Protein Dynamics and Stability:
Universality vs. Specificity
Rony Granek
The Stella and Avram Goren-Goldstein Department of Biotechnology Engineering, BenGurion University of The Negev, Beer Sheva 84105, Israel
Tel-Aviv University
Ben-Gurion University
Joseph Klafter
Shlomi Reuveni
RG
Marina de Leeuw
Roee Ben-Halevi
Amit Srivastava
Natural Proteins
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.).
Problem –
a folded protein has less internal motion than an unfolded
Protein.
PROTEIN VIBRATIONS:
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
Calculation of cumulated Density of States using the GNM

G( )   g ( )d ~  d s
ds
-
Spectral dimension
0
Are proteins fractal-like?
slope  d s  1.93
1.73
1.52
N  1184
N  505
N  190
M. de Leeuw et al., PLOS-ONE (2009); Reuveni et al., PRL (2007)
Fractal nature of proteins?
Mass fractal dimension
df
:
M ~r
df
The atoms enclosed in spheres of different radii (pdb: 1OCP )
D. M. Leitner
and coworkers
N  1184
N  505
slope  d f  2.66
2.51
2.50
M. de Leeuw et al., PLOS-ONE (2009); Reuveni et al., PRL (2007)
N  190
Modeling a Protein as a Fractal – A Step Towards Universality
Replace with an
abstract
representation of a
protein
M (r) ~ r
df
g ( ) ~ 
d s 1
2  df  3
1  ds  2
Protein Stability & Unfolding

u
– Amplitude of a normal mode

u2
Equipartition

 (l )
T

3k BT
2
m
Thermal fluctuations of the displacements ( d s  2 )
2
u
T


min ~ Rg  d
f
2
u
/ ds
o
T
2
  d g ( ) u
min
~N
If d s  2 ,
1 / d s
2
u
T
T
~ min
( 2 d s )
~ N ( 2 / d s 1)
Landau-Peierls Instability
N
– # of amino acids (“polymer index”)
increases with increasing
N !
Large fluctuations may assist enzymatic/biological
activity.

u2
But
1/ 2
should not exceed the mean inter-amino acid distance,
otherwise protein must unfold (or not fold).
 Marginal stability. To have large amplitude motion but
remain folded:
Proteins can “live” in the “twilight” zone: Folded-Unfolded !
 To keep proteins folded,
ds
ds
should depend on N :
should approach the value of
2 for large proteins.
Instability threshold: Universal relation between exponents
Cluster melting analog:
Unfolding/Melting occurs from the surface inward
2
1
b

 1
ds d f
ln N
 mo2 Rc2 

b  ln 
 k BT 
Reuveni et al., PRL (2007)
4249 proteins
Colored histogram (100X100 bins)
fit to
2

ds
fit to
M. de Leeuw et al., PLOS-ONE,
2009
2
ds
1
df

1
df
b
 1
ln( N )
a
b
ln(N )
a=0.95
cc=0.58
4249 proteins
X-axis separated into 100 bins
fit to
2

ds
fit to
M. de Leeuw et al., PLOS-ONE,
2009
2
ds
1
df

1
df
b
 1
ln( N )
a
b
ln(N )
a=1.065
cc=0.945
Dynamics
Fluctuations in distance between two amino acids
x(t )  X (t )  X eq
1 d / 2
short time s :  (t )  r


1  const. t s
x (t )  x (0) ~  ( d d  d 21)
t s l s
long times : ξ(t)  r
Granek & Klafter, PRL (2005)
Static variance

d ( 2 d 1)
x2 ~ r f s
Reuveni et al., PNAS (2010)
Propagation length
 (t ) 2 ~ t
ds d f
Photo-induced electron transfer,
single molecule experiments,
Xie and coworkers
Random walk probability of return to the origin on the same network
Vibrational mean square displacement
2
ui (t )
P0 (t ) ~ t  d s / 2
t
T i
  dt P0 (t ) ~ t1d s / 2
0
Reuveni et al., PNAS (2010)
Conclusions:

Novel approach for vibrations in folded proteins based 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 to anomalous dynamics/ strange kinetics:
autocorrelation of separation; vibrational MSD; random walk MSD, return probability & mean first passage time, dynamic structure
factor.