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Hydration of hydrophobic solutes treated by the fundamental measure approach

We have developed a method to calculate the hydration of hydrophobic solutes by the fundamental measure theory. This method allows us to carry out calculations of the density profile and the hydration energy for hydrophobic molecules. An additional benefit of the method is the possibility to calculate interaction forces between solvated nanoparticles. On the basis of the designed method we calculate hydration of spherical solutes of various sizes from one angstrom up to several nanometers. We have applied method to evaluate the free energies, the enthalpies, and the entropies of hydrated rare gases and hydrocarbons. The obtained results are in agreement with available experimental data and simulations.


I. INTRODUCTION
Hydrophobic interactions play an important role in stabilization of various biomacromolecular complexes including nucleic acids,proteins,and lipids,because these complexes contain a large number of nonpolar groups [1,2]. Despite the long history of studies of hydrophobic interactions, the theoretical treatment of the nature of hydrophobic interactions is still incomplete. The main problem of the interactions is a complicated multiscale character of these effects which can reveal as microscopic changes of water structure near small hydrophobic groups, as well as conformations and aggregation of biomacromolecules at mesoscopic scales up to several tens of angstroms [3]. The Monte Carlo (MC) and molecular dynamics (MD) methods are most frequently used for modelling molecular interactions in solutions [4]. However, their application to the problem of hydrophobic interactions of macromolecules demands huge computing expenses and in some cases it is essentially limited because of the specified multiscale character of these interactions.New methods based on a statistical treatment have been actively developed in last decade [3,5,6,7,8]. The most suitable method for the specified effects seems to be the density functional theory (DFT) [9,10]. The basic purpose of the approach is to construct the free energy functional of the system which depends on density distribution of liquids particles and intermolecular interaction potentials. Within the framework of this approach there are a lot of various models connected with a concrete choice of the density functional [11,12,13,14] .
In our opinion, one of the most perspective DFT models for calculation of solvation phenomena is the fundamental measure theory (FMT) [12,13,15,16]  This approach is intimately related to the scaled-particle theory (SPT) [17,18] for the homogeneous hard-sphere fluid, and thus one expects that in the uniform limit it should reproduce the SPT results. The current status of this theory includes various generalizations of the scheme to different inter-particle interactions, binary mixtures and polydisperse systems, application to interfaces, wetting, confined geometries, porous media, and dynamical problems as well (see, review [15]). In this work we will calculate the radial distribution functions for rare gases and hydrocarbons, and thermodynamic parameters of their hydrophobic solvation on the basis of the modified FMT which treats the pressure of the liquid system correctly as distinct from the original FMT. In particular, we will determine the size dependence of the solvation energy and calculate interaction forces between two solutes within the limit of low concentration of the dissolved particles. The rest of this paper is organized as follows.
The theory is described in section II, the obtained results and the concluding remarks are presented in section III.

II. FUNDAMENTAL MEASURE THEORY
The DFT is based on the unequivocal dependence between equilibrium density distribution n eq (r) and external potential u ext (r) acting on a system [19]. The free energy of the system F [n] is related with the thermodynamic (Ω) and the chemical (µ) potentials: where u(r) = µ − u ext (r). The equilibrium density n eq (r) is determined by the minimum u] for the given temperature T : In turn chemical potential µ is determined from boundary conditions, i.e., the equilibrium density should tend to the average density of homogeneous liquid n eq (r → ∞) → ρ b . Thus, having the information about functionalΩ[n, u] and the method for evaluation of n(r), we can calculate all necessary equilibrium characteristics of the system. The density functional theory [9] solves the problem by searching the free energy F [n], which consists of two contributions: the ideal (F id [n]) and the excess (F ex [n]) free energies: where Λ is the de Broglie wave length, and β = (k B T ) −1 is the inverse temperature. This free energy is related with the grand canonical functional, minimization of the functional leads to the equilibrium density: Thus, if the functional F ex [n] is known we can calculate the density profile n(r) and then all required characteristics of the solvation.
There are many ways for constructing F ex [n], most of them use the data on functional derivatives ∂F ex /∂n(r) and ∂ 2 F ex /∂n(r)∂n(r ′ ). For Lennard-Jones (LJ) fluids the excess free energy is decomposed into the contribution from a reference system of hard spheres, and the free energy due to attractive interactions The attractive interactions are usually treated by the first-order or the second-order perturbation theories [20,21]. In the mean field approximation the attraction potential u att (r) is considered as a perturbation which gives the contribution to the free energy Various functionals for inhomogeneous hard-sphere (HS) fluids have been developed. One of such methods is the FMT [12,13,15,16] in which the excess free energy is calculated by the use of coarse-grained or smoothed densities. We note that originally this method has been formulated for HS liquids. But later this method has been applied to spheroids of rotation [22] and also for various repulsive and attractive potentials [23,24,25]. In HS liquids the repulsive contribution to the excess free energy is written as where variables n i (r) are determined as weights of the density n(r ′ ) averaged with weight factors w (i) (r − r ′ ) depending on the fundamental geometrical measures of fluid particles, such as volume, surface, etc. The original Rosenfeld formulation [12,13] utilizes the following weighting functions where δ(r) and Θ(r) are the Dirac delta-function and the Heaviside function, respectively, σ is the diameter of a solvent particle. These weight factors determine weighted densities n 0 (r), n v1 (r), n 3 (r), and latter is the local factor of packing.
Within the FMT framework [12,13,15,16] the function Φ[n i ] is determined through six weight densities n i (r): Using these equations and (4) we obtain the equilibrium density Although the original FMT and the SPT is restricted by the HS fluids, there are no limitations for solute-solvent potential u ext (r). Hence we may apply the FMT to treat the LJ solutes.The information on the weighted densities also allows us to calculate the mean force potential W (r) determining the interaction force between two solutes in an infinitely diluted solution where V uu (r) is the direct intermolecular interaction potential. In turn the excess part of thermodynamic potential Ω ex determining the solvation energy is calculated as Since experiments are most commonly done at fixed pressure p, it is convenient to introduce the decomposition of the hydration chemical potential into the excess solvation entropy ∆S and the excess solvation enthalpy ∆H achieved by the use of an isobaric temperature derivative [26], III. RESULTS AND DISCUSSION

A. Application to bulk water
The FMT reduces to the SPT in the limiting case of homogeneous liquid, determining the pressure p and the surface tension γ ∞ of the liquid at a planar wall [27] as In the general case the pressure p hs is too high to describe fluids under normal conditions, for example, it yields 8000 atm for the effective HS diameter σ = σ w = 2.77Å corresponding to water at 25 0 C [28,29]. On the other hand, formula (15) underestimates the surface tension γ ∞ with respect to the experimental values. To exclude these drawbacks both the thermodynamic parameters are considered as the fitting ones in the modified SPT models (see, for example [30,31]). Thus we are to modify the FMT to obtain the realistic estimate for the surface tension. There are a lot of ways to do it, we use the simplest one by exclusion small distances at r < r cut in integrals (6), (7), and (8). As a result, the formulas for the thermodynamic parameters are also modified: We chose r cut to fitγ ∞ by the experimental value γ exp . Below we will indicate that the modified pressurep hs also reduces significantly by three orders and plays a minor role in the hydration of hydrophobic solutes.
First, we employ the method to evaluate the thermodynamic parameters of bulk water.
The HS diameter for water was chosen σ w = 2.77Å and density of water was the ρ b σ 3 w = 0.7. Such choice of the water diameter has been motivated by the solubility experiments of Pierotti [28]. To estimate the attractive contribution (6) we have used the LJ parameters corresponding to the SPC/E model of water [32]. For the solution of the equation (11) we used the Picard iterative algorithm. On every k-step of iteration it was necessary to calculate weight densities n i (r), which are related with n eq (r) through integrals (8). For improvement of convergence we used the algorithm based on mixing of parts of the previous and new iterations n k+1 in = λn k in + (1 − λ)n k out , where λ is the mixing parameter dependent on the bulk density. The step of integration made 0.01σ w , and the number of points of integration is N = 2 12 . These numerical parameters provide the relative precision of density profiles up to 10 −6 and for the chemical potential up to 0.1 kcal/mole. Using relations (13) with the cut-off radius r cut we have evaluated the excess chemical potential depending on the diameter of the HS solute: where σ u is solute diameter. We used r cut = 0.42σ w to fit the calculated surface tension to the experimental value γ exp = 102 cal/(moleÅ 2 ) [33]. Figure 1 represents the dependence of the excess chemical potential ∆µ ex (σ u )/4πσ 2 u obtained by this fitting. Apparently from the figure our calculations are very well compared with the MC results [34] and the calculations received on the basis of the information theory [35]. The figure shows that on the site from zero up to 4Å the curve behaves almost in the linear fashion. It indicates that at such solute sizes the surface effects do not yield the appreciable contribution to the excess chemical potential and the volume contribution has crucial importance. Whereas for the particles, whose radius is more than 10Å, the contribution of the volume component decreases and the surface component grows.
We have also calculated the Tolman length δ, which is the surface thermodynamic property of the water vapor-liquid interface (the distance between the equimolar surface and the surface of tension). For this purpose we use the relation where R = σ u /2 is the solute radius. As a result, we have obtained δ = 0.92Å which is agreed the MC simulations of the SPC/E water [34]. This value is a little bit more than that calculated in [30] (δ = 0.9Å). The pressure of bulk water is estimated as Table 1 lists the data on the above thermodynamic parameters obtained by the FMT with and without cut of the integration range, as well as the data derived from MC simulations [34]. Although the pressure obtained by our procedure exceeds the simulated one by an order, it does not yield the significant effect on the dependence ∆µ ex (σ u ), since the pressure effect is minor for water under normal conditions. These results hold, however, only at temperatures near 25 o C, since the vapor-liquid interfacial tension of water decreases monotonically with temperature increasing, while the excess chemical potential of hard sphere solutes exhibits a maximum with temperature increasing.

B. HS and LJ solutes
On the basis of the designed method we have carried out calculations for various hydrophobic objects. We have considered the two models of hydrophobic solvation. The first of them consists in that the solute is modeled as hard sphere. In framework of this model we have obtained the dependence of the excess chemical potential on the radius of HS solute. Figure 2a shows the comparison of the results received by FMT and by Monte Carlo simulation [29]. The discrepancy between two results are practically missing. The FMT results a bit underestimate the excess chemical potential for hard-sphere solute which radius more than 4Å. To justify our results we have also calculated the enthalpic and entropic contribution into the chemical potential for the HS solute. Figure 2b shows the comparison of the results calculated by the modified FMT with MC results [29]. The main difference between the FMT and the MC results is that the FMT as the SPT yields a monotonic decrease of entropy versus the HS diameter, while the MC data indicate more complicated behavior.
On the other hand, the magnitude of the excess chemical potential for the HS solutes which radius is a less 6Å received by MC a little bit exceed the FMT results.
The LJ model consists in that interaction between solute and solvent molecules is realized by the LJ potential where ε uv and σ uv are constructed from the corresponding parameters ε u , σ u , ε v and σ v by Lorentz-Berhlot mixing rules. Figure 3 shows the two examples of the radial distribution functions (methane and neon) calculated by the modified FMT and derived from the simulations [30,36]. The discrepancies between the first maxima of the peaks do not exceed 10%and the widths of the peaks are practically coincide. The reason of this difference underlies that the water molecules are modeled as hard spheres and the peaks become higher and in many instances a bit narrower. Since we have modeled the solutes as spheres this model seems to be more essential for calculation the excess chemical potential of the atomic solutes. Besides we will apply this approximation to treat the hydration of hydrocarbons also.

C. Inert gases and hydrocarbons
For the evidence of the efficiency of the FMT we have calculated radial distribution functions for linear, branched, cyclic hydrocarbons (methane, ethane, butane and et al.) and for rare gases in water. The LJ parameters are presented in Table 2. On the basis of HS model we have calculated radial distribution function and evaluated the excess chemical potential for the hydrocarbons and rare gases. The calculation of the excess chemical potential has been carried out with the use of the perturbation theory (PT) to take into account the attractive contribution of the solute-solvent interactions. In this case we have calculated it by using equation (6). The attractive solute-solvent contribution to the hydration free energy has been estimated as where g hs (r) is correlation function of water molecules around the HS solute [37]. For this contribution we have used the Weeks-Chandler-Anderson decomposition for the LJ potential into the attractive and the repulsive parts [20]: where σ uv and ε uv are the LJ diameter and well depth, respectively. We denote this approximation as the LJPT model.
Note that the attractive contribution of solvent-solvent interactions energy decreases from 15 to 1 percent with increasing the solute radius from 0.5 to 20Å. Opposite, the attractive part of solute-solvent energy increase with increasing the solute radius. Its magnitude is about 80% of the excess chemical potential. Table 2 Figure 4 shows the insignificant difference between the experimental values of the excess chemical potential and received by the modified FMT. The agreement between the experimental and the calculated data is strongly quantitative. Table 2 shows that the excess chemical potential with corrections of attractive contributions of the solvent-solute and solvent-solvent energy depends on solute radius in the "U" form [38]. The left and right parts of sign "U" put together the rare gases and the hydrocarbons, respectively.
We have also applied the LJ model and utilized the Lennard-Jones potential for solutesolvent interactions. The discrepancy between the experimental and the calculated values of the excess chemical potential for the rare gases and hydrocarbons is practically missing. Table 2 shows the LJ parameters of the rare gases and hydrocarbons and the results of the calculations. We have carried out the analysis of decomposition the excess chemical potential for hydrocarbons and rare gases on entropic and enthalpic parts. Using Eqn. (14), we have derived the excess chemical potential to two parts ∆H and −T ∆S. Table 2 lists the data on the calculated and the experimental excess chemical potential, the enthalpies, and the entropies of hydration of the hydrocarbons from methane to hexane, the branched hydrocarbons (2-methylpropane, 2-methylbutane, and neopentane), and the cyclic hydrocarbons (cyclopentane and cyclohexane). The FMT calculations by the LJPT and the LJ models are labeled as LJPT and LJ, respectively. We note similarly [39] that the obvious feature of the entropic and enthalpic terms that its are greatly larger in absolute value than the excess chemical potential. The hydration enthalpies are large and auspicious, and the hydration entropies are large and unfavorable. The entropic terms are marginally larger in absolute value than the enthalpic terms resulting in the unfavorable but small hydration free energies of the hydrocarbons. It is recognized that this behavior is typical of hydrophobic hydration.
Solvation of apolar compounds in most other solvents, in fact, is usually accompanied by smaller enthalpic and entropic changes. Table 2  Similarly such decomposition the excess chemical potential into entropic and enthalpic part has been carried out for rare gases. Table 2 shows the discrepancy of solvation free energy for all rare gases to be negligible. The difference between the calculated and the experimental excess chemical potential for all atoms does not exceed 0.33 kcal/mol at tem- The hydrophobic effect is frequently connected to characteristic temperature dependences [41,42]. One of the most surprising observations is the entropy of transition convergence of nonpolar molecules from gas phase or nonpolar solvent into water at temperature about 400 K. We have made the calculations to show that the FMT is able to predict the temperature convergence of entropy both qualitatively, and quantitatively correctly. The calculations have been carried out at several temperatures along the experimental saturation curve of water. Using (14) we have obtained the solvation entropy by taking the derivative of the chemical potential along the saturation curve. Figure 5 shows the temperature dependence of the entropy for the different solutes for two cases of calculation. In the first case (Fig.5a) we calculated the excess chemical potential without the fact that the diameter of water decreases with temperature increasing. The entropies are large and negative at room temperature for all the solutes and decrease in magnitude with increasing temperature. The temperature dependence of entropies is approximately linear with slopes increasing with the increasing solute size. Moreover, the entropies converge at about 400 K to approximately zero entropy, although at closer inspection the temperature range of the convergence region is several 10 K and the entropy is not exactly zero at convergence. In the second case, we have taken into account the dependence solvent diameter on temperature [43]. Following this approach, the depletion potential W and the depletion force F between two macroparticles have been calculated depending on the distance h between the solutes [44]: We have also compared the results of MD simulations [45] and that obtained by the FMT. Figure 6 shows that the first peak of FMT results is a bit narrower and higher than MD simulation but localization of curves' zeros almost coincides. The reason of narrowing of the peaks seems to be the modelling of water molecules as hard spheres not as particles interacting via Lennard-Jones potential.

IV. SUMMARY
In this work we have used the FMT for the quantitative description of the hydrophobic phenomena on the basis of the density functional theory. As a result, we have received profiles of radial distribution functions for isolated solutes in a hard-sphere fluid interacting with solute by Lennard-Jones potential. Using the distribution function profile, we have constructed the dependence of the excess chemical potential on the radius of the spherical solutes. To fit the properties of bulk water, namely its surface tension, we have modified the FMT by cutting the radius of integration at small distances. The excess chemical potential has been calculated for several systems, i.e., when interaction between the solute and solvent is simulated as the hard sphere or as the Lennard-Jones potential. We also have shown that the obtained distribution functions reproduce with good precision the oscillating behavior of depletion forces for the particles dissolved in fluid. We have applied method to evaluate the free energies, the enthalpies, and the entropies of hydrated rare gases and hydrocarbons.
The obtained results are in agreement with available experimental data and simulations. the interactions are described by HS repulsive potentials is also well relevant for realistic water-solute potentials. However, such success requires the hard sphere water diameter to be used as an adjustable parameter. The recent extensions of the FMT to soft potentials indicate that the soft FMT is capable to predict the solvent structure for soft repulsive and attractive interaction potentials [46,47]. In this case the FMT yields a systematic way to generalize the treatment for hard bodies to soft interactions. Another bottleneck of the current implementation is the spherical shape of solutes, since the realistic applications should treat the three-dimensional solute structure. There are no restrictions to generalize the above approach to the three-dimensional case, the examples of such generalization are presented in [48,49,50], however such calculations demand special algorithms and more sophisticated computations. Thus, we think that the FMT can provide a promising basis for the accurate study of hydrophobic molecular solutes.  Table 2. LJ parameters of rare gases and hydrocarbons [38,39], the experimental [40] and calculated data on of the excess chemical potential, the enthalpies, and the entropies of their hydration.  lines for the modified SPC [30] and the IT [35], respectively, the triangles correspond to MC simulations (SPC/E water) [34].  Ne [36] and the MC data for Me [30], respectively.