Splitting probabilities as a test of reaction coordinate choice in single-molecule experiments

On the Mechanism of Polaritonic Rate Suppression from Quantum Transition Paths

Polariton chemistry holds promise for facilitating mode-selective chemical reactions, but the underlying mechanism behind the rate modifications observed under strong vibrational coupling is not well understood. Using the recently developed quantum transition path theory, we have uncovered a mechanism of resonant suppression of a thermal reaction rate in a simple model polaritonic system consisting of a reactive mode in a bath confined to a lossless microcavity with a single photon mode. We observed the formation of a polariton during rate-limiting transitions on reactive pathways and identified the concomitant rate suppression as being due to hybridization between the reactive mode and the cavity mode, which inhibits bath-mediated tunneling. The transition probabilities that define the quantum master equation can be directly translated into a visualization of the corresponding polariton energy landscape. This landscape exhibits a double funnel structure with a large barrier between the initial and final states.

Nonadiabatic transition paths from quantum jump trajectories

We present a means of studying rare reactive pathways in open quantum systems using transition path theory and ensembles of quantum jump trajectories. This approach allows for the elucidation of reactive paths for dissipative, nonadiabatic dynamics when the system is embedded in a Markovian environment. We detail the dominant pathways and rates of thermally activated processes and the relaxation pathways and photoyields following vertical excitation in a minimal model of a conical intersection. We find that the geometry of the conical intersection affects the electronic character of the transition state as defined through a generalization of a committor function for a thermal barrier crossing event. Similarly, the geometry changes the mechanism of relaxation following a vertical excitation. Relaxation in models resulting from small diabatic coupling proceeds through pathways dominated by pure dephasing, while those with large diabatic coupling proceed through pathways limited by dissipation. The perspective introduced here for the nonadiabatic dynamics of open quantum systems generalizes classical notions of reactive paths to fundamentally quantum mechanical processes.

Stochastic Model of Solvent Exchange in the First Coordination Shell of Aqua Ions

Ion microsolvation is a basic, yet fundamental, process of ionic solutions underlying many relevant phenomena in either biological or nanotechnological applications, such as solvent reorganization energy, ion transport, catalytic activity, and so on. As a consequence, it is a topic of extensive investigations by various experimental techniques, ranging from X-ray diffraction to NMR relaxation and from calorimetry to vibrational spectroscopy, and theoretical approaches, especially those based on molecular dynamics (MD) simulations. The conventional microscopic view of ion solvation is usually provided by a "static" cluster model representing the first ion-solvent coordination shell. Despite the merits of such a simple model, however, ion coordination in solution should be better regarded as a complex population of dynamically interchanging molecular configurations. Such a more comprehensive view is more subtle to characterize and often elusive to standard approaches. In this work, we report on an effective computational strategy aiming at providing a detailed picture of solvent coordination and exchange around aqua ions, thus including the main structural, thermodynamic, and dynamic properties of ion microsolvation, such as the most probable first-shell complex structures, the corresponding free energies, the interchanging energy barriers, and the solvent-exchange rates. Assuming the solvent coordination number as an effective reaction coordinate and combining MD simulations with enhanced sampling and master-equation approaches, we propose a stochastic model suitable for properly describing, at the same time, the thermodynamics and kinetics of ion-water coordination. The model is successfully tested toward various divalent ions (Ca²⁺, Zn²⁺, Hg²⁺, and Cd²⁺) in aqueous solution, considering also the case of a high ionic concentration. Results show a very good agreement with those issuing from brute-force MD simulations, when available, and support the reliable prediction of rare ion-water complexes and slow water exchange rates not easily accessible to usual computational methods.

A Bayesian approach to extracting free-energy profiles from cryo-electron microscopy experiments

Cryo-electron microscopy (cryo-EM) extracts single-particle density projections of individual biomolecules. Although cryo-EM is widely used for 3D reconstruction, due to its single-particle nature it has the potential to provide information about a biomolecule's conformational variability and underlying free-energy landscape. However, treating cryo-EM as a single-molecule technique is challenging because of the low signal-to-noise ratio (SNR) in individual particles. In this work, we propose the cryo-BIFE method (cryo-EM Bayesian Inference of Free-Energy profiles), which uses a path collective variable to extract free-energy profiles and their uncertainties from cryo-EM images. We test the framework on several synthetic systems where the imaging parameters and conditions were controlled. We found that for realistic cryo-EM environments and relevant biomolecular systems, it is possible to recover the underlying free energy, with the pose accuracy and SNR as crucial determinants. We then use the method to study the conformational transitions of a calcium-activated channel with real cryo-EM particles. Interestingly, we recover not only the most probable conformation (used to generate a high-resolution reconstruction of the calcium-bound state) but also a metastable state that corresponds to the calcium-unbound conformation. As expected for turnover transitions within the same sample, the activation barriers are on the order of [Formula: see text]. We expect our tool for extracting free-energy profiles from cryo-EM images to enable more complete characterization of the thermodynamic ensemble of biomolecules.

Proton transfer from water to aromatic N-heterocyclic anions from DFT-MD simulations

The phenomenon of proton transfer from water to six N-heterocyclic anions and free energy landscapes of this process are studied using both electronic structure calculations and first principles molecular metadynamics simulations. Our investigation involves microhydrated and aqueous phase interaction of water with six aromatic heterocyclic anions relevant to chemistry and biology: imidazolide, pyrrolide, benzimidazolide, 2-cyanopyrrolide, indolide, and indazolide. The basic structures of all these heterocyclic anions differ by substituted functional groups as well as fused rings. We study the proton transfer reaction and the minimum number of required water molecules for the reaction in hydrated microclusters. We find out that at least four water molecules are necessary for hydrated clusters to facilitate the intracluster proton transfer reaction from water to anions except for pyrrolide, for which this magic number is 3. To obtain the reaction free energy and activation barrier of the proton transfer process in an aqueous solution, the metadynamics method based first principles molecular dynamics simulations were performed. The complete proton transfer was observed in aqueous solutions for all the anions. The water molecule directly involved in proton transfer becomes acidic due to the cooperative effect of neighboring water molecules. From the metadynamics simulation, we obtain the values of activation barrier for the proton transfer processes from neutral water to anions, and the highest activation barrier is obtained for benzimidazolide, whereas the lowest activation barrier is obtained for pyrrolide. The structures and free energy profiles of the process for all the anions are discussed, and a comparative outlook of the study is presented here.

Exploring the sequence fitness landscape of a bridge between protein folds

Most foldable protein sequences adopt only a single native fold. Recent protein design studies have, however, created protein sequences which fold into different structures apon changes of environment, or single point mutation, the best characterized example being the switch between the folds of the GA and GB binding domains of streptococcal protein G. To obtain further insight into the design of sequences which can switch folds, we have used a computational model for the fitness landscape of a single fold, built from the observed sequence variation of protein homologues. We have recently shown that such coevolutionary models can be used to design novel foldable sequences. By appropriately combining two of these models to describe the joint fitness landscape of GA and GB, we are able to describe the propensity of a given sequence for each of the two folds. We have successfully tested the combined model against the known series of designed GA/GB hybrids. Using Monte Carlo simulations on this landscape, we are able to identify pathways of mutations connecting the two folds. In the absence of a requirement for domain stability, the most frequent paths go via sequences in which neither domain is stably folded, reminiscent of the propensity for certain intrinsically disordered proteins to fold into different structures according to context. Even if the folded state is required to be stable, we find that there is nonetheless still a wide range of sequences which are close to the transition region and therefore likely fold switches, consistent with recent estimates that fold switching may be more widespread than had been thought.

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.

Molecular free energy profiles from force spectroscopy experiments by inversion of observed committors

In single-molecule force spectroscopy experiments, a biomolecule is attached to a force probe via polymer linkers and the total extension of the molecule plus apparatus is monitored as a function of time. In a typical unfolding experiment at constant force, the total extension jumps between two values that correspond to the folded and unfolded states of the molecule. For several biomolecular systems, the committor, which is the probability to fold starting from a given extension, has been used to extract the molecular activation barrier (a technique known as "committor inversion"). In this work, we study the influence of the force probe, which is much larger than the molecule being measured, on the activation barrier obtained by committor inversion. We use a two-dimensional framework in which the diffusion coefficient of the molecule and of the pulling device can differ. We systematically study the free energy profile along the total extension obtained from the committor by numerically solving the Onsager equation and using Brownian dynamics simulations. We analyze the dependence of the extracted barrier on the linker stiffness, molecular barrier height, and diffusion anisotropy and, thus, establish the range of validity of committor inversion. Along the way, we showcase the committor of 2-dimensional diffusive models and illustrate how it is affected by barrier asymmetry and diffusion anisotropy.

Modeling of Multicolor Single-Molecule Förster Resonance Energy-Transfer Experiments on Protein Folding

Using a coarse-grained, C_α-model of BBL protein, a multicolor single-molecule Förster resonance energy transfer (FRET) experiment is modeled. Three fluorophores are introduced, which, for simplicity, are associated with C_α beads. Two fluorophores are placed at the ends of protein chain and the third one at the middle of the chain. The free-energy surfaces (FESs) depending on the interfluorophore distances and on the FRET efficiencies corresponding to these distances have been constructed and compared with the FESs depending on the conventional collective variables, such as the fraction of native contacts and radius of gyration. It has been found that multicolor experiments can successfully resolve all essential BBL states that are revealed by the conventional FESs. The resolution of these states with the FRET-efficiency histogram is found to be successful if the energy transfer is measured between the fluorophores at the BBL ends. We also show that, although the present model construct of BBL is very simple, it captures some characteristic features of the single-molecule FRET experiments, such as the pattern of the FRET-efficiency histograms and their evolution with the denaturant concentration.

Crystallization of hard spheres revisited. I. Extracting kinetics and free energy landscape from forward flux sampling

We investigate the kinetics and the free energy landscape of the crystallization of hard spheres from a supersaturated metastable liquid though direct simulations and forward flux sampling. In this first paper, we describe and test two different ways to reconstruct the free energy barriers from the sampled steady state probability distribution of cluster sizes without sampling the equilibrium distribution. The first method is based on mean first passage times, and the second method is based on splitting probabilities. We verify both methods for a single particle moving in a double-well potential. For the nucleation of hard spheres, these methods allow us to probe a wide range of supersaturations and to reconstruct the kinetics and the free energy landscape from the same simulation. Results are consistent with the scaling predicted by classical nucleation theory although a quantitative fit requires a rather large effective interfacial tension.

Caliber Corrected Markov Modeling (C2M2): Correcting Equilibrium Markov Models

Rate processes are often modeled using Markov State Models (MSMs). Suppose you know a prior MSM and then learn that your prediction of some particular observable rate is wrong. What is the best way to correct the whole MSM? For example, molecular dynamics simulations of protein folding may sample many microstates, possibly giving correct pathways through them while also giving the wrong overall folding rate when compared to experiment. Here, we describe Caliber Corrected Markov Modeling (C₂M₂), an approach based on the principle of maximum entropy for updating a Markov model by imposing state- and trajectory-based constraints. We show that such corrections are equivalent to asserting position-dependent diffusion coefficients in continuous-time continuous-space Markov processes modeled by a Smoluchowski equation. We derive the functional form of the diffusion coefficient explicitly in terms of the trajectory-based constraints. We illustrate with examples of 2D particle diffusion and an overdamped harmonic oscillator.

Direct measurement of sequence-dependent transition path times and conformational diffusion in DNA duplex formation

The conformational diffusion coefficient, D, sets the timescale for microscopic structural changes during folding transitions in biomolecules like nucleic acids and proteins. D encodes significant information about the folding dynamics such as the roughness of the energy landscape governing the folding and the level of internal friction in the molecule, but it is challenging to measure. The most sensitive measure of D is the time required to cross the energy barrier that dominates folding kinetics, known as the transition path time. To investigate the sequence dependence of D in DNA duplex formation, we measured individual transition paths from equilibrium folding trajectories of single DNA hairpins held under tension in high-resolution optical tweezers. Studying hairpins with the same helix length but with G:C base-pair content varying from 0 to 100%, we determined both the average time to cross the transition paths, τ_tp, and the distribution of individual transit times, P_TP(t). We then estimated D from both τ_tp and P_TP(t) from theories assuming one-dimensional diffusive motion over a harmonic barrier. τ_tp decreased roughly linearly with the G:C content of the hairpin helix, being 50% longer for hairpins with only A:T base pairs than for those with only G:C base pairs. Conversely, D increased linearly with helix G:C content, roughly doubling as the G:C content increased from 0 to 100%. These results reveal that G:C base pairs form faster than A:T base pairs because of faster conformational diffusion, possibly reflecting lower torsional barriers, and demonstrate the power of transition path measurements for elucidating the microscopic determinants of folding.

Single-File Escape of Colloidal Particles from Microfluidic Channels

Single-file diffusion is a ubiquitous physical process exploited by living and synthetic systems to exchange molecules with their environment. It is paramount to quantify the escape time needed for single files of particles to exit from constraining synthetic channels and biological pores. This quantity depends on complex cooperative effects, whose predominance can only be established through a strict comparison between theory and experiments. By using colloidal particles, optical manipulation, microfluidics, digital microscopy, and theoretical analysis we uncover the self-similar character of the escape process and provide closed-formula evaluations of the escape time. We find that the escape time scales inversely with the diffusion coefficient of the last particle to leave the channel. Importantly, we find that at the investigated microscale, bias forces as tiny as 10^{-15}  N determine the magnitude of the escape time by drastically reducing interparticle collisions. Our findings provide crucial guidelines to optimize the design of micro- and nanodevices for a variety of applications including drug delivery, particle filtering, and transport in geometrical constrictions.

-1 Programmed Ribosomal Frameshifting as a Force-Dependent Process

-1 Programmed ribosomal frameshifting is a translational recoding event in which ribosomes slip backward along messenger RNA presumably due to increased tension disrupting the codon-anticodon interaction at the ribosome's coding site. Single-molecule physical methods and recent experiments characterizing the physical properties of mRNA's slippery sequence as well as the mechanical stability of downstream mRNA structure motifs that give rise to frameshifting are discussed. Progress in technology, experimental assays, and data analysis methods hold promise for accurate physical modeling and quantitative understanding of -1 programmed ribosomal frameshifting.

High-resolution free energy landscape analysis of protein folding

The free energy landscape can provide a quantitative description of folding dynamics, if determined as a function of an optimally chosen reaction coordinate. The profile together with the optimal coordinate allows one to directly determine such basic properties of folding dynamics as the configurations of the minima and transition states, the heights of the barriers, the value of the pre-exponential factor and its relation to the transition path times. In the present study, we review the framework, in particular, the approach to determine such an optimal coordinate, and its application to the analysis of simulated protein folding dynamics.

Reconstructing folding energy landscapes from splitting probability analysis of single-molecule trajectories

Structural self-assembly in biopolymers, such as proteins and nucleic acids, involves a diffusive search for the minimum-energy state in a conformational free-energy landscape. The likelihood of folding proceeding to completion, as a function of the reaction coordinate used to monitor the transition, can be described by the splitting probability, p(fold)(x). P(fold) encodes information about the underlying energy landscape, and it is often used to judge the quality of the reaction coordinate. Here, we show how p(fold) can be used to reconstruct energy landscapes from single-molecule folding trajectories, using force spectroscopy measurements of single DNA hairpins. Calculating p(fold)(x) directly from trajectories of the molecular extension measured for hairpins fluctuating in equilibrium between folded and unfolded states, we inverted the result expected from diffusion over a 1D energy landscape to obtain the implied landscape profile. The results agreed well with the landscapes reconstructed by established methods, but, remarkably, without the need to deconvolve instrumental effects on the landscape, such as tether compliance. The same approach was also applied to hairpins with multistate folding pathways. The relative insensitivity of the method to the instrumental compliance was confirmed by simulations of folding measured with different tether stiffnesses. This work confirms that the molecular extension is a good reaction coordinate for these measurements, and validates a powerful yet simple method for reconstructing landscapes from single-molecule trajectories.

Transition-Path Probability as a Test of Reaction-Coordinate Quality Reveals DNA Hairpin Folding Is a One-Dimensional Diffusive Process

Chemical reactions are typically described in terms of progress along a reaction coordinate. However, the quality of reaction coordinates for describing reaction dynamics is seldom tested experimentally. We applied a framework for gauging reaction-coordinate quality based on transition-path analysis to experimental data for the first time, looking at folding trajectories of single DNA hairpin molecules measured under tension applied by optical tweezers. The conditional probability for being on a reactive transition path was compared with the probability expected for ideal diffusion over a 1D energy landscape based on the committor function. Analyzing measurements and simulations of hairpin folding where end-to-end extension is the reaction coordinate, after accounting for instrumental effects on the analysis, we found good agreement between transition-path and committor analyses for model two-state hairpins, demonstrating that folding is well-described by 1D diffusion. This work establishes transition-path analysis as a powerful new tool for testing experimental reaction-coordinate quality.

Markov state models provide insights into dynamic modulation of protein function

Protein function is inextricably linked to protein dynamics. As we move from a static structural picture to a dynamic ensemble view of protein structure and function, novel computational paradigms are required for observing and understanding conformational dynamics of proteins and its functional implications. In principle, molecular dynamics simulations can provide the time evolution of atomistic models of proteins, but the long time scales associated with functional dynamics make it difficult to observe rare dynamical transitions. The issue of extracting essential functional components of protein dynamics from noisy simulation data presents another set of challenges in obtaining an unbiased understanding of protein motions. Therefore, a methodology that provides a statistical framework for efficient sampling and a human-readable view of the key aspects of functional dynamics from data analysis is required. The Markov state model (MSM), which has recently become popular worldwide for studying protein dynamics, is an example of such a framework.

In this Account, we review the use of Markov state models for efficient sampling of the hierarchy of time scales associated with protein dynamics, automatic identification of key conformational states, and the degrees of freedom associated with slow dynamical processes. Applications of MSMs for studying long time scale phenomena such as activation mechanisms of cellular signaling proteins has yielded novel insights into protein function. In particular, from MSMs built using large-scale simulations of GPCRs and kinases, we have shown that complex conformational changes in proteins can be described in terms of structural changes in key structural motifs or “molecular switches” within the protein, the transitions between functionally active and inactive states of proteins proceed via multiple pathways, and ligand or substrate binding modulates the flux through these pathways. Finally, MSMs also provide a theoretical toolbox for studying the effect of nonequilibrium perturbations on conformational dynamics. Considering that protein dynamics in vivo occur under nonequilibrium conditions, MSMs coupled with nonequilibrium statistical mechanics provide a way to connect cellular components to their functional environments. Nonequilibrium perturbations of protein folding MSMs reveal the presence of dynamically frozen glass-like states in their conformational landscape. These frozen states are also observed to be rich in β-sheets, which indicates their possible role in the nucleation of β-sheet rich aggregates such as those observed in amyloid-fibril formation. Finally, we describe how MSMs have been used to understand the dynamical behavior of intrinsically disordered proteins such as amyloid-β, human islet amyloid polypeptide, and p53. While certainly not a panacea for studying functional dynamics, MSMs provide a rigorous theoretical foundation for understanding complex entropically dominated processes and a convenient lens for viewing protein motions.

Reconstructing folding energy landscapes by single-molecule force spectroscopy

Folding may be described conceptually in terms of trajectories over a landscape of free energies corresponding to different molecular configurations. In practice, energy landscapes can be difficult to measure. Single-molecule force spectroscopy (SMFS), whereby structural changes are monitored in molecules subjected to controlled forces, has emerged as a powerful tool for probing energy landscapes. We summarize methods for reconstructing landscapes from force spectroscopy measurements under both equilibrium and nonequilibrium conditions. Other complementary, but technically less demanding, methods provide a model-dependent characterization of key features of the landscape. Once reconstructed, energy landscapes can be used to study critical folding parameters, such as the characteristic transition times required for structural changes and the effective diffusion coefficient setting the timescale for motions over the landscape. We also discuss issues that complicate measurement and interpretation, including the possibility of multiple states or pathways and the effects of projecting multiple dimensions onto a single coordinate.

Optimal reaction coordinate as a biomarker for the dynamics of recovery from kidney transplant

The evolution of disease or the progress of recovery of a patient is a complex process, which depends on many factors. A quantitative description of this process in real-time by a single, clinically measurable parameter (biomarker) would be helpful for early, informed and targeted treatment. Organ transplantation is an eminent case in which the evolution of the post-operative clinical condition is highly dependent on the individual case. The quality of management and monitoring of patients after kidney transplant often determines the long-term outcome of the graft. Using NMR spectra of blood samples, taken at different time points from just before to a week after surgery, we have shown that a biomarker can be found that quantitatively monitors the evolution of a clinical condition. We demonstrate that this is possible if the dynamics of the process is considered explicitly: the biomarker is defined and determined as an optimal reaction coordinate that provides a quantitatively accurate description of the stochastic recovery dynamics. The method, originally developed for the analysis of protein folding dynamics, is rigorous, robust and general, i.e., it can be applied in principle to analyze any type of biological dynamics. Such predictive biomarkers will promote improvement of long-term graft survival after renal transplantation, and have potentially unlimited applications as diagnostic tools.

The evolution of disease or the progress of recovery of a patient is usually monitored by collecting physical parameters, which may be simply the body temperature for a common cold or properties of tissue samples for e.g., cancer. Most often clinical decisions are taken based on the current value or because of a sizable change of a relevant parameter. As more advanced diagnostic tools become available, and huge numbers of parameters can be collected at short, frequent time intervals, two related questions arise. The first is, which of the parameters provides relevant information on the progress of disease or recovery as opposed to noise? Is there more information that can be obtained from the history of the evolution of such parameters? Here we propose a novel approach that leads, for the specific case of recovery from kidney transplant, to a positive answer.

Anomalous dynamics of DNA hairpin folding

By means of computer simulations of a coarse-grained DNA model we show that the DNA hairpin zippering dynamics is anomalous; i.e., the characteristic time τ scales nonlinearly with N, the hairpin length, τ ∼ N(α) with α>1. This is in sharp contrast to the prediction of the zipper model for which τ ∼ N. We show that the anomalous dynamics originates from an increase in the friction during zippering due to the tension built in the closing strands. From a simple polymer model we get α = 1+ν ≈ 1.59 with ν being the Flory exponent, a result which is in agreement with the simulations. We discuss transition path times data where such effects should be detected.

Protein folding: complex potential for the driving force in a two-dimensional space of collective variables

Using the Helmholtz decomposition of the vector field of folding fluxes in a two-dimensional space of collective variables, a potential of the driving force for protein folding is introduced. The potential has two components. One component is responsible for the source and sink of the folding flows, which represent respectively, the unfolded states and the native state of the protein, and the other, which accounts for the flow vorticity inherently generated at the periphery of the flow field, is responsible for the canalization of the flow between the source and sink. The theoretical consideration is illustrated by calculations for a model β-hairpin protein.

Expectation-maximization of the potential of mean force and diffusion coefficient in Langevin dynamics from single molecule FRET data photon by photon

The dynamics of a protein along a well-defined coordinate can be formally projected onto the form of an overdamped Lagevin equation. Here, we present a comprehensive statistical-learning framework for simultaneously quantifying the deterministic force (the potential of mean force, PMF) and the stochastic force (characterized by the diffusion coefficient, D) from single-molecule Förster-type resonance energy transfer (smFRET) experiments. The likelihood functional of the Langevin parameters, PMF and D, is expressed by a path integral of the latent smFRET distance that follows Langevin dynamics and realized by the donor and the acceptor photon emissions. The solution is made possible by an eigen decomposition of the time-symmetrized form of the corresponding Fokker-Planck equation coupled with photon statistics. To extract the Langevin parameters from photon arrival time data, we advance the expectation-maximization algorithm in statistical learning, originally developed for and mostly used in discrete-state systems, to a general form in the continuous space that allows for a variational calculus on the continuous PMF function. We also introduce the regularization of the solution space in this Bayesian inference based on a maximum trajectory-entropy principle. We use a highly nontrivial example with realistically simulated smFRET data to illustrate the application of this new method.

Transition pathway and its free-energy profile: a protocol for protein folding simulations

We propose a protocol that provides a systematic definition of reaction coordinate and related free-energy profile as the function of temperature for the protein-folding simulation. First, using action-derived molecular dynamics (ADMD), we investigate the dynamic folding pathway model of a protein between a fixed extended conformation and a compact conformation. We choose the pathway model to be the reaction coordinate, and the folding and unfolding processes are characterized by the ADMD step index, in contrast to the common a priori reaction coordinate as used in conventional studies. Second, we calculate free-energy profile as the function of temperature, by employing the replica-exchange molecular dynamics (REMD) method. The current method provides efficient exploration of conformational space and proper characterization of protein folding/unfolding dynamics from/to an arbitrary extended conformation. We demonstrate that combination of the two simulation methods, ADMD and REMD, provides understanding on molecular conformational changes in proteins. The protocol is tested on a small protein, penta-peptide of met-enkephalin. For the neuropeptide met-enkephalin system, folded, extended, and intermediate sates are well-defined through the free-energy profile over the reaction coordinate. Results are consistent with those in the literature.

Reaction coordinates, one-dimensional Smoluchowski equations, and a test for dynamical self-consistency

We propose a method for identifying accurate reaction coordinates among a set of trial coordinates. The method applies to special cases where motion along the reaction coordinate follows a one-dimensional Smoluchowski equation. In these cases the reaction coordinate can predict its own short-time dynamical evolution, i.e., the dynamics projected from multiple dimensions onto the reaction coordinate depend only on the reaction coordinate itself. To test whether this property holds, we project an ensemble of short trajectory swarms onto trial coordinates and compare projections of individual swarms to projections of the ensemble of swarms. The comparison, quantified by the Kullback-Leibler divergence, is numerically performed for each isosurface of each trial coordinate. The ensemble of short dynamical trajectories is generated only once by sampling along an initial order parameter. The initial order parameter should separate the reactants and products with a free energy barrier, and distributions on isosurfaces of the initial parameter should be unimodal. The method is illustrated for three model free energy landscapes with anisotropic diffusion. Where exact coordinates can be obtained from Kramers-Langer-Berezhkovskii-Szabo theory, results from the new method agree with the exact results. We also examine characteristics of systems where the proposed method fails. We show how dynamical self-consistency is related (through the Chapman-Kolmogorov equation) to the earlier isocommittor criterion, which is based on longer paths.

To milliseconds and beyond: challenges in the simulation of protein folding

Quantitatively accurate all-atom molecular dynamics (MD) simulations of protein folding have long been considered a holy grail of computational biology. Due to the large system sizes and long timescales involved, such a pursuit was for many years computationally intractable. Further, sufficiently accurate forcefields needed to be developed in order to realistically model folding. This decade, however, saw the first reports of folding simulations describing kinetics on the order of milliseconds, placing many proteins firmly within reach of these methods. Progress in sampling and forcefield accuracy, however, presents a new challenge: how to turn huge MD datasets into scientific understanding. Here, we review recent progress in MD simulation techniques and show how the vast datasets generated by such techniques present new challenges for analysis. We critically discuss the state of the art, including reaction coordinate and Markov state model (MSM) methods, and provide a perspective for the future.

Microscopically computing free-energy profiles and transition path time of rare macromolecular transitions

We introduce a rigorous method to microscopically compute the observables which characterize the thermodynamics and kinetics of rare macromolecular transitions for which it is possible to identify a priori a slow reaction coordinate. In order to sample the ensemble of statistically significant reaction pathways, we define a biased molecular dynamics (MD) in which barrier-crossing transitions are accelerated without introducing any unphysical external force. In contrast to other biased MD methods, in the present approach the systematic errors which are generated in order to accelerate the transition can be analytically calculated and therefore can be corrected for. This allows for a computationally efficient reconstruction of the free-energy profile as a function of the reaction coordinate and for the calculation of the corresponding diffusion coefficient. The transition path time can then be readily evaluated within the dominant reaction pathways approach. We illustrate and test this method by characterizing a thermally activated transition on a two-dimensional energy surface and the folding of a small protein fragment within a coarse-grained model.

Universal relaxation governs the nonequilibrium elasticity of biomolecules

Experimental and computational dynamic force spectroscopy is widely used to determine the mechanical properties of single biomolecules. Whereas so far the focus has mainly been on rupture or unfolding forces, recent force-probe molecular dynamics simulations have revealed a strong loading rate dependence of biomolecular elasticities, which cannot be explained by the established one-dimensional transition-state treatments. We show that this nonequilibrium behavior can be explained by a theory that includes relaxation effects. For three structurally and mechanically quite diverse systems, a single relaxation mode suffices to quantitatively describe their loading-rate-dependent elastic behavior. Atomistic simulations of these systems revealed the microscopic nature of the respective relaxation modes. This result suggests a new type of "elasticity spectroscopy" experiment, which should render nonequilibrium properties of structured macromolecules accessible to single-molecule force spectroscopy.

Characterization of a dynamic string method for the construction of transition pathways in molecular reactions

We explore the theoretical foundation of different string methods used to find dominant reaction pathways in high-dimensional configuration spaces. Pathways are assessed by the amount of reactive flux they carry and by their orientation relative to the committor function. By examining the effects of transforming between different collective coordinates that span the same underlying space, we unmask artificial coordinate dependences in strings optimized to follow the free energy gradient. In contrast, strings optimized to follow the drift vector produce reaction pathways that are significantly less sensitive to reparameterizations of the collective coordinates. The differences in these paths arise because the drift vector depends on both the free energy gradient and the diffusion tensor of the coarse collective variables. Anisotropy and position dependence of diffusion tensors arise commonly in spaces of coarse variables, whose generally slow dynamics are obtained by nonlinear projections of the strongly coupled atomic motions. We show here that transition paths constructed to account for dynamics by following the drift vector will (to a close approximation) carry the maximum reactive flux both in systems with isotropic position dependent diffusion and in systems with constant but anisotropic diffusion. We derive a simple method for calculating the committor function along paths that follow the reactive flux. Lastly, we provide guidance for the practical implementation of the dynamic string method.

The molten globule state is unusually deformable under mechanical force

Recently, the role of force in cellular processes has become more evident, and now with advances in force spectroscopy, the response of proteins to force can be directly studied. Such studies have found that native proteins are brittle, and thus not very deformable. Here, we examine the mechanical properties of a class of intermediates referred to as the molten globule state. Using optical trap force spectroscopy, we investigated the response to force of the native and molten globule states of apomyoglobin along different pulling axes. Unlike natively folded proteins, the molten globule state of apomyoglobin is compliant (large distance to the transition state); this large compliance means that the molten globule is more deformable and the unfolding rate is more sensitive to force (the application of force or tension will have a more dramatic effect on the unfolding rate). Our studies suggest that these are general properties of molten globules and could have important implications for mechanical processes in the cell.