114
7 Fluctuations of the Proton Electromotive Force Across Inner Mitochondrial. . .
q ω = α(ω)(P MF ) ω
(7.6)
The impedances of the membrane, Z m , and of the buffer system ,Z b , can be written
respectively as:
1
Z m
=
1
R m
+ iωC m ,
1
Z b
= R a +
1
iωC b
(7.7)
where i is the imaginary unit. The total impedance Z T is given by: Z T = Z m + Z b .
From the relation between the susceptibility and the impedance, α(ω) =
i
ωZ ω
, we
get for the real, α (ω), and imaginary, α (ω), parts of the susceptibility:
α
(ω) = −
ωR m τ m
1+(ωτ m ) 2 +
R a
ωτ b
ωD(ω)
(7.8)
α
(ω) = −
R m
1+(ωτ m ) 2 + R a
ωD(ω)
(7.9)
where D(ω) is given by:
D(ω) =
ωR m τ m
1 + (ωτ m ) 2 +
R a
ωτ b
2
+
R m
1 + (ωτ m ) 2 + R a
2
(7.10)
where τ m = R m C m and τ b = R a C b are the corresponding relaxation times of both
systems.
Then the corresponding spectral density of the mean square of the fluctuational
proton-motive force, [(P MF ) 2 ] ω will be given by [3]:
[(P MF )
2
] ω =
α (ω)
| α(ω) | 2
2kT
ω
(7.11)
where | α(ω) | 2 = [α (ω)] 2 + [α (ω)] 2 . The mean square of the fluctuating protonmotive force acting across an I MM patch is given by the integral:
< (P MF )
2 >=
1
π
∞
0
[(P MF )
2
] ω dω
(7.12)
7.4 Parameter Definitions
In order to estimate the I MM patch capacitance we consider an idealized mitochondrion with an inter-membrane space about 15–20 nm wide. This minimum