The outer hair cell (OHC) of the organ of Corti underlies a process that enhances
hearing, termed cochlear amplification. The cell possesses a unique
voltage-sensing protein, prestin, that changes conformation to cause cell length
changes, a process termed electromotility (eM). The prestin voltage sensor
generates a capacitance that is both voltage- and frequency-dependent, peaking
at a characteristic membrane voltage (V_{h}), which can be greater than
the linear capacitance of the OHC. Accordingly, the OHC membrane time constant
depends upon resting potential and the frequency of AC stimulation. The
confounding influence of this multifarious time constant on eM frequency
response has never been addressed. After correcting for this influence on the
whole-cell voltage clamp time constant, we find that both guinea pig and mouse
OHC eM is low pass, substantially attenuating in magnitude within the frequency
bandwidth of human speech. The frequency response is slowest at V_{h},
with a cut-off, approximated by single Lorentzian fits within that bandwidth,
near 1.5 kHz for the guinea pig OHC and near 4.3 kHz for the mouse OHC, each
increasing in a U-shaped manner as holding voltage deviates from V_{h}.
Nonlinear capacitance (NLC) measurements follow this pattern, with cut-offs
about double that for eM. Macro-patch experiments on OHC lateral membranes,
where voltage delivery has high fidelity, confirms low pass roll-off for NLC.
The U-shaped voltage dependence of the eM roll-off frequency is consistent with
prestin’s voltage-dependent transition rates. Modeling indicates that the
disparity in frequency cut-offs between eM and NLC may be attributed to
viscoelastic coupling between prestin’s molecular conformations and
nanoscale movements of the cell, possibly via the cytoskeleton, indicating that
eM is limited by the OHC’s internal environment, as well as the external
environment. Our data suggest that the influence of OHC eM on cochlear
amplification at higher frequencies needs reassessment.

## Introduction

Outer hair cell (OHC) electromotility (eM) underlies cochlear amplification in
mammals, where in its absence, hearing deficits amount to 40–60 dB (Dallos et al., 2008; Ashmore et al., 2010). The molecular basis of OHC eM is the
membrane-bound protein prestin (SLC26a5), an anion transporter family member that
has evolved to work as a voltage-dependent motor protein in these cells (Zheng et al., 2000). The protein’s
voltage-sensor activity presents as a voltage-dependent (or nonlinear) capacitance
(NLC), obeying Boltzmann statistics (Ashmore,
1990; Santos-Sacchi, 1991), and
whose peak magnitude corresponds to the voltage (V_{h}) where sensor charge
is equally displaced to either side of the membrane. Voltage-dependent
conformational change in prestin is believed to form the basis of precise phase
differences between OHC activity and basilar membrane motion that leads to
amplification, indicative of local cycle-by-cycle feedback (Dallos et al., 2008). Recent measurements of OHC extracellular
voltage and basilar membrane motion have observed the predicted amplifying phase
differences (Dong and Olson, 2013). However,
other recent measurements did not find appropriate phase or timing differences in
the motions within the organ of Corti, and have challenged the standard view of
amplification (Ren et al., 2016; He et al., 2018). Additionally, it is
commonly accepted that unconstrained (load-free) OHC eM magnitude is invariant
across stimulating frequency, having been measured out beyond 70 kHz (Frank et al., 1999); the flat frequency
response is another important element in cycle-by-cycle feedback theory. However,
these data were obtained at voltage offsets far removed from V_{h}, which
has questionable physiological significance. Based on our measures (Santos-Sacchi and Tan, 2018), we estimate
that their depolarized offset from V_{h} would have been ∼65 mV.
Indeed, by exploring at an offset potential near V_{h}, we recently
demonstrated that eM measured with the microchamber displays significant low-pass
behavior (Santos-Sacchi and Tan, 2018).
Here we further explore the low-pass nature of eM under whole-cell voltage clamp out
to 6.25 kHz in the guinea pig and mouse with ramped changes in holding potential
that provide shorter, more stationary-in-time measures compared with our previous
experiments. Furthermore, whole-cell voltage clamp provides far better voltage
control than the microchamber, but patch electrode series resistance effects must be
considered. Following precise corrections for NLC-induced, voltage- and
frequency-dependent voltage roll-off under whole-cell voltage clamp, never
previously done, we find that eM and NLC show similarities in their nonlinear
voltage and frequency dependence, both frequency cut-offs being U-shaped functions
of holding voltage, with minima near V_{h}. Macro-patch experiments on the
OHC lateral membrane, which provide robust voltage control, confirm our low-pass
whole-cell measures of NLC within the speech frequency range. We further find that
the frequency cut-offs of eM and NLC are disparate. Modeling suggests that both
internal (e.g., cytoskeletal interactions, membrane lipid interactions) and external
(e.g., viscous environment of the cell) loads are influential in effecting this
disparity.

## Materials and methods

Whole-cell recordings were made from single isolated OHCs from the apical two turns
of organ of Corti of guinea pigs. An inverted Nikon Eclipse TI-2000 microscope with
a 40× lens was used to observe cells during voltage clamp. Experiments were
performed at room temperature. Blocking solutions were used to remove ionic
currents, limiting confounding effects on NLC determination and voltage delivery
under voltage clamp (Santos-Sacchi, 1991; Santos-Sacchi and Song, 2016).
Extracellular solution was (in mM): NaCl 100, TEA-Cl 20, CsCl 20, CoCl_{2} 2, MgCl_{2} 1, CaCl_{2} 1, and HEPES 10. Intracellular solution was
(in mM): CsCl 140, MgCl_{2} 2, HEPES 10, and EGTA 10. All chemicals were
purchased from Sigma-Aldrich.

An Axon 200B amplifier was used for whole-cell recording with jClamp software
(http://www.scisoftco.com). An Axon Digidata 1440 was used for
digitizing. AC (frequency) analysis of membrane currents (I_{m}) and eM were
made by stimulating cells with a voltage ramp from 100 to −110 mV (nominal),
superimposed with summed AC voltages at harmonically related frequencies of 195.3,
390.6, 781.3, 1,562.5, 3,125, and 6,250 Hz, with a 10-µs sample clock. Currents
were filtered at 10 kHz with a four-pole Bessel filter. Corrections for series
resistance were made during analysis. Capacitance was measured using dual-sine
analysis at harmonic frequencies (Santos-Sacchi et
al., 1998; Santos-Sacchi, 2004).
Briefly, real and imaginary components of membrane current at harmonic frequencies
were determined by Fast Fourier Transform (FFT) in jClamp, corrected for the
roll-off of recording system admittance (Gillis,
1995) and stray capacitance (Santos-Sacchi, 2018). Series resistance (R_{s}), membrane
resistance (R_{m}), and membrane capacitance (C_{m}) were extracted
using the dual-sine, three-parameter solution of the standard patch clamp model
(Santos-Sacchi et al., 1998; Santos-Sacchi, 2004), based on the original
single sine solution (Pusch and Neher,
1988). To extract Boltzmann parameters, capacitance-voltage data were fit to
the first derivative of a two-state Boltzmann function.

where

Q_{max} is the maximum nonlinear charge moved, *V*_{h} is voltage at peak capacitance
or equivalently, at half-maximum charge transfer, *V*_{m} is membrane potential, *z* is valence, C_{lin} is linear
membrane capacitance, *e* is electron charge, *k _{B}* is Boltzmann’s constant, and T is absolute
temperature.

As noted above, whole-cell NLC determination was made following stray capacitance
removal (Santos-Sacchi et al., 1998; Santos-Sacchi and Song, 2016; Santos-Sacchi, 2018; Santos-Sacchi and Tan, 2018). Patch pipettes were coated with
M-coat (Micro Measurements) to reduce stray capacitance. Remaining stray capacitance
was removed by amplifier compensation circuitry before establishing whole-cell
configuration, and if necessary, additional compensation was applied under
whole-cell voltage-clamp conditions (Schnee et
al., 2011a,b) and/or through a
software algorithm within jClamp software to ensure expected frequency-independent
linear capacitance (Santos-Sacchi and Song,
2016; Santos-Sacchi, 2018). The
latter approach simply performs single tau capacitance compensation mathematically
on collected data, analogous to the electronic compensation with the amplifier.
R_{s} was determined from voltage step–induced whole-cell
currents before AC measures, the derivation provided in the appendix of (Huang and Santos-Sacchi, 1993).
R_{m}(*v*) across ramp voltage was determined from
R_{s}-corrected ramp voltage and generated ramp currents with AC
components removed, ΔV_{m}/(ΔI_{Rs}).

Simultaneous and synchronous eM measures were made with fast video recording. A
Phantom 110 or 310 camera (Vision Research) was used for video measures at a frame
rate of 25 kHz. Magnification was set to provide 176 or 106 nm/pixel. A method was
developed to track the apical image of the OHC, providing subpixel resolution of
movements (see Fig. 1). Video frames were
filtered with a Gaussian Blur filter (http://www.gimp.org) before
measurement. The patch electrode provided a fixed point near the basal end of the
cell. Since movements were measured at the apical pole of the cell and the cell was
held physically at the point of electrode insertion, estimates of full whole-cell
movements were obtained from the ratio of apical/basal partitioning at the point of
electrode insertion as in microchamber experiments (Frank et al., 1999; Santos-Sacchi and Tan, 2018). Resultant movements were analyzed by FFT
in MATLAB. The colors in surface plots were generated in the MATLAB plotting routine *surf* with shading set to <interp>. This
procedure allows contours to be readily observable. Although eM was measured at all
six AC frequencies, only five estimates of NLC were possible using the dual-sine
approach. To obtain τ_{clamp} estimates at 6.25 kHz, the magnitude of
NLC at 3.125 kHz was used, factored by 0.748, in line with the slope of its decrease
across frequency.

Models were implemented in MATLAB Simulink and Simscape, as detailed previously
(Song and Santos-Sacchi, 2013; Santos-Sacchi and Song, 2014b). Model
R_{s} and R_{m} were from the averages of guinea pig and mouse
OHC parameters obtained (see below). R_{s}-corrected V_{m} vs. eM
data were fit with the first derivative of a two-state Boltzmann function (Santos-Sacchi, 1991). We processed
biophysical data as either group-averaged (whole-cell currents and eM averaged
before analysis) or individual cell analysis followed by averaging, thus providing
statistics (mean ± SEM).

Under voltage clamp, the voltage delivered across the cell membrane depends on the
electrode R_{s}. In the absence of R_{s}, the command voltage is
delivered faithfully to the cell membrane in magnitude and time (phase). Otherwise,
the voltage delivered to the membrane will suffer from voltage drops across
R_{s}, depending on the magnitude and time (phase) of evoked currents.
In any analysis of voltage-dependent cellular processes, such as eM (Santos-Sacchi and Dilger, 1988; Iwasa and Kachar, 1989), it is important to
accurately assess membrane voltage. Actual membrane potential under voltage clamp
can be exactly determined by subtraction of the voltage drop across R_{s},
i.e., I_{Rs} * R_{s}, with I_{Rs} being the sum of resistive
and capacitive components of the cell membrane current. For sinusoidal stimulation
across the voltage ramp, the AC command voltage V_{c} and evoked currents
I_{Rs}, are evaluated as complex values at each excitation frequency and
ramp offset voltage, (A + *j*B), where A and B are the real and
imaginary components, obtained by FFT, thereby supplying $Vc(f,v)$ and $IRs(f,v).$ Our goal is to accurately determine the frequency response of eM, given a nonzero
series resistance that imposes its own influence on the frequency response of
voltage-driven eM data. The method essentially seeks to identify the true excitation
voltage, $Vm(f,v),$ i.e., the drive for eM, supplied to the plasma membrane.

Before we present an analysis of averaged eM data, we illustrate the approaches
available for R_{s} correction in a MATLAB Simulink model. In the model, we
are able to directly measure imposed membrane voltage so we can compare results from
these approaches to actual values. The generated charge in the model is taken as the
equivalent of eM, since we have shown that OHC eM and prestin charge movement are
directly coupled (Santos-Sacchi and Tan,
2018). The model we use to confirm our voltage corrections has been fully
described before (Santos-Sacchi and Song,
2014a), and includes a slow, stretched exponential intermediary
transition between chloride binding transitions and voltage-dependent charge
transitions, the latter corresponding to eM. Modifications of the model were made to
correspond to the biophysical data. Chloride was 140 mM. For the model, the number
of prestin particles was set to 25.92e6, and *z* = 0.92 for the
guinea pig, and correspondingly 5.62e6, and 1 for the mouse. The forward
(*α*) and backward (*β*)
voltage-dependent transition rate constants for the electromechanical component of
the model were 1.2947e6 and 1.1558e4, respectively. Both forward
(*α _{m}*) and backward
(

*β*) parallel intermediary transition rate constants (which are equal to each other, but separately labeled to distinguish direction—incidentally, the equivalence is the basis of detailed balance in the model) were defined as

_{m}where *b* = −0.4663 and *A* = 3.0398e4 for the
guinea pig data comparisons, or *A* = 15.199 for the mouse data
comparisons to account for the difference in NLC and eM frequency response of the
species that we find. Units for rate constants and A are in s^{−1}; b
is unitless.

For the model, there are three ways to correct the frequency response of eM magnitude
for the confounding effects of R_{s}-induced membrane voltage roll-off.

### Method 1

The complex ratio of AC command voltage (located at differing ramp voltage
offsets [*v*]) to the directly measured membrane voltage within
the model at each excitation frequency can be used to correct eM magnitude
frequency response, eM being both a function of frequency and holding voltage
(Santos-Sacchi and Tan, 2018),
namely, eM $(f,v),$ akin to NLC $(f,v)$ (Santos-Sacchi and Song, 2016).
Parallel bars indicate absolute values:

### Method 2

The complex ratio of command voltage to calculated V_{m $(f,v),$} namely, $IRs(f,v)\u22c5Rs,$ can be used for correction.

### Method 3

Finally, the multifarious clamp time constant, $\tau clamp(f,v),$ determined at each excitation frequency and ramp voltage offset, can be used to
correct, via a Lorentzian function (*A* =1/[1 +
(2π*f*τ)^{2}]^{1/2}), the
magnitude of eM, scaled to DC (steady state) levels [$1\u2212Rs/Rm(f0,v)],$$f0=0\u2248$ ramp frequency.

where

For the model, Fig. S1 (guinea pig parameters) and Fig. S2 (mouse parameters)
show that each method of R_{s} correction gives the same results,
revealing the intrinsic low pass eM response of the meno presto model that
derives from its stretched-exponential intermediary transition kinetics (Song and Santos-Sacchi, 2013; Santos-Sacchi and Tan, 2018).

Because R_{s} was measured before running the ramp protocols, there is
always a possibility that it changed during recording. The possibility of slight
changes in R_{s} is well established in the literature. Fortunately, the
currents generated simultaneously with eM must also report on R_{s}.
Thus, we plot frequency cut-offs of eM based on Methods 2 and 3, each showing
overlap (see Fig. 6), and indicating that
our estimates of R_{s} are indeed accurate, with only minor corrections
of the initial estimates (0.95–1.05 × initial estimates).

For the on-cell macro-patch approach, we used pipette inner diameters of 3.31
± 0.24 µm (electrode resistance in bath 1.49 ± 0.04 MΩ, *n* = 10), with M-coat applied within ∼20 µm of
the tip. Extracellular solution was in the pipette. To establish Gohm seals (2.7
± 0.24 GΩ, *n* = 10) we supplemented extracellular
solution with 5–7.5 µm Gd^{+3}; we have shown previously
that these low concentrations help to form seals without affecting NLC (Santos-Sacchi and Song, 2016). Ramps with
superimposed sinusoids were used, as above. Subtraction of currents at very
depolarized potentials, where predominantly linear membrane capacitance and
stray capacitance contribute, provided prestin-associated nonlinear currents
(see Fig. 7). Subsequently, these
nonlinear capacitive currents were used for dual-sine capacitance
estimation.

Data points from previous publications were extracted from plots using the application Grabit (written by Jiro Doke) in MATLAB.

### Online supplemental material

Figs. S1 and S2 depict model data of the meno presto model for guinea pig and mouse. Each illustrates that voltage corrections used on the collected cell data are accurate and appropriate.

## Results

Fig. 1 depicts the method used to measure eM.
Movement of a piezo-driven AFM tip (0.2 N/m) confirmed the fidelity of the measuring
technique out to 6,250 Hz (Fig. 1 B, panel
4). To obtain accurate FFT results, the ramp-induced movement was detrended by
subtracting a linear fit for the AFM tip measures or a sigmoidal fit for the guinea
pig or mouse OHC eM measures (Fig. 1, B panel
3; D, panel 3; and F, panel 3). Subsequent analyses followed this detrending
approach. In Fig. 1, D and F, panel 4, an FFT
of the whole ramped eM response was made, and shows that unlike the piezo-driven
response (Fig. 1 B, panel 4), the magnitude
of both guinea pig and mouse OHC eM falls precipitously with frequency. This
roll-off arises from both the voltage-filtering effects of R_{s} and the
kinetics of prestin and/or mechanical impediments to cell movements. In the
following analysis, we restrict FFT inspection to defined integral segments (see the
red and blue highlighted example regions in Fig. S1, Fig. S2, Fig. 3, and Fig. 4) of
the eM ramp response to assess voltage dependence of the frequency response.
Furthermore, we detail methods (see Materials and methods) to remove the effects of
series resistance interference, thus revealing true eM frequency response as a
function of true membrane voltage.

For the OHC under voltage clamp, we do not have direct access to the membrane voltage
as in the model (see Materials and methods), so only two methods for eM correction
are available—that using estimates of R_{s} and NLC to gauge *τ _{clamp}*, and that using direct measures of
whole-cell currents. In Fig. 2, complex
whole-cell currents or calculated multifarious clamp time constants were used to
predict the roll-off in AC command voltage (see equations in Materials and methods).
Corrections will precisely account for the frequency-dependent roll-off in AC
voltages and subsequently, the same corrections are applied to voltage-dependent eM
measures.

We analyzed eight guinea pig OHCs. eM was commensurate with previously reported
average eM gains of 15–19 nm/mV (Ashmore,
1987; Santos-Sacchi, 1989).
Average (±SEM) Boltzmann parameters of NLC at the lowest frequency of 195 Hz
are Q_{max}: 3.18 ± 0.06 pC, V_{h}: −46.1 ± 1.1 mV, *z*: 0.92 ± 0.02, and C_{lin}: 22.58 ± 0.51 pF.
R_{s} was 9.1 ± 0.8 MΩ. R_{m} at zero holding
potential was 401 ± 75 MΩ. Average eM gain evoked by the ramp protocol
(Fig. 3 B) was 22.8 ± 2.8 nm/mV.
Maximum eM was 2.54 ± 0.32 µm; equivalent *z* was 0.94
± 0.02. Fig. 3, E–G, surface and
two-dimensional (2-D) plots, show average guinea pig OHC eM before and after
correcting for R_{s}-induced voltage errors.

We also analyzed four mouse OHCs. Average (±SEM) Boltzmann parameters of NLC at
the lowest frequency of 195 Hz are Q_{max}: 0.74 ± 0.02 pC,
V_{h}: −47.2 ± 3.3 mV, *z*: 0.82 ± 0.01,
and C_{lin}: 7.60 ± 0.13 pF. R_{s} was 5.7 ± 0.1 MΩ.
R_{m} at zero holding potential was 239 ± 53 MΩ. Average eM
gain evoked by the ramp protocol (Fig. 4 B)
was 7.0 ± 0.6 nm/mV. Maximum eM was 1.06 ± 0.08 µm; equivalent *z* was 0.70 ± 0.03. Fig. 4, E–G, surface and 2-D plots, show average mouse OHC eM before and
after correcting for R_{s}-induced voltage errors.

Fig. 3, E–G, and Fig. 4, E–G, illustrate that using either of the two possible correction approaches (Method 2 and Method 3) produces essentially the same results (in line with modeling in the figures in the Online supplemental material) and confirming the low pass nature of both guinea pig and mouse eM. Notably, the equivalence of the two methods also confirms the validity of the dual sine approach to measure high frequency capacitance, whose multifarious influence on clamp time constant must be considered in order to properly evaluate eM frequency response under voltage clamp.

In Fig. 5, A and B, we plot, at all
interrogated frequencies, the voltage dependence of group-averaged eM gain and NLC
for mouse and guinea pig OHCs. Guinea pig responses are lower pass than mouse
responses. We additionally analyzed OHC response data individually, followed by
averaging. Fig. 5, C and D, show mean and SEM
of eM at V_{h}, and Boltzmann parameters of fits based on single cell
analysis at frequencies spanning 195 to 3,125 Hz (mouse eM was additionally measured
at 6,250 Hz). It is clear that the guinea pig eM frequency response is lower pass
than the mouse response (quantification is shown in Fig. 6). It is noteworthy that guinea pig OHCs show a disparity between
eM and NLC V_{h} that is not found in the mouse. Such disparity has been
observed previously for guinea pig OHCs (Song and
Santos-Sacchi, 2013; Duret et al.,
2017), and we previously found that the disparity is diminished with
increased turgor pressure for the guinea pig. For our mouse OHCs, we note that the
cells appear substantially turgid (see Fig. 1 E), so this may be the reason for the absence in disparity in the
mouse.

Fig. 6 shows eM and NLC cut-off frequencies
(*Fc*) for guinea pig and mouse. To approximate differences in
frequency responses between NLC and eM for the mouse and guinea pig, we apply single
Lorentzian fits to group-averaged eM and NLC within a restricted bandwidth
(195–3,125 Hz) at each incremental holding potential of the ramped AC
response. The cut-off frequencies display U-shaped dependencies on holding
potential, AC voltage stimulation at V_{h} providing the slowest response.
eM corrections based on Methods 2 and 3 are overlapped, indicating accurate
estimates of R_{s}. Individually analyzed cell averages (filled symbols with
SEM, using Method 2) show good agreement with the group-averaged data. The *Fc* at V_{h} for guinea pig OHC eM is 1.47 ± 0.065
kHz, and that for mouse OHC eM is 4.26 ± 0.74 kHz. NLC measures follow this
pattern, with cut-offs faster than that for eM. The *Fc* at
V_{h} for guinea pig OHC NLC is 2.47 ± 0.091 kHz, and that for
mouse OHC NLC is 6.83 ± 0.46 kHz.

To assess the frequency response of NLC under near ideal voltage clamp, we employed
macro-patches of the guinea pig OHC lateral membrane. NLC measures were made using
the ramp protocol, as above. Our video analysis setup was insufficient to measure
patch movements induced by our ramped 20 mV AC stimuli. However, as with
voltage-corrected whole-cell measures, patch NLC shows low-pass behavior (Fig. 7). At the lowest frequency of 195 Hz,
fitted Boltzmann parameters of NLC are Q_{max}: 38.6 ± 6.1 fC,
V_{h}: −38.1 ± 4.4 mV, and *z*: 0.67 ±
0.03. No correction for resting potential was made for on-cell patch V_{h}. *z* is lower than whole-cell measures. For comparison, Gale and Ashmore (1997b), using a lock-in
amplifier to measure NLC with 1 kHz sinusoids, found *z* to be 0.65
(slope factor β = 0.04). Given our *z* and a hemispherical
patch surface area (diameter equal to 3.5–4 µm), we calculate
3,851–5,030/µm^{2} elementary prestin motor units, in line
with previous estimates.

Fig. 8 compares peak NLC from on-cell macro-patches with scaled values of whole-cell eM, NLC, and data from Gale and Ashmore (1997a). For frequency roll-off comparison, values were scaled to coincide with our macro-patch NLC values at 390.6 Hz. The roll-off of whole-cell eM and that of the patch movements measured by Gale and Ashmore are similar. The measures of peak NLC of whole-cell, macro-patch, and Gale and Ashmore’s patch NLC (points within our recording bandwidth are shown) are similar. The cut-off frequencies obtained with Lorentzian fits (either whole-cell or macro-patch data) show that eM and NLC frequency responses differ, with eM being slower, as noted above. We emphasize that the single Lorentzian fits here do not describe the full frequency response of NLC, but only that within our recording bandwidth. These measures are intended to make comparisons with our eM measures within the same recorded bandwidth. Thus, we conclude that the degree of coupling between prestin conformational changes (i.e., NLC) and eM must underlie differences between their frequency responses.

Finally, we observe that fits of the square root of eM provide cut-offs comparable to
those for NLC for both mouse and guinea pig OHCs. The *Fc* at
V_{h} for guinea pig OHC eM^{1/2} is 2.77 ± 0.11 kHz, and
that for mouse OHC eM^{1/2} is 7.69 ± 0.81 kHz. *t* test
comparisons to the cut-offs of NLC (2.47 ± 0.091 and 6.83 ± 0.46,
respectively) show no statistically significant differences (P = 0.1092 and P =
0.4948, respectively). Below we consider the basis of such disparity.

## Discussion

Prestin works by sensing voltage with consequent alterations in its conformational
state, leading to contractions and elongations of the cylindrical cell that provides
enhancement of auditory threshold (Ashmore,
2008; Santos-Sacchi et al.,
2017). Because of the large restricted prestin-generated charge movements
evoked by voltage (Ashmore, 1990; Santos-Sacchi, 1991), a substantial
alteration in membrane capacitance ensues, paradoxically altering the very voltage
that the sensor senses. This is even expected to occur during normal acoustic
stimulation, where the receptor potential will suffer from a multifarious membrane
resister-capacitor (RC) filter. For example, the receptor potential cut-off
frequencies estimated using linear capacitance alone (Johnson et al., 2011) would be greatly reduced at
V_{h}, where NLC can be as great as linear capacitance. In our study
under voltage clamp, series resistance and membrane capacitance conspire to limit
the imposition of voltage across the membrane, interfering in both time and
magnitude. To evaluate effects of membrane potential across frequency, it is
required to alleviate this interference. Here we have compensated for the
multifarious time constants induced by R_{s} and NLC(*v,f*)
in order to reveal the true voltage-induced frequency response of both guinea pig
and mouse OHC eM across an array of holding potentials. Before proceeding, we review
the literature on measures of eM frequency response.

### eM bandwidth studies

A review of OHC eM bandwidth studies has been published recently (Santos-Sacchi, 2019). Here we briefly
review those studies with the aid of Table 1. Over the years, better techniques that have extended voltage
delivery capabilities have provided increasingly higher estimates for eM cut-off
frequencies. Of course, R_{s} of the patch pipette or the microchamber
pipette, in combination with OHC capacitance, ultimately controls frequency
delivery bandwidth under voltage clamp. Typically, the time course of the clamp
can be garnered from exponential decays of current induced by voltage steps, but
AC currents can also be used as we have done in our present work. Prior to Frank et al. (1999), eM cut-offs were
typically below 10 kHz. Interestingly, Gale
and Ashmore (1997a) measured both patch NLC and membrane movements,
and found a NLC cut-off near 10 kHz, but a patch movement cut-off significantly
lower (see Fig. 8). The reduced cut-off
was attributed to mechanical impediments to patch movements, despite the
expected coupling between charge movement and mechanical response. The
introduction of the partitioning microchamber by Dallos and Evans essentially
reduced membrane capacitance by delivering the voltage stimulus across the
reduced series combination of the partitioned cell membrane capacitance (Dallos and Evans, 1995). Given membrane
time constants that are equal for each partition, each will experience a flat
delivery of a command voltage across the cell. However, the command voltage
frequency response itself will be attenuated by the R_{s} *
C_{in} filter, C_{in} being the input capacitance resulting
from the series combination of partitioned capacitances (see microchamber
schematic in Fig. 9 of Santos-Sacchi and Tan,
2018). Thus, Dallos and Evans, even though they did not monitor
membrane currents to determine actual voltage roll-off, did estimate, given
published OHC characteristics, the voltage delivery cut-off to be ∼30 kHz
in their experiments. This cut-off, as noted above, depended on their
microchamber R_{s} and the input membrane capacitance. Of note for the
studies by both Dallos and Evans and Frank et al., their zero microchamber
offsets interrogated eM at holding membrane potentials far removed from
V_{h}; this is apparent from the small eM gains reported compared
with established eM gains between 15–19 nm/mV. Santos-Sacchi and Tan (2018) measured eM frequency
response at two microchamber offsets, one near V_{h} and the other well
offset from V_{h} (we estimate ∼65 mV away). By measuring clamp
tau from exponential decays at those two offsets, they found that eM is lower
pass at V_{h}. Unfortunately, none of the previous studies have actually
accounted for the voltage- and frequency-dependent roll-off of voltage delivery
to the OHC, and without that correction, the true voltage-dependent eM frequency
response remained unclear.

### Similarities and differences between voltage-sensor movement (NLC) and eM

The electromechanical results we report on now explore the component of eM within
0.195–6.3 kHz, corresponding to human speech frequencies, and are made
under whole-cell voltage clamp where, unlike the microchamber, precise DC and AC
voltage delivery to the membrane is possible and predictable. Thus, we find with
whole-cell recording, following voltage corrections due to the multifarious
clamp time constant, that prestin’s eM and NLC frequency response, in
both mouse and guinea pig, exhibits low-pass electromechanical behavior within
our measurement bandwidth. The frequency response is slowest at V_{h},
with a cut-off, approximated by single Lorentzian fits within that bandwidth,
near 1.5 kHz for the guinea pig OHC and near 4.3 kHz for the mouse OHC, each
increasing in a U-shaped manner away from V_{h}. NLC measures follow
this U-shaped pattern.

By using macro-patch measurements, where voltage control is near ideal, we find
that NLC frequency response is similar in roll-off to corresponding whole-cell
measures within the speech frequency bandwidth, indicating our whole-cell
voltage corrections were effective. Nevertheless, eM frequency cut-offs differ
from those of NLC, which are faster. In OHC patches, Gale and Ashmore found that
patch movement *Fc* was only 0.19 times that of NLC *Fc* (using the ratio of their reported corresponding tau
values of 16 and 83.5 µs for NLC and movements, respectively), which they
attributed to interactions of prestin with cytoskeletal elements and viscous
damping (Gale and Ashmore 1997a). That
is, factors that influence prestin electromechanical behavior are present within
the on-cell patch, i.e., intrinsic to the local environment of prestin within
the membrane. Indeed, we have found physical and functional interactions of
MAP1S, a small actin-binding protein, with prestin (Bai et al., 2010). However, such influential effects on eM
might be expected to work on both eM and NLC, since charge movement and
mechanics are coupled (Dong and Iwasa,
2004; Santos-Sacchi and Tan,
2018). Consequently, the discrepancy between eM and NLC frequency
response that is observed must point to variability of coupling charge movement
to eM. Of course, uncoupling is clearly observable by deflating the OHC, which
preserves NLC while abolishing whole-cell eM (Santos-Sacchi, 1991). Thus, we suggest that a variable coupling
between whole-cell mechanics and prestin activity limits OHC influences on
cochlear amplification at high frequencies. To this point, Vavakou et al., 2019 have recently observed limited,
low-pass estimates of OHC eM frequency response using optical coherence
microscopy vibrometry in the high-frequency region of the living gerbil.

### What factors may influence OHC eM frequency response?

Within our interrogation bandwidth, NLC and eM are low pass. Since we accurately
corrected for voltage delivery roll-off, the differential eM roll-off must
result from other factors intrinsic and/or extrinsic to the cell. Two features
require exploration not previously considered. One is the mechanism for voltage
dependence of the cut-off frequency, *Fc* (Fig. 6). The other is a discrepancy in *Fc* for NLC and eM. The values of *Fc* for NLC are higher than those
of eM. That is, eM appears to be low-pass filtered.

There are two factors that contribute to the frequency dependence that we measure. One is due to the intrinsic transition rates between prestin’s conformational states (Iwasa, 1997). Another factor, which is extrinsic to the motile element itself, is due to mechanical loads imposed on both the cell and the motile element. In the absence of mass, the characteristic frequency is determined by viscoelastic relaxation.

Mechanical factors could lead to a voltage dependence of *Fc* if
the axial stiffness of the cell is itself voltage dependent, because the
characteristic frequency of viscoelastic relaxation is determined by $k/\eta ,$ where *k* is the axial stiffness and η the drag coefficient.
Indeed, it has been reported that the axial stiffness significantly decreases on
deep depolarization (He and Dallos,
1999, 2000). Such a
reduction in stiffness leads to a lower viscoelastic frequency as holding
voltage is moved away from V_{h}, and is inconsistent with our
observations. On the other hand, an alteration of axial stiffness has been
theoretically predicted as an analogue of “gating compliance,” in
which conformational transitions contribute to length changes of the cell (Iwasa, 2000). The predicted voltage
dependence of the compliance, the inverse of stiffness, is bell-shaped, similar
to NLC and qualitatively similar to our observed *Fc* values.
Nevertheless, this effect is much too small to account for our observations.
Actually, the magnitude of the “gating compliance” is
quantitatively consistent with a report that voltage dependence of axial OHC
stiffness is absent (Hallworth, 2007).
Indeed, a previous treatment of a viscoelastic process involving the OHC does
not lead to voltage dependence of the viscoelastic frequency for a voltage
driven stimulus, while it does for a force stimulus (Iwasa, 2016).

A stochastic transition model, even the simplest two-state model, inherently
predicts voltage dependence because the characteristic frequency is expressed as
a sum of transition rates in opposite directions from the distribution’s
midpoint (Iwasa, 1997). In one
direction, the rates increase exponentially with larger depolarizations, and in
the other direction with larger hyperpolarizations, similar to the *Fc* values in Fig. 6.

Disparate frequency dependence for NLC and eM is a new observation, and there is
no previous theoretical prediction or explanation. This is because it has been
assumed that the cell undergoes the same mode of motion as the motile element.
For a model in which eM is the low-pass filtered output of the motile element,
it is possible to obtain different values of *Fc* for NLC and
eM.

To capture the essential features of the observed high-frequency behavior of NLC and that of eM, we examined various mechanical loads assuming two conformations for the motile element (see Appendix). Virtually all kinetic models of prestin thus far reported, including the meno presto model (Santos-Sacchi and Song, 2014a), use this two-state formalism for eM and NLC generation.

We found that our experimentally observed eM and NLC frequency dependence can be explained by assuming that the motile element, which undergoes stochastic transitions, drives a mechanical system with two modes of motion. More specifically, cell displacement is the low-pass filtered output of the motile element displacement (see Appendix). The schematic configuration of the mechanical model is shown in Fig. 9. The model is constructed based on likely mechanical interactions within and external to the OHC. Thus, we view the elements $k2, $$\eta 2,$ and $k1$ as representing the local environment, consisting of the lateral plasma membrane and cytoskeletal structures, surrounding the motile element (P). The elements $k0$ and $\eta 0$ represent parts of the cell distant from the given motile element, primarily involved in energy dissipation due to the drag against the external medium. The existence of the elastic element $k0$may indicate that the drag is not concentrated at a single location, but it is distributed along the cellular axis.

Small mechanical displacements, x and y, elicited by small sinusoidal voltages with frequency $f$ applied on the motile element, can be expressed as (see derivation in Appendix),

where $fg$ is the gating frequency, $k02=k0/k2,$$k12=k1/k2,$$2\pi f0=\eta 0/k2,$ and $2\pi f2=\eta 0/k2.$ Here, $i=\u22121.$ The voltage dependence of the magnitude of these quantities is determined by $P\xaf\xb1,$ which is proportional to NLC at low frequencies (see Appendix). Notice here that
NLC is associated with *x*, and eM with *y*. Eq. 10a indeed shows that *x* is a low-pass filtered output of *y*.

Importantly, the model captures two features of our experimental observations.
One is the voltage dependence of *Fc* due to gating frequency $fg.$ The other is that the roll-off of eM takes place at lower frequency than that of
NLC.

Fig. 10 shows model fits of
experimentally determined eM and NLC frequency responses at three holding
potentials, passing through V_{h}. The parameter values obtained by
simultaneous data fit at each holding voltage appear reasonable. However,
standard errors are large, mainly due to the dependency between the substantial
number of parameters. Such an example is shown in Fig. 10 A, which shows plots using two sets of parameter
values, where the stiffness ratios of one set are larger than the other by about
fivefold. For this reason, it is unlikely that the present analysis can provide
definitive information regarding the axial stiffness of OHCs. Nevertheless, Fig. 10 indicates that the gating
frequency $fg$ obtained at each holding voltage is consistent with the stochastic model with
the lowest value, namely near V_{h} (at –45.8 mV). That is, the
values of $fg$ distinctly characterize the plots at each holding voltage, while the values for
the remaining parameters remain essentially the same.

### Summary

The relationship between OHC resting potential and NLC V_{h} will govern
the frequency response of eM. We previously proposed that a mismatch between
resting potential and NLC V_{h} could enhance the frequency response of
eM through a gain-bandwidth adjustment (Santos-Sacchi and Tan, 2018). Our new data are in line with this
proposal and provide a detailed description of the OHC’s
electromechanical cut-off frequency across voltage in both guinea pig and mouse
OHCs. This cut-off frequency is U-shaped about V_{h}, and increases as
holding voltage deviates from V_{h}. Importantly, the eM voltage
dependence is mirrored by that of NLC, though frequency cut-offs differ. The
cut-off disparity likely results from the influences of internal and external
loads upon the motile mechanism. In this regard, we should comment on the
observed differences in *Fc* values for mouse and guinea pig. Do
differences in intrinsic prestin kinetics exist between mouse and guinea pig, as
the viscoelastic model might suggest? While preliminary, we have macro-patch
measures out to 20 kHz that indicate similar prestin kinetics in the two
species. That is, we explored the frequency response of NLC in patches from
mouse and guinea pig OHC lateral membranes, each showing low-pass, stretched
exponential behavior with roll-offs at half magnitude near 10–12 kHz at
room temperature (unpublished data). In this light, we expect that the
structural differences between the two species’ cells (e.g., length,
diameter) may have provided differing loads on prestin with consequential
differences in eM and NLC, given that the piezoelectric nature of prestin will
be affected by load. In sum, all of our observations indicate that the influence
of OHC activity on cochlear amplification is more complicated than has been
envisioned.

Finally, we note that the kinetics of prestin are readily monitored through
shifts in V_{h}, a reflection of the ratio of forward to backward
transition rates. These kinetics, a determining factor in eM frequency response,
depend on a host of other factors, including intracellular chloride, membrane
tension and thickness, and temperature (Iwasa,
1993; Gale and Ashmore,
1994; Meltzer and Santos-Sacchi,
2001; Oliver et al., 2001; Santos-Sacchi et al., 2001; Izumi et al., 2011; Santos-Sacchi and Song, 2016). Indeed, alterations in
prestin kinetics have been found in mutations in prestin (Homma et al., 2013). We suspect that eM frequency
response is not a static feature in vivo.

## Appendix

### Rate equations for membrane molecules with mechanoelectric coupling: Introduction

There are two kinds of theories for describing the frequency dependence of cells with motile elements based on mechanoelectric coupling. One of them is based on kinetic equations with intrinsic transition rates, which have been used to describe conformational changes of proteins, such as ion transporters (Kolb and Läuger, 1978), ignoring the effect of mechanical load imposed on those cells (Iwasa, 1997).

Another kind of treatment assumes that intrinsic conformational transitions are infinitely fast. Conformational transitions are determined by the equation of motion for these cells, on which mechanical load is imposed (Iwasa, 2016). The resulting frequency responses of these cells are characterized by mechanical factors, such as the resonance frequency and the frequency for viscoelastic relaxation.

The present treatment introduces the dependence on mechanical load into the conformational transition rates, enabling description of the general case, where the intrinsic transition rates are finite. Two previous theories are recovered as special cases. In addition, we examine how mechanical complexity of the system can give rise to discrepancy in the frequency dependence of nonlinear capacitance NLC and that of motile response. For simplicity, we assume that the motile molecule has two discrete conformational states. The terms “motile molecules” and “motile element” are used interchangeably.

### Motile element with two states

Consider a membrane molecule with two discrete conformational states, $C0$ and $C1,$ and let the transition rates $k+$ and *k _{−}* between them be schematically
expressed as

Let $P1$ be the probability that the molecule in state $C1$. Then, the probability $P1$ can be expressed by the transition rates

where *q* is the charge transferred across the membrane during
conformational changes, *V* the membrane potential, *V*_{1/2} the
half-point voltage of the transition, and *β* =
1/*k*_{B}*T* with the Boltzmann constant
and *T* the temperature.

If $q$ is positive, the energy
level of the state $C1$ is
higher, reducing $P1$ as the membrane potential *V* rises. For prestin in outer hair
cells, which shorten on depolarization, if we choose $C1$ as
the shortened state, the unit length change *a* on conformational
change is negative, and then we have $q<0.$ Notice that the quantity *a* does not appear in Eq. A1.

The transition rates that satisfy Eq. A1 can be given by

Here, *α* is a constant between 0 and 1 and $k\xaf+$ and $k\xaf\u2212$ are transition rates at the operating voltage *V*_{ 0 } , around which *V* changes with time. The exponential function
can be linearized because we assume *V* − *V*_{ 0 } is small. These rates at the operation point are expressed by

where the transition rate $k\xaf$ is due to an energy barrier between the two states, excluding the difference in the energy levels, which are voltage dependent.

The time dependence of $P1$ can be expressed by the rate equation

Now we introduce sinusoidal voltage changes of small amplitude υ on top of
constant voltage $V\xaf,$ i.e., $V=V\xaf+v\u2009exp[i\omega t],$ where *ω* is the angular frequency and $i=\u22121.$ Then the transition rates are time-dependent due to the voltage dependence Eq. A1. They satisfy

Notice $k\xaf+$ and $k\xaf\u2212$ are time independent, and we assume that υ is small so that $\beta qv\u226a1.$ A set of $k+$ and *k _{−}* that satisfies Eq. A5 can be expressed

If we express $P1=P\xaf1+p1exp[i\omega t],$ we have respectively for the 0th and first order terms (Iwasa, 1997)

Notice that $p1$ does not depend on the factor $\alpha .$

Eq. A9 leads to voltage-driven mechanical displacement $ap1exp[i\omega t]$ with

where $P\xaf\xb1=P1\xaf1\u2212P\xaf1.$ The amplitude $|x|$ of the motile response is given by

Charge displacement is expressed by $qp1$ and the contribution to complex admittance $Y(\omega )$ is given by $(q/v)(d/dt)p1exp[i\omega t]$ (Iwasa, 1997). The contribution to the membrane capacitance is $Cnl(\omega )=Im[Y(\omega )]/\omega $ and therefore

This contribution to the membrane capacitance is commonly referred to as NLC because it shows marked voltage dependence. Notice also that the above derivation evaluates the contribution of a single unit of motile element. For a cell that contains $N$ motile units, both $|x|$ and $Cnl$ need to be multiplied by $N$.

The roll-off frequency $\omega g$ due to gating is expressed by

With Eqs. A6 and A7, this means that $1/\omega g$ rises at both ends of the membrane potential because $\alpha $ can take any value between 0 and 1. That means $\omega r$ can be asymmetric unless $\alpha =1/2.$

In the special case of α = 1/2, $k\xaf+$ = 1/$k\xaf\u2212\u2009$. If we define $b(V)=exp[\u2212\beta q(V\xaf\u2212V0)/2],$ then

which resembles the bell-shaped voltage dependence of nonlinear capacitance at low frequencies $(\omega \u21920).$

### Mechanoelastic coupling

For motile membrane proteins based on mechanoelectric coupling, charge transfer is affected by mechanical factors. Here, we assume the cell is cylindrical as in the case of cochlear outer hair cells and approximate it as a one-dimensional object (Fig. A1).

Supposing charge transfer *q* is associated with a change $a$ in the length of the
cell, Eq. A1 should be replaced
by

*V*

_{1/2}is the midpoint voltage of the Boltzmann function and

*F*is the axial force. Eq. A16 and A17 for the transition rates should be

where *f* is a small change in the axial force *F* that corresponds to a small voltage change *V* − *V*_{0}. The
transition rates $k\xaf+$ and $k\xaf\u2212$ are redefined by including the effect of the axial force *F*_{0} of the resting
condition. For the rest of the present paper, the dependence on the value of the parameter *α* does not appear except for $\omega g.$

With a shorthand notation $P\xaf\xb1(=P\xaf1(1\u2212P\xaf1)),$ the change of the conformational probability $p1$ can be driven either by changes in the voltage as well force:

If the motile element is driven by voltage changes, $p1$ is proportional to *v* and mechanical displacement is given by $ap1.$

### Effect of viscous drag

Movement is driven by a deviation from Boltzmann distribution. When voltage
changes with amplitude *v* imposed, $p1$ as expressed by Eq. A18 is the
goal of the drive. Since this force is countered by viscous drag (with drag
coefficient $\eta ),$ the equation of motion in the frequency domain can be expressed by

Notice here that the equilibrium transition rates here depend not only on $V\xaf$ but also on $F\xaf$ because the motile element based on piezoelectricity is sensitive to mechanical force as well as the membrane potential.

Eq. A19 leads to

similar to the previous treatment for the special case without inertial loading (Iwasa, 2016). Here, the viscoelastic relaxation frequency is defined by $\omega \eta =k/\eta .$ It is essentially an equation for viscoelastic relaxation, adding a low pass filter to the motile mechanism. It is consistent with previous expressions in both extremes, i.e., $\omega g\u2192\u221e$ and $\omega \eta \u2192\u221e.$

The voltage dependence of NLC and that of motile response are identical. In the following, we show that mechanical load with complex relaxation can lead to discrepancy in their frequency dependences.

### Complex mechanical relaxation

Let *Χ* represent the point that links a spring $k1$ with a dashpot $\eta 1$.
Let *Υ* represent the point that joins the spring *k*_{1} with the rest, which includes a spring $k2$,
a dashpot $\eta 2$,
and a driver (Fig. A2). The equations of
motion of this system driven by force *F* generated at the
location *P* can be expressed

If the force generator operates at a frequency *ω* with
small amplitude on top of its steady value $F\xaf,$$F=F\xaf+fexp[i\omega t].$ By letting the small amplitude components of *X* and *Y* with frequency *ω* be represented by *x* and *y*, Eqs. A21 and A22 turn into

Eq. A23 can be rewritten as

which indicates that the quantity *x* is obtained by low-pass
filtering *y* with roll-off frequency of $\omega 1(=k1/\eta 1).$

By introducing a characteristic frequency, $\omega 2(=(k1+k2)/\eta 2), $Eq. A24 can be transformed into

An approach analogous to those in the previous sections lead to an equation

Since we have $y=ap,$ this equation leads to

Eqs. A25 and A30 show that the relationship
between *y* and υ has three adjustable parameters, $\omega 1,\omega 2,$ and $kr(=k1/k2).$ For an example of the frequency dependence of *y*, see Fig. A3.

The frequency dependence of NLC is the same as that of *y*. Motile
response *x* is obtained by low-pass filtering *y*. The roll-off frequency of *y* is
voltage-dependent due to the voltage dependence of *ω _{g}.*

With the connectivity of Fig. A2, it is
difficult to make high frequency roll off of both quantities as similar as the
experimental data. For *y* to roll off at relatively high
frequency, *ω _{1}* has to be small and

*ω*has to be large because

_{2}*G*must be small as required by Eq. A30. This requirement makes $x$ roll off at a frequency much lower than

_{1}*y*does (Fig. A3).

### Modified complex mechanical relaxation

The model described above predicts a difference between *x* and *y* much larger than the experimentally observed frequency
dependence. Let us add a spring across the upper dashpot (Fig. A4).

The set of equations that describe this configuration are

If force $F$ is driven at angular
frequency *ω* with amplitude *f*, the
equation is transformed into

where variables in the lower case *x* and *y* are
the complex amplitude of frequency *ω*.

which corresponds to $G1(\omega )$ in the previous case.

Since force generation is associated with spring *k*_{2} in the manner
similar to the previous case, we obtain

and *x* is obtained with Eq. A35.

If the cell contains $N$ motile units, *a* should be replaced by $aN.$ For numerical analysis, the number of parameters can be reduced by introducing
the ratios $k02(=k0/k2),k12(=k1/k2),\omega 0(=k2/\eta 0),$ and $\omega 2(=k2/\eta 2)$ and *x* and *y* are expressed by

The corresponding equations (Eq.
10) in the main text are expressed with linear frequency *f* instead of angular frequency *ω*.
Because these equations depend only on frequency ratios, no extra factor
appears.

## Acknowledgments

Richard W. Aldrich served as editor.

This research was supported by National Institute on Deafness and Other Communication Disorders, National Institutes of Health grants R01 DC000273, R01 DC016318, and R01 DC008130 to J. Santos-Sacchi.

The authors declare no competing financial interests.

Author contributions: J. Santos-Sacchi conceived and performed experiments and wrote paper. W. Tan performed experiments. K.H. Iwasa did the modeling for the Appendix and wrote the Appendix and parts of the paper concerning this modeling.