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 ~ daltons (compact structure)

Introduction N = 100 # of amino acid residues (for a single domain protein) = 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 N = 1000 other configurations Modeling for folding kinetics

Concepts from chemical reactions Transition state theory F Reaction coordinate Unfolded Transition state Folded  F* Arrhenius relation : k AB ~ 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 continuous 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 N 0 native residues, E(N 0 ) 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 Folding (or unfolding) kinetics can be treated as random walks on the network (energy landscape) generated from the random energy model ( R 0 ~ 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 walker is at  probability for a random walker 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 : ~ 4 x configurations 2 x 2 x 2 cube : 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)

A possible order parameter : depth  (minimal # of steps from the native state) 

 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’ : R 1 >>1 ‘uphill’ : R 1 <<1

Summary Random walks on a complex energy landscape statistical 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)