The voltage and concentration dependence of extracellular DHS blockage and permeation of the hair cell transducer channel can be quantitatively described by a two barrier–one binding site model (Fig. 1; Woodhull, 1973; cf. Hille, 2001). The reaction scheme, assuming a Hill coefficient nH= 1 (cf. Fig. 2D) is:
where C indicates the unblocked transducer channels, CD the blocked channels, Do and Di the extra- and intracellular blocker, k1 and k2 the forward and k−1 and k−2 the reverse rate constants, which are dependent on the membrane voltage, Vm. Different from what has been previously suggested in bullfrog sacculus (Kroese et al. 1989), DHS was found to block the transducer channel from the intracellular side (Fig. 3) although it was two orders of magnitude less effective than extracellular DHS. This much lower sensitivity of the transducer channel to intracellular DHS can be most parsimoniously modelled by assuming that the concentration of the blocker at the intracellular face of the transducer channels is negligible, i.e. [D]i= 0, which effectively imposes a zero backward flow of DHS. The transducer current that remains in the presence of the blocker, IDHS, normalized to the control current Ic then depends on the extracellular drug concentration [Do], and applied voltage Vm, according to (e.g. Woodhull, 1973; Lane et al. 1991):
which equals the Hill equation (eqn (4)) with nH= 1 and half-blocking concentration KD (same unit as [D]o):
The voltage dependence of the block is thus expressed in terms of KD, which depends on the free energy of the binding site (Eb), the free energy difference of the two barriers (ΔE=E2−E1) at Vm= 0, and the related fractional electrical position of the binding site across the membrane, δb, and distance between the barriers, Δδ=δ2−δ1 (Fig. 1), while the slope factor:
if kT= 4.1 × 10−21 J (thermal noise energy at room temperature) and z= 2 (valence of DHS), where e is the unitary charge. Fits made using eqns (1) and (2) thus yield values for ΔE, Eb, Δδ and δb.
Figure 1. Energy profile of the two barrier–one binding site model used to describe the blockage and permeation of the hair cell transducer channel by dihydrostreptomycin In the absence of a voltage across the membrane (Vm= 0), the two barriers have estimated free energies E1 (11.05 kT) and E2 (15.68 kT) above the free energy level of the minima at the extra- and intracellular sides of the membrane. The barriers are located at relative electrical distances δ1 (range 0–0.09) and δ2 (range 0.91–1), as measured across the membrane from the extracellular side. The two barriers sandwich the binding site of DHS with a minimum in free energy, Eb (−8.27 kT), below zero. The binding site is located at a relative electrical distance δb of 0.79, measured from the extracellular side. Relative electrical positions across the membrane of the two barriers and the binding site, as well as their respective energies are drawn according to values found from fitting these model parameters in 1.3 mm Ca2+.
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Figure 2. Extracellular dihydrostreptomycin blocks the OHC transducer channel A and B, transducer currents recorded from an apical P7 OHC before (A) and during (B) the superfusion of 50 μm dihydrostreptomycin (DHS) when sinusoidal force stimuli of 45 Hz were used. The cell was held at −84 mV and the membrane potential was stepped, in 20 mV increments, between −144 mV and +96 mV. For clarity only responses to every other voltage step are shown. Driver voltage (DV, amplitude 35 V) to the fluid jet is shown above the traces. Positive DVs are excitatory. Membrane potentials are shown next to some of the traces. Recordings in A and B are the average of 4 repetitions and are offset so that the zero-transducer current levels (responses to inhibitory stimuli) are equally spaced. Cm 6.0 pF; Rs 0.8 MΩ. C, average normalized current–voltage curves for the control transducer currents (n= 30) and the current recorded during the superfusion of 3 μm (n= 9), 10 μm (n= 8) and 50 μm (n= 13) DHS (1.3 mm extracellular Ca2+), when both sine waves and voltage step stimuli were used. Note that the currents have been normalized to the maximal current recorded at +96 mV (Control: 878 ± 36 pA; 3 μm: 980 ± 55 pA; 10 μm: 708 ± 73 pA; 50 μm: 924 ± 61 pA). D, dose–response curves for the block of the transducer current by DHS at three different membrane potentials obtained from 35 apical-coil OHCs. Continuous lines are the fits through the data using eqn (4). Fit at −164 mV: half blocking concentration (KD) = 11.4 ± 0.6 μm, Hill coefficient (nH) = 0.90 ± 0.03 (number of measurements from left to right: 1, 10, 9, 4, 1, 10); −84 mV: KD= 7.0 ± 0.2 μm, nH= 0.96 ± 0.03 (2, 3, 12, 9, 4, 8, 1, 5, 13, 6, 1); −44 mV: KD= 21.2 ± 0.9 μm, nH= 0.96 ± 0.04 (number of measurements as for −84 mV). E, average KD and nH plotted as a function of the membrane potential. Number of measurements: 35 at −164 mV; 50 at −144 mV and −24 mV and 64 for all other potentials tested. The fit through the KD data points is according to eqn (2) with: ΔE=E2−E1= 4.63 kT, Δδ=δ2−δ1= 0.91, Eb=−8.27 kT and δb= 0.79.
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Figure 3. Block of the transducer channel by intracellular dihydrostreptomycin A and B, transducer currents recorded from apical P7 OHCs in the absence (A) and presence (B) of 10 mm DHS in the intracellular solution and 1.3 mm Ca2+ in the extracellular solution. The cells were held at −84 mV and the membrane potential was stepped, in 20 mV increments, between −164 mV and +156 mV. For clarity only responses to every other voltage step are shown. Driver voltage to the fluid jet was 35 V. Recordings in A and B are averaged from 2 and 3 repetitions, respectively, and are offset so that the zero-transducer current levels (responses to inhibitory stimuli) are equally spaced. A: Cm 5.9 pF; Rs 1.6 MΩ. B: Cm 5.5 pF; Rs 2.0 MΩ. C, normalized current–voltage curves for the transducer currents recorded in the absence (control, n= 7) and in the presence of DHS (10 μm: n= 5; 100 μm: n= 5; 1 mm: n= 6; 10 mm: n= 3) in the intracellular solution, including those shown in A and B. Currents have been normalized to the maximal current recorded at −164 mV (Control: −1417 ± 79 pA; 10 μm: −1437 ± 141 pA; 100 μm: −1367 ± 107 pA; 1 mm: −1294 ± 124 pA; 10 mm: −1271 ± 76 pA). D, dose–response curves for the block of the transducer current by intracellular DHS at three different membrane potentials obtained from 19 apical OHCs. Continuous lines are fits through the data using eqn (4). Fit at +156 mV: KD= 277 ± 44 μm, nH= 0.88 ± 0.10; +76 mV: KD= 742 ± 131 μm, nH= 0.73 ± 0.09; +36 mV: KD= 3567 ± 881 μm, nH= 0.71 ± 0.12 (number of measurements from left to right: 5, 5, 4, 3). E, average KD and nH plotted as a function of the membrane potential including those shown in D. Number of measurements as in D.
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Since the drug's binding time constants (Fig. 9A) were found to be at least an order of magnitude slower (∼0.5–3 ms) than the presumed time constant of activation of the transducer current (< 50 μs, Kennedy et al. 2003) absolute rate constants could be determined. First-order kinetics of the reaction scheme (see above, with [D]i= 0) leads to a drug binding time constant, τ, related to the rate constants and extracellular concentration [D]o according to:
At low concentrations (< 5 μm) the observed behaviour of 1/τversus[D]o (Fig. 9B) shows a linear relationship of which the slope was taken to define k1, according to eqn (3) (continuous lines, Vm=−84 mV). From k1 the absolute values of the energy barriers E1 and E2 were calculated using Eyring's rate theory (cf. Hille, 2001) and the values obtained for Eb, ΔE, δb and Δδ. The values of δ2 and δ1, required for this calculation, are restricted by the fitted value obtained for Δδ (i.e. 0 ≤δ1≤ 1 −Δδ= 0.09; 0.91 =Δδ≤δ2≤ 1), and thus lead to a range of values for both E1 and E2 (see Results). Combining these values allows for calculation of k−1 and k2 so that the drug's rate of entry into the hair cells, determined by k2, could also be quantified.