Supporting Information Selective ion-binding by protein probed by the 3D-RISM theory Norio Yoshida † , Saree Phongphanphanee ‡ , Yutaka Maruyama † , Takashi Imai § , and Fumio Hirata †‡ † Department of Theoretical Molecular Science, Institute for Molecular Science, Okazaki 444-8585, Japan, ‡ Department of Fundamental Molecular Science, The Graduate University for Advanced Studies, Okazaki 444-8585, Japan, § Ritsumeikan Univ, Dept Biosci & Bioinformat, Shiga, 525-8577 Japan E-mail: hirata@ims.ac.jp Computational Methods The distribution of water, cation and anion around lysozyme and its mutants in electrolyte solution are calculated by reference interaction site model (RISM) integral equation theory, the best known statistical mechanical theories for liquids. 1 Since the RISM theory has already been described in detail, the formalism is briefly outlined here. The Ornstein-Zernike (OZ) integral equation for a mixture is 2 h (12) = c (12) + c (13)h (32)d (3) All species , (1) where super scripts in a Greek letter indicate the species that compose the mixture, and is the number density of the species . h(12) and c(12) denote the total and direct correlation functions, respectively. The summation in the right hand side of above equation runs over all species in the mixture. In this study, we consider the lysozyme as a solute, which is infinitely diluted in electrolyte solutions, therefore the number density of lysozyme can be regarded as zero. This assumption allows us to split the OZ equation to two sets of equations that are for solute-solvent and solvent-solvent systems. The OZ equation for the solute-solvent system is as follows: h s (12) = c s (12) + s c s (13)h ss (32)d (3) s Solvent species , (2) and the OZ equation for solvent-solvent system is h s s (12) = c s s (12) + s c s s (13)h s s (32)d (3) s Solvent species , (3) where super scripts s, s' and s'' denote solvent species. In the RISM theory, it is an essential step to express the pair correlation functions between a pair of molecules in terms of those between interaction sites, which can be accomplished by averaging the functions over orientations fixing the distance between the interaction sites: h ab s s (r ) = 1 2 d (1)d (2) (R 1 + l 1 a ) (R 2 + l 2 b r )h s s (12) , (4) where R 1 and l 1 a indicate position of molecule 1 in laboratory frame and position of site a of molecule 1 in molecular frame. The RISM equation can be derived from Eq. (2) and Eq. (4) making some approximation for the direct correlation function. 3 The resulting equations is, h ab s s (r ) = ac s c cd s s (r ) db s c, d Site on s, s + s ac s c cd s s (r cd ) h db s s (r db ) c, d Site on s, s s Solvent species , (5) where is an intramolecular correlation function and the asterisk denotes the convolution integrals. On the other hand, because the protein is a fully anisotropic molecule, orientation dependency of the solute-solvent correlation functions cannot be ignored. Therefore, we employ the three dimensional (3D) RISM theory for solute-solvent system. 4 As distinct from usual (1D) RISM theory, the 3D-RISM approaches averages out just the solvent molecular orientations but keeps the orientational description of the solute molecule. This averaging reduces Eq. (2) to the 3D- RISM equation: h a s (r ) = c c s (r c ) ca s + s h ca ss (r ca ) cSite on s s Solvent species . (6) In order to complement these equations, we employ the Kovalenko-Hirata (KH) closure for both solute-solvent and solvent-solvent systems, because it improves numerical convergence rather dramatically. 5 The KH closure for solute- solvent system is expressed as: g a s (r ) = exp(d a s (r )) for d a s (r ) 0 1 + d a s (r ) for d a s (r ) > 0 (7a) d a s (r ) = u a s (r ) + h a s (r ) c a s (r ), (7b) where u a (r) and g a (r)=h a (r)+1 denotes the interaction potential and three-dimensional distribution function (3D-DF) between solute molecule and solvent site a at position r, respectively. For solvent-solvent system, a similar expression of the KH closure has been also devised. Although the approximation is rather heuristic, this closure turns out to be quite successful in explaining a variety of solution processes including vapor-liquid phase transition and ionic product of water. 5,6 The interaction potential is described as sum of the electrostatic interaction and Lennard-Jones potential as follows: u a s (r ) = q b u q a s r r b u bsolute + 4 ab ab r ab 12 ab r ab 6 bsolute , (8) where q a denotes a partial charge on site a, and are Lennard- Jones parameters of common expression. We employed the