Ion channels are embedded in the plasma membrane, a compositionally diverse two-dimensional liquid that has the potential to exert profound influence on their function. Recent experiments suggest that this membrane is poised close to an Ising critical point, below which cell-derived plasma membrane vesicles phase separate into coexisting liquid phases. Related critical points have long been the focus of study in simplified physical systems, but their potential roles in biological function have been underexplored. Here we apply both exact and stochastic techniques to the lattice Ising model to study several ramifications of proximity to criticality for idealized lattice channels, whose function is coupled through boundary interactions to critical fluctuations of membrane composition. Because of diverging susceptibilities of system properties to thermodynamic parameters near a critical point, such a lattice channel’s activity becomes strongly influenced by perturbations that affect the critical temperature of the underlying Ising model. In addition, its kinetics acquire a range of time scales from its surrounding membrane, naturally leading to non-Markovian dynamics. Our model may help to unify existing experimental results relating the effects of small-molecule perturbations on membrane properties and ion channel function. We also suggest ways in which the role of this mechanism in regulating real ion channels and other membrane-bound proteins could be tested in the future.

## Introduction

In addition to separating the cell from its surroundings, the plasma membrane is home to diverse functional processes. Membrane-bound ion channels sense chemical and electrical signals and control conductance to specific ions, leading to the complex dynamics that underlie neural function. Although most ion channels are broadly classified as ligand gated or voltage gated, many are also sensitive to a wide range of modulators including calcium levels, pH, temperature, and lipids (Kinnunen, 1991). Ion channel function can depend on the 2-D solvent properties of the membrane in which they are embedded, in both reconstituted (Bristow and Martin, 1987; Rankin et al., 1997) and in vivo assays (Sooksawate and Simmonds, 2001; Allen et al., 2007). Furthermore, structural studies have demonstrated close association between particular lipids and ion channels (Barrantes, 2004; Zhu et al., 2018), sometimes dependent on their functional state (Gao et al., 2016).

Although early efforts assumed the membrane to be a homogenous 2-D solvent for embedded proteins, it is now thought that the membrane is heterogeneous, with liquid structures often termed “rafts” at length scales of 10−100 nm, much larger than the 1-nm scale of individual lipids (Simons and Toomre, 2000; Dart, 2010). Neurotransmitter receptors are often found to be associated with ordered membrane domains when probed with biochemical methods (Allen et al., 2007), sometimes in a subtype-specific manner (Li et al., 2007). Many channels and channel scaffolding elements are posttranslationally lipidated with palmitoyl groups (Fukata and Fukata, 2010; Fukata et al., 2013; Borroni et al., 2016; Globa and Bamji, 2017), and mutation of palmitoylated cysteines reduces their localization to synapses (Christopherson et al., 2003; Rathenberg et al., 2004; Delint-Ramirez et al., 2011). Protein palmitoylation is highly correlated with partitioning into ordered membrane domains in both intact and isolated biological membranes (Levental et al., 2010; Lorent et al., 2018).

Experiments have suggested a physical mechanism that may underlie these structures. Vesicles isolated from mammalian cell lines have membranes tuned close to a liquid–liquid miscibility critical point (Veatch et al., 2008). When cooled below their critical temperature, Tc, these vesicles macroscopically phase separate into two liquid phases termed liquid ordered (lo) and liquid disordered (ld), which differ in the partitioning of lipids, proteins, and a fluorescent dye (Baumgart et al., 2007). Experiments suggest that in physiological conditions, plasma membranes exist close to but above a miscibility criticality point. In this nearly critical region, they are expected to display a range of critical phenomena that arise because the free energy cost of making relatively large domains becomes comparable to the thermal energy. Components of a fluid membrane are expected to exhibit long-range correlations in space, with a correlation length ξ that diverges as the critical point is approached, ξ ∼ (TTc)−ν (Cardy, 1996). Critical systems also have a long memory, with a correlation time τ that diverges as the critical point is approached, τ ∼ ξz with z a dynamic critical exponent (Hohenberg and Halperin, 1977; Honerkamp-Smith et al., 2012). In addition, changing control parameters such as temperature and the addition of small molecules to the chemical environment leads to large changes to system properties. The influence of a small change in system parameters on physical properties (“susceptibility”) diverges as the critical point is approached, leaving membrane properties particularly sensitive to external perturbations in their critical region. Systems near criticality exhibit “universal” properties that are largely independent of the microscopic properties of the particular system under consideration, making them highly amenable to quantitative description with highly simplified models, a property we take advantage of here.

Previously, we have argued that proximity to this critical point is likely to underlie much of the raft heterogeneity seen in diverse membrane systems (Machta et al., 2011), with embedded proteins subject to long-range critical Casimir forces (Machta et al., 2012). More recently, we have shown that n-alcohol anesthetics take membrane-derived vesicles away from criticality by lowering Tc (Gray et al., 2013). Despite structural diversity, n-alcohol anesthetics are known to exert similar effects on diverse ion channels (Franks and Lieb, 1994), leading us to speculate that these effects might arise because anesthetics mimic or interfere with native regulation of channels by their surrounding membrane. In support of this, we found that several conditions that reverse anesthetic effects on ion channels and organisms also raise critical temperatures in vesicles (Machta et al., 2016).

In this study, we explore several consequences of thermodynamic criticality for a membrane-bound protein whose internal state is coupled to the state of its surrounding membrane. This study is motivated in part by the observation that many hydrophobic compounds exert influence both on the critical temperature of membrane de-mixing and on ion channel function. Although some of these compounds likely interact directly with hydrophobic regions of ion channels (Mihic et al., 1997; Borghese et al., 2006; Nury et al., 2011; LeBard et al., 2012; Yip et al., 2013), this work focuses on what influence they could have indirectly, through their effects on membrane thermodynamics.

We develop a simple model for an ion channel regulated by its surrounding membrane by coupling a lattice Ising model of a membrane to a highly abstracted lattice ion channel. We show that when the membrane is held close to a critical point, this type of coupling leads to a qualitatively distinct regime including strong responses to perturbations that influence Tc, a hierarchy of time scales, and non-Markovian dynamics. Additionally, we show that these effects are critical phenomena, arising from the diverging generalized susceptibilities and correlation times that emerge near the critical point. This demonstrates that the effects we observe arise not from details inherent to our model, but from the proximity to a critical de-mixing transition. Even though our model is a drastic simplification of real membrane/protein systems, the conclusions drawn are expected to be relevant to these systems when membranes exhibit critical behavior and when proteins exhibit multiple functional states with altered domain affinity.

## Materials and methods

Two classes of computer simulations are used in this manuscript, each simulating a 2-D lattice Ising model decorated with a single lattice channel as described in Results. For measurements of mean conductance, we made use of a recently developed algorithm that deterministically computes exact free energies for 2-D lattice spin models (Thomas and Middleton, 2013) with user-specified numerical precision. All results presented from this algorithm were computed on 128 × 128 lattices using 1,028-bit numbers. Lattice parameters and lattice channel boundary conditions were determined as described in Results and Discussion. Couplings between sites that are plus or minus infinity are set to ±10 in our implementation because computations become very slow when couplings are larger and because this leads to acceptably small errors. We expect errors of order e−20, corresponding to the ratio of Boltzmann factors between correct and incorrect couplings. For dynamic simulations, we used the Kawasaki algorithm, a Markov-chain Monte Carlo procedure whose dynamics are in the model B universality class (Hohenberg and Halperin, 1977). All dynamic simulations were performed on 128 × 128 lattices with temperature and lattice channel boundary conditions as described in Results. We define a sweep to be a time step in which 1282 moves are attempted, so that every spin is attempted to move on average twice. Every sweep, we supplement these dynamics with a single attempt to flip the lattice channel to satisfy detailed balance according to the Metropolis probability.

### Online supplemental material

In the supplemental material, we first provide a more detailed argument for the form of the scaling function needed to capture fixed/free boundary conditions used to make Fig. 2 D. We then discuss the scaling form of the dynamic correlation function χ(τ,r,ξ) and the effect of our choice of JR → ∞. Finally, we discuss subtleties in our Gaussian blurring procedure used to make Fig. 3 A. Figure S1 shows that results are robust to changes in JR. Figure S2 shows results are robust to resolution of Gaussian blurring.

## Results and discussion

We consider a lattice system consisting of membrane degrees of freedom {s} and a single hypothetical lattice channel (Fig. 1 A) with state R and Hamiltonian $H tot ( { s },R )$. The Hamiltonian of our combined system comprises three components: where $H mem$ describes interactions of membrane lipids with one another, ER measures the energy of the lattice channel in state R without considering membrane interactions, and $H int$ describes the interaction between the lattice channel in state R and the components that it contacts at its boundary, which we notate by $s∈∂.$

(1)

As in previous work, we model the membrane as a square lattice of spins s1 = ±1 with the usual Ising Hamiltonian

$Hmem=−J∑i,jsisj,$

where the sum runs over all nearest neighbor pairs, neither of which are contained in the lattice channel. In our model, up/down spins roughly correspond to more ordered/disordered domain partitioning components, with a lattice constant corresponding to l ∼ 1–2 nm of membrane. This scale is based both on the approximate size of a lipid of ∼0.8 nm2 (Chiu et al., 2002; Alwarawrah et al., 2010) and on measurements of the correlation length near the critical point of cell-derived vesicles that suggest an analogy to a lattice model with l ∼ 2 nm (Veatch et al., 2008). The lattice channel contained in our model has an internal state that can take two values, R = ±1, which could correspond to open/closed. This state can be affected by a chemical potential (perhaps modulated by a ligand) that in our model contributes a term in the Hamiltonian ER = −µR. Crucially, we also include an interaction between membrane and lattice channel of the form

$Hint=∑i∈∂hi(R)si,$

where this sum is over spins that border the lattice channel, whose boundary conditions in state R are determined by hi(R) (Fig. 1 B). These interactions between our lattice channel’s state R and surrounding spins mimic a coupling between a protein’s functional state and the membrane’s state as seen experimentally (Bristow and Martin, 1987; Sooksawate and Simmonds, 2001; Allen et al., 2007). They could arise from hydrophobic mismatch (Andersen and Koeppe, 2007; Soubias et al., 2008), distinct lateral pressure profiles in the different phases (Gruner and Shyamsunder, 1991), or specific interactions with particular components that partition strongly into distinct phases (Levitan et al., 2014).

In previous work, we have considered a similar model for a protein’s fixed interaction with its surrounding membrane to understand the membrane-mediated critical Casimir forces that act on a pair of proteins (Machta et al., 2012). Previous authors have also considered the role of single-site proteins in an Ising model which are driven out of equilibrium and which thereby prevent phase separation (Sabra and Mouritsen, 1998). To understand how criticality could directly influence protein conformational equilibria, we consider three types of boundary conditions: fixed boundary conditions in which hi(R) = ±JR, free boundary conditions in which hi(R) = 0, and Janus boundary conditions in which hi(R) = +JR on one side of the lattice channel and hi(R) = −JR on the other side. We consider these simplest boundary conditions to gain a qualitative understanding of the effects of different boundary conditions on the lattice channel’s static properties. For our studies of mean activity, we use JR → ∞ for consistency with our past work (Machta et al., 2012), corresponding directly to a physical system in which a lattice channel of slightly larger radius has interaction energy hi = J. In the supplemental text, we explore the dependence of our results on the strength of the coupling JR. We find that the large interaction energies we use will give quantitatively similar results to a system in which a lattice channel of the same radius has interactions at least on the order of kBT.

### Lattice channel average conductance

We first investigated how a change in the membrane’s properties might influence average channel activity, as would be measured by a whole-cell recording or other technique probing the response of many channels together. In our model, the membrane and the lattice channel embedded within it are in thermal equilibrium. Membrane (Ising spin) degrees of freedom are tuned close to a liquid–liquid miscibility critical point, and a single lattice channel interacts with the membrane by preferring different local liquid environments when in different functional states. In equilibrium, the system consisting of lattice channel and membrane will be in a given state with probability determined by an appropriate Boltzmann distribution,

$P({s},R)=e−βHtot({s},R)/Z,$

where Z is defined such that the sum of probabilities is unity.

We can integrate over membrane degrees of freedom to isolate the lattice channel whose internal states are occupied according to $P(R)=exp(−βF R )/Z,$ where

$exp( −βF R )=∑ {s} exp[ −βH tot ( { s },R ) ]$
(2)

defines $F R$, the free energy of state R.

In the context of our model, changes to membrane miscibility influence lattice channel activity if P(R) depends on the details of $H mem$. Changes to $H mem$ can lead to a change in a state’s free energy $ΔF R$, with resulting changes in P(R). In this two-state example, the membrane influences the activity of the lattice channel, P(R = ±1), through changes to $ΔFdiff=ΔF+1−ΔF−1$. In the case where P(R = ±1) $≪$ 1, P(R = +1) increases by a factor of $exp( −βΔF diff )$. In the general case, the ratio P(R = +1)/P(R = −1) increases by $exp( −βΔF diff )$. Thus $exp( −βΔF diff )$ measures the degree to which a change in $H mem$ potentiates the + state (Fig. 2 A).

It is often difficult to estimate free energies from simulation, but for 2-D lattice systems, there has been substantial recent progress (Thomas and Middleton, 2013), which we can use for our lattice membrane/channel system. To study the static properties of our equilibrium system, we make use of a remarkable algorithm developed for spin glasses that uses Pfaffian elimination to exactly calculate partition functions to user-specified precision on lattices with arbitrary nearest neighbor couplings Jij in zero field (Thomas and Middleton, 2013). We implement the Hamiltonian defined previously by setting all interactions Jij between spins to βJ, interactions between lattice sites internal to the lattice channel to minus infinity, and interactions between lattice sites contained in the lattice channel and spins to hi(R) (Fig. 1 B).

We estimate the potentiation of function that accompanies a 1% change in Tc, comparable both to the natural variation seen in cell-derived vesicles (Veatch et al., 2008) and to the magnitude of the effect seen with physiologically relevant concentrations of n-alcohols (Gray et al., 2013). We examine boundary conditions in which the R = −1 state has fixed boundary conditions (hi = +∞), whereas the R = +1 state has either free boundary conditions (hi = 0) or Janus boundary conditions. The results are shown in Fig. 2 (B–D). In Fig. 2 B, this potentiation is shown for lattice channels in which the R = +1 state has free boundary conditions, at T = 1.05Tc, for channels with a range of radii (the lattice channels are displayed in Fig. 4 and in the supplemental material, and the radii are calculated as the radius of the circle circumscribing each lattice channel). Larger lattice channels see more pronounced potentiation owing to the increased boundary length of interaction. In Fig. 2 C, the potentiation is shown for the same lattice channels over a range of temperatures. Away from the critical point, this change has only a small effect on the state occupancy—the potentiation is small. However, near the critical point, the potentiation becomes much larger, even for the modest 1% change in Tc explored here. This change in Tc is comparable to that seen with addition of clinically relevant concentrations of anesthetic (Gray et al., 2013; Machta et al., 2016). For the largest lattice channel sizes explored here, our observed potentiation is similar to that observed for the GABAA channel (see Fig. 2 of Franks and Lieb [1994]) whose radius is ∼6 nm. Although we do not know details of GABAA’s interaction with its surrounding membrane, our results imply that anesthetic effects on membrane Tc could in principle lead to observed changes in channel function even in the absence of direct binding interactions between compound and channel.

For channels regulated through this mechanism, we expect that, for a given channel and given boundary conditions, different perturbations should have effects on channel activity predicted solely through that perturbation’s influence on Tc. Experimental results on three different channels that respond differently to n-alcohols are consistent with this prediction. In Fig. 3, we plot results from whole-cell channel recordings from Mascia et al. (1996), Nakahiro et al. (1996), and Zuo et al. (2001) for several channels in the presence of n-alcohols plotted against extrapolated ΔTc as investigated in Gray et al. (2013). As shown in the figure, for any given channel, the effects of these n-alcohols are well predicted by their effect on Tc. Although this collapse of the data is certainly suggestive, future work is needed to exclude a more mundane possibility: that it arises because of a mutual correlation of both assays with the hydrophobicity of different n-alcohols.

An Ising lattice at fixed chemical potential for up spins (not included in our model) has a diverging susceptibility; a small change in this chemical potential leads to a diverging change in the number of up spins near the critical point, which would result in large changes to the activity of our lattice channel. However, for most components, biological membranes are held at fixed particle number rather than chemical potential, equivalent to fixed number (not chemical potential) of up spins in our model. Changes to the number of up spins, as would accompany biological changes in most lipid levels, do not lead to large changes to the lattice channel’s activity. One exception to this could be cholesterol which can be rapidly transferred between internal membranes and the plasma membrane so that its plasma membrane concentration under perturbation is fully described by neither a fixed chemical potential nor a fixed particle number. More broadly, the relationship between particle number and chemical potential requires care to interpret (Ayuyan and Cohen, 2018).

### Lattice channel kinetics

We next looked at the role a critical membrane could play in shaping the kinetics of ion channel function as would be measured in single-channel voltage clamp experiments. Real ion channels have kinetics that span a wide array of time scales that are typically understood as arising from internal states of the channel itself (Mortensen and Smart, 2007). Here we consider a hypothetical lattice channel that contains only a single time scale, but that couples to its surrounding lipids, acquiring slower time scales from the critical membrane in which it is embedded. In our studies of dynamics, we consider fixed boundary conditions and use hi(R) = RJ, the same as the interaction between neighboring lipids (R = ±1 is the state of the lattice channel).

We set ER = 0 to treat both states of the lattice channel symmetrically. We use the local Kawasaki algorithm (Kawasaki, 1972) as we have done previously (Machta et al., 2011) which implements diffusive model B dynamics (Hohenberg and Halperin, 1977). This algorithm exchanges neighboring spins with probabilities chosen to maintain detailed balance. Using these dynamics, a sweep (in which every spin is proposed to exchange with its neighbors once) corresponds to roughly a microsecond of real time, because each lattice site corresponds to the size of a lipid (∼1 nm2) and lipids diffuse at a rate of ∼1 μm2/s (Machta et al., 2011). One caveat is that cells are likely in the regime where hydrodynamic relaxation (Honerkamp-Smith et al., 2012) competes with diffusive dynamics. Our Kawasaki dynamics forbid exchange with spins in the lattice channel, but every sweep the lattice channel changes state with the Metropolis probability to ensure detailed balance.

Although our lattice channel attempts to change its state R during every sweep, its apparent kinetics are much slower. Representative traces of R(τ) are shown in Fig. 4 A after convolution with a Gaussian of width σ = 103 sweeps. Because patch-clamp recordings measure times on the order of 50–100 µs (Hamill et al., 1981; Kasianowicz et al., 1996; Sakmann and Neher, 2009), we plot R(τ) after convolution with the Gaussian $k( τ )=e −τ 2 /σ 2$, where σ = 103 (see supplemental material for bare curves and a discussion). Thus, although our lattice channel has only two states, the recording shows short spikes reaching levels in between the two, a feature common to real recordings that presumably has a similar explanation (Hamill et al., 1981). The traces shown in Fig. 4 A share qualitative features with real ion channel traces, spanning many time scales (Mortensen and Smart, 2007): “flickers” shorter than σ, single “openings” (which often contain many short flickers), and “bursts” in which a series of openings occur close to each other. The time scale of these features increases as the critical point is approached (Fig. 4 A). Similar changes in the time scale of ion channel dynamics arising from the addition of some anesthetics have been observed experimentally (Wachtel, 1995). Our findings indicate that these could result from perturbations to the proximity of the membrane to criticality, although they are typically interpreted as arising from direct binding interactions between anesthetic and channel. It is interesting to note that synthetic membranes brought close to a possibly analogous liquid–gel critical point can display currents in the absence of proteins that bear a striking resemblance to those mediated by ion channels (Antonov et al., 1980; Wodzinska et al., 2009). Our results suggest that this commonality may arise because both protein channel and membrane-mediated currents acquire their long time scales from critical fluctuations of the membrane itself.

We quantify these findings by looking at the time autocorrelation of the lattice channel, $χ( τ )=〈 R( 0 )R( τ ) 〉$ for a range of temperatures (Fig. 4 B), demonstrating that correlations in state persist much longer as the critical point is approached. We also perform simulations for lattice channels of different radii, all at T = 1.05Tc (Fig. 4 C), demonstrating that larger lattice channels have slower kinetics owing to the increased area for interaction with surrounding membrane. These curves quantify the many time scales that can be seen qualitatively in Fig. 4 A; they are not closely approximated by single exponentials.

Although the combined system of lattice channel and membrane taken together is Markovian, when considered in isolation, the lattice channel displays non-Markovian dynamics, as evidenced by the nonexponential autocorrelation function. This implies that some memory of its history is stored in the membrane degrees of freedom surrounding it. Far from criticality, where no long time scales are inherent to the membrane, this effect is small, and lattice channel kinetics, even when considered in isolation, appear almost Markovian. However, close to the critical point, the lattice channels acquire a range of time scales from the membrane in which they are embedded and have dynamics that are highly non-Markovian. The non-Markovian behavior of several real ion channels and other membrane-embedded proteins has been well characterized, and various internal states of these proteins have been hypothesized (Rankin et al., 1997; Yamashita et al., 2005; Schmauder et al., 2011). It is certainly likely that much of this observed non-Markovian behavior does indeed arise because these channels contain internal states not directly probed with conductance. However, our results suggest that, in some cases, the history of the protein’s state may be stored in the membrane rather than in internal states of the protein itself.

### Scaling collapse of average conductance

We next wanted to verify that our observed effects could be understood as a critical phenomenon. Properties that arise due to proximity to criticality are often universal, applying to surprisingly diverse real systems as well as simplified minimal models. Scaling functions are a useful tool to probe universality. These functions only describe the universal properties of a system and do not contain the specific details that distinguish different real systems, or even real systems from models within the same universality class. As a result, scaling collapses are important tests of which model predictions are expected to apply to real systems that are qualitatively different in their microscopic details. According to scaling, the free energy associated with inserting a lattice channel with particular boundary conditions should have a universal contribution that depends on lattice channel size, boundary conditions, and the distance to the critical point:

(3)

where yt = ν = 1 and yh = γ = 15/8 are the scaling dimension of the fields t and h, r is the radius of the lattice channel, and $U 1$ is a universal function. Eq. 3 represents our knowledge of the functional form of $F R$: although we do not know the specifics of its dependence on t, r, and h, we know from scaling that it does not depend on each independently, but rather on the combinations $rt 1/y t$ and $h/t y h /y t$, as well as on other parameters represented by the ellipsis.

Taking a derivative of Eq. 3 with respect to t, we find that

(4)

where $U 2$ is another universal function that is an appropriate sum of derivatives of $U 1 .$

We verify that our measured potentiation can indeed be collapsed in the manner predicted by Eq. 4 in Fig. 2 D for the fixed/Janus boundary conditions. However, for the fixed/free systems, we find an additional contribution that arises more directly from the boundary. This term is proportional to the length of the boundary and should scale as

(5)

leading to a similar collapse but with $l y t t$ on the x axis (see supplemental material for discussion). Because the microscopics of our model do not enter into the derivations of the above equations, the collapse to these universal functions depicted in Fig. 2 for our simulations demonstrates that it is the proximity to criticality of our simulations, rather than their microscopic details, that dominate our results.

### Scaling collapse of lattice channel kinetics

We expect that the long time scales seen in Fig. 4 A are inherited from the surrounding membrane. Near the Ising critical point, the correlation length diverges as $ξ∝t −ν$ (where the rescaled temperature is t = (TTc)/Tc), and the correlation time diverges as $τ cor ∝ξ z$ where a dynamic exponent is z = 4 − η = 3.75 for the Kawasaki dynamics used here (Hohenberg and Halperin, 1977). If the curves of R(τ) reflect properties of these slow critical fluctuations, then we might expect that autocorrelation functions in systems with different radii and correlation lengths χ(τ,r,ξ) might take a universal form:

$χ( τ,r,ξ )=r λ U( r/ξ ,τ/ξ ),$
(6)

with λ = 2yb = 1 where yb is the scaling dimension of the boundary operator, which is expected to be 1/2 from conformal field theory arguments (Cardy, 2004; see supplemental material for a brief discussion). To check this, we performed simulations for the same lattice channels shown in Fig. 4 C but now at different temperatures chosen so that $r/ξ$ is fixed. Although the bare curves decay over time scales that differ by a factor of 102, when we plot $χ( τ )/r$ versus $τ/ξ z$, we see collapse onto a single curve within stochastic error (Fig. 4 D) as predicted by Eq. 6, thus demonstrating that the dynamics observed are critical phenomena.

### Conclusions

Regulation by the nearly critical membrane might be very widespread for both ion channels and other membrane-bound proteins. There are hints that the membrane may be playing an important role in localization (Li et al., 2007) and in some cases in direct regulation (Allen et al., 2007) of channels. Our model could account for the sensitivity of many diverse ion channels to chemically diverse anesthetics (Franks and Lieb, 1994), which we have demonstrated lower the critical temperature of cell-derived vesicles by ∼4°K, just over 1% (Gray et al., 2013). Although this model remains to be critically tested, this work suggests that it is thermodynamically feasible for n-alcohols and other hydrophobic components to influence channel function without directly binding to channels. This explanation is appealing in that it might also explain why cholesterol depletion, which changes the membrane’s “magnetization” (Zhao et al., 2013), also affects many anesthetic-sensitive channels (Sooksawate and Simmonds, 2001). Recent lipidomics work in neuronal and synaptically derived vesicles have shown that membrane composition undergoes regulated changes during development (Tulodziecka et al., 2016), possibly in part to modulate channel function through a mechanism like the one explored here.

Recent work has demonstrated that anesthetics have little influence on the mean conductance of gramicidin channels, whose gating is quite sensitive to some combination of features of their local membrane environment (Herold et al., 2017). Although those authors interpret this as evidence that anesthetics do not influence membrane properties, our results naturally predict that gramicidin, a small peptide, should be relatively insensitive to changes in Tc. Gramicidin channels dimerize across leaflets, forming a pore and leading to a deformation of the surrounding membrane. The energetic cost of this deformation depends strongly on the hydrophobic thickness of the membrane, leading to dependence of function on some membrane properties. Our work suggests that these relatively small channels would not be particularly sensitive to perturbations that only influence Tc. Perturbations that influence Tc alone do not change the local structure of the membrane dramatically, but instead influence the sizes of typical domains. As such, channels sensitive to Tc must be relatively large. Gramicidin takes up only a slightly larger area than a lipid and is therefore much smaller than anesthetic-sensitive channels, which tend to be ∼10 nm across.

In Weinrich et al. (2017), the authors show that when the membrane is in a single phase state, gramicidin channels have kinetics described by a single exponential for the channel dwell time, consistent with the predictions of our model in the absence of membrane coupling. However, in membranes with two coexisting phases, the channel adopts kinetics that cannot be fitted with a single exponential, but are reasonably well described by a fit to the sum of two exponentials. The authors left open the possibility that the two time scales come from different kinetics in the two phases, or from a mechanism where channels acquired more complex kinetics from switching between the phases. The latter is similar to our model.

Our results suggest that channels modulated in the manner considered here should have their partitioning into small domains modulated by the addition of ligand, a prediction that could be tested using superresolution techniques. In cells, partitioning into larger domains below the critical temperature is likely to be complicated by interactions with the cortical cytoskeleton, which interfere with macroscopic phase separation (Machta et al., 2011). Although macroscopic domains are observed in plasma membrane vesicles isolated from cortical cytoskeleton and viewed at low temperature, it is not known whether these membranes preserve the physical and chemical properties of domains important for channel regulation by this mechanism. Although most previous studies, both by others (Reynwar and Deserno, 2008; Katira et al., 2016) and by ourselves (Machta et al., 2011, 2012), have focused on the role of membrane thermodynamics in localizing proteins into domains, this work suggests that the same domains could couple more directly to function. We predict that when a receptor binds ligand, it will change its preference for its surrounding lipid environment, leading to different interaction partners and an imprint on the state of the local membrane that will persist even after the ligand has dispersed. We have highlighted a specific impact of thermodynamic criticality on a single ion channel, and further work will clarify how this mechanism contributes to neural function.

## Acknowledgments

We thank Bill Bialek, Colin Clement, Jim Sethna, and Ned Wingreen for useful discussions. B.B. Machta thanks C.K. Thomas and A.A. Middleton for making their code available on their website.

This work was supported by an National Defense Science and Engineering Graduate Fellowship, National Institute of General Medical Sciences, National Institutes of Health (T32 GM008313 to O. Kimchi and R01 GM110052 to S.L. Veatch), National Science Foundation (MCB 1552439 to S.L. Veatch), and a Lewis-Sigler Fellowship (B.B. Machta). B.B. Machta acknowledges the Aspen Center for Physics and National Science Foundation PHY-1607611.

The authors declare no competing interests.

Author contributions: O. Kimchi, S.L. Veatch, and B.B. Machta designed research, performed research, and wrote the paper.

José D. Faraldo-Gómez served as editor.

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