Quarterly Reviews of Biophysics



Review Article

Understanding protein folding with energy landscape theory Part II: Quantitative aspects


Steven S.  Plotkin  a1 c1 and José N.  Onuchic  a2 c1
a1 Department of Physics and Astronomy, University of British Columbia, Vancouver, Canada V6T 1Z1
a2 Department of Physics, University of California at San Diego, La Jolla 92093, USA

Abstract

1. Introduction 206

2. Quantifying the notions behind the energy landscape 206

2.1 Basic concepts of the Random Energy Model (REM) 206

2.2 Replica-symmetric partition functions and densities of states 209

2.3 The RHP phase diagram and avoided phase transitions 210

2.4 Basic concepts of the entropy of topologically constrained polymers 212

3. Beyond the Random Energy Model 219

3.1 The GREM and the glass transition in a finite RHP 222

4. Basics of configurational diffusion for RHPs and proteins 227

4.1 Kinetics on a correlated energy landscape 231

5. Thermodynamics and kinetics of protein folding 234

5.1 A protein Hamiltonian with cooperative interactions 234

5.2 Variance of native contact energies 235

5.3 Thermodynamics of protein folding 236

5.4 Free-energy surfaces and dynamics for a Hamiltonian with pair-wise interactions 240

5.5 The effects of cooperativity on folding 242

5.6 Transition-state drift 242

5.7 Phase diagram for a model protein 245

5.8 A non-Arrhenius folding-rate curve for proteins 246

6. Non-Markovian configurational diffusion and reaction coordinates in protein folding 247

6.1 An illustrative example 250

6.2 Non-Markovian rate theory and reaction surfaces 251

6.3 Application of non-Markovian rate theory to simulation data 257

7. Structural and energetic heterogeneity in the folding mechanism 259

7.1 The general strategy 261

7.2 An illustrative example 263

7.3 Free-energy functional 264

7.4 Dependence of the barrier height on mean loop length (contact order) and structural variance 268

7.5 Illustrations using lattice model proteins and functional theory 269

7.6 Connections of functional theory with experiments 271

8. Conclusions and future prospects 273

9. Acknowledgments 274

10. Appendices

A. Table of common symbols 275

B. GREM construction for the glass transition 276

C. Effect of a Q-dependent diffusion coefficient 279

D. A frequency-dependent Einstein relation 279

11. References 281

We have seen in Part I of this review that the energy landscape theory of protein folding is a statistical description of a protein's complex potential energy surface, where individual folding events are sampled from an ensemble of possible routes on the landscape. We found that the most likely global structure for the landscape of a protein can be described as that of a partially random heteropolymer with a rugged, yet funneled landscape towards the native structure. Here we develop some quantitative aspects of folding using tools from the statistical mechanics of disordered systems, polymers, and phase transitions in finite-sized systems. Throughout the text we will refer to concepts and equations developed in Part I of the review, and the reader is advised to at least survey its contents before proceeding here. Sections, figures or equations from Part I are often prefixed with I- [e.g. Section I-1.1, Fig. I-1, Eq. (I-1.1)].


Correspondence:
c1 Correspondence may be addressed to either author. S. Plotkin: Tel.: 604-822-8813; Fax: 604-822-5324; E-mail: steve@physics.ubc.ca J. Onuchic: Tel.: 858-534-7067; Fax: 858-822-4560; E-mail: jonuchic@ucsd.edu