## Abstract

Phase space has been used to visualize and analyze the dynamic behavior of stochastic and chaotic systems. We applied this concept to maxi channels recorded from excised inside-out patches of in situ interstitial cells of Cajal. Pseudo-phase portraits of channel current were fairly homogeneous from patch to patch. They showed three main peaks, α, β, and γ, in increasing conductance. These represented single or near aggregated states. The α-peak was the closed state. The β-peak was small, consisting of a single conductance state, or in some cases two (a doublet). The β-peak state(s) had a long lifetime and displayed a characteristic behavior of frequent short transitions to γ but not to α. It was always preceded by a short series of α/γ-transitions. The γ-peak was the largest and consisted of a large number of conductance states with fast state transitions, sometimes to the extent of causing a diffusive-type behavior. Phase portraits allowed us to construct a provisional gating scheme for the maxi channel and suggest that further analysis of recordings in higher dimensional phase space and with related techniques may be promising.

- ion channel
- maxi channel
- kinetics
- phase portrait

the maxi channel is frequently observed in patch-clamp recordings of interstitial cells of Cajal (ICC), the pacemaker cells of the gut (11, 33, 34, 46, 47). When patches are excised from the cell, the maxi channel often activates within a minute or two. This excision activation is a characteristic property of the channel, as is a large conductance of ∼210 pS in culture and ∼320 pS in situ (hence, the maxi in maxi channel); multiple and complex subconductance states; a bell-shaped open probability-potential relationship, peaking at 0 mV; slow inactivation at potentials away from 0 mV; a low discrimination between anions and cations; and activation by protein kinase A. These properties are shared with a class of ion channels called maxi-anion or maxi-chloride channels (36). As we found poor discrimination between cations and anions, we thought it more appropriate to name our channel simply the maxi channel (33).

It is our general aim to obtain a detailed kinetic model of the maxi channel using traditional kinetic techniques such as dwell time distributions and Hidden Markov Models. However, a first step is to obtain a low-resolution, rough grain view of its kinetics, so that two questions can be answered. First, are the maxi channels likely a homogeneous group according to their kinetics? This will determine if the maxi channels should be split into groups or if indeed a detailed model is at all suitable. Secondly, what are the likely complexities of their kinetics? This will determine the precise approach to a more detailed analysis. To gain this rough grain view we found pseudo-phase portraits to be very useful. Phase space is a central concept in nonlinear dynamics with applications in all branches of science (7, 8). The etymology of “phase space” is very much a historical contingency (31), but in outline it is a rather simple concept. A time series of *n* variables can be thought of as a trajectory through an *n*-dimensional space, a phase space. If there is only one variable, then the trajectory can be mapped onto a multidimensional space by having each dimension represent the value of the variable at a time separated from each other dimension by some lag (2, 38). This is often called pseudo-phase space. Visualization of the *n* dimensional phase space, or pseudo-phase space, is simple for *n* of two or three. The resulting graph of the trajectory is often called a phase portrait. However, for *n* greater than three there are methods to find the most interesting region of the space and project this onto two or three dimensions. This is called embedding, with specific methods including principle component analysis, normal mode analysis, and Poincare sections.

Here we show how two-dimensional pseudo-phase portraits of patch-clamp recordings of the maxi channel can be used to answer our two questions about its kinetics. This should provide a basis for detailed kinetic modeling.

## METHODS

#### Patch clamp of the maxi channel.

The maxi channel was recorded from excised inside-out patches of small intestinal ICC in situ as described previously (33). All procedures were approved and carried out in accordance with regulations from the Animal Ethics Board of McMaster University. Maxi channels were identified in excised patches from voltage ramps with inactivation at negative potentials and activation at potentials near to 0 mV, as described previously (33). The same voltage-clamp protocol was used for all recordings presented and analyzed here: +20 mV for 2 s followed by a step to −40 mV for 4 s. The initial +20-mV step was required to keep the channel open at −40 mV, as the maxi channel inactivates at potentials away from 0 mV (33, 34). For each patch, this 6-s protocol was repeated 100 times (i.e., for t10 min). The pipette solution throughout consisted of 150 mM NaCl and 10 mM HEPES adjusted to pH 7.35 with Tris base [2-amino-2-(hydroxymethyl)-1,3-propanediol]. The bath solution was 95% O_{2}-5% CO_{2} bubbled Krebs containing the following (mM): 120.3 NaCl, 5.9 KCl, 15.5 NaHCO_{3}, 1.2 NaH_{2}PO_{4}, 0.1 citric acid, 0.1 aspartic acid, 2.5 CaCl_{2}, 1.2 MgCl_{2}, and 11.5 glucose.

#### Pseudo-phase portraits.

A pseudo-phase portrait is the graph of *v*_{x+τ}(*v*_{x}) where *v*_{x} is the magnitude of the measured variable at time *x* and *v*_{x+τ} is its value at some later time. This is usually plotted as a scatter of {*v*_{x+τ}, *v*_{x}} pairs or a connected trajectory. Due to the near instantaneous changes in conductance of an ion channel, {*v*_{x+τ}, *v*_{x}} pairs should cluster at spots (Fig. 1) and with a large number of pairs the density of these spots will be great. Therefore, instead of plotting the *v*_{x+τ}(*v*_{x}) trajectory or scatter, we plotted the log frequency of {*v*_{x+τ}, *v*_{x}} pairs as a two-dimensional image,
*a* and *b* are bins of *v*, each bin represented by a pixel in the *F*(*a*, *b*) image. A similar binning of pseudo-phase space has been employed by Janssen and Nyberg (12); they just did not log the count. Unless otherwise stated bins were 0.1-pA wide. Also. we summated *F*(*a*, *b*) with several different values of τ. These pseudo-phase portraits are denoted as *F*(*a*, *b*)_{p,q,r...} where *p*, *q*, *r*, etc. are lag values. These summated images produce a more populated and therefore smoother portrait. Some simple facts of interpretation of pseudo-phase portraits are illustrated in Fig. 1.

## RESULTS

Pseudo-phase portraits of single maxi channels were diverse in their appearance (Fig. 2*A*). Most, if not all, shared two or more features of the scheme in Fig. 2*C*. In this scheme, there are three main peaks along the principle diagonal, α, β, and γ, in increasing conductance, with γ much larger than the others. Many portraits has just these three peaks (Fig. 2*A, d*, *g*, and *h*), but often there were more. In these cases, peaks were lumped into α, β, or γ by the following operational definition: β-peaks were small and associated with long transition “streaks” extending from their origin to near the maximum conductance of the channel; α-peaks were any peaks smaller in conductance than β; and γ-peaks were any peaks larger in conductance than β. The α included the closed state but was stretched along the principle diagonal to varying degrees (not circular, as would be expected for just noise), suggesting the additional presence of low conductance states. As with α, the extension of the β-peak along the principle was variable to the extent that sometimes it split into a doublet (Fig. 2*A*, *b*, *f*, *i*, *j*, and *p*). The γ was the largest peak, suggesting either a large number of near aggregate (near the same conductance) states or diffusive conductance changes (see below). In some cases, there was a partial splitting of a small lower γ conductance peak (Fig. 2*A*, *m*, *o*, and *p*). There were strong α/γ-transition peaks, less so for α/β.

Of course this scheme is just an attempt at generalization. In some cases, it is questionable whether the β-peak was actually present at all (Fig. 2*A*, *k*). Some maxi channels could clearly not be fit into our scheme, even at a stretch (Fig. 2*A*, *e* and *f*). Excluding these latter two channels we subjectively quantified the peak conductances (Table 1). Specifically, we wanted to test the hypothesis that β was a simple fraction of the total conductance, as has been shown for the maxi channel in cultured ICC (11, 34); thus the β/total conductance ratios were calculated. The β was around a quarter or a third of the full conductance of the channel depending on the method of calculation.

The features of our scheme could be identified and explained from observation of current traces (Fig. 3). The large size of the γ-peak was explained by *1*) a large number of quickly transitioning, near aggregate states, almost leading to the appearance of noise but with clear step like transitions (Fig. 3, *hU* and *pU*); *2*) relaxations in current, likely reflecting changes in the lipid environment of the channel (3, 6) (grey arrow lines in Fig. 3, *iL* and *pU*); and *3*) transitions to lower conductance levels that were so short as to be cut in apparent amplitude by the band-width of the recording apparatus (Fig. 3). The β-peak state tended to be long lived, interrupted only by a propensity of short, band-width limited β/γ-transitions (Fig. 3, *hL*, *iL*, and *pL*). This explains the long β/γ-transition streak in many of the pseudo-phase portraits. However, there were almost no α/β-transitions, explaining why there were generally only weak α/β-transition streaks. Another curious feature of the β-peak state was that it was always preceded by one or more α/γ-transitions before a final α/β-transition (Fig. 3, *hL*, *iL*, and *pL*). With pseudo-phase portraits where the γ-peak was very large, this resulted from either trains of short, band-width limited α/γ-transitions (Fig. 3, *dU*) or a diffusive pattern of conductance change (Fig. 3, *dL*). It might be possible that if the size of the γ-peak was only due to very fast, band-width limited transitions (rather than a multiplicity of near aggregate states), then changing the filtering of the data or portrait lag might resolve γ into smaller peaks. However, we found this not to be the case (Fig. 4).

A number of patches contained more than one maxi channel (Fig. 5). In some of their pseudo-phase portraits, the sojourns to subconductance states were limited enough so that the number of channels can be clearly counted as stepped blocks (Fig. 5*A*, *e*, *f*, *i*, *j*, and *B*, *q*). In others, however, the number of subconductances and/or extent of the γ-peak meant that the blocks smeared to varying extents into a smear along the principle diagonal (Fig. 5, compare with Fig. 1).

## DISCUSSION

Analysis of maxi channel recordings with pseudo-phase portraits has allowed us to answer our two questions. Most of the single maxi channels seemed to form a kinetically homogeneous group, with phase portraits conforming to the scheme in Fig. 2*C*. This together with observations of the traces (Fig. 3) allows us to construct a provisional gating scheme (Fig. 6). This scheme can be partitioned into three modes, normal, β, and β-entry. For extended periods of time, the channel stays in either the normal or β-mode. The normal mode as the name suggests is what the channels exists in most of the time. The β-mode appears to represent a voltage-dependent inactivation that occurs sometimes with the step to −40 mV from +20 mV and is always preceded by the β-entry mode. The presence in some phase portraits of a β-doublet (Fig. 2*A*, *l* and *p*), each peak of which contains a β/γ-streak (thereby confirming that both are indeed β-peaks), could be due to two possibilities. First, there may be two “parallel” β-modes. This is not to suggest that there are two completely different conformational states; that would be taking a too literal view of what a gating scheme represents, for it does only represent something physical. Instead there must be parts of the channel that are conformationally independent of each other, while both effecting conductance. One part causes the β-mode pattern, and the other, some much more long-lived conformation, either increases or decrease the overall conductance (causing the β-mode doublet). The second possibility is that the doublet is caused by a leak current (rather than a modulating independent conformation). This is possibly less likely as it would cause all the other peaks to double.

The large size and principle extension of the γ-peak and inspection of recordings suggest that the γ-peak consists of a large number of near aggregate states. Aggregated states are those states with the same conductance (45). By near aggregated we mean a collection of states with nearly the same conductance. If there are enough of these states and there is some noise, then they will form one indistinguishable peak. What might be the possible solution to the complexity of the γ-peak states? In the traditional approach to ion channel kinetic analysis, aggregated states are distinguished by plotting the distribution of dwell times in the aggregate conductance (26, 41). Each state should contribute a single exponential to the dwell time distribution with a time constant proportional to the microscopic rate constant(s) of escape from that state. In this way the number of states can be found. Of course as the number of states goes up, their exponents become indistinguishable, usually forming a power-law distribution. Some information can still be gained from such a distribution, although not of the detail as obtained from exponentials (9, 29, 32, 37, 40, 44). Both the exponential and power-law distribution approaches have been applied to properly aggregated states (i.e., a single open or closed conductance). However, in the case of the γ-peak the states are near aggregated. This fact may open an avenue to its kinetic analysis using pseudo-phase space, as follows.

The pseudo-phase portrait is a two-dimensional space where each point is the value of the time series at time *x* and time *x* + τ. This can be extended to the concept of an *n*-dimensional pseudo-phase space where each point is the value of the time series at time *x*, time *x* + τ, time *x* +2τ,....*x* + (*n* − 1)τ. As *n* increases more of the local trajectory will be described by the position of a single point and so more near aggregated states should be distinguished as point clusters as they will tend to have different local trajectories. Of course the crunch comes when trying to visualize this *n*-dimensional space and identify clusters, but there are several established methods to do this coming under the rubric of “embedding” or “embedology” (38). Phase space and embedding have been applied to molecular dynamics simulations of proteins (2, 13, 15, 30, 35, 39, 43), but as far as we know they have not been applied to a recording of a physical molecule. By distinguishing near aggregated states, they could be useful for the maxi channel.

Komatsuzaki and colleagues (16) developed a theory similar to phase space called the “multiscale state space network.” Like phase space its abstraction and distance from more traditional modes of analysis could be a deterrent. However, they have also developed a related concept called “local equilibrium states” (LES) that is more relatable to traditional approaches of single channel analysis (1). In traditional analysis, all the samples in a time series are binned in a single amplitude histogram and the bumps in that histogram are used to distinguish observable states (mixture modeling). In the LES approach, distributions are calculated for small segments of the time series instead and these distributions clustered by statistical similarity to discover LES. It can be seen that the LES approach is related to phase space, with the length of the LES segment equivalent to the dimensionality of the phase space. The major difference is that LES does not take into account the time order of samples, whereas phase space does.

Clustering of trajectories in phase space should allow the separation (segmentation in time) and identification of near aggregated states such as constitute the γ-peak. Once this is done more traditional methods can then be applied, such as dwell time analysis and Hidden Markov Modeling. Pseudo-phase space probably cannot be used to replace these traditional methods, as although rough guesses of transition rates (Fig. 6*B*) can be made from the intensity of transition peaks, these will only ever be rough guesses due to the fact of the lag used in construction of the space and that a single transition peak can be aggregated (represent more than one conformational transition). Phase space should be seen as one extra tool for kinetic analysis, not an alternative to those already established.

#### Summary.

Liebovitch and colleagues (5, 17–25, 42) have been proponents of the application of fractals and other aspects of chaos theory and nonlinear dynamics to ion channels. This advocacy proved rather controversial for some in the traditional Markovian camp (see 4, 10, 14, 19, 27, 28, 37). As the interest of Liebovitch and colleagues (23, 24) has been in chaos theory, they have employed pseudo-phase portraits in at least some of their work on ion channels. A pseudo-phase portrait has also been made of the locust muscle BK channel by Usherwood and colleagues (5). We have shown here using pseudo-phase portraits that there is a degree of homogeneity in the kinetics of maxi channels so that a partial gating scheme could be constructed. It should be possible to further refine and expand this scheme with phase space and related methods that can handle the many near aggregated states in the maxi channel. However, this can only be as a complement to traditional kinetic techniques, it cannot replace them, as such the gating scheme we present is only provisional (“rough grain” as we say in the Introduction). Also, the gating scheme corresponds to only one set of experimental conditions, voltage protocol, ionic conditions, etc. Under different conditions (such as changing permeant ions), different trajectories will occur at different rates and these will be a subject of future studies.

## GRANTS

This work was supported by a Canadian Institutes of Health Research Operating Grant MOP12874 and Natural Sciences and Engineering Research Council Discovery Grant 386877.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

## AUTHOR CONTRIBUTIONS

Author contributions: S.P.P. conception and design of research; S.P.P. performed experiments; S.P.P. analyzed data; S.P.P. interpreted results of experiments; S.P.P. prepared figures; S.P.P. drafted manuscript; S.P.P. and J.D.H. edited and revised manuscript; S.P.P. and J.D.H. approved final version of manuscript.

- Copyright © 2013 the American Physiological Society