fractons in proteins: can they lead to anomalously decaying time-autocorrelations? r. granek, dept....
Post on 20-Dec-2015
214 views
TRANSCRIPT
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:
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
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?
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