Per|Mut: Spatially Resolved Hydration Entropies from Atomistic Simulations

Free Energy Density of a Fluid and Its Role in Solvation and Binding

The concept that a fluid has a position-dependent free energy density appears in the literature but has not been fully developed or accepted. We set this concept on an unambiguous theoretical footing via the following strategy. First, we set forth four desiderata that should be satisfied by any definition of the position-dependent free energy density, f(R), in a system comprising only a fluid and a rigid solute: its volume integral, plus the fixed internal energy of the solute, should be the system free energy; it deviates from its bulk value, fbulk, near a solute but should asymptotically approach fbulk with increasing distance from the solute; it should go to zero where the solvent density goes to zero; and it should be well-defined in the most general case of a fluid made up of flexible molecules with an arbitrary interaction potential. Second, we use statistical thermodynamics to formulate a definition of the free energy density that satisfies these desiderata. Third, we show how any free energy density satisfying the desiderata may be used to analyze molecular processes in solution. In particular, because the spatial integral of f(R) equals the free energy of the system, it can be used to compute free energy changes that result from the rearrangement of solutes as well as the forces exerted on the solutes by the solvent. This enables the use of a thermodynamic analysis of water in protein binding sites to inform ligand design. Finally, we discuss related literature and address published concerns regarding the thermodynamic plausibility of a position-dependent free energy density. The theory presented here has applications in theoretical and computational chemistry and may be further generalizable beyond fluids, such as to solids and macromolecules.

Why Solvent Response Contributions to Solvation Free Energies Are Compatible with Ben-Naim's Theorem

We resolve a seeming paradox arising from a common misinterpretation of Ben-Naim's theorem, which rests on the decomposition of the Hamiltonian of a molecular solute/solvent system into solute-solvent and solvent-solvent interactions. According to this theorem, the solvation entropy can also be decomposed into a solute-solvent term and a remaining solvent-solvent term that is commonly referred to as the solvent reorganization term. Crucially, the latter equals the average solvent-solvent interaction energy such that these two solvent-solvent terms cancel and thus do not change the total solvation free energy. This analytical result implies that changes in the solvent-solvent interactions cannot contribute to any thermodynamic driving force. The solvent reorganization term is often identified with the contribution of many-body solvent correlations to the solvation entropy, which seems to imply that these correlations, too, cannot contribute to solvation. However, recent calculations based on atomistic simulations of a solvated globular protein and spatially resolved mutual information expansions revealed substantial contributions of many-body solvent correlations to the solvation free energy, which are not canceled by the enthalpy change of the solvent. Here, we resolved this seeming contradiction and illustrate by two examples─a simple Ising model and a solvated Lennard-Jones particle─that the solvent reorganization entropy and the actual entropy contribution arising from many-body solvent correlations differ both conceptually and numerically. Whereas the solvent reorganization entropy in fact arises from both solvent-solvent as well as solute-solvent interactions and thus has no straightforward intuitive interpretation, the mutual information expansion permits an interpretation in terms of the entropy contribution of solvent-solvent correlations to the solvation free energy.

Pure Graphene Acts as an "Entropic Surfactant" at the Octanol-Water Interface

Graphene oxide (GO) has proved itself as a nanomaterial capable of acting as a surfactant by lowering the interfacial tension of the oil-water interface due to its polar oxygen groups. However, the surfactant behavior of the pure graphene sheet─since prevention of edge oxidation in experimental setups is nontrivial─is still an unresolved issue in graphene research despite significant progress in the field in recent years. Here, we conduct both atomistic and coarse-grained simulations to demonstrate that─surprisingly─even pristine graphene, which only consists of hydrophobic carbon atoms, is attracted to the octanol-water interface and consequently reduces its surface tension by 2.3 kBT/nm2 or about 10 mN/m. Interestingly, the location of the free energy minimum is not precisely at the oil-water interface itself but is rather buried about two octanol layers into the octanol phase, being about 0.9 nm from the water phase. We demonstrate that the observed surfactant behavior is purely entropically driven and can be attributed to the unfavorable lipid-like structuring of octanol molecules at the free octanol-water interface. In essence, graphene enhances the inherent lipid-like behavior of octanol at the water interface rather than directly acting as a surfactant. Importantly, graphene does not display surfactant-like behavior in corresponding Martini coarse-grained simulations of the octanol-water system since the free liquid-liquid interface loses essential structure at the lower coarse-grained resolution. However, a similar surfactant behavior is recovered in coarse-grained simulations of longer alcohols such as dodecan-1-ol and hexadecan-1-ol. The observed discrepancies at different model resolutions enable us to construct a comprehensive model of the surfactant behavior of graphene at the octanol-water interface. The here-gained insights may facilitate the broader utilization of graphene in numerous domains of nanotechnology. Furthermore, since a drug's octanol-water partition coefficient is a crucial physicochemical parameter in rational drug discovery, we also believe that the universality of the here-illustrated entropic surfactant behavior of planar molecules deserves special attention in the drug design and development field.

Spatial Layouts of Low-Entropy Hydration Shells Guide Protein Binding

Protein-protein binding enables orderly biological self-organization and is therefore considered a miracle of nature. Protein‒protein binding is driven by electrostatic forces, hydrogen bonding, van der Waals force, and hydrophobic interactions. Among these physical forces, only hydrophobic interactions can be considered long-range intermolecular attractions between proteins due to the electrostatic shielding of surrounding water molecules. Low-entropy hydration shells around proteins drive hydrophobic attraction among them that essentially coordinate protein‒protein binding. Here, an innovative method is developed for identifying low-entropy regions of hydration shells of proteins by screening off pseudohydrophilic groups on protein surfaces and revealing that large low-entropy regions of the hydration shells typically cover the binding sites of individual proteins. According to an analysis of determined protein complex structures, shape matching between a large low-entropy hydration shell region of a protein and that of its partner at the binding sites is revealed as a universal law. Protein‒protein binding is thus found to be mainly guided by hydrophobic collapse between the shape-matched low-entropy hydration shells that is verified by bioinformatics analyses of hundreds of structures of protein complexes, which cover four test systems. A simple algorithm is proposed to accurately predict protein binding sites.

Quantifying the Molecular Polarization Response of Liquid Water Interfaces at Heterogeneously Charged Surfaces

The hydration shells of proteins mediate interactions, such as small molecule binding, that are vital to their biological function or in some cases their dysfunction. However, even when the structure of a protein is known, the properties of its hydration environment cannot be easily predicted due to the complex interplay between protein surface heterogeneity and the collective structure of water's hydrogen bonding network. This manuscript presents a theoretical study of the influence of surface charge heterogeneity on the polarization response of the liquid water interface. We focus our attention on classical point charge models of water, where the polarization response is limited to molecular reorientation. We introduce a new computational method for analyzing simulation data that is capable of quantifying water's collective polarization response and determining the effective surface charge distribution of hydrated surfaces over atomistic length scales. To illustrate the utility of this method, we present the results of molecular dynamics simulations of liquid water in contact with a heterogeneous model surface and the CheY protein.

Total free energy analysis of fully hydrated proteins

The total free energy of a hydrated biomolecule and its corresponding decomposition of energy and entropy provides detailed information about regions of thermodynamic stability or instability. The free energies of four hydrated globular proteins with different net charges are calculated from a molecular dynamics simulation, with the energy coming from the system Hamiltonian and entropy using multiscale cell correlation. Water is found to be most stable around anionic residues, intermediate around cationic and polar residues, and least stable near hydrophobic residues, especially when more buried, with stability displaying moderate entropy-enthalpy compensation. Conversely, anionic residues in the proteins are energetically destabilized relative to singly solvated amino acids, while trends for other residues are less clear-cut. Almost all residues lose intraresidue entropy when in the protein, enthalpy changes are negative on average but may be positive or negative, and the resulting overall stability is moderate for some proteins and negligible for others. The free energy of water around single amino acids is found to closely match existing hydrophobicity scales. Regarding the effect of secondary structure, water is slightly more stable around loops, of intermediate stability around β strands and turns, and least stable around helices. An interesting asymmetry observed is that cationic residues stabilize a residue when bonded to its N-terminal side but destabilize it when on the C-terminal side, with a weaker reversed trend for anionic residues.

Data Structures and Algorithms for k-th Nearest Neighbours Conformational Entropy Estimation

Entropy of multivariate distributions may be estimated based on the distances of nearest neighbours from each sample from a statistical ensemble. This technique has been applied on biomolecular systems for estimating both conformational and translational/rotational entropy. The degrees of freedom which mostly define conformational entropy are torsion angles with their periodicity. In this work, tree structures and algorithms to quickly generate lists of nearest neighbours for periodic and non-periodic data are reviewed and applied to biomolecular conformations as described by torsion angles. The effect of dimensionality, number of samples, and number of neighbours on the computational time is assessed. The main conclusion is that using proper data structures and algorithms can greatly reduce the complexity of nearest neighbours lists generation, which is the bottleneck step in nearest neighbours entropy estimation.

Spatially Resolved Hydration Thermodynamics in Biomolecular Systems

Water is essential for the structure, dynamics, energetics, and thus the function of biomolecules. It is a formidable challenge to elicit, in microscopic detail, the role of the solvation-related driving forces of biomolecular processes, such as the enthalpy and entropy contributions to the underlying free-energy landscape. In this Perspective, we discuss recent developments and applications of computational methods that provide a spatially resolved map of hydration thermodynamics in biomolecular systems and thus yield atomic-level insights to guide the interpretation of experimental observations. An emphasis is on the challenge of quantifying the hydration entropy, which requires characterization of both the motions of the biomolecules and of the water molecules in their surrounding.

Spatially resolved free-energy contributions of native fold and molten-globule-like Crambin

The folding stability of a protein is governed by the free-energy difference between its folded and unfolded states, which results from a delicate balance of much larger but almost compensating enthalpic and entropic contributions. The balance can therefore easily be shifted by an external disturbance, such as a mutation of a single amino acid or a change of temperature, in which case the protein unfolds. Effects such as cold denaturation, in which a protein unfolds because of cooling, provide evidence that proteins are strongly stabilized by the solvent entropy contribution to the free-energy balance. However, the molecular mechanisms behind this solvent-driven stability, their quantitative contribution in relation to other free-energy contributions, and how the involved solvent thermodynamics is affected by individual amino acids are largely unclear. Therefore, we addressed these questions using atomistic molecular dynamics simulations of the small protein Crambin in its native fold and a molten-globule-like conformation, which here served as a model for the unfolded state. The free-energy difference between these conformations was decomposed into enthalpic and entropic contributions from the protein and spatially resolved solvent contributions using the nonparametric method Per|Mut. From the spatial resolution, we quantified the local effects on the solvent free-energy difference at each amino acid and identified dependencies of the local enthalpy and entropy on the protein curvature. We identified a strong stabilization of the native fold by almost 500 kJ mol-1 due to the solvent entropy, revealing it as an essential contribution to the total free-energy difference of (53 ± 84) kJ mol-1. Remarkably, more than half of the solvent entropy contribution arose from induced water correlations.

Spectrally Resolved Estimation of Water Entropy in the Active Site of Human Carbonic Anhydrase II

A major challenge in understanding ligand binding to biomacromolecules lies in dissecting the underlying thermodynamic driving forces at the atomic level. Quantifying the contributions of water molecules is often especially demanding, although they can play important roles in biomolecular recognition and binding processes. One example is human carbonic anhydrase II, whose active site harbors a conserved network of structural water molecules that are essential for enzymatic catalysis. Inhibitor binding disrupts this water network and changes the hydrogen-bonding patterns in the active site. Here, we use atomistic molecular dynamics simulations to compute the absolute entropy of the individual water molecules confined in the active site of hCAII using a spectrally resolved estimation (SRE) approach. The entropy decrease of water molecules that remain in the active site upon binding of a dorzolamide inhibitor is caused by changes in hydrogen bonding and stiffening of the hydrogen-bonding network. Overall, this entropy decrease is overcompensated by the gain due to the release of three water molecules from the active site upon inhibitor binding. The spectral density calculations enable the assignment of the changes to certain vibrational modes. In addition, the range of applicability of the SRE approximation is systematically explored by exploiting the gradually changing degree of immobilization of water molecules as a function of the distance to a phospholipid bilayer surface, which defines an "entropy ruler". These results demonstrate the applicability of SRE to biomolecular solvation, and we expect it to become a useful method for entropy calculations in biomolecular systems.