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Mitochondrial cristae modeled as an out-of-equilibrium membrane driven by a proton field Nirbhay Patil, Stéphanie Bonneau, Frederic Joubert, Anne-Florence Bitbol, Hélène Berthoumieux

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Nirbhay Patil, Stéphanie Bonneau, Frederic Joubert, Anne-Florence Bitbol, Hélène Berthoumieux. Mitochondrial cristae modeled as an out-of-equilibrium membrane driven by a proton field. Phys- ical Review E , American Physical Society (APS), 2020, 102 (2-1), pp.022401. 10.1103/Phys- RevE.102.022401. hal-02936356

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HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. PHYSICAL REVIEW E 102, 022401 (2020)

Mitochondrial cristae modeled as an out-of-equilibrium membrane driven by a proton ﬁeld

Nirbhay Patil,1,2 Stéphanie Bonneau,2 Fréderic Joubert ,2 Anne-Florence Bitbol,2,3 and Hélène Berthoumieux1 1Laboratoire de Physique Théorique de la Matière Condensée (LPTMC, UMR 7600), Sorbonne Université, CNRS, F-75005 Paris, France 2Laboratoire Jean Perrin (UMR 8237), Institut de Biologie Paris-Seine, Sorbonne Université, CNRS, F-75005 Paris, France 3School of Life Sciences, Institute of Bioengineering, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

(Received 6 December 2019; revised 25 June 2020; accepted 27 June 2020; published 5 August 2020)

As the places where most of the fuel of the cell, namely, ATP, is synthesized, mitochondria are crucial organelles in eukaryotic cells. The shape of the invaginations of the mitochondria inner membrane, known as a crista, has been identified as a signature of the energetic state of the organelle. However, the interplay between the rate of ATP synthesis and the crista shape remains unclear. In this work, we investigate the crista membrane deformations using a pH-dependent Helfrich model, maintained out of equilibrium by a diffusive flux of protons. This model gives rise to shape changes of a cylindrical invagination, in particular to the formation of necks between wider zones under variable, and especially oscillating, proton flux.

DOI: 10.1103/PhysRevE.102.022401

I. INTRODUCTION potential difference between the cristae and the matrix, as energy supply [8]. More recently, it has been established that Mitochondria are important organelles of eukaryotic cells the proton flux going down the gradient and allowing the often called the “powerhouses of the cell,” due to their role in rotor to turn with respect to the stator in the ATP synthase the synthesis of adenosine triphosphate (ATP) from adenosine enzyme is probably localized on the surface of the crista diphosphate (ADP) and an inorganic phosphate (Pi). These organelles of micrometric size comprise an inner membrane membrane [9,10]. This flux is established and maintained by (IM), which delimits a region called the matrix, and an the respiratory chain, which injects protons from the matrix outer membrane (OM) [1,2]. The volume between the IM on the IMS part of the crista membrane. The proteins of the and the OM is called the intermembrane space (IMS). The respiratory chain are located in the weakly curved zones of inner membrane presents numerous tubular invaginations of the invagination and thus are spatially separated from the nanometric size, called cristae, where ATP synthesis takes ATP synthases. Recent in vivo pH measurement show that the place. The liquid inside the cristae is isolated from the IMS pH decreases along the crista membrane between the proton by the crista junction, an aggregate of proteins that limit the source (the respiratory chain proteins) and the proton sink (the diffusion [3]. Recently, it has been shown that cristae have ATP synthase) [11]. a higher membrane potential than the intervening boundary Membrane deformation driven by out-of-equilibrium membranes, involving confined proton loops and individual chemical dynamics is a ubiquitous phenomenon in living functioning of each crista within the same mitochondrion [4]. cells. A mechanism of hydro-osmotic instabilities generated It has been observed experimentally in isolated mitochondria by ion pumps has been recently suggested to describe the that the cristae assume different shapes depending on the state dynamics of the contractile vacuole complex [12]. Surface of ATP production. Five stationary states (state I to state V) deformation driven by diffusion of an “active” species is have been introduced to describe the energy status of isolated commonly observed in vivo such as the division of eukaryotic mitochondria [5], but most present research focuses on state cells by accumulations of myosin motors at the cell ring [13]. III and state IV, since it allows one to mimic the in vivo A phenomenological model has been proposed to study the situation where an increase of energy demand and energy coupling between diffusion of active agents and surface shape production occurs. Here we will consider only these two by introducing a modified Helfrich model associated with an states. A high rate of proton injection by the respiratory chain active tension coupled to the two-dimensional diffusion of the and of ATP production (state III) is associated with bumpy and chemical regulator on the deforming surface [14]. Note that wide crista tubules, while a low rate of proton injection and the coupling between a diffusive active agent and the bending ATP production state IV) is associated with a more regular of a membrane has not been included in the model [15]. cylindrical shape as illustrated by Fig. 1. Moreover, recent The dynamical coupling between crista shape and ATP experiments employing superresolution imaging techniques production rate is a recent discovery and the physicochemical have directly evidenced the dynamical deformations of cristae mechanism at the origin of this coupling still needs to be in cultivated cells [6]. characterized. The lipid composition of the IM has been + → The endothermic reaction ADP Pi ATP, is catalyzed pointed out as a key point for the ATP-synthesis machinery. by the ATP synthase, which is located in the curved zone Indeed, the crista membrane is enriched in cardiolipin, and of the crista membrane [7]. Traditionally, the ATP synthase loss of mature cardiolipins affects the shape of the crista enzyme was supposed to use the bulk proton electrochemical and perturbs its function [16]. These lipids possess a protic potential gradient, involving both the bulk pH and the electric hydrophilic head, and in vitro experiments have shown that

FIG. 1. Cross section of a mitochondrion in different states of ATP production [2]. (a) State III, for which sugar in excess is available leading to a high rate of ATP production. (b) State IV, for which no sugar is available leading to a vanishing ATP production rate. Mitochondria (left to right) have diameters of 1500 nm and 500 nm. tubular invaginations can be created by an externally con- FIG. 2. Schematic representation of a crista. (a) The plots rep- trolled pH gradient in giant vesicles comprising cardiolipins resent the proton concentration on the surface in states III and IV. [17]. A theoretical description modeling these giant vesicles The tubes represent the shape of the invagination in state III and IV. as locally planar bilayer membranes with lipid density and (b) Deformation fields of the membrane and intrinsic basis of the composition heterogeneities in each monolayer [18] has suc- deformed surface. cessfully reproduced the dynamics of the membrane in the regime of small deformations [19]. In this model, composition can represent, e.g., the acid and the basic form of cardiolipins, In experimental observations [20], cristae feature different which is controlled by the local pH field. However, because shapes, the most common being an elongated pancake, with the crista membrane is enriched in proteins, representing up to rows of ATP synthase situated at the rim of the protrusion 50% of its mass, a detailed model describing a pure lipid bi- and respiratory chain proteins located in the flat zone. We layer and including the slippage between the two monolayers nevertheless chose to work in the cylindrical geometry, which may not be necessary to describe this system. Therefore, here is a simple special case of this pancake shape. Indeed, this we consider a simpler and more phenomenological Helfrich geometry is analytically tractable, and despite its simplicity, model with pH-dependent parameters. it captures several key ingredients of the system, such as the This work proposes a model for the dynamics of the nanometric confinement and the presence of zones of various deformation for the crista membrane between state IV and curvatures. The protons diffuse on the crista surface at the state III. We start with a reaction-diffusion system describing concentration the proton flux on a cylinder (representing the crista mem- + + brane), which contains a proton source, a proton sink, and a [H ](s) = [H ]IV + h(s), (1) reflecting barrier. The resulting proton concentration field will be considered as the driving force inducing the membrane de- where s is a coordinate parametrizing the position along formation. We then propose a pH-dependent Helfrich model, + the tube, while [H ]IV represents the constant concentration in which the bending modulus, spontaneous curvature, and in state IV taken as a reference, and h(s) is the variation tension depend on the local proton concentration, assuming in the proton concentration induced by the functioning of small variations of this concentration. We derive the Green the respiratory chain and the ATP synthases. Note that the function of the system and study the phase diagram of the concentration is expressed as a number of protons per unit crista shape in this model. Finally we solve the hydrodynamic of length, taking advantage of the one-dimensional symmetry equations of the system for a proton field oscillating between of the problem. At one end of the cylinder, s = 0, one finds a state III and state IV and show that such a model generates reflecting barrier for the protons modeling the crista junction, dynamical deformations of the membrane, as well as the while at the other end, s = L, a ring-shaped proton sink formation of necks and bumps, along the cylinder leading to models the ATP synthases. Between the two, at s = Ls,a a rougher surface in state IV. The last part is devoted to the ring-shaped proton source models the respiratory chain. This conclusion. model for the geometrical confinement of the ATP synthesis machinery takes into account the spatial separation of the II. MODEL OF A MITOCHONDRIAL CRISTA proton source and sink and their respective localization in zones of low and high curvature [20]. A. Proton field along the crista + We assume that h(s) is small, i.e., h(s)/[H ]IV 1, and We model the crista as an axisymmetric cylinder of mem- that the tube shape does not deviate much from a regular brane of finite length L closed by a spherical cap (see Fig. 2). cylinder. In this framework, the equation governing the proton

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TABLE I. Parameter values estimated from recent experimental measurements.

L 150 nm R 10 nm −1 4 −1 ksc 600 s [11] ksk 2.7 × 10 s [11] D 10−7 cm2 s−1 [9] ω 0.63 s−1 −7 −1 −19 σ0 10 Nm [19] κ0 10 Nm −9 −1 −10 −1 ηb 10 Nsm [19] ηs 5 × 10 Nsm [19] concentration on the surface can be written as follows: ∂h(s, t ) ∂2h(s, t ) = D + S (s, t ) − S (s, t )(2) ∂t ∂s2 in out FIG. 3. Dynamics of the proton field along the cristae. The plots , ∂h(s, t ) represent the field h(s t ) solution of Eq. (3) for parameter values = = / = / = 0, (3) given in Table I and for tIV 0[T ], tIV→III T 4[T ], tIII T 2[T ], ∂s = s 0 tIII→IV = 3T/4[T ]. The concentration h(s, t )isexpressedinproton where Eq. (3) illustrates the reflecting barrier, D is the dif- per nm. fusion coefficient, and Sin(s, t ) and Sout (s, t ) are the source and the sink of the proton, respectively. We consider a system concentration of 8 × 10−2 proton per nanometer. These plots oscillating with a period T = 2π/ω between a state IV of are obtained in the case of an oscillating period much longer homogeneous proton concentration and a state III associated than the typical diffusion time along the tube, 2π/ω L2/D. with a maximal proton flux and write k [1 − cos(ωt )] −(s − L )2 B. Model of the membrane S (s, t ) = sc exp s , (4) in 2 To model the membrane of a mitochondrial crista, we start 2 2π2 2 1 1 from the Helfrich model for elastic membranes. Developed by − ω ksk[1 cos( t )] + Wolfgang Helfrich in 1973 [21], this effective energy func- Sout (s, t ) = [[H ]IV + h(s, t )] π2 tional of a membrane takes into account molecular properties 2 2 2 of lipid membranes such as the fluidity and the absence of −(s − L)2 in-plane shear stress but is written at a continuous coarse- × exp , (5) 22 grained scale. We modify the standard model by introducing 2 pH-dependent parameters, with ksc the maximal rate of injection of the proton source, + k ([H ] + h(s, t )) the maximal rate of the proton sink. The 1 2 sk IV H = κ(s)[C − C0(s)] + σ (s) dA, (6) spatial extensions of the source and of the sink are modeled 2 by two Gaussian functions of widths 1 and 2, respec- where C is the local curvature of the surface and dA the area , = tively. The system oscillates between state IV (Sin(s t ) 0, of a surface element. The surface tension σ (s), the bending S (s, t ) = 0) for t = 0 modulo T, noted [T ], and state III out IV modulus κ(s), and the spontaneous curvature parameter C0(s) (where the source and the sink function at their top rates) for are assumed to depend linearly on the surface proton concen- = / ω tIII T 2[T ] with a frequency . We assume that there is tration as follows: no proton accumulation in the cristae during a period, i.e., L 2π/ω L 2π/ω , = , κ(s) = κ0 + h(s)δκ, (7) 0 ds 0 dtSin(s t ) 0 ds 0 dtSout (s t ), which sets the value of the ratio k /k . sc sk σ = σ + δσ, The proton concentration h(s, t ) is obtained by solving nu- (s) 0 h(s) (8) merically Eq. (3) with an initial vanishing concentration field = + δ , h(s, 0) = 0, using the parameter values given in Table I.The C0(s) C00 h(s) C0 (9) profiles of the proton concentration along the tube at different with h(s) defined in Eq. (1). For simplicity the Gaussian times, shown in Fig. 3, are obtained assuming a diffusion curvature is not considered here. In this chemico-mechanical = −7 2 −1 coefficient of D 10 cm s [9], which corresponds to model, the chemical reaction of ATP synthesis generates a the estimated diffusion coefficient of protons along a lipid dynamical field h(s, t ) on the membrane and will drive the membrane. The system oscillates between state IV, in which deformation of the cylinder. the source Sin(s, t ) and the sink Sout (s, t ) vanish and the field h(s, t ) is uniform and equal to zero along the cylinder, and IV III. DYNAMICAL EQUATIONS OF THE SURFACE state III, for which the proton concentration is approximately DEFORMATION homogeneous between the junction (L = 0) and the source (L = Ls) and decreases between the source and the sink. Note We consider an initial equilibrium state defined by a van- 2 that protons diffuse with a characteristic time τD = L /D = ishing field h(s, t ) = 0 and a finite cylinder of length L and 0.25 ms between the source and the sink. An injection rate ksc of radius R. In this case, the Hamiltonian given in Eq. (A1) equal to 600 protons per second at the maximum rate, which is restricted to the Helfrich model in which the spontaneous is a reasonable value for a crista of this size [11], leads to a curvature C00 is a phenomenological parameter, illustrating an

022401-3 NIRBHAY PATIL et al. PHYSICAL REVIEW E 102, 022401 (2020) asymmetry in the membrane. A nonvanishing, nonhomoge- The curvature tensor, also known as the second fundamen- neous proton field h(s) will lead to a deformed cylinder. The tal form, is defined as Kab = ea · ∂b n using the convention that initial surface and the deformed surface, can be respectively for a pointing outward normal vector, the curvature is positive parametrized by the three-dimensional vectors X0 and Xt , [22]. It gives defined as follows: R + un 0 = + δ , K = . (15) Xt X0 Xt (10) ab − 0 un with = s + θ ⎛ ⎞ ⎛ ⎞ Finally, the sum C Ks Kθ of the principal curvatures can R cos(θ ) un(z, t ) cos(θ ) be written as ⎜ ⎟ ⎜ ⎟ X0 = ⎝R sin(θ )⎠,δXt = ⎝un(z, t )sin(θ )⎠ 1 u C = − n + u . (16) 2 n z us(z, t ) R R with z ∈ [0, L],θ∈ [0, 2π]. (11) a = kb ab = −1 using Kb Kakg , with g (gab) . Finally, we recall the expression of the covariant derivative θ The surface is thus parametrized by the variables z, such of a tangential vector xae , that any point on this surface can be uniquely represented by a a value of each of these parameters: Xt (s,θ). The deformed ∇ b = ∂ b + b c, ax ax acx (17) state Xt is characterized by two fields un(z, t ) and us(z, t ) ab defined in Fig. 2(b). In this work, we consider only small and of a tensor t ea ⊗ eb, deformations, i.e., us(z, t )/R, un(z, t )/R much smaller than 1. ∇ bc = ∂ bc + b dc + c bd , In the following, we will work in the intrinsic basis of the de- at at adt adt (18) formed surface represented in Fig. 2(b) and use the curvilinear abscissa s. To first order, the derivatives with respect to z and s where the Christoffel symbols can be written as of the deformation fields are equal: ∂zui(z, t ) = ∂sui(s, t ), with −Ru 0 0 un i = s, n, and we will use the second ones. s = n , θ = R . ab ab (19) 0 u un s R 0 A. Some elements of differential geometry for an axisymmetric To characterize the dynamical deformation of the tube, we membrane introduce the flow velocity of the surface elements of the We wish to describe the dynamics of the axisymmetric membrane, membrane Xt driven by the concentration field h(t ). Let us first introduce some terminology and results from differential v(s, t ) = vs(s, t )es + vn(s, t )n, (20) geometry and their expressions to the first order in the deformation field. Note that to simplify the notations, we denote with vs = ∂t us and vn = ∂t un. It is composed of an in-plane = the fields un(s, t ) and us(s, t )asun and us and the spatial flow vs vses and a term describing the deformation of the ∂ , = = , surface v = v n. derivative sui(s t ) ui, with i s n. The tangent vectors n n on the surface are defined as ⎛ ⎞ θ B. Stress tensor acting on the deforming surface un cos( ) ⎜ ⎟ Next, we determine the stress tensor f acting on the surface es = ∂sXt = ⎝u sin(θ )⎠, n X . General surface stresses are complex objects that one can 1 + u (s) t s grasp by asking the question: “What forces should be exerted onto a membrane edge with unitary length to prevent it from −(R + un )sin(θ )$R + un ) cos(θ ) eθ = ∂θ Xt = . shrinking?” [22]. In this case, the stress tensor is the sum of 0 two contributions: a mechanical stress tensor fH deriving from (12) the Helfrich energy given in Eq. (A1) and a viscous stress tensor fη. The normal vector can be expressed as The mechanical stress tensor fH can be written as a 3 × 2 ⎛ ⎞ tensor [23,24] that can be decomposed into a surface stress cos(θ ) tensor f ab generating forces tangent to the surface and a ∧ H eθ es ⎜ ⎟ × n = an × n = = ⎝sin(θ )⎠, (13) 2 1 tensor fH fH n ea generating forces normal to the | eθ ∧ es | surface. To derive the expression of f, we follow the approach −u n developed by Guven and coworkers, presented in the Ap- X → and the metric of the surface is defined as g = e · e , with pendix. Instead of varying the shape of the membrane( t ab a b X + δX (a = (θ,s), b = (θ,s)) and is equal to t t ) and explicitly tracking the changes of the intrinsic basis, the metric, the curvature, and the energy of the 2 deformed membrane, this elegant approach enforces the geo- R + 2Run 0 g ≈ . (14) metric relations associated with the fundamental forms of the ab + 012us membrane by introducing Lagrange multipliers. We introduce

022401-4 MITOCHONDRIAL CRISTAE MODELED AS AN … PHYSICAL REVIEW E 102, 022401 (2020) the extended functional H = 1 κ − 2 + σ with 2 (s)[C C0(s)] (s), the mechanical energy density of the membrane. (Details of the calculation are given = + λab − · in the Appendix). Hc H (gab ea eb) dA In our case, the mechanical stress associated with the surface Xt depends both on the deformation fields ab a , , , , + (Kab−ea · ∇bn) dA+ f · (ea − ∇aX) dA (us(s t ) un(s t )) and on the concentration field h(s t ). The ab an surface stress tensor fH and the normal stress tensor fH a 2 expanded to first order in the fields can be expressed as + λ⊥(ea · n) dA + λn(n − 1)dA, (21) ab = ab + ab , fH fH0 fH1 (26)

λab ab an = an + an , in which H is given in Eq. (A1), the matrices and fH fH0 fH1 (27) enforce the definition of metric and the curvature, fa pins the a with basis vector to the tangent of the surface, λ⊥ enforces the κ − 2 − σ 2 λ 0 (1 X ) 2 0R normal vector to be perpendicular, and n its normalization. 4 0 f ab = 2R , The minimization of the Hamiltonian Eq. (21) with respect to H0 − 2κ (28) 0 − (1 X ) 0 − σ the, now, 12 independent functions gives [25] 2R2 0 an = and fH0 0, and where the first-order part, fab = T ab − HacKb, ab = ab + ab, H c (22) fH1 fh fM (29) f an =−∇ Hab , an = an + an, H ( b ) (23) fH1 fh fM (30) ab is the sum of a term f1h depending on the concentration field with , ab h(s t ) and of a term f1M depending on the deformation fields un(s, t ), us(s, t ). δH Using the expression of the Hamiltonian given in Eq. (A1), Hab = , (24) the dependences of κ, C00, and σ on h(s, t )giveninEqs.(7)– δKab √ (9) and the expression of the stress tensor given in Eqs. (29) 2 δ gH T ab =−√ , and (A13), we derive the expression of the stress tensor to first δ (25) g gab order in the fields and find

2σ + − 2 κ − 2 κ [2R 0 (2 X ) 0]un (s) R X 0un (s) 0 f ab = R5 , (31) 1M − − 2κ + 2σ 1 X κ + (1 X ) 0 2R 0 0 R3 0un(s) R2 us(s) κ f an = 0, 0 u (s) + κ u(s) , (32) 1M 2 n 0 n R 2δσ+ κ δ − − 2 δκ − 2R 2RX 0 C0 (1 X ) Rh(s)0 f ab = 2R5 , 1h − κ δ − δσ − − 2δκ (33) 0 2R[(1 X ) 0 C0 R ] (1 X ) h(s) 2R2 Rκ δC − (1 − X )δκ f an = 0, 0 0 h(s) . (34) 1h R

The stress tensor depending on the proton field involves where v is given in Eq. (20), which is derived from the strain , ab both the field h(s t ) in its tangential component f1h and its tensor of a three-dimensional fluid shell taken in the limit of a , an spatial derivative h (s t ) in its normal component f1h .The small thickness [26]. The viscous stress in compressible thin latter contribution will vanish for a constant field h. In fact, films can be expressed as h a constant field simply induces a renormalization of the 1 c c ,η = η − + η , parameters of the Helfrich model. By contrast, a spatially fab 2 s vab 2 vc gab bvc gab (36) , nonhomogeneous field h(s t ) will generate forces on the which involves the two-dimensional strain rate v and phe- normal direction that tend to pinch or expand the tube. ab nomenological coefficients ηs and ηb that are the shear and The viscous stress tensor for a deforming membrane is a a b bulk viscosity of the film. The viscous stress fη = fab,ηe × e surface tensor that can be obtained from the two-dimensional can be written as 2 × 2 matrix, whose components are given strain rate, in covariant coordinates in Eq. (36)[22]. It yields the force generated by the flow within the membrane surface. Note that we do not account for the bulk viscosity of the fluid = 1 ∇ + ∇ + , vab 2 ( avb bva ) Kabvn (35) surrounding the membrane in our force balance. Indeed, its

022401-5 NIRBHAY PATIL et al. PHYSICAL REVIEW E 102, 022401 (2020) contribution can be neglected for strongly curved membrane (a) buds [27] and tubes [28] with characteristic sizes smaller than a few micrometers, which is the appropriate regime for cristae. In the absence of deformation, i.e., when the ﬁelds vn, vs, un, us, h vanish, the viscous stress also vanishes and the equilibrium shape of the surface can be obtained by setting to (b) ab zero the divergence of the stress tensor fH,0 given in Eq. (28).

C. Hydrodynamic equations for the tubular membrane We now consider the response of the membrane to a time- varying ﬁeld, h(s, t ). In the absence of inertia, corresponding to the low-Reynolds number regime which is appropriate at the length scales considered, the dynamics of the system (c) derives from the force balance, which can be written in the tangential basis of the surface as ∇ a + a = , a fb Kab fn 0 (37)

an ab ∇a f − Kab f = 0, (38) where the stress tensor f = fH + fη is the sum of the viscous and the mechanical stress tensor. The first equation, Eq. (37), FIG. 4. (a) Stability diagram of a Helfrich cylinder as a function of X = C00R. (b) 1. Gn(s)/R, Normalized Green functions for differ- corresponds to the force balance on the surface along es and ent values of X =−0.9, 0, 0.9andforδσ = 0.15 and δC0 = 0. 2. eθ . The second equation, Eq. (38), often called the shape Characteristic lengths of oscillation λ0 and decay λd as a function equation of the surface, is the force balance in the normal of X. 3. Maximal amplitude of the deformations driven respectively direction. Expanding the covariant derivatives in Eqs. (37) and by a variation of σ , Anσ andbyavariationofC0, AnC . (c) Three- (38) and collecting the first-order terms in the fields vn, vs, un, dimensional representations of a cylinder (top) deformed by h(s) = u , h, we derive the hydrodynamic equations governing the s δ(s)for(X = 0.96, δσ = 0.2, δκ = 0, δC0 = 0) (middle), and for time evolution of the system (X = 0.96, δσ =−0.2, δκ = 0, δC0 = 0) (bottom). All quantities η + η ∂ s + η − η ∂ θ + ∂ s are dimensionless. ( s b) svs ( b s ) svθ s fH1,s θ The shape equation for this system, given in Eq. (38), can be + ( f s − f ) s + Rfns = 0, (39) H0,s H0,θ sθ written as s θ − 2 κ vs vθ ns ss ss (1 X ) 0 (ηs + ηb) + (ηb − ηs ) + ∂s f − us f − Rf σ − = 0, (43) R R H1 H0 H1 0 2R2 +u f θθ = . ab n H0 0 (40) where we have employed the expression of fH0 given in Eq. (28). Given that R, σ and κ are necessarily positive, this The force balance along eθ vanishes for symmetry reasons. ab ab an equation admits a solution for Replacing the mechanical tensors f0H , f1H , f1H by their expressions given in Eqs. (28), (29), and (A13), we finally X = C00R ∈ ] − 1, 1[. (44) obtain the dynamical equations of the system as a function of the velocity, displacement, and concentration fields: In this range [see Fig. 4(a)], the radius of the equilibrium cylinder R can be expressed as a function of the parameters + + = , of the nonperturbed Helfrich model, a1 vn a2vs a3h 0 (41) 1 + + + + + + = , R = . (45) b1vn b2vs b3un b4un b5un b6h b7h 0 σ 2 + 2 0 C κ (42) 00 0 The stationary shape of a deformed cylinder associated with a where the coefficients ai,(i = 1,...,7) and bi (i = 1,...,3) perturbation field h(s) = 0 is given by the displacement fields are given in Appendix A2 and where ui and vi denote the (u (s), u (s)), which are solutions of Eqs. (41) and (42) where spatial derivative u = ∂ u (s, t ) and v = ∂ v (s, t )fori = n s i s i i s i v and v are set to zero. Replacing the coefficients (a , b )by s, n. n s i j their expressions given in Appendix A2, we obtain 4Rσ δC δκ(1 − X )2 IV. STATIC GREEN FUNCTION OF AN INFINITE 0 0 − 2δσ − h(s) = 0 (46) MEMBRANE CYLINDER 1 + X R2 2σ To gain physical insight into the model and the static solu- − 0 u (s) + 2R2Xu(s) + R4u(s) 2 − 2 n n n tions of Eqs. (39) and (40), we derive the static Green function R (1 X ) of the system. We consider as a reference equilibrium state an 2[(1 + X )δσ − Rσ δC ] = 0 0 [h(s) + R2h(s)]. (47) infinite cylinder of radius R and a vanishing field h(s) = 0. R(1 − X 2 )

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The first equation corresponds to the force balance along λo and decay λd are given, respectively, by the vector es. The second equation is the shape equation. We observe that this system of equations does not depend on λ = 2 , o R 1+X (53) us. A nonhomogeneous perturbation field h(s), which yields nonhomogeneous tension, bending rigidity, and spontaneous λ = 2 , d R − (54) curvature, can lead to a stationary shape in the cylinder 1 X geometry only if the condition for X satisfying the condition given in Eq. (44). Figure 4(b), panel 2, represents λo and λd as functions of X for the range of possible equilibrium cylinders. The decay 2 4Rσ0δC0 δκ(1 − X ) λ − 2δσ − = 0 (48) length d is an increasing function of X that diverges for 1 + X R2 X = 1. Thus, cylinders with larger spontaneous curvatures are deformed on a longer range by a heterogeneous h.The oscillation length λo, on the contrary, is a decreasing function between the perturbation parameters, (δκ, δσ, δC0), is satis- of X. fied. Assuming that this condition holds, we express δκ as a The limit X = 1 is associated with a phenomenon of buck- δσ δ function of and X, ling (λd = 0, λo = R), i.e., an infinite oscillating deformation wave. Indeed, X = 1 corresponds to a cylinder with curvature along eφ equal to the spontaneous curvature of the membrane. 2 4Rσ0 2R A sphere of radius R is an equilibrium shape of such a δκ = δC0 − δσ. (49) (1 − X )2(1 + X ) (1 − X )2 membrane. A perturbation of the cylinder will tend toward such shapes by forming a succession of drops. Conversely, the limit X =−1 corresponds to a cylinder folded in the Let us now consider a localized perturbation in the pro- direction opposite to that of the spontaneous curvature. The ton concentration: h(s) = h0δ(s) with h0 = 1 introduced to deformation induced by a perturbation will then be purely λ = dimension h(s), and derive the Green function Gn(s) yielding decaying with a characteristic length d R, thus minimizing the deformation field un(s) in response to this perturbation. the energy of deformation. Performing a Fourier transform of the shape equation given The respective amplitudes Anσ , AnC of the deformations iqs σ in Eq. (47) and introducingu ñ(q) = 1/2π dse un(s), the induced by perturbations in or in C0, and defined as normal deformation field in the Fourier space, we find Gn(sm ) Anσ (X ) = Gnσ (X ) , (55) Gn 2 2 − + δσ − σ δ G (s ) R(q R 1)[(1 X ) R 0 C0] A X = G X n m uñ(q) ≡ G˜ n(q) = √ . nC ( ) nC ( ) (56) 2 2 4 4 Gn σ0 2π(1 − 2Xq R + q R ) (50) are plotted with respect to√X √in Fig. 4(b), panel√ 3.√ Here Performing the inverse Fourier transform of G˜ given in − n we have introduced s = 2/ 1 + X sin 1 ( X + 1/ 2), Eq. (50), we obtain the expression of the Green function in m which is the coordinate of the maximum of G (s)/G .The real space, n n absolute values of the amplitudes Anσ , AnC are increasing functions of X but possess opposite signs. An increase of the tension [δσ > 0, h(s) > 0] leads to a constriction of the | | + | | − s 1 X s 1 X cylinder (since Anσ < 0). On the other hand, an increase Gn(s) = GnR sin exp − , R 2 R 2 of the spontaneous curvature (δC0 > 0, h(s) > 0) leads to > (51) a dilatation of the cylinder (since AnC 0). A perturbation δ = . where we have introduced in the spontaneous curvature C0R 0 1 (respectively in the surface tension δσ/σ = 0.1) will induce a variation of the radius of 5% (respectively, 10% ), for X → 1. For X negative δσ or 0 < X 1, the deformation induced by a variation of h is G = G σ (X ) + G (X )RδC , n n σ nC 0 negligible. Figure 4(c) shows three-dimensional representations of the nonperturbed and of the perturbed cylinder for X = 0.9, and with with δσ > 0 and δσ < 0, respectively. This illustrates that the model proposed in Eq. (A1) can generate tubular membrane shapes of various curvature that can resemble mitochondrial (1 + X ) 1 Gnσ (X ) =−√ √ , GnC (X ) = √ √ . (52) cristae. 2 1 + X 2 1 + X Using the expression of the Green function given in Eq. (51), we can find the stationary shapes generated by any proton concentration field h(s) through The deformation induced by a localized perturbation is thus ∞ an oscillating and exponentially decaying function, as shown un(s) = h(x)Gn(s − x) dx. (57) in Fig. 4(b), panel 1. The characteristic lengths of oscillation −∞

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(a) vanish at the boundary of the tubule, and the ﬁrst-order stress tensors satisfy s , = , s , = , f1,s(0 t ) 0 f1,s(L t ) 0 (60) sn , = , sn , = , fH1(0 t ) 0 fH1(L t ) 0 (61) s = s + s where f1,s fH1,s fη,s, and where the expression of the (b) stress is given in Eqs. (29), (A13), and (36). Moreover, the edges of the cylinder are assumed to be pinned in s = 0 and s = L, which leads to a vanishing tangential velocity vs in s = 0 and s = L,

vs(0, t ) = 0, vs(L, t ) = 0. (62)

(c) In order to facilitate the numerical resolution of the hydrody- namical equations given in Eqs.(41) and (42), the deformation fields un(s, t ), us(s, t ) are expressed as function of the velocity fields vn(s, t ) and vs(s, t ) and of the deformation field at the time t − dt, using the backward Euler method and a FIG. 5. Infinite cylinder submitted to a step of proton concen- discretization of the time with a time step dt, tration. (a) Proton concentration h(s) along the cylinder. (b) De- t = t−dt + × t , formation field un(s) induced by a step of proton for X = 0.9, us us dt vs (63) δσ = 0.1, δC0 =−0.1. (c) Three-dimensional representation of the t = t−dt + × t , corresponding shape. All quantities are dimensionless. un un dt vn (64) t , t where we have introduced the notation (ui vi )for , , , As a practical illustration, in Fig. 5, we consider a step (ui(s t ) ui(s t )). Inserting the expressions in Eqs. (63) and (64) the deformation fields into Eqs. (41) and (42) leads function for h(s), t t to the following coupled system of equations for vs and vn: h(s) =−1s< 0, = 1s> 0 (58) ∂ t + ∂2 t + ∂ t = , a1 svn a2 s vs a3 sh 0 (65)

[see Fig. 5(a)]. The corresponding deformation field u (s) t t 4 n (b1 + b3 × dt)v + b4 × dt∂sv + b5 × dt∂ vn obtained using Eq. (57) is continuous, unlike the input step n n s + ∂ t + t−dt + ∂2 t−dt function, and is an odd function of s that oscillates before b2 svs b3un b4 s un reaching a plateau of constant value, as shown in Fig. 5(b). t−dt t 2 t + b5∂su + a6h + a7∂ h = 0. (66) The resulting stationary shape of the tube, represented in Fig. n s 5(c), corresponds to two cylinders of different radii welded Assuming that the geometry of the system at the time together through an oscillating neck. t − dt is known, the system to solve is a couple of ordinary t t differential equations in space for vs and vn for which the six necessary boundary conditions are given in Eqs. (60)–(62). V. APPLICATION TO MITOCHONDRIA CRISTAE The complete dynamic of the system is obtained by solving = In this section, we describe the dynamical deformation of Eqs. (65) and (66) starting at t 0, when all the fields, 0, 0, 0 a finite tube of membrane of length L submitted to the proton h ui vi , vanish. Then, knowing the geometry of the system − t−dt field h(s, t ) solution of Eq. (3) and represented in Fig. 3, which at the time t dt, i.e., the deformation fields ui , and the t t , t models a crista oscillating between state IV and state III. concentration field at the time t, h , the velocities (vs(s) vn(s)) To do so, we first specify the external mechanical forces are determined as solutions of Eqs. (65) and (66). Next, the de- t t exerted on the axisymmetric membrane. Here we do not take formation fields un, us at time t are calculated using Eqs. (63) into account extra external pressures or viscous forces that and (64). The procedure can then be iterated to obtain the state + could be applied to the membrane and we consider that a of the system at time t dt, and so forth. Further details on the numerical resolution are given in Appendix A3. constant tension fext is exerted by the rest of the mitochondria on the tubule boundary rings in s = 0 and in s = L. This force, Figure 6(a) represents the shape of the tube for the oscillating proton field given in Fig. 3. The tube alternates between a κ quasinondeformed cylinder for a quasivanishing proton field f (0) =− σ + (1 − X 2 ) 0 e , ext 2 s (see Fig. 3) and a tubular invagination presenting a bump 2R κ and a neck for the top proton flux. These oscillations are f (L) = σ + (1 − X 2 ) 0 e (59) ext 2R2 s obtained in the regime of a fast mechanical relaxation time τ0 = 2ηsηb/σ0(ηs + ηb) compared to the oscillation period balances the effective tension of the undeformed Helfrich between state III and IV, T = 2π/ω. It is thus the dynamics of cylinder, defined by X0 in Eq. (11), and derives from the the diffusive process that governs the shape change dynamics zero-order stress tensor fH0 given in Eq. (28). Consequently, in this case. Figure 6(b) represents the deformation fields the first-order forces deriving from the cylinder deformation associated with these shapes. We compute the bump-to-neck

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(a) for appropriate values of the parameters, it can reproduce the salient features of the experimentally observed shapes of the mitochondrial cristae, oscillating between regular tubes in state IV and bumpy tubular protrusions in state III. As a next step, for more realism, less symmetric mor- phologies closer to experimental observations of crista shapes, such as flat balloons with proton sinks on the rim and proton sources on flat zones, could be considered [20]. The phenomelogical model introduced here is a coarse- (b) grained description of a membrane containing different lipids, proteins, etc. It will be interesting to consider a more mi- croscopic model of the membrane to gain insight into the values of the parameters δκ, δC0, δσ of this model, and test whether realistic values yield membrane shapes resembling experimental observations. Finally, a coupling between the shape and the concentration field could be introduced by considering the advection of the protons on the deformed membrane. FIG. 6. Model crista oscillating between states IV and III. A In vivo, the model could be tested with super-resolution finite membrane tube is submitted to the proton field h(s, t )repre- imaging techniques by changing the energy demand and by sented in Fig. 3. (a) Representation of the shape of the cylinder at monitoring crista morphology variation within mitochondria. different times of the period (the same ones as in Fig. 3). (b) Defor- Alternatively, experiments on biomimetic membranes can be , , mation fields un(s t )andus(s t ) associated with these shapes (same performed where a local proton flux is introduced and mem- colors as in Fig. 3). The plots are obtained for values given in Table I, brane fluctuations are followed. ATP synthase and complexes = . δσ =− . σ , δ = . , X 0 8, 0 35 per h(s t ), C0 1 05 per h(s t ). of the respiratory chain can be co-reconstituted in liposomes to closely mimic the functioning of the respiratory chain ratio obtained in this framework and obtain 1.6. Measuring the [29]. Finally, proton diffusion along membranes mimicking projected sizes of the bumps and necks in state III on the eight the lipid composition of inner mitochondrial membranes can best visible cristae on Fig. 1 yields an average bump-to-neck be estimated using lipid-anchored pH sensor fluorescent dye. ratio of 1.9 (standard error of the mean: 0.3). The theoretical Such experiments should bring measurements allowing us to prediction is in agreement with the measures. This shows test the ability of our model to predict crista morphology. that the model developed in this work captures the salient features of the shape of the cristae in state III and IV observed APPENDIX experimentally and represented in Fig. 1 (regular tubes in state IV, bumpy tubes in state III). 1. Covariant stress tensor for a membrane In this subsection we derive the stress tensor associated VI. CONCLUSION with the Hamiltonian Mitochondrial cristae are membrane protrusions that con- 1 = κ − 2 + σ . fine the ATP-synthesis machinery. Experimental observations H (s)[C C0(s)] (s) dA (A1) 2 have shown that the shape of these protrusions is coupled to the energetic state of mitochondria, as seen in Fig. 1. However, We consider an infinitesimal deformation of the surface the underlying physical mechanisms controlling this coupling X → X + δX. Instead of tracking the variation of the intrinsic remain to be elucidated. basis and the fundamental forms of the surface induce by We considered the hypothesis that the shape of the in- this deformation, we enforce these geometrical constraints vagination is driven by the flux of protons diffusing on by introducing Lagrange multipliers, and we work with a the membrane. Indeed, the absence of cardiolipins in the generalized Hamiltonian Hc, crista membrane induces anomalous crista shapes, and local pH heterogenities induce crista-like deformations of GUVs ab ab Hc = H + λ (gab − ea · eb)dA+ (Kab − ea · ∇bn) comprising cardiolipins [17]. We described the mechanical properties of the mitochondrial crista membrane using a a pH-dependent Helfrich model coupled to a diffusive proton × dA + f · (ea − ∇aX) dA (A2) concentration field on the surface. We first derived the stationary Green function of this system and showed that it can a 2 + λ⊥(ea · n) dA + λn(n − 1)dA, (A3) qualitatively reproduce the bumpy shapes of the mitochondrial invaginations observed experimentally. We then studied the dynamical shape change of the invagination for a membrane in which the matrices λab and ab enforce the definition of tubule subjected to an oscillating proton concentration field, metric and the curvature, fa pins the basis vector to the tangent a modeling the oscillations between state IV (no ATP synthesis) of the surface, λ⊥ enforces the normal vector to be perpen- and state III (high rate of ATP synthesis). We showed that, dicular, and λn is its normalization. The minimization of the

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Hamiltonian (A3) with respect to the, now, 12 independent (1 − X )2 2Rσ a =− δκ − δσ + 0 δC , (A14c) functions gives the following equations: 3 2R2 1 + X 0 δ Hc a ηb + ηs = ∇af , (A4) b = , (A14d) δX 1 R2 δH η − η c =− a + ac b + λab − λa , b s f Kc 2 eb ⊥n (A5) b2 = , (A14e) δea R δ Hc ab a ab 2σ0 = (∇b + λ⊥)ea + (2λn − Kab)n, (A6) b = , (A14f) δn 3 R2(1 − X 2 ) δ Hc ab 4Xσ = ab + H , (A7) 0 δ b4 = , (A14g) Kab 1 − X 2 δ √ ab √ ab Hc ab g g 1 ab 2 = λ + λρ ρ g + λφρφ g − T , (A8) 2R σ0 δg 2 2 2 b5 = , (A14h) ab 1 − X 2 Hab ab (1 − X 2 ) 1 2Xσ where and T are defined as the functional derivatives of b =− δκ + δσ + 0 δC , (A14i) H = 1 κ − 2 + σ 6 3 − 2 0 the Hamiltonian density, 2 (s)[C C0(s)] (s) with 2R R 1 X respect to the two fundamental forms: 1 b = [2R3δC σ − (1 − X )2(1 + X )δκ]. δH 7 − 2 0 0 Hab = , R(1 X ) δ (A9) Kab√ (A14j) 2 δ gH T ab =−√ . (A10) Here κ has already been substituted in terms of σ , R, g δg 0 0 ab and X using the relation between the Helfrich parameters a We notice that the Lagrange multiplier f obeys a conser- (45). If we set a1, a2, b1, and b2 to zero (i.e., no viscosities), a vation law (∇af = 0). Using that Eqs. (A4)–(A8) vanish, we we obtain the steady-state force balance equations for the eliminate the Lagrange multipliers present in Eq. (A5) and Helfrich cylinder model. find a = ab − Hac b − ∇ Hab . f T Kc eb b n (A11) 3. Details of the numerical resolution of hydrodynamics equations The quantity fa can be identified as the surface stress tensor [22,30]. The surface stress tensor is a 2 × 3 matrix and its The system of equations (65) and (66) is solved numeri- coordinates in the intrinsic basis of the surface can be written cally at time t using the command NDSolve of the Mathe- t t as matica. The solutions obtained for vs and vn are discretized in space on a regular grid with a step equal to 10−2 for a ab = ab − Hac b, fH T Kc (A12) cylinder with radius R = 1. The discrete spatial derivatives of (i),vect ( j),vect the velocities [v , (i = 1,...,4), vs , ( j = 1, 2)] are an =−∇ Hab . n fH ( b ) (A13) derived on this grid. The vectors are interpolated using polynomials of de- ∂i pol ,∂j pol , = 2. Coefficients for dynamical force balance equations gree 18 to obtain analytical functions ( svn s vs i ,..., , = ,..., , The full expression for the dynamical force balance equa- 1 4 j 1 2) which are used to derive the defor- tions Eqs. (41) and (42) can be obtained by inserting the mation fields and their derivatives at the time step t, coefficients, which are given as follows: ∂iut (s) = ∂iut−dt + dt × ∂iv pol , i = 1,...,2, (A15) η − η s s s s s s a = b s , (A14a) 1 R ∂ j t = ∂ j t−dt + × ∂ j pol , = ,..., . s un(s) s un dt s vn j 1 4 (A16) a2 = ηb + ηs, (A14b)

[1] T. G. Frey and C. A. Manella, The internal structure [4] D. M. Wolf, M. Segaxa, A. K. Kondadi, R. Anand, S. T. of mitochondria, Trends Biochem. Sci. 25, 319 Bailey, A. S. Reichert, A. M. van der Bliek, D. B. Schackelford, (2000). M. Liesa, and O. S. Shirihai, Individual cristae within the [2] C. A. Mannella, Structure and dynamics of the mitochondrial same mitochondrion display different membrane potentials inner membrane cristae, Biochim. Biophys. Acta (BBA)-Mol. and are functionally independent, EMBO J 38, e101056 Cell Res. 1763, 542 (2006). (2019). [3] S. Koob and A. S. Reichert, Novel intracellular func- [5] B. Chance and G. R. Williams, Respiratory enzymes in oxida- tions of apolipoproteins: The ApoO protein family as tive phosphorylation, III the steady state, JBC 217, 409 (1955). constituents of the Mitoﬁlin/MINOS complex determines [6] C. Wang, M. Taki, Y. Sato, Y. Tamura, H. Yaginuma, Y. Okada, cristae morphology in mitochondria, Biol. Chem. 395, 285 and S. Yamaguchi, A photostable ﬂuorescent marker for the (2014). superresolution live imaging of the dynamic structure of the

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mitochondrial cristae, Proc. Natl. Acad. Sci. USA 116, 15817 [18] A.-F. Bitbol, L. Peliti, and J.-B. Fournier, Membrane stress (2019). tensor in the presence of lipid density and composition inho- [7] S. Cogliati, J. Enriquez, and L. Scorrano, Mitochondrial cristae: mogeneities, Eur. Phys. J. E 34, 53 (2011). Where beauty meets functionality, Trends Biochem. Sci. 41, [19] A.-F. Bitbol, N. Puff, J.-B. Fournier Y. Sakuma, M. Imai, and 261 (2016). M. I. Angelova, Lipid membrane deformation in response to a [8] P. Mitchell, Coupling of phosphorylation to electron and hy- local pH modiﬁcation: Theory and experiments, Soft Matter 8, drogen transfer by a chemi-osmotic type of mechanism, Nature 6073 (2012). (London) 191, 144 (1961). [20] K. M. Davies, M. Strauss, B. Daum, J. H. Heinz, D. Osiewacz, [9] R. B. Gennis, Proton dynamics at the membrane surface, A. Rycovska, V. Zickermann, and W. Kühlbrandt, Macromolec- Biophysical J. 110, 1909 (2016). ular organisation of ATP synthase and complex I in whole [10] A. M. Morelli, S. Ravera, D. Calzia, and I. Panfoli, An update mitochondria, Proc. Natl. Acad. Sci. USA 108, 14121 (2011). of the chemiosmotic theory as suggested by possible proton [21] W. Helfrich, Elastic properties of lipid bilayers—Theory and currents inside the coupling membrane, Open Biol. 9, 180221 possible experiments, Z. Naturforsch. C 28, 693 (1973). (2019). [22] M. Deserno, Fluid lipid membranes: From differential geometry [11] B. Rieger, W. Junge, and K. B. Busch, Lateral ph gradient be- to curvature stresses, Chem. Phys. Lipids 185, 11 (2015).

tween OXPHOS complex IV and F0F1 ATP-synthase in folded [23] R. Capovilla and J. Guven, Stresses in lipid membranes, J. Phys. mitochondrial membranes, Nat. Commun. 5, 3103 (2014). A 35, 6233 (2002). [12] S. C. Al-Izzy, G. Rowlands, P. Sens, and M. S. Turner, Hydro- [24] J.-B. Fournier, On the stress and torque tensors in fluid mem- Osmotic Instabilities in Active Membrane Tubes, Phys. Rev. branes, Soft Matter 3, 883 (2007). Lett. 120, 138102 (2018). [25] M. M. Müller, Theoretical studies of fluid membrane me- [13] H. Turlier, B. Audoly, J. Prost, and J.-F. Joanny, Furrow con- chanics, Ph.D. thesis, Johannes Gutenberg-Universität Mainz striction in animal cell cytokinesis, Biophys. J. 106, 114 (2014). (2007). [14] A. Mietke, F. Jülicher, and I. F. Sbalzarini, Self-organized shape [26] A. Mietke, Dynamics of active surfaces, Ph.D. thesis, TU dynamics of active surfaces, Proc. Natl. Acad. Sci. USA 116,29 Dresden (2018). (2019). [27] M. Arroyo and A. De Simone, Relaxation dynamics of fluid [15] H. Berthoumieux, J.-L. Maître, C.-P. Heisenberg, E. K. Paluch, membranes, Phys. Rev. E 79, 031915 (2009). F. Jülicher, and G. Salbreux, Active elastic thin shell theory for [28] M. Rahimi and M. Arroyo, Shape dynamics, lipid hydrody- cellular deformations, New J. Phys. 16, 065005 (2014). namics, and the complex viscoelasticity of bilayer membranes, [16] D. Acehan, A. Malhotra, Y. Xu, M. Ren, D. L. Stokes, and M. Phys. Rev. E 86, 011932 (2012). Schlame, Cardiolipin affects the supramolecular organization of [29] J. Sjöholm, J. Bergstrand, T. Nilsson, R. Šachl, C. von ATP synthase in mitochondria, Biophys. J. 100, 2184 (2011). Ballmoos, J. Widengren, and P. Brzezinski, The lateral distance [17] N. Khalifat, N. Puff, S. Bonneau, J.-B. Fournier, and M. I. between a proton pump and ATP synthase determines the ATP- Angelova, Membrane deformation under local pH gradient: synthesis rate, Sci. Rep. 7, 2926 (2017). Mimicking mitochondrial cristae dynamics, Biophys. J. 95, [30] J. Guven, Membrane geometry with auxiliary variables and 4924 (2008). quadratic constraints, J. Phys. A: Math. Gen. 37, L313 (2004).

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