Coordinate and time-dependent diffusion dynamics in protein folding

Quantifying the energy landscape in weakly and strongly disordered frictional media

We investigate the "roughness" of the energy landscape of a system that diffuses in a heterogeneous medium with a random position-dependent friction coefficient α(x). This random friction acting on the system stems from spatial inhomogeneity in the surrounding medium and is modeled using the generalized Caldira-Leggett model. For a weakly disordered medium exhibiting a Gaussian random diffusivity D(x) = kBT/α(x) characterized by its average value ⟨D(x)⟩ and a pair-correlation function ⟨D(x1)D(x2)⟩, we find that the renormalized intrinsic diffusion coefficient is lower than the average one due to the fluctuations in diffusivity. The induced weak internal friction leads to increased roughness in the energy landscape. When applying this idea to diffusive motion in liquid water, the dissociation energy for a hydrogen bond gradually approaches experimental findings as fluctuation parameters increase. Conversely, for a strongly disordered medium (i.e., ultrafast-folding proteins), the energy landscape ranges from a few to a few kcal/mol, depending on the strength of the disorder. By fitting protein folding dynamics to the escape process from a metastable potential, the decreased escape rate conceptualizes the role of strong internal friction. Studying the energy landscape in complex systems is helpful because it has implications for the dynamics of biological, soft, and active matter systems.

Exploring the Folding Mechanism of Dimeric Superoxide Dismutase

The Cu/Zn Human Superoxide Dismutase (SOD1) is a dimeric metalloenzyme whose genetic mutations are directly related to amyotrophic lateral sclerosis (ALS), so understanding its folding mechanism is of fundamental importance. Currently, the SOD1 dimer formation is studied via molecular dynamics simulations using a simplified structure-based model and an all-atom model. Results from the simplified model reveal a mechanism dependent on distances between monomers, which are limited by constraints to mimic concentration dependence. The stability of intermediates (during the int state) is significantly affected by this distance, as well as by the presence of two folded monomers prior to dimer formation. The kinetics of interface formation are also highly dependent on the separation distance. The folding temperature of the dimer is about 4.2% higher than that of the monomer, a value not too different from experimental data. All-atom simulations on the apo dimer give binding free energy between monomers similar to experimental values. An intermediate state is evident for the apo form at a separation distance between monomers slightly larger than the native distance which has little formed interface between monomers. We have shown that this intermediate is stabilized by non-native intra- and intercontacts.

Coordinate-Dependent Drift-Diffusion Reveals the Kinetic Intermediate Traps of Top7-Based Proteins

The computer-designed Top7 served as a scaffold to produce immunoreactive proteins by grafting of the 2F5 HIV-1 antibody epitope (Top7-2F5) followed by biotinylation (Top7-2F5-biotin). The resulting nonimmunoglobulin affinity proteins were effective in inducing and detecting the HIV-1 antibody. However, the grafted Top7-2F5 design led to protein aggregation, as opposed to the soluble biotinylated Top7-2F5-biotin. The structure-based model predicted that the thermodynamic cooperativity of Top7 increases after grafting and biotin-labeling, reducing their intermediate state populations. In this work, the folding kinetic traps that might contribute to the aggregation propensity are investigated by the diffusion theory. Since the engineered proteins have similar sequence and structural homology, they served as protein models to study the kinetic intermediate traps that were uncovered by characterizing the position-dependent drift-velocity (v(Q)) and the diffusion (D(Q)) coefficients. These coordinate-dependent coefficients were taken into account to obtain the folding and transition path times over the free energy transition states containing the intermediate kinetic traps. This analysis may be useful to predict the aggregated kinetic traps of scaffold-epitope proteins that might compose novel diagnostic and therapeutic platforms.

Biotinylation Eliminates the Intermediate State of Top7 Designed with an HIV-1 Epitope

Broadly neutralizing antibodies against HIV-1 are rare with the 2F5 antibody being one of the most protective. Insertion of an antibody epitope into a stable and small protein scaffold overcomes many of the obstacles found to produce antibodies. However, the design leads to grafting of epitopes that may cause protein aggregation. Here, I investigated the 2F5 epitope grafted into the Top7 as the scaffold in which the resulting immunoreactive protein precipitates along the storage time, as opposed to its completely soluble biotinylated version. Molecular dynamics showed that biotinylation eliminates the intermediate state of the scaffold-epitope Top7-2F5 by switching a noncooperative to a cooperative folding. The aggregation propensity of the Top7-designed proteins is examined in light of thermodynamic cooperativity and kinetic traps along the decreasing depth of the intermediate ensemble in the free energy landscape. This protocol may predict stable and soluble scaffold-epitopes with the purpose of composing novel therapeutic and diagnostic platforms.

Biotin-painted proteins have thermodynamic stability switched by kinetic folding routes

Biotin-labeled proteins are widely used as tools to study protein-protein interactions and proximity in living cells. Proteomic methods broadly employ proximity-labeling technologies based on protein biotinylation in order to investigate the transient encounters of biomolecules in subcellular compartments. Biotinylation is a post-translation modification in which the biotin molecule is attached to lysine or tyrosine residues. So far, biotin-based technologies proved to be effective instruments as affinity and proximity tags. However, the influence of biotinylation on aspects such as folding, binding, mobility, thermodynamic stability, and kinetics needs to be investigated. Here, we selected two proteins [biotin carboxyl carrier protein (BCCP) and FKBP3] to test the influence of biotinylation on thermodynamic and kinetic properties. Apo (without biotin) and holo (biotinylated) protein structures were used separately to generate all-atom structure-based model simulations in a wide range of temperatures. Holo BCCP contains one biotinylation site, and FKBP3 was modeled with up to 23 biotinylated lysines. The two proteins had their estimated thermodynamic stability changed by altering their energy landscape. In all cases, after comparison between the apo and holo simulations, differences were observed on the free-energy profiles and folding routes. Energetic barriers were altered with the density of states clearly showing changes in the transition state. This study suggests that analysis of large-scale datasets of biotinylation-based proximity experiments might consider possible alterations in thermostability and folding mechanisms imposed by the attached biotins.

Coarse-Grained Simulations of Protein Folding: Bridging Theory and Experiments

Computational coarse-grained models play a fundamental role as a research tool in protein folding, and they are important in bridging theory and experiments. Folding mechanisms are generally discussed using the energy landscape framework, which is well mapped within a class of simplified structure-based models. In this chapter, simplified computer models are discussed with special focus on structure-based ones.

Physics of biomolecular recognition and conformational dynamics

Biomolecular recognition usually leads to the formation of binding complexes, often accompanied by large-scale conformational changes. This process is fundamental to biological functions at the molecular and cellular levels. Uncovering the physical mechanisms of biomolecular recognition and quantifying the key biomolecular interactions are vital to understand these functions. The recently developed energy landscape theory has been successful in quantifying recognition processes and revealing the underlying mechanisms. Recent studies have shown that in addition to affinity, specificity is also crucial for biomolecular recognition. The proposed physical concept of intrinsic specificity based on the underlying energy landscape theory provides a practical way to quantify the specificity. Optimization of affinity and specificity can be adopted as a principle to guide the evolution and design of molecular recognition. This approach can also be used in practice for drug discovery using multidimensional screening to identify lead compounds. The energy landscape topography of molecular recognition is important for revealing the underlying flexible binding or binding-folding mechanisms. In this review, we first introduce the energy landscape theory for molecular recognition and then address four critical issues related to biomolecular recognition and conformational dynamics: (1) specificity quantification of molecular recognition; (2) evolution and design in molecular recognition; (3) flexible molecular recognition; (4) chromosome structural dynamics. The results described here and the discussions of the insights gained from the energy landscape topography can provide valuable guidance for further computational and experimental investigations of biomolecular recognition and conformational dynamics.

Extension-Dependent Drift Velocity and Diffusion (DrDiff) Directly Reconstructs the Folding Free Energy Landscape of Atomic Force Microscopy Experiments

Two equilibrium force microscopy trajectories [q(t)] of high-precision single-molecule spectroscopy assays were analyzed: the pulling of an HIV RNA hairpin and of a 3-aa sequence of the bacteriorhodopsin membrane protein. Both present hundreds of two-state folding transitions, and their free-energy [F(q)] landscapes were previously obtained by deconvolving time signals with the inverse Boltzmann and p_fold methods. In this letter, the two F profiles were reconstructed directly from the measured time-series by the drift-diffusion (DrDiff) framework that characterized the effective conformational drift-velocity [v(q)] and diffusion [D(q)] coefficients. The two thermodynamic F profiles reconstructed with DrDiff directly from q(t) were in good agreement with those previously obtained from the deconvolved time signals. q(t) trajectories simulated with a two-dimensional framework in which the diffusion coefficient of the pulling setup (q coordinate) differed from the molecule (x coordinate) were also analyzed by DrDiff. The performance in reconstructing F was investigated in different conditions of diffusion anisotropy in the simulated time-series using Brownian dynamics. In addition, recently developed theories were used in order to evaluate the quality of the analysis performed in the experimental time series: the memory effects and the intrinsic biomolecular dynamic properties after connecting the probe to the molecule. With the 2-dimensional diffusive models and the additional analyses, it is proposed that the different physical regimes imposed by the stiffer probes of the two biomolecules will have an impact in the measured extension-dependent D and, thus, in the reconstruction of F by DrDiff. Stiffer AFM probes may reflect the molecular behavior more faithfully and reconstruction of F might be more successful. The reported quantities extracted directly from q(t) highlights the current state of the biomolecule characterization by force spectroscopy experiments: it is still challenging despite the recent advances, yet it is very promising.

The Role of Electrostatics and Folding Kinetics on the Thermostability of Homologous Cold Shock Proteins

Understanding which aspects contribute to the thermostability of proteins is a challenge that has persisted for decades, and it is of great relevance for protein engineering. Several types of interactions can influence the thermostability of a protein. Among them, the electrostatic interactions have been a target of particular attention. Aiming to explore how this type of interaction can affect protein thermostability, this paper investigated four homologous cold shock proteins from psychrophilic, mesophilic, thermophilic, and hyperthermophilic organisms using a set of theoretical methodologies. It is well-known that electrostatics as well as hydrophobicity are key-elements for the stabilization of these proteins. Therefore, both interactions were initially analyzed in the native structure of each protein. Electrostatic interactions present in the native structures were calculated with the Tanford-Kirkwood model with solvent accessibility, and the amount of hydrophobic surface area buried upon folding was estimated by measuring both folded and extended structures. On the basis of Energy Landscape Theory, the local frustration and the simplified alpha-carbon structure-based model were modeled with a Debye-Hückel potential to take into account the electrostatics and the effects of an implicit solvent. Thermodynamic data for the structure-based model simulations were collected and analyzed using the Weighted Histogram Analysis and Stochastic Diffusion methods. Kinetic quantities including folding times, transition path times, folding routes, and Φ values were also obtained. As a result, we found that the methods are able to qualitatively infer that electrostatic interactions play an important role on the stabilization of the most stable thermophilic cold shock proteins, showing agreement with the experimental data.

A side-chain engineering strategy for constructing fluorescent dyes with direct and ultrafast self-delivery to living cells

Organic fluorescent dyes with excellent self-delivery to living cells are always difficult to find due to the limitation of the plasma membrane having rigorous selectivity. Herein, in order to improve the permeability of dyes, we utilize a side-chain engineering strategy (SCES): adjusting the side-chain length of dyes to fine-tune the adsorption and desorption processes on the membrane–aqueous phase interfaces of the outer and inner leaflets of the plasma membrane. For this, a family of fluorescent derivatives (SPs) was prepared by functionalizing a styryl-pyridinium fluorophore with alkyl side-chains containing a different carbon number from 1 to 22. Systematic experimental investigations and simulated calculations demonstrate that the self-delivery rate of SPs with a suitable length side-chain is about 22-fold higher in SiHa cells and 76-fold higher in mesenchymal stem cells than that of unmodified SP-1, enabling cell-imaging at an ultralow loading concentration of 1 nM and deep penetration in turbid tissue and in vivo. Moreover, the SCES can even endow a membrane-impermeable fluorescent scaffold with good permeability. Further, quantitative research on the relationship between Clog P and cell permeability shows that when Clog P is in the range of 1.3–2.5, dyes possess optimal permeability. Therefore, this work not only systematically reports the effect of side-chain length on dye delivery for the first time, but also provides some ideal fluorescent probes. At the same time, it gives a suitable Clog P range for efficient cellular delivery, which can serve as a guide for designing cell-permeant dyes. In a word, all the results reveal that the SCES is an effective strategy to dramatically improve dye permeability.

A side-chain engineering strategy can dramatically improve dye delivery by fine-tuning the adsorption and desorption abilities of the transmembrane process, enabling ultralow loading bioimaging and deep tissue penetration.

Position-, disorder-, and salt-dependent diffusion in binding-coupled-folding of intrinsically disordered proteins

Successful extensions of protein-folding energy landscape theory to intrinsically disordered proteins' (IDPs') binding-coupled-folding transition can enormously simplify this biomolecular process into diffusion along a limited number of reaction coordinates, and the dynamics subsequently is described by Kramers' rate theory. As the critical pre-factor, the diffusion coefficient D has direct implications on the binding kinetics. Here, we employ a structure-based model (SBM) to calculate D in the binding-folding of an IDP prototype. We identify a strong position-dependent D during binding by applying a reaction coordinate that directly measures the fluctuations in a Cartesian configuration space. Using the malleability of the SBM, we modulate the degree of conformational disorder in an isolated IDP and determine complex effects of intrinsic disorder on D varying for different binding stages. Here, D tends to increase with disorder during initial binding but shows a non-monotonic relationship with disorder in terms of a decrease-followed-by-increase in D during the late binding stage. The salt concentration, which correlates with electrostatic interactions via Debye-Hückel theory in our SBM, also modulates D in a stepwise way. The speeding up of diffusion by electrostatic interactions is observed during the formation of the encounter complex at the beginning of binding, while the last diffusive binding dynamics is hindered by non-native salt bridges. Because D describes the diffusive speed locally, which implicitly reflects the roughness of the energy landscape, we are eventually able to portray the binding energy landscape, including that from IDPs' binding, then to binding with partial folding, and finally to rigid docking, as well as that under different environmental salt concentrations. Our theoretical results provide key mechanistic insights into IDPs' binding-folding, which is internally conformation- and externally salt-controlled with respect to diffusion.

A Dynamic Hydrophobic Core and Surface Salt Bridges Thermostabilize a Designed Three-Helix Bundle

Thermostable proteins are advantageous in industrial applications, as pharmaceuticals or biosensors, and as templates for directed evolution. As protein-design methodologies improve, bioengineers are able to design proteins to perform a desired function. Although many rationally designed proteins end up being thermostable, how to intentionally design de novo, thermostable proteins is less clear. UVF is a de novo-designed protein based on the backbone structure of the Engrailed homeodomain (EnHD) and is highly thermostable (T_m > 99°C vs. 52°C for EnHD). Although most proteins generally have polar amino acids on their surfaces and hydrophobic amino acids buried in their cores, protein engineers followed this rule exactly when designing UVF. To investigate the contributions of the fully hydrophobic core versus the fully polar surface to UVF’s thermostability, we built two hybrid, chimeric proteins combining the sets of buried and surface residues from UVF and EnHD. Here, we determined a structural, dynamic, and thermodynamic explanation for UVF’s thermostability by performing 4 μs of all-atom, explicit-solvent molecular dynamics simulations at 25 and 100°C, Tanford-Kirkwood solvent accessibility Monte Carlo electrostatic calculations, and a thermodynamic analysis of 40 temperature runs by the weighted-histogram analysis method of heavy-atom, structure-based models of UVF, EnHD, and both chimeric proteins. Our models showed that UVF was highly dynamic because of its fully hydrophobic core, leading to a smaller loss of entropy upon folding. The charged residues on its surface made favorable electrostatic interactions that contributed enthalpically to its thermostability. In the chimeric proteins, both the hydrophobic core and charged surface independently imparted thermostability.

Stochastic diffusion framework determines the free-energy landscape and rate from single-molecule trajectory

A theoretical stochastic diffusion framework is developed that characterizes the position-dependent diffusion coefficient [D(Q)] and drift velocity [ v (Q)] by analysing single-molecule time traces [Q(t)]. The free-energy landscape [F(Q)] that governs the dynamics is reconstructed with the calculated D and v . There are many computational tools that perform this task in which some are computationaly demanding, difficult to run, and, most of the time, not directly available to the community. This is a first attempt to implement the simplified stochastic diffusion framework that is fast, easy to run in a Python environment, and available to be extended as needed. It does not require adjustable parameters, inference methods, or sampling bias such as Monte Carlo Bayesian estimators or umbrella samplings. The stochastic framework was applied in the protein-like lattice model with Monte Carlo simulations, which accurately predicted the folding rates with the coordinate-dependent D and F plugged into Kramers' theory. The results were compared with two other independently developed methodologies (the Bayesian analysis and fep1D algorithm) presenting a good match, which confirms its validity. This theoretical framework might be useful in determining the free-energy and rates by providing time series only from biological or condensed-phase systems. The code is freely available at https://github.com/ronaldolab/stochastic_diffusion.

A critical comparison of coarse-grained structure-based approaches and atomic models of protein folding

Structure-based coarse-grained Gō-like models have been used extensively in deciphering protein folding mechanisms because of their simplicity and tractability. Meanwhile, explicit-solvent molecular dynamics (MD) simulations with physics-based all-atom force fields have been applied successfully to simulate folding/unfolding transitions for several small, fast-folding proteins. To explore the degree to which coarse-grained Gō-like models and their extensions to incorporate nonnative interactions are capable of producing folding processes similar to those in all-atom MD simulations, here we systematically compare the computed unfolded states, transition states, and transition paths obtained using coarse-grained models and all-atom explicit-solvent MD simulations. The conformations in the unfolded state in common Gō models are more extended, and are thus more in line with experiment, than those from all-atom MD simulations. Nevertheless, the structural features of transition states obtained by the two types of models are largely similar. In contrast, the folding transition paths are significantly more sensitive to modeling details. In particular, when common Gō-like models are augmented with nonnative interactions, the predicted dimensions of the unfolded conformations become similar to those computed using all-atom MD. With this connection, the large deviations of all-atom MD from simple diffusion theory are likely caused in part by the presence of significant nonnative effects in folding processes modelled by current atomic force fields. The ramifications of our findings to the application of coarse-grained modeling to more complex biomolecular systems are discussed.

Molecular Simulations Suggest a Force-Dependent Mechanism of Vinculin Activation

Focal adhesions are dynamic constructs at the leading edge of migrating cells, linking them to the extracellular matrix and enabling force sensing and transmission. The lifecycle of a focal adhesion is a highly coordinated process involving spatial and temporal variations of protein composition, interaction, and cellular tension. The assembly of focal adhesions requires the recruitment and activation of vinculin. Vinculin is present in the cytoplasm in an autoinhibited conformation in which its tail is held pincerlike by its head domains, further stabilized by two high-affinity head-tail interfaces. Vinculin has binding sites for talin and F-actin, but effective binding requires vinculin activation to release its head-tail associations. In migrating cells, it has been shown that the locations of vinculin activation coincide with areas of high cellular tension, and that the highest recorded tensions across vinculin are associated with adhesion assembly. Here, we use a structure-based model to investigate vinculin activation by talin modulated by tensile force generated by transient associations with F-actin. We show that vinculin activation may proceed from an intermediate state stabilized by partial talin-vinculin association. There is a low-force regime and a high-force regime where vinculin activation is dominated by two different pathways with distinct responses to force. Specifically, at zero or low forces, vinculin activation requires substantial destabilization of the main head-tail interface, which is rigid and undergoes very limited fluctuations, despite the other being relatively flexible. This pathway is not significantly affected by force; instead, higher forces favor an alternative pathway, which seeks to release the vinculin tail from its pincerlike head domains before destabilizing the head-tail interfaces. This pathway has a force-sensitive activation barrier and is significantly accelerated by force. Experimental data of vinculin during various stages of the focal adhesion lifecycle are consistent with the proposed force-regulated activation pathway.

Quantifying Nonnative Interactions in the Protein-Folding Free-Energy Landscape

Protein folding is a central problem in biological physics. Energetic roughness is an important aspect that controls protein-folding stability and kinetics. The roughness is associated with conflicting interactions in the protein and is also known as frustration. Recent studies indicate that an addition of a small amount of energetic frustration may enhance folding speed for certain proteins. In this study, we have investigated the conditions under which frustration increases the folding rate. We used a Cα structure-based model to simulate a group of proteins. We found that the free-energy barrier at the transition state (ΔF) correlates with nonnative-contact variation (ΔA), and the simulated proteins are clustered according to their fold motifs. These findings are corroborated by the Clementi-Plotkin analytical model. As a consequence, the optimum frustration regime for protein folding can be predicted analytically.

Connecting thermal and mechanical protein (un)folding landscapes

Molecular dynamics simulations supplement single-molecule pulling experiments by providing the possibility of examining the full free energy landscape using many coordinates. Here, we use an all-atom structure-based model to study the force and temperature dependence of the unfolding of the protein filamin by applying force at both termini. The unfolding time-force relation τ(F) indicates that the force-induced unfolding behavior of filamin can be characterized into three regimes: barrier-limited low- and intermediate-force regimes, and a barrierless high-force regime. Slope changes of τ(F) separate the three regimes. We show that the behavior of τ(F) can be understood from a two-dimensional free energy landscape projected onto the extension X and the fraction of native contacts Q. In the low-force regime, the unfolding rate is roughly force-independent due to the small (even negative) separation in X between the native ensemble and transition state ensemble (TSE). In the intermediate-force regime, force sufficiently separates the TSE from the native ensemble such that τ(F) roughly follows an exponential relation. This regime is typically explored by pulling experiments. While X may fail to resolve the TSE due to overlap with the unfolded ensemble just below the folding temperature, the overlap is minimal at lower temperatures where experiments are likely to be conducted. The TSE becomes increasingly structured with force, whereas the average order of structural events during unfolding remains roughly unchanged. The high-force regime is characterized by barrierless unfolding, and the unfolding time approaches a limit of ∼10 μs for the highest forces we studied. Finally, a combination of X and Q is shown to be a good reaction coordinate for almost the entire force range.

Protein misfolding occurs by slow diffusion across multiple barriers in a rough energy landscape

The timescale for the microscopic dynamics of proteins during conformational transitions is set by the intrachain diffusion coefficient, D. Despite the central role of protein misfolding and aggregation in many diseases, it has proven challenging to measure D for these processes because of their heterogeneity. We used single-molecule force spectroscopy to overcome these challenges and determine D for misfolding of the prion protein PrP. Observing directly the misfolding of individual dimers into minimal aggregates, we reconstructed the energy landscape governing nonnative structure formation. Remarkably, rather than displaying multiple pathways, as typically expected for aggregation, PrP dimers were funneled into a thermodynamically stable misfolded state along a single pathway containing several intermediates, one of which blocked native folding. Using Kramers' rate theory, D was found to be 1,000-fold slower for misfolding than for native folding, reflecting local roughening of the misfolding landscape, likely due to increased internal friction. The slow diffusion also led to much longer transit times for barrier crossing, allowing transition paths to be observed directly for the first time to our knowledge. These results open a new window onto the microscopic mechanisms governing protein misfolding.

Exploring multi-dimensional coordinate-dependent diffusion dynamics on the energy landscape of protein conformation change

We explore the multi-dimensional diffusion dynamics of protein conformational change. We found in general that the diffusion is anisotropic and inhomogeneous. The directional and positional dependence of diffusion have significant impacts on the protein conformational kinetics: the dominant kinetic path of conformational change is shifted from the naively expected steepest decent gradient paths. The kinetic transition state is shifted away from the transition state. The effective kinetic free energy barrier height, determining the kinetic rate of the conformational change, is shifted away from the one estimated from the thermodynamic free energy barrier. The shift of the transition state in position and value will modify the phi value analysis for identification of hot residues and interactions responsible for conformational dynamics. Ongoing and future experiments can test the predictions of the model.

Analyzing the effect of homogeneous frustration in protein folding

The energy landscape theory has been an invaluable theoretical framework in the understanding of biological processes such as protein folding, oligomerization, and functional transitions. According to the theory, the energy landscape of protein folding is funneled toward the native state, a conformational state that is consistent with the principle of minimal frustration. It has been accepted that real proteins are selected through natural evolution, satisfying the minimum frustration criterion. However, there is evidence that a low degree of frustration accelerates folding. We examined the interplay between topological and energetic protein frustration. We employed a Cα structure-based model for simulations with a controlled nonspecific energetic frustration added to the potential energy function. Thermodynamics and kinetics of a group of 19 proteins are completely characterized as a function of increasing level of energetic frustration. We observed two well-separated groups of proteins: one group where a little frustration enhances folding rates to an optimal value and another where any energetic frustration slows down folding. Protein energetic frustration regimes and their mechanisms are explained by the role of non-native contact interactions in different folding scenarios. These findings strongly correlate with the protein free-energy folding barrier and the absolute contact order parameters. These computational results are corroborated by principal component analysis and partial least square techniques. One simple theoretical model is proposed as a useful tool for experimentalists to predict the limits of improvements in real proteins.

Localizing internal friction along the reaction coordinate of protein folding by combining ensemble and single-molecule fluorescence spectroscopy

Theory, simulations and experimental results have suggested an important role of internal friction in the kinetics of protein folding. Recent experiments on spectrin domains provided the first evidence for a pronounced contribution of internal friction in proteins that fold on the millisecond timescale. However, it has remained unclear how this contribution is distributed along the reaction and what influence it has on the folding dynamics. Here we use a combination of single-molecule Förster resonance energy transfer, nanosecond fluorescence correlation spectroscopy, microfluidic mixing and denaturant- and viscosity-dependent protein-folding kinetics to probe internal friction in the unfolded state and at the early and late transition states of slow- and fast-folding spectrin domains. We find that the internal friction affecting the folding rates of spectrin domains is highly localized to the early transition state, suggesting an important role of rather specific interactions in the rate-limiting conformational changes.

Connecting the kinetics and energy landscape of tRNA translocation on the ribosome

Functional rearrangements in biomolecular assemblies result from diffusion across an underlying energy landscape. While bulk kinetic measurements rely on discrete state-like approximations to the energy landscape, single-molecule methods can project the free energy onto specific coordinates. With measures of the diffusion, one may establish a quantitative bridge between state-like kinetic measurements and the continuous energy landscape. We used an all-atom molecular dynamics simulation of the 70S ribosome (2.1 million atoms; 1.3 microseconds) to provide this bridge for specific conformational events associated with the process of tRNA translocation. Starting from a pre-translocation configuration, we identified sets of residues that collectively undergo rotary rearrangements implicated in ribosome function. Estimates of the diffusion coefficients along these collective coordinates for translocation were then used to interconvert between experimental rates and measures of the energy landscape. This analysis, in conjunction with previously reported experimental rates of translocation, provides an upper-bound estimate of the free-energy barriers associated with translocation. While this analysis was performed for a particular kinetic scheme of translocation, the quantitative framework is general and may be applied to energetic and kinetic descriptions that include any number of intermediates and transition states.

Subdiffusion of proteins and oligomers on membranes

Diffusion of proteins on lipid membranes plays a central role in cell signaling processes. From a mathematical perspective, most membrane diffusion processes are explained by the Saffman-Delbrück theory. However, recent studies have suggested a major limitation in the theoretical framework, the lack of complexity in the modeled lipid membrane. Lipid domains (sometimes termed membrane rafts) are known to slow protein diffusion, but there have been no quantitative theoretical examinations of how much diffusion is slowed in a general case. We provide an overall theoretical framework for confined-domain ("corralled") diffusion. Further, there have been multiple apparent contradictions of the basic conclusions of Saffman and Delbrück, each involving cases in which a single protein or an oligomer has multiple transmembrane regions passing through a lipid phase barrier. We present a set of corrections to the Saffman-Delbrück theory to account for these experimental observations. Our corrections are able to provide a quantitative explanation of numerous cellular signaling processes that have been considered beyond the scope of the Saffman-Delbrück theory, and may be extendable to other forms of subdiffusion.

Transition paths, diffusive processes, and preequilibria of protein folding

Fundamental relationships between the thermodynamics and kinetics of protein folding were investigated using chain models of natural proteins with diverse folding rates by extensive comparisons between the distribution of conformations in thermodynamic equilibrium and the distribution of conformations sampled along folding trajectories. Consistent with theory and single-molecule experiment, duration of the folding transition paths exhibits only a weak correlation with overall folding time. Conformational distributions of folding trajectories near the overall thermodynamic folding/unfolding barrier show significant deviations from preequilibrium. These deviations, the distribution of transition path times, and the variation of mean transition path time for different proteins can all be rationalized by a diffusive process that we modeled using simple Monte Carlo algorithms with an effective coordinate-independent diffusion coefficient. Conformations in the initial stages of transition paths tend to form more nonlocal contacts than typical conformations with the same number of native contacts. This statistical bias, which is indicative of preferred folding pathways, should be amenable to future single-molecule measurements. We found that the preexponential factor defined in the transition state theory of folding varies from protein to protein and that this variation can be rationalized by our Monte Carlo diffusion model. Thus, protein folding physics is different in certain fundamental respects from the physics envisioned by a simple transition-state picture. Nonetheless, transition state theory can be a useful approximate predictor of cooperative folding speed, because the height of the overall folding barrier is apparently a proxy for related rate-determining physical properties.

Biomolecular dynamics: order-disorder transitions and energy landscapes

While the energy landscape theory of protein folding is now a widely accepted view for understanding how relatively weak molecular interactions lead to rapid and cooperative protein folding, such a framework must be extended to describe the large-scale functional motions observed in molecular machines. In this review, we discuss (1) the development of the energy landscape theory of biomolecular folding, (2) recent advances toward establishing a consistent understanding of folding and function and (3) emerging themes in the functional motions of enzymes, biomolecular motors and other biomolecular machines. Recent theoretical, computational and experimental lines of investigation have provided a very dynamic picture of biomolecular motion. In contrast to earlier ideas, where molecular machines were thought to function similarly to macroscopic machines, with rigid components that move along a few degrees of freedom in a deterministic fashion, biomolecular complexes are only marginally stable. Since the stabilizing contribution of each atomic interaction is on the order of the thermal fluctuations in solution, the rigid body description of molecular function must be revisited. An emerging theme is that functional motions encompass order-disorder transitions and structural flexibility provides significant contributions to the free energy. In this review, we describe the biological importance of order-disorder transitions and discuss the statistical-mechanical foundation of theoretical approaches that can characterize such transitions.

Effects of mutation, truncation, and temperature on the folding kinetics of a WW domain

The purpose of this work is to show how mutation, truncation, and change of temperature can influence the folding kinetics of a protein. This is accomplished by principal component analysis of molecular-dynamics-generated folding trajectories of the triple β-strand WW domain from formin binding protein 28 (FBP28) (Protein Data Bank ID: 1E0L) and its full-size, and singly- and doubly-truncated mutants at temperatures below and very close to the melting point. The reasons for biphasic folding kinetics [i.e., coexistence of slow (three-state) and fast (two-state) phases], including the involvement of a solvent-exposed hydrophobic cluster and another delocalized hydrophobic core in the folding kinetics, are discussed. New folding pathways are identified in free-energy landscapes determined in terms of principal components for full-size mutants. Three-state folding is found to be a main mechanism for folding the FBP28 WW domain and most of the full-size and truncated mutants. The results from the theoretical analysis are compared to those from experiment. Agreements and discrepancies between the theoretical and experimental results are discussed. Because of its importance in understanding protein kinetics and function, the diffusive mechanism by which the FBP28 WW domain and its full-size and truncated mutants explore their conformational space is examined in terms of the mean-square displacement and principal component analysis eigenvalue spectrum analyses. Subdiffusive behavior is observed for all studied systems.

Configuration-dependent diffusion dynamics of downhill and two-state protein folding

Configuration-dependent diffusion (CDD) is important for protein folding kinetics with small thermodynamic barriers. CDD can be even more crucial in downhill folding without thermodynamic barriers. We explored the CDD of a downhill protein (BBL), and a two-state protein (CI2). The hidden kinetic barriers due to CDD were revealed. The increased ~1 k(B)T kinetic barrier is in line with experimental value based on other fast folding proteins. Compared to that of CI2, the effective free-energy profile of BBL is found to be significantly influenced by CDD, and the kinetics are totally determined by diffusion. These findings are consistent with both earlier bulk and single-molecule fluorescence measurements. In addition, we found the temperature dependence of CDD. We also found that the ratio of folding transition temperature against optimal kinetic folding temperature can provide both a quantitative measure for the underlying landscape topography and an indicator for the possible appearance of downhill folding. Our study can help for a better understanding of the role of diffusion in protein folding dynamics.

Quantifying internal friction in unfolded and intrinsically disordered proteins with single-molecule spectroscopy

Internal friction, which reflects the "roughness" of the energy landscape, plays an important role for proteins by modulating the dynamics of their folding and other conformational changes. However, the experimental quantification of internal friction and its contribution to folding dynamics has remained challenging. Here we use the combination of single-molecule Förster resonance energy transfer, nanosecond fluorescence correlation spectroscopy, and microfluidic mixing to determine the reconfiguration times of unfolded proteins and investigate the mechanisms of internal friction contributing to their dynamics. Using concepts from polymer dynamics, we determine internal friction with three complementary, largely independent, and consistent approaches as an additive contribution to the reconfiguration time of the unfolded state. We find that the magnitude of internal friction correlates with the compactness of the unfolded protein: its contribution dominates the reconfiguration time of approximately 100 ns of the compact unfolded state of a small cold shock protein under native conditions, but decreases for more expanded chains, and approaches zero both at high denaturant concentrations and in intrinsically disordered proteins that are expanded due to intramolecular charge repulsion. Our results suggest that internal friction in the unfolded state will be particularly relevant for the kinetics of proteins that fold in the microsecond range or faster. The low internal friction in expanded intrinsically disordered proteins may have implications for the dynamics of their interactions with cellular binding partners.

Diffusion models of protein folding

In theory and in the analysis of experiments, protein folding is often described as diffusion along a single coordinate. We explore here the application of a one-dimensional diffusion model to interpret simulations of protein folding, where the parameters of a model that "best" describes the simulation trajectories are determined using a Bayesian analysis. We discuss the requirements for such a model to be a good approximation to the global dynamics, and several methods for testing its accuracy. For example, one test considers the effect of an added bias potential on the fitted free energies and diffusion coefficients. Such a bias may also be used to extend our approach to determining parameters for the model to systems that would not normally explore the full coordinate range on accessible time scales. Alternatively, the propagators predicted from the model at different "lag" times may be compared with observations from simulation. We then present some applications of the model to protein folding, including Kramers-like turnover in folding rates of coarse-grained models, the effect of non-native interactions on folding, and the effect of the chosen coordinate on the observed position-dependence of the diffusion coefficients. Lastly, we consider how our results are useful for the interpretation of experiments, and how this type of Bayesian analysis may eventually be applied directly to analyse experimental data.