, Rapid comm. (2011)

MSD of dihedral angles

http://upload.wikimedia.org/wikipedia/commons/4/42/Bond_dihedral_angle.png

II) Vibrational mean square displacement (MSD)

TT

ututu22

)0()()(

TT

tuN

tu )(1

)(22

A specific alpha-carbon

Average over all alpha-carbons

In the fractal model:

2/12

~)( sd

T

ttu

Slope=1-ds/2=0.30

Slope=1-ds/2=0.29 Vibrational MSD

Dynamics

III) Random Walk on the GNM network

Probability of return to the origin2/

0~)( s

dttP

Mean square displacement (MSD) sfw

ddddttr w 2 ~)(

/22

Mean first passage time (MFPT) fw

dd

r

~MFPT

On Fractals

)(0

tP

)(2

tr

Note equivalence to vibrations: )(2

tr

MFPT

fsdd

tt/2

~)(

)12(2~

sf

dd

rx

)(0

tPT

tudt

d)(

2

Probability of return to the origin2/

0~)( s

dttP

Reuveni et al., PNAS (2010)

1 4 7N

4 9 2N

1 7 6 7N

1 .6 4R W

sd

1 .7 4R W

sd

2R W

sd

0 .0 0 8 0 .9 5D O S R W

s sd d

. 0 .8 6c c

2197N

0.3RW

sd

RW Mean square displacement (MSD) w

dttr

/22~)(


1 4 7N

4 9 2N

1 7 6 7N

2 .4 1w

d

2 .4 2w

d

2 .4 0w

d

2 .1 1w

d 2197N

Propagation length (vibrations) w

dtt

/22~)(

Different proteins

Mean first passage timefw

ddr

~MFPT


1 4 7N

4 9 2N

1 7 6 7N

0 .7 5

0 .5 1

0 .3 0 5

r

N

MFPTx ~~

2

fwdd

rx

~2

Variance of distance fluctuations


?What about specificity

214 Adenylate kinase, N=

but205)A 29( , (a.u.) rMFPT

337)127,53( T

Structure with bound ATP and AMP (substrate)

Dynamic Structure Factor

HbMb

Dynamic Structure Factor–Sierpinski gasket

)(~

kS

),( tkS

- “Forzen network” static structure factor

- Dynamic structure factor

No translational and rotational diffusion

ps 53.100Reuveni, Klafter, Granek, to be published

Stretched exponential!

Dynamic Structure Factor–Sierpinski gasket

)()0,( kSkS

),( tkS

- Static structure factor



ps 53.100Reuveni, Klafter, Granek, to be published

Dynamic Structure Factor–Proteins

)()0,( kSkS

),( tkS

- Static structure factor



Reuveni, Klafter, Granek, to be published

o

A 5 , 5 bkb

ps 22

ps 32

ps 170

N

kji

ikjjkik

N

ji

ijjiijoTEMBruumH

,,

2

,

22

2

1ˆ

2

1

Tensorial Elasticity Model

ikj

k

i

j

Advantage:Allows to compute anisotropic displacement fluctuations.Disadvantage:Complicated.Involves an additional free parameters (B).

Ben-Halevi et al., to be published

Bond-bendingAnisotropic network model (ANM)

Anisotropic Network Model (ANM) for Methyltransferase (2NQ5).


Tensorial Model for Methyltransferase (2NQ5).

-22A 1/ Bm

o

Slope=1.508

0

1

2

3

4

5

6

7

8

9

-4.00E+00 -3.00E+00 -2.00E+00 -1.00E+00 0.00E+00 1.00E+00 2.00E+00

ln

)(ln G


Slope=1.63

Unfolding by Pulling Force

Cymotrypsin Inhibitor (CI2)

GNM of CI2

Srivastava & Granek, to be published

force at the end pair

Conclusions:

Novel approach for vibrations in folded proteinsbased on their fractal nature Provides a description

on a universal level.

Folded proteins are marginally stable: they exist in a thermodynamic state close to unfolding, which allows for large scale motion without unfolding.

The above criterion leads to a universal “equation of state”, verified for about 5,000 proteins.

The fractal-like properties of proteins lead toAnomalous dynamics/ strange kinetics:autocorrelation of separation; vibrational MSD; random walk MSD, return probability & mean first passage time, dynamic structure factor.

Questions to address:

Relation to enzymatic/biological activity.

Role of water.

More realistic potentials, e.g.: Tensorial elasticity, distinguishing between backbone bonds, H-bonds, electrostatic bonds, etc.

Response and unfolding under force.

Coworkers

Aviv University-Tel

Shlomi Reuveni

Joseph Klafter

Gurion University-Ben

Marina de Leeuw

Roee Ben-Halevi

Amit Srivastava

RG

Funding: ISF, DIP

Thank You!

protein dynamics and stability: universality vs....

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