Transcript long talk

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
Dynamic Force Spectroscopy:
Force(pN)
Force (pN)
2250
2350
32 00
33 00
34 00
35 00
36 00
2450
0
20
40
60
80
10 0
Distance (nm)
2550
2650
2750
0
20
40
60
Distance (nm)
80
100
Processes:
Mechanical response:
Maximal spring force:
Fmax
=>
  k T 2 3   U 3 K   2 3 
 Fc 1   B  ln  c 2 x V   
  U c    k BT c MFc   
F(V) ~ (lnV)2/3
as compared with
F (V )  const  ln(V )
Single Stranded DNA translocation through
a nanopore: One polymer at a time
Relevant systems
Individual membrane channels:
ion flux & biopolymers translocation
A. Meller, L. Nivon, and D. Branton. Phys. Rev. Lett. 86 (2001)
3
2
1
J. Li and H. A. Lester. Mol. Pharmacol. 55 (1999).
J. J. Kasianowicz, E. Brandin, D. Branton and D. W. Deamer
Proc. Natl. Acad. Sci. USA 93 (1996)
Translocation and conformational fluctuation
O. Flomenbom and J. Klafter Biophys. J. 86 (2004).
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 beam
is focused on the enzyme, and the
fluorescent state of a single enzyme
is monitored.
•
The Michaelis-Menten reaction
Relevant systems
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)

2
1
k1
r1
1

r2
N
k2
2
rN
kN
N
Single molecule experiments in proteins:
Fractons in proteins
• Fluorescence resonant energy transfer (tens of angstroms).
• Photo-induced electron transfer (a few angstroms)
x(t )  X (t )  X eq
S. C. Kou and X. S. Xie, PRL (2004)
W. Min et al., PRL (2005)
R. Granek and J. Klafter, PRL (2005)
Autocorrelation function
Cx (t )  x(t ) x(0)
1  const. t 1/ 2 t  1 s
C x (t ) ~  1/ 2
t
t  1 s
Small scale motion – VIBRATIONS?
Fluctuating Enzymes
Fractal nature of proteins.
Mass fractality of proteins:
M ~R
df
Mass enclosed by concentric
spheres of radius R centered
at a backbone atom, in a
single protein (1MZ5).
Analysis covered over 200 proteins (!):
d f  2 .5  0 .2
M. B. Enright and D. M. Leitner, PRE (2005)
Manifold dimension D
Linear polymers D=1
Membranes D=2
• Chemical length
springs.
l
– the length of the minimal path along the connecting
• Chemical length exponent
• Or Flory exponent
dmin

l~r
Real space
M ~l ~r
d min
l
r
Manifold space
D

1
d


 min

df
d min  d f D
Density of (eigen) states:
N ( ) ~ 
d s 1
d s – Spectral dimension
Experiments (electron spin relaxation):
d s  1.3  1.7
for ~200-300 amino acids
N ( )
Computational studies involving ~60 proteins ->
Molecular weight dependent
For ~100 amino acids
ds :
1.3  d s  2
For over 2000-3000 amino acids
A. Vulpiani and coworkers (2002,2004)
Fractons
m
Vibrations of the fractal

 
 
d  
2
m 2 u (l , t )  mo 
u (l ' , t )  u (l , t )
 
dt
l 'l
2

Normal modes (eigenmodes, eigenstates) – Fractons:

 i t
u(l , t )   (l )e




2
   (l )  o 

(
l
'
)


(
l
)


 
2

l 'l
Strongly localized eigenstates

 (l ) !
Yakubo and Nakayama (1989)
S. Alexander and R. Orbach (1982)
mass
natural
o Spring
frequency

u displacement
 “name” of a point
l mass
Localization length in real space
r ( ) ~ 
d s d f
Localization length in manifold space
 ( ) ~   d
s
D
Disorder averaged eigenstate – Averaging over different realizations
of the fractal, or over many localization centers:

 (l )  f  o 
1  const. y 2
f ( y) ~   y
e
ds D
l
for
y  1
for
y  1

Inequalities between the different broken dimensions:
1  ds  D  d f  3
ds
– Spectral dimension
D
– Manifold dimension
df
– Fractal dimension
Remark:
For folded proteins D  1 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 D  d. f
Landau-Peirels Instability

u

 (l )
– Amplitude of a normal mode
Equipartition theorem
2
u
T
3k BT

2
m
Thermal fluctuations of the displacements ( d s  2 )
2
u
T

min ~ Rg

2
u
d f / ds
If d s  2 ,
o

u2
T
2
  d N ( ) u
~ No
T
min
1/ d s
T
~ min
( 2 d s )
~ No
( 2 / d s 1)
N o – # of amino acids (“polymer index”)
increases with increasing
No !
Large fluctuations may assist enzymatic/biological activity.
2
u
But:
1/ 2
should not exceed the mean inter-amino acid distance,
otherwise protein must unfold (or not fold).
If evolution designed only folded proteins,
ds
d s should depend on N o
.
should approach the value of 2 for large proteins !
2
ds
A. Vulpiani
and coworkers
(2002,2004)
Displacement difference time-autocorrelation function


Two point masses, l and l ' .
 
 
Positions in space R (l , t ) and R(l ' , t ) .

 
 
Separation vector X (t )  R (l , t )  R (l ' , t )

Equilibrium spacing X eq



Displacement difference vector x (t )  X (t )  X eq

 
 
x (t )  u (l , t )  u (l ' , t )
Expansion in normal modes

 

u (l , t )   u (t ) (l )

+ disorder averaging


 




x (t )  x (0)  2 1   (| l  l ' |) u (t )  u (0)

Two limits:
1) Undamped fractons (pure vibrations)


2
u (t )  u (0)  u
T
cos( t )
The calculation involves a time-dependent propagation length
(t ) ~ t d s / D
 
If (t ) | l  l ' | , motion of the two particles is uncorrelated.
 
If (t ) | l  l ' | , motion of the two particles is strongly correlated.
2 d s

1

const.
t



x (t )  x (0) ~  ( 2d D  d  2)
s
t s
for
t   1
for
t   1
  D/ d

d f / ds
1 
s
1   | l  l '|  o | r  r '| b
1
o
more precisely, for
t  1 :


2
k BT
2 d s
x (t )  x (0)  x  C
(ot )
2
mo
2
x
  d f ( 2 d s 1)
k BT  | r  r ' | 


2 
mo  b 
numbers:
Short-time exponent
0.1  2  d s  0.7
Long-time exponent
2d s
0.3 
 ds  1
D
C  numerical constant
2) Strongly overdamped fractons


2
u (t )  u (0)  u e
2
 t
T
where
m
is the friction. Therefore, the propagation length is
(t ) ~ t
ds 2D
1 d s / 2

1

const.
t



x (t )  x (0) ~  ( d D  d 21)
s
t s
  
2d
 2  2 | r  r ' | b 
o
f
for t   2
for t   2
/ ds
Conclusions
1. Novel approach for vibrations in folded proteins based on their
fractal nature  Provides a description on a universal level, yet still
microscopic in essence.
2. Slow power law decay of the autocorrelation function of the
distance between two associated groups, even for pure vibrations.
3. In the case of pure vibrations, this powerlaw decay requires broken
dimensions that obey the inequalities
ds  2
2d s
2
 ds  3
D
These inequalities do not hold for uniform lattices in all dimensions.