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

Abstract

1. Introduction 2062. Quantifying the notions behind the energy landscape 2062.1 Basic concepts of the Random Energy Model (REM) 2062.2 Replica-symmetric partition functions and densities of states 2092.3 The RHP phase diagram and avoided phase transitions 2102.4 Basic concepts of the entropy of topologically constrained polymers 2123. Beyond the Random Energy Model 2193.1 The GREM and the glass transition in a finite RHP 2224. Basics of configurational diffusion for RHPs and proteins 2274.1 Kinetics on a correlated energy landscape 2315. Thermodynamics and kinetics of protein folding 2345.1 A protein Hamiltonian with cooperative interactions 2345.2 Variance of native contact energies 2355.3 Thermodynamics of protein folding 2365.4 Free-energy surfaces and dynamics for a Hamiltonian with pair-wise interactions 2405.5 The effects of cooperativity on folding 2425.6 Transition-state drift 2425.7 Phase diagram for a model protein 2455.8 A non-Arrhenius folding-rate curve for proteins 2466. Non-Markovian configurational diffusion and reaction coordinates in protein folding 2476.1 An illustrative example 2506.2 Non-Markovian rate theory and reaction surfaces 2516.3 Application of non-Markovian rate theory to simulation data 2577. Structural and energetic heterogeneity in the folding mechanism 2597.1 The general strategy 2617.2 An illustrative example 2637.3 Free-energy functional 2647.4 Dependence of the barrier height on mean loop length (contact order) and structural variance 2687.5 Illustrations using lattice model proteins and functional theory 2697.6 Connections of functional theory with experiments 2718. Conclusions and future prospects 2739. Acknowledgments 27410. AppendicesA. Table of common symbols 275B. GREM construction for the glass transition 276C. Effect of a Q-dependent diffusion coefficient 279D. A frequency-dependent Einstein relation 27911. References 281We 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)].