Ammm:Nonpolar solvation

Hydrophobicity

Hydrophobicity

• The hydrophobic effect - the tendency for oil and water to segregate- is important in diverse phenomena. Oil and water molecules actually attract each other, but not nearly as strongly as water attracts itself.

• The fact that hydrophobic interaction seems to cause clustering of hydrophobic units was first noted by Walter Kauzmann in 1959.

• Hydrophobicity plays an important role in a wide variety of chemical phenomena
  • protein folding
  • the self-assembly into micelles and membranes
  • the gating of ion channels

Contrasting length scales:Hydration of small and large cavities

It is now generally recognized that hydrophobicity manifests itself differently on small and large length scales.

• Hydrogen bonding is maintained near a small hydrophobic region and not maintained near a large hydrophobic region (Frank Stillinger, 1973), which provides the physical basis for understanding hydrophobic effects.

The following picture shows a disordered hydrogen bonding network typical of liquid water. The blue spheres are oxygen and the white spheres are hydrogen. The dotted lines show the hydrogen bonds with which the water molecules attract one another.

The blue and white particles represent the oxygen (O) and hydrogen (H) atoms, respectively, of the water molecules. The dashed lines indicate hydrogen bonds (that is, O-HO within 35° of being linear and O-to-O bonds of no more than 0.35 nm in length). The space-filling size of the hydrophobic (red) particle in a is similar to that of a methane molecule. The hydrophobic cluster in b contains 135 methane-like particles that are hexagonally close-packed to form a roughly spherical unit of radius larger than 1 nm. For the single cavity pictured in a, each water molecule can readily participate in four hydrogen bonds. (Owing to thermal motions, hydrogen bonding in liquid water is disordered.) Water molecules in a are typical of the bulk liquid where most molecules participate in four hydrogen bonds. The water molecules shown in b, however, are not typical of the bulk. Here, the cluster is sufficiently large that hydrogen bonds cannot simply go around the hydrophobic region. In this case, water molecules near the hydrophobic cluster have typically three or fewer hydrogen bonds.

• On small length scale:
  • In the hydrophobic hydration of small hydrophobic molecules, such as methane, the molecules can fit into the water hydrogen bond network without destroying any hydrogen bonds. Because no H-bonds are broken, the enthalpy of solution (dH) is small.
  • The formation of small cavities in the solvent to accommodate the solute is an entropically dominated process; and the presence of the solute constraints the orientational and translational degrees of freedom of the neighboring water molecules.
  • The standard entropy of solvation for argon, methane is negative and is proportional to the molar excluded volume of the solute.
  • This means that the free energy of the solution is positive and increases both with temperature and with the excluded volume of the solute, and the dissolution of the small apolar solutes is entropically driven.

• On large length scale:
  • When large hydrophobic solutes were inserted into water, they must break some H-bonds at the interface. The missing interfacial H-bonds give rise to a large positive enthalpy of solvation and correspondingly to a free energy change that is proportional to the solute's surface area A, as opposed to being proportional to volume for small hydrophobes. (The pressure under most relevant conditions is small enough to be neglected).
  • Fewer water H-bonds will have to be broken when two large hydrophobes are in contact than when they are apart, so there is a negative enthalpy change when two or more such solutes are brought into contact from larger separations. Since the free energy change is dominated by the enthalpy change, it will be negative too, and there will be a thermodynamic driving force towards aggregation. Thus large-scale hydrophobicity is expected to be enthalpically driven.

Kim Shaap (J. Phys. Chem. 1996, 100, 7713-7721) argued, based on the increasingly positive heat capacity (cp=(dh/dT)p, the the low solubility of apolar groups in water at higher temperatures (>90 °C) is caused by unfavorable enthalpic interactions,not unfavorable entropy changes.

Thermodynamics

Free energy for small cavities in water

For a cavity with volume v(which need not be spherical), the difference in microstate energy $\Delta E$ is infinite whenever a solvent particle is in the cavity, and zero otherwise. This implies that the solvation energy $\Delta G$ of the cavity depends on P (N), the probability of finding in a volume v of pure solvent, N solvent molecules. $\Delta {G}_{v}=-{k}_{b}Tln{P}_{v}(0)$ . For small volumes, th probability Pv(N) is almost exactly gaussian and delta Gv can therefore be expressed analytically in terms of the cavity volume v, the mean number of molecules that occupy that volume in the pure liquid ${<N>}_{v}$(=$N\rho$ , where $\rho$ is solvent density), and the mean-square fluctuation in the number of molecules ${\chi }_{v}$. In particular we can find that $\Delta G\approx {\frac {{k}_{b}T{\rho }^{2}{v}^{2}}{2{\chi }_{v}}}+{\frac {{k}_{b}T\left(ln2\pi {\chi }_{v}\right)}{2}}$ where ${\chi }_{v}={<{\left(\delta N\right)}^{2}>}_{v}=\rho v+{\rho }^{2}\int _{v}^{}dr\int _{v}^{}dr'\left[g\left(|r-r'|\right)-1\right]$ The radial distribution function g(r) is the unity beyond the correlation length of the liquid, therefore $\Delta {G}_{v}$ for small hydrophobic solutes is approximtely linear in solute volume.

Interfaces and size scaling of dG

At ambient conditions (room temperature and 1 atm pressure), liquid water lies close to phase coexistence with its vapor. This condition ensures that large cavities in water are accompanied by an interface like that between liquid and vapor, as suggested by Stillinger, and confirmed by theoretical analysis and simulation. The cost to hydrate the large spherical cavity of radius R is thus $\Delta G=4\pi {R}^{2}\gamma +{\frac {4\pi {R}^{3}p}{3}}\approx 4\pi {R}^{2}\gamma$ ,where $\gamma$ refers to the liquid–vapor surface tension, and p to the pressure, both at the temperature considered.

How solvation free energy changes with solute size

The results are for ambient conditions (room temperature and 1 atm pressure). The circles show the results of detailed microscopic calculations. The liquid–vapor surface tension is shown by gamma. The solid lines show the approximate scaling behavior of G/4R2 for small R, and the asymptotic behavior for large R. This approach can be used to infer the typical length characterizing the crossover behavior, but not the quantitative behavior of G in the crossover regime.

Driving force of assembly

• • When n solutes cluster together to form a hydrophobic unit with an extended surface, the overall solvation free energy changes from growing linearly with solvated volume to growing linearly with solvated surface area. The figure below illustrates that if n is large enough, the solutes can form a cluster with sufficiently large volume to surface ratio that its solvaton free energy is lower than the overall solvation free energy of individual solutes. This effect results in a favoroable driving force for cluster assembly.
  • The diving force gets stronger with increasing T. Entropy does contribute,but the assembly process is drive by the difference between the entropically dominated solvation free energy of small molecules and the enthalpically dominated solvation free energy of large surfaces.

The driving force, dG, for assembling a cluster of small hydrophobic particles. Red lines indicate the free energies at a higher liquid T; blue lines indicate the free energies at a lower T.

Curvature effects play an important role on small(~ 0.1-2 nm) scales

It is known that for systems of nanometer scale, the surface tension γ is no longer a constant. Corrections with curvature effect must be added. For a special case of a spherical solute, Tolman proposed that a surface area function.

Non-polar solvation

The implicit solvent model review

• The cavity is a basic concept in all continuum models. The concept of cavity was first introduced by Onsagers in 1936.

• Polarizable Continuum Model (PCM) was developed by Tomasi in 1981, and subsquently implemented in local and official versions of various QM packages.
  • The total solvation free energy (${G}_{solv}$ ) is given as the sum of a solvent-solvent cavity term (${G}_{cav}$ ) , a solute-solvent repulsive-dispersion term (${G}_{disp}$ ), and a solute-solvent electrostatics (polar) term (${G}_{pol}$ ): ${G}_{solv}={G}_{cav}+{G}_{disp}+{G}_{pol}$ (1)

• From early 1990s (till now),empirical surface area models for the nonpolar component of the solvation free energy are widely used in molecular mechanics/chemistry/biochemistry.
  • The GB/SA model computes ${G}_{cav}+{G}_{disp}$ together as a linear function of the solvent-accessible surface areas: ${G}_{cav}+{G}_{disp}=\gamma \cdot SA(k)$ (2) where SA(k) is the total solvent accessible surface area of atom k and $\gamma$ is an empirically determined atomic surface tension parameter.

• According to Chandler and co-workers Gatt(Gdisp)can be approximated by the van der Waals attractive interaction potential energy between solute and solvent, which has subsequently been confirmed in the simulations by Levy and co-workers.

Analytical Generalized Born plus Nonpolar Model (AGBNP1 & AGBNP2 ) (Ronald Levy, Rutgers)

Overview of AGBNP1:

• Geometrical Estimators.

• Electrostatic Models.

• Nonpolar Models.

Improvement of AGBNP2

• Pairwise Descreening Model Using the Solvent Excluded Volume.

• Short-range Hydrogen Bonding Correction Function.

• Results and Discussion

• With new surface area implementation and without corrections, the hydration free energies of the normal alkanes are too small compared to experiments, this is due to the rate of increase of positive cavity term with increasing alkane size which is insufficient to offset the solute-solvent van der Waals interaction energy term, which becomes more negative with increasing solute size.
• They limited the increase of the surface tension parameters and achieved better results(colume 4 in Table2).

Solvent-accessible surface area model with additional volume and dispersion integral terms (Nathan Baker, WUSTL)

For the repulsive solvation interactions:

• The use of both area and volume terms differs from the more popular area-only treatment of most nonpolar implicit solvent models
• The use of these per-atom gammas des not significantly improve the accuracy of this models.
• The solvent-accessibl definition is from Lee and Richards and Richmonds.
• The Shrake-Rupley SASA algorithm is used to construct solvent-accessible surfaces (discretized with 2000 surface quandrature points). The SA derivatives were calculated by a crude numerical finite differencing scheme through 0.05A displacements of atomic positions.

Repulsive and attractive free energy decomposition (Ray Luo, UC Irvine)

• Surface- or Volume-Dependence for Repulsive Component?

Surprisingly, both molecular surfaces nd volumes can be used as estimators of repulsive ( cavity) solvation free energies with very similar high accuracies, even if the tested monomer molecules are all within the previously reported switching regions ( spherical radii around 10A). A probable reason for the aparent discrpancy between their simulation and literature is that the simulations are on realistic molecules in this study, but simulatios were preformed on ideal spherical cavities in previous studies.

• Compare the parameters optimized here with those used by Baker and co-worker. Since different radius set, different benchmarks (free energies versus mean forces) are used.
  • For the repulsive componenet, a 1.14A water probe was used for the WCA scheme by Baker. In Luo's study, the water probes are uniformly larger with SAV: 1.3A for the sigma scheme and 1.37A for the WCA scheme. Interestingly, the water probes are uniformly smaller with SAS:0.28Afor the sigma scheme an 0.31A for the WCA scheme. However, the monomer test sets would perform equally excellently with the water probes found by Baker.

A Level-Set Variational Implicit Solvent Approach for Nonpolar Molecules (Andrew McCammons, UCSD)

• There are systems, for which established implicit models conceptually fail. Established models fail as interface location is predefined. Geometric, dispersion, and electrostatic contributions have to be coupled/balanced somehow.

• Variational implicit solvent model. Basic idea: write the free energy G as a functional of r

and obtain the solute/water interface by minimization:

Next work:

• add the electrostatic part of the free energy into the model and develop a corresponding level-set method. The electrostatic free energy is often described by the Poisson-Boltzmann (PB) or Generalized Born (GB) method in which the solute-solvent interface is used as the dielectric boundary.
• apply the model and methods to larger systems of polymers and biomolecules.
• apply theory and methods to the calculation of surface forces of solute-solvent interfaces that can be used in Brownian dynamics simulations of biomolecules.

Futher reading

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References