116
7 Fluctuations of the Proton Electromotive Force Across Inner Mitochondrial. . .
1 pSiemens at pH approximately 7.4 (Lauger [4], Lill et al. [5]), what corresponds
to a resistance R m = 1 × 10 12 .
7.7 Relaxation Times of the Electrical and Buffer Reservoirs
Coupling between the buffer and the membrane systems is expected to be of importance when the characteristic relaxation times (as defined by their corresponding
RC) of both systems, are of the same order.
We use this condition to calculate the access resistance R a of the electrical link
between the I MM and the buffer compartment, namely, τ = τ m = τ b , assuming
that proton movement between the membrane and buffer compartments is fast
relative to the proton translocation across the I MM (see Discussion). From this
condition we obtain:
R a =
τ
C b
(7.17)
Figures 7.2 and 7.3 summarize our results. The coupling condition of the
relaxation times of the membrane and buffer systems (Eq. (7.17)) makes the
“spectral density” function (Eq. (7.11)) to decrease assimptotically with increasing
radial frequency (see Fig. 6.2). This creates a convergence condition for the integral
of Eq. (7.13), allowing for the determination of the mean square P MF fluctuation
for a range of fluctuational domain sizes (Fig. 7.3).
Figure 7.3 spans a relatively wide range of possible sizes of fluctuational domains
and displays the corresponding mean P MF fluctuation and associated relaxation
times. The factors limiting the size of a fluctuational domain were defined by its
physical barriers (delimiting membranes, etc.) and the lack of correlation with
neighbouring domains (distance factor). In the present case the physical limits
are defined by the 2 mitochondrial membranes, limiting the extension of the
intermembrane space. We chose this as a maximum size for a fluctuational domain,
defining thus the value of 20–30 nm as an upper size limit for a fluctuational domain,
with 10 nm being a definite size possibility in many types of mitochondria. Taking
15 nm as “typical” width for the I MS, Fig. 7.3 indicates a mean P MF fluctuation of
100 mV at pH 7.4 with a corresponding relaxation time of 1 μs. On another extreme,
for a fluctuational domain size of 70 nm we obtain a mean P MF fluctuation of about
40 mV with a corresponding relaxation time of 18 μs.
Both the amplitude of the mean P MF fluctuation and its characteristic relaxation
time are relevant in any attempt at estimating the effects of a fluctuating P MF on
the molecular machinery resident in the I MM.
In order to be of relevance such fluctuational force is required to have a minimum
amplitude and duration which are to be compared with the dynamic characteristics
of the specific system upon which it acts, in this case, protons interacting with
ATP synthase molecules. We shall consider two main effects of the fluctuating
7 Fluctuations of the Proton Electromotive Force Across Inner Mitochondrial. . .
1 pSiemens at pH approximately 7.4 (Lauger [4], Lill et al. [5]), what corresponds
to a resistance R m = 1 × 10 12 .
7.7 Relaxation Times of the Electrical and Buffer Reservoirs
Coupling between the buffer and the membrane systems is expected to be of importance when the characteristic relaxation times (as defined by their corresponding
RC) of both systems, are of the same order.
We use this condition to calculate the access resistance R a of the electrical link
between the I MM and the buffer compartment, namely, τ = τ m = τ b , assuming
that proton movement between the membrane and buffer compartments is fast
relative to the proton translocation across the I MM (see Discussion). From this
condition we obtain:
R a =
τ
C b
(7.17)
Figures 7.2 and 7.3 summarize our results. The coupling condition of the
relaxation times of the membrane and buffer systems (Eq. (7.17)) makes the
“spectral density” function (Eq. (7.11)) to decrease assimptotically with increasing
radial frequency (see Fig. 6.2). This creates a convergence condition for the integral
of Eq. (7.13), allowing for the determination of the mean square P MF fluctuation
for a range of fluctuational domain sizes (Fig. 7.3).
Figure 7.3 spans a relatively wide range of possible sizes of fluctuational domains
and displays the corresponding mean P MF fluctuation and associated relaxation
times. The factors limiting the size of a fluctuational domain were defined by its
physical barriers (delimiting membranes, etc.) and the lack of correlation with
neighbouring domains (distance factor). In the present case the physical limits
are defined by the 2 mitochondrial membranes, limiting the extension of the
intermembrane space. We chose this as a maximum size for a fluctuational domain,
defining thus the value of 20–30 nm as an upper size limit for a fluctuational domain,
with 10 nm being a definite size possibility in many types of mitochondria. Taking
15 nm as “typical” width for the I MS, Fig. 7.3 indicates a mean P MF fluctuation of
100 mV at pH 7.4 with a corresponding relaxation time of 1 μs. On another extreme,
for a fluctuational domain size of 70 nm we obtain a mean P MF fluctuation of about
40 mV with a corresponding relaxation time of 18 μs.
Both the amplitude of the mean P MF fluctuation and its characteristic relaxation
time are relevant in any attempt at estimating the effects of a fluctuating P MF on
the molecular machinery resident in the I MM.
In order to be of relevance such fluctuational force is required to have a minimum
amplitude and duration which are to be compared with the dynamic characteristics
of the specific system upon which it acts, in this case, protons interacting with
ATP synthase molecules. We shall consider two main effects of the fluctuating