Using Protein Folding Rates to Test

The fastest simple, kinetically two-state protein folds a million times more rapidly than the slowest. Here we review many recent theories of protein folding kinetics in terms of their ability to qualitatively rationalize, if not quantitatively predict, this fundamental experimental observation. INTRODUCTION: TWO-STATE FOLDING RATES AS AN EXPERIMENTAL BENCHMARK Simple, single-domain proteins typically fold via a process that lacks well-populated intermediates [reviewed in (1)]. Despite the potential simplification afforded by the absence of kinetic intermediates, reported two-state folding rates span a remarkable six order of magnitude range (Figure 1) (2, 3). When coupled with the appealing simplicity of two-state behavior, the broad range of two-state rates provides a potentially straightforward and quantitative opportunity to test theories of the folding process. Theoretical models of protein folding kinetics can, admittedly somewhat artificially, be grouped into two broad classes. Perhaps the more prominent class consists of theories emerging from observations of the simulated folding of simple onand off-lattice models in silico. The second class consists of theoretical models emerging from the experimental observation of protein folding in vitro. In this article, we broadly review many recent simulationand experiment based theories of folding kinetics and critically evaluate these theories in terms of their ability to qualitatively rationalize, if not quantitatively predict, the vast range of folding rates observed for two-state proteins. Figure 1 The simple, single-domain proteins cytochrome b562 and muscle acylphosphatase are of similar size and stability, yet the former folds in microseconds, and the latter folds in seconds (2, 3). Here we critically review the ability of various theories of protein folding to rationalize qualitatively, if not predict quantitatively, this six order of magnitude range of rates. SIMULATION-DERIVED THEORIES OF FOLDING Perhaps the majority of contemporary theories of protein folding kinetics are based on observations of the simulated folding of simple, computational models. In an ideal world, the relevant simulations would entail a level of detail commensurate with the complex, atomistic structure of a fully solvated polypeptide. In reality, however, the simulation of protein folding in atomistic detail has proven computationally overwhelming. The difficulty is twofold. First, computational times scale strongly with the number of atoms, and even the smallest proteins are composed of thousands of atoms solvated by thousands of water molecules. Second, even the most rapidly folding proteins fold extremely slowly relative to the femtosecond time step of fully detailed molecular dynamics simulations [for a recent, partial solution to this dilemma, see Pande and coworkers folding@home project (4)]. Faced with this computational obstacle, the large majority of the theoretical literature in protein folding has been based on observations not of fully detailed protein models but on highly simplified representations of the polypeptide chain. Lattice polymers, perhaps the most popular computational model, simplify the description of the polypeptide chain by distilling each amino acid into a single bead and simplify folding dynamics by limiting the moves of each bead to hops between discrete points on a coarse lattice. Although lattice polymers and many off-lattice computational models are highly simplified, they capture many of the potentially relevant aspects of real proteins. For example, lattice polymers, similar to proteins, are sequence-specific heteropolymers that can encode a unique native fold, and though the coarse lattice significantly reduces the entropy of the unfolded state, its entropy is still sufficiently large that folding would be slow were it a fully random search process. Similarly, some lattice polymer sequences surmount this entropic barrier (the Levinthal Paradox) much more efficiently than others, and thus lattice polymers exhibit a wide range of folding rates. To date a major goal of computational folding studies has been the identification of the equilibrium properties that uniquely identify those rare lattice polymer sequences that fold rapidly, under the assumption that similar behavior will underlie the rapid folding of real proteins. The criteria predicted to distinguish between the rapidly and slowly folding lattice polymer sequences thus provide a clear opportunity for evaluating the correspondence between theory and experiment in protein folding. In the last decade alone, more than 700 papers on the folding kinetics of simple onand off-lattice protein models have appeared in the literature. Although it is impossible to accurately distill such a large, diverse, and often conflicting literature into a few brief conclusions, much of this literature can be classified by the criterion that is predicted to separate rapidly folding sequences from slowly folding sequences. Here we briefly describe the three major criteria that have been suggested as potential determinants of the relative folding rates of simplified computational models and review the experimental literature for clues as to whether similar criteria are responsible for the broad range of rates observed for the folding of two-state proteins. Figure 2 Rapid folding will occur when the energy of each unfolded or partially folded conformation decreases more or less monotonically [along some simple, but often difficult to define, order parameter(s)] as conformations become more native-like (5, 6). Roughness on this energy landscape can produce complex, stretched-exponential kinetics in lattice polymers (9) and may also account for part of the broad range of observed two-state folding rates. Smooth Energy Landscapes A large body of theoretical work suggests that the rapid folding of lattice polymers (5, 6) and simplified off-lattice polymers (7, 8) is associated with smooth, funnel-shaped energy landscapes (Figure 2). That is, rapid folding will occur when the energy of each unfolded or partially folded conformation decreases more or less monotonically as conformations become more native-like along some reaction coordinate(s). If this energetic guidance does not occur, or if the landscape is rough (contains local minima deeper than a few kBT that act as traps), folding becomes glassy, dominated by multiple kinetic traps, and slows dramatically. Differences in the roughness of the energy landscape can lead to orders of magnitude changes in the folding rates of simplified computational models (9). B How can we determine if variations in energy-landscape roughness contribute significantly to the relative folding rates of small proteins? If energetic roughness dominates the energy landscape, slow, nonsingle-exponential kinetics will be observed (5, 9, 10). That is, at temperatures at which kBT is small relative to the myriad kinetic barriers on a rough landscape (near the so-called glass transition temperature, T B g), the myriad of local kinetic barriers will begin to retard the folding process, switching the kinetics from single exponential (h = 1) to a slower, stretched exponential (h > 1): S(t) = Sn + A0exp(-(kft)), 1. where kf is the folding rate, A0 denotes the amplitude change upon folding, and Sn and S(t) respectively denote the signal of the native state and that observed at time t (5, 8–10). We can determine the extent to which roughness defines relative folding rates in the laboratory by employing this metric to measure the relative energy-landscape roughness of both rapidly and slowly folding proteins. Do differences in energy-landscape roughness account for a significant fraction of the 1,000,000-fold range of rates observed for two-state protein folding? Although indications of landscape roughness have been reported for the single-domain protein ubiquitin [(11), but see commentary in (12)] at low temperatures, energetic roughness does not generally appear to account for the vast range of folding rates observed under more physiological conditions. Once the complication of proline isomerization is taken into account, the large majority of simple, single domain proteins appear to fold with single exponential kinetics (1, 11, 12). Examples include the folding of the 62-residue protein L, which appears perfectly single exponential down to the experimental limit of 15°C (12), and the pI3k-SH3 domain, which also exhibits a smooth energy landscape despite being the second most slowly folding two-state protein reported to date (13) (Figure 3). It thus appears that although theory is correct in associating smooth energy landscapes with the rapid folding of naturally occurring proteins (5–8), differences in the roughness of the energy landscape do not play a significant role in defining the six order of magnitude range of observed two-state folding rates. Figure 3 The folding energy landscapes of even the most slowly folding single-domain proteins are exceptionally smooth. Shown here is the refolding of the pI3 k-SH3 domain, one of the most slowly folding of all two-state proteins (13), after folding is initiated by rapid dilution at 5°C. There is no statistically significant evidence of deviation from the fitted single-exponential kinetics (B. Gillespie and K.W. Plaxco, unpublished observations), even at this low temperature, where kBT is reduced and the effects of landscape roughness enhanced. It appears that energetic roughness plays no significant role in slowing the folding of even the most slowly folding two-state proteins (12). The Energy Gap Hypothesis Karplus and coworkers have reported that a “necessary and sufficient criterion to ensure [rapid] folding is that the ground state be a pronounced energy minimum (14)” relative to all other maximally compact states (Figure 4). More recently these and other researchers have identified a number of related measures of energetic gap between the ground state and other maximally compact states. These include the Z score, which is a measure of the statistical significance of the size of the gap between the energy of the native state and the mean energy of all other maximally compact states (15). This measure of the energy gap is reported to correlate significantly with the folding rates of a number of simple lattice polymers (16) and may account for the vast dispersion observed in the folding rates of real proteins. Unfortunately, we cannot measure the size of the energy gap experimentally, and thus, we cannot directly determine whether differences in the energy gap account for the wide range of observed protein folding rates. The difficulty is that, as demonstrated by both solution-phase and crystallographic structural studies, the vast majority of proteins populate only a single maximally compact state, the native state, and therefore, the energy gap between the ground state and all other maximally compact states is too large to measure. An indirect experimental test of the energy gap hypothesis may be provided, however, by the empirical observation that, for some lattice polymer models, the melting temperature (Tm, often denoted Tf in the theoretical literature) is correlated with the magnitude of the energy gap. This leads to the (indirect, empirically based) prediction that folding rates should also correlate with Tm (16). To the extent that this has been monitored experimentally, however, it appears that folding rates are generally uncorrelated with Tm. For example, although folding rates may be modestly correlated with Tm across a set of mutants of a single protein (A. Fersht, personal communication), a survey of the experimental literature suggests that Tm is effectively completely uncorrelated with folding rates across a set of nonhomologous proteins (Figure 5). Even for sets of closely related proteins, the correlation between Tm and folding rates is often nonexistent. A set of homeodomain sequences provides an extreme example: Despite Tm ranging from 54°C to 116°C, the folding rates of the three characterized homeodomains are effectively within error of one another (12, 17; B. Gillespie and K.W. Plaxco, unpublished data). Similarly, there is no correlation between Tm and folding rates across mesophile-, thermophile-, and hyperthermophile-derived cold shock proteins (18). Thus, although theory is correct in associating a large energy gap with the rapid folding of naturally occurring proteins, it appears that there is no evidence in favor of the hypothesis that the size of the energy gap is a significant determinant of relative protein folding rates.