Protein folding kinetics and more

Chi-Lun Lee (李紀倫 )

Department of Physics

National Central University

Introduction

Protein (polypeptide chain): chain of amino

acid residues

Primary structure : sequence of amino acid residues

Secondary structure : locally folded three-dimensional structure ( helix, sheet, etc.)

Tertiary structure : fully-folded compact structure

For a single domain globular protein (~100 amid acid residues), its diameter ~ 5nm and molecular mass ~ 10000 daltons (compact structure)

Features of protein folding :

• Volume exclusion and chain connectivity

• Van der Waals interactions

• Hydrogen bond

• Hydrophobic interactions

• …

Peak in specific heat vs. T

c

T

Resemblance with first order transitions

Cooperativity in folding

Concepts from chemical reactions

Transition state theory

F

Reaction coordinate

Unfolded

Transition state

Folded

F*

Arrhenius relation : kAB ~ exp(-F*/T)

foldedunfolded

0 1 (order parameter)

The real world is much more complicated

Energy surface may be rough at times…

• Traps from local minima

• Non-Arrenhius relation

• Non-exponential relaxation

• Glassy dynamics?

• Defining an order parameter

• Specifying a network

• Assigning energy distribution P(E,)

• Projecting the network on the order parameter c

ontinuous time random walk (CTRW)

Statistical Energy Landscape Theory

Generalized master equation

Kinetics : Metropolis dynamics+CTRW

Transition rate between two conformations

( R0 ~ 1 ns )

Results : mean first passage time (MFPT)

Results : second moments

Poisson

long-time relaxation

Results : a dynamic ‘phase diagram’

exponential relaxation

power-law relaxation

A fantasy from the protein folding problem…

A ‘toy’ model : Rubik’s cube

1.E+05

1.E+06

1.E+07

1.E+08

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

1/T

MFPT

Monte Carlo simulations

Summary

• Random walks on a complex energy landscape

• Exponential nonexponential kinetics

• Nonexponential kinetics can happen for a downhi

ll folding process (cf. experimental work by Grue

bele et al., PNAS 96, 6031(1999))