Protein Folding in the Landscape Perspective: Chevron Plots and Non-Arrhenius Kinetics Hue Sun Chan and Ken A. Dill* Department of Pharmaceutical Chemistry, University of California, San Francisco, California ABSTRACT We use two simple models and the energy landscape perspective to study pro- tein folding kinetics. A major challenge has been to use the landscape perspective to inter- pret experimental data, which requires en- semble averaging over the microscopic trajec- tories usually observed in such models. Here, because of the simplicity of the model, this can be achieved. The kinetics of protein folding falls into two classes: multiple-exponential and two-state (single-exponential) kinetics. Experi- ments show that two-state relaxation times have ‘‘chevron plot’’ dependences on denatur- ant and non-Arrhenius dependences on tem- perature. We find that HP and HP models can account for these behaviors. The HP model often gives bumpy landscapes with many ki- netic traps and multiple-exponental behavior, whereas the HP model gives more smooth funnels and two-state behavior. Multiple-expo- nential kinetics often involves fast collapse into kinetic traps and slower barrier climbing out of the traps. Two-state kinetics often in- volves entropic barriers where conformational searching limits the folding speed. Transition states and activation barriers need not define a single conformation; they can involve a broad ensemble of the conformations searched on the way to the native state. We find that unfolding is not always a direct reversal of the folding process. Proteins 30:2–33, 1998.  1998 Wiley-Liss, Inc. Key words: chevron plot; energy landscape; folding funnel; kinetic trap; lattice models; non-Arrhenius behavior TWO PERSPECTIVES: ENERGY LANDSCAPES VS. FOLDING PATHWAYS Our aim here is to show that the energy landscape perspective of protein folding, and some simple mod- els on which it is based, can explain some of the main classes of experimental folding data—single- and multiple-exponential relaxations, chevron plots, and non-Arrhenius kinetics. Recently, a so-called ‘‘New View’’ 1,2 of protein folding kinetics has emerged. The New View replaces the concept of ‘‘folding pathway’’ with ‘‘energy landscapes.’’ 3–8 Anfinsen 9,10 and others showed that proteins can fold reversibly to stable states, which was taken as evidence that folding is ‘‘path independent.’’ In contrast, Levinthal 11,12 ar- gued that proteins would fold too slowly by undi- rected random searching of conformations. So he concluded that protein folding must follow a ‘‘path- way,’’ which is ‘‘a well-defined sequence of events which follow one another so as to carry the protein from the unfolded random coil to a uniquely folded metastable state.’’ 11 These two lines of argument have led to the ‘‘Levinthal paradox’’ 13,14 : How could folding be pathway dependent and pathway indepen- dent at the same time? But according to the energy landscape perspective, there is no paradox because the new view recognizes that ‘‘folding pathways’’ are not the correct solution to the kinetic problem Levinthal posed. 8 The land- scape perspective readily explains the process of reaching a global minimum in free energy (satisfying Anfinsen’s experiments) and doing so quickly (satis- fying Levinthal’s concerns) by multiple folding routes on funnel-like energy landscapes 15,16 (reviewed in Refs. 3–8). Instead of viewing folding as a process in which all chains perform essentially the same se- quence of events to reach the native state, the new view envisions folding as representing the ensemble average of a process that is microscopically more heterogeneous. In this process, each individual pro- tein molecule may follow its own trajectory, but just like skiers down a mountain, they all may eventually reach the same point at the bottom, the native state. How might experiments distinguish pathways from more complex landscapes? Experiments in the 1970s began to explore folding pathways. 17–24 But the pathways that are found by experimentalists are very different from the microscopic pathways defined by Levinthal. The main observables in experiments have been exponential decays of optical signals. Mass-action models are used to fit the time constants Contract grant sponsor: NIH. Parts of this paper were given as the First Anfinsen Memo- rial Lecture by K.A.D. at the Johns Hopkins Protein Folding Meeting, Coolfont, West Virginia, March 1996. Electronic ver- sion of the energy landscape drawings in this paper are available at the website http://laplace.ucsf.edu. H.S.C.’s E- mail: chan@maxwell.ucsf.edu *Correspondence to: Ken A. Dill, Department of Pharmaceu- tical Chemistry, University of California, San Francisco, CA 94143-1204. E-mail: dill@maxwell.ucsf.edu Received 11 March 1997; Accepted 30 June 1997 PROTEINS: Structure, Function, and Genetics 30:2–33 (1998)  1998 WILEY-LISS, INC.