58
3 Mechanism of Unidirectional Rotation of γ Subunit in F 1 -ATPase
vanishes. The ATP hydrolysis and synthesis reactions occur with the same frequency.
On an average, the net change in system free energy per ATP cycle is zero.
Case III. ΔG + W >0. The external force is stronger than the entropic force by
water. The γ subunit rotates in the inverse direction. More strictly, the frequency of
the inverse rotation is higher than that of the normal rotation, and the inverse rotation
dominates when ΔG + W is not close to zero. The overall reaction occurring is the
ATP synthesis, ADP + Pi→ATP + H 2 O. On an average, the system free energy
increases by ΔG + W per ATP cycle. This is not thermodynamically inconsistent,
because a large energy of W is given to the system.
3.8.3 Comparison with Experimental Results Observed When
External Torque is Applied
It was experimentally observed in case I that as the external torque becomes stronger,
the apparent rotation rate decreases [34]. This result suggests that the normal rotation
occurs almost exclusively for sufficiently weak torque but the frequencies of no rotation and the inverse rotation increase as the torque becomes stronger. Let us assume
that ΔG + W = −16k B T (T = 298 K). Suppose that the aqueous solution is under
the physiological condition and ΔG = −20k B T (T = 298 K). Since states (a) and
(b) shown in Fig. 3.22 share the same value of the hydration entropy, the free-energy
balance can be considered as follows. The energy given to the system per ATP cycle
is W = 4k B T. Here, we introduce two possible occasions. In the first occasion, the
normal and inverse rotations occur 9x/10 times and x/10 times, respectively (x is
a large, positive value). The normal and inverse rotations are accompanied by the
ATP hydrolysis and synthesis, respectively. The change in system free energy upon
the hydrolysis and synthesis are −20k B T and 20k B T, respectively. The free-energy
balance is expressed as {(9x/10)(−20k B T ) + (x/10)(20k B T )}/x = −16k B T, which
coincides with ΔG + W = −16k B T. That is, the system free energy decreases by
−16k B T per ATP cycle on an average. In the second occasion, the normal rotation occurs 4x/5 times and the rotation is stopped x/5 times. When the rotation
is stopped, neither the ATP hydrolysis nor synthesis occurs with the system free
energy being unchanged. The free-energy balance is expressed as {(4x/5)(−20k B T )
+ (x/5)(0k B T )}/x = −16k B T. In the real system, all the normal, inverse, and stopped
rotations should occur. When the torque is very weak, the normal rotation occurs
almost exclusively. However, with an increase in torque strength, the proportion of
the stopped rotation becomes higher. With a further increase in torque strength toward
ΔG + W =0, the proportion of the inverse rotation also becomes higher.
In case II where the rotation essentially vanishes, the normal and inverse rotations occur with the same frequency, and there should be significantly many stopped
rotations as well. In fact, it was experimentally observed that the frequency of the
normal and inverse rotations is quite low in the case of ΔG + W =0 [34]. When the
ATP, ADP, and Pi concentrations in the aqueous solution are changed (i.e., they are
3 Mechanism of Unidirectional Rotation of γ Subunit in F 1 -ATPase
vanishes. The ATP hydrolysis and synthesis reactions occur with the same frequency.
On an average, the net change in system free energy per ATP cycle is zero.
Case III. ΔG + W >0. The external force is stronger than the entropic force by
water. The γ subunit rotates in the inverse direction. More strictly, the frequency of
the inverse rotation is higher than that of the normal rotation, and the inverse rotation
dominates when ΔG + W is not close to zero. The overall reaction occurring is the
ATP synthesis, ADP + Pi→ATP + H 2 O. On an average, the system free energy
increases by ΔG + W per ATP cycle. This is not thermodynamically inconsistent,
because a large energy of W is given to the system.
3.8.3 Comparison with Experimental Results Observed When
External Torque is Applied
It was experimentally observed in case I that as the external torque becomes stronger,
the apparent rotation rate decreases [34]. This result suggests that the normal rotation
occurs almost exclusively for sufficiently weak torque but the frequencies of no rotation and the inverse rotation increase as the torque becomes stronger. Let us assume
that ΔG + W = −16k B T (T = 298 K). Suppose that the aqueous solution is under
the physiological condition and ΔG = −20k B T (T = 298 K). Since states (a) and
(b) shown in Fig. 3.22 share the same value of the hydration entropy, the free-energy
balance can be considered as follows. The energy given to the system per ATP cycle
is W = 4k B T. Here, we introduce two possible occasions. In the first occasion, the
normal and inverse rotations occur 9x/10 times and x/10 times, respectively (x is
a large, positive value). The normal and inverse rotations are accompanied by the
ATP hydrolysis and synthesis, respectively. The change in system free energy upon
the hydrolysis and synthesis are −20k B T and 20k B T, respectively. The free-energy
balance is expressed as {(9x/10)(−20k B T ) + (x/10)(20k B T )}/x = −16k B T, which
coincides with ΔG + W = −16k B T. That is, the system free energy decreases by
−16k B T per ATP cycle on an average. In the second occasion, the normal rotation occurs 4x/5 times and the rotation is stopped x/5 times. When the rotation
is stopped, neither the ATP hydrolysis nor synthesis occurs with the system free
energy being unchanged. The free-energy balance is expressed as {(4x/5)(−20k B T )
+ (x/5)(0k B T )}/x = −16k B T. In the real system, all the normal, inverse, and stopped
rotations should occur. When the torque is very weak, the normal rotation occurs
almost exclusively. However, with an increase in torque strength, the proportion of
the stopped rotation becomes higher. With a further increase in torque strength toward
ΔG + W =0, the proportion of the inverse rotation also becomes higher.
In case II where the rotation essentially vanishes, the normal and inverse rotations occur with the same frequency, and there should be significantly many stopped
rotations as well. In fact, it was experimentally observed that the frequency of the
normal and inverse rotations is quite low in the case of ΔG + W =0 [34]. When the
ATP, ADP, and Pi concentrations in the aqueous solution are changed (i.e., they are