Protein folding in the landscape perspective: chevron plots and non-Arrhenius kinetics

The Streptococcus phage protein paratox is an intrinsically disordered protein

The bacteriophage protein paratox (Prx) blocks quorum sensing in its streptococcal host by directly binding the signal receptor and transcription factor ComR. This reduces the ability of Streptococcus to uptake environmental DNA and protects phage DNA from damage by recombination. Past work characterizing the Prx:ComR molecular interaction revealed that paratox adopts a well-ordered globular fold when bound to ComR. However, solution-state biophysical measurements suggested that Prx may be conformationally dynamic. To address this discrepancy, we investigated the stability and dynamic properties of Prx in solution using circular dichroism, nuclear magnetic resonance, and several fluorescence-based protein folding assays. Our work shows that under dilute buffer conditions Prx is intrinsically disordered. We also show that the addition of kosmotropic salts or protein stabilizing osmolytes induces Prx folding. However, the solute stabilized fold is different from the conformation Prx adopts when it is bound to ComR. Furthermore, we have characterized Prx folding thermodynamics and folding kinetics through steady-state fluorescence and stopped flow kinetic measurements. Our results show that Prx is a highly dynamic protein in dilute solution, folding and refolding within the 10 ms timescale. Overall, our results demonstrate that the streptococcal phage protein Prx is an intrinsically disordered protein in a two-state equilibrium with a solute-stabilized folded form. Furthermore, the solute-stabilized fold is likely the predominant form of Prx in a solute-crowded bacterial cell. Finally, our work suggests that Prx binds and inhibits ComR, and thus quorum sensing in Streptococcus, by a combination of conformational selection and induced-fit binding mechanisms.

Pathways of amyloid fibril formation and protein aggregation

The main cause of many neurodegenerative diseases and systemic amyloidoses is protein and peptide aggregation and the formation of amyloid fibrils. The study of aggregation mechanisms, the discovery and description of aggregate structures, and a comprehensive understanding of the molecular mechanisms of amyloid formation are of great importance for the diagnostic processes at the molecular level and for the development of therapeutic strategies to counter aggregation-associated disorders. Given that understanding protein misfolding phenomena is directly related to the protein folding process, we will briefly explain the protein folding mechanism and then discuss the important factors involved in protein aggregation. In the following, we review different mechanisms of amyloid formation and finally represent the current knowledge on how amyloid fibrils are formed based on kinetic and thermodynamic factors.

Molecular Crowding: The History and Development of a Scientific Paradigm

It is now generally accepted that macromolecules do not act in isolation but "live" in a crowded environment, that is, an environment populated by numerous different molecules. The field of molecular crowding has its origins in the far 80s but became accepted only by the end of the 90s. In the present issue, we discuss various aspects that are influenced by crowding and need to consider its effects. This Review is meant as an introduction to the theme and an analysis of the evolution of the crowding concept through time from colloidal and polymer physics to a more biological perspective. We introduce themes that will be more thoroughly treated in other Reviews of the present issue. In our intentions, each Review may stand by itself, but the complete collection has the aspiration to provide different but complementary perspectives to propose a more holistic view of molecular crowding.

Molecular dynamics simulations at high temperatures of the Aeropyrum pernix L7Ae thermostable protein: Insight into the unfolding pathway

Most life forms on earth live at temperatures below 50 °C. Within these organisms are proteins that form the three-dimensional structures essential to their biological activity and function. However, some thermophilic life forms can resist higher temperatures and have corresponding adaptations to preserve protein function at these high temperatures. Among the structural factors responsible for this resistance of thermophilic proteins to high temperatures is the presence of additional hydrogen bonds in the thermophilic proteins, which means that the structure of the protein is more resistant to unfolding. Similarly, thermostable proteins are rich in structure-stabilizing salt bridges and/or disulfide bridges. In this context, we perform multiple replica molecular dynamics simulations at different temperatures on the Aeropyrum pernix (L7Ae) protein (from the crenarchaeal species A. pernix), known for its high melting temperature, and this in the aim to elucidate the structural factors responsible for its high thermostability. The results reveal that between the most sensitive regions of the protein to the increase of temperature are the loops L1, and L5, which surround the hydrophobic core region of the protein, besides the loop L9, and the C-terminal α5 region. This latter is the longer alpha helix of the protein secondary structure motifs and it is the first to be denaturated at 450 K, while the rest of the protein secondary structure motifs at this temperature were intact. The mechanism of unfolding that follows this protein at 550 K is similar to other thermophile proteins found in literature, with the opening of the loops that surround the hydrophobic core of the protein. So, the latter is completely exposed to the solvent, and partially denatured. The total denaturation process of the protein takes an average time of 40 ns to be achieved. Our investigation also shows that all the calculated salt bridges, with distances less than or equal to 6 A°, are on the periphery part of the protein, exposed to the solvent. However, the hydrophobic core of the protein is not involved in the formation of salt bridges, but rather with formation of some important hydrogen bondings that still persist even at 450 K. So, optimizing hydrogen bonding, near or within the core region, at high temperatures is a strategy that follows this thermostable protein to protect its hydrophobic core from denaturation, and ensure the thermal stability of the protein.

Hysteretic Pressure Dependence of Ca2+ Binding in LH1 Bacterial Membrane Chromoproteins

Much of the thermodynamic parameter values that support life are set by the properties of proteins. While the denaturing effects of pressure and temperature on proteins are well documented, their precise structural nature is rarely revealed. This work investigates the destabilization of multiple Ca²⁺ binding sites in the cyclic LH1 light-harvesting membrane chromoprotein complexes from two Ca-containing sulfur purple bacteria by hydrostatic high-pressure perturbation spectroscopy. The native (Ca-saturated) and denatured (Ca-depleted) phases of these complexes are well distinguishable by much-shifted bacteriochlorophyll a exciton absorption bands serving as innate optical probes in this study. The pressure-induced denaturation of the complexes related to the failure of the protein Ca-binding pockets and the concomitant breakage of hydrogen bonds between the pigment chromophores and protein environment were found cooperative, involving all or most of the Ca²⁺ binding sites, but irreversible. The strong hysteresis observed in the spectral and kinetic characteristics of phase transitions along the compression and decompression pathways implies asymmetry in the relevant free energy landscapes and activation free energy distributions. A phase transition pressure equal to about 1.9 kbar was evaluated for the complexes from Thiorhodovibrio strain 970 from the pressure dependence of biphasic kinetics observed in the minutes to 100 h time range.

From data to noise to data for mixing physics across temperatures with generative artificial intelligence

Using simulations or experiments performed at some set of temperatures to learn about the physics or chemistry at some other arbitrary temperature is a problem of immense practical and theoretical relevance. Here we develop a framework based on statistical mechanics and generative artificial intelligence that allows solving this problem. Specifically, we work with denoising diffusion probabilistic models and show how these models in combination with replica exchange molecular dynamics achieve superior sampling of the biomolecular energy landscape at temperatures that were never simulated without assuming any particular slow degrees of freedom. The key idea is to treat the temperature as a fluctuating random variable and not a control parameter as is usually done. This allows us to directly sample from the joint probability distribution in configuration and temperature space. The results here are demonstrated for a chirally symmetric peptide and single-strand RNA undergoing conformational transitions in all-atom water. We demonstrate how we can discover transition states and metastable states that were previously unseen at the temperature of interest and even bypass the need to perform further simulations for a wide range of temperatures. At the same time, any unphysical states are easily identifiable through very low Boltzmann weights. The procedure while shown here for a class of molecular simulations should be more generally applicable to mixing information across simulations and experiments with varying control parameters.

Data Collection for Dilute Protein Solutions via a Neutron Backscattering Spectrometer

Understanding protein functions requires not only static but also dynamic structural information. Incoherent quasi-elastic neutron scattering (QENS), which utilizes the highly incoherent scattering ability of hydrogen, is a powerful technique for revealing the dynamics of proteins in deuterium oxide (D2O) buffer solutions. The background scattering of sample cells suitable for aqueous protein solution samples, conducted with a neutron backscattering spectrometer, was evaluated. It was found that the scattering intensity of an aluminum sample cell coated with boehmite using D2O was lower than that of a sample cell coated with regular water (H2O). The D2O-Boehmite coated cell was used for the QENS measurement of a 0.8 wt.% aqueous solution of an intrinsically disordered protein in an intrinsically disordered region of a helicase-associated endonuclease for a fork-structured type of DNA. The cell was inert against aqueous samples at 283–363 K. In addition, meticulous attention to cells with small individual weight differences and the positional reproducibility of the sample cell relative to the spectrometer neutron beam position enabled the accurate subtraction of the scattering profiles of the D2O buffer and the sample container. Consequently, high-quality information on protein dynamics could be extracted from dilute protein solutions.

Chaperones and Catalysts: How Antigen Presentation Pathways Cope With Biological Necessity

Immune recognition by T lymphocytes and natural killer (NK) cells is in large part dependent on the identification of cell surface MHC molecules bearing peptides generated from either endogenous (MHC I) or exogenous (MHC II) dependent pathways. This review focuses on MHC I molecules that coordinately fold to bind self or foreign peptides for such surface display. Peptide loading occurs in an antigen presentation pathway that includes either the multimolecular peptide loading complex (PLC) or a single chain chaperone/catalyst, TAP binding protein, related, TAPBPR, that mimics a key component of the PLC, tapasin. Recent structural and dynamic studies of TAPBPR reveal details of its function and reflect on mechanisms common to tapasin. Regions of structural conservation among species suggest that TAPBPR and tapasin have evolved to satisfy functional complexities demanded by the enormous polymorphism of MHC I molecules. Recent studies suggest that these two chaperone/catalysts exploit structural flexibility and dynamics to stabilize MHC molecules and facilitate peptide loading.

Dissecting Multiple Pathways in the Relaxation Dynamics of Helix <==> Coil Transitions with Optimum Dimensionality Reduction

Fast kinetic experiments with dramatically improved time resolution have contributed significantly to understanding the fundamental processes in protein folding pathways involving the formation of a-helices and b-hairpin, contact formation, and overall collapse of the peptide chain. Interpretation of experimental results through application of a simple statistical mechanical model was key to this understanding. Atomistic description of all events observed in the experimental findings was challenging. Recent advancements in theory, more sophisticated algorithms, and a true long-term trajectory made way for an atomically detailed description of kinetics, examining folding pathways, validating experimental results, and reporting new findings for a wide range of molecular processes in biophysical chemistry. This review describes how optimum dimensionality reduction theory can construct a simplified coarse-grained model with low dimensionality involving a kinetic matrix that captures novel insights into folding pathways. A set of metastable states derived from molecular dynamics analysis generate an optimally reduced dimensionality rate matrix following transition pathway analysis. Analysis of the actual long-term simulation trajectory extracts a relaxation time directly comparable to the experimental results and confirms the validity of the combined approach. The application of the theory is discussed and illustrated using several examples of helix <==> coil transition pathways. This paper focuses primarily on a combined approach of time-resolved experiments and long-term molecular dynamics simulation from our ongoing work.

The Right-Handed Parallel β-Helix Topology of Erwinia chrysanthemi Pectin Methylesterase Is Intimately Associated with Both Sequential Folding and Resistance to High Pressure

The complex topologies of large multi-domain globular proteins make the study of their folding and assembly particularly demanding. It is often characterized by complex kinetics and undesired side reactions, such as aggregation. The structural simplicity of tandem-repeat proteins, which are characterized by the repetition of a basic structural motif and are stabilized exclusively by sequentially localized contacts, has provided opportunities for dissecting their folding landscapes. In this study, we focus on the Erwinia chrysanthemi pectin methylesterase (342 residues), an all-β pectinolytic enzyme with a right-handed parallel β-helix structure. Chemicals and pressure were chosen as denaturants and a variety of optical techniques were used in conjunction with stopped-flow equipment to investigate the folding mechanism of the enzyme at 25 °C. Under equilibrium conditions, both chemical- and pressure-induced unfolding show two-state transitions, with average conformational stability (ΔG° = 35 ± 5 kJ·mol⁻¹) but exceptionally high resistance to pressure (P_m = 800 ± 7 MPa). Stopped-flow kinetic experiments revealed a very rapid (τ < 1 ms) hydrophobic collapse accompanied by the formation of an extended secondary structure but did not reveal stable tertiary contacts. This is followed by three distinct cooperative phases and the significant population of two intermediate species. The kinetics followed by intrinsic fluorescence shows a lag phase, strongly indicating that these intermediates are productive species on a sequential folding pathway, for which we propose a plausible model. These combined data demonstrate that even a large repeat protein can fold in a highly cooperative manner.

Optimal proteome allocation and the temperature dependence of microbial growth laws

Although the effect of temperature on microbial growth has been widely studied, the role of proteome allocation in bringing about temperature-induced changes remains elusive. To tackle this problem, we propose a coarse-grained model of microbial growth, including the processes of temperature-sensitive protein unfolding and chaperone-assisted (re)folding. We determine the proteome sector allocation that maximizes balanced growth rate as a function of nutrient limitation and temperature. Calibrated with quantitative proteomic data for Escherichia coli, the model allows us to clarify general principles of temperature-dependent proteome allocation and formulate generalized growth laws. The same activation energy for metabolic enzymes and ribosomes leads to an Arrhenius increase in growth rate at constant proteome composition over a large range of temperatures, whereas at extreme temperatures resources are diverted away from growth to chaperone-mediated stress responses. Our approach points at risks and possible remedies for the use of ribosome content to characterize complex ecosystems with temperature variation.

Half a century of amyloids: past, present and future

Amyloid diseases are global epidemics with profound health, social and economic implications and yet remain without a cure. This dire situation calls for research into the origin and pathological manifestations of amyloidosis to stimulate continued development of new therapeutics. In basic science and engineering, the cross-β architecture has been a constant thread underlying the structural characteristics of pathological and functional amyloids, and realizing that amyloid structures can be both pathological and functional in nature has fuelled innovations in artificial amyloids, whose use today ranges from water purification to 3D printing. At the conclusion of a half century since Eanes and Glenner's seminal study of amyloids in humans, this review commemorates the occasion by documenting the major milestones in amyloid research to date, from the perspectives of structural biology, biophysics, medicine, microbiology, engineering and nanotechnology. We also discuss new challenges and opportunities to drive this interdisciplinary field moving forward.

Simple Model of Protein Energetics To Identify Ab Initio Folding Transitions from All-Atom MD Simulations of Proteins

A fundamental requirement to predict the native conformation, address questions of sequence design and optimization, and gain insights into the folding mechanisms of proteins lies in the definition of an unbiased reaction coordinate that reports on the folding state without the need to compare it to reference values, which might be unavailable for new (designed) sequences. Here, we introduce such a reaction coordinate, which does not depend on previous structural knowledge of the native state but relies solely on the energy partition within the protein: the spectral gap of the pair nonbonded energy matrix (ENergy Gap, ENG). This quantity can be simply calculated along unbiased MD trajectories. We show that upon folding the gap increases significantly, while its fluctuations are reduced to a minimum. This is consistently observed for a diverse set of systems and trajectories. Our approach allows one to promptly identify residues that belong to the folding core as well as residues involved in non-native contacts that need to be disrupted to guide polypeptides to the folded state. The energy gap and fluctuations criteria are then used to develop an automatic detection system which allows us to extract and analyze folding transitions from a generic MD trajectory. We speculate that our method can be used to detect conformational ensembles in dynamic and intrinsically disordered proteins, revealing potential preorganization for binding.

The hydrophobic effect characterises the thermodynamic signature of amyloid fibril growth

Many proteins have the potential to aggregate into amyloid fibrils, protein polymers associated with a wide range of human disorders such as Alzheimer’s and Parkinson’s disease. The thermodynamic stability of amyloid fibrils, in contrast to that of folded proteins, is not well understood: the balance between entropic and enthalpic terms, including the chain entropy and the hydrophobic effect, are poorly characterised. Using a combination of theory, in vitro experiments, simulations of a coarse-grained protein model and meta-data analysis, we delineate the enthalpic and entropic contributions that dominate amyloid fibril elongation. Our prediction of a characteristic temperature-dependent enthalpic signature is confirmed by the performed calorimetric experiments and a meta-analysis over published data. From these results we are able to define the necessary conditions to observe cold denaturation of amyloid fibrils. Overall, we show that amyloid fibril elongation is associated with a negative heat capacity, the magnitude of which correlates closely with the hydrophobic surface area that is buried upon fibril formation, highlighting the importance of hydrophobicity for fibril stability.

Most proteins fold in the cell into stable, compact structures. Nevertheless, many proteins also have the ability to stick together, forming long fibrillar structures that are associated with a wide range of human disorders including Alzheimer’s and Parkinson’s disease. The exact nature of the amyloid-causing stickiness is not well understood, nevertheless amyloid fibrils show some very specific thermodynamic characteristics. Some fibrils even destabilise at low temperatures. In this work we translate hydrophobic theory previously used to model protein folding to fibril formation. We combine this theory with experimental measurements, simulations and meta-data analysis of different types of fibrils. This allowed us to unravel the nature of the stickiness in amyloid fibrils by observing the effect of temperature changes, specifically at low temperatures, on hydrophobicity.

Scale-consistent approach to the derivation of coarse-grained force fields for simulating structure, dynamics, and thermodynamics of biopolymers

In this chapter the scale-consistent approach to the derivation of coarse-grained force fields developed in our laboratory is presented, in which the effective energy function originates from the potential of mean force of the system under consideration and embeds atomistically detailed interactions in the resulting energy terms through use of Kubo's cluster-cumulant expansion, appropriate selection of the major degrees of freedom to be averaged out in the derivation of analytical approximations to the energy terms, and appropriate expression of the interaction energies at the all-atom level in these degrees of freedom. Our approach enables the developers to find correct functional forms of the effective coarse-grained energy terms, without having to import them from all-atom force fields or deriving them on a heuristic basis. In particular, the energy terms derived in such a way exhibit correct dependence on coarse-grained geometry, in particular on site orientation. Moreover, analytical formulas for the multibody (correlation) terms, which appear to be crucial for coarse-grained modeling of many of the regular structures such as, e.g., protein α-helices and β-sheets, can be derived in a systematic way. Implementation of the developed theory to the UNIfied COarse-gRaiNed (UNICORN) model of biological macromolecules, which consists of the UNRES (for proteins), NARES-2P (for nucleic acids), and SUGRES-1P (for polysaccharides) components, and is being developed in our laboratory is described. Successful applications of UNICORN to the prediction of protein structure, simulating the folding and stability of proteins and nucleic acids, and solving biological problems are discussed.

Computational methods in the study of self-entangled proteins: a critical appraisal

The existence of self-entangled proteins, the native structure of which features a complex topology, unveils puzzling, and thus fascinating, aspects of protein biology and evolution. The discovery that a polypeptide chain can encode the capability to self-entangle in an efficient and reproducible way during folding, has raised many questions, regarding the possible function of these knots, their conservation along evolution, and their role in the folding paradigm. Understanding the function and origin of these entanglements would lead to deep implications in protein science, and this has stimulated the scientific community to investigate self-entangled proteins for decades by now. In this endeavour, advanced experimental techniques are more and more supported by computational approaches, that can provide theoretical guidelines for the interpretation of experimental results, and for the effective design of new experiments. In this review we provide an introduction to the computational study of self-entangled proteins, focusing in particular on the methodological developments related to this research field. A comprehensive collection of techniques is gathered, ranging from knot theory algorithms, that allow detection and classification of protein topology, to Monte Carlo or molecular dynamics strategies, that constitute crucial instruments for investigating thermodynamics and kinetics of this class of proteins.

Temperature, Hydrostatic Pressure, and Osmolyte Effects on Liquid-Liquid Phase Separation in Protein Condensates: Physical Chemistry and Biological Implications

Liquid-liquid phase separation (LLPS) of proteins and other biomolecules play a critical role in the organization of extracellular materials and membrane-less compartmentalization of intra-organismal spaces through the formation of condensates. Structural properties of such mesoscopic droplet-like states were studied by spectroscopy, microscopy, and other biophysical techniques. The temperature dependence of biomolecular LLPS has been studied extensively, indicating that phase-separated condensed states of proteins can be stabilized or destabilized by increasing temperature. In contrast, the physical and biological significance of hydrostatic pressure on LLPS is less appreciated. Summarized here are recent investigations of protein LLPS under pressures up to the kbar-regime. Strikingly, for the cases studied thus far, LLPSs of both globular proteins and intrinsically disordered proteins/regions are typically more sensitive to pressure than the folding of proteins, suggesting that organisms inhabiting the deep sea and sub-seafloor sediments, under pressures up to 1 kbar and beyond, have to mitigate this pressure-sensitivity to avoid unwanted destabilization of their functional biomolecular condensates. Interestingly, we found that trimethylamine-N-oxide (TMAO), an osmolyte upregulated in deep-sea fish, can significantly stabilize protein droplets under pressure, pointing to another adaptive advantage for increased TMAO concentrations in deep-sea organisms besides the osmolyte's stabilizing effect against protein unfolding. As life on Earth might have originated in the deep sea, pressure-dependent LLPS is pertinent to questions regarding prebiotic proto-cells. Herein, we offer a conceptual framework for rationalizing the recent experimental findings and present an outline of the basic thermodynamics of temperature-, pressure-, and osmolyte-dependent LLPS as well as a molecular-level statistical mechanics picture in terms of solvent-mediated interactions and void volumes.

Temperature Dependence of Rate Processes Beyond Arrhenius and Eyring: Activation and Transitivity

Advances in the understanding of the dependence of reaction rates from temperature, as motivated from progress in experiments and theoretical tools (e. g., molecular dynamics), are needed for the modeling of extreme environmental conditions (e.g., in astrochemistry and in the chemistry of plasmas). While investigating statistical mechanics perspectives (Aquilanti et al., 2017b, 2018), the concept of transitivity was introduced as a measure for the propensity for a reaction to occur. The Transitivity plot is here defined as the reciprocal of the apparent activation energy vs. reciprocal absolute temperature. Since the transitivity function regulates transit in physicochemical transformations, not necessarily involving reference to transition-state hypothesis of Eyring, an extended version is here proposed to cope with general types of transformations. The transitivity plot permits a representation where deviations from Arrhenius behavior are given a geometrical meaning and make explicit a positive or negative linear dependence of transitivity for sub- and super-Arrhenius cases, respectively. To first-order in reciprocal temperature, the transitivity function models deviations from linearity in Arrhenius plots as originally proposed by Aquilanti and Mundim: when deviations are increasingly larger, other phenomenological formulas, such as Vogel-Fulcher-Tammann, Nakamura-Takayanagi-Sato, and Aquilanti-Sanches-Coutinho-Carvalho are here rediscussed from the transitivity concept perspective and with in a general context. Emphasized is the interest of introducing into this context modifications to a very successful tool of theoretical kinetics, Eyring's Transition-State Theory: considering the behavior of the transitivity function at low temperatures, in order to describe deviation from Arrhenius behavior under the quantum tunneling regime, a "d-TST" formulation was previously introduced (Carvalho-Silva et al., 2017). In this paper, a special attention is dedicated to a derivation of the temperature dependence of viscosity, making explicit reference to feature of the transitivity function, which in this case generally exhibits a super-Arrhenius behavior. This is of relevance also for advantages of using the transitivity function for diffusion-controlled phenomena.

Interrogating the Structural Dynamics and Energetics of Biomolecular Systems with Pressure Modulation

High hydrostatic pressure affects the structure, dynamics, and stability of biomolecular systems and is a key parameter in the context of the exploration of the origin and the physical limits of life. This review lays out the conceptual framework for exploring the conformational fluctuations, dynamical properties, and activity of biomolecular systems using pressure perturbation. Complementary pressure-jump relaxation studies are useful tools to study the kinetics and mechanisms of biomolecular phase transitions and structural transformations, such as membrane fusion or protein and nucleic acid folding. Finally, the advantages of using pressure to explore biomolecular assemblies and modulate enzymatic reactions are discussed.

The folding pathways and thermodynamics of semiflexible polymers

Inspired by the protein folding and DNA packing, we have systematically studied the thermodynamic and kinetic behaviors of single semiflexible homopolymers by Langevin dynamics simulations. In line with experiments, a rich variety of folding products, such as rod-like bundles, hairpins, toroids, and a mixture of them, are observed in the complete diagram of states. Moreover, knotted structures with a significant population are found in a certain range of bending stiffness in thermal equilibrium. As the solvent quality becomes poorer, the population of the intermediate occurring in the folding process increases, which leads to a severe chevron rollover for the folding arm. However, the population of the intermediates in the unfolding process is very low, insufficient to induce unfolding arm rollover. The total types of folding pathways from the coil state to the toroidal state for a semiflexible polymer chain remain unchanged by varying the solvent quality or temperature, whereas the kinetic partitioning into different folding events can be tuned significantly. In the process of knotting, three types of mechanisms, namely, plugging, slipknotting, and sliding, are discovered. Along the folding evolution, a semiflexible homopolymer chain can knot at any stage of folding upon leaving the extended coil state, and the probability to find a knot increases with chain compactness. In addition, we find rich types of knotted topologies during the folding of a semiflexible homopolymer chain. This study should be helpful in gaining insight into the general principles of biopolymer folding.

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.

Energy Landscape and Pathways for Transitions between Watson-Crick and Hoogsteen Base Pairing in DNA

The recent discovery that Hoogsteen (HG) base pairs are widespread in DNA across diverse sequences and positional contexts could have important implications for understanding DNA replication and DNA-protein recognition. While evidence is emerging that the Hoogsteen conformation could be a thermodynamically accessible conformation of the DNA duplex and provide a means to expand its functionality, relatively little is known about the molecular mechanism underlying the Watson-Crick (WC) to HG transition. In this Perspective, we describe pathways and kinetics for this transition at an atomic level of detail, using the energy landscape perspective. We show that competition between the duplex conformations results in a double funnel landscape, which explains some recent experimental observations. The interconversion pathways feature a number of intermediates, with a variable number of WC and HG base pairs. The relatively slow kinetics, with possible deviations from two-state behavior, suggest that this conformational switch is likely to be a challenging target for both simulation and experiment.

Site-specific time-resolved FRET reveals local variations in the unfolding mechanism in an apparently two-state protein unfolding transition

Using multi-site time-resolved FRET, it is shown that equilibrium unfolding of monellin is not only heterogeneous, but that the degree of non-cooperativity differs between the sole α-helix and different parts of the β-sheet.

BayesWHAM: A Bayesian approach for free energy estimation, reweighting, and uncertainty quantification in the weighted histogram analysis method

The weighted histogram analysis method (WHAM) is a powerful approach to estimate molecular free energy surfaces (FES) from biased simulation data. Bayesian reformulations of WHAM are valuable in proving statistically optimal use of the data and providing a transparent means to incorporate regularizing priors and estimate statistical uncertainties. In this work, we develop a fully Bayesian treatment of WHAM to generate statistically optimal FES estimates in any number of biasing dimensions under arbitrary choices of the Bayes prior. Rigorous uncertainty estimates are generated by Metropolis-Hastings sampling from the Bayes posterior. We also report a means to project the FES and its uncertainties into arbitrary auxiliary order parameters beyond those in which biased sampling was conducted. We demonstrate the approaches in applications of alanine dipeptide and the unthreading of a synthetic mimic of the astexin-3 lasso peptide. Open-source MATLAB and Python implementations of our codes are available for free public download. © 2017 Wiley Periodicals, Inc.

On the indirect relationship between protein dynamics and enzyme activity

The behaviors of simple thermal systems have been well studied in physical chemistry and the principles obtained from such studies have been applied to complex thermal systems, such as proteins and enzymes. But the simple application of such principles is questionable and may lead to mistakes under some circumstances. In enzymology, the transition state theory of chemical reactions has been accepted as a fundamental theory, but the role of protein dynamics in enzyme catalysis is controversial in the context of transition state theory. By studying behaviors of complex thermal systems, we have revised the Arrhenius equation and transition state theory and our model is validated in enzymology. Formally speaking, the revised Arrhenius equation is apparently similar to a conventional Arrhenius equation, but the physical meanings of its parameters differ from that of traditional forms in principle. Within this model, the role of protein dynamics in enzyme catalysis is well defined and quantified.

Distributions of experimental protein structures on coarse-grained free energy landscapes

Predicting conformational changes of proteins is needed in order to fully comprehend functional mechanisms. With the large number of available structures in sets of related proteins, it is now possible to directly visualize the clusters of conformations and their conformational transitions through the use of principal component analysis. The most striking observation about the distributions of the structures along the principal components is their highly non-uniform distributions. In this work, we use principal component analysis of experimental structures of 50 diverse proteins to extract the most important directions of their motions, sample structures along these directions, and estimate their free energy landscapes by combining knowledge-based potentials and entropy computed from elastic network models. When these resulting motions are visualized upon their coarse-grained free energy landscapes, the basis for conformational pathways becomes readily apparent. Using three well-studied proteins, T4 lysozyme, serum albumin, and sarco-endoplasmic reticular Ca(2+) adenosine triphosphatase (SERCA), as examples, we show that such free energy landscapes of conformational changes provide meaningful insights into the functional dynamics and suggest transition pathways between different conformational states. As a further example, we also show that Monte Carlo simulations on the coarse-grained landscape of HIV-1 protease can directly yield pathways for force-driven conformational changes.

A growing toolbox of techniques for studying β-barrel outer membrane protein folding and biogenesis

Great strides into understanding protein folding have been made since the seminal work of Anfinsen over 40 years ago, but progress in the study of membrane protein folding has lagged behind that of their water soluble counterparts. Researchers in these fields continue to turn to more advanced techniques such as NMR, mass spectrometry, molecular dynamics (MD) and single molecule methods to interrogate how proteins fold. Our understanding of β-barrel outer membrane protein (OMP) folding has benefited from these advances in the last decade. This class of proteins must traverse the periplasm and then insert into an asymmetric lipid membrane in the absence of a chemical energy source. In this review we discuss old, new and emerging techniques used to examine the process of OMP folding and biogenesis in vitro and describe some of the insights and new questions these techniques have revealed.

Determination of structural elements on the folding reaction of mnemiopsin by spectroscopic techniques

Mnemiopsin 1 is a member of photoprotein family, made up of 206 amino acid residues. These Ca(2+)-regulated photoproteins are responsible for light emission in a variety of marine cnidarians and ctenophores. They composed of an apoprotein, a single polypeptide chain of 25 kDa, molecular oxygen and the non-covalently bound chromophore. In this study, we examined whether three mutations, namely R39K, S128G and V183T affect the thermodynamic stability as well as refolding and unfolding kinetics of mnemiopsin 1. Conformational stability measurements using fluorescence and far-UV CD spectroscopies revealed that all variants unfold in multi-step manner in which the secondary and tertiary structures are lost in different steps. However kinetic studies showed that point mutation S128G destabilizes both kinetic intermediate and native conformation; while, these structural elements are stabilized in V183T. We also found that the stability of folded and intermediate states increases in R39K. We concluded that the initial packing of helical segments within the protein structure is more facilitated when Lys with smaller side chain is present in the protein chain.

Adaptive Steered Molecular Dynamics of the Long-Distance Unfolding of Neuropeptide Y

Neuropeptide Y (NPY) has been found to adopt two stable conformations in vivo: (1) a monomeric form called the PP-fold in which a polyproline tail is folded onto an α-helix via a β-turn and (2) a dimeric form of the unfolded proteins in which the α-helices interact with each other via side chains. The transition pathway and rates between the two conformations remain unknown and are important to the nature of the binding of the protein. Toward addressing this question, the present work suggests that the unfolding of the PP-fold is too slow to play a role in NPY monomeric binding unless the receptor catalyzes it to do so. Specifically, the dynamics and structural changes of the unfolding of a monomeric NPY protein have been investigated in this work. Temperature accelerated molecular dynamics (MD) simulations at 500 K under constant (N,V,E) conditions suggests a hinge-like unraveling of the tail rather than a random unfolding. The free energetics of the proposed unfolding pathway have been described using an adaptive steered MD (SMD) approach at various temperatures. This approach generalizes the use of Jarzynski's equality through a series of stages that allows for better convergence along nonlinear and long-distance pathways. Results acquired using this approach provide a potential of mean force (PMF) with narrower error bars and are consistent with some of the earlier reports on the qualitative behavior of NPY binding.

Knotted proteins: A tangled tale of Structural Biology

Knotted proteins have their native structures arranged in the form of an open knot. In the last ten years researchers have been making significant efforts to reveal their folding mechanism and understand which functional advantage(s) knots convey to their carriers. Molecular simulations have been playing a fundamental role in this endeavor, and early computational predictions about the knotting mechanism have just been confirmed in wet lab experiments. Here we review a collection of simulation results that allow outlining the current status of the field of knotted proteins, and discuss directions for future research.

Decoy Database Improvement for Protein Folding

Predicting protein structures and simulating protein folding are two of the most important problems in computational biology today. Simulation methods rely on a scoring function to distinguish the native structure (the most energetically stable) from non-native structures. Decoy databases are collections of non-native structures used to test and verify these functions. We present a method to evaluate and improve the quality of decoy databases by adding novel structures and removing redundant structures. We test our approach on 20 different decoy databases of varying size and type and show significant improvement across a variety of metrics. We also test our improved databases on two popular modern scoring functions and show that for most cases they contain a greater or equal number of native-like structures than the original databases, thereby producing a more rigorous database for testing scoring functions.

Efficient maximum likelihood parameterization of continuous-time Markov processes

Continuous-time Markov processes over finite state-spaces are widely used to model dynamical processes in many fields of natural and social science. Here, we introduce a maximum likelihood estimator for constructing such models from data observed at a finite time interval. This estimator is dramatically more efficient than prior approaches, enables the calculation of deterministic confidence intervals in all model parameters, and can easily enforce important physical constraints on the models such as detailed balance. We demonstrate and discuss the advantages of these models over existing discrete-time Markov models for the analysis of molecular dynamics simulations.

Quantitative assessments of the distinct contributions of polypeptide backbone amides versus side chain groups to chain expansion via chemical denaturation

In aqueous solutions with high concentrations of chemical denaturants such as urea and guanidinium chloride (GdmCl) proteins expand to populate heterogeneous conformational ensembles. These denaturing environments are thought to be good solvents for generic protein sequences because properties of conformational distributions align with those of canonical random coils. Previous studies showed that water is a poor solvent for polypeptide backbones, and therefore, backbones form collapsed globular structures in aqueous solvents. Here, we ask if polypeptide backbones can intrinsically undergo the requisite chain expansion in aqueous solutions with high concentrations of urea and GdmCl. We answer this question using a combination of molecular dynamics simulations and fluorescence correlation spectroscopy. We find that the degree of backbone expansion is minimal in aqueous solutions with high concentrations of denaturants. Instead, polypeptide backbones sample conformations that are denaturant-specific mixtures of coils and globules, with a persistent preference for globules. Therefore, typical denaturing environments cannot be classified as good solvents for polypeptide backbones. How then do generic protein sequences expand in denaturing environments? To answer this question, we investigated the effects of side chains using simulations of two archetypal sequences with amino acid compositions that are mixtures of charged, hydrophobic, and polar groups. We find that side chains lower the effective concentration of backbone amides in water leading to an intrinsic expansion of polypeptide backbones in the absence of denaturants. Additional dilution of the effective concentration of backbone amides is achieved through preferential interactions with denaturants. These effects lead to conformational statistics in denaturing environments that are congruent with those of canonical random coils. Our results highlight the role of side chain-mediated interactions as determinants of the conformational properties of unfolded states in water and in influencing chain expansion upon denaturation.

Tuning the Attempt Frequency of Protein Folding Dynamics via Transition-State Rigidification: Application to Trp-Cage

The attempt frequency or prefactor (k0) of the transition-state rate equation of protein folding kinetics has been estimated to be on the order of 10(6) s(-1), which is many orders of magnitude smaller than that of chemical reactions. Herein we use the mini-protein Trp-cage to show that it is possible to significantly increase the value of k0 for a protein folding reaction by rigidifying the transition state. This is achieved by reducing the conformational flexibility of a key structural element (i.e., an α-helix) formed in the transition state via photoisomerization of an azobenzene cross-linker. We find that this strategy not only decreases the folding time of the Trp-cage peptide by more than an order of magnitude (to ∼100 ns at 25°C) but also exposes parallel folding pathways, allowing us to provide, to the best of our knowledge, the first quantitative assessment of the curvature of the transition-state free-energy surface of a protein.

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.

Theoretical perspectives on nonnative interactions and intrinsic disorder in protein folding and binding

The diverse biological functions of intrinsically disordered proteins (IDPs) have markedly raised our appreciation of protein conformational versatility, whereas the existence of energetically favorable yet functional detrimental nonnative interactions underscores the physical limitations of evolutionary optimization. Here we survey recent advances in using biophysical modeling to gain insight into experimentally observed nonnative behaviors and IDP properties. Simulations of IDP interactions to date focus mostly on coupled folding-binding, which follows essentially the same organizing principle as the local-nonlocal coupling mechanism in cooperative folding of monomeric globular proteins. By contrast, more innovative theories of electrostatic and aromatic interactions are needed for the conceptually novel but less-explored 'fuzzy' complexes in which the functionally bound IDPs remain largely disordered.

The role of high-dimensional diffusive search, stabilization, and frustration in protein folding

Proteins are polymeric molecules with many degrees of conformational freedom whose internal energetic interactions are typically screened to small distances. Therefore, in the high-dimensional conformation space of a protein, the energy landscape is locally relatively flat, in contrast to low-dimensional representations, where, because of the induced entropic contribution to the full free energy, it appears funnel-like. Proteins explore the conformation space by searching these flat subspaces to find a narrow energetic alley that we call a hypergutter and then explore the next, lower-dimensional, subspace. Such a framework provides an effective representation of the energy landscape and folding kinetics that does justice to the essential characteristic of high-dimensionality of the search-space. It also illuminates the important role of nonnative interactions in defining folding pathways. This principle is here illustrated using a coarse-grained model of a family of three-helix bundle proteins whose conformations, once secondary structure has formed, can be defined by six rotational degrees of freedom. Two folding mechanisms are possible, one of which involves an intermediate. The stabilization of intermediate subspaces (or states in low-dimensional projection) in protein folding can either speed up or slow down the folding rate depending on the amount of native and nonnative contacts made in those subspaces. The folding rate increases due to reduced-dimension pathways arising from the mere presence of intermediate states, but decreases if the contacts in the intermediate are very stable and introduce sizeable topological or energetic frustration that needs to be overcome. Remarkably, the hypergutter framework, although depending on just a few physically meaningful parameters, can reproduce all the types of experimentally observed curvature in chevron plots for realizations of this fold.

Statistical analyses of protein folding rates from the view of quantum transition

Understanding protein folding rate is the primary key to unlock the fundamental physics underlying protein structure and its folding mechanism. Especially, the temperature dependence of the folding rate remains unsolved in the literature. Starting from the assumption that protein folding is an event of quantum transition between molecular conformations, we calculated the folding rate for all two-state proteins in a database and studied their temperature dependencies. The non-Arrhenius temperature relation for 16 proteins, whose experimental data had previously been available, was successfully interpreted by comparing the Arrhenius plot with the first-principle calculation. A statistical formula for the prediction of two-state protein folding rate was proposed based on quantum folding theory. The statistical comparisons of the folding rates for 65 two-state proteins were carried out, and the theoretical vs. experimental correlation coefficient was 0.73. Moreover, the maximum and the minimum folding rates given by the theory were consistent with the experimental results.

Quantifying the Sources of Kinetic Frustration in Folding Simulations of Small Proteins

Experiments and atomistic simulations of polypeptides have revealed structural intermediates that promote or inhibit conformational transitions to the native state during folding. We invoke a concept of “kinetic frustration” to quantify the prevalence and impact of these behaviors on folding rates within a large set of atomistic simulation data for 10 fast-folding proteins, where each protein’s conformational space is represented as a Markov state model of conformational transitions. Our graph theoretic approach addresses what conformational features correlate with folding inhibition and therefore permits comparison among features within a single protein network and also more generally between proteins. Nonnative contacts and nonnative secondary structure formation can thus be quantitatively implicated in inhibiting folding for several of the tested peptides.

High-pressure fluorescence applications

Fluorescence is the most widely used technique to study the effect of pressure on biochemical systems. The use of pressure as a physical variable sheds light into volumetric characteristics of reactions. Here we focus on the effect of pressure on protein solutions using a simple unfolding example in order to illustrate the applications of the methodology. Topics covered in this review include the relationships between practical aspects and technical limitations; the effect of pressure and the study of protein cavities; the interpretation of thermodynamic and relaxation kinetics; and the study of relaxation amplitudes. Finally, we discuss the insights available from the combination of fluorescence and other methods adapted to high pressure, such as SAXS or NMR. Because of the simplicity and accessibility of high-pressure fluorescence, the technique is a starting point that complements appropriately multi-methodological approaches related to understanding protein function, disfunction, and folding from the volumetric point of view.