Chapter 6
Appendix 2: Morphometric Approach
Abstract The hydration of a biomolecule with a fixed structure can be decomposed
into the following two processes: process 1, the hydrophobic hydration, where a
cavity matching the polyatomic structure of the biomolecule is created; and process
2 where biomolecule-water van der Waals and electrostatic interaction potentials
are taken into account. In process 1, the cavity is treated as a solute in the angledependent integral equation (ADIE) theory, the most reliable statistical-mechanical
theory for solute hydration. However, such a complexly shaped solute cannot directly
be handled by the ADIE theory because of the mathematical complications encountered. We have solved this problem by combining the ADIE theory with the morphometric approach (MA). In this chapter, we briefly summarize the basic characteristics of the MA. Our new hybrid method where this ADIE-MA theory and the
three-dimensional reference interaction site model (3D-RISM) theory are applied to
processes 1 and 2, respectively, is capable of calculating the hydration free energy,
energy, and entropy of a large polyatomic solute like a protein with sufficient accuracy
and high speed.
Keywords Cavity creation · Hydrophobic effect · van der Waals potential ·
Electrostatic potential · Morphometric approach · Excluded volume ·
Water-accessible surface area
The cavity created in process 1, which matches the polyatomic structure of a solute
molecule, is modeled as a set of fused, neutral hard spheres. In the morphometric
approach (MA) [1–3], the cavity is geometrically characterized by the excluded
volume V ex , water-accessible surface area A, and integrated mean and Gaussian
curvatures of the accessible surface denoted by Y and Z, respectively. A thermodynamic quantity of hydration in process 1, X p1 , is expressed as the linear combination
of the four geometric measures (i.e., V ex , A, Y, and Z):
X p1 = C 1 (X p1 )V ex + C 2 (X p1 )A + C 3 (X p1 )Y + C 4 (X p1 )Z .
(6.1)
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
M. Kinoshita, Mechanism of Functional Expression of F 1 -ATPase,
SpringerBriefs in Molecular Science,
https://doi.org/10.1007/978-981-33-6232-1_6
77
Appendix 2: Morphometric Approach
Abstract The hydration of a biomolecule with a fixed structure can be decomposed
into the following two processes: process 1, the hydrophobic hydration, where a
cavity matching the polyatomic structure of the biomolecule is created; and process
2 where biomolecule-water van der Waals and electrostatic interaction potentials
are taken into account. In process 1, the cavity is treated as a solute in the angledependent integral equation (ADIE) theory, the most reliable statistical-mechanical
theory for solute hydration. However, such a complexly shaped solute cannot directly
be handled by the ADIE theory because of the mathematical complications encountered. We have solved this problem by combining the ADIE theory with the morphometric approach (MA). In this chapter, we briefly summarize the basic characteristics of the MA. Our new hybrid method where this ADIE-MA theory and the
three-dimensional reference interaction site model (3D-RISM) theory are applied to
processes 1 and 2, respectively, is capable of calculating the hydration free energy,
energy, and entropy of a large polyatomic solute like a protein with sufficient accuracy
and high speed.
Keywords Cavity creation · Hydrophobic effect · van der Waals potential ·
Electrostatic potential · Morphometric approach · Excluded volume ·
Water-accessible surface area
The cavity created in process 1, which matches the polyatomic structure of a solute
molecule, is modeled as a set of fused, neutral hard spheres. In the morphometric
approach (MA) [1–3], the cavity is geometrically characterized by the excluded
volume V ex , water-accessible surface area A, and integrated mean and Gaussian
curvatures of the accessible surface denoted by Y and Z, respectively. A thermodynamic quantity of hydration in process 1, X p1 , is expressed as the linear combination
of the four geometric measures (i.e., V ex , A, Y, and Z):
X p1 = C 1 (X p1 )V ex + C 2 (X p1 )A + C 3 (X p1 )Y + C 4 (X p1 )Z .
(6.1)
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
M. Kinoshita, Mechanism of Functional Expression of F 1 -ATPase,
SpringerBriefs in Molecular Science,
https://doi.org/10.1007/978-981-33-6232-1_6
77