NMR 1H-1H Dipole Relaxation in Fluids: Relaxation of Individual 1H-1H Pairs versus Relaxation of Molecular Modes.

The intramolecular 1H NMR dipole-dipole relaxation of molecular fluids has traditionally been interpreted within the Bloembergen-Purcell-Pound (BPP) theory of NMR intramolecular relaxation. The BPP theory draws upon Debye's theory for describing the rotational diffusion of the 1H-1H pair and predicts a monoexponential decay of the 1H-1H dipole-dipole autocorrelation function between distinct spin pairs. Using molecular dynamics (MD) simulations, we show that for both n-heptane and water this is not the case. In particular, the autocorrelation function of individual 1H-1H intramolecular pairs itself evinces a rich stretched-exponential behavior, implying a distribution in rotational correlation times. However, for the high-symmetry molecule neopentane, the individual 1H-1H intramolecular pairs do conform to the BPP description, suggesting an important role of molecular symmetry in aiding agreement with the BPP model. The intermolecular autocorrelation functions for n-heptane, water, and neopentane also do not admit a monoexponential behavior of individual 1H-1H intermolecular pairs at distinct initial separations. We suggest expanding the autocorrelation function in terms of modes, provisionally termed molecular modes, that do have an exponential relaxation behavior. With care, the resulting Fredholm integral equation of the first kind can be inverted to recover the probability distribution of the molecular modes. The advantages and limitations of this approach are noted.

A cluster size distribution theory to study the thermodynamics and phase behavior of multi-bonding single site solutes in patchy colloidal mixtures.

We study binary mixtures of multi-bonding single site solute particles in a solvent comprising patchy colloid particles. The particles in the mixture interact by very short-ranged attraction and hard-sphere repulsion. The attractive patch on the solute can bond with multiple solvent particles, whereas the patch on the solvent is restricted to bond only once. From a quasi-chemical analysis of association, in the hard-sphere reference we develop an accurate multi-body correlation information for the distribution of solvent particles over the patch region of the solute. We use this information within Wertheim's multi-density formalism to develop a cluster size distribution theory that is capable of capturing the physics of multi-body association for any geometry of association sites on the solute. We use this general framework to study a mixture containing Janus solutes and one- or two-patch solvent particles over a range of concentration of the solute and association strengths. We find that a mixture of two-patch solvent (with both patches of the same kind) and multi-bonding solutes with different patch geometries can have a vapor-liquid equilibrium, although the pure components themselves cannot phase separate. The liquid state occurs at very low densities, forming a so-called empty liquid. For the relative association strengths studied in this work, we observe that the vapor-liquid coexistence curve broadens as the concentration of the patchy solvent particles in the liquid phase is increased. The pressure-composition phase equilibrium curves show negative azeotropes for these mixtures. We also observe that, for these mixtures, as the size of the patch on the solute particles is decreased, the critical temperature and the critical packing fraction decreases.

Solvophobic and solvophilic contributions in the water-to-aqueous guanidinium chloride transfer free energy of model peptides.

We study the solvation free energy of two different conformations (helix and extended) of two different peptides (deca-alanine and deca-glycine) in two different solvents (water and aqueous guanidinium chloride, GdmCl). The free energies are obtained using the quasichemical organization of the potential distribution theorem, an approach that naturally provides the repulsive (solvophobic or cavity) and attractive (solvophilic) contributions to solvation. The solvophilic contribution is further parsed into a chemistry contribution arising from solute interaction with the solvent in the first solvation shell and a long-range contribution arising from non-specific interactions between the solute and the solvent beyond the first solvation shell. The cavity contribution is obtained for two different envelopes, ΣSE, which theory helps identify as the solvent excluded volume, and ΣG, a larger envelope beyond which solute-solvent interactions are Gaussian. The ΣSE envelope is independent of the solvent, as expected on the basis of the insensitivity to the solvent type of the distance of closest approach between protein heavy atoms and solvent heavy atoms, but contrary to the intuition based on treating solvent constituents as spheres of some effective radii. For both envelopes, the cavity contribution in water is proportional to the surface area of the envelope. The same does not hold for GdmCl(aq), revealing the limitation of using molecular area to assess solvation energetics. The ΣG-cavity contribution predicts that GdmCl(aq) should favor the more compact state, contrary to the role of GdmCl in unfolding proteins. The chemistry contribution attenuates this effect, but still the net local (chemistry plus ΣG-packing) contribution is inadequate in capturing the role of GdmCl. With the inclusion of the long-range contribution, which is dominated by van der Waals interaction, aqueous GdmCl favors the extended conformation over the compact conformation. Our finding emphasizes the importance of weak, but attractive, long-range dispersion interactions in protein solution thermodynamics.

NMR spin-rotation relaxation and diffusion of methane.

The translational diffusion-coefficient and the spin-rotation contribution to the 1H NMR relaxation rate for methane (CH4) are investigated using MD (molecular dynamics) simulations, over a wide range of densities and temperatures, spanning the liquid, supercritical, and gas phases. The simulated diffusion-coefficients agree well with measurements, without any adjustable parameters in the interpretation of the simulations. A minimization technique is developed to compute the angular velocity for non-rigid spherical molecules, which is used to simulate the autocorrelation function for spin-rotation interactions. With increasing diffusivity, the autocorrelation function shows increasing deviations from the single-exponential decay predicted by the Langevin theory for rigid spheres, and the deviations are quantified using inverse Laplace transforms. The 1H spin-rotation relaxation rate derived from the autocorrelation function using the "kinetic model" agrees well with measurements in the supercritical/gas phase, while the relaxation rate derived using the "diffusion model" agrees well with measurements in the liquid phase. 1H spin-rotation relaxation is shown to dominate over the MD-simulated 1H-1H dipole-dipole relaxation at high diffusivity, while the opposite is found at low diffusivity. At high diffusivity, the simulated spin-rotation correlation time agrees with the kinetic collision time for gases, which is used to derive a new expression for 1H spin-rotation relaxation, without any adjustable parameters.

Role of Solute Attractive Forces in the Atomic-Scale Theory of Hydrophobic Effects.

The role that van der Waals (vdW) attractive forces play in the hydration and association of atomic hydrophobic solutes such as argon (Ar) in water is reanalyzed using the local molecular field (LMF) theory of those interactions. In this problem, solute vdW attractive forces can reduce or mask hydrophobic interactions as measured by contact peak heights of the ArAr correlation function compared to reference results for purely repulsive core solutes. Nevertheless, both systems exhibit a characteristic hydrophobic inverse temperature behavior in which hydrophobic association becomes stronger with increasing temperature through a moderate temperature range. The new theoretical approximation obtained here is remarkably simple and faithful to the statistical mechanical LMF assessment of the necessary force balance. Our results extend and significantly revise approximations made in a recent application of the LMF approach to this problem and, unexpectedly, support a theory of nearly 40 years ago.

Role of internal motions and molecular geometry on the NMR relaxation of hydrocarbons.

The role of internal motions and molecular geometry on 1H NMR relaxation rates in liquid-state hydrocarbons is investigated using MD (molecular dynamics) simulations of the autocorrelation functions for intramolecular and intermolecular 1H-1H dipole-dipole interactions. The effects of molecular geometry and internal motions on the functional form of the autocorrelation functions are studied by comparing symmetric molecules such as neopentane and benzene to corresponding straight-chain alkanes n-pentane and n-hexane, respectively. Comparison of rigid versus flexible molecules shows that internal motions cause the intramolecular and intermolecular correlation-times to get significantly shorter, and the corresponding relaxation rates to get significantly smaller, especially for longer-chain n-alkanes. Site-by-site simulations of 1H's across the chains indicate significant variations in correlation times and relaxation rates across the molecule, and comparison with measurements reveals insights into cross-relaxation effects. Furthermore, the simulations reveal new insights into the relative strength of intramolecular versus intermolecular relaxation as a function of internal motions, as a function of molecular geometry, and on a site-by-site basis across the chain.

Quasichemical theory and the description of associating fluids relative to a reference: Multiple bonding of a single site solute.

We derive an expression for the chemical potential of an associating solute in a solvent relative to the value in a reference fluid using the quasichemical organization of the potential distribution theorem. The fraction of times the solute is not associated with the solvent, the monomer fraction, is expressed in terms of (a) the statistics of occupancy of the solvent around the solute in the reference fluid and (b) the Widom factors that arise because of turning on solute-solvent association. Assuming pair-additivity, we expand the Widom factor into a product of Mayer f-functions and the resulting expression is rearranged to reveal a form of the monomer fraction that is analogous to that used within the statistical associating fluid theory (SAFT). The present formulation avoids all graph-theoretic arguments and provides a fresh, more intuitive, perspective on Wertheim's theory and SAFT. Importantly, multi-body effects are transparently incorporated into the very foundations of the theory. We illustrate the generality of the present approach by considering examples of multiple solvent association to a colloid solute with bonding domains that range from a small patch on the sphere to a Janus particle to a solute whose entire surface is available for association.

Electrostatic and induction effects in the solubility of water in alkanes.

Experiments show that at 298 K and 1 atm pressure, the transfer free energy, µex, of water from its vapor to liquid normal alkanes CnH2n+2 (n=5…12) is negative. Earlier it was found that with the united-atom TraPPE model for alkanes and the SPC/E model for water, one had to artificially enhance the attractive alkane-water cross interaction to capture this behavior. Here we revisit the calculation of µex using the polarizable AMOEBA and the non-polarizable Charmm General (CGenFF) forcefields. We test both the AMOEBA03 and AMOEBA14 water models; the former has been validated with the AMOEBA alkane model while the latter is a revision of AMOEBA03 to better describe liquid water. We calculate µex using the test particle method. With CGenFF, µex is positive and the error relative to experiments is about 1.5 kBT. With AMOEBA, µex is negative and deviations relative to experiments are between 0.25 kBT (AMOEBA14) and 0.5 kBT (AMOEBA03). Quantum chemical calculations in a continuum solvent suggest that zero point effects may account for some of the deviation. Forcefield limitations notwithstanding, electrostatic and induction effects, commonly ignored in consideration of water-alkane interactions, appear to be decisive in the solubility of water in alkanes.

Intramolecular Interactions Overcome Hydration to Drive the Collapse Transition of Gly15.

Simulations and experiments show oligo-glycines, polypeptides lacking any side chains, can collapse in water. We assess the hydration thermodynamics of this collapse by calculating the hydration free energy at each of the end points of the reaction coordinate, here taken as the end-to-end distance (r) in the chain. To examine the role of the various conformations for a given r, we study the conditional distribution, P(Rg|r), of the radius of gyration for a given value of r. The free energy change versus Rg, -kBT ln P(Rg|r), is found to vary more gently compared to the corresponding variation in the excess hydration free energy. Using this observation within a multistate generalization of the potential distribution theorem, we calculate a tight upper bound for the hydration free energy of the peptide for a given r. On this basis, we find that peptide hydration greatly favors the expanded state of the chain, despite primitive hydrophobic effects favoring chain collapse. The net free energy of collapse is seen to be a delicate balance between opposing intrapeptide and hydration effects, with intrapeptide contributions favoring collapse.

Thermodynamics of mixtures of patchy and spherical colloids of different sizes: A multi-body association theory with complete reference fluid information.

We present a theory to predict the structure and thermodynamics of mixtures of colloids of different diameters, building on our earlier work [A. Bansal et al., J. Chem. Phys. 145, 074904 (2016)] that considered mixtures with all particles constrained to have the same size. The patchy, solvent particles have short-range directional interactions, while the solute particles have short-range isotropic interactions. The hard-sphere mixture without any association site forms the reference fluid. An important ingredient within the multi-body association theory is the description of clustering of the reference solvent around the reference solute. Here we account for the physical, multi-body clusters of the reference solvent around the reference solute in terms of occupancy statistics in a defined observation volume. These occupancy probabilities are obtained from enhanced sampling simulations, but we also present statistical mechanical models to estimate these probabilities with limited simulation data. Relative to an approach that describes only up to three-body correlations in the reference, incorporating the complete reference information better predicts the bonding state and thermodynamics of the physical solute for a wide range of system conditions. Importantly, analysis of the residual chemical potential of the infinitely dilute solute from molecular simulation and theory shows that whereas the chemical potential is somewhat insensitive to the description of the structure of the reference fluid, the energetic and entropic contributions are not, with the results from the complete reference approach being in better agreement with particle simulations.

Structure and thermodynamics of a mixture of patchy and spherical colloids: A multi-body association theory with complete reference fluid information.

A mixture of solvent particles with short-range, directional interactions and solute particles with short-range, isotropic interactions that can bond multiple times is of fundamental interest in understanding liquids and colloidal mixtures. Because of multi-body correlations, predicting the structure and thermodynamics of such systems remains a challenge. Earlier Marshall and Chapman [J. Chem. Phys. 139, 104904 (2013)] developed a theory wherein association effects due to interactions multiply the partition function for clustering of particles in a reference hard-sphere system. The multi-body effects are incorporated in the clustering process, which in their work was obtained in the absence of the bulk medium. The bulk solvent effects were then modeled approximately within a second order perturbation approach. However, their approach is inadequate at high densities and for large association strengths. Based on the idea that the clustering of solvent in a defined coordination volume around the solute is related to occupancy statistics in that defined coordination volume, we develop an approach to incorporate the complete information about hard-sphere clustering in a bulk solvent at the density of interest. The occupancy probabilities are obtained from enhanced sampling simulations but we also develop a concise parametric form to model these probabilities using the quasichemical theory of solutions. We show that incorporating the complete reference information results in an approach that can predict the bonding state and thermodynamics of the colloidal solute for a wide range of system conditions.

Understanding the Thermodynamics of Hydrogen Bonding in Alcohol-Containing Mixtures: Cross-Association.

The thermodynamics of hydrogen bonding in 1-alcohol + water binary mixtures is studied using molecular dynamic (MD) simulation and the polar and perturbed chain form of the statistical associating fluid theory (polar PC-SAFT). The fraction of free monomers in pure saturated liquid water is computed using both TIP4P/2005 and iAMOEBA simulation water models. Results are compared to spectroscopic data available in the literature as well as to polar PC-SAFT. Polar PC-SAFT models hydrogen bonds using single bondable association sites representing electron donors and electron acceptors. The distribution of hydrogen bonds in pure alcohols is computed using the OPLS-AA force field. Results are compared to Monte Carlo (MC) simulations available in the literature as well as to polar PC-SAFT. The analysis shows that hydrogen bonding in pure alcohols is best predicted using a two-site model within the SAFT framework. On the other hand, molecular simulations show that increasing the concentration of water in the mixture increases the average number of hydrogen bonds formed by an alcohol molecule. As a result, a transition in association scheme occurs at high water concentrations where hydrogen bonding is better captured within the SAFT framework using a three-site alcohol model. The knowledge gained in understanding hydrogen bonding is applied to model vapor-liquid equilibrium (VLE) and liquid-liquid equilibrium (LLE) of mixtures using polar PC-SAFT. Predictions are in good agreement with experimental data, establishing the predictive power of the equation of state.

Importance of Hydrophilic Hydration and Intramolecular Interactions in the Thermodynamics of Helix-Coil Transition and Helix-Helix Assembly in a Deca-Alanine Peptide.

For a model deca-alanine peptide the cavity (ideal hydrophobic) contribution to hydration favors the helix state over extended states and the paired helix bundle in the assembly of two helices. The energetic contributions of attractive protein-solvent interactions are separated into quasi-chemical components consisting of a short-range part arising from interactions with solvent in the first hydration shell and the remaining long-range part that is well described by a Gaussian. In the helix-coil transition, short-range attractive protein-solvent interactions outweigh hydrophobic hydration and favor the extended coil states. Analysis of enthalpic effects shows that it is the favorable hydration of the peptide backbone that favors the unfolded state. Protein intramolecular interactions favor the helix state and are decisive in favoring folding. In the pairing of two helices, the cavity contribution outweighs the short-range attractive protein-water interactions. However, long-range, protein-solvent attractive interactions can either enhance or reverse this trend depending on the mutual orientation of the helices. In helix-helix assembly, change in enthalpy arising from change in attractive protein-solvent interactions favors disassembly. In helix pairing as well, favorable protein intramolecular interactions are found to be as important as hydration effects. Overall, hydrophilic protein-solvent interactions and protein intramolecular interactions are found to play a significant role in the thermodynamics of folding and assembly in the system studied.

Molecular Theory and the Effects of Solute Attractive Forces on Hydrophobic Interactions.

The role of solute attractive forces on hydrophobic interactions is studied by coordinated development of theory and simulation results for Ar atoms in water. We present a concise derivation of the local molecular field (LMF) theory for the effects of solute attractive forces on hydrophobic interactions, a derivation that clarifies the close relation of LMF theory to the EXP approximation applied to this problem long ago. The simulation results show that change from purely repulsive atomic solute interactions to include realistic attractive interactions diminishes the strength of hydrophobic bonds. For the Ar-Ar rdfs considered pointwise, the numerical results for the effects of solute attractive forces on hydrophobic interactions are opposite in sign and larger in magnitude than predicted by LMF theory. That comparison is discussed from the point of view of quasichemical theory, and it is suggested that the first reason for this difference is the incomplete evaluation within LMF theory of the hydration energy of the Ar pair. With a recent suggestion for the system-size extrapolation of the required correlation function integrals, the Ar-Ar rdfs permit evaluation of osmotic second virial coefficients B2. Those B2's also show that incorporation of attractive interactions leads to more positive (repulsive) values. With attractive interactions in play, B2 can change from positive to negative values with increasing temperatures. This is consistent with the puzzling suggestions of decades ago that B2 ≈ 0 for intermediate cases of temperature or solute size. In all cases here, B2 becomes more attractive with increasing temperature.

Polymorphic Protein Crystal Growth: Influence of Hydration and Ions in Glucose Isomerase.

Crystal polymorphs of glucose isomerase were examined to characterize the properties and to quantify the energetics of protein crystal growth. Transitions of polymorph stability were measured in poly(ethylene glycol)/NaCl solutions, and one transition point was singled out for more detailed quantitative analysis. Single crystal x-ray diffraction was used to confirm space groups and identify complementary crystal structures. Crystal polymorph stability was found to depend on the NaCl concentration, with stability transitions requiring > 1 M NaCl combined with a low concentration of PEG. Both salting-in and salting-out behavior was observed and was found to differ for the two polymorphs. For NaCl concentrations above the observed polymorph transition, the increase in solubility of the less stable polymorph together with an increase in the osmotic second virial coefficient suggests that changes in protein hydration upon addition of salt may explain the experimental trends. A combination of atomistic and continuum models was employed to dissect this behavior. Molecular dynamics simulations of the solvent environment were interpreted using quasi-chemical theory to understand changes in protein hydration as a function of NaCl concentration. The results suggest that protein surface hydration and Na+ binding may introduce steric barriers to contact formation, resulting in polymorph selection.

Conditional solvation thermodynamics of isoleucine in model peptides and the limitations of the group-transfer model.

The hydration thermodynamics of the amino acid X relative to the reference G (glycine) or the hydration thermodynamics of a small-molecule analog of the side chain of X is often used to model the contribution of X to protein stability and solution thermodynamics. We consider the reasons for successes and limitations of this approach by calculating and comparing the conditional excess free energy, enthalpy, and entropy of hydration of the isoleucine side chain in zwitterionic isoleucine, in extended penta-peptides, and in helical deca-peptides. Butane in gauche conformation serves as a small-molecule analog for the isoleucine side chain. Parsing the hydrophobic and hydrophilic contributions to hydration for the side chain shows that both of these aspects of hydration are context-sensitive. Furthermore, analyzing the solute-solvent interaction contribution to the conditional excess enthalpy of the side chain shows that what is nominally considered a property of the side chain includes entirely nonobvious contributions of the background. The context-sensitivity of hydrophobic and hydrophilic hydration and the conflation of background contributions with energetics attributed to the side chain limit the ability of a single scaling factor, such as the fractional solvent exposure of the group in the protein, to map the component energetic contributions of the model-compound data to their value in the protein. But ignoring the origin of cancellations in the underlying components the group-transfer model may appear to provide a reasonable estimate of the free energy for a given error tolerance.

Solvation free energy of the peptide group: its model dependence and implications for the additive-transfer free-energy model of protein stability.

The group-additive decomposition of the unfolding free energy of a protein in an osmolyte solution relative to that in water poses a fundamental paradox: whereas the decomposition describes the experimental results rather well, theory suggests that a group-additive decomposition of free energies is, in general, not valid. In a step toward resolving this paradox, here we study the peptide-group transfer free energy. We calculate the vacuum-to-solvent (solvation) free energies of (Gly)n and cyclic diglycine (cGG) and analyze the data according to experimental protocol. The solvation free energies of (Gly)n are linear in n, suggesting group additivity. However, the slope interpreted as the free energy of a peptide unit differs from that for cGG scaled by a factor of half, emphasizing the context dependence of solvation. However, the water-to-osmolyte transfer free energies of the peptide unit are relatively independent of the peptide model, as observed experimentally. To understand these observations, a way to assess the contribution to the solvation free energy of solvent-mediated correlation between distinct groups is developed. We show that linearity of solvation free energy with n is a consequence of uniformity of the correlation contributions, with apparent group-additive behavior in the water-to-osmolyte transfer arising due to their cancellation. Implications for inferring molecular mechanisms of solvent effects on protein stability on the basis of the group-additive transfer model are suggested.

Regularizing Binding Energy Distributions and the Hydration Free Energy of Protein Cytochrome C from All-Atom Simulations.

By introducing an external field to temper short-range protein water interactions, we regularize the statistical problem of calculating the hydration free energy, µ(ex), of the protein cytochrome C using the potential distribution theorem. Using this approach, we calculate the nonelectrostatic (dispersion) and electrostatic contributions to µ(ex). The nonelectrostatic contribution interpreted within an accessible surface area approach leads to a surface energy parameter that is about twice the value based on the hydration of small alkanes: at the size scale of the protein, hydrophobic hydration is more stronger relative to small alkanes. The electrostatic contribution does not obey linear response behavior. Further, depending on the choice of the protein dielectric constant, continuum dielectric calculations of the electrostatic contribution differ from the all-atom result by between 6%-12% (in a net value of about -2000 kcal/mol). We conclude by indicating potential applications of the present physically transparent approach toward illuminating the role of water, ions, and osmolytes in protein solution thermodynamics, including in protein folding and aggregation.