Dill 1999 - Paper - Lecture 4

REVIEW Polymer principles and protein folding KEN A. DILL 1 University of California, San Francisco, 3333 California Street, Ste. 415, San Francisco, California 94118 ~Received January 25, 1999; Accepted March 4, 1999! Abstract This paper surveys the emerging role of statistical mechanics and polymer theory in protein folding. In the polymer perspective, the folding code is more a solvation code than a code of local fc propensities. The polymer perspective resolves two classic puzzles: ~1! the Blind Watchmaker’s Paradox that biological proteins could not have originated from random sequences, and ~2! Levinthal’s Paradox that the folded state of a protein cannot be found by random search. Both paradoxes are traditionally framed in terms of random unguided searches through vast spaces, and vastness is equated with impossibility. But both processes are partly guided. The searches are more akin to balls rolling down funnels than balls rolling aimlessly on flat surfaces. In both cases, the vastness of the search is largely irrelevant to the search time and success. These ideas are captured by energy and fitness landscapes. Energy landscapes give a language for bridging between microscopics and macroscopics, for relating folding kinetics to equilibrium fluctuations, and for developing new and faster computational search strategies. Keywords: new view; polymer; protein folding; statistical mechanics This paper describes a perspective on protein folding that derives in part from simple statistical mechanical and polymer models. As with any perspective, this one is a personal opinion, with all the limitations that implies. The first part of this paper explores the folding code. ~1! Structure: How is the native structure encoded in the amino acid sequence? ~2! Thermodynamics: Why is folding so cooperative? ~3! Kinetics: What determines the speed and the rate- limiting steps of folding? Polymer modeling suggests that the fold- ing code is more a solvation code and less a linear encoding of torsion angles along the peptide bond, even though the latter is not negligible. The second part explores the energy landscape perspec- tive on folding kinetics. Polymer modeling suggests that the fold- ing process more closely resembles balls rolling down bumpy funnels than balls rolling aimlessly on flat surfaces or rolling sin- gle file along identical trajectories. DISCUSSION Side-chain interactions contribute to architecture, just as backbone interactions do The backbone forces of folding Table 1 compares two different perspectives on the folding code. A backbone-centric, helix-centric perspective arose over the 50 year time frame, from the 1930s to 1980s. It originated with Mir- sky and Pauling in 1936 ~Mirsky & Pauling, 1936!, who proposed that backbone hydrogen bonding is a prominent folding force. During the next 15 years, Pauling’s group used the structures of small molecule hydrogen-bonding compounds to predict that folded proteins would have a-helical and b-sheet structures ~Pauling & Corey, 1951a, 1951b, 1951c, 1951d; Pauling et al., 1951!. The first X-ray crystal structures of globular proteins gave strong support to this view by confirming the existence of the predicted a-helices and b-sheets ~Kendrew et al., 1958!. Hydrogen bonding was seen to be an important structure-causing force in proteins. During the same period, a step was taken toward understanding folding cooperativity through an understanding of the helix-coil transition. For many years it had been known that protein folding is cooperative, i.e., that there is a dramatic transition from dena- tured to native states upon only small changes in solvent, pH, or temperature. In the 1950s and 1960s, theoretical work particularly of Schellman ~1958!, Zimm and Bragg ~1959!, Poland and Scheraga ~1970!, and experiments ~Doty & Yang, 1956; Doty et al., 1956! showed that long peptide chains can undergo a helix-coil transition that is cooperative. The helix-coil transition is driven by hydrogen bonding and fc propensities among near-neighbor groups along the chain. For many years, this has been the main model for con- formational cooperativity in biomolecules. To complete the picture of structure, thermodynamics, and ki- netics, experiments beginning in the 1970s showed that helices can form rapidly ~Kim & Baldwin, 1982; Williams et al., 1996!. One inference was that folding is hierarchical and can be explained by a scheme 18 r 28 r 38: the primary structure leads to secondary structure ~fast!, which is then assembled into tertiary structure ~slower!. Hierarchical assembly was seen as a solution to the prob- Reprint requests to: Ken A. Dill, University of California, San Francisco, 3333 California Street, Ste. 415, San Francisco, California 94118; e-mail: dill@maxwell.ucsf.edu. 1 The author is grateful to Hans Neurath, the Protein Society, and Protein Science, for the opportunity to present this overview, which is largely taken from a talk given on the occasion of the Hans Neurath Award lecture, at the Protein Society meeting, July 27, 1998. Protein Science ~1999!, 8:1166–1180. Cambridge University Press. Printed in the USA. Copyright © 1999 The Protein Society 1166 lem of how the protein sorts through conformational space hay- stack quickly on its way to finding the native state needle. The same hierarchy has been widely explored as a computational strat- egy for predicting native states from amino acid sequences: use local helix and sheet propensities to predict secondary structures, then assemble them into tertiary structures. The upshot was a perspective in which the backbone inter- actions—hydrogen bonding and fc propensities—have been seen as a large part of the explanation of the structures, thermodynam- ics, and kinetics of protein folding ~Honig & Cohen, 1996; Aurora et al., 1997; Baldwin & Rose, 1999a, 1999b!. The fc propensities are not equivalent to hydrogen bonding, since hydrogen bonds are involved in nonlocal interactions, whereas fc interactions, by def- inition, are not. Nevertheless, from the perspective of the sequence- dependent interactions, and sequence-structure relationships, a backbone-centric view is largely a fc-centric view, since there has been little basis for believing one amide-carbonyl backbone hy- drogen bond has a substantially different strength than another in a sequence-dependent way. On the other hand, hydrophobic inter- actions, which were first identified as important for protein folding by Kauzmann ~1959!, were seen as a nonspecific glue that aided collapse but otherwise played little role in dictating the specific architectures of native proteins ~Anfinsen & Scheraga, 1975!. Hy- drophobic interactions are mainly expressed by the side chains. They are “through-space” and solvent-mediated contact inter- actions, rather than “through-nearest-neighbor-bonds,” as are fc interactions. This distinction between torsion-based nearest-neighbor “through-chain” interactions that involve fc angles, and contact- based “through-space” interactions, that involve displacement or exchange of solvent, seems less ambiguous than distinctions be- tween secondary vs. tertiary forces, or local vs. nonlocal forces. The fc interactions are mainly steric torsional constraints cap- tured in Ramachandran plots. Contact interactions, such as side- chain contacts, include hydrogen bonding and hydrophobic interactions and van der Waals interactions among non-neighboring monomers. The fc perspective does not address a key issue. As with most other polymers, a large conformational space is a consequence of weak preferences of each monomer unit for one region of torsion- angle space relative to another region. But if they are to account for the folding code, fc propensities must be different in the native state than in the denatured state. In particular, fc interactions must change when folding conditions are turned on. But there is little evidence that tri- and tetra-peptides adopt native-like conforma- tions and overcome the chain entropy, when the solvent or tem- perature are changed. The side-chain forces of folding A different perspective has developed from polymer modeling over about the past 15 years. The polymer perspective is side-chain centric, rather than backbone centric. The idea is that folding is dictated not so much by the propensities for nearest neighbor amino acids to favor particular fc values ~a-helix or b-sheet propensi- ties!, even though there is abundant evidence for such preferences ~Honig & Cohen, 1996; Aurora et al., 1997!. Rather, in the side- chain-centric view the greater contribution to the free energy of folding is encoded in a more delocalized “solvation” code: there are very few conformations of the full chain that can bury nonpolar amino acids to the greatest possible degree ~Dill, 1985; Dill et al., 1995!. Even short peptides, such as amphipathic helices, can be driven by solvation. Hydrophobic interactions, however they are defined, are among the strongest interactions among amino acids in water. And in large proteins, there are many of them. In this view, hydrophobic interactions are not nonspecific glue, but a cru- cial structure-determining driving force. In this view, folding co- operativity more closely resembles a process of polymer collapse in a poor solvent than a helix-coil transformation. In this view, fast secondary structure formation is less a consequence of strong helix propensities, and more an indirect consequence of a drive toward nonpolar desolvation. The true balance between side-chain and backbone forces is not yet known. The side-chain-centric view has been based on the following logic. Simplified models that include side-chain inter- actions, but have the fc preferences “turned off,” predict many properties of globular proteins. In contrast, models that keep fc propensities and turn off side-chain interactions predict only heli- ces or strands and no compact folded state ~Thomas & Dill, 1993!. Indeed, helix-coil experiments show that fc propensities control structures for sequences that are unable to collapse. For example highly charged poly-benzyl-l-glutamate is the classical helix- former. It follows that a minimal model of globular protein behavior can be constructed from a side-chain-centric perspective but not from a backbone-centric perspective. This means that it may be possible to design polymers that could fold and perform protein-like func- tions, even without peptide backbones. RNA molecules already provide some proof of this principle. Minimal models are guides for such general principles. But minimal models do not tell us the actual balance of forces in real proteins. If our goal is an accurate model of proteins, we undoubtedly cannot ignore backbone interactions ~Honig & Cohen, 1996! or details of steric packing, or the different fc interactions among the amino acids. In the end, since protein stability is a small difference of large interactions, all interactions can contribute to structure, thermodynamics, and kinetics. What is the evidence for the side-chain-centric view? ~1! A backbone centric view does not predict collapse. A coordination of fc choices to cause collapse would be extraordinarily fortuitous. ~2! Helix and strand propensities tend to be weak. Excepting poly- alanine-based sequences ~Scholtz & Baldwin, 1992!, peptides that are found to be in helices or strands in globular proteins are un- stable when isolated in solution. Moreover, most helices and strands are amphipathic ~Eisenberg et al., 1984; Bowie et al., 1990; Branden & Tooze, 1999!, implicating solvation forces. The a-helical and b-strand propensities are context dependent ~Kabsch & Sander, 1984; Minor & Kim, 1994!, and the nonlocal interactions in b-sheets are large ~Smith & Regan, 1995! and numerous. ~3! In a globular protein, the number of local interactions is proportional to the Table 1. Backbone-centric view Side-chain centric view Dominant force FC, Hydrogen bonds Hydrophobicity, hydrogen bonds Thermodynamic cooperativity Helix-coil transition Collapse transition Kinetics Helix formation is fast Desolvation is fast Role of hydrophobicity Nonspecific Drives specific architecture Folding code FC-centric ~18 r 28 r 38! Solvation code Polymer principles and protein folding 1167 number of amino acids N, but the number of nonlocal interactions is proportional to about 2N, so the latter should dominate in larger proteins. ~4! Helices and strands often take their conformational instructions from their context or from the solvent ~Kuroda et al., 1996; Predki et al., 1996!. ~5! To a first approximation, a fold is determined by the binary sequence of hydrophobic0polar mono- mers, even when fc propensities are largely chosen randomly ~Reidhaar-Olson & Sauer, 1988; Bowie et al., 1990; Lim & Sauer, 1991; Gassner et al., 1992; Lim et al., 1992; Kamtekar et al., 1993; Matthews, 1993; Munson et al., 1994, 1996; Lazar et al., 1997; Roy et al., 1997; Schafmeister et al., 1997; Wu & Kim, 1997!. ~6! Protein folds are less affected by mutations on their surfaces than in their hydrophobic cores ~Lim & Sauer, 1991; Matthews, 1993!. ~7! Some experiments show that protein folding is not hierarchical, implying that secondary structures are not pre- assembled and used as building blocks in tertiary assembly. For example, a b-sheet protein can fold via a helical intermediate ~Shiraki et al., 1995; Hamada et al., 1996!. ~8! Hydrophobic clus- tering, like secondary structure formation, can be very fast ~Chan et al., 1997; Ramachandra Shastry & Roder, 1998; Ramachandra Shastry et al., 1998!, and it can drive helix and sheet formation. Simplified models are hypothesis generators The predictions described above come, in part, from models that involve considerable simplification. An example is the HP model, in which each amino acid is represented as a bead, each bond is a straight line, bond angles are a few discrete options rather than a continuum, different conformations conform to lattices in two or three dimensions, and the 20 amino acids are condensed into a two-letter alphabet: H ~hydrophobic! or P ~polar!~Dill, 1985; Dill et al., 1995!. While statistical mechanical models are simplified in their rep- resentation of energies and atomic details, they are more refined in other respects ~Camacho & Thirumalai, 1993; Bryngelson et al., 1995; Dill et al., 1995; Karplus, 1997; Onuchic et al., 1997!: ~1! their full conformational space can be explored extensively, sometimes without sampling or approximation, and ~2! sometimes the full sequence space can be explored. For some questions, it is more important to get right the representation of conformational or sequence spaces than it is to get right the atomic details. Many questions of structure, stability, and kinetics are not about the locations of the hydrogen bonds in native lysozyme. They are not questions that are answerable by crystallography. They are about distributions and ensembles, flexibilities and entropies, energy land- scapes, folding kinetics, big conformational changes, or sequence space. They are low-resolution questions that have low-resolution answers. Apart from X-ray crystallography and NMR, the work- horses of biomolecule science for many years have been low- resolution experiments—CD, fluorescence, small-angle scattering, some NMR experiments, calorimetry, chromatography, ANS bind- ing, melting curves, etc. For questions involving conformational ensembles, conforma- tional entropies, sequence space, long time and large spatial scales, ensemble averaging, or non-native states like transition states, mol- ten globules, intermediates or denatured states, there is currently little alternative to some degree of simplification in models. It is sometimes helpful not to have atomic details, picosecond by pico- second, because it is hard to see the forest of principles through the trees of detail. It would be a mistake to believe that any model is “improvable” by adding structural detail. Sometimes details are the problem, not the solution. This is a key message from the successes of the two-dimensional lattice Ising model in the revolution that took place in understanding critical phenomena ~Stanley, 1971!. The inability of earlier models of phase transitions to capture sub- tle critical behavior was attributable, not to the lack of realism and atomic detail, but to a lack of rigor in the mathematics of the models. Mathematician Mark Kac once said that the purpose of models is “to polarize our thinking,” to help us formulate questions. A model manifests a point of view; it regards certain components of a problem as relevant, important, or dominant, and other compo- nents as irrelevant, unimportant, or negligible, and then devises a chain of logic leading to predictions from those premises. Most broadly, the point of a model is to make decisive and testable predictions, regardless of whether its fine structure looks realistic. A key advantage of simplified models is that their parameters are physical and minimal in number. The chain of logic from premises to conclusions is direct. Simplified models serve to generate hy- potheses that often cannot be generated in any other way, but that can then be tested by experiments or refined simulations. Simplified models have been useful for exploring entropies and combinatoric principles of conformational and sequence spaces. Two problematic paradoxes of protein science have been shown by polymer modeling to be neither problematic nor paradoxical. ~1! The Blind Watchmaker Paradox: The probability that natural proteins could be found in a random search of sequence space was seen to be impossibly small. ~2! The Levinthal Paradox: The prob- ability that a protein could find its native state by random search was seen to be impossibly small. Both paradoxes have been framed in terms of random unguided processes that search for a single point, the endstate, in a vast space. Biological evolution searches through sequence space; the endstate is a single protein having a particular function. Protein folding searches through conforma- tional space; the endstate is the single native structure of a given protein. In both cases, the vastness of the search ~i.e., the size of the search space! is taken, according to the paradox, to be the key to the impossibility of reaching the endstate. Creationists, evolutionists, and blind watchmakers: Can proteins arise from random sequences? Sequence space is a large place. For protein chains of 100 amino acids, the number of possible sequences of the 20 different amino acids is 20 100 5 10 130 ~see Fig. 1!. Creationists have used such numbers to argue the impossibility that proteins, and life, could Fig. 1. Sequence space is large. There are 20 100 different 100-mer se- quences. The probability of finding one particular sequence is 20 2100 , but the probability of finding any sequence that folds to a particular structure is predicted to be more than 100 orders of magnitude larger. 1168 K.A. Dill have arisen from the random sequences that were plausibly on the prebiotic earth. Creationists solve the large numbers problem by invoking divine intervention. Evolutionists solve the large num- bers problem instead by the accretion of advantage that happens through natural selection ~Dawkins, 1996!. But both evolutionists and creationists start from the same premise, the large numbers problem. Evolutionists too assume that natural proteins are infin- itesimal specks in an impossibly vast and meaningless sequence space, as indicated by the following quotes: “Only a very small fraction of this unimaginably large number of polypeptide chains would adopt a single stable three-dimensional conformation” ~Al- berts et al., 1998!, and “It is certain that we need a hefty measure of cumulative selection in our explanations of life” ~Dawkins, 1996!. It is this numbers problem that I refer to, with the help of the wonderful metaphor of Richard Dawkins, as the Blind Watch- maker’s paradox. But statistical mechanical modeling ~Lau & Dill, 1990; Chan & Dill, 1991; Lipman & Wilbur, 1991! shows that there is very little numbers problem in the first place. Reach into a soup of random amino acid sequences. The chance of pulling out a biologically important molecule depends on what is meant by “biologically important.” The following two questions are vastly different ~see Fig. 1!: ~1! What is the probability of pulling from that soup a specific sequence? ~2! What is the probability of pulling from that soup any sequence that folds to a specific structure? The answer to question ~1! is 10 2130 for a 100-mer. The chance of pulling out a polypeptide having, say, the lysozyme sequence, is essentially zero. But to achieve biological function, we care only about finding a particular fold, not a particular sequence. And modeling shows that the probability of finding a structure is likely to be more than 100 orders of magnitude larger than the probability of finding a se- quence ~Lau & Dill, 1990; Chan & Dill, 1991!. The chance of pulling out any sequence that folds to roughly lysozyme’s structure is closer to 10 210 to 10 220 . While this number too may seem impossibly small, nature works with these sorts of numbers all the time. These numbers would imply about one such sequence in a liter of random sequences at nanomolar concentrations! And the probability of finding any chain fold, not just lysozyme, is even higher. Why does the numbers problem disappear when we seek struc- tures rather than sequences? There is an enormous “degeneracy” in sequence space: many different sequences can fold to the same native structure. A protein can be mutated substantially without changing its fold. The explanation for this is simple. If, as noted above, a fold is primarily determined by the binary sequence of hydrophobic0polar monomers, then the essential features of the full 20 100 sequence space are found by searching a space of only about the 2 100 5 10 30 sequences that are written in a binary al- phabet ~H 5 hydrophobic, P 5 polar!, a reduction of 100 orders of magnitude. Degeneracy means that hydrophobic monomers are largely interchangeable with each other, and polar monomers are interchangeable with each other, for determining a fold. ~Function may have additional requirements, but estimates indicate that these do not change the numbers much ~Lau & Dill, 1990!.! Moreover, modeling ~Lau & Dill, 1990; Chan & Dill, 1991! and experiments ~Reidhaar-Olson & Sauer, 1988; Matthews, 1993! show that the relevant space is even smaller, because only about N03of the residues are crucial for folding—those that define the hydro- phobic core. To first approximation, most surface sites can be mutated without changing structure or function. ~The factor of N03 comes from the geometry of surface0volume ratios. Pack 100 amino acid spheres together to make a protein; 67 of them will be on the surface and 33 will be in the core.! Therefore, the real search for protein structure takes place, not in a space of 20 N , but in a space nearer in size to 2 N03 5 2 33 5 10 10 for N 5 100. The other 120 orders of magnitude in sequence space are highly de- generate; the folded states of those sequences will look much like ones already found in the search of the smaller space. Sequence space is therefore not likely to be vast darkness with infinitesimal specks of protein-like light spots. It is not perfectly light either. On a logarithmic scale, sequence space is predicted to be more like a beige sea in which virtually all molecules are “nearly folded.” A typical random chain of 100 amino acids is predicted to be highly compact in water, have considerable sec- ondary structure, and be structured much like a molten globule ~Lau & Dill, 1990; Chan & Dill, 1991!. This is a far better starting point for natural selection than are specks in astronomically large sequence spaces. But whatever the starting point, there remains the need for a process of improvement. Evolution can improve proteins by nat- ural selection. Richard Dawkins has explained natural selection using the metaphor of a Blind Watchmaker. By invoking “Watch- maker,” he means the endstate has the appearance of having been designed. The traditional inference is that an object that appears designed was built by a systematic step by step procedure, as in building a watch. But Dawkins’ term “Blind” means that, on the contrary, the natural selection process is not so systematic and does not involve a specific pre-ordained sequence of events. Natural selection is a Blind Watchmaker that improves proteins through incremental steps, each of which involves some bias, however small, at the same time it also involves considerable random choice among alternatives. The Blind Watchmaker is also a useful metaphor for protein folding kinetics, the systematic accretion of native structure over the time course of a folding experiment ~Zwanzig et al., 1992!.A native protein has the appearance of design. For example, the steric fit of side chains in a protein core is as precise as that of a jigsaw puzzle. But tight packing of irregular objects can also be achieved by shaking up nuts and bolts in a jar, with no design involved ~Bromberg & Dill, 1994!. The appearance of design does not mean that folding happens in serial step by step fashion. Even a jigsaw puzzle can be constructed through different parallel se- quences of events ~Harrison & Durbin, 1985!. The native structure can be reached, over the time course of folding, by a process that ~1! starts from different initial conformations and ~2! proceeds by incremental improvements, each of which has some bias but also involves considerable random choice among alternatives. Even a very small bias ~deviation from randomness! in choosing among alternatives can speed up the search time ~compared to a random search! by tens to hundreds of orders of magnitude ~see below!. When perfect randomness is not the driver, the vastness of the search becomes irrelevant to the search time ~Dill, 1993!. These ideas are captured in landscapes: fitness landscapes in sequence space or energy landscapes in conformational space. We now focus on the latter. The old and new views of folding kinetics: Different questions Protein folding kinetics has been described in terms of so-called Old and New Views ~Baldwin, 1994, 1995; Dill & Chan, 1997!.To define these views, we first distinguish models, old and new, from Polymer principles and protein folding 1169 views of microscopic folding processes, old and new. For models, the terms old and new is too stark a contrast. It leads to the perception that we should be asking: Why do we need new mod- els? What was wrong with the old ones? Which models are better? But these questions miss the mark. Old and new models do not address the same questions. The old models are mass action mod- els used to fit experimental data on folding relaxation times and amplitudes. The new view is not a denial of these models. Mass action models remain valid for representing such data. While mass action modeling gives a macroscopic description of experimental data, the new statistical mechanical models give a microscopic framework to explain that data. Where old and new views differ, however, is in their interpre- tation of the microscopic processes of folding. In the hope that changing terminology can help untangle some confusion, I will replace “old view” with “Sequential Micropath view” and “New view” with “Ensemble view.” Table 2 summarizes the differences between the two views of folding kinetics ~Dill & Chan, 1997!. The language of the Sequen- tial Micropath view—pathways, transition states, reaction coordi- nates, on-path and off-path intermediates—is intended to explain what exponentials do ~i.e., what you see in experiments!. Experi- mental relaxation data are interpreted in terms of mass-action di- agrams having arrows that connect symbols like D ~denatured!, I ~intermediate!, and N ~native!. Nothing in this language says what any one molecule is doing at any given time, or how the kinetics of folding is related to the monomer sequence, or how to assign microscopic chain conformations to labels I or D or transition state, etc. But the language of the Ensemble view—landscapes, folding funnels—is intended to describe what molecules do ~i.e., how individual molecules progress toward the folded state!, and how different monomer sequences lead to different kinetics. Sequential micropath perspective The main problem, according to the Sequential Micropath view, was the search problem, which has been called the Levinthal par- adox. As Levinthal posed it ~Levinthal, 1968; Wetlaufer, 1973!,a random search of conformations would take a protein forever. Levinthal saw folding as a search through a vast conformational space, the haystack, for the native structure, the needle. Suppose the conformational space is represented by four preferred fc an- gles for each peptide bond: a-helical, b-strand, and two others. In terms of those discrete options, the size of the space for a 100 residue chain is 4 100 ’ 10 60 chain conformations. Only one of these is the native structure. Levinthal’s proposed solution for find- ing the needle in the haystack was that all chains must follow the same microscopic pathway, like ants single file on a trail ~Levinthal, 1968!. By “same pathway,” he specified that every chain follows the same sequence of bond angle changes, in the same order, to reach the native state. In the Sequential Micropath view, kinetic intermediates ~if they were on-pathway! were seen as helpful mile- posts because they would show what route was taken, and there- fore what routes were avoided, and therefore how the haystack was searched efficiently. Two-state kinetics was seen as uninformative about the mileposts of folding. Ensemble perspective In the Ensemble view, the vastness of the search is largely irrelevant. The more important problem is kinetic traps ~Chan & Dill, 1994, 1998!. Chains can sort very quickly through vast stretches of conformational space. In this view, chains fall energetically downhill, as when balls roll down bumpy funnels. Chains do not fold by random searches on level energy landscapes. In this view, two-state kinetics often means the chain is folding at nearly its maximum possible diffusion-limited speed, without kinetic traps. In this view, stable intermediates are mainly seen as kinetic traps that slow down the folding process. Here’s why the vastness of the search is irrelevant. Even a very small bias, in the form of the forces of protein folding, can be the difference between folding times measured in “lifetimes of the universe” vs. milliseconds ~Bryngelson & Wolynes, 1987; Dill, 1987; Zwanzig et al., 1992!. ~On a large flat golf course, a golf ball will “never” find the hole by random processes, but if the golf course has even a small tilt that funnels toward the hole, no problem!! What causes the funnel-like tilt on a folding landscape? The first estimates of the shapes of folding energy landscapes were based on mean-field theories ~see Fig. 2!~Dill, 1985; Bryngelson & Wolynes, 1987!. Hydrophobic collapse leads to compact chain conforma- tions. The funnel arises because the drive to collapse is also a drive toward a reduced ensemble of conformations. There are many non-native states ~high energy!, but only one native state ~low energy!. The fraction of conformations that are compact is infin- itesimal compared to the total conformational space. If there are 4 100 conformations of a chain, Flory–Huggins-like theories predict that only about ~40e! 100 ’ 10 17 of those conformations are com- pact ~Dill, 1985!. More accurate recent estimates predict a number that is even smaller ~Yue & Dill, 1995!: the number of compact conformations having a hydrophobic core may even be as small as 1 for some sequences. This estimate is supported by experiments on reduced alphabets based on hydrophobicity codes; a small frac- tion of sequences appear to fold relatively uniquely ~Riddle et al., 1997; Roy et al., 1997; Schafmeister et al., 1997!. In short, the Ensemble view is a reversal of the sequential micropath view. What was seen as the slow step—the search through the huge haystack of non-native chain conformations—is now seen as happening at near diffusion-limited speed. Collapse can be fast. Sifting through most of the haystack is fast; the slow part is the endgame of reconfiguring a very small set of near-native con- formations. In the past few years, new fast experimental methods ~Burton et al., 1997; Callendar et al., 1999! have shown that pro- teins can fold at nearly diffusion-limited rates, on submillisecond time scales ~Huang & Oas, 1995; Ballew et al., 1996a, 1996b; Pascher et al., 1996; Burton et al., 1997; Chan C-K et al., 1997; Table 2. Sequential micropath view Ensemble view Language Paths, intermediates, transition states, reaction coordinates Landscapes, funnels Explains What exponentials do ~what you see! What molecules do ~how it works! Main problem Search problem Trap problem Proposed solution Sequential pathways Funnels Intermediates Mileposts Traps Two-state kinetics No information Implies fast folding 1170 K.A. Dill Gilmanshin et al., 1997a, 1997b, 1998; Ramachandra Shastry & Roder, 1998; Ramachandra Shastry et al., 1998!. Energy land- scapes provide the language that can help describe folding events at any level, from the microscopic to the macroscopic. Energy landscapes connect single-chain microscopics to experimental macroscopics One long-term goal of protein folding experiments has been to help understand the microscopic basis for the folding code. But this remains a promise, not a reality. Why? Prior to energy land- scapes, there has been no way to connect the macroscopics that experiments measure to the microscopics that are needed in com- putational folding algorithms. Here is the problem. Figure 3 illustrates the kind of folding kinetics data that is tra- ditionally measured, and the mass-action models that are used to interpret them. When a single exponential decay is observed in both folding and unfolding directions, it is described as “two-state” kinetics, because two mass action symbols, such as N ~native! and D ~denatured!, and an arrow interconnecting them, provide the simplest scheme that can model the data. But when multiple ex- ponentials are observed, at least one additional symbol must be invoked in a mass-action law. When there are three such symbols, say N, D, and I ~intermediate!, there are two main ways those symbols have been interconnected by arrows: I is called an “On- pathway” intermediate or I is an “Off-pathway” intermediate. But even when experiments provide a perfectly accurate mass action model for the folding and unfolding kinetics of a particular protein, it does not give enough information to make a microscopic model of how folding takes place. Experimental data are too av- eraged to inform the local decisions that must be made in confor- mational searching. A microstate is a single chain conformation. A macrostate—such as the unfolded state U, an intermediate state I, a molten globule M, or a transition state T—is some collection of individual conformations. The native state N is often appropriately regarded as both a microstate and a macrostate. To construct a folding algorithm requires a computational recipe that will begin with a microstate—some particular chain conformation—then eval- uate its energy, then choose which specific bonds to change and by how much, in order to take a computational step to make it a more native conformation. But experimentally obtained mass action models give only rec- ipes for dealing with macrostates, such as I ~intermediate!, D ~de- natured state!, T ~transition state!, etc., and not for dealing with microstates. Here are macrorecipes for how to move a chain con- formation toward the native structure. From an on-pathway inter- mediate state, move uphill along the reaction coordinate in the forward direction. From a transition state conformation, move down- hill. From an off-pathway intermediate, go back to the denatured state and try again to go forward along the reaction coordinate. But these macrorecipes do not answer the following questions. ~1! What is the macrostate to which a particular chain conforma- Fig. 2. Smooth funnel landscape ~bottom!. Denatured conformations fol- low different folding routes to the native state. The top figure shows the Flory–Huggins excluded volume estimate for the landscape shape ~turned sideways!: V;~N0r!!0@~N0r! N ~N0r 2 N!!#, where V is the number of chain conformations, N is chain length, and 0 # r # 1 is the compactness of the chain, an approximate measure of the depth on the landscape ~Dill, 1985!. Fig. 3. Simple mass-action schemes describe observed relaxation rates and amplitudes, using symbols such as N ~native!, D ~denatured!, and I ~intermediate!. Polymer principles and protein folding 1171 tion should be assigned? Or choose a macrostate: what micro- scopic conformations are in it? What is the ensemble called “intermediate state,” or “denatured state,” or “transition state”? Currently, such assignments must be made arbitrarily. Macrostates are averages over many microscopic conformations; they are not descriptions of single chain conformations. ~2! What series of chain conformations defines the reaction coordinate? A reaction coordi- nate is a macrovariable, not a microvariable ~see below!. For pro- tein folding, the reaction coordinate is not known in microscopic terms. ~3! Even if we knew the reaction coordinate, how do we know which way is forward? Which specific bond angles should we change to progress toward the native state? Every protein fold- ing algorithm must make these kinds of microdecisions at every step. But no experiment yet gives such microinformation. Energy landscapes can provide the common language to bridge between micro- and macrodescriptions. What is an energy landscape? According to the principles of thermodynamics, if a system has n degrees of freedom f 5 @f 1 , f 2 ,..., f n #, the stable state of the system can be found by determining the set of values f * 5 @f * 1 , f * 2 ,..., f * n # that gives the minimum value of the free energy func- tion F~f! 5 F~f 1 , f 2 ,..., f n !, when explored over all possible values of f ~see Fig. 4!. Such functions F~f! are called energy landscapes. Energy landscapes, per se, are neither new, nor con- troversial, nor limited to proteins. “Energy landscape” is nothing more than a name for this function. For protein folding, f may be the backbone and sidechain bond angles, for example. Distinguishing between microscopics and macroscopics The distinction between the old and new views is the distinction between an energy landscape and a reaction coordinate diagram, which, in turn, is a distinction between microscopics and macro- scopics ~see Fig. 5!. The Sequential Micropath view postulates a simple relationship between these two types of diagrams. In the Ensemble view, the relationship can be complex, but, in general, is not known. A microstate is a single point on an energy landscape and has free energy F micro 5 F~f!, which is also called the internal free energy ~Dill & Chan, 1997!. A macrostate has free energy F macro 5 F~j!, where j is just a scalar quantity, such as a reaction coordinate or a progress variable. A given value of j represents some particular ensemble of microscopic conformations. Figure 6 illustrates the difference between F micro and F macro ,in a simple model. Suppose we choose as a progress variable the number of hydrophobic contacts, j 5 0, 1, 2,..., m, to reflect the extent of folding. This is just one of many possible progress vari- ables; it is just chosen here for illustration because it simplifies the math. The density of states g~j! is a count of the number of different microstates f that define a particular macrostate j. Fig- ure 6 shows one of the g~0! ’ 500,000 conformations that have j 5 0 hydrophobic contacts, one of the g~4! 5 67 conformations that have j 5 4 hydrophobic contacts, and the g~6! 5 1 confor- mation that has j 5 6 hydrophobic contacts; this is the native structure in this model. To determine F micro , focus on a particular conformation. For that conformation, sum all the energies due to bond angles, torsions, stretches, van der Waals interactions, hydrogen bonds, electrostat- ics, and include the solvation free energy due to the relative amounts of buried and exposed hydrophobic and polar surface. F micro is a free energy, rather than just an energy, because it includes solvation and desolvation entropies and the hydrophobic effect. F micro is not the total free energy, however, because it does not include the chain conformational entropy: it treats only a single conformation. In the HP model, in which hydrophobicity dominates, a given chain conformation has j hydrophobic contacts, so F micro 5 jE, where E , 0 is the free energy of desolvating two nonpolar groups and bringing them into contact. The relationship between energy landscapes and reaction dia- grams is a relationship between F macro and F micro . F macro does include the chain conformational entropy, F macro ~j! 52kT ln V52kT ln@g~j!e 2F micro 0kT # 5 F micro ~j! 2 kT ln g~j!, ~1! where V is the partition function, F micro ~j! is the internal free energy for each conformation that has j hydrophobic contacts, and g~j! is the number of conformations having j hydrophobic con- tacts. Other progress variables can be more complex, but this sim- ple model is sufficient for present purposes. If we express the conformational entropy of the macrostate j as S conformational ~j! 5 klng~j!, then Equation 1 becomes F macro ~j! 5 F micro ~j! 2 TS conformational ~j!, ~2! The main point is that F micro is the free energy of a single chain conformation whereas F macro is the free energy of some ensemble of conformations that collectively have some macroscopic mean- ing, such as an intermediate, transition state, molten globule, or the denatured state. F macro includes a conformational entropy ~k ln g~j!!, due to the number of microscopic conformations in the particular Fig. 4. Energy landscapes are free energies, F micro ~f 1 , f 2 ,...!, as a func- tion of the degrees of freedom, f 1 , f 2 ,...,such as backbone and side-chain bond angles. 1172 K.A. Dill macrostate. F macro ~j! is a function of a single variable j, and therefore it corresponds to just an ordinary two-dimensional plot, of the folding free energy vs. reaction coordinate j. This is the traditional reaction coordinate diagram ~see Fig. 5B!. In contrast, F micro ~f! is the energy landscape; it is a function of many degrees of freedom. Landscapes are usually plotted in three dimensions, as a simplification, since it is impossible to draw high-dimensional surfaces. Traditional terms such as intermediate state, pathway, transition state, and free energy barrier refer to F macro ~j!. In con- trast, computer simulations usually explore F micro ~f!. What are “folding pathways”? Micropaths and microbarriers vs. macropaths and macrobarriers The distinction between micro and macro also applies to folding kinetics. A micropath is one trajectory that one protein follows as it folds. At time t, the degrees of freedom have the value f~t!. That is, f~0! 5 @f 1 ~0!, f 2 ~0!,..., f n ~0!# at time t 5 0, then f~t 1 ! 5 @f 1 ~t 1 !, f 2 ~t 1 !,..., f n ~t 1 !# at time t 5 t 1 , etc. f~t! describes the path a fly might take in a multidimensional space. Most computer simulations have explored one or a few micropaths, although a few modeling efforts have been able to explore more complete ensem- ble averages ~Chan & Dill, 1994, 1998!. Because proteins are subject to Brownian motion, a micropath involves much motion that would seem pointless to an observer. For example, a chain can pass back and forth through a given configuration many times. In contrast, a macropath describes some progress variable j~t!, which involves different ensembles at different times during the folding or unfolding process. Experiments have given information only about macropaths, whereas simulations usually give only informa- tion about micropaths. The key to the Sequential Micropath view is an implicit assump- tion of equivalence between macropaths and micropaths. The prem- ise of the Sequential Micropath view is that there is a simple and direct relationship between f~t! and j~t!, just as there is in tradi- tional chemical kinetics ~see Fig. 7!. If an energy landscape has an energy well corresponding to reactants A, another energy well corresponding to products B, and a lowest-energy “superhighway,” which defines the route that most molecules take from A to B, then a one-dimensional reaction pathway j can be obtained by painting a stripe along the centerline of the superhighway through the multi- AB Fig. 5. ~A! Energy landscape vs. ~B! reaction diagram. A landscape is a free energy F micro of each individual chain conformation vs. the many microscopic degrees of freedom. A reaction diagram is a free energy F macro of an ensemble of molecules, and includes the chain conformational entropy. Here F macro is a function of a single variable, j, such as a reaction coordinate. The reaction coordinate is usually not known for protein folding. The red arrow on the landscape indicates a possible micropath, an individual folding trajectory. In this case, the micropath never involves an uphill step, and yet the reaction diagram has a free energy barrier. The barrier is due to the slow entropic search of many different chains seeking the entry to the central steep funnel. Fig. 6. The density of states g~j! is a count of the number of chain con- formations, in this case having j hydrophobic contacts. On the energy ladder, more hydrophobic contacts corresponds to lower energy. Polymer principles and protein folding 1173 dimensional f space from A to B. In some cases, particularly near the end of the folding process, this way of defining reaction path- way may be useful and adequate. But this direct relationship be- tween f and j is valid only when molecules, like ants along a single file trail, all follow essentially the same route. That is, if one mol- ecule folds by first forming a helix at the N-terminal end, then form- ing a contact between residues 1 and 27, then undoing the helix, then forming a contact between monomers 3 and 18, etc., then the equiv- alence of micropaths and macropaths would mean that all the other molecules will undergo exactly the same sequence of events too. But while micropaths in chemical reaction kinetics overwhelm- ingly overlap with each other, the micropaths in protein folding can be very different. Chemical bonding involves energies much greater than kT, whereas each interaction in a folding process is not much larger than kT, so thermal motions can cause much larger variations in folding than in chemical reactions. One molecule may form its N-terminal helix first, while another molecule in solution, bombarded differently by Brownian motion, may form its C-terminal contacts first. In the end, both molecules will fold, but each follows a different micropath. Simulations usually show some degree of preference among microroutes, particularly in late stages of fold- ing, but there cannot be perfect registry in the early stages of the micropaths because the starting points ~the denatured conforma- tions! are so different. Figure 8A illustrates that for traditional chemical reactions ~Fig. 7!, there is a direct correspondence between the macrolevels ~reaction coordinate diagram! and microlevels ~energy landscape!. Peaks and valleys along the reaction profile represent microscopic milestones along the energy landscape. But Figure 8B illustrates that sometimes there may be no such correspondence for some folding processes. For folding, a given experimental observation ~as manifested in the reaction coordinate profile! can arise from many different landscape shapes. A landscape uniquely specifies a reaction diagram, but a reaction diagram does not uniquely specify a landscape. Figure 8B illustrates that folding becomes increas- ingly pathway-like at late stages, because the molecules become localized near the native state in conformational space. When chem- ical reactions have a single exponential time dependence, it implies an identifiable energy barrier. But for a single exponential, or any other particular time dependence in folding processes, no direct inference about microscopic bottlenecks is possible, as shown below. Sometimes micropaths can be very different from macropaths. Figures 5 and 9 show two landscape features in which micropaths do not coincide with macropaths. ~1! A downhill micropath con- tributes to an uphill macropath. A downhill micropath means that the chain does not break favorable contacts, say hydrophobic con- tacts. But this can involve a barrier on a reaction diagram because the microscopic meandering on flat plains on an energy diagram can be slow ~Fig. 5!. This will be manifested as a conformational entropy barrier ~uphill! on the reaction diagram but only as a slight downhill slope on the energy landscape. ~2! A downhill macropath can include some uphill micropaths. An uphill micropath can arise when one chain breaks favorable contacts, while most other chains find lower energy routes that avoid breaking contacts ~Fig. 9!.Itis because we do not yet know the relationships between micropaths and macropaths that we cannot use experimental data and mass- action models ~macropaths!, to help us forge folding algorithms, which require knowledge of microscopic details. Energy landscapes are funnels: The bottom is smaller than the top While the shapes of folding energy landscapes are not yet known in detail, it is uncontestable that they are funnel like, in the sense of the term that we use here ~Dill & Chan, 1997!. Here, “funnel” means that many conformations have high energy and few have low energy. More specifically, conformations having high F micro ~denatured states! have high conformational entropy and states having low F micro ~native state and other deep minima! have low conformational entropy. ~By some definitions, “funnel” also car- ries the connotation of smooth landscapes, so it also has implica- tions about dynamics and time dependence, namely that barriers are small so the process happens quickly. Here, the term funnel carries no such implication about kinetics or barrier heights or smoothness or any landscape shape feature other than: there are many conformations of high free energy ~F micro ! and few confor- mations of low free energy.! Energy landscapes also have funnel-like shapes for processes of ligand binding to biomolecules: there are few tightly bound con- formations, and many unbound or weakly bound conformations ~Frauenfelder et al., 1991; Miller & Dill, 1997; Tsai et al., 1999!. The chicken–egg problem: Collapse first or secondary structure first? Which comes first in the folding process, collapse or secondary structure? Just as the answer to where chickens come from is more complex than “chicken” or “egg,” so also folding is undoubtedly Fig. 7. Classical energy landscape for chemical reactions. Reactants, prod- ucts, and intermediate states are low-energy depressions. The reaction path- way is a lowest energy highway from reactants to products. Transition states are peaks along the pathway. For simple chemical reactions, most molecules follow essentially the same reaction path. 1174 K.A. Dill more complex than “collapse first” or “secondary structure first.” “Collapse,” “secondary structure,” and “hierarchical” assembly ~Baldwin & Rose, 1999a, 1999b! are macroterms, like “reaction coordinate,” since each describes an ensemble property. Hierarchic folding has been recently defined “as a process in which folding begins with structures that are local in sequence and marginal in stability; these local structures interact to produce intermediates of ever-increasing complexity and grow, ultimately, into the native conformation” ~Baldwin & Rose, 1999a, 1999b!. It is proposed that hierarchical folding involves multiple folding routes, rather than a unique sequential pathway. By these criteria, there is little to distinguish hierarchical folding from the Ensemble view. Growing stability corresponds to a downhill flow on a landscape, and the early preference for local contacts is similar to that found in energy- based microscopic models, such as the following. The diffusion- collision model is based on assuming that fc preferences are established early, then secondary structures assemble into tertiary structures ~Karplus & Weaver, 1976; Lee et al., 1987; Yapa et al., 1992!. A zippers model also proposes that local contacts form earlier, on average, than the nonlocal contacts. But the zippers model supposes that structure development is driven by solvation forces ~Dill et al., 1993; Fiebig & Dill, 1993!. A B Fig. 8. A: For chemical reactions ~energies .. kT !, the macrostates on reaction coordinate diagrams correspond to the time series of microstates on the energy landscape. B: For folding processes ~energies per interaction ’ kT !, the observed macrostates may not uniquely specify the time series of microstates on the energy landscape. Fig. 9. An uphill micropath ~red line! is surrounded by more favorable routes that do not involve uphill steps to reach the native state. Polymer principles and protein folding 1175 Alternatives to the sequential micropath and ensemble views? There have been efforts to marry old and new views ~Pande et al., 1998; Pande & Rokhsar, 1999!. Those efforts aim to recon- cile how there can be preferred folding routes at the same time that individual chains follow different micropaths. But no marriage is needed. Preferred routes and states are part and parcel of the En- semble perspective. In my opinion, the Ensemble perspective is not one model, one result, or one energy landscape shape. It is not a denial of patterns, pathways, uniqueness, or structure. It is just a perspective based on recognizing the general funnel-like nature of the energy function F micro ~f! with bumps and wiggles and shapes that have yet to be determined. The funnel perspective is univer- sally captured in many different models, monomer sequences, po- tential functions, move sets, and definitions of transition states. While particular results can depend on model details, the funnel concept is a broad brush picture of how a large ensemble leads to a small ensemble, how an unstructured population changes through time to become a single structure, and how the degrees of freedom diminish from being many and uncoupled and unsynchronized to being few and coupled and synchronized. This process is bound to involve preferences. Indeed, at the end of the folding process, it would be remarkable— and maybe impossible—to have large diversity in conformations or trajectories. Most of the simulations that have led to the En- semble perspective have found preferred folding routes in the late stages of folding ~Miller et al., 1992; Lazaridis & Karplus, 1997!. What is new in the new view, and what was the essence of Levinthal’s problem, was what happens in the early stages of folding, not the late stages. Levinthal’s concern was how to search the huge space of denatured conformations. The Ensemble view merely asserts that molecules cannot be synchronized at the be- ginning of folding because different chains have such different unfolded conformations. Although the denatured state is a single macrostate, it is a very heterogeneous collection of microstates. It will surely remain a matter of opinion for any given simula- tion whether what is interesting is the pathway or the variance from it. In either case, energy landscapes provide the basis for calculating any property of interest. Why do we need energy landscapes? Once energy landscapes are better understood, particularly for more realistic models of proteins, they should be able to serve several purposes. First, they should provide a consistent and rigorous lan- guage for interrelating macroscopics to microscopics. Simulators’ micropaths, when properly averaged, can teach us about experi- mentalists’ macropaths, and experimentalists’ macropaths can test simulators’ models. Second, landscapes provide a link between thermodynamics and kinetics, described below. And third, land- scapes may provide the bridge so that folding kinetics can be brought to bear on speeding up conformational search strategies, also described below. Relating thermodynamics and kinetics: A fluctuation– dissipation relationship for proteins? A most remarkable result in statistical mechanics is the fluctuation–dissipation theorem ~Chandler, 1987!. This theorem relates a kinetic property of systems ~the rate of approach to equi- librium!, to an equilibrium property ~the nature of the equilibrium fluctuations!. Energy landscapes provide the framework for relat- ing the thermodynamics and kinetics of protein folding. Figure 10 shows two landscapes: one is a smooth funnel, the other is a rugged funnel. For the smooth funnel, folding kinetics should be fast and two state. For the rugged landscape, folding kinetics will be slower and more complex. The shape of the landscape also describes the fluctuations at equilibrium. Fluctuations are interesting for two reasons. First, these are the motions that are important for protein function, such as when an enzyme enters a transition state for catalyzing a reac- tion. Second, fluctuations can be measured by NMR or thermal factors in X-ray crystallography. The fluctuations are those con- formations having energies only one or two kT higher than the native conformation and are therefore transiently populated due to occasional Brownian bombardments. If a protein has a smooth landscape, the motions of the protein are mostly small wiggles, never deviating much from the native structure because to do so would require a high energy. But for a bumpy landscape, very non-native-like conformations can occasionally be populated un- der native conditions because the energies of such conformations are not much higher than those of the native molecule ~Miller & Dill, 1995; Tang & Dill, 1998!. During those fluctuations, protons or ligands could exchange in or out, or the protein could have other transiently different properties than the native molecule. If we knew the shapes of energy landscapes, we could better understand the relationship between folding kinetics and equilibrium fluctua- tions around the native state. Landscape-ology can help in developing new conformational search strategies Knowing the shapes of energy landscapes should also help to create faster computer conformational search methods. In the Se- quential Micropath view, on-pathway intermediates are held in special regard because of how they might illuminate the folding Fig. 10. Comparing fluctuations on smooth vs. rugged landscapes. The state of lowest free energy is native ~N!, indicated as the lower tick mark on the y-axis. Normal fluctuations increase the energy, as indicated by the higher tick mark. Thermal fluctuations lead to only small conformational deviations from the native structure on smooth landscapes, but can lead to larger deviations on rough landscapes. The native lattice conformation has six hydrophobic contacts, whereas a conformation having only one unit higher energy ~five hydrophobic contacts!, has a completely different con- formation. Rugged landscapes mean that small excursions in energy ~from native! can lead to large excursions in structure. 1176 K.A. Dill path. But, as noted above, computer folding algorithms have not been able to use such macro-information. The landscape view is more egalitarian. Every conformation, no matter how distant from the native state, can give some useful information about the native state, as described below. New and faster conformational search strategies are emerging that are based on what rudimentary knowl- edge is currently available of the shapes of energy landscapes. Current search methods, such as Monte Carlo ~MC!, Simulated Annealing ~SA!, and Molecular Dynamics ~MD!, explore energy surfaces and are slow because they get caught in kinetic traps. We call these local search methods; they do not make use of global information about the shape of the underlying energy landscape. In a local search method, some very small change in a conformation is considered. Such changes are highly localized on the energy landscape. An energy is evaluated, and some decision is made whether to take that step uphill or downhill. The move is accepted or rejected, usually either based on Metropolis criteria or Newton’s laws. Such strategies are very slow because they are unguided by global information, they involve much randomness, and they usu- ally terminate in kinetic traps. Here is an analogy. One way to find the lowest point on the Himalayan mountains is to always walk downhill until you can go down no further. Then go uphill until you can go down again. This is the Monte Carlo and SA approach. Such random walking is much slower and more haphazard than if you used a contour map to guide your journey. For example, if a protein folding algorithm creates a structure that does not have a hydrophobic core, it should not keep changing one bond angle at a time as many current methods do; it should stop wasting its time and move to a very different part of conformation space. New methods are developing for speeding up conformational searching, based on emerging knowledge of the shapes of energy landscapes ~Maranas et al., 1995; Phillips et al., 1995!. For exam- ple, the idea behind the Convex Global Underestimator ~CGU! method ~Phillips et al., 1995; Dill et al., 1997; K.W. Foreman, A.J. Phillips, J.B. Rosen, & K.A. Dill, unpubl. comm.! is to sample a few conformations chosen randomly from the conformational space, find the nearest local energy minimum for each one, then construct a multi-dimensional parabolic “underestimator” surface U~f! un- derneath all the minima that are known so far ~see Fig. 11!. U~f! serves as a predictor for where the global minimum might be found, if the energy landscape is funnel-like. Subsequent under- estimator surfaces are constructed iteratively for increasingly nar- rowed regions around the native state. In this way, every chain conformation that is sampled—no matter how non-native— contributes some information about the landscape shape, and con- tributes to an estimate of where the native state will be found. In contrast, local search methods make no such use of collective information about all other conformations that have been sampled before a given step. The CGU and other underestimator methods look promising, on the following bases. ~1! Starting from different initial starting points on the landscape, the CGU usually reaches the same final point, indicating that it finds global minima and does not get stuck in kinetic traps. ~2! An advantage of the CGU over MC and SA is that no problem-dependent adjustment is required, as when devising temperature schedules or proper move sets. ~3! Tests so far in a simple protein folding model and on van der Waals clusters up to 21 atoms shows that CGU reaches much lower on energy land- scapes in a given time than MC or SA ~see Fig. 12!, and the ad- vantage increases with chain length ~K.W. Foreman, A.J. Phillips, J.B. Rosen, & K.A. Dill, unpubl. comm.!. The only knowledge the CGU currently uses is just that landscapes are funnel-like. As we learn more about the shapes of protein energy landscapes, it should be possible to create faster conformational search strategies. Historical parallels with polymer science? For the past 40 years, a defining paradigm of protein science has been Structural Biology. Structural Biology has provided a frame- work for deciding what questions are important and how to answer them. Two key imperatives of Structural Biology are ~1! high resolution, the importance of atomic detail and ~2! unique archi- tectures, the importance of specific geometric interrelationships among atoms. Protein structures have atomic resolution, and every Fig. 11. Convex global underestimator ~CGU! conformational search strat- egy. Traditional methods, such as Monte Carlo ~MC!, molecular dynamics ~MD!, and simulated annealing ~SA!, search over the tops of energy land- scapes and can get caught in kinetic traps. The CGU searches underneath the landscape instead by using a few sampled local minima ~indicated by dots! to generate a series of underestimating parabolic surfaces to locate the global minimum ~Dill et al., 1997!. Fig. 12. Relative search depth ~energy! of simulated annealing compared to the CGU, for different lengths of model proteins, up to 36 amino acids ~K.W. Foreman et al., in prep.!. For short chains, SA reaches the same depth on energy landscapes as the CGU, but for longer chains, SA gets stuck at increasingly higher altitudes on the energy landscape, where the relative depth is indicated by the cartoon on the right. Polymer principles and protein folding 1177 atom has its place. It has been considered important to know THE native structure, THE transition state, or THE intermediate structure. Of course, it is clear that proteins wiggle and move; they are not perfectly static ~Karplus, 1997!. But even so, such fluctuations are often regarded as a sort of footnote to the main message, much like error bars in experimental data. According to this logic, folding pathways are less like a perfect train track, where no lateral vari- ation is allowed, and more like a highway, where some small degree of weaving and lateral meandering can take place. But in the polymer view, statistics can play a fundamentally different and deeper role. It is more like replacing a train track, not with a highway, but with a ski bowl. Driving on a highway from point A to point B can be described by average velocities, posi- tions, and altitudes along the “reaction coordinate,” the highway. But tracking an ensemble of skiers is quite a different business than tracking the flow of cars on a highway. Skiers can take different routes. The average position of skiers on a mountainside is a much more heterogeneous property, with less apparent meaning. What is THE structure, or even THE averaged structure, at any given time, is not yet clear, or necessarily always meaningful. The importance of one particular structure and the neglect of statistics has a parallel in the history of polymer science. The breakthrough that founded polymer science was the macromolec- ular hypothesis, the idea that there were long chains covalently linked together ~Flory, 1953; Morawetz, 1985!. The huge resis- tance to this idea prior to the 1920s was due to a faith in the importance of specific structures and a reluctance to fully appre- ciate the statistics. According to Flory ~1953!: “Organic chemists were motivated by the desire to devise con- cise formulas and to isolate pure substances, the term pure ... invariably implying a formula of convenient size. Hence the quest for the cellulose molecule or the rubber molecule contin- ued....Bytheturn of the century this objective had crystallized to a discipline which dominated synthetic organic chemistry. To be eligible for acceptance in the chemical kingdom, a newly created substance . . . had to be separated in such a state that it could be characterized by a molecular formula. The investigator was obliged to adduce elementary analyses to confirm the com- position, and to supplement these with molecular weight deter- minations for the purpose of showing that the substance was neither more nor less complex than the formula proposed. Other- wise the fruits of his labors would not be elevated to an honored place in the immortal pages of the chemical compendiums. The successes of synthetic organic chemistry in creating the hun- dreds of thousands of different combinations and permutations of atoms must not be discounted. In magnitude of creative achievement, they are scarcely surpassed in any other field of science. While this discipline was strikingly successful, it also tended to narrow the outlook of contemporary researchers. They came to believe that every definable substance could be classi- fied in terms of a single definite molecule capable of being represented by a concise formula.” With the macromolecular hypothesis came the recognition that N polymer molecules in solution, even when they are called by the same name, such as polyethylene, are not identical to each other. Each molecule in solution can have a different conformation and even, for synthetic polymers, a different chain length. Hence dif- ferent experiments see different ensemble averages and give dif- ferent perspectives on the same “molecule,” polyethylene. Within only about a decade after the macromolecular hypothesis was ac- cepted, quantitative statistical mechanical models began to suc- cessfully explain rubber elasticity, the viscosities and viscoelasticities of chain molecule liquids, the dependence of the physical proper- ties of polymeric materials on molecular weight distributions, and the unusual thermodynamics of polymer solutions. Such statistical ideas now provide the foundation of modern polymer theory. For many properties of proteins, too, it seems clear that statistics is not just a caveat about small details but is at the very heart of the problems that proteins must solve. Conclusions Statistical mechanical models can give useful insights about pro- teins. While all-atom models sacrifice conformational sampling to gain atomic detail, statistical mechanical models do the reverse. Because simple models explore non-native states so effectively, have few parameters, and cost little computer time, they have been useful for exploring folding forces and principles. They have led to the perspective that the folding code is primarily a solvation code, rather than a local propensities code. Statistical mechanical models are well suited to addressing combinatoric problems, such as the Levinthal and Blind Watchmaker paradoxes. The conclusion is that we should beware of needle in a haystack arguments, because nature does not seem to work that way. Each step is not unguided. Conformational and sequence spaces are more like landscapes. Landscapes are funnel like, wide at the top and narrow at the bottom, sometimes with hills and valleys. All conformations—not just on-pathway intermediates for example—can give some guid- ance toward the global minimum. New computational search meth- ods are drawing on this information to make better folding and docking algorithms. 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