6
2 A New View on Mechanism of Functional Expression …
constant and T is the absolute temperature (k B T = 0.592 kcal/mol at T = 298 K).
G° is “the formation free energy of 1 mol of ADP plus that of 1 mol of Pi” minus
“the formation free energy of 1 mol of ATP plus that of 1 mol of H 2 O” in the standard
state where the concentrations of ATP, ADP, and Pi equal 1 mol/L, T = 298 K, and
the pressure P is 1 atm. The free-energy change upon the chemical reaction G is
approximately related to G° through
G = G
◦
+ RT ln(Z ), Z = [ADP][Pi]
[ATP]
(2.1)
where R is the gas constant and [X] is the dimensionless concentration of X (i.e.,
the concentration of X in mol/L divided by 1 mol/L) in the aqueous solution. The
chemical potential of water in the aqueous solution is assumed to be equal to that of
pure water. A value of Z is valid only for given values of pH and [Mg
2+ ] [2, 3].
When [ATP] is sufficiently high and [ADP] and [Pi] are sufficiently low such that
ln(Z) takes a negative or small, positive value and G < 0, the reaction frequency
in the right direction (ATP hydrolysis: ATP + H 2 O → ADP + Pi) is much higher
than that in the left direction (ATP synthesis: ADP + Pi → ATP + H 2 O). That is, the
overall reaction under this solution condition is the ATP hydrolysis. G is dependent
on the concentrations of ATP, ADP, and Pi but roughly equal to −20k B T (T = 298 K)
in aqueous solution under the physiological condition [4]. When [ATP] is sufficiently
low and [ADP] and [Pi] are sufficiently high such that ln(Z) is positive and |ΔG°|
< RT ln(Z), G > 0. Under this solution condition, the overall reaction is the ATP
synthesis. In the chemical equilibrium state, G = 0 and the reaction frequencies in
the right and left directions are the same.
2.1.2 ATP Hydrolysis Cycle Where a Molecular Motor Acts
as a Catalyst for Hydrolysis Reaction
Without a catalyst, the reaction rate of the chemical reaction, ATP + H 2 O → ADP +
Pi, in bulk aqueous solution is extremely low. Importantly, an ATP-driven molecular
motor acts as the catalyst. In other words, the molecular motor is coupled with the
reaction. The reaction proceeds as the ATP hydrolysis cycle comprising the following
three events: (1) ATP binding to the molecular motor, (2) hydrolysis of ATP into ADP
and Pi, and (3) dissociation of ADP and Pi from the molecular motor.
As explained in Fig. 2.1, the binding free energy of A (A is ATP or ADP) and
the molecular motor M, G B °(A), is “the free energy of 1 mol of MA” minus “the
free energy of 1 mol of M plus that of 1 mol of A” in the standard state. G B °(ATP)
and G B °(ADP) are both negative. Let G B (A) be the free-energy change upon the
binding of A to M. Under the solution condition that [ATP] and [ADP] are sufficiently
high and low, respectively, G B (ATP) < 0 and G B (ADP) > 0: ATP binds to M (M +
ATP → MATP) and ADP dissociates from M (MADP → M + ADP). The discussion
to be made for Pi is similar to that for ADP.
2 A New View on Mechanism of Functional Expression …
constant and T is the absolute temperature (k B T = 0.592 kcal/mol at T = 298 K).
G° is “the formation free energy of 1 mol of ADP plus that of 1 mol of Pi” minus
“the formation free energy of 1 mol of ATP plus that of 1 mol of H 2 O” in the standard
state where the concentrations of ATP, ADP, and Pi equal 1 mol/L, T = 298 K, and
the pressure P is 1 atm. The free-energy change upon the chemical reaction G is
approximately related to G° through
G = G
◦
+ RT ln(Z ), Z = [ADP][Pi]
[ATP]
(2.1)
where R is the gas constant and [X] is the dimensionless concentration of X (i.e.,
the concentration of X in mol/L divided by 1 mol/L) in the aqueous solution. The
chemical potential of water in the aqueous solution is assumed to be equal to that of
pure water. A value of Z is valid only for given values of pH and [Mg
2+ ] [2, 3].
When [ATP] is sufficiently high and [ADP] and [Pi] are sufficiently low such that
ln(Z) takes a negative or small, positive value and G < 0, the reaction frequency
in the right direction (ATP hydrolysis: ATP + H 2 O → ADP + Pi) is much higher
than that in the left direction (ATP synthesis: ADP + Pi → ATP + H 2 O). That is, the
overall reaction under this solution condition is the ATP hydrolysis. G is dependent
on the concentrations of ATP, ADP, and Pi but roughly equal to −20k B T (T = 298 K)
in aqueous solution under the physiological condition [4]. When [ATP] is sufficiently
low and [ADP] and [Pi] are sufficiently high such that ln(Z) is positive and |ΔG°|
< RT ln(Z), G > 0. Under this solution condition, the overall reaction is the ATP
synthesis. In the chemical equilibrium state, G = 0 and the reaction frequencies in
the right and left directions are the same.
2.1.2 ATP Hydrolysis Cycle Where a Molecular Motor Acts
as a Catalyst for Hydrolysis Reaction
Without a catalyst, the reaction rate of the chemical reaction, ATP + H 2 O → ADP +
Pi, in bulk aqueous solution is extremely low. Importantly, an ATP-driven molecular
motor acts as the catalyst. In other words, the molecular motor is coupled with the
reaction. The reaction proceeds as the ATP hydrolysis cycle comprising the following
three events: (1) ATP binding to the molecular motor, (2) hydrolysis of ATP into ADP
and Pi, and (3) dissociation of ADP and Pi from the molecular motor.
As explained in Fig. 2.1, the binding free energy of A (A is ATP or ADP) and
the molecular motor M, G B °(A), is “the free energy of 1 mol of MA” minus “the
free energy of 1 mol of M plus that of 1 mol of A” in the standard state. G B °(ATP)
and G B °(ADP) are both negative. Let G B (A) be the free-energy change upon the
binding of A to M. Under the solution condition that [ATP] and [ADP] are sufficiently
high and low, respectively, G B (ATP) < 0 and G B (ADP) > 0: ATP binds to M (M +
ATP → MATP) and ADP dissociates from M (MADP → M + ADP). The discussion
to be made for Pi is similar to that for ADP.