Finite Temperature String Method Research Articles

Finite Temperature String with Order Parameter as Collective Variables for Molecular Crystal: A Case of Polymorphic Transformation of TNT under External Electric Field.

An external electric field is an effective tool to induce the polymorphic transformation of molecular crystals, which is important practically in the chemical, material, and energy storage industries. However, the understanding of this mechanism is poor at the molecular level. In this work, two types of order parameters (OPs) were constructed for the molecular crystal based on the intermolecular distance, bond orientation, and molecular orientation. Using the K-means clustering algorithm for the sampling of OPs based on the Euclidean distance and density weight, the polymorphic transformation of TNT was investigated using a finite temperature string (FTS) under external electric fields. The potential of mean force (PMF) was obtained, and the essence of the polymorphic transformation between o-TNT and m-TNT was revealed, which verified the effectiveness of the FTS method based on K-means clustering to OPs. The differences in PMFs between the o-TNT and transition state were decreased under external electric fields in comparison with those in no field. The fields parallel to the c-axis obviously affected the difference in PMF, and the relationship between the changes in PMFs and field strengths was found. Although the external electric field did not promote the convergence, the time of the polymorphic transformation was reduced under the external electric field in comparison to its absence. Moreover, under the external electric field, the polymorphic transformation from o-TNT to m-TNT occurred while that from m-TNT to o-TNT was prevented, which was explained by the dipole moment of molecule, relative permittivity, chemical potential difference, nucleation work and nucleation rate. This confirmed that the polymorphic transformation orientation of the molecular crystal could be controlled by the external electric field. This work provides an effective way to explore the polymorphic transformation of the molecular crystals at a molecular level, and it is useful to control the production process and improve the performance of energetic materials by using the external electric fields.

Transition Path Theory for Langevin Dynamics on Manifolds: Optimal Control and Data-Driven Solver

We present a data-driven point of view for rare events, which represent conformational transitions in biochemical reactions modeled by overdamped Langevin dynamics on manifolds in high dimensions. We first reinterpret the transition state theory and the transition path theory from the optimal control viewpoint. Given a point cloud probing the manifold, we construct a discrete Markov chain with a -matrix computed from an approximated Voronoi tesselation via the point cloud. We use this -matrix to compute a discrete committor function whose level set automatically orders the point cloud. Then based on the committor function, an optimally controlled random walk on point clouds is constructed and utilized to efficiently sample transition paths, which become an almost sure event in time instead of a rare event in the original reaction dynamics. To compute the mean transition path efficiently, a local averaging algorithm based on the optimally controlled random walk is developed, which adapts the finite temperature string method to the controlled Monte Carlo samples. Numerical examples on sphere/torus including a conformational transition for the alanine dipeptide in vacuum are conducted to illustrate the data-driven solver for the transition path theory on point clouds. The mean transition path obtained via the controlled Monte Carlo simulations highly coincides with the computed dominant transition path in the transition path theory.

Supervised learning and the finite-temperature string method for computing committor functions and reaction rates.

A central object in the computational studies of rare events is the committor function. Though costly to compute, the committor function encodes complete mechanistic information of the processes involving rare events, including reaction rates and transition-state ensembles. Under the framework of transition path theory, Rotskoff et al. [Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference, Proceedings of Machine Learning Research (PLMR, 2022), Vol. 145, pp. 757-780] proposes an algorithm where a feedback loop couples a neural network that models the committor function with importance sampling, mainly umbrella sampling, which collects data needed for adaptive training. In this work, we show additional modifications are needed to improve the accuracy of the algorithm. The first modification adds elements of supervised learning, which allows the neural network to improve its prediction by fitting to sample-mean estimates of committor values obtained from short molecular dynamics trajectories. The second modification replaces the committor-based umbrella sampling with the finite-temperature string (FTS) method, which enables homogeneous sampling in regions where transition pathways are located. We test our modifications on low-dimensional systems with non-convex potential energy where reference solutions can be found via analytical or finite element methods, and show how combining supervised learning and the FTS method yields accurate computation of committor functions and reaction rates. We also provide an error analysis for algorithms that use the FTS method, using which reaction rates can be accurately estimated during training with a small number of samples. The methods are then applied to a molecular system in which no reference solution is known, where accurate computations of committor functions and reaction rates can still be obtained.

Finite temperature string method with umbrella sampling using path collective variables: application to secondary structure change in a protein.

The transition of an α-helix to a β-sheet in proteins is among the most complex conformational changes seen in biomolecular systems. Due to long time scales involved in the transition, it is challenging to study such protein conformational changes using direct molecular dynamics simulations. This limitation is typically overcome using an indirect approach wherein one computes the free energy landscape associated with the transition. Computation of free energy landscapes, however, requires a suitable set of collective variables that describe the transition. In this work, we demonstrate the use of path collective variables [D. Branduardi, F. L. Gervasio and M. Parrinello, J. Chem. Phys., 2007, 126, 054103] and combine it with the finite temperature string (FTS) method [E. Weinan, W. Ren and E. Vanden-Eijnden, J. Phys. Chem. B, 2005, 109, 6688-6693] to determine the molecular mechanisms involved during the structural transition of the mini G-protein from an α-helix to a β-hairpin. The transition from the α-helix proceeds via unfolding of the terminal residues, giving rise to a β-turn unfolded intermediate to eventually form the β-hairpin. Our proposed algorithm uses umbrella sampling simulations to simulate images along the string and the weighted histogram analysis to compute the free energy along the computed transition path. This work demonstrates that the string method in combination with path collective variables can be exploited to study complex protein conformational changes such as a complete change in the secondary structure.

Prediction of multiple dry-wet transition pathways with a mesoscale variational approach.

Water fluctuates in a hydrophobic confinement, forming multiple dry and wet hydration states through evaporation and condensation. Transitions between such states are critical to both thermodynamics and kinetics of solute molecular processes, such as protein folding and protein-ligand binding and unbinding. To efficiently predict such dry-wet transition paths, we develop a hybrid approach that combines a variational implicit solvation model, a generalized string method for minimum free-energy paths, and the level-set numerical implementation. This approach is applied to three molecular systems: two hydrophobic plates, a carbon nanotube, and a synthetic host molecule Cucurbit[7]uril. Without an explicit description of individual water molecules, our mesoscale approach effectively captures multiple dry and wet hydration states, multiple dry-wet transition paths, such as those geometrically symmetric and asymmetric paths, and transition states, providing activation energy barriers between different states. Further analysis shows that energy barriers depend on mesoscopic lengths, such as the separation distance between the two plates and the cross section diameter of the nanotube, and that the electrostatic interactions strongly influence the dry-wet transitions. With the inclusion of solute atomic motion, general collective variables as reaction coordinates, and the finite-temperature string method, together with an improved treatment of continuum electrostatics, our approach can be further developed to sample an ensemble of transition paths, providing more accurate predictions of the transition kinetics.

The Role of Conserved Residues in the DEDDh Motif: the Proton-Transfer Mechanism of HIV-1 RNase H

RNase H is a prototypical example for two-metal-ion catalysis in enzymes. An RNase H activity cleaving the ribonucleic acid (RNA) backbone of a DNA/RNA hybrid is present not only in important drug targets, such as the HIV-1 reverse transcriptase, but also in many other nucleases, such as Homo sapiens (Hs) and Escherichia coli (Ec) RNase H or, notably, in enzymes that are part of the CRISPR gene editing molecular machinery. Despite its importance, the reaction mechanism uncovering the proton-transfer events is not yet understood. In particular, it is not known, which group is the proton donor for the leaving group. Moreover, several different proton acceptors were proposed, and the exact identity of the proton acceptor is also elusive. Here, we revisit the mechanism for RNAse H, whereby we find that the highly conserved Glu residue of the DDE motif acts as a proton donor via a mechanism further stabilized by the 2′O atom of the sugar. Additionally, we also describe an alternative proton-transfer mechanism via a conserved catalytic His residue to deprotonate the attacking water molecule. Furthermore, our quantum mechanics/molecular mechanics (QM/MM) calculations combining Hamiltonian replica exchange with a finite-temperature string method provide an accurate free-energy profile for the reaction catalyzed by the HIV-1 RNase H. Our reported pathway is consistent with kinetic data obtained for mutant HIV-1, Hs, and Ec RNase H, with the calculated pKa values of the DEDD residues and with crystallographic studies. The overall reaction barrier of ∼19 kcal mol–1, encountered in the phosphate-cleavage step, matches the slow experimental rate of ∼1–100 min–1. Additionally, using molecular dynamics (MD) calculations, we sample the recently identified binding site for a third transient divalent metal ion in the vicinity of the scissile phosphate in the product complex. Our results account for the experimental observation of a third metal ion facilitating product release in an Aquifex aeolicus RNase III crystal structure and the Bh RNase H in crystallo reaction. Taken together, we provide a molecular mechanism of the nuclease catalytic reaction that is likely common for the broad family of two-metal-ion catalytic phosphate-cleaving enzymes with a DDE motif.

Examining the Mechanism of Phosphite Dehydrogenase with Quantum Mechanical/Molecular Mechanical Free Energy Simulations

The projected decline of available phosphorus necessitates alternative methods to derive usable phosphate for fertilizer and other applications. Phosphite dehydrogenase oxidizes phosphite to phosphate with the cofactor NAD+ serving as the hydride acceptor. In addition to producing phosphate, this enzyme plays an important role in NADH cofactor regeneration processes. Mixed quantum mechanical/molecular mechanical free energy simulations were performed to elucidate the mechanism of this enzyme and to identify the protonation states of the substrate and product. Specifically, the finite temperature string method with umbrella sampling was used to generate the free energy surfaces and determine the minimum free energy paths for six different initial conditions that varied in the protonation state of the substrate and the position of the nucleophilic water molecule. In contrast to previous studies, the mechanism predicted by all six independent strings is a concerted but asynchronous dissociative mechanism in which hydride transfer from the phosphite substrate to NAD+ occurs prior to attack by the nucleophilic water molecule. His292 is identified as the most likely general base that deprotonates the attacking water molecule. However, Arg237 could also serve as this base if it were deprotonated and His292 were protonated prior to the main chemical transformation, although this scenario is less probable. The simulations indicate that the phosphite substrate is monoanionic in its active form and that the most likely product is dihydrogen phosphate. These mechanistic insights may be helpful for designing mutant enzymes or artificial constructs that convert phosphite to phosphate and NAD+ to NADH more effectively.

Early-stage human islet amyloid polypeptide aggregation: Mechanisms behind dimer formation.

Amyloid aggregates of human islet amyloid polypeptide (hIAPP or human amylin) have long been implicated in the development of type II diabetes. While hIAPP is known to aggregate into amyloid fibrils, it is the early-stage prefibrillar species that have been proposed to be cytotoxic. A detailed picture of the early-stage aggregation process and relevant intermediates would be valuable in the development of effective therapeutics. Here, we use atomistic molecular dynamics simulations with a combination of enhanced sampling methods to examine the formation of the hIAPP dimer in water. Bias-exchange metadynamics calculations reveal relative conformational stabilities of the hIAPP dimer. Finite temperature string method calculations identify pathways for dimer formation, along with relevant free energy barriers and intermediate structures. We show that the initial stages of dimerization involve crossing a substantial free energy barrier to form an intermediate structure exhibiting transient β-sheet character, before proceeding to form an entropically stabilized dimer structure.

String Methods for Stochastic Image and Shape Matching

Matching of images and analysis of shape differences is traditionally pursued by energy minimization of paths of deformations acting to match the shape objects. In the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework, iterative gradient descents on the matching functional lead to matching algorithms informally known as Beg algorithms. When stochasticity is introduced to model stochastic variability of shapes and to provide more realistic models of observed shape data, the corresponding matching problem can be solved with a stochastic Beg algorithm, similar to the finite temperature string method used in rare event sampling. In this paper, we apply a stochastic model compatible with the geometry of the LDDMM framework to obtain a stochastic model of images and we derive the stochastic version of the Beg algorithm which we compare with the string method and an expectation-maximization optimization of posterior likelihoods. The algorithm and its use for statistical inference is tested on stochastic LDDMM landmarks and images.

Proton‐Coupled Electron Transfer in Soybean Lipoxygenase: Hydrogen Tunneling, Electrostatics, and Conformational Motions

The enzyme soybean lipoxygenase (SLO) serves as a prototype for hydrogen tunneling reactions because of its unusually large deuterium kinetic isotope effect (KIE) of 80 for the wild‐type enzyme and over 500 for a double mutant. The proton‐coupled electron transfer (PCET) reaction catalyzed by SLO has been studied with a vibronically nonadiabatic PCET theory that includes the quantum mechanical effects of the electrons and the transferring proton, as well as the proton donor‐acceptor and solvent motions. The calculations reproduced the experimentally observed magnitude and temperature dependence of the KIE for wild‐type SLO and a series of single mutants. The large KIE was found to arise from the small overlap between the reactant and product proton vibrational wave functions. Theoretical modeling of the double mutant illustrated that the colossal KIE of over 500 arises from a combination of a non‐optimal active site conformation corresponding to longer equilibrium donor‐acceptor distances and reduced sampling of shorter donor‐acceptor distances. Recent quantum mechanical/molecular mechanical (QM/MM) free energy simulations using a finite temperature string method combined with umbrella sampling have provided additional insights about the conformational motions, as well as the significance of electrostatics in this enzymatic reaction. These calculations highlight the critical roles of hydrogen tunneling, electrostatics, and protein conformational motions in enzyme catalysis.Support or Funding InformationNational Institutes of Health Grant GM056207This abstract is from the Experimental Biology 2018 Meeting. There is no full text article associated with this abstract published in The FASEB Journal.

Human Islet Amyloid Polypeptide: Identifying Early-Stage Aggregation Mechanisms through Molecular Simulation

Human islet amyloid polypeptide (hIAPP) is implicated in the onset of type II diabetes and is known to aggregate into amyloid fibrils. However, it is prefibrillar species, not mature fibrils, that are proposed to be cytotoxic. In order to better understand the role of hIAPP aggregation in the onset of disease, as well as to design effective therapeutics, it is crucial to understand the mechanism of early-stage hIAPP aggregation. In this work, we use atomistic molecular dynamics simulations combined with multiple advanced sampling techniques to examine the formation of the hIAPP dimer and trimer. Metadynamics calculations reveal a free energy landscape for hIAPP dimer formation, which suggests multiple possible transition pathways. We employ the finite temperature string method to identify favorable pathways for dimer and trimer formation, along with relevant free energy barriers and intermediate structures. Results provide valuable insights into mechanisms and energetics of hIAPP aggregation.

Finite Temperature String Method with Umbrella Sampling: Application on a Side Chain Flipping in Mhp1 Transporter.

Protein conformational change is of central importance in molecular biology. Here we demonstrate a computational approach to characterize the transition between two metastable conformations in all-atom simulations. Our approach is based on the finite temperature string method, and the implementation is essentially a generalization of umbrella sampling simulations with Hamiltonian replica exchange. We represent the transition pathway by a curve in the conformational space, with the curve parameter taken as the reaction coordinate. Our approach can efficiently refine a transition pathway and compute a one-dimensional free energy as a function of the reaction coordinate. A diffusion model can then be used to calculate the forward and backward transition rates, the major kinetic quantities for the transition. We applied the approach on a local transition in the ligand-free Mhp1 transporter, between its outward-facing conformation and an intermediate conformation with the side chain of Phe305 flipped to the outside of the protein. Our simulations predict that the flipped-out position of this side chain has a free energy 6.5 kcal/mol higher than the original position in the crystal structure, and that the forward and backward transition rates are in the millisecond and submicrosecond time scales, respectively.

A minimization principle for transition paths of maximum flux for collective variables.

Considered is the construction of transition paths of conformational changes for proteins and other macromolecules, using methods that do not require the generation of dynamics trajectories. Special attention is given to the use of a reduced set of collective variables for describing such paths. A favored way to define transition paths is to seek channels through the transition state having cross sections with a high reactive flux (density of last hitting points of reactive trajectories). Given here is a formula for reactive flux that is independent of the parameterization of "collective variable space." This formula is needed for the principal curve of the reactive flux (as in the revised finite temperature string method) and for the maximum flux transition (MaxFlux) path. Additionally, a resistance functional is derived for narrow tubes, which when minimized yields a MaxFlux path. A strategy for minimization is outlined in the spirit of the string method. Finally, alternative approaches based on determining trajectories of high probability are considered, and it is observed that they yield paths that depend on the parameterization of collective variable space, except in the case of zero temperature, where such a path coincides with a MaxFlux path.

Investigations of α-helix↔β-sheet transition pathways in a miniprotein using the finite-temperature string method

A parallel implementation of the finite-temperature string method is described, which takes into account the invariance of coordinates with respect to rigid-body motions. The method is applied to the complex α-helix↔β-sheet transition in a β-hairpin miniprotein in implicit solvent, which exhibits much of the complexity of conformational changes in proteins. Two transition paths are considered, one derived from a linear interpolant between the endpoint structures and the other derived from a targeted dynamics simulation. Two methods for computing the conformational free energy (FE) along the string are compared, a restrained method, and a tessellation method introduced by E. Vanden-Eijnden and M. Venturoli [J. Chem. Phys. 130, 194103 (2009)]. It is found that obtaining meaningful free energy profiles using the present atom-based coordinates requires restricting sampling to a vicinity of the converged path, where the hyperplanar approximation to the isocommittor surface is sufficiently accurate. This sampling restriction can be easily achieved using restraints or constraints. The endpoint FE differences computed from the FE profiles are validated by comparison with previous calculations using a path-independent confinement method. The FE profiles are decomposed into the enthalpic and entropic contributions, and it is shown that the entropy difference contribution can be as large as 10 kcal/mol for intermediate regions along the path, compared to 15-20 kcal/mol for the enthalpy contribution. This result demonstrates that enthalpic barriers for transitions are offset by entropic contributions arising from the existence of different paths across a barrier. The possibility of using systematically coarse-grained representations of amino acids, in the spirit of multiple interaction site residue models, is proposed as a means to avoid ad hoc sampling restrictions to narrow transition tubes.

Optimization-Based String Method for Finding Minimum Energy Path

AbstractWe present an efficient algorithm for calculating the minimum energy path (MEP) and energy barriers between local minima on a multidimensional potential energy surface (PES). Such paths play a central role in the understanding of transition pathways between metastable states. Our method relies on the original formulation of the string method [Phys. Rev. B, 66,052301 (2002)], i.e. to evolve a smooth curve along a direction normal to the curve. The algorithm works by performing minimization steps on hyperplanes normal to the curve. Therefore the problem of finding MEP on the PES is remodeled as a set of constrained minimization problems. This provides the flexibility of using minimization algorithms faster than the steepest descent method used in the simplified string method [J. Chem. Phys., 126(16), 164103 (2007)]. At the same time, it provides a more direct analog of the finite temperature string method. The applicability of the algorithm is demonstrated using various examples.

Insight into the molecular mechanism of water evaporation via the finite temperature string method

The process of water's evaporation at its liquid/air interface has proven challenging to study experimentally and, because it constitutes a rare event on molecular time scales, presents a challenge for computer simulations as well. In this work, we simulated water's evaporation using the classical extended simple point charge model water model, and identified a minimum free energy path for this process in terms of 10 descriptive order parameters. The measured free energy change was 7.4 kcal/mol at 298 K, in reasonable agreement with the experimental value of 6.3 kcal/mol, and the mean first-passage time was 1375 ns for a single molecule, corresponding to an evaporation coefficient of 0.25. In the observed minimum free energy process, the water molecule diffuses to the surface, and tends to rotate so that its dipole and one O-H bond are oriented outward as it crosses the Gibbs dividing surface. As the water molecule moves further outward through the interfacial region, its local density is higher than the time-averaged density, indicating a local solvation shell that protrudes from the interface. The water molecule loses donor and acceptor hydrogen bonds, and then, with its dipole nearly normal to the interface, stops donating its remaining hydrogen bond. At that point, when the final, accepted hydrogen bond is broken, the water molecule is free. We also analyzed which order parameters are most important in the process and in reactive trajectories, and found that the relative orientation of water molecules near the evaporating molecule, and the number of accepted hydrogen bonds, were important variables in reactive trajectories and in kinetic descriptions of the process.

Simulating rare events using a weighted ensemble-based string method.

We introduce an extension to the weighted ensemble (WE) path sampling method to restrict sampling to a one-dimensional path through a high dimensional phase space. Our method, which is based on the finite-temperature string method, permits efficient sampling of both equilibrium and non-equilibrium systems. Sampling obtained from the WE method guides the adaptive refinement of a Voronoi tessellation of order parameter space, whose generating points, upon convergence, coincide with the principle reaction pathway. We demonstrate the application of this method to several simple, two-dimensional models of driven Brownian motion and to the conformational change of the nitrogen regulatory protein C receiver domain using an elastic network model. The simplicity of the two-dimensional models allows us to directly compare the efficiency of the WE method to conventional brute force simulations and other path sampling algorithms, while the example of protein conformational change demonstrates how the method can be used to efficiently study transitions in the space of many collective variables.

Theoretical Study of the Mechanism of the Hydride Transfer between Ferredoxin–NADP+Reductase and NADP+: The Role of Tyr303

During photosynthesis, ferredoxin-NADP(+) reductase (FNR) catalyzes the electron transfer from ferredoxin to NADP(+) via its FAD cofactor. The final hydride transfer event between FNR and the nucleotide is a reversible process. Two different transient charge-transfer complexes form prior to and upon hydride transfer, FNR(rd)-NADP(+) and FNR(ox)-NADPH, regardless of the hydride transfer direction. Experimental structures of the FNR(ox):NADP(+) interaction have suggested a series of conformational rearrangements that might contribute to attaining the catalytically competent complex, but to date, no direct experimental information about the structure of this complex is available. Recently, a molecular dynamics (MD) theoretical approach was used to provide a putative organization of the active site that might represent a structure close to the transient catalytically competent interaction of Anabaena FNR with its coenzyme, NADP(+). Using this structure, we performed fully microscopic simulations of the hydride transfer processes between Anabaena FNR(rd)/FNR(ox) and NADP(+)/H, accounting also for the solvation. A dual-level QM/MM hybrid approach was used to describe the potential energy surface of the whole system. MD calculations using the finite-temperature string method combined with the WHAM method provided the potential of mean force for the hydride transfer processes. The results confirmed that the structural model of the reactants evolves to a catalytically competent transition state through very similar free energy barriers for both the forward and reverse reactions, in good agreement with the experimental hydride transfer rate constants reported for this system. This theoretical approach additionally provides subtle structural details of the mechanism in wild-type FNR and provides an explanation why Tyr303 makes possible the photosynthetic reaction, a process that cannot occur when this Tyr is replaced by a Ser.

Energetics and Mechanism of the Normal-to-Amyloidogenic Isomerization of β2-Microglobulin: On-the-Fly String Method Calculations

We use on-the-fly finite temperature string method in collective variables to study the transition from a normal to an amyloidogenic conformation of β2-microglobulin. We show that the protonation state of two histidine residues is of key importance and that under acidic (protonating) conditions, the transition to the amyloidgenic form is facilitated by both displacement of N-terminal residues to disrupt a hydrophobic pocket and by side-chain/side-chain electrostatic attraction, both of which facilitate a cis-trans prolyl isomerization. The free energy barriers for the normal-to-amyloidogenic isomerization are found to be 14.9 and 7.1 kcal/mol for the neutral and protonated cases, respectively, consistent with enhanced amyloidgenesis at low pH observed both in vitro and in hemodialysis-associated amyloidosis, and somewhat lower than experimentally determined barriers for bare prolyl cis-trans isomerization. We suggest specific mutagenesis experiments which could be used to further validate the mechanism observed.

Analysis and elimination of a bias in targeted molecular dynamics simulations of conformational transitions: application to calmodulin.

The popular targeted molecular dynamics (TMD) method for generating transition paths in complex biomolecular systems is revisited. In a typical TMD transition path, the large-scale changes occur early and the small-scale changes tend to occur later. As a result, the order of events in the computed paths depends on the direction in which the simulations are performed. To identify the origin of this bias, and to propose a method in which the bias is absent, variants of TMD in the restraint formulation are introduced and applied to the complex open ↔ closed transition in the protein calmodulin. Due to the global best-fit rotation that is typically part of the TMD method, the simulated system is guided implicitly along the lowest-frequency normal modes, until the large spatial scales associated with these modes are near the target conformation. The remaining portion of the transition is described progressively by higher-frequency modes, which correspond to smaller-scale rearrangements. A straightforward modification of TMD that avoids the global best-fit rotation is the locally restrained TMD (LRTMD) method, in which the biasing potential is constructed from a number of TMD potentials, each acting on a small connected portion of the protein sequence. With a uniform distribution of these elements, transition paths that lack the length-scale bias are obtained. Trajectories generated by steered MD in dihedral angle space (DSMD), a method that avoids best-fit rotations altogether, also lack the length-scale bias. To examine the importance of the paths generated by TMD, LRTMD, and DSMD in the actual transition, we use the finite-temperature string method to compute the free energy profile associated with a transition tube around a path generated by each algorithm. The free energy barriers associated with the paths are comparable, suggesting that transitions can occur along each route with similar probabilities. This result indicates that a broad ensemble of paths needs to be calculated to obtain a full description of conformational changes in biomolecules. The breadth of the contributing ensemble suggests that energetic barriers for conformational transitions in proteins are offset by entropic contributions that arise from a large number of possible paths.