Piezo’s membrane footprint and its contribution to mechanosensitivity

1Piezo ion channels transduce mechanical stimuli into electrical activity. 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. 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.

2Studies 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. 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, or in cell excised patches with applied positive or negative pressure, 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.

3The reconstitution and excised patch 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’. 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. And more directly, Piezo gating is altered by applying force to a tether artificially attached to the channel. 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.

4A partial molecular structure of a Piezo channel has been determined. 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.

5On 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. 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.

6However, 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.

7Figure 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. 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).

8Since 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. 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.

9Of 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 energywhere 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. The latter contribution to the membrane bending energy may need to be considered if the bilayer contains lipids that induce intrinsic curvature.

10Next, 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. 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 and one numerical – 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.

11The 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. 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, and values of relevant to living cells and required to activate Piezo have been described. Therefore, calculation of Piezo’s membrane footprint and its associated energy is a well-defined mechanics problem involving no free parameters.

12The 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. 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.

13The 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, 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.

14Piezo’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.

15The 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: 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

16is 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. In addition to internal protein interactions, may include a contribution to the membrane bending energy due to the Gaussian curvature of the membrane. 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.

17Now, 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.

18The 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.

19Figure 5 presents open probability and gating sensitivity curves for the energy values in Figure 4, applied to a 2-state gating model, for which

20The 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.

21Next, 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.

22The 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.

23Finally, 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. 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.

24We 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.

25In Figure 6 we neglected the contribution to the membrane bending energy due to the Gaussian curvature of the membrane. 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. 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.

26While 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. 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.

27Depending 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.

28In 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.

29Piezo’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, while other data show the importance of tether attachments (e.g., the cytoskeleton) to the channel or the membrane. A large membrane footprint essentially demands that both contributions be energetically important.

30Piezo 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.

31We used Equation 1 to analytically and numerically calculate Piezo’s membrane footprint, and its associated energy, through the Monge and arclength 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.