Cooperativity and the origins of rapid, single-exponential kinetics in protein folding

Water as a Good Solvent for Unfolded Proteins: Folding and Collapse are Fundamentally Different

The argument that the hydrophobic effect is the primary effect driving the folding of globular proteins is nearly universally accepted (including by the authors). But does this view also imply that water is a "poor" solvent for the unfolded states of these same proteins? Here we argue that the answer is "no," that is, folding to a well-packed, extensively hydrogen-bonded native structure differs fundamentally from the nonspecific chain collapse that defines a poor solvent. Thus, the observation that a protein folds in water does not necessitate that water is a poor solvent for its unfolded state. Indeed, chain-solvent interactions that are marginally more favorable than nonspecific intrachain interactions are beneficial to protein function because they destabilize deleterious misfolded conformations and inter-chain interactions.

Steric confinement and enhanced local flexibility assist knotting in simple models of protein folding

The chaperonin complex GroEL-GroES is able to accelerate the folding process of knotted proteins considerably. However, the folding mechanism inside the chaperonin cage is elusive. Here we use a combination of lattice and off-lattice Monte Carlo simulations of simple Gō models to study the effect of physical confinement and local flexibility on the folding process of protein model systems embedding a trefoil knot in their native structure. This study predicts that steric confinement plays a specific role in the folding of knotted proteins by increasing the knotting probability for very high degrees of confinement. This effect is observed for protein MJ0366 even above the melting temperature for confinement sizes compatible with the size of the GroEL/GroES chaperonin cage. An enhanced local flexibility produces the same qualitative effects on the folding process. In particular, we observe that knotting probability increases up to 40% in the transition state of protein MJ0366 when flexibility is enhanced. This is underlined by a structural change in the transition state, which becomes devoid of helical content. No relation between the knotting mechanism and flexibility was found in the context of the off-lattice model adopted in this work.

How determinant is N-terminal to C-terminal coupling for protein folding?

The existence of native interactions between the protein termini is a major determinant of the free energy barrier in a two-state folding transition being therefore a critical modulator of protein folding cooperativity.

Probing the origins of two-state folding

Many protein systems fold in a two-state manner. Random models, however, rarely display two-state kinetics and thus such behavior should not be accepted as a default. While theories for the prevalence of two-state kinetics have been presented, none sufficiently explain the breadth of experimental observations. A model, making minimal assumptions, is introduced that suggests two-state behavior is likely for any system with an overwhelmingly populated native state. We show two-state folding is a natural consequence of such two-state thermodynamics, and is strengthened by increasing the population of the native state. Further, the model exhibits hub-like behavior, with slow interconversions between unfolded states. Despite this, the unfolded state equilibrates quickly relative to the folding time. This apparent paradox is readily understood through this model. Finally, our results compare favorable with measurements of folding rates as a function of chain length and Keq, providing new insight into these relations.

A firefly-inspired method for protein structure prediction in lattice models

We introduce a Firefly-inspired algorithmic approach for protein structure prediction over two different lattice models in three-dimensional space. In particular, we consider three-dimensional cubic and three-dimensional face-centred-cubic (FCC) lattices. The underlying energy models are the Hydrophobic-Polar (H-P) model, the Miyazawa–Jernigan (M-J) model and a related matrix model. The implementation of our approach is tested on ten H-P benchmark problems of a length of 48 and ten M-J benchmark problems of a length ranging from 48 until 61. The key complexity parameter we investigate is the total number of objective function evaluations required to achieve the optimum energy values for the H-P model or competitive results in comparison to published values for the M-J model. For H-P instances and cubic lattices, where data for comparison are available, we obtain an average speed-up over eight instances of 2.1, leaving out two extreme values (otherwise, 8.8). For six M-J instances, data for comparison are available for cubic lattices and runs with a population size of 100, where, a priori, the minimum free energy is a termination criterion. The average speed-up over four instances is 1.2 (leaving out two extreme values, otherwise 1.1), which is achieved for a population size of only eight instances. The present study is a test case with initial results for ad hoc parameter settings, with the aim of justifying future research on larger instances within lattice model settings, eventually leading to the ultimate goal of implementations for off-lattice models.

Helix/coil nucleation: a local response to global demands

A complete description of protein structure and function must include a proper treatment of mechanisms that lead to cooperativity. The helix/coil transition serves as a simple example of a cooperative folding process, commonly described by a nucleation-propagation mechanism. The prevalent view is that coil structure must first form a short segment of helix in a localized region despite paying a free energy cost (nucleation). Afterward, helical structure propagates outward from the nucleation site. Both processes entail enthalpy-entropy compensation that derives from the loss in conformational entropy on helix formation with concomitant gain in favorable interactions. Nucleation-propagation models inherently assume that cooperativity arises from a sequential series of local events. An alternative distance constraint model asserts there is a direct link between available degrees of freedom and cooperativity through the nonadditivity in conformational entropy. That is, helix nucleation is a concerted manifestation of rigidity propagating through atomic structure. The link between network rigidity and nonadditivity of conformational entropy is shown in this study by solving the distance constraint model using a simple global constraint counting approximation. Cooperativity arises from competition between excess and deficiency in available degrees of freedom in the coil and helix states respectively.

The folding of knotted proteins: insights from lattice simulations

We carry out systematic Monte Carlo simulations of Gō lattice proteins to investigate and compare the folding processes of two model proteins whose native structures differ from each other due to the presence of a trefoil knot located near the terminus of one of the protein chains. We show that the folding time of the knotted fold is larger than that of the unknotted protein and that this difference in folding time is particularly striking in the temperature region below the optimal folding temperature. Both proteins display similar folding transition temperatures, which is indicative of similar thermal stabilities. By using the folding probability reaction coordinate as an estimator of folding progression we have found out that the formation of the knot is mainly a late folding event in our shallow knot system.

Effective knowledge-based potentials

Empirical or knowledge-based potentials have many applications in structural biology such as the prediction of protein structure, protein-protein, and protein-ligand interactions and in the evaluation of stability for mutant proteins, the assessment of errors in experimentally solved structures, and the design of new proteins. Here, we describe a simple procedure to derive and use pairwise distance-dependent potentials that rely on the definition of effective atomic interactions, which attempt to capture interactions that are more likely to be physically relevant. Based on a difficult benchmark test composed of proteins with different secondary structure composition and representing many different folds, we show that the use of effective atomic interactions significantly improves the performance of potentials at discriminating between native and near-native conformations. We also found that, in agreement with previous reports, the potentials derived from the observed effective atomic interactions in native protein structures contain a larger amount of mutual information. A detailed analysis of the effective energy functions shows that atom connectivity effects, which mostly arise when deriving the potential by the incorporation of those indirect atomic interactions occurring beyond the first atomic shell, are clearly filtered out. The shape of the energy functions for direct atomic interactions representing hydrogen bonding and disulfide and salt bridges formation is almost unaffected when effective interactions are taken into account. On the contrary, the shape of the energy functions for indirect atom interactions (i.e., those describing the interaction between two atoms bound to a direct interacting pair) is clearly different when effective interactions are considered. Effective energy functions for indirect interacting atom pairs are not influenced by the shape or the energy minimum observed for the corresponding direct interacting atom pair. Our results suggest that the dependency between the signals in different energy functions is a key aspect that need to be addressed when empirical energy functions are derived and used, and also highlight the importance of additivity assumptions in the use of potential energy functions.

The nucleation mechanism of protein folding: a survey of computer simulation studies

The nucleation mechanism of protein folding, originally proposed by Baldwin in the early 1970s, was firstly observed by Shakhnovich and co-workers two decades later in the context of Monte Carlo simulations of a simple lattice model. At about the same time the extensive use of φ-value analysis provided the first experimental evidence that the folding of Chymotrypsin-inhibitor 2, a small single-domain protein, which folds with two-state kinetics, is also driven by a nucleation mechanism. Since then, the nucleation mechanism is generally considered the most common form of folding mechanism amongst two-state proteins. However, recent experimental data has put forward the idea that this may not necessarily be so, since the accuracy of the experimentally determined φ values, which are used to identify the critical (i.e. nucleating) residues, is typically poor. Here, we provide a survey of in silico results on the nucleation mechanism, ranging from simple lattice Monte Carlo to more sophisticated off-lattice molecular dynamics simulations, and discuss them in light of experimental data.

Population-based local search for protein folding simulation in the MJ energy model and cubic lattices

We present experimental results on benchmark problems in 3D cubic lattice structures with the Miyazawa-Jernigan energy function for two local search procedures that utilise the pull-move set: (i) population-based local search (PLS) that traverses the energy landscape with greedy steps towards (potential) local minima followed by upward steps up to a certain level of the objective function; (ii) simulated annealing with a logarithmic cooling schedule (LSA). The parameter settings for PLS are derived from short LSA-runs executed in pre-processing and the procedure utilises tabu lists generated for each member of the population. In terms of the total number of energy function evaluations both methods perform equally well, however, PLS has the potential of being parallelised with an expected speed-up in the region of the population size. Furthermore, both methods require a significant smaller number of function evaluations when compared to Monte Carlo simulations with kink-jump moves.

What have we learned from the studies of two-state folders, and what are the unanswered questions about two-state protein folding?

Small proteins with globular structures often fold by simple all-or-none mechanisms, both in an equilibrium and a kinetic sense, despite the very large number of partly folded conformations available. This type of 'two-state' folding will be discussed in terms of experimental tests, underlying molecular mechanisms, and limits to two-state behavior. Factors that appear to be important for two-state folding include topology (sequence distance of contacts in the native structure), molecular cooperativity and local energy distribution. Because their local stability distributions and cooperativities can be dissected and analyzed separately from topological features, recent studies of the folding of symmetric proteins will be discussed as a means to better understand the origins of two-state folding.

Direct measurement of tertiary contact cooperativity in RNA folding

All structured biological macromolecules must overcome the thermodynamic folding problem to populate a unique functional state among a vast ensemble of unfolded and alternate conformations. The exploration of cooperativity in protein folding has helped reveal and distinguish the underlying mechanistic solutions to this folding problem. Analogous dissections of RNA tertiary stability remain elusive, however, despite the central biological importance of folded RNA molecules and the potential to reveal fundamental properties of structured macromolecules via comparisons of protein and RNA folding. We report a direct quantitative measure of tertiary contact cooperativity in a folded RNA. We precisely measured the stability of an independently folding P4-P6 domain from the Tetrahymena thermophila group I intron by single molecule fluorescence resonance energy transfer (smFRET). Using wild-type and mutant RNAs, we found that cooperativity between the two tertiary contacts enhances P4-P6 stability by 3.2 +/- 0.2 kcal/mol.

Predicting coupling limits from an experimentally determined energy landscape

Repeat proteins are composed of tandem structural modules in which close contacts do not extend beyond adjacent repeats. Despite the local nature of these close contacts, repeat proteins often unfold as a single, highly coupled unit. Previous studies on the Notch ankyrin domain suggest that this lack of equilibrium unfolding intermediates results both from stabilizing interfaces between each repeat and from a roughly uniform distribution of stability across the folding energy landscape. To investigate this idea, we have generated 15 variants of the Notch ankyrin domain with single and multiple destabilizing substitutions that make the energy landscape uneven. By applying a free energy additivity analysis to these variants, we quantified the destabilization threshold over which repeats 6 and 7 decouple from repeats 1-5. The free energy coupling limit suggested by this additivity analysis ( approximately 4 kcal/mol) is also reflected in m-value analysis and in differences among equilibrium unfolding transitions as monitored by CD versus fluorescence for all 15 variants. All of these observations are quantitatively predicted by analyzing the response of the experimentally determined energy landscape to increasing unevenness. These results highlight the importance of a uniform distribution of local stability in achieving cooperative unfolding.