10
2 A New View on Mechanism of Functional Expression …
The ATP hydrolysis cycle is repeated as an irreversible process during which
myosin performs unidirectional movement along F-actin, diverse substrates are transported across the membrane by ABC transporter, and the γ subunit performs unidirectional rotation in F 1 -ATPase. In cases of actomyosin and F 1 -ATPase, the cycle is
accompanied by a decrease in system free energy of G ~ −20k B T (T = 298 K).
F is composed of the conformational (intramolecular) energy, conformational
entropy, and hydration free energy of solutes 1 and 2. The hydration free energy
is the sum of the hydration energy and entropy. The absolute value of hydration
entropy of a solute is the magnitude of water-entropy loss caused by the insertion
of the solute. In general, a change in conformational energy and that in hydration
energy are compensating. That is, the latter is positive when the former is negative
and the latter is negative when the former is positive. The sum of the two energies
remains roughly constant, i.e., the change in this sum is rather small. (In a strict
sense, it is often that the change in the hydration energy is larger.) On the other hand,
a conformational-entropy change is significantly smaller than the hydration-entropy
change. (See Sects. 2.6–2.8 for more details.) Therefore, the hydration entropy can
be treated as a principal component of F. It follows that F and the water entropy
S Water is approximately related through
F ∼ −T S Water .
(2.2)
In system (I) shown in Fig. 2.3, for example, we can state that the structures of
the two solutes and the position of solute 2 are determined so that the water entropy
can be maximized.
2.4 Mechanism of Force Generation by Water for Moving
or Rotating a Protein
Let us consider solutes 1 and 2 in system (I) (see Fig. 2.3). We assume that the
structures of the two solutes are fixed for simplicity. F is a function of the Cartesian
coordinates of the center of gravity of solute 2, (x, y, z). The origin of the coordinate
system (0, 0, 0) is taken to be, for example, the left edge of solute 1. “F(x, y, z) − F(+
∞, + ∞, + ∞)” represents the spatial distribution of the potential of mean force
(PMF: the water-mediated interaction) between solutes 1 and 2. The mean force
acting on solute 2, f , is expressed as
f = f x i + f y j + f z k, f x = − ∂ F
∂ x, f y = −∂ F
∂ y, f z = −∂ F
∂z
(2.3)
where i, j, and k denote the direction unit vectors. f (x 0 , y 0 , z 0 ) represents the force
induced between solutes 1 and 2 averaged over all the possible configurations of
water molecules in the entire system with (x, y, z) being fixed at (x 0 , y 0 , z 0 ). We
can take the view that a potential or force field acts on solute 2 near solute 1 (or
2 A New View on Mechanism of Functional Expression …
The ATP hydrolysis cycle is repeated as an irreversible process during which
myosin performs unidirectional movement along F-actin, diverse substrates are transported across the membrane by ABC transporter, and the γ subunit performs unidirectional rotation in F 1 -ATPase. In cases of actomyosin and F 1 -ATPase, the cycle is
accompanied by a decrease in system free energy of G ~ −20k B T (T = 298 K).
F is composed of the conformational (intramolecular) energy, conformational
entropy, and hydration free energy of solutes 1 and 2. The hydration free energy
is the sum of the hydration energy and entropy. The absolute value of hydration
entropy of a solute is the magnitude of water-entropy loss caused by the insertion
of the solute. In general, a change in conformational energy and that in hydration
energy are compensating. That is, the latter is positive when the former is negative
and the latter is negative when the former is positive. The sum of the two energies
remains roughly constant, i.e., the change in this sum is rather small. (In a strict
sense, it is often that the change in the hydration energy is larger.) On the other hand,
a conformational-entropy change is significantly smaller than the hydration-entropy
change. (See Sects. 2.6–2.8 for more details.) Therefore, the hydration entropy can
be treated as a principal component of F. It follows that F and the water entropy
S Water is approximately related through
F ∼ −T S Water .
(2.2)
In system (I) shown in Fig. 2.3, for example, we can state that the structures of
the two solutes and the position of solute 2 are determined so that the water entropy
can be maximized.
2.4 Mechanism of Force Generation by Water for Moving
or Rotating a Protein
Let us consider solutes 1 and 2 in system (I) (see Fig. 2.3). We assume that the
structures of the two solutes are fixed for simplicity. F is a function of the Cartesian
coordinates of the center of gravity of solute 2, (x, y, z). The origin of the coordinate
system (0, 0, 0) is taken to be, for example, the left edge of solute 1. “F(x, y, z) − F(+
∞, + ∞, + ∞)” represents the spatial distribution of the potential of mean force
(PMF: the water-mediated interaction) between solutes 1 and 2. The mean force
acting on solute 2, f , is expressed as
f = f x i + f y j + f z k, f x = − ∂ F
∂ x, f y = −∂ F
∂ y, f z = −∂ F
∂z
(2.3)
where i, j, and k denote the direction unit vectors. f (x 0 , y 0 , z 0 ) represents the force
induced between solutes 1 and 2 averaged over all the possible configurations of
water molecules in the entire system with (x, y, z) being fixed at (x 0 , y 0 , z 0 ). We
can take the view that a potential or force field acts on solute 2 near solute 1 (or