State dependent effects on the frequency response of prestin’s real and imaginary components of nonlinear capacitance

The outer hair cell (OHC) membrane harbors a voltage-dependent protein, prestin (SLC26a5), in high density, whose charge movement is evidenced as a nonlinear capacitance (NLC). NLC is bell-shaped, with its peak occurring at a voltage, Vh, where sensor charge is equally distributed across the plasma membrane. Thus, Vh provides information on the conformational state of prestin. Vh is sensitive to membrane tension, shifting to positive voltage as tension increases and is the basis for considering prestin piezoelectric (PZE). NLC can be deconstructed into real and imaginary components that report on charge movements in phase or 90 degrees out of phase with AC voltage. Here we show in membrane macro-patches of the OHC that there is a partial trade-off in the magnitude of real and imaginary components as interrogation frequency increases, as predicted by a recent PZE model (Rabbitt in Proc Natl Acad Sci USA 17:21880–21888, 2020). However, we find similar behavior in a simple 2-state voltage-dependent kinetic model of prestin that lacks piezoelectric coupling. At a particular frequency, Fis, the complex component magnitudes intersect. Using this metric, Fis, which depends on the frequency response of each complex component, we find that initial Vh influences Fis; thus, by categorizing patches into groups of different Vh, (above and below − 30 mV) we find that Fis is lower for the negative Vh group. We also find that the effect of membrane tension on complex NLC is dependent, but differentially so, on initial Vh. Whereas the negative group exhibits shifts to higher frequencies for increasing tension, the opposite occurs for the positive group. Despite complex component trade-offs, the low-pass roll-off in absolute magnitude of NLC, which varies little with our perturbations and is indicative of diminishing total charge movement, poses a challenge for a role of voltage-driven prestin in cochlear amplification at very high frequencies.


Results
modeled prestin and its NLC as a piezoelectric process based on first principles. NLC can be deconstructed into real and imaginary components 23 , since it is defined as the frequency-dependent membrane admittance (Y* m (ω)) divided by iω, where ω = 2πf and i = √ −1, providing a real capacitive component and an imaginary conductive component. On removing stray capacitance (by subtraction, see "Methods" section) and linear capacitance (by Boltzmann fitting, see "Methods" section) from C * m (ω), prestin-associated complex NLC(ω) is obtained. With his PZE model, Rabbitt predicted that the imaginary component of complex NLC should increase over frequency, as the real component decreases. We previously reported on the real and imaginary components of complex NLC 7 and concluded that the imaginary component was small and essentially frequency independent. However, we did not fit both complex components to Boltzmann functions to unambiguously extract details of those components. In that paper, we focused our attention on the absolute magnitude of complex NLC, i.e., Re(NLC) 2 + Im(NLC) 2 , which is comparable to all previous measures of OHC/prestin NLC.
Here we explore in detail both components in 25 macro-patches of the OHC lateral membrane, first under zero pipette pressure, namely in the absence of externally applied membrane tension. For the 3D plots, black dots depict average data, and magenta dots depict standard error (+ SE). The multicolored shading is provided by the interp surface plot function in Matlab, and the red and blue solid lines depict Boltzmann fits (see "Methods" section) at selected frequencies. Figure 1 shows that indeed, Rabbitt's modelling prediction concerning our data that he references is borne out. There appears to be a trade-off between real (Fig. 1A) and imaginary (Fig. 1B) component peak magnitude across frequency, especially revealed in the extracted 2-state fits of NLC (Fig. 1C). Superimposed lines in green and magenta are the PZE model responses at V h from Rabbitt 21 , which shows general agreement within this bandwidth. Figure 1D shows the absolute magnitude of NLC that continuously decreases across frequency. Thus, while there are reciprocal trade-offs between the real and imaginary NLC components, the trade-off is not fully reciprocal; that is, the overall kinetics of prestin (governed by a host of molecular loads) slows. Figure 2 (top panel) highlights the relationship between real and imaginary components of NLC. Here we plot extracted Q max from fitted NLC components across frequency. The trade-off in component magnitude shows that at a particular frequency (F is ), the two intersect, and above that frequency the imaginary component will dominate. Rabbit 21 modeled that the imaginary component will eventually reduce to zero after peaking, but here (because of our limited frequency interrogation range) we use F is to explore influences on NLC frequency response (see below). Figure 2 (middle panel) also shows that the voltage sensitivity (z) of the two components decreases in parallel across frequency, possibly indicating an increasingly impeded sensor charge movement within the membrane plane. In other words, voltage-induced conformational switching of prestin is depressed as frequency increases. www.nature.com/scientificreports/ Another interesting observation that this fitting exercise exposes relates to the offset (ΔC sa ) in capacitance at negative voltages, which is uniformly observed in all previous studies reporting on OHC NLC. Indeed, when we first observed this offset and developed the method used to fit the response 24 , we thought it was solely due to surface area alterations as prestin changed states from expanded to contracted. Surface area change in prestin embedded membrane was first reported by Kalinec et al. 25 . However, the magnitude of ΔC sa is greater than that expected for surface area changes alone in prestin. Thus, it likely includes changes to overall dielectric properties of the membrane, as well. Figure 2 (bottom panel) compares the offsets in real and imaginary component fits by plotting the difference between the two. The magnitude of ΔC sa is little affected across frequency, and, in fact, it can be seen in Fig. 1A,B that the offset is virtually absent in the imaginary component of NLC. Thus, the charge movement associated with the offset is solely capacitive in nature.
Do these real and imaginary component trade-off observations necessarily indicate that prestin is piezoelectric? Figure 3 shows that the same basic behavior observed in our data is recapitulated by a simple 2-state kinetic model with voltage-dependence only, i.e., both real and imaginary NLC components are observed, with trade-offs in magnitude across frequency.
This simple 2-state model for charged prestin particles possesses an expanded (X) and compact (C) state, with the population of prestin molecules redistributing during changes in membrane voltage.
The forward and backward rates (α, β, respectively) are governed solely by changes in membrane voltage (V m ) about a characteristic potential, V h , where particles (charges) are distributed equally on either side of the membrane field. z denotes the voltage sensitivity or unitary particle charge (e -X distance travelled perpendicular to the membrane field). F, R and T have their usual meanings.
In Fig. 3A we show results from simulations with both rate constants (α 0 = β 0 ) equal to 45,000/s, and in Fig. 3B equal to 90,000/s. V h was set to − 40 mV. These rate constants provide characteristic time constants (tau = 1/(α 0 + β 0 )) for charge movement in the time domain at the potential V h . In the frequency domain, the magnitude response is a single Lorentzian (equivalent to a first-order RC response, whose transfer function displays the www.nature.com/scientificreports/ same complex behavior); the cut-off frequency at V h is defined as F c = 1/(tau × 2π), where the response falls to 0.707 of the initial low frequency magnitude (or correspondingly where a phase lag of 45 degrees develops). In Fig. 3A,B, in addition to the complex component plots, the absolute magnitude of NLC at V h is shown with red lines, and the red circles mark the measured cut-off frequency, which agrees with that calculated from the utilized rate constants. The real component of complex NLC is lower pass than the absolute value. Thus, it is the absolute value of NLC that appropriately characterizes the model's frequency response, and also that of the biophysical data. We have previously used the absolute magnitude of complex NLC to characterize prestin frequency response in macro-patches 7 . Additionally, in Fig. 3 we illustrate the validity of our method of subtraction of a nonlinear DC conductance from our patch admittance data prior to subsequent analysis. With a nonlinear DC conductance included in the model (Fig. 3C), like that we find in membrane patches, the imaginary component at low frequencies is distorted. Following removal (Fig. 3D), complex NLC is equivalent to the model without such conductance (Fig. 3A). In the 2-state model, overall NLC behavior arises due to the exponential voltage-dependent transition rates between prestin's expanded and contracted states. Altering the rate constants will shift the cut-off and the component magnitudes (compare Fig. 3A,B), like alterations of the time constant for an RC response. There is no intrinsic piezoelectric coupling between voltage and tension embodied in this simplest 2-state model, nor in a first order RC circuit. But, without the delays introduced in those conformational transitions (i.e., in a simple ultra-fast 2-state model), no imaginary component exists within our interrogation bandwidth. Thus, the requirement of piezoelectricity in prestin is not firmly established by the trade-off in real and imaginary components of NLC that we find across stimulus frequency.
The data we have presented thus far are from averaged NLC, but we know that NLC V h can vary among cells (and patches) due to a variety of reasons, e.g., membrane tension 15,17,18 , anions [26][27][28] , temperature [29][30][31] , initial voltage conditions 32,33 , and membrane cholesterol content 34,35 . Indeed, prestin likely presents characteristics that vary along the length of the cell 20 , where functional expression (e.g., V h ) can be non-homogenous 36,37 . Consequently, we sought to characterize the dependence of NLC real and imaginary component behavior on prestin's initial  Figure 1A,B indicate that ΔC sa is largely absent in the imaginary component. www.nature.com/scientificreports/ with both rate constants equal to 90,000/s. In addition to the complex component plots, the absolute magnitude of NLC at V h is shown with red lines. The rates provide characteristic time constants (tau = 1/(α 0 + β 0 ); 11.1 μs and 5.6 μs, respectively) for charge movement in the time domain at the potential V h . In the frequency domain, the cut-off frequency at V h is defined as F c = 1/(tau × 2π), 14.3 and 28.6 kHz, respectively. The filled red circles mark the cut-off frequencies.
(C) Same model as in (A), but with an additional nonlinear DC conductance (designed to be like that in our patch data) which distorts the low frequency response of the imaginary component. (D) After removing the DC conductance with the approach detailed in the "Methods" section, the response is essentially the same as in (A), where no DC conductance is included, confirming the validity of our approach to remove any residual DC conductance in our patch data. www.nature.com/scientificreports/ state; namely, as typified by initial V h , which reports on the distribution of proteins in either the expanded or contracted states. First, we categorized NLC data into groups possessing V h values above and below − 30 mV.   www.nature.com/scientificreports/ Membrane tension is well known to shift V h of OHC NLC. To investigate the influence of tension-induced V h shift on real and imaginary components of NLC, we altered the tension delivered to the macro-patch membrane (Fig. 7). In 7 patches we were able to incrementally alter pipette pressure (thus, membrane tension) from 0 to − 2, − 4, − 6, − 8, and − 10 mm Hg (0 to − 1.22 kPa). Though substantial shifts in average V h occur (from − 42.8 ± 8 to − 9.1 ± 11.8 mV), only slight changes in the real or imaginary components of complex NLC are readily visible. However, upon plotting intersecting frequencies (lower panels), clear changes in F is are obvious. Increasing tension causes a shift in F is to higher frequencies. For 0 mmHg, F is is 12.9 kHz, whereas − 10 mmHg pressure shifts F is to 28.3 kHz. Does initial V h factor into these responses?
As with our V h categorization observations made above, we sought to determine the influence of initial V h on tension effects. To this end, we categorized the patches into groups with initial V h above and below − 30 mV. First, Fig. 8 illustrates that tension susceptibility depends on initial V h , with the positive group (red symbols; 4.28 mV/mmHg) showing V h shift sensitivity nearly twice that of the negative group (blue symbols; 2.54 mV/ mmHg). For all averaged patch responses (magenta symbols), the response was intermediate with standard errors the largest, as predicted from the categorization results.
In Fig. 9, we plot average NLC for the negative initial V h group. Figure 9A shows results at 0 mmHg and Fig. 9B shows results at − 10 mmHg. Differences are apparent in the frequency responses, with Fig. 9C highlighting the influence of increasing membrane tension on the intersection frequency, F is . Increasing tension induces a shift in F is to higher frequencies. For 0 mmHg, F is is 11 kHz, whereas − 10 mmHg pressure shifts F is to 16.2 kHz.
A similar analysis is shown in Fig. 10 for the positive initial V h group. Surprisingly, in this case, increasing tension induces a shift in F is to lower frequencies. The results for both groups are summarized in Fig. 11A. This plot shows that the two initial V h group responses converge towards a common F is at − 10 mmHg, namely about 16 kHz. It may not be a coincidence that this pressure value is close to the turgor pressure (~ 1 kPa) of the native OHC 38 . Finally, in Fig. 11B we plot the absolute magnitude of NLC for the two initial V h groups. The negative group shows a very similar frequency response regardless of tension, and is in line with our previous observation on the immutable frequency response of the absolute magnitude of complex NLC during tension changes 7 . However, the positive group shows altered frequency responses, with increases in tension slowing the frequency response. Our data thus indicate an interaction between initial prestin state and tension sensitivity.

Discussion
OHC NLC is frequency dependent and low pass. We first demonstrated this in guinea pig OHCs using AC voltage chirp stimuli to assess the real component of complex NLC nearly 30 years ago 13 . Over the years this has been confirmed across species 9,39-42 . Indeed, combining measures across a number of studies, we arrived at a collective estimate of NLC (absolute magnitude) frequency response that led us to suggest that prestin activity could not drive eM to sufficiently influence cochlear mechanics at very high acoustic frequencies (> 50 kHz), where cochlear amplification is expected to work best 7 . Recently, Rabbitt 21 , expanding on earlier modelling investigations 16,[43][44][45] , has modelled OHC NLC as a piezoelectric process whose imaginary complex component takes on special significance, namely, signifying power output associated with prestin charge displacement. He modelled that as the real component of NLC decreases across frequency, the imaginary component increases-a frequency-dependent, reciprocal trade-off in magnitudes. Thus, he suggested that dielectric loss in voltage-driven prestin charge movement is indicative of considerable influence at high frequencies, thereby overriding our suggestions that prestin is limited in its high frequency effectiveness 7,46 .
Here we find experimentally that prestin's complex NLC displays a partially reciprocal trade-off between real and imaginary components across interrogating frequency that follows predictions based on the PZE model  Fig. 1   . Shift in V h due to tension versus initial V h . This plot shows that sensitivity to tension differs depending on initial V h . Upon categorization of initial V h with values greater than and less than − 30 mV, differences in sensitivity to tension are exposed. In those patches with initial V h greater than − 30 mV (red symbols; − 23.0 ± 2.6 mV; 4.28 mV/mmHg; n = 3), sensitivity is nearly double that of the group with initial V h less than − 30 (blue symbols; − 57.6 ± 5.4 mV; 2.536 mV/mmHg; n = 4). For comparison, the relationship for all average patches is shown (magenta symbols), where a sensitivity to tensions is 3.3 mV/mmHg. This latter relationship is derived from data in Fig. 7. Linear fits provide the slope sensitivity. www.nature.com/scientificreports/ to piezoelectricity in prestin, and whether other predictions of the PZE model correspond to the observed prestin behavior that we find remains to be seen. To be sure, we are not claiming that prestin is not piezoelectric-like, since we have substantially contributed to the experimental evidence that it is, but only that non PZE models can generate real and imaginary components of NLC. For decades, we and others 9,13,39,44,48 have suggested that NLC and eM behavior are governed by the conformational kinetics of OHC motor (prestin) transitions, in line with traditional biophysical concepts of voltagedependent protein behavior. Indeed, when we initially realized that prestin kinetics were influenced by chloride ions 49,50 , we suggested that delays introduced by stretched exponential transition rates in prestin would impact the phase of eM. Subsequently, we found that an eM phase lag (re voltage) develops across frequency and this could be influenced by chloride anions 51 . It is likely that the frequency dependent imaginary component of NLC we observe here correlates with that out of phase mechanical response. Thus, we agree with Rabbitt 21 that the imaginary component of NLC may have special significance in prestin function, regardless of whether prestin works as a piezoelectric device or not. Nevertheless, our data show that absolute magnitude of NLC (representing total charge, including in and out of phase components) decreases as a power function of frequency, being 40 dB down at 77 kHz 7 . This is a minuscule fraction of voltage-driven prestin activity that exists at low frequencies.
Based on PZE models 16,21,47 , mechanical loads can influence the frequency response of prestin. We agree that NLC and eM are susceptible to mechanical load, but the frequency response of whole-cell eM is slower than that of NLC, and likely corresponds to both cellular (external) and molecular (intrinsic) load components 8 . Thus, with minimal influence of external loads, we view our macro-patch data as providing the best estimate of prestin's intrinsic conformational switching limit (including all influential molecular impedances), as originally espoused by Gale and Ashmore 39 .
Characteristics of OHC NLC have been known for some time to depend on initial conditions; for example, V h depends on initial holding potential and anion binding is state-dependent 33,52,53 . In an effort to determine whether complex NLC shows sensitivity to initial conditions, we categorized our data into classes that had initial V h values above and below − 30 mV. We found effects of initial conditions on the frequency response of real and imaginary components, where a metric of this relationship, F is , the intersection frequency of real and imaginary magnitude components varies with initial V h . As initial V h shifts positively, F is increases nonlinearly (see Fig. 6).
We also found initial V h effects on the influence of static membrane tension on F is . Whereas the group with initial V h positive to − 30 mV shows a decrease in F is with increasing membrane tension, the opposite is found in the group with initial V h negative to − 30 mV. At − 10 mmHg (1.22 kPa), each group's F is intersects near 16 kHz (see Fig. 11A). Interestingly, this pressure value is close to the turgor pressure (about 1 kPa) of the native cylindrical OHC 38 .
Lastly, we found that the group with initial V h negative to − 30 mV shows little variation in the frequency response of the absolute magnitude of NLC with membrane tension, similar to what we observed previously 7 . www.nature.com/scientificreports/ However, the frequency response of the group with initial V h positive to − 30 mV shows a decrease in the frequency response as membrane tension increases. From all these observations, it appears that the frequency response/magnitude of the imaginary component of NLC is mainly sensitive to initial V h . Though adherence to a minimum phase system response might predict a complimentary relationship between real and imaginary components, viscoelastic non-minimum phase systems are known in biology 54,55 . Consequently, our data indicate that the state of prestin, characterized by initial V h , influences frequency relationships between components of complex NLC, and thus, according to Rabbitt 21 , the power output of prestin activity. Nevertheless, changes in the frequency response are relatively small with our perturbations. The fact that the individual components of complex NLC are differentially susceptible indicates that frequency-dependent phase differences occur between elements of sensor charge movement, possibly indicating that widely distributed sensor charge residues 56 may independently interact within local intra-protein, nanoenvironments. Supporting this view, recent cryo-EM determined structural features of the mechano-sensitive channel MscS point to differential interactions of the protein with lipids surrounding and within the protein itself 57 . Thus, the study of complex NLC of our sensor charge residue mutations 56 in HEK cell macro-patches, may assign particular residues responsible for the complex component responses. For example, we may find that certain mutations may remove the imaginary component without altering the real component.
As we noted above, there are several factors that influence V h . In the case of membrane tension, what could govern V h ? There are numerous descriptions of membrane proteins/channels that are influenced by membrane tension 58 , with some possessing voltage-dependence; yet, interestingly, none have been deemed piezoelectric, per se. Rather, standard biophysical influences of load on a protein's conformational state are espoused. For example, some Kv channels are voltage and tension sensitive, and simple models where changes in the equilibrium constant for channel opening, or equivalently the ratio of forward to backward transition rates, can account for tension effects on channel activity 59 . What other influences could there be in the absence of direct PZE effects or direct biophysical mechanical influences on prestin?
Several ideas come to mind. Perhaps the degree of cooperativity among prestin units alters with membrane tension. We recently provided evidence for negative cooperativity in prestin that was related to the density of prestin within the membrane 60 . Additionally, we previously modelled initial voltage influences on NLC as resulting from OHC molecular motor-motor (prestin) interactions 33 . For Kv 7.4 channels, cooperativity has been found to impart mechanical sensitivity to the channel in OHCs 61 . Another possibility is the presence of a conductive Figure 11. Summary of initial V h influence on tension effects. (A) The direction of change in F is is dependent on initial V h . Analyses were made with stray capacitance removal using either the + 160 mV (circles) or − 160 mV (asterisks) holding admittance. Results are identical, as expected. The two initial V h group responses converge until a common F is vs. tension is reached at − 10 mmHg, namely about 16 kHz. (B) Plot of the absolute magnitude of NLC for the two initial V h groups. The negative group shows a very similar frequency response regardless of tension, but the positive group shows altered frequency responses, with increases in tension slowing the frequency response. www.nature.com/scientificreports/ element influencing the interaction between voltage sensor charge and the membrane field, where that element alters with tension. Under this circumstance, the voltage sensed by the prestin charge may change during alterations in tension, leading to apparent V h shifts relative to voltage clamp commands. Could this be a viscoelastic coupled conductive element, thus providing time dependent changes in voltage sensed? We have observed multi-exponential time-dependent behavior in NLC at fixed voltages 32,33 . Is G metL , the tension-dependent prestin leakage conductance we have identified 62 , the conductive component? We are currently testing complex NLC in prestin mutants that have reduced G metL conductance. Finally, the shift in V h could additionally result from changes in the membrane surface charge or dipole potential that may accompany changes in membrane tension. Warshaviak et al. 63 have found that physiologically relevant changes in membrane tension could shift that potential by tens of millivolts. Such effects could produce changes in the distribution of voltage-dependent protein conformations that would be evident as altered V h , despite effective voltage clamp. Interestingly, changes in temperature could conceivably alter the membrane dipole (in addition to many other properties of the membrane), since electrical breakdown in membrane is temperature dependent 64 . Thus, our observation of shifts in V h due to temperature [29][30][31] could also be independent of direct action on prestin. Of course, in this case of these "charge screening" effects, alterations in transition rates would not underlie V h shifts, so this is unlikely to fully account for prestin's tension responsiveness since we do find changes in the frequency response of real and imaginary components of complex NLC. Hence, transition rates are indeed altered.
To summarize our observations, we find that prestin's complex NLC displays partial reciprocal trade-offs in real and imaginary component magnitude across frequency. The trade-off is prestin state-dependent in that the frequency response of the components alter with initial V h and membrane tension. However, these observations do not nullify our initial observations 7,46 that absolute complex NLC, signifying total charge moved, decreases precipitously as a power function of frequency; thus, prestin charge displacement that correlates with electromotility 9 is expected to have limited physiological influence at very high frequencies in excess of 50 kHz.
Some final comments on the OHC's ability to drive cochlear amplification at very high frequencies in a hypothetical cycle-by-cycle manner are in order. The membrane RC filter, which limits high frequency AC receptor potential generation, has been considered a potential problem for decades 65,66 , and while there are a number of publications that have suggested that the problem could be alleviated 17,[67][68][69] , none have been directly tested in vivo. Thus, the RC problem may remain, and indeed arguments have been made for its effect on eM measured in vivo 10 . Both the potential RC problem and the low-pass nature of prestin charge movement pose serious obstacles for proposed theoretical mechanisms that offer to circumvent an OHC speed limit 8 .

Methods
Detailed methods, including specifics of our voltage chirp stimulus protocol and extraction of complex NLC from patch admittance, are available in Refs. 7,46 . Briefly, extracellular solution was (in mM): NaCl 100, TEA-Cl 20, CsCl 20, CoCl 2 2, MgCl 2 1, CaCl 2 1, Hepes 10, pH 7.2. Experiments were performed at room temperature. Extracellular solution was in the patch pipette. On-cell macro-patches on the guinea pig OHC lateral membrane were made near the middle of the cylindrical cell, where prestin resides at high density 70,71 . Pipette inner tip size was about 3.5 μm, series resistance (R s ) estimated to be below 1 MΩ, and seals at 0 mV near 5 GΩ (see Ref. 7 ). Under these conditions, R s effects are minimal 7 and were not corrected for in this report. Furthermore, in this report we make relative assessments of NLC among groups of macro-patches. Chirp voltage stimuli (10 mV pk) were superimposed on step holding potentials from − 160 to + 160 mV, in 40 mV increments. As detailed previously 7 , the FFT derived admittance at + 160 mV, where NLC is absent, is subtracted from admittance at all other step potentials, thereby removing stray capacitance. Additionally, because we focus on effects of V h on complex NLC behavior, we confirmed that subtractions for stray capacitance using either + 160 or − 160 mV responses give the exact same Boltzmann fits, as expected since stray capacitance will not influence prestin charge movement/ NLC. A residual DC nonlinear conductance was also removed. DC conductance was determined from the DC component of FFT current response at each stepped voltage. Conductance, ΔI(0)/ΔV hold , was gauged by spline interpolating between each step voltage response, and differentiating digitally. Since, such an approach can have untoward effects at the end point voltage extremes, we fit the conductance linearly between 0 and 396 Hz (conductance at each frequency determined from the real components), and used the resultant fitted conductance at zero Hz for subtraction of the real components of admittance across frequencies. Figure 3 illustrates, in a simple 2-state kinetic model with solely voltage-dependent transition rates, the necessity of such nonlinear DC conductance subtractions to validly extract the imaginary component of NLC at low frequencies. The model is fully described in the "Results" Section. Patch membrane tension was imposed by changing pipette pressure. All data collection and analyses were performed with the software programs jClamp (www. sciso ftco. com) and Matlab (www. mathw orks. com). All means and standard errors (SE) are from individually analyzed patch data. Plots were made in Matlab.
In order to extract Boltzmann parameters, capacitance-voltage data were fit to the first derivative of a twostate Boltzmann function 13,39 , with an component of capacitance that characterizes sigmoidal changes in specific membrane capacitance 24,72 . We refer to this as the "C sa fit" in text and figures.
where b = exp −ze V m −V h k B T , C sa = �C sa (1+b −1 ) . Q max is the maximum nonlinear charge moved, V h is voltage at peak capacitance or equivalently, at halfmaximum 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. C sa is a component of capacitance that www.nature.com/scientificreports/ characterizes sigmoidal changes in specific membrane capacitance, with ΔC sa referring to the maximal change at very negative voltages 24,72 . Qmax is the maximum nonlinear charge moved, Vh is voltage at peak capacitance or equivalently, at halfmaximum charge transfer, Vm is membrane potential, z is valence, Clin is linear membrane capacitance, e is electron charge, kB is Boltzmann's constant, and T is absolute temperature. Csa is a component of capacitance that characterizes sigmoidal changes in specific membrane capacitance, with ΔCsa referring to the maximal change at very negative voltages 24,72 .
For some data, a power fit as a function of frequency (f) was performed 46,73 .
where C 0 is the asymptotic component, and a and b control the frequency response.