Fractons in proteins: can they lead to anomalously decaying time-autocorrelations?

R. Granek, Dept. of Biotech. Eng., BGU J. Klafter, School of Chemistry, TAU

Outline

• Single molecule experiments on proteins.

• Fractal nature of proteins. Fractons – the vibrational normal modes of a fractal.

• Time-autocorrelation function of the distance between two associated groups.

• Conclusions

• Single molecule techniques offer a possibility to follow real-time dynamics of individual molecules.

• For some biological systems it is possible to probe the dynamics of conformational changes and follow reactivities.

• Distributions rather than ensemble averages (adhesion forces, translocation times, reactivities)

Processes on the level of a single molecule

• Dynamic Force Spectroscopy (DFS) of Adhesion Bonds

• Translocation of ssDNA through a nanopore

• Enzymatic activity(in collaboration with the groups of de Schryver and Nolte)

• Protein vibrations

2250

2350

2450

2550

2650

27500 20 40 60 80 100

Fo

rce

(pN

)

Distance (nm)

Distance (nm)

For

ce(p

N)

3200

3300

3400

3500

3600

0 20 40 60 80 100

Dynamic Force Spectroscopy:

Mechanical response:

Processes:

Maximal spring force:

2 32 3

max 2

31 ln c xB

cc B c c

U Kk TF F V

U k T MF

=> F(V) ~ (lnV)2/3

as compared with

( ) ln( )F V const V

Single Stranded DNA translocation through a nanopore: One polymer at a time

A. Meller, L. Nivon, and D. Branton. Phys. Rev. Lett. 86 (2001)

J. J. Kasianowicz, E. Brandin, D. Branton and D. W. DeamerProc. Natl. Acad. Sci. USA 93 (1996)

Individual membrane channels: : ion flux & & biopolymers translocation

Relevant systems

O. Flomenbom and J. Klafter Biophys. J. 86 (2004).

Translocation and conformational fluctuation J. Li and H. A. Lester. Mol. Pharmacol. 55 (1999).

3 2 1

Lipase B From Candida Antarctica (CALB) Activity(The groups of de Schryver and Nolte)

• The enzyme (CALB) is immobilized.

• The substrate diffuses in the solution

• During the experiment, a laser beamis focused on the enzyme, and the fluorescent state of a single enzyme is monitored.

• The Michaelis-Menten reaction

Chemical activity

K. Velonia, etn al., Angew. Chem. (2005)

O. Flomenbom, et al., PNAS (2005)

L. Edman, & R. Rigler, Proc. Natl. Acad. Sci. U.S.A., 97 (2000) H. Lu, L. Xun, X. S. Xie, Science, 282 (1998)

Relevant systems

1 2 N

1 2 N

rNr2r1 k1 k2 kN

Single molecule experiments in proteins:Fractons in proteins

• Fluorescence resonant energy transfer (tens of angstroms).• Photo-induced electron transfer (a few angstroms)

eqXtXtx )()(

S. C. Kou and X. S. Xie, PRL (2004)W. Min et al., PRL (2005)R. Granek and J. Klafter, PRL (2005)

Autocorrelation function )0()()( xtxtCx

stt

stttCx

1

11~)(

2/1

2/1 const.

Small scale motion – VIBRATIONS?

Fluctuating Enzymes

Mass fractality of proteins: fdRM ~

Mass enclosed by concentricspheres of radius R centeredat a backbone atom, in a single protein (1MZ5).

Analysis covered over 200 proteins:(!)

2.05.2 fdM. B. Enright and D. M. Leitner, PRE (2005)

Fractal nature of proteins.

Linear polymers D=1Membranes D=2

• Chemical length – the length of the minimal path along the connecting springs.

• Chemical length exponent

• Or Flory exponent

mind

1

min

d

l

min~ drl

fdD rlM ~~

Real space

Manifold space lr

Ddd fmin

Manifold dimension D

Density of (eigen) states:

1~)( sdN sd – Spectral dimension

Computational studies involving ~60 proteins ->

Molecular weight dependent :

23.1 sd For over 2000-3000 amino acidsFor ~100 amino acids

sd

Experiments (electron spin relaxation):

for ~200-300 amino acids7.13.1 sd

)(N

A. Vulpiani and coworkers (2002,2004)

Fractons

ll

o tlutlumtludt

dm

'

22

2

),(),'(),(

Vibrations of the fractal

Normal modes (eigenmodes, eigenstates) – Fractons:

tieltlu )(),(

l

u

m

o

mass

Spring natural frequency

displacement

“name” of a point mass

ll

o lll

'

22 )()'()(

Strongly localized eigenstates ! )(l

Yakubo and Nakayama (1989)S. Alexander and R. Orbach (1982)

Disorder averaged eigenstate – Averaging over different realizations of the fractal, or over many localization centers:

Localization length in real space

fs ddr

~)(

Localization length in manifold space

Dd s ~)(

lfl Ddo

s )(

1

11~)(

2

ye

yyyf

y for

forconst.

Inequalities between the different broken dimensions:

31 fs dDd

f

s

d

D

d – Spectral dimension

– Manifold dimension

– Fractal dimension

Remark :For folded proteins although the backbone is 1-dim.There are strong inter amino acid interactions, i.e. new “springs” connecting nearest-neighbor amino acids (in real space), even if they are distant along the backbone.

Moreover, for the same reason we expect.

1D

fdD

Landau-Peirels Instability

u

– Amplitude of a normal mode )(l

Equipartition theorem 2

2 3

m

Tku B

T

Thermal fluctuations of the displacements ( )

)1/2()2(min

222 ~~)(min

ss

od

od

TTTNuNduu

ssf do

ddg NR /1/

min ~~ oN – # of amino acids (“polymer index”)

If , increases with increasing !

Large fluctuations may assist enzymatic/biological activity.

oNT

u 22sd

2sd

If evolution designed only folded proteins, should depend on .

should approach the value of 2 for large proteins !

sd oN

sd

sd2

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

otherwise protein must unfold (or not fold).

2/12u

A. Vulpianiand coworkers

)2002,2004(

Displacement difference time-autocorrelation function

eqXtXtx

)()(

Two point masses, 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()(|)'(|12)0()( utullxtx

Two limits:

1 (Undamped fractons (pure vibrations)

)cos()0()( 2 tuutuT

The calculation involves a time-dependent propagation lengthDd stt /~)(

If , motion of the two particles is uncorrelated.|'|)( llt

If , motion of the two particles is strongly correlated.|'|)( llt

1)22(

12

for

forconst.1~)0()(

tt

ttxtx

ss

s

dDd

d

sfs ddo

dDo brrll /1/1

1 |'||'|

sdo

o

B tm

TkCxxtx 2

22 )()0()(

constantnumericalC

)12(

22 |'|

sf dd

o

B

b

rr

m

Tkx

1tmore precisely, for :

numbers:

Short-time exponent

Long-time exponent

7.021.0 sd

12

3.0 ss d

D

d

2 (Strongly overdamped fractons

t

Tuutu

2

2)0()(

e

where is the friction. Therefore, the propagation length ism

2)12(

22/11

~)0()(

tt

ttxtx

ss

s

dDd

d

for

forconst.

Dsdtt

2~)(

sf dd

o

brr /2

22 |'|

Conclusions

1.1. Novel approach for Novel approach for vibrations in in folded proteins based on their based on their fractal nature nature Provides a description on a universal level, yet still Provides a description on a universal level, yet still microscopic in essence.microscopic in essence.

2.2. Slow Slow power law decay of the decay of the autocorrelation function of the of the distance between two associated groups, even for distance between two associated groups, even for pure vibrations..

3.3. In the case of pure vibrations, this powerlaw decay requires broken In the case of pure vibrations, this powerlaw decay requires broken dimensions that obey the inequalitiesdimensions that obey the inequalities

These inequalities do These inequalities do not hold for uniform lattices in all dimensions. hold for uniform lattices in all dimensions.

32

2 ss d

D

d

2sd