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

Albumin/Mitotane Interaction Affects Drug Activity in Adrenocortical Carcinoma Cells: Smoke and Mirrors on Mitotane Effect with Possible Implications for Patients' Management

BACKGROUND

Mitotane is the only drug approved for the treatment of adrenocortical carcinoma (ACC). Although it has been used for many years, its mechanism of action remains elusive. H295R cells are, in ACC, an essential tool to evaluate drug mechanisms, although they often lead to conflicting results.

METHODS

Using different in vitro biomolecular technologies and biochemical/biophysical experiments, we evaluated how the presence of "confounding factors" in culture media and patient sera could reduce the pharmacological effect of mitotane and its metabolites.

RESULTS

We discovered that albumin, the most abundant protein in the blood, was able to bind mitotane. This interaction altered the effect of the drug by blocking its biological activity. This blocking effect was independent of the albumin source or methodology used and altered the assessment of drug sensitivity of the cell lines.

CONCLUSIONS

In conclusion, we have for the first time demonstrated that albumin does not only act as an inert drug carrier when mitotane or its metabolites are present. Indeed, our experiments clearly indicated that both albumin and human serum were able to suppress the pharmacological effect of mitotane in vitro. These experiments could represent a first step towards the individualization of mitotane treatment in this rare tumor.

Assessing Site-specific PEGylation of TEM-1 β-lactamase with Cell-free Protein Synthesis and Coarse-grained Simulation

PEGylation is a broadly used strategy to enhance the pharmacokinetic properties of therapeutic proteins. It is well established that the location and extent of PEGylation have a significant impact on protein properties. However, conventional PEGylation techniques have limited control over PEGylation sites. Emerging site-specific PEGylation technology provides control of PEG placement by conjugating PEG polymers via click chemistry reaction to genetically encoded non-canonical amino acids. Unfortunately, a method to rapidly determine the optimal PEGylation location has yet to be established. Here we seek to address this challenge. In this work, coarse-grained molecular dynamic simulations are paired with high-throughput experimental screening utilizing cell-free protein synthesis to investigate the effect of site-specific PEGylation on the two-state folder protein TEM-1 β-lactamase. Specifically, the conjugation efficiency, thermal stability, and enzymatic activity are studied for the enzyme PEGylated at several different locations. The results of this analysis confirm that the physical properties of the PEGylated protein vary considerably with PEGylation site and that traditional design recommendations are insufficient to predict favorable PEGylation sites. In this study, the best predictor of the most favorable conjugation site is coarse-grained simulation. Thus, we propose a dual combinatorial screening approach in which coarse-grained molecular simulation informs site selection for high-throughput experimental verification.

Topological principles of protein folding

What is the topology of a protein and what governs protein folding to a specific topology? This is a fundamental question in biology. The protein folding reaction is a critically important cellular process, which is failing in many prevalent diseases. Understanding protein folding is also key to the design of new proteins for applications. However, our ability to predict the folding of a protein chain is quite limited and much is still unknown about the topological principles of folding. Current predictors of folding kinetics, including the contact order and size, present a limited predictive power, suggesting that these models are fundamentally incomplete. Here, we use a newly developed mathematical framework to define and extract the topology of a native protein conformation beyond knot theory, and investigate the relationship between native topology and folding kinetics in experimentally characterized proteins. We show that not only the folding rate, but also the mechanistic insight into folding mechanisms can be inferred from topological parameters. We identify basic topological features that speed up or slow down the folding process. The approach enabled the decomposition of protein 3D conformation into topologically independent elementary folding units, called circuits. The number of circuits correlates significantly with the folding rate, offering not only an efficient kinetic predictor, but also a tool for a deeper understanding of theoretical folding models. This study contributes to recent work that reveals the critical relevance of topology to protein folding with a new, contact-based, mathematically rigorous perspective. We show that topology can predict folding kinetics when geometry-based predictors like contact order and size fail.

First-passage times in protein folding: exploring the native-like states vs. overcoming the free energy barrier

Using a model β-hairpin protein as a representative example of simple two-state folders whose kinetics are uncomplicated by the presence of on- and off-pathway intermediates, it is studied how the search for the protein's native state among native-like states affects the folding kinetics. It is revealed that the first-passage time (FPT) distributions are essentially single-exponential not only for the times to overcome the free energy barrier between the unfolded and native-like states but also for the times to find the native state among the native-like ones. The FPT distributions of this type are observed through all studied two-state-like regimes of protein folding, varying from a regime close to two-state folding to a regime close to downhill folding. If the protein explores native-like states for a time much longer than the time to overcome the free energy barrier, which is characteristic of high temperatures, the resulting FPT distribution to reach the native state remains close to exponential but the mean FPT (MFPT) is determined not by the height of the free energy barrier but by the time to explore native-like states. In particular, the mean time to overcome the free energy barrier is in reasonable agreement with the Kramers rate formula and generally far shorter than the overall MFPT to reach the native state. The observed increase of the overall MFPT, as a result of longer exploration of native-like states, may lead to an overestimate of the height of the free energy barrier between the unfolded and folded states when it is calculated from the overall MFPT.

The Molten Globule, and Two-State vs. Non-Two-State Folding of Globular Proteins

From experimental studies of protein folding, it is now clear that there are two types of folding behavior, i.e., two-state folding and non-two-state folding, and understanding the relationships between these apparently different folding behaviors is essential for fully elucidating the molecular mechanisms of protein folding. This article describes how the presence of the two types of folding behavior has been confirmed experimentally, and discusses the relationships between the two-state and the non-two-state folding reactions, on the basis of available data on the correlations of the folding rate constant with various structure-based properties, which are determined primarily by the backbone topology of proteins. Finally, a two-stage hierarchical model is proposed as a general mechanism of protein folding. In this model, protein folding occurs in a hierarchical manner, reflecting the hierarchy of the native three-dimensional structure, as embodied in the case of non-two-state folding with an accumulation of the molten globule state as a folding intermediate. The two-state folding is thus merely a simplified version of the hierarchical folding caused either by an alteration in the rate-limiting step of folding or by destabilization of the intermediate.

Modulating long-range energetics via helix stabilization: A case study using T4 lysozyme

Cooperative protein folding requires distant regions of a protein to interact and provide mutual stabilization. The mechanism of this long-distance coupling remains poorly understood. Here, we use T4 lysozyme (T4L*) as a model to investigate long-range communications across two subdomains of a globular protein. T4L* is composed of two structurally distinct subdomains, although it behaves in a two-state manner at equilibrium. The subdomains of T4L* are connected via two topological connections: the N-terminal helix that is structurally part of the C-terminal subdomain (the A-helix) and a long helix that spans both subdomains (the C-helix). To understand the role that the C-helix plays in cooperative folding, we analyzed a circularly permuted version of T4L* (CP13*), whose subdomains are connected only by the C-helix. We demonstrate that when isolated as individual fragments, both subdomains of CP13* can fold autonomously into marginally stable conformations. The energetics of the N-terminal subdomain depend on the formation of a salt bridge known to be important for stability in the full-length protein. We show that the energetic contribution of the salt bridge to the stability of the N-terminal fragment increases when the C-helix is stabilized, such as occurs upon folding of the C-terminal subdomain. These results suggest a model where long-range energetic coupling is mediated by helix stabilization and not specific tertiary interactions.

Heterogeneity in the Folding of Villin Headpiece Subdomain HP36

Small single domain proteins that fold on the microsecond time scale have been the subject of intense interest as models for probing the complexity of folding energy landscapes. The villin headpiece subdomain (HP36) has been extensively studied because of its simple three helix structure, ultrafast folding lifetime of a few microseconds, and stable native fold. We have previously shown that folding as measured by a single ¹³C═¹⁸O isotopic label on residue A57 in helix 2 occurs at a different rate than that measured by global probes of folding, indicating noncooperative complexity in the folding of HP36. In order to determine whether this complexity reflects intermediates or parallel pathways over a small activation barrier, ¹³C═¹⁸O labels were individually incorporated at six different positions in HP36, including into all 3 helices. The equilibrium thermal unfolding transitions and the folding/unfolding dynamics were monitored using the unique IR signature of the ¹³C═¹⁸O label by temperature dependent FTIR and temperature jump IR spectroscopy, respectively. Equilibrium experiments reveal that the ¹³C═¹⁸O labels at different positions in HP36 show drastic differences in the midpoint of their transitions ( T_m), ranging from 45 to 67 °C. Heterogeneity is also observed in the relaxation kinetics; there are differences in the microsecond phase when different labeled positions are probed. At a final temperature of 45 °C, the relaxation rate for ¹³C═¹⁸O A57 is 2.4e + 05 s^-1 whereas for ¹³C═¹⁸O L69 HP36 the relaxation rate is 5.1e + 05 s^-1, two times faster. The observation of site-dependent midpoints for the equilibrium unfolding transitions and differences in the relaxation rates of the labeled positions enables us to probe the progressive accumulation of the folded structure, providing insight into the microscopic details of the folding mechanism.

Estimating the mean first passage time of protein misfolding

Most theoretical and experimental studies confirm that proteins fold in the time scale of microseconds to milliseconds, but the kinetics of the protein misfolding remains largely unexplored. The kinetics of unfolding-folding-misfolding equilibrium in proteins is formulated in the analytical framework of the Master equation. The folded, unfolded and the misfolded state are characterized in terms of their respective contacts. The Mean First Passage Time (MFPT) to acquire the misfolded conformation from the native or folded state is derived from this equation with different boundary conditions. The MFPT is found to be practically independent of the length of the protein, the number of native contacts and the rate constant for the misfolded to the folded state. The results obtained from the survival probability are directly correlated to the age of onset and appearance of misfolding diseases in humans.

Identification and Characterization of an Inside-Out Folding Intermediate of T4 Phage Sliding Clamp

Protein folding process involves formation of transiently occurring intermediates that are difficult to isolate and characterize. It is both necessary and interesting to characterize the structural conformations adopted by these intermediates, also called molten globules (MG), to understand protein folding. Here, we investigated the equilibrium (un)folding intermediate state of T4 phage gene product 45 (gp45, also known as DNA polymerase processivity factor or sliding clamp) obtained during chemical denaturation. We show that gp45 undergoes substantial conformational rearrangement during unfolding and forms an expanded dry-MG. By monitoring the fluorescence of tryptophans that were strategically introduced at various sites, we demonstrate that the urea-treated molecule has its surface residues flip inside the core, and closely placed residues move farther. We were also able to isolate and purify the MG form of gp45 in native condition (i.e., nondenaturing buffer, at physiological pH and temperature); characteristics of this purified molecule substantially match with urea-treated wild-type gp45. To the best of our knowledge, this is one of the few reports that demonstrate the isolation and purification of a protein folding intermediate in native condition. We believe that our work not only allows us to dissect the process of protein folding, but will also help in the designing of folding inhibitors against sliding clamps to treat a wide variety of diseases from bacterial infection to cancer, due to the vast presence of clamps in all the domains of life.

Globular Protein Folding In Vitro and In Vivo

In vitro, computational, and theoretical studies of protein folding have converged to paint a rich and complex energy landscape. This landscape is sensitively modulated by environmental conditions and subject to evolutionary pressure on protein function. Of these environments, none is more complex than the cell itself, where proteins function in the cytosol, in membranes, and in different compartments. A wide variety of kinetic and thermodynamics experiments, ranging from single-molecule studies to jump kinetics and from nuclear magnetic resonance to imaging on the microscope, have elucidated how protein energy landscapes facilitate folding and how they are subject to evolutionary constraints and environmental perturbation. Here we review some recent developments in the field and refer the reader to some original work and additional reviews that cover this broad topic in protein science.

Equalizing the information amounts of protein and mRNA by information theory

Based on the Shannon's information communication theory, information amount of the entire length of a polymeric macromolecule can be calculated in bits through adding the entropies of each building block. Proteins, DNA and RNA are such macromolecules. When only the building blocks' variation is considered as the source of entropy, there is seemingly lower information in case of the protein if this approach is applied directly on a protein of specific size and the coding sequence size of the mRNA corresponding to the particular length of the protein. This decrease in the information amount seems contradictory but this apparent conflict is resolved by considering the conformational variations in proteins as a new variable in the calculation and balancing the approximated entropy of the coding part of the mRNA and the protein. Probabilities can change therefore we also assigned hypothetical probabilities to the conformational states, which represent the uneven distribution as the time spent in one conformation, providing the probability of the presence in either or one of the possible conformations. Results that are obtained by using hypothetical probabilities are in line with the experimental values of variations in the conformational-state of protein populations. This equalization approach has further biological relevance that it compensates for the degeneracy in the codon usage during protein translation and it leads to the conclusion that the alphabet size for the protein is rather optimal for the proper protein functioning within the thermodynamic milieu of the cell. The findings were also discussed in relation to the codon bias and have implications in relation to the codon evolution concept. Eventually, this work brings the fields of protein structural studies and molecular protein translation processes together with a novel approach.

How cooperative are protein folding and unfolding transitions?

A thermodynamically and kinetically simple picture of protein folding envisages only two states, native (N) and unfolded (U), separated by a single activation free energy barrier, and interconverting by cooperative two-state transitions. The folding/unfolding transitions of many proteins occur, however, in multiple discrete steps associated with the formation of intermediates, which is indicative of reduced cooperativity. Furthermore, much advancement in experimental and computational approaches has demonstrated entirely non-cooperative (gradual) transitions via a continuum of states and a multitude of small energetic barriers between the N and U states of some proteins. These findings have been instrumental towards providing a structural rationale for cooperative versus noncooperative transitions, based on the coupling between interaction networks in proteins. The cooperativity inherent in a folding/unfolding reaction appears to be context dependent, and can be tuned via experimental conditions which change the stabilities of N and U. The evolution of cooperativity in protein folding transitions is linked closely to the evolution of function as well as the aggregation propensity of the protein. A large activation energy barrier in a fully cooperative transition can provide the kinetic control required to prevent the accumulation of partially unfolded forms, which may promote aggregation. Nevertheless, increasing evidence for barrier-less "downhill" folding, as well as for continuous "uphill" unfolding transitions, indicate that gradual non-cooperative processes may be ubiquitous features on the free energy landscape of protein folding.

The Folding of de Novo Designed Protein DS119 via Molecular Dynamics Simulations

As they are not subjected to natural selection process, de novo designed proteins usually fold in a manner different from natural proteins. Recently, a de novo designed mini-protein DS119, with a βαβ motif and 36 amino acids, has folded unusually slowly in experiments, and transient dimers have been detected in the folding process. Here, by means of all-atom replica exchange molecular dynamics (REMD) simulations, several comparably stable intermediate states were observed on the folding free-energy landscape of DS119. Conventional molecular dynamics (CMD) simulations showed that when two unfolded DS119 proteins bound together, most binding sites of dimeric aggregates were located at the N-terminal segment, especially residues 5-10, which were supposed to form β-sheet with its own C-terminal segment. Furthermore, a large percentage of individual proteins in the dimeric aggregates adopted conformations similar to those in the intermediate states observed in REMD simulations. These results indicate that, during the folding process, DS119 can easily become trapped in intermediate states. Then, with diffusion, a transient dimer would be formed and stabilized with the binding interface located at N-terminals. This means that it could not quickly fold to the native structure. The complicated folding manner of DS119 implies the important influence of natural selection on protein-folding kinetics, and more improvement should be achieved in rational protein design.

Complexity of the folding transition of the B domain of protein A revealed by the high-speed tracking of single-molecule fluorescence time series

The equilibrium unfolding transition of the B domain of protein A (BdpA) was investigated by using single-molecule fluorescence spectroscopy based on line-confocal detection of fast-flowing samples. The method achieved the time resolution of 120 μs and the observation time of a few milliseconds in the single-molecule time-series measurements of fluorescence resonance energy transfer (FRET). Two samples of BdpA doubly labeled with donor and acceptor fluorophores, the first possessing fluorophores at residues 22 and 55 (sample 1) and the second at residues 5 and 55 (sample 2), were prepared. The equilibrium unfolding transition induced by guanidium chloride (GdmCl) was monitored by bulk measurements and demonstrated that the both samples obey the apparent two-state unfolding. In the absence of GdmCl, the single-molecule FRET measurements for the both samples showed a single peak assignable to the native state (N). The FRET efficiency for N shifts to lower values as the increase of GdmCl concentration, suggesting the swelling of the native state structure. At the higher concentration of GdmCl, the both samples convert to the unfolded state (U). Near the unfolding midpoint for sample 1, the kinetic exchange between N and U causes the averaging of the two states and the higher values of the relative fluctuation. The time series for different molecules in U showed slightly different FRET efficiencies, suggesting the apparent heterogeneity. Thus, the high-speed tracking of fluorescence signals from single molecules revealed a complexity and heterogeneity hidden in the apparent two-state behavior of protein folding.

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.

Effects of desolvation barriers and sidechains on local–nonlocal coupling and chevron behaviors in coarse-grained models of protein folding

Coarse-grained protein chain models with desolvation barriers or sidechains lead to stronger local–nonlocal coupling and more linear chevron plots.

Symmetric protein architecture in protein design: top-down symmetric deconstruction

Top-down symmetric deconstruction (TDSD) is a joint experimental and computational approach to generate a highly stable, functionally benign protein scaffold for intended application in subsequent functional design studies. By focusing on symmetric protein folds, TDSD can leverage the dramatic reduction in sequence space achieved by applying a primary structure symmetric constraint to the design process. Fundamentally, TDSD is an iterative symmetrization process, in which the goal is to maintain or improve properties of thermodynamic stability and folding cooperativity inherent to a starting sequence (the "proxy"). As such, TDSD does not attempt to solve the inverse protein folding problem directly, which is computationally intractable. The present chapter will take the reader through all of the primary steps of TDSD-selecting a proxy, identifying potential mutations, establishing a stability/folding cooperativity screen-relying heavily on a successful TDSD solution for the common β-trefoil fold.

The RING domain of the scaffold protein Ste5 adopts a molten globular character with high thermal and chemical stability

Ste5 is a scaffold protein that controls the pheromone response of the MAP-kinase cascade in yeast cells. Upon pheromone stimulation, Ste5 (through its RING-H2 domain) interacts with the β and γ subunits of an activated heterodimeric G protein and promotes activation of the MAP-kinase cascade. With structural and biophysical studies, we show that the Ste5 RING-H2 domain exists as a molten globule under native buffer conditions, in yeast extracts, and even in denaturing conditions containing urea (7 M). Furthermore, it exhibits high thermal stability in native conditions. Binding of the Ste5 RING-H2 domain to the physiological Gβ/γ (Ste4/Ste18) ligand is accompanied by a conformational transition into a better folded, more globular structure. This study reveals novel insights into the folding mechanism and recruitment of binding partners by the Ste5 RING-H2 domain. We speculate that many RING domains may share a similar mechanism of substrate recognition and molten-globule-like character.

Protein domain definition should allow for conditional disorder

Proteins are often classified in a binary fashion as either structured or disordered. However this approach has several deficits. Firstly, protein folding is always conditional on the physiochemical environment. A protein which is structured in some circumstances will be disordered in others. Secondly, it hides a fundamental asymmetry in behavior. While all structured proteins can be unfolded through a change in environment, not all disordered proteins have the capacity for folding. Failure to accommodate these complexities confuses the definition of both protein structural domains and intrinsically disordered regions. We illustrate these points with an experimental study of a family of small binding domains, drawn from the RNA polymerase of mumps virus and its closest relatives. Assessed at face value the domains fall on a structural continuum, with folded, partially folded, and near unstructured members. Yet the disorder present in the family is conditional, and these closely related polypeptides can access the same folded state under appropriate conditions. Any heuristic definition of the protein domain emphasizing conformational stability divides this domain family in two, in a way that makes no biological sense. Structural domains would be better defined by their ability to adopt a specific tertiary structure: a structure that may or may not be realized, dependent on the circumstances. This explicitly allows for the conditional nature of protein folding, and more clearly demarcates structural domains from intrinsically disordered regions that may function without folding.

How long does it take to equilibrate the unfolded state of a protein?

How long does it take to equilibrate the unfolded state of a protein? The answer to this question has important implications for our understanding of why many small proteins fold with two state kinetics. When the equilibration within the unfolded state U is much faster than the folding, the folding kinetics will be two state even if there are many folding pathways with different barriers. Yet the mean first passage times (MFPTs) between different regions of the unfolded state can be much longer than the folding time. This seems to imply that the equilibration within U is much slower than the folding. In this communication we resolve this paradox. We present a formula for estimating the time to equilibrate the unfolded state of a protein. We also present a formula for the MFPT to any state within U, which is proportional to the average lifetime of that state divided by the state population. This relation is valid when the equilibration within U is very fast as compared with folding as it often is for small proteins. To illustrate the concepts, we apply the formulas to estimate the time to equilibrate the unfolded state of Trp-cage and MFPTs within the unfolded state based on a Markov State Model using an ultra-long 208 microsecond trajectory of the miniprotein to parameterize the model. The time to equilibrate the unfolded state of Trp-cage is ∼100 ns while the typical MFPTs within U are tens of microseconds or longer.

Reassessing the folding of the KIX domain: evidence for a two-state mechanism

The debate about the presence and role of intermediates in the folding of proteins has been a critical issue, especially for fast folders. One of the classical methodologies to identify such metastable species is the "burst-phase analysis," whereby the observed signal amplitude from stopped-flow traces is determined as a function of denaturant concentration. However, a complication may arise when folding is sufficiently fast to jeopardize the reliability of the stopped-flow technique. In this study, we reassessed the folding of the KIX domain from cAMP Response Element-Binding (CREB)-binding protein, which has been proposed to involve the formation of an intermediate that accumulates in the dead time of the stopped flow. By using an in-house-built capillary continuous flow with a 50-μs dead time, we demonstrate that this intermediate is not present; the problem arose because of the instrumental limitation of the standard stopped flow to assess very fast refolding rate constants (e.g., ≥ 500 s⁻¹).

Thermodynamics of protein folding using a modified Wako-Saitô-Muñoz-Eaton model

Herein, we propose a modified version of the Wako-Saitô-Muñoz-Eaton (WSME) model. The proposed model introduces an empirical temperature parameter for the hypothetical structural units (i.e., foldons) in proteins to include site-dependent thermodynamic behavior. The thermodynamics for both our proposed model and the original WSME model were investigated. For a system with beta-hairpin topology, a mathematical treatment (contact-pair treatment) to facilitate the calculation of its partition function was developed. The results show that the proposed model provides better insight into the site-dependent thermodynamic behavior of the system, compared with the original WSME model. From this site-dependent point of view, the relationship between probe-dependent experimental results and model's thermodynamic predictions can be explained. The model allows for suggesting a general principle to identify foldon behavior. We also find that the backbone hydrogen bonds may play a role of structural constraints in modulating the cooperative system. Thus, our study may contribute to the understanding of the fundamental principles for the thermodynamics of protein folding.

Why do protein folding rates correlate with metrics of native topology?

For almost 15 years, the experimental correlation between protein folding rates and the contact order parameter has been under scrutiny. Here, we use a simple simulation model combined with a native-centric interaction potential to investigate the physical roots of this empirical observation. We simulate a large set of circular permutants, thus eliminating dependencies of the folding rate on other protein properties (e.g. stability). We show that the rate-contact order correlation is a consequence of the fact that, in high contact order structures, the contact order of the transition state ensemble closely mirrors the contact order of the native state. This happens because, in these structures, the native topology is represented in the transition state through the formation of a network of tertiary interactions that are distinctively long-ranged.

Cooperativity, local-nonlocal coupling, and nonnative interactions: principles of protein folding from coarse-grained models

Coarse-grained, self-contained polymer models are powerful tools in the study of protein folding. They are also essential to assess predictions from less rigorous theoretical approaches that lack an explicit-chain representation. Here we review advances in coarse-grained modeling of cooperative protein folding, noting in particular that the Levinthal paradox was raised in response to the experimental discovery of two-state-like folding in the late 1960s, rather than to the problem of conformational search per se. Comparisons between theory and experiment indicate a prominent role of desolvation barriers in cooperative folding, which likely emerges generally from a coupling between local conformational preferences and nonlocal packing interactions. Many of these principles have been elucidated by native-centric models, wherein nonnative interactions may be treated perturbatively. We discuss these developments as well as recent applications of coarse-grained chain modeling to knotted proteins and to intrinsically disordered proteins.

Unifying model for two-state and downhill protein folding

A protein-folding model is proposed at the amino acid level, in which the folding process is divided into two successive stages: the rate-determining step, dominated by the "stochastic interactions"of solvent molecules, and the rapid phase, dominated by the "order interactions"among atoms in polypeptide. The master equation approach is used to investigate the folding kinetics, and an analytical treatment of the master equation yields a simple three-parameter expression for folding time. It is found that both two-state and downhill protein-folding kinetics can be described by a unifying model.

Protein folded states are kinetic hubs

Understanding molecular kinetics, and particularly protein folding, is a classic grand challenge in molecular biophysics. Network models, such as Markov state models (MSMs), are one potential solution to this problem. MSMs have recently yielded quantitative agreement with experimentally derived structures and folding rates for specific systems, leaving them positioned to potentially provide a deeper understanding of molecular kinetics that can lead to experimentally testable hypotheses. Here we use existing MSMs for the villin headpiece and NTL9, which were constructed from atomistic simulations, to accomplish this goal. In addition, we provide simpler, humanly comprehensible networks that capture the essence of molecular kinetics and reproduce qualitative phenomena like the apparent two-state folding often seen in experiments. Together, these models show that protein dynamics are dominated by stochastic jumps between numerous metastable states and that proteins have heterogeneous unfolded states (many unfolded basins that interconvert more rapidly with the native state than with one another) yet often still appear two-state. Most importantly, we find that protein native states are hubs that can be reached quickly from any other state. However, metastability and a web of nonnative states slow the average folding rate. Experimental tests for these findings and their implications for other fields, like protein design, are also discussed.

Bayesian single-exponential kinetics in single-molecule experiments and simulations

In this work, we develop a fully Bayesian method for the calculation of probability distributions of single-exponential rates for any single-molecule process. These distributions can even be derived when no transitions from one state to another have been observed, since in that case the data can be used to estimate a lower bound on the rate. Using a Bayesian hypothesis test, one can easily test whether a transition occurs at the same rate or at different rates in two data sets. We illustrate these methods with molecular dynamics simulations of the folding of a beta-sheet protein. However, the theory presented here can be used on any data from simulation or experiment for which a two-state description is appropriate.