Dewetting-controlled binding of ligands to hydrophobic pockets

Do All Paths Lead to Rome? How Reliable is Umbrella Sampling Along a Single Path?

Molecular dynamics (MD) simulations are widely applied to estimate absolute binding free energies of protein-ligand and protein-protein complexes. A routinely used method for binding free energy calculations with MD is umbrella sampling (US), which calculates the potential of mean force (PMF) along a single reaction coordinate. Surprisingly, in spite of its widespread use, few validation studies have focused on the convergence of the free energy computed along a single path for specific cases, not addressing the reproducibility of such calculations in general. In this work, we therefore investigate the reproducibility and convergence of US along a standard distance-based reaction coordinate for various protein-protein and protein-ligand complexes, following commonly used guidelines for the setup. We show that repeating the complete US workflow can lead to differences of 2-20 kcal/mol in computed binding free energies. We attribute those discrepancies to small differences in the binding pathways. While these differences are unavoidable in the established US protocol, the popularity of the latter could hint at a lack of awareness of such reproducibility problems. To test if the convergence of PMF profiles can be improved if multiple pathways are sampled simultaneously, we performed additional simulations with an adaptive-biasing method, here the accelerated weight histogram (AWH) approach. Indeed, the PMFs obtained from AHW simulations are consistent and reproducible for the systems tested. To the best of our knowledge, our work is the first to attempt a systematic assessment of the pitfalls in one the most widely used protocols for computing binding affinities. We anticipate therefore that our results will provide an incentive for a critical reassessment of the validity of PMFs computed with US, and make a strong case to further benchmark the performance of adaptive-biasing methods for computing binding affinities.

Understanding the "Berg limit": the 65° contact angle as the universal adhesion threshold of biomatter

Surface phenomena in aqueous environments such as long-range hydrophobic attraction, macromolecular adhesion, and even biofouling are predominantly influenced by a fundamental parameter-the water contact angle. The minimal contact angle required for these and related phenomena to occur has been repeatedly reported to be around 65° and is commonly referred to as the "Berg limit." However, the universality of this specific threshold across diverse contexts has remained puzzling. In this perspective article, we aim to rationalize the reoccurrence of this enigmatic contact angle. We show that the relevant scenarios can be effectively conceptualized as three-phase problems involving the surface of interest, water, and a generic oil-like material that is representative of the nonpolar constituents within interacting entities. Our analysis reveals that attraction and adhesion emerge when substrates display an underwater oleophilic character, corresponding to a "hydrophobicity under oil", which occurs for contact angles above approximately 65°. This streamlined view provides valuable insights into macromolecular interactions and holds implications for technological applications.

Interfacial water molecules contribute to antibody binding to the receptor-binding domain of SARS-CoV-2 spike protein

Antibodies that recognize the spike protein of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), especially the neutralizing antibodies, carry great hope in the treatment and final elimination of COVID-19. Driven by a synchronized global effort, thousands of antibodies against the spike protein have been identified during the past two years, with the structural information available at atomistic detail for hundreds of these antibodies. We developed an improved molecular mechanics/Poisson-Boltzmann surface area (MM/PBSA) method including explicitly treated interfacial water to calculate the binding free energy between representative antibodies and the receptor binding domain (RBD) domain of SARS-COV-2 spike proteins. We discovered that explicit treatment of water molecules located at the interface between RBD and antibody effectively improves the results for the WT and variants of concern (VOC) systems. Interfacial water molecules, together with surface and internal water molecules, behave drastically from bulk water and exert peculiar impacts on protein dynamics and energy, and thus warrant explicit treatment to complement implicit solvent models. Our results illustrate the importance of including interfacial water molecules to approach efficient and reliable prediction of binding free energy.Communicated by Ramaswamy H. Sarma.

On the Nature of Guest Complexation in Water: Triggered Wetting-Water-Mediated Binding

The complexity of macromolecular surfaces means that there are still many open questions regarding how specific areas are solvated and how this might affect the complexation of guests. Contributing to the identification and classification of the different possible mechanisms of complexation events in aqueous solution, and as part of the recent SAMPL8 exercise, we report here on the synthesis and conformational properties of TEEtOA 2, a cavitand with conformationally flexible ethyl groups at its portal. Using a combination of ITC and NMR spectroscopy, we report the binding affinities of a series of carboxylates to 2 and compare it to a related cavitand TEMOA 1. Additionally, we report MD simulations revealing how the wetting of the pocket of 2 is controlled by the conformation of its rim ethyl groups and, correspondingly, a novel triggered wetting, guest complexation mechanism, whereby the approaching guest opens up the pocket of the host, inducing its wetting and ultimately allows the formation of a hydrated host-guest complex (H·G·H2O). A general classification of complexation mechanisms is also suggested.

Solubility of Hydrophobes into Macrocyclic Hosts

The intrinsic nature of macrocyclic molecules to preferentially absorb a specific solute has been opening up supramolecular chemistry. Nevertheless, the determinant factor with molecular perspectives in promoting host-guest complexations remains inconclusive, due to the lack of rigorous thermodynamic examination on the guest solubility inside the host. Here, we quantify the solute-solvent energetic and entropic contributions between the end states and on the docking route during inclusion of noble gases in cucurbit[5]uril, cucurbit[6]uril, and α-cyclodextrin, using molecular dynamics simulations in combination with the potential distribution theorem. Results show that in all of the pairs examined both the solute-solvent energy and entropy favor the inclusion, while the former is rather dominant. The frequency of interior drying, which pertains to the entropic contribution, differs between the hosts and is controlled by the existence of lid water at portal and the flexibility of host framework. Moreover, the hosts exhibit various types of absorption manners, involving non-, single-, and double-free-energy barriers.

Cavitand Complexes in Aqueous Solution: Collaborative Experimental and Computational Studies of the Wetting, Assembly, and Function of Nanoscopic Bowls in Water

Water is the dominant liquid on Earth. Despite this, the main focus of supramolecular chemistry research has been on binding and assembly events in organic solvents. This arose because it is more straightforward to synthesize organic-media-soluble hosts and because of the relative simplicity of organic solvents compared to water. Nature, however, relies on water as a solvent, and spurred by this fact, supramolecular chemists have recently been making forays into the aqueous domain to understand water-mediated non-covalent interactions. These studies can benefit from the substantial understanding of the hydrophobic effect and electrostatic interactions developed by physical chemists. Nearly 20 years ago, the Gibb group first synthesized a class of water-soluble host molecules, the deep-cavity cavitands, that possess non-polar pockets that readily bind non-polar moieties in aqueous solution and are capable of assembling into a wide range of complexes with distinct stoichiometries. As such, these amphipathic host species are ideal platforms for studying the role of negatively curved features on guest complexation and the structural requirements for guided assembly processes driven by the hydrophobic effect. Here we review the collaborative experimental and computational investigations between Gibb and Ashbaugh over the past 10 years exploring questions including the following: How does water wet/solvate the non-polar surfaces of non-polar pockets? How does this wetting control the binding of non-polar guests? How does wetting affect the binding of anionic species? How does the nature and size of a guest size impact the assembly of cavitand hosts into multimeric capsular complexes? What are the conformational motifs of guests packed within the confines of capsular complexes? How might the electrostatic environment engendered by hosts impact the properties and reactivity of internalized guests?

Protein flexibility reduces solvent-mediated friction barriers of ligand binding to a hydrophobic surface patch

Solvent fluctuations have been explored in detail for idealized and rigid hydrophobic model systems, but so far it has remained unclear how internal protein motions and their coupling to the surrounding solvent affect the dynamics of ligand binding to biomolecular surfaces. Here, molecular dynamics simulations were used to elucidate the solvent-mediated binding of a model ligand to the hydrophobic surface patch of ubiquitin. The ligand's friction profiles reveal pronounced long-time correlations and enhanced friction in the vicinity of the protein, similar to idealized hydrophobic surfaces. Interestingly, these effects are shaped by internal protein motions. Protein flexibility modulates water density fluctuations near the hydrophobic surface patch and smooths out the friction profile of ligand binding.

Coupling Monte Carlo, Variational Implicit Solvation, and Binary Level-Set for Simulations of Biomolecular Binding

We develop a hybrid approach that combines the Monte Carlo (MC) method, a variational implicit-solvent model (VISM), and a binary level-set method for the simulation of biomolecular binding in an aqueous solvent. The solvation free energy for the biomolecular complex is estimated by minimizing the VISM free-energy functional of all possible solute-solvent interfaces that are used as dielectric boundaries. This functional consists of the solute volumetric, solute-solvent interfacial, solute-solvent van der Waals interaction, and electrostatic free energy. A technique of shifting the dielectric boundary is used to accurately predict the electrostatic part of the solvation free energy. Minimizing such a functional in each MC move is made possible by our new and fast binary level-set method. This method is based on the approximation of surface area by the convolution of an indicator function with a compactly supported kernel and is implemented by simple flips of numerical grid cells locally around the solute-solvent interface. We apply our approach to the p53-MDM2 system for which the two molecules are approximated by rigid bodies. Our efficient approach captures some of the poses before the final bound state. All-atom molecular dynamics simulations with most of such poses quickly reach the final bound state. Our work is a new step toward realistic simulations of biomolecular interactions. With further improvement of coarse graining and MC sampling, and combined with other models, our hybrid approach can be used to study the free-energy landscape and kinetic pathways of ligand binding to proteins.

Pressure Induced Wetting and Dewetting of the Nonpolar Pocket of Deep-Cavity Cavitands in Water

Hydrophobic interactions drive the binding of nonpolar ligands to the oily pockets of proteins and supramolecular species in aqueous solution. As such, the wetting of host pockets is expected to play a critical role in determining the thermodynamics of guest binding. Here we use molecular simulations to examine the impact of pressure on the wetting and dewetting of the nonpolar pockets of a series of deep-cavity cavitands in water. The portals to the cavitand pockets are functionalized with both nonpolar (methyl) and polar (hydroxyl) groups oriented pointing either upward or inward toward the pocket. We find wetting of the pocket is favored by the hydroxyl groups and dewetting is favored by the methyl groups. The distribution of waters in the pocket is found to exhibit a two-state-like equilibrium between wet and dry states with a free energy barrier between the two states. Moreover, we demonstrate that the pocket hydration of the cavitands can be collapsed onto a unified adsorption isotherm by assuming the effective pressures within each cavitand pocket differ by a shift pressure that depends on the chemical identity and number of functional groups placed about the portal. These observations support the development of a two-state capillary evaporation model that accurately describes the equilibrium between states and naturally gives rise to the effective shift pressures observed from simulation. This work demonstrates that the hydration of host pockets can be tuned following simple design rules that in turn are expected to impact the thermodynamics of guest complexation.

[Ligand-receptor binding kinetics in drug design]

Traditionally, the thermodynamic values of affinity are considered as the main criterion for the development of new drugs. Usually, these values for drugs are measured <i>in vitro</i> at steady concentrations of the receptor and ligand, which are differed from <i>in vivo</i> environment. Recent studies have shown that the kinetics of the process of drug binding to its receptor make significant contribution in the drug effectiveness. This has increased attention in characterizing and predicting the rate constants of association and dissociation of the receptor ligand at the stage of preclinical studies of drug candidates. A drug with a long residence time can determine ligand-receptor selectivity (kinetic selectivity), maintain pharmacological activity of the drug at its low concentration in vivo. The paper discusses the theoretical basis of protein-ligand binding, molecular determinants that control the kinetics of the drug-receptor binding. Understanding the molecular features underlying the kinetics of receptor-ligand binding will contribute to the rational design of drugs with desired properties.

Role of Water Hydrogen Bonding on Transport of Small Molecules inside Hydrophobic Channels

We conducted a systematic analysis of water networking inside smooth hyperboloid hydrophobic structures (cylindrical, barrel, and hourglass shapes) to elucidate the role of water hydrogen bonding on the transport of small hydrophobic molecules (ligands). Through a series of molecular dynamics simulations, we established that a hydrogen-bonded network forming along the centerline resulted in a water exclusion zone adjacent to the walls. The size of the exclusion zone is a function of the geometry and the nonbonded interaction strength, defining the effective hydrophobicity of the structure. Exclusion of water molecules from this zone results in lower apparent viscosity, leading to acceleration of ligand transport up to 7 times faster than that measured in the bulk. Transport of ligands into and out of the hydrophobic structures was shown to be controlled by a single water molecule that capped the narrow regions in the structure. This mechanism provides physical insights into the behavior and role of water in the bottleneck regions of hydrophobic enzyme channels. These findings were then used in a sister publication [ Escalante , D. E. , Comput. Struct. Biotechnol. J. 2019 17 757 760 ] to develop a model that can accurately predict the transport of ligands along nanochannels of broad-substrate specificity enzymes.

Variational implicit-solvent predictions of the dry-wet transition pathways for ligand-receptor binding and unbinding kinetics

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, multimethod 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 multistate continuous-time Markov chain Brownian dynamics simulation 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 semiquantitative agreement with existing explicit-water simulations, but significantly accelerate the unbinding process. Moreover, our methods allow the characterization of nonequilibrium hydration states of pocket and ligand during the ligand movement, for which we find substantial memory and hysteresis effects for binding vs. unbinding. Our study thus provides a significant step forward toward efficient, physics-based interpretation and predictions of the complex kinetics in realistic ligand-receptor systems.

Tailoring the Variational Implicit Solvent Method for New Challenges: Biomolecular Recognition and Assembly

Predicting solvation free energies and describing the complex water behavior that plays an important role in essentially all biological processes is a major challenge from the computational standpoint. While an atomistic, explicit description of the solvent can turn out to be too expensive in large biomolecular systems, most implicit solvent methods fail to capture “dewetting” effects and heterogeneous hydration by relying on a pre-established (i.e., guessed) solvation interface. Here we focus on the Variational Implicit Solvent Method, an implicit solvent method that adds water “plasticity” back to the picture by formulating the solvation free energy as a functional of all possible solvation interfaces. We survey VISM's applications to the problem of molecular recognition and report some of the most recent efforts to tailor VISM for more challenging scenarios, with the ultimate goal of including thermal fluctuations into the framework. The advances reported herein pave the way to make VISM a uniquely successful approach to characterize complex solvation properties in the recognition and binding of large-scale biomolecular complexes.

"Martinizing" the Variational Implicit Solvent Method (VISM): Solvation Free Energy for Coarse-Grained Proteins

Solvation is a fundamental driving force in many biological processes including biomolecular recognition and self-assembly, not to mention protein folding, dynamics, and function. The variational implicit solvent method (VISM) is a theoretical tool currently developed and optimized to estimate solvation free energies for systems of very complex topology, such as biomolecules. VISM's theoretical framework makes it unique because it couples hydrophobic, van der Waals, and electrostatic interactions as a functional of the solvation interface. By minimizing this functional, VISM produces the solvation interface as an output of the theory. In this work, we push VISM to larger scale applications by combining it with coarse-grained solute Hamiltonians adapted from the MARTINI framework, a well-established mesoscale force field for modeling large-scale biomolecule assemblies. We show how MARTINI-VISM (^MVISM) compares with atomistic VISM (^AVISM) for a small set of proteins differing in size, shape, and charge distribution. We also demonstrate ^MVISM's suitability to study the solvation properties of an interesting encounter complex, barnase-barstar. The promising results suggest that coarse-graining the protein with the MARTINI force field is indeed a valuable step to broaden VISM's and MARTINI's applications in the near future.

Recent developments in the theoretical, simulational, and experimental studies of the role of water hydrogen bonding in hydrophobic phenomena

Hydrophobic effects (hydrophobic hydration and hydrophobic interaction) constitute an important element of a wide variety of phenomena relevant to biological, physical, chemical, environmental, engineering, and pharmaceutical sciences, such as the immiscibility of oil and water, self-assembly of amphiphiles leading to micelle and membrane formation, folding and stability and unfolding of the native structure of a biologically active protein, gating of ion channels, wetting, froth floatation, and adhesion. On the other hand, the hydrogen bonding ability of water plays a major (if not crucial) role in hydrophobic phenomena. We present a review of most important and relatively recent experimental, simulational, and theoretical research on hydrophobic phenomena in various systems. With a particular interest we survey investigations clarifying the role of water hydrogen bonding therein, because it has been the main object of our own recent research. We have developed a probabilistic hydrogen bond (PHB) model that allows one to obtain an analytic expression for the number of bonds per water molecule as a function of its distance to a hydrophobe, hydrophobe radius, and temperature. Knowing that function, one can explicitly identify a water hydrogen bond contribution to the external potential whereto a water molecule is subjected near a hydrophobe. Combining the PHB model with the classical density functional theory (DFT), one can examine the contribution of water hydrogen bonding to the temperature and lengthscale effects on the hydration of particles and on their solvent-mediated interactions over the entire low-to-high temperature and small-to-large lengthscale ranges. We applied the combined DFT/PHB model to study a variety of hydrophobic phenomena such as (liquid) water in contact with a hydrophobic plate, solvation of spherical solutes of various radii in associated and non-associated liquids at various temperatures, the solvent-mediated interaction of spherical solutes and its temperature dependence, interaction of C60 fullerenes in water, temperature effect on the evaporation lengthscale of water confined between two hydrophobes, temperature dependence of the effective width of the solute-solvent transition layer and average density therein. These applications demonstrated that the DFT/PHB model can serve as a valuable tool in studying hydrophobic phenomena because it constitutes a balanced combination of simplicity, accuracy, and detail. The predictions of the combined DFT/PHB approach for the solvent density profiles and thermodynamic aspects of hydrophobic phenomena are generally in good agreement with experiments and simulations. For example, it predicts the small-to-large crossover lengthscale of its mechanism to be approximately in the range from 1nm to 4nm, and decreasing with increasing temperature. It also suggests that, in terms of the average fluid density in the solute-solvent transition layer, the transition layer for small hydrophobes (of radii ≲2 nm) becomes enriched with rather than depleted of fluid when both the solvent-solute affinity and hb-energy alteration ratio become large enough. The boundary values of these parameters, needed for the depletion-to-enrichment crossover, are predicted to decrease with increasing temperature.

Stochastic level-set variational implicit-solvent approach to solute-solvent interfacial fluctuations

Recent years have seen the initial success of a variational implicit-solvent model (VISM), implemented with a robust level-set method, in capturing efficiently different hydration states and providing quantitatively good estimation of solvation free energies of biomolecules. The level-set minimization of the VISM solvation free-energy functional of all possible solute-solvent interfaces or dielectric boundaries predicts an equilibrium biomolecular conformation that is often close to an initial guess. In this work, we develop a theory in the form of Langevin geometrical flow to incorporate solute-solvent interfacial fluctuations into the VISM. Such fluctuations are crucial to biomolecular conformational changes and binding process. We also develop a stochastic level-set method to numerically implement such a theory. We describe the interfacial fluctuation through the "normal velocity" that is the solute-solvent interfacial force, derive the corresponding stochastic level-set equation in the sense of Stratonovich so that the surface representation is independent of the choice of implicit function, and develop numerical techniques for solving such an equation and processing the numerical data. We apply our computational method to study the dewetting transition in the system of two hydrophobic plates and a hydrophobic cavity of a synthetic host molecule cucurbit[7]uril. Numerical simulations demonstrate that our approach can describe an underlying system jumping out of a local minimum of the free-energy functional and can capture dewetting transitions of hydrophobic systems. In the case of two hydrophobic plates, we find that the wavelength of interfacial fluctuations has a strong influence to the dewetting transition. In addition, we find that the estimated energy barrier of the dewetting transition scales quadratically with the inter-plate distance, agreeing well with existing studies of molecular dynamics simulations. Our work is a first step toward the inclusion of fluctuations into the VISM and understanding the impact of interfacial fluctuations on biomolecular solvation with an implicit-solvent approach.

Numerical Treatment of Stokes Solvent Flow and Solute-Solvent Interfacial Dynamics for Nonpolar Molecules

We design and implement numerical methods for the incompressible Stokes solvent flow and solute-solvent interface motion for nonpolar molecules in aqueous solvent. The balance of viscous force, surface tension, and van der Waals type dispersive force leads to a traction boundary condition on the solute-solvent interface. To allow the change of solute volume, we design special numerical boundary conditions on the boundary of a computational domain through a consistency condition. We use a finite difference ghost fluid scheme to discretize the Stokes equation with such boundary conditions. The method is tested to have a second-order accuracy. We combine this ghost fluid method with the level-set method to simulate the motion of the solute-solvent interface that is governed by the solvent fluid velocity. Numerical examples show that our method can predict accurately the blow up time for a test example of curvature flow and reproduce the polymodal (e.g., dry and wet) states of hydration of some simple model molecular systems.

Molecular modeling of ions at interfaces: exploring similarities to hydrophobic solvation through the lens of induced aqueous interfacial fluctuations

Ion specific effects are ubiquitous in chemistry and biology.

A self-consistent phase-field approach to implicit solvation of charged molecules with Poisson-Boltzmann electrostatics

Dielectric boundary based implicit-solvent models provide efficient descriptions of coarse-grained effects, particularly the electrostatic effect, of aqueous solvent. Recent years have seen the initial success of a new such model, variational implicit-solvent model (VISM) [Dzubiella, Swanson, and McCammon Phys. Rev. Lett. 96, 087802 (2006) and J. Chem. Phys. 124, 084905 (2006)], in capturing multiple dry and wet hydration states, describing the subtle electrostatic effect in hydrophobic interactions, and providing qualitatively good estimates of solvation free energies. Here, we develop a phase-field VISM to the solvation of charged molecules in aqueous solvent to include more flexibility. In this approach, a stable equilibrium molecular system is described by a phase field that takes one constant value in the solute region and a different constant value in the solvent region, and smoothly changes its value on a thin transition layer representing a smeared solute-solvent interface or dielectric boundary. Such a phase field minimizes an effective solvation free-energy functional that consists of the solute-solvent interfacial energy, solute-solvent van der Waals interaction energy, and electrostatic free energy described by the Poisson-Boltzmann theory. We apply our model and methods to the solvation of single ions, two parallel plates, and protein complexes BphC and p53/MDM2 to demonstrate the capability and efficiency of our approach at different levels. With a diffuse dielectric boundary, our new approach can describe the dielectric asymmetry in the solute-solvent interfacial region. Our theory is developed based on rigorous mathematical studies and is also connected to the Lum-Chandler-Weeks theory (1999). We discuss these connections and possible extensions of our theory and methods.

LS-VISM: A software package for analysis of biomolecular solvation

We introduce a software package for the analysis of biomolecular solvation. The package collects computer codes that implement numerical methods for a variational implicit-solvent model (VISM). The input of the package includes the atomic data of biomolecules under consideration and the macroscopic parameters such as solute-solvent surface tension, bulk solvent density and ionic concentrations, and the dielectric coefficients. The output includes estimated solvation free energies and optimal macroscopic solute-solvent interfaces that are obtained by minimizing the VISM solvation free-energy functional among all possible solute-solvent interfaces enclosing the solute atoms. We review the VISM with various descriptions of electrostatics. We also review our numerical methods that consist mainly of the level-set method for relaxing the VISM free-energy functional and a compact coupling interface method for the dielectric Poisson-Boltzmann equation. Such numerical methods and algorithms constitute the central modules of the software package. We detail the structure of the package, format of input and output files, workflow of the codes, and the postprocessing of output data. Our demo application to a host-guest system illustrates how to use the package to perform solvation analysis for biomolecules, including ligand-receptor binding systems. The package is simple and flexible with respect to minimum adjustable parameters and a wide range of applications. Future extensions of the package use can include the efficient identification of ligand binding pockets on protein surfaces.

STABILITY OF A CYLINDRICAL SOLUTE-SOLVENT INTERFACE: EFFECT OF GEOMETRY, ELECTROSTATICS, AND HYDRODYNAMICS

The solute-solvent interface that separates biological molecules from their surrounding aqueous solvent characterizes the conformation and dynamics of such molecules. In this work, we construct a solvent fluid dielectric boundary model for the solvation of charged molecules and apply it to study the stability of a model cylindrical solute-solvent interface. The motion of the solute-solvent interface is defined to be the same as that of solvent fluid at the interface. The solvent fluid is assumed to be incompressible and is described by the Stokes equation. The solute is modeled simply by the ideal-gas law. All the viscous force, hydrostatic pressure, solute-solvent van der Waals interaction, surface tension, and electrostatic force are balanced at the solute-solvent interface. We model the electrostatics by Poisson's equation in which the solute-solvent interface is treated as a dielectric boundary that separates the low-dielectric solute from the high-dielectric solvent. For a cylindrical geometry, we find multiple cylindrically shaped equilibrium interfaces that describe polymodal (e.g., dry and wet) states of hydration of an underlying molecular system. These steady-state solutions exhibit bifurcation behavior with respect to the charge density. For their linearized systems, we use the projection method to solve the fluid equation and find the dispersion relation. Our asymptotic analysis shows that, for large wavenumbers, the decay rate is proportional to wavenumber with the proportionality half of the ratio of surface tension to solvent viscosity, indicating that the solvent viscosity does affect the stability of a solute-solvent interface. Consequences of our analysis in the context of biomolecular interactions are discussed.

Electrostatic contribution from solvent in modulating single-walled carbon nanotube association

We perform all-atom molecular dynamics simulations to compute the potential of mean force (PMF) between two (10,10) single-walled carbon nanotubes solvated in pure nonpolarizable SPC/E and polarizable TIP4P-FQ water, at various temperatures. In general, the reversible work required to bring two nanotubes from a dissociated state (free energy reference) to contact state (free energy minimum) is more favorable and less temperature-dependent in TIP4P-FQ than in SPC/E water models. In contrast, molecular properties and behavior of water such as the spatially-resolved water number density (intertube, intratube, or outer regions), for TIP4P-FQ are more sensitive to temperature than SPC/E. Decomposition of the solvent-induced PMF into different spatial regions suggests that TIP4P-FQ has stronger temperature dependence; the opposing destabilizing/stabilizing contributions from intertube water and more distal water balance each other and suppress the temperature dependence of total association free energy. Further investigation of hydrogen bonding network in intertube water reveals that TIP4P-FQ retains fewer hydrogen bonds than SPC/E, which correlates with the lower water number density in this region. This reduction of hydrogen bonds affects the intertube water dipoles. As the intertube volume decreases, TIP4P-FQ dipole moment approaches the gas phase value; the distribution of dipole magnitude also becomes narrower due to less average polarization/perturbation from other water molecules. Our results imply that the reduction of water under confinement may seem trivial, but underlying effects to structure and free energetics are non-negligible.

Solvent cavitation under solvophobic confinement

The stability of liquids under solvophobic confinement can tip in favor of the vapor phase, nucleating a liquid-to-vapor phase transition that induces attractive forces between confining surfaces. In the case of water adjacent to hydrophobic surfaces, experimental and theoretical evidence support confinement-mediated evaporation stabilization of biomolecular and colloidal assemblies. The macroscopic thermodynamic theory of cavitation under confinement establishes the connection between the size of the confining surfaces, interfacial free energies, and bulk solvent pressure with the critical evaporation separation and interfacial forces. While molecular simulations have confirmed the broad theoretical trends, a quantitative comparison based on independent measurements of the interfacial free energies and liquid-vapor coexistence properties has, to the best of our knowledge, not yet been performed. To overcome the challenges of simulating a large number of systems to validate scaling predictions for a three-dimensional fluid, we simulate both the forces and liquid-vapor coexistence properties of a two-dimensional Lennard-Jones fluid confined between solvophobic plates over a range of plate sizes and reservoir pressures. Our simulations quantitatively agree with theoretical predictions for solvent-mediated forces and critical evaporation separations once the length dependence of the solvation free energy of an individual confining plate is taken into account. The effective solid-liquid line tension length dependence results from molecular scale correlations for solvating microscopic plates and asymptotically decays to the macroscopic value for plates longer than 150 solvent diameters. The success of the macroscopic thermodynamic theory at describing two-dimensional liquids suggests application to surfactant monolayers to experimentally confirm confinement-mediated cavitation.

Simulation optimization of spherical non-polar guest recognition by deep-cavity cavitands

Biomimetic deep-cavity cavitand hosts possess unique recognition and encapsulation properties that make them capable of selectively binding a range of non-polar guests within their hydrophobic pocket. Adamantane based derivatives which snuggly fit within the pocket of octa-acid deep cavity cavitands exhibit some of the strongest host binding. Here we explore the roles of guest size and attractiveness on optimizing guest binding to form 1:1 complexes with octa-acid cavitands in water. Specifically we simulate the water-mediated interactions of the cavitand with adamantane and a range of simple Lennard-Jones guests of varying diameter and attractive well-depth. Initial simulations performed with methane indicate hydrated methanes preferentially reside within the host pocket, although these guests frequently trade places with water and other methanes in bulk solution. The interaction strength of hydrophobic guests increases with increasing size from sizes slightly smaller than methane to Lennard-Jones guests comparable in size to adamantane. Over this guest size range the preferential guest binding location migrates from the bottom of the host pocket upwards. For guests larger than adamantane, however, binding becomes less favorable as the minimum in the potential-of-mean force shifts to the cavitand face around the portal. For a fixed guest diameter, the Lennard-Jones well-depth is found to systematically shift the guest-host potential-of-mean force to lower free energies, however, the optimal guest size is found to be insensitive to increasing well-depth. Ultimately our simulations show that adamantane lies within the optimal range of guest sizes with significant attractive interactions to match the most tightly bound Lennard-Jones guests studied.

Water repellency in hydrophobic nanocapsules--molecular view on dewetting

The hydrophobic effect plays a major role in a variety of important phenomena in chemistry, materials science and biology, for instance in protein folding and protein-ligand interactions. Studies--performed within cavities of the unique metal oxide based porous capsules of the type {(pentagon)12(linker)30}≡{(W)W5}12{Mo2(ligand)}30 with different acetate/water ligand ratios--have provided unprecedented results revealing segregation/repellency of the encapsulated "water" from the internal hydrophobic ligand walls of the capsules, while the disordered water molecules, interacting strongly with each other via hydrogen bonding, form in all investigated cases the same type of spherical shell. The present results can be (formally) compared--but only regarding the repellency effect--with the amazing "action" of the (super)hydrophobic Lotus (Nelumbo) leaves, which are self-cleaning based on water repellency resulting in the formation of water droplets picking up dirt. The present results were obtained by constructing deliberately suitable hydrophobic interiors within the mentioned capsules.

Variational Implicit Solvation with Poisson-Boltzmann Theory

We incorporate the Poisson-Boltzmann (PB) theory of electrostatics into our variational implicit-solvent model (VISM) for the solvation of charged molecules in an aqueous solvent. In order to numerically relax the VISM free-energy functional by our level-set method, we develop highly accurate methods for solving the dielectric PB equation and for computing the dielectric boundary force. We also apply our VISM-PB theory to analyze the solvent potentials of mean force and the effect of charges on the hydrophobic hydration for some selected molecular systems. These include some single ions, two charged particles, two charged plates, and the host-guest system Cucurbit[7]uril and Bicyclo[2.2.2]octane. Our computational results show that VISM with PB theory can capture well the sensitive response of capillary evaporation to the charge in hydrophobic confinement and the polymodal hydration behavior and can provide accurate estimates of binding affinity of the host-guest system. We finally discuss several issues for further improvement of VISM.

Heterogeneous Hydration of p53/MDM2 Complex

Water-mediated interactions play critical roles in biomolecular recognition processes. Explicit solvent molecular dynamics (MD) simulations and the variational implicit-solvent model (VISM) are used to study those hydration properties during binding for the biologically important p53/MDM2 complex. Unlike simple model solutes, in such a realistic and heterogeneous solute–solvent system with both geometrical and chemical complexity, the local water distribution sensitively depends on nearby amino acid properties and the geometric shape of the protein. We show that the VISM can accurately describe the locations of high and low density solvation shells identified by the MD simulations and can explain them by a local coupling balance of solvent–solute interaction potentials and curvature. In particular, capillary transitions between local dry and wet hydration states in the binding pocket are captured for interdomain distance between 4 to 6 Å, right at the onset of binding. The underlying physical connection between geometry and polarity is illustrated and quantified. Our study offers a microscopic and physical insight into the heterogeneous hydration behavior of the biologically highly relevant p53/MDM2 system and demonstrates the fundamental importance of hydrophobic effects for biological binding processes. We hope our study can help to establish new design rules for drugs and medical substances.

Variational Implicit-Solvent Modeling of Host-Guest Binding: A Case Study on Cucurbit[7]uril|

The synthetic host cucurbit[7]uril (CB[7]) binds aromatic guests or metal complexes with ultrahigh affinity compared with that typically displayed in protein–ligand binding. Due to its small size, CB[7] serves as an ideal receptor–ligand system for developing computational methods for molecular recognition. Here, we apply the recently developed variational implicit-solvent model (VISM), numerically evaluated by the level-set method, to study hydration effects in the high-affinity binding of the B2 bicyclo[2.2.2]octane derivative to CB[7]. For the unbound host, we find that the host cavity favors the hydrated state over the dry state due to electrostatic effects. For the guest binding, we find reasonable agreement to experimental binding affinities. Dissection of the individual VISM free-energy contributions shows that the major driving forces are water-mediated hydrophobic interactions and the intrinsic (vacuum) host–guest van der Waals interactions. These findings are in line with recent experiments and molecular dynamics simulations with explicit solvent. It is expected that the level-set VISM, with further refinement on the electrostatic descriptions, can efficiently predict molecular binding and recognition in a wide range of future applications.

Phase-field approach to implicit solvation of biomolecules with Coulomb-field approximation

A phase-field variational implicit-solvent approach is developed for the solvation of charged molecules. The starting point of such an approach is the representation of a solute-solvent interface by a phase field that takes one value in the solute region and another in the solvent region, with a smooth transition from one to the other on a small transition layer. The minimization of an effective free-energy functional of all possible phase fields determines the equilibrium conformations and free energies of an underlying molecular system. All the surface energy, the solute-solvent van der Waals interaction, and the electrostatic interaction are coupled together self-consistently through a phase field. The surface energy results from the minimization of a double-well potential and the gradient of a field. The electrostatic interaction is described by the Coulomb-field approximation. Accurate and efficient methods are designed and implemented to numerically relax an underlying charged molecular system. Applications to single ions, a two-plate system, and a two-domain protein reveal that the new theory and methods can capture capillary evaporation in hydrophobic confinement and corresponding multiple equilibrium states as found in molecular dynamics simulations. Comparisons of the phase-field and the original sharp-interface variational approaches are discussed.

Evaluation of Hydration Free Energy by Level-Set Variational Implicit-Solvent Model with Coulomb-Field Approximation

In this article, we systematically apply a novel implicit-solvent model, the variational implicit-solvent model (VISM) together with the Coulomb-Field Approximation (CFA), to calculate the hydration free energy of a large set of small organic molecules. Because these molecules have been studied in detail by molecular dynamics simulations and other implicit-solvent models, they provide a good benchmark for evaluating the performance of VISM-CFA. With all-atom Amber force field parameters, VISM-CFA is able to reproduce well not only the experimental and MD simulated total hydration free energy but also the polar and nonpolar contributions individually. The correlation between VISM-CFA and experiments is R² = 0.763 for the total hydration free energy, with a root-mean-square deviation (RMSD) of 1.83 kcal/mol, and the correlation to results from TIP3P explicit water MD simulations is R² = 0.839 with a RMSD = 1.36 kcal/mol. In addition, we demonstrate that VISM captures dewetting phenomena in the p53/MDM2 complex and hydrophobic characteristics in the system. This work demonstrates that the level-set VISM-CFA can be used to study the energetic behavior of realistic molecular systems with complicated geometries in solvation, protein–ligand binding, protein–protein association, and protein folding processes.

Solvent fluctuations in hydrophobic cavity-ligand binding kinetics

Water plays a crucial part in virtually all protein-ligand binding processes in and out of equilibrium. Here, we investigate the role of water in the binding kinetics of a ligand to a prototypical hydrophobic pocket by explicit-water molecular dynamics (MD) simulations and implicit diffusional approaches. The concave pocket in the unbound state exhibits wet/dry hydration oscillations whose magnitude and time scale are significantly amplified by the approaching ligand. In turn, the ligand's stochastic motion intimately couples to the slow hydration fluctuations, leading to a sixfold-enhanced friction in the vicinity of the pocket entrance. The increased friction considerably decelerates association in the otherwise barrierless system, indicating the importance of molecular-scale hydrodynamic effects in cavity-ligand binding arising due to capillary fluctuations. We derive and analyze the diffusivity profile and show that the mean first passage time distribution from the MD simulation can be accurately reproduced by a standard Brownian dynamics simulation if the appropriate position-dependent friction profile is included. However, long-time decays in the water-ligand (random) force autocorrelation demonstrate violation of the Markovian assumption, challenging standard diffusive approaches for rate prediction. Remarkably, the static friction profile derived from the force correlations strongly resembles the profile derived on the Markovian assumption apart from a simple shift in space, which can be rationalized by a time-space retardation in the ligand's downhill dynamics toward the pocket. The observed spatiotemporal hydrodynamic coupling may be of biological importance providing the time needed for conformational receptor-ligand adjustments, typical of the induced-fit paradigm.

Variational Implicit Solvation with Solute Molecular Mechanics: From Diffuse-Interface to Sharp-Interface Models

Central in a variational implicit-solvent description of biomolecular solvation is an effective free-energy functional of the solute atomic positions and the solute-solvent interface (i.e., the dielectric boundary). The free-energy functional couples together the solute molecular mechanical interaction energy, the solute-solvent interfacial energy, the solute-solvent van der Waals interaction energy, and the electrostatic energy. In recent years, the sharp-interface version of the variational implicit-solvent model has been developed and used for numerical computations of molecular solvation. In this work, we propose a diffuse-interface version of the variational implicit-solvent model with solute molecular mechanics. We also analyze both the sharp-interface and diffuse-interface models. We prove the existence of free-energy minimizers and obtain their bounds. We also prove the convergence of the diffuse-interface model to the sharp-interface model in the sense of Γ-convergence. We further discuss properties of sharp-interface free-energy minimizers, the boundary conditions and the coupling of the Poisson-Boltzmann equation in the diffuse-interface model, and the convergence of forces from diffuse-interface to sharp-interface descriptions. Our analysis relies on the previous works on the problem of minimizing surface areas and on our observations on the coupling between solute molecular mechanical interactions with the continuum solvent. Our studies justify rigorously the self consistency of the proposed diffuse-interface variational models of implicit solvation.

Motion of a Cylindrical Dielectric Boundary

The interplay between geometry and electrostatics contributes significantly to hydrophobic interactions of biomolecules in an aqueous solution. With an implicit solvent, such a system can be described macroscopically by the dielectric boundary that separates the high-dielectric solvent from low-dielectric solutes. This work concerns the motion of a model cylindrical dielectric boundary as the steepest descent of a free-energy functional that consists of both the surface and electrostatic energies. The effective dielectric boundary force is defined and an explicit formula of the force is obtained. It is found that such a force always points from the solvent region to solute region. In the case that the interior of a cylinder is of a lower dielectric, the motion of the dielectric boundary is initially driven dominantly by the surface force but is then driven inward quickly to the cylindrical axis by both the surface and electrostatic forces. In the case that the interior of a cylinder is of a higher dielectric, the competition between the geometrical and electrostatic contributions leads to the existence of equilibrium boundaries that are circular cylinders. Linear stability analysis is presented to show that such an equilibrium is only stable for a perturbation with a wavenumber larger than a critical value. Numerical simulations are reported for both of the cases, confirming the analysis on the role of each component of the driving force. Implications of the mathematical findings to the understanding of charged molecular systems are discussed.

Hydrophobicity of proteins and interfaces: insights from density fluctuations

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.

Shape effect on nanoparticle solvation: a comparison of morphometric thermodynamics and microscopic theories

Conventional wisdom for controlling the nanoparticle size and shape during synthesis is that particle growth favors the direction of a facet with the highest surface energy. However, the particle solvation free energy, which dictates the particle stability and growth, depends not only on the surface area and surface free energy but also on other geometric measures such as the solvent excluded volume and the surface curvature and their affiliated thermodynamic properties. In this work, we study the geometrical effects on the solvation free energies of nonspherical nanoparticles using morphometric thermodynamics and density functional theories. For idealized systems that account for only molecular excluded-volume interactions, morphometric thermodynamics yields a reliable solvation free energy when the particle size is significantly larger than the solvent correlation length. However, noticeable deviations can be identified in comparison to the microscopic theories for predicting the solvation free energies of small nanoparticles. This conclusion also holds for predicting the potential of mean force underlying the colloidal "key-and-lock" interactions. Complementary to the microscopic theories, morphometric thermodynamics requires negligible computational cost, therefore making it very appealing for a broad range of practical applications.