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Interplay between Hydrodynamics and the Free Energy Surface in the Assembly of Nanoscale Hydrophobes
Abstract
Solvent plays an important role in the relative motion of nanoscopic bodies, and the study of such phenomena can help elucidate the mechanism of hydrophobic assembly, as well as the influence of solvent-mediated effects on in vivo motion in crowded cellular environments. Here we study important aspects of this problem within the framework of Brownian dynamics. We compute the free energy surface that the Brownian particles experience and their hydrodynamic interactions from molecular dynamics simulations in explicit solvent. We find that molecular scale effects dominate at short distances, thus giving rise to deviations from the predictions of continuum hydrodynamic theory. Drying phenomena, solvent layering, and fluctuations engender distinct signatures of the molecular scale. The rate of assembly in the diffusion-controlled limit is found to decrease from molecular scale hydrodynamic interactions, in opposition to the free energy driving force for hydrophobic assembly, and act to reinforce the influence of the free energy surface on the association of more hydrophilic bodies.
... The complications are mainly two fold. First, as has been seen in studies of model systems [3][4][5][6][7][8][9][10], various proteins [11][12][13], human immunodeficiency virus [14] and actual anticancer drugs [15][16][17], the solvent often manifests itself at the molecular scale. While coarse-grained models can be fit to explicit solvent molecular dynamics (MD) simulations [3], predictive power can be attained only by performing all-atom MD. ...
... In this paper we consider a popular prototypical cavityligand system in explicit water where the attraction between water and the two nanoscale objects, namely a fullerene molecule and a spherical cavity, is weak [3,4,[6][7][8][9][10]. We provide a full dynamical picture of the unbinding process demonstrating the clear role of water. ...
... Previous pioneering studies [6-8, 10, 19-21] involving explicit all-atom MD, brownian dynamics, transition path sampling and other approaches have clearly shown that the association in such systems has a clear signature of solvent fluctuations and a sharp dewetting transition as the nanoscale objects approach each other -if sterically constrained to move along the axis of symmetry. This is a popular setup that has been considered in numerous studies over the years [3,[6][7][8]10], and is suggestive of biological systems where steric hindrances in the binding pocket do not allow the ligand to roll or move in a free manner [12,15]. Using unbiased MD and brownian dynamics tools, it was previously possible to calculate the timescales of association or binding for such systems that explicitly accounted for the dewetting transition [3]. ...
A key factor influencing a drug's efficacy is its residence time in the
binding pocket of the host protein. Using atomistic computer simulation to
predict this residence time and the associated dissociation process is a
desirable but extremely difficult task due to the long timescales involved.
This gets further complicated by the presence of biophysical factors such as
steric and solvation effects. In this work, we perform molecular dynamics (MD)
simulations of the unbinding of a popular prototypical hydrophobic
cavity-ligand system using a metadynamics based approach that allows direct
assessment of kinetic pathways and parameters. When constrained to move in an
axial manner, we find the unbinding time to be on the order of 4000 sec. In
accordance with previous studies, we find that the ligand must pass through a
region of sharp dewetting transition manifested by sudden and high fluctuations
in solvent density in the cavity. When we remove the steric constraints on
ligand, the unbinding happens predominantly by an alternate pathway, where the
unbinding becomes 20 times faster, and the sharp dewetting transition instead
becomes continuous. We validate the unbinding timescales from metadynamics
through a Poisson analysis, and by comparison through detailed balance to
binding timescale estimates from unbiased MD. This work demonstrates that
enhanced sampling can be used to perform explicit solvent molecular dynamics
studies at timescales previously unattainable, obtaining direct and reliable
pictures of the underlying physio-chemical factors including free energies and
rate constants.
... Several previous works addressed the estimation of position-dependent friction coefficients in underdamped Langevin models: 38,40,81 typically, the dissociation process of small species in water appears to proceed with an increase of the friction towards the transition state. Specifically for the case of the fullerene dimer in water, Morrone and co-workers investigated the position dependence of the friction, albeit in the overdamped limit, 82 finding a similar trend. Unfortunately, the umbrella sampling approach based on CV autocorrelation that was shown effective in the overdamped regime in Refs. ...
Rare events include many of the most interesting transformation processes in condensed matter, from phase transitions to biomolecular conformational changes to chemical reactions. Access to the corresponding mechanisms, free-energy landscapes and kinetic rates can in principle be obtained by different techniques after projecting the high-dimensional atomic dynamics on one (or a few) collective variable. Even though it is well-known that the projected dynamics approximately follows – in a statistical sense – the generalized, underdamped or overdamped Langevin equations (depending on the time resolution), to date it is nontrivial to parameterize such equations starting from a limited, practically accessible amount of non-ergodic trajectories. In this work we focus on Markovian, underdamped Langevin equations, that arise naturally when considering, e.g., numerous water-solution processes at sub-picosecond resolution. After contrasting the advantages and pitfalls of different numerical approaches, we present an efficient parametrization strategy based on a limited set of molecular dynamics data, including equilibrium trajectories confined to minima and few hundreds transition path sampling-like trajectories. Employing velocity autocorrelation or memory kernel information for learning the friction and likelihood maximization for learning the free-energy landscape, we demonstrate the possibility to reconstruct accurate barriers and rates both for a benchmark system and for the interaction of carbon nanoparticles in water.
... Despite its simplicity (and the uncertainty on the value of N u ), this geometrical approach provides a simple rationalization and scaling for the surprisingly low critical volume fraction required for obtaining shear thickening with these highly textured particles. Furthermore, the proposed criterion for CST being based on frictional contact between particles [15,27,37,38], our observations, and scaling are consistent with a mechanism involving solid friction that is amplified by short-range (<1 nm) reversible physico-chemical interactions [56] (such as hydrogen bonds [57]) rather than a scenario based purely on hydrodynamic forces, which might be expected to dominate at long range (>100 nm) [11,[58][59][60]. Although moderate CST can be obtained in the absence of hydrogen bond interactions ( in Fig. 4), we find that it is fully suppressed over the range of volume fractions studied by reducing the nanometric roughness of the particles ( in Fig. 4), thus providing a key way to tune the solid friction. ...
Shear thickening denotes the reversible increase in viscosity of a suspension of rigid particles under external shear. This ubiquitous phenomenon has been documented in a broad variety of multiphase particulate systems, while its microscopic origin has been successively attributed to hydrodynamic interactions and frictional contact between particles. The relative contribution of these two phenomena to the magnitude of shear thickening is still highly debated, and we report here a discriminating experimental study using a model shear-thickening suspension that allows us to independently tune both the surface chemistry and the surface roughness of the particles. We show here that both properties matter when it comes to continuous shear thickening (CST) and that the presence of hydrogen bonds between the particles is essential to achieve discontinuous shear thickening (DST) by enhancing solid friction between closely contacting particles. Moreover, a simple argument allows us to predict the onset of CST, which for these very rough particles occurs at a critical volume fraction much lower than that previously reported in the literature. Finally, we demonstrate how mixtures of particles with opposing surface chemistry make it possible to finely tune the shear-thickening response of the suspension at a fixed volume fraction, paving the way for a fine control of the shear-thickening transition in engineering applications.
... 41) which only deviates by the constraint f c from the Markovian LE in the sections above. Note that the constant velocityẋ(t) = v c (preserved by f c ) allows us to setẍ = 0. To derive an estimate for the free energy at point x, it is possible to perform an ensemble average over numerous dcTMD runs which yields based on ξ(t) = 0[42] F (x) = W (x) − v c x x 0 Γ(y)dy + const(4.42) ...
Understanding the dynamical behavior of proteins is a highly challenging area of current research. Based on the progress in algorithmic methods and the increase of computational power in the recent years, molecular dynamics simulations have emerged as powerful tool to access molecular motions on time scales from femto- to milliseconds. However, the resulting data is so overwhelming that a suitable interpretation framework is needed in order to detect and analyse the essential dynamics of the system under study. Frequently, following a dimensionality reduction to identify collective variables, the dynamics are described in terms of a diffusive motion on a low-dimensional free energy landscape. By using projection operator approaches, such as developed by Zwanzig, it is possible to derive coarse-grained equations of motions for the collective variables, such as the generalized Langevin equation. Going further, by assuming a time scale separation between the slow dynamics along the system coordinate and the fast fluctuations of the bath, this equation can be simplified to the (memory-less) Markovian Langevin equation, which describes the system dynamics in terms of a deterministic drift, a Stokes’ friction and a stochastic force. Alternatively, an additional step of coarse graining can be applied in order to account for the dynamics in terms of jumps between metastable conformational states. By furthermore assuming that those jumps are memory-free, a so-called Markov state model can be constructed.
In this thesis the virtues and shortcomings of data-based Markovian modeling are investigated. In particular, two modifications of the data-driven Langevin equation are presented: the rescaled and the binned data-driven Langevin equation. While the former approach allows for the rescaling of the dissipative force of the model, the latter concept enables the analysis of extensive MD data. In addition, it is investigated under which conditions the data-driven Langevin equation can be applied in the nonequilibrium regime. By considering molecular dynamics simulations of several systems with varying complexity it is shown that Markovian models can serve as powerful system descriptions of nontrivial dynamics. First, an one-dimensional model of sodium chloride in water and a five-dimensional model of the small Aib 9 peptide are constructed. Then, the Markovian framework is challenged by considering the dynamics of the 164-residue T4 lysozyme, the unbinding of benzamidine from trypsin and the unbinding of a resorcinol scaffold-based inhibitor from the N-terminal domain of heat shock protein 90. The latter two systems exhibit dynamics on the order of milliseconds or even seconds. To investigate the nonequilibrium regime, the enforced dissociation of sodium chloride in water and the pressure-jump induced nucleation and growth process in a liquid of hard spheres are considered.
... [1][2][3][4][5] For example, it has been recognized that hydrophobic interaction as the major driving force plays a key role in folding of proteins and self-assembly of biomolecules in cells. [1,[6][7][8][9] A fascinating effect appreciated recently is that, when two nanoscale hydrophobic objects approach each other and reach a critical separation, though large enough to accommodate water molecules, a dewetting (drying) transition will occur in the inter-object region. This behavior will lead to long-range hydrophobic attraction, resulting in hydrophobic collapse, [7,10] which is possibly relevant to collapse of multi-domain proteins and folding of heterogeneous globular proteins. ...
We investigate the influence of an external electric field on the dewetting behavior of nitrogen-water systems between two hydrophobic plates using molecular dynamics simulations. It is found that the critical distance of dewetting increases obviously with the electric field strength, indicating that the effective range of hydrophobic attraction is extended. The mechanism behind this interesting phenomenon is related to the rearrangement of hydrogen bond networks between water molecules induced by the external electric field. Changes in the hydrogen bond networks and in the dipole orientation of the water molecules result in the redistribution of the neutral nitrogen molecules, especially in the region close to the hydrophobic plates. Our findings may be helpful for understanding the effects of the electric field on the long-range hydrophobic interactions.
... Berne and co-workers studied extensively the dewetting transition leading to self-assembly process of hydrophobic moieties, protein folding (15), and protein collapse (2,16). Recently, they demonstrated that solvent fluctuation triggers crucial hydrodynamic interactions in hydrophobic collapse (17,18). ...
Significance
Despite its paramount importance, a microscopic characterization of protein association/dissociation has remained a nontrivial challenge. We have carried out atomistic simulation (biased and unbiased) to characterize insulin dimerization and dissociation process fully with a special attention to the role of water. Insulin is our system of interest because its monomer is the biologically active species but it remains in the inactive hexameric and dimeric storage forms. Our study reveals that at larger separation of two monomers (R MM ∼ 5 nm), dynamical properties of confined water molecules exhibit considerable deviation from bulklike characteristics, although structure remains bulklike. Analysis of both the large-scale water density fluctuation and protein conformational changes allows us to provide a complete picture of protein dissociation–association mechanism.
... In particular, it has been demonstrated that the dry-wet transitions are a precursor of the ligand-receptor binding and unbinding (17,21,22). Besides being the origin for the thermodynamically driven forces, water fluctuations also modify the friction and kinetics of associating hydrophobic molecules (23)(24)(25)(26)(27), slowing down the binding kinetics and giving rise to local non-Markovian effects (18,27). ...
Ligand-receptor binding and unbinding are fundamental biomolecular processes and particularly essential to drug efficacy. Environmental water fluctuations, however, impact the corresponding thermodynamics and kinetics and thereby challenge theoretical descriptions. Here, we devise a holistic, implicit-solvent, multi-method approach to predict the (un)binding kinetics for a generic ligand-pocket model. We use the variational implicit-solvent model (VISM) to calculate the solute-solvent interfacial structures and the corresponding free energies, and combine the VISM with the string method to obtain the minimum energy paths and transition states between the various metastable ('dry' and 'wet') hydration states. The resulting dry-wet transition rates are then used in a spatially-dependent multi-state continuous-time Markov chain Brownian dynamics simulations, and the related Fokker-Planck equation calculations, of the ligand stochastic motion, providing the mean first-passage times for binding and unbinding. We find the hydration transitions to significantly slow down the binding process, in semi-quantitative agreement with existing explicit-water simulations, but significantly accelerate the unbinding process. Moreover, our methods allow the characterization of non-equilibrium hydration states of pocket and ligand during the ligand movement, for which we find substantial memory and hysteresis effects for binding versus unbinding. Our study thus provides a significant step forward towards efficient, physics-based interpretation and predictions of the complex kinetics in realistic ligand-receptor systems.
... Solvent-mediated collective phenomena, such as hydrophobic effects, are important in a wide variety of contexts [1][2][3][4][5][6] ranging from detergency [7,8] and colloidal assembly [9,10], to protein folding [11,12] and aggregation [13,14]. Consequently, numerous theoretical, simulation, and experimental studies have been devoted to uncovering the molecular underpinnings of hydrophobic hydration and interactions [15][16][17][18]. ...
Hydrophobic effects drive diverse aqueous assemblies, such as micelle formation or protein folding, wherein the solvent plays an important role. Consequently, characterizing the free energetics of solvent density fluctuations can lead to important insights into these processes. Although techniques such as the indirect umbrella sampling (INDUS) method (Patel et al. J. Stat. Phys. 2011, 145, 265-275) can be used to characterize solvent fluctuations in static observation volumes of various sizes and shapes, characterizing how the solvent mediates inherently dynamic processes, such as self-assembly or conformational change, remains a challenge. In this work, we generalize the INDUS method to facilitate the enhanced sampling of solvent fluctuations in dynamical observation volumes, whose positions and shapes can evolve. We illustrate the usefulness of this generalization by characterizing water density fluctuations in dynamic volumes pertaining to the hydration of flexible solutes, the assembly of small hydrophobes, and conformational transitions in a model peptide. We also use the method to probe the dynamics of hard spheres.
... The major distinguishing feature of RAVE is that the RC is learnt together with its Boltzmann probability distribution, 18 which can then serve as the ideal bias potential and be leveraged outside pre-existing biasing frameworks such as metadynamics or umbrella sampling. [4][5][6]10 In the original proof-of-concept paper, RAVE was applied to model potentials including a fullerene-nanopocket [19][20][21][22][23] unbinding test case where it was demonstrated that sampling in simulations could indeed be enhanced with the simultaneous on-the-fly learning of the RC and bias potential. 18 We demonstrated that RAVE could reproduce the dissociation free energy profile for the unbinding of a fullerene from a nanopocket in much less computational time than using the popular metadynamics and umbrella sampling methods. ...
In this work we demonstrate how to leverage our recent iterative deep learning-all atom molecular dynamics (MD) technique 'Reweighted autoencoded variational Bayes for enhanced sampling (RAVE)' (Ribeiro, Bravo, Wang, Tiwary, J. Chem. Phys. 149, 072301 (2018)) for sampling protein-ligand unbinding mechanisms and calculating absolute binding affinities when plagued with difficult to sample rare events. RAVE iterates between rounds of MD and deep learning, and unlike other enhanced sampling methods, it stands out in simultaneously learning both a low-dimensional physically interpretable reaction coordinate (RC) and associated free energy. Here, we introduce a simple but powerful extension to RAVE which allows learning a position-dependent RC expressed as a superposition of piecewise linear RCs valid in different metastable states. With this approach, we retain the original physical interpretability of a RAVE-derived RC while making it applicable to a wider range of complex systems. We demonstrate how in its multi-dimensional form introduced here, RAVE can efficiently simulate the unbinding of the tightly bound benzene-lysozyme (L99A variant) complex, in all atom-precision and with minimal use of human intuition except for the choice of a larger dictionary of order parameters. These simulations had a 100% success rate, and took between 3-50 nanoseconds for a process that takes on an average close to few hundred milliseconds, thereby reflecting a seven order of magnitude acceleration relative to straightforward MD. Furthermore, without any time-dependent biasing, the trajectories display clear back-and-forth movement between various metastable intermediates, demonstrating the reliability of the RC and its probability distribution learnt in RAVE. Our binding free energy is in good agreement with other reported simulation results. We thus believe that RAVE, especially in its multi-dimensional variant introduced here, will be a useful tool for simulating the dissociation process of practical biophysical systems with rare events in an automated manner with minimal use of human intuition.
... Does the solvent perfectly stick on the surface, or partially slip [8][9][10][11] ? If it sticks, is there an immobile layer of molecules at the surface of the nanoparticle that modifies the effective hydrodynamic size of the particle ? ...
We have used non-equilibrium molecular dynamics to simulate the flow of water molecules around a charged nanoparticle described at the atomic scale. These non-equilibrium simulations allowed us to compute the friction coefficient of the nanoparticle and then to deduce its hydrodynamic radius. We have compared two different strategies to thermostat the simulation box, since the low symmetry of the flow field renders the control of temperature non trivial. We show that both lead to an adequate control of the temperature of the system. To deduce the hydrodynamic radius of the nanoparticle we have employed a partial thermostat, which exploits the cylindrical symmetry of the flow field. Thereby, only a part of the simulation box far from the nanoparticle is thermostated. We have taken into account the finite concentration of the nanoparticle when calculating the friction force acting on it. We have focused on the case of polyoxometalate ions, which are inorganic charged nanoparticles. It appears that, for a given structure of the nanoparticle at the atomic level, the hydrodynamic radius significantly increases with the nanoparticle’s charge, a phenomenon that had not been quantified so far using molecular dynamics. The presence of an added salt only slightly modifies the hydrodynamic radius.
... 7 It is broadly accepted that hydrophobic interactions push oily molecules together and play a fundamental role in, e.g., self-assembly processes, protein folding, and protein−protein and protein−ligand interactions in water. Counterintuitively, recent studies have indicated that weakly attractive van der Waals interactions of large nonpolar solutes such as neopentane, 8 adamantane, 8 fullerene, 9 and small alcohol molecules like tert-butanol 10 with water are sufficiently strong to drive these solutes apart in aqueous solution. 1,7,10,11 Repulsive water-mediated hydrophobic interactions, which are also suggested by positive osmotic second virial coefficients, 6,12−14 may thus overcompensate the direct, attractive, van der Waals interactions between the nonpolar solutes, preventing their association. ...
... [1][2][3][4] The thermodynamics and kinetics of confinement induced evaporation have continued to receive much attention as they are believed to underlie hydrophobic self-assembly. [4][5][6][7] In addition, such transitions are suspected to have significant biophysical consequences, underlying the function of membrane bound proteins such as ion-channels 8,9 and G-protein coupled receptors, 10 both of which are common drug targets due to their role in signaling. ...
Liquid water confined between nanoscale hydrophobic objects can become metastable with respect to its vapor at nanoscale separations. While the separations are only several molecular diameters, macroscopic theories are often invoked to interpret the thermodynamics and kinetics of water under confinement. We perform detailed rate and free energy calculations via molecular simulations in order to assess the dependence of the rate of evaporation, free energy barriers, and free energy differences between confined liquid and vapor upon object separation and compare them to the relevant macroscopic theories. At small enough separations, the rate of evaporation appears to deviate significantly from the predictions of classical nucleation theory, and we attribute such deviations to changes in the structure of the confined liquid film. However, the free energy difference between the confined liquid and vapor phases agrees quantitatively with macroscopic theory, and the free energy barrier to condensation displays qualitative agreement. Overall, the present work suggests that theories attempting to capture the kinetic behavior of nanoscale systems should incorporate structural details rather than treating it as a continuum.
... 5 Numerous papers report calculations of the density fluctuation near solid/liquid and protein/liquid interfaces. [1][2][3]6,7 These results indicate that the P(n) distribution is wider near the hydrophobic surfaces and becomes narrower with increasing hydrophilicity. For example, if n ̅ is the average number of water molecules in the probe volume, then both P(n > n ̅ ) and P(n < n ̅ ) decrease with increasing hydrophilicity. ...
Using molecular dynamics simulation, we studied the density fluctuations and cavity formation probabilities in aqueous solutions and their effect on the hydration of CO2. With increasing salt concentration, we report an increased probability of observing a larger than the average number of species in the probe volume. Our energetic analyses indicate that the van der Waals and electrostatic interactions between CO2 and aqueous solutions become more favorable with increasing salt concentration, favoring the solubility of CO2 (salting in). However, due to the decreasing number of cavities forming when salt concentration is increased, the solubility of CO2 decreases. The formation of cavities was found to be the primary control on the dissolution of gas, and is responsible for the observed CO2 salting-out effect. Our results provide the fundamental understanding of the density fluctuation in aqueous solutions and the molecular origin of the salting-out effect for real gas.
... In an important study, Morrone et al. 94 examined the spatial dependence of the friction coefficient in the Brownian limit for two nonpolar bodies. They find that the friction coefficient deviates from continuum hydrodynamic predictions at small separations and depends on the nature of solute-solvent interactions. ...
An outstanding challenge in computational biophysics is the simulation of a living cell at molecular detail. Over the past several years, using Stokesian dynamics, progress has been made in simulating coarse grained molecular models of the cytoplasm. Since macromolecules comprise 20%-40% of the volume of a cell, one would expect that steric interactions dominate macromolecular diffusion. However, the reduction in cellular diffusion rates relative to infinite dilution is due, roughly equally, to steric and hydrodynamic interactions, HI, with nonspecific attractive interactions likely playing rather a minor role. HI not only serve to slow down long time diffusion rates but also cause a considerable reduction in the magnitude of the short time diffusion coefficient relative to that at infinite dilution. More importantly, the long range contribution of the Rotne-Prager-Yamakawa diffusion tensor results in temporal and spatial correlations that persist up to microseconds and for intermolecular distances on the order of protein radii. While HI slow down the bimolecular association rate in the early stages of lipid bilayer formation, they accelerate the rate of large scale assembly of lipid aggregates. This is suggestive of an important role for HI in the self-assembly kinetics of large macromolecular complexes such as tubulin. Since HI are important, questions as to whether continuum models of HI are adequate as well as improved simulation methodologies that will make simulations of more complex cellular processes practical need to be addressed. Nevertheless, the stage is set for the molecular simulations of ever more complex subcellular processes.
... where Einstein's convention for summation over repeated indices is assumed. The particle mobility tensor in the present geometry can be written as an algebraic sum of two distinct contributions µ γλ αβ (r γ , r λ , ω) = b γλ αβ (r γ , r λ ) + ∆µ γλ αβ (r γ , r λ , ω) , (1) where b γλ αβ is the pair-mobility in an unbounded geometry (bulk flow), and ∆µ γλ αβ is the frequency-dependent correction due to the presence of the elastic membrane. An analogous relation holds for µ γγ αβ . ...
We present an analytical calculation of the hydrodynamic interaction between two spherical particles near an elastic interface such as a cell membrane. The theory predicts the frequency dependent self- and pair-mobilities accounting for the finite particle size up to the 5th order in the ratio between particle diameter and wall distance as well as between diameter and interparticle distance. We find that particle motion towards a membrane with pure bending resistance always leads to mutual repulsion similar as in the well-known case of a hard-wall. In the vicinity of a membrane with shearing resistance, however, we observe an attractive interaction in a certain parameter range which is in contrast to the behavior near a hard wall. This attraction might facilitate surface chemical reactions. Furthermore, we show that there exists a frequency range in which the pair-mobility for perpendicular motion exceeds its bulk value, leading to short-lived superdiffusive behavior. Using the analytical particle mobilities we compute collective and relative diffusion coefficients. The appropriateness of the approximations in our analytical results is demonstrated by corresponding boundary integral simulations which are in excellent agreement with the theoretical predictions.
... The physics behind such a phenomenon has previously been modelled to simulate the evaporation of water from a confined space between 'small' (1 × 1 nm 2 ) or 'large' (3 × 3 nm 2 ) hydrophobic surfaces, specifically modelling the free energy profiles and the rate of evaporation as functions of diameter (which we define here as the spacing, d space ) between the walls of a capillary 31,42,43 . It has been shown that, with respect to larger surfaces, the liquid inside the confined space near a d critical = 15 Å becomes metastable, whereby condensed water can evaporate 31,43 . ...
Three water adsorption-desorption mechanisms are common in inorganic materials: chemisorption, which can lead to the modification of the first coordination sphere; simple adsorption, which is reversible; and condensation, which is irreversible. Regardless of the sorption mechanism, all known materials exhibit an isotherm in which the quantity of water adsorbed increases with an increase in relative humidity. Here, we show that carbon-based rods can adsorb water at low humidity and spontaneously expel about half of the adsorbed water when the relative humidity exceeds a 50-80% threshold. The water expulsion is reversible, and is attributed to the interfacial forces between the confined rod surfaces. At wide rod spacings, a monolayer of water can form on the surface of the carbon-based rods, which subsequently leads to condensation in the confined space between adjacent rods. As the relative humidity increases, adjacent rods (confining surfaces) in the bundles are drawn closer together via capillary forces. At high relative humidity, and once the size of the confining surfaces has decreased to a critical length, a surface-induced evaporation phenomenon known as solvent cavitation occurs and water that had condensed inside the confined area is released as a vapour.
... The hydrophobic effect is the phenomenon of solvent induced interaction between two or more apolar molecules [30][31][32][33][34][35]. These interactions are the main driving forces responsible of protein folding, biopolymer interaction, stability of micelles and membranes in aqueous solutions. ...
... Identification of good low dimensional CVs is in fact useful not just for enhanced sampling simula-tions such as umbrella sampling and metadynamics but also for distributed computing techniques like Markov State Models (MSM) 13 , allowing one to significantly improve the quality and reliability of the constructed kinetic models. Last but not the least, having an optimal low dimensional CV can also help in the building of Brownian dynamics type models 14,15 . Indeed, given the importance of this problem, there exists a range of methods that have been proposed to solve it [16][17][18][19][20][21][22][23] . ...
Significance
Molecular-dynamics (MD) simulations have become a versatile tool for exploration of complex molecular systems. However, they are limited in the timescales that can be reached. Thus, over the years, a suite of enhanced-sampling algorithms have been proposed that assist MD to transcend the timescale limitation, with diverse applications across physical and life sciences. A continuing grand challenge in the success of many such sampling methods pertains to a judicious choice of order parameters. In this work, we propose a new method for designing order parameters that minimizes the role played by human intuition and makes the progress significantly more automated than before. We expect this algorithm to be of great use in furthering the success of enhanced sampling.
... The velocity-rescaled Berendsen thermostat at a 0.1 ps relaxation time is used in this work. 44 We also perform the simulation in NVE ensemble to calculate the dynamic behavior of interfacial water. The NVE run is performed using the method of our previous studies. ...
As a chemical and structural simple hydrophilic material, Mg(OH)2 exhibits great potential in water environment remediation. In this work, we use molecular dynamics (MD) simulation to investigate the distribution of water molecules on the Mg(OH)2 (001) surface as well as the dynamic behaviors of interfacial water. Even though Mg(OH)2 substrate can considerably affect the density profile of water molecules as well as the water dipole orientations, no specific adsorption sites for water can be observed on the surface of Mg(OH)2. Meanwhile, the interaction of water molecules with Mg(OH)2 substrate does not disturb the hydrogen bonds between interfacial water molecules. More interestingly, the substrate has modest effect on the dynamic behaviors of interfacial water, e.g., the residence time, hydrogen bond lifetime, and self-diffusion coefficient, which is in sharp contrast to many other hydrophilic materials.
... However, the RPY tensor only includes the far-field part of hydrodynamic effects [23,24]. A recent simulation study of the association of two non-polar model objects clearly showed that at short distances (,1-2 nm) molecular scale effects dominate, giving rise to deviations from continuum hydrodynamic theory [40][41][42]. Even though a more sophisticated hydrodynamic model was used, the deviation from the atomistic simulation results was not eliminated. ...
DNA binding proteins efficiently search for their cognitive sites on long genomic DNA by combining 3D diffusion and 1D diffusion (sliding) along the DNA. Recent experimental results and theoretical analyses revealed that the proteins show a rotation-coupled sliding along DNA helical pitch. Here, we performed Brownian dynamics simulations using newly developed coarse-grained protein and DNA models for evaluating how hydrodynamic interactions between the protein and DNA molecules, binding affinity of the protein to DNA, and DNA fluctuations affect the one dimensional diffusion of the protein on the DNA. Our results indicate that intermolecular hydrodynamic interactions reduce 1D diffusivity by 30%. On the other hand, structural fluctuations of DNA give rise to steric collisions between the CG-proteins and DNA, resulting in faster 1D sliding of the protein. Proteins with low binding affinities consistent with experimental estimates of non-specific DNA binding show hopping along the CG-DNA. This hopping significantly increases sliding speed. These simulation studies provide additional insights into the mechanism of how DNA binding proteins find their target sites on the genome.
... The predictions of the uncertainties in both aggregation and breakup rates are quite sizable, unsuitable for firm predictions in applications such as drug delivery. An alternative way to estimate these rates could be found in the recent work of Morrone et al. 43 The dependence of these estimates though also hinge on the PMF and will thus still mitigate large uncertainty to the predictions. ...
For over five decades, Molecular Dynamics (MD) simulations have helped to elucidate critical mechanisms in a broad range of physiological systems and technological innovations. MD simulations are synergetic with experiments, relying on measurements to calibrate their parameters and probing "what if scenarios" for systems that are difficult to investigate experimentally. Yet in certain systems, such as nanofluidics, the results of experiments and MD simulations differ by several orders of magnitude. This discrepancy may be attributed to the spatiotemporal scales and structural information accessible by experiments and simulations. Furthermore, MD simulations rely on parameters that are often calibrated semi-empirically while the effects of their computational implementation on their predictive capabilities, have only been sporadically probed. In this work, we show that experimental and MD investigations can be consolidated through a rigorous Uncertainty Quantification framework. We employ a Bayesian probabilistic framework for large scale MD simulations of graphitic nano-structures in aqueous environments. We assess the uncertainties in the MD predictions for quantities of interest regarding wetting behavior and hydrophobicity. We focus on three representative systems: water wetting of graphene, the aggregation of fullerenes in aqueous solution, and the water transport across carbon nanotubes. We demonstrate that the dominant mode of calibrating MD potentials in nanoscale fluid mechanics, through single values of water contact angle on graphene, leads to large uncertainties and fallible quantitative predictions. We demonstrate that the use of additional experimental data reduces uncertainty, improves the predictive accuracy of MD models and consolidates the results of experiments and simulations.
... In the calculation of τ r , two limiting boundary conditions for the shear regimes of the surrounding fluid at the molecular surface can be adopted, namely 'stick' and 'slip'. The standard choice is stick conditions, although slip has been suggested to be appropriate for almost spherical molecules such as fullerenes [41]. A trial slip calculation for 1, the closest to a spherical object, leads however to a shorter T 1,p (8.3 rather than 10.2 ms), thus making the agreement with experiment worse. ...
We have investigated the structure and nuclear magnetic resonance (NMR) spectroscopic properties of some dihydrogen endofullerene nitroxides by means of density-functional theory (DFT) calculations. Quantum versus classical roto-translational dynamics of H2 have been characterized and compared. Geometrical parameters and hyperfine couplings calculated by DFT have been input to the Solomon-Bloembergen equations to predict the enhancement of the NMR longitudinal relaxation of H2 due to coupling with the unpaired electron. Estimating the rotational correlation time via computed molecular volumes leads to a fair agreement with experiment for the simplest derivative; the estimate is considerably improved by recourse to the calculation of the diffusion tensor. For the other more flexible congeners, the agreement is less good, which may be due to an insufficient sampling of the conformational space. In all cases, relaxation by Fermi contact and Curie mechanisms is predicted to be negligible.
We introduce a machine-learning-based coarse-grained molecular dynamics model that faithfully retains the many-body nature of the intermolecular dissipative interactions. Unlike the common empirical coarse-grained models, the present model is constructed based on the Mori-Zwanzig formalism and naturally inherits the heterogeneous state-dependent memory term rather than matching the mean-field metrics such as the velocity autocorrelation function. Numerical results show that preserving the many-body nature of the memory term is crucial for predicting the collective transport and diffusion processes, where empirical forms generally show limitations.
The water dispersity of nanoparticles (NPs) with net charges have been investigated using molecular dynamics simulations. The net charge enhances the repulsive interaction between the like charged NPs greatly, and hence improves the NP’s water dispersity significantly. Unexpectedly, the water dispersity of the positively charged NP is significantly better than that of the NP carrying an equal but negative net charge, though the interaction between + n e charged NPs is very close to that between -n e charged NPs. On the one hand, hydration water of a positively charged NP shows a slight stronger interaction with the surrounding molecules relative to the hydration water of the NP carrying an equal but negative net charge, which should facilitate the dissolution of the positively charged NPs. On the other hand, aggregated NPs each with a net charge of -n e interact more strongly with the water molecules simultaneously contacting multiple aggregated NPs than the aggregated NPs each with a net charge of + n e do, which proved can lead to the better dispersities of the positively charged NPs. Furthermore, the orientation of hydration water of a positively charged NP is more disordered than that of the NP carrying an equal but negative net charge. In terms of entropy, a more disordered orientation of hydration water should help the NPs disperse into the solvent. Our study may help to better understand the dissolution and hydration mechanisms of nanoscale proteins and NPs, which might be helpful in designs of NPs with high water solubility.
Markov processes provide a popular approach to construct low-dimensional dynamical models of a complex biomolecular system. By partitioning the conformational space into metastable states, protein dynamics can be approximated in terms of memory-less jumps between these states, resulting in a Markov state model (MSM). Alternatively, suitable low-dimensional collective variables may be identified to construct a data-driven Langevin equation (dLE). In both cases, the underlying Markovian approximation requires a propagation time step (or lag time) δt that is longer than the memory time τM of the system. On the other hand, δt needs to be chosen short enough to resolve the system timescale τS of interest. If these conditions are in conflict (i.e., τM > τS), one may opt for a short time step δt = τS and try to account for the residual non-Markovianity of the data by optimizing the transition matrix or the Langevin fields such that the resulting model best reproduces the observables of interest. In this work, rescaling the friction tensor of the dLE based on short-time information in order to obtain the correct long-time behavior of the system is suggested. Adopting various model problems of increasing complexity, including a double-well system, the dissociation of solvated sodium chloride, and the functional dynamics of T4 lysozyme, the virtues and shortcomings of the rescaled dLE are discussed and compared to the corresponding MSMs.
Significance
The kinetics of ligand–receptor (un)binding—how fast a ligand binds into and resides in a receptor—cannot be inferred solely from the binding affinity which describes the thermodynamic stability of the bound complex. A bottleneck in understanding such kinetics, which is critical to drug efficacy, lies in the modeling of the collective water fluctuations in apolar confinement. We develop a theoretical approach that couples a variational implicit-solvent model with the string method to describe the dry–wet transition pathways, which then serve as input for the ligand multistate Brownian dynamics. Without explicit descriptions of individual water molecules, our theory predicts the key thermodynamic and kinetic properties of unbinding and binding, the latter in quantitative agreement with explicit-water molecular dynamics simulations.
This chapter outlines what people do and do not know about supramolecular interactions in aqueous solution. While the customary viewpoint of supramolecular chemists is from the perspective of a molecular host, water scientists consider water to be the host of interest. Thus, rather than considering host and guest desolvation and host–guest complex formation, water scientists consider how water (the host) responds to the addition of a solute (the guest). The chapter aims to takes this “reverse” perspective of the water scientist and discusses what happens when a solute is taken from the gas phase into the aqueous phase. In aqueous solutions chemistry, there are three effects, the hydrophobic effect, the Hofmeister effect, and the reverse Hofmeister effect, that are intrinsically linked to each other by relatively weak solvation of solutes and the high cohesiveness of water. In studying the hydrophobic effect, scientists look to describe the attraction between nonpolar molecules in an aqueous environment.
Through molecular dynamics simulations, we examined the hydrodynamic behavior of the Brownian motion of fullerene particles based on molecular interactions. The solvation free energy and the velocity autocorrelation function (VACF) were calculated by using the Lennard-Jones (LJ) and Weeks-Chandler-Andersen (WCA) potentials for the solute-solvent and solvent-solvent interactions and by changing the size of the fullerene particles. We also measured the diffusion constant of the fullerene particles and the shear viscosity of the host fluid, and then the hydrodynamic radius HD was quantified from the Stokes-Einstein relation. The HD value exceeds that of the gyration radius of the fullerene when the solvation free energy exhibits largely negative values using the LJ potential. In contrast, aHD becomes comparable to the size of bare fullerene, when the solvation free energy is positive using the WCA potential. Furthermore, the VACF of the fullerene particles is directly comparable with the analytical expressions utilizing the Navier-Stokes equations both in incompressible and compressible forms. Hydrodynamic long-time tail t-3/2 is demonstrated for timescales longer than the kinematic time of the momentum diffusion over the particles' size. However, the VACF in shorter timescales deviates from the hydrodynamic description, particularly for smaller fullerene particles and for the LJ potential. This occurs even though the compressible effect is considered when characterizing the decay of VACF around the sound propagation time scale over the particles' size. These results indicate that the nanoscale Brownian motion is influenced by the solvation structure around the solute particles originating from the molecular interaction.
Hydrophobic effects drive diverse aqueous assemblies, such as micelle formation or protein folding, wherein the solvent plays an important role. Consequently, characterizing the free energetics of solvent density fluctuations can lead to important insights into these processes. Although techniques such as the indirect umbrella sampling (INDUS) method can be used to characterize solvent fluctuations in static observation volumes of various sizes and shapes, characterizing how the solvent mediates inherently dynamic processes, such as self-assembly or conformational change, remains a challenge. In this work, we generalize the INDUS method to facilitate the enhanced sampling of solvent fluctuations in dynamical observation volumes, whose positions and shapes can evolve. We illustrate the usefulness of this generalization by characterizing water density fluctuations in dynamic volumes pertaining to the hydration of flexible solutes, the assembly of small hydrophobes, and conformational transitions in a model peptide. We also use the method to probe the dynamics of hard spheres.
In this work we demonstrate how to leverage our recent iterative deep learning-all atom molecular dynamics (MD) technique "Reweighted autoencoded variational Bayes for enhanced sampling (RAVE)" (Ribeiro, Bravo, Wang, Tiwary, J. Chem. Phys. 149, 072301 (2018)) for investigating ligand-protein unbinding mechanisms and calculating absolute binding free energies, ΔGb, when plagued with difficult to sample rare events. In order to do so, we introduce a simple but powerful extension to RAVE that allows learning a reaction coordinate expressed as a piecewise function that is linear over all intervals. Such an approach allows us to retain the physical interpretation of a RAVE-derived reaction coordinate while making the method more applicable to a wider range of complex biophysical problems. As we will demonstrate, using as our test-case the slow dissociation of benzene from the L99A variant of lysozyme, the RAVE extension led to observing an unbinding event in 100% of the independent all-atom MD simulations, all within 3-50 nanoseconds for a process that takes on an average close to few hundred milliseconds, which reflects a seven order of magnitude acceleration relative to straightforward MD. Furthermore, we will show that without the use of time-dependent biasing, clear back-and-forth movement between metastable intermediates was achieved during the various simulations, demonstrating the caliber of the RAVE-derived piecewise reaction coordinate and bias potential, which together drive efficient and accurate sampling of the ligand-protein dissociation event. Last, we report the results for ΔGb, which via very short MD simulations, can form a strict lower-bound that is ∽2-3 kcal/mol off from experiments. We believe that RAVE, together with its multi-dimensional extension that we introduce here, will be a useful tool for simulating the slow unbinding process of practical ligand-protein complexes in an automated manner with minimal use of human intuition.
As standard unbiased molecular dynamics (MD) simulations become impractical for sampling rare events, ``targeted MD'' employs a moving distance constraint to enforce rare transitions along some reaction coordinate $x$. To calculate free energy profiles from these nonequilibrium simulations via $\Delta G(x) = W(x) - W_{\rm diss}(x)$, apart from the (readily obtained) work $W(x)$ performed on the system also the dissipated work $W_{\rm diss}(x)$ is required. By employing a second-order cumulant expansion of Jarzynski's equality combined with an analysis within Langevin theory, the dissipated work can be expressed via a nonequilibrium friction coefficient $\Gamma_{\rm NEQ}(x)$ that may be calculated on-the-fly from constraint force fluctuations. Adopting the ion dissociation of NaCl in water as test system, this friction correction is shown to result in accurate free energy profiles, even for a modest number of simulations and at high constraint velocities. As a bonus, the analysis of $\Gamma_{\rm NEQ}(x)$ may yield valuable insight into the microscopic mechanism of friction.
Using explicit-water molecular dynamics simulations of a generic pocket-ligand model, we investigate how chemical and shape anisotropy of small ligands influences the affinities, kinetic rates, and pathways for their association with hydrophobic binding sites. In particular, we investigate aromatic compounds, all of similar molecular size, but distinct by various hydrophilic or hydrophobic residues. We demonstrate that the most hydrophobic sections are in general desolvated primarily upon binding to the cavity, suggesting that specific hydration of the different chemical units can steer the orientation pathways via a “hydrophobic torque.” Moreover, we find that ligands with bimodal orientation fluctuations have significantly increased kinetic barriers for binding compared to the kinetic barriers previously observed for spherical ligands due to translational fluctuations. We exemplify that these kinetic barriers, which are ligand specific, impact both binding and unbinding times for which we observe considerable differences between our studied ligands.
We investigate how to tune the rate of hydrophobic ligand-receptor association due to the role of solvent in adjustable receptor pockets by explicit-water molecular dynamics (MD) simulations. Our model considers the binding of a spherical ligand (key/guest) to a concave surface recess in a non-polar wall as receptor (lock/host). We systematically modify the receptor's physicochemical properties in terms of geometry and dispersion attraction which, in turn, alter the water occupancy and fluctuations within the pocket. We demonstrate that even minor pocket modifications can lead to a significant acceleration of the water-mediated association. For example, the binding switches from comparably slow to fast if the binding pocket becomes only slightly deeper. We find that the degree of hydrophobicity, characterized by hydration occupancy and its fluctuations, clearly correlates with the binding times and, for instance, links the sudden acceleration to an abrupt increase in hydrophobicity. For a deeper analysis based on passage time theory, we quantify the intimate coupling between solvent fluctuations and the ligand's local dynamics and friction. The coupling exhibits substantial non-equilibrium effects and maximizes shortly before binding which slows down the binding kinetics in all cases. In sum we rationalize how the physicochemical properties of a nonpolar, concave binding site tune key-lock binding kinetics due to water-mediated forces and fluctuations. Our study thus complements the profound understanding of the solvent's influence in host-guest binding which is essential for tailored solutions in catalysis and pharmaceutical applications.
Molecular dynamics (MD) simulations have become a tool of immense use and popularity for simulating a variety of systems. With the advent of massively parallel computer resources, one now routinely sees applications of MD to systems as large as hundreds of thousands to even several million atoms, which is almost the size of most nanomaterials. However, it is not yet possible to reach laboratory timescales of milliseconds and beyond with MD simulations. Due to the essentially sequential nature of time, parallel computers have been of limited use in solving this so-called timescale problem. Instead, over the years a large range of statistical mechanics based enhanced sampling approaches have been proposed for accelerating molecular dynamics, and accessing timescales that are well beyond the reach of the fastest computers. In this review we provide an overview of these approaches, including the underlying theory, typical applications, and publicly available software resources to implement them.
Hydrophobic interactions are driven by the combined influence of the direct attraction between oily solutes and an additional water-mediated interaction whose magnitude (and sign) depends sensitively on both solute size and attraction. The resulting delicate balance can lead to a slightly repulsive water-mediated interaction that drives oily molecules apart rather than pushing them together and thus opposes their direct (van der Waals) attraction for each other. As a consequence, competing solute size-dependent crossovers weaken hydrophobic interactions sufficiently that they are only expected to significantly exceed random thermal energy fluctuations for processes that bury more than ∼1 nm of water-exposed area.
We use a recently proposed method called Spectral Gap Optimization of Order Parameters (SGOOP) (Tiwary and Berne, Proc. Natl. Acad. Sci 2016, 113, 2839 (2016)), to determine an optimal 1-dimensional reaction coordinate (RC) for the unbinding of a bucky-ball from a pocket in explicit water. This RC is estimated as a linear combination of the multiple available order parameters that collectively can be used to distinguish the various stable states relevant for unbinding. We pay special attention to determining and quantifying the degree to which water molecules should be included in the RC. Using SGOOP with under-sampled biased simulations, we predict that water plays a distinct role in the reaction coordinate for unbinding in the case when the ligand is sterically constrained to move along an axis of symmetry. This prediction is validated through extensive calculations of the unbinding times through metadynamics, and by comparison through detailed balance with unbiased molecular dynamics estimate of the binding time. However when the steric constraint is removed, we find that the role of water in the reaction coordinate diminishes. Here instead SGOOP identifies a good one-dimensional RC involving various motional degrees of freedom.
The free energetics of water density fluctuations near a surface, and the
rare low-density fluctuations in particular, serve as reliable indicators of
surface hydrophobicity; the easier it is to displace the interfacial waters,
the more hydrophobic the underlying surface. However, characterizing the free
energetics of such rare fluctuations requires computationally expensive,
non-Boltzmann sampling methods like umbrella sampling. This inherent
computational expense associated with umbrella sampling makes it challenging to
investigate the role of polarizability or electronic structure effects in
influencing interfacial fluctuations. Importantly, it also limits the size of
the volume, which can be used to probe interfacial fluctuations. The latter can
be particularly important in characterizing the hydrophobicity of large
surfaces with molecular-level heterogeneities, such as those presented by
proteins. To overcome these challenges, here we present a method for the sparse
sampling of water density fluctuations, which is roughly two orders of
magnitude more efficient than umbrella sampling. We employ thermodynamic
integration to estimate the free energy differences between biased ensembles,
thereby circumventing the umbrella sampling requirement of overlap between
adjacent biased distributions. Further, a judicious choice of the biasing
potential allows such free energy differences to be estimated using short
simulations, so that the free energetics of water density fluctuations are
obtained using only a few, short simulations. Leveraging the efficiency of the
method, we characterize water density fluctuations in the entire hydration
shell of the protein, ubiquitin; a large volume containing an average of more
than six hundred waters.
The ability to correlate fullerene solubility with experimentally or computationally accessible parameters can significantly facilitate the nanotechnology nowadays for a wide range of applications, while providing crucial insight into optimum design of future fullerene species. To date, there has been no single relationship that satisfactorily describes the existing data clearly manifesting the effects of solvent species, system temperature, and isomer. Using atomistic molecular dynamics simulations on two standard fullerene species, C60 and PCBM ([6,6]-phenyl-C61-butyric acid methyl ester), in a representative series of organic solvent media (i.e., chloroform, toluene, chlorobenzene, 1,3-dichlorobenzene, and 1,2-dichlorobenzene), we show that a single time constant characterizing the dynamic stability of a tiny (angstrom-sized) solvation shell encompassing the fullerene particle can be utilized to well capture the known trends of fullerene solubility as reported in the literature. The underlying physics differs substantially between the two fullerene species, however. Whereas C60 was previously shown to be dictated by a diffusion-limited aggregation mechanism, the side-chain substituted PCBM is demonstrated herein to proceed with an analogous reaction-limited aggregation with the "reaction rate" set by the fullerene rotational diffusivity in the medium. The present results suggest that dynamic quantities-in contrast to the more often employed, static ones-may provide an excellent means to characterize the complex (entropic and enthalpic) interplay between fullerene species and the solvent medium, shed light on the factors determining the solvent quality of a nanoparticle solution, and, in particular, offer a practical pathway to foreseeing optimum fullerene design and fullerene-solvent interactions.
Fullerene C60 sub-colloidal particle with diameter ∼1 nm represents a boundary case between small and large hydrophobic solutes on the length scale of hydrophobic hydration. In the present paper, a molecular dynamics simulation is performed to investigate this complex phenomenon for bare C60
fullerene and its amphiphilic/charged derivatives, so called shape amphiphiles. Since most of the unique properties of water originate from the pattern of hydrogen bond network and its dynamics, spatial, and orientational aspects of water in solvation shells around the solute surface having hydrophilic and hydrophobic regions are analyzed. Dynamical properties such as translational-rotational mobility, reorientational correlation and occupation time correlation functions of water molecules, and diffusion coefficients are also calculated. Slower dynamics of solvent molecules—water retardation—in the vicinity of the solutes is observed. Both the topological properties of hydrogen bond pattern and the “dangling” –OH groups that represent surface defects in water network are monitored. The fraction of such defect structures is increased near the hydrophobic cap of fullerenes. Some “dry” regions of C60 are observed which can be considered as signatures of surface dewetting. In an effort to provide molecular level insight into the thermodynamics of hydration, the free energy of solvation is determined for a family of fullerene particles using thermodynamic integration technique.
Processes ranging from oil-water phase separation to the formation of solid clathrate hydrates send mixed messages regarding whether oil molecules hate or love to be surrounded by water. Recent experimental and theoretical results help decipher these mixed messages by illuminating the conditions under which the stability of a hydrophobic contact is expected to exceed thermal energy fluctuations - thus facilitating hydrophobic self-assembly and the emergences of structure from randomness. Important open questions remain regarding the dependence of hydrophobic interactions on molecular size and temperature, as well as the balance of direct and water-mediated interactions.
Given the importance of water-mediated hydrophobic interactions in a wide range of biological and synthetic self-assembly processes, it is remarkable that both the sign and the magnitude of the hydrophobic interactions between simple amphiphiles, such as alcohols, remain unresolved. To address this question, we have performed Raman hydration-shell vibrational spectroscopy and polarization-resolved femtosecond infrared experiments, as well as random mixing and molecular dynamics simulations. Our results indicate that there are no more hydrophobic contacts in aqueous solutions of alcohols ranging from methanol to tertiary butyl alcohol than in random mixtures of the same concentration. This implies that the interaction between small hydrophobic groups is weaker than thermal energy fluctuations. Thus, the corresponding water-mediated hydrophobic interaction must be repulsive, with a magnitude sufficient to negate the attractive direct van der Waals interaction between the hydrophobic groups.Keywords: Raman spectroscopy; femtosecond IR spectroscopy; hydrophobic interactions; random mixing
Liquid water confined between hydrophobic objects of sufficient size becomes metastable with respect to its vapor at separations smaller than a critical drying distance. Macroscopic thermodynamic arguments predicting this distance have been restricted to the limit of perfectly rigid confining materials. However, no material is perfectly rigid and it is of interest to account for this fact in the thermodynamic analysis. We present a theory that combines the current macroscopic theory with the thermodynamics of elasticity to derive an expression for the critical drying distance for liquids confined between flexible materials. The resulting expression is the sum of the well-known drying distance for perfectly rigid confining materials and a new term that accounts for flexibility. Thermodynamic arguments show that this new term is necessarily positive, meaning that flexibility increases the critical drying distance. To study the expected magnitude and scaling behavior of the flexible term, we consider the specific case of water and present an example of drying between thin square elastic plates that are simply supported along two opposite edges and free at the remaining two. We find that the flexible term can be the same order of magnitude or greater than the rigid solution for materials of biological interest at ambient conditions. In addition, we find that when the rigid solution scales with the characteristic size of the immersed objects, the flexible term is independent of size and vice versa. Thus, the scaling behavior of the overall drying distance will depend on the relative weights of the rigid and flexible contributions.
An understanding of density fluctuations in bulk water has made significant
contributions to our understanding of the hydration and interactions of
idealized, purely repulsive hydrophobic solutes. To similarly inform the
hydration of realistic hydrophobic solutes that have dispersive interactions
with water, here we characterize water density fluctuations in the presence of
attractive fields that correspond to solute-water attractions. We find that
when the attractive field acts only in the solute hydration shell, but not in
the solute core, it does not significantly alter water density fluctuations in
the solute core region. We further find that for a wide range of solute sizes
and attraction strengths, the free energetics of turning on the attractive
fields in bulk water are accurately captured by linear response theory. Our
results also suggest strategies for more efficiently estimating hydration free
energies of realistic solutes in bulk water and at interfaces.
The notion of (static) solvation shells has recently proved fruitful in revealing key molecular factors that dictate the solubility and aggregation properties of fullerene species in polar or ionic solvent media. Using molecular dynamics schemes with carefully evaluated force fields, we have scrutinized both the static and the dynamic features of the solvation shells of single C60 particle for three non-polar organic solvents (i.e., chloroform, toluene, and chlorobenzene) and a range of system temperatures (i.e., T = 250-330 K). The central findings have been that, while the static structures of the solvation shell remain, in general, insensitive to the effects of changing solvent type or system temperature, the dynamic behavior of solvent molecules within the shell exhibits prominent dependence on both factors. Detailed analyses led us to propose the notion of dynamically stable solvation shell, effectiveness of which can be characterized by a new physical parameter defined as the ratio of two fundamental time constants representing, respectively, the solvent relaxation (or residence) time within the first solvation shell and the characteristic time required for the fullerene particle to diffuse a distance comparable to the shell thickness. We show that, for the five (two from the literature) different solvent media and the range of system temperatures examined herein, this parameter bears a value around unity and, in particular, correlates intimately with known trends of solubility for C60 solutions. We also provide evidence revealing that, in addition to fullerene-solvent interactions, solvent-solvent interactions play an important role, too, in shaping the dynamic solvation shell, as implied by recent experimental trends.
The macroscopic diffusion constant for a charged diffuser is in part dependent on (1) the volume excluded by solute "obstacles" and (2) long-range interactions between those obstacles and the diffuser. Increasing excluded volume reduces transport of the diffuser, while long-range interactions can either increase or decrease diffusivity, depending on the nature of the potential. We previously demonstrated [P. M. Kekenes-Huskey et al., Biophys. J. 105, 2130 (2013)] using homogenization theory that the configuration of molecular-scale obstacles can both hinder diffusion and induce diffusional anisotropy for small ions. As the density of molecular obstacles increases, van der Waals (vdW) and electrostatic interactions between obstacle and a diffuser become significant and can strongly influence the latter's diffusivity, which was neglected in our original model. Here, we extend this methodology to include a fixed (time-independent) potential of mean force, through homogenization of the Smoluchowski equation. We consider the diffusion of ions in crowded, hydrophilic environments at physiological ionic strengths and find that electrostatic and vdW interactions can enhance or depress effective diffusion rates for attractive or repulsive forces, respectively. Additionally, we show that the observed diffusion rate may be reduced independent of non-specific electrostatic and vdW interactions by treating obstacles that exhibit specific binding interactions as "buffers" that absorb free diffusers. Finally, we demonstrate that effective diffusion rates are sensitive to distribution of surface charge on a globular protein, Troponin C, suggesting that the use of molecular structures with atomistic-scale resolution can account for electrostatic influences on substrate transport. This approach offers new insight into the influence of molecular-scale, long-range interactions on transport of charged species, particularly for diffusion-influenced signaling events occurring in crowded cellular environments.
The dynamics of colloids and proteins in dense suspensions is of fundamental importance, from a standpoint of understanding the biophysics of proteins in the cytoplasm and for the many interesting physical phenomena in colloidal dispersions. Recent experiments and simulations have raised questions about our understanding of the dynamics of these systems. Experiments on vesicles in nematic fluids and colloids in an actin network have shown that the dynamics of particles can be ``non-Gaussian", i.e., the self-part of van Hove correlation function, Gs(r,t) is an exponential rather than Gaussian function of r, in regimes where the mean-square displacement is linear in t. It is usually assumed that a linear mean-square displacement implies a Gaussian Gs(r,t). In a different result, simulations of a mixture of proteins, aimed at mimicking the cytoplasm of E. coli, have shown that hydrodynamic interactions (HI) play a key role in slowing down the dynamics of proteins in concentrated (relative to dilute) solutions. In this work we study a simple system, a dilute tracer colloidal particle immersed in a concentrated solution of larger spheres, using simulations with and without HI. The simulations reproduce the non-Gaussian Brownian diffusion of the tracer, implying that this behavior is a general feature of colloidal dynamics, and is a consequence of local heterogeneities on intermediate timescales. Although HI results in a lower diffusion constant, Gs(r,t) is very similar with and without HI, provided they are compared at the same value of the mean-square displacement.
We present the first nearly atomistic molecular dynamics study of nanorod-nanorod association in explicit solvent, showing that inter-rod forces can be dominated by microscopic factors absent in common continuum descriptions. Specifically, we find that alkane ligands on faceted CdS nanorods in n-hexane undergo a temperature-dependent order-disorder transition akin to that of self-assembled monolayers on macroscopic substrates. This collective ligand alignment organizes nearby solvent molecules, strongly influencing the statistics of rod-rod separation. The strong temperature-dependence of this mechanism could be exploited in the laboratory to manipulate and optimize the assembly of ordered structures.
We formulate and apply a microscopic statistical-mechanical theory for the non-hydrodynamic relative diffusion coefficient of a pair of spherical nanoparticles in entangled polymer melts based on a combination of Brownian motion, mode-coupling, and polymer physics ideas. The focus is on the mesoscopic regime where particles are larger than the entanglement spacing. The dependence of the non-hydrodynamic friction on interparticle separation, degree of entanglement, and tube diameter is systematically studied. The overall magnitude of the relative diffusivity is controlled by the ratio of the particle to tube diameter and the number of entanglements in a manner reminiscent of single-particle self-diffusion and Stokes-Einstein violations. A rich spatial separation dependence of mobility enhancement relative to the hydrodynamic behavior is predicted even for very large particles, and the asymptotic dependence is derived analytically in the small and large separation limits. Particle separations in excess of 100 nm are sometimes required to recover the hydrodynamic limit. The effects of local polymer-particle packing correlations are found to be weak, and the non-hydrodynamic effects are also small for unentangled melts.
A calculation is presented for the effect of the hydrodynamic interaction on the diffusion‐controlled rate coefficient for particles that coalesce by diffusion under the influence of an interaction potential. The hydrodynamic effect is found to make a substantial reduction in the rate compared to the Debye model which includes only the effects of diffusion and the forces between the reacting particles. For the case where the particles are hard spheres the reduction is 46%. For ionic species the reduction varies between 25% and 60% depending on the extent of attraction or repulsion.
The effects of short range solvent structure and short range dynamical correlations are investigated for the steady state rate constant k for solution reactions influenced by diffusion. The description is in terms of a Smoluchowski equation describing relative motion of two molecules in an outer spatial translational region, supplemented by a sink term that accounts for dynamics in an inner reaction zone. In the outer region, solvent structural effects are included by a potential of mean force, which exhibits a short range well and barrier combination leading to ’’potential caging.’’ Outer region short range dynamical correlations are included via a separation‐dependent diffusion coefficient, leading to ’’dynamical caging’’ as relative motion is showed at small separations. These two short range effects are neglected in standard diffusion treatments. We find that k is only modestly influenced by the above short range effects. In order to expose short range structure and correlation influence in a more sensitive way, we formulate the short range caging picture. This information focuses on events occuring at the small separations usually implied by the term ’’solvent cage.’’ At this level of description, short range effects are clearly revealed. We compare our formulation with the encounter formulation of Noyes.
Spatially periodic fundamental solutions of the Stokes equations of motion for a viscous fluid past a periodic array of obstacles are obtained by use of Fourier series. It is made clear that the divergence of the lattice sums pointed out by Burgers may be rescued by taking into account the presence of the mean pressure gradient. As an application of these solutions the force acting on any one of the small spheres forming a periodic array is considered. Cases for three special types of cubic lattice are investigated in detail. It is found that the ratios of the values of this force to that given by the Stokes formula for an isolated sphere are larger than 1 and do not differ so much among these three types provided that the volume concentration of the spheres is the same and small. The method is also applied to the two-dimensional flow past a square array of circular cylinders, and the drag on one of the cylinders is found to agree with that calculated by the use of elliptic functions.
The friction coefficient exerted by a hard-sphere fluid on an infinitely massive Brownian sphere is calculated for several size ratios , where and are the diameters of the Brownian and fluid spheres, respectively. The exact microscopic expression derived in part I of this work from kinetic theory is transformed and shown to be proportional to the time integral of the autocorrelation function of the momentum transferred from the fluid to the Brownian sphere during instantaneous collisions. Three different methods are described to extract the friction coefficient from molecular dynamics simulations carried out onfinite systems. The three independent methods lead to estimates of which agree within statisticalerrors (typically 5%). The results are compared to the predictions of Enskog theory and of the hydrodynamic Stokes law. The former breaks down as the size ratio and/or the packing fraction of the fluid increase. Somewhat surprisingly, Stokes' law is found to hold withstick boundary conditions, in the range 1/4.5 explored in the present simulations, with a hydrodynamic diameterd=. The analysis of the moleuclar dynamics data on the basis of Stokes' law withslip boundary conditions is less conclusive, although the right trend is found as / increases.
In molecular dynamics (MD) simulations the need often arises to maintain such parameters as temperature or pressure rather than energy and volume, or to impose gradients for studying transport properties in nonequilibrium MD. A method is described to realize coupling to an external bath with constant temperature or pressure with adjustable time constants for the coupling. The method is easily extendable to other variables and to gradients, and can be applied also to polyatomic molecules involving internal constraints. The influence of coupling time constants on dynamical variables is evaluated. A leap‐frog algorithm is presented for the general case involving constraints with coupling to both a constant temperature and a constant pressure bath.
In this work, the shear viscosity at ambient conditions of several water models (SPC/E, TIP4P, TIP5P, and TIP4P/2005) is evaluated using the Green-Kubo formalism. The performance of TIP4P/2005 is excellent, that of SPC/E and TIP5P is more or less acceptable, whereas TIP4P and especially TIP3P give a poor agreement with experiment. Further calculations have been carried out for TIP4P/2005 to provide a wider assessment of its performance. In accordance with experimental data, TIP4P/2005 predicts a minimum in the shear viscosity for the 273 K isotherm, a shift in the minimum toward lower pressures at 298 K, and its disappearance at 373 K.
Nanofluidics has emerged recently in the footsteps of microfluidics, following the quest for scale reduction inherent to nanotechnologies. By definition, nanofluidics explores transport phenomena of fluids at nanometer scales. Why is the nanometer scale specific? What fluid properties are probed at nanometric scales? In other words, why does 'nanofluidics' deserve its own brand name? In this critical review, we will explore the vast manifold of length scales emerging for fluid behavior at the nanoscale, as well as the associated mechanisms and corresponding applications. We will in particular explore the interplay between bulk and interface phenomena. The limit of validity of the continuum approaches will be discussed, as well as the numerous surface induced effects occurring at these scales, from hydrodynamic slippage to the various electro-kinetic phenomena originating from the couplings between hydrodynamics and electrostatics. An enlightening analogy between ion transport in nanochannels and transport in doped semi-conductors will be discussed (156 references).
Molecular theory of Brownian motion of a single heavy par-ticle in a fluid had received a considerable attention in earlier years. 1-7 After molecular dynamics (MD) simulation technique was utilized, this subject has been widely studied by a variety of MD simulation methods. 8-23 The common issues here were about the long time behavior of the force and velocity auto-correlation functions, the system size dependent friction co-efficient of a massive Brownian particle, and test of the Stokes-Einstein law. For the case of two heavy particles in a fluid, the generalized Langevin equation, obtained by Deutch and Oppenheim, 24 re-sulted in a Fokker-Planck equation more general than the equ-ation obtained by Mazo. 25 For long times their result reduces to that found by Mazo. They evaluated the molecular expression obtained for the friction tensor by modification of an approxi-mate hydrodynamic fluctuation theory argument developed by Zwanaig. 26 The dependence of the friction tensors on inter-par-ticle separations, R12, is obtained to the lowest order in (a/R12) where a is the radius of the Brownian particles. Recently, The effects of hydrodynamic interactions on the friction tensors for two particles in solution were studied. 27 The particles have linear dimensions on nanometer scales and are either simple spherical particles interacting with the solvent through repulsive Lennard-Jones forces or are composite clu-ster particles whose atomic components interact with the solvent through repulsive Lennard-Jones forces. The solvent dynamics is modelled at a mesoscopic level through multi-particle col-lisions that conserve mass, momentum and energy. 28 The de-pendence of the two-particle relative friction tensors on the inter-particle separation indicated the importance of hydro-dynamic interactions for these nano-particles. In this work we focus on the calculation of the friction coe-fficients between two very massive Brownian particles in a Lennard-Jones solvent using MD algorithm of Newton's equ-ations of motion. Our goal is to obtain the friction coefficient as function of the distance between the two Brownian particles and to compare with the hydrodynamic friction tensors evaluated by a molecular theory. 24 This work is related to and motivated by our previous work 22 in which equilibrium MD simulations in a microcanonical ensemble were performed to evaluate the friction coefficient of a Brownian particle (BP) in a Lennard-Jones (LJ) solvent from the time dependent friction coefficients and the momentum autocorrelation functions of the BP with its infinite mass at various ratios of LJ parameters of the BP and solvent, σB/σs.
A spectrally accurate method for the fast evaluation of N-particle sums of the periodic Stokeslet is presented. Two different decomposition methods, leading to one sum in real space and one in reciprocal space, are considered. An FFT based method is applied to the reciprocal part of the sum, invoking the equivalence of multiplications in reciprocal space to convolutions in real space, thus using convolutions with a Gaussian function to place the point sources on a grid. Due to the spectral accuracy of the method, the grid size needed is low and also in practice, for a fixed domain size, independent of N. The leading cost, which is linear in N, arises from the to-grid and from-grid operations. Combining this FFT based method for the reciprocal sum with the direct evaluation of the real space sum, a spectrally accurate algorithm with a total complexity of O(NlogN)O(NlogN) is obtained. This has been shown numerically as the system is scaled up at constant density.
The different roles the attractive and repulsive forces play in forming the equilibrium structure of a Lennard-Jones liquid are discussed. It is found that the effects of these forces are most easily separated by considering the structure factor (or equivalently, the Fourier transform of the pair-correlation function) rather than the pair-correlation function itself. At intermediate and large wave vectors, the repulsive forces dominate the quantitative behavior of the liquid structure factor. The attractions are manifested primarily in the small wave vector part of the structure factor; but this effect decreases as the density increases and is almost negligible at reduced densities higher than 0.65. These conclusions are established by considering the structure factor of a hypothetical reference system in which the intermolecular forces are entirely repulsive and identical to the repulsive forces in a Lennard-Jones fluid. This reference system structure factor is calculated with the aid of a simple but accurate approximation described herein. The conclusions lead to a very simple prescription for calculating the radial distribution function of dense liquids which is more accurate than that obtained by any previously reported theory. The thermodynamic ramifications of the conclusions are presented in the form of calculations of the free energy, the internal energy (from the energy equation), and the pressure (from the virial equation). The implications of our conclusions to perturbation theories for liquids and to the interpretation of x-ray scattering experiments are discussed.
Molecular dynamics simulations are used to determine the time‐dependent friction for pair diffusion in an isotropic Lennard‐Jones fluid as a function of the separation between two diffusing particles. A numerical method proposed by Straub, Borkovec and Berne is used. It is found that both the initial value and the detailed time‐dependence of the friction are dependent on the interparticle separation. The dependence of the pair diffusion coefficient on separation is determined. Comparisons are made with various hydrodynamic and collision theories. The rate constant for diffusion controlled reactions is discussed.
A series of Monte Carlo simulations has been carried out to characterize the temperature and size dependence of the results for liquid water using the TIP4P potential function. Five temperatures from -25 to 100°C and four system sizes from 64 to 512 molecules have been studied. Comparisons are made with experimental thermodynamic and structural data as well as results of prior simulations.
Phenomena and processes related to the behavior of fullerenes in solutions are reviewed. Data on the solubility of C60 and C70 fullerenes in a large number of solvents at various temperatures are presented as well as on diffusion coefficient of fullerenes in solutions. The relation between the factors controlling the behavior of dissolved fullerenes and the clustering tendency they show is analyzed. This tendency, which sets fullerenes apart from other large molecules, underlies many aspects of fullerene behavior in solutions, such as the recently discovered nonmonotone temperature dependence of fullerene solubility in various solvents, the nonlinear concentration dependence of nonlinear optical susceptibility, the sharp dependence of the color of a fullerene solution on the solution composition (the solvatochromatic effect), the concentration dependence of the heat of solution of fullerenes in organic solvents, etc. Growth mechanisms of fractal clusters in fullerene solutions are analyzed along with similarity laws determining the thermodynamic characteristics of fullerite crystals.
The lattice sum of the Rotne–Prager hydrodynamic mobility tensor is cast into a rapidly converging form by an Ewald summation technique. The result has a direct application to the problem of how to deal with the long range of hydrodynamic interactions in computer simulations of macromolecular solutions.
The structure of ST2 water around a pair of spherical nonpolar (Lennard‐Jones) particles, A1–A2, is studied as a function of pair separation, r, using a force‐bias Monte Carlo technique with importance sampling. The change in the water structure (or equivalently the hydrophobic hydration) is correlated with the potential of mean force, WAA (r), determined in a previous study. It is found that the second minimum in WAA (r) corresponds to a water molecule lying between the two apolar particles. The water always maintains a ’’linear hydrogen bond’’ network and is more ordered in the neighborhood of the two spheres except possibly when they are separated by a distance corresponding to the position of the free energy barrier, that is the maximum in WAA (r).
Approximate calculations of the hydrodynamic effect on diffusion controlled reaction rates yield finite corrections to the Smoluchowski law. However, exact calculations using hydrodynamics with stick boundary conditions have the unsatisfactory feature of predicting a vanishing coagulation rate. In this paper it is demonstrated that this failure of the theory is due to the stick boundary conditions. It is shown that hydrodynamic calculations with slip boundary conditions yield a finite coagulation rate. The exact slip hydrodynamics result indicates a reduction of 29% in the coagulation rate of neutral particles relative to the classical Smoluchowski theory. This is a smaller result than the 46% reduction predicted by the approximate Deutch–Felderhof theory for particles with stick boundary conditions.
The force autocorrelation function of an infinitely massive Brownian particle is studied with a molecular dynamics simulation. The plateau time problem, the calculation of the friction coefficient, and the relationship between the stochastic and real force are discussed.
▪ Abstract We present an overview of the lattice Boltzmann method (LBM), a parallel and efficient algorithm for simulating single-phase and multiphase fluid flows and for incorporating additional physical complexities. The LBM is especially useful for modeling complicated boundary conditions and multiphase interfaces. Recent extensions of this method are described, including simulations of fluid turbulence, suspension flows, and reaction diffusion systems.
A general method for computing the hydrodynamic interactions among an infinite suspension of particles, under the condition of vanishingly small particle Reynolds number, is presented. The method follows the procedure developed by O'Brien (1979) for constructing absolutely convergent expressions for particle interactions. For use in dynamic simulation, the convergence of these expressions is accelerated by application of the Ewald summation technique. The resulting hydrodynamic mobility and/or resistance matrices correctly include all far-field non-convergent interactions. Near-field lubrication interactions are incorporated into the resistance matrix using the technique developed by Durlofsky, Brady & Bossis (1987). The method is rigorous, accurate and computationally efficient, and forms the basis of the Stokesian-dynamics simulation method. The method is completely general and allows such diverse suspension problems as self-diffusion, sedimentation, rheology and flow in porous media to be treated within the same formulation for any microstructural arrangement of particles. The accuracy of the Stokesian-dynamics method is illustrated by comparing with the known exact results for spatially periodic suspensions.
Two unequal rigid spheres are immersed in unbounded fluid and are acted on by externally applied forces and couples. The Reynolds number of the flow around them is assumed to be small, with the consequence that the hydrodynamic interactions between the spheres can be described by a set of linear relations between, on the one hand, the forces and couples exerted by the spheres on the fluid and, on the other, the translational and rotational velocities of the spheres. These relations may be represented completely by either a set of 10 resistance functions or a set of 10 mobility functions. When non-dimensionalized, each function depends on two variables, the non-dimensionalized centre-to-centre separation s and the ratio of the spheres’ radii λ. Two expressions are given for each function, one a power series in s−1 and the other an asymptotic expression valid when the spheres are close to touching.
We develop a unified and generally applicable theory of solvation of small and large apolar species in water. In the former, hydrogen bonding of water is hindered yet persists near the solutes. In the latter, hydrogen bonding is depleted, leading to drying of extended apolar surfaces, large forces of attraction, and hysteresis on mesoscopic length scales. The crossover occurs on nanometer length scales, when the local concentration of apolar units is sufficiently high, or when an apolar surface is sufficiently large. Our theory for the crossover has implications concerning the stability of protein assemblies and protein folding.
We study the system-size dependence of translational diffusion coefficients and viscosities in molecular dynamics simulations under periodic boundary conditions. Simulations of water under ambient conditions and a Lennard-Jones (LJ) fluid show that the diffusion coefficients increase strongly as the system size increases. We test a simple analytic correction for the system-size effects that is based on hydrodynamic arguments. This correction scales as N-1/3, where N is the number of particles. For a cubic simulation box of length L, the diffusion coefficient corrected for system-size effects is D0 = DPBC + 2.837297kBT/(6πηL), where DPBC is the diffusion coefficient calculated in the simulation, kB the Boltzmann constant, T the absolute temperature, and η the shear viscosity of the solvent. For water, LJ fluids, and hard-sphere fluids, this correction quantitatively accounts for the system-size dependence of the calculated self-diffusion coefficients. In contrast to diffusion coefficients, the shear viscosities of water and the LJ fluid show no significant system-size dependences.
Using molecular dynamics computer simulation, we have calculated the velocity autocorrelation function and diffusion constant for a variety of solutes in a dense fluid of spherical solvent particles. We explore the effects of surface roughness of the solute on the resulting hydrodynamic boundary condition as we naturally approach the Brownian limit (when the solute becomes much larger and more massive than the solvent particles). We find that when the solute and solvent interact through a purely repulsive isotropic potential, in the Brownian limit the Stokes−Einstein law is satisfied with slip boundary conditions. However, when surface roughness is introduced through an anisotropic solute−solvent interaction potential, we find that the Stokes−Einstein law is satisfied with stick boundary conditions. In addition, when the attractive strength of a short-range isotropic solute−solvent potential is increased, the solute becomes dressed with solvent particles, making it effectively rough, and so stick boundary conditions are again recovered.
A simple method is presented for determining the dynamic (frequency-dependent) friction experienced by solute intramolecular coordinates due to the solvent molecules. This method is applied to the case of a diatomic molecule dissolved in Lennard-Jonesium.
The potential of mean force between two large parallel hydrophobic oblate ellipsoidal plates in liquid water is determined by molecular dynamics. Each ellipsoid displaces approximately 40 water molecules and has major and minor axes of 3.1 and 9.3 A, respectively, has a surface area of 650 A*, and interacts repulsively with the solvent water molecules. The potential of mean force is calculated from thermodynamic perturbation theory for a series of decreasing plate separations, using constant-pressure molecular dynamics. As the plates are moved together, they are first separated by three water layers and then by two, but for shorter distances, a dewetting transition occurs, and one water layer is never observed despite the fact that one can fit. As the plates are brought together, there is a corresponding weak oscillation in the potential of mean force corresponding to the removal of each water layer until the dewetting transition takes place, and for closer separations, the surrounding water molecules induce a constant average attractive force of 25 (kJ/mol)/A between the plates. This hydrophobic attraction is largely entropic in character, and the potential of mean force is found to be proportional to the area of the water-vacuum surface in this dewetting regime. The constant of proportionality is found to be smaller than the gas-liquid surface tension of the water model used. There is a very strong short-range driving force toward contact pairing. ~~~~~~~~~~~ ~ work.7,10-22 Similarly, an inert gas particle prefers to be separated from a flat wall by a layer of sol~ent.~~~~~ Thus, the simple picture alluded to above does not appear to give the whole truth. It should be noted that the simulations were performed using fixed charge models of water (like ST2,25 TIP4P,26 SPCZ7 etc.).
Multiple time scale methodologies have gained widespread use in molecular dynamics simulations and are implemented in a variety of ways across numerous packages. However, performance of the algorithms depends upon the details of the implementation. This is particularly important in the way in which the nonbonded interactions are partitioned. In this work, we show why some previous implementations give rise to energy drifts, and how this can be corrected. We also provide a recipe for using multiple time step methods to generate stable trajectories in large scale biomolecular simulations, where long trajectories are needed.
In this work, we show thatin any finite system, the binary friction tensor for two Brownian particlescannot be directly estimated from an evaluation of the microscopic Green-Kubo formula, involving the time integral of force-force
autocorrelation functions. This pitfall is associated with a subtle inversion of the thermodynamic and long-time limits and
leads to spurious results for the estimates of the friction matrix based on molecular dynamics simulations. Starting from
a careful analysis of the coupled Langevin equations for two interacting Brownian particles, we derive a method to circumvent
these effects and extract the binary friction tensor from the correlation function matrix of the instantaneous forces exerted
by the bath particles on the fixed Brownian particles, and from the relaxation of the total momentum of the bath in afinite system. The general methodology is applied to the case of two hard or soft Brownian spheres in a bath of light particles.
Numerical estimates of the relevant correlation functions and of the resulting self and mutual components of the matrix of
friction tensors are obtained by molecular dynamics simulations for various spacings between the Brownian particles.
We use molecular dynamics simulations to study thermal sliding of two nanostructured surfaces separated by nanoscale water
films. We find that friction at molecular separations is determined primarily by the effective free energy landscape for motion
in the plane of sliding, which depends sensitively on the surface character and the molecular structure of the confined water.
Small changes in the surface nanostructure can have dramatic effects on the apparent rheology. Whereas porous and molecularly
rough interfaces of open carbon nanotube membranes are found to glide with little friction, a comparably smooth interface
of end-capped nanotubes is effectively stuck. The addition of salt to the water layer is found to reduce the sliding friction.
Surprisingly, the intervening layers of water remain fluid in all cases, even in the case of high apparent friction between
the two membranes.
Macroscopic characterizations of hydrophobicity (e.g., contact angle measurements) do not extend to the surfaces of proteins and nanoparticles. Molecular measures of hydrophobicity of such surfaces need to account for the behavior of hydration water. Theory and state-of-the-art simulations suggest that water density fluctuations provide such a measure; fluctuations are enhanced near hydrophobic surfaces and quenched with increasing surface hydrophilicity. Fluctuations affect conformational equilibria and dynamics of molecules at interfaces. Enhanced fluctuations are reflected in enhanced cavity formation, more favorable binding of hydrophobic solutes, increased compressibility of hydration water, and enhanced water-water correlations at hydrophobic surfaces. These density fluctuation-based measures can be used to develop practical methods to map the hydrophobicity/philicity of heterogeneous surfaces including those of proteins. They highlight that the hydrophobicity of a group is context dependent and is significantly affected by its environment (e.g., chemistry and topography) and especially by confinement. The ability to include information about hydration water in mapping hydrophobicity is expected to significantly impact our understanding of protein-protein interactions as well as improve drug design and discovery methods and bioseparation processes.
Using molecular dynamics computer simulation, we have calculated the velocity autocorrelation function and diffusion constant for a spherical solute in a dense fluid of spherical solvent particles. The size and mass of the solute particle are related in such a way that we can naturally approach the Brownian limit (when the solute becomes much larger and more massive than the solvent particles). We find that as long as the solute radius is interpreted as an effective hydrodynamic radius, the Stokes-Einstein law with slip boundary conditions is satisfied as the Brownian limit is approached (specifically, when the solute is roughly 100 times more massive than the solvent particles). In contrast, the Stokes-Einstein law is not satisfied for a tagged particle of the neat solvent. We also find that in the Brownian limit the amplitude of the long-time tail of the solute's velocity autocorrelation function is in good agreement with theoretical hydrodynamic predictions. When the solvent density is substantially lower than the triple density, the Stokes-Einstein law is no longer satisfied, and the amplitude of the long-time tail is not in good agreement with theoretical predictions, signaling the breakdown of hydrodynamics. (C) 2003 American Institute of Physics.
A method for simulating the Brownian dynamics of N particles with the inclusion of hydrodynamic interactions is described. The particles may also be subject to the usual interparticle or external forces (e.g., electrostatic) which have been included in previous methods for simulating Brownian dynamics of particles in the absence of hydrodynamic interactions. The present method is derived from the Langevin equations for the N particle assembly, and the results are shown to be consistent with the corresponding Fokker--Planck results. Sample calculations on small systems illustrate the importance of including hydrodynamic interactions in Brownian dynamics simulations. The method should be useful for simulation studies of diffusion limited reactions, polymer dynamics, protein folding, particle coagulation, and other phenomena in solution.
We study by molecular dynamics simulations the driving force for the hydrophobic interaction between graphene sheets of different sizes down to the atomic scale. Similar to the prediction by Lum, Chandler, and Weeks for hard-sphere solvation [J. Phys. Chem. B 1999, 103, 4570-4577], we find the driving force to be length-scale dependent, despite the fact that our model systems do not exhibit dewetting. For small hydrophobic solutes, the association is purely entropic, while enthalpy favors dissociation. The latter is demonstrated to arise from the enhancement of hydrogen bonding between the water molecules around small hydrophobes. On the other hand, the attraction between large graphene sheets is dominated by enthalpy which mainly originates from direct solute-solute interactions. The crossover length is found to be inside the range of 0.3-1.5 nm(2) of the surface area of the hydrophobe that is eliminated in the association process. In the large-scale regime, different thermodynamic properties are scalable with this change of surface area. In particular, upon dimerization, a total and a water-induced stabilization of approximately 65 and 12 kJ/mol/nm(2) are obtained, respectively, and on average around one hydrogen bond is gained per 1 nm(2) of graphene sheet association. Furthermore, the potential of mean force between the sheets is also scalable except for interplate distances smaller than 0.64 nm which corresponds to the region around the barrier for removing the last layer of water. It turns out that, as the surface area increases, the relative height of the barrier for association decreases and the range of attraction increases. It is also shown that, around small hydrophobic solutes, the lifetime of the hydrogen bonds is longer than in the bulk, while around large hydrophobes it is the same. Nevertheless, the rearrangement of the hydrogen-bond network for both length-scale regimes is slower than in bulk water.
Theory and computation have long been used to rationalize the experimental association rate constants of protein-protein complexes, and Brownian dynamics (BD) simulations, in particular, have been successful in reproducing the relative rate constants of wild-type and mutant protein pairs. Missing from previous BD studies of association kinetics, however, has been the description of hydrodynamic interactions (HIs) between, and within, the diffusing proteins. Here we address this issue by rigorously including HIs in BD simulations of the barnase-barstar association reaction. We first show that even very simplified representations of the proteins--involving approximately one pseudoatom for every three residues in the protein--can provide excellent reproduction of the absolute association rate constants of wild-type and mutant protein pairs. We then show that simulations that include intermolecular HIs also produce excellent estimates of association rate constants, but, for a given reaction criterion, yield values that are decreased by ∼35-80% relative to those obtained in the absence of intermolecular HIs. The neglect of intermolecular HIs in previous BD simulation studies, therefore, is likely to have contributed to the somewhat overestimated absolute rate constants previously obtained. Consequently, intermolecular HIs could be an important component to include in accurate modeling of the kinetics of macromolecular association events.
To begin to elucidate the principles of intermolecular dynamics in the crowded environment of cells, employing brownian dynamics (BD) simulations, we examined possible mechanism(s) responsible for the great reduction in diffusion constants of macromolecules in vivo from that at infinite dilution. In an Escherichia coli cytoplasm model comprised of 15 different macromolecule types at physiological concentrations, BD simulations of molecular-shaped and equivalent sphere representations were performed with a soft repulsive potential. At cellular concentrations, the calculated diffusion constant of GFP is much larger than experiment, with no significant shape dependence. Next, using the equivalent sphere system, hydrodynamic interactions (HI) were considered. Without adjustable parameters, the in vivo experimental GFP diffusion constant was reproduced. Finally, the effects of nonspecific attractive interactions were examined. The reduction in diffusivity is very sensitive to macromolecular radius with the motion of the largest macromolecules dramatically slowed down; this is not seen if HI dominate. In addition, long-lived clusters involving the largest macromolecules form if attractions dominate, whereas HI give rise to significant, size independent intermolecular dynamic correlations. These qualitative differences provide a testable means of differentiating the importance of HI vs. nonspecific attractive interactions on macromolecular motion in cells.
In this paper, we study the interfacial friction of water at graphitic interfaces with various topologies, water between planar graphene sheets, inside and outside carbon nanotubes, with the goal to disentangle confinement and curvature effects on friction. We show that the friction coefficient exhibits a strong curvature dependence; while friction is independent of confinement for the graphene slab, it decreases with carbon nanotube radius for water inside, but increases for water outside. As a paradigm the friction coefficient is found to vanish below a threshold diameter for armchair nanotubes. Using a statistical description of the interfacial friction, we highlight here a structural origin of this curvature dependence, mainly associated with a curvature-induced incommensurability between the water and carbon structures. These results support the recent experiments reporting fast transport of water in nanometric carbon nanotube membranes.
The validity of the Stokes-Einstein (SE) relation for particle diffusion in the nano- and molecular scales has attracted much interest, but the results in the literature are controversial. In this work, it is shown that there exists a critical particle size where the SE relation breaks down by comparing particle transport in the macro- and molecular scales. Using molecular-dynamics simulations, we study the critical size and find that the van der Waals force plays an important role in particle diffusion as the particle size approaches molecular scale. Due to the limitations of computing facilities, we could not find exactly where the critical particle size is, but the simulation results qualitatively predict that this critical size is of a few nanometers.
We use molecular dynamics simulations of the SPC-E model of liquid water to derive probability distributions for water density fluctuations in probe volumes of different shapes and sizes, both in the bulk as well as near hydrophobic and hydrophilic surfaces. Our results are obtained with a biased sampling of coarse-grained densities that is easily combined with molecular dynamics integration algorithms. Our principal result is that the probability for density fluctuations of water near a hydrophobic surface, with or without surface water attractions, is akin to density fluctuations at the water-vapor interface. Specifically, the probability of density depletion near the surface is significantly larger than that in the bulk, and this enhanced probability is responsible for hydrophobic forces of assembly. In contrast, we find that the statistics of water density fluctuations near a model hydrophilic surface are similar to that in the bulk.
The potentials of mean force (PMFs) were determined, in both water with the TIP3P water model and in vacuo, for systems involving formation of nonpolar dimers composed of bicyclooctane, adamantane (both an all-atom model and a sphere with the radius of 3.4 A representing adamantane), and fullerene, respectively. A series of umbrella-sampling molecular dynamics simulations with the AMBER force field were carried out for each pair under both environmental conditions. The PMFs were calculated by using the weighted histogram analysis method. The results were compared with our previously determined PMF for neopentane. The shape of the PMFs for dimers of all four nonpolar molecules is characteristic of hydrophobic interactions with contact and solvent-separated minima and desolvation maxima. The positions of all these minima and maxima change with the size of the nonpolar molecule; for larger molecules they shift toward larger distances. Comparison of the PMFs of the bicyclooctane, adamantane, and fullerene dimers in water and in vacuo shows that hydrophobic interactions in each dimer are different from that for the dimer of neopentane. Interactions in the bicyclooctane, adamantane, and fullerene dimers are stronger in vacuo than in water. These dimers cannot be treated as classical, spherical, hydrophobic objects. The solvent contribution to the PMF was also computed by subtracting the PMF determined in vacuo from that in explicit solvent. The solvent contribution to the PMFs of bicyclooctane, adamantane, and fullerene is positive, as opposed to that of neopentane. The water molecules in the first solvation sphere of both adamantane and neopentane dimers are more ordered as compared to bulk water, with their dipole moments pointing away from the surface of the dimers. The average number of hydrogen bonds per water molecule in the first hydration shell of adamantane is smaller compared to that in bulk water, but this shell is thicker for all-atom adamantane than for neopentane or a spherical model of adamantane. In the second hydration shell, the average number of hydrogen bonds is greater compared to that in bulk water only for neopentane and a spherical model of adamantane but not for the all-atom model. The strength of the hydrophobic interactions shows a linear dependence on the number of carbon atoms both in water and in vacuo. Smaller nonpolar particles interact more strongly in water than in vacuo. For larger molecules, such as bicyclooctane, adamantane and fullerene, the reversed tendency is observed.
Particles suspended or dispersed in a fluid medium occur in a wide variety of natural and man-made settings, e.g. slurries, composite materials, ceramics, colloids, polymers, proteins, etc. The central theoretical and practical problem is to understand and predict the macroscopic equilibrium and transport properties of these multiphase materials from their microstructural mechanics. The macroscopic properties might be the sedimentation or aggregation rate, self-diffusion coefficient, thermal conductivity, or rheology of a suspension of particles. The microstructural mechanics entails the Brownian, interparticle, external, and hydrodynamic forces acting on the particles, as well as their spatial and temporal distribution, which is commonly referred to as the microstructure. If the distribution of particles were given, as well as the location and motion of any boundaries and the physical properties of the particles and suspending fluid, one would simply have to solve (in principle, not necessarily in practice) a well-posed boundary-value problem to determine the behavior of the material. Averaging this solution over a large volume or over many different configurations, the macroscopic or averaged properties could be determined. The two key steps in this approach, the solution of the many-body problem and the determination of the microstructure, are formidable but essential tasks for understanding suspension behavior.
This article discusses a new, molecular-dynamics-like approach, which we have named Stokesian dynamics, for dynamically simulating the behavior of many particles suspended or dispersed in a fluid medium. Particles in suspension may interact through both hydrodynamic and nonhydrodynamic forces, where the latter may be any type of Brownian, colloidal, interparticle, or external force. The simulation method is capable of predicting both static (i.e. configuration-specific) and dynamic microstructural properties, as well as macroscopic properties in either dilute or concentrated systems. Applications of Stokesian dynamics are widespread; problems of sedimentation, flocculation, diffusion, polymer rheology, and transport in porous media all fall within its domain. Stokesian dynamics is designed to provide the same theoretical and computational basis for multiphase, dispersed systems as does molecular dynamics for statistical theories of matter.
This review focuses on the simulation method, not on the areas in which Stokesian dynamics can be used. For a discussion of some of these many different areas, the reader is referred to the excellent reviews and proceedings of topical conferences that have appeared (e.g. Batchelor 1976a, Dickinson 1983, Faraday Discussions 1983, 1987, Family & Landau 1984). Before embarking on a description of Stokesian dynamics, we pause here to discuss some of the relevant theoretical literature on suspensions, and dynamic simulation in general, in order to put Stokesian dynamics in perspective.
Hydrophobicity is often characterized macroscopically by the droplet contact angle. Molecular signatures of hydrophobicity have, however, remained elusive. Successful theories predict a drying transition leading to a vapor-like region near large hard-sphere solutes and interfaces. Adding attractions wets the interface with local density increasing with attractions. Here we present extensive molecular simulation studies of hydration of realistic surfaces with a wide range of chemistries from hydrophobic (-CF(3), -CH(3)) to hydrophilic (-OH, -CONH(2)). We show that the water density near weakly attractive hydrophobic surfaces (e.g., -CF(3)) can be bulk-like or larger, and provides a poor quantification of surface hydrophobicity. In contrast, the probability of cavity formation or the free energy of binding of hydrophobic solutes to interfaces correlates quantitatively with the macroscopic wetting properties and serves as an excellent signature of hydrophobicity. Specifically, the probability of cavity formation is enhanced in the vicinity of hydrophobic surfaces, and water-water correlations correspondingly display characteristics similar to those near a vapor-liquid interface. Hydrophilic surfaces suppress cavity formation and reduce the water-water correlation length. Our results suggest a potentially robust approach for characterizing hydrophobicity of more complex and heterogeneous surfaces of proteins and biomolecules, and other nanoscopic objects.
We probe the effects of solute length scale, attractions, and hydrostatic pressure on hydrophobic hydration shells using extensive molecular simulations. The hydration shell compressibility and water fluctuations both display a nonmonotonic dependence on solute size, with a minimum near molecular solutes and enhanced fluctuations for larger ones. These results and calculations on proteins suggest that the hydration shells of unfolded proteins are more compressible than of folded ones contributing to pressure denaturation. More importantly, the nonmonotonicity implies a solute curvature-dependent pressure sensitivity for interactions between hydrophobic solutes.
The dynamics and structure of water at hydrophobic and hydrophilic diamond surfaces is examined via non-equilibrium Molecular Dynamics simulations. For hydrophobic surfaces under shearing conditions, the general hydrodynamic boundary condition involves a finite surface slip. The value of the slip length depends sensitively on the surface water interaction strength and the surface roughness; heuristic scaling relations between slip length, contact angle, and depletion layer thickness are proposed. Inert gas in the aqueous phase exhibits pronounced surface activity but only mildly increases the slip length. On polar hydrophilic surfaces, in contrast, slip is absent, but the water viscosity is found to be increased within a thin surface layer. The viscosity and the thickness of this surface layer depend on the density of polar surface groups. The dynamics of single water molecules in the surface layer exhibits a similar distinction: on hydrophobic surfaces the dynamics is purely diffusive, while close to a hydrophilic surface transient binding or trapping of water molecules over times of the order of hundreds of picoseconds occurs. We also discuss in detail the effect of the Lennard-Jones cutoff length on the interfacial properties.
The structure and flow of water inside 75 and 150 nm-long carbon nanotubes with diameters ranging from 0.83 to 1.66 nm are examined using molecular dynamics simulation. The flow rate enhancement, defined as the ratio of the observed flow rate to that predicted from the no-slip Poiseuille relation, is calculated for each tube and the liquid structure is examined using an axial distribution function. The relationship between the intermolecular water structure and water flow is quantified and differences between continuum and subcontinuum flow are discussed.
A new multiscale coarse-graining (CG) methodology is developed to bridge molecular and hydrodynamic models of a fluid. The hydrodynamic representation considered in this work is based on the equations of fluctuating hydrodynamics (FH). The essence of this method is a mapping from the position and velocity vectors of a snapshot of a molecular dynamics (MD) simulation to the field variables on Eulerian cells of a hydrodynamic representation. By explicit consideration of the effective lengthscale d(mol) that characterizes the volume of a molecule, the computed density fluctuations from MD via our mapping procedure have volume dependence that corresponds to a grand canonical ensemble of a cold liquid even when a small cell length (5-10 A) is used in a hydrodynamic representation. For TIP3P water at 300 K and 1 atm, d(mol) is found to be 2.4 A, corresponding to the excluded radius of a water molecule as revealed by its center-of-mass radial distribution function. By matching the density fluctuations and autocorrelation functions of momentum fields computed from solving the FH equations with those computed from MD simulation, the sound velocity and shear and bulk viscosities of a CG hydrodynamic model can be determined directly from MD. Furthermore, a novel staggered discretization scheme is developed for solving the FH equations of an isothermal compressive fluid in a three dimensional space with a central difference method. This scheme demonstrates high accuracy in satisfying the fluctuation-dissipation theorem. Since the causative relationship between field variables and fluxes is captured, we demonstrate that the staggered discretization scheme also predicts correct physical behaviors in simulating transient fluid flows. The techniques presented in this work may also be employed to design multiscale strategies for modeling complex fluids and macromolecules in solution.