pH gradient-driven deformation of a crista-like vesicle

H\'el\`ene Berthoumieux; Yorgos Chatziantoniou

REVIEW 4 major objections 6 minor 37 references

pH gradient-driven deformation of a crista-like vesicle

T0 review · 4 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read A purely diffusive proton gradient from pole to equator can deform a spherical, pH-sensitive vesicle into a crista-like shape with flat zones at the respiratory complexes and strongly curved zones at the ATP synthase, and the model's…

desk verdict Careful derivation, but the proton-concentration fit that drives the deformation is reversed, so the crista-like shape and the functional phase diagram are not established for a real RC-to-ATP-S gradient. read the letter →

arxiv 2502.06582 v1 pith:KS5HRSZC submitted 2025-02-10 physics.bio-ph

Yorgos Chatziantoniou , Hélène Berthoumieux This is my paper · ORCID

classification physics.bio-ph

keywords mitochondrialcristaepHgradientspontaneouscurvatureHelfrichHamiltonianactivemembranevesicledeformationprotondiffusioncardiolipin

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Mitochondrial cristae are folds of the inner membrane where ATP is made, and their shape—flat zones hosting respiratory complexes and tightly curved zones hosting ATP synthase—is part of how they work. This paper asks whether the proton current that drives ATP synthesis can itself sculpt that shape. The authors model a crista as a spherical vesicle whose membrane has a spontaneous curvature that depends linearly on the local proton concentration, with the protons diffusing passively from the poles to the equator. In the small-deformation regime they solve the resulting shape equation and find a parameter region in which the vesicle flattens at the poles and becomes highly curved at the equator, exactly the morphology of a functioning crista. If correct, the result shows that no protein scaffolding is needed: a pure diffusive pH gradient plus a pH-sensitive lipid is sufficient to organize the crista geometry that maximizes ATP output.

What carries the argument

The machinery is a pH-dependent Helfrich Hamiltonian, $\mathcal{H} = \int_S d^2X \,[\tfrac{1}{2}\kappa(C-C_0[h])^2 + \sigma_0] - PV$, with the spontaneous curvature coupled linearly to the local proton concentration, $C_0(h)=C_0^0 + C_1^0 h(\theta)$. The proton field $h(\theta)$ is prescribed by the spherical Laplace equation $\Delta_S h=0$ between the respiratory-complex boundary at $\theta=\theta_0$ and the ATP-synthase boundary at $\theta=\pi/2$, giving $h(\theta)=-\alpha_0 \ln(\tan(\theta/2))+\alpha_1$ fitted to measured crista pH values. The normal force balance reduces to a fourth-order linear ODE for the radial deformation $u(\theta)$; its solution is the sum of a particular solution proportional to $h(\theta)$ and kernel modes built from the functions $C_{A,b}(\theta)$, with coefficients fixed by five biological boundary conditions (flat RC patch, ATP-S opening angle, and force balances at both protein sites) plus volume conservation. A functionality score $S=S_1-S_2-S_3$, combining equatorial perimeter, proton path length, and pole flatness, classifies the resulting shapes and locates the well-functioning crista region in the $(A,\xi)$ plane.

What would settle it

Measure the deformation profile of a cardiolipin-containing giant unilamellar vesicle under a controlled pole-to-equator pH gradient and compare it with Eq. (26) using independently measured values of the bending modulus, surface tension, and pH–curvature coupling: if the vesicle does not develop a flatter polar region and a more curved equator as the gradient is increased, or if the measured proton profile near the deformed membrane deviates strongly from the Laplace solution of Eq. (1), the central claim fails.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that a closed spherical membrane with a pH-dependent spontaneous curvature, driven only by a diffusive proton concentration field, can adopt shapes with strongly inhomogeneous curvature. The authors state in the Discussion that 'we show that such a system can adopt shapes with inhomogeneous curvature. In the relevant parameter space, we identify the zone leading to both highly curved in the ATP-S location and flat zones in the RC location. These shapes correspond to a well-functioning crista.' Concretely, the shape equation $(\Delta_S+2)(\Delta_S+A)u = -[\Delta_S+2(\xi-1)]\beta h + \delta p$, solved with boundary conditions encoding the flat respiratory-complex patch and the opening angle of ATP synthase, yields a deformation with near-zero mean curvature near the pole and enhanced curvature at the equator; the model's functionality score peaks at $A=6$, $\xi=-1$, corresponding to a typical crista radius $R\approx 100$ nm and a spontaneous curvature $C_0\approx -0.01$ nm$^{-1}$.

Load-bearing premise

The load-bearing assumption is that the proton concentration is set by pure diffusion on the undeformed spherical surface and does not change when the membrane deforms, with the pH–curvature coupling strength β chosen by hand so that the induced spontaneous curvature is about 10% of the sphere's curvature; if the shape feeds back on the proton distribution, or if β is much weaker, the predicted flat-pole/curved-equator shape need not appear.

Editorial extensions

If this is right

A purely diffusive proton field is sufficient to create the flat-pole/curved-equator anatomy of a crista; the respiratory complexes and ATP synthase act only as boundary conditions, not as scaffolds that hold the shape.
The model predicts an optimal operating point ($A=6$, $\xi=-1$) that corresponds, for $R\approx 100$ nm, to a spontaneous curvature $C_0\approx -0.01$ nm$^{-1}$, a value compatible with molecular-dynamics estimates for cardiolipin-containing membranes.
The phase diagram of the functionality score delimits, in the space of reduced pressure and spontaneous curvature, which parameter combinations yield well-functioning cristae; the measured lateral pH gradient (pH 6.4 near the complexes, pH 7.1 near ATP synthase) is used as input, so the diagram can be tested against pH-sensitive vesicle experiments.
Without a proton flux the vesicle stays spherical; the proton field is the sole driver of deformation, so the model makes the testable prediction that turning off the pH gradient erases the crista-like shape.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves open how cardiolipin sorting would feed back on the pH field; coupling diffusion to curvature would likely amplify the predicted flat-pole/curved-equator split, since cardiolipin concentrates in curved zones and raises local pH sensitivity.
The optimal operating point $A=6$ sits at an eigenvalue of the operator that controls spherical stability, so the model implicitly predicts that the best-functioning crista is near a shape instability; varying $A$ across this value in the same calculation would test how robust the well-functioning zone is.
A direct experiment the paper does not propose: reverse the proton source and sink, placing RC at the equator and ATP-S at the poles; the symmetry of the equations then predicts a shape with the flat and curved zones swapped, which would distinguish this passive-diffusion mechanism from protein-scaffolding models.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The stress-test note holds up, and it hits the load-bearing part of the paper. Eq. (2) with the reported α0 = −6×10⁻⁸ mol/L and α1 = 7.9×10⁻⁸ mol/L gives a proton concentration that increases from the RC pole (θ≈0.1) to the ATP-synthase equator (θ=π/2), exactly opposite to the decrease the text describes and the biology requires. At the RC, h is about −1×10⁻⁷, so the total proton concentration nearly vanishes and pH is undefined; at the equator h is positive, giving pH ≈ 6.75 rather than the cited 7.1. So the fitted profile is both inconsistent with the experimental values and reversed relative to the physical gradient. Since the active stress enters as βh, the driving force in Eq. (17) is flipped. The flat-pole/curved-equator morphology and the optimal region near (A=6, ξ=−1) are computed for a gradient flowing from the equator to the pole, not from RC to ATP-S. The central biological claim is therefore not established for the real system.

The derivation itself is the strong part. The stress-tensor formalism is applied consistently, the linearized shape equation is correct, and the appendices give explicit solutions for the degenerate A=0 and A=2 cases, plus a bound on the series truncation. Extending the group's earlier flat/cylindrical active-membrane work to a closed spherical vesicle with protein-mimicking boundary conditions is a legitimate step. The authors also honestly acknowledge the missing cardiolipin-sorting feedback.

The functionality score is the next soft spot. S is built from the exact features the model is designed to produce—large equator, short proton path, flat RC zone—so maximizing S in parameter space is partly circular. The weights and cutoffs are arbitrary, and the claimed comparison to experimental measurements is qualitative rather than a quantitative overlay. The coupling β is chosen by hand to give a 10% curvature change, with no sensitivity analysis.

Because the calibration error is correctable and the mathematical core is sound, this deserves referee time despite the flawed input. But I would not cite it in its current form, and the reading group will get more from it as a case study in how a sign error can invert a conclusion than as a reliable result. My recommendation: send to peer review, and require the authors to refit h to ref. 29, verify the sign of α0, and show what changes in Fig. 5 under the correct gradient.

Referee Report

4 major / 6 minor

Summary. The paper models a mitochondrion crista as a spherical vesicle whose membrane has a pH-dependent spontaneous curvature. A proton concentration field h(θ) is taken to be purely diffusive between the respiratory complexes at the poles and ATP synthase at the equator, and the resulting linearized Helfrich shape equation is solved in the small-deformation regime. The authors define a scalar functionality score S based on geometric features (equator size, proton path length, flatness near the RC) and map the (A, ξ) parameter space to identify a 'well-functioning crista' region, with an optimum at (A=6, ξ=−1).

Significance. The manuscript provides a detailed derivation of the linearized shape equation for a closed spherical vesicle with a position-dependent spontaneous curvature, including explicit treatments of the degenerate cases A=0 and A=2 and error bounds for the truncation of the kernel functions. This is a useful methodological contribution for active-membrane models. If the pH-gradient calibration and the sign of the driving force were correct, the model would offer a minimal mechanism by which a proton gradient alone could produce inhomogeneous curvature with flat polar regions and a curved equator. However, the central quantitative claims are currently compromised by an internal inconsistency in the proton-field fit, and the 'well-functioning crista' conclusion rests on an ad hoc score. The paper is therefore promising but not yet established.

major comments (4)

[Section II, Eq. (2) and Fig. 2] The fitted proton profile is internally inconsistent with the stated experimental pH values and has the wrong sign. With α0=−6.0×10−8 mol L−1 and α1=7.9×10−8 mol L−1, Eq. (2) gives h(π/2)=α1=7.9×10−8, so with h0=10−7 mol L−1 the pH at the ATP-S location is −log10(1.79×10−7)≈6.75, not 7.1 as claimed; at the RC location θ0≈0.1, h≈−1.0×10−7, making h0+h≈0 and pH undefined. Moreover, since dh/dθ=−α0/sinθ, the stated negative α0 makes h increase from RC to ATP-S, contradicting the text's statement that h decreases monotonically from RC to ATP-S and the pH gradient direction cited from Ref. 29. Because the active stress enters through δξ=βh in Eq. (17), this sign error reverses the direction of the driving force. The predicted flat-pole/curved-equator morphology and the optimum in Fig. 5 are therefore not established for a proton gradient flowing from RC to ATP-S.
[Section IV, Eq. (34)] The functionality score S = S1 − S2 − S3 uses equal weights of 1 for the three geometric contributions, with no biophysical justification for these weights or for the chosen normalizations, and no sensitivity analysis with respect to the weighting. The paper's central claim that shapes in the identified parameter region 'correspond to a well-functioning crista' depends entirely on this score. Since S1, S2, and S3 are precisely the geometric features the model is designed to produce, the maximization of S in parameter space is partly self-referential: the score rewards the exact outputs of the mechanism being tested. The authors should either derive the score from an independent biophysical argument or demonstrate robustness of the optimal region to reasonable weight variations.
[Section III, after Eq. (19)] The coupling parameter β = R C1^0 is chosen by hand so that the variation of the natural curvature is 'of the order of 10% of the curvature of the system' (caption of Fig. 4). No experimental or simulation constraint is given for β, and the paper does not test whether the predicted shapes persist for smaller β. Since the deformation is linearly proportional to β through Eq. (17) (and the particular solution in Eq. (22)), the existence of the observed flat-pole/curved-equator morphology is conditional on this arbitrary choice. A sensitivity analysis over β, or a measurement-based estimate, is required to support the claim that the proton gradient suffices to deform the vesicle.
[Abstract and Discussion] The abstract states that the phase diagram is 'compared to experimental measurements,' but the paper does not make a quantitative comparison to experimental crista shapes or pH measurements. The only experimental inputs are the pH values used (incorrectly, as noted above) and parameter values cited for R, κ, and σ0. The Discussion's statement that the optimum gives C0≈−0.01 nm 'in agreement with recent simulations' is not a comparison to measurements. To support the claimed comparison, the authors should plot the model's deformed shapes against experimental images or quantitative morphometric data for cristae, or at minimum against the in vitro vesicle data of Refs. 14 and 15.

minor comments (6)

[Section II] The symbol h0 is written as '10 × 10−7 mol L−1', which is ambiguous: it could mean 10−7 or 10−6 mol L−1. Since the pH calibration depends on h0, this must be stated unambiguously.
[Section III, Eq. (28)] The angle γ is used in the boundary condition but is defined only in the caption of Fig. 1. Define γ explicitly in the main text when it first appears.
[Section IV, Eq. (31)] The volume constraint δV = ∫ sin(θ)dθ u(θ) = 0 is missing the factor 2πR^3 that appears in the full volume integral. While this factor cancels in the constraint, the equation as written is dimensionally inconsistent with δV.
[Fig. 2 caption] The caption states that h(θ) decreases monotonically from RC to ATP-S, but for the given α0 and α1, dh/dθ>0, so it increases. This is the same sign issue as in the main text; the caption and text should be corrected together.
[Appendix D, matrix M] The third row of the matrix M in Eq. (D4) appears garbled: it reads '1 cot( γ/2) 1 cot( γ/2) 1 cot( γ/2)' and is missing apparent entries for the C0,0 and C0,1 columns. Please check the typesetting.
[Appendix E, Eq. (E4)] The expression for N(A) contains 'ln ϵ + ln(1−cos²θ0) − ln|bm0|' inside a fraction; verify that the division by cos(θ0) applies as intended and that the bracketing in the floor expression is correct. Also, the product in Eq. (23) for n=0 is an empty product; define it as 1 for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the pH-driven shape is solved from a stated Helfrich model and boundary conditions, and the functionality score is an evaluation metric rather than a fitted input.

full rationale

The paper's derivation is self-contained. The input proton field h(theta) is defined by solving Laplace's equation (Eq. 1) and calibrated to two experimental pH values from ref. 29; this is a data input, not an output of the model. The Helfrich energy with a linear, pH-dependent spontaneous curvature (Eqs. 4, 5) is a stated constitutive assumption, not a disguised consequence of the target shape. The shape equation (Eq. 17) and the stress-tensor terms are derived within the paper from the Guven variational formalism; the prior work cited in refs. 26, 27 is motivational, and the same-group citations are not load-bearing, because the equations are re-derived here and no unverified uniqueness claim is invoked. The deformation u(theta) is obtained by solving the fourth-order ODE with boundary conditions (Eqs. 27-30) that encode protein geometry; the flat-at-the-pole and curved-at-the-equator morphology is therefore a computed consequence of these inputs. The functionality score (Eq. 34) is defined after the shape is solved and is used to rank shapes; maximizing it is a model analysis, not a way of fitting the deformation, so it does not make the shape prediction circular. The known calibration inconsistency in the Fig. 2 caption values (alpha0<0 and alpha1=7.9e-8 do not reproduce the cited pH=6.4 and 7.1, and reverse the stated monotonic decrease of h) is a correctness or robustness defect, not a circularity: the model would still be a well-posed input-output computation if the fit were corrected. No step in the derivation is equivalent by construction to its own conclusion.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The model relies on a small number of fitted constants (α0, α1 from pH data, β by hand) and several domain assumptions about diffusion, linear coupling, and closed geometry. The functionality score is an ad hoc construction with arbitrary weights. No invented entities are introduced.

free parameters (6)

α0 = -6.0e-8 mol/L
Fit to experimental pH values at RC (6.4) and ATP-S (7.1) from ref 29, setting the amplitude of the surface proton concentration profile.
α1 = 7.9e-8 mol/L
Fit to the same two experimental pH values; determines the offset of the concentration profile.
β = R C1^0 = not stated in text (chosen so |C1^0 h| ≈ 0.1/R)
Hand-picked to give a 10% variation of spontaneous curvature; no experimental or simulation value is used.
Γ0 (RC spring constant) = 100 σ_typ = 1e-5 J/m^2
Chosen as a typical membrane surface tension multiplied by 100; not derived from data.
k (ATP-S bending elasticity) = 100 κ = 6e-18 J
Chosen as 100 times the bending modulus; representative value, not measured.
functionality score weights and cutoff = weights all 1, S3 averaged over [θ0, π/16]
Ad hoc choices defining the score; no sensitivity analysis is provided.

assumptions (7)

domain assumption Proton concentration obeys the 2D Laplace equation Δ_S h=0 with no feedback from membrane deformation
Eq. (1) in Section II; the shape dependence of proton trapping is neglected, as acknowledged in the Discussion.
domain assumption Spontaneous curvature depends linearly on local proton concentration: C0(h) = C0^0 + C1^0 h
Eq. (5) in Section III; linearization justified by small h, but no molecular derivation is given.
domain assumption Small deformation regime u(θ) << 1 and linear response to the proton field
Section III, after Eq. (3); all results are first-order in u and h.
domain assumption Gaussian curvature term is neglected in the Helfrich energy
Section III, 'The Gaussian bending rigidity has been neglected for the sake of simplicity.'
domain assumption The vesicle remains closed with constant volume (δV=0)
Eq. (31) in Section III; in vivo cristae are connected to the inner membrane, not closed spheres.
standard math Stress tensor formalism of Guven and Deserno is valid for the Helfrich Hamiltonian
Refs 23,24; used in Eqs. (6)-(9).
domain assumption Helfrich model with pH-dependent spontaneous curvature is a valid effective energy for the crista membrane
Eq. (4); assumes homogeneous composition and continuum mechanics at the nanoscale.

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Pith. "Pith review of pH gradient-driven deformation of a crista-like vesicle." pith.science (2026). https://pith.science/paper/KS5HRSZC

@misc{pith2026250206582,
  author       = {Pith},
  title        = {Pith review of: pH gradient-driven deformation of a crista-like vesicle},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KS5HRSZC}},
  note         = {Machine review of arXiv:2502.06582}
}

read the original abstract

The inner membrane of mitochondria presents folds, the cristae, which are the production place of ATP. This synthesis is driven by a flow of protons confined to the surface of the membrane, which also shapes the crista to ensure a high synthesis rate. We model a crista as a spherical vesicle submitted to a diffusive proton gradient flowing from the poles to the equator. Using Helfrich model, we introduce a pH-dependent spontaneous curvature for the membrane and determine the shape of the vesicle, when submitted to the pH gradient, in the regime of small deformations. Based on biophysical arguments, we define a functionality score for the vesicle and construct a phase diagram identifying the zones of "well-functioning" cristae, which we compare to experimental measurements.

Figures

Figures reproduced from arXiv: 2502.06582 by the authors.

**Figure 1.** FIG. 1. Sketch and parameterization of the system. (a) Sketch of a functioning crista. The white ellipsoid represents the crista [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** FIG. 2. Proton concentration along the membrane. We plot [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Generating functions of (∆ [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. Profiles of the deformed crista as a function of ( [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. Functionality score for the crista. We represent the functionality score defined in Eq. (34) as a density map, function [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: FIG. 6. 3D shape of an active vesicle with optimal functional [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

37 extracted references · 37 canonical work pages

[1]

P \'a nek, M

T. P \'a nek, M. Eli \'a s, M. Vancov \'a , J. Luke s , and H. Hashimi. Returning to the fold for lessons in mitochondrial crista diversity and evolution. Current Biology , 30:575--588, 2020

work page 2020
[2]

K. M. Davies, M. Strauss, B. Daum, J. H. Heinz, D. Osiewacz, A. Rycovska, V. Zickermann, and W. K \"u hlbrandt. Macromolecular organisation of atp synthase and complex i in whole mitochondria. Proc. Nat. Acad. of Sci. , 108:14121--14126, 2011

work page 2011
[3]

Serowy, S

S. Serowy, S. M. Saparov, Y. N. Antonenko, W. Kozlovsky, V. Hagen, and P. Pohl. Structural proton diffusion along lipid bilayers. Biophysical Journal , 84:1031--1037, 2003

work page 2003
[4]

D. G. Knyazev, T. P. Silverstein, S. Brescia, A. Maznichenko, and P. Pohl. A new theory about interfacial proton diffusion revisited: The commonly accepted laws of electrostatics and diffusion prevail. Biomolecules , 13:1641, 2023

work page 2023
[5]

Heberle, G

J. Heberle, G. Riesle, J. Thiedemann, D. Oesterhelt, and N. A. Dencher. Proton migration along the membrane surface and retarded surface to bulk transfer. Nature , 370, 1994

work page 1994
[6]

J. Heberle. Proton transfer reactions across bacteriorhodopsin and along the membrane. BBA-Bioenergetics , 1458, 2000

work page 2000
[7]

R. B. Gennis. Proton dynamics at the membrane surface. Biophysical Journal , 110, 2016

work page 2016
[8]

D. A. Cherepanov, B. Feniouk, , W. Junge, and Mulkidjanian A. Y. Low dielectric permittivity of water at the membrane interface: effect on the energy coupling mechanism in biological membranes. Biophysical journal , 85:1307--1316, 2003

work page 2003

Show all 37 references

[9]

Paradies, V

G. Paradies, V. Paradies, V. De Benedictis, F. M. Ruggiero, and G. Petrosillo. Functional role of cardiolipin in mitochondrial bioenergetics. BBA , 1837:408--417, 2014

2014
[10]

Ikon and R

N. Ikon and R. O.. Ryan. Cardiolipin and mitochondrial cristae organization. BBA , 1859:1156--1163, 2017

2017
[11]

Beltrán-Heredia, F-C

E. Beltrán-Heredia, F-C. Tsai, S. Salinas-Almaguer, F. J. Cao, P. Bassereau, and F. Monroy. Membrane curvature induces cardiolipin sorting. Comm. Biol , 2:225, 2019

2019
[12]

N. A. Dencher T. H. Haines. Cardiolipin: a proton trap for oxidative phosphorylation. FEBS , 528, 2002

2002
[13]

A. R. Variyam, M. Rzycki, A. Yucknovsky, A. A. Stuchebrukhov, and N. Drabik, D. Amdursky. Proton diffusion on the surface of mixed lipid membranes highlights the role of membrane composition. Biophysical Journal , 123:1--11, 2024

2024
[14]

Khalifat, N

N. Khalifat, N. Puff, S. Bonneau, J.-B. Fournier, and M. I. Angelova. Membrane deformation under local ph gradient: mimicking mitochondrial cristae dynamics. Biophysical journal , 95:4924--4933, 2008

2008
[15]

Khalifat, J-B

N. Khalifat, J-B. Fournier, M. I. Angelova, and N. Puff. Lipid packing variations induced by ph in cardiolipin-containing bilayers: The driving force for the cristae-like shape instability. BBA-Biomenbranes , 1808:2724--2723, 2011

2011
[16]

Allolio and D

C. Allolio and D. Harries. Calcium ions promote membrane fusion by forming negative curvature inducing clusters on specific anionic lipids. ACS nano , 15:12880--12887, 2021

2021
[17]

F.Joubert and N. Puff. Mitochondrial cristae architecture and functions: lessons from minimal model systems. Membranes , 11:11070465, 2021

2021
[18]

Helfrich

W. Helfrich. Elastic properties of lipid bilayers—theory and possible experi- ments. Z. Naturforsch. , 33:305--315, 1973

1973
[19]

Helfrich

Ou-Yang Zhong-can and W. Helfrich. Bending energy of vesicle membranes: General expressions for the first, second and third variation of the shape energy and applications to spheres and cylinder. Phys. Rev. A , 39:5280--5288, 1989

1989
[20]

U. Seifert. Configurations of fluid membranes and vesicles. Advances in Physics , 46:13--137, 1997

1997
[21]

Mietke, F

A. Mietke, F. J \"u licher, and I. F. Sbalzarini. Self-organized shape dynamics of active surfaces. Proc. Natl. Acad. Sci. U. S. A. , 116:29--34, 2019

2019
[22]

A. Mietke. Dynamics of active surfaces . PhD thesis, TU Dresden, 2018

2018
[23]

J. Guven. Membrane geometry with auxiliary variables and quadratic constraints. Journal of Physics A: Mathematical and General , 37:L313, 2004

2004
[24]

M. Deserno. Fluid lipid membranes: From differential geometry to curvature stresses. Chemistry and physics of lipids , 185:11--45, 2015

2015
[25]

Bitbol, N

A-F. Bitbol, N. Puff, Y. Sakuma, M. Imai, J-B. Fournier, and M. I. Angelova. Lipid membrane deformation in response to a local ph modification: theory and experiments. Soft Matter , 8:6073--6082, 2012

2012
[26]

Patil, S

N. Patil, S. Bonneau, A-F. Bitbol F. Joubert, and H. Berthoumieux. Mitochondrial cristae modeled as an out-of-equilibrium membrane driven by a proton field. Phys. Rev. E , 102:0224401, 2020

2020
[27]

T. V. Mendes, J. Ranft, and H. Berthoumieux. Model of membrane deformations driven by a surface gradient. Phys. Rev. E , 108:014113, 2023

2023
[28]

T. B. Blum, A. Hahn, T. Meier, K. M. Davies, and W. K \"u hlbrandt. Dimers of mitochondrial atp synthase induce membrane curvature and self-assemble into rows. Proc. Natl. Acad. Sci. U. S. A. , 116, 2019

2019
[29]

Rieger, W

B. Rieger, W. Junge, and K. B Busch. Lateral ph gradient between oxphos complex iv and f0f1 atp-synthase in folded mitochondrial membranes. Nature communications , 5:3103, 2014

2014
[30]

u ller. Theoretical studies of fluid membrane mechanics . PhD thesis, Johannes Gutenberg-Universit \

M. M. M \"u ller. Theoretical studies of fluid membrane mechanics . PhD thesis, Johannes Gutenberg-Universit \"a t Mainz, 2007

2007
[31]

Fournier

J.-B. Fournier. On the stress and torque tensors in fluid membranes. Soft Matter , 3:883--888, 2007

2007
[32]

Zhong-can and W

O. Zhong-can and W. Helfrich. Instability and deformation of a spherical vesicle by pressure. Phys. Rev. Lett. , 59:5280--5288, 1987

1987
[33]

Konar, H

S. Konar, H. Arif, and C. Allolio. Mitochondrial membranes: model lipid compositions, material properties and the changing curvature of cardiolipin. Biophysical Journal , 122:4274--4287, 2023

2023
[34]

Sj \"o holm, J

J. Sj \"o holm, J. Bergstrand, T. Nilsson, R. Šachl, C. von Ballmoos, J. Widengren, and P. Brzezinski. The lateral distance between a proton pump and atp synthase determines the atp-synthesis rate. Scientific Reports , 7:2926, 2017

2017
[35]

Salbreux and F

G. Salbreux and F. Jülicher. Mechanics of active surfaces. Phys. rev. E , 96:032404, 2017

2017
[36]

Berthoumieux, J.-L

H. Berthoumieux, J.-L. Maître, C.-P. Heisenberg, E. K. Paluch, F. J \"u licher, and G. Salbreux. Active elastic thin shell theory for cellular deformations. New Journal of Physics , 16:065005, 2014

2014
[37]

K. J. Boyd, N. N. Alder, and E. R. May. Buckling under pressure: Curvature-based lipid segregation and stability modulation in cardiolipin-containing bilaye. Langmuir , 33:6937–6946., 2017

2017

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

P \'a nek, M

T. P \'a nek, M. Eli \'a s, M. Vancov \'a , J. Luke s , and H. Hashimi. Returning to the fold for lessons in mitochondrial crista diversity and evolution. Current Biology , 30:575--588, 2020

work page 2020

[2] [2]

K. M. Davies, M. Strauss, B. Daum, J. H. Heinz, D. Osiewacz, A. Rycovska, V. Zickermann, and W. K \"u hlbrandt. Macromolecular organisation of atp synthase and complex i in whole mitochondria. Proc. Nat. Acad. of Sci. , 108:14121--14126, 2011

work page 2011

[3] [3]

Serowy, S

S. Serowy, S. M. Saparov, Y. N. Antonenko, W. Kozlovsky, V. Hagen, and P. Pohl. Structural proton diffusion along lipid bilayers. Biophysical Journal , 84:1031--1037, 2003

work page 2003

[4] [4]

D. G. Knyazev, T. P. Silverstein, S. Brescia, A. Maznichenko, and P. Pohl. A new theory about interfacial proton diffusion revisited: The commonly accepted laws of electrostatics and diffusion prevail. Biomolecules , 13:1641, 2023

work page 2023

[5] [5]

Heberle, G

J. Heberle, G. Riesle, J. Thiedemann, D. Oesterhelt, and N. A. Dencher. Proton migration along the membrane surface and retarded surface to bulk transfer. Nature , 370, 1994

work page 1994

[6] [6]

J. Heberle. Proton transfer reactions across bacteriorhodopsin and along the membrane. BBA-Bioenergetics , 1458, 2000

work page 2000

[7] [7]

R. B. Gennis. Proton dynamics at the membrane surface. Biophysical Journal , 110, 2016

work page 2016

[8] [8]

D. A. Cherepanov, B. Feniouk, , W. Junge, and Mulkidjanian A. Y. Low dielectric permittivity of water at the membrane interface: effect on the energy coupling mechanism in biological membranes. Biophysical journal , 85:1307--1316, 2003

work page 2003