Brownian dynamics simulations of protein folding: Access to milliseconds time scale and beyond

Protein folding occurs on a time scale ranging from milliseconds to minutes for a majority of proteins. Computer simulation of protein folding, from a random configuration to the native structure, is nontrivial owing to the large disparity between the simulation and folding time scales. As an effort to overcome this limitation, simple models with idealized protein subdomains, e.g., the diffusion–collision model of Karplus and Weaver, have gained some popularity. We present here new results for the folding of a four-helix bundle within the framework of the diffusion–collision model. Even with such simplifying assumptions, a direct application of standard Brownian dynamics methods would consume 10,000 processor-years on current supercomputers. We circumvent this difficulty by invoking a special Brownian dynamics simulation. The method features the calculation of the mean passage time of an event from the flux overpopulation method and the sampling of events that lead to productive collisions even if their probability is extremely small (because of large free-energy barriers that separate them from the higher probability events). Using these developments, we demonstrate that a coarse-grained model of the four-helix bundle can be simulated in several days on current supercomputers. Furthermore, such simulations yield folding times that are in the range of time scales observed in experiments.

Definition of amide protection factors for early kinetic intermediates in protein folding

Hydrogen–deuterium exchange experiments have been used previously to investigate the structures of well defined states of a given protein. These include the native state, the unfolded state, and any intermediates that can be stably populated at equilibrium. More recently, the hydrogen–deuterium exchange technique has been applied in kinetic labeling experiments to probe the structures of transiently formed intermediates on the kinetic folding pathway of a given protein. From these equilibrium and nonequilibrium studies, protection factors are usually obtained. These protection factors are defined as the ratio of the rate of exchange of a given backbone amide when it is in a fully solvent-exposed state (usually obtained from model peptides) to the rate of exchange of that amide in some state of the protein or in some intermediate on the folding pathway of the protein. This definition is straightforward for the case of equilibrium studies; however, it is less clear-cut for the case of transient kinetic intermediates. To clarify the concept for the case of burst-phase intermediates, we have introduced and mathematically defined two different types of protection factors: one is Pstruc, which is more related to the structure of the intermediate, and the other is Papp, which is more related to the stability of the intermediate. Kinetic hydrogen–deuterium exchange data from disulfide-intact ribonuclease A and from cytochrome c are discussed to explain the use and implications of these two definitions.

GroEL-GroES-mediated protein folding requires an intact central cavity

The chaperonin GroEL is an oligomeric double ring structure that, together with the cochaperonin GroES, assists protein folding. Biochemical analyses indicate that folding occurs in a cis ternary complex in which substrate is sequestered within the GroEL central cavity underneath GroES. Recently, however, studies of GroEL “minichaperones” containing only the apical substrate binding subdomain have questioned the functional importance of substrate encapsulation within GroEL-GroES complexes. Minichaperones were reported to assist folding despite the fact that they are monomeric and therefore cannot form a central cavity. Here we compare directly the folding activity of minichaperones with that of the full GroEL-GroES system. In agreement with earlier studies, minichaperones assist folding of some proteins. However, this effect is observed only under conditions where substantial spontaneous folding is also observed and is indistinguishable from that resulting from addition of the nonchaperone protein α-casein. By contrast, the full GroE system efficiently promotes folding of several substrates under conditions where essentially no spontaneous folding is observed. These data argue that the full GroEL folding activity requires the intact GroEL-GroES complex, and in light of previous studies, underscore the importance of substrate encapsulation for providing a folding environment distinct from the bulk solution.

Ultrafast signals in protein folding and the polypeptide contracted state

To test the significance of ultrafast protein folding signals (≪1 msec), we studied cytochrome c (Cyt c) and two Cyt c fragments with major C-terminal segments deleted. The fragments remain unfolded under all conditions and so could be used to define the unfolded baselines for protein fluorescence and circular dichroism (CD) as a function of denaturant concentration. When diluted from high to low denaturant in kinetic folding experiments, the fragments readjust to their new baseline values in a “burst phase” within the mixing dead time. The fragment burst phase reflects a contraction of the polypeptide from a more extended unfolded condition at high denaturant to a more contracted unfolded condition in the poorer, low denaturant solvent. Holo Cyt c exhibits fluorescence and CD burst phase signals that are essentially identical to the fragment signals over the whole range of final denaturant concentrations, evidently reflecting the same solvent-dependent, relatively nonspecific contraction and not the formation of a specific folding intermediate. The significance of fast folding signals in Cyt c and other proteins is discussed in relation to the hypothesis of an initial rate-limiting search-nucleation-collapse step in protein folding [Sosnick, T. R., Mayne, L. & Englander, S. W. (1996) Proteins Struct. Funct. Genet. 24, 413–426].

Protein folding kinetics exhibit an Arrhenius temperature dependence when corrected for the temperature dependence of protein stability

The anomalous temperature dependence of protein folding has received considerable attention. Here we show that the temperature dependence of the folding of protein L becomes extremely simple when the effects of temperature on protein stability are corrected for; the logarithm of the folding rate is a linear function of 1/T on constant stability contours in the temperature–denaturant plane. This convincingly demonstrates that the anomalous temperature dependence of folding derives from the temperature dependence of the interactions that stabilize proteins, rather than from the super Arrhenius temperature dependence predicted for the configurational diffusion constant on a rough energy landscape. However, because of the limited temperature range accessible to experiment, the results do not rule out models with higher order temperature dependences. The significance of the slope of the stability-corrected Arrhenius plots is discussed.

Detection of residue contacts in a protein folding intermediate

Protein folding can be described in terms of the development of specific contacts between residues as a highly disordered polypeptide chain converts into the native state. Here we describe an NMR based strategy designed to detect such contacts by observation of nuclear Overhauser effects (NOEs). Experiments with α-lactalbumin reveal the existence of extensive NOEs between aromatic and aliphatic protons in the archetypal molten globule formed by this protein at low pH. Analysis of their time development provides direct evidence for near-native compactness of this state. Through a rapid refolding procedure the NOE intensity can be transferred efficiently into the resolved and assigned spectrum of the native state. This demonstrates the viability of using this approach to map out time-averaged interactions between residues in a partially folded protein.

From residue matching patterns to protein folding topographies: General model and bovine pancreatic trypsin inhibitor

A coarse-grained model for protein-folding dynamics is introduced based on a discretized representation of torsional modes. The model, based on the Ramachandran map of the local torsional potential surface and the class (hydrophobic/polar/neutral) of each residue, recognizes patterns of both torsional conformations and hydrophobic-polar contacts, with tolerance for imperfect patterns. It incorporates empirical rates for formation of secondary and tertiary structure. The method yields a topological representation of the evolving local torsional configuration of the folding protein, modulo the basins of the Ramachandran map. The folding process is modeled as a sequence of transitions from one contact pattern to another, as the torsional patterns evolve. We test the model by applying it to the folding process of bovine pancreatic trypsin inhibitor, obtaining a kinetic description of the transitions between the contact patterns visited by the protein along the dominant folding pathway. The kinetics and detailed balance make it possible to invert the result to obtain a coarse topographic description of the potential energy surface along the dominant folding pathway, in effect to go backward or forward between a topological representation of the chain conformation and a topographical description of the potential energy surface governing the folding process. As a result, the strong structure-seeking character of bovine pancreatic trypsin inhibitor and the principal features of its folding pathway are reproduced in a reasonably quantitative way.

Conformational influences of glycosylation of a peptide: A possible model for the effect of glycosylation on the rate of protein folding

Improved strategies for synthesis make it possible to expand the range of glycopeptides available for detailed conformational studies. The glycopeptide 1 was synthesized using a new solid phase synthesis of carbohydrates and a convergent coupling to peptide followed by deprotection. Its conformational properties were subjected to NMR analysis and compared with a control peptide 2 prepared by conventional solid phase methods. Whereas peptide 2 fails to manifest any appreciable secondary structure, the glycopeptide 1 does show considerable conformational bias suggestive of an equilibrium between an ordered and a random state. The implications of this ordering effect for the larger issue of protein folding are considered.

Pressure-induced protein-folding/unfolding kinetics

We use an off-lattice minimalist model to describe the effects of pressure in slowing down the folding/unfolding kinetics of proteins when subjected to increasingly larger pressures. The potential energy function used to describe the interactions between beads in the model includes the effects of pressure on the pairwise interaction of hydrophobic groups in water. We show that pressure affects the participation of contacts in the transition state. More significantly, pressure exponentially decreases the chain reconfigurational diffusion coefficient. These results are consistent with experimental results on the kinetics of pressure-denaturation of staphylococcal nuclease.

From snapshot to movie: φ 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 φ 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.

Limited internal friction in the rate-limiting step of a two-state protein folding reaction

Small, single-domain proteins typically fold via a compact transition-state ensemble in a process well fitted by a simple, two-state model. To characterize the rate-limiting conformational changes that underlie two-state folding, we have investigated experimentally the effects of changing solvent viscosity on the refolding of the IgG binding domain of protein L. In conjunction with numerical simulations, our results indicate that the rate-limiting conformational changes of the folding of this domain are strongly coupled to solvent viscosity and lack any significant “internal friction” arising from intrachain collisions. When compared with the previously determined solvent viscosity dependencies of other, more restricted conformational changes, our results suggest that the rate-limiting folding transition involves conformational fluctuations that displace considerable amounts of solvent. Reconciling evidence that the folding transition state ensemble is comprised of highly collapsed species with these and similar, previously reported results should provide a significant constraint for theoretical models of the folding process.

Single-molecule protein folding: Diffusion fluorescence resonance energy transfer studies of the denaturation of chymotrypsin inhibitor 2

We report single-molecule folding studies of a small, single-domain protein, chymotrypsin inhibitor 2 (CI2). CI2 is an excellent model system for protein folding studies and has been extensively studied, both experimentally (at the ensemble level) and theoretically. Conformationally assisted ligation methodology was used to synthesize the proteins and site-specifically label them with donor and acceptor dyes. Folded and denatured subpopulations were observed by fluorescence resonance energy transfer (FRET) measurements on freely diffusing single protein molecules. Properties of these subpopulations were directly monitored as a function of guanidinium chloride concentration. It is shown that new information about different aspects of the protein folding reaction can be extracted from such subpopulation properties. Shifts in the mean transfer efficiencies are discussed, FRET efficiency distributions are translated into potentials, and denaturation curves are directly plotted from the areas of the FRET peaks. Changes in stability caused by mutation also are measured by comparing pseudo wild-type CI2 with a destabilized mutant (K17G). Current limitations and future possibilities and prospects for single-pair FRET protein folding investigations are discussed.

Coupling between protein folding and allostery in the GroE chaperonin system

GroEL is an allosteric protein that facilitates protein folding in an ATP-dependent manner. Herein, the relationship between cooperative ATP binding by GroEL and the kinetics of GroE-assisted folding of two substrates with different GroES dependence, mouse dihydrofolate reductase (mDHFR) and mitochondrial malate dehydrogenase, is examined by using cooperativity mutants of GroEL. Strong intra-ring positive cooperativity in ATP binding by GroEL decreases the rate of GroEL-assisted mDHFR folding owing to a slow rate of the ATP-induced transition from the protein-acceptor state to the protein-release state. Inter-ring negative cooperativity in ATP binding by GroEL is found to affect the kinetic partitioning of mDHFR, but not of mitochondrial malate dehydrogenase, between folding in solution and folding in the cavity underneath GroES. Our results show that protein folding by this “two-stroke motor” is coupled to cooperative ATP binding.

Transition-state structure as a unifying basis in protein-folding mechanisms: Contact order, chain topology, stability, and the extended nucleus mechanism

I attempt to reconcile apparently conflicting factors and mechanisms that have been proposed to determine the rate constant for two-state folding of small proteins, on the basis of general features of the structures of transition states. Φ-Value analysis implies a transition state for folding that resembles an expanded and distorted native structure, which is built around an extended nucleus. The nucleus is composed predominantly of elements of partly or well-formed native secondary structure that are stabilized by local and long-range tertiary interactions. These long-range interactions give rise to connecting loops, frequently containing the native loops that are poorly structured. I derive an equation that relates differences in the contact order of a protein to changes in the length of linking loops, which, in turn, is directly related to the unfavorable free energy of the loops in the transition state. Kinetic data on loop extension mutants of CI2 and α-spectrin SH3 domain fit the equation qualitatively. The rate of folding depends primarily on the interactions that directly stabilize the nucleus, especially those in native-like secondary structure and those resulting from the entropy loss from the connecting loops, which vary with contact order. This partitioning of energy accounts for the success of some algorithms that predict folding rates, because they use these principles either explicitly or implicitly. The extended nucleus model thus unifies the observations of rate depending on both stability and topology.

Denaturant-induced movement of the transition state of protein folding revealed by high-pressure stopped-flow measurements

The small all-β protein tendamistat folds and unfolds with two-state kinetics. We determined the volume changes associated with the folding process by performing kinetic and equilibrium measurements at variable pressure between 0.1 and 100 MPa (1 to 1,000 bar). GdmCl-induced equilibrium unfolding transitions reveal that the volume of the native state is increased by 41.4 ± 2.0 cm3/mol relative to the unfolded state. This value is virtually independent of denaturant concentration. The use of a high-pressure stopped-flow instrument enabled us to measure the activation volumes for the refolding (ΔVf0‡) and unfolding reaction (ΔVu0‡) over a broad range of GdmCl concentrations. The volume of the transition state is 60% native-like (ΔVf0‡ = 25.0 ± 1.2 cm3/mol) in the absence of denaturant, indicating partial solvent accessibility of the core residues. The volume of the transition state increases linearly with denaturant concentration and exceeds the volume of the native state above 6 M GdmCl. This result argues for a largely desolvated transition state with packing deficiencies at high denaturant concentrations and shows that the structure of the transition state depends strongly on the experimental conditions.