60
3 Mechanism of Unidirectional Rotation of γ Subunit in F 1 -ATPase
for the mutant and the wild type are the same, W Wild = −ΔG (ΔG + W Wild =
0) means ΔG + W Mutant < 0. Hence, even when the rotation essentially vanishes
for the mutant, the overall reaction occurring is still the ATP hydrolysis. When
the external torque is further increased but ΔG + W Mutant < 0 still holds (i.e.,
the torque is not very strong), the γ subunit rotates in the inverse direction but
the overall reaction occurring is still the ATP hydrolysis.
(iv) When the external torque is applied, the rotation becomes quite irregular and
the rotation rate frequently exhibits an abrupt increase or decrease [35].
It is not straightforward to interpret the aforementioned results because no information on the mutant structure is experimentally available. However, we can give a
speculative discussion in what follows.
The wild type fortuitously meets all the following requirements for the functional
expression: (1) The packing efficiency of a β subunit, which is strongly dependent
on the chemical compound bound to the β subunit, influences the packing efficiency
of the subcomplex including the β subunit; (2) the nonuniformity of the packing
structure of the α 3 β 3 complex is sufficiently high; (3) the hydration entropy of the
α 3 β 3 γ complex is quite variable depending on the orientation of the γ subunit; and
(4) a very large increase in the water-entropy loss for one of the three subcomplexes
is almost cancelled out by large decreases in the water-entropy loss for the other two
subcomplexes, causing no free-energy barrier for the rotation of the γ subunit.
None of the four requirements is completely met by the mutant. The absolute
value of hydration entropy of state (c) or (d) in Fig. 3.23 is not much larger than
that of state (a) or (b) shown in Fig. 3.22. In other words, for the mutant, the loss
of water entropy caused in state change (a)→(c) or (a)→(d) is much smaller, and
factor 2 is as significant as or more significant than factor 1 (see Sect. 3.8.1 for the
definition of factors 1 and 2). Therefore, the ATP hydrolysis can persist even when
the inverse rotation of the γ subunit is forcibly executed. Therefore, state change
(a)→(c), where the ATP binding, ATP hydrolysis, and dissociation of ADP and Pi
lead a decrease in system free energy, is often favored. For the wild type, during
the rotation, a very large increase in the water-entropy loss for one of the three
subcomplexes is almost cancelled out by large decreases in the water-entropy loss
for the other two subcomplexes, giving rise to no free-energy barrier for the rotation
of the γ subunit. This may not be the case for the mutant, resulting in a nontrivial freeenergy barrier. For the mutant, overcoming this free-energy barrier by the thermal
fluctuation is a stochastic process, causing the quite irregular rotational behavior
where an abrupt decrease in the rotation rate is often encountered. It is not rare that
the state change does not reach a stable state when the external torque is applied, and
the inverse rotation can be accompanied by no reactions with the occurrence of state
change (a)→(d) (see Figs. 3.22 and 3.23). Taken together, state changes (a)→(c)
and (a)→(d) dominate for the mutant. The free-energy balance as that considered in
Sect. 3.8.3 is no more valid. We can state for the mutant that the entropic force by
water driving the normal rotation is considerably weaker and readily yields to the
external force driving the inverse rotation.
3 Mechanism of Unidirectional Rotation of γ Subunit in F 1 -ATPase
for the mutant and the wild type are the same, W Wild = −ΔG (ΔG + W Wild =
0) means ΔG + W Mutant < 0. Hence, even when the rotation essentially vanishes
for the mutant, the overall reaction occurring is still the ATP hydrolysis. When
the external torque is further increased but ΔG + W Mutant < 0 still holds (i.e.,
the torque is not very strong), the γ subunit rotates in the inverse direction but
the overall reaction occurring is still the ATP hydrolysis.
(iv) When the external torque is applied, the rotation becomes quite irregular and
the rotation rate frequently exhibits an abrupt increase or decrease [35].
It is not straightforward to interpret the aforementioned results because no information on the mutant structure is experimentally available. However, we can give a
speculative discussion in what follows.
The wild type fortuitously meets all the following requirements for the functional
expression: (1) The packing efficiency of a β subunit, which is strongly dependent
on the chemical compound bound to the β subunit, influences the packing efficiency
of the subcomplex including the β subunit; (2) the nonuniformity of the packing
structure of the α 3 β 3 complex is sufficiently high; (3) the hydration entropy of the
α 3 β 3 γ complex is quite variable depending on the orientation of the γ subunit; and
(4) a very large increase in the water-entropy loss for one of the three subcomplexes
is almost cancelled out by large decreases in the water-entropy loss for the other two
subcomplexes, causing no free-energy barrier for the rotation of the γ subunit.
None of the four requirements is completely met by the mutant. The absolute
value of hydration entropy of state (c) or (d) in Fig. 3.23 is not much larger than
that of state (a) or (b) shown in Fig. 3.22. In other words, for the mutant, the loss
of water entropy caused in state change (a)→(c) or (a)→(d) is much smaller, and
factor 2 is as significant as or more significant than factor 1 (see Sect. 3.8.1 for the
definition of factors 1 and 2). Therefore, the ATP hydrolysis can persist even when
the inverse rotation of the γ subunit is forcibly executed. Therefore, state change
(a)→(c), where the ATP binding, ATP hydrolysis, and dissociation of ADP and Pi
lead a decrease in system free energy, is often favored. For the wild type, during
the rotation, a very large increase in the water-entropy loss for one of the three
subcomplexes is almost cancelled out by large decreases in the water-entropy loss
for the other two subcomplexes, giving rise to no free-energy barrier for the rotation
of the γ subunit. This may not be the case for the mutant, resulting in a nontrivial freeenergy barrier. For the mutant, overcoming this free-energy barrier by the thermal
fluctuation is a stochastic process, causing the quite irregular rotational behavior
where an abrupt decrease in the rotation rate is often encountered. It is not rare that
the state change does not reach a stable state when the external torque is applied, and
the inverse rotation can be accompanied by no reactions with the occurrence of state
change (a)→(d) (see Figs. 3.22 and 3.23). Taken together, state changes (a)→(c)
and (a)→(d) dominate for the mutant. The free-energy balance as that considered in
Sect. 3.8.3 is no more valid. We can state for the mutant that the entropic force by
water driving the normal rotation is considerably weaker and readily yields to the
external force driving the inverse rotation.