Frame tension governs the thermal fluctuations of a fluid membrane: new evidence

Two different tensions can be defined for a fluid membrane: the internal tension, $\gamma$, conjugated to the real membrane area in the Hamiltonian, and the frame tension, $\tau$, conjugated to the projected (or frame) area. According to the standard statistical description of a membrane, the fluctuation spectrum is governed by $\gamma$. However, using rotational invariance arguments, several studies argued that fluctuation spectrum must be governed by the frame tension $\tau$ instead. These studies disagree on the origin of the result obtained with the standard description yet: either a miscounting of configurations, quantified with the integration measure, or the use of a quadratic approximation of the Helfrich Hamiltonian. Analyzing the simplest case of a one-dimensional membrane, for which arc length offers a natural parametrization, we give a new proof that the fluctuations are driven by $\tau$, and show that the origin of the issue with the standard description is a miscounting of membrane configurations. The origin itself of this miscounting depends on the thermodynamic ensemble in which calculations are made.


Introduction
Bilayer membranes are quasi-two-dimensional fluid sheets formed by spontaneous self-assembly of lipid molecules in water. They are of the utmost importance for the proper function of living cells, as they spatially separate intracellular compartments and present a boundary to the extracellular environment. Their mechanical properties play a key role in many cellular processes like motility, proliferation and endocytosis. Hence, a physical understanding of the bilayer membrane and its mechanics is necessary for the proper understanding of biological cells, and has motivated a long series of investigations. In particular, a number of questions have been raised regarding the definition of the tension of a fluid membrane, stirring up much debate among physicists.
The elasticity of a fluid membrane is traditionally studied in the framework of the Helfrich effective Hamiltonian 1,2 : Here the integral runs over the surface area A of the membrane, H and G are the mean and Gaussian curvatures, while κ and κ are the bending rigidity and the Gaussian rigidity, respectively. According to Gauss-Bonnet theorem, the surface integral over the Gaussian curvature G can be written as a constant plus a bound- ary term which, for larger membranes, is negligible in comparison with the γA term. It must be noted that Eq. 1 applies to a fluid membrane with fluctuating area A: either the membrane is incompressible but exchanges lipid molecules with a reservoir, and then the intrinsic (or internal) tension γ is essentially the chemical potential of the reservoir of molecules 3 ; or the membrane is compressible but contains a fixed number of molecules, and then γ is the mechanical tension associated with a departure of the lipid surface concentration from the equilibrium configuration 4 .
The intrinsic tension γ must not be confused with the frame tension τ, which corresponds to the force per unit length that the interface would exert on a surrounding frame whose area A p is kept fixed, and which is the actual mechanically accessible quantity. Intrinsic and frame tensions are clearly different for a fluctuation area: γ is the conjugate variable to the surface area A, while τ is the conjugate variable to its area projected on the frame A p = L p × L p . The latter is then a thermodynamic quantity which depends on the entropy of the membrane, and can be seen as the renormalized version of the former.
When the membrane fluctuates only weakly about a plane, its statistical properties are treated analytically, based on the following set of assumptions: and y, thereby excluding conformations in which the surface forms overhangs. The area of the membrane is then A = A p 1 + (∇h) 2 dxdy.
2. Assuming small fluctuations, the energy of the membrane is described with the quadratic approximation of the Helfrich Hamiltonian Eq. 1 (omitting the contribution of the Gaussian curvature): 3. Monge integration measure: the membrane partition function Ξ is obtained by summing over all interface configurations, which mathematically expresses as a functional integral: Defining an integration measure D[h] requires a discretization of the interface 5 . Here, the membrane is discretized in N p × N p patches with constant dimension a p (with N p = L p /a p 1) on the reference frame, and the integration mea- m,n=1 dh mn / where , the quantum of height fluctuation, is a characteristic length introduced to render the integration measure properly dimensionless.
In the rest of the present paper, we will refer to this set of assumptions as the Monge model (although this is more than just the Monge parametrization, obviously).
A third tension has been introduced in literature, named fluctuation tension r, and defined as the q 2 −coefficient in the membrane fluctuation spectrum: Using very general arguments based on rotational invariance of the free energy of an incompressible membrane, Cai et al. 3 came to the conclusion that the fluctuation tension must coincide with the frame tension: r = τ. This result was then extended to the case of a compressible membrane by Farago 4 3 , the latter derive the corrective terms to the integration measure in order to recover r = τ, while still using the approximated Hamiltonian Eq. 2. In spite of the results presented in these different studies, expressions of Eq 4 with both r = τ 3,6-12 and r = γ 13-16 are still commonly found in the devoted literature, and still nowadays the exact relation between the different membrane tensions is the object of numerical investigations 17,18 .
Elucidating whether the fluctuation tension r matches the frame tension τ, or the intrinsic tension γ, or none of them, is not just a purely academic question: it is also crucial for experimental [19][20][21] and particle-based numerical 13,[22][23][24][25] investigations dealing with membrane fluctuations. The intrinsic tension, introduced within the Helfrich Hamiltonian, is not a quantity which is directly measurable; its determination relies on accurate theoretical relationship with the frame or fluctuation tension.
In the present paper, focusing on the simpler case of a onedimensional (1D) membrane embedded in the plane (or equivalently, a two-dimensional membrane which fluctuates along one direction only), we provide a new demonstration that the membrane fluctuations are governed by the frame tension. This demonstration enforces previous arguments advanced to justify that r = τ, and then increases our confidence in the final result. Moreover, our analysis allows to clearly point out the inconsistencies in the Monge model: in agreement with Cai et al. , we show that they are caused by an inaccurate integration measure, rather than the use of the approximated Hamiltonian. Nonetheless, restoring rotational invariance of the integration measure is not mandatory to recover r = τ from calculations: a consistent expansion of the Hamiltonian and the measure leads to the correct result.
The outline of the paper is as follows: in Section 2, we give one more evidence than the Monge model has some consistency issues: analyzing the fluctuations of a 1D incompressible membrane either with fluctuating physical length L but fixed projected length on the frame L p , or with fixed L but fluctuating L p , we show that Monge model does not satisfy equivalence of thermodynamic ensembles, in contradiction with fundamentals of statistical physics. Specifically, we obtain r = γ in the first ensemble, while r = τ in the second ensemble. In Section 3 we discuss the coarse-graining procedure upon which the Monge model is built, highlighting the origin of the troubles with this model: the choices made for the expression of the coarse-grained Hamiltonian (or its quadratic approximation) and that of the integration measure are not compatible choices to build a free energy that satisfies rotational invariance. In Section 4 we develop a coarsegrained model for a 1D membrane, for which a rotationally invariant integration measure is built from first principles. Using a consistent expansion of this measure an the Hamiltonian, we obtain r = τ for a membrane with fluctuating length and then recover ensemble equivalence. In Section 5 we show that to obtain this result, the only correction needed to the Monge model is to take into account the fluctuating character of the number of patches in the integration measure. We then discuss the situation of a compressible membrane with a fixed number of molecules. We show that here again, the Monge integration measure must be modified to recover r = τ. We also discuss the case of an interface between two fluids, which has no bending rigidity and so is stabilized by surface tension only. In that case intrinsic tension and frame tension cannot be distinguised for small fluctuations.
2 One more evidence of Monge model inconsistency: non-equivalence of ensembles We start our analysis by calculating the average square height and average length of a 1D incompressible membrane within two different statistical ensembles using the Monge model, and show that the expressions are different in the thermodynamic limit, in contradiction with fundamentals of statistical physics.

(T, L p , γ) ensemble
We first consider an incompressible membrane in the ensemble (T, L p , γ) where temperature T , projected length L p and intrinsic tension γ are fixed (see Fig. 1a). The 1D version of Helfrich Hamiltonian Eq. 1 is (where C(s) is the local curvature along arc length s, and L the length of the membrane), and its quadratic approximation writes In agreement with the Monge model, we consider a discretized version of the membrane made of N p 1 patches whose x coordinates are distributed evenly, with projected distance a p = L p /N p . Assuming fixed boundary conditions h(x = 0) = h(x = L p ) = 0, we expand h(x) as a sine Fourier series: The Hamiltonian Eq. (6) rewrites in terms of theh n as: where q n = nπ/L p .
We then evaluate the partition function of the membrane We introduced in the last equality J h→h , the Jacobian associated with the change of variables from the real to reciprocal space. Because of the linear relationship between the two sets of variables (Eq. 7), we know that J h→h is constant. Its value can be obtained, e.g., by calculating a simple Gaussian integral in both direct and reciprocal spaces. We show in Appendix A that, with our choice for normalization of Fourier coefficients, one has J h→h = ((N p − 1)/2) N p /2 . Summing over all Fourier componentsh n finally yields Let us emphasize again that the number of molecules that compose the interface fluctuates, so Ξ Monge is really a grand-canonical partition function. Assuming moderate tension and/or thermodynamic limit ( κ/γ L p ), and using a continuous description of Fourier modes, it comes Note that the introduction of the Jacobian J h→h ensures that ln Ξ Monge is extensive at the thermodynamic limit: ln Ξ Monge ∝ L p . It will also affect the expression of the frame tension τ. We then obtain the average squared amplitude of height straightforwardly: where α n = L p (γq 2 n + κq 4 n )/4. Actually, defining the fluctuation tension from the mean square amplitude of Fourier mode is not very convenient when it comes to compare between different thermodynamic ensembles, because the wavenumber does not necessarily follows the same discretization in different ensembles (e.g, multiples of π/L p in ensemble where L p is fixed, and multiples of π/L in ensemble where L is fixed). As a consequence, the variableh n does not correspond to a same wavenumber value in different ensembles. To avoid this complication, we express the average of real variables such as the average squared height h 2 or equivalently, the average membrane length L = L p 1 + h 2 /2 , which can be deduced directly from the partition function as: Eqs 11 and 13 are the 1D versions of the expressions of h 2 n and A derived for a 2D membrane with the Monge model 3,26 . As announced above, within this description the fluctuation tension identifies with the intrinsic tension: r = γ. In ref. 6 , expression of L has been derived assuming κ/γ a p , whereas Eq. 13 assumes κ/γ L p . Hence, both expressions converge to the same following formula within the range a p κ/γ L p : The frame tension is given by In the range a p κ/γ L p , verified by most membranes, this expression simplifies to: Although the condition κ/γ a p is not required for our purpose, it is easy to see in this regime that γ differs from τ: according to Eq. 14, the small fluctuations approximation is satisfied whenever k B T / √ κγ 1. Since κ/γ a p , k B T /a p γ can have any finite value. According to Eq. 18, one then have τ = γ in general (anticipating that ∼ a p , as we will argue in Section 4, the argument of the ln function is 1, and so τ > γ). The specific case of an interface between two immiscible fluids, for which κ = 0, will be discussed in Section 5.3.
Note that including quartic terms in the expansion of the Helfrich Hamiltonian would lead to corrective terms into the fluctuation spectrum and thus in the expression of r. Using Wick's theorem, it comes that r = γ (1 + O (k B T /γL)) 3,27 . Clearly, these corrective terms cannot yields r = τ, as this was already emphasized by Cai et al.. In the rest of the manuscript we will not consider these corrections coming from the quartic terms in H .

(T, τ, L) ensemble
The same incompressible membrane is now treated within the (T, τ, L) ensemble, in which the frame tension τ and the membrane length L are fixed, while the projected length L p fluctuates. The effective Hamiltonian associated with this ensemble is derived from H by a double Legendre transform: and its quadratic approximation writes: Note that we have parametrized the height with the arc length s rather than x, because L -and not L p -is fixed here. Still using fixed boundary conditions h(s = 0) = h(s = L) = 0, we expand the membrane height as h(s) = ∑ N−1 n=1h n sin (nπs/L). The approximated Hamiltonian then rewrites: where q n = nπ/L. After integrating the canonical partition func- and still assuming moderate tension or thermodynamic limit ( κ/γ L), we obtain where a = L/N is the patch length. It comes straightforwardly Note that this expression is identical to Eq. 11 with the substitution γ → τ and L p → L. From this observation, ref. 6 (see footnote 5 herein) argued that the fluctuation tension should be renormalized as r = (L p /L)τ. However, as we already emphasized above, we are comparing the square amplitude of height for two different wavenumbers in the two ensembles, since here q n = nπ/L = nπ/L p . Hence, h 2 n does not correspond to the same quantity in both ensembles. Instead, we compare the relation between membrane and frame length in both ensembles: which can be inverted, in the regime of small fluctuations (L − L p )/L p 1, as: The internal tension γ = k B T ∂ ln Z/∂ L| T,τ expresses here as Eq. 27 then implies that here r = τ ! Moreover, following the same arguments as in Section 2.1, one can easily realize from Eq. 28 that γ = τ for finite values of k B T /aτ.
Note that we took care to distinguish the patch length (named a p and a, respectively) in the two ensembles. It could be argued that the difference between these two cutoffs exactly compensate for the apparent difference in the expression of the average squared height or average length obtained in the two ensembles. A relation between a and a p can be set by imposing ensemble equivalence for the average energy H : according to equipartition theorem, each quadratic term in Eqs. 8 and 22 contributes k B T /2 to the average energy. Thus, equivalence of ensemble requires that the number of modes in both ensembles, respectively N p = L p /a p and N = L/a, are equal. Clearly, with this relation Eqs. 24 and 27 do not match Eqs. 11 and 13. Therefore, ensemble equivalence is not satisfied for the average squared height or average length, in the thermodynamic limit (L, L p → ∞), in contradiction with fundamentals of statistical physics.
Let us emphasize that we assumed L p γ/κ 1 in the calculations, so we are not in the vanishing tension regime, and the equivalence of ensembles is not questionable here 6 . Indeed, Eq. 27 (or 13) shows that in this regime, L and L p are extensive vari- ables which are linearly related for a fixed tension. In the next two sections we identify the origins of the paradox we pointed out, and show that r = τ is the correct result indeed.

Tracking the Monge model inconsistencies
The description of a membrane at the mesoscopic scale -where it appears as a continuous sheet -relies on a coarse-graining construction from its description at the atomic level. To understand the origin of the issue pointed above, it is useful to come back to the coarse-graining construction at the roots of the Monge model. As before, we consider a 1D incompressible membrane with fluctuating number of molecules. Our starting point is then the classical grand-canonical partition function of the lipid molecules that compose the membrane. After integrating over the molecules' momenta, this writes: where N is the (fluctuating) number of molecules, the r α = (x α , y α ) are their position vectors, µ 0 the chemical potential of the reservoir of molecules, U ({r α }) the intermolecular potential, and λ the thermal de Broglie length.
We now want to sum over all microstates corresponding to a same membrane shape. In a first step, we define a discrete interface by introducing N p N points (but N p 1), whose locations are the coarse-grained degrees of freedom (dof) that define the membrane shape. In the Monge model, these points are distributed evenly on the reference axis x, with fixed projected distance a p , and the height h n of point n (1 ≤ n ≤ N p ) is defined as the height of the center of mass of the molecules whose x coordinate lies within na p , (n + 1)a p . Note that this is not the only coarse-grained model we could build, e.g., rather than points evenly distributed along the reference frame, we could have chosen them evenly distributed along the membrane geometric support. Grouping all microscopic configurations with same heights {h n }, the partition function rewrites: where H ({h n }) is defined through: In this equation y n ({r α }) is the height of the center of mass of the the patch n, expressed in terms of the molecules coordinates r α : where Π n is the boxcar function, defined as Within the context of a continuum model, the quantum of height fluctuations introduced in Eq. 29 may be thought of as an arbitrary coarse-graining length for vertical displacements, so that the height of a patch center must be increased or decreased by an amount or greater before its new height is treated as distinct from its old one in the sum over membrane configurations 28 . At the microscopic level, is actually related to the range of the interaction potential U ({r α }) between molecules.
In the continuous limit (N p → ∞, a p → 0), Ξ is expressed as a functional integral  To restore the rotational invariance of Ξ, Cai et al. 3 introduced two corrective terms to the Monge measure. The first one, named Faddeev-Popov term, corrects the fact that the Monge measure D[h] Monge induces over-counting of configurations of a fluid membrane. Indeed, the infinitesimal vertical displacement dh n of point n has a component which is locally tangent to the bilayer. At first order, this locally tangent displacement does not change the shape of the membrane, and then should be disregarded to avoid multiple counting of a same configuration 29 . Faddeev-Popov term regularizes this overcounting by removing the locally tangential component from the measure. However, Cai et al. acknowledged that this term is still not sufficient to restore the rotational invariance of Ξ. They introduced a second term, named Liouville term, to meet this requirement. The interpretation of this term is less obvious, but the above derivation of the coarse-graining procedure helps us to gain insight on its origin: we passed from a fined-grained description where the number of molecules fluctuates, to a coarse-grained description, where the number N p of membrane patches is fixed (while their masses fluctuate). But N p -or equivalently the projected distance a p = L p /N p -has been chosen arbitrarily. Its value should be related to the average of the fluctuating number of molecules that compose the interface. An alternative approach, which we will develop in the next section, is to let this number of patches fluctuate by grouping a fixed number of molecules on every patch.
Note that Cai et al. determine the expression of the Liouville term by imposing r = τ, which is required by the rotational invariance of Ξ. Hence, the equality r = τ is not an outcome of the correction, but a requisite instead. In the next Sections we show that a consistent expansion of the Hamiltonian and the measure -what requires to take the fluctuating character of the number of patches into account -allows us to obtain r = τ.

Fixing Monge model: a self-similar coarsegraining
The previous section showed that the Monge model is built on the passage from a fined-grained description in which the number of molecules (with fixed masses) fluctuates to a coarse-grained description in which the number of patches (with fluctuating masses) is fixed. No relation is set on the number of such patches and the average number of molecules in the membrane. We develop here an alternate coarse-graining in which each patch contains a fixed number of molecules. The number of patches N is then a fluctuating variable, and considering that the membrane is incompressible, the average number of patches is here trivially related to the average number of molecules in the membrane. This "self-similar" coarse-graining does not alter the system description: like in the fined-grained description, the membrane is made of a fluctuating number of particles (the patches) with fixed masses, and distributed evenly along the membrane physical length. Furthermore, since patches are all identical, a chemical potential of the patches µ can be defined unambiguously, and is related to the chemical potential of molecules µ 0 through µ = µ 0 Γa, where a is the size of a patch and Γ the linear density of molecules. The definition of a unique chemical potential is less obvious within the Monge model, where patches do not have constant sizes. The intrinsic tension γ of the membrane is defined as For a 1D incompressible membrane, patches can only rotate and translate. This is why the 1D case is much simpler than its two-dimensional counterpart, whose patches also deform from one configuration to another, as changes in membrane conformation are generally not isometric. A coarse-grained configuration of the 1D membrane is then specified by the number of patches it contains N, and the orientation θ n of each one of them. However, the variables θ n cannot vary independently as the projected membrane length of the x and y axes are fixed. Each coarsegrained conformation of the membrane is counted only once in the evaluation of the grand-canonical partition function by summing uniformly over the patch orientations which are compatible with these constraints. Grouping all fine-grained configurations corresponding to a given value of the variables θ n , the grandcanonical partition function becomes: where N 0 = L p /a is the minimal number of patches, and In the continuous limit (N → ∞, a → 0, with Na = L), the membrane is parametrized by the local tangent angle θ (s) along the curvilinear coordinate s (s ∈ [0, L]). Clearly, the measure D N [θ ] is invariant by rotation of the reference frame. Subsequently the coarse-grained Hamiltonian H ({θ n }) is also rotationally invariant and can be identified with the Helfrich Hamiltonian Eq. 5, which rewrites in terms of θ (s) as In order to evaluate Ξ, we now assume small fluctuations : cos θ k 1 − θ 2 k /2, sin θ k θ k . In the continuous limit, the two constraints rewrite as Note that this expansion breaks the rotational invariance of the measure D N [θ ], but is consistent with the expression of the Helfrich Hamiltonian Eq. 36, which is limited to quadratic terms in θ (s). Expressing the first constraint with help of the identity the partition function rewrites: Introducing the Fourier expansion θ (s) = ∑ N−1 p=1θ p cos (pπs/L), in which the zeroth order term has been eliminated to satisfy the second constraint, it comes: We show in Appendix B that the Jacobian associated with the change of variables is still J θ →θ (N/2) N/2 when N = L/a 1. Here again, taking the Jacobian into account ensures that ln Ξ is an extensive quantity for large membranes, and will also affect the expression of the frame tension τ. After integrating over theθ p variables, we get: where g(ω, L) = −β γL + ω(L − L p ) − Lu(ω), with u(ω) = π/a 0 ln a 2π (ω + κβ q 2 ) dq/2π. Here again, we assumed κ/γ L so that the lower bound in the integral is 0.
This double integral can be evaluated in the thermodynamic limit (L, L p → ∞) thanks to the saddle point approximation: Ξ ∝ e g(ω ,L ) where ω and L , the values of ω and L that extremize g(ω, L), are given implicitly by: and ω = u(ω ) + β γ.
Using this last equation, it comes: ln Ξ = g(ω , L ) = −ω L p . From Eq. 42, we also see that ω is independent of L p , so the frame tension is τ = −k B T ∂ ln Ξ/∂ L p = k B T ω . Injecting this relation into Eq. 42 finally yields: and Eq. 44 is identical to Eq. 10 derived with the Monge model, just by replacing γ with τ and taking = ea/π. The quantum of height fluctuations is then of the order of magnitude of a patch size (which is few molecule sizes), while Morse et al. 3,30 argued it should be of the order of the de Broglie length (which is smaller than molecule size). However, if we had introduced a quantum of angle fluctuations Θ in the integration measure, i.e.: D N [θ (s)] ≡ δ (∑ n θ n ) ∏ N n=1 (dθ n /Θ), we then would end up with = Θea/π. Equation 43 can be inverted for small fluctuations (k B T / √ κγ 1). The frame tension is thus in agreement with previous studies 3,4,7,31 . Deriving Eq.
42 with respect to γ gives: ∂ ω /∂ γ = u (ω )∂ ω /∂ γ + β . Together with Eq. 41 this yields ∂ ω /∂ γ = β L /L p , and thus L = L : the average and most probable values coincide, as expected. Equation 41 then rewrites This equation is identical to Eq. 26 derived in the (T, τ, L) ensemble. Assuming small fluctuations, the relation is inverted as Eq. 47 is then identical to Eq. 12 with the substitution γ → τ (and the undetermined length a p replaced with the physical patch length a). Using the same substitution in Eq. 11, the height fluctuations are characterized by: These results prove that the fluctuation tension r does coincide with the frame tension τ, in agreement with Cai et al. It is also worth noticing that Eq. 46 suggests a linear scaling of L − L p with L rather than L p when departing from the regime of small fluctuations, in agreement with the numerical findings of ref. 6 .

Varying patch number is mandatory to recover r = τ
Starting from a coarse-grained description in which both the Hamiltonian and the measure are rotationnally invariant, we were able to recover r = τ for an incompressible membrane within the (T, L p , γ) ensemble. However, let us emphasize again that to obtain this result, we used a quadratic expansion of the measure D N [θ ], thereby breaking its rotational invariance. Moreover, the approximation sin θ (s) θ (s), cos θ (s) 1 − θ (s) 2 /2 used in the measure is equivalent to assume that θ (s) and h(s) are linearly related: θ (s) dh/ds. Therefore, the associated measures are also linearly related: (the factor N has been introduced to make the measure dimensionless), with J θ →h = 1/a N (the derivation of this Jacobian is reported in Appendix C). Accordingly, the Faddeev-Popov corrective term, which is a correction to the linear relationship between the two measures, and whose expression is derived in Appendix C, is a second order correction to the expression of τ, in agreement with Cai et al. 3 Moreover, inserting θ (s) dh/ds in Eq. 36 breaks the rotational invariance of the Hamiltonian. Yet, substituting θ (s) → dh/ds (and using = ea/π) in the grand-canonical partition function (Eq. 38) will not change its final expression (Eq. 44). Hence, the use of an approximated Hamiltonian cannot explain the Monge model inaccuracies and the non-equivalence of ensembles discussed in Section 2 either. This should come as no surprise: ensemble equivalence is a very general feature of standard statistical physics and does not makes any assumption on the realism of the Hamiltonian. It can also be noticed that in the ensemble (T, τ, L), we obtained (Sect. 2.2) r = τ while using both an approximated measure and an approximated Hamiltonian.
Therefore, the only remaining reason why the Monge model in the (T, L p , γ) does not give the right value for r is because it disregards the inherent fluctuation of the number of patches. This correction can be identified with the Liouville corrective term introduced in 3 . Note that the saddle point approximation we used in our derivation allows us to replace the sum over the patch number with a fixed ad-hoc patch number N , and can be interpreted as a correction (so-called Liouville correction) to the Monge measure. However, for calculating accurately other quantities, such as h 4 , the other corrections to the Monge model (expansion of H to higher terms, and Faddeev-Popov term) might be mandatory.

The case of a compressible membrane
At this point, one question still remains: we mentioned in the introduction that the same Helfrich Hamiltonian also applies to compressible membranes with fluctuating surface area but fixed number of molecules; the term γA then corresponds to a stretching energy 32 . This is the point of view adopted by Farago and Pincus 4,26 . For a 1D membrane, the relevant thermodynamic ensemble is the ensemble of fixed temperature, number of patches and projected length, (T, N, L p ). How can we then explain that the result obtained with the Monge model (r = γ) is still not correct in this ensemble ? Clearly, the issue cannot be associated with a fluctuating number of patches as in the (T, γ, L) ensemble. Farago 4 , then Schmid 6 invoked the lack of rotational invariance of the approximated Hamiltonian to explain the inaccuracies in the Monge model. Yet, we obtained r = τ in the (T, L, τ) and (T, L p , γ) ensembles while using the very same approximated Hamiltonian, and all three ensembles should be equivalent in the thermodynamic limit. More likely, the issue originates from the measure D[h] Monge that still does not properly enumerates the membrane configurations: because the allowed displacements of the patches are those which keep uniform patch length, we should consider all the possible membrane conformations for a given patch length a, and sum over all possible values of a. The partition function then writes: Taking the fluctuating character of the patch size a is thus essential to get the correct result, the same way that the fluctuation of patch number is essential to recover r = τ in the (T, L p , γ). Indeed, in both cases this boils down to sum over membrane length L, and the partition function 49 is in fact identical to Eq. 38.

The case of a fluid interface
We finally focus our attention to the case of an interface between two immiscible fluids, and ask ourselves whether internal and frame tensions should be distinguished. At the difference of a membrane, an interface between two immiscible fluids has no bending rigidity (κ = 0) 2 . The interface is stabilized by the surface tension only, whose origin comes from a non-affinity between molecules of the two bulk phases 33 , and therefore is very different from -and usually much larger than -typical tension value in a membrane. The high tension criterion k B T /τa 1, which is required to be in the small fluctuations regime (Eq. 27) when κ → 0, is then satisfied. According to Eq. 45, τ and γ are then equal up to O((k B T /τa)) 2 and so can be used indifferently in the expressions of the height and length fluctuations of a fluid interface.

Conclusion
In summary, starting from the description of a membrane at the molecular level, we first revealed the inconsistencies lying in the foundations of the Monge model to describe the statistics of an incompressible membrane: the choices made for the expression of the coarse-grained Hamiltonian (or its quadratic approximation) and that of the integration measure are not compatible choices to build a free energy that satisfies rotational invariance. By analyzing the simpler case of a 1D membrane, for which an integration measure that satisfies rotational invariance can be built from first principles, we found in agreement with previous studies that the lack of rotational invariance of the measure used in the Monge model has two origins itself: first, it assumes a fixed number of patches, while for an incompressible membrane with fluctuating length the number of patches must fluctuate; and second, for a given number of membrane patches, there is an over-counting of membrane configurations because the size of a patch is allowed to vary with this linear measure. Finally, using a consistent expansion of the measure and the Hamiltonian, we were able to prove that the frame tension τ drives the thermal fluctuations of a membrane. Our analysis then shows that to obtain this result, rotational invariance of the Hamiltonian is not necessary, and one can use its quadratic approximation. Only the measure used in the Monge model must be amended, in agreement with Cai et al. , although its rotational invariance is not required either. In fact, the only correction that must be taken into account is the fluctuating character of the membrane length. We believe that the multiplicity of the arguments advanced to justify that r = τ will reinforce the conviction in these results, and hence will encourage the use of the correct expressions in future studies. As a final note, let us emphasize that the Helfrich model represents a simplified version of a real membrane; in particular it is described as an infinitely thin sheet, and energy does not depend on the local lipid concentration along the membrane. All the discussion about whether r = τ or not is restricted in the context of this specific theoretical framework. But real membranes are more complex, and recent particle-based numerical simulations taking into account the fluctuations of lipid concentration and membrane thickness have been performed in recent years 22,23 . These simulations have shown that for membranes under very strong stretching, the fluctuation tension deviates from the frame tension. A possible explanation for this result is the existence of a coupling term between membrane height and membrane thickness.

B Expression of J θ →θ
We evaluate the Jacobian J θ →θ associated with the linear change of variables {θ n } → θ p , with θ n = ∑ N−1 p=0θ p cos (pπn/N), n ∈ {0, ..., N − 1}. As in Appendix A, J θ →θ is obtained by evaluating a simple integral expressed with both sets of variables. Note however that here the N variables θ n are not independent, but are constrained by ∑ N−1 n=0 sin θ n ∑ N−1 n=0 θ n = 0. This constraint is properly taken into account by introducing a Dirac delta in the integration. For the variables {θ p } on the other hand, this constraint simply readsθ 0 = 0. The second constraint of fixed projected length a ∑ n cos θ n = L p does not need to be considered to evaluate J θ →θ , since its value does not depend on the choice of the integral to evaluate. Let us then evaluate the integral with D N [θ ] ≡ δ (∑ N−1 n=0 θ n ) ∏ N−1 n=0 dθ n . Using the identities δ (x) = +∞ −∞ e iqx dq/2π, and +∞ −∞ exp iqθ n − θ 2 n dθ n = e −q 2 /4 √ π, it comes: The integral J rewrites with variablesθ p as where we used the equality ∑ N−1 n=0 θ 2 n = N 2 ∑ N−1