Concerted Dihedral Rotations Give Rise to Internal Friction in Unfolded Proteins

The role of memory-dependent friction and solvent viscosity in isomerization kinetics in viscogenic media

Molecular isomerization kinetics in liquid solvent depends on a complex interplay between the solvent friction acting on the molecule, internal dissipation effects (also known as internal friction), the viscosity of the solvent, and the dihedral free energy profile. Due to the absence of accurate techniques to directly evaluate isomerization friction, it has not been possible to explore these relationships in full. By combining extensive molecular dynamics simulations with friction memory-kernel extraction techniques we consider a variety of small, isomerising molecules under a range of different viscogenic conditions and directly evaluate the viscosity dependence of the friction acting on a rotating dihedral. We reveal that the influence of different viscogenic media on isomerization kinetics can be dramatically different, even when measured at the same viscosity. This is due to the dynamic solute-solvent coupling, mediated by time-dependent friction memory kernels. We also show that deviations from the linear dependence of isomerization rates on solvent viscosity, which are often simply attributed to internal friction effects, are due to the simultaneous violation of two fundamental relationships: the Stokes-Einstein relation and the overdamped Kramers prediction for the barrier-crossing rate, both of which require explicit knowledge of friction.

Effects of Proline on Internal Friction in Simulated Folding Dynamics of Several Alanine-Based α-Helical Peptides

We have studied in silico the effect of proline, a model cosolvent, on local and global friction coefficients in (un)folding of several typical alanine-based α-helical peptides. Local friction is related to dwell times of a single, ensemble-averaged hydrogen bond (HB) within each peptide. Global friction is related to energy dissipated in a series of configurational changes of each peptide experienced by increasing the number of HBs during folding. Both of these approaches are important in relation to future atomic force microscopic-based measurements of internal friction via force-clamp single-molecule force spectroscopy. Molecular dynamics (MD) simulations for six peptides, namely, ALA5, ALA8, ALA15, ALA21, (AAQAA)3, and H2N-GN(AAQAA)2G-COONH2, have been conducted at 2 and 5 M proline solutions in water. Using previously obtained MD data for these peptides in pure water as well as upgraded theoretical models, we obtained variations of local and global internal friction coefficients as a function of solution viscosity. The results showed the substantial role of proline in stabilizing the folded state and slowing the overall folding dynamics. Consequently, larger friction coefficients were obtained at larger viscosities. The local and global internal friction, i.e., respective, friction coefficients approximated to zero viscosity, was also obtained. The evolution of friction coefficients with viscosity was weakly dependent on the number of concurrent folding pathways but was rather dominated by a stabilizing effect of proline on the folded states. Obtained values of local and global internal friction showed qualitatively similar results and a clear dependency on the structure of the studied peptide.

Activation and friction in enzymatic loop opening and closing dynamics

Protein loop dynamics have recently been recognized as central to enzymatic activity, specificity and stability. However, the factors controlling loop opening and closing kinetics have remained elusive. Here, we combine molecular dynamics simulations with string-method determination of complex reaction coordinates to elucidate the molecular mechanism and rate-limiting step for WPD-loop dynamics in the PTP1B enzyme. While protein conformational dynamics is often represented as diffusive motion hindered by solvent viscosity and internal friction, we demonstrate that loop opening and closing is activated. It is governed by torsional rearrangement around a single loop peptide group and by significant friction caused by backbone adjustments, which can dynamically trap the loop. Considering both torsional barrier and time-dependent friction, our calculated rate constants exhibit very good agreement with experimental measurements, reproducing the change in loop opening kinetics between proteins. Furthermore, we demonstrate the applicability of our results to other enzymatic loops, including the M20 DHFR loop, thereby offering prospects for loop engineering potentially leading to enhanced designs.

Log-periodic oscillations as real-time signatures of hierarchical dynamics in proteins

The time-dependent relaxation of a dynamical system may exhibit a power-law behavior that is superimposed by log-periodic oscillations. D. Sornette [Phys. Rep. 297, 239 (1998)] showed that this behavior can be explained by a discrete scale invariance of the system, which is associated with discrete and equidistant timescales on a logarithmic scale. Examples include such diverse fields as financial crashes, random diffusion, and quantum topological materials. Recent time-resolved experiments and molecular dynamics simulations suggest that discrete scale invariance may also apply to hierarchical dynamics in proteins, where several fast local conformational changes are a prerequisite for a slow global transition to occur. Employing entropy-based timescale analysis and Markov state modeling to a simple one-dimensional hierarchical model and biomolecular simulation data, it is found that hierarchical systems quite generally give rise to logarithmically spaced discrete timescales. By introducing a one-dimensional reaction coordinate that collectively accounts for the hierarchically coupled degrees of freedom, the free energy landscape exhibits a characteristic staircase shape with two metastable end states, which causes the log-periodic time evolution of the system. The period of the log-oscillations reflects the effective roughness of the energy landscape and can, in simple cases, be interpreted in terms of the barriers of the staircase landscape.

Electrostatics and hydrophobicity in the dynamics of intrinsically disordered proteins

Internal friction is a major contribution to the dynamics of intrinsically disordered proteins (IDPs). Yet, the molecular origin of internal friction has so far been elusive. Here, we investigate whether attractive electrostatic interactions in IDPs modulate internal friction differently than the hydrophobic effect. To this end, we used nanosecond fluorescence correlation spectroscopy (nsFCS) and single-molecule Förster resonance energy transfer (FRET) to quantify the conformation and dynamics of the disordered DNA-binding domains Myc, Max and Mad at different salt concentrations. We find that internal friction effects are stronger when the chain is compacted by electrostatic attractions compared to the hydrophobic effect. Although the effect is moderate, the results show that the heteropolymeric nature of IDPs is reflected in their dynamics.

Investigation of Rare Protein Conformational Transitions via Dissipation-Corrected Targeted Molecular Dynamics

To sample rare events, dissipation-corrected targeted molecular dynamics (dcTMD) applies a constant velocity constraint along a one-dimensional reaction coordinate s, which drives an atomistic system from an initial state into a target state. Employing a cumulant approximation of Jarzynski's identity, the free energy ΔG(s) is calculated from the mean external work and dissipated work of the process. By calculating the friction coefficient Γ(s) from the dissipated work, in a second step, the equilibrium dynamics of the process can be studied by propagating a Langevin equation. While so far dcTMD has been mostly applied to study the unbinding of protein-ligand complexes, here its applicability to rare conformational transitions within a protein and the prediction of their kinetics are investigated. As this typically requires the introduction of multiple collective variables {xj} = x, a theoretical framework is outlined to calculate the associated free energy ΔG(x) and friction Γ(x) from dcTMD simulations along coordinate s. Adopting the α-β transition of alanine dipeptide as well as the open-closed transition of T4 lysozyme as representative examples, the virtues and shortcomings of dcTMD to predict protein conformational transitions and the related kinetics are studied.

Anisotropic Friction in a Ligand-Protein Complex

The effect of an externally applied directional force on molecular friction is so far poorly understood. Here, we study the force-driven dissociation of the ligand-protein complex biotin-streptavidin and identify anisotropic friction as a not yet described type of molecular friction. Using AFM-based stereographic single molecule force spectroscopy and targeted molecular dynamics simulations, we find that the rupture force and friction for biotin-streptavidin vary with the pulling angle. This observation holds true for friction extracted from Kramers' rate expression and by dissipation-corrected targeted molecular dynamics simulations based on Jarzynski's identity. We rule out ligand solvation and protein-internal friction as sources of the angle-dependent friction. Instead, we observe a heterogeneity in free energy barriers along an experimentally uncontrolled orientation parameter, which increases the rupture force variance and therefore the overall friction. We anticipate that anisotropic friction needs to be accounted for in a complete understanding of friction in biomolecular dynamics and anisotropic mechanical environments.

Probing the Energy Landscape of Spectrin R15 and R16 and the Effects of Non-native Interactions

Understanding the details of a protein folding mechanism can be a challenging and complex task. One system with an interesting folding behavior is the α-spectrin domain, where the R15 folds three-orders of magnitude faster than its homologues R16 and R17, despite having similar structures. The molecular origins that explain these folding rate differences remain unclear, but our previous work revealed that a combined effect produced by non-native interactions could be a reasonable cause for these differences. In this study, we explore further the folding process by identifying the molecular paths, metastable states, and the collective motions that lead these unfolded proteins to their native state conformation. Our results uncovered the differences between the folding pathways for the wild-type R15 and R16 and an R16 mutant. The metastable ensembles that speed down the folding were identified using an energy landscape visualization method (ELViM). These ensembles correspond to similar experimentally reported configurations. Our observations indicate that the non-native interactions are also associated with secondary structure misdocking. This computational methodology can be used as a fast, straightforward protocol for shedding light on systems with unclear folding or conformational traps.

Effects of Sequence Composition, Patterning and Hydrodynamics on the Conformation and Dynamics of Intrinsically Disordered Proteins

Intrinsically disordered proteins (IDPs) and intrinsically disordered regions (IDRs) perform diverse functions in cellular organization, transport and signaling. Unlike the well-defined structures of the classical natively folded proteins, IDPs and IDRs dynamically span large conformational and structural ensembles. This dynamic disorder impedes the study of the relationship between the amino acid sequences of the IDPs and their spatial structures and dynamics, with different experimental techniques often offering seemingly contradictory results. Although experimental and theoretical evidence indicates that some IDP properties can be understood based on their average biophysical properties and amino acid composition, other aspects of IDP function are dictated by the specifics of the amino acid sequence. We investigate the effects of several key variables on the dimensions and the dynamics of IDPs using coarse-grained polymer models. We focus on the sequence "patchiness" informed by the sequence and biophysical properties of different classes of IDPs-and in particular FG nucleoporins of the nuclear pore complex (NPC). We show that the sequence composition and patterning are well reflected in the global conformational variables such as the radius of gyration and hydrodynamic radius, while the end-to-end distance and dynamics are highly sequence-specific. We find that in good solvent conditions highly heterogeneous sequences of IDPs can be well mapped onto averaged minimal polymer models for the purpose of prediction of the IDPs dimensions and dynamic relaxation times. The coarse-grained simulations are in a good agreement with the results of atomistic MD. We discuss the implications of these results for the interpretation of the recent experimental measurements, and for the further applications of mesoscopic models of FG nucleoporins and IDPs more broadly.

Molecular Origin of Internal Friction in Intrinsically Disordered Proteins

Protein folding and dynamics are controlled by an interplay of thermal and viscosity effects. The effect of viscous drag through the solvent molecules is described by the classic Kramers theory in the high friction limit, which considers the dampening of the reactant molecules in the solution and quantifies the dependence of the reaction rate on the frictional drag. In addition to the external energy dissipation originating from the surrounding solvent molecules, there is an additional mode of internal energy dissipative force operative within the polypeptide chain reflecting the internal resistance of the chain to its conformational alterations. This dry, solvent-independent intrinsic frictional drag is termed internal friction. In the case of natively folded proteins, the physical origin of internal friction is primarily attributed to the intrachain interactions or other nonnative interactions in their compact states. However, the molecular origin of internal friction in intrinsically disordered proteins (IDPs) remains elusive.In this Account, we address this fundamental issue: what are the principal drivers of viscosity-independent (dry) friction in highly solvated, expanded, conformationally flexible, rapidly fluctuating IDPs that do not possess persistent intrachain interactions? IDPs exhibit diffusive conformational dynamics that is predominantly dominated by the segmental motion of the backbone arising due to the dihedral rotations in the Ramachandran Φ-Ψ space. The physical origin of friction in a complex biopolymeric system such as IDPs can be described by classic polymer models, namely, Rouse/Zimm models with internal friction. These one-dimensional models do not invoke torsional fluctuation components. Kuhn's classic description includes the connection between internal friction and microscopic dihedral hopping. Based on our time-resolved fluorescence anisotropy results, we describe that the sequence-dependent, collective, short-range backbone dihedral rotations govern localized internal friction in an archetypal IDP, namely, α-synuclein. The highly sensitive, residue-specific fluorescence depolarization kinetics offers a potent methodology to characterize and quantify the directional decorrelation engendered due to the short-range dihedral relaxation of the polypeptide backbone in the dihedral space. We utilized this characteristic relaxation time scale as our dynamic readout to quantify the site-specific frictional component. Our linear viscosity-dependent model of torsional relaxation time scale furnished a finite nonzero time constant at the zero solvent viscosity representing the solvent-independent internal friction. These results unveil the effect of the degree of dihedral restraining parameter on the internal friction component by showing that a restrained proline residue imparts higher torsional stiffness in the chain segments and, therefore, exhibits higher internal friction. This Account sheds light on the molecular underpinning of the sequence-specific internal friction in IDPs and will be of interest to unmask the role of internal friction in a diverse range of biomolecular processes involving binding-induced folding, allosteric interaction, protein misfolding and aggregation, and biomolecular condensation via phase separation.

The biophysics of disordered proteins from the point of view of single-molecule fluorescence spectroscopy

Intrinsically disordered proteins (IDPs) and regions (IDRs) have emerged as key players across many biological functions and diseases. Differently from structured proteins, disordered proteins lack stable structure and are particularly sensitive to changes in the surrounding environment. Investigation of disordered ensembles requires new approaches and concepts for quantifying conformations, dynamics, and interactions. Here, we provide a short description of the fundamental biophysical properties of disordered proteins as understood through the lens of single-molecule fluorescence observations. Single-molecule Förster resonance energy transfer (FRET) and fluorescence correlation spectroscopy (FCS) provides an extensive and versatile toolbox for quantifying the characteristics of conformational distributions and the dynamics of disordered proteins across many different solution conditions, both in vitro and in living cells.

Analytical Approaches for Deriving Friction Coefficients for Selected α-Helical Peptides Based Entirely on Molecular Dynamics Simulations

In this paper we derive analytically from molecular dynamics (MD) simulations the friction coefficients related to conformational transitions within several model peptides with α-helical structures. We study a series of alanine peptides with various length from ALA₅ to ALA₂₁ as well as their two derivatives, the (AAQAA)₃ peptide and a 13-residue KR1 peptide that is a derivative of the (AAQAA)₂ peptide with the formula GN(AAQAA)₂G. We use two kinds of approaches to derive their friction coefficients. In the local approach, friction associated with fluctuations of single hydrogen bonds are studied. In the second approach, friction coefficients associated with a folding transitions within the studied peptides are obtained. In both cases, the respective friction coefficients differentiated very well the subtle structural changes between studied peptides and compared favorably to experimentally available data.

Variation of Structural and Dynamical Flexibility of Myelin Basic Protein in Response to Guanidinium Chloride

Myelin basic protein (MBP) is intrinsically disordered in solution and is considered as a conformationally flexible biomacromolecule. Here, we present a study on perturbation of MBP structure and dynamics by the denaturant guanidinium chloride (GndCl) using small-angle scattering and neutron spin–echo spectroscopy (NSE). A concentration of 0.2 M GndCl causes charge screening in MBP resulting in a compact, but still disordered protein conformation, while GndCl concentrations above 1 M lead to structural expansion and swelling of MBP. NSE data of MBP were analyzed using the Zimm model with internal friction (ZIF) and normal mode (NM) analysis. A significant contribution of internal friction was found in compact states of MBP that approaches a non-vanishing internal friction relaxation time of approximately 40 ns at high GndCl concentrations. NM analysis demonstrates that the relaxation rates of internal modes of MBP remain unaffected by GndCl, while structural expansion due to GndCl results in increased amplitudes of internal motions. Within the model of the Brownian oscillator our observations can be rationalized by a loss of friction within the protein due to structural expansion. Our study highlights the intimate coupling of structural and dynamical plasticity of MBP, and its fundamental difference to the behavior of ideal polymers in solution.

Molecular Origin of Driving-Dependent Friction in Fluids

The friction coefficient of fluids may become a function of the velocity at increased external driving. This non-Newtonian behavior is of general theoretical interest and of great practical importance, for example, for the design of lubricants. Although the effect has been observed in large-scale atomistic simulations of bulk liquids, its theoretical formulation and microscopic origin are not well understood. Here, we use dissipation-corrected targeted molecular dynamics, which pulls apart two tagged liquid molecules in the presence of surrounding molecules, and analyze this nonequilibrium process via a generalized Langevin equation. The approach is based on a second-order cumulant expansion of Jarzynski's identity, which is shown to be valid for fluids and therefore allows for an exact computation of the friction profile as well of the underlying memory kernel. We show that velocity-dependent friction in fluids results from an intricate interplay of near-order structural effects and the non-Markovian behavior of the friction memory kernel. For complex fluids such as the model lubricant C₄₀H₈₂, the memory kernel exhibits a stretched-exponential long-time decay, which reflects the multitude of timescales of the system.

NMR Provides Unique Insight into the Functional Dynamics and Interactions of Intrinsically Disordered Proteins

Intrinsically disordered proteins are ubiquitous throughout all known proteomes, playing essential roles in all aspects of cellular and extracellular biochemistry. To understand their function, it is necessary to determine their structural and dynamic behavior and to describe the physical chemistry of their interaction trajectories. Nuclear magnetic resonance is perfectly adapted to this task, providing ensemble averaged structural and dynamic parameters that report on each assigned resonance in the molecule, unveiling otherwise inaccessible insight into the reaction kinetics and thermodynamics that are essential for function. In this review, we describe recent applications of NMR-based approaches to understanding the conformational energy landscape, the nature and time scales of local and long-range dynamics and how they depend on the environment, even in the cell. Finally, we illustrate the ability of NMR to uncover the mechanistic basis of functional disordered molecular assemblies that are important for human health.

Distant residues modulate conformational opening in SARS-CoV-2 spike protein

The novel coronavirus (SARS-CoV-2) pandemic resulted in the largest public health crisis in recent times. Significant drug design effort against SARS-CoV-2 is focused on the receptor-binding domain (RBD) of the spike protein, although this region is highly prone to mutations causing therapeutic resistance. We applied deep data analysis methods on all-atom molecular dynamics simulations to identify key non-RBD residues that play a crucial role in spike−receptor binding and infection. Because the non-RBD residues are typically conserved across multiple coronaviruses, they can be targeted by broad-spectrum antibodies and drugs to treat infections from new strains that might appear during future epidemics.

Infection by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) involves the attachment of the receptor-binding domain (RBD) of its spike proteins to the ACE2 receptors on the peripheral membrane of host cells. Binding is initiated by a down-to-up conformational change in the spike protein, the change that presents the RBD to the receptor. To date, computational and experimental studies that search for therapeutics have concentrated, for good reason, on the RBD. However, the RBD region is highly prone to mutations, and is therefore a hotspot for drug resistance. In contrast, we here focus on the correlations between the RBD and residues distant to it in the spike protein. This allows for a deeper understanding of the underlying molecular recognition events and prediction of the highest-effect key mutations in distant, allosteric sites, with implications for therapeutics. Also, these sites can appear in emerging mutants with possibly higher transmissibility and virulence, and preidentifying them can give clues for designing pan-coronavirus vaccines against future outbreaks. Our model, based on time-lagged independent component analysis (tICA) and protein graph connectivity network, is able to identify multiple residues that exhibit long-distance coupling with the RBD opening. Residues involved in the most ubiquitous D614G mutation and the A570D mutation of the highly contagious UK SARS-CoV-2 variant are predicted ab initio from our model. Conversely, broad-spectrum therapeutics like drugs and monoclonal antibodies can target these key distant-but-conserved regions of the spike protein.

Fluorescence Depolarization Kinetics Captures Short-Range Backbone Dihedral Rotations and Long-Range Correlated Dynamics of an Intrinsically Disordered Protein

Intrinsically disordered proteins (IDPs) do not autonomously fold into well-defined three-dimensional structures and are best described as a heterogeneous ensemble of rapidly interconverting conformers. It is challenging to elucidate their complex dynamic signatures using a single technique. In this study, we employed sensitive fluorescence depolarization kinetics by following picosecond time-resolved fluorescence anisotropy decays to directly capture the essential dynamical features of intrinsically disordered α-synuclein (α-syn) site-specifically labeled with thiol-active fluorophores. By utilizing a long-lifetime (≥10 ns) anisotropic label, we were able to discern three distinct rotational components of α-syn. The subnanosecond component represents the local wobbling-in-cone motion of the fluorophore, whereas the slower (∼1.4 ns) component corresponds to the short-range backbone dynamics governed by collective torsional fluctuations in the Ramachandran Φ-Ψ dihedral space. This backbone dihedral rotational time scale is sensitive to the local chain stiffness and slows down in the presence of an adjacent proline residue. We also observed a small-amplitude (≤10%) slower rotational correlation time (6-10 ns) that represents the long-range correlated dynamics involving a much longer segment of the polypeptide chain. These intrinsic dynamic signatures of IDPs will provide critical mechanistic underpinnings in a mosaic of biophysical phenomena involving internal friction, allosteric interactions, and phase separation.

Integrating single-molecule spectroscopy and simulations for the study of intrinsically disordered proteins

Over the last two decades, intrinsically disordered proteins and protein regions (IDRs) have emerged from a niche corner of biophysics to be recognized as essential drivers of cellular function. Various techniques have provided fundamental insight into the function and dysfunction of IDRs. Among these techniques, single-molecule fluorescence spectroscopy and molecular simulations have played a major role in shaping our modern understanding of the sequence-encoded conformational behavior of disordered proteins. While both techniques are frequently used in isolation, when combined they offer synergistic and complementary information that can help uncover complex molecular details. Here we offer an overview of single-molecule fluorescence spectroscopy and molecular simulations in the context of studying disordered proteins. We discuss the various means in which simulations and single-molecule spectroscopy can be integrated, and consider a number of studies in which this integration has uncovered biological and biophysical mechanisms.

Temperature Dependence of Internal Friction of Peptides

Internal friction is a valuable concept to describe the kinetics of proteins. As is well known, internal friction can be modulated by solvent features (such as viscosity). How can internal friction be affected by environmental temperature? The answer to this question is not evident. In the present work, we approach this problem with simulations on two model peptides. The thermodynamics and relaxation kinetics are characterized through long molecular dynamics simulations, with the viscosity modulated by varying the mass of solvent molecules. Based on the extrapolation to zero viscosity together with scaling of the relaxation time scales, we discover that internal friction is almost invariant at various temperatures. Controlled simulations further support the idea that internal friction is independent of environmental temperature. Comparisons between the two model peptides help us to understand the diverse phenomena in experiments.

Structural and dynamic characterization of the C-terminal tail of ErbB2: Disordered but not random

ErbB2 (or HER2) is a receptor tyrosine kinase overexpressed in some breast cancers and associated with poor prognosis. Treatments targeting the receptor extracellular and kinase domains have greatly improved disease outcome in the last 20 years. In parallel, the structures of these domains have been described, enabling better mechanistic understanding of the receptor function and targeted inhibition. However, the ErbB2 disordered C-terminal cytoplasmic tail (CtErbB2) remains very poorly characterized in terms of structure, dynamics, and detailed functional mechanism. Yet, it is where signal transduction is triggered via phosphorylation of tyrosine residues and carried out via interaction with adaptor proteins. Here, we report the first description, to our knowledge, of the ErbB2 disordered tail at atomic resolution using NMR, complemented by small-angle x-ray scattering. We show that although no part of CtErbB2 has any fully populated secondary or tertiary structure, it contains several transient α-helices and numerous transient polyproline II helices, populated up to 20 and 40%, respectively, and low but significant compaction. The presence of some structural elements suggests, along the lines of the results obtained for EGFR (ErbB1), that they may have a functional role in ErbB2's autoregulation processes. In addition, the transient formation of polyproline II helices is compliant with previously suggested interactions with SH3 domains. All in all, our in-depth structural study opens perspectives in the mechanistic understanding of ErbB2.

Piecewise All-Atom SMD Simulations Reveal Key Secondary Structures in Luciferase Unfolding Pathway

Although the folding of single-domain proteins is well characterized theoretically and experimentally, the folding of large multidomain proteins is less well known. Firefly luciferase, a 550 residue three-domain protein, has been commonly used as a substrate to study chaperone reactions and as a model system for the study of folding of long polypeptide chains, including related phenomena such as cotranslational folding. Despite being characterized by various experimental techniques, the atomic-level contributions of various secondary structures of luciferase to its fold's mechanical stability remain unknown. Here, we developed a piecewise approach for all-atom steered molecular dynamics simulations to examine specific secondary structures that resist mechanical unfolding while minimizing the amount of computational resources required by the large water box of standard all-atom steered molecular dynamics simulations. We validated the robustness of this approach with a small NI3C protein and used our approach to elucidate the specific secondary structures that provide the largest contributions to luciferase mechanostability. In doing so, we show that piecewise all-atom steered molecular dynamics simulations can provide novel atomic resolution details regarding mechanostability and can serve as a platform for novel mutagenesis studies as well as a point for comparison with high-resolution force spectroscopy experiments.

Slow Folding of a Helical Protein: Large Barriers, Strong Internal Friction, or a Shallow, Bumpy Landscape?

The rate at which a protein molecule folds is determined by opposing energetic and entropic contributions to the free energy that shape the folding landscape. Delineating the extent to which they impact the diffusional barrier-crossing events, including the magnitude of internal friction and barrier height, has largely been a challenging task. In this work, we extract the underlying thermodynamic and dynamic contributions to the folding rate of an unusually slow-folding helical DNA-binding domain, PurR, which shares the characteristics of ultrafast downhill-folding proteins but nonetheless appears to exhibit an apparent two-state equilibrium. We combine equilibrium spectroscopy, temperature-viscosity-dependent kinetics, statistical mechanical modeling, and coarse-grained simulations to show that the conformational behavior of PurR is highly heterogeneous characterized by a large spread in melting temperatures, marginal thermodynamic barriers, and populated partially structured states. PurR appears to be at the threshold of disorder arising from frustrated electrostatics and weak packing that in turn slows down folding due to a shallow, bumpy landscape and not due to large thermodynamic barriers or strong internal friction. Our work highlights how a strong temperature dependence on the pre-exponential could signal a shallow landscape and not necessarily a slow-folding diffusion coefficient, thus determining the folding timescales of even millisecond folding proteins and hints at possible structural origins for the shallow landscape.

The non-classical kinetics and the mutual information of polymer loop formation

The loop formation of a single polymer chain has served as a model system for various biological and chemical processes. Theories based on the Smoluchowski equation proposed that the rate constant (k_loop) of the loop formation would be inversely proportional to viscosity (η), i.e., k_loop ∼ η^-1. Experiments and simulations showed, however, that k_loop showed the fractional viscosity dependence of k_loop ∼ η^-β with β < 1 either in glasses or in low-viscosity solutions. The origin of the fractional viscosity dependence remains elusive and has been attributed to phenomenological aspects. In this paper, we illustrate that the well-known failure of classical kinetics of the loop formation results from the breakdown of the local thermal equilibrium (LTE) approximation and that the mutual information can quantify the breakdown of the LTE successfully.

Intraresidual Correlated Motions in Peptide Chains

We investigate the interresidual and intraresidual correlations between dihedral displacements of adjacent residues within model polyalanine peptides by analyzing extensive molecular dynamics trajectories. Correlations are evaluated individually at different residue conformations covering the whole (ϕ_i,ψ_i)-space. From these, we draw maps that unveil an unprecedented strong intramolecular correlation displaying opposite (correlated/anticorrelated) behaviors at different conformations. Both interresidual and intraresidual correlations arise from the propensity of the peptide to minimize the overall atomic displacements.

A Unified Description of Intrinsically Disordered Protein Dynamics under Physiological Conditions Using NMR Spectroscopy

Intrinsically disordered proteins (IDPs) are flexible biomolecules whose essential functions are defined by their dynamic nature. Nuclear magnetic resonance (NMR) spectroscopy is ideally suited to the investigation of this behavior at atomic resolution. NMR relaxation is increasingly used to detect conformational dynamics in free and bound forms of IDPs under conditions approaching physiological, although a general framework providing a quantitative interpretation of these exquisitely sensitive probes as a function of experimental conditions is still lacking. Here, measuring an extensive set of relaxation rates sampling multiple-time-scale dynamics over a broad range of crowding conditions, we develop and test an integrated analytical description that accurately portrays the motion of IDPs as a function of the intrinsic properties of the crowded molecular environment. In particular we observe a strong dependence of both short-range and long-range motional time scales of the protein on the friction of the solvent. This tight coupling between the dynamic behavior of the IDP and its environment allows us to develop analytical expressions for protein motions and NMR relaxation properties that can be accurately applied over a vast range of experimental conditions. This unified dynamic description provides new insight into the physical behavior of IDPs, extending our ability to quantitatively investigate their conformational dynamics under complex environmental conditions, and accurately predicting relaxation rates reporting on motions on time scales up to tens of nanoseconds, both in vitro and in cellulo.

Energetic Self-Folding Mechanism in α-Helices

A novel energetic route driving the folding of a polyalanine peptide from an extended conformation to its α-helix native conformation is described, supported by a new method to compute mean potential energy surfaces accurately in terms of the dihedral angles of the peptide chain from extensive molecular dynamics simulations. The energetic self-folding (ESF) route arises specifically from the balance between the intrinsic propensity of alanine residues toward the α_R conformation and two, opposite, effects: the destabilizing interaction with neighbor residues and the stabilizing formation of native hydrogen bonds, with the latter being dominant for large peptide lengths. The ESF mechanism provides simple but robust support to the nucleation-elongation or zipper models and offers a quantitative energetic funnel picture of the folding process. The mechanism is validated by the reasonable agreement between the computed folding energies and the experimental values.

Mass-Dependent Solvent Friction of a Hydrophobic Molecule

We show by molecular dynamics simulations that the friction constant of a freely diffusing methane molecule in explicit water increases with the methane mass by a factor of up to 1.8 in the infinite mass limit compared to the massless limit. This effect is rationalized by the mass dependence of the friction memory kernel which is extracted from the simulation data by mapping on the generalized Langevin equation. On the basis of the mass-dependent memory kernels, we obtain perfect agreement between simulation results and analytic predictions for both mean-square displacements and force autocorrelation functions. The memory kernels, which account for methane interactions with the solvent, decay significantly slower with increasing methane mass. The mass-dependent friction is correlated with the mean escape time of water molecules from the first hydration shell, which increases monotonically with the methane mass by a factor of 3 over the mass range considered. Our proposed scaling expression allows the direct prediction of diffusion constants for stable methane isotopes, which will help to better understand bacterial methane isotope fractionation. Our scaling analysis suggests that mass-dependent friction will be relevant also for larger solutes in sufficiently viscous solvents.

Structural and dynamic responses of calcium ion binding loop residues in metallo-proteins

Conformational changes in bio-molecular systems are fundamental to several biological processes. It is important to study changes in responses of underlying microscopic variables, like dihedral angles as conformational change takes place. We perform all-atom simulations and modelling via Langevin equation to illustrate the changes in structural and dynamic responses of dihedral angles of calcium ion binding residues of different proteins in metal ion free (apo) and bound (holo) states. The equilibrium distributions of dihedral angles in apo- and holo-states represent structural response. Our studies show the presence of dihedrals with multiple peaks (isomeric states) separated by barrier heights is more frequent in apo- than in holo-state. The relaxation time-scale of dihedral fluctuations is found to increase linearly with decreasing barrier height due to more frequent barrier re-crossing events. The slow kinetic response of the dihedrals also contributes to slowing down of macro-scale fluctuations, which may be useful to understand kinetics of various bio-molecular processes.

Dynamics of proteins in solution

The dynamics of proteins in solution includes a variety of processes, such as backbone and side-chain fluctuations, interdomain motions, as well as global rotational and translational (i.e. center of mass) diffusion. Since protein dynamics is related to protein function and essential transport processes, a detailed mechanistic understanding and monitoring of protein dynamics in solution is highly desirable. The hierarchical character of protein dynamics requires experimental tools addressing a broad range of time- and length scales. We discuss how different techniques contribute to a comprehensive picture of protein dynamics, and focus in particular on results from neutron spectroscopy. We outline the underlying principles and review available instrumentation as well as related analysis frameworks.

Principal component analysis of nonequilibrium molecular dynamics simulations

Principal component analysis (PCA) represents a standard approach to identify collective variables {x_i} = x, which can be used to construct the free energy landscape ΔG(x) of a molecular system. While PCA is routinely applied to equilibrium molecular dynamics (MD) simulations, it is less obvious as to how to extend the approach to nonequilibrium simulation techniques. This includes, e.g., the definition of the statistical averages employed in PCA as well as the relation between the equilibrium free energy landscape ΔG(x) and the energy landscapes ΔG(x) obtained from nonequilibrium MD. As an example for a nonequilibrium method, "targeted MD" is considered which employs a moving distance constraint to enforce rare transitions along some biasing coordinate s. The introduced bias can be described by a weighting function P(s), which provides a direct relation between equilibrium and nonequilibrium data, and thus establishes a well-defined way to perform PCA on nonequilibrium data. While the resulting distribution P(x) and energy ΔG∝lnP will not reflect the equilibrium state of the system, the nonequilibrium energy landscape ΔG(x) may directly reveal the molecular reaction mechanism. Applied to targeted MD simulations of the unfolding of decaalanine, for example, a PCA performed on backbone dihedral angles is shown to discriminate several unfolding pathways. Although the formulation is in principle exact, its practical use depends critically on the choice of the biasing coordinate s, which should account for a naturally occurring motion between two well-defined end-states of the system.

The combined force field-sampling problem in simulations of disordered amyloid-β peptides

Molecular dynamics simulations of intrinsically disordered proteins (IDPs) can provide high resolution structural ensembles if the force field is accurate enough and if the simulation sufficiently samples the conformational space of the IDP with the correct weighting of sub-populations. Here, we investigate the combined force field-sampling problem by testing a standard force field as well as newer fixed charge force fields, the latter specifically motivated for better description of unfolded states and IDPs, and comparing them with a standard temperature replica exchange (TREx) protocol and a non-equilibrium Temperature Cool Walking (TCW) sampling algorithm. The force field and sampling combinations are used to characterize the structural ensembles of the amyloid-beta peptides Aβ42 and Aβ43, which both should be random coils as shown recently by experimental nuclear magnetic resonance (NMR) and 2D Förster resonance energy transfer (FRET) experiments. The results illustrate the key importance of the sampling algorithm: while the standard force field using TREx is in poor agreement with the NMR J-coupling and nuclear Overhauser effect and 2D FRET data, when using the TCW method, the standard and optimized protein-water force field combinations are in very good agreement with the same experimental data since the TCW sampling method produces qualitatively different ensembles than TREx. We also discuss the relative merit of the 2D FRET data when validating structural ensembles using the different force fields and sampling protocols investigated in this work for small IDPs such as the Aβ42 and Aβ43 peptides.

Internal friction in an intrinsically disordered protein-Comparing Rouse-like models with experiments

Internal friction is frequently found in protein dynamics. Its molecular origin however is difficult to conceptualize. Even unfolded and intrinsically disordered polypeptide chains exhibit signs of internal friction despite their enormous solvent accessibility. Here, we compare four polymer theories of internal friction with experimental results on the intrinsically disordered protein ACTR (activator of thyroid hormone receptor). Using nanosecond fluorescence correlation spectroscopy combined with single-molecule Förster resonance energy transfer (smFRET), we determine the time scales of the diffusive chain dynamics of ACTR at different solvent viscosities and varying degrees of compaction. Despite pronounced differences between the theories, we find that all models can capture the experimental viscosity-dependence of the chain relaxation time. In contrast, the observed slowdown upon chain collapse of ACTR is not captured by any of the theories and a mechanistic link between chain dimension and internal friction is still missing, implying that the current theories are incomplete. In addition, a discrepancy between early results on homopolymer solutions and recent single-molecule experiments on unfolded and disordered proteins suggests that internal friction is likely to be a composite phenomenon caused by a variety of processes.

Origin of Internal Friction in Disordered Proteins Depends on Solvent Quality

Protein dynamics often exhibit internal friction; i.e., contributions to friction that cannot solely be attributed to the viscosity of the solvent. Remarkably, even unfolded and intrinsically disordered proteins (IDPs) exhibit this behavior, despite typically being solvent-exposed. Several competing molecular mechanisms have been suggested to underlie this phenomenon, in particular dihedral relaxation and intrachain interactions. It has also recently been shown that single-molecule data reflecting internal friction in the disordered protein ACTR cannot be explained using polymer models unless this friction is dependent on protein collapse. However, the connection between the collapse of the chain and the underlying mechanism of internal friction has been unclear. To address this issue, we combine molecular simulation and single-molecule experimental data to investigate how chain compaction affects protein dynamics in the context of ACTR. Chain reconfiguration times and internal friction estimated from all-atom simulations are in semiquantitative agreement with experimental data. We dissect the underlying molecular mechanism with all-atom and coarse-grained simulations and clearly identify both intrachain interactions and dihedral angle transitions as contributions to internal friction. However, their relative contribution is strongly dependent on the compactness of the IDP; while dihedral relaxation dominates internal friction in expanded configurations, intrachain interactions dominate for more compact chains. Our results thus imply a continuous transition between mechanisms and provide a link between internal friction in IDPs and that in more compact and folded states of proteins.

Frictional Effects on RNA Folding: Speed Limit and Kramers Turnover

We investigated frictional effects on the folding rates of a human telomerase hairpin (hTR HP) and H-type pseudoknot from the Beet Western Yellow Virus (BWYV PK) using simulations of the Three Interaction Site (TIS) model for RNA. The heat capacity from TIS model simulations, calculated using temperature replica exchange simulations, reproduces nearly quantitatively the available experimental data for the hTR HP. The corresponding results for BWYV PK serve as predictions. We calculated the folding rates ( k_F) from more than 100 folding trajectories for each value of the solvent viscosity (η) at a fixed salt concentration of 200 mM. By using the theoretical estimate (∝ √N where N is the number of nucleotides) for folding free energy barrier, k_F data for both the RNAs are quantitatively fit using one-dimensional Kramers's theory with two parameters specifying the curvatures in the unfolded basin and the barrier top. In the high-friction regime (η ≳ 10^-5 Pa·s), for both HP and PK, k_F values decrease as 1/η, whereas in the low friction regime, k_F values increase as η increases, leading to a maximum folding rate at a moderate viscosity (∼10^-6 Pa·s), which is the Kramers turnover. From the fits, we find that the speed limit to RNA folding at water viscosity is between 1 and 4 μs, which is in accord with our previous theoretical prediction as well as results from several single molecule experiments. Both the RNA constructs fold by parallel pathways. Surprisingly, we find that the flux through the pathways could be altered by changing solvent viscosity, a prediction that is more easily testable in RNA than in proteins.

Dynamics of Disordered Proteins under Confinement: Memory Effects and Internal Friction

Many proteins are disordered under physiological conditions. How efficiently they can search for their cellular targets and how fast they can fold upon target binding is determined by their intrinsic dynamics, which have thus attracted much recent attention. Experiments and molecular simulations show that the inherent reconfiguration timescale for unfolded proteins has a solvent friction component and an internal friction component, and the microscopic origin of the latter, along with its proper mathematical description, has been a topic of considerable debate. Internal friction varies across different proteins of comparable length and increases with decreasing denaturant concentration, showing that it depends on how compact the protein is. Here we report on a systematic atomistic simulation study of how confinement, which induces a more compact unfolded state, affects dynamics and friction in disordered peptides. We find that the average reconfiguration timescales increase exponentially as the peptide's spatial dimensions are reduced; at the same time, confinement broadens the spectrum of relaxation timescales exhibited by the peptide. There are two important implications of this broadening: First, it limits applicability of the common Rouse and Zimm models with internal friction, as those models attempt to capture internal friction effects by introducing a single internal friction timescale. Second, the long-tailed distribution of relaxation times leads to anomalous diffusion effects in the dynamics of intramolecular distances. Analysis and interpretation of experimental signals from various measurements that probe intramolecular protein dynamics (such as single-molecule fluorescence correlation spectroscopy and single-molecule force spectroscopy) rely on the assumption of diffusive dynamics along the distances being probed; hence, our results suggest the need for more general models allowing for anomalous diffusion effects.

Perspective: Chain dynamics of unfolded and intrinsically disordered proteins from nanosecond fluorescence correlation spectroscopy combined with single-molecule FRET

The dynamics of unfolded proteins are important both for the process of protein folding and for the behavior of intrinsically disordered proteins. However, methods for investigating the global chain dynamics of these structurally diverse systems have been limited. A versatile experimental approach is single-molecule spectroscopy in combination with Förster resonance energy transfer and nanosecond fluorescence correlation spectroscopy. The concepts of polymer physics offer a powerful framework both for interpreting the results and for understanding and classifying the properties of unfolded and intrinsically disordered proteins. This information on long-range chain dynamics can be complemented with spectroscopic techniques that probe different length scales and time scales, and integration of these results greatly benefits from recent advances in molecular simulations. This increasing convergence between the experiment, theory, and simulation is thus starting to enable an increasingly detailed view of the dynamics of disordered proteins.

Transition path times reveal memory effects and anomalous diffusion in the dynamics of protein folding

Recent single-molecule experiments probed transition paths of biomolecular folding and, in particular, measured the time biomolecules spend while crossing their free energy barriers. A surprising finding from these studies is that the transition barriers crossed by transition paths, as inferred from experimentally observed transition path times, are often lower than the independently determined free energy barriers. Here we explore memory effects leading to anomalous diffusion as a possible origin of this discrepancy. Our analysis of several molecular dynamics trajectories shows that the dynamics of common reaction coordinates used to describe protein folding is subdiffusive, at least at sufficiently short times. We capture this effect using a one-dimensional fractional Brownian motion (FBM) model, in which the system undergoes a subdiffusive process in the presence of a potential of mean force, and show that this model yields much broader distributions of transition path times with stretched exponential long-time tails. Without any adjustable parameters, these distributions agree well with the transition path times computed directly from protein trajectories. We further discuss how the FBM model can be tested experimentally.

Relevance of Internal Friction and Structural Constraints for the Dynamics of Denatured Bovine Serum Albumin

A general property of disordered proteins is their structural expansion that results in a high molecular flexibility. The structure and dynamics of bovine serum albumin (BSA) denatured by guanidinium hydrochloride (GndCl) were investigated using small-angle neutron scattering (SANS) and neutron spin-echo spectroscopy (NSE). SANS experiments demonstrated the relevance of intrachain interactions for structural expansion. Using NSE experiments, we observed a high internal flexibility of denatured BSA in addition to center-of-mass diffusion detected by dynamic light scattering. Internal motions measured by NSE were described using concepts based on polymer theory. The contribution of residue-solvent friction was accounted for using the Zimm model including internal friction (ZIF). Disulfide bonds forming loops of amino acids of the peptide backbone have a major impact on internal dynamics that can be interpreted with a reduced set of Zimm modes.

Butane dihedral angle dynamics in water is dominated by internal friction

The dihedral dynamics of butane in water is known to be rather insensitive to the water viscosity; possible explanations for this involve inertial effects or Kramers' turnover, the finite memory time of friction, and the presence of so-called internal friction. To disentangle these factors, we introduce a method to directly extract the friction memory function from unconstrained simulations in the presence of an arbitrary free-energy landscape. By analysis of the dihedral friction in butane for varying water viscosity, we demonstrate the existence of an internal friction contribution that does not scale linearly with water viscosity. At normal water viscosity, the internal friction turns out to be eight times larger than the solvent friction and thus completely dominates the effective friction. By comparison with simulations of a constrained butane molecule that has the dihedral as the only degree of freedom, we show that internal friction comes from the six additional degrees of freedom in unconstrained butane that are orthogonal to the dihedral angle reaction coordinate. While the insensitivity of butane's dihedral dynamics to water viscosity is solely due to the presence of internal friction, inertial effects nevertheless crucially influence the resultant transition rates. In contrast, non-Markovian effects due to the finite memory time are present but do not significantly influence the dihedral barrier-crossing rate of butane. These results not only settle the character of dihedral dynamics in small solvated molecular systems such as butane, they also have important implications for the folding of polymers and proteins.

Homogeneous and heterogeneous dynamics in native and denatured bovine serum albumin

Quasielastic incoherent neutron spectroscopy experiments reveal that chemical denaturation significantly modifies the internal dynamics of bovine serum albumin.

Predicting reaction coordinates in energy landscapes with diffusion anisotropy

We consider a range of model potentials with metastable states undergoing molecular dynamics coupled to a thermal bath in the high friction regime and consider how the optimal reaction coordinate depends on the diffusion anisotropy. For this we use our recently proposed method "spectral gap optimization of order parameters (SGOOP)" [P. Tiwary and B. J. Berne, Proc. Natl. Acad. Sci. U. S. A. 113, 2839 (2016)]. We show how available information about dynamical observables in addition to static information can be incorporated into SGOOP, which can then be used to accurately determine the "best" reaction coordinate for arbitrary anisotropies. We compare our results with transmission coefficient calculations and published benchmarks wherever applicable or available, respectively.

Time-resolved observation of protein allosteric communication

Allostery represents a fundamental mechanism of biological regulation that is mediated via long-range communication between distant protein sites. Although little is known about the underlying dynamical process, recent time-resolved infrared spectroscopy experiments on a photoswitchable PDZ domain (PDZ2S) have indicated that the allosteric transition occurs on multiple timescales. Here, using extensive nonequilibrium molecular dynamics simulations, a time-dependent picture of the allosteric communication in PDZ2S is developed. The simulations reveal that allostery amounts to the propagation of structural and dynamical changes that are genuinely nonlinear and can occur in a nonlocal fashion. A dynamic network model is constructed that illustrates the hierarchy and exceeding structural heterogeneity of the process. In compelling agreement with experiment, three physically distinct phases of the time evolution are identified, describing elastic response ([Formula: see text] ns), inelastic reorganization ([Formula: see text] ns), and structural relaxation ([Formula: see text]s). Issues such as the similarity to downhill folding as well as the interpretation of allosteric pathways are discussed.