Decoding How Proteins Fold

Correlation between Protein Stability Cores and Protein Folding Kinetics: A Case Study on Pseudomonas aeruginosa Apo-Azurin

The folding pathways of some proteins include the population of partially structured species en route to the native state. Identification and characterization of these folding intermediates are particularly difficult as they are often only transiently populated and play different mechanistic roles, being either on-pathway productive species or off-pathway kinetic traps. To define the role of folding intermediates, a quantitative analysis of the folding and unfolding rate constants over a wide range of denaturant concentration is often required. Such a task is further complicated by the reversible nature of the folding reaction, which implies the observed kinetics to be governed by a complex combination of different microscopic rate constants. Here, we tackled this problem by measuring directly the folding rate constant under highly denaturing conditions, namely by inducing the folding of a PDZ domain through a quasi-irreversible binding reaction with a specific peptide. In analogy with previous works based on hydrogen exchange experiments, we present evidence that the folding pathway of the PDZ domain involves the formation of an obligatory on-pathway intermediate. The results presented exemplify a novel type of kinetic test to detect an on-pathway folding intermediate.

Review of the fourth Johns Hopkins Protein Folding Meeting

In appropriate physiological milieux proteins spontaneously fold into their functional three-dimensional structures. The amino acid sequences of functional proteins contain all the information necessary to specify the folds. This remarkable observation has spawned research aimed at answering two major questions. (1) Of all the conceivable structures that a protein can adopt, why is the ensemble of native-like structures the most favorable? (2) What are the paths by which proteins manage to robustly and reproducibly fold into their native structures? Anfinsen's thermodynamic hypothesis has guided the pursuit of answers to the first question whereas Levinthal's paradox has influenced the development of models for protein folding dynamics.

The first part of this paper contains an overview of protein structures, their spontaneous formation ("folding"), and the thermodynamic and kinetic aspects of this phenomenon, as revealed by in vitro experiments. It is stressed that universal features of folding are observed near the point of thermodynamic equilibrium between the native and denatured states of the protein. Here the "two-state" ("denatured state" <--> "native state") transition proceeds without accumulation of metastable intermediates, but includes only the unstable "transition state". This state, which is the most unstable in the folding pathway, and its structured core (a "nucleus") are distinguished by their essential influence on the folding/unfolding kinetics. In the second part of the paper, a theory of protein folding rates and related phenomena is presented. First, it is shown that the protein size determines the range of a protein's folding rates in the vicinity of the point of thermodynamic equilibrium between the native and denatured states of the protein. Then, we present methods for calculating folding and unfolding rates of globular proteins from their sizes, stabilities and either 3D structures or amino acid sequences. Finally, we show that the same theory outlines the location of the protein folding nucleus (i.e., the structured part of the transition state) in reasonable agreement with experimental data.

In the first part of this paper we overview protein structures, their spontaneous formation (“folding”) and thermodynamic and kinetic aspects of this phenomenon. It is stressed that universal features of folding are observed near the point of thermodynamic equilibrium between the native and denatured states of the protein. Here the “two-state” (“denatured state” ↔ “native state”) transition proceeds without accumulation of metastable intermediates, and only the transition state, i.e., the most unstable state in the folding pathway, is outlined by its essential influence on the folding/unfolding kinetics. In the second part of the paper, a theory of protein folding rates and related phenomena is presented. First, it is shown that the protein size determines the range of protein’s folding rates in the vicinity of the point of thermodynamic equilibrium between the native and denatured states of the protein. Then we present methods for calculating folding and unfolding rates of globular proteins from their sizes, stabilities and either 3D structures or amino acid sequences. And, at last, we show that the same theory outlines the location of the protein folding nucleus (i.e., the structured part of transition state) in a reasonable concordance with experimental data.

Metaphysics of the principle of least action

The first part of this paper contains an overview of protein structures, their spontaneous formation (&quot;folding&quot;), and the thermodynamic and kinetic aspects of this phenomenon, as revealed by in vitro and in vivo experiments. It is stressed that universal features of the in vitro folding are observed near the point of thermodynamic equilibrium between the native and denatured states of the protein. Here the &quot;two-state&quot; (&quot;denatured state&quot; &amp;#8596; &quot;native state&quot;) transition proceeds without accumulation of metastable intermediates; this facilitates investigation of the &quot;transition state&quot;. This state, which is the most unstable in the folding pathway, and its structured core (a &quot;nucleus&quot;) are distinguished by their essential influence on the folding/unfolding kinetics. In the second part of the paper, a theory of protein folding rates and related phenomena is presented. First, it is shown that the protein size determines the range of a protein’s folding rates in the vicinity of the point of thermodynamic equilibrium between the native and denatured states of the protein. Then, we present methods for calculating folding and unfolding rates of globular proteins from their sizes, stabilities and either 3D structures or amino acid sequences. Finally, we show that the same theory outlines the location of the protein folding nucleus (i.e., the structured part of the transition state) in reasonable agreement with experimental data.

The Structure of a Two-Disulfide Intermediate Assists in Elucidating the Oxidative Folding Pathway of a Cyclic Cystine Knot Protein

The paper delves into the methodological aspects of how foundational mathematical and physical tenets, most notably the principle of least action, are interpreted and assimilated within humanities discourse. The pursuit of the article’s objectives is driven by the necessity for a philosophical and methodological analysis of the current conceptual status of the principle of least action. This analysis is informed by cognitive-axiological and teleological imperatives of a “synthetic” development program for the principle. Any fundamental principle will not have a definitive explanation, as otherwise it would not be fundamental, but in this case, some justification can be given based on deeper grounds discussed in the article. Drawing on the epistemological frameworks of French philosophers P. Hadot, E. Levinas, J. Bouveresse, and J.-T. Desanti, the article weaves together mathematical abstractions with human experience and philosophical doctrines with physical theories. J.-T. Desanti’s perspective on mathematical objects as an outcome of human activity is examined. Also scrutinized is B. Nicolescu’s concept of transdisciplinarity, which challenges the traditional subject-object dualism in science. The author’s methodological stance emerges from a dialectical viewpoint, one that eschews a simplistic dichotomy between materialism and idealism and is grounded in rigorous scientific inquiry and an exhaustive examination of the subject matter itself. The principle of least action, as a paramount principle in physics, is shown to exemplify a legacy of innovation, positioning it as both methodologically insightful and heuristically valuable. The paper also highlights how this principle diverges from the classical principle of economy. The broader goal – within the context of “sustainable development,” transdisciplinary studies, and creative industries – is to establish the principle of least action as a paradigmatic imperative for interaction within social and economic systems. The conclusions drawn from this study contribute to a deeper understanding of the principle’s essence, the nature of transdisciplinarity, and confront the vestiges of scientism in the humanities.

Protein folding is the process that a protein molecule transforms from the linear polymer of peptides to a three-dimensional native structure with specific biological function. By now, the protein folding problem has been studied for more than 50 years and already became a broad and active research field. To answer the 58th question raised by Science in 2005, in this article we briefly reviewed the background and research history of the protein folding problem, and introduced the progresses of protein folding prediction research from four aspects: the protein folding process prediction (protein folding simulation), the folding process related parameter prediction, the protein folding result prediction (protein structure prediction), and the folding result related parameter prediction. The studies on the protein folding problem began in the 60s of 20th century, with the efforts to seek a solution to the paradox that a protein can actually form a native 3D structure in only several seconds but the time scale estimated by a thermodynamic ergodic hypothesis would be longer than the age of universe. Computer simulation is an important approach for protein folding study. The protein models can be classified into 3 categories: lattice model, off-lattice model and all-atom model. The current knowledge about protein folding mechanism is based on the concept of folding funnel on a free-energy landscape, and the current opinion is that the protein folding mechanism is not unique for the whole protein universe and that there may exist a continuum between the two extreme ends of hierarchical folding and nucleation folding scenarios. The hardware for protein folding simulation was becoming more powerful; distributed systems (e.g, Folding@home), special-purpose machines (e.g, ANTON), and GPU-based platforms have been developed for protein folding simulation. Meanwhile, the folding simulation software was continuously enhanced. An important issue in protein folding simulation is to overcome the local energy barrier to find the global energy minimum; several approaches such as replica-exchange, multi-scale modeling and Modeling Employing Limited Data (MELD) were developed to tackle this issue; human intelligence involvement (e.g, “Foldit” Game) is another interesting effort. During the past two decades, the ability of protein folding simulation was continuously rising. For now, the folding simulation for the proteins with dozens of amino acids can reach a time scale of millisecond, while the protein size able to do effective folding simulation is around 100 amino acids. The targets of protein folding simulation have been largely expanded and now include both the in vitro and the in vivo folding such as co-translational folding, chaperone-assistant folding, small-molecule- induced folding and metal-coupled folding. Folding rate and folding type are two important parameters related with the protein folding process and now they can be predicted by statistical and machine-learning approaches based on different levels of structural features such as the topological properties of tertiary structure, the contents of secondary structure and the amino acid frequencies of primary structure. The result of a protein folding process is the formation of a protein structure. According to the hierarchy of structural organization, the protein structure prediction problem includes secondary structure prediction, tertiary structure prediction and quaternary structure prediction. By now, the secondary structure prediction algorithm has experienced five generations and the current accuracy is about 80% for 3-classes prediction. The tertiary structure prediction approaches mainly include two categories: template-based modeling and free modeling, with the former having higher accuracy and the latter having larger application scope. The quaternary structure prediction includes the prediction of complex structure and the prediction of the possibility of protein-protein interaction, and these predictions can be performed based on protein 3D structure or merely amino acid sequence. Structure related parameter prediction also attracted research interests, including the predictions of protein structural classes, secondary structure contents, disordered regions, solvent accessible surface region and the amino acid contacting pairs in the interface of protein-protein interaction. In the end, some possible development directions worth noticing in the future of protein folding research were suggested and they are: the coupling between protein folding and binding, the fusion of protein folding research with systems biology and the application of deep-learning techniques in the field of protein folding prediction.

Relativistic non-Hamiltonian mechanics

Understanding the relationship between amino acid sequences and folding rate of proteins is a challenging task similar to protein folding problem. In this work, we have analyzed the relative importance of protein sequence and structure for predicting the protein folding rates in terms of amino acid properties and contact distances, respectively. We found that the parameters derived with protein sequence (physical-chemical, energetic, and conformational properties of amino acid residues) show very weak correlation (|r| < 0.39) with folding rates of 28 two-state proteins, indicating that the sequence information alone is not sufficient to understand the folding rates of two-state proteins. However, the maximum positive correlation obtained for the properties, number of medium-range contacts, and alpha-helical tendency reveals the importance of local interactions to initiate protein folding. On the other hand, a remarkable correlation (r varies from -0.74 to -0.88) has been obtained between structural parameters (contact order, long-range order, and total contact distance) and protein folding rates. Further, we found that the secondary structure content and solvent accessibility play a marginal role in determining the folding rates of two-state proteins. Multiple regression analysis carried out with the combination of three properties, beta-strand tendency, enthalpy change, and total contact distance improved the correlation to 0.92 with protein folding rates. The relative importance of existing methods along with multiple-regression model proposed in this work will be discussed. Our results demonstrate that the native-state topology is the major determinant for the folding rates of two-state proteins.

In extracting folding and unfolding rate information from our apparent rate constant data, and in fitting the extracted folding rates as a function of temperature to Kramer's model, we inadvertently used the pre-jump equilibrium temperature instead of the post-jump temperature: a difference of 12 °C. The absolute folding and unfolding rate values for each WW variant in Tables 3 and ​and44 are different than reported in the original paper by 2-fold, but most of the folding and unfolding rate ratios (comparing the folding or unfolding rates of the glycosylated vs. non-glycosylated WW variants) are similar to the values presented in the original paper. The changes to the data do not change the conclusion of our paper that specific, evolved protein-glycan contacts must also play a role in mediating the beneficial energetic effects on protein folding that glycosylation can confer. Table 3 Experimentally Measured and Computationally Predicted Folding Rates for Pin WW Variants Having Either Asn or Asn-GlcNAc at the Indicated Positionsa Table 4 Experimentally Measured and Computationally Predicted Unfolding Rates for Pin WW Variants Having Either Asn or Asn-GlcNAc at the Indicated Positionsa The corrected data appear in Tables 3 and ​and44 below. The corrected versions of Figures S44 to S55 (the data that Tables 3 and ​and44 summarize) are provided in the Supporting Information. The following sentences in the Results section text referring to our kinetic data (pp. 15364): The modest stabilizing effect of the Asn to Asn-GlcNAc substitution at positions 20 and 30 appears to be a result, in part, of an increased folding rate. Glycosylation at position 20 (compare 20 with 20g) increases the folding rate ⍰1.1-fold (Table 3). Glycosylation at position 30 (compare 30 with 30g) has a similar effect. These small folding rate increases agree with the predictions of the computational model and could be consistent with a small amount of denatured-state destabilization as a consequence of glycosylation. However, the decreased unfolding rate of 20g relative to 20 (also predicted by the model) could indicate that glycosylation at position 20 actually stabilizes the folding transition state and native state simultaneously. are changed to: The modest stabilizing effect of the Asn to Asn-GlcNAc substitution at position 20 appears to be primarily due to an increased folding rate (20g folds 1.5 times faster than 20). This folding rate increase agrees with the predictions of the computational model, and could be consistent with a small amount of denatured-state destabilization as a consequence of glycosylation. However, the unfolding rates of 20g and 20 are indistinguishable, which disagrees with the predicted decrease in unfolding rate upon glycosylation. The stabilizing effect of glycosylation at position 30 appears to come primarily from a decrease in unfolding rate (30g unfolds 0.7 times as fast as non-glycosylated Pin WW), as predicted by the model. This decrease compensates for an unexpected decrease in folding rate: 30g folds 0.8 times as fast as Pin WW whereas the model predicted a large increase in folding rate. Results for 33 and 33g are similarly inconsistent with the predictions of the model. In addition, the following sentence in the Discussion section text about our kinetic data (pp. 15366): For example, the observed increased stability and increased folding rate of 20g relative to 20, and of 30g relative to 30, could be the result of simultaneous transition state and native state stabilization (rather than destabilization of the denatured ensemble), reflecting the presence of favorable glycan-protein contacts at these positions. is changed to: For example, the observed increased stability of 20g and 30g relative to 20 and 30, respectively, could be the result of favorable native-state GlcNAc-protein contacts at these positions.

The dominant fish populations in undisturbed arctic lakes are characterized as being in a state of "least specific dissipation": the greatest biomass attainable for a given energy input. A survey of autonomous ecosystems in various parts of the world indicated that this pattern is widespread. It is concluded that ecosystems are formed at the point of intersection of two established physical principles: the "principle of most action" (≈least dissipation or conservation of free energy) and the "principle of least action". "Action" is defined as the product of energy times time (joule-seconds). The trend to "most action" necessitates deceleration of energy flow: "least action" accelerates energy flow. For an ecosystem to survive over ecological time, the principle of most action must override the principle of least action. In that different species of organism have different capacities to conserve free energy (increase action), a hierarchy is formed locally in which action increases at each hierarchical level. Over the long term, as a result of genetic instability, both principles induce change, but the principle of least action dominates system behaviour causing increasingly rapid energy dissipation. Evolution is the resultant of these two countervailing forces.