2.4 Mechanism of Force Generation by Water for Moving or Rotating a Protein
11
equivalently, solute 2 feels a potential or force field due to the presence of solute 1
near it) [11]. Since S Water in Eq. (2.2) is quite large and strongly dependent on (x, y,
z), the entropic force expressed by Eqs. (2.2) and (2.3) plays substantial roles.
When the structures of solutes 1 and 2 are made variable as in the real system,
the PMF should shift downward (i.e., in the negative direction) especially in regions
where the PMF is positive and large. This is because the solutes always exhibit
structural changes so that F can become as low as possible.
We then consider solutes 1 and 2 in system (III) (see Fig. 2.3). We assume that the
structures of the two solutes are fixed for simplicity. F is a function of the rotation
angle θ defined for solute 2. The mean torque acting on solute 2, τ (θ ), is expresses
as
τ = −∂ F
∂θ.
(2.4)
τ (θ 0 ) represents the torque acting on solute 2 averaged over all the possible configurations of water molecules in the entire system with θ being fixed at θ 0 . In this case,
F becomes lowest at θ = θ min and τ (θ min ) = 0. Since S Water in Eq. (2.2) is quite large
and strongly dependent on θ, the entropic torque expressed by Eqs. (2.2) and (2.4)
plays substantial roles.
2.5 Entropic Excluded-Volume Effect and Entropic Force
and Potential Generated by Water
2.5.1 Entropy-Driven Formation of Ordered Structure
The concept referred to as the “entropic excluded-volume effect” or the “solvententropy effect” (“water-entropy effect” when the solvent is water) is crucially important in colloidal and biological systems [11, 12]. As illustrated in Fig. 2.4a, the
insertion of a solute into water causes the generation of a space which is inaccessible
to the centers of water molecules (the space occupied by the solute itself plus the
space shown in gray). If water molecules are spheres with diameter d S and the solute
is a sphere with diameter d L , the excluded space is a sphere with diameter “d S +
d L ”. The volume of the excluded space is the “excluded volume (EV)”. Upon the
contact of two solutes (see Fig. 2.4b), the two excluded spaces overlap, the total
EV reduces by the volume of the overlapping space marked in black, and the total
volume available for the translational displacement of water molecules increases
by the same volume. The contact leads to an increase in the number of accessible
translational configurations of water molecules (i.e., the number of possible coordinates of the centers of water molecules). This increase is followed by a gain of
the configurational entropy of water. An interaction driving the solutes to contact
each other is thus induced [11, 12]. Suppose that the solutes and water molecules are
11
equivalently, solute 2 feels a potential or force field due to the presence of solute 1
near it) [11]. Since S Water in Eq. (2.2) is quite large and strongly dependent on (x, y,
z), the entropic force expressed by Eqs. (2.2) and (2.3) plays substantial roles.
When the structures of solutes 1 and 2 are made variable as in the real system,
the PMF should shift downward (i.e., in the negative direction) especially in regions
where the PMF is positive and large. This is because the solutes always exhibit
structural changes so that F can become as low as possible.
We then consider solutes 1 and 2 in system (III) (see Fig. 2.3). We assume that the
structures of the two solutes are fixed for simplicity. F is a function of the rotation
angle θ defined for solute 2. The mean torque acting on solute 2, τ (θ ), is expresses
as
τ = −∂ F
∂θ.
(2.4)
τ (θ 0 ) represents the torque acting on solute 2 averaged over all the possible configurations of water molecules in the entire system with θ being fixed at θ 0 . In this case,
F becomes lowest at θ = θ min and τ (θ min ) = 0. Since S Water in Eq. (2.2) is quite large
and strongly dependent on θ, the entropic torque expressed by Eqs. (2.2) and (2.4)
plays substantial roles.
2.5 Entropic Excluded-Volume Effect and Entropic Force
and Potential Generated by Water
2.5.1 Entropy-Driven Formation of Ordered Structure
The concept referred to as the “entropic excluded-volume effect” or the “solvententropy effect” (“water-entropy effect” when the solvent is water) is crucially important in colloidal and biological systems [11, 12]. As illustrated in Fig. 2.4a, the
insertion of a solute into water causes the generation of a space which is inaccessible
to the centers of water molecules (the space occupied by the solute itself plus the
space shown in gray). If water molecules are spheres with diameter d S and the solute
is a sphere with diameter d L , the excluded space is a sphere with diameter “d S +
d L ”. The volume of the excluded space is the “excluded volume (EV)”. Upon the
contact of two solutes (see Fig. 2.4b), the two excluded spaces overlap, the total
EV reduces by the volume of the overlapping space marked in black, and the total
volume available for the translational displacement of water molecules increases
by the same volume. The contact leads to an increase in the number of accessible
translational configurations of water molecules (i.e., the number of possible coordinates of the centers of water molecules). This increase is followed by a gain of
the configurational entropy of water. An interaction driving the solutes to contact
each other is thus induced [11, 12]. Suppose that the solutes and water molecules are