Noura Dawass

2 24 S HAPE E FFECTS 2.1. I NTRODUCTION In chapter 1, the method of Krüger and co–workers [74] was presented, where KBIs of finite subvolumes are used to estimate KBIs in the thermodynamic limit. Considering the shape of the subvolume, the sphere is the natural choice in sim- ple liquids, but other shapes may be more convenient for specific applications. For example, the KB theory was previously applied to study the interactions be- tween large biomolecules and the surrounding solvent molecules [99, 114, 115] . Giambasu et al. [116] used KBIs to study the ionic atmosphere surrounding nu- cleic acids. In their work, selecting the shape of the subvolume depended on the inhomogeneous region surrounding the nucleic acids [116] . For instance, hexag- onal prisms were used to study the fluctuations of solvent molecules around DNA. In principle, it is possible to compute KBIs using the right hand side (R.H.S) of Eq. (1.18) for any shape of the subvolume V . The size of the subvolume can be gradually increased as shown in Figure 1.2, and the number of particles in each subvolume is then used to compute G V αβ using the R.H.S of Eq. (1.18) . Cu- bic subvolumes have been used in the works of Schnell et al. [16] , Cortes-Huerto et al. [83] and others [70, 113, 117] in combination with the R.H.S of Eq. (1.18) . The advantage of using cubic subvolumes is that one does not need to compute distances between molecules and the center of the subvolume. The alternative formulation of finite-size KBIs (Eq. (1.25) ), i.e. direct integration over the RDF, has only been applied to spherical subvolumes [74] . It is important to note that Eq. (1.25) is valid for subvolumes of any shape, provided the geometrical function w ( x ) is known for that shape. The objective of this chapter is to present a unified framework for computing KBIs for subvolumes of arbitrary convex shape. We provide a numerical method to compute the function w ( x ) based on umbrella sampling MC. Once the func- tion w ( x ) is computed for a specific shape, it can be used for any size of the sub- volume. We compute the function w ( x ) for the following shapes: square, cube, and spheroids and cuboids with different aspect ratios. Numerical tables of these functions are provided in a data repository (see Ref. [118] ). We also investigate the effect of the shape of the subvolume on the computation of KBIs. We will show that using a cubic or spherical subvolume leads to the same KB integral in the thermodynamic limit, and that for large subvolumes KBIs scale as area over volume, independent of the shape of the subvolume. This scaling will also ex- plain why truncation of KBIs (i.e. truncating the integral of Eq. (1.3) ) leads to nonphysical results. The chapter is organized as follows. In section 2.2, the numerical method used to compute w ( x ) is introduced. The method is verified by comparing our numerical results to the analytic expressions for a sphere (3 D ), circle (2 D ), and line (1 D ). In section 2.3.2, the function w ( x ) is computed numerically for a cube