Chapter 5
Appendix 1: Angle-Dependent Integral
Equation Theory
Abstract By virtue of the recently achieved, remarkable progress of the integral
equation theories for solute hydration, we can now handle a large biomolecule like
a protein or protein complex immersed in water by means of statistical mechanics
in which the biomolecule structure is taken into account at the atomic level and
a molecular model is employed for water. The angle-dependent integral equation
(ADIE) theory is best suited to the assessment of the hydrophobic hydration, the
most important physical factor in hydration thermodynamics. In this chapter, we
briefly summarize the basic characteristics of the ADIE theory.
Keywords Integral equation theory · Ornstein-Zernike relation · Closure
equation · Angle-dependent integral equation theory · Hydration
thermodynamics · Hydrophobic hydration
The Ornstein-Zernike (OZ) relation and a closure equation are the two basic equations
in an integral equation theory (IET) [1]. They are derived from the system partition
function using a variety of correlation and distribution functions defined on the basis
of classical statistical mechanics. Not only the solvent structure near a solute but also
thermodynamic quantities of solvation (e.g., the solvation free energy, energy, and
entropy) can be calculated via the following two steps:
Step 1. The bulk solvent is treated. In the case of a single component, the temperature, number density, and solvent-solvent interaction potential form the input data.
By numerically solving the two basic equations mentioned above, we can calculate the solvent-solvent correlation functions, microscopic density and orientational
structures of the solvent, and thermodynamic quantities. Unlike in the molecular
dynamics (MD) and Monte Carlo (MC) simulations, the average value of a physical
quantity can be calculated, in essence, for an infinitely large system and an infinitely
large number of system configurations. The IET is free from the problems of too
small a system size and statistical errors, which often arise in the MD and MC simulations. Not only a simple-fluid solvent but also a molecular liquid such as water
can be treated by the IET. For water, the angle-dependent integral equation (ADIE)
theory [2–6] is the most reliable tool.
© 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_5
71
Appendix 1: Angle-Dependent Integral
Equation Theory
Abstract By virtue of the recently achieved, remarkable progress of the integral
equation theories for solute hydration, we can now handle a large biomolecule like
a protein or protein complex immersed in water by means of statistical mechanics
in which the biomolecule structure is taken into account at the atomic level and
a molecular model is employed for water. The angle-dependent integral equation
(ADIE) theory is best suited to the assessment of the hydrophobic hydration, the
most important physical factor in hydration thermodynamics. In this chapter, we
briefly summarize the basic characteristics of the ADIE theory.
Keywords Integral equation theory · Ornstein-Zernike relation · Closure
equation · Angle-dependent integral equation theory · Hydration
thermodynamics · Hydrophobic hydration
The Ornstein-Zernike (OZ) relation and a closure equation are the two basic equations
in an integral equation theory (IET) [1]. They are derived from the system partition
function using a variety of correlation and distribution functions defined on the basis
of classical statistical mechanics. Not only the solvent structure near a solute but also
thermodynamic quantities of solvation (e.g., the solvation free energy, energy, and
entropy) can be calculated via the following two steps:
Step 1. The bulk solvent is treated. In the case of a single component, the temperature, number density, and solvent-solvent interaction potential form the input data.
By numerically solving the two basic equations mentioned above, we can calculate the solvent-solvent correlation functions, microscopic density and orientational
structures of the solvent, and thermodynamic quantities. Unlike in the molecular
dynamics (MD) and Monte Carlo (MC) simulations, the average value of a physical
quantity can be calculated, in essence, for an infinitely large system and an infinitely
large number of system configurations. The IET is free from the problems of too
small a system size and statistical errors, which often arise in the MD and MC simulations. Not only a simple-fluid solvent but also a molecular liquid such as water
can be treated by the IET. For water, the angle-dependent integral equation (ADIE)
theory [2–6] is the most reliable tool.
© 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_5
71