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
• N = 100 # of amino acid residues (for a single domain pro
tein)
• = 10 # of allowed conformations for each amino acid re
sidue
• For each time only one amino acid residue is allowed to c
hange its state
• A single configuration is connected to N = 1000 other co
nfigurations
Modeling for folding kinetics
Concepts from chemical reactions
Transition state theory
F
Reaction coordinate
Unfolded
Transition state
Folded
F*
Arrhenius relation : kAB ~ exp(-F*/T)
foldedunfolded
(order parameter)
For complex kinetics, the stories can be much more complicated
Statistical energy landscape theory
Energy surface may be rough at times…
• Traps from local minima
• Non-Arrenhius relation
• Non-exponential relaxation
• Glassy dynamics
Peak in specific heat vs. T
c
T
Resemblance with first order transitions (nucleation)?
Cooperativity in folding
• Defining an order parameter
• Specifying a network
• Assigning energy distribution P(E,)
• Projecting the network on the order parameter continuou
s time random walk (CTRW)
Theory : to build up and categorize an energy landscape
Generalized master equation
Random energy model
i =
– 0 , when the ith residue is in its native state.
a Gaussian random variable with mean – and variance when the residue is non-native.
– 0 native
– non-native
Bryngelson and Wolynes, JPC 93, 6902(1989)
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 energ
y of the system:
Kinetics : Metropolis dynamics+CTRW
Transition rate between two conformations
Folding (or unfolding) kinetics can be treated as random
walks on the network (energy landscape) generated from
the random energy model
( R0 ~ 1 ns )
Random walks on a network (Markovian)
One-dimensional CTRW (non-Markovian)
Two major ingredients for CTRW :
•Waiting time distribution function
•Jumping probabilities
after mapping on
can be derived from statistics of the escape rate :
And can be derived from the
equilibrium condition
equilibrium distribution :
probability density that at time a random w
alker is at
probability for a random w
alker to stay at for at least time
probability to jump from to ’ in one step after time
Let us define
A ‘toy’ model : Rubik’s cube
3 x 3 x 3 cube : ~ 4x1019 configurations2 x 2 x 2 cube : 88179840 configurations
Energy : -(total # of patches coinciding with their central-face color)
0.E+00
2.E+06
4.E+06
6.E+06
8.E+06
1.E+07
1.E+07
1.E+07
2.E+07
2.E+07
2.E+07
E
Num
ber of
sta
tes
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+07
1.0E+08
0 5 10 15
Depth
Num
ber
of c
onfi
gura
tion
s
A possible order parameter : depth (minimal # of steps from the native state)
Two timing in the ‘folding’ process : 1 , 2
Anomalous diffusion
Rolling along the order parameter
‘downhill’ : R1 >>1
‘uphill’ : R1 <<1
Summary
• Random walks on a complex energy landscape statistic
al energy landscape theory (possibly non-Markovian)
• 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
U
F
1 , 2
T
F
3 , 4
U
1 , 2
•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
short-time scale : exponential decay
long-time scale : power-law decay
Waiting time distribution function