The persistent question of potassium channel permeation mechanisms

Potassium channels play critical roles in many physiological processes, providing a selective permeation route for K+ ions in and out of a cell, by employing a carefully designed selectivity filter, evolutionary conserved from viruses to mammals. The structure of the selectivity filter was determined at atomic resolution, showing a tight coordination of desolvated K+ ions by the channel. However, the molecular mechanism of K+ ions permeation through potassium channels is still not resolved, with structural, functional and computational studies often providing conflicting data and interpretations. In this review, we will present the proposed mechanisms, discuss their origins, and critically assess them against all available data. The general properties shared by all potassium channels are introduced first, followed by the introduction of two main mechanisms of ion permeation - soft and direct knock-on. Then, we will discuss computational and experimental studies that shaped the field and provided crucial insights. A special focus is placed on Molecular Dynamics simulations, that heralded the field by providing mechanistic and energetic aspects of K+ permeation, but at the same time created long-standing controversies. Further challenges and possible solutions are presented as well.


INTRODUCTION
Potassium channels (K + channels), the most widely distributed ion channels [1] , are transmembrane proteins found in almost all organisms [2] and several virus families [3,4] . K + channels enable the flux of K + ions across plasma and organelle membranes down the electrochemical gradient.
Potassium channels are characterized by high transport rates (~10 8 K + ions per second, or 1 ion per 10 ns, near diffusion-limited rates), exquisite K + selectivity, especially over other monovalent ions such Na + (~100-1000 times more selective for [5] ) and intricate gating mechanisms, that allow for current regulation by opening and closing the channel. K + currents mediated by potassium channels establish the membrane potential, and terminate action potentials in electrically excitable cells such as neurons and cardiac myocytes.
Structurally, a given K + channel can be divided into a mandatory pore-forming domain and optional regulatory domains (Figure 1 A). The pore-forming domain overall structure is conserved in all potassium channels, as its primary function is to permeate K + ions. The regulatory domains are responsible for sensing external stimuli, such as voltage or ligand binding, and allow the channel to gate in response to them, although it is worth noting that pore-forming domain-only K + channels exist and are capable of gating as well. The variety of regulatory domains and gating mechanisms gives rise to the large size of the potassium channel [6] . However, the most fundamental function of K + channels -rapid and selective permeation of potassium ions -is expected to occur in a similar fashion in all K channels, and its molecular underpinnings are the main point of this review. In the 1990s, mutagenesis work on several potassium channels led to the identification of pore loops and the signature sequence [12,13] , culminating in the first structures of the KcsA channel solved in MacKinnon's laboratory [14] . Visualising, for the first time, the SF (Figure 1 C), occupied by four potassium ions (S1-S4), quickly led to questions about the number of potassium ions simultaneously bound to the filter, and consequently, the ion permeation mechanism under electrochemical gradients. Initially, it was assumed that the configuration of four potassium ions lined up in a row would not be stable in the filter, as the distance between the charges would be only~3.3 Å. This assumption was reinforced by further work by MacKinnon and coworkers, that showed ion occupancies obtained with x-ray crystallography by replacing K + ions with Rb + or Tl + , which suggested that only two ions occupy the ion binding sites S1 to S4 simultaneously: the ions would reside either in sites S1 and S3 or in sites S2 and S4 ( Figure 2, left [15] ). The remaining binding sites were proposed to be occupied by water molecules, to prevent direct repulsive electrostatic interactions between neighbouring ions. The crystal structure of the KcsA channel (PDB ID: 1K4C [16] ), with four ion binding sites in the filter, was thus proposed to be a superposition of these two occupancy states of the filter. The two discrete occupancy patterns of the SF: potassium ions in sites S1 and S3, water in sites S2 and S4 (called KWKW or S1,S3), and potassium ions in sites S2 and S4, water in sites S1 and S3 (called WKWK or S2,S4) ( Figure 2) immediately suggested an intuitive "knock-on" ion permeation mechanism (later termed also as "soft knock-on" [17] ) in the presence of a driving force for ion permeation [18] . In this mechanism, the ion-water pairs would move in concert upon the entry of an additional K + ion from either side of the membrane, leading to a co-permeation of potassium ions and water molecules through the channel, with a ratio of ion to water permeation of 1. Figure 2. Overview of proposed ion permeation mechanisms in potassium channels. In the soft knock-on (knock-on) mechanism (left), the occupancy pattern of the selectivity filter seen in the x-ray structure is interpreted as a combination of two isoenergetic, coexisting states, both containing two bound K + ions and two water molecules. With the ionic gradient present, outward ion permeation occurs by a shifting between KWKW and WKWK patterns, upon the arrival and binding of a potassium ion and water molecule. Consequently, the KWKW and WKWK states are thought to be predominant also at the ion permeation conditions, and one water molecule is postulated to be permeating per one permeation K + ion. In contrast, in the direct (hard) knock-on mechanism (right), the SF is postulated to be occupied exclusively by potassium ions (the KKKK occupancy pattern, and at physiological temperatures certain dynamics of bound potassium ions is expected. Consequently, ion permeation occurs via short-lived, strong interactions between potassium ions occupying neighbouring ion binding sites. The water molecules do not play a major role in ion permeation, apart from occasional presence in the water-exposed ion binding sites (S1 and S4), and do not frequently co-permeate with K + ions, in contrast to the soft-knock on mechanism. Importantly, several different ion permeation pathways, stemming from various occupancy patterns of the SF, have been observed to date -two possibilities are shown in the Figure, however several others are possible.
The soft knock-on mechanism was widely accepted, making its way to several popular biology textbooks, and formed the basis for almost all subsequent studies on ion conduction in potassium channels for at least 10 years [9,19,20] . Early computational studies, which heavily relied on the initial specification of the selectivity filter occupancy, all started from the KWKW and/or WKWK patterns (see next section). Similarly, many experimental results -e.g. electrophysiological recordings [21][22][23] , isothermal titration calorimetry (ITC) [9,19,20] , structures of channels with mutated SF filters [24] -were rationalized and analyzed in the framework proposed by the soft knock-on mechanism.
Subsequent computational studies however, with longer sampling and other potassium channels, started suggesting a more complicated picture, with direct ion-ion contacts (without intervening water molecules) being energetically accessible [25] . In 2014, a combined computational and experimental effort proposed these direct ion-ion contacts as a dominant contribution to ion permeation, coining the term direct knock-on (or hard knock-on [26] ). In the direct knock-on mechanism, water molecules do not co-permeate with potassium ions -instead, water is mostly excluded from the SF, apart from occasional presence in the sites S1 and S4 ( Figure 2, right). Consequently, potassium ions are occupying neighbouring ion binding sites, which leads to strong electrostatic repulsion between the ions. The strong repulsion has been proposed to contribute to the high permeation rate of potassium channels.
The debate which of the ion permeation mechanisms predominates under physiological conditions is not, as of now, settled. As we are going to critically discuss in this review, various experimental and computational approaches suggest support for either one of the two mechanisms. It is also important to mention here that the mechanisms presented in Figure 2 present somewhat idealized, schematic situations. The soft knock-on mechanism, for example, suggests two specific occupancies of the filter to be dominant (KWKW and WKWK) and a strict ion-water co-permeation ratio of 1. Similarly, direct knock-on, although possibly realized through several distinct permeation pathways, imposes an ion-water co-permeation ratio of (or close to) 0. We cannot exclude an existence of mixed mechanisms, which can be further dependent on the conditions (potassium concentration, the magnitude of membrane voltage or temperature) and the specific potassium channel. Some more exotic mechanisms, involving e.g. even fewer than 2 K + ions in the SF at a specific time and increased water fluxes [27,28] , have been proposed as well, and will be discussed here accordingly.

COMPUTATIONAL STUDIES ON PERMEATION MECHANISMS
As mentioned already, computational studies played a pivotal role in proposing both soft and direct knock-on permeation mechanisms. It is indeed particularly interesting that similar approaches led to very different answers. In this part, by careful examination of simulation concepts and protocols, we will show potential reasons for these discrepancies, and provide an overview of the current status in state-of-the-art computer simulations of potassium channels.

Early studies
The availability of the first crystal structure [14] of a potassium channel -KcsA -greatly expanded the possibilities of computational studies of potassium channels. Using the Poisson-Boltzmann (PB) equation it was discovered that the cavity and pore helices are electrostatically tuned so that occupancy by monovalent cations is preferred [29] .
The first molecular dynamics studies of the KcsA channel were focused at the dynamics of water and potassium in the SF. They showed spontaneous transitions between several SF configurations, in particular the transition of potassium ions from S1,S3 to S2,S4 occupancies was observed [30,31] ; as well as a concerted transition of a configuration with K + in Scav, S3, S1 to S4, S2, S0, with water molecules in between K + ions [32] . These simulations also provided insight into the basis for K + over Na + selectivity -as the fluctuations of the SF were larger than the difference between radii of K + and Na + with RMS fluctuations on the order of 1 Å, so the rigidity of the SF deemed to not explain the ion selectivity of KcsA [32] .
Despite their success, the limited computational power at the time precluded the observation of complete permeation events using unbiased MD simulations of KcsA. Instead, most of the early studies were performed using free energy calculations based on predefined ion positions. These comprise two main methods: 1) alchemical free energy calculations or 2) potential of mean force (PMF) based calculations. In alchemical free energy calculations [33] , a given part of the system (e.g. a K + ion) is transformed to a different one (e.g. a water molecule) via a non-physical path.
Then, the free energy difference between these states is calculated, e.g. using the Zwanzig equation [34] if the free energy perturbation method (FEP) is used. In PMF calculations, a biasing potential is used to increase the sampling of the phase space. For instance, in the umbrella sampling method [35] a path along chosen coordinates (e.g. position of ions along the channel's pore) is divided into a series of overlapping 'windows', which are sampled separately in MD simulations, and then the free energy distributions are 'glued' together to yield a complete energy surface along the chosen coordinates (or 'PMF').
The seminal work by Åqvist and Luzhkov [36] , in which the relative free energies of multiple ion/water configurations in the SF were studied using FEP -with 1,2,3 or 4 K + in the filter, and the remaining sites occupied by water molecules -proposed a preference for two potassium ions in the filter -either in S2,S4 or in S1,S3. Another crucial study was carried out by Berneche & Roux [37] , where umbrella sampling was used to calculate the free energy (PMF) surface for positions of three ions in SF along the pore axis. The study suggested that permeation in KcsA can occur via two pathways. The first one, reminiscent of the 'knock-on' mechanism proposed originally by Hodgkin and Keynes [38] , started from a configuration with K + in S1, S3, separated by a single water molecule. An ion in the cavity then approached the intracellular entrance to SF, pushing the ion pair to S2,S4, resulting in the exit of the outermost ion to the extracellular side. Then, the two ions remaining in SF were able to move back and forth in SF with a small barrier of 1 kcal mol -1 . The largest free energy barrier for this pathway was on the order of 2-3 kcal mol -1 , thus the process is essentially diffusion-limited. The second pathway happened with the ion pair in SF moving to the extracellular side, thus leaving a vacancy on the intracellular side of SF that was then filled by an incoming ion. The barrier in this case was 3-4 kcal mol -1 .
Importantly, both pathways were essentially transitions between states with 2 and 3 potassium ions in S1,S3 and S0,S2,S4 respectively. Each ion pair was separated by a single water molecule, with each ion pair being separated by a single water molecule, thus being essentially small variations of the soft knock-on mechanism ( Figure 1). The existence of two distinct pathways was then proposed to complement the small free energy barriers, leading the high conductivity of K + channels.
However, later, in umbrella sampling calculations by Furini and Domene [25] it was revealed that other K + permeation pathways are also possible. Simulations were initiated in three different SF configurations -in one of them with K+ ions were separated by water molecules (KWKW, Figure 2), but in two other configurations either vacancies or direct contacts between ions were considered. Moreover, the simulations were done for two K + channels, namely KcsA and KirBac.
Two permeation pathways were observed -a previously described water-mediated knock-on [37] , as well as a novel, water-free permeation mechanism which involved formation of direct ion-ion contacts. Importantly, the free energy barriers for along these two mechanisms were essentially the same -on the order of 2-3 kcal mol -1 for outward permeation. Moreover, as revealed by FEP calculations, the energy cost to arrive from a given 'water-mediated' configuration of SF to an analogous 'water-free' configuration, was rather low -less than 2 kcal/mol, indicating a similar plausibility of both pathways. describing interactions between K+ ions and carbonyl oxygens (the NBFIX correction). We will discuss this issue later in this review.

Direct simulations of permeation events
In 2006, Khalili-Araghi et al. described the first molecular dynamics simulations of ion permeation in a K + channel. For the Kv1.2 channel, using its then-recently determined crystal structure in the open state (PDB ID: 2A79) [40] , they observed several permeation events in 25-ns MD simulations under applied electric field, corresponding to a transmembrane voltage of 1 V. Permeations occurred via the water mediated knock-on mechanism, in agreement with [37,39] . However, such high voltages may affect the conduction mechanism, which was acknowledged by the authors. In addition, to ensure SF stability, its backbone dihedral angles were restrained to the values seen in the X-ray structure, possibly affecting the dynamics of the channel. The possible issues of using high voltages, as well as effects of the SF structure on the permeation mechanism will be discussed in more detail below.
With advances in computing power and development of specialized hardware, observing multiple permeation events on a microsecond scale in MD simulations of potassium channels became possible. In 2009, the Shaw group published a study of permeation of about 500 ions through the Kv1.2 pore domain (PDB IDs: 2A79 [40] , 2R9R [41] ), over the course of~30 μ s under applied electric field [42] . The soft knock-on permeation mechanism was observed.
Potassium ions showed a preference for binding at sites S2,S4 over S1,S3, with water having the opposite preference, and the formation of the knock-on intermediate with two ions forming a direct contact -i.e. the [S6, S4, S2] -> [S5, S4, S2] transition -was the rate-limiting step. The voltage range was reported as -180 mV to 180 mV. However, in the method of computing voltage, V = E z, where E is the electric field, z was taken as the length of the SF instead of the length of the simulation box along the membrane normal, which produced underestimated voltages, obscuring the interpretation of these results. The issue was further clarified by Roux and coworkers, who rigorously showed that the membrane voltage in periodic systems should be calculated with z being the length of the box in the direction of the membrane normal [43] .
In the follow-up work the authors thus repeated the simulations of Kv1.2 with the desired voltage, as well as performing similar simulations of the KcsA channel [44] . For Kv1.2, the simulated conductance at around 300 mV was much lower (~2 pS) than the experimental one (73 pS). One of the possible reasons considered by the authors was an overestimated interaction energy of K + with the SF carbonyl groups in the used CHARMM force field (inspired by earlier PMF-based works); surprisingly however, reducing the corresponding LJ interactions via NBFIX correction did not increase the conductance. On the other hand, this change revealed the dependence of the permeation mechanism on the force field: weakening the interactions in Kv1.2 led to a switch from the water-mediated knock-on to a different mechanism, where SF occupancy by K + changed from~2.6 to <2 due to K + dissociating more readily from S0 and S5. In 2014, an alternative method of generating a transmembrane voltage was applied for the first time to study potassium channels [26] -computational electrophysiology (CompEL). In this method, instead of applying an electric field, the system is separated into two compartments by a double membrane setup, and a charge imbalance is created in each compartment, thus directly simulating an electrochemical gradient [46,47] . This charge imbalance, calibrated to correspond to a predefined transmembrane voltage across both membranes, is maintained during the simulation by monitoring ion permeation events and swapping ions in solution with water molecules between the two compartments. Compared to an applied electric field, CompEL, despite an increase in system size after adding a second lipid bilayer, is only slightly more computationally expensive, due to the possibility to insert an ion channel into the second bilayer, that either doubles the effective sampling or allows to study study rectification properties of a given channel in one simulation (for detailed discussion see Kutzner et al. [47] ). Moreover, CompEL does not suffer from potential issues related to simulations with an external electric field. Periodic boundary conditions (PBC) combined with the electric field may complicate a quantitative comparison to experiment; besides, the relation between the applied electric field and the locally acting field being not known a priori [48] -it depends on the used electrostatics scheme used (cut-off, reaction field, Ewald summation, and boundary conditions), as well as on the dielectric distribution across the simulation box. However, the combination of the commonly used PME electrostatics scheme, a 'tinfoil' PBC, and single lipid bilayer was shown to follow V = E z , and only substantial fluctuations of the membrane dielectric properties may cause artifacts (e.g. in electroporation [49] , which is beyond the scope of this review). Therefore, both approaches -the applied external field and CompEL -are acceptable for simulations of ion permeation in K + channels at physiological and near-physiological voltages, at which they produce nearly identical results, as verified by us for the MthK channel [50,51] .
Using CompEL, currents through KcsA, Kv1.2, and MthK were simulated at physiological voltages, with a total of more than 1300 permeation events over the course of~50 μs [26] .
Permeation in all three channels occured via a water-free mechanism with formation of direct ion-ion contacts. At the core of this mechanism lies a concerted transition of K + from [Scav, S3, S2] to an intermediate [S4, S3, S2], with a following 'knock-on' of the central ion pair to S2,S1, and finally restoration of the central ion pair at S3,S2, via the exit of K + at S1 to S0, and shift of S4 to S3. Other, similar pathways are also possible, united by lack of water co-permeation: water molecules were found to frequently occupy the S1 and S4 sites but not the central binding sites and therefore did not co-permeate ( Figure 2, right panel). The rate of K + ions leaving the extracellular site was found to be determined by the rate with which incoming intracellular ions arrive at the SF, thus the model inherently implies that K + permeation is diffusion-limited as long as K+ ions occupy S2 and S3. This mechanism was observed independently of the force field and water model used (i.e. CHARMM36 with the CHARMM-specific TIP3P water model, AMBER99sb with both SPC/E and TIP3P water models). Only supraphysiological voltages (~1 V) led to the co-permeation of water, although position restraints had to be applied to the SF to preserve its structure under these conditions, because it experienced instabilities at such high voltages. The conductance observed was in agreement with experimental results (up to a factor of ~2, similar to the experimental range of variation) [5] .
The direct knock-on mechanism was then consistently observed for several potassium channels and simulation conditions [50][51][52][53][54] . Ion permeation in some K2P channels (TRAAK, TREK-1, TREK-2) occurred via this mechanism [53][54][55] . In Kopec et al. [50] , the origins of the exquisite K+/Na+ selectivity was studied, in KcsA, MthK, Kv1.2 W362Y and NaK2K channels. Binding of Na + drastically reduced permeation in all K channels studied, and substantial Na + currents (albeit several times smaller than maximum K + currents) were observed only in KcsA and NaK2K in the complete absence of K + , in accordance with the known robust selectivity of K + channels. The direct knock-on mechanism suggests that the complete desolvation of ions underpins selectivity, since the desolvation penalty for K + is much smaller than for Na + . Indeed, ion selectivity, as well as permeation rates, of K + channels observed in the study were found to be drastically diminished with any level of water co-permeation. The water-free ion permeation was found to be in agreement with recalculated IR spectra of ion-occupied KcsA SFs (see below), as well as measured ion interaction energies [56] , while the calculated Na + /K + permeability ratio of 0.02-0.04 is in good agreement with the experimental values of 0.006-0.4 [5,57,58] .
Of particular interest here are the MD simulations of the pore domain of the Kv1.2 channel, started from the 2.4 Å crystal structure of the Kv1.2-Kv2.1 paddle chimera channel (PDB ID: 2R9R). As described above, previous long simulations on specialized hardware showed the soft knock-on mechanism in this channel. The electrostatic profile of the Kv1.2 SF was however markedly different than the one for KcsA, which was explained by different orientations of conserved aspartate residues behind the SF (see Figure 1 C & D). Driving the side chains of these residues (D80) in KcsA, from initially oriented towards the SF center to extracellular-facing ('flipped' orientation, observed experimentally for the KcsA E71A mutant at low potassium concentration), drew the profile closer to that of Kv1.2 [44] . Indeed, in Kv1.2, the corresponding D375 residues were almost exclusively found in the "flipped" orientation. As shown in Kopec et al., such "flipped" aspartates in Kv1.2 lead to the unstable SF, allowing water to enter the SF, and subsequently reducing both potassium current and selectivity, in good agreement with experimental studies of mutated channels at corresponding aspartates [59,60] . Kopec et al proposed the W362Y mutation, that strengthened this residue's hydrogen bond to D375 and reduced the flipping rate, or position restraints on D375 that kept the residue in a non-flipped orientation. Both approaches resulted in reduced water-permeation and increased direct ion-ion contacts, which in turn increased the current and selectivity to the levels observed for other channels in this study [44] . These observations strongly suggest that the "flipped" aspartate conformation, attained in e.g. unrestrained MD simulations of Kv1.2 started from PDB: 2R9R, is likely not representative for the conductive SF.
Alongside the two mechanisms described above -water-mediated and direct knock-on -other alternative mechanisms of ion permeation were observed in MD simulations as well. Recently, an uncommon permeation mechanism was revealed in a MD study of the hERG1 channel by Miranda et al [61] . There, very low flickering outward conductance (0.8±0.2 pS, in reasonable agreement with the 3-5 pS flickering conductance reported for open-state hERG1 from experiments [62][63][64][65][66] and conserved ion selectivity was accompanied by very low stability and high distortion of the SF. The SF was occupied by only~1 K + ion, and rare permeation events were water-mediated. Potential reasons for this behavior, might be (1) the starting CryoEM structure, that is of relatively low resolution (3.8 A, PDB ID 5VA2 [7] ) coupled with the absence of water molecules behind the SF, usually observed in high resolution crystal structures of potassium channels (see Figure 3 and the discussion in the NMR part), (2) high voltage used in simulations (500 and750 mV) (3) the sequence of the SF of hERG, namely 624 SVGFGN 629 , with residues 624 S and 629 N differing from the more common T and D, respectively ( 629 N replaces the aspartates discussed above for Kv1.2; the corresponding T->S mutation in MthK drastically reduces both open probability and single channel conductance [67] ). It is important to note that non-selective K channels such as NaK [10] and NaK2CNG-N [50] are also characterised by distorted and water-permeating SF, thus the question of mechanism of ion permeation and selectivity of such SFs should be addressed in future studies. Another example of a non-traditional mechanism was observed in work by Sumikama & Oiki [27,28] , where Kv1.2 [27] displayed a mechanism reminiscent of the earlier proposed vacancy model [68] -an ion pair in S3, S1 with a water molecule in between shifts to S2,S0, leaving a vacancy at S3 behind, which is then filled by a water molecule from S4 upon the entry of K + to this site. KcsA in their studies [28] showed two modes of permeation, one of which resembled a water-mediated knock-on, and another consisted of spontaneous exit of a K + from a 2-K + -occupied SF, followed by another ion entering the SF from the cavity. It should be noted, however, that both the simulations of the hERG1 channel [61] and Sumikama & Oiki's [24,25] work featured supraphysiological voltages, which may have affected the channels, and specifically, the SF structure [26] , which in turn could lead to a different preferred mechanism. Together, these data indicate a potential variability of permeation mechanisms for channels with different structures, as well as for different simulation conditions.

Computational studies of K + channels: technical aspects
These observations bring us to the discussion concerning simulation parameters and conditions one might use in order to obtain reliable results in MD simulations of ion permeation in potassium channels. One of the critical choices in any MD study is the force field choice. A controversy in the field of MD simulations of potassium channels has been the use of force fields with less attractive Lennard-Jones parameters for ion-backbone carbonyl interactions, through the so-called 'NBFIX' correction (historically, potassium channels simulations were among first ones with such modified interactions, thus the correction was often called simply 'NBFIX'. Since then however, the approach gained popularity for fine-tuning many other Lennard-Jones parameters [69] . In this review, we use the 'NBFIX' term exclusively when referring to modified potassium ion-backbone carbonyl interactions). Conductance and selectivity are largely governed by these interactions, therefore several studies aimed at improving them. A popular approach involves using the energetics and thermodynamics of K + interaction with N-methylacetamide (NMA) -a model of the protein backbone carbonyls. Initially, in 2002, it was postulated that in traditional force fields, e.g. CHARMM22 and its further iterations, K + binds too strongly to the SF carbonyls, and introducing a NBFIX correction weaking these interactions leads to a more accurate representation of K + -NMA solvation free energies [70,71] . However, the experimental solvation free energy of K + in NMA was not known at that time. Only in 2010, a combined experimental and computational effort led to the measurement of the KCl solvation free energy. The individual contributions from K + and Clwere then approached computationally, since measurements of single ion solvation free energies is often a daunting task [72] . However, some variations due to the force field family were observed, which is expected because of the different energetics of potassium-backbone carbonyl interactions. For instance, in recent work by Ocello et al. [74] for the TRAAK channel it was observed that a combination of AMBERff14SB force field and SF occupied initially only by K + is required to observe substantial conductance (on the order of 2-9 pS) at 100 mV and 200 mV, while water between K + ions led to filter instabilities producing reduced currents, and permeation events being possible only after water was expelled from the SF. Switching to the CHARMM36m force field reduced the probability of finding SF in native conformation by half, with water-mediated configurations being non-permeable (stability issues were observed for KcsA [75] with CHARMM36m as well. Gating of TRAAK through bending of transmembrane helices around a conserved G268 residue was also affected by force-field choice, with AMBER producing a better preserved initial conductive state of the channel, and in CHARMM transitions between conductive and non-conductive were observed more frequently. As mentioned previously, the structure of K + channels -and SF in particular -also affects their permeation properties. Distortions in SF (e.g. flipping of SF residues, SF widening and/or collapse) caused e.g. by high voltages [26] and/or simulation artifacts stemming from the usage of low resolution initial structures (e.g. with absent water molecules behind SF), may explain the unusual mechanisms observed in such conditions [27,28] . Notably, high voltages in general tend to promote water-mediated mechanisms [26,50,61] , while the direct knock-on was consistently observed even at low voltages -see [51] (150 mV, MthK), [74] (100 mV, TRAAK), [54] (40 mV, TREK-2) -further suggesting the impact of voltage on permeation mechanism.
Of interest, increased dynamics and plasticity of the SF were repeatedly observed in MD simulations of nonselective channels, such as NaK and NaK2CNG, that allow both K + and Na + to permeate at similar rates (see Figure 1 D for the structural details of their SFs). In these channels, ion permeation occurs together with high water co-permeation, which has been suggested as a factor underlying the lack of K+ selectivity [50] . Therefore, at this stage, any ion permeation mechanism in K+ channels that simultaneously predicts high levels of water permeation, seems incompatible with high potassium selectivity observed in these channels.
To summarize, despite more than two decades of computational studies of K + channels, a number of controversies and discrepancies between different studies still exist. We were able to trace down several causes of such discrepancies, such as initial structures and/or force field parameters as well as the strength of the applied electric field. It is encouraging that recent simulations of several potassium channels, carried out by several research groups, using modern force fields, are mostly in broad agreement. These simulations show predominantly water-free ion permeation enabled via direct ion-ion contacts. However, MD simulations of K + channels still systematically produce currents lower than experimental values [23,41,73]. One of the likely reasons for that is the use of force fields that do not explicitly account for induced electronic polarization [70,74] , thus the development of polarizable force fields for K + channels presents a promising area of research.

X-ray crystallography
As mentioned in the introduction, a careful analysis of ion occupancy in the crystal structure of KcsA, aided by anomalous diffraction of thallium ions, a mimic for potassium, led to the interpretation of alternating ion/water occupancy in the selectivity filter [15] . Supported by electrophysiological streaming potential measurements (see next section) that imply ion/water co-permeation in a one-to-one ratio and the consideration that ions would be unlikely to occupy to neighbouring binding sites to to electrostatic repulsion, the crystallographic data were interpreted as an overlay between KWKW and WKWK occupancies at comparable populations, although it was noted that the estimated thallium occupancy of 2.5-3 ions for the four binding sites would imply that ions sometimes occupy adjacent binding sites. Later efforts re-refining the same data and analysing high-resolution data from the MthK channel [26] , as well as anomalous scattering of potassium in the engineered NaK2K channel [76] and K2P TREK-1 channel [54] consistently found ion occupancies of near unity for all four binding sites in the selectivity filter in the conductive conformation of the SF, implying a vanishingly small water occupancy in the filter and a high fraction of direct ion-ion contacts. A very recent, crystallographic dataset for the MthK channel suggested a slightly lower occupancy of about 3.2 ions in the SF (with the lowest occupancy for the S2 binding site), similar to the number of Tl + ions originally reported for KcsA [77] . Although these crystallographic studies were carried out under cryogenic conditions (low temperature) and in the absence of an electric field or concentration gradient -hence in a non-conducting state -the filter occupancy under these conditions still represents low-energy states that can be assumed to be informative on the configurations involved in ion permeation. Hence, these high-resolution analyses of multiple potassium channels show a high, nearing full ion occupancy, indicating a high abundance of direct ion-ion contacts and therefore being more consistent with the direct knock-on permeation mechanism than with soft knock-on.
This picture was recently challenged by a crystallographic study of a selectivity filter mutant of KcsA [78] . In this study, the conserved glycine (G77) contributing with its carbonyl to the S2 and S1 ion binding sites was mutated to alanine and cysteine (G77A, G77C). In the crystal structures, the mutant proteins were shown to have ions bound to the S2 and S4 binding sites.
As this is one of the configurations (namely WKWK or [S2, S4]) proposed to predominate in the soft knock-on mechanism, this was taken as evidence to support it, together with the previously known T75C (threonine forming the S4 ion binding site) mutant [79] , that seems to prefer ions bound to sites S1 and S3 (i.e. KWKW, [S1, S3]). The low single channel conductance of these mutants as well as the reduced K + /Na + selectivity of the mutant equivalent to G77C in the Shaker K + channel [13] however question the assumption of an unchanged permeation mechanism as compared to wild-type channels. The T75A mutant in KcsA seems to retain its K + selectivity [80] , but the equivalent mutation in both MthK and NaK2K yields non-selective channels [8] , thus calling for additional electrophysiological, structural and computational characterization. Furthermore, the difference in ion occupancy in structures of the G77A and G77C KcsA mutants, as compared to the wild type channels showing full ion occupancy, would need to be reconciled in order to support the assumption of an unaltered permeation mechanism in these mutants. It thus remains unclear if these mutated channels are informative on the ion occupancy and permeation mechanism of canonical, wild type potassium channels.

Cryo-EM
In principle, a structural determination of a potassium channel with cryo-electron microscopy (cryoEM) should provide direct insights into the occupancy of the SF through ion densities, complementary to those obtained through X-ray crystallography, with the additional advantage of retaining the membrane environment around the channel. Indeed, structures of multiple members of the potassium channel family have been obtained in lipid nanodiscs, including voltage gated human channels (hERG, Kv11.1) [7] , Ca2+ and Na+ gated big conductance (BK) channels [81] , the voltage gated Kv1.2-2.1 paddle chimera channel [82] , the voltage gated KAT1 channel [83] , the voltage and cyclic-nucleotide gated SthK channel [84] , the voltage gated Eag channel (Kv10.1) [85] and the pH gated K2P TASK2 channel [83]. However, none of these structures reached a resolution higher than 3.0 Å, especially in the transmembrane part (where the SF is located), and therefore, although in many of these structures the SF is seen in the conductive conformation, non distinguishable from x-ray structures, an equivocal determination of the SF occupancy is precluded. We expect, however, that with the recent breakthroughs in cryo-EM, that enable atomic level resolution for both soluble and membrane proteins [86,87] , a glimpse into the SF content is around the corner.

Nuclear Magnetic Resonance (NMR)
NMR spectroscopy offers an appealing experimental way to study the dynamics of proteins in  [88] . In the context of potassium channels, many NMR experiments have been published, focusing mostly on the gating processes that occur both at the main activation gate and at the selectivity filter [89][90][91] , as well as on the coupling between these two gates. Even showed signals only for the upper part of the SF (Gly67 and Tyr66 in NaK2K, forming ion binding sites S1 and S2). The signal from Gly65 was weak, and signals from the bottom part of the SF (Val64 and Thr63) were absent. This observation indicates that the ion binding sites S4 and S3 are not in contact with water. As a control, removing all potassium ions from the sample led to clear H/D signals from Val64 and Thr63, thus being exposed to water in potassium-free conditions (i.e. water molecules occupy the SF when no ions are available). Re-introduction of potassium ions into the sample caused these signals to disappear, suggesting that K+ ions are indeed replacing all water molecules in at least ion binding sites S3 and S4. In a second experiment, the magnetization transfer from water to protein was used to provide another way of measuring which residues are in contact with water, importantly on a shorter time scale of 1 to 100 ms. The efficiency of the transfer depends on the distance between the bound water molecules and the protein (amide protons). Residues from the SF that showed the magnetization transfer were GYG (65-67) residues ( Figure 1). These residues are in contact with two structural water molecules (in the NaK2K channel), necessary for filter stability [92,93] .
Indeed, in all high resolution structures of potassium channels, such water molecules are resolved ( Figure 3) and their presence stabilizes the SF in the conductive conformation (see the discussion in the computational part as well). Accordingly, in the ssNMR experiment, the signal buildup behavior for magnetization transfer was followed (based on different experimental transfer times). The resulting signal buildup rate shows a clear dependence on the distance to the structural waters bound behind the SF, that is the rate is the highest for Gly67 (top of the SF), followed by Tyr66 and lowest for Gly65 (middle of the SF), and not detectable for either Val64 nor Thr6. Consequently, if there were any water molecules bound in the filter (i.e. in the ion binding sites, as postulated by the soft knock-on mechanism), they should affect such a magnetization transfer buildup rate. These two experiments, together with MD simulations carried out in the absence of membrane voltage (i.e. same as the experiment), led to the conclusion that the SF of potassium channels, in its conductive conformation, is water-free under near-physiological conditions (i.e. in a lipid membrane and at room temperature). Such a conclusion strongly supports the direct knock-on mechanism, although these ssNMR experiments did not address the actual occupancy of the filter by K+ ions, and how many K+ ions are present in the filter at a given time (instantaneous occupancy). Even more recently, a similar lack of water accessibility to the inner ion binding sites has been later observed, also via ssNMR, in the inward-rectifier KirBac1.1 channel [94] . In another study, Eichmann et al. used solution state NMR to study binding of NH4+ ions to the KcsA channel, embedded in detergent micelles. NH4 + ions are considered as good surrogates of K + ions, both from the physicochemical perspective (ionic radii differ only by~0.15 Å and both ions have comparable hydration free energies [95] ), as well as from their ability to permeate potassium channels [5] , suggesting a similar mode of binding to the SF and the mechanism of ion permeation shared by these two ions (although currents carried by NH4+ are typically markedly lower than K + currents). Crucially, NH4 + ions containing the 15 N isotope provide high sensitivity for NMR investigations. The NMR spectra revealed 5 signals indicative of NH4 + ions bound to the channel (ascribed to the ion binding sites S0 -S4). The signals disappeared after addition of K + ions to the sample, suggesting a competition between K + and NH4 + ions for the same ion binding sites, most likely at the SF. The disappearance of signals in a control experiment with the scorpion toxin agitoxin II, that plugs the SF and removes ions from it [96] , reinforced the idea that the observed NH4 + signals indeed come from the ammonium ions bound to the SF. By integrating 15NH + cross peaks and comparing their volumes with cross peak volumes of 15N-1H (amide) moieties, the authors were able to estimate the number of NH4 + ions in the SF at a given time, to be in the range of 3-4, strongly suggesting direct ion-ion contacts between the ions and lack of water molecules in the filter at zero voltage and 36 degrees.

Functional measurements
Early experimental work by Hodgkin and Keynes in 1955 determining the ion flux-ratio of K + channels arrived at the result that K + ions do not traverse the channels independently, but that about three ions moved "in lockstep" with each other [38 . Flux-ratio measurements performed later with technology of increased precision [97 recorded the flux-ratio exponent of Shaker K + channels as 3.4 and interpreted this result as indicating that "the pore in these channels can simultaneously accommodate at least four K+ ions" [97,98] .
The most direct, albeit difficult to interpret, experiments regarding the ratio of water to ion flux during K + channel permeation are streaming potential measurements. Streaming potentials are recorded when ion channels experience a strong osmotic gradient from one side of the membrane to the other under otherwise symmetrical ionic concentrations. The osmotic pressure drives water through the pore, which sweeps ions from the ionic solution with it, building up a small voltage difference across the membrane as a consequence. The literature cites a small number of key papers, which determine streaming potentials in the Ca 2+ coupled large conductance K + channel [99] , the K + channel from sarcoplasmic reticulum [100] , the human Ether-a-go-go-Related (hERG) [101] , and most recently, KcsA [102] . Due to the small magnitude of the streaming potentials, the measurements are however difficult to carry out, rather noisy, and the results are often described as hard to interpret, partly due to the necessary use of correction factors in the conversion to water-ion flux ratios. For instance, potassium permeation mediated by the selective membrane carrier valinomycin generates a measurable streaming potential, although it is assumed not to be associated with water co-translocation [99] .
The valinomycin signal is hence assumed to stem from the direct effect of the osmotic gradient on the ion activity coefficient, not part of the actual streaming potential.Thus, valinomycin control experiments are frequently carried out as control to derive the channel related streaming potential.
The original papers report that per K + ion, between 2 and 4 water molecules co-permeate in the Ca 2+ coupled K + channel, inferred from a (corrected) streaming potential of 1-2 mV [99] . For the K + channel from sarcoplasmic reticulum, the water-to-ion ratio was found to lie between 2-3 [100] . In 2005, Ando et al. introduced a new approach to recording streaming potentials using an osmotic-pulse methodology [101] . In hERG, the authors derived a water-ion flux ratio between 0.9-1.4, depending on K + concentration, while for KcsA, a ratio between 1.0-2.2 emerged, where the higher numbers are associated with more dilute ionic solutions.
Remarkably, the upper limit in KcsA is obtained from recording streaming potentials due to ion flux at 3mM [K + ], a concentration at which the channel normally inactivates and becomes non-conductive to K + ions [18] .
In further experiments, Pohl and colleagues recorded water flux through KcsA channels by determining the dilution of ionic solutions by inflowing water near the membrane [103] , and by stopped-flow measurements of the volume of proteoliposomes under osmotic gradients [104] . In both studies, the thus deduced diffusion coefficient of water inside KcsA exceeded that of bulk water at low K + concentrations, whereas K + concentrations above 300 mM were found to completely block water permeation through the channels. Inactivated KcsA channels, impermeable to K + ions, were still observed to allow water passage at similar rates, while open channels under 200 mM K + displayed an intermediate osmotic permeability in "qualitative agreement with single file movement of K + and water" [104 ].
A molecular mechanistic interpretation of these findings is not straightforward. First, the results imply that water flux through KcsA is 100x faster than K + permeation at low [K + ], and that at 200mM K+, the rate of water permeation still far exceeds that of K+ conduction, which is difficult to reconcile with a 1:1 co-permeation mechanism. Secondly, raised K + levels reduce water flux to a non-detectable level, leaving open the possibility of K + permeation without water. The simplest explanation that might resolve this puzzle is to assume that water flow occurs through a subset of KcsA channels unoccupied with K + , since most channels do not contain an ion at the given experimental conditions [104] -whereas open, active channels filled with K + ions at high concentration are less or indeed non-permeable to water.
Rauh et al. [105] used an electrophysiological approach in a study that originally aimed to characterize an unusual voltage gating in viral potassium channels. In contrast to classical voltage sensing and gating, that is mediated by voltage-sensing domains (VSDs), some potassium channels show a distinct voltage gated process, which is thought to be mediated by changes of K + ion occupancy in the selectivity filter. As both ion permeation and such voltage gating occur at the filter and involve K + ions moving between different sites, it was assumed that atomistic models of ion permeation would provide insights into voltage gating at the filter as well.
The authors used single-channel data, namely IV curves, to test three different mechanisms of ion permeation, two of which resembled the soft knock-on mechanism (no simultaneous occupancy of adjacent binding sites by K + in the SF), whereas the third mechanism involved some states characteristic for the direct knock-on mechanism. The considered mechanisms varied in the number of discrete states and rate constants between them, in ion occupancy of such states and the voltage (in)dependence of certain transitions. The mechanisms were subsequently tested by a global fit of 36 experimentally derived curves obtained at varying K + concentration and membrane voltages. The only mechanism that fit all the electrophysiological data was one of the variants of the knock-on mechanism, the so-called '5 state model', therefore supporting the original knock-on mechanism. It is however important to note that the mechanism related to the direct knock-on considered by Rauh et al. was only a one possible variant (see Figure 2). The approach did not provide any information regarding the water presence in the SF, nor the instantaneous number of K + in the filter during permeation, which is expected to be close to 2 for the knock-on mechanism, but higher for direct knock-on.

Two-dimensional infrared spectroscopy (2D IR)
Another experimental technique proposed to study the ion permeation mechanism in potassium channels is two-dimensional infrared spectroscopy (2D IR), which, in principle, allows studying the SF in a membrane-like environment with native K + ions and at zero voltage. The chemical bond vibrations are sensitive to the electrostatic environment, therefore the vibration frequencies will be markedly different depending on the presence of ions and/or water nearby [106] . Of particular interest here is the amide I stretching mode, dominated by the stretching motion of the carbonyl group (C=O). As the ion binding sites S1-S3 are formed exclusively by carbonyl oxygens, the 2D IR technique seems to be well suited to experimentally probe the occupancy and dynamics of the SF of potassium channels. However, to isolate the stretching motion of the specific carbonyl groups, isotope labeling (13C and 18O) is necessary, that redshifts the amide I frequency. Moreover, such a 2D IR spectrum reports on an ensemble average of ion occupancy states of the SF. To interpret the spectrum, i. e. to decompose it into the contributions from specific occupancy states, MD-based theoretical spectra are used: when a theoretical spectrum, or a combination of thereof, agrees with the experimental spectrum, the occupancy state(s) used to calculate the theoretical spectra is deemed as compatible with the one present under experimental conditions.
In 2016, Kratochvil et al. used, for the first time, the 2D IR approach, together with the semisynthesis of the isotopically-labeled KcsA channel and MD-derived spectra, to probe the occupancy of the SF of KcsA [107] . It was found that only spectra that were calculated from simulations of water-containing states of the SF, namely soft knock-on signature states KWKW and WKWK (Figure 2) with the addition of the "carbonyl flipped" state of the S3 valine ( Figure 1 D) , were in agreement with the experimental spectrum, after fitting the relative population of each state guided by the experimental spectrum. In contrast, the populations of spectra calculated from simulations of states characteristic for the direct knock-on (e.g. 0KK0 and KK0K) were found to match the experimental spectra less well. However, unlike the snapshots representing the soft knock-on, the states representing the direct knock-on were not fitted to the experimental data, but were rather taken directly from computational electrophysiology simulations by Köpfer et al., i.e. at non-zero voltage. As mentioned, the 2D IR experiment is performed at 0 voltage. Nevertheless, Kratochivil et al. concluded that only the water containing soft knock-on states KWKW and WKWK (and presumably the "carbonyl flipped" conformation) are in line with the 2D IR spectra.
We have later repeated the calculations but used the same procedure of fitting of MD-derived spectra for both mechanisms (soft and direct knock-on) [50] using the same protocol as the one used in Kratochvil et al. For the direct knock-on mechanism, we have also included a larger number of the SF occupancy states, to reflect those commonly seen in our simulations (e.g. WKK0 and WKKK). In our analysis, we were able to reproduce the theoretical spectra of Kratochvil et al. for the soft knock-on mechanism, indicating the robustness of the method. In contrast however, when the direct knock-on states were treated with the same fitting protocol, also these yielded a very good agreement with the experimental spectra, in fact as good as the soft knock-on states. This observation led us to conclude that such an analysis, based on the experimental spectra from Kratochvil et al., cannot differentiate between the occupancy states characteristic for soft and direct knock-on mechanisms, and thus could not discriminate between both mechanisms of ion permeation.
It is worth noting that the experimental spectra for the labelled KcsA discussed here come from one specific isotope-labelling pattern, namely 13 C 18 O labels simultaneously introduced on the carbonyl groups of Val76 (S3 binding site), G77 (S2 binding site), G79 (S1 binding site) (see Figure 1 C). The simultaneous IR signal from all three labelled residues can obstruct the interpretation of the experimental spectra. It has been recently argued, based on theoretical calculations alone, that isotope labeling of single residues, namely V76 and G77, in independent measurements, should be sensitive to the water presence and/or direct contacts between K + ions in sites S2 and S3, thus distinguishing between the soft and direct knock-on mechanisms [108] . However, as of now (early 2021), the corresponding experimental spectra have not been published.

Concluding remarks
The ion permeation mechanism in potassium channels has been intensively studied in the past decades by an impressive number and combinations of experimental, computational and theoretical approaches. Yet, as reviewed above, there is no agreement on what mechanism is dominant, especially under physiological conditions. For convenience, we gathered outcomes of the studies reviewed here in Table 1. Possible reconciling scenarios would include the possibility that under different conditions such as channel composition or ion concentration, a different mechanism prevails, or that particular experiments primarily "see" a particular filter conformation or subpopulation, which may or may not be representative for the major, ion conductive state. Table 1. Agreement between various computational and experimental studies and proposed ion permeation mechanisms in potassium channels. Name of the channel for which the given observation was recorded is given in the parenthesis. ✔ means that the main observations are compatible with a given mechanism, whereas ✗ means it would be very hard to reconcile the two. ? stands for observations that can be interpreted differently or potential issues with the employed method. See text for the full discussion