RISM theories

The assumption of Gaussian fluctuations gives the PY approximation for hard sphere fluids and tire MS approximation on addition of an attractive potential. The RISM theory for molecular fluids can also be derived from the same model. 

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],... 

The integral equation method is free of the disadvantages of the continuum model and simulation techniques mentioned in the foregoing, and it gives a microscopic picture of the solvent effect within a reasonable computational time. Since details of the RISM-SCF/ MCSCF method are discussed in the following section we here briefly sketch the reference interaction site model (RISM) theory. 

Essentially, the RISM and extended RISM theories can provide infonnation equivalent to that obtained from simulation techniques, namely, thermodynamic properties, microscopic liquid structure, and so on. But it is noteworthy that the computational cost is dramatically reduced by this analytical treatment, which can be combined with the computationally expensive ab initio MO theory. Another aspect of such treatment is the transparent logic that enables phenomena to be understood in terms of statistical mechanics. Many applications have been based on the RISM and extended RISM theories [10,11]. 

We recently proposed a new method referred to as RISM-SCF/MCSCF based on the ab initio electronic structure theory and the integral equation theory of molecular liquids (RISM). Ten-no et al. [12,13] proposed the original RISM-SCF method in 1993. The basic idea of the method is to replace the reaction field in the continuum models with a microscopic expression in terms of the site-site radial distribution functions between solute and solvent, which can be calculated from the RISM theory. Exploiting the microscopic reaction field, the Fock operator of a molecule in solution can be expressed by... 

One of the most efficient ways to treat this problem is to combine the ab initio MO method and the RISM theory, and this has been achieved by a slight modification of the original RISM-SCF method. Effective atomic charges in liquid water are determined such that the electronic structure and the liquid properties become self-consistent, and along the route of convergence the polarization effect can be naturally incorporated. 

The solute solvent contribution to the free energy stabilizing the DNA poly ion can be calculated within the polymer RISM theory by a charging up process " ... 

If each polymer is modeled as being composed of N beads (or sites) and the interaction potential between polymers can be written as the sum of site-site interactions, then generalizations of the OZ equation to polymers are possible. One approach is the polymer reference interaction site model (PRISM) theory [90] (based on the RISM theory [91]) which results in a nonlinear integral equation given by... 

To describe the equilibrium structure of the system, one can use the polymer integral equation RISM theory [140,141], which allows one to find collective correlation functions. For AB copolymers, the polymer RISM equation is represented in the matrix form [142,143]... 

What is most important for our discussion is the fact that the spatial scale r of the segregated structure for protein-like copolymers is appreciably larger than that for random-block copolymers with the same composition and the same average block length. Also, MIST in the protein-like copolymer system occurs at a temperature higher than that of the random-block system, which is in agreement with the prediction of the polymer RISM theory [153]. 

The original OZ equation can be applied only to liquids composed of spherically symmetrical particles, namely atoms. Chandler and Andersen proposed one of the possible extensions to treat general polyatomic cases, referred to as the reference interaction site model (RISM) or site-site OZ (SSOZ) theory [4], This has been further extended by Hirata and Rossky to be applicable to polar molecules such as water [5], In the RISM theory,... 

RISM theory can be regarded as an alternative to the molecular simulation method, while the RISM-SCF/MCSCF method can be considered as an alternative to the QM/MM method. It is important to note that the method is derived from a natural extension of the RISM theory as well as the ab initio theory. In this section, after introducing the method, we will show some representative examples. 

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. 

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

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...

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. 

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