A semi-analytical description of protein folding that incorporates detailed geometrical information

Much has been done to study the interplay between geometric and energetic effects on the protein folding energy landscape. Numerical techniques such as molecular dynamics simulations are able to maintain a precise geometrical representation of the protein. Analytical approaches, however, often focus on the energetic aspects of folding, including geometrical information only in an average way. Here, we investigate a semi-analytical expression of folding that explicitly includes geometrical effects. We consider a Hamiltonian corresponding to a Gaussian filament with structure-based interactions. The model captures local features of protein folding often averaged over by mean-field theories, for example, loop contact formation and excluded volume. We explore the thermodynamics and folding mechanisms of beta-hairpin and alpha-helical structures as functions of temperature and Q, the fraction of native contacts formed. Excluded volume is shown to be an important component of a protein Hamiltonian, since it both dominates the cooperativity of the folding transition and alters folding mechanisms. Understanding geometrical effects in analytical formulae will help illuminate the consequences of the approximations required for the study of larger proteins.     

Frustration Sculpts the Early Stages of Protein Folding

The funneled energy landscape theory implies that protein structures are minimally frustrated. Yet, because of the divergent demands between folding and function, regions of frustrated patterns are present at the active site of proteins. To understand the effects of such local frustration in dictating the energy landscape of proteins, here we compare the folding mechanisms of the two alternative spliced forms of a PDZ domain (PDZ2 and PDZ2as) that share a nearly identical sequence and structure, while displaying different frustration patterns. The analysis, based on the kinetic characterization of a large number of site‐directed mutants, reveals that although the late stages for folding are very robust and biased by native topology, the early stages are more malleable and dominated by local frustration. The results are briefly discussed in the context of the energy‐landscape theory.     

Folding behavior of model proteins with weak energetic frustration

The native structure of fast-folding proteins, albeit a deep local free-energy minimum, may involve a relatively small energetic penalty due to nonoptimal, though favorable, contacts between amino acid residues. The weak energetic frustration that such contacts represent varies among different proteins and may account for folding behavior not seen in unfrustrated models. Minimalist model proteins with heterogeneous contacts--as represented by lattice heteropolymers consisting of three types of monomers--also give rise to weak energetic frustration in their corresponding native structures, and the present study of their equilibrium and nonequilibrium properties reveals some of the breadth in their behavior. In order to capture this range within a detailed study of only a few proteins, four candidate protein structures (with their cognate sequences) have been selected according to a figure of merit called the winding index--a characteristic of the number of turns the protein winds about an axis. The temperature-dependent heat capacities reveal a high-temperature collapse transition, and an infrequently observed low-temperature rearrangement transition that arises because of the presence of weak energetic frustration. Simulation results motivate the definition of a new measure of folding affinity as a sequence-dependent free energy--a function of both a reduced stability gap and high accessibility to non-native structures--that correlates strongly with folding rates.     

Folding dynamics of Trp-cage in the presence of chemical interference and macromolecular crowding. I

Proteins fold and function in the crowded environment of the cell's interior. In the recent years it has been well established that the so-called "macromolecular crowding" effect enhances the folding stability of proteins by destabilizing their unfolded states for selected proteins. On the other hand, chemical and thermal denaturation is often used in experiments as a tool to destabilize a protein by populating the unfolded states when probing its folding landscape and thermodynamic properties. However, little is known about the complicated effects of these synergistic perturbations acting on the kinetic properties of proteins, particularly when large structural fluctuations, such as protein folding, have been involved. In this study, we have first investigated the folding mechanism of Trp-cage dependent on urea concentration by coarse-grained molecular simulations where the impact of urea is implemented into an energy function of the side chain and/or backbone interactions derived from the all-atomistic molecular dynamics simulations with urea through a Boltzmann inversion method. In urea solution, the folding rates of a model miniprotein Trp-cage decrease and the folded state slightly swells due to a lack of contact formation between side chains at the terminal regions. In addition, the equilibrium m-values of Trp-cage from the computer simulations are in agreement with experimental measurements. We have further investigated the combined effects of urea denaturation and macromolecular crowding on Trp-cage's folding mechanism where crowding agents are modeled as hard-spheres. The enhancement of folding rates of Trp-cage is most pronounced by macromolecular crowding effect when the extended conformations of Trp-cast dominate at high urea concentration. Our study makes quantitatively testable predictions on protein folding dynamics in a complex environment involving both chemical denaturation and macromolecular crowding effects.     

Exploring the parameter space of the coarse‐grained UNRES force field by random search: Selecting a transferable medium‐resolution force field

We explored the energy‐parameter space of our coarse‐grained UNRES force field for large‐scale ab initio simulations of protein folding, to obtain good initial approximations for hierarchical optimization of the force field with new virtual‐bond‐angle bending and side‐chain‐rotamer potentials which we recently introduced to replace the statistical potentials. 100 sets of energy‐term weights were generated randomly, and good sets were selected by carrying out replica‐exchange molecular dynamics simulations of two peptides with a minimal α‐helical and a minimal β‐hairpin fold, respectively: the tryptophan cage (PDB code: 1L2Y) and tryptophan zipper (PDB code: 1LE1). Eight sets of parameters produced native‐like structures of these two peptides. These eight sets were tested on two larger proteins: the engrailed homeodomain (PDB code: 1ENH) and FBP WW domain (PDB code: 1E0L); two sets were found to produce native‐like conformations of these proteins. These two sets were tested further on a larger set of nine proteins with α or α + β structure and found to locate native‐like structures of most of them. These results demonstrate that, in addition to finding reasonable initial starting points for optimization, an extensive search of parameter space is a powerful method to produce a transferable force field. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

Understanding the role of the topology in protein folding by computational inverse folding experiments

Recent studies suggest that protein folding should be revisited as the emergent property of a complex system and that the nature allows only a very limited number of folds that seem to be strongly influenced by geometrical properties. In this work we explore the principles underlying this new view and show how helical protein conformations can be obtained starting from simple geometric considerations. We generated a large data set of C-alpha traces made of 65 points, by computationally solving a backbone model that takes into account only topological features of the all-alpha proteins; then, we built corresponding tertiary structures, by using the sequences associated to the crystallographic structures of four small globular all-alpha proteins from PDB, and analysed them in terms of structural and energetic properties. In this way we obtained four poorly populated sets of structures that are reasonably similar to the conformational states typical of the experimental PDB structures. These results show that our computational approach can capture the native topology of all-alpha proteins; furthermore, it generates backbone folds without the influence of the side chains and uses the protein sequence to select a specific fold among the generated folds. This agrees with the recent view that the backbone plays an important role in the protein folding process and that the amino acid sequence chooses its own fold within a limited total number of folds.     

Statistical theory for protein ensembles with designed energy landscapes

Combinatorial protein libraries provide a promising route to investigate the determinants and features of protein folding and to identify novel folding amino acid sequences. A library of sequences based on a pool of different monomer types are screened for folding molecules, consistent with a particular foldability criterion. The number of sequences grows exponentially with the length of the polymer, making both experimental and computational tabulations of sequences infeasible. Herein a statistical theory is extended to specify the properties of sequences having particular values of global energetic quantities that specify their energy landscape. The theory yields the site-specific monomer probabilities. A foldability criterion is derived that characterizes the properties of sequences by quantifying the energetic separation of the target state from low-energy states in the unfolded ensemble and the fluctuations of the energies in the unfolded state ensemble. For a simple lattice model of proteins, excellent agreement is observed between the theory and the results of exact enumeration. The theory may be used to provide a quantitative framework for the design and interpretation of combinatorial experiments.     

Crystallographic B-factors highlight energetic frustration in aldolase folding

Kinetic simulations of the folding and unfolding of the mammalian TIM barrel protein aldolase were conducted to determine if a minimalist monomeric Gō model, using the native structure to determine attractive energies in the protein chain, could capture the experimentally determined folding pathway. The folding order, that is, the order in which different secondary structures fold, between the Gō model simulations and that from hydrogen-deuterium exchange experiments, did not agree. To explain this discrepancy, two alternate variant of the basic Gō model were simulated: (1) a monomer Gō model with native contact energies weighted by a statistical potential (SP model) and (2) a monomer Gō model with native contact energies inversely weighted by crystallographic B factors (B model). The B model demonstrated the best agreement between simulation and experiments. The success of the B model is attributed to the ability of B factors to highlight local energetic frustration in the aldolase structure which results in weaker native contacts in these frustrated regions. The predictive success of the B model also reveals the potential use of B factor information for energetic weighting in general protein modeling.     

Modeling the interplay between geometrical and energetic effects in protein folding

A theoretical framework is constructed with the aid of a free-energy functional method that is capable of describing the interplay between geometrical and energetic effects on protein folding. In this paper, we generalize a free-energy functional model based on polymer theory to make it more appropriate for comparison with protein folding simulations and experiments. This generalization is made by introducing cooperativity into the configurational entropy and the internal energy. Modifications to configurational entropy enable the model to account for the loop-loop interactions, a contribution neglected in the original model. Modifications to the internal energy introduce many-body corrections, which are needed to establish quantitative contact to simulations as well as experimental observations. To demonstrate the efficiency of the modified analytical model, we compare our results with C(alpha) structure-based (Go) model simulations of chymotrypsin inhibitor II and the SH3 domain of src.     

Protein mechanical unfolding: a model with binary variables

A simple model, recently introduced as a generalization of the Wako-Saito; model of protein folding, is used to investigate the properties of widely studied molecules under external forces. The equilibrium properties of the model proteins, together with their energy landscape, are studied on the basis of the exact solution of the model. Afterwards, the kinetic response of the molecules to a force is considered, discussing both force clamp and dynamic loading protocols and showing that theoretical expectations are verified. The kinetic parameters characterizing the protein unfolding are evaluated by using computer simulations and agree nicely with experimental results, when these are available. Finally, the extended Jarzynski equality is exploited to investigate the possibility of reconstructing the free energy landscape of proteins with pulling experiments.     

On the influence of charged side chains on the folding-unfolding equilibrium of beta-peptides: a molecular dynamics simulation study

The influence of charged side chains on the folding-unfolding equilibrium of beta-peptides was investigated by means of molecular dynamics simulations. Four different peptides containing only negatively charged side chains, positively charged side chains, both types of charged side chains (with the ability to form stabilizing salt bridges) or no charged side chains were studied under various conditions (different simulation temperatures, starting structures and solvent environment). The NMR solution structure in methanol of one of the peptides (A) has already been published; the synthesis and NMR analysis of another peptide (B) is described here. The other peptides (C and D) studied herein have hitherto not been synthesized. All four peptides A-D are expected to adopt a left-handed 3(14)-helix in solution as well as in the simulations. The resulting ensembles of structures were analyzed in terms of conformational space sampled by the peptides, folding behavior, structural properties such as hydrogen bonding, side chain-side chain and side chain-backbone interactions and in terms of the level of agreement with the NMR data available for two of the peptides. It was found that the presence of charged side chains significantly slows down the folding process in methanol solution due to the stabilization of intermediate conformers with side chain-backbone interactions. In water, where the solvent competes with the solute-solute polar interactions, the folding process to the 3(14)-helix is faster in the simulations.     

Stability of a protein tethered to a surface

Surface-tethered proteins are increasingly being used in a variety of experimental situations, and they are the basis for many new technologies. Nevertheless, a thorough understanding of how a surface can impact the native state stability of an attached protein is lacking. In this work, the authors use molecular dynamics simulations of a model beta-barrel protein to investigate how surface tethering influences native state stability. They find that stability, as measured by the folding temperature Tf, can be either increased, decreased, or remain unchanged as a result of tethering. Observed shifts are highly dependent on the location of residue used as the tether point, and stability is influenced by a number of factors, both energetic and entropic. These factors include native state vibrations, loss of bulk unfolded conformations, changes to the unfolded state ensemble, and the emergence of an entropic term not present for the bulk protein. They discuss each of these contributions in detail and comment on their relative importance and connection to experiment.     

Using one‐step perturbation to predict the effect of changing force‐field parameters on the simulated folding equilibrium of a β‐peptide in solution

Computer simulation using molecular dynamics is increasingly used to simulate the folding equilibria of peptides and small proteins. Yet, the quality of the obtained results depends largely on the quality of the force field used. This comprises the solute as well as the solvent model and their energetic and entropic compatibility. It is, however, computational very expensive to perform test simulations for each combination of force‐field parameters. Here, we use the one‐step perturbation technique to predict the change of the free enthalpy of folding of a β‐peptide in methanol solution due to changing a variety of force‐field parameters. The results show that changing the solute backbone partial charges affects the folding equilibrium, whereas this is relatively insensitive to changes in the force constants of the torsional energy terms of the force field. Extending the cut‐off distance for nonbonded interactions beyond 1.4 nm does not affect the folding equilibrium. The same result is found for a change of the reaction‐field permittivity for methanol from 17.7 to 30. The results are not sensitive to the criterion, e.g., atom‐positional RMSD or number of hydrogen bonds, that is used to distinguish folded and unfolded conformations. Control simulations with perturbed Hamiltonians followed by backward one‐step perturbation indicated that quite large perturbations still yield reliable results. Yet, perturbing all solvent molecules showed where the limitations of the one‐step perturbation technique are met. The evaluated methodology constitutes an efficient tool in force‐field development for molecular simulation by reducing the number of required separate simulations by orders of magnitude. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Synthesis and Dynamics of Nanosized Phenylene–Ethynylene–Butadiynylene Rotaxanes and the Role of Shape Persistence

Phenylacetylene‐based [2]rotaxanes were synthesized by a covalent‐template approach by aminolysis of the corresponding prerotaxanes. The wheel and the bulky stoppers are made of phenylene–ethynylene–butadiynylene macrocycles of the same size. The stoppers are large enough to enable the synthesis and purification of the rotaxane. However, the wheel unthreads from the axle at elevated temperatures. The deslipping kinetics and the activation parameters were determined. We described theoretically the unthreading by state‐of‐the‐art DFT‐based molecular‐mechanics models and a string method for the simulation of rare events. This approach enabled us to characterize in detail the unthreading mechanism, which involves the folding of the stopper during its passage through the wheel opening, a process that defies intuitive geometrical considerations. The conformational and energetic features of the transition allowed us to infer the molecular residues controlling the disassembly timescale.     

Probing the formation of stable tertiary structure in a model miniprotein at atomic resolution: determinants of stability of a helical hairpin

The minimal model system to study the basic principles of protein folding is the hairpin. The formation of beta-hairpins, which are the basic components of antiparallel beta-sheets, has been studied extensively in the past decade, but much less is known about helical hairpins. Here, we probe hairpin formation between a polyproline type-II helix and an alpha-helix as present in the natural miniprotein peptide YY (PYY). Both turn sequence and interactions of aromatic side chains from the C-terminal alpha-helix with the pockets formed by N-terminal Pro residues are shown by site-directed mutagenesis and solution NMR spectroscopy in different solvent systems to be important determinants of backbone dynamics and hairpin stability, suggesting a close analogy with some beta-hairpin structures. It is shown that multiple relatively weak contacts between the helices are necessary for the formation of the helical hairpin studied here, whereas the type-I beta-turn acts like a hinge, which through certain single amino acid substitutions is destabilized such that hairpin formation is completely abolished. Denaturation and renaturation of tertiary structure by temperature or cosolvents were probed by measuring changes of chemical shifts. Folding of PYY is both reversible and cooperative as inferred from the sigmoidal denaturation curves displayed by residues at the interface of the helical hairpin. Such miniproteins thus feature an important hallmark of globular proteins and should provide a convenient system to study basic aspects of helical hairpin folding that are complementary to those derived from studies of beta-hairpins.     

Nanosecond folding dynamics of a three-stranded beta-sheet

We studied conformational stability and folding kinetics of a three-stranded beta-sheet containing two rigid turns. Static infrared measurements indicate that this beta-sheet undergoes a broad but cooperative thermal unfolding transition with a midpoint at approximately 53 degrees C. Interestingly, time-resolved infrared experiments show that its relaxation kinetics in response to a temperature-jump (T-jump) occur on the nanosecond time scale (e.g., the relaxation time is approximately 140 ns at 35.0 degrees C), thereby suggesting that the conformational relaxation encounters only a small free energy barrier or even proceeds in a downhill manner. Further Langevin dynamics simulations suggest that the observed T-jump relaxation kinetics could be modeled by a conformational diffusion process along a single-well free energy profile, which allowed us to determine the effective diffusion constant and also the roughness of the folding energy landscape.     

Evaluating force field accuracy with long-time simulations of a β-hairpin tryptophan zipper peptide

We have combined graphics processing unit-accelerated all-atom molecular dynamics with parallel tempering to explore the folding properties of small peptides in implicit solvent on the time scale of microseconds. We applied this methodology to the synthetic β-hairpin, trpzip2, and one of its sequence variants, W2W9. Each simulation consisted of over 8 μs of aggregated virtual time. Several measures of folding behavior showed good convergence, allowing comparison with experimental equilibrium properties. Our simulations suggest that the intramolecular interactions of tryptophan side chains are responsible for much of the stability of the native fold. We conclude that the ff99 force field combined with ff96 φ and ψ dihedral energies and an implicit solvent can reproduce plausible folding behavior in both trpzip2 and W2W9.     

Structural Insights into the Trp‐Cage Folding Intermediate Formation

The 20 residue long Trp‐cage is the smallest protein known, and thus has been the subject of several in vitro and in silico folding studies. Here, we report the multistate folding scenario of the miniprotein in atomic detail. We detected and characterized different intermediate states by temperature dependent NMR measurements of the ¹⁵N and ¹³C/¹⁵N labeled protein, both at neutral and acidic pH values. We developed a deconvolution technique to characterize the invisible—fully folded, unfolded and intermediate—fast exchanging states. Using nonlinear fitting methods we can obtain both the thermodynamic parameters (ΔH^F–I, T_m^F–I, ΔC_p^F–I and ΔH^I–U, T_m^I–U, ΔC_p^I–U) and the NMR chemical shifts of the conformers of the multistate unfolding process. During the unfolding of Trp‐cage distinct intermediates evolve: a fast‐exchanging intermediate is present under neutral conditions, whereas a slow‐exchanging intermediate‐pair emerges at acidic pH. The fast‐exchanging intermediate has a native‐like structure with a short α‐helix in the G¹¹–G¹⁵ segment, whereas the slow‐exchanging intermediate‐pair presents elevated dynamics, with no detectable native‐like residue contacts in which the G¹¹? P¹² peptide bond has either cis or trans conformation. Heteronuclear relaxation studies combined with MD simulations revealed the source of backbone mobility and the nature of structural rearrangements during these transitions. The ability to detect structural and dynamic information about folding intermediates in vitro provides an excellent opportunity to gain new insights into the energetic aspects of the energy landscape of protein folding. Our new experimental data offer exceptional testing ground for further computational simulations.