78
6 Appendix 2: Morphometric Approach
Here, X p1 is μ H,1 /(k B T ), ε VH,1 /(k B T ), or S VH,1 /k B (μ H,1 , ε VH,1 , and S VH,1 are the
hydration free energy, energy, and entropy, respectively, k B is the Boltzmann constant,
and T is the absolute temperature). Equation (6.1) is referred to as the “morphometric
form”. The four coefficients, C 1 (X p1 )–C 4 (X p1 ), are dependent only on the thermodynamic state of bulk water. Hence, they are determined for isolated, spherical cavities
possessing much simpler geometric properties as described below.
Beforehand, we calculate values of X p1 for isolated neutral hard spheres with
sufficiently many different diameters using the ADIE theory [4–8]. By applying
the morphometric form for the isolated neutral hard spheres to the sufficiently many
combinations of the values of X p1 calculated and the diameter, we determine C 1 (X p1 )–
C 4 (X p1 ) by means of the least-squares method. Once C 1 (X p1 )–C 4 (X p1 ) are determined, all one has to do is to calculate V ex , A, Y, and Z of the cavity from the
Cartesian coordinates of the center of each neutral hard sphere in the cavity and its
diameter σ (σ is a Lennard-Jones potential parameter assigned to each neutral hard
sphere which corresponds to each atom in the polyatomic solute). X p1 of the cavity is
then obtained from Eq. (6.1). The MA is advantageous in the following two respects:
(i) X p1 can be calculated with very high speed (in ~1 s even for a large protein with
a prescribed structure on a standard workstation); and (ii) X p1 can be decomposed
into a variety of physically insightful components, and by assessing their signs and
relative magnitudes we can disclose the dependences of hydration properties on the
geometric characteristics of a polyatomic solute, temperature, and pressure. Consult
our earlier publications [2, 9–13] for more details.
We have recently developed a new method [9, 13] for calculating the hydration
free energy, energy, and entropy of a solute molecule. The solute hydration can be
decomposed into processes 1 and 2 explained in Chap. 5. In the new method, the
ADIE theory combined with the MA is employed for process 1, and the 3D-RISM
theory is applied to process 2. The new method enables us to finish the calculation
for a large polyatomic solute like a protein with sufficient accuracy and high speed.
(Most of the computation time is consumed for process 2 where the 3D-RISM theory
is employed.) Solutes with a wide range of sizes can be handled in the same manner.
Neither a stage of training nor parameterization is necessitated. A solute possessing
a significantly large total charge can be handled without difficulty.
We note that the free-energy perturbation [14–16] and thermodynamic integration
[17] methods using the molecular dynamics (MD) simulations suffer an unacceptably
heavy computational burden [18, 19]. This problem was solved by the development of
a novel MD simulation method based on solution theory in energy representation (the
so-called ER method) [20–22]. However, the ER method is inapplicable to a solute
possessing a significantly large total charge. We also note that the decomposition of
the hydration free energy, energy, and entropy into a variety of physically insightful
components is impractical in the MD simulations. It is worthwhile to add that values
of the hydration free energy calculated by our new method for many different proteins
with zero total charge were compared to those obtained using the ER method, and
the agreement was just excellent.
We conclude this book by noting the following. In our new method, a thermodynamic quantity of hydration is calculated for a fixed structure of the solute. However,
6 Appendix 2: Morphometric Approach
Here, X p1 is μ H,1 /(k B T ), ε VH,1 /(k B T ), or S VH,1 /k B (μ H,1 , ε VH,1 , and S VH,1 are the
hydration free energy, energy, and entropy, respectively, k B is the Boltzmann constant,
and T is the absolute temperature). Equation (6.1) is referred to as the “morphometric
form”. The four coefficients, C 1 (X p1 )–C 4 (X p1 ), are dependent only on the thermodynamic state of bulk water. Hence, they are determined for isolated, spherical cavities
possessing much simpler geometric properties as described below.
Beforehand, we calculate values of X p1 for isolated neutral hard spheres with
sufficiently many different diameters using the ADIE theory [4–8]. By applying
the morphometric form for the isolated neutral hard spheres to the sufficiently many
combinations of the values of X p1 calculated and the diameter, we determine C 1 (X p1 )–
C 4 (X p1 ) by means of the least-squares method. Once C 1 (X p1 )–C 4 (X p1 ) are determined, all one has to do is to calculate V ex , A, Y, and Z of the cavity from the
Cartesian coordinates of the center of each neutral hard sphere in the cavity and its
diameter σ (σ is a Lennard-Jones potential parameter assigned to each neutral hard
sphere which corresponds to each atom in the polyatomic solute). X p1 of the cavity is
then obtained from Eq. (6.1). The MA is advantageous in the following two respects:
(i) X p1 can be calculated with very high speed (in ~1 s even for a large protein with
a prescribed structure on a standard workstation); and (ii) X p1 can be decomposed
into a variety of physically insightful components, and by assessing their signs and
relative magnitudes we can disclose the dependences of hydration properties on the
geometric characteristics of a polyatomic solute, temperature, and pressure. Consult
our earlier publications [2, 9–13] for more details.
We have recently developed a new method [9, 13] for calculating the hydration
free energy, energy, and entropy of a solute molecule. The solute hydration can be
decomposed into processes 1 and 2 explained in Chap. 5. In the new method, the
ADIE theory combined with the MA is employed for process 1, and the 3D-RISM
theory is applied to process 2. The new method enables us to finish the calculation
for a large polyatomic solute like a protein with sufficient accuracy and high speed.
(Most of the computation time is consumed for process 2 where the 3D-RISM theory
is employed.) Solutes with a wide range of sizes can be handled in the same manner.
Neither a stage of training nor parameterization is necessitated. A solute possessing
a significantly large total charge can be handled without difficulty.
We note that the free-energy perturbation [14–16] and thermodynamic integration
[17] methods using the molecular dynamics (MD) simulations suffer an unacceptably
heavy computational burden [18, 19]. This problem was solved by the development of
a novel MD simulation method based on solution theory in energy representation (the
so-called ER method) [20–22]. However, the ER method is inapplicable to a solute
possessing a significantly large total charge. We also note that the decomposition of
the hydration free energy, energy, and entropy into a variety of physically insightful
components is impractical in the MD simulations. It is worthwhile to add that values
of the hydration free energy calculated by our new method for many different proteins
with zero total charge were compared to those obtained using the ER method, and
the agreement was just excellent.
We conclude this book by noting the following. In our new method, a thermodynamic quantity of hydration is calculated for a fixed structure of the solute. However,