Transcript 961101

Protein folding dynamics
and more
Chi-Lun Lee (李紀倫)
Department of Physics
National Central University
For a single domain globular protein (~100 amid acid
residues), its diameter ~ several nanometers and molecular
mass ~ 10000 daltons (compact structure)
Introduction
Modeling for folding kinetics
• N = 100 # of amino acid residues (for a single domain
protein)
•
n = 10 # of allowed conformations for each amino acid
residue
• For each time only one amino acid residue is allowed to
change its state
• A single configuration is connected to Nn = 1000 other
configurations
Concepts from chemical reactions
Transition state theory
Transition state
F
DF*
Unfolded
Folded
Reaction coordinate
Arrhenius relation : kAB ~ exp(-DF*/T)
For complex kinetics, the stories can be much more
complicated
Statistical energy landscape
theory
unfolded
folded
r (order parameter)
Energy surface may be rough at times…
• Traps from local minima
• Non-Arrenhius relation
• Non-exponential
relaxation
• Glassy dynamics
Cooperativity in folding
Peak in specific heat vs. T
c
T
Resemblance with first order transitions (nucleation)?
Theory : to build up and categorize an
energy landscape
• Defining an order parameter r
• Specifying a network
• Assigning energy distribution P(E,r)
• Projecting the network on the order parameter
continuous time random walk (CTRW)
Generalized master equation
Random energy model
Bryngelson and Wolynes,
JPC 93, 6902(1989)
– e0 , when the ith residue is in its native state.
ei =
De
a Gaussian random variable with mean –e and
variance De when the residue is non-native.
– e non-native
de
– e0 native
Random energy model
•Another important assumption : random erergy
approximation (energies for different configurations are
uncorrelated)
•This assumption was speculated by the fact that one
conformational change often results in the
rearrangements of the whole polypeptide chain.
Random energy model
•For a model protein with N0 native residues, E(N0) is a
Gaussian random variable with mean
and variance
order parameter
Random energy model
Using a microcanonical ensemble analysis, one can
derive expressions for the entropy and therefore the free
energy of the system:
Kinetics : Metropolis dynamics+CTRW
Transition rate between two conformations
( R0 ~ 1 ns )
Folding (or unfolding) kinetics can be treated as random
walks on the network (energy landscape) generated from
the random energy model
Random walks on a network (Markovian)
after mapping on r
One-dimensional CTRW (non-Markovian)
Two major ingredients for CTRW :
•Waiting time distribution function
•Jumping probabilities
can be derived from statistics of the
escape rate :
And
can be derived from the
equilibrium condition
equilibrium distribution :
Let us define
probability density that at time t a random
walker is at r
probability for a random
walker to stay at r for at least time t
probability to jump from r to r’ in one step after time t
Therefore
0 jump
1 jump
2 jumps
or
Generalized Fokker-Planck equation
Results : mean first passage time (MFPT)
Results : second moments
long-time
relaxation
Poisson
Results : first passage time (FPT) distribution
0<a<1
Lévy distribution
Power-law exponents for the FPT distribution
Locating the folding transition
folding transition
cf. simulations (Kaya and Chan, JMB 315, 899 (2002))
Results : a dynamic ‘phase diagram’
(power-law decay)
(exponential decay)
A fantasy from the protein folding problem…
A ‘toy’ model : Rubik’s cube
3 x 3 x 3 cube : ~ 4x1019 configurations
2 x 2 x 2 cube : 88179840 configurations
Metropolis dynamics (on a 2 x 2 x 2 cube)
Transition rate between two conformations
Monte Carlo simulations
Energy : -(total # of patches coinciding with
their central-face color)
2.E+07
2.E+07
1.E+07
1.E+07
1.E+07
8.E+06
6.E+06
4.E+06
2.E+06
E
-2
4
-2
2
-2
0
-1
8
-1
6
-1
4
-1
2
-1
0
-8
-6
-4
-2
0.E+00
0
Number of states
2.E+07
0
2
4
6
8
10
0
-4
-8
E -12
35
30
-16
25
-20
20
Cv
-24
15
T
10
5
0
0
2
4
6
T
8
10
12
A possible order parameter : depth r (minimal
# of steps from the native state)
Number of configurations
1.0E+08
1.0E+07
1.0E+06
1.0E+05
1.0E+04
1.0E+03
1.0E+02
1.0E+01
1.0E+00
0
5
10
Depth
r
15
Funnel-like energy landscape
0
2
4
6
8
10
0
-5
-10
E
-15
-20
-25
-30
Depth
r
12
14
16
Free energy
Energy fluctuations (T=1.3)
• A strectched exponential relaxation
Two timing in the ‘folding’ process : t1 , t2
Rolling along the Anomalous
order parameter diffusion
‘downhill’ : R1 >>1
‘uphill’ : R1 <<1
Summary
• Random walks on a complex energy landscape
statistical energy landscape theory (possibly nonMarkovian)
• Local minima (misfolded states)
• Exponential
nonexponential kinetics
• Nonexponential kinetics can happen even for a ‘downhill’
folding process (cf. experimental work by Gruebele et al.,
PNAS 96, 6031(1999))
Acknowledgment : Jin Wang, George Stell
T
U
t1 , t2
t1 , t2
t3 , t4
U
F
F
•If T is high (e.g., entropy associated
with transition state ensemble is small)
exponential kinetics likely
•If T is low or there is no T
nonexponential kinetics
Waiting time distribution function
long-time scale :
power-law decay
short-time scale :
exponential decay
Results : diffusion parameter
Lee, Stell, and Wang, JCP 118, 959 (2003)