Hydrophobicity within the three-dimensional Mercedes-Benz model: Potential of mean force

The Magnetic Field Freezes the Mercedes-Benz Water Model

In this study, we investigate the impact of magnetic fields on the structural and thermodynamic properties of water. To accomplish this, we employed the Mercedes-Benz (MB) model, a two-dimensional representation of water using Lennard-Jones disks with angle-dependent interactions that closely mimic hydrogen bond formation. We extended the MB model by introducing two charges to enable interaction with the magnetic field. Employing molecular dynamics simulations, we thoroughly explored the thermodynamic properties concerning various magnetic flux intensities. As a result, we observed that under a weak magnetic flux, the property of water remained unaltered, while a stronger flux astonishingly led to the freezing of water molecules. Furthermore, our study revealed that once a specific flux magnitude was reached, the density anomaly disappeared, and an increase in flux caused the MB particles to form a glassy state.

Nucleation and Growth of Amyloid Fibrils

The formation of amyloid fibrils is a complex phenomenon that remains poorly understood at the atomic scale. Herein, we perform extended unbiased all-atom simulations in explicit solvent of a short amphipathic peptide to shed light on the three mechanisms accounting for fibril formation, namely, nucleation via primary and secondary mechanisms, and fibril growth. We find that primary nucleation takes place via the formation of an intermediate state made of two laminated β-sheets oriented perpendicular to each other. The amyloid fibril spine subsequently emerges from the rotation of these β-sheets to account for peptides that are parallel to each other and perpendicular to the main axis of the fibril. Growth of this spine, in turn, takes place via a dock-and-lock mechanism. We find that peptides dock onto the fibril tip either from bulk solution or after diffusing on the fibril surface. The latter docking pathway contributes significantly to populate the fibril tip with peptides. We also find that side chain interactions drive the motion of peptides in the lock phase during growth, enabling them to adopt the structure imposed by the fibril tip with atomic fidelity. Conversely, the docked peptide becomes trapped in a local free energy minimum when docked-conformations are sampled randomly. Our simulations also highlight the role played by nonpolar fibril surface patches in catalyzing and orienting the formation of small cross-β structures. More broadly, our simulations provide important new insights into the pathways and interactions accounting for primary and secondary nucleation as well as the growth of amyloid fibrils.

Simple rose model of water in constant electric field

A simple two-dimensional statistical mechanical water model, called the rose model, was used in this work. We studied how a homogeneous constant electric field affects the properties of water. The rose model is a very simple model that helps explain the anomalous properties of water. Rose water molecules are represented as two-dimensional Lennard-Jones disks with potentials for orientation-dependent pairwise interactions mimicking formations of hydrogen bonds. The original model is modified by addition of charges for interaction with the electric field. We studied what kind of influence the electric field strength has on the model's properties. To determine the structure and thermodynamics of the rose model under the influence of the electric field we used Monte Carlo simulations. Under the influence of a weak electric field the anomalous properties and phase transitions of the water do not change. On the other hand, the strong fields shift the phase transition points as well as the position of the density maximum.

The electric field changes the anomalous properties of the Mercedes Benz water model

The influence of a homogeneous constant electric field on water properties was assessed. We used a simple two-dimensional statistical mechanical model called the Mercedes-Benz (MB) model of water in the study. The MB water molecules are two-dimensional disks with Gaussian arms that mimic the formation of hydrogen bonds. The model is modified with added charges for interaction with the electric field. The influence of the strength of the electric field on the water's properties was studied using Monte Carlo simulations. The structure and thermodynamics of the water were determined as a function of the strength of the electric field. We observed that the properties and phase transitions of the water in the low strength electric field does not change. In contrast, the high strength electric field shifts boiling and melting points as well as the position of the density maxima. After further increasing the strength of the electric field the density anomaly disappears.

Monte Carlo simulations of simple two dimensional water-alcohol mixtures

Simple alcohols such as methanol and ethanol, are organic chemicals that can be used to store energy, which can be used as an alternative to fossil fuels. Each alcohol has at least one hydroxyl group attached to a carbon atom of an alkyl group. They can be considered as organic derivatives of water in which one of the hydrogen atoms is replaced by an alkyl group. In this work, we determined the thermodynamic and structural properties of two dimensional water-alcohol mixtures using the Monte Carlo method. We used two-dimensional Mercedes-Benz (MB) model for water and MB based models for lower alcohols. The structural and thermodynamic properties of the mixtures were studied by Monte Carlo simulations in the isothermal-isobaric ensemble. We show that 2D models display similar trends in the density maxima as in real water-alcohol mixtures. With increasing content of alcohols, the temperature of maxima increases and upon further increase starts to decrease and at high concentrations, the density maxima disappears.

Statistical-mechanical liquid theories reproduce anomalous thermodynamic properties of explicit two-dimensional water models

We have developed an analytical theory for a simple model of liquid water. We apply Wertheim's thermodynamic perturbation theory (TPT) and integral equation theory (IET) for associative liquids to the rose model, which is among the simplest models of water. The particles interact through rose potentials for orientation dependent pairwise interactions. Modifying both the shape and range of a three-petal rose function, we construct an efficient and dynamical mimic of the two-dimensional (2D) Mercedes-Benz (MB) water model. The particles in 2D MB are 2D Lennard-Jones disks with three hydrogen bonding arms arranged symmetrically, resembling the Mercedes-Benz logo. Both models qualitatively predict both the anomalous properties of pure water and the anomalous solvation thermodynamics of nonpolar molecules. The IET is based on the orientationally averaged version of the Ornstein-Zernike equation. This is one of the main approximations in the present work. IET correctly predicts the pair correlation functions at high temperatures. Both TPT and IET are in semi-quantitative agreement with the Monte Carlo values of the molar volume, isothermal compressibility, thermal expansion coefficient, and heat capacity. A major advantage of these theories is that they require orders of magnitude less computer time than the Monte Carlo simulations.

Simple two-dimensional models of alcohols

Alcohols are organic compounds characterized by one or more hydroxyl groups attached to a carbon atom of an alkyl group. They can be considered as organic derivatives of water in which one of the hydrogen atoms is replaced by an alkyl group. In this work, the Mercedes-Benz model of water is used to design simple two-dimensional (2D) models of lower alcohols. The structural and thermodynamic properties of the constructed simple models are studied by conducting Monte Carlo simulations in the isothermal-isobaric ensemble. We show that 2D models display similar trends in structuring and thermodynamics as in experiments. The present work on the smallest amphiphilc organic solutes provides a simple testing ground to study the competition between polar and non-polar effects within the molecule and physical properties.

Isothermal-isobaric algorithm to study the effects of rotational degrees of freedom-Benz water model

We have developed isothermal-isobaric algorithm for non-equilibrium Monte Carlo simulations. As first we have shown that the new method correctly predict density by comparing it to the density determined in canonical Monte Carlo simulations through the virial pressure. The new method was then used to study the effect of translational and rotational degrees of freedom on the structural and thermodynamic properties of the simple Mercedes-Benz water model. By holding one of the temperatures constant and varying the other one, we investigated how the position of the density maxima changes. We have observed that upon increase of rotational temperature the fluid become more Lennard-Jones like and the density maxima disappears.

Methane Clathrate Formation is Catalyzed and Kinetically Inhibited by the Same Molecule: Two Facets of Methanol

Here, we perform molecular dynamics simulations to provide atomic-level insights into the dual roles of methanol in enhancing and delaying the rate of methane clathrate hydrate nucleation. Consistent with experiments, we find that methanol slows clathrate hydrate nucleation above 250 K but promotes clathrate formation at temperatures below 250 K. We show that this behavior can be rationalized by the unusual temperature dependence of the methane-methanol interaction in an aqueous solution, which emerges due to the hydrophobic effect. In addition to its antifreeze properties at temperatures above 250 K, methanol competes with water to interact with methane prior to the formation of clathrate nuclei. Below 250 K, methanol encourages water to occupy the space between methane molecules favoring clathrate formation and it may additionally promote water mobility.

Factors Promoting the Formation of Clathrate-Like Ordering of Water in Biomolecular Structure at Ambient Temperature and Pressure

Clathrate hydrate forms when a hydrophobic molecule is entrapped inside a water cage or cavity. Although biomolecular structures also have hydrophobic patches, clathrate-like water is found in only a limited number of biomolecules. Also, while clathrate hydrates form at low temperature and moderately higher pressure, clathrate-like water is observed in biomolecular structure at ambient temperature and pressure. These indicate presence of other factors along with hydrophobic environment behind the formation of clathrate-like water in biomolecules. In the current study, we presented a systematic approach to explore the factors behind the formation of clathrate-like water in biomolecules by means of molecular dynamics simulation of a model protein, maxi, which is a naturally occurring nanopore and has clathrate-like water inside the pore. Removal of either confinement or hydrophobic environment results in the disappearance of clathrate-like water ordering, indicating a coupled role of these two factors. Apart from these two factors, clathrate-like water ordering also requires anchoring groups that can stabilize the clathrate-like water through hydrogen bonding. Our results uncover crucial factors for the stabilization of clathrate-like ordering in biomolecular structure which can be used for the development of new biomolecular structure promoting clathrate formation.

Influence of glycerol on the cooling effect of pair hydrophobicity in water: relevance to proteins’ stabilization at low temperature

Glycerol reduces the cooling effect of pair hydrophobicity (reduction of hydrophobicity with decreasing temperature) in water.

Molecular Insights into the Unusual Structure of an Antifreeze Protein with a Hydrated Core

The primary driving force for protein folding is the formation of a well-packed, anhydrous core. However, recently, the crystal structure of an antifreeze protein, maxi, has been resolved where the core of the protein is filled with water, which apparently contradicts the existing notion of protein folding. Here, we have performed standard molecular dynamics (MD) simulation, replica exchange MD (REMD) simulation, and umbrella sampling using TIP4P water at various temperatures (300, 260, and 240 K) to explore the origin of this unusual structural feature. It is evident from standard MD and REMD simulations that the protein is found to be stable at 240 K in its unusual state. The core of protein has two layers of semi-clathrate water separating the methyl groups of alanine residues from different helical strands. However, with increasing temperature (260 and 300 K), the stability decreases as the core becomes dehydrated, and methyl groups of alanine are tightly packed driven by hydrophobic interactions. Calculation of the potential of mean force by an umbrella sampling technique between a pair of model hydrophobes resembling maxi protein at 240 K shows the stabilization of second solvent-separated minima (SSM), which provides a thermodynamic rationale of the unusual structural feature in terms of weakening of the hydrophobic interaction. Because the stabilization of SSMs is implicated for cold denaturation, it suggests that the maxi protein is so designed by nature where the cold denatured-like state becomes the biologically active form as it works near or below the freezing point of water.

Hierarchy of anomalies in the two-dimensional Mercedes-Benz model of water

We investigate by Monte Carlo simulations density, diffusion, and structural anomalies of the simple two-dimensional Mercedes-Benz (MB) model of water, which is a very simple toy model for explaining the origin of water properties. MB water molecules are modeled as two-dimensional Lennard-Jones disks, with three orientation-dependent hydrogen-bonding arms, arranged as in the MB logo. The model is in a way also a variance of silica-like models. Beside the known thermodynamic anomaly for the model we also found diffusion and structural anomalies and map out the cascade of density, structural, pair entropy, and diffusivity anomalies for MB model. The orientational order parameters with three and six-fold symmetry were determined and maximum for each one observed. The anomalies occur in hierarchy order, which is a slight variation of the hierarchy order in real water. The diffusion anomaly region is the innermost in the hierarchy while for water it is the density anomaly region.

Modelling water with simple Mercedes-Benz models

The structures and properties of biomolecules like proteins, nucleic acids, and membranes depend on water. Water is also very important in industry. Overall, water is unusual substance with more than 70 anomalous properties. The understanding of water is advancing significantly due to theoretical and computational modeling. There are different kind of models, models with fine-scale properties and increasing structural detail with increasing computational expense and simple models which focus on global properties of water like thermodynamics, phase diagram and are less computational expensive. Simplified models give a better understanding of water in ways that complement more complex models. Here, we review a simple model, the two dimensional Mercedes-Benz (MB) model of water. We present results by Monte Carlo simulations for anomalies and phase diagram and application of various theoretical methods.

A Simple Two Dimensional Model of Methanol

Methanol is the simplest alcohol and possible energy carrier because it is easier to store than hydrogen and burns cleaner than fossil fuels. It is a colorless liquid, completely miscible with water and organic solvents and is very hygroscopic. Here, simple two-dimensional models of methanol, based on Mercedes-Benz (MB) model of water, are examined by Monte Carlo simulations. Methanol particles are modeled as dimers formed by an apolar Lennard-Jones disk, mimicking the methyl group, and a sphere with two hydrogen bonding arms for the hydroxyl group. The used models are the one proposed by Hribar-Lee and Dill (Acta Chimica Slovenica, 53:257, 2006.) with the overlapping discs and a new model with tangentially fused dimers. The comparison was done between the models, in connection to the MB water, as well as with experimental results and with new simulations done for 3D models of methanol. Both 2D models show similar trends in structuring and thermodynamics. The difference is the most pronounced at lower temperatures, where the smaller model exhibits spontaneous crystallization, while the larger model shows metastable states. The 2D structural organization represents well the clustering tendency observed in 3D models, as well as in experiments. The models qualitatively agree with the bulk methanol thermodynamic properties like density and isothermal compressibility, however, heat capacity at the constant pressure shows trend more similar to the water behavior. This work on the smallest amphiphilic organic solute provides a simple testing ground to study the competition between polar and non-polar effects within the molecule and physical properties.

Liquid part of the phase diagram and percolation line for two-dimensional Mercedes-Benz water

Monte Carlo simulations and Wertheim's thermodynamic perturbation theory (TPT) are used to predict the phase diagram and percolation curve for the simple two-dimensional Mercedes-Benz (MB) model of water. The MB model of water is quite popular for explaining water properties, but the phase diagram has not been reported till now. In the MB model, water molecules are modeled as two-dimensional Lennard-Jones disks, with three orientation-dependent hydrogen-bonding arms, arranged as in the MB logo. The liquid part of the phase space is explored using grand canonical Monte Carlo simulations and two versions of Wertheim's TPT for associative fluids, which have been used before to predict the properties of the simple MB model. We find that the theory reproduces well the physical properties of hot water but is less successful at capturing the more structured hydrogen bonding that occurs in cold water. In addition to reporting the phase diagram and percolation curve of the model, it is shown that the improved TPT predicts the phase diagram rather well, while the standard one predicts a phase transition at lower temperatures. For the percolation line, both versions have problems predicting the correct position of the line at high temperatures.

Analytical theory of the hydrophobic effect of solutes in water

We develop an analytical statistical-mechanical model for hydrophobic solvation in water. In this three-dimensional Mercedes-Benz-like model, two neighboring waters have three possible interaction states: a radial van der Waals interaction, a tetrahedral orientation-dependent hydrogen-bonding interaction, or no interaction. Nonpolar solutes are modeled as van der Waals particles of different radii. The model is sufficiently simple that we can calculate the partition function and thermal and volumetric properties of solvation versus temperature, pressure, and solute radius. Predictions are in good agreement with results of Monte Carlo simulations. And their trends agree with experiments on hydrophobic solute insertion. The theory shows that first-shell waters are more highly structured than bulk waters, because of hydrogen bonding, and that that structure melts out faster with temperature than it does in bulk waters. Because the theory is analytical, it can explore a broad range of solvation properties and anomalies of water, at minimal computational expense.

Integral equation and thermodynamic perturbation theory for a two-dimensional model of chain-forming fluid

In this paper we applied analytical theories for the two dimensional chain-forming fluid. Wertheims thermodynamic perturbation theory (TPT) and integral equation theory (IET) for associative liquids were used to study thermodynamical and structural properties of the chain-forming model. The model has polymerizing points at arbitrary position from center of the particles. Calculated analytical results were tested against corresponding results obtained by Monte Carlo computer simulations to check the accuracy of the theories. The theories are accurate for the different positions of patches of the model at all values of the temperature and density studied. The IET's pair correlation functions of the model agree well with computer simulations. Both TPT and IET are in good agreement with the Monte Carlo values of the energy, chemical potential and ratios of free, once and twice bonded particles.

Liquid-liquid critical point in a simple analytical model of water

A statistical model for a simple three-dimensional Mercedes-Benz model of water was used to study phase diagrams. This model on a simple level describes the thermal and volumetric properties of waterlike molecules. A molecule is presented as a soft sphere with four directions in which hydrogen bonds can be formed. Two neighboring waters can interact through a van der Waals interaction or an orientation-dependent hydrogen-bonding interaction. For pure water, we explored properties such as molar volume, density, heat capacity, thermal expansion coefficient, and isothermal compressibility and found that the volumetric and thermal properties follow the same trends with temperature as in real water and are in good general agreement with Monte Carlo simulations. The model exhibits also two critical points for liquid-gas transition and transition between low-density and high-density fluid. Coexistence curves and a Widom line for the maximum and minimum in thermal expansion coefficient divides the phase space of the model into three parts: in one part we have gas region, in the second a high-density liquid, and the third region contains low-density liquid.

Existence of a liquid-liquid phase transition in methanol

A simple model is constructed to study the phase diagram and thermodynamic properties of methanol, which is described as a dimer of an apolar sphere mimicking the methyl group and a sphere with core-softened potential as the hydroxyl group. Performing classical Monte Carlo simulations, we obtained the phase diagram, showing a second critical point between two different liquid phases. Evaluating systems with a different number of particles, we extrapolate to infinite size in accordance with Ising universality class to obtain bulk values for critical temperature, pressure, and density. Strong evidence that the structure of the liquid changes upon transition from high- to low-density phase was provided. From the experimentally determined hydrogen bond strength and length in methanol and water, we propose where the second critical point of methanol should be.

Thermodynamics and the hydrophobic effect in a core-softened model and comparison with experiments

A simple and computationally inexpensive core-softened model, originally proposed by Franzese [G. Franzese, J. Mol. Liq. 136, 267 (2007)], was adopted to show that it exhibits properties of waterlike fluid and hydrophobic effect. The potential used between particles is spherically symmetric with two characteristic lengths. Thermodynamics of nonpolar solvation were modeled as an insertion of a modified Lennard-Jones particle. It was investigated how the anomalous predictions of the model as well as the nonpolar solvation compare with the experimental data for water anomalies and the temperature dependence of noble gases hydration. It was shown that the model qualitatively follows the same trends as water. The model is able to reproduce waterlike anomalous properties (density maximum, heat capacity minimum, isothermal compressibility, etc.) and hydrophobic effect (minimum solubility for nonpolar solutes near ambient conditions, increased solubility of larger noble gases, etc.). It is argued that the model yields similar results as more complex and computationally expensive models.

The application of the integral equation theory to study the hydrophobic interaction

The Wertheim's integral equation theory was tested against newly obtained Monte Carlo computer simulations to describe the potential of mean force between two hydrophobic particles. An excellent agreement was obtained between the theoretical and simulation results. Further, the Wertheim's integral equation theory with polymer Percus-Yevick closure qualitatively correctly (with respect to the experimental data) describes the solvation structure under conditions where the simulation results are difficult to obtain with good enough accuracy.

The application of the thermodynamic perturbation theory to study the hydrophobic hydration

The thermodynamic perturbation theory was tested against newly obtained Monte Carlo computer simulations to describe the major features of the hydrophobic effect in a simple 3D-Mercedes-Benz water model: the temperature and hydrophobe size dependence on entropy, enthalpy, and free energy of transfer of a simple hydrophobic solute into water. An excellent agreement was obtained between the theoretical and simulation results. Further, the thermodynamic perturbation theory qualitatively correctly (with respect to the experimental data) describes the solvation thermodynamics under conditions where the simulation results are difficult to obtain with good enough accuracy, e.g., at high pressures.

Ice polyamorphism in the minimal Mercedes-Benz model of water

We investigate ice polyamorphism in the context of the two-dimensional Mercedes-Benz model of water. We find a first-order phase transition between a crystalline phase and a high-density amorphous phase. Furthermore, we find a reversible transformation between two amorphous structures of high and low density; however, we find this to be a continuous and not an abrupt transition, as the low-density amorphous phase does not show structural stability. We discuss the origin of this behavior and its implications with regard to the minimal generic modeling of polyamorphism.

Unifying microscopic mechanism for pressure and cold denaturations of proteins

We study the stability of globular proteins as a function of temperature and pressure through NPT simulations of a coarse-grained model. We reproduce the elliptical stability of proteins and highlight a unifying microscopic mechanism for pressure and cold denaturations. The mechanism involves the solvation of nonpolar residues with a thin layer of water. These solvated states have lower volume and lower hydrogen-bond energy compared to other conformations of nonpolar solutes. Hence, these solvated states are favorable at high pressure and low temperature, and they facilitate protein unfolding under these thermodynamical conditions.

Analytical model for three-dimensional Mercedes-Benz water molecules

We developed a statistical model which describes the thermal and volumetric properties of water-like molecules. A molecule is presented as a three-dimensional sphere with four hydrogen-bonding arms. Each water molecule interacts with its neighboring waters through a van der Waals interaction and an orientation-dependent hydrogen-bonding interaction. This model, which is largely analytical, is a variant of a model developed before for a two-dimensional Mercedes-Benz model of water. We explored properties such as molar volume, density, heat capacity, thermal expansion coefficient, and isothermal compressibility as a function of temperature and pressure. We found that the volumetric and thermal properties follow the same trends with temperature as in real water and are in good general agreement with Monte Carlo simulations, including the density anomaly, the minimum in the isothermal compressibility, and the decreased number of hydrogen bonds upon increasing the temperature.

Simple model of hydrophobic hydration

Water is an unusual liquid in its solvation properties. Here, we model the process of transferring a nonpolar solute into water. Our goal was to capture the physical balance between water's hydrogen bonding and van der Waals interactions in a model that is simple enough to be nearly analytical and not heavily computational. We develop a 2-dimensional Mercedes-Benz-like model of water with which we compute the free energy, enthalpy, entropy, and the heat capacity of transfer as a function of temperature, pressure, and solute size. As validation, we find that this model gives the same trends as Monte Carlo simulations of the underlying 2D model and gives qualitative agreement with experiments. The advantages of this model are that it gives simple insights and that computational time is negligible. It may provide a useful starting point for developing more efficient and more realistic 3D models of aqueous solvation.

Hydrophobic interactions in the formation of secondary structures in small peptides

Effects of the attractive and repulsive parts of hydrophobic interactions on α helices and β sheets in small peptides are investigated using a simple atomic potential. Typically, a physical spatial range of attraction tends to favor β sheets, but α helices would be favored if the attractive range were more extended. We also found that desolvation barriers favor β sheets in collapsed conformations of polyalanine, polyvaline, polyleucine, and three fragments of amyloid peptides tested in this study. Our results provide insight into the multifaceted role of hydrophobicity in secondary structure formation, including the α to β transitions in certain amyloid peptides.