Integrative Approach with Electrophysiological and Theoretical Methods Reveals a New Role of S4 Positively Charged Residues in PKD2L1 Channel Voltage-Sensing

Numerical model-based simulations provide important insights into ion channel gating when experimental limitations exist. Here, a novel strategy combining numerical simulations with patch clamp experiments was used to investigate the net positive charges in the putative transmembrane segment 4 (S4) of the atypical, positively-shifted voltage-dependence of polycystic kidney disease 2-like 1 (PKD2L1) channel. Charge-neutralising mutations (K452Q, K455Q and K461Q) in S4 reduced gating charges, positively shifted the Boltzmann-type activation curve [i.e., open probability (P open)-V curve] and altered the time-courses of activation/deactivation of PKD2L1, indicating that this region constitutes part of a voltage sensor. Numerical reconstruction of wild-type (WT) and mutant PKD2L1-mediated currents necessitated, besides their voltage-dependent gating parameters, a scaling factor that describes the voltage-dependence of maximal conductance, G max. Subsequent single-channel conductance (γ) measurements revealed that voltage-dependence of G max in WT can be explained by the inward-rectifying property of γ, which is greatly changed in PKD2L1 mutants. Homology modelling based on PKD2 and NaVAb structures suggest that such voltage dependence of P open and γ in PKD2L1 could both reflect the charged state of the S4 domain. The present conjunctive experimental and theoretical approaches provide a framework to explore the undetermined mechanism(s) regulating TRP channels that possess non-classical voltage-dependent properties.

has been extended to explore the fusion between theoretical equations and biological systems 1, 2 . Representative examples involve the modelling of excitable cells and systems in the squid giant axon 3 and the use of a model of cardiac electrical activity 4 .
Transient receptor potential (TRP) channels are activated by a broad spectrum of physical and chemical stimuli generated from external sources and the interior of cells, and contribute to a variety of biological functions and the pathogenesis of many diseased states 5,6 . Although TRP channels are generally considered to be non-voltage-gated, activation of several TRP channels show intrinsic voltage-dependency 7 , including TRPV1 8-10 , TRPV3 11 , TRPV5 12 , TRPV6 13 , TRPM3 14 , TRPM4 15 , TRPM5 16 , TRPM8 17 , TRPC5 18 , PKD2L1 19 and TRPA1 20 . However, the mechanism(s) describing how these channels exert voltage-sensing and gating remains largely unknown.
The key structural components of classical voltage-gated ion channels are the voltage-sensing (S1-S4) and pore (S5 and S6) domains, which are linked by a short S4-S5 linker. The pore domain serves as an ion conduction pathway with a well-defined selectivity filter for specific ions. Importantly, S4 has a cluster of positively charged amino acids (AA) spaced every three residues, which are highly conserved and thought to play a key role in converting the change to the transmembrane potential to the conformational change of the pore domain that causes open-close transitions of the channels 21,22 . AA sequence alignment between PKD2L1 channels and classical voltage-gated channels, i.e. voltage gated calcium (Ca V ) channels, voltage-gated potassium (K V ) channels and voltage gated sodium (Na V ) channels, reveals that PKD2L1 has a few AA residues reminiscent of a voltage-sensor 23 . Indeed, recent structural analyses of K V 1.2 and PKD2 suggest that the S4-S5 linker interacts with the C-terminus of S6, and mechanical movement of this linker leads to constriction or dilatation of the channel pore 24,25 .
One common approach to investigate the voltage dependency is to construct the chord conductance-voltage relationship (G-V curve), which is expressed in the form of the Boltzmann equation. However, voltage-dependency of PKD2L1 causes a time-dependent shift and evaluation of its maximal activation, which only occurs at extremely positive potentials and shows varying extents of inactivation, is technically difficult to perform 19 .
In the present study, we undertake a complementary approach combining mutagenesis experiments and mathematical model-based simulations to more precisely understand the molecular mechanism underlying the voltage dependency of the PKD2L1 channel. To this end, we first characterised both wild-type (WT) and mutant PKD2L1 channels to compare their differences in voltage-dependent sensing/gating. The results indicated that the voltage dependence of these channels might be attributable to a similar voltage-sensing mechanism involved in classical voltage gated channels. We then attempted to faithfully construct a numerical model that reproduces these channel-gating properties based on experimental data obtained by voltage jump experiments. However, this modelling required an additional scaling factor to adjust the maximal conductance, which depends on the polarity of the membrane potential. The biophysical significance of this factor was further sought in terms of single channel recording. Finally, a possible structural basis for the observed voltage-dependence of PKD2L1 was explored by homology modelling based on the atomic structures of PKD2 and Na V Ab.

Result
The PKD2L1 channel has voltage dependency. Previous studies 19,26,27 have found that the voltage dependence of the PKD2L1 channel unstably varies from measurement to measurement. This instability in measurement arises partly because of slow voltage-dependent inactivation at relatively depolarised holding potentials unnoticeably adopted in previous studies, as well as the difficulty of attaining the maximal channel activation, which needs extremely strong depolarisations 19 . To more precisely analyse the voltage dependency of the PKD2L1 channel under stable conditions, we designed a multi-step pulse protocol; the cell was first held at −100 mV to eliminate inactivation and subjected to different levels of depolarisation (3 s, −90 to + 140 mV) to produce varying extents of inactivation. Then, a depolarising pulse of 50 ms to + 120 mV was applied to activate the channel (Fig. 1a). The extent of inactivation was evaluated as the magnitude of inward-going tail currents at −100 mV immediately after the depolarisation to + 120 mV (Fig. 1b).
The steady-state inactivation relationship of WT PKD2L1 evaluated in this way is well described by the Boltzmann equation with a half-maximal inactivation voltage (V 0.5 ) of + 42.2 ± 3.3 mV and slope factor k of + 20.3 ± 4.7 mV (Fig. 1c). The PKD2L1 channel did not appear to be fully inactivated even at very strong depolarisation levels (e.g., > + 100 mV), whereas this channel is almost 100% available at potentials as negative as −100 mV. The relevance of this protocol was confirmed by a very slow recovery process of the channel from inactivation, which occurs in tens of seconds ( Fig. 1d-f). Recovery from inactivation is > 100 times slower than the activation process ( < 100 ms; see the right inset of Fig. 1b and below). Thus, the above-used pulse protocol would not affect the extent of inactivation. Based on the above results, we next investigated the voltage dependency of the activation of the PKD2L1 channel by applying a set of short (40 ms) depolarising pulses (−100 to +300 mV with a 20 mV increment, 30 s interval, holding potential of −100 mV). PKD2L1-expressing cells exhibited rapidly growing currents in response to depolarising pulses (outward-going currents in Fig. 2a). The currents were slowly deactivated upon repolarisation (inward-going tail currents in Fig. 2a). Note that the pulse duration was sufficiently long to fully activate the channel, but short enough not to cause detectable inactivation. The current-voltage (I-V) relationships constructed from the instantaneous peak and steady-state amplitudes of tail currents are shown in Fig. 2b. The steady state activation curve (P open -V curve) calculated from them (for details, see the figure legend) is well fitted by the Boltzmann equation, yielding a V 0.5 of + 132.1 ± 1.3 mV and a k of + 47.2 ± 1.1 mV (Fig. 2c).
These data demonstrate that human PKD2L1 channels have intrinsic voltage-dependency 19, 27 that obeys classical statistical mechanics 28 like other voltage-dependent channels.
Mutations in S4 alter the voltage dependence. In voltage-dependent Na + , K + , or Ca 2+ channels, the probability of channel opening (P open ) is modified by the membrane potential. This is achieved through the S4 voltage sensor that detects changes in the transmembrane potential and transfers this energy to the pore domain to control channel gating 21 . Importantly, PKD2L1 has an AA sequence reminiscent of the S4 domain of a voltage-gated cation channel 23,29 . Alignment of the S4 AA sequence of PKD2L1 with the counterparts of TRPC3, and Ca V 1.2, K V 1.2 and Na V 1.1 reveals the presence of positively charged AA residues conserved in S4 (Fig. 3a). Notably, we found that the 2nd, 3rd and 5th Lys/Arg residues critical for the gating of voltage-gated channels are conserved in PKD2L1 (i.e., K452, K455 and K461; Fig. 3a). Therefore, to investigate the role(s) of these AA To determine the voltage dependence of inactivation, currents were evoked by a 40 ms test pulse to +120 mV after 3 s V h displacements (conditioning pulses) from −100 to +140 mV with 20 mV increments. (b) Representative superimposed current traces for steady state inactivation. (c) Steady state inactivation curves. Amplitudes of tail currents are normalised to the maximal peak tail amplitude and plotted against the membrane potential. Points are fitted with a Boltzmann function to yield the voltage for halfmaximal inactivation (V 0.5 ) of 14.7 mV (n = 6). (d) The time course of recovery of peak whole-cell current from inactivation. The pulse protocol was composed of two pulses, one 1 s pre-pulse from −100 to +140 mV, followed by a depolarising pulse to +140 mV with progressively prolonged durations from 0.1 to 20.1 s and then a 40 ms test pulse from −100 mV to +140 mV. The V h was at −100 mV. (e) Representative superimposed current traces for recovery of whole-cell current from inactivation. (f) The normalised value is the result of dividing the test peak currents (I) by the corresponding control peak current (I max ). Each symbol represents the mean value of five experiments. Data points are the means ± SEM. residues in PKD2L1 gating, we next neutralised them (Fig. 3a) by glutamine substitutions and tested their effects on the voltage sensitivity of PKD2L1. Western blot analysis confirmed robust expression of WT and the mutants K452Q, K455Q and K461Q in membrane fractions extracted from transfected HEK293T cells (Fig. 3b). The neutralised mutants K452Q and K455Q showed reduced expression levels (Fig. 3b). Results from confocal microscopy indicated that part of PKD2L1 immunoreactivity was localised in the plasma membrane of HEK293T cells transfected with either WT or the mutants (Fig. 3c).
To investigate the functional impact of charge-neutralisation, we tested the voltage dependency of mutant PKD2L1 channels using the same step-pulse protocol as above (Fig. 4a). All charge-neutralising mutants expressed in HEK293T cells formed functional channels. Amongst them, K461Q showed Boltzmann-type voltage-dependent activation similar to WT, but with a pronounced positive shift in V 0.5 ( Fig. 4b and c) and decreased voltage-sensitivity (i.e., reduced valence Z app ; Fig. 4d). These results strongly suggest that the charged state of the S4 region is crucial for PKD2L1 gating. The K452Q and K455Q mutants did not maximally activate even upon very strong depolarisation levels (up to + 300 mV), and thus estimation of V 0.5 and Z app values was not feasible.
To more explicitly estimate the number of charges involved in the activation of these channels, we next employed the 'limiting slope analysis' 30 . Specifically, the chord conductance obtained from macroscopic currents as shown in Fig. 4 was logarithmically plotted against the membrane potential and this relationship was subjected to linear regression to assess the effective gating charge Zδ (Fig. 5a). The value of Zδ, which was 0.51 ± 0.01 for PKD2L1-WT, decreased to 0.44 ± 0.08, 0.41 ± 0.08 and 0.34 ± 0.11 for the mutants K452Q, K455Q and K461Q, respectively (Fig. 5b). Taken together, these data indicate that neutralisation of positively charged AAs at positions 452, 455 and 461 effectively reduces the number of charges apparently associated with PKD2L1 activation by Mathematical reconstruction of macroscopic PKD2L1 currents. Simulation with mathematical models is often instrumental in defining the gating kinetics of ion channels over extreme voltage ranges where electrophysiological measurements are practically unfeasible 31 . Taking advantage of such an approach, we next constructed a mathematical model appropriately describing the voltage-dependent gating kinetics of the WT-PKD2L1 channel. The model was then modified to fit the voltage-dependence of mutant PKD2L1 channels that could not be fully activated even by extreme depolarisation (Fig. 5).
The time constants of activation and deactivation were obtained by voltage-jump experiments by fitting the time courses of current growth or decline with appropriate mathematical functions (Fig. 6a). Then, rate constants ( Fig. 4) calculated from the time constants and P open values were used as the initial estimates for numerical optimisation, constrained by microscopic reversibility and subjected to sensitivity analysis. After optimising the mathematical formulation of the rate constants for open and close transitions, macroscopic PKD2L1 currents activated by respective depolarising pulses were numerically calculated, according to eqs 1-5.
As shown in Fig. 6b and c, simulated currents reproduced reasonably the observed voltage-dependence of PKD2L1 currents in WT, as well as in the K452Q, K455Q and K461Q mutants. However, it was necessary to introduce an additional scaling factor (SF) to adjust the chord conductance for respective mutant channels, depending on the direction of ionic flow (Fig. 6a, right panel). As a result, G TRP in eq. 1 was re-written as G TRP = G TRP,max ·SF (V m ), where G TRP,max is the maximal conductance for each channel and SF(V m ) is a function of the membrane potential.
One plausible mechanism for this apparent voltage-dependence of maximal conductance is the intrinsic rectifying property or partial opening (or dilation) of an ion conductive pore [32][33][34] . To examine this possibility more directly, we next measured the unitary conductance (γ) of constitutively active WT and mutant PKD2L1 channels by means of single channel recording. As summarised in Fig. 7a and Supplementary Fig. S7, the reversal potential of the unitary current does not change for the mutants. However, the value of γ is larger for inward than outward directions in WT and the K461Q mutant ( Fig. 7c and Table 1). This disparity of γ, i.e., inward-rectifying property, is markedly attenuated in K452Q and K455Q mutants. In comparison with WT, the γ value is larger for the outward direction in K452Q but decreased for both directions in K455Q ( Fig. 7b and d). By incorporating these results in the mathematical model, we could precisely reproduce the time-dependent changes of WT and mutant PKD2L1 currents observed in patch clamp experiments (Fig. 7e).
The above results imply that the three positively charged K452, K455 and K461 residues in the putative S4 region of the PKD2L1 channel may differentially contribute to its voltage-dependent activation through at least two distinct mechanisms, i.e., alteration of P open and rectification of unitary ionic flow or the voltage-dependence of γ.

Discussion
PKD2L1 has been reported to form a nonselective cation channel when it is expressed in Xenopus oocytes [35][36][37][38][39] , planar lipid bilayers 40 or mammalian cells 19,26,27,41 . Since its maximum activation requires extremely strong depolarisation and inactivation proceeds time-dependently, it was technically difficult to accurately evaluate the voltage-dependent kinetics of PKD2L1 19 . However, by applying the optimised voltage-pulse protocol, we could clearly demonstrate three important features of this channel. First, the expressed PKD2L1 channel has constitutive activity and the membrane potential (i.e., holding potential) only modulates the channel by changing the extent of slow inactivation. Second, the voltage-dependent development/decline (or activation/deactivation) of the PKD2L1 channel by depolarisation/repolarisation follows fast kinetics of tens of milliseconds, while its inactivation/recovery requires more than 100-fold longer time courses (Figs 1 and 2). Third, the large unitary conductance of the PKD2L1 channel (162 pS) is inward rectifying, but this property is independent of divalent cation blockade or other conduction block mechanisms (Table 1). This result is consistent with those of previous studies 19,27 . These findings together suggest that: (1) membrane depolarisation per se is not mandatory to activate the PKD2L1 channel but strongly modulatory, and, when it prolongs, induces the slow inactivation of the channel; and (2) ion conduction through the pore of PKD2L1 may also be intrinsically voltage-dependent.
Comparison of PKD2L1 with other voltage-dependent channels. The PKD2L1 channel has constitutive activity around the resting membrane potential, but it is voltage-dependently enhanced, which follows a Boltzmann-type relationship with an apparent V 0.5 of + 132.1 ± 1.3 mV and a k of + 47.2 ± 1.1 mV (Fig. 2c). However, these values are largely variable from study to study. It seems likely that this discrepancy arises from variable extents of voltage-dependent inactivation involved therein. Indeed, by applying a sufficiently long and negative holding potential (−100 mV), we could successfully remove inactivation and reproducibly obtain the voltage-dependent gating parameters (Fig. 1).
In most isoforms of TRP channels, the midpoint of the voltage-dependent activation curve is very positive under normal conditions. This makes the channel activity very small at physiological membrane potentials, as observed in TRPV1 17, 42 , TRPV3 43 , TRPM3 44 , TRPM4 7 , TRPM5 45 and TRPM8 17, 42 channels. However, application of specific ligands or modulators often causes a dramatic negative shift of the activation curve, inducing substantial channel activities at physiological membrane potentials. This holds true for the PKD2L1 channel, in which the activation curve undergoes a significant negative shift by acidic extracellular pH, hypotonicity-induced cell swelling, or elevated extracellular Ca 2+ concentration 19,27 .
The voltage sensor domain of the voltage-gated K channel (K V ) is localised to the fourth transmembrane S4 bearing an array of positive charges, and conformational change is transmitted via the S4-S5 linker to gate the pore domain 21 . A homologous structure exists in weakly voltage-dependent (or modulated) channels such as TRP channels and some of them (e.g., TRPM8) appear to gate through a similar voltage-dependent mechanism 46 . In our present study, the neutralising mutations of positively charged AA residues K452Q, K455Q and K461Q in S4 weakened the voltage dependence of PKD2L1, accompanying a positive shift of the G-V curve (Fig. 4) with reduced gating charge (Figs 4d and 5). The total gating charge was previously assessed for several voltage-sensitive TRP channels by the limiting slope method. However, the obtained values were much smaller than those of voltage-gated channels (e.g., ~7 for Kv) and well below 1.0, e.g., 0.89 (TRPM8) 46 , 0.7 (TRPM4) 7 and 0.375 (TRPA1) 20 . We also obtained a similar value for the total gating charge of PKD2L1 (0.51; Fig. 5). Importantly, however, the gating charge was reduced efficiently by single mutations in the putative S4 domain of the PKD2L1 channel, i.e., K452Q, K455Q or K461Q (reduced by 0.07-0.17), which resulted in significantly reduced voltage sensitivity. This strongly indicates that these lysine residues may somehow contribute to the voltage sensing/ gating of the PKD2L1 channel.
The positive charges in the S4 region are usually formed as the cluster of arginine residues. The rationale behind it may be the energetically lower cost for arginines (by about 1.0 kcal mol −1 ) than lysines to transfer their diffuse positive charges from water to the lipid phase 47 . Because PKD2L1 has three uniquely lined lysines in S4, the activation of its voltage sensor may require an additional 3 kcal per one subunit. This could explain in part why the V 0.5 value of the PKD2L1 channel is more positive (by ~100 mV) than those of typical voltage-dependent channels.

Possible mechanisms involved in the voltage dependence of PKD2L1.
To gain more insights about the role of the predicted voltage sensor of PKD2L1, we performed homology modelling based on two atomic structures recently resolved by CryoEM: PKD2 (PDBID: 5T4D) 25 and Na V Ab (PDBID: 3RVY) 48 . The former channel has the highest degree of overall AA sequence identity (61.63%) to PKD2L1 among the polycystin subfamily. The structure of PKD2 in a lipid bilayer resolved at 3.0 Å is presumed to represent a closed or resting state characterised by two narrowed parts in the selectivity filter region (due to L641, G642, D643) and lower gate (due to L677 and N681) that restrict ionic flow. Na V Ab, a bacterial tetrameric voltage-gated Na channel,  appears to have similarities to PKD2L1 in the arrangement of critical AA pairs involved in electrostatic and hydrogen-bonding interactions during voltage sensor movement 21,22 , as shown by AA alignment ( Supplementary  Fig. S1). The resolved structure of Na V Ab provides a pre-open state that is undergoing partial transition from resting to fully-open configurations.
In the modelled structure of PKD2L1 based on PKD2, which is presumed to show the resting state, the side chains of K452 in S4 (K405 in PKD2) and D390 in S3 (D343 in PKD2) are located sufficiently close (4.3 Å) to interact with each other, and those of K455 in S4 (K408 in PKD2) and D390 in S3 (D343 in PKD2) are even in closer apposition (3.6 Å). The locations of these two AA pairs also appear to exist in Na V Ab (R2 and D80, R3 and D80). At the position of D390 in PKD2L1, two other adjacent negative AA residues E369 (6.4 Å) and E373 (5.1 Å) of S2 may also contribute to counter act K455 (K408 in PKD2). (Supplementary Fig. S2) Homology modelling by substituting the S2-S4 domains of Na V Ab with those of PKD2L1 or its mutant K452Q, which is presumed to represent a pre-activated state ( Supplementary Figs S3 and S4), gives a different picture. In the WT-PKD2L1 model, the upward-translocated and clockwise-rotated voltage sensor is stabilised via both electrostatic and hydrogen-bonded interactions of K452 (with V439: 3.44 Å), K455 (with S458: 2.81 Å) and K461 (with E413: 2.98 Å, D430: 2.97 Å, and N427: 2.90 Å). In the modelled structure of the K452Q-mutant ( Supplementary Fig. S4), however, these three lysine residues interact differently forming new pairs: K455 and S458 (3.04 Å), K461 and E413 (2.96 Å), and K461 and N427 (3.40 Å). The neutralised K452 by glutamine substitution no longer contributes to voltage sensor stabilisation. Probably because of these changes voltage-dependent translocation/rotation of the voltage sensor of the K452Q mutant would be greatly suppressed. In the absence of K452-mediated hydrogen bond formation, the rotation of the voltage sensor is restricted by ~12° compared with that of the WT. This suppression may cause a more positive shift of the G-V curve in the K452Q mutant than in WT.
Markedly attenuated voltage sensor movement in the K452Q mutant may also affect the inward rectifying property of unitary conductance (Fig. 7a). By analysing and comparing the pore structures of PKD2-based PKD2L1 and PKD2L1-Na V Ab hybrid models with the HOLE software, it is suggested that, compared with the resting state, the pore diameter of the pre-activated PKD2L1 channel may be greatly distended, in particular at the position of F502 (which corresponds to T224 in Na V Ab). This AA is located at the central cavity near the entrance to function as a selectivity filter and thus might influence the ionic flow. (Supplementary Fig. S5) Consistent with this concept, it has been suggested that a wide inner cavity of BK channels could provide a low access-resistance pathway allowing its high single channel conductance, and that F380 located at the inner cavity is a critical determinant of potassium permeation, mutation to Ile or Trp has been shown to reduce greatly the conductance 49 . Therefore, it is conceivable that voltage-dependent dilation of the central cavity (and translocation of F502) affects the inward rectifying property of the PKD2L1 channel. This structural change might be more prominent in the K452Q mutant, which shows an almost doubled unitary conductance for the outward direction. Obviously, further systematic mutational and structural studies are required to corroborate these possibilities.
In conclusion, we have found that positive charges in S4 critically affect the voltage-dependence of not only the open probability (P open ) but also the unitary conductance (γ) of the PKD2L1 channel by combined applications of patch clamp experiments and numerical model simulations. The present theoretical approach can be generalised to channels that possess non-classical voltage-dependent properties.

Methods
Molecular cloning and plasmid construction. Human PKD2L1 (GenBank accession No. NM_016112.2) was cloned from human brain, whole marathon-ready cDNA (BD Biosciences, San Jose, CA, USA) by applying a PCR-based approach designed to contain the untranslated leader sequence from the alfalfa mosaic virus 50 and consensus sequence from the translation initiation 51 , and was subcloned into the expression vector pCI-neo (Promega, Madison, WI, USA). PKD2L1 glutamine mutants were constructed from PKD2L1-pCI-neo using overlap extension PCR 52 . The primer pairs used for K452Q, K455Q, and K461Q are summarised in Table 2.
Cell cultures and cDNA expression. HEK293T     The free Ca 2+ concentration was 100 nM calculated with the CaBuf software 53 . Data was analysed using IGOR (Wavemetrics, Lake Oswego, OR, USA) and Origin (OriginLab Corp., Northampton, MA, USA) software. The tail currents measured during the current-voltage relationship were fitted using the Boltzmann equation: I/ I max = 1/(1 + exp((V 0.5 − V m )/k)), where V m is the membrane potential, V 0.5 is the potential to give a half-value of conductance, k is the slope factor and I max is the saturating current. The apparent channel open probability (P open ) was determined as G/G max , where G represents the steady state conductance and G max represents the saturating conductance measured at strongly depolarised potentials. Accordingly, we fitted activation curves using the Boltzmann function: where z is the gating charge (in elementary charge units: e o = 1.6 × 10 −19 C), V is voltage, V 0.5 is the voltage for half-maximal activation, k B is the Boltzmann constant (1.38 × 10 −23 J/K) and T is the absolute temperature. To obtain a gating charge, we used the limiting slope conductance method 28 . Briefly, the valence of the apparent single-gate charge for activation (Zδ) was calculated from the k a value, using the equation Zδ = k a k B T/e. The k a value was obtained from conductance versus voltage relationships.

Formulations of kinetic properties for PKD2L1 currents and simulation protocol.
To simulate the voltage dependence in PKD2L1, we first constructed the macroscopic PKD2L1 current (I PKD2L1 ) model, which is restricted to the activation-deactivation kinetics, because in a short period of time (<100 ms) slow transition to inactivation is almost negligible (Fig. 1). The I PKD2L1 was defined as  where G TRP is the maximal conductance (nS/pF), and x i is the activation variable in I PKD2L1 . The reversal potential of 11.64 mV is obtained from macroscopic PKD2L1 current-voltage relationships ( Supplementary Fig. S6). The voltage-dependence of the activation variable in the I PKD2K1 was calculated with the following first-order differential equation where i represents the channel type, where the channel type can be PKD2L1WT, K452Q, K455Q, or K461Q, x i is the state variable of I PKD2L1 for the channel types of i, x ∞,i is the steady-state value of x i , and τ ι is the time constant for x i . From our experimental voltage-clamp measurements, x ∞,i was fitted as a single Boltzmann function: ,WT m 0 5 B For x ∞,i in each type of channel, the half-maximal activation voltages and slope factors are listed in Table 3. Each time constant τ ι , for i = WT, K452Q, K455Q and K461Q, was derived from current activation (typically V m greater than 100 mV) and current deactivation experiments (typically V m less than 0 mV), and was expressed as functions of V m as follows: The values in eqs 4 and 5 were determined from our experimental data by nonlinear least-squares fitting. All simulations for calculating I PKD2L1 for each clamped voltage pulse with 40 ms duration were performed using the forward Euler method with a 0.01 ms time step. The activation variable (x i ) was set equal to zero as an initial condition. Simulations were encoded in C/C + + , and run on an IBM-compatible computer with the Intel ICC compiler version 15.0.1.

Bioinformatics.
Sequence alignments were done using the Clustal Omega multiple sequence alignment tool 54 with the following Uniprot codes: PKD2L1 (Q9P0L9), PKD2 (Q13563), TRPC3 (Q13507), Ca V 1.2 (Q13936), K V 1.2 (P16389), Na V 1.1 (P35498) and Na V Ab (A8EVM5). Structural alignments and figures were generated using SWISS-MODEL 55 . The following PDB entries were used for structural comparisons: Na V Ab (3RVY) and PKD2 (5T4D). Inner pore measurements were calculated using the HOLE suite of programmes 56 and visualised with the Visual Molecular Dynamics (VMD) 57 or PyMOL 58 programs. Hydrogen bonding analysis was used using the Hydrogen Bond Calculation program 59 .
Statistical analyses. All data are expressed as means ± SEM. We accumulated the data for each condition from at least three independent experiments. The statistical analyses were performed using Student's t-test. A value of P < 0.05 was considered significant. Data availability. All relevant data are included within the paper.