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

0 jump 1 jump

2 jumps

Therefore

or

Generalized Fokker-Planck equation

Results : mean first passage time (MFPT)

Results : second moments

Poisson

long-time relaxation

Results : first passage time (FPT) distribution

0 < < 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 configurations2 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)

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

-24

-20

-16

-12

-8

-4

0

0 2 4 6 8 10

T

E

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12

T

Cv

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)

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8 10 12 14 16

Depth

E

Funnel-like energy landscape

Free energy

Energy fluctuations (T=1.3)

• A strectched exponential relaxation

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

Results : diffusion parameter

Lee, Stell, and Wang, JCP 118, 959 (2003)