Water diffusion through a membrane protein channel: A first passage time approach

Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime

We derive general bounds on the probability that the empirical first-passage time τ[over ¯]_{n}≡∑_{i=1}^{n}τ_{i}/n of a reversible ergodic Markov process inferred from a sample of n independent realizations deviates from the true mean first-passage time by more than any given amount in either direction. We construct nonasymptotic confidence intervals that hold in the elusive small-sample regime and thus fill the gap between asymptotic methods and the Bayesian approach that is known to be sensitive to prior belief and tends to underestimate uncertainty in the small-sample setting. We prove sharp bounds on extreme first-passage times that control uncertainty even in cases where the mean alone does not sufficiently characterize the statistics. Our concentration-of-measure-based results allow for model-free error control and reliable error estimation in kinetic inference, and are thus important for the analysis of experimental and simulation data in the presence of limited sampling.

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.

Conformational Dynamics of Loop L3 in OmpF: Implications toward Antibiotic Translocation and Voltage Gating

In the present work, we delineate the molecular mechanism of a bulky antibiotic permeating through a bacterial channel and uncover the role of conformational dynamics of the constriction loop in this process. Using the temperature accelerated sliced sampling approach, we shed light onto the dynamics of the L3 loop, in particular the F118 to S125 segment, at the constriction regions of the OmpF porin. We complement the findings with single channel electrophysiology experiments and applied-field simulations, and we demonstrate the role of hydrogen-bond stabilization in the conformational dynamics of the L3 loop. A molecular mechanism of permeation is put forward wherein charged antibiotics perturb the network of stabilizing hydrogen-bond interactions and induce conformational changes in the L3 segment, thereby aiding the accommodation and permeation of bulky antibiotic molecules across the constriction region. We complement the findings with single channel electrophysiology experiments and demonstrate the importance of the hydrogen-bond stabilization in the conformational dynamics of the L3 loop. The generality of the present observations and experimental results regarding the L3 dynamics enables us to identify this L3 segment as the source of gating. We propose a mechanism of OmpF gating that is in agreement with previous experimental data that showed the noninfluence of cysteine double mutants that tethered the L3 tip to the barrel wall on the OmpF gating behavior. The presence of similar loop stabilization networks in porins of other clinically relevant pathogens suggests that the conformational dynamics of the constriction loop is possibly of general importance in the context of antibiotic permeation through porins.

Analysis of the Passage Times for Unfolding/Folding of the Adenine Riboswitch Aptamer

The conformational transitions of the adenosine deaminase A-riboswitch aptamer both with and without ligand binding are investigated within the tenets of the generalized Langevin equation in a complex viscoelastic cellular environment. Steered molecular dynamics (SMD) simulations are performed to evaluate and compare the results of the first passage times (FPTs) with those obtained from the theory for the unfold and fold transitions of the aptamer. The results of the distribution of Kramers's FPT reveal that the unfold-fold transitions are faster and hence more probable as compared to the fold-unfold transitions of the riboswitch aptamer for both ligand-bound and -unbound states. The transition path time is lower than Kramers's FPT for the riboswitch aptamer as the transition path times for the unfold-fold transition of both without and with ligand binding are insensitive to the details of the exact mechanism of the transition events. However, Kramers's FPTs show varied distributions which correspond to different transition pathways, unlike the transition path times. The mean FPT increases with an increase in the complexity of the cellular environment. The results of Kramers's FPT, transition path time distribution, and mean FPT obtained from our calculations qualitatively match with those obtained from the SMD simulations. Analytically derived values of the mean transition path time show good quantitative agreement with those estimated from the single-molecule force spectroscopy experiments for higher barrier heights.

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.

Water in Nanopores and Biological Channels: A Molecular Simulation Perspective

This Review explores the dynamic behavior of water within nanopores and biological channels in lipid bilayer membranes. We focus on molecular simulation studies, alongside selected structural and other experimental investigations. Structures of biological nanopores and channels are reviewed, emphasizing those high-resolution crystal structures, which reveal water molecules within the transmembrane pores, which can be used to aid the interpretation of simulation studies. Different levels of molecular simulations of water within nanopores are described, with a focus on molecular dynamics (MD). In particular, models of water for MD simulations are discussed in detail to provide an evaluation of their use in simulations of water in nanopores. Simulation studies of the behavior of water in idealized models of nanopores have revealed aspects of the organization and dynamics of nanoconfined water, including wetting/dewetting in narrow hydrophobic nanopores. A survey of simulation studies in a range of nonbiological nanopores is presented, including carbon nanotubes, synthetic nanopores, model peptide nanopores, track-etched nanopores in polymer membranes, and hydroxylated and functionalized nanoporous silica. These reveal a complex relationship between pore size/geometry, the nature of the pore lining, and rates of water transport. Wider nanopores with hydrophobic linings favor water flow whereas narrower hydrophobic pores may show dewetting. Simulation studies over the past decade of the behavior of water in a range of biological nanopores are described, including porins and β-barrel protein nanopores, aquaporins and related polar solute pores, and a number of different classes of ion channels. Water is shown to play a key role in proton transport in biological channels and in hydrophobic gating of ion channels. An overall picture emerges, whereby the behavior of water in a nanopore may be predicted as a function of its hydrophobicity and radius. This informs our understanding of the functions of diverse channel structures and will aid the design of novel nanopores. Thus, our current level of understanding allows for the design of a nanopore which promotes wetting over dewetting or vice versa. However, to design a novel nanopore, which enables fast, selective, and gated flow of water de novo would remain challenging, suggesting a need for further detailed simulations alongside experimental evaluation of more complex nanopore systems.

Shape-selective adsorption mechanism of CS-Z1 microporous molecular sieve for organic pollutants

A self-made microporous molecular sieve CS-Z1 has been found to have excellent adsorption performance for small molecular nitrile and pyridine pollutants in acrylonitrile production wastewater. In order to explore its adsorption mechanism, the adsorption kinetics, isotherms and thermodynamics of CS-Z1 for eight nitrile and pyridine organic pollutants with different structures and properties were investigated. Meanwhile, the analysis of molecular dynamics simulation based on density functional theory was conducted to revel the adsorption-diffusion process of different organic pollutants on the surface and in the pores of CS-Z1. Both the experimental and simulated results verified the shape-selective adsorption mechanism of CS-Z1 for these organic pollutants. The adsorption processes of CS-Z1 for these pollutants were spontaneous physical adsorption, and the adsorption efficiency of CS-Z1 mainly depended on the molecular size of pollutant. Benefitting from the flexible crystalline structure of CS-Z1 and the breathing vibration of CS-Z1 orifices, it could adsorb some pollutants with slightly larger size than its pore diameter. Molecular dynamics simulation results visually display the shape-selective adsorption process of CS-Z1 for these pollutants through the respiratory effect of CS-Z1 molecular sieve orifices.

Molecular mechanisms of pore formation and membrane disruption by the antimicrobial lantibiotic peptide Mutacin 1140

The emergence of antibiotic-resistance is a major concern to global human health and identification of novel antibiotics is critical to mitigate the threat.

Single-molecule electrochemistry in nanochannels: probing the time of first passage

The diffusive mass transport of individual redox molecules was probed experimentally in microfabricated nanogap electrodes. The residence times for molecules inside a well-defined detection volume were extracted and the resulting distribution was compared with quantitative analytical predictions from random-walk theory for the time of first passage. The results suggest that a small number of strongly adsorbing sites strongly influence mass transport at trace analyte levels.

Hydration water dynamics around a protein surface: a first passage time approach

A stochastic noise-driven dynamic model is proposed to study the diffusion of water molecules around a protein surface, under the effect of thermal fluctuations that arise due to the collision of water molecules with the surrounding environment. The underlying dynamics of such a system may be described in the framework of the generalized Langevin equation, where the thermal fluctuations are assumed to be algebraically correlated in time, which governs the non-Markovian behavior of the system. Results of the calculations of mean-square displacement and the velocity autocorrelation function reveal that the hydration water around the protein surface follows subdiffusive dynamics at long times. Analytical expressions for the first passage time distribution, survival probability, mean residence time and mean first passage time of water molecules are derived for different boundary conditions, to analyze hydration water dynamics under the effect of thermally correlated noise. The results depict a unimodal distribution of the first passage time unlike Brownian motion. The survival probability of hydration water follows a stretched exponential decay for both boundary conditions. The mean residence time of the hydration water molecule for different initial positions increases with increase in the complexity/heterogeneity of the surrounding environment for both boundary conditions. The mean first passage time of the water molecule to reach the absorbing/reflecting boundary follows an asymptotic power law with respect to the thickness of the hydration layer, and increases with increase in the complexity/heterogeneity of the environment.

Layer-by-layer Assembled Membranes with Immobilized Porins

With the synthesis and functionalization of membranes for selective separations, reactivity, and stimuli responsive behavior arises new and advanced opportunities. The integration of bio-based channels is one of these advancements in membrane technologies. By a layer-by-layer (LbL) assembly of polyelectrolytes, outer membrane protein F trimers (OmpF) or "porins" from Escherichia coli with a central pore of ~2 nm diameter at its opening and ~0.7 × 1.1 nm at its constricted region are immobilized within the pores of poly(vinylidene fluoride) microfiltration membranes, as opposed to traditional ruptured lipid bilayer or vesicles processes. These OmpF-membranes demonstrate selective rejections of non-charged organics over ionic solutes, allowing the passage of salts up to 2 times higher than traditional nanofiltration membranes starting with rejections of 84% for 0.4-1.0 kDa organics. The presence of charged groups in OmpF membranes also leads to pH-dependent salt rejection through Donnan exclusion. These OmpF-membranes also show exceptional durability and stability, delivering consistent and constant permeability and recovery for over 160 h of operation. Characterization of solutions containing OmpF, and membranes were conducted during each stage of the process, including detection by fluorescence labelling (FITC), zeta potential, pH responsiveness, flux changes, and rejections of organic-inorganic solutions.

Molecular dynamics simulation of water permeation through the alpha-hemolysin channel

The alpha-hemolysin (AHL) nanochannel is a non-selective channel that allows for uncontrolled transport of small molecules across membranes leading to cell death. Although it is a bacterial toxin, it has promising applications, ranging from drug delivery systems to nano-sensing devices. This study focuses on the transport of water molecules through an AHL nanochannel using molecular dynamics (MD) simulations. Our results show that AHL can quickly transport water across membranes. The first-passage time approach was used to estimate the diffusion coefficient and the mean exit time. To study the energetics of transport, the potential of mean force (PMF) of a water molecule along the AHL nanochannel was calculated. The results show that the energy barriers of water permeation across a nanopore are always positive along the channel and the values are close to thermal energy (kBT). These findings suggest that the observed quick permeation of water is due to small energy barriers and a hydrophobic inner channel surface resulting in smaller friction. We speculate that these physical mechanisms are important in how AHL causes cell death.

Origin of subdiffusion of water molecules on cell membrane surfaces

Water molecules play an important role in providing unique environments for biological reactions on cell membranes. It is widely believed that water molecules form bridges that connect lipid molecules and stabilize cell membranes. Using all-atom molecular dynamics simulations, we show that translational and rotational diffusion of water molecules on lipid membrane surfaces exhibit subdiffusion and aging. Moreover, we provide evidence that both divergent mean trapping time (continuous-time random walk) and long-correlated noise (fractional Brownian motion) contribute to this subdiffusion. These results suggest that subdiffusion on cell membranes causes the water retardation, an enhancement of cell membrane stability, and a higher reaction efficiency.

1/ f Fluctuations of amino acids regulate water transportation in aquaporin 1

Aquaporins (AQPs), which transport water molecules across cell membranes, are involved in many physiological processes. Recently, it is reported that the water-water interactions within the channel are broken at the aromatic/arginine selectivity filter (ar/R region), which prevents proton transportation [U. K. Eriksson et al., Science 340, 1346 (2013)]. However, the effects of the conformational fluctuations of amino acids on water transportation remain unclear. Using all-atom molecular dynamics simulations, we analyze water transportation and fluctuations of amino acids within AQP1. The amino acids exhibit 1/f fluctuations, indicating possession of long-term memory. Moreover, we find that water molecules crossing the ar/R region obey a non-Poisson process. To investigate the effect of 1/f fluctuations on water transportation, we perform restrained molecular dynamics simulations of AQP1 and simple Langevin stochastic simulations. As a result, we confirm that 1/f fluctuations of amino acids contribute to water transportation in AQP1. These findings appreciably enhance our understanding of AQPs and suggest possibilities for developing biomimetic nanopores.

Power-law trapping of water molecules on the lipid-membrane surface induces water retardation

Cell membranes provide unique local environments for biological reactions, where the diffusion of biomolecules as well as water molecules plays critical roles. Translational and rotational motions of water molecules near membranes are known to be slower than those in bulk. Using all-atom molecular dynamics simulations of a membrane, we show that the temperature dependence of the water molecular motions on the membrane surface is different from that in bulk. Decreasing temperature enhances the water retardation on the membrane surface, and the lateral motions of water molecules are correlated with the vertical motions. We find that trapping times of water molecules onto membrane surfaces are distributed according to a power-law distribution and that the power-law exponents depend linearly on temperature, suggesting a random energy landscape picture. Moreover, we find that water molecules on the membrane surfaces exhibit subdiffusions in translational motions.

Decay of unstable states driven by colored noise in an electromagnetic field

The statistics of the first passage time in connection with the quasideterministic (QD) approach is used to characterize the non-Markovian decay process of the unstable state of an electrically charged Brownian particle under the influence of an electromagnetic field. We consider a constant magnetic field and a fluctuating electric field, which satisfies the properties of a Gaussian exponentially correlated noise. It is shown that at the beginning of the decay process, the magnetic field is strongly coupled to the noise correlation time and thus the requirements of the QD approach are not satisfied. Only in the approximation of a weak coupling between both parameters can the time characterization of the decay process be successfully achieved. Our theoretical approach relies on a Langevin equation for the charged particle in an arbitrary two-dimensional unstable potential and applies to a bistable potential as a particular case.

First-passage-time analysis of atomic-resolution simulations of the ionic transport in a bacterial porin

We have studied the dynamics of chloride and potassium ions in the interior of the Outer membrane porin F (OmpF) under the influence of an external electric field. From the results of extensive all-atom molecular dynamics (MD) simulations of the system, we computed several first-passage-time (FPT) quantities to characterize the dynamics of the ions in the interior of the channel. Such FPT quantities obtained from MD simulations demonstrate that it is not possible to describe the dynamics of chloride and potassium ions inside the whole channel with a single constant diffusion coefficient. However, we showed that a valid, statistically rigorous description in terms of a constant diffusion coefficient D and an effective deterministic force F(eff) can be obtained after appropriate subdivision of the channel in different regions suggested by the x-ray structure. These results have important implications for popular simplified descriptions of channels based on the one-dimensional Poisson-Nernst-Planck equations. Also, the effect of entropic barriers on the diffusion of the ions is identified and briefly discussed.

Anomalous dynamics of water confined in MCM-41 at different hydrations

We present the results of molecular dynamics simulations on water confined in a silica pore of 15 Å modelled to mimic the MCM-41 structure. We focus on the dynamical properties of water for different hydration levels of the pore. Density profiles show a well-defined double-layer structure close to the surface. From the analysis of the layers we find that water molecules close to the substrate show an anomalous diffusion which is connected to the behaviour of the residence time distribution. The interaction with the substrate induces temporal disorder. As a consequence strong deviations are found from the Markovian processes that usually determine the long-time diffusion properties. The residence time distribution of the water molecules in the inner part of the pore, far from the hydrophilic surface, is compatible with a Markovian process.

Detection of weak and large electric fields through the transient dynamics of a Brownian particle in an electromagnetic field

In this work we present a mechanism to detect the presence of an external electric field of either weak or large amplitude by means of the decay process from an unstable state, described by a bistable potential, of an electrically charged Brownian particle embedded in a uniform electromagnetic field. Since the detection process takes place around the initial unstable state of the bistable potential, our theoretical description is given in the linear approximation of the aforementioned potential. The decay process is characterized through the statistics of the passage time distribution calculated by means of two theoretical approaches relying on the overdamped Langevin equation: one is the quasideterministic approach valid for large times and used for the detection of weak signals, whereas the other one is the rotational approach, valid for intermediate times and adequate for the detection of large electric-field amplitudes.

Reply to Comment on 'Diffusion of water and sodium counter-ions in nanopores of β-lactoglobulin crystal: a molecular simulation study'

The analysis in Hu and Jiang's Comment to our paper cannot reveal long-time diffusion, and incorrectly led the authors to conclude that the diffusion in beta-lactoglobuline is anomalous. In this context, the limitations of applying a mean-square displacement analysis to short, heterogeneous pore channels are discussed. A more appropriate approach based on first-passage time analysis is illustrated by a detailed analysis of water motion in a natural membrane protein channel. The partitioning and the motion of water molecules between core and surface hydration layers is discussed. Finally, the calculation of the water density profile is commented upon.