THEFASTPROTEINFOLDINGPROBLEM

Combining temperature perturbations with X-ray crystallography to study dynamic macromolecules: A thorough discussion of experimental methods

Temperature is an important state variable that governs the behavior of microscopic systems, yet crystallographers rarely exploit temperature changes to study the structure and dynamics of biological macromolecules. In fact, approximately 90% of crystal structures in the Protein Data Bank were determined under cryogenic conditions, because sample cryocooling makes crystals robust to X-ray radiation damage and facilitates data collection. On the other hand, cryocooling can introduce artifacts into macromolecular structures, and can suppress conformational dynamics that are critical for function. Fortunately, recent advances in X-ray detector technology, X-ray sources, and computational data processing algorithms make non-cryogenic X-ray crystallography easier and more broadly applicable than ever before. Without the reliance on cryocooling, high-resolution crystallography can be combined with various temperature perturbations to gain deep insight into the conformational landscapes of macromolecules. This Chapter reviews the historical reasons for the prevalence of cryocooling in macromolecular crystallography, and discusses its potential drawbacks. Next, the Chapter summarizes technological developments and methodologies that facilitate non-cryogenic crystallography experiments. Finally, the chapter discusses the theoretical underpinnings and practical aspects of multi-temperature and temperature-jump crystallography experiments, which are powerful tools for understanding the relationship between the structure, dynamics, and function of proteins and other biological macromolecules.

The Role of Transient Intermediate Structures in the Unfolding of the Trp-Cage Fast-Folding Protein: Generating Ensembles from Time-Resolved X-ray Solution Scattering with Genetic Algorithms

The Trp-cage miniprotein is one of the smallest systems to exhibit a stable secondary structure and fast-folding dynamics, serving as an apt model system to study transient intermediates with both experimental and computational analyses. Previous spectroscopic characterizations that have been done on Trp-cage have inferred a single stable intermediate on a pathway from folded to unfolded basins. We aim to bridge the understanding of Trp-cage structural folding dynamics on microsecond-time scales, by utilizing time-resolved X-ray solution scattering to probe the temperature-induced unfolding pathway. Our results indicate the formation of a conformationally extended intermediate on the time scale of 1 μs, which undergoes complete unfolding within 5 μs. We further investigated the atomistic structural details of the unfolding pathway using a genetic algorithm to generate ensemble model fits to the scattering profiles. This analysis paves the way for direct benchmarking of theoretical models of protein folding ensembles produced with molecular dynamics simulations.

Super-resolution fluorescence imaging of extracellular environments

The extracellular matrix (ECM) is an important biophysical environment that plays a role in a number of physiological processes. The ECM is highly dynamic, with changes occurring as local, nanoscale, physicochemical variations in physical confinement and chemistry from the perspective of biological molecules. The length and time scale of ECM dynamics are challenging to measure with current spectroscopic techniques. Super-resolution fluorescence microscopy has the potential to probe local, nanoscale, physicochemical variations in the ECM. Here, we review super-resolution imaging and analysis methods and their application to study model nanoparticles and biomolecules within synthetic ECM hydrogels and the brain extracellular space (ECS). We provide a perspective of future directions for the field that can move super-resolution imaging of the ECM towards more biomedically-relevant samples. Overall, super-resolution imaging is a powerful tool that can increase our understanding of extracellular environments at new spatiotemporal scales to reveal ECM processes at the molecular-level.

The concept of protein folding/unfolding and its impacts on human health

Proteins have evolved in specific 3D structures and play different functions in cells and determine various reactions and pathways. The newly synthesized amino acid chains once depart ribosome must crumple into three-dimensional structures so can be biologically active. This process of protein that makes a functional molecule is called protein folding. The protein folding is both a biological and a physicochemical process that depends on the sequence of it. In fact, this process occurs more complicated and in some cases and in exposure to some molecules like glucose (glycation), mistaken folding leads to amyloid structures and fatal disorders called conformational diseases. Such conditions are detected by the quality control system of the cell and these abnormal proteins undergo renovation or degradation. This scenario takes place by the chaperones, chaperonins, and Ubiquitin-proteasome complex. Understanding of protein folding mechanisms from different views including experimental and computational approaches has revealed some intermediate ensembles such as molten globule and has been subjected to biophysical and molecular biology attempts to know more about prevalent conformational diseases.

Revealing the Dynamical Role of Co-solvents in the Coupled Folding and Dimerization of Insulin

Solvent-protein interactions are important for protein biological functions, especially for a coupled folding and binding system such as insulin. By monitoring the change in the conformation of insulin dimers during dissociation with temperature-jump infrared spectroscopy, we show that co-solvents can significantly destabilize the dimers by perturbing their hydrophobic center. The transition from the native to intermediate dimer state is observed as the buried residues are exposed to solvents in the presence of 10% dimethyl sulfoxide and with α-helices unfolding when ethanol is present, which reduces the dissociation time dramatically to 50% and 20% of the value in a D₂O solution, respectively. We propose a self-consistent analysis using complementary methods to resolve this coupled folding and binding process and obtain a much higher rate of monomer association than of intermediate folding. Our results demonstrate that the conformational changes are critical in dimer formation and strongly affected by co-solvents.

Arguments for an additional long-lived intermediate in the photocycle of the full-length aureochrome 1c receptor: A time-resolved small-angle X-ray scattering study

Aureochromes (AUREO) act as blue-light photoreceptors in algae. They consist of a light-, oxygen-, voltage-sensitive (LOV) domain and a DNA-binding basic region/leucine zipper. Illumination of the flavin cofactor in LOV leads to the formation of an adduct, followed by global structural changes. Here, we first applied UV/vis spectroscopy to characterize the photocycle of full-length aureochrome 1c (PtAUREO1c) from the diatom Phaeodactylum tricornutum. With a time constant of 850 s and a quantum yield of 23%, PtAUREO1c reveals a faster recovery time and a much lower sensitivity toward light than PtAUREO1a, pointing to its role as a high light sensor in vivo. UV/vis spectroscopy offers details on the local recovery of the flavin chromophore. However, kinetic information on the global structural recovery of full-length AUREO or any other multidomain LOV protein is missing. This information is essential not least for the photoreceptors' applications as optogenetic devices. Therefore, we established a procedure to apply small-angle X-ray scattering on PtAUREO1c in a time-resolved manner employing an in-house setup. In combination with UV/vis spectroscopy under similar conditions, we revealed a discrepancy between the recovery of the global protein structure and the adduct lifetime. Accordingly, we propose to supplement the photocycle by an intermediate state (I447), which decays with a time constant of about 800 s and prolongs the lifetime of the signaling state.

Insulin hexamer dissociation dynamics revealed by photoinduced T-jumps and time-resolved X-ray solution scattering

The structural dynamics of insulin hexamer dissociation were studied by the photoinduced temperature jump technique and monitored by time-resolved X-ray scattering. The process of hexamer dissociation was found to involve several transient intermediates, including an expanded hexamer and an unstable tetramer. Our findings provide insights into the mechanisms of protien-protein association.

Protein folding transition path times from single molecule FRET

The transition path is the tiny segment of a single molecule trajectory when the free energy barrier between states is crossed and for protein folding contains all of the information about the self-assembly mechanism. As a first step toward obtaining structural information during the transition path from experiments, single molecule FRET spectroscopy has been used to determine average transition path times from a photon-by-photon analysis of fluorescence trajectories. These results, obtained for several different proteins, have already provided new and demanding tests that support both the accuracy of all-atom molecular dynamics simulations and the basic postulates of energy landscape theory of protein folding.

Implications of short time scale dynamics on long time processes

This review provides a comprehensive overview of the structural dynamics in topical gas- and condensed-phase systems on multiple length and time scales. Starting from vibrationally induced dissociation of small molecules in the gas phase, the question of vibrational and internal energy redistribution through conformational dynamics is further developed by considering coupled electron/proton transfer in a model peptide over many orders of magnitude. The influence of the surrounding solvent is probed for electron transfer to the solvent in hydrated I-. Next, the dynamics of a modified PDZ domain over many time scales is analyzed following activation of a photoswitch. The hydration dynamics around halogenated amino acid side chains and their structural dynamics in proteins are relevant for iodinated TyrB26 insulin. Binding of nitric oxide to myoglobin is a process for which experimental and computational analyses have converged to a common view which connects rebinding time scales and the underlying dynamics. Finally, rhodopsin is a paradigmatic system for multiple length- and time-scale processes for which experimental and computational methods provide valuable insights into the functional dynamics. The systems discussed here highlight that for a comprehensive understanding of how structure, flexibility, energetics, and dynamics contribute to functional dynamics, experimental studies in multiple wavelength regions and computational studies including quantum, classical, and more coarse grained levels are required.

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.

Plasticity in the Oxidative Folding Pathway of the High Affinity Nerita Versicolor Carboxypeptidase Inhibitor (NvCI)

Nerita Versicolor carboxypeptidase inhibitor (NvCI) is the strongest inhibitor reported so far for the M14A subfamily of carboxypeptidases. It comprises 53 residues and a protein fold composed of a two-stranded antiparallel β sheet connected by three loops and stabilized by three disulfide bridges. Here we report the oxidative folding and reductive unfolding pathways of NvCI. Much debate has gone on whether protein conformational folding guides disulfide bond formation or instead they are disulfide bonds that favour the arrangement of local or global structural elements. We show here that for NvCI both possibilities apply. Under physiological conditions, this protein folds trough a funnelled pathway involving a network of kinetically connected native-like intermediates, all sharing the disulfide bond connecting the two β-strands. In contrast, under denaturing conditions, the folding of NvCI is under thermodynamic control and follows a "trial and error" mechanism, in which an initial quasi-stochastic population of intermediates rearrange their disulfide bonds to attain the stable native topology. Despite their striking mechanistic differences, the efficiency of both folding routes is similar. The present study illustrates thus a surprising plasticity in the folding of this extremely stable small disulfide-rich inhibitor and provides the basis for its redesign for biomedical applications.

Optimized parameter selection reveals trends in Markov state models for protein folding

As molecular dynamics simulations access increasingly longer time scales, complementary advances in the analysis of biomolecular time-series data are necessary. Markov state models offer a powerful framework for this analysis by describing a system's states and the transitions between them. A recently established variational theorem for Markov state models now enables modelers to systematically determine the best way to describe a system's dynamics. In the context of the variational theorem, we analyze ultra-long folding simulations for a canonical set of twelve proteins [K. Lindorff-Larsen et al., Science 334, 517 (2011)] by creating and evaluating many types of Markov state models. We present a set of guidelines for constructing Markov state models of protein folding; namely, we recommend the use of cross-validation and a kinetically motivated dimensionality reduction step for improved descriptions of folding dynamics. We also warn that precise kinetics predictions rely on the features chosen to describe the system and pose the description of kinetic uncertainty across ensembles of models as an open issue.

Time resolved calorimetry of photo-induced folding in horse heart cytochrome c at high pH

Here the molar volume and enthalpy changes associated with the early events in the folding of ferrocytochrome c (Cc) at high pH have been examined using time resolved photoacoustic calorimetry (PAC). The data reveal an overall volume change of 1.3 ± 0.3 mL mol^-1 and an enthalpy change of 13 ± 7 kcal mol ^-1 occurring subsequent to photodissociation of the unfolded CO bound Cc species in <∼20 ns. Two additional kinetic phases are observed that are associated with non-native His binding (ΔH and ΔV of 2 ± 4 kcal mol^-1 and -0.5 mL mol^-1, τ ∼ 2.5 μs ) and Met binding (ΔH and ΔV -0.4 ± 2 kcal mol^-1 and -0.1 ± 0.1 mL mol^-1, τ∼ 660 ns). Considering only protein conformational changes (excluding volume and enthalpies associated with heme ligation events) the initial conformational event exhibits a ΔH and ΔV of 6 ± 3 kcal mol^-1 and -3±0.1 mL mol^-1, respectively, that are attributed to a small contraction of the unfolded protein. The corresponding enthalpy associated with both native and non-native ligand binding are found to be -5±4 kcal mol^-1 (Fe-Met) and +20 ± 4 kcal mol^-1 (Fe-His) with the change in volume for both phases being essential negligible. This would indicate that non-native ligand binding likely occurs from an already collapsed conformation.

Studying Protein-Protein Binding through T-Jump Induced Dissociation: Transient 2D IR Spectroscopy of Insulin Dimer

Insulin homodimer associates through the coupled folding and binding of two partially disordered monomers. We aim to understand this dynamics by observing insulin dimer dissociation initiated with a nanosecond temperature jump using transient two-dimensional infrared spectroscopy (2D IR) of amide I vibrations. With the help of equilibrium FTIR and 2D IR spectra, and through a systematic study of the dependence of dissociation kinetics on temperature and insulin concentration, we are able to decompose and analyze the spectral evolution associated with different secondary structures. We find that the dissociation under all conditions is characterized by two processes whose influence on the kinetics varies with temperature: the unfolding of the β sheet at the dimer interface observed as exponential kinetics between 250 and 1000 μs and nonexponential kinetics between 5 and 150 μs that we attribute to monomer disordering. Microscopic reversibility arguments lead us to conclude that dimer association requires significant conformational changes within the monomer in concert with the folding of the interfacial β sheet. While our data indicates a more complex kinetics, we apply a two-state model to the β-sheet unfolding kinetics to extract thermodynamic parameters and kinetic rate constants. The association rate constant, ka (23 °C) = 8.8 × 10(5) M(-1) s(-1) (pH 0, 20% EtOD), is approximately 3 orders of magnitude slower than the calculated diffusion limited association rate, which is explained by the significant destabilizing effect of ethanol on the dimer state and the highly positive charge of the monomers at this pH.

Excited-state deactivation mechanisms of protonated and neutral phenylalanine: a theoretical study

Minimum energy paths (MEPs) of protonated phenylalanine (PheH⁺) at the electronic ground and S₁ (¹ππ*) excited states along the C_α–C_β bond stretching coordinate, following proton transfer to the aromatic chromophore.

Evolution, energy landscapes and the paradoxes of protein folding

Protein folding has been viewed as a difficult problem of molecular self-organization. The search problem involved in folding however has been simplified through the evolution of folding energy landscapes that are funneled. The funnel hypothesis can be quantified using energy landscape theory based on the minimal frustration principle. Strong quantitative predictions that follow from energy landscape theory have been widely confirmed both through laboratory folding experiments and from detailed simulations. Energy landscape ideas also have allowed successful protein structure prediction algorithms to be developed. The selection constraint of having funneled folding landscapes has left its imprint on the sequences of existing protein structural families. Quantitative analysis of co-evolution patterns allows us to infer the statistical characteristics of the folding landscape. These turn out to be consistent with what has been obtained from laboratory physicochemical folding experiments signaling a beautiful confluence of genomics and chemical physics.

A molecular interpretation of 2D IR protein folding experiments with Markov state models

The folding mechanism of the N-terminal domain of ribosomal protein L9 (NTL91-39) is studied using temperature-jump (T-jump) amide I' two-dimensional infrared (2D IR) spectroscopy in combination with spectral simulations based on a Markov state model (MSM) built from millisecond-long molecular dynamics trajectories. The results provide evidence for a compact well-structured folded state and a heterogeneous fast-exchanging denatured state ensemble exhibiting residual secondary structure. The folding rate of 26.4 μs(-1) (at 80°C), extracted from the T-jump response of NTL91-39, compares favorably with the 18 μs(-1) obtained from the MSM. Structural decomposition of the MSM and analysis along the folding coordinate indicates that helix-formation nucleates the global folding. Simulated difference spectra, corresponding to the global folding transition of the MSM, are in qualitative agreement with measured T-jump 2D IR spectra. The experiments demonstrate the use of T-jump 2D IR spectroscopy as a valuable tool for studying protein folding, with direct connections to simulations. The results suggest that in addition to predicting the correct native structure and folding time constant, molecular dynamics simulations carried out with modern force fields provide an accurate description of folding mechanisms in small proteins.

First passage analysis of the folding of a β-sheet miniprotein: is it more realistic than the standard equilibrium approach?

Simulations of first-passage folding of the antiparallel β-sheet miniprotein beta3s, which has been intensively studied under equilibrium conditions by A. Caflisch and co-workers, show that the kinetics and dynamics are significantly different from those for equilibrium folding. Because the folding of a protein in a living system generally corresponds to the former (i.e., the folded protein is stable and unfolding is a rare event), the difference is of interest. In contrast to equilibrium folding, the Ch-curl conformations become very rare because they contain unfavorable parallel β-strand arrangements, which are difficult to form dynamically due to the distant N- and C-terminal strands. At the same time, the formation of helical conformations becomes much easier (particularly in the early stage of folding) due to short-range contacts. The hydrodynamic descriptions of the folding reaction have also revealed that while the equilibrium flow field presented a collection of local vortices with closed ”streamlines”, the first-passage folding is characterized by a pronounced overall flow from the unfolded states to the native state. The flows through the locally stable structures Cs-or and Ns-or, which are conformationally close to the native state, are negligible due to detailed balance established between these structures and the native state. Although there are significant differences in the general picture of the folding process from the equilibrium and first-passage folding simulations, some aspects of the two are in agreement. The rate of transitions between the clusters of characteristic protein conformations in both cases decreases approximately exponentially with the distance between the clusters in the hydrogen bond distance space of collective variables, and the folding time distribution in the first-passage segments of the equilibrium trajectory is in good agreement with that for the first-passage folding simulations.

Fast protein folding kinetics

Fast-folding proteins have been a major focus of computational and experimental study because they are accessible to both techniques: they are small and fast enough to be reasonably simulated with current computational power, but have dynamics slow enough to be observed with specially developed experimental techniques. This coupled study of fast-folding proteins has provided insight into the mechanisms, which allow some proteins to find their native conformation well <1 ms and has uncovered examples of theoretically predicted phenomena such as downhill folding. The study of fast folders also informs our understanding of even 'slow' folding processes: fast folders are small; relatively simple protein domains and the principles that govern their folding also govern the folding of more complex systems. This review summarizes the major theoretical and experimental techniques used to study fast-folding proteins and provides an overview of the major findings of fast-folding research. Finally, we examine the themes that have emerged from studying fast folders and briefly summarize their application to protein folding in general, as well as some work that is left to do.

Discriminating trpzip2 and trpzip4 peptides' folding landscape using the two-dimensional infrared spectroscopy: a simulation study

We analyzed, based on the theoretical spectroscopic modeling, how the differences in the folding landscapes of two β-hairpin peptides trpzip2 and trpzip4 are reflected in their thermal unfolding infrared measurements. The isotope-edited equilibrium FTIR and two dimensional infrared spectra of the two peptides were calculated, using the nonlinear exciton propagation method, at a series of temperatures. The spectra calculations were based on the configuration distributions generated using the GB(OBC) implicit solvent MD simulation and the integrated tempering sampling technique. Conformational analysis revealed the different local thermal stabilities for these two peptides, which suggested the different folding landscapes. Our study further suggested that the ellipticities of the isotope peaks in the coherent IR signals are more sensitive to these local stability differences compared with other spectral features such as the peak intensities. Our technique can thus be combined with the relevant experimental measurements to achieve a better understanding of the peptide folding behaviors.

Folding dynamics of the Trp-cage miniprotein: evidence for a native-like intermediate from combined time-resolved vibrational spectroscopy and molecular dynamics simulations

Trp-cage is a synthetic 20-residue miniprotein which folds rapidly and spontaneously to a well-defined globular structure more typical of larger proteins. Due to its small size and fast folding, it is an ideal model system for experimental and theoretical investigations of protein folding mechanisms. However, Trp-cage's exact folding mechanism is still a matter of debate. Here we investigate Trp-cage's relaxation dynamics in the amide I' spectral region (1530-1700 cm(-1)) using time-resolved infrared spectroscopy. Residue-specific information was obtained by incorporating an isotopic label ((13)C═(18)O) into the amide carbonyl group of residue Gly11, thereby spectrally isolating an individual 310-helical residue. The folding-unfolding equilibrium is perturbed using a nanosecond temperature-jump (T-jump), and the subsequent re-equilibration is probed by observing the time-dependent vibrational response in the amide I' region. We observe bimodal relaxation kinetics with time constants of 100 ± 10 and 770 ± 40 ns at 322 K, suggesting that the folding involves an intermediate state, the character of which can be determined from the time- and frequency-resolved data. We find that the relaxation dynamics close to the melting temperature involve fast fluctuations in the polyproline II region, whereas the slower process can be attributed to conformational rearrangements due to the global (un)folding transition of the protein. Combined analysis of our T-jump data and molecular dynamics simulations indicates that the formation of a well-defined α-helix precedes the rapid formation of the hydrophobic cage structure, implying a native-like folding intermediate, that mainly differs from the folded conformation in the orientation of the C-terminal polyproline II helix relative to the N-terminal part of the backbone. We find that the main free-energy barrier is positioned between the folding intermediate and the unfolded state ensemble, and that it involves the formation of the α-helix, the 310-helix, and the Asp9-Arg16 salt bridge. Our results suggest that at low temperature (T ≪ Tm) a folding path via formation of α-helical contacts followed by hydrophobic clustering becomes more important.

Modeling the thermal unfolding 2DIR spectra of a β-hairpin peptide based on the implicit solvent MD simulation

We simulated the equilibrium isotope-edited FTIR and 2DIR spectra of a β-hairpin peptide trpzip2 at a series of temperatures. The simulation was based on the configuration distributions generated using the GB(OBC) implicit solvent model and the integrated tempering sampling (ITS) technique. A soaking procedure was adapted to generate the peptide in explicit solvent configurations for the spectroscopy calculations. The nonlinear exciton propagation (NEP) method was then used to calculate the spectra. Agreeing with the experiments, the intensities and ellipticities of the isotope-shifted peaks in our simulated signals have the site-specific temperature dependences, which suggest the inhomogeneous local thermal stabilities along the peptide chain. Our simulation thus proposes a cost-effective means to understand a peptide's conformational change and related IR spectra across its thermal unfolding transition.

Effect of coil-globule transition on the single-chain crystallization

The folding process of a single chain including coil-globule transition and crystallization has been investigated through dynamic Monte Carlo simulations. The results based upon ensemble averaging illustrated three distinct states: coil, molten globule, and globule states. Furthermore, the crystallization process from these collapsed states demonstrated various characteristics and it also verified the thermodynamic partitions. The isothermal crystallization in the three states showed the folding rates, and the final crystallite morphologies strongly depended on the collapsed states. Especially, the onset temperature of crystallization in the intermediate molten globule state demonstrated the strongest sensitivity to the solvent qualities in the three different states. Moreover, the crystallization in this intermediate state illustrated a two-step folding mechanism with the prior dense core serving as a precursor to induce the subsequent crystallization. Our observations would help in understanding the thermodynamics and kinetics of phase transition of a single macromolecule. Possible relations to the protein folding were also discussed.

Microsecond folding experiments and simulations: a match is made

For the past two decades, protein folding experiments have been speeding up from the second or millisecond time scale to the microsecond time scale, and full-atom simulations have been extended from the nanosecond to the microsecond and even millisecond time scale. Where the two meet, it is now possible to compare results directly, allowing force fields to be validated and refined, and allowing experimental data to be interpreted in atomistic detail. In this perspective we compare recent experiments and simulations on the microsecond time scale, pointing out the progress that has been made in determining native structures from physics-based simulations, refining experiments and simulations to provide more quantitative underlying mechanisms, and tackling the problems of multiple reaction coordinates, downhill folding, and complex underlying structure of unfolded or misfolded states.

Energy landscape and multiroute folding of topologically complex proteins adenylate kinase and 2ouf-knot

While fast folding of small proteins has been relatively well characterized by experiments and theories, much less is known for slow folding of larger proteins, for which recent experiments suggested quite complex and rich folding behaviors. Here, we address how the energy landscape theory can be applied to these slow folding reactions. Combining the perfect-funnel approximation with a multiscale method, we first extended our previous atomic-interaction based coarse grained (AICG) model to take into account local flexibility of protein molecules. Using this model, we then investigated the energy landscapes and folding routes of two proteins with complex topologies: a multidomain protein adenylate kinase (AKE) and a knotted protein 2ouf-knot. In the AKE folding, consistent with experimental results, the kinetic free energy surface showed several substates between the fully unfolded and native states. We characterized the structural features of these substates and transitions among them, finding temperature-dependent multiroute folding. For protein 2ouf-knot, we found that the improved atomic-interaction based coarse-grained model can spontaneously tie a knot and fold the protein with a probability up to 96%. The computed folding rate of the knotted protein was much slower than that of its unknotted counterpart, in agreement with experimental findings. Similar to the AKE case, the 2ouf-knot folding exhibited several substates and transitions among them. Interestingly, we found a dead-end substate that lacks the knot, thus suggesting backtracking mechanisms.

Markov processes follow from the principle of maximum caliber

Markov models are widely used to describe stochastic dynamics. Here, we show that Markov models follow directly from the dynamical principle of maximum caliber (Max Cal). Max Cal is a method of deriving dynamical models based on maximizing the path entropy subject to dynamical constraints. We give three different cases. First, we show that if constraints (or data) are given in the form of singlet statistics (average occupation probabilities), then maximizing the caliber predicts a time-independent process that is modeled by identical, independently distributed random variables. Second, we show that if constraints are given in the form of sequential pairwise statistics, then maximizing the caliber dictates that the kinetic process will be Markovian with a uniform initial distribution. Third, if the initial distribution is known and is not uniform we show that the only process that maximizes the path entropy is still the Markov process. We give an example of how Max Cal can be used to discriminate between different dynamical models given data.

Microsecond folding dynamics of apomyoglobin at acidic pH

Apomyolgobin (apoMb) is an important model for understanding the folding mechanism of helical proteins. This study focuses on a partially structured state of sperm whale apoMb populated at pH 4.2 (M-state), which structurally resembles a late kinetic intermediate in the formation of the native state (N) at higher pH. The thermodynamics and cooperativity of apoMb folding at pH 4.2 and 6.2 were studied by global analysis of the urea-induced unfolding transitions monitored by tryptophan fluorescence and circular dichroism. The kinetics of folding and unfolding of apoMb at pH 4.2 was measured over a time window from 40 to 850 μs, using fluorescence-detected continuous-flow measurements. Our observation of biphasic kinetics provides clear evidence for rapid (<100 μs) accumulation of previously unresolved intermediate states in both refolding and unfolding experiments. Quantitative kinetic modeling of the results, using a four-state mechanism with two intermediates on a direct route between the unfolded and folded states (U↔I↔L↔M), gave new insight into the conformational states and barriers that precede the rate-limiting step in the formation of the N-state of apoMb.

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

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

Single Molecule Analysis Research Tool (SMART): an integrated approach for analyzing single molecule data

Single molecule studies have expanded rapidly over the past decade and have the ability to provide an unprecedented level of understanding of biological systems. A common challenge upon introduction of novel, data-rich approaches is the management, processing, and analysis of the complex data sets that are generated. We provide a standardized approach for analyzing these data in the freely available software package SMART: Single Molecule Analysis Research Tool. SMART provides a format for organizing and easily accessing single molecule data, a general hidden Markov modeling algorithm for fitting an array of possible models specified by the user, a standardized data structure and graphical user interfaces to streamline the analysis and visualization of data. This approach guides experimental design, facilitating acquisition of the maximal information from single molecule experiments. SMART also provides a standardized format to allow dissemination of single molecule data and transparency in the analysis of reported data.

Spectroscopic studies of protein folding: linear and nonlinear methods

Although protein folding is a simple outcome of the underlying thermodynamics, arriving at a quantitative and predictive understanding of how proteins fold nevertheless poses huge challenges. Therefore, both advanced experimental and computational methods are continuously being developed and refined to probe and reveal the atomistic details of protein folding dynamics and mechanisms. Herein, we provide a concise review of recent developments in spectroscopic studies of protein folding, with a focus on new triggering and probing methods. In particular, we describe several laser-based techniques for triggering protein folding/unfolding on the picosecond and/or nanosecond timescales and various linear and nonlinear spectroscopic techniques for interrogating protein conformations, conformational transitions, and dynamics.

UV resonance Raman studies of the NaClO4 dependence of poly-L-lysine conformation and hydrogen exchange kinetics

We used 204 nm excitation UV Resonance Raman (UVRR) spectroscopy to examine the effects of NaClO(4) on the conformation of poly-L-lysine (PLL). The presence of NaClO(4) induces the formation of α-helix, π-helix/bulge, and turn conformations. The dependence of the AmIII(3) frequency on the peptide Ψ Ramachandran angle allows us to experimentally determine the conformational population distributions and the energy landscape of PLL along the Ramachandran Ψ angle. We also used UVRR to measure the NaClO(4) concentration dependence of PLL amide hydrogen exchange kinetics. Exchange rates were determined by fitting the D(2)O exchanging PLL UVRR AmII' band time evolution. Hydrogen exchange is slowed at high NaClO(4) concentrations. The PLL AmII' band exchange kinetics at 0.0, 0.2, and 0.35 M NaClO(4) can be fit by single exponentials, but the AmII' band kinetics of PLL at 0.8 M NaClO(4) requires a double exponential fit. The exchange rates for the extended conformations were monitored by measuring the C(α)-H band kinetics. These kinetics are identical to those of the AmII' band until 0.8 M NaClO(4) whereupon the extended conformation exchange becomes clearly faster than that of the α-helix-like conformations. Our results indicate that ClO(4)(-) binds to the PLL backbone to protect it from OH(-) exchange catalysis. In addition, ClO(4)(-) binding also slows the conformational exchange between the extended and α-helix-like conformations, probably by increasing the activation barriers for conformational interchanges.

Making connections between ultrafast protein folding kinetics and molecular dynamics simulations

Determining the rate of forming the truly folded conformation of ultrafast folding proteins is an important issue for both experiments and simulations. The double-norleucine mutant of the 35-residue villin subdomain is the focus of recent computer simulations with atomistic molecular dynamics because it is currently the fastest folding protein. The folding kinetics of this protein have been measured in laser temperature-jump experiments using tryptophan fluorescence as a probe of overall folding. The conclusion from the simulations, however, is that the rate determined by fluorescence is significantly larger than the rate of overall folding. We have therefore employed an independent experimental method to determine the folding rate. The decay of the tryptophan triplet-state in photoselection experiments was used to monitor the change in the unfolded population for a sequence of the villin subdomain with one amino acid difference from that of the laser temperature-jump experiments, but with almost identical equilibrium properties. Folding times obtained in a two-state analysis of the results from the two methods at denaturant concentrations varying from 1.5-6.0 M guanidinium chloride are in excellent agreement, with an average difference of only 20%. Polynomial extrapolation of all the data to zero denaturant yields a folding time of 220 (+100,-70) ns at 283 K, suggesting that under these conditions the barrier between folded and unfolded states has effectively disappeared--the so-called "downhill scenario."

Simulating the T-jump-triggered unfolding dynamics of trpzip2 peptide and its time-resolved IR and two-dimensional IR signals using the Markov state model approach

We proposed a computational protocol of simulating the T-jump peptide unfolding experiments and the related transient IR and two-dimensional IR (2DIR) spectra based on the Markov state model (MSM) and nonlinear exciton propagation (NEP) methods. MSMs partition the conformation space into a set of nonoverlapping metastable states, and we can calculate spectra signal for each of these states using the NEP method. Thus the overall spectroscopic observable for a given system is simply the sum of spectra of different metastable states weighted by their populations. We show that results from MSMs constructed from a large number of simulations have a much better agreement with the equilibrium experimental 2DIR spectra compared to that generated from straightforward MD simulations starting from the folded state. This indicates that a sufficient sampling of important relevant conformational states is critical for calculating the accurate spectroscopic observables. MSMs are also capable of simulating the unfolding relaxation dynamics upon the temperature jump. The agreement of the simulation using MSMs and NEP with the experiment not only provides a justification for our protocol, but also provides the physical insight of the underlying spectroscopic observables. The protocol we developed has the potential to be extended to simulate a wide range of fast triggering plus optical detection experiments for biomolecules.

Effects of bounded space in the solutions of time-space fractional diffusion equation

By using a recently proposed numerical method, the fractional diffusion equation with memory in a finite domain is solved for different asymmetry parameters and fractional orders. Some scaling laws are revisited in this condition, such as growth rate in a distance from pulse perturbation, the time when the perturbative peak reaches the other points, and advectionlike behavior as a result of asymmetry and memory. Conditions for negativity and instability of solutions are shown. Also up-hill transport and its time-space region are studied.

Dynamics of protein folding: probing the kinetic network of folding-unfolding transitions with experiment and theory

The problem of spontaneous folding of amino acid chains into highly organized, biologically functional three-dimensional protein structures continues to challenge the modern science. Understanding how proteins fold requires characterization of the underlying energy landscapes as well as the dynamics of the polypeptide chains in all stages of the folding process. In recent years, important advances toward these goals have been achieved owing to the rapidly growing interdisciplinary interest and significant progress in both experimental techniques and theoretical methods. Improvements in the experimental time resolution led to determination of the timescales of the important elementary events in folding, such as formation of secondary structure and tertiary contacts. Sensitive single molecule methods made possible probing the distributions of the unfolded and folded states and following the folding reaction of individual protein molecules. Discovery of proteins that fold in microseconds opened the possibility of atomic-level theoretical simulations of folding and their direct comparisons with experimental data, as well as of direct experimental observation of the barrier-less folding transition. The ultra-fast folding also brought new questions, concerning the intrinsic limits of the folding rates and experimental signatures of barrier-less "downhill" folding. These problems will require novel approaches for even more detailed experimental investigations of the folding dynamics as well as for the analysis of the folding kinetic data. For theoretical simulations of folding, a main challenge is how to extract the relevant information from overwhelmingly detailed atomistic trajectories. New theoretical methods have been devised to allow a systematic approach towards a quantitative analysis of the kinetic network of folding-unfolding transitions between various configuration states of a protein, revealing the transition states and the associated folding pathways at multiple levels, from atomistic to coarse-grained representations. This article is part of a Special Issue entitled: Protein Dynamics: Experimental and Computational Approaches.

Nanosecond time-resolved polarization spectroscopies: tools for probing protein reaction mechanisms

Polarization methods, introduced in the 1800s, offered one of the earliest ways to examine protein structure. Since then, many other structure-sensitive probes have been developed, but circular dichroism (CD) remains a powerful technique because of its versatility and the specificity of protein structural information that can be explored. With improvements in time resolution, from millisecond to picosecond CD measurements, it has proven to be an important tool for studying the mechanism of folding and function in many biomolecules. For example, nanosecond time-resolved CD (TRCD) studies of the sub-microsecond events of reduced cytochrome c folding have provided direct experimental evidence of kinetic heterogeneity, which is an inherent property of the diffusional nature of early folding dynamics on the energy landscape. In addition, TRCD has been applied to the study of many biochemical processes, such as ligand rebinding in hemoglobin and myoglobin and signaling state formation in photoactive yellow protein and prototropin 1 LOV2. The basic approach to TRCD has also been extended to include a repertoire of nanosecond polarization spectroscopies: optical rotatory dispersion (ORD), magnetic CD and ORD, and linear dichroism. This article will discuss the details of the polarization methods used in this laboratory, as well as the coupling of time-resolved ORD with the temperature-jump trigger so that protein folding can be studied in a larger number of proteins.

Mass spectrometry combined with oxidative labeling for exploring protein structure and folding

This review discusses various mass spectrometry (MS)-based approaches for exploring structural aspects of proteins in solution. Electrospray ionization (ESI)-MS, in particular, has found fascinating applications in this area. For example, when used in conjunction with solution-phase hydrogen/deuterium exchange (HDX), ESI-MS is a highly sensitive tool for probing conformational dynamics. The main focus of this article is a technique that is complementary to HDX, that is, the covalent labeling of proteins by hydroxyl radicals. The reactivity of individual amino acid side chains with *OH is strongly affected by their degree of solvent exposure. Thus, analysis of the oxidative labeling pattern by peptide mapping and tandem mass spectrometry provides detailed structural information. A convenient method for *OH production is the photolysis of H(2)O(2) by a pulsed UV laser, resulting in oxidative labeling on the microsecond time scale. Selected examples demonstrate the use of this technique for structural studies on membrane proteins, and the combination with rapid mixing devices for characterizing the properties of short-lived protein (un)folding intermediates.

Investigation of protein folding by coarse-grained molecular dynamics with the UNRES force field

Coarse-grained molecular dynamics simulations offer a dramatic extension of the time-scale of simulations compared to all-atom approaches. In this article, we describe the use of the physics-based united-residue (UNRES) force field, developed in our laboratory, in protein-structure simulations. We demonstrate that this force field offers about a 4000-times extension of the simulation time scale; this feature arises both from averaging out the fast-moving degrees of freedom and reduction of the cost of energy and force calculations compared to all-atom approaches with explicit solvent. With massively parallel computers, microsecond folding simulation times of proteins containing about 1000 residues can be obtained in days. A straightforward application of canonical UNRES/MD simulations, demonstrated with the example of the N-terminal part of the B-domain of staphylococcal protein A (PDB code: 1BDD, a three-alpha-helix bundle), discerns the folding mechanism and determines kinetic parameters by parallel simulations of several hundred or more trajectories. Use of generalized-ensemble techniques, of which the multiplexed replica exchange method proved to be the most effective, enables us to compute thermodynamics of folding and carry out fully physics-based prediction of protein structure, in which the predicted structure is determined as a mean over the most populated ensemble below the folding-transition temperature. By using principal component analysis of the UNRES folding trajectories of the formin-binding protein WW domain (PDB code: 1E0L; a three-stranded antiparallel beta-sheet) and 1BDD, we identified representative structures along the folding pathways and demonstrated that only a few (low-indexed) principal components can capture the main structural features of a protein-folding trajectory; the potentials of mean force calculated along these essential modes exhibit multiple minima, as opposed to those along the remaining modes that are unimodal. In addition, a comparison between the structures that are representative of the minima in the free-energy profile along the essential collective coordinates of protein folding (computed by principal component analysis) and the free-energy profile projected along the virtual-bond dihedral angles gamma of the backbone revealed the key residues involved in the transitions between the different basins of the folding free-energy profile, in agreement with existing experimental data for 1E0L .

Modulation of calmodulin plasticity by the effect of macromolecular crowding

In vitro biochemical reactions are most often studied in dilute solution, a poor mimic of the intracellular space of eukaryotic cells, which are crowded with mobile and immobile macromolecules. Such crowded conditions exert volume exclusion and other entropic forces that have the potential to impact chemical equilibria and reaction rates. In this article, we used the well-characterized and ubiquitous molecule calmodulin (CaM) and a combination of theoretical and experimental approaches to address how crowding impacts CaM's conformational plasticity. CaM is a dumbbell-shaped molecule that contains four EF hands (two in the N-lobe and two in the C-lobe) that each could bind Ca(2+), leading to stabilization of certain substates that favor interactions with other target proteins. Using coarse-grained molecular simulations, we explored the distribution of CaM conformations in the presence of crowding agents. These predictions, in which crowding effects enhance the population of compact structures, were then confirmed in experimental measurements using fluorescence resonance energy transfer techniques of donor- and acceptor-labeled CaM under normal and crowded conditions. Using protein reconstruction methods, we further explored the folding-energy landscape and examined the structural characteristics of CaM at free-energy basins. We discovered that crowding stabilizes several different compact conformations, which reflects the inherent plasticity in CaM's structure. From these results, we suggest that the EF hands in the C-lobe are flexible and can be thought of as a switch, while those in the N-lobe are stiff, analogous to a rheostat. New combinatorial signaling properties may arise from the product of the differential plasticity of the two distinct lobes of CaM in the presence of crowding. We discuss the implications of these results for modulating CaM's ability to bind Ca(2+) and target proteins.