Slow domain reconfiguration causes power-law kinetics in a two-state enzyme

Main role of fractal-like nature of conformational space in subdiffusion in proteins

Protein dynamics involves a myriad of mechanical movements happening at different time and space scales, which make it highly complex. One of the less understood features of protein dynamics is subdiffusivity, defined as sublinear dependence between displacement and time. Here, we use all-atoms molecular dynamics (MD) simulations to directly interrogate an already well-established theory and demonstrate that subdiffusivity arises from the fractal nature of the network of metastable conformations over which the dynamics, thought of as a diffusion process, takes place.

Perspective: How Fast Dynamics Affect Slow Function in Protein Machines

Internal motions in proteins take place on a broad range of time- and space-scales. The potential roles of these dynamics in the biochemical functions of proteins have intrigued biophysicists for many years, and multiple mechanisms to couple motions to function have been proposed. Some of these mechanisms have relied on equilibrium concepts. For example, the modulation of dynamics was proposed to change the entropy of a protein, hence affecting processes such as binding. This so-called dynamic allostery scenario has been demonstrated in several recent experiments. Perhaps even more intriguing may be models that involve out-of-equilibrium operation, which by necessity require the input of energy. We discuss several recent experimental studies that expose such potential mechanisms for coupling dynamics and function. In Brownian ratchets, for example, directional motion is promoted by switching a protein between two free energy surfaces. An additional example involves the effect of microsecond domain-closure dynamics of an enzyme on its much slower chemical cycle. These observations lead us to propose a novel two-time-scale paradigm for the activity of protein machines: fast equilibrium fluctuations take place on the microsecond-millisecond time scale, while on a slower time scale, free energy is invested in order to push the system out of equilibrium and drive functional transitions. Motions on the two time scales affect each other and are essential for the overall function of these machines.

Dynamic Heterogeneity in the Optical Signals from Single Nano-Objects

In contrast to ensemble-averaged measurements, single-molecule experiments directly display the heterogeneity of molecular properties in space and time. In many complex systems, spatial heterogeneity is regularly accompanied by temporal or dynamic heterogeneity; if a property differs from molecule to molecule, it will often vary in time for one and the same molecule. In this short paper, we discuss a few examples of complex systems where dynamical heterogeneity was observed in single-molecule or single-particle optical signals. For single biomolecules, the first demonstration of dynamic heterogeneity in a single enzyme was provided by Xie and colleagues. Other examples are found in glassy systems, and very recently in the magnetic relaxation of single superparamagnetic nanoparticles. The ubiquity of this phenomenon suggests that, rather than an exception, dynamic heterogeneity is the rule in complex systems with multiple degrees of freedom.

Heterogeneity in Protein Folding and Unfolding Reactions

Proteins have dynamic structures that undergo chain motions on time scales spanning from picoseconds to seconds. Resolving the resultant conformational heterogeneity is essential for gaining accurate insight into fundamental mechanistic aspects of the protein folding reaction. The use of high-resolution structural probes, sensitive to population distributions, has begun to enable the resolution of site-specific conformational heterogeneity at different stages of the folding reaction. Different states populated during protein folding, including the unfolded state, collapsed intermediate states, and even the native state, are found to possess significant conformational heterogeneity. Heterogeneity in protein folding and unfolding reactions originates from the reduced cooperativity of various kinds of physicochemical interactions between various structural elements of a protein, and between a protein and solvent. Heterogeneity may arise because of functional or evolutionary constraints. Conformational substates within the unfolded state and the collapsed intermediates that exchange at rates slower than the subsequent folding steps give rise to heterogeneity on the protein folding pathways. Multiple folding pathways are likely to represent distinct sequences of structure formation. Insight into the nature of the energy barriers separating different conformational states populated during (un)folding can also be obtained by resolving heterogeneity.

Single-molecule conformational dynamics of a transcription factor reveals a continuum of binding modes controlling association and dissociation

Binding and unbinding of transcription factors to DNA are kinetically controlled to regulate the transcriptional outcome. Control of the release of the transcription factor NF-κB from DNA is achieved through accelerated dissociation by the inhibitor protein IκBα. Using single-molecule FRET, we observed a continuum of conformations of NF-κB in free and DNA-bound states interconverting on the subseconds to minutes timescale, comparable to in vivo binding on the seconds timescale, suggesting that structural dynamics directly control binding kinetics. Much of the DNA-bound NF-κB is partially bound, allowing IκBα invasion to facilitate DNA dissociation. IκBα induces a locked conformation where the DNA-binding domains of NF-κB are too far apart to bind DNA, whereas a loss-of-function IκBα mutant retains the NF-κB conformational ensemble. Overall, our results suggest a novel mechanism with a continuum of binding modes for controlling association and dissociation of transcription factors.

Interplay of Polarity and Confinement in Asymmetric Catalysis with Chiral Rh Diene Complexes in Microemulsions

Microemulsions provide a unique opportunity to tailor the polarity and liquid confinement in asymmetric catalysis via nanoscale polar and nonpolar domains separated by a surfactant film. For chiral diene Rh complexes, the influence of counterion and surfactant film on the catalytic activity and enantioselectivity remained elusive. To explore this issue chiral norbornadiene Rh(X) complexes (X=OTf, OTs, OAc, PO₂F₂) were synthesized and characterized by X‐ray crystallography and theoretical calculations. These complexes were used in Rh‐catalyzed 1,2‐additions of phenylboroxine to N‐tosylimine in microemulsions stabilized either exclusively by n‐octyl‐β‐D‐glucopyranoside (C₈G₁) or a C₈G₁‐film doped with anionic or cationic surfactants (AOT, SDS and DTAB). The Rh(OAc) complex showed the largest dependence on the composition of the microemulsion, yielding up to 59 % (90 %ee) for the surfactant film doped with 5 wt% of AOT as compared to 52 % (58 %ee) for neat C₈G₁ at constant surfactant concentration. Larger domains, determined by SAXS analysis, enabled further increase in yield and selectivity while the reaction rate almost remained constant according to kinetic studies.

Diffusion of a disordered protein on its folded ligand

Flexibility in complexes between intrinsically disordered proteins and folded ligands is widespread in nature. However, timescales and spatial amplitudes of such dynamics remained unexplored for most systems. Our results show that the disordered cytoplasmic tail of the cell adhesion protein E-cadherin diffuses across the entire surface of its folded binding partner β-catenin at fast submillisecond timescales. The nanometer amplitude of these motions could allow kinases to access their recognition motifs without requiring a dissociation of the complex. We expect that the rugged energy landscape found in the E-cadherin/β-catenin complex is a defining feature of dynamic and partially disordered protein complexes.

Intrinsically disordered proteins often form dynamic complexes with their ligands. Yet, the speed and amplitude of these motions are hidden in classical binding kinetics. Here, we directly measure the dynamics in an exceptionally mobile, high-affinity complex. We show that the disordered tail of the cell adhesion protein E-cadherin dynamically samples a large surface area of the protooncogene β-catenin. Single-molecule experiments and molecular simulations resolve these motions with high resolution in space and time. Contacts break and form within hundreds of microseconds without a dissociation of the complex. The energy landscape of this complex is rugged with many small barriers (3 to 4 k_BT) and reconciles specificity, high affinity, and extreme disorder. A few persistent contacts provide specificity, whereas unspecific interactions boost affinity.

Factors underlying asymmetric pore dynamics of disaggregase and microtubule-severing AAA+ machines

Disaggregation and microtubule-severing nanomachines from the AAA+ (ATPases associated with various cellular activities) superfamily assemble into ring-shaped hexamers that enable protein remodeling by coupling large-scale conformational changes with application of mechanical forces within a central pore by loops protruding within the pore. We probed the asymmetric pore motions and intraring interactions that support them by performing extensive molecular dynamics simulations of single-ring severing proteins and the double-ring disaggregase ClpB. Simulations reveal that dynamic stability of hexameric pores of severing proteins and of the nucleotide-binding domain 1 (NBD1) ring of ClpB, which belong to the same clade, involves a network of salt bridges that connect conserved motifs of central pore loops. Clustering analysis of ClpB highlights correlated motions of domains of neighboring protomers supporting strong interprotomer collaboration. Severing proteins have weaker interprotomer coupling and stronger intraprotomer stabilization through salt bridges involving pore loops. Distinct mechanisms are identified in the NBD2 ring of ClpB involving weaker interprotomer coupling through salt bridges formed by noncanonical loops and stronger intraprotomer coupling. Analysis of collective motions of PL1 loops indicates that the largest amplitude motions in the spiral complex of spastin and ClpB involve axial excursions of the loops, whereas for katanin they involve opening and closing of the central pore. All three motors execute primarily axial excursions in the ring complex. These results suggest distinct substrate processing mechanisms of remodeling and translocation by ClpB and spastin compared to katanin, thus providing dynamic support for the differential action of the two severing proteins. Relaxation dynamics of the distance between the PL1 loops and the center of mass of protomers reveals observation-time-dependent dynamics, leading to predicted relaxation times of tens to hundreds of microseconds on millisecond experimental timescales. For ClpB, the predicted relaxation time is in excellent agreement with the extracted time from smFRET experiments.

Effect of Memory and Inertial Contribution on Transition-Time Distributions: Theory and Simulations

Transition paths refer to the time taken by molecules to cross a barrier separating two molecular conformations. In this work, we study how memory, as well as inertial contribution in the dynamics along a reaction coordinate, can affect the distribution of the transition-path time. We use a simple model of dynamics governed by a generalized Langevin equation with a power-law memory along with the inertial term, which was neglected in previous studies, where memory effects were explored only in the overdamped limit. We derive an approximate expression for the transit-time distribution and discuss our results for the short- and long-time limits and also compare it with known results in the high friction (overdamped) limit as well as in the Markovian limit. We have developed a numerical algorithm to test our theoretical results against extensive numerical simulations.

Membrane Chemistry Tunes the Structure of a Peptide Transporter

Membrane proteins require lipid bilayers for function. While lipid compositions reach enormous complexities, high-resolution structures are usually obtained in artificial detergents. To understand whether and how lipids guide membrane protein function, we use single-molecule FRET to probe the dynamics of DtpA, a member of the proton-coupled oligopeptide transporter (POT) family, in various lipid environments. We show that detergents trap DtpA in a dynamic ensemble with cytoplasmic opening. Only reconstitutions in more native environments restore cooperativity, allowing an opening to the extracellular side and a sampling of all relevant states. Bilayer compositions tune the abundance of these states. A novel state with an extreme cytoplasmic opening is accessible in bilayers with anionic head groups. Hence, chemical diversity of membranes translates into structural diversity, with the current POT structures only sampling a portion of the full structural space.

Non-Markov bond model for dynamic force spectroscopy

Single-molecule force spectroscopy data are conventionally analyzed using a schematic model, wherein a molecular bond is represented as a virtual particle diffusing in a one-dimensional free-energy landscape. However, this simple and efficient approach is unable to account for the "anomalous" bond-breaking kinetics increasingly observed in force spectroscopy experiments and simulations, e.g., in the form of non-exponential distributions of bond lifetimes under constant load. Here, we show that such characteristic traits arise naturally in a rigorous extension of the one-dimensional theory that accounts for the transient dynamics of a generic set of coupled degrees of freedom. These "hidden modes" affect the reaction dynamics in various ways, depending on their relaxation spectrum and the loading protocol, giving rise, in particular, to apparent static and dynamic disorder. In two complementary asymptotic limits, we are able to find exact analytical expressions for pertinent experimental observables, such as the mean rupture force and the rupture-force distribution. Intriguingly, our asymptotic results become unconditionally exact at high loading rates, thus providing us with a microscopically consistent theory of rapid force spectroscopy that avoids the usual Markov assumption.

Generalized Langevin Equation as a Model for Barrier Crossing Dynamics in Biomolecular Folding

Conformational memory in single-molecule dynamics has attracted recent attention and, in particular, has been invoked as a possible explanation of some of the intriguing properties of transition paths observed in single-molecule force spectroscopy (SMFS) studies. Here we study one candidate for a non-Markovian model that can account for conformational memory, the generalized Langevin equation with a friction force that depends not only on the instantaneous velocity but also on the velocities in the past. The memory in this model is determined by a time-dependent friction memory kernel. We propose a method for extracting this kernel directly from an experimental signal and illustrate its feasibility by applying it to a generalized Rouse model of a SMFS experiment, where the memory kernel is known exactly. Using the same model, we further study how memory affects various statistical properties of transition paths observed in SMFS experiments and evaluate the performance of recent approximate analytical theories of non-Markovian dynamics of barrier crossing. We argue that the same type of analysis can be applied to recent single-molecule observations of transition paths in protein and DNA folding.

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.

Transient antibody-antigen interactions mediate the strain-specific recognition of a conserved malaria epitope

Transient interactions in which binding partners retain substantial conformational disorder play an essential role in regulating biological networks, challenging the expectation that specificity demands structurally defined and unambiguous molecular interactions. The monoclonal antibody 6D8 recognises a completely conserved continuous nine-residue epitope within the intrinsically disordered malaria antigen, MSP2, yet it has different affinities for the two allelic forms of this antigen. NMR chemical shift perturbations, relaxation rates and paramagnetic relaxation enhancements reveal the presence of transient interactions involving polymorphic residues immediately C-terminal to the structurally defined epitope. A combination of these experimental data with molecular dynamics simulations shows clearly that the polymorphic C-terminal extension engages in multiple transient interactions distributed across much of the accessible antibody surface. These interactions are determined more by topographical features of the antibody surface than by sequence-specific interactions. Thus, specificity arises as a consequence of subtle differences in what are highly dynamic and essentially non-specific interactions.

Single-Molecule Test for Markovianity of the Dynamics along a Reaction Coordinate

In an effort to answer the much-debated question of whether the time evolution of common experimental observables can be described as one-dimensional diffusion in the potential of mean force, we propose a simple criterion that allows one to test whether the Markov assumption is applicable to a single-molecule trajectory x( t). This test does not involve fitting of the data to any presupposed model and can be applied to experimental data with relatively low temporal resolution.

Multiple Pathways and Time Scales for Conformational Transitions in apo-Adenylate Kinase

The open/close transition in adenylate kinase (AK) is regarded as a representative example for large-scale conformational transition in proteins, yet its mechanism remains unclear despite numerous experimental and computational studies. Using extensive (∼50 μs) explicit solvent atomistic simulations and Markov state analysis, we shed new lights on the mechanism of this transition in the apo form of AK. The closed basin of apo AK features an open NMP domain while the LID domain closes and rotates toward it. Therefore, although the computed structural properties of the closed ensemble are consistent with previously reported FRET and PRE measurements, our simulations suggest that NMP closure is likely to follow AMP binding, in contrast to the previous interpretation of FRET and PRE data that the apo state was able to sample the fully closed conformation for "ligand selection". The closed state ensemble is found to be kinetically heterogeneous; multiple pathways and time scales are associated with the open/close transition, providing new clues to the disparate time scales observed in different experiments. Besides interdomain interactions, a novel mutual information analysis identifies specific intradomain interactions that correlate strongly to transition kinetics, supporting observations from previous chimera experiments. While our results underscore the role of internal domain properties in determining the kinetics of open/close transition in apo AK, no evidence is observed for any significant degree of local unfolding during the transition. These observations about AK have general implications to our view of conformational states, transition pathways, and time scales of conformational changes in proteins. The key features and time scales of observed transition pathways are robust and similar from simulations using two popular fixed charge force fields.