Coordinate-dependent diffusion in protein folding

Spectral Map for Slow Collective Variables, Markovian Dynamics, and Transition State Ensembles

Understanding the behavior of complex molecular systems is a fundamental problem in physical chemistry. To describe the long-time dynamics of such systems, which is responsible for their most informative characteristics, we can identify a few slow collective variables (CVs) while treating the remaining fast variables as thermal noise. This enables us to simplify the dynamics and treat it as diffusion in a free-energy landscape spanned by slow CVs, effectively rendering the dynamics Markovian. Our recent statistical learning technique, spectral map [Rydzewski, J. J. Phys. Chem. Lett. 2023, 14(22), 5216-5220], explores this strategy to learn slow CVs by maximizing a spectral gap of a transition matrix. In this work, we introduce several advancements into our framework, using a high-dimensional reversible folding process of a protein as an example. We implement an algorithm for coarse-graining Markov transition matrices to partition the reduced space of slow CVs kinetically and use it to define a transition state ensemble. We show that slow CVs learned by spectral map closely approach the Markovian limit for an overdamped diffusion. We demonstrate that coordinate-dependent diffusion coefficients only slightly affect the constructed free-energy landscapes. Finally, we present how spectral maps can be used to quantify the importance of features and compare slow CVs with structural descriptors commonly used in protein folding. Overall, we demonstrate that a single slow CV learned by spectral map can be used as a physical reaction coordinate to capture essential characteristics of protein folding.

Antioxidant capacity of xylooligosaccharides generated from beechwood xylan by recombinant family GH10 Aspergillus niger xylanase A and insights into the enzyme's competitive inhibition by riceXIP

In this study, the family GH10 xylanase AnXylA10 derived from Aspergillus niger JL15 strain was expressed in Pichia pastoris X33. The recombinant xylanase, reAnXylA10 exhibited optimal activity at 40 ℃ and pH 5.0. The hydrolysates generated from beechwood xylan using reAnXylA10 primarily consisted of xylobiose (X2) to xylohexaose (X6) and demonstrated remarkable antioxidant capacity. Furthermore, the rice xylanase inhibitory protein (riceXIP) was observed to competitively inhibit reAnXylA10, exhibiting an inhibition constant (Ki) of 140.6 nM. Molecular dynamics (MD) simulations of AnXylA10-riceXIP complex revealed that the α-7 helix (Q225-S238) of riceXIP intruded into the catalytic pocket of AnXylA10, thereby obstructing substrate access to the active site. Specifically, residue K226 of riceXIP formed robust interactions with E136 and E242, the two catalytic sites of AnXylA10, predominantly through high-occupied hydrogen bonds. Based on QTAIM, electron densities for the atom pairs K226riceXIP@HZ1-E136AnXylA10@OE2 and K226riceXIP@HZ3-E242AnXylA10@OE1 were determined to be 0.04628 and 0.02914 a.u., respectively. Binding free energy of AnXylA10-riceXIP complex was -59.0±7.6 kcal/mol, significantly driven by electrostatic and van der Waals forces. Gaining insights into the interaction between xylanase and its inhibitors, and mining the inhibition mechanism in depth, will facilitate the design of innovative GH10 family xylanases that are both highly efficient and resistant to inhibitors.

Data-Driven Learning of the Generalized Langevin Equation with State-Dependent Memory

We present a data-driven method to learn stochastic reduced models of complex systems that retain a state-dependent memory beyond the standard generalized Langevin equation with a homogeneous kernel. The constructed model naturally encodes the heterogeneous energy dissipation by jointly learning a set of state features and the non-Markovian coupling among the features. Numerical results demonstrate the limitation of the standard generalized Langevin equation and the essential role of the broadly overlooked state-dependency nature in predicting molecule kinetics related to conformation relaxation and transition.

Estimating Position-Dependent and Anisotropic Diffusivity Tensors from Molecular Dynamics Trajectories: Existing Methods and Future Outlook

Confinement can substantially alter the physicochemical properties of materials by breaking translational isotropy and rendering all physical properties position-dependent. Molecular dynamics (MD) simulations have proven instrumental in characterizing such spatial heterogeneities and probing the impact of confinement on materials' properties. For static properties, this is a straightforward task and can be achieved via simple spatial binning. Such an approach, however, cannot be readily applied to transport coefficients due to lack of natural extensions of autocorrelations used for their calculation in the bulk. The prime example of this challenge is diffusivity, which, in the bulk, can be readily estimated from the particles' mobility statistics, which satisfy the Fokker-Planck equation. Under confinement, however, such statistics will follow the Smoluchowski equation, which lacks a closed-form analytical solution. This brief review explores the rich history of estimating profiles of the diffusivity tensor from MD simulations and discusses various approximate methods and algorithms developed for this purpose. Besides discussing heuristic extensions of bulk methods, we overview more rigorous algorithms, including kernel-based methods, Bayesian approaches, and operator discretization techniques. Additionally, we outline methods based on applying biasing potentials or imposing constraints on tracer particles. Finally, we discuss approaches that estimate diffusivity from mean first passage time or committor probability profiles, a conceptual framework originally developed in the context of collective variable spaces describing rare events in computational chemistry and biology. In summary, this paper offers a concise survey of diverse approaches for estimating diffusivity from MD trajectories, highlighting challenges and opportunities in this area.

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.

Accurate Memory Kernel Extraction from Discretized Time-Series Data

Memory effects emerge as a fundamental consequence of dimensionality reduction when low-dimensional observables are used to describe the dynamics of complex many-body systems. In the context of molecular dynamics (MD) data analysis, accounting for memory effects using the framework of the generalized Langevin equation (GLE) has proven efficient, accurate, and insightful, particularly when working with high-resolution time series data. However, in experimental systems, high-resolution data are often unavailable, raising questions about the impact of the data resolution on the estimated GLE parameters. This study demonstrates that direct memory extraction from time series data remains accurate when the discretization time is below the memory time. To obtain memory functions reliably, even when the discretization time exceeds the memory time, we introduce a Gaussian Process Optimization (GPO) scheme. This scheme minimizes the deviation of discretized two-point correlation functions between time series data and GLE simulations and is able to estimate accurate memory kernels as long as the discretization time stays below the longest time scale in the data, typically the barrier crossing time.

Quantifying the energy landscape in weakly and strongly disordered frictional media

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

Trajectory Statistical Learning of the Potential Mean of Force and Diffusion Coefficient from Molecular Dynamics Simulations

Central to studying the conformational changes of a complex protein is understanding the dynamics and energetics involved. Phenomenologically, structural dynamics can be formulated using an overdamped Langevin model along an observable, e.g., the distance between two residues in the protein. The Langevin model is specified by the deterministic force (the potential of mean force, PMF) and stochastic force (characterized by the diffusion coefficient, D). It is therefore of great interest to be able to extract both PMF and D from an observable time series but under the same computational framework. Here, we approach this challenge in molecular dynamics (MD) simulations by treating it as a missing-data Bayesian estimation problem. An important distinction in our methodology is that the entire MD trajectory, as opposed to the individual data elements, is used as the statistical variable in Bayesian imputation. This idea is implemented through an eigen-decomposition procedure for a time-symmetrized Fokker-Planck equation, followed by maximizing the likelihood for parameter estimation. The mathematical expressions for the functional derivatives used in learning PMF and D also provide new physical insights for the manner by which the information on both the deterministic and stochastic forces is encoded in the dynamics data. An all-atom MD simulation of a nontrivial biomolecule case is used to illustrate the application of this approach. We show that, interestingly, the results of trajectory statistical learning can motivate new order parameters for an improved description of the kinetic bottlenecks in conformational changes. Complementing purely data-driven or black-box methods, this work underscores the advantages of physics-based machine learning in gaining chemical insights from quantitative parameter estimation.

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 molecular diffusion in confinement I: Transport of small particles in potential and density gradients

HYPOTHESIS

Diffusion in confinement is an important fundamental problem with significant implications for applications of supported liquid phases. However, resolving the spatially dependent diffusion coefficient, parallel and perpendicular to interfaces, has been a standing issue. In the vicinity of interfaces, density fluctuations as a consequence of layering locally impose statistical drift, which impedes the analysis of spatially dependent diffusion coefficients even further. We hypothesise, that we can derive a model to spatially resolve interface-perpendicular diffusion coefficients based on local lifetime statistics with an extension to explicitly account for the effect of local drift using the Smoluchowski equation, that allows us to resolve anisotropic and spatially dependent diffusivity landscapes at interfaces.

METHODS AND SIMULATIONS

An analytic relation between local crossing times in system slices and diffusivity as well as an explicit term for calculating drift-induced systematic errors is presented. The method is validated on Molecular Dynamics simulations of bulk water and applied to simulations of water in slit pores.

FINDINGS

After validation on bulk liquids, we clearly demonstrate the anisotropic nature of diffusion coefficients at interfaces. Significant spatial variations in the diffusivities correlate with interface-induced structuring but cannot be solely attributed to the drift induced by local density fluctuations.

Construction of Coarse-Grained Molecular Dynamics with Many-Body Non-Markovian Memory

We introduce a machine-learning-based coarse-grained molecular dynamics model that faithfully retains the many-body nature of the intermolecular dissipative interactions. Unlike the common empirical coarse-grained models, the present model is constructed based on the Mori-Zwanzig formalism and naturally inherits the heterogeneous state-dependent memory term rather than matching the mean-field metrics such as the velocity autocorrelation function. Numerical results show that preserving the many-body nature of the memory term is crucial for predicting the collective transport and diffusion processes, where empirical forms generally show limitations.

Toward a Benchmark for Markov State Models: The Folding of HP35

Adopting a 300 μs long MD trajectory of the folding of villin headpiece (HP35) by D. E. Shaw Research, we recently constructed a Markov state model (MSM) based on inter-residue contacts. The model reproduces the folding time and predicts that the native basin and unfolded region consist of metastable substates that are structurally well-characterized. Recognizing the need to establish well-defined benchmark problems, we study to what extent and in what sense this MSM can be employed as a reference model. Hence, we test the robustness of the MSM by comparing it to models that use alternative combinations of features, dimensionality reduction methods, and clustering schemes. The study suggests some main characteristics of the folding of HP35 that should be reproduced by other competitive models. Moreover, the discussion reveals which parts of the MSM workflow matter most for the considered problem and illustrates the promises and pitfalls of state-based models for the interpretation of biomolecular simulations.

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.

Selecting Features for Markov Modeling: A Case Study on HP35

Markov state models represent a popular means to interpret molecular dynamics trajectories in terms of memoryless transitions between metastable conformational states. To provide a mechanistic understanding of the considered biomolecular process, these states should reflect structurally distinct conformations and ensure a time scale separation between fast intrastate and slow interstate dynamics. Adopting the folding of villin headpiece (HP35) as a well-established model problem, here we discuss the selection of suitable input coordinates or "features", such as backbone dihedral angles and interresidue distances. We show that dihedral angles account accurately for the structure of the native energy basin of HP35, while the unfolded region of the free energy landscape and the folding process are best described by tertiary contacts of the protein. To construct a contact-based model, we consider various ways to define and select contact distances and introduce a low-pass filtering of the feature trajectory as well as a correlation-based characterization of states. Relying on input data that faithfully account for the mechanistic origin of the studied process, the states of the resulting Markov model are clearly discriminated by the features, describe consistently the hierarchical structure of the free energy landscape, and─as a consequence─correctly reproduce the slow time scales of the process.

Graph identification of proteins in tomograms (GRIP-Tomo)

In this study, we present a method of pattern mining based on network theory that enables the identification of protein structures or complexes from synthetic volume densities, without the knowledge of predefined templates or human biases for refinement. We hypothesized that the topological connectivity of protein structures is invariant, and they are distinctive for the purpose of protein identification from distorted data presented in volume densities. Three-dimensional densities of a protein or a complex from simulated tomographic volumes were transformed into mathematical graphs as observables. We systematically introduced data distortion or defects such as missing fullness of data, the tumbling effect, and the missing wedge effect into the simulated volumes, and varied the distance cutoffs in pixels to capture the varying connectivity between the density cluster centroids in the presence of defects. A similarity score between the graphs from the simulated volumes and the graphs transformed from the physical protein structures in point data was calculated by comparing their network theory order parameters including node degrees, betweenness centrality, and graph densities. By capturing the essential topological features defining the heterogeneous morphologies of a network, we were able to accurately identify proteins and homo-multimeric complexes from 10 topologically distinctive samples without realistic noise added. Our approach empowers future developments of tomogram processing by providing pattern mining with interpretability, to enable the classification of single-domain protein native topologies as well as distinct single-domain proteins from multimeric complexes within noisy volumes.

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

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

Assessing transition rates as functions of environmental variables

We present a method to estimate the transition rates of molecular systems under different environmental conditions that cause the formation or the breaking of bonds and require the sampling of the Grand Canonical Ensemble. For this purpose, we model the molecular system in terms of probable "scenarios," governed by different potential energy functions, which are separately sampled by classical MD simulations. Reweighting the canonical distribution of each scenario according to specific environmental variables, we estimate the grand canonical distribution, then use the Square Root Approximation method to discretize the Fokker-Planck operator into a rate matrix and the robust Perron Cluster Cluster Analysis method to coarse-grain the kinetic model. This permits efficiently estimating the transition rates of conformational states as functions of environmental variables, for example, the local pH at a cell membrane. In this work, we formalize the theoretical framework of the procedure, and we present a numerical experiment comparing the results with those provided by a constant-pH method based on non-equilibrium Molecular Dynamics Monte Carlo simulations. The method is relevant for the development of new drug design strategies that take into account how the cellular environment influences biochemical processes.

Computing committors via Mahalanobis diffusion maps with enhanced sampling data

The study of phenomena such as protein folding and conformational changes in molecules is a central theme in chemical physics. Molecular dynamics (MD) simulation is the primary tool for the study of transition processes in biomolecules, but it is hampered by a huge timescale gap between the processes of interest and atomic vibrations that dictate the time step size. Therefore, it is imperative to combine MD simulations with other techniques in order to quantify the transition processes taking place on large timescales. In this work, the diffusion map with Mahalanobis kernel, a meshless approach for approximating the Backward Kolmogorov Operator (BKO) in collective variables, is upgraded to incorporate standard enhanced sampling techniques, such as metadynamics. The resulting algorithm, which we call the target measure Mahalanobis diffusion map (tm-mmap), is suitable for a moderate number of collective variables in which one can approximate the diffusion tensor and free energy. Imposing appropriate boundary conditions allows use of the approximated BKO to solve for the committor function and utilization of transition path theory to find the reactive current delineating the transition channels and the transition rate. The proposed algorithm, tm-mmap, is tested on the two-dimensional Moro-Cardin two-well system with position-dependent diffusion coefficient and on alanine dipeptide in two collective variables where the committor, the reactive current, and the transition rate are compared to those computed by the finite element method (FEM). Finally, tm-mmap is applied to alanine dipeptide in four collective variables where the use of finite elements is infeasible.

Identifying Strategies to Experimentally Probe Multidimensional Dynamics in the Ribosome

The ribosome is a complex biomolecular machine that utilizes large-scale conformational rearrangements to synthesize proteins. For example, during the elongation cycle, the "head" domain of the ribosomal small subunit (SSU) is known to undergo transient rotation events that allow for movement of tRNA molecules (i.e., translocation). While the head may exhibit rigid-body-like properties, the precise relationship between experimentally accessible probes and multidimensional rotations has yet to be established. To address this gap, we perform molecular dynamics simulations of the translocation step of the elongation cycle in the ribosome, where the SSU head spontaneously undergoes rotation and tilt-like motions. With this data set (1250 simulated events), we used statistical and information-theory-based measures to identify possible single-molecule probes that can isolate SSU head rotation and head tilting. This analysis provides a molecular interpretation for previous single-molecule measurements, while establishing a framework for the design of next-generation experiments that may precisely probe the mechanistic and kinetic aspects of the ribosome.

Statistical physics of inhomogeneous transport: Unification of diffusion laws and inference from first-passage statistics

Characterization of composite materials, whose properties vary in space over microscopic scales, has become a problem of broad interdisciplinary interest. In particular, estimation of the inhomogeneous transport coefficients, e.g., the diffusion coefficient or the heat conductivity, which shape important processes in biology and engineering, is a challenging task. The analysis of such systems is further complicated because two alternative formulations of the inhomogeneous transport equations exist in the literature-the Smoluchowski and Fokker-Planck equations, which are also related to the so-called Ito-Stratonovich dilemma. Using the theory of statistical physics, we show that the two formulations, usually regarded as distinct models, are physically equivalent. From this result we develop efficient estimates for the transverse space-dependent diffusion coefficient in fluids near a phase boundary. Our method requires only measurements of escape probabilities and mean exit times of molecules leaving a narrow spatial region. We test our estimates in three case studies: (i) a Langevin model of a Büttikker-Landauer ratchet; atomistic molecular-dynamics simulations of liquid-water molecules in contact with (ii) vapor, and (iii) soap (surfactant) film which has promising applications in physical chemistry. Our analysis reveals that near the surfactant monolayer the mobility of water molecules is slowed down almost twice with respect to the bulk liquid. Moreover, the diffusion coefficient of water correlates with the transition from hydrophilic to hydrophobic parts of the film.

Cooperative Protein Allosteric Transition Mediated by a Fluctuating Transmission Network

Allosteric communication between distant protein sites represents a key mechanism of biomolecular regulation and signal transduction. Compared to other processes such as protein folding, however, the dynamical evolution of allosteric transitions is still not well understood. As an example of allosteric coupling between distant protein regions, we consider the global open-closed motion of the two domains of T4 lysozyme, which is triggered by local motion in the hinge region. Combining extensive molecular dynamics simulations with a correlation analysis of interresidue contacts, we identify a network of interresidue distances that move in a concerted manner. The cooperative process originates from a cogwheel-like motion of the hydrophobic core in the hinge region, which constitutes an evolutionary conserved and flexible transmission network. Through rigid contacts and the protein backbone, the small local changes of the hydrophobic core are passed on to the distant terminal domains and lead to the emergence of a rare global conformational transition. As in an Ising-type model, the cooperativity of the allosteric transition can be explained via the interaction of local fluctuations.

De novo protein folding on computers. Benefits and challenges

There has been recent success in prediction of the three-dimensional folded native structures of proteins, most famously by the AlphaFold Algorithm running on Google's/Alphabet's DeepMind computer. However, this largely involves machine learning of protein structures and is not a de novo protein structure prediction method for predicting three-dimensional structures from amino acid residue sequences. A de novo approach would be based almost entirely on general principles of energy and entropy that govern protein folding energetics, and importantly do so without the use of the amino acid sequences and structural features of other proteins. Most consider that problem as still unsolved even though it has occupied leading scientists for decades. Many consider that it remains one of the major outstanding issues in modern science. There is crucial continuing help from experimental findings on protein unfolding and refolding in the laboratory, but only to a limited extent because many researchers consider that the speed by which real proteins folds themselves, often from milliseconds to minutes, is itself still not fully understood. This is unfortunate, because a practical solution to the problem would probably have a major effect on personalized medicine, the pharmaceutical industry, biotechnology, and nanotechnology, including for example "smaller" tasks such as better modeling of flexible "unfolded" regions of the SARS-COV-2 spike glycoprotein when interacting with its cell receptor, antibodies, and therapeutic agents. Some important ideas from earlier studies are given before moving on to lessons from periodic and aperiodic crystals, and a possible role for quantum phenomena. The conclusion is that better computation of entropy should be the priority, though that is presented guardedly.

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

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

Observing the base-by-base search for native structure along transition paths during the folding of single nucleic acid hairpins

Biomolecular folding involves searching among myriad possibilities for the native conformation, but the elementary steps expected from theory for this search have never been detected directly. We probed the dynamics of folding at high resolution using optical tweezers, measuring individual trajectories as nucleic acid hairpins passed through the high-energy transition states that dominate kinetics and define folding mechanisms. We observed brief but ubiquitous pauses in the transition states, with a dwell time distribution that matched microscopic theories of folding quantitatively. The sequence dependence suggested that pauses were dominated by microbarriers from nonnative conformations during the search by each nucleotide residue for the native base-pairing conformation. Furthermore, the pauses were position dependent, revealing subtle local variations in energy-landscape roughness and allowing the diffusion coefficient describing the microscopic dynamics within the barrier to be found without reconstructing the shape of the energy landscape. These results show how high-resolution measurements can elucidate key microscopic events during folding to test fundamental theories of folding.

Protein Folding Dynamics as Diffusion on a Free Energy Surface: Rate Equation Terms, Transition Paths, and Analysis of Single-Molecule Photon Trajectories

The rates of protein (un)folding are often described as diffusion on the projection of a hyperdimensional energy landscape onto a few (ideally one) order parameters. Testing such an approximation by experiment requires resolving the reactive transition paths of individual molecules, which is now becoming feasible with advanced single-molecule spectroscopic techniques. This has also sparked the interest of theorists in better understanding reactive transition paths. Here we focus on these issues aiming to establish (i) practical guidelines for the mechanistic interpretation of transition path times (TPT) and (ii) methods to extract the free energy surface and protein dynamics from the maximum likelihood analysis of photon trajectories (MLA-PT). We represent the (un)folding rates as diffusion on a 1D free energy surface with the FRET efficiency as a reaction coordinate proxy. We then perform diffusive kinetic simulations on surfaces with two minima and a barrier, but with different shapes (curvatures, barrier height, and symmetry), coupled to stochastic simulations of photon emissions that reproduce current SM-FRET experiments. From the analysis of transition paths, we find that the TPT is inversely proportional to the barrier height (difference in free energy between minimum and barrier top) for any given surface shape, and that dividing the TPT into climb and descent segments provides key information about the barrier's symmetry. We also find that the original MLA-PT procedure used to determine the TPT from experiments underestimates its value, particularly for the cases with smaller barriers (e.g., fast folders), and we suggest a simple strategy to correct for this bias. Importantly, we also demonstrate that photon trajectories contain enough information to extract the 1D free energy surface's shape and dynamics (if TPT is >4-5-fold longer than the interphoton time) using the MLA-PT directly implemented with a diffusive free energy surface model. When dealing with real (unknown) experimental data, the comparison between the likelihoods of the free energy surface and discrete kinetic three-state models can be used to evaluate the statistical significance of the estimated free energy surface.

Probing the Internal Dynamics and Shape of Simple Peptides in Urea, Guanidinium Hydrochloride, and Proline Solutions with Time-Resolved Fluorescence Anisotropy and Atomistic Cosolvent Simulations

Picosecond time-resolved fluorescence anisotropy was used to measure the effect of denaturants and osmolytes on the reorientation dynamics of the simplest dipeptide. The solvent denaturants guanidinium hydrochloride (gdm), urea, and the osmolyte proline were used at several concentrations. Analysis of the concentration dependence of denaturants at a fixed temperature showed faster and slower reorientation time in two different denaturants at a nearly identical solvent viscosity (η). The reorientation time τ significantly deviates from Kramers' theory (τ ∝ η¹) in the high friction limit for guanidinium and urea with r ≈ 0.4 and r ≈ 0.6 at pH 7.2, respectively. In proline, τ is nearly proportional to η. Atomistic molecular dynamics simulations of the dipeptide in identical cosolvents showed excellent agreement with the measured rotational orientation time. The dipeptide dihedral (ϕ, ψ) isomerization times in water and 6 M urea are almost identical and significantly slower in guanidinium. If a faster and slower reorientation time can be associated with the compact and expanded shapes, the fractional viscosity dependence for guanidinium and urea may result from the fact that internal dynamics of peptides in these cosolvents involve higher and lower internal friction within the dynamic elements.

Out-of-Equilibrium Biophysical Chemistry: The Case for Multidimensional, Integrated Single-Molecule Approaches

Out-of-equilibrium processes are ubiquitous across living organisms and all structural hierarchies of life. At the molecular scale, out-of-equilibrium processes (for example, enzyme catalysis, gene regulation, and motor protein functions) cause biological macromolecules to sample an ensemble of conformations over a wide range of time scales. Quantifying and conceptualizing the structure–dynamics to function relationship is challenging because continuously evolving multidimensional energy landscapes are necessary to describe nonequilibrium biological processes in biological macromolecules. In this perspective, we explore the challenges associated with state-of-the-art experimental techniques to understanding biological macromolecular function. We argue that it is time to revisit how we probe and model functional out-of-equilibrium biomolecular dynamics. We suggest that developing integrated single-molecule multiparametric force–fluorescence instruments and using advanced molecular dynamics simulations to study out-of-equilibrium biomolecules will provide a path towards understanding the principles of and mechanisms behind the structure–dynamics to function paradigm in biological macromolecules.

First-passage times in protein folding: exploring the native-like states vs. overcoming the free energy barrier

Using a model β-hairpin protein as a representative example of simple two-state folders whose kinetics are uncomplicated by the presence of on- and off-pathway intermediates, it is studied how the search for the protein's native state among native-like states affects the folding kinetics. It is revealed that the first-passage time (FPT) distributions are essentially single-exponential not only for the times to overcome the free energy barrier between the unfolded and native-like states but also for the times to find the native state among the native-like ones. The FPT distributions of this type are observed through all studied two-state-like regimes of protein folding, varying from a regime close to two-state folding to a regime close to downhill folding. If the protein explores native-like states for a time much longer than the time to overcome the free energy barrier, which is characteristic of high temperatures, the resulting FPT distribution to reach the native state remains close to exponential but the mean FPT (MFPT) is determined not by the height of the free energy barrier but by the time to explore native-like states. In particular, the mean time to overcome the free energy barrier is in reasonable agreement with the Kramers rate formula and generally far shorter than the overall MFPT to reach the native state. The observed increase of the overall MFPT, as a result of longer exploration of native-like states, may lead to an overestimate of the height of the free energy barrier between the unfolded and folded states when it is calculated from the overall MFPT.

Non-Markovian modeling of protein folding

Protein-folding kinetics is often described as Markovian (i.e., memoryless) diffusion in a one-dimensional free energy landscape, governed by an instantaneous friction coefficient that is fitted to reproduce experimental or simulated folding times. For the α-helix forming polypeptide alanine9 and a specific reaction coordinate that consists of the summed native hydrogen-bond lengths, we demonstrate that the friction extracted from molecular dynamics simulations exhibits significant memory with a decay time that is in the nanosecond range and thus, of the same order as the folding and unfolding times. Our non-Markovian modeling not only reproduces the molecular dynamics simulations accurately but also demonstrates that memory friction effects lead to anomalous and drastically accelerated protein kinetics.

We extract the folding free energy landscape and the time-dependent friction function, the two ingredients of the generalized Langevin equation (GLE), from explicit-water molecular dynamics (MD) simulations of the α-helix forming polypeptide alanine9 for a one-dimensional reaction coordinate based on the sum of the native H-bond distances. Folding and unfolding times from numerical integration of the GLE agree accurately with MD results, which demonstrate the robustness of our GLE-based non-Markovian model. In contrast, Markovian models do not accurately describe the peptide kinetics and in particular, cannot reproduce the folding and unfolding kinetics simultaneously, even if a spatially dependent friction profile is used. Analysis of the GLE demonstrates that memory effects in the friction significantly speed up peptide folding and unfolding kinetics, as predicted by the Grote–Hynes theory, and are the cause of anomalous diffusion in configuration space. Our methods are applicable to any reaction coordinate and in principle, also to experimental trajectories from single-molecule experiments. Our results demonstrate that a consistent description of protein-folding dynamics must account for memory friction effects.

SlyD Accelerates trans-to-cis Prolyl Isomerization in a Mechanosignaling Protein under Load

Prolyl isomerization is recognized as one of the key regulatory mechanisms, which plays a crucial role in cell signaling, ion channel gating, phage virus infection, and molecular timing. This isomerization is usually slow but often accelerated by an enzyme, called peptidyl-prolyl isomerase (PPIase). In the current project, we investigate using single-molecule force spectroscopy (SMFS) the impact of a bacterial PPIase, SlyD, on the cis-trans isomerization of the proline 2225 (P2225) in an isolated 20th domain of a cytoskeletal mechanosensing protein filamin-A (FlnA20). To explore the FlnA20-PPIase interaction, we have used multiple SMFS modes, like constant velocity, constant distance, and jumping trap experiments. In our previous study, we reported the unique nature of the P2225, which is conserved in all naturally occurring filamins and can slowly (minutes) interconvert between cis-trans isomers, in absence of any PPIase. Our current results show a staggering 25-fold acceleration of the trans-to-cis isomerization rate in the presence of saturating SlyD concentration (7.25 μM) compared to the unenzymatic condition. A SlyD concentration-dependent depletion of the trans isomeric lifetime was also observed. Additionally, we observed that SlyD stabilizes the cis-isomer in the native state of FlnA20 by ∼2 k_BT. This is the first single-molecule observation of the cis-trans isomerization catalysis by a PPIase in a mechanosensing protein.

Estimating transition path times and shapes from single-molecule photon trajectories: A simulation analysis

In a two-state molecular system, transition paths comprise the portions of trajectories during which the system transits from one stable state to the other. Because of their low population, it is essentially impossible to obtain information on transition paths from experiments on a large sample of molecules. However, single-molecule experiments such as laser optical tweezers or Förster resonance energy transfer (FRET) spectroscopy have allowed transition-path durations to be estimated. Here, we use molecular simulations to test the methodology for obtaining information on transition paths in single-molecule FRET by generating photon trajectories from the distance trajectories obtained in the simulation. Encouragingly, we find that this maximum likelihood analysis yields transition-path times within a factor of 2-4 of the values estimated using a good coordinate for folding, but tends to systematically underestimate them. The underestimation can be attributed partly to the fact that the large changes in the end-end distance occur mostly early in a folding trajectory. However, even if the transfer efficiency is a good reaction coordinate for folding, the assumption that the transition-path shape is a step function still leads to an underestimation of the transition-path time as defined here. We find that allowing more flexibility in the form of the transition path model allows more accurate transition-path times to be extracted and points the way toward further improvements in methods for estimating transition-path time and transition-path shape.

First-passage fingerprints of water diffusion near glutamine surfaces

The extent to which biological interfaces affect the dynamics of water plays a key role in the exchange of matter and chemical interactions that are essential for life. The density and the mobility of water molecules depend on their proximity to biological interfaces and can play an important role in processes such as protein folding and aggregation. In this work, we study the dynamics of water near glutamine surfaces-a system of interest in studies of neurodegenerative diseases. Combining molecular-dynamics simulations and stochastic modelling, we study how the mean first-passage time and related statistics of water molecules escaping subnanometer-sized regions vary from the interface to the bulk. Our analysis reveals a dynamical complexity that reflects underlying chemical and geometrical properties of the glutamine surfaces. From the first-passage time statistics of water molecules, we infer their space-dependent diffusion coefficient in directions normal to the surfaces. Interestingly, our results suggest that the mobility of water varies over a longer length scale than the chemical potential associated with the water-protein interactions. The synergy of molecular dynamics and first-passage techniques opens the possibility for extracting space-dependent diffusion coefficients in more complex, inhomogeneous environments that are commonplace in living matter.

From Nonequilibrium Single-Molecule Trajectories to Underlying Dynamics

Single-molecule observations of biomolecular dynamics and folding are commonly rationalized using the model of diffusive dynamics on a free-energy landscape, which is inferred via the Boltzmann inversion of the equilibrium distribution of the experimental observable. Can the same model be applied to high-resolution single-molecule trajectories of molecular machines that lack thermal equilibrium so that the Boltzmann inversion method is inapplicable? In this Letter, we discuss two approaches to reconstructing the underlying free-energy landscape in such nonequilibrium systems and explore the performance of this model in application to trajectories with complex underlying dynamics.

Statistical Tests for Force Inference in Heterogeneous Environments

We devise a method to detect and estimate forces in a heterogeneous environment based on experimentally recorded stochastic trajectories. In particular, we focus on systems modeled by the heterogeneous overdamped Langevin equation. Here, the observed drift includes a "spurious” force term when the diffusivity varies in space. We show how Bayesian inference can be leveraged to reliably infer forces by taking into account such spurious forces of unknown amplitude as well as experimental sources of error. The method is based on marginalizing the force posterior over all possible spurious force contributions. The approach is combined with a Bayes factor statistical test for the presence of forces. The performance of our method is investigated analytically, numerically and tested on experimental data sets. The main results are obtained in a closed form allowing for direct exploration of their properties and fast computation. The method is incorporated into TRamWAy, an open-source software platform for automated analysis of biomolecule trajectories.

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.

Effects of Topology and Sequence in Protein Folding Linked via Conformational Fluctuations

Experiments have compared the folding of proteins with different amino acid sequences but the same basic structure, or fold. Results indicate that folding is robust to sequence variations for proteins with some nonlocal folds, such as all-β, whereas the folding of more local, all-α proteins typically exhibits a stronger sequence dependence. Here, we use a coarse-grained model to systematically study how variations in sequence perturb the folding energy landscapes of three model sequences with 3α, 4β + α, and β-barrel folds, respectively. These three proteins exhibit folding features in line with experiments, including expected rank order in the cooperativity of the folding transition and stability-dependent shifts in the location of the free-energy barrier to folding. Using a generalized-ensemble simulation approach, we determine the thermodynamics of around 2000 sequence variants representing all possible hydrophobic or polar single- and double-point mutations. From an analysis of the subset of stability-neutral mutations, we find that folding is perturbed in a topology-dependent manner, with the β-barrel protein being the most robust. Our analysis shows, in particular, that the magnitude of mutational perturbations of the transition state is controlled in part by the size or "width" of the underlying conformational ensemble. This result suggests that the mutational robustness of the folding of the β-barrel protein is underpinned by its conformationally restricted transition state ensemble, revealing a link between sequence and topological effects in protein folding.

Competitive binding of HIF-1α and CITED2 to the TAZ1 domain of CBP from molecular simulations

Many intrinsically disordered proteins (IDPs) are involved in complex signalling networks inside the cell.

Understanding biochemical processes in the presence of sub-diffusive behavior of biomolecules in solution and living cells

In order to maintain cellular function, biomolecules like protein, DNA, and RNAs have to diffuse to the target spaces within the cell. Changes in the cytosolic microenvironment or in the nucleus during the fulfillment of these cellular processes affect their mobility, folding, and stability thereby impacting the transient or stable interactions with their adjacent neighbors in the organized and dynamic cellular interior. Using classical Brownian motion to elucidate the diffusion behavior of these biomolecules is hard considering their complex nature. The understanding of biomolecular diffusion inside cells still remains elusive due to the lack of a proper model that can be extrapolated to these cases. In this review, we have comprehensively addressed the progresses in this field, laying emphasis on the different aspects of anomalous diffusion in the different biochemical reactions in cell interior. These experiment-based models help to explain the diffusion behavior of biomolecules in the cytosolic and nuclear microenvironment. Moreover, since understanding of biochemical reactions within living cellular system is our main focus, we coupled the experimental observations with the concept of sub-diffusion from in vitro to in vivo condition. We believe that the pairing between the understanding of complex behavior and structure-function paradigm of biological molecules would take us forward by one step in order to solve the puzzle around diseases caused by cellular dysfunction.

Probing microscopic conformational dynamics in folding reactions by measuring transition paths

Transition paths comprise those parts of a folding trajectory where the molecule passes through the high-energy transition states separating folded and unfolded conformations. The transition states determine the folding kinetics and mechanism but are difficult to observe because of their brief duration. Single-molecule experiments have in recent years begun to characterize transition paths in folding reactions, allowing the microscopic conformational dynamics that occur as a molecule traverses the energy barriers to be probed directly. Here we review single-molecule fluorescence and force spectroscopy measurements of transition-path properties, including the time taken to traverse the paths, the local velocity along them, the path shapes, and the variability within these measurements reflecting differences between individual barrier crossings. We discuss how these measurements have been related to theories of folding as diffusion over an energy landscape to deduce properties such as the diffusion coefficient, and how they are being combined with simulations to obtain enhanced atomistic understanding of folding. The richly detailed information available from transition path measurements holds great promise for improved understanding of microscopic mechanisms in folding.

Recent Advances in Coarse-Grained Models for Biomolecules and Their Applications

Molecular dynamics simulations have emerged as a powerful tool to study biological systems at varied length and timescales. The conventional all-atom molecular dynamics simulations are being used by the wider scientific community in routine to capture the conformational dynamics and local motions. In addition, recent developments in coarse-grained models have opened the way to study the macromolecular complexes for time scales up to milliseconds. In this review, we have discussed the principle, applicability and recent development in coarse-grained models for biological systems. The potential of coarse-grained simulation has been reviewed through state-of-the-art examples of protein folding and structure prediction, self-assembly of complexes, membrane systems and carbohydrates fiber models. The multiscale simulation approaches have also been discussed in the context of their emerging role in unravelling hierarchical level information of biosystems. We conclude this review with the future scope of coarse-grained simulations as a constantly evolving tool to capture the dynamics of biosystems.

Modeling of Drug Diffusion Based on Concentration Profiles in Healthy and Damaged Human Skin

Based on experimental drug concentration profiles in healthy as well as tape-stripped ex vivo human skin, we model the penetration of the antiinflammatory drug dexamethasone into the skin layers by the one-dimensional generalized diffusion equation. We estimate the position-dependent free-energy and diffusivity profiles by solving the conjugated minimization problem, in which the only inputs are concentration profiles of dexamethasone in skin at three consecutive penetration times. The resulting free-energy profiles for damaged and healthy skin show only minor differences. In contrast, the drug diffusivity in the first 10 μm of the upper skin layer of damaged skin is 200-fold increased compared to healthy skin, which reflects the corrupted barrier function of tape-stripped skin. For the case of healthy skin, we examine the robustness of our method by analyzing the behavior of the extracted skin parameters when the number of input and output parameters are reduced. We also discuss techniques for the regularization of our parameter extraction method.

Unfolded states under folding conditions accommodate sequence-specific conformational preferences with random coil-like dimensions

Proteins are marginally stable molecules that fluctuate between folded and unfolded states. Here, we provide a high-resolution description of unfolded states under refolding conditions for the N-terminal domain of the L9 protein (NTL9). We use a combination of time-resolved Förster resonance energy transfer (FRET) based on multiple pairs of minimally perturbing labels, time-resolved small-angle X-ray scattering (SAXS), all-atom simulations, and polymer theory. Upon dilution from high denaturant, the unfolded state undergoes rapid contraction. Although this contraction occurs before the folding transition, the unfolded state remains considerably more expanded than the folded state and accommodates a range of local and nonlocal contacts, including secondary structures and native and nonnative interactions. Paradoxically, despite discernible sequence-specific conformational preferences, the ensemble-averaged properties of unfolded states are consistent with those of canonical random coils, namely polymers in indifferent (theta) solvents. These findings are concordant with theoretical predictions based on coarse-grained models and inferences drawn from single-molecule experiments regarding the sequence-specific scaling behavior of unfolded proteins under folding conditions.

Ultrafast folding kinetics of WW domains reveal how the amino acid sequence determines the speed limit to protein folding

Protein (un)folding rates depend on the free-energy barrier separating the native and unfolded states and a prefactor term, which sets the timescale for crossing such barrier or folding speed limit. Because extricating these two factors is usually unfeasible, it has been common to assume a constant prefactor and assign all rate variability to the barrier. However, theory and simulations postulate a protein-specific prefactor that contains key mechanistic information. Here, we exploit the special properties of fast-folding proteins to experimentally resolve the folding rate prefactor and investigate how much it varies among structural homologs. We measure the ultrafast (un)folding kinetics of five natural WW domains using nanosecond laser-induced temperature jumps. All five WW domains fold in microseconds, but with a 10-fold difference between fastest and slowest. Interestingly, they all produce biphasic kinetics in which the slower phase corresponds to reequilibration over the small barrier (<3 RT) and the faster phase to the downhill relaxation of the minor population residing at the barrier top [transition state ensemble (TSE)]. The fast rate recapitulates the 10-fold range, demonstrating that the folding speed limit of even the simplest all-β fold strongly depends on the amino acid sequence. Given this fold's simplicity, the most plausible source for such prefactor differences is the presence of nonnative interactions that stabilize the TSE but need to break up before folding resumes. Our results confirm long-standing theoretical predictions and bring into focus the rate prefactor as an essential element for understanding the mechanisms of folding.

Measuring the average shape of transition paths during the folding of a single biological molecule

Transition paths represent the parts of a reaction where the energy barrier separating products and reactants is crossed. They are essential to understanding reaction mechanisms, yet many of their properties remain unstudied. Here, we report measurements of the average shape of transition paths, studying the folding of DNA hairpins as a model system for folding reactions. Individual transition paths were detected in the folding trajectories of hairpins with different sequences held under tension in optical tweezers, and path shapes were computed by averaging all transitions in the time domain, 〈t(x)〉, or by averaging transitions of a given duration in the extension domain, 〈x(t|τ)〉 _τ Whereas 〈t(x)〉 was close to straight, with only a subtle curvature, 〈x(t|τ)〉 _τ had more pronounced curvature that fit well to theoretical expectations for the dominant transition path, returning diffusion coefficients similar to values obtained previously from independent methods. Simulations suggested that 〈t(x)〉 provided a less reliable representation of the path shape than 〈x(t|τ)〉 _τ , because it was far more sensitive to the effects of coupling the molecule to the experimental force probe. Intriguingly, the path shape variance was larger for some hairpins than others, indicating sequence-dependent changes in the diversity of transition paths reflective of differences in the character of the energy barriers, such as the width of the barrier saddle-point or the presence of parallel paths through multiple barriers between the folded and unfolded states. These studies of average path shapes point the way forward for probing the rich information contained in path shape fluctuations.