Barrier Crossings Research Articles

A Method for Treating Significant Conformational Changes in Alchemical Free Energy Simulations of Protein-Ligand Binding.

Relative binding free energy (RBFE) simulation is a rigorous approach to the calculation of quantitatively accurate binding free energy values for protein-ligand binding in which a reference binder is gradually converted to a target binder through alchemical transformation during the simulation. The success of such simulations relies on being able to accurately sample the correct conformational phase space for each alchemical state; however, this becomes a challenge when a significant conformation change occurs between the reference and target binder-receptor complexes. Increasing the simulation time and using enhanced sampling methods can be helpful, but effects can be limited, especially when the free energy barrier between conformations is high or when the correct target complex conformation is difficult to find and maintain. Current RBFE protocols seed the reference complex structure into every alchemical window of the simulation. In our study, we describe an improved protocol in which the reference structure is seeded into the first half of the alchemical windows, and the target structure is seeded into the second half of the alchemical windows. By applying information about the relevant correct end point conformations to different simulation windows from the beginning, the need for large barrier crossings or simulation prediction of the correct structures during an alchemical simulation is in many cases obviated. In the diverse cases we examine below, the simulations yielded free energy predictions that are satisfactory as compared to experiment and superior to running the simulations utilizing the conventional protocol. The method is straightforward to implement for publicly available FEP workflows.

Publisher’s Note: “Stochastic model for barrier crossings and fluctuations in local timescale” [J. Math. Phys. 65, 023302 (2024)

Publisher’s Note: “Stochastic model for barrier crossings and fluctuations in local timescale” [J. Math. Phys. 65, 023302 (2024)

Failed Barrier Crossings Tell a Story

Failed Barrier Crossings Tell a Story

Stochastic model for barrier crossings and fluctuations in local timescale

The problem of computing the rate of diffusion-aided activated barrier crossings between metastable states is one of broad relevance in physical sciences. The transition path formalism aims to compute the rate of these events by analysing the statistical properties of the transition path between the two metastable regions concerned. In this paper, we show that the transition path process is a unique solution to an associated stochastic differential equation, with a discontinuous and singular drift term. The singularity arises from a local time contribution, which accounts for the fluctuations at the boundaries of the metastable regions. The presence of fluctuations at the local time scale calls for an excursion theoretic consideration of barrier crossing events. We show that the rate of such events, as computed from excursion theory, factorizes into a local time term and an excursion measure term, which bears empirical similarity to the transition state theory rate expression. Since excursion theory makes no assumption about the presence of a transition state in the potential energy landscape, the mathematical structure underlying this factorization ought to be general. We hence expect excursion theory (and local times) to provide some physical and mathematical insights in generic barrier crossing problems.

Regular reaction dynamics in analytical form in the vicinity of symmetrical transition states. Central barrier crossings in SN2 reactions.

When an activated complex, as defined in transition state theory (TST), has a polyhedral shape, its kinetic energy is found to be diagonal in a system of spherical polar coordinates. If, in addition, the polyhedron is characterized by a high symmetry, then its dynamics considerably simplifies. An application of this approach to the most symmetrical TS known to date, i.e., that which controls the Cl- + CH3Cl → ClCH3 + Cl- SN2 nucleophilic substitution, is presented and an analytical expression of its potential energy surface is provided. In a substantial range around the saddle point, approximate equations of motion for the two components of the reaction coordinate, i.e., the antisymmetrical stretching motion of the ClCCl core and the wagging motion of the hydrogen triad, can be derived in an analytical form. During an extensive period of time, the main component of the reaction coordinate is governed by an unexpectedly simple equation of motion that depends on a single initial condition, irrespective of the other ones and of the internal energy. Reactive trajectories are observed to form a perfectly collimated bundle characterized by undetectable dispersion, thereby giving a spectacular example of regular dynamics in an anharmonic potential. Regularity and collimation are brought about by local symmetry, which is a widespread feature of potential energy surfaces. Anharmonicity is observed to influence the dynamics only at a late stage. As energy increases, trajectories tend to fan out and to deviate from the analytical equation. For the wagging motion, chaos sets in at much lower energies.

Remeshing flexible membranes under the control of free energy.

Cell membranes are flexible and often undergo large-scale morphological changes during processes like mitosis, protrusion and retraction, or vesicle fusion. Mathematical modeling of cell membranes depends on a representation of the free-form surface by discrete meshes. During morphological changes, these meshes must be adjusted under the minimization of the total free energy. Current methodology for meshing is limited in one of two ways: 1) Free energy-dependent methods have no restriction on the mesh geometry. The resulting irregular meshes cause artifacts in follow-up models of morphodynamics. 2) Geometry-dependent methods maintain mesh quality but violate the physics of free energy minimization. To fill this gap, we regulate mesh geometries via a free-energy-determined remeshing process: adding and removing mesh elements upon morphological changes based on barrier crossings in a double-barrier potential between neighboring vertices in the meshes. We test the method's robustness by reproducing the morphodynamics of red blood cells and vesicle fusions; and we demonstrate the method's adaptability by simulating the formation of filopodia, lamellipodia and invaginations. Finally, we use the method to study a mechanical decoupling effect of two connected membrane tethers that has been recently observed experimentally, but has not been mechanistically explained in the context of a complete membrane surface. We propose a biophysical model that strengthens the decoupling effect and broadens the original interpretation of the experiment. The method is developed in C/Matlab and distributed via https://github.com/DanuserLab/biophysicsModels.

Excitations follow (or lead?) density scaling in propylene carbonate.

Structural excitations that enable interbasin (IB) barrier crossings on a potential energy landscape are thought to play a facilitating role in the relaxation of liquids. Here, we show that the population of these excitations exhibits the same density scaling observed for α relaxation in propylene carbonate, even though they are heavily influenced by intramolecular modes. We also find that IB crossing modes exhibit a Grüneisen parameter (γG) that is approximately equivalent to the density scaling parameter γTS. These observations suggest that the well-documented relationship between γG and γTS may be a direct result of the pressure dependence of the frequency of unstable (relaxation) modes associated with IB motion.

Sea crossings of migratory pink‐footed geese: seasonal effects of winds on flying and stopping behaviour

Migratory birds may need to cross barriers such as seas, without opportunities to rest or refuel. Waterbirds, unlike land birds, can stop at sea to rest or wait for better winds and thus may be less selective for supportive winds at departure and tolerate larger drift. However, pay‐offs of waiting are likely to depend on circumstances (e.g. pressure for well‐timed arrival, wind availability and travelling with/without juvenile brood), thus migratory behaviour during barrier crossings is expected to differ between seasons. We studied pink‐footed geese Anser brachyrhynchus crossing the Barents Sea (ca 650 km), in spring and autumn during 2018–2020, using 94 GPS‐tracks of 38 individuals, with annotated ERA5 weather data. We found that 1) especially in autumn, geese selected supportive winds for departure; 2) in spring, geese experienced lower wind support and more crosswinds than in autumn, leading to 23% longer routes, 60% longer durations, 93% longer air distances and 45% higher ratios of air‐to‐ground distances; 3) in both seasons, geese had more tailwinds in the first part of crossings, and in spring when deviating more from the shortest route; 4) geese stopped at sea more often in spring (mean 11×) than autumn (3×), in spring during earlier stages of crossings, but in both seasons, spent half of the crossing time at sea, during which they still continued to approach their destination slowly; 5) stops at sea happened mostly in adverse winds, warmer air, higher air humidity and on calmer water and, in autumn, took longer without juvenile brood. We conclude that for migrating pink‐footed geese, Arctic capital breeders, the importance of time and energy can shift en route and that seasonal differences in wind support, flying and stopping behaviour and the pressure for a well‐timed arrival cause the Barents Sea to be a larger barrier in spring than in autumn.

Efficient two-dimensional control of barrier crossing

Driven barrier crossings are pervasive in optical-trapping experiments and steered molecular-dynamics simulations. Despite the high fidelity of control, the freedom in the choice of driving protocol is rarely exploited to improve efficiency. We design protocols that reduce dissipation for rapidly driven barrier crossing under two-dimensional control of a harmonic trapping potential, controlling both trap center and stiffness. For fast driving, the minimum-dissipation protocol jumps halfway between the control-parameter endpoints. For slow driving, the minimum-dissipation protocol generically slows down and tightens the trap as it crosses the barrier, resulting in both significant energy savings and increased flux compared to naive and one-dimensional protocols (that only change trap center). Combining fast and slow results, we design protocols that improve performance at all speeds.

Barrier crossings and winds shape daily travel schedules and speeds of a flight generalist

External factors such as geography and weather strongly affect bird migration influencing daily travel schedules and flight speeds. For strictly thermal-soaring migrants, weather explains most seasonal and regional differences in speed. Flight generalists, which alternate between soaring and flapping flight, are expected to be less dependent on weather, and daily travel schedules are likely to be strongly influenced by geography and internal factors such as sex. We GPS-tracked the migration of 70 lesser kestrels (Falco naumanni) to estimate the relative importance of external factors (wind, geography), internal factors (sex) and season, and the extent to which they explain variation in travel speed, distance, and duration. Our results show that geography and tailwind are important factors in explaining variation in daily travel schedules and speeds. We found that wind explained most of the seasonal differences in travel speed. In both seasons, lesser kestrels sprinted across ecological barriers and frequently migrated during the day and night. Conversely, they travelled at a slower pace and mainly during the day over non-barriers. Our results highlighted that external factors far outweighed internal factors and season in explaining variation in migratory behaviour of a flight generalist, despite its ability to switch between flight modes.

The use of nocturnal flights for barrier crossing in a diurnally migrating songbird

BackgroundThe migration patterns of land birds can generally be divided into those species that migrate principally during the day and those that migrate during the night. Some species may show individual plasticity in the use of day or night flight, particularly when crossing large, open-water or desert barriers. However, individual plasticity in circadian patterns of migratory flights in diurnally migrating songbirds has never been investigated.MethodsWe used high precision GPS tracking of a diurnal, migratory swallow, the purple martin (Progne subis), to determine whether individuals were flexible in their spring migration strategies to include some night flight, particularly at barrier crossing.ResultsMost (91%) of individuals made large (sometimes > 1000 km), open-water crossings of the Caribbean Sea and the Gulf of Mexico that included the use of night flight. 32% of all water crossings were initiated at night, demonstrating that night flight is not only used to complete large crossings but may confer other advantages for diurnal birds. Birds were not more likely to initiate crossings with supportive winds, however crossings were more likely when they reduced travel distances. Our results are consistent with diurnal birds using night flight to help achieve time- and energy-savings through ‘short cuts’ at barrier crossings, at times and locations when foraging opportunities are not available.ConclusionsOverall, our results demonstrate the use of nocturnal flight and a high degree of individual plasticity in migration strategies on a circadian scale in a species generally considered to be a diurnal migrant. Nocturnal flights at barrier crossing may provide time and energy savings where foraging opportunities are low in an otherwise diurnal strategy. Future research should target how diel foraging and refueling strategies support nocturnal flights and barrier crossing in this and other diurnal species.

Dissipation bounds the amplification of transition rates far from equilibrium

Complex systems can convert energy imparted by nonequilibrium forces to regulate how quickly they transition between long-lived states. While such behavior is ubiquitous in natural and synthetic systems, currently there is no general framework to relate the enhancement of a transition rate to the energy dissipated or to bound the enhancement achievable for a given energy expenditure. We employ recent advances in stochastic thermodynamics to build such a framework, which can be used to gain mechanistic insight into transitions far from equilibrium. We show that under general conditions, there is a basic speed limit relating the typical excess heat dissipated throughout a transition and the rate amplification achievable. We illustrate this tradeoff in canonical examples of diffusive barrier crossings in systems driven with autonomous and deterministic external forcing protocols. In both cases, we find that our speed limit tightly constrains the rate enhancement.

Hydrogen-Deuterium Exchange within Adenosine Deaminase, a TIM Barrel Hydrolase, Identifies Networks for Thermal Activation of Catalysis.

Proteins are intrinsically flexible macromolecules that undergo internal motions with time scales spanning femtoseconds to milliseconds. These fluctuations are implicated in the optimization of reaction barriers for enzyme catalyzed reactions. Time, temperature, and mutation dependent hydrogen-deuterium exchange coupled to mass spectrometry (HDX-MS) has been previously employed to identify spatially resolved, catalysis-linked dynamical regions of enzymes. We now extend this technique to pursue the correlation of protein flexibility and chemical reactivity within the diverse and widespread TIM barrel proteins, targeting murine adenosine deaminase (mADA) that catalyzes the irreversible deamination of adenosine to inosine and ammonia. Following a structure-function analysis of rate and activation energy for a series of mutations at a second sphere phenylalanine positioned in proximity to the bound substrate, the catalytically impaired Phe61Ala with an elevated activation energy (Ea = 7.5 kcal/mol) and the wild type (WT) mADA (Ea = 5.0 kcal/mol) were selected for HDX-MS experiments. The rate constants and activation energies of HDX for peptide segments are quantified and used to assess mutation-dependent changes in local and distal motions. Analyses reveal that approximately 50% of the protein sequence of Phe61Ala displays significant changes in the temperature dependence of HDX behaviors, with the dominant change being an increase in protein flexibility. Utilizing Phe61Ile, which displays the same activation energy for kcat as WT, as a control, we were able to further refine the HDX analysis, highlighting the regions of mADA that are altered in a functionally relevant manner. A map is constructed that illustrates the regions of protein that are proposed to be essential for the thermal optimization of active site configurations that dominate reaction barrier crossings in the native enzyme.

Diffusible Cross-linkers Cause Superexponential Friction Forces.

The friction between cytoskeletal filaments is of central importance for the formation of cellular structures such as the mitotic spindle and the cytokinetic ring. This friction is caused by passive cross-linkers, yet the underlying mechanism and the dependence on cross-linker density are poorly understood. Here, we use theory and computer simulations to study the friction between two filaments that are cross-linked by passive proteins, which can hop between discrete binding sites while physically excluding each other. The simulations reveal that filaments move via rare discrete jumps, which are associated with free-energy barrier crossings. We identify the reaction coordinate that governs the relative microtubule movement and derive an exact analytical expression for the free-energy barrier and the friction coefficient. Our analysis not only elucidates the molecular mechanism underlying cross-linker-induced filament friction, but also predicts that the friction coefficient scales superexponentially with the density of cross-linkers.

Studying rare events using forward-flux sampling: Recent breakthroughs and future outlook.

Rare events are processes that occur upon the emergence of unlikely fluctuations. Unlike what their name suggests, rare events are fairly ubiquitous in nature, as the occurrence of many structural transformations in biology and material sciences is predicated upon crossing large free energy barriers. Probing the kinetics and uncovering the molecular mechanisms of possible barrier crossings in a system is critical to predicting and controlling its structural and functional properties. Due to their activated nature, however, rare events are exceptionally difficult to study using conventional experimental and computational techniques. In recent decades, a wide variety of specialized computational techniques-known as advanced sampling techniques-have been developed to systematically capture improbable fluctuations relevant to rare events. In this perspective, we focus on a technique called forward flux sampling [Allen et al., J. Chem. Phys. 124, 024102 (2006)] and overview its recent methodological variants and extensions. We also provide a detailed overview of its application to study a wide variety of rare events and map out potential avenues for further explorations.

Transition path times in asymmetric barriers.

Biomolecular conformational transitions are usually modeled as barrier crossings in a free energy landscape. The transition paths connect two local free energy minima and transition path times (TPT) are the actual durations of the crossing events. The simplest model employed to analyze TPT and to fit empirical data is that of a stochastic particle crossing a parabolic barrier. Motivated by some disagreement between the value of the barrier height obtained from the TPT distributions as compared to the value obtained from kinetic and thermodynamic analyses, we investigate here TPT for barriers which deviate from the symmetric parabolic shape. We introduce a continuous set of potentials, that starting from a parabolic shape, can be made increasingly asymmetric by tuning a single parameter. The TPT distributions obtained in the asymmetric case are very well-fitted by distributions generated by parabolic barriers. The fits, however, provide values for the barrier heights and diffusion coefficients which deviate from the original input values. We show how these findings can be understood from the analysis of the eigenvalues spectrum of the Fokker-Planck equation and highlight connections with experimental results.

Optimizing Brownian escape rates by potential shaping

Brownian escape is key to a wealth of physico-chemical processes, including polymer folding and information storage. The frequency of thermally activated energy barrier crossings is assumed to generally decrease exponentially with increasing barrier height. Here, we show experimentally that higher, fine-tuned barrier profiles result in significantly enhanced escape rates, in breach of the intuition relying on the above scaling law, and address in theory the corresponding conditions for maximum speed-up. Importantly, our barriers end on the same energy on which they start. For overdamped dynamics, the achievable boost of escape rates is, in principle, unbounded so that the barrier optimization has to be regularized. We derive optimal profiles under 2 different regularizations and uncover the efficiency of N-shaped barriers. We then demonstrate the viability of such a potential in automated microfluidic Brownian dynamics experiments using holographic optical tweezers and achieve a doubling of escape rates compared to unhindered Brownian motion. Finally, we show that this escape rate boost extends into the low-friction inertial regime.

Wind-associated detours promote seasonal migratory connectivity in a flapping flying long-distance avian migrant.

It is essential to gain knowledge about the causes and extent of migratory connectivity between stationary periods of migrants to further the understanding of processes affecting populations, and to allow efficient implementation of conservation efforts throughout the annual cycle. Avian migrants likely use optimal routes with respect to mode of locomotion, orientation and migration strategy, influenced by external factors such as wind and topography. In self-powered flapping flying birds, any increases in fuel loads are associated with added flight costs. Energy-minimizing migrants are therefore predicted to trade-off extended detours against reduced travel across ecological barriers with no or limited foraging opportunities. Here, we quantify the extent of detours taken by different populations of European nightjars Caprimulgus europaeus, to test our predictions that they used routes beneficial according to energetic principles and evaluate the effect of route shape on seasonal migratory connectivity. We combined data on birds tracked from breeding sites along a longitudinal gradient from England to Sweden. We analysed the migratory connectivity between breeding and main non-breeding sites, and en route stopover sites just south of the Sahara desert. We quantified each track's route extension relative to the direct route between breeding and wintering sites, respectively, and contrasted it to the potential detour derived from the barrier reduction along the track while accounting for potential wind effects. Nightjars extended their tracks from the direct route between breeding and main non-breeding sites as they crossed the Mediterranean Sea-Sahara desert, the major ecological barrier in the Palaearctic-African migration system. These clockwise detours were small for birds from eastern sites but increased from east to west breeding longitude. Routes of the tracked birds were associated with partial reduction in the barrier crossing resulting in a trade-off between route extension and barrier reduction, as expected in an energy-minimizing migrant. This study demonstrates how the costs of barrier crossings in prevailing winds can disrupt migratory routes towards slightly different goals, and thereby promote migratory connectivity. This is an important link between individual migration strategies in association with an ecological barrier, and both spatially and demographic population patterns.

Blinded by the light? Circadian partitioning of migratory flights in a nightjar species

Circadian migration patterns for most migratory birds can be partitioned into day and night flights, but can be flexible at barrier crossings. Whether obligate nocturnal species opportunely use some daylight during migration has not been previously investigated. We tested the night constraint hypothesis using autumn Global Positioning System (GPS) tracking of a migratory nightjar species (Eastern Whip-poor-will, Antrostomus vociferus), and found that a minimum of 89% of travel occurred in darkness. All seven GPS-tracked individuals avoided crossing the Gulf of Mexico, which would have required some daytime flight. Future research should investigate the mechanisms and implications of light restrictions on the migratory movements of nocturnal species.

A Human Blood-Brain Interface Model to Study Barrier Crossings by Pathogens or Medicines and Their Interactions with the Brain

A Human Blood-Brain Interface Model to Study Barrier Crossings by Pathogens or Medicines and Their Interactions with the Brain