Cooperative response and clustering: consequences of membrane-mediated interactions among mechanosensitive channels

TL;DR

This paper investigates how membrane-mediated interactions among mechanosensitive channels cause collective behaviors, phase transitions, and delays in cellular responses to osmotic shocks, with implications for E. coli.

Contribution

It reveals the impact of inter-channel interactions on collective dynamics and response delays, highlighting phase transitions in channel states.

Findings

Inter-channel interactions induce phase transitions in channel states.

Collective dynamics cause delays in cellular response to osmotic shocks.

Cell-to-cell variability in response times is significantly affected.

Abstract

Mechanosensitive (MS) channels are ion channels which act as cells' safety valves, opening when the osmotic pressure becomes too high and making cells avoid damage by releasing ions. They are found on the cellular membrane of a large number of organisms. They interact with each other by means of deformations they induce in the membrane. We show that collective dynamics arising from the inter-channel interactions lead to first and second-order phase transitions in the fraction of open channels in equilibrium relating to the formation of channel clusters. We show that this results in a considerable delay of the response of cells to osmotic shocks, and to an extreme cell-to-cell stochastic variations in their response times, despite the large numbers of channels present in each cell. We discuss how our results are relevant for {\it E. coli}.

Equations

$$\frac{F}{N}=(e_{int}+e_{mem})-Ts,$$

$$s=$$

$$f\ln \left[\frac{\rho (1-f)(1-{\varphi }^{\mathrm{out}})}{(1-\rho f)(1-{\varphi }^{\mathrm{in}})}\right]+\frac{1}{\rho }\ln \left[\frac{(1-\rho f)}{(1-\rho )}\right]+$$

$${\varphi }^{\mathrm{out}}(1-f)\ln \left[\frac{(1-{\varphi }^{\mathrm{out}})}{{\varphi }^{\mathrm{out}}}\right]+{\varphi }^{\mathrm{in}}f\ln \left[\frac{(1-{\varphi }^{\mathrm{in}})}{{\varphi }^{\mathrm{in}}}\right]\};$$

$$e_{int}=2f({\epsilon }_{cc}+2({\epsilon }_{co}-{\epsilon }_{cc}){\varphi }^{\mathrm{in}}+$$

$$+({\epsilon }_{cc}-2{\epsilon }_{co}+{\epsilon }_{oo})({\varphi }^{\mathrm{in}}{)}^2);$$

$$e_{mem}=\frac{(\Delta G_0-\tau \Delta A)}{2}(2f{\varphi }^{\mathrm{in}}+2(1-f){\varphi }^{\mathrm{out}}-1).$$

Full text

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Abrupt changes in the osmolarity of the environment is a hazard most organisms are subject to at one time or another Poolman et al. (2004); Wood et al. (2001); Roberts (2000); Wegmann (1986); Brown et al. (1986); Blomberg (1997). A sudden drop in osmolarity (an osmotic shock) will cause water to rush into a living cell, and requires an immediate response by the cell to prevent it from getting damaged or undergoing lysis from the increased tension on the cellular membrane. Mechanosensitive channels (or MS channels) are ion channels located on the cell membrane, which open when the membrane tension becomes too high Martinac and Buechner (1987); Sukharev et al. (1993), and play a crucial role in the cell’s defence mechanism against osmotic shocks Cui et al. (1995); Levina et al. (1999). They act as safety valves, releasing ions and decreasing the osmotic pressure and the membrane tension. Mechanosensitive channels are found in many organisms, and have been well characterised in the bacterium E. coli Kung (2005); Booth et al. (2007); Pivetti et al. (2003).

The cellular membrane in which the mechanosensitive channels are inserted is a lipid bilayer. The interior of the bilayer is hydrophobic, making it energetically favourable for it to thicken or compress to match the hydrophobic parts of the channel proteins inserted in the membrane Phillips et al. (2009). This results in a deformation profile around each channel, with the thickness of the bilayer being a function of position. This deformation mediates a short-range effective force between two neighbouring channels, similar to the force between two nearby corks floating on water, which interact through the deformation they induce on the surface of water. This interaction can be attractive or repulsive, depending on the shapes of the two molecules. Furthermore, a theoretical analysis suggests that the interaction between two neighbouring channels lowers the tension needed to open them during an osmotic shock Ursell et al. (2007), raising the possibility that their function could be influenced by their spatial distribution on the membrane (as already noticed for other membrane proteins Botelho et al. (2006); Skoge et al. (2006)). This is reinforced by the fact that the channels’ attractive forces suggest that they may agglomerate into clusters. Our goal in this paper is to determine the consequences that the inter-channel interaction has on the dynamics of this system, focusing in particular on channel clustering and its consequences for the cell’s response to osmotic shocks.

A preliminary study of mechanosensitive channel clustering was done in Guseva et al. (2011); Guseva (2011). In that work, diffusion, leading to the formation of clusters of channels, and opening were considered two separate processes. This assumption made the model easier to analyse, but it is hard to justify: in reality, diffusion and gating take place simultaneously.

In this work, we formulate a model of the collective dynamics of mechanosensitive channels, where diffusion and gating are considered simultaneous, and no assumption of time separation between clustering and gating is made. Using a combination of analytic techniques and numerical simulations, we analyse the equilibrium and the dynamics of the system, focusing in particular on the response of the channels to osmotic shocks. We find that the interplay between the spatial and the internal degrees of freedom of the channels leads to unexpected collective phenomena, with possible implications for their biological function. We show that the fraction of open channels undergoes a phase transition as the membrane tension increases; and this transition changes from second-order to first-order as the density of channels crosses a critical value. We explain this change in the nature of the transition as the result of collective gating induced by a cluster of channels which appears for high densities. Studying the time evolution of the system after applying an osmotic shock, we find that clustering leads to dramatic changes in the channels’ response, slowing down considerably their gating. In addition, clustered channels show extreme ensemble variations in their response times, despite the large numbers of channels present in each cell, what could translate into large stochastic cell-to-cell differences in response times in a population of cells. Finally, we discuss how our results are relevant for the stress response of E. coli and other organisms.

In our model, we regard the cellular membrane as a two dimensional square lattice of size $L$, where each of the sites can be either empty or occupied by a channel. We focus on the Mechanosensitive Channels of Large Conductance (MscL), which have been well studied in this context Batiza et al. (2002); Kloda et al. (2008); Chang et al. (1998). The total number $N$ of channels is regarded as constant, so that the density $\rho$ of channels (mean number of channels per lattice site) is fixed—we are in the canonical ensemble. We consider that MscL can be in one of two states, closed or open. The interaction energies between two channels have been obtained by minimising the energy functional defined by the deformation profile Ursell et al. (2007). For small distances (a few nanometers) between channels, if $\varepsilon_{oo}$, $\varepsilon_{cc}$ and $\varepsilon_{oc}$ are the interaction energies between a pair of open channels, a pair of closed channels, and an open and a closed channel, respectively, we have $\varepsilon_{oo}<\varepsilon_{cc}<0$ and $\varepsilon_{oc}>0$: two open channels and two closed channels attract each other, the attraction being stronger in the first case; and a closed channel repels an open channel (see Guseva et al. (2011) for the complete energy profiles). The diagram in Fig. 1 illustrates our lattice model.

If we consider that all channels are in the same state, without the possibility of gating, this model is exactly the lattice gas model. The 2D lattice gas model is exactly solvable, due to its equivalence to the Ising model Newman and Barkema (1999), and presents a phase transition from a homogeneous to a clustered channel distribution as the density of channels increases Guseva et al. (2011).

The model we describe shares some similarities with the spin-1 Ising model analysed in a mean-field approximation in the grand canonical ensemble in Sivardière and Lajzerowicz (1975), but our approach has the advantage of yielding more information about the spatial distribution of channels.

We start by studying the equilibrium properties of the system, as defined by our lattice model. We use a mean-field approximation, which will allow us to write explicit expressions for the energy and entropy of the system, from which we can find its free energy. As a simplifying assumption, we assume the existence of at most one cluster. For the energy values we consider, the existence of a single cluster in equilibrium is reasonable and supported by test simulations. Let $f$ be the fraction of channels in the cluster; the other channels are spread throughout the rest of the membrane. Furthermore, let $\phi^{\rm in}$ be the fraction of open channels within the cluster, and $\phi^{\rm out}$ the fraction of open channels outside the cluster. The three quantities $f$, $\phi^{\rm in}$ and $\phi^{\rm out}$ are the thermodynamic variables of our model. Our next job is to write the free energy of the system in terms of these variables. Cluster formation and channel gating are then studied by finding the global minimum of the free energy. For example, a cluster is present if $f>0$ in the state of minimum free energy.

The free energy per channel, $F/N$, for a given temperature $T$, can be written as

[TABLE]

where the entropy per channel, $s$, can be estimated via combinatorial analysis, calculating the number of configurations that channels can assume. The energy per channel is divided into two terms: the interaction among channels, $e_{int}$, and the interaction of each channel with the membrane, $e_{mem}$. For each of the configurations devised in the preceding calculation, the interaction among channels can be estimated considering that channels only interact with nearest neighbours. The interaction with the membrane depends on the difference of energies between closed and open states and the work due to the variation on the channel’s area in the gating process. In the mean-field approximation, we find (see Supplementary Material for complete derivation of the results):

[TABLE]

[TABLE]

[TABLE]

Here the parameters $\Delta G_{0}$, $\Delta A$ and $\tau$ are the difference between the energies of open and closed states, the difference in membrane areas between the open and closed configurations of a channel, and the membrane tension, respectively. The term $\Delta G_{0}$ covers both the energetic cost of membrane deformation and the cost of changing the internal structure of the channel. We used $\Delta G_{0}=50\ k_{B}T$ and $\Delta A=20\ \mbox{nm}^{2}$, following Chiang et al. (2004). For these parameters, a single non-interacting channel has a 50% opening probability at the tension $\tau=2.5\ k_{b}T/nm^{2}$ Ursell et al. (2007).

The equilibrium distribution for this system is then given by the values $f_{eq}$, $\phi^{\rm out}_{eq}$ and $\phi^{\rm in}_{eq}$ which minimize the free energy, for given values of $\rho$ and $\tau$ (in the following, the subscript eq will be omitted). The fraction of open channels on the whole lattice is given by $P_{o}=f\phi^{\rm in}+(1-f)\phi^{\rm out}$. Figure 2 shows how $f$ and $P_{o}$ vary as functions of the membrane tension, $\tau$, for different values of the density $\rho$.

We see in Fig. 2(a) that the fraction of open channels $P_{o}$ undergoes a transition from nearly zero (all channels closed) to non-zero values as the membrane tension $\tau$ increases. The nature of the transition depends on the channel density $\rho$. For small values of $\rho$, the transition is continuous: $P_{o}$ increases smoothly from zero beyond a critical value of $\tau$. For $\rho$ greater than a critical value $\rho_{c}$, however, the transition is discontinuous, with $P_{o}$ jumping abruptly to a positive value at the critical tension.

The key to explaining this phenomenon is in the corresponding behaviour of the cluster fraction $f$, depicted in Fig. 2(b). At low tensions, the channels are closed. Since the force between two closed channels is attractive, they can form a cluster if their density is high enough; this is the case for the two upper curves in Fig. 2(b). Comparing with Fig. 2(a), we see that these correspond to the densities for which the transition in $P_{o}$ is abrupt: if a cluster already exists at low tension, $P_{o}$ has a discontinuous transition. The reason for this comes from the fact that the interaction energy between two open channels is much greater than any other combination of channels, and this becomes more and more so as the tension increases, since high tensions favour the opening of the channels. In equilibrium, if one of the channels in the cluster is open, all the others are open as well, because any mixture of open and closed channels incurs a heavy cost in free energy. So at a critical tension, the whole cluster opens, and since the cluster contains a finite fraction of the channels in the cell, this results in the abrupt jump in $P_{o}$ seen in Fig. 2(a).

For lower channel densities, on the other hand, there is no cluster at lower tensions. As the tension is increased, it eventually becomes favourable for channels to open, and as they do, they will tend to bunch together in a cluster, because of the high open-open interaction energy. But because in this case there was no cluster to start with, the number of open channels will increase gradually as the tension rises, and so will the cluster size. This predicts that the cluster size $f$ and the fraction of open channels $P_{o}$ will undergo a continuous transition, and increase in tandem. This is exactly what we see in Fig. 2(a). In both the low-density and high-density regimes, the clustering reduces considerably the threshold for channel opening—see Fig. 2(a)—, which might have implications for the response of the cell to osmotic shock, as we shall see in the following. These collective phenomena are a direct consequence of the inextricable link between the spatial distribution of channels and their internal gating dynamics.

In order to understand the response of the channels to an osmotic shock, we have to go beyond the equilibrium theory and look at their time-dependent activation dynamics. To study the coupled gating and diffusion dynamics, we use a Monte Carlo simulation scheme with two possible actions in each step: (i) with probability $p_{G}$, a randomly chosen channel attempts changing its state (closed/open); or, (ii) with probability $1-p_{G}$, it attempts to move to one of its four neighbouring sites, if it is vacant. The attempts succeed with a probability of acceptance, $A$, according to the criterion: $A=e^{-\beta\Delta E}$ if $\Delta E>0$, or $A=1$ if $\Delta E\leq 0$, where $\Delta E$ is the change in energy between final and initial configurations of the system following the attempt. Thus, the algorithm is a variation of the Kawasaki dynamics, for which the position updates are local, making it suitable for non-equilibrium simulations of the lattice gas Newman and Barkema (1999). The probability $p_{G}$ is determined by the ratio of the rates of diffusion and gating: $p_{G}=\lambda_{G}/(\lambda_{G}+\lambda_{D})$, where $\lambda_{G}=1/\Delta t_{G}$ and $\lambda_{D}=1/\Delta t_{D}$ are the rates of gating and diffusion, given by the experimentally measured characteristic times of gating and diffusion, $\Delta t_{G}$ and $\Delta t_{D}$, respectively. Each Monte Carlo step is given after $N$ random choices of channels to attempt change of state or diffusion, where $N$ is the total number of channels. We relate a Monte Carlo step, $\Delta t_{MC}$, to a real time interval using the weighted average $\Delta t_{MC}=p_{G}\Delta t_{G}+(1-p_{G})\Delta t_{D}$. In our simulations, we have used $\Delta t_{G}=4\ \mu s$ and $\Delta t_{D}=208\ \mu s$ Booth (2014); Kumar et al. (2010) (see Supplementary material), for which we have $\Delta t_{MC}\cong 8\ \mu s$. Since the increase in channel area during the gating process precludes the determination of a single value for the lattice constant, we had to choose it in a range of reasonable biological values. We use $L=400$ and $\rho=0.002$ for lattice size and channel density, respectively, in accordance to typical values for E. coli (see Supplementary material). In all our simulations, we start the system from an equilibrium situation at low membrane tension $\tau$. We then increase $\tau$ abruptly, mimicking an osmotic shock, and follow the dynamics of the channels using the algorithm described above. The value of $\tau$ is kept fixed throughout the simulation

It is instructive to compare the time evolution of a hypothetical system of non-interacting channels to that of the real system of interacting channels. After approximately $10$ MC steps per channel, all the non-interacting channels are open and stay in this state until the end of the simulation (black squares, Fig. 3a). The behaviour of the system of interacting channels, in contrast, is governed by two processes acting on vastly different time scales (black circles, Fig. 3a): (i) the fast opening of the isolated channels outside the cluster; (ii) and the much slower opening of the channels in the cluster. The most striking aspect of the dynamics shown in Fig. 3a is the dramatic variability of the opening times of the cluster: in one run of the simulation, the cluster may open in a few microseconds, and in another it may take $100$ milliseconds to open. This massive variation is a result of the long-range correlations created by the interactions between channels. The stochastic nature of the cluster is a direct effect of the nontrivial collective behavior of the interactive channels.

The variation is further highlighted by the histogram of the fraction of open channels in a cell some (long) time after the osmotic shock is applied, in $500$ independent runs of the Monte-Carlo simulation (see Fig. 3b). We see that the distribution is bimodal, with roughly similar numbers of cells with open and closed clusters, even after very long times after the shock. This means that in a population of cells subjected to osmotic shock, there will be massive differences in the response times from one cell to another, even if the cells are genetically identical and even though they feel exactly the same stress. In essence, the collective dynamics that emerged from the channel interactions amplifies stochastic fluctuations at the molecular scale to the “macroscopic”, population scale, making them potentially detectable by population assays.

We note that this large variability disappears once the tension becomes strong enough. For $\tau=4.0\ k_{b}T/nm^{2}$ and $5.0\ k_{b}T/nm^{2}$ (respectively, up and down-triangles in Fig. 3a), both clustered and freely-diffusing channels respond very quickly, with all the channels in the system opening after only 10 MC steps.

Finally, we would like to emphasize that the membranes are a crowded environment and that the tight packing of channels withing this environment may introduce additional effects on gating. The most important effect appears due to the packing frustration and entropic tension. Packing frustration may lead to a decrease in the tendency to gate due to space limitation created by neighbors. Additionally, entropic tension originated by environmental crowding may have a significant influence on MS channels’ conformational change due to volume exclusion Lindén et al. (2012). These effects could be considered, as a first approximation, accounting to a different choice of parameters in a simple extension of our model (specifically the energy difference between open and closed states).

Using fluorescence microscopy and Western blot analysis Bialecka-Fornal et al. (2012), the average number of MscLs in native E. coli cells have been estimated between $300$ and $1000$ channels. These results are similar to the one obtained with ribosome profiling Li et al. (2014) that measure $360$ to $560$ channels per cell. These numbers take the channel density close to or above the threshold for cluster formation at low tensions. Patch-clamp experiments complemented with fluorescent and atomic force microscopy show evidence for crowding and collective response of channels in liposomes Grage et al. (2011). Other studies have shown non-homogeneous distributions of overexpressed MS channels in live bacterial cells  Wahome et al. (2009); Norman et al. (2005); Bialecka-Fornal et al. (2012). Although all these studies suggest cluster formation for native channels, the debate around this question is still open. A recent study, through use of PALM (photo-activated localization microscopy) and SPT (single particle tracking), had shown strong indications that labeling with fluorescent molecules predisposes MscL channels to form clusters van den Berg et al. (2016). In either case, if there is any form of channel aggregation in bacterial cells, the collective phenomena we describe here may be directly relevant for the osmotic response of bacteria. Furthermore this model can be extended to other types of channels, such as electrically sensitive ion channels, which are also expected to react cooperatively to external stimuli Cervera et al. (2014).

Possible evidence for the large variability in channel activation predicted by our analysis is the recent observation of very late channel gating activity in E. coli cells subject to osmotic shock Booth (2014): gating was seen as long as $100\ ms$ after the shock. Since isolated MscL channels are known to gate within a few microseconds after their tension threshold is passed, it is difficult to explain this observation if the channels do not interact. This is naturally explained by the variability of channel activation, however: Fig. 3 shows that a cluster could take a time of the order of $100\ ms$ to open. Another recent work Buda et al. (2016) also highlights the large cell-to-cell variability of the downshock responses. Furthermore this work shows a very slow cell volume recovery, which may also indicate channels cooperative activity.

Calculations based on the ionic flux through single open channels suggest that as few as $5$ to $10$ channels would be enough to protect a cell Booth (2014). This contrasts with the recent measurements of MscL numbers on native cells, which indicate numbers of channels up to two orders of magnitude greater than this estimate. With so many channels in a native E. coli cell, simultaneous opening of all channels would lead to a drastic release of intracellular material, as well as depolarisation of the membrane potential Booth (2014), with potentially fatal consequences for the cell. Hence this high expression level of channels is still a mystery. As seen in Fig. 3, the presence of the cluster significantly delays the opening of the whole system of channels, compared with the non-interacting case, for a shock with smaller membrane tension. Thus, clustering could provide a mean to self-regulate the simultaneous opening of a large number of channels, in order to restore the osmotic equilibrium of the cell and function as a channel reservoir if more of them are needed in case of a severe shock. This is an admittedly speculative, but plausible fitness advantage for the large numbers of channels found in E. coli.

I Acknowledgments

It is a pleasure to thank Ian Booth, Heloisa Galbiati and Samantha Miller for important discussions. This work was supported by the European Union Seventh Framework Programme [FP7/2007-2013] (NICHE; grant agreement 289384).