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This paper proposes a generalization of the well-known folding funnel concept of proteins. In the funnel model the polypeptide chain is treated as an individual object not interacting with other proteins. Since biological systems are considerably crowded, protein-protein interaction is a fundamental feature during the life cycle of proteins. The folding superfunnel proposed here describes the folding process of interacting proteins in various situations. The first example discussed is the folding of the freshly synthesized protein with the aid of chaperones. Another important aspect of protein-protein interactions is the folding of the recently characterized intrinsically disordered proteins, where binding to target proteins plays a crucial role in the completion of the folding process. The third scenario where the folding superfunnel is used is the formation of aggregates from destabilized proteins, which is an important factor in case of several conformational diseases. The folding superfunnel constructed here with the minimal assumption about the interaction potential explains all three cases mentioned above. Proteins 2016; 84:1009-1016. © 2016 Wiley Periodicals, Inc.

We have investigated effects of salt ions on folding events of a helical miniprotein chicken villin headpiece subdomain HP36. Low concentrations of ions alter electrostatic interactions between charged groups of a protein and can change the populations of conformers. Here, we compare two data sets of folding simulations of HP36 in explicit water solvent with or without ions. For efficient sampling of the conformational space of HP36, the multicanonical replica-exchange molecular dynamics method was employed. Our analyses suggest that salt alters salt-bridging nature of the protein at later stages of folding at room temperature. Especially, more nonnative, nonlocal salt bridges are formed at near-native conformations in pure water. Our analyses also show that such salt-bridge formation hinders the fully native hydrophobic-core packing at the final stages of folding. Proteins 2014; 82:933–943. © 2013 Wiley Periodicals, Inc.

Optimization of surface exposed charge–charge interactions in the native state has emerged as an effective means to enhance protein stability; but the effect of electrostatic interactions on the kinetics of protein folding is not well understood. To investigate the kinetic consequences of surface charge optimization, we characterized the folding kinetics of a Fyn SH3 domain variant containing five amino acid substitutions that was computationally designed to optimize surface charge–charge interactions. Our results demonstrate that this optimized Fyn SH3 domain is stabilized primarily through an eight-fold acceleration in the folding rate. Analyses of the constituent single amino acid substitutions indicate that the effects of optimization of charge–charge interactions on folding rate are additive. This is in contrast to the trend seen in folded state stability, and suggests that electrostatic interactions are less specific in the transition state compared to the folded state. Simulations of the transition state using a coarse-grained chain model show that native electrostatic contacts are weakly formed, thereby making the transition state conducive to nonspecific, or even nonnative, electrostatic interactions. Because folding from the unfolded state to the folding transition state for small proteins is accompanied by an increase in charge density, nonspecific electrostatic interactions, that is, generic charge density effects can have a significant contribution to the kinetics of protein folding. Thus, the interpretation of the effects of amino acid substitutions at surface charged positions may be complicated and consideration of only native-state interactions may fail to provide an adequate picture. Proteins 2011. © 2012 Wiley Periodicals, Inc.

Protein folding into tertiary structures is controlled by an interplay of attractive contact interactions and steric effects. We investigate the balance between these contributions using structure-based models using an all-atom representation of the structure combined with a coarse-grained contact potential. Tertiary contact interactions between atoms are collected into a single broad attractive well between the C(β) atoms between each residue pair in a native contact. Through the width of these contact potentials we control their tolerance for deviations from the ideal structure and the spatial range of attractive interactions. In the compact native state dominant packing constraints limit the effects of a coarse-grained contact potential. During folding, however, the broad attractive potentials allow an early collapse that starts before the native local structure is completely adopted. As a consequence the folding transition is broadened and the free energy barrier is decreased. Eventually two-state folding behavior is lost completely for systems with very broad attractive potentials. The stabilization of native-like residue interactions in non-perfect geometries early in the folding process frequently leads to structural traps. Global mirror images are a notable example. These traps are penalized by the details of the repulsive interactions only after further collapse. Successful folding to the native state requires simultaneous guidance from both attractive and repulsive interactions.

Flavodoxins are single domain proteins with an alpha/beta structure, whose function and folding have been well studied. Detailed experiments have shown that several members of this protein family present a stable intermediate, which accumulates along the folding process. In this work, we use a coarse-grained model for protein folding, whose interactions are based on the topology of the native state, to analyze the thermodynamic characteristics of the folding of Anabaena apoflavodoxin. Our model shows evidence for the existence of a thermodynamic folding intermediate, which reaches a significant population along the thermal transition. According to our simulation results, the intermediate is compact, well packed, and involves distortions of the native structure similar to those experimentally found. These mainly affect the long loop in the protein surface comprising residues 120-139. Although the agreement between simulation and experiment is not perfect, something impossible for a crude model, our results show that the topology of the native state is able to dictate a folding process which includes the presence of an intermediate for this protein.

We have characterized here the structural and dynamics properties of urea-denatured state of dSmt3 by multidimensional NMR at 27 degrees C and pH 5.6. The various results suggest that hydrophobic clusters as well as different native and non-native secondary structural elements are transiently formed. The backbone in the regions Gln26-Lys31 and Gly47-Gln60 shows conformationally restricted motions. The AABUF profile of the sequence reflected that this region has the highest tendency to undergo hydrophobic clustering and may thus assist the formation of transient structures. The secondary chemical shifts and coupling constants indicated that this region has strong tendency to occupy the broad beta-domain of (phi,psi) space. A number of NMR parameters indicated that the region Asp58-Gln60 (corresponding to beta3-beta4 turn in the folded state) has residual turn-like structure. The present structural and dynamics results on urea-denatured dSmt3 have been compared with the previously published results on denatured states of similar fold proteins e.g. human SUMO-1 (55% homologous), ubiquitin (13.8% homologous) and GB1. Although the sequence homology is rather poor between them, the residual structure in all cases seems to be largely native type. The implications of these to sequence-folding paradigm and initial folding processes have been discussed.

The effects of a single-point mutation on folding thermodynamics and kinetics are usually interpreted by focusing on the native structure and the transition state. Here, the entire conformational spaces of a 20-residue three-stranded antiparallel beta-sheet peptide (double hairpin) and of its single-point mutant W10V are sampled close to the melting temperature by equilibrium folding-unfolding molecular dynamics simulations for a total of 40 micros. The folded state as well as the most populated free energy basins in the denatured state are isolated by grouping conformations according to fast relaxation at equilibrium. Such kinetic analysis provides more detailed and useful information than a simple projection of the free energy. The W10V mutant has the same native structure as the wild type peptide, and similar folding rate and stability. In the denatured state, the N-terminal hairpin is about 20% more structured in W10V than the wild type mainly because of van der Waals interactions. Notably, the W10V mutation influences also the van der Waals energy at the transition state ensemble causing a shift in the ratio of fluxes between two different transition state regions on parallel folding pathways corresponding to nucleation at either of the two beta-hairpins. Previous experimental studies have focused on the effects of denaturant-dependent or temperature-dependent changes in the structure of the denatured state. The atomistic simulations show that a single-point mutation in the central strand of a beta-sheet peptide results in remarkable changes in the topography of the denatured state ensemble. These changes modulate the relative accessibility of parallel folding pathways because of kinetic partitioning of the denatured state. Therefore, the observed dependence of the folding process on the starting ensemble raises questions on the biological significance of in vitro folding studies under strongly denaturing conditions.

We study the dependence of folding time, nucleation site, and stability of a model β-hairpin on the location of a cross-strand hydrophobic pair, using a coarse-grained off-lattice model with the aid of Monte Carlo simulations. Our simulations have produced 6500 independent folding trajectories dynamically, forming the basis for extensive statistical analysis. Four folding pathways, zipping-out, middle-out, zipping-in, and reptation, have been closely monitored and discussed in all seven sequences studied. A hydrophobic pair placed near the β-turn or in the middle section effectively speed up folding; a hydrophobic pair placed close to the terminal ends or next to the β-turn encourages stability of the entire chain. Proteins 2006. © 2006 Wiley-Liss, Inc.

Chaperonin-mediated protein fold- ing is complex. There have been diverse results on folding behavior, and the chaperonin molecules have been investigated as enhancing or retarding the folding rate. To understand the diversity of chaperonin-mediated protein folding, we report a study based on simulations using a simplified Go - type model. By considering effects of affinity be- tween the substrate protein and the chaperonin wall and spatial confinement of the chaperonin cavity, we study the thermodynamics and kinetics of folding of an unfrustrated substrate protein en- capsulated in a chaperonin cavity. The affinity makes the hydrophobic residues of the protein bind to the chaperonin wall, and a strong (or weak) affinity results in a large (or small) effect of binding. Compared with the folding in bulk, the folding in chaperonin cavity with different strengths of affin- ity shows two kinds of behaviors: one with less dependence on the affinity but more reliance on the spatial confinement effect and the other relying strongly on the affinity. It is found that the enhance- ment or retardation of the folding rate depends on the competition between the spatial confinement and the affinity due to the chaperonin cavity, and a strong affinity produces a slow folding while a weak affinity induces a fast folding. The crossover be- tween two kinds of folding behaviors happens in the case that the favorable effect of confinement is balanced by the unfavorable effect of the affinity, and a critical affinity strength is roughly defined. By analyzing the contacts formed between the residues of the protein and the chaperonin wall and between the residues of the protein themselves, the role of the affinity in the folding processes is studied. The binding of the residues with the chaperonin wall reduces the formation of both native contacts and nonnative contact or mis-contacts, providing a loose structure for further folding after allosteric change of the chaperonin cavity. In addition, 15 single-site- mutated mutants are simulated in order to test the validity of our model and to investigate the impor- tance of affinity. Inspiringly, our results of the folding rates have a good correlation with those obtained from experiments. The folding rates are inversely correlated with the strength of the bind- ing interactions, i.e., the weaker the binding, the faster the folding. We also find that the inner hydro- phobic residues have larger effects on the folding kinetics than those of the exterior hydrophobic residues. We suggest that, besides the confinement effect, the affinity acts as another important factor to affect the folding of the substrate proteins in chaperonin systems, providing an understanding of the folding mechanism of the molecular chaperonin systems. Proteins 2005;61:777-794.

The folding of an oligomeric protein poses an extra challenge to the folding problem because the protein not only has to fold correctly; it has to avoid nonproductive aggregation. We have carried out over 100 molecular dynamics simulations using an implicit solvation model at different temperatures to study the unfolding of one of the smallest known tetramers, p53 tetramerization domain (p53tet). We found that unfolding started with disruption of the native tetrameric hydrophobic core. The transition state for the tetramer to dimer transition was characterized as a diverse ensemble of different structures using Φ value analysis in quantitative agreement with experimental data. Despite the diversity, the ensemble was still native-like with common features such as partially exposed tetramer hydrophobic core and shifts in the dimer–dimer arrangements. After passing the transition state, the secondary and tertiary structures continued to unfold until the primary dimers broke free. The free dimer had little secondary structure left and the final free monomers were random-coil like. Both the transition states and the unfolding pathways from these trajectories were very diverse, in agreement with the new view of protein folding. The multiple simulations showed that the folding of p53tet is a mixture of the framework and nucleation-condensation mechanisms and the folding is coupled to the complex formation. We have also calculated the entropy and effective energy for the different states along the unfolding pathway and found that the tetramerization is stabilized by hydrophobic interactions. Proteins 2005. © 2005 Wiley-Liss, Inc.

Native-state hydrogen exchange experiments on several proteins have revealed par- tially unfolded conformations with diverse stabili- ties. These equilibrium observations have been used to support kinetic arguments that folding proceeds via a sequential "pathway." This interpretative logic is evaluated here by analyzing the relationship between thermodynamic behavior and folding kinet- ics in a class of simplified lattice protein models. The chain models studied have varying degrees of coop- erative interplay (coupling) between local helical conformational preference and favorable nonlocal interactions. When model cooperativity is high, as native conditions are weakened, "isotherms" of free energy of exchange for residues belonging to the same helix merge together before global unfolding. The point of merger depends on the model energetic favorability of the helix. This trend is similar to the corresponding experimental observations. Kineti- cally, we find that the ordering of helix formation in the very last stage of native core assembly tends to follow the stabilities of their converged isotherms. In a majority (but not all) of folding trajectories, the final assembly of helices that are thermodynami- cally more stable against exchange precedes that of helices that are less stable against exchange. These model features are in partial agreement with com- mon experimental interpretations. However, the model results also underscore the ensemble nature of the folding process: the kinetics of helix forma- tion is not a discrete, strictly "all-or-none" process as that envisioned by certain non-explicit-chain models. Helices generally undergo many cycles of partial formation and dissolution before their con- formations are fixed in the final assembly stage of folding, a kinetic stage that takes up only 2% of the average folding time in the present model; and the ordering of the helices' final assembly in some trajec- tories can be different from the dominant ordering stipulated by the exchange isotherms. Proteins 2005

We introduce a simple procedure to analyze the temperature dependence of the folding and unfolding rates of two-state proteins. We start from the simple transition-state-like rate expression: k = D(eff)exp(-DeltaG(TS)/RT), in which upper and lower bounds for the intra-chain effective diffusion coefficient (D(eff)) are obtained empirically using the timescales of elementary processes in protein folding. From the changes in DeltaG(TS) as a function of temperature, we calculate enthalpies and heat capacities of activation, together with the more elusive entropies of activation. We then estimate the conformational entropy of the transition state by extrapolation to the temperature at which the solvation entropy vanishes by cancellation between polar and apolar terms. This approach is based on the convergence temperatures for the entropy of solvating apolar (approximately 385 K) and polar groups (approximately 335 K), the assumption that the structural properties of the transition state are somewhere in between the unfolded and folded states, and the established relationship between observed heat capacity and solvent accessibility.1 To circumvent the lack of structural information about transition states, we use the empirically determined heat capacities of activation as constraints to identify the extreme values of the transition state conformational entropy that are consistent with experiment. The application of this simple approach to six two-state folding proteins for which there is temperature-dependent data available in the literature provides important clues about protein folding. For these six proteins, we obtain an average equilibrium cost in conformational entropy of -4.3 cal x mol(-1)K(-1)per residue, which is in close agreement to previous empirical and computational estimates of the same quantity. Furthermore, we find that all these proteins have a conformationally diverse transition state, with more than half of the conformational entropy of the unfolded state. In agreement with predictions from theory and computer simulations, the transition state signals the change from a regime dominated by loss in conformational entropy to one driven by the gain in stabilization free energy (i.e., including protein interactions and solvation effects). Moreover, the height of the barrier is determined by how much stabilization free energy is realized at that point, which is related to the relative contribution of local versus non-local interactions. A remarkable observation is that the fraction of conformational entropy per residue that is present in the transition state is very similar for the six proteins in this study. Based on this commonality, we propose that the observed change in thermodynamic regime is connected to a change in the pattern of structure formation: from one driven by formation of pairwise interactions to one dominated by coupling of the networks of interactions involved in forming the protein core. In this framework, the barrier to two-state folding is crossed when the folding protein reaches a "critical native density" that allows expulsion of remaining interstitial water and consolidation of the core. The principle of critical native density should be general for all two-state proteins, but can accommodate different folding mechanisms depending on the particularities of the structure and sequence.

The temperature dependence of the free energy landscape of the src-SH3 protein domain is investigated through fully atomic simulations in explicit solvent. Simulations are performed above and below the folding transition temperature, enabling an analysis of both protein folding and unfolding. The transition state for folding and unfolding, identified from the free energy surfaces, is found to be very similar, with structure in the central hydrophobic sheet and little structure throughout the rest of the protein. This is a result of a polarized folding (unfolding) mechanism involving early formation (late loss) of the central hydrophobic sheet at the transition state. Unfolding simulations map qualitatively well onto low-temperature free energy surfaces but appear, however, to miss important features observed in folding simulations. In particular, details of the folding mechanism involving the opening and closing of the hydrophobic core are not captured by unfolding simulations performed under strongly denaturing conditions. In addition, free energy surfaces at high temperatures do not display a desolvation barrier found at lower temperatures, involving the expulsion of water molecules from the hydrophobic core.

By Monte Carlo simulations, we explored the effect of single mutations on the thermodynamics and kinetics of the folding of a two-dimensional, energetically frustrated, hydrophobic protein model. Φ-Value analysis, corroborated by simulations beginning from given sets of judiciously chosen initial contacts, suggests that the transition state of the model consists of a limited region of the native structure, that is, a folding nucleus. It seems that the most important contacts in the transition state (large and positive Φ) are not the ones with the highest contact order, because in this case the entropic cost of their formation would be too high, but exactly the ones that decrease the entropic cost of difficult contacts, reducing their effective contact order. Mutations of internal monomers involved in high-order contacts were actually the ones resulting in the fastest kinetics (and Φ < 0), indicating they tend to make low order, non-native contacts of low entropic cost that stabilize the unfolded state with respect to the transition state. Folding acceleration by other non-native interactions was also observed and a simple general mechanism is proposed according to which non-native contacts can act indirectly over the folding nucleus, “chelating” out potentially harmful contacts. The polymer graph of our model, which facilitates the visualization of effective contact orders, successfully suggests the relative kinetic importance of different contacts and is reasonably consistent with analogous graphs for the well characterized family of SH3 domains. Proteins 2002;49:167–180. © 2002 Wiley-Liss, Inc.

When a denatured polypeptide is put into refolding conditions, it undergoes conformational changes on a variety of times scales. We set out here to distinguish the fast events that promote productive folding from other processes that may be generic to any non-folding polypeptide. We have apply an ab initio folding algorithm to model the folding of various proteins and their compositionally identical, random-sequence analogues. In the earliest stages, proteins and their scrambled-sequence counterparts undergo indistinguishable reductions in the extent to which they explore conformation space. For both polypeptides, an early contraction occurs but does not involve the formation of a distinct intermediate. Following this phase, however, the naturally-occurring sequences are distinguished by an increase in the formation of three-body correlations wherein a hydrophobic group desolvates and protects an intra-molecular hydrogen bond. These correlations are manifested in a mild but measurable reduction of the accessible configuration space beyond that of the random-sequence peptides, and portend the folding to the native structure. Hence, early events reflect a generic response of the denatured ensemble to a change in solvent condition, but the wild-type sequence develops additional correlations as its structure evolves that can reveal the protein's foldability.

We report a simple method for measuring the accessible conformational space explored by an ensemble of protein structures. The method is useful for diverse ensembles derived from molecular dynamics trajectories, molecular modeling, and molecular structure determinations. It can be used to examine a wide range of time scales. The central tactic we use, which has been previously employed, is to replace the true mechanical degrees of freedom of a molecular system with the conformationally effective degrees of freedom as measured by the root-mean squared cartesian distances among all pairs of conformations. Each protein conformation is treated as a point in a high dimensional euclidean space. In this article, we model this space in a novel way by representing it as an N-dimensional hypercube, describable with only two parameters: the number of dimensions and the edge length. To validate this approach, we provide a number of elementary test cases and then use the N-cube method for measuring the size and shape of conformational space covered by molecular dynamics trajectories spanning 10 orders of magnitude in time. These calculations were performed on a small protein, the villin headpiece subdomain, exploring both the native state and the misfolded/folding regime. Distinct features include single, vibrationally averaged, substate minima on the 0.1–1-ps time scale, thermally averaged conformational states that persist for 1–100 ps and transitions between these local minima on nanosecond time scales. Large-scale refolding modes appear to become uncorrelated on the microsecond time scale. Associated length scales for these events are 0.2 A for the vibrational minima; 0.5 A for the conformational minima; and 1–2 A for the nanosecond events. We find that the conformational space that is dynamically accessible during folding of villin has enough volume for ∼109 minima of the variety that persist for picoseconds. Molecular dynamics trajectories of the native protein and experimentally derived solution ensembles suggest the native state to be composed of ∼102 of these thermally accessible minima. Thus, based on random exploration of accessible folding space alone, protein folding for a small protein is predicted to be a milliseconds time scale event. This time can be compared with the experimental folding time for villin of 10–100 μs. One possible explanation for the 10–100-fold discrepancy is that the slope of the “folding funnel” increases the rate 1–2 orders of magnitude above random exploration of substates. Proteins 2001;42:495–511. © 2001 Wiley-Liss, Inc.

We propose a coarse-grained lattice model for Monte Carlo simulations of folding of proteins consisting of several alpha-helices. A chain representing a protein is considered to contain A and B monomers forming relatively stiff A subchains, mimicking helices, and flexible B links between these subchains, respectively. Using this model, we simulate (1) folding of four-helix proteins in solution; (2) folding of membrane proteins containing one, two, or four helices; and (3) refolding of four-helix proteins adsorbed at the liquid-solid interface. For these cases, we show typical scenarios of protein folding and refolding and study the dependence of the folding time on the chain length. Combining the latter results with those already available in the literature, we discuss the relative rates of folding of proteins belonging to different classes.