Molecular basis of cooperativity in protein folding IV. Core: A general cooperative folding model

The cooperative nature of the protein folding process is independent of the characteristic fold and the specific secondary structure attributes of a globular protein. A general folding/unfolding model should, therefore, be based upon structural features that transcend the peculiarities of α-helices, β-sheets, and other structural motifs found in proteins. The studies presented in this paper suggest that a single structural characteristic common to all globular proteins is essential for cooperative folding. The formation of a partly folded state from the native state results in the exposure to solvent of two distinct regions: (1) the portions of the protein that are unfolded; and (2) the “complementary surfaces,” located in the regions of the protein that remain folded. The cooperative character of the folding/unfolding transition is determined largely by the energetics of exposing complementary surface regions to the solvent. By definition, complementary regions are present only in partly folded states; they are absent from the native and unfolded states. An unfavorable free energy lowers the probability of partly folded states and increases the cooperativity of the transition. In this paper we present a mathematical formulation of this behavior and develop a general cooperative folding/unfolding model, termed the “complementary region” (CORE) model. This model successfully reproduces the main properties of folding/unfolding transitions without limiting the number of partly folded states accessible to the protein, thereby permitting a systematic examination of the structural and solvent conditions under which intermediates become populated. It is shown that the CORE model predicts two-state folding/unfolding behavior, even though the two-state character is not assumed in the model. © 1993 Wiley-Liss, Inc.     

Evidence for sequential barriers and obligatory intermediates in apparent two-state protein folding

Many small proteins fold fast and without detectable intermediates. This is frequently taken as evidence against the importance of partially folded states, which often transiently accumulate during folding of larger proteins. To get insight into the properties of free energy barriers in protein folding we analyzed experimental data from 23 proteins that were reported to show non-linear activation free-energy relationships. These non-linearities are generally interpreted in terms of broad transition barrier regions with a large number of energetically similar states. Our results argue against the presence of a single broad barrier region. They rather indicate that the non-linearities are caused by sequential folding pathways with consecutive distinct barriers and a few obligatory high-energy intermediates. In contrast to a broad barrier model the sequential model gives a consistent picture of the folding barriers for different variants of the same protein and when folding of a single protein is analyzed under different solvent conditions. The sequential model is also able to explain changes from linear to non-linear free energy relationships and from apparent two-state folding to folding through populated intermediates upon single point mutations or changes in the experimental conditions. These results suggest that the apparent discrepancy between two-state and multi-state folding originates in the relative stability of the intermediates, which argues for the importance of partially folded states in protein folding.     

Folding-unfolding of FN-III domains in tenascin: an elastically coupled two-state system

In a single-molecule atomic force microscopy (AFM) experiment, the tenascin molecule is stretched by an external force causing an elongation which is due to the unfolding of the FN-III modules. The features of the force-extension curves depend on the pulling speed and show a saw-tooth pattern (lower speeds) or a smooth pattern (higher speeds). In any case, the unfolded domains are elastically coupled to the unfolded modules, acting as transmitters of the external force. In this communication, the folding-unfolding process of the FN-III domains in tenascin is studied using reaction rate theory and a simple two-state model. The main hypothesis of the study is that, at microscopic level, the force needed to unfold a domain and the unfolding rate (unfolding velocity) can mimic the macroscopic process of measurement by AFM. As the external force is applied, the probability of unfolding increases as dictated by the reaction rate theory. Within this context, a relationship between the unfolding force and the unfolding velocity is obtained. The latter relation will describe microscopically the process in a phenomenological fashion. Moreover, while relating the results of this study with other experimental (AFM measurements) and theoretical (Monte Carlo simulations) data, we found that the graph of unfolding force-unfolding velocity is similar to that of external force-pulling velocity. The refolding process can also be studied within this model and the results show similar trends. The latter suggests a generic and universal behavior of such kind of molecular domains at least in the light of the proposed model.     

Protein folding cooperativity: basic insights from minimalist models

Basic concepts about two-state, cooperative protein folding and its relation to first-order phase transitions are reviewed. Minimalist models capable of reproducing the required free energy barrier between folded and unfolded macroscopic states are described. A significantly more restrictive "calorimetric" criterion is also discussed, which is based on direct comparison between model and experimental heat capacities with additional assumptions about conformational enthalpy variation in the unfolded state.     

Probing the energy landscape of the membrane protein bacteriorhodopsin

The folding and stability of transmembrane proteins is a fundamental and unsolved biological problem. Here, single bacteriorhodopsin molecules were mechanically unfolded from native purple membranes using atomic force microscopy and force spectroscopy. The energy landscape of individual transmembrane alpha helices and polypeptide loops was mapped by monitoring the pulling speed dependence of the unfolding forces and applying Monte Carlo simulations. Single helices formed independently stable units stabilized by a single potential barrier. Mechanical unfolding of the helices was triggered by 3.9-7.7 A extension, while natural unfolding rates were of the order of 10(-3) s(-1). Besides acting as individually stable units, helices associated pairwise, establishing a collective potential barrier. The unfolding pathways of individual proteins reflect distinct pulling speed-dependent unfolding routes in their energy landscapes. These observations support the two-stage model of membrane protein folding in which alpha helices insert into the membrane as stable units and then assemble into the functional protein.     

The effect of temperature on mechanical resistance of the native and intermediate states of I27

We investigated the effect of temperature on the mechanical unfolding of I27 from human cardiac titin, employing a custom-built temperature control device for single-molecule atomic force microscopy measurement. A sawtooth pattern was observed in the force curves where each force peak reports on the unfolding of an I27 domain. In early unfolding events, we observed a “hump-like” deviation due to the detachment of β-strand A from each I27 domain. The force at which the humps appear was ∼130 pN and showed no temperature dependence, at least in the temperature range of 2°C-30°C. The hump structure was successfully analyzed with a two-state worm-like chain model, and the Gibbs free energy difference of the detachment reaction was estimated to be 11.6 ± 0.58 kcal/mol and found to be temperature independent. By contrast, upon lowering the temperature, the mean unfolding force from the partly unfolded intermediate state was found to markedly increase and the unfolding force distribution to broaden significantly, suggesting that the distance (x_u) between the folded and transition states in the energy landscape along the pulling direction is decreased. These results suggest that the local structure of β-strand A are stabilized by topologically simple local hydrogen-bond network and that the temperature does not affect the detachment reaction thermodynamically and kinetically, whereas the interaction between the β-strands A′ and G, which is a critical region for its mechanical stability, is strongly dependent on the temperature.     

Distance-dependent atomic knowledge-based force in protein fold recognition

We have recently introduced a novel model for discriminating the correctly folded proteins from well-designed decoy structures using mechanical interatomic forces. In the model, we considered a protein as a collection of springs and the force imposed to each atom was calculated by using the relation between the potential energy and the force. A mean force potential function is obtained from statistical contact preferences within the known protein structures. In this article, the interatomic forces are calculated by numerical derivation of the potential function. For assessing the knowledge-based force function we consider an optimal structure and define a score function on the 3D structure of a protein. We compare the force imposed to each atom of a protein with the corresponding atom in the optimum structure. Afterwards we assign larger scores to those atoms with the lower forces. The total score is the sum of partial scores of atoms. The optimal structure is assumed to be the one with the highest score in the dataset. Finally, several decoy sets are applied in order to evaluate the performance of our model.     

Apparent two-state tendamistat folding is a sequential process along a defined route

The small all-beta-sheet protein tendamistat folds and unfolds rapidly in apparent two-state reactions. Kinetic measurements of two tendamistat variants under various solvent conditions reveal, however, that folding occurs in at least two sequential steps through a metastable obligatory intermediate. Depending on the solvent conditions either step can become rate limiting. The activation parameters indicate that the first step represents an enthalpic barrier whereas the second step is an entropic barrier at 25 degrees C. Our results suggest that initial non-specific collapse precedes formation of native secondary and tertiary structure in tendamistat folding. This points at a distinct route in tendamistat folding and indicates that partially folded metastable intermediates might play an important role in the mechanism of apparent two-state folding.     

The Folding Trajectory of RNase H Is Dominated by Its Topology and Not Local Stability: A Protein Engineering Study of Variants that Fold via Two-State and Three-State Mechanisms

Proteins can sample a variety of partially folded conformations during the transition between the unfolded and native states. Some proteins never significantly populate these high-energy states and fold by an apparently two-state process. However, many proteins populate detectable, partially folded forms during the folding process. The role of such intermediates is a matter of considerable debate. A single amino acid change can convert Escherichia coli ribonuclease H from a three-state folder that populates a kinetic intermediate to one that folds in an apparent two-state fashion. We have compared the folding trajectories of the three-state RNase H and the two-state RNase H, proteins with the same native-state topology but altered regional stability, using a protein engineering approach. Our data suggest that both versions of RNase H fold through a similar trajectory with similar high-energy conformations. Mutations in the core and the periphery of the protein affect similar aspects of folding for both variants, suggesting a common trajectory with folding of the core region followed by the folding of the periphery. Our results suggest that formation of specific partially folded conformations may be a general feature of protein folding that can promote, rather than hinder, efficient folding.     

Hopping around an entropic barrier created by force

We use Langevin dynamics to investigate the role played by the recently discovered force-induced entropic energy barrier on the two-state hopping phenomena that has been observed in single RNA, DNA and protein molecules placed under a stretching force. Simple considerations about the free energy of a molecule readily show that the application of force introduces an entropic barrier separating the collapsed state of the molecule, from a force-driven extended conformation. A notable characteristic of the force induced barrier is its long distances to transition state, up to tens of nanometers, which renders the kinetics of crossing this barrier highly sensitive to an applied force. Langevin dynamics across such force induced barriers readily demonstrates the hopping behavior observed for a variety of single molecules placed under force. Such hopping is frequently interpreted as a manifestation of two-state folding/unfolding reactions observed in bulk experiments. However, given that such barriers do not exist at zero force these reactions do not take place at all in bulk.     

Minimizing pulling geometry errors in atomic force microscope single molecule force spectroscopy

In atomic force microscopy-based single molecule force spectroscopy (AFM-SMFS), it is assumed that the pulling angle is negligible and that the force applied to the molecule is equivalent to the force measured by the instrument. Recent studies, however, have indicated that the pulling geometry errors can drastically alter the measured force-extension relationship of molecules. Here we describe a software-based alignment method that repositions the cantilever such that it is located directly above the molecule's substrate attachment site. By aligning the applied force with the measurement axis, the molecule is no longer undergoing combined loading, and the full force can be measured by the cantilever. Simulations and experimental results verify the ability of the alignment program to minimize pulling geometry errors in AFM-SMFS studies.     

分子马达定向运动的两态模型

采用非对称周期势来描述马达蛋白与具有周期性和极性的微丝轨道之间的相互作用,计算了马达蛋白两态模型的几率流和有效势之间的关系。计算结果表明：马达蛋白的定向运动不仅与有效势的整体倾斜密切相关,还与有效势的势垒高度有关。有效势倾斜等效于一个平均力的作用,而这一平均力的存在体现了两态跃迁细致平衡的破坏。同时将不同ATP浓度下力与速度的关系曲线与实验作了比较,这些曲线与实验定性吻合。     

A steered molecular dynamics method with adaptive direction adjustments

In this paper, a new steered molecular dynamics (SMD) method with adjusting pulling direction is proposed to search an optimum trajectory of ligand dissociation. A multiobjective model and a searching technique based on information entropy with multi-population are developed to optimize the pulling direction. The improved method has been used to dissociate the substrate-bound complex structure of cytochrome P450 3A4-metyrapone. A more favorable dissociation pathway can be gained. The results show that the new pathway obtained by the proposed method has less dissociation time, smaller rupture force and lower energy barrier than that by the conventional SMD.     

Heterogeneous folding of the trpzip hairpin: full atom simulation and experiment

The beta-hairpin trpzip2 can be tuned continuously from a two-state folder to folding on a rough energy landscape without a dominant refolding barrier. At high denaturant concentration, this extremely stable peptide exhibits a single apparent "two-state" transition temperature when monitored by different spectroscopic techniques. However, under optimal folding conditions the hairpin undergoes an unusual folding process with three clusters of melting transitions ranging from 15 degrees C to 160 degrees C, as monitored by 12 different experimental and computational observables. We explain this behavior in terms of a rough free energy landscape of the unfolded peptide caused by multiple tryptophan interactions and alternative backbone conformations. The landscape is mapped out by potentials of mean force derived from replica-exchange molecular dynamics simulations. Implications for deducing cooperativity from denaturant titrations, for the origin of folding cooperativity, and for the folding of thermophilic proteins are pointed out. trpzip is an excellent small tunable model system for the glass-like folding transitions predicted by landscape theory.     

Collapse Dynamics of Single Proteins Extended by Force

Single-molecule force spectroscopy has opened up new approaches to the study of protein dynamics. For example, an extended protein folding after an abrupt quench in the pulling force was shown to follow variable collapse trajectories marked by well-defined stages that departed from the expected two-state folding behavior that is commonly observed in bulk. Here, we explain these observations by developing a simple approach that models the free energy of a mechanically extended protein as a combination of an entropic elasticity term and a short-range potential representing enthalpic hydrophobic interactions. The resulting free energy of the molecule shows a force-dependent energy barrier of magnitude, ΔE = ɛ(F − F_c)^3/2, separating the enthalpic and entropic minima that vanishes at a critical force F_c. By solving the Langevin equation under conditions of a force quench, we generate folding trajectories corresponding to the diffusional collapse of an extended polypeptide. The predicted trajectories reproduce the different stages of collapse, as well as the magnitude and time course of the collapse trajectories observed experimentally in ubiquitin and I27 protein monomers. Our observations validate the force-clamp technique as a powerful approach to determining the free-energy landscape of proteins collapsing and folding from extended states.     

Hydrophobic folding units at protein-protein interfaces: implications to protein folding and to protein-protein association.

A hydrophobic folding unit cutting algorithm, originally developed for dissecting single-chain proteins, has been applied to a dataset of dissimilar two-chain protein-protein interfaces. Rather than consider each individual chain separately, the two-chain complex has been treated as a single chain. The two-chain parsing results presented in this work show hydrophobicity to be a critical attribute of two-state versus three-state protein-protein complexes. The hydrophobic folding units at the interfaces of two-state complexes suggest that the cooperative nature of the two-chain protein folding is the outcome of the hydrophobic effect, similar to its being the driving force in a single-chain folding. In analogy to the protein-folding process, the two-chain, two-state model complex may correspond to the formation of compact, hydrophobic nuclei. On the other hand, the three-state model complex involves binding of already folded monomers, similar to the association of the hydrophobic folding units within a single chain. The similarity between folding entities in protein cores and in two-state protein-protein interfaces, despite the absence of some chain connectivities in the latter, indicates that chain linkage does not necessarily affect the native conformation. This further substantiates the notion that tertiary, non-local interactions play a critical role in protein folding. These compact, hydrophobic, two-chain folding units, derived from structurally dissimilar protein-protein interfaces, provide a rich set of data useful in investigations of the role played by chain connectivity and by tertiary interactions in studies of binding and of folding. Since they are composed of non-contiguous pieces of protein backbones, they may also aid in defining folding nuclei.     

Force-Induced Lysozyme—HyHEL5 Antibody Dissociation and Its Analysis by Means of a Cooperative Binding Model

Dynamic force spectroscopy probes the kinetic properties of molecules interacting with each other such as antibody-antigen, receptor-ligand, etc. In this article, a statistical model for the dissociation of such cooperative systems is presented. The partner molecules are assumed to be linked by a number of relatively weak bonds that can be grouped together into cooperative units. Single bonds are assumed to open and close statistically. Our model was used to analyze molecular recognition experiments of single receptor-ligand pairs in which the two molecules are brought into contact using an atomic force microscope, which leads to the formation of a strong and specific bond. Then a prescribed time-dependent force is applied to the complex and the statistical distribution of forces needed to pull the molecules completely apart is measured. This quantity is also calculated from our model. Furthermore, its dependence on the model parameters, such as binding free energy, number of bonds and groups, number of cooperative elementary bonds and degree of cooperativity within a group, influence of the force on the binding free energy, and the rate of change of the pulling force, is determined.     

Changing the Mechanical Unfolding Pathway of FnIII10 by Tuning the Pulling Strength

We investigate the mechanical unfolding of the tenth type III domain from fibronectin (FnIII₁₀) both at constant force and at constant pulling velocity, by all-atom Monte Carlo simulations. We observe both apparent two-state unfolding and several unfolding pathways involving one of three major, mutually exclusive intermediate states. All three major intermediates lack two of seven native β-strands, and share a quite similar extension. The unfolding behavior is found to depend strongly on the pulling conditions. In particular, we observe large variations in the relative frequencies of occurrence for the intermediates. At low constant force or low constant velocity, all three major intermediates occur with a significant frequency. At high constant force or high constant velocity, one of them, with the N- and C-terminal β-strands detached, dominates over the other two. Using the extended Jarzynski equality, we also estimate the equilibrium free-energy landscape, calculated as a function of chain extension. The application of a constant pulling force leads to a free-energy profile with three major local minima. Two of these correspond to the native and fully unfolded states, respectively, whereas the third one can be associated with the major unfolding intermediates.     

Energy barriers, cooperativity, and hidden intermediates in the folding of small proteins

Current theoretical views of the folding process of small proteins (< approximately 100 amino acids) postulate that the landscape of potential mean force (PMF) for the formation of the native state has a funnel shape and that the free energy barrier to folding arises from the chain configurational entropy only. However, recent theoretical studies on the formation of hydrophobic clusters with explicit water suggest that a barrier should exist on the PMF of folding, consistent with the fact that protein folding generally involves a large positive activation enthalpy at room temperature. In addition, high-resolution structural studies of the hidden partially unfolded intermediates have revealed the existence of non-native interactions, suggesting that the correction of the non-native interactions during folding should also lead to barriers on PMF. To explore the effect of a PMF barrier on the folding behavior of proteins, we modified Zwanzig's model for protein folding with an uphill landscape of PMF for the formation of transition states. We found that the modified model for short peptide segments can satisfy the thermodynamic and kinetic criteria for an apparently two-state folding. Since the Levinthal paradox can be solved by a stepwise folding of short peptide segments, a landscape of PMF with a locally uphill search for the transition state and cooperative stabilization of folding intermediates/native state is able to explain the available experimental results for small proteins. We speculate that the existence of cooperative hidden folding intermediates in small proteins could be the consequence of the highly specific structures of the native state, which are selected by evolution to perform specific functions and fold in a biologically meaningful time scale.     

Mitochondrial import driving forces: enhanced trapping by matrix Hsp70 stimulates translocation and reduces the membrane potential dependence of loosely folded preproteins

The mitochondrial heat shock protein Hsp70 (mtHsp70) is essential for driving translocation of preproteins into the matrix. Two models, trapping and pulling by mtHsp70, are discussed, but positive evidence for either model has not been found so far. We have analyzed a mutant mtHsp70, Ssc1-2, that shows a reduced interaction with the membrane anchor Tim44, but an enhanced trapping of preproteins. Unexpectedly, at a low inner membrane potential, ssc1-2 mitochondria imported loosely folded preproteins more efficiently than wild-type mitochondria. The import of a tightly folded preprotein, however, was not increased in ssc1-2 mitochondria. Thus, enhanced trapping by mtHsp70 stimulates the import of loosely folded preproteins and reduces the dependence on the import-driving activity of the membrane potential, directly demonstrating that trapping is one of the molecular mechanisms of mtHsp70 action.