Asymmetric Interplay Between K+ and Blocker and Atomistic Parameters From Physiological Experiments Quantify K+ Channel Blocker Release

Modulating the activity of ion channels by blockers yields information on both the mode of drug action and on the biophysics of ion transport. Here we investigate the interplay between ions in the selectivity filter (SF) of K+ channels and the release kinetics of the blocker tetrapropylammonium in the model channel KcvNTS. A quantitative expression calculates blocker release rate constants directly from voltage-dependent ion occupation probabilities in the SF. The latter are obtained by a kinetic model of single-channel currents recorded in the absence of the blocker. The resulting model contains only two adjustable parameters of ion-blocker interaction and holds for both symmetric and asymmetric ionic conditions. This data-derived model is corroborated by 3D reference interaction site model (3D RISM) calculations on several model systems, which show that the K+ occupation probability is unaffected by the blocker, a direct consequence of the strength of the ion-carbonyl attraction in the SF, independent of the specific protein background. Hence, KcvNTS channel blocker release kinetics can be reduced to a small number of system-specific parameters. The pore-independent asymmetric interplay between K+ and blocker ions potentially allows for generalizing these results to similar potassium channels.

At positive voltages, state B could possibly be confused with state M because of the similar dwell times. However, the rate constant of blocking k OB (blue circles in Fig. S2B) is by far higher than that of k OM (blue and black diamonds) even at low TPrA concentrations. This leads to dominance of B over M, making its determination insensitive to interference by the O-M gating. At negative voltages, only higher concentrations were analyzed. Here k OB is much faster even than k OF (orange circles in Fig. S2B). Under these conditions, the rate constants of blocking override those of the intrinsic gating. This makes the determination of the rate constants of blocking reliable, but those of O-F gating unreliable. Thus only the values of the O-F gating without TPrA are shown (black triangles in Fig. S2A,B). O-F gating is not of interest here, and any error in its determination would not significantly affect the values of the rate constants of blocking. In contrast, the error sum of the fits was still slightly sensitive to the values of the O-M gating (diamonds in Fig. S2A,B). However, even in this case, any error in the O-M rate constants would not significantly affect the values of blocking because k OB is much faster than k OM at both negative and positive voltages (for those concentrations analyzed in the respective voltage ranges).
However, two effects limit the range where the rate constants of blocking can be determined with high accuracy. First, due to their voltage dependence, k OB and k OM at 0.05 mM TPrA (blue) meet each other at about -80 mV. At more negative voltages, k OM starts to override k OB . However, the range of determination of blocker kinetics is extended with increasing blocker concentration, which increases k OB (Fig. 2C), thus moving the point of intersection to more negative potentials. Consequently, the curve of the concentration-independent O off k (Fig. 2B) could be composed from the results obtained at different TPrA concentrations.
Second, high blocker concentrations decrease the amplitude of the apparent single-channel current (Fig. S1B). This lowers the signal-to-noise ratio (the limiting factor for the analysis by extended beta distributions 1 ) as it merges the characteristic features of the amplitude histograms. Because of the voltage dependence of blocking (Fig. 2B,C), this causes a limit with increasing positive voltages, as indicated by the orange circles (5 mM TPrA) in Fig. S2A,B, ending at -80 mV.

The rate constants of the intrinsic gating in the presence of TPrA
The rate constants of the intrinsic gating of the Kcv NTS channel without TPrA are those of the 4-state sub model consisting of states O, F, M and S in Fig. 2A. They have already been determined at various cytosolic and external K + concentrations in the range of -160 mV to +160 mV 3 . Here, they are not of interest for the kinetics of blocking. Nevertheless, it may be interesting, whether the blocker also influences the single-channel current or the rate constants of the intrinsic gating.
There is no influence of the blocker on I true (Fig. S3A), the true open-channel current. This is the hypothetical open-channel current obtained from sojourns in the open state, which are not interrupted by gating or blocking events. It can be determined from extended beta distribution analysis as in Fig. S1 1 .
As mentioned above, the O-F gating cannot be determined with high accuracy in the presence of the blocker. Nevertheless, in the presence of TPrA, a change of these rate constants by a factor 2 to 3 decreased the quality of the fits. Thus, it can be assumed that these rate constants are also nearly voltage-independent like the control values in Fig. S2. Their exact values are not of interest here, and their choice does not influence the values of the rate constants of blocking. However, as found previously (e.g. ref. 6 ), their ratio is much more reliable than their individual values. The current I OF is calculated by means of the ratio I OF is the current that could be recorded with a setup that can resolve O-S and O-M gating, but not O-F gating. This was in a previous work achieved in high K + concentrations with the VC100 lownoise, high bandwidth patch-clamp amplifier of Chimera Instruments, LLC, New York, NY, USA 2 . Figure S3B shows that I OF is unchanged by TPrA. This is an indication that the O-F gating is not influenced by the blocker. The values of k OM (Figs. S2B and S3C) are not significantly different from those found without blocker. We have previously identified k OM as the rate constant of the closure of the selectivity filter, which is influenced by the ion occupation of the filter binding sites 3 . However, the reverse reaction k MO at negative voltages is decreased by the blocker leading to a stronger voltage dependence (Fig. S3D). Since the mechanism of k MO is currently still unknown, we will not speculate here on the mechanistic details of the influence of the blocker on k MO. The O-S gating showed higher values of the rate constants in the presence of TPrA. However, these rate constants are too slow for a meaningful evaluation by fitting amplitude histograms, so we refrain from showing or interpreting them. For the more adequate dwell time analysis, the measured time series are too short. Similarly, any statements regarding the O-S rate constants were also avoided in the previous study without blocker 3 .

Binding of the blocker
Since only O off k is in the focus of this investigation, the discussion of OB k (Fig. 2C,D) is given here. The rate constant of blocker binding, OB k , increases with the TPrA concentration (Fig. 2C). The a-priori expectation that OB k is proportional to the concentration of TPrA is verified in Fig Fig. S4A). The blue line represents the blocking, the measured current is drawn in red.
For simplicity, all blocking events are shown with dwell times of equal length representing the related inverse average rate constant k av in the dissociated state (D) and the blocked state (B) for the clear identification of the events. In reality (and in the fitting routine), the durations are determined by a random generator selecting the actual time of a transition from the exponential function of the probability of that transition calculated for the average rate constant k av . In the resulting amplitude histogram, the average of thousands of transitions are included leading to the average k av 7 .
The errors in determining k BO caused by the use of the model in Fig. 2A instead of the correct one in Fig. S4A are discussed now. The difference between the two models results from the fact that the model in Fig 7,8 . k OB , k OF and k OM are additionally determined in the fit by the number of transitions into the B, F or M state. At -160 mV, the ratios r FB = k OF /k OB and r MB = k OM /k OB are less than 0.1, and k OS is negligible. In order to model the measured blocking kinetics correctly, the fit has to use a k OB,fit = k OB -k OF -k OM , i.e. approximately 0.8 k OB . For the present analysis, this is of no consequence, because we are interested only in k BO, the dissociation in the open state.   ) and (b)). O-M gating is not discussed here, because the durations in the closed state M are much longer than those of B and thus the related events will not influence the fitted rate constants of blocking because they are assigned to M.
At the O-F transitions in the real system (Fig. S4B, black trace), three possible scenarios have to be discussed. On the left-hand side (label (a)), the transition of the dissociation of the blocker coincides with the F state. The apparent dwell time in B is prolonged, on average by 1.5 µs (half dwell time in F) as compared to 3 µs (average dwell time in B). This has a negligible effect on the determination of k BO , because the chance that this occurs (k OF /k OB ) is small. On the right-hand side (label (b)), blocking (O-B transition) occurs during the F state. Again, the apparent blocked state is prolonged, and this sojourn in state F is lost. The shortened open events next to the prolonged Bstates may have a minor influence on the averages value of the open time 1/k av considered above. As a summary, it can be stated that because of the numerical situation the influence of the effects at the O-F transitions on the determination of k BO can be ignored, since they are not larger than the fitting scatter.
At positive and low negative voltages, the rate constants of blocking become slower and are in the same range as k OF (Fig. S2). At these voltages, also the blocking events are of similar length as the dwell time in the open state of the O-F gating (Fig. S4C). Very similar scenarios as in Fig. S4B have to be considered here. The red trace shows the situation that blocking occurs during state F (label (c)). As in the red trace of Fig. S2B, the dwell time in state B is prolonged. It is still expected that it rarely occurs that B just coincides with the O-F transition (label (c)). More often is the occurrence of state F during the open (dissociated) or closed (associated) state of blocking (labelled by (d) and (e) in Fig, S4C). This has no effect on the dwell time in B. The effects on the dwell time in the open state are not of interest here, because we are interested only in k BO . In addition, the relative prolongation of the individual sojourn in B in case (d) is very small. On average, it is 1/k BO,fit = 1/k BO + ½ 1/k FO about 25 µs + 1.5 µs. The same holds for a scenario corresponding to Fig. S2B (a) where the blocker dissociation occurs during state F (not shown) and consequently the apparent sojourn in B is prolonged. The interference with O-M gating can be ignored because k OM is so small (Fig. S2B) that these events are extremely rare. Thus, Figs. S4B and S4C show that fitting on the basis of the model in Fig. 2A yields reliable results for k BO .
The rate constants of ion hopping, which are not affected by the presence of the blocker at 0 mV, are also not affected in the presence of transmembrane voltage.
The evidence for the absence of an effect of the blocker on the ion distribution from crystal structure analysis cited in the main text and by the 3D RISM calculations is obtained for a membrane voltage of 0 mV. The following arguments underpin that this conclusion also holds under conditions of a membrane voltage: The rate constants k ij in Fig. 3A are a constant term k ij,0 , which is multiplied by an exponential term in the case of the voltage-dependent rate constants (see also Eqs. S8 and S9, below). Thus, the ion occupation probabilities in the filter are completely determined by k ij,0 and the exponential term. The k ij,0 are unaffected by the blocker, an assumption which is confirmed by unchanged ion occupations in the crystal structures 19,22,23 (at 0 mV) and the 3D RISM calculations. This causes the occupation probabilities predicted by the flux model in Fig. 3A being unaffected by the blocker at 0 mV (inset in Fig. 3C), except that the "forbidden" state 2 is now absorbed by state 1.
The other question regarding the exponential term is whether the Eyring barriers imposing the voltage dependency represented by s and V ij are modified by the blocker. The basic message is that several mutations of the channel protein in the cavity and mutations affecting the anchoring of the selectivity filter do not modify the voltage dependency of the sub-millisecond channel closure 21,25 (k OM, Fig. S3C). Since k OM is determined by the ion occupations of states 3 and state 4 in the selectivity filter (Fig. 3A, and ref. 21 ), this indicates that the selectivity filter is so stable that external influences cannot change the voltage-dependent exponential terms of the rate constants of ion hopping.

Calculation of flux rates and occupation probabilities from the flux model
The cyclic model of ion transport in Fig. 3A has five different states of ion occupancy (clockwise numbered from 1 to 5). The open-channel current can be calculated from the rate constants k ij between these states. (The indices of the rate constants of the ion hopping model are given as numbers or by i,j if not specified, those of the gating process ( Fig. 2A) by letters, excluding i,j). Clockwise motion in the model represents outward flux, anticlockwise motion equals inward flux; the net single-channel current is the difference of the two fluxes. The equation for the fitting of measured IV curves in the previous work 3 has been set up with the so-called "arrow scheme" 9  e is the unit charge, the numbers 1-5 represent the states in the ion-hopping model (Fig. 4A). An arrow between the numbers i and j stands for the rate constant k ij or k ji depending on the direction of the arrow. Each row in square brackets gives the product of the rate constants. The matrices C5 i D (C5 means "cyclic 5-state model") are defined as follows: The rate constants in a row represented by an arrow are multiplied, and the products of each row are added. The indices are cyclic, i.e., the index following 5 is not 6, but 1. For the matrices C5 2 D to C5 5 D , the indices are rotated starting with the subscript of the matrix. For details, see ref. 9 . Ion activities a K (instead of concentrations, see ref. 3 ) are inserted into the binding reactions by 12 12,1 Kin s ij are the locations of the Eyring barriers and V ij the characteristic voltages (causing an e-fold increase) of the respective state transitions. The rate constants with index "0" are the rate constants at 0 mV. In previous work, the parameters of this model were determined for Kcv NTS under control conditions without blocker 3 . The parameters are shown in Table S1. Supplementary Table S1. Parameters for the model in Fig. 3A as obtained previously 3 from the analysis of IV curves and gating. The error is given as geometrical error (scatter factor, which gives the standard deviation as a factor). "calc" indicates that k 21 is not a fit parameter as it can be calculated from micro reversibility at 1 mM K + on either side and V = 0 mV: k 21 = k 12,1 k 23,0 k 34 k 45 k 51,0 / (k 32,0 k 43 k 54,1 k 15,0 ). The rate constants k 12,1 and k 21 are labelled by *, because these values are not well determined. Only the ratio k 12,1 /k 21 (last row) is reliable.

Parameter
Unit Geometric mean Scatter factor

Ion occupancy measures from 3D RISM (reference interaction site model) calculations
The thermodynamic constants are obtained from the molecule-solvent site pair distribution function g(r) on a grid r as a result of a solute-solvent interaction model taken from an established force field as in molecular dynamics (MD) simulations. Unlike MD, we quickly obtain from these calculations the equilibrium populations, i.e. the ion occupation probabilities under the many conditions that were used in the experiments. At the price of neglected solute flexibility this analysis offers reliable trends on ion occupations in the SF under noise-free controlled conditions 10 . The equilibrium occupancy of K + in the conductive pathway can be calculated by two thermodynamic routes: In a first scenario the channel is exposed to a finite concentration solvent (as in ref. 10 ; so-called solute-solvent (uv) calculations) which provides directly a partitioning constant K c (mass action law) for species i according to with c ch being the concentration within the channel, c bulk the reference pure solvent ion concentration, N ch the number of particles in the channel volume V ch , and ρ the bulk density. If the integration is performed over two-dimensional slices orthogonal to the z coordinate, the integration over z can be done cumulatively starting at a reference point to yield the total partitioning constant. In a second scenario, we can also start from the perspective of a reaction between an infinitely diluted (single hydrated) ion and the hydrated channel to derive a thermodynamic binding constant K according to ch 0 () from a so-called solute-solute (uu) calculation 11,12 where ρ 0 = 6.02214 10 -4 Å -3 corresponds to the formal standard concentration c 0 = 1 mol l -1 .

The influence of the blocker on the ion distribution in the selectivity filter
3D RISM calculations were performed with standard force field parameters (see main text) for the channel protein as well as with scaled carbonyl charges. TBA parameters are collected in Table  S2. The calculations with reduced filter charges were performed by halving the carbonyl atom charges while keeping the total charge of the protein constant. This leads to a partial charge of the carbonyl C atom from original 0.51 to 0.255 and the carbonyl O atom from -0.51 to -0.255 (Table S3). Radial integration was performed based on grid data sliced by applying the HOLE algorithm 13 .
Supplementary Table S2. TBA parameters used for the calculations, atom notation following the pdb file.