Pair-Reaction Dynamics in Water: Competition of Memory, Potential Shape, and Inertial Effects.

When described by a one-dimensional reaction coordinate, pair-reaction rates in a solvent depend, in addition to the potential barrier height and the friction coefficient, on the potential shape, the effective mass, and the friction relaxation spectrum, but a rate theory that accurately accounts for all of these effects does not exist. After a review of classical reaction-rate theories, we show how to extract all parameters of the generalized Langevin equation (GLE) and, in particular, the friction memory function from molecular dynamics (MD) simulations of two prototypical pair reactions in water, the dissociation of NaCl and of two methane molecules. The memory exhibits multiple time scales and, for NaCl, pronounced oscillatory components. Simulations of the GLE by Markovian embedding techniques accurately reproduce the pair-reaction kinetics from MD simulations without any fitting parameters, which confirms the accuracy of the approximative form of the GLE and of the parameter extraction techniques. By modification of the GLE parameters, we investigate the relative importance of memory, mass, and potential shape effects. Neglect of memory slows down NaCl and methane dissociation by roughly a factor of 2; neglect of mass accelerates reactions by a similar factor, and the harmonic approximation of the potential shape gives rise to slight acceleration. This partial error cancellation explains why Kramers' theory, which neglects memory effects and treats the potential shape in harmonic approximation, describes reaction rates better than more sophisticated theories. In essence, all three effects, friction memory, inertia, and the potential shape nonharmonicity, are important to quantitatively describe pair-reaction kinetics in water.

Short-Range Cooperative Slow-down of Water Solvation Dynamics Around SO4 2--Mg2+ Ion Pairs.

The presence of ions affects the structure and dynamics of water on a multitude of length and time scales. In this context, pairs of Mg2+ and SO4 2- ions in water constitute a prototypical system for which conflicting pictures of hydration geometries and dynamics have been reported. Key issues are the molecular pair and solvation shell geometries, the spatial range of electric interactions, and their impact on solvation dynamics. Here, we introduce asymmetric SO4 2- stretching vibrations as new and most specific local probes of solvation dynamics that allow to access ion hydration dynamics at the dilute concentration (0.2 M) of a native electrolyte environment. Highly sensitive heterodyne 2D-IR spectroscopy in the fingerprint region of the SO4 2- ions around 1100 cm-1 reveals a specific slow-down of solvation dynamics for hydrated MgSO4 and for Na2SO4 in the presence of Mg2+ ions, which manifests as a retardation of spectral diffusion compared to aqueous Na2SO4 solutions in the absence of Mg2+ ions. Extensive molecular dynamics and density functional theory QM/MM simulations provide a microscopic view of the observed ultrafast dephasing and hydration dynamics. They suggest a molecular picture where the slow-down of hydration dynamics arises from the structural peculiarities of solvent-shared SO4 2--Mg2+ ion pairs.

Proton-transfer spectroscopy beyond the normal-mode scenario.

A stochastic theory is developed to predict the spectral signature of proton-transfer processes and is applied to infrared spectra computed from ab initio molecular-dynamics simulations of a single H5O2 + cation. By constraining the oxygen atoms to a fixed distance, this system serves as a tunable model for general proton-transfer processes with variable barrier height. Three spectral contributions at distinct frequencies are identified and analytically predicted: the quasi-harmonic motion around the most probable configuration, amenable to normal-mode analysis, the contribution due to transfer paths when the proton moves over the barrier, and a shoulder for low frequencies stemming from the stochastic transfer-waiting-time distribution; the latter two contributions are not captured by normal-mode analysis but exclusively reported on the proton-transfer kinetics. In accordance with reaction rate theory, the transfer-waiting-contribution frequency depends inversely exponentially on the barrier height, whereas the transfer-path-contribution frequency is rather insensitive to the barrier height.

Barrier-crossing times for different non-Markovian friction in well and barrier: A numerical study.

We introduce a generalized Langevin model system for different non-Markovian effects in the well and barrier regions of a potential, and use it to numerically study the barrier-crossing time. In the appropriate limits, our model interpolates between the theoretical barrier-crossing-time predictions by Grote and Hynes (GH), as well as by Pollak et al., which for a single barrier memory time can differ by several orders of magnitude. Our model furthermore allows one to test an analytic rate theory for space-inhomogeneous memory, which disagrees with our numerical results in the long well-memory regime. In this regime, we find that short barrier memory decreases the barrier-crossing time as compared to long barrier memory. This is in contrast with the short well-memory regime, where both our numerical results and the GH theory predict an acceleration of the barrier crossing time with increasing barrier memory time. Both effects, the "Markovian-barrier acceleration" and GH "non-Markovian-barrier acceleration," can be understood from a committor analysis. Our model combines finite relaxation times of orthogonal degrees of freedom with a space-inhomogeneous coupling to such degrees and represents a step towards more realistic modeling of reaction coordinates.

Spectral signatures of excess-proton waiting and transfer-path dynamics in aqueous hydrochloric acid solutions.

The theoretical basis for linking spectral signatures of hydrated excess protons with microscopic proton-transfer mechanisms has so far relied on normal-mode analysis. We introduce trajectory-decomposition techniques to analyze the excess-proton dynamics in ab initio molecular-dynamics simulations of aqueous hydrochloric-acid solutions beyond the normal-mode scenario. We show that the actual proton transfer between two water molecules involves for relatively large water-water separations crossing of a free-energy barrier and thus is not a normal mode, rather it is characterized by two non-vibrational time scales: Firstly, the broadly distributed waiting time for transfer to occur with a mean value of 200-300 fs, which leads to a broad and weak shoulder in the absorption spectrum around 100 cm-1, consistent with our experimental THz spectra. Secondly, the mean duration of a transfer event of about 14 fs, which produces a rather well-defined spectral contribution around 1200 cm-1 and agrees in location and width with previous experimental mid-infrared spectra.

Time-Dependent Friction Effects on Vibrational Infrared Frequencies and Line Shapes of Liquid Water.

From ab initio simulations of liquid water, the time-dependent friction functions and time-averaged nonlinear effective bond potentials for the OH stretch and HOH bend vibrations are extracted. The obtained friction exhibits not only adiabatic contributions at and below the vibrational time scales but also much slower nonadiabatic contributions, reflecting homogeneous and inhomogeneous line broadening mechanisms, respectively. Intermolecular interactions in liquid water soften both stretch and bend potentials compared to the gas phase, which by itself would lead to a red-shift of the corresponding vibrational bands. In contrast, nonadiabatic friction contributions cause a spectral blue shift. For the stretch mode, the potential effect dominates, and thus, a significant red shift when going from gas to the liquid phase results. For the bend mode, potential and nonadiabatic friction effects are of comparable magnitude, so that a slight blue shift results, in agreement with well-known but puzzling experimental findings. The observed line broadening is shown to be roughly equally caused by adiabatic and nonadiabatic friction contributions for both the stretch and bend modes in liquid water. Thus, the quantitative analysis of the time-dependent friction that acts on vibrational modes in liquids advances the understanding of infrared vibrational frequencies and line shapes.

Hydrophobicity of Self-Assembled Monolayers of Alkanes: Fluorination, Density, Roughness, and Lennard-Jones Cutoffs.

The interplay of fluorination and structure of alkane self-assembled monolayers and how these affect hydrophobicity are explored via molecular dynamics simulations, contact angle goniometry, and surface-enhanced infrared absorption spectroscopy. Wetting coefficients are found to grow linearly in the monolayer density for both alkane and perfluoroalkane monolayers. The larger contact angles of monolayers of perfluorinated alkanes are shown to be primarily caused by their larger molecular volume, which leads to a larger nearest-neighbor grafting distance and smaller tilt angle. Increasing the Lennard-Jones force cutoff in simulations is found to increase hydrophilicity. Specifically, wetting coefficients scale like the inverse square of the cutoff, and when extrapolated to the infinite cutoff limit, they yield contact angles that compare favorably to experimental values. Nanoscale roughness is also found to reliably increase monolayer hydrophobicity, mostly via the reduction of the entropic part of the work of adhesion. Analysis of depletion lengths shows that droplets on nanorough surfaces partially penetrate the surface, intermediate between Wenzel and Cassie-Baxter states.

Non-Markovian modeling of protein folding.

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 [Formula: see text] 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.

Semi-automatic stitching of filamentous structures in image stacks from serial-section electron tomography.

We present a software-assisted workflow for the alignment and matching of filamentous structures across a three-dimensional (3D) stack of serial images. This is achieved by combining automatic methods, visual validation, and interactive correction. After the computation of an initial automatic matching, the user can continuously improve the result by interactively correcting landmarks or matches of filaments. Supported by a visual quality assessment of regions that have been already inspected, this allows a trade-off between quality and manual labour. The software tool was developed in an interdisciplinary collaboration between computer scientists and cell biologists to investigate cell division by quantitative 3D analysis of microtubules (MTs) in both mitotic and meiotic spindles. For this, each spindle is cut into a series of semi-thick physical sections, of which electron tomograms are acquired. The serial tomograms are then stitched and non-rigidly aligned to allow tracing and connecting of MTs across tomogram boundaries. In practice, automatic stitching alone provides only an incomplete solution, because large physical distortions and a low signal-to-noise ratio often cause experimental difficulties. To derive 3D models of spindles despite dealing with imperfect data related to sample preparation and subsequent data collection, semi-automatic validation and correction is required to remove stitching mistakes. However, due to the large number of MTs in spindles (up to 30k) and their resulting dense spatial arrangement, a naive inspection of each MT is too time-consuming. Furthermore, an interactive visualisation of the full image stack is hampered by the size of the data (up to 100 GB). Here, we present a specialised, interactive, semi-automatic solution that considers all requirements for large-scale stitching of filamentous structures in serial-section image stacks. To the best of our knowledge, it is the only currently available tool which is able to process data of the type and size presented here. The key to our solution is a careful design of the visualisation and interaction tools for each processing step to guarantee real-time response, and an optimised workflow that efficiently guides the user through datasets. The final solution presented here is the result of an iterative process with tight feedback loops between the involved computer scientists and cell biologists. LAY DESCRIPTION: Electron tomography of biological samples is used for a three-dimensional (3D) reconstruction of filamentous structures, such as microtubules (MTs) in mitotic and meiotic spindles. Large-scale electron tomography can be applied to increase the reconstructed volume for the visualisation of full spindles. For this, each spindle is cut into a series of semi-thick physical sections, from which electron tomograms are acquired. The serial tomograms are then stitched and non-rigidly aligned to allow tracing and connecting of MTs across tomogram boundaries. Previously, we presented fully automatic approaches for this 3D reconstruction pipeline. However, large volumes often suffer from imperfections (ie physical distortions) caused by the image acquisition process, making it difficult to apply fully automatic approaches for matching and stitching of numerous tomograms. Therefore, we developed an interactive, semi-automatic solution that considers all requirements for large-scale stitching of microtubules in image stacks of consecutive sections. We achieved this by combining automatic methods, visual validation and interactive error correction, thus allowing the user to continuously improve the result by interactively correcting landmarks or matches of filaments. We present large-scale reconstructions of spindles in which the automatic workflow failed and where different steps of manual corrections were needed. Our approach is also applicable to other biological samples showing 3D distributions of MTs in a number of different cellular contexts.

Exploring the Absorption Spectrum of Simulated Water from MHz to Infrared.

Absorption spectra of liquid water at 300 K are calculated from both classical and density functional theory molecular dynamics simulation data, which together span from 1 MHz to hundreds of THz, agreeing well with experimental data qualitatively and quantitatively over the entire range, including the IR modes, the microwave peak, and the intermediate THz bands. The spectra are decomposed into single-molecular and collective components, as well as into components due to molecular reorientations and changes in induced molecular dipole moments. These decompositions shed light on the motions underlying the librational and translational (hydrogen-bond stretching) bands at 20 and 5 THz, respectively; interactions between donor protons and acceptor lone pair electrons are shown to be important for the line shape in both librational and translational regimes, and in- and out-of-phase librational dimer modes are observed and explored.

Collective exchange processes reveal an active site proton cage in bacteriorhodopsin.

Proton translocation across membranes is vital to all kingdoms of life. Mechanistically, it relies on characteristic proton flows and modifications of hydrogen bonding patterns, termed protonation dynamics, which can be directly observed by fast magic angle spinning (MAS) NMR. Here, we demonstrate that reversible proton displacement in the active site of bacteriorhodopsin already takes place in its equilibrated dark-state, providing new information on the underlying hydrogen exchange processes. In particular, MAS NMR reveals proton exchange at D85 and the retinal Schiff base, suggesting a tautomeric equilibrium and thus partial ionization of D85. We provide evidence for a proton cage and detect a preformed proton path between D85 and the proton shuttle R82. The protons at D96 and D85 exchange with water, in line with ab initio molecular dynamics simulations. We propose that retinal isomerization makes the observed proton exchange processes irreversible and delivers a proton towards the extracellular release site.

Butane dihedral angle dynamics in water is dominated by internal friction.

The dihedral dynamics of butane in water is known to be rather insensitive to the water viscosity; possible explanations for this involve inertial effects or Kramers' turnover, the finite memory time of friction, and the presence of so-called internal friction. To disentangle these factors, we introduce a method to directly extract the friction memory function from unconstrained simulations in the presence of an arbitrary free-energy landscape. By analysis of the dihedral friction in butane for varying water viscosity, we demonstrate the existence of an internal friction contribution that does not scale linearly with water viscosity. At normal water viscosity, the internal friction turns out to be eight times larger than the solvent friction and thus completely dominates the effective friction. By comparison with simulations of a constrained butane molecule that has the dihedral as the only degree of freedom, we show that internal friction comes from the six additional degrees of freedom in unconstrained butane that are orthogonal to the dihedral angle reaction coordinate. While the insensitivity of butane's dihedral dynamics to water viscosity is solely due to the presence of internal friction, inertial effects nevertheless crucially influence the resultant transition rates. In contrast, non-Markovian effects due to the finite memory time are present but do not significantly influence the dihedral barrier-crossing rate of butane. These results not only settle the character of dihedral dynamics in small solvated molecular systems such as butane, they also have important implications for the folding of polymers and proteins.

Memory-induced acceleration and slowdown of barrier crossing.

We study the mean first-passage time τMFP for the barrier crossing of a single massive particle with non-Markovian memory by Langevin simulations in one dimension. In the Markovian limit of short memory time τΓ, the expected Kramers turnover between the overdamped (high-friction) and the inertial (low-friction) limits is recovered. Compared to the Markovian case, we find barrier crossing to be accelerated for intermediate memory time, while for long memory time, barrier crossing is slowed down and τMFP increases with τΓ as a power law τMFP∼τΓ2. Both effects are derived from an asymptotic propagator analysis: while barrier crossing acceleration at intermediate memory can be understood as an effective particle mass reduction, slowing down for long memory is caused by the slow kinetics of energy diffusion. A simple and globally accurate heuristic formula for τMFP in terms of all relevant time scales of the system is presented and used to establish a scaling diagram featuring the Markovian overdamped and the Markovian inertial regimes, as well as the non-Markovian intermediate memory time regime where barrier crossing is accelerated and the non-Markovian long memory time regime where barrier crossing is slowed down.