Piezo's membrane footprint and its contribution to mechanosensitivity.

Piezo1 is an ion channel that gates open when mechanical force is applied to a cell membrane, thus allowing cells to detect and respond to mechanical stimulation. Molecular structures of Piezo1 reveal a large ion channel with an unusually curved shape. This study analyzes how such a curved ion channel interacts energetically with the cell membrane. Through membrane mechanical calculations, we show that Piezo1 deforms the membrane shape outside the perimeter of the channel into a curved 'membrane footprint'. This membrane footprint amplifies the sensitivity of Piezo1 to changes in membrane tension, rendering it exquisitely responsive. We assert that the shape of the Piezo channel is an elegant example of molecular form evolved to optimize a specific function, in this case tension sensitivity. Furthermore, the predicted influence of the membrane footprint on Piezo gating is consistent with the demonstrated importance of membrane-cytoskeletal attachments to Piezo gating.

Introduction

Piezo ion channels transduce mechanical stimuli into electrical activity (Coste et al., 2010). These channels – Piezo1 and Piezo2 in mammals – underlie many important processes in biology, including cell volume regulation in erythrocytes, cardiovascular system development, and touch sensation (Maksimovic et al., 2014; Ranade et al., 2014a; Ranade et al., 2014b; Cahalan et al., 2015). In electrophysiological experiments Piezo channels seem to be exquisitely sensitive to applied mechanical force: when the membrane of a cell is poked gently with a probe, or when pressure is applied to stretch a small patch of cell membrane on a gigaseal pipette, Piezo channels open (Coste et al., 2010; Lewis and Grandl, 2015).

Studies have addressed how Piezo channels ‘sense’ and open in response to mechanical force. In one approach Piezo channels, purified and reconstituted into droplet bilayers, opened when force was applied by swelling a droplet (Syeda et al., 2016). This observation implies that Piezo needs only the cell membrane to couple mechanical forces to pore opening. In another approach, Piezo channels in patches excised from cell membrane blebs (Cox et al., 2016), or in cell excised patches with applied positive or negative pressure (Lewis and Grandl, 2015), open in response to pressure application. These observations also support the notion that Piezo only needs an intact lipid membrane to transduce force into pore opening.

The reconstitution (Syeda et al., 2016) and excised patch (Lewis and Grandl, 2015; Cox et al., 2016) experiments suggest the ‘force-from-membrane’ hypothesis for mechanosensitive gating, which, in its simplest form, invokes lateral membrane tension as the origin of the ‘opening force’ (Sukharev et al., 1999; Perozo et al., 2002; Chiang et al., 2004; Teng et al., 2015). But other experiments suggest additional possibilities for force exertion. When blebs are formed on the surface of a cell by removing local cytoskeletal attachments, certain properties of Piezo mechanosensitive gating change (Cox et al., 2016). And more directly, Piezo gating is altered by applying force to a tether artificially attached to the channel (Wu et al., 2016). Therefore, while membrane-mediated forces alone appear to be sufficient to open Piezo, tethers attached to the membrane or to the channel itself also seem to play a role in Piezo gating.

A partial molecular structure of a Piezo channel has been determined (Guo and MacKinnon, 2017; Saotome et al., 2018; Zhao et al., 2018). Piezo is a trimer of 3 identical subunits that form one central pore and three long arms that extend away from the center. A peculiar aspect of the structure is that the extended arms, which are made of transmembrane helices, do not lie in a plane as would be expected if Piezo normally resides in a planar membrane like most other ion channels. This property of the structure implied that Piezo likely curves the cell membrane locally into a spherical dome (projecting into the cell), which was confirmed by electron micrographs of small unilamellar lipid vesicles (Guo and MacKinnon, 2017).

On the basis of Piezo’s demonstrated ability to curve lipid membranes locally into a dome, a mechanism for membrane tension sensitivity – called the membrane dome mechanism – was proposed (Guo and MacKinnon, 2017). Simply stated, the dome shape provides a source of potential energy for gating – in the form of excess membrane area 'stored' by curving the membrane – when the membrane comes under tension. If the Piezo dome becomes flatter when Piezo opens, then the projected (in-plane) area of the dome will expand, that is, the available in-plane area of the membrane-Piezo system will increase. Under tension , the flatter shape will be favored by energy , where  is here the change in the projected area of the Piezo dome. Therefore, this model rationalized Piezo’s peculiar shape as a means to utilize, for gating purposes, the energy stored in a curved membrane under tension.

However, the membrane dome model of Piezo gating only considered the shape of the membrane within Piezo’s perimeter and not the shape of the surrounding membrane, which is necessarily coupled to the curvature of the Piezo dome. In the present analysis we study the energetic contribution to Piezo gating provided by the shape of the surrounding membrane. Through membrane mechanical calculations, we show that the Piezo dome can strongly curve the surrounding membrane. We find that the energetic coupling between the shape of the Piezo dome and the surrounding membrane amplifies Piezo’s tension sensitivity, and may explain the experimentally observed regulation of Piezo gating by membrane-cytoskeletal attachments.

Results

System of Piezo plus membrane

Figure 1A and B show two orientations of the molecular model of Piezo1 in yellow, which from here on we refer to as Piezo. Shown in grey, a spherical cap is placed such that it intersects the protein near the middle of the transmembrane helices. This grey surface therefore corresponds to the mid-bilayer surface of the membrane. We call the grey spherical cap, with its embedded Piezo channel, a mid-bilayer representation of the Piezo dome. This dome shape, produced by curved Piezo channels embedded in lipid bilayer membranes, has been confirmed experimentally (Guo and MacKinnon, 2017). The intersection of the grey surface and the Piezo channel, shown in cyan, informs that the dome surface area is covered by approximately 20% protein and 80% lipid membrane. Note that, if the unperturbed configuration of the lipid membrane is planar, the Piezo protein must apply, through its curved structure, a distorting force on the membrane to locally bend the membrane into a dome shape. And, of course, the membrane applies an opposing force on the protein. The result is a stable, non-planar equilibrium configuration of the membrane-Piezo system with zero net force, in which the sum energy of the channel and the membrane is minimized. In the present analysis we do not consider the flexing of Piezo. Instead, we focus on the membrane shape associated with a particular (e.g., closed) Piezo configuration (Figure 1A and B).

Figure 1.

Piezo curves the membrane.

(A) Side and (B) top-down (projecting into the cell) views of the Piezo dome. The approximate position of the curved mid-bilayer surface of the Piezo dome is indicated in grey, with the cyan regions corresponding to the intersection of the mid-bilayer surface and the Piezo protein. (C) The curved shape of the mid-bilayer surface of the Piezo dome (indicated in grey) deforms the mid-bilayer surface of the surrounding lipid membrane (indicated in blue) and results in a membrane footprint of Piezo that extends beyond the size of the dome (see Figure 2A for further details). [The atomic structure of the Piezo protein in (A) and (B) corresponds to mPiezo1 with Protein Data Bank (http://www.rcsb.org) ID 6B3R.].

Figure 2.

Membrane footprint of the Piezo dome.

The shape of the Piezo membrane footprint depends on (A) the radius of curvature of the Piezo dome , (B) the membrane bending modulus (membrane bending stiffness) , and (C) the membrane tension . All curves show the cross section of the mid-bilayer surface and its intersection with the Piezo protein. Unless indicated otherwise, we calculated the Piezo membrane footprint using the value observed for Piezo in a closed conformation (Guo and MacKinnon, 2017) with and . For Figure 1C we used the same parameter values as in the left panel of Figure 2A. The range of considered in (B) corresponds to the approximate range of measured for phosphatidylcholine bilayers with different acyl-chain lengths and degrees of unsaturation (Rawicz et al., 2000). Scale bars, 4 .

Since the surrounding lipid membrane connects smoothly to the Piezo dome, the curved shape of Piezo is expected to induce membrane curvature beyond the perimeter of the Piezo dome. The fundamental reason for this is, the energetic cost to curve a membrane contains a term proportional to the membrane’s mean curvature squared. As a result, a sharp transition from the curved dome shape to a planar membrane is associated with a higher energy than a gradual transition. This effect is shown in Figure 1C: the grey surface corresponds to the mid-bilayer surface of the dome in Figure 1A and B and the blue surface to the mid-bilayer surface of the surrounding membrane. We refer to the region of deformed lipid membrane outside the perimeter of the Piezo dome as Piezo’s membrane footprint (Phillips et al., 2009). The total energy of the membrane-Piezo system therefore has to include Piezo’s membrane footprint in addition to the Piezo dome. As we will show, Piezo’s membrane footprint not only influences the total energy of the membrane-Piezo system, but it also has a very large influence on Piezo’s ability to sense changes in membrane tension.

Shape and energy of the membrane footprint

Of all the possible shapes Piezo’s membrane footprint may adopt, we assume that the dominant shape corresponds to that associated with the lowest energy. To calculate this lowest energy membrane footprint, we begin with a well-known expression for the lipid membrane deformation energy (Helfrich, 1973)where is the membrane bending modulus (membrane bending stiffness),  is the membrane tension,  and are the principal curvatures of the mid-bilayer surface (which are functions of position on the membrane), and is the decrease in in-plane area associated with deforming the membrane out of its unperturbed (planar) configuration. The integration is carried out over the surface of the membrane footprint (see Appendix 1). In the integrand of the membrane bending energy in Equation 1 we did not include a contribution  due to the Gaussian curvature of the membrane, which is independent of the shape of the membrane footprint, and a contribution due to the membrane spontaneous curvature (Helfrich, 1973). The latter contribution to the membrane bending energy may need to be considered if the bilayer contains lipids that induce intrinsic curvature.

Next, we minimize by solving a differential equation corresponding to the first variation of Equation 1 set equal to zero – the Euler-Lagrange equation – subject to specific boundary conditions (Fox, 1987). This solution yields the shape of the lipid membrane when its energy is minimal. Using this shape, we calculate by evaluating Equation 1. We used two separate, previously developed methods – one analytical (Weikl et al., 1998; Turner and Sens, 2004; Wiggins and Phillips, 2005; Li et al., 2017) and one numerical (Peterson, 1985; Seifert et al., 1991; Deserno, 2004; Bahrami et al., 2016) – to carry out these calculations. The analytical solutions involve a 'small gradient' approximation of Equation 1 and are therefore only accurate for cases in which the membrane curvature is small. Nevertheless, the analytical solutions provide an important check (see Materials and methods) on the numerical solutions, which are not limited to membranes with small curvatures. Because Piezo can be highly curved, the solutions shown in the main text figures were calculated numerically.

The shape of Piezo’s membrane footprint – and therefore its associated energy – depends on three key physical properties of the membrane-Piezo system: the basic shape of the Piezo dome, the membrane bending modulus , and the membrane tension . The general shape of Piezo in a closed conformation is well defined and approximated here as a dome, or spherical cap, of area and radius of curvature  with  (Guo and MacKinnon, 2017). We assume that the area of the Piezo dome stays approximately constant independent of the conformational state of Piezo. The value of for membranes with lipid compositions common to cell membranes is well documented, around (Rawicz et al., 2000), and values of relevant to living cells and required to activate Piezo have been described (Lewis and Grandl, 2015; Cox et al., 2016). Therefore, calculation of Piezo’s membrane footprint and its associated energy is a well-defined mechanics problem involving no free parameters.

The left panel of Figure 2A shows a cross section through the surface displayed in Figure 1C, calculated as described above, corresponding to , , and ( at ). For context on this value of the membrane tension, commonly studied membranes undergo lysis at around (Rawicz et al., 2000). Thus, is a modest value of the membrane tension, likely experienced by cell membranes under non-pathological stress. The left panel of Figure 2A illustrates that, if one includes the membrane footprint, then Piezo has an extensive reach and, as we will show, this reach has significant functional consequences. But first we inspect how the three physical properties , , and of the membrane-Piezo system affect the size and shape of Piezo’s membrane footprint. If  were to be increased (i.e., if Piezo were to become flatter) then the deformation footprint would become less pronounced and smaller in height (right panel of Figure 2A). The magnitudes of and change the reach of the membrane footprint: larger and smaller values produce a more gradual approach to the plane of the membrane (Figure 2B and C). This relationship is expressed by the characteristic decay length of membrane shape deformations,which appears in the analytical solution to the Euler-Lagrange equation associated with Equation 1 (Appendix 1, Equations A6 and A7). Substituting and yields  , which means that under these conditions Piezo’s membrane footprint is much larger than the Piezo protein itself.

The membrane footprint energy, , is graphed in Figure 3 as a function of Piezo’s radius of curvature.  is greater than or equal to zero because this energy represents the work required to deform the membrane from a plane into the shape of Piezo’s membrane footprint. Figure 3A shows the energetic consequence if Piezo could undergo a conformational transition that changes its radius of curvature: a highly curved Piezo (small ) is associated with a large . We also see that is a sensitive function of membrane tension. If Piezo becomes flatter when it opens, as was proposed in the membrane dome mechanism (Guo and MacKinnon, 2017), then the deformation footprint will contribute to the energetics of gating, as shown (Figure 3B). We denote here the radii of curvature of the Piezo dome in the closed and open conformational states of Piezo by and , with . Under finite membrane tension () Piezo flattening (i.e., a transition from to ) will reduce and thus stabilize the flatter, open conformation relative to the closed conformation. In the absence of membrane tension () the membrane footprint is of no energetic consequence. Thus, Piezo’s membrane footprint would impose a tension-dependent bias, favoring the open conformation of Piezo only when tension is applied, and more so when tension is greater.

Figure 3.

Energy of the Piezo membrane footprint.

(A) Energy cost of the Piezo membrane footprint as a function of the radius of curvature of the Piezo dome . We calculated by minimizing Equation 1 with the membrane bending rigidity  and the indicated values of the membrane tension . (B) Schematic of the proposed mechanism for the mechanical activation of Piezo through membrane tension, for which we assume that the radius of curvature of the Piezo dome in the closed conformational state, , takes a smaller value than in the open conformational state, .

Piezo’s membrane footprint in the absence of applied tension, which is associated with , deserves a comment because the membrane is still highly curved here (see Figure 2C as  becomes smaller). If represents the work required to deform the membrane from a plane into the shape of Piezo’s membrane footprint, and Piezo’s membrane footprint is curved, how can be zero? The explanation is that, in the limit , the membrane curves in a special way around the Piezo dome such that the principal curvatures and in Equation 1 sum to zero. This special surface, called a catenoid, would never truly be achieved in this physical system because thermal fluctuations will not permit zero tension and, potentially, because of deviations of the Piezo dome from a perfect spherical cap. Nevertheless, in the absence of applied tension the deformation footprint should approach the approximate shape of a catenoid. As we demonstrate below, this behavior yields fascinating consequences for Piezo’s mechanosensitivity.

Influence of the membrane footprint on gating

The above analysis suggests that , the energy required to form Piezo’s membrane footprint, should influence the gating properties of Piezo. To investigate the nature of this influence, we add to the Piezo dome energy the energetic contribution due to Piezo’s membrane footprint. The dome energy, , has three additive contributions (Guo and MacKinnon, 2017): the protein energy , in which we include all contributions to the energy of the membrane-Piezo system that do not depend on the membrane tension or the membrane shape, the energy required to bend the membrane in between Piezo’s arms (still part of the dome) against membrane bending stiffness, , and the work required to form the dome against membrane tension, . The total energy of the membrane-Piezo system is therefore given by

is the work required to form both the Piezo dome (i.e., the curved Piezo protein and the curved membrane between the arms) and Piezo’s membrane footprint, starting from a hypothetically planar standard state. The value of is unknown,  was estimated previously to be (approximating all of the dome area to be occupied by lipids), and  with, similarly as above,  being the decrease in the in-plane area of the Piezo dome compared to the planar state (Guo and MacKinnon, 2017). In addition to internal protein interactions,  may include a contribution to the membrane bending energy due to the Gaussian curvature of the membrane (Helfrich, 1973). The Gauss-Bonnet theorem mandates that, for a fixed membrane topology, this contribution to only depends on the boundaries of the membrane, and hence takes a constant value for a given Piezo conformational state and membrane composition (Weikl et al., 1998; Wiggins and Phillips, 2005).

Now, if the dome increases its radius of curvature when Piezo opens, then the total energy difference between the open and closed conformations, , is obtained by applying Equation 3 to each conformation and taking the difference. The upper panel of Figure 4A shows this difference for the tension-dependent components of , and , for a closed to open transition if and (i.e., Piezo being flat in the open conformation), as a function of . and are plotted separately for comparison. It is immediately clear that  is expected to contribute substantially to Piezo’s tension-dependent gating. Two other possible geometries, corresponding to a smaller degree of flattening (Figure 4B), or to flattening from a less curved closed state (Figure 4C), are also shown. We consider the former geometry to explore the decrease in Piezo curvature required for mechanosensitivity, and the latter geometry because the curvature of the Piezo dome may be reduced in cellular membrane environments. In all three cases, for tension values likely relevant to Piezo gating, the contribution to the Piezo gating energy due to Piezo’s membrane footprint is approximately equal to or greater than the tension-dependent contribution due to the Piezo dome itself.

Figure 4.

Energy of Piezo gating.

Tension-dependent contributions to the Piezo gating energy (upper panels) and associated tension sensitivity (lower panels) due to the Piezo dome, , and the Piezo membrane footprint, , as a function of membrane tension for the Piezo dome radii of curvature (A)  and , (B)  and , and (C)  and in the closed and open conformational states of Piezo, respectively. For all calculations, we set the membrane bending rigidity .

The tension sensitivity of Piezo gating depends on how steeply  changes with respect to a change in , . We graph the predicted tension sensitivity of Piezo gating in the lower panels of Figure 4A–C, again with the contributions due to the Piezo dome and Piezo’s membrane footprint separated for comparison. The negative sign indicates that increasing  favors the open conformation. For the dome, sensitivity is constant, equal to a constant change in . For the membrane footprint, the magnitude of the sensitivity is not constant and very large for small . In fact, using the analytical approach for calculating Piezo’s membrane footprint it can be shown that the tension sensitivity grows without bound as the membrane tension approaches zero. This remarkable result means that Piezo’s membrane footprint renders Piezo exquisitely sensitive in the low-tension regime; most sensitive to the smallest perturbations around zero tension. The diverging tension sensitivity as  is a consequence of the idealized catenoidal membrane footprint that is formed at zero tension. The membrane footprint is large and curved, but in a special manner. Once an incrementally small value of membrane tension is applied, this large, previously energy-free, membrane footprint is available to release in-plane area and to unbend, reducing the free energy of the expanded (open) conformation relative to the closed conformation.

Figure 5 presents open probability () and gating sensitivity () curves for the energy values in Figure 4, applied to a 2-state gating model, for which

Figure 5.

Piezo activation through membrane tension.

Piezo activation curves (upper panels) and associated tension sensitivity (lower panels) resulting solely from the gating energy due to the Piezo dome, , and from the gating energy due to the Piezo dome together with the Piezo membrane footprint, , as a function of membrane tension for the Piezo dome radii of curvature (A)  and , (B)  and , and (C)  and in the closed and open conformational states of Piezo, respectively. For all calculations, we set the membrane bending rigidity . We used the values (A) , (B) , and (C)  for the (unknown) contribution of the protein energy to the Piezo gating energy such that gating occurs within the indicated tension range.

The unknown values of were chosen so that opening occurs within the tension range shown. Since is unknown, the Piezo gating tension is not a model prediction. The solid and dashed curves correspond to the gating response with and without inclusion of Piezo’s membrane footprint energy. The membrane footprint energy shifts the curve in the direction of smaller tension values and steepens it (i.e., increases its sensitivity). The particular gating curves shown here depend on a specific, simple gating equilibrium scheme and an unknown value of . Because the contribution of Piezo’s membrane footprint to the Piezo gating energy is so large, however, the conclusion that the position and steepness of the curve should exhibit a strong dependence on Piezo’s membrane footprint will apply to a wide range of possible gating schemes.

Modulation of gating through the membrane

Next, we consider the influence of membrane bending stiffness on Piezo gating. We quantify the magnitude of the membrane bending stiffness by the membrane bending modulus . We are interested in this dependence because membrane bending stiffness is a function of lipid composition, which could vary among different cell types and possibly even between different regions within the same cell. To what extent might membrane bending stiffness influence Piezo’s response to membrane tension? Membrane bending stiffness enters the Piezo gating energy through the dome contribution and the footprint contribution . Figure 6A shows the sum of these two membrane bending stiffness-dependent contributions to the Piezo gating energy and associated and sensitivity curves for three different values of . Note that contributes as a tension-independent constant, whereas the contribution depends on membrane tension. Together,  and contribute significantly to  and thus to gating. This implies that Piezo channels in different cell types and possibly different locations within a cell will exhibit different gating characteristics.

Figure 6.

Modulation of Piezo gating through the membrane.

(A) Membrane bending stiffness-dependent contribution to the Piezo gating energy (left panel) and associated Piezo activation and tension sensitivity curves (middle and right panels) as a function of membrane tension for the indicated values of the membrane bending stiffness . (B) Membrane tension-dependent contribution to the Piezo gating energy (left panel) and associated Piezo activation and tension sensitivity curves (middle and right panels) as a function of membrane tension for infinite and finite membrane compartments. For both (A) and (B) we employed the Piezo dome radii of curvature  and in the closed and open conformational states of Piezo, respectively. For (B) we used the unconstrained membrane arc lengths  and  separating the border of the Piezo dome and the border of the membrane compartment along the membrane in the radial direction, which correspond to the membrane compartment diameters  and , respectively, and set . We calculated the curves in the middle and right panels of (A) and (B) from the total energy of the membrane-Piezo system in Equation 3, with the values (A)  and (B)  for the (unknown) contribution of the protein energy to the Piezo gating energy such that gating occurs within the indicated tension ranges.

The membrane footprint induced by Piezo is expected to influence the distribution of molecules – both lipids and proteins – in the surrounding membrane. Piezo’s membrane footprint should attract lipids and proteins that exhibit an energetic preference for the curved shape of the membrane footprint, and repel molecules that ‘prefer’ other membrane shapes. Conversely, our model of Piezo’s membrane footprint implies that the composition of the surrounding membrane should influence the energetics of Piezo gating. This model prediction raises interesting possibilities for the regulation of Piezo gating in different membrane environments. The membrane footprint induced by Piezo also implies that Piezo channels should interact with each other through the membrane, and hence influence each other’s local distribution in the membrane and gating properties.

Finally, we consider the effect of membrane compartment size on Piezo gating. In the calculations presented so far Piezo was assumed to reside in an infinite membrane that approaches a planar configuration far from the channel. But real cell membranes are compartmentalized. For example, cytoskeletal attachments, which occur at spatial frequencies of up to tens of nanometers, can restrict the shapes a membrane can take (Kusumi et al., 2014). Figure 6B shows the sum of the tension-dependent contributions to the Piezo gating energy, and , and associated and sensitivity curves, for different compartmental restrictions. Membrane compartments with diameters approximately equal to and are compared to an infinite membrane. These compartments restrict the distance between Piezo’s outer perimeter and the edge of the membrane compartment to distances of and  along the membrane in the radial direction, respectively. In general, the effects of membrane compartmentalization are greater in the low-tension regime. This result can be understood in terms of the characteristic decay length of membrane shape deformations in Equation 2: larger values of reduce the size of Piezo’s membrane footprint so that it fits better into the membrane compartment.

We also note that the smaller the membrane compartment, the greater influence it has on Piezo gating. This is because in these particular calculations the membrane is constrained to planarity at the edge of the membrane compartment, but the effect will in general also depend on the membrane slope constraint at the edge of the membrane compartment. The important point is that membrane compartmentalization can have a large effect on Piezo gating because membrane compartments can alter the shape and therefore the energy of Piezo’s membrane footprint. Experimentally observed effects of cytoskeletal removal on some properties of Piezo gating could reflect the importance of Piezo’s membrane footprint for Piezo gating (Cox et al., 2016).

In Figure 6 we neglected the contribution to the membrane bending energy due to the Gaussian curvature of the membrane (Helfrich, 1973). While being independent of the shape of the membrane footprint, the Gaussian contribution to the membrane bending energy depends on the membrane composition and on how the membrane is constrained at the Piezo and membrane compartment boundaries (Weikl et al., 1998; Wiggins and Phillips, 2005). Contributions to the membrane bending energy due to the Gaussian membrane curvature may therefore further modulate Piezo gating in compartmentalized membranes with heterogeneous lipid compositions.

Discussion

While Piezo channels can exhibit complex gating properties, including inactivation and voltage dependence, their dominant functional characteristic is that they open in response to mechanical force (Coste et al., 2010; Lewis and Grandl, 2015). This paper analyzes the influence of Piezo’s unusual dome shape on the lipid bilayer membrane that surrounds the channel. The results depend on three key properties of the membrane-Piezo system and they are known: Piezo’s shape, the lipid bilayer bending modulus, and the levels of tension that can be applied to a lipid membrane. Finding the shape of the lipid membrane surrounding Piezo, and its associated energy, amounts to solving a simple mechanics problem. And the inescapable conclusion is that Piezo, owing to its unusual shape, imposes a large structural perturbation – a deformation called a membrane footprint – on its surrounding membrane.

Depending on the applied membrane tension, Piezo’s membrane footprint can come with a large energetic cost. Consequently, if Piezo changes its shape, for example if it becomes flatter upon opening, then the surrounding membrane will weigh in prominently in an energetic sense to Piezo’s tension sensitivity. Moreover, Piezo’s membrane footprint weighs in in such a way that the tension sensitivity of Piezo gating is greatest in the low-tension regime. This property would seem to render Piezo poised to respond to the slightest changes in cell membrane tension.

In our analysis of Piezo’s membrane footprint we used a spherical dome shape to approximate a more complex underlying geometry of Piezo. Deviations from a spherical dome shape will alter the shape and energetics of the membrane footprint. But the basic idea that Piezo’s curved shape will create an energetically important membrane footprint will still apply.

Piezo’s large membrane footprint rationalizes what at first seemed to be a contradiction in the experimental literature. Certain data show the clear importance of the membrane in mediating Piezo’s mechanosensitivity (Lewis and Grandl, 2015; Syeda et al., 2016; Cox et al., 2016), while other data show the importance of tether attachments (e.g., the cytoskeleton) to the channel or the membrane (Cox et al., 2016; Wu et al., 2016). A large membrane footprint essentially demands that both contributions be energetically important.

Piezo is a very uniquely shaped membrane protein. We think that this shape evolved specifically to exploit the physical properties of the lipid membrane to create a large membrane footprint, enabling exquisite tension sensitivity.

Materials and methods

We used Equation 1 to analytically and numerically calculate Piezo’s membrane footprint, and its associated energy, through the Monge (Weikl et al., 1998; Turner and Sens, 2004; Wiggins and Phillips, 2005; Li et al., 2017) and arclength (Peterson, 1985; Seifert et al., 1991; Deserno, 2004; Bahrami et al., 2016) parametrizations of surfaces, respectively. Appendix 1-sections 1 and 2 provide a detailed discussion of these Monge and arclength solutions. All of the results shown in the main text figures were calculated numerically using the arclength parametrization of surfaces, which allows for large membrane curvatures. In Appendix 1-section 3 we compare the analytical and numerical solutions obtained using the Monge and arclength parametrizations of surfaces. We find that the Monge parametrization of surfaces tends to overestimate the magnitudes of Piezo’s membrane footprint and its associated membrane deformation energy but yields, for the scenarios considered in the main text figures, qualitatively similar predictions as the arclength parameterization of surfaces.

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Thank you for submitting your article "Piezo's membrane footprint and its contribution to mechanosensitivity" for consideration by eLife. Your article has been reviewed by three peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Richard Aldrich as the Senior Editor. The following individuals involved in review of your submission have agreed to reveal their identity: Ardem Patapoutian (Reviewer #1); Olaf S Andersen (Reviewer #2).

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Summary:

This study explores the physical basis of mechanosensitivity of Piezo1 ion channels. Structural studies have shown that Piezo1 channels have a unique curved shape in detergent micelles. This curved architecture is largely retained in liposomes/lipid membranes such that the lipid membranes are significantly deformed upon incorporation of Piezo1 channels. This paper examines the importance of this membrane deformation on the overall gating and mechanosensitivity of the Piezo1 channels from an analytical standpoint. The overall conclusion is that Piezo, owing to its unusual shape, imposes a large structural perturbation (which the authors refer to as membrane footprint) on its surrounding membrane. This membrane footprint weighs such that the tension sensitivity of Piezo gating is greatest in the low-tension regime rendering Piezo poised to respond to the slightest changes in cell membrane tension. The authors' treatment of mechanosensitivity is elegant – it eschews details such as "structural domains" and "conformational changes" and yet is able to incorporate structural information to offer mechanistic insight into how Piezo channels work.

Essential revisions:

1) As the authors note, only partial molecular structures of Piezo have been determined. 12 predicted TM helices per promoter are conspicuously absent, which could contribute significantly to the shape of Piezo. Do the authors think the additional TMs extend the Piezo "dome" or are they part of the membrane footprint? How might the additional TMs affect the parameters in the mechanical calculations?

2) The authors only consider a single aspect observed in the Piezo structure – its unique curvature – in constructing and describing the model of mechanosensitivity. In the context of the dome/membrane footprint gating models, what other aspects of the structure could be important? For example, as seen in Figure 1B, the "arms" of Piezo are "tucked" inward, creating a relatively compact triskelion. Might the three armed architecture afford the ability of Piezo to "curl" outward in an expanded/open state?

3) What would be the curvature of Piezo1, in the open state, which would enable it to be sensitive to mechanical stimuli? This might allow one to rationalize the pretty large (~47kT) energy difference between open and closed states (used for many of the simulations) presented.

4) Would the authors predict that Piezo1 is osmosensitive as well? Change in osmotic pressure would change the intrinsic curvature of the membrane and thus the membrane deformation energy which according to the authors' model would affect gating and mechanosensitivity of the channel. To the best of my knowledge, Piezo1 channels show very little osmosensitivity. It might serve the paper if the authors briefly discuss the different modes of mechanosensitivity (osmosensitive, shear-sensitive, mechano- or stretch sensitive, etc.) and how their model relates to Piezo1's unique mode of mechanical response.

5) Two related points raised by the reviewers:

Throughout the paper the authors consider the Piezo-membrane system as consisting of a single Piezo trimer and its surrounding membrane, and that the "membrane footprint" of Piezo can extend tens of nanometers beyond the actual surface area occupied by the protein. But cellular membranes are highly heterogenous and is occupied by many membrane proteins, some of which may interact with Piezo and modulate function. It seems that the deformed membrane surrounding Piezo would be energetically unfavorable to be occupied by other membrane proteins, at least those that prefer planar membranes. A further consequence of the membrane footprint model is that two neighboring Piezo molecules would repel each other to prevent overlap of deformation footprints (Philips, Ursell et al., Nature 2009). Can the authors discuss how the membrane footprint might effect interaction of Piezos with other membrane proteins, and whether their model precludes the formation of Piezo clustering (a behavior seen in various other ion channel families)?

The final question which I have is whether the authors think that if their proposed mechanism is 'correct' and the membrane deformation energy is a crucial component of gating energetics, would there be interchannel cooperativity – i.e. cooperativity between (two or more) neighboring channels? The authors describe the membrane footprint of a single channel and how it relates to its gating. However, if there are two or more channels in the vicinity, then the overall membrane footprint of the multiple channels might (and in all likelihood will) be different from the sum of individual channels and therefore the energetics and mechanosensitivity would also be different. Furthermore, the nature of the membrane (intrinsic curvature and composition) would also in likelihood affect such cooperativity. I would think that this is an important point and wonder what the authors think about it. It might be worthwhile to discuss this aspect, even if briefly.

Essential revisions:

1) As the authors note, only partial molecular structures of Piezo have been determined. 12 predicted TM helices per promoter are conspicuously absent, which could contribute significantly to the shape of Piezo. Do the authors think the additional TMs extend the Piezo "dome" or are they part of the membrane footprint? How might the additional TMs affect the parameters in the mechanical calculations?

The presence of the additional TMs would make the arms longer and the area of Piezo larger. However, because the first 12 helices corresponding to the first 3 domains are not visualized in any of the structures that have been solved, we suspect that there is likely a flexible connection between the first 3 domains and the others, so that the first 3 can adopt many orientations with respect to the remainder. If this is the case, then they would be structurally passive and not contribute to the formation of the dome (i.e., they might extend into the membrane footprint but presumably would not affect its shape much).

In summary, the dome – defined geometrically by its area and radius of curvature – is formed by the rigid components of Piezo. The first 3 domains do not seem rigidly attached to Piezo. If, in a particular conformation of the channel, the first 3 domains adopt a more rigidly attached configuration, then they could increase the area and alter the energetics of Piezo. For now, the model is based on what we have been able to see.

2) The authors only consider a single aspect observed in the Piezo structure – its unique curvature – in constructing and describing the model of mechanosensitivity. In the context of the dome/membrane footprint gating models, what other aspects of the structure could be important? For example, as seen in Figure 1B, the "arms" of Piezo are "tucked" inward, creating a relatively compact triskelion. Might the three armed architecture afford the ability of Piezo to "curl" outward in an expanded/open state?

Viewed down the 3-fold axis the arms are indeed tucked inward. Viewed from other angles one can see that each arm adopts a left-handed helical twist such that it curls around an approximately spherical dome-shaped membrane surface. In other words, curving of the arms is actually consistent with a spherical dome-like shape. If one imagines that the arms remain adherent to the membrane surface as the dome decreases its radius of curvature (i.e., flattens), then the arms would indeed straighten out (i.e., become less tucked inward) as the dome flattens. Thus, straightening of the arms, at least to some extent, is expected.

We want to emphasize, however, that the spherical dome model is an approximation of some more complex geometry. This approximation captures the essence of a system that we think works by pulling membrane area out of the membrane plane in the closed conformation and permitting that area to re-enter the membrane plane upon opening (i.e., by flattening). In detail, aspects of the structure of Piezo that cause deviations from the spherical dome shape (e.g., by breaking rotational symmetry about the central pore axis of Piezo) will perturb the shape of the membrane footprint and the magnitude of the energy. Nevertheless, the membrane footprint will contribute to the energetics of channel gating.

To highlight this important point raised, we have added emphasis in the third paragraph of the subsection “Shape and energy of the membrane footprint” that the spherical dome is an approximation. We further address this point in the last paragraph of the aforementioned subsection and in the third paragraph of the Discussion section.

3) What would be the curvature of Piezo1, in the open state, which would enable it to be sensitive to mechanical stimuli? This might allow one to rationalize the pretty large (~47kT) energy difference between open and closed states (used for many of the simulations) presented.

This question was addressed in the calculation of membrane shapes and energies for three different gating scenarios in Figures 4 and 5. For the closed and opened radius of curvature, Rc and Ro respectively, we plot energy (Figure 4) and open probability (Figure 5) – and their slopes (sensitivity) – for values Rc→Ro (nm) corresponding to 10.2→∞, 10.2→11.2, and 20→∞. Thus, a change in radius of curvature as small as 10.2→11.2 nm (Figures 4B and 5B), for example, will in theory render Piezo quite sensitive to membrane tension. In Figure 5, the unknown protein energy difference between closed and open states (ΔGDP) was selected for each gating geometry so as to place the tension curve within the ranges shown (similar to experimental observation). In other words, we do show that the curvature changes rationalize those large energies. Incidentally, all of the analysis presented here entailed analytic or numerical calculations – in particular, solution of the Euler-Lagrange or Hamilton equations associated with equation 1 – rather than simulations.

We have revised our manuscript to emphasize, in the second paragraph of the subsection “Influence of the membrane footprint on gating”, the rationale behind the gating geometries considered in Figures 4 and 5.

4) Would the authors predict that Piezo1 is osmosensitive as well? Change in osmotic pressure would change the intrinsic curvature of the membrane and thus the membrane deformation energy which according to the authors' model would affect gating and mechanosensitivity of the channel. To the best of my knowledge, Piezo1 channels show very little osmosensitivity. It might serve the paper if the authors briefly discuss the different modes of mechanosensitivity (osmosensitive, shear-sensitive, mechano- or stretch sensitive, etc.) and how their model relates to Piezo1's unique mode of mechanical response.

Whether osmotic pressure is predicted by the model to activate Piezo1 depends on the scenario considered. In short, to the extent that an osmotic pressure difference across a cell membrane produces membrane unfolding and cell swelling – a regime in which the Young-Laplace equation will apply – the model predicts that an osmotic pressure difference will activate the channel through increases in membrane tension. That said, eukaryotic cells seem to have many compensatory mechanisms that minimize increases in membrane tension over a slow timescale. For example, caveolea appear to serve as membrane area buffers, allowing the membrane to ‘unfold’ when required. Thus, whether or not Piezo1 opens would seem to depend on the timescale over which a tension-inducing membrane perturbation occurs relative to the timescale of compensatory mechanisms (i.e., membrane unfolding, ion and water transport, etc.). To speculate with a specific example, rapid poking of the cell membrane opens Piezo1 channels (transiently), but an osmotic stress that would take more time to evolve might not. In summary, experimental tests to see whether Piezo1 can be activated by an osmotic pressure difference have to ensure that the Young-Laplace regime is achieved.

As alluded to by the reviewer, chemical differences between the two lipid bilayer leaflets of the membrane may induce a spontaneous curvature of the membrane. We have revised our manuscript to discuss, in the first paragraph of the subsection “Shape and energy of the membrane footprint”, how spontaneous curvature of the membrane would enter our model of Piezo’s membrane footprint.

5) Two related points raised by the reviewers:

Throughout the paper the authors consider the Piezo-membrane system as consisting of a single Piezo trimer and its surrounding membrane, and that the "membrane footprint" of Piezo can extend tens of nanometers beyond the actual surface area occupied by the protein. But cellular membranes are highly heterogenous and is occupied by many membrane proteins, some of which may interact with Piezo and modulate function. It seems that the deformed membrane surrounding Piezo would be energetically unfavorable to be occupied by other membrane proteins, at least those that prefer planar membranes. A further consequence of the membrane footprint model is that two neighboring Piezo molecules would repel each other to prevent overlap of deformation footprints (Philips, Ursell et al., Nature 2009). Can the authors discuss how the membrane footprint might effect interaction of Piezos with other membrane proteins, and whether their model precludes the formation of Piezo clustering (a behavior seen in various other ion channel families)?

The final question which I have is whether the authors think that if their proposed mechanism is 'correct' and the membrane deformation energy is a crucial component of gating energetics, would there be interchannel cooperativity – i.e. cooperativity between (two or more) neighboring channels? The authors describe the membrane footprint of a single channel and how it relates to its gating. However, if there are two or more channels in the vicinity, then the overall membrane footprint of the multiple channels might (and in all likelihood will) be different from the sum of individual channels and therefore the energetics and mechanosensitivity would also be different. Furthermore, the nature of the membrane (intrinsic curvature and composition) would also in likelihood affect such cooperativity. I would think that this is an important point and wonder what the authors think about it. It might be worthwhile to discuss this aspect, even if briefly.

To the first point, yes, Piezo’s membrane footprint should influence the distribution of molecules – both lipids and proteins – in the surrounding membrane. In particular, Piezo’s membrane footprint should attract lipids and proteins that exhibit an energetic preference for the curved shape of the membrane footprint, and repel molecules that “prefer” other membrane shapes. Thus, a prediction of the model is that molecules that couple to the membrane shape should show a distribution in the vicinity of Piezo that is different from their bulk distribution. Experiments to test this prediction are underway.

The specific case of Piezo surrounding Piezo is queried. Yes, Piezo channels should interact with nearby Piezo channels through their respective membrane footprints. We agree with the reviewers and with the cited reference that Piezo pairs, it would seem, should repel each other. The possible influence that groupings of Piezo could have on each other is more complex and an interesting topic for further study. We note also that clustering of membrane proteins is often mediated by cytoplasmic anchoring proteins.

The second point is a corollary of the first. Just as Piezo should influence the composition of its surrounding membrane, the composition of the surrounding membrane should influence the energetics of Piezo gating. That is to say, lipids or other proteins that integrate favorably into Piezo’s curved membrane footprint should alter the energetics of the footprint and thus alter the functional behavior of Piezo. This raises interesting possibilities for the regulation of Piezo gating in different membrane environments. The reviewers raise the special case of the ‘other protein’ being Piezo. Yes, we would predict that Piezo channels at a certain density should influence each other’s gating. Experiments to test this prediction are also underway.

We have added these important aspects to the second paragraph of the subsection “Modulation of gating through the membrane”.