Although we are unable to determine the H+
conduction rate we can ask to what degree the time course of fluorescence decay is at least qualitatively consistent with our expectation based on theory. We simulated H+
flux into a population of vesicles with a mean number of μ channels per vesicle assuming that a vesicle will have n channels with subset m facing outside-in with a frequency
Upon addition of valinomycin the membrane potential is driven to very near the Nernst potential for K+
(about −60 mV inside) because the K+
conductance exceeds H+
conductance under all conditions in the simulation. We model Hv voltage-dependent gating with a two-state Boltzmann function with midpoint activation voltage Vmid
= 40 mV and valence 3 5; 15
graphs simulation results for populations of vesicles with channels distributed according to equation 2
. If one focuses on a single vesicle a value 1.0 on the y-axis corresponds to a free (unbound) internal H+
concentration of 10−7
M; a value 0 corresponds to a free internal H+
concentration of 10−6
M. During the simulation the free internal H+
concentration in the given vesicle changes from 10−7
(approximately) following a time course that depends on the number of channels and their orientation in the vesicle. The graph shows the weighted sum of time courses for all vesicles (including empty) in the population for a 200 s interval. The curves show a fast followed by a slower decay at smaller values of μ: the slower component is attributable mainly to a fraction of vesicles with only outside-out channels, which have a very low open probability. The curves also show a negative second derivative (curvature) at early time points due to H+
buffering inside the vesicles. These same qualitative features are observed in fluorescence decay data ().
Fig. 3 Comparison of dilution series data with theory. (A) Theoretical curves for the decrease of internal pH over time at the equivalent protein to lipid ratios as in , scaled with the theoretical fraction of empty vesicles. Curves are colored to match (more ...)
The theoretical and experimental curves are different in two obvious respects. As a function of μ the curves do not exhibit the same spacing between them. We think this most likely reflects the nonlinear (and unknown) relationship between free internal H+ concentration and fluorescence. We emphasize that this unknown relationship prevents us from determining H+ flux rates, but it does not prevent us from determining the fraction of vesicles with no channels versus vesicles with at least one channel (). The second difference between theory and experiment is a more prominent slow component of fluorescence change in the data, which is consistent with channel-independent H+ leak in the vesicles, which we have not included in the model.
In the simulation the curves correspond to an open channel conductance of 0.1 fS. This value should not be taken as an accurate determination of Hv channel conductance for the reasons discussed above. However, this value is smaller than the reported conductances measured electrophysiologically – 10–100 fS 15
– by a factor too large to be accounted for by the unknown relationship between H+
concentration and fluorescence.