3D-RISM

One important class of integral equation theories is based on the reference interaction site model (RISM) proposed by Chandler [77]. These RISM theories have been used to smdy the confonnation of small peptides in liquid water [78-80]. However, the approach is not appropriate for large molecular solutes such as proteins and nucleic acids. Because RISM is based on a reduction to site-site, solute-solvent radially symmetrical distribution functions, there is a loss of infonnation about the tliree-dimensional spatial organization of the solvent density around a macromolecular solute of irregular shape. To circumvent this limitation, extensions of RISM-like theories for tliree-dimensional space (3d-RISM) have been proposed [81,82],... 

A different approach to mention here because it has some similarity to QM/MM is called RISM-SCF [5], It is based on a QM description of the solute, and makes use of some expressions of the integral equation of liquids (a physical approach that for reasons of space we cannot present here) to obtain in a simpler way the information encoded in the solvent distribution function used by MM and QM/MM methods. Both RISM-SCF and QM/MM use this information to define an effective Hamiltonian for the solute and both proceed step by step in improving the description of the solute electronic distribution and solvent distribution function, which in both methods are two coupled quantities. There is in this book a contribution by Sato dedicated to RISM-SCF to which the reader is referred. Sato also includes a mention of the 3D-RISM approach [6] which introduces important features in the physics of the model. In fact the simulation-based methods we have thus far mentioned use a spherically averaged radial distribution function, p(r) instead of a full position dependent function p(r) expression. For molecules of irregular shape and with groups of different polarity on the molecular periphery the examination of the averaged p(r) may lead to erroneous conclusions which have to be corrected in some way [7], The 3D version we have mentioned partly eliminates these artifacts. 

In this chapter, we review our recent studies on molecular recognition by protein based on the RISM and 3D-RISM theories, which have been carried out as a part of the Scientific Research in Priority Areas Water and Biomolecules during last 5 years. 

The 3D-RISM equation for the solute-solvent system at infinite dilution can be derived from (10.8) by taking the statistical average over the angular coordinate of solvent, but not for that of solute [10,11,17]. The equation reads... 

The procedure of solving the equations consists of two steps. We first solve the RISM equation (10.11) for hy (r) of solvent or a mixture of solvents in cases of solutions. Then, we solve the 3D-RISM equation (10.14) for h1 r) of a protein-solvent (solution) system, inserting hy (r) for the solvent into (10.14), which has been calculated in the first step. Considering the definition g(r) = h(r) + 1, g(r) thus obtained is the three-dimensional distribution of solvent molecules around a protein in terms of the interaction site density representation of the solvent or a mixture of solvents in case of solutions. 

The example is the partial molar volume of protein, which can be calculated using (10.19) from h(r), or equivalently from c(r) obtained from the 3D-RISM equation. The partial molar volume of several proteins in water which appear frequently in the literature of protein research is plotted against the molecular weight in Fig. 10.1. [23] By comparing the results with the experimental ones plotted in the same figure, one can readily see that the theory is capable of reproducing the experimental results in quantitative level. At a glance, the results seem to be reproduced by just simple consideration... 

In the following sections, we will demonstrate how the 3D-RISM theory is capable of describing molecular recognition processes. 

In this study [15,29], we have carried out the 3D-RISM calculation for a hen egg-white lysozyme immersed in water and obtained the 3D-distribution function of oxygen and hydrogen of water molecules around and inside the protein. The native 3D structure of the protein is taken from the protein data bank (PDB). The protein is known to have a cavity composed of the residues from Y53 to 158 and from A82 to S91, in which four water molecules have been determined by means of the X-ray diffraction measurement [30]. In our calculation, those water molecules are not included explicitly. 

It is interesting to compare the hydration structure obtained by the 3D-RISM theory with crystallographic water sites of X-ray structure [30]. The crystallographic water molecules in the cavity are depicted in the right of Fig. 10.3, showing four water sites in the cavity, much as the 3D-RISM theory has detected. Moreover, the water distributions obtained from the theory and experiment are quite similar to each other. Thus the 3D-RISM theory can predict the water-binding sites with great success. 

Fig. 10.2. Isosurface representation of the 3D distribution function g(r) of water oxygen around lysozyme calculated by the 3D-RISM theory. Green surfaces or spots show the area where the distribution function is larger than 2 (left), 4 (center), and 8 (right)...

Figure 10.5 shows the size dependence of the coordination number of noble gases at the two binding sites, which is calculated at the concentration of 0.001 M. At the substrate binding site, the coordination number becomes exponentially larger as the size of gas increases (Fig. 10.5a). At the internal site, the coordination number becomes larger with increase in the gas size up to a k, 3.4 A, while it decreases in the region where a > 3.4 A (Fig. 10.5b). As a result, argon has the largest binding affinity to the internal site. These results demonstrate that the 3D-RISM theory has the ability to describe ligand-size...

The 3D-RISM calculation was carried out for aqueous solutions of three different electrolytes, CaCp, NaCl, and KC1, and for four different mutants of the protein, wild type, Q86D, A92D, Q86D/A92D that have been studied experimentally by Kuroki and Yutani [46]. 

Fig. 10.T. Selective ion binding by human lysozyme upper left, wild type upper middle, Q86D upper right, A92D lower left, Q86D/A92D. The lower middle picture shows the calcium binding site in the Q86D/A92D mutant detected by X-ray, while the picture in lower right exhibits the binding-site found by the 3D-RISM theory...

Pressure denaturation of protein has been one of the problems in the focus of protein research due not only to its significance in science [47-49], but also to its importance in industrial applications, including food processing [50], The molecular mechanism of the process has not been clarified for a long time, especially concerning the role played by water or hydration. We have applied the RISM/3D-RISM theory to this problem to clarify the molecular mechanism behind the thermodynamics process [51]. 

The data shown in Fig. 10.8 are the PMV change upon the structural transition and its decomposition into different contributions obtained by the 3D-RISM theory [53]. The decomposition is made by the following equation, which was proposed first by Chalikian and Breslauer [54] and later redefined theoretically by us [23,55],... 

The relation between the thermodynamics and the molecular process of pressure denaturation, clarified by the 3D-RISM theory, is as follows. At the low pressure condition in which all the calculations have been carried out, HPS is not the equilibrium conformation but is one of the fluctuating structures. 

There are several directions to extend the RISM-SCF/MCSCF method that are not described in the present chapter. One of such directions is a combination of ab initio MO theory with 3D-RISM, which is explained in chapter 4. The site-site treatment of the solute-solvent correlations involving the approximation of radial averaging constitutes a bottleneck of the RISM-SCF method, and thus lacks a 3D picture of the solvation structure for complex solutes. The SCF theory combined with the 3D-RISM is free from such a bottleneck. The test computation on the carbon monoxide in water provides a detailed hydration structure of water solvent as well as polarized CO electronic structure. [25] It is also found that the results from the original RISM-SCF/MCSCF method are in reasonable accord with those following from the 3D-RISM-SCF approach after reduction of the orientational dependence. This shows the RISM-SCF/MCSCF approach gives a proper picture for a solvation process. 

In principle, the three-dimensional RISM (3D-RISM) theory described in Chapter 4 is significantly more accurate than the RISM theory employed so far which can be distinguished from the 3D-RISM theory by calling it the one-dimensional RISM (ID-RISM) theory. This is because the 3D-RISM theory, in contrast to the ID-RISM theory, takes orientational average of the molecular Ornstein-Zernike (OZ) equation for solvent molecules only, keeping full description of the shape and orientation of the solute molecule. In reality, a solvent site cannot access to a completely buried atom in the solute molecule. Even if is... 

Table 3.1. Total Energies (kcal/ml) of Five Conformations of Met-enkephalin in Ambient Water, Following from the ID- and 3D-RISM/HNC Theories and from those with the RBC(Repulsive Bridge Correction)-TPT(Thermodynamic Perturbation Theory) Treatment...

Conf. Econi ID-RISM/ HNC 3D-RISM/ HNC ID-RISM/ HNC-hRBC-TPT 3D-RISM/ HNC-fRBC-TPT ... 

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