Phi values in protein-folding kinetics have energetic and structural components

Phi values are experimental measures of how the kinetics of protein folding is changed by single-site mutations. Phi values measure energetic quantities, but they are often interpreted in terms of the structures of the transition-state ensemble. Here, we describe a simple analytical model of the folding kinetics in terms of the formation of protein substructures. The model shows that Phi values have both structural and energetic components. It also provides a natural and general interpretation of "nonclassical" Phi values (i.e., < 0 or > 1). The model reproduces the Phi values for 20 single-residue mutations in the alpha-helix of the protein CI2, including several nonclassical Phi values, in good agreement with experiments.     

Phi-value analysis and the nature of protein-folding transition states

Phi values are used to map structures of protein-folding transition states from changes in free energies of denaturation (DeltaDeltaG(D-N)) and activation on mutation. A recent reappraisal proposed that Phi values for DeltaDeltaG(D-N) < 1.7 kcal/mol are artifactual. On discarding such derived Phi values from published studies, the authors concluded that there are no high Phi values in diffuse transition states, which are consequently uniformly diffuse with no evidence for nucleation. However, values of DeltaDeltaG(D-N) > 1.7 kcal/mol are often found for large side chains that make dispersed tertiary interactions, especially in hydrophobic cores that are in the process of being formed in the transition state. Conversely, specific local interactions that probe secondary structure tend to have DeltaDeltaG(D-N) approximately 0.5-2 kcal/mol. Discarding Phi values from lower-energy changes discards the crucial information about local interactions and makes transition states appear uniformly diffuse by overemphasizing the dispersed tertiary interactions. The evidence for the 1.7 kcal/mol cutoff was based on mutations that had been deliberately designed to be unsuitable for Phi-value analysis because they are structurally disruptive. We confirm that reliable Phi values can be derived from the recommended mutations in suitable proteins with 0.6 < DeltaDeltaG(D-N) < 1.7 kcal/mol, and there are many reliable high Phi values. Transition states vary from being rather diffuse to being well formed with islands of near-complete secondary structure. We also confirm that the structures of transition-state ensembles can be perturbed by mutations with DeltaDeltaG(D-N) > 2 kcal/mol and that protein-folding transition states do move on the energy surface on mutation.     

Simple model of protein folding kinetics.

A simple model of the kinetics of protein folding is presented. The reaction coordinate is the "correctness" of a configuration compared with the native state. The model has a gap in the energy spectrum, a large configurational entropy, a free energy barrier between folded and partially folded states, and a good thermodynamic folding transition. Folding kinetics is described by a master equation. The folding time is estimated by means of a local thermodynamic equilibrium assumption and then is calculated both numerically and analytically by solving the master equation. The folding time has a maximum near the folding transition temperature and can have a minimum at a lower temperature.     

Landscape approaches for determining the ensemble of folding transition states: success and failure hinge on the degree of frustration

We present a method for determining structural properties of the ensemble of folding transition states from protein simulations. This method relies on thermodynamic quantities (free energies as a function of global reaction coordinates, such as the percentage of native contacts) and not on "kinetic" measurements (rates, transmission coefficients, complete trajectories); consequently, it requires fewer computational resources compared with other approaches, making it more suited to large and complex models. We explain the theoretical framework that underlies this method and use it to clarify the connection between the experimentally determined Phi value, a quantity determined by the ratio of rate and stability changes due to point mutations, and the average structure of the transition state ensemble. To determine the accuracy of this thermodynamic approach, we apply it to minimalist protein models and compare these results with the ones obtained by using the standard experimental procedure for determining Phi values. We show that the accuracy of both methods depends sensitively on the amount of frustration. In particular, the results are similar when applied to models with minimal amounts of frustration, characteristic of rapid-folding, single-domain globular proteins.     

Phi-value analysis of a three-state protein folding pathway by NMR relaxation dispersion spectroscopy

Experimental studies of protein folding frequently are consistent with two-state folding kinetics. However, recent NMR relaxation dispersion studies of several fast-folding mutants of the Fyn Src homology 3 (SH3) domain have established that folding proceeds through a low-populated on-pathway intermediate, which could not be detected with stopped-flow experiments. The dispersion experiments provide precise kinetic and thermodynamic parameters that describe the folding pathway, along with a detailed site-specific structural characterization of both the intermediate and unfolded states from the NMR chemical shifts that are extracted. Here we describe NMR relaxation dispersion Phi-value analysis of the A39V/N53P/V55L Fyn SH3 domain, where the effects of suitable point mutations on the energy landscape are quantified, providing additional insight into the structure of the folding intermediate along with per-residue structural information of both rate-limiting transition states that was not available from previous studies. In addition to the advantage of delineating the full three-state folding pathway, the use of NMR relaxation dispersion as opposed to stopped-flow kinetics to quantify Phi values facilitates their interpretation because the obtained chemical shifts monitor any potential structural changes along the folding pathway that might be introduced by mutation, a significant concern in their analysis. Phi-Value analysis of several point mutations of A39V/N53P/V55L Fyn SH3 establishes that the beta(3)-beta(4)-hairpin already is formed in the first transition state, whereas strand beta(1), which forms nonnative interactions in the intermediate, does not fully adopt its native conformation until after the final transition state. The results further support the notion that on-pathway intermediates can be stabilized by nonnative contacts.     

A two-dimensional view of the folding energy landscape of cytochrome c

Time-correlated single photon counting (TCSPC) was combined with fluorescence correlation spectroscopy (FCS) to study the transition between acid-denatured states and the native structure of cytochrome c (Cyt c) from Saccharomyces cerevisiae. The use of these techniques in concert proved to be more powerful than either alone, yielding a two-dimensional picture of the folding energy landscape of Cyt c. TCSPC measured the distribution of distances between the heme of the protein and a covalently attached dye molecule at residue C102 (one folding reaction coordinate), whereas FCS measured the hydrodynamic radius (a second folding reaction coordinate) of the protein over a range of pH values. These two independent measurements provide complimentary information regarding protein conformation. We see evidence for a well defined folding intermediate in the acid renaturation folding pathway of this protein reflected in the distribution of lifetimes needed to fit the TCSPC data. Moreover, FCS studies revealed this intermediate state to be in dynamic equilibrium with unfolded structures, with conformational fluctuations into and out of this intermediate state occurring on an approximately 30-micros time scale.     

Folding a protein in a computer: an atomic description of the folding/unfolding of protein A

We study the folding mechanism of a three-helix bundle protein at atomic resolution, including effects of explicit water. Using replica exchange molecular dynamics we perform enough sampling over a wide range of temperatures to obtain the free energy, entropy, and enthalpy surfaces as a function of structural reaction coordinates. Simulations were started from different configurations covering the folded and unfolded states. Because many transitions between all minima at the free energy surface are observed, a quantitative determination of the free energy barriers and the ensemble of configurations associated with them is now possible. The kinetic bottlenecks for folding can be determined from the thermal ensembles of structures on the free energy barriers, provided the kinetically determined transition-state ensembles are similar to those determined from free energy barriers. A mechanism incorporating the interplay among backbone ordering, sidechain packing, and desolvation arises from these calculations. Large Phi values arise not only from native contacts, which mostly form at the transition state, but also from contacts already present in the unfolded state that are partially destroyed at the transition.     

Finding transition states for crystalline solid-solid phase transformations

We present a method to identify transition states and minimum energy paths for martensitic solid-solid phase transformations, thereby allowing quantification of the activation energies of such transformations. Our approach is a generalization of a previous method for identifying transition states for chemical reactions, namely the climbing image-nudged elastic band algorithm, where here the global deformation of the crystalline lattice (volume and shape fluctuations) becomes the reaction coordinate instead of atomic motion. We also introduce an analogue to the Born-Oppenheimer approximation that allows a decoupling of nuclear motion and lattice deformation, where the nuclear positions along the path are determined variationally according to current deformation state. We then apply this technique to characterize the energetics of elemental lithium phase transformations as a function of applied pressure, where we see a validation of the Born-Oppenheimer-like approximation, small energy barriers (expected for martensitic transformations), and a pronounced pressure dependence of various properties characterizing the phase transitions.     

Application of physical organic chemistry to engineered mutants of proteins: Hammond postulate behavior in the transition state of protein folding.

Transition states in protein folding may be analyzed by linear free-energy relationships (LFERs) analogous to the Brønsted equation for changes in reactivity with changes in structure. There is an additional source of LFERs in protein folding: the perturbation of the equilibrium and rate constants by denaturants. These LFERs give a measure of the position of the transition state along the reaction coordinate. The transition state for folding/unfolding of barnase has been analyzed by both types of LFERs: changing the structure by protein engineering and perturbation by denaturants. The combination has allowed the direct monitoring of Hammond postulate behavior of the transition state on the reaction pathway. Movement of the transition state has been found and analyzed to give further details of the order of events in protein folding.     

Reaction coordinates and rates from transition paths

The molecular mechanism of a reaction in solution is reflected in its transition-state ensemble and transition paths. We use a Bayesian formula relating the equilibrium and transition-path ensembles to identify transition states, rank reaction coordinates, and estimate rate coefficients. We also introduce a variational procedure to optimize reaction coordinates. The theory is illustrated with applications to protein folding and the dipole reorientation of an ordered water chain inside a carbon nanotube. To describe the folding of a simple model of a three-helix bundle protein, we variationally optimize the weights of a projection onto the matrix of native and nonnative amino acid contacts. The resulting one-dimensional reaction coordinate captures the folding transition state, with formation and packing of helix 2 and 3 constituting the bottleneck for folding.     

Sampling the multiple folding mechanisms of Trp-cage in explicit solvent

We investigate the kinetic pathways of folding and unfolding of the designed miniprotein Trp- cage in explicit solvent. Straightforward molecular dynamics and replica exchange methods both have severe convergence problems, whereas transition path sampling allows us to sample unbiased dynamical pathways between folded and unfolded states and leads to deeper understanding of the mechanisms of (un)folding. In contrast to previous predictions employing an implicit solvent, we find that Trp-cage folds primarily (80% of the paths) via a pathway forming the tertiary contacts and the salt bridge, before helix formation. The remaining 20% of the paths occur in the opposite order, by first forming the helix. The transition states of the rate-limiting steps are solvated native-like structures. Water expulsion is found to be the last step upon folding for each route. Committor analysis suggests that the dynamics of the solvent is not part of the reaction coordinate. Nevertheless, during the transition, specific water molecules are strongly bound and can play a structural role in the folding.     

A simple method for probing the mechanical unfolding pathway of proteins in detail

Atomic force microscopy is an exciting new single-molecule technique to add to the toolbox of protein (un)folding methods. However, detailed analysis of the unfolding of proteins on application of force has, to date, relied on protein molecular dynamics simulations or a qualitative interpretation of mutant data. Here we describe how protein engineering Phi value analysis can be adapted to characterize the transition states for mechanical unfolding of proteins. Single-molecule studies also have an advantage over bulk experiments, in that partial Phi values arising from partial structure in the transition state can be clearly distinguished from those averaged over alternate pathways. We show that unfolding rate constants derived in the standard way by using Monte Carlo simulations are not reliable because of the errors involved. However, it is possible to circumvent these problems, providing the unfolding mechanism is not changed by mutation, either by a modification of the Monte Carlo procedure or by comparing mutant and wild-type data directly. The applicability of the method is tested on simulated data sets and experimental data for mutants of titin I27.     

Phi-value analysis by molecular dynamics simulations of reversible folding

In Phi-value analysis, the effects of mutations on the folding kinetics are compared with the corresponding effects on thermodynamic stability to investigate the structure of the protein-folding transition state (TS). Here, molecular dynamics (MD) simulations (totaling 0.65 ms) have been performed for a large set of single-point mutants of a 20-residue three-stranded antiparallel beta-sheet peptide. Between 57 and 120 folding events were sampled at near equilibrium for each mutant, allowing for accurate estimates of folding/unfolding rates and stability changes. The Phi values calculated from folding and unfolding rates extracted from the MD trajectories are reliable if the stability loss upon mutation is larger than approximately 0.6 kcal/mol, which is observed for 8 of the 32 single-point mutants. The same heterogeneity of the TS of the wild type was found in the mutated peptides, showing two possible pathways for folding. Single-point mutations can induce significant TS shifts not always detected by Phi-value analysis. Specific nonnative interactions at the TS were observed in most of the peptides studied here. The interpretation of Phi values based on the ratio of atomic contacts at the TS over the native state, which has been used in the past in MD and Monte Carlo simulations, is in agreement with the TS structures of wild-type peptide. However, Phi values tend to overestimate the nativeness of the TS ensemble, when interpreted neglecting the nonnative interactions.     

Reconstructing folding energy landscapes from splitting probability analysis of single-molecule trajectories

Structural self-assembly in biopolymers, such as proteins and nucleic acids, involves a diffusive search for the minimum-energy state in a conformational free-energy landscape. The likelihood of folding proceeding to completion, as a function of the reaction coordinate used to monitor the transition, can be described by the splitting probability, p_fold(x). P_fold encodes information about the underlying energy landscape, and it is often used to judge the quality of the reaction coordinate. Here, we show how p_fold can be used to reconstruct energy landscapes from single-molecule folding trajectories, using force spectroscopy measurements of single DNA hairpins. Calculating p_fold(x) directly from trajectories of the molecular extension measured for hairpins fluctuating in equilibrium between folded and unfolded states, we inverted the result expected from diffusion over a 1D energy landscape to obtain the implied landscape profile. The results agreed well with the landscapes reconstructed by established methods, but, remarkably, without the need to deconvolve instrumental effects on the landscape, such as tether compliance. The same approach was also applied to hairpins with multistate folding pathways. The relative insensitivity of the method to the instrumental compliance was confirmed by simulations of folding measured with different tether stiffnesses. This work confirms that the molecular extension is a good reaction coordinate for these measurements, and validates a powerful yet simple method for reconstructing landscapes from single-molecule trajectories.Structure formation by biological polymers like proteins and nucleic acids, an essential process linked to biological function, is typically described by energy landscape theory, in which folding is viewed as a diffusive search through the conformational space of the molecule for the minimum-energy structure (1, 2). This search takes place on an energy landscape, with the surface describing the energy of the molecule as a function of all possible conformations. Because of the many conformational degrees of freedom in even a small biopolymer, folding landscapes are inherently multidimensional, forming a hypersurface. Experimental measurements, however, typically follow some observable that is used to monitor the progress of the reaction. As a result, the full energy hypersurface is projected onto a 1D profile along the chosen reaction coordinate. There is great interest in measuring such energy landscape profiles directly, because they provide a fundamental basis for understanding folding phenomena.Recent advances in single-molecule approaches have provided powerful tools for measuring landscapes. Most notably, 1D landscape profiles can be reconstructed in several ways from force spectroscopy measurements, wherein tension is applied to a molecule and the resulting changes in the molecular extension, the reaction coordinate, are measured (3). Landscapes may be reconstructed from equilibrium measurements, based on fluctuations in the extension (4–6); from force jumps, based on nonequilibrium distributions of the extension (7, 8); and from force ramps, using fluctuation theorems (9–11). However, these methods are influenced strongly by the characteristics of the experimental apparatus, such as the stiffness and/or size of the force probe and the properties of any molecular handles used to attach to the molecule of interest, whose effects must be removed to recover the intrinsic landscape (4, 12, 13).Here, we describe a new approach to landscape reconstruction that makes use of the splitting probability, p_fold(x), which measures the likelihood that the molecule goes to the folded state as a function of its position along the reaction coordinate, x (14). We demonstrate this method using folding trajectories of single DNA hairpins measured under tension in an optical trap (15), where the reaction coordinate is the end-to-end extension of the molecule (3). We find that the landscape recovered by this method agrees well with the results using other approaches but does not require deconvolution of instrumental effects.     

Gating of acetylcholine receptor channels: brownian motion across a broad transition state

Acetylcholine receptor channels (AChRs) are proteins that switch between stable "closed" and "open" conformations. In patch clamp recordings, diliganded AChR gating appears to be a simple, two-state reaction. However, mutagenesis studies indicate that during gating dozens of residues across the protein move asynchronously and are organized into rigid body gating domains ("blocks"). Moreover, there is an upper limit to the apparent channel opening rate constant. These observations suggest that the gating reaction has a broad, corrugated transition state region, with the maximum opening rate reflecting, in part, the mean first-passage time across this ensemble. Simulations reveal that a flat, isotropic energy profile for the transition state can account for many of the essential features of AChR gating. With this mechanism, concerted, local structural transitions that occur on the broad transition state ensemble give rise to fractional measures of reaction progress (Phi values) determined by rate-equilibrium free energy relationship analysis. The results suggest that the coarse-grained AChR gating conformational change propagates through the protein with dynamics that are governed by the Brownian motion of individual gating blocks.     

From snapshot to movie: phi analysis of protein folding transition states taken one step further

Kinetic anomalies in protein folding can result from changes of the kinetic ground states (D, I, and N), changes of the protein folding transition state, or both. The 102-residue protein U1A has a symmetrically curved chevron plot which seems to result mainly from changes of the transition state. At low concentrations of denaturant the transition state occurs early in the folding reaction, whereas at high denaturant concentration it moves close to the native structure. In this study we use this movement to follow continuously the formation and growth of U1A's folding nucleus by phi analysis. Although U1A's transition state structure is generally delocalized and displays a typical nucleation-condensation pattern, we can still resolve a sequence of folding events. However, these events are sufficiently coupled to start almost simultaneously throughout the transition state structure.     

Comparison of successive transition states for folding reveals alternative early folding pathways of two homologous proteins

The energy landscape theory provides a general framework for describing protein folding reactions. Because a large number of studies, however, have focused on two-state proteins with single well-defined folding pathways and without detectable intermediates, the extent to which free energy landscapes are shaped up by the native topology at the early stages of the folding process has not been fully characterized experimentally. To this end, we have investigated the folding mechanisms of two homologous three-state proteins, PTP-BL PDZ2 and PSD-95 PDZ3, and compared the early and late transition states on their folding pathways. Through a combination of Φ value analysis and molecular dynamics simulations we obtained atomic-level structures of the transition states of these homologous three-state proteins and found that the late transition states are much more structurally similar than the early ones. Our findings thus reveal that, while the native state topology defines essentially in a unique way the late stages of folding, it leaves significant freedom to the early events, a result that reflects the funneling of the free energy landscape toward the native state.     

Single-molecule transition-state analysis of RNA folding

How RNA molecules fold into functional structures is a problem of great significance given the expanding list of essential cellular RNA enzymes and the increasing number of applications of RNA in biotechnology and medicine. A critical step toward solving the RNA folding problem is the characterization of the associated transition states. This is a challenging task in part because the rugged energy landscape of RNA often leads to the coexistence of multiple distinct structural transitions. Here, we exploit single-molecule fluorescence spectroscopy to follow in real time the equilibrium transitions between conformational states of a model RNA enzyme, the hairpin ribozyme. We clearly distinguish structural transitions between effectively noninterchanging sets of unfolded and folded states and characterize key factors defining the transition state of an elementary folding reaction where the hairpin ribozyme's two helical domains dock to make several tertiary contacts. Our single-molecule experiments in conjunction with site-specific mutations and metal ion titrations show that the two RNA domains are in a contact or close-to-contact configuration in the transition state even though the native tertiary contacts are at most partially formed. Such a compact transition state without well formed tertiary contacts may be a general property of elementary RNA folding reactions.