CClosed, Two Dimensional Surface Dynamics
David V. Svintradze ∗ Department of Physics, Tbilisi State University, Chavchavadze Ave. 03, 0179 Tbilisi, Georgia (Dated: February 21, 2018)We present dynamic equations for two dimensional closed surfaces and analytically solve it forsome simplified cases. We derive final equations for surface normal motions by two different ways.The solution of the equations of motions in normal direction indicates that any closed, two di-mensional, homogeneous surface with time invariable surface energy density adopts constant meancurvature shape when it comes in equilibrium with environment. As an example, we apply theformalism to analyze equilibrium shapes of micelles and explain why they adopt spherical, lamellarand cylindrical shapes. We show that theoretical calculation for micellar optimal radius is in goodagreement with all atom simulations and experiments.
I. INTRODUCTION
Biological systems exhibit a variety of morphologiesand experience large shape deformations during a mo-tion. Such ’choreography’ of shape motility is charac-teristic not only for all living organisms and cells [1]but also for proteins, nucleic acids and to all biomacro-molecules in general. Shape motility, which is a motionof two-dimensional surfaces, may be a result of active(by consuming energy) or passive (without consumingenergy) processes. The time scale for shape dynamicsmay vary from slow (nanometer per nanoseconds) to veryfast (nanometer per femtosecond) [2, 3]. Slowly movingsurfaces are considered as over-damped systems. An ex-ample is cell motility. In that case one may use welldeveloped the Helfrich formalism to describe the motion.This is a coarse-grained description of membranes withan expansion of the free energy in powers of the curvaturetensor [4]. However, while the formalism [4] are applica-ble to slowly moving surfaces they are not applicable tofast moving surfaces, where biomolecules maybe fitted.Surface dynamics for proteins or DNA [2, 3] may reach nm/f s range. So that surfaces may be represented asvirtual three dimensional pseudo Riemannian manifolds.We derived fully generic equations of motions for threemanifolds [5], but purposefully omitted lengthy discus-sion about motion of two-dimensional surfaces, which isa topic for this paper.Currently, significant progress on fluidic models ofmembrane dynamics has already been made. The roleof geometric constraints in self-assembly have been elu-cidated by linking together thermodynamics, interactionfree energies and geometry [6, 7]. The Helfrich formal-ism provides the foundation for a purely differential ge-ometric approach whereby the membrane surface poten-tial energy density is considered as a functional of thestatic curvature [4], see also review papers[8–10]. Themodel has been improved by adding force and torque ∗ Present address: Max Plank Institute for the Physics of ComplexSystem, 01187 Dresden, [email protected], [email protected] balance equations [11, 12]. Specific dynamical equationsaccounting for bending as well as electrodynamic effectshave also been reported [13–15]. Furthermore, activemembrane theories have extended our understanding ofpassive membranes. Active membrane theories includeexternal forces [16–19] and provide a framework for thestudy of active biological or chemical processes at sur-faces, such as the cell cortex, the mechanics of epithelialtissues, or reconstituted active systems on surfaces [16].Among the remarkable aspects of fluid lipid mem-branes deduced from the large body of theoretical work[8–10], is that the physical behavior of a membrane on thelength scale not much bigger than its own thickness, canbe described with high accuracy by a purely geometricHamiltonian [4, 20, 21]. Associated Euler-Lagrange equa-tions [22, 23], so called shape equations, are fourth orderpartial nonlinear differential equations, and finding a gen-eral analytical solution is typically difficult, even thoughit has been analytically [24] and numerically solved forsome specific [25–32] and general cases [33, 34].In fluid dynamics, material particles can be treated asa vertex of geometric figure and virtual layers as sur-faces and equations of motion for such surfaces can besearched. We refer to the formalism as differentially vari-ational surfaces (DVS) (or DVS formalism) [5].In this paper, we propose different approach to the’shape choreography’ problem. We use DVS formal-ism, tensor calculus of moving surfaces and the first lawof thermodynamics to derive the final equation for theclosed 2D surface dynamics (later on referred as surface)and to solve it analytically for the equilibrium case. Inother words, we derive generic equations of motions forclosed two-dimensional surfaces and without any a pri-ori symmetric assumptions, we show that constant meancurvature shapes are equilibrium solutions. In contrastto the Young-Laplace law these solutions, are univer-sally correct descriptions of capillary surfaces as well asmolecular surfaces. In addition, our equations of mo-tions (20-25) are generic and exact. It advances our un-derstanding of fluid dynamics because generalizes idealmagneto-hydrodynamic and Naiver-Stokes equations [5]and in contrast to Navier-Stokes, as we demonstrate inthis paper, are trivially solvable for equilibrium shapes.To demonstrate the validity of these equations and their a r X i v : . [ phy s i c s . b i o - ph ] F e b AVID V. SVINTRADZEanalytical solutions we apply them to micelles. Withinour formalism it becomes simple task to show micelleslamellar, cylindrical, spherical shapes and assert their op-timal spherical radius.For clarity, we shall give brief description of micellesand their structures. A micelle consists of monolayerof lipid molecules containing hydrophilic head and hy-drophobic tail. These amphiphilic molecules, in aqueousenvironment, aggregate spontaneously into a monomolec-ular layer held together due to a hydrophobic effect[35, 36] (see also [5, 37–40]) by weak non-covalent forces[41]. They form flexible surfaces that show variety ofshapes of different topology, but remarkably in thermo-dynamic equilibrium conditions they are spherical, lamel-lar (plane) or cylindrical in shape.
II. METHODS
In the section we provide basics of tensor calculusfor moving surfaces and summarize the theorems weused directly or indirectly to derive equations for two-dimensional surface dynamics. Differential geometry pre-liminaries we used here are available in tensor calculustextbook [42] and in our work [5].
A. Basics of differential geometry.
Suppose that S i ( i = 1 ,
2) are the surface coordinatesof the moving manifold (or the surface) S and the ambi-ent Euclidean space is referred to coordinates X α (Fig-ure 1). Coordinates S i , X α are arbitrarily chosen so that FIG. 1.
Graphical illustration of the arbitrary sur-face and its local tangent plane. (cid:126)S , (cid:126)S , (cid:126)N are localtangent plane base vectors and local surface normal respec-tively. (cid:126)X , (cid:126)X , (cid:126)X are arbitrary base vectors of the ambientEuclidean space and (cid:126)R = (cid:126)R ( X ) = (cid:126)R ( t, S ) is radius vectorof the point. (cid:126)V is arbitrary surface velocity and C, V , V display projection of the velocity to the (cid:126)N, (cid:126)S , (cid:126)S directionsrespectively. sufficient differentiability is achieved in both, space andtime. Surface equation in ambient coordinates can be written as X α = X α ( t, S i ). Let the position vector (cid:126)R beexpressed in coordinates as (cid:126)R = (cid:126)R ( X α ) = (cid:126)R ( t, S i ) (1)Latin letters in indexes indicate surface related tensors.Greek letters in indexes show tensors related to Euclideanambient space. All equations are fully tensorial and fol-low the Einstein summation convention. Covariant basesfor the ambient space are introduced as (cid:126)X α = ∂ α (cid:126)R , where ∂ α = ∂/∂X α . The covariant metric tensor is the dotproduct of covariant bases X αβ = (cid:126)X α (cid:126)X β (2)The contravariant metric tensor is defined as the matrixinverse of the covariant metric tensor, so that X αβ X βγ = δ αγ , where δ αγ is the Kronecker delta. As far as the ambi-ent space is set to be Euclidean, the covariant bases arelinearly independent, so that the square root of the met-ric tensor determinant is unit. Furthermore, the Christof-fel symbols given by Γ αβγ = (cid:126)X α · ∂ β (cid:126)X γ vanish and setthe equality between partial and curvilinear derivatives ∂ α = ∇ α .Now let’s discuss tensors on the embedded surface witharbitrary coordinates S i . Latin indexes throughout thetext are used exclusively for curved surfaces and curvi-linear derivative ∇ i is no longer the same as the partialderivative ∂ i = ∂/∂S i . Similar to the bases of ambientspace, covariant bases of an embedded manifold are de-fined as (cid:126)S i = ∂ i (cid:126)R and the covariant surface metric tensoris the dot product of the covariant surface bases: S ij = (cid:126)S i · (cid:126)S j (3)The contravariant metric tensor is the matrix inverseof the covariant one. The matrix inverse nature ofcovariant-contravariant metrics gives possibilities to raiseand lower indexes of tensors defined on the manifold. Thesurface Christoffel symbols are given by Γ ijk = (cid:126)S i · ∂ j (cid:126)S k and along with Christoffel symbols of the ambient spaceprovide all the necessary tools for covariant derivatives tobe defined at tensors with mixed space/surface indexes: ∇ i T αjβk = ∂ i T αjβk + X γi Γ αγν T νjβk − X γi Γ µγβ T αjµk +Γ jim T αmβk − Γ mik T αjβm (4)where X γi is the shift tensor which reciprocally shiftsspace bases to surface bases, as well as space metricto surface metric; for instance, (cid:126)S i = X αi (cid:126)X α and S ij = (cid:126)S i · (cid:126)S j = X αi (cid:126)X α X βj (cid:126)X β = X αi X βj X αβ . Note that in (4)Christoffel symbols with Greek indexes are zeros.Using (2,4), one may directly prove metrilinic prop-erty of the surface metric tensor ∇ i S mn = 0, from wherefollows (cid:126)S m · ∇ i (cid:126)S n = 0, meaning that (cid:126)S m ⊥∇ i (cid:126)S n are or-thogonal vectors and as so ∇ i (cid:126)S n must be parallel to (cid:126)N the surface normal ∇ i (cid:126)S j = (cid:126)N B ij (5)2OVING MANIFOLDS ... (cid:126)N is a surface normal vector with unit length and B ij is the tensorial coefficient of the relationship and is gen-erally referred as the symmetric curvature tensor. Thetrace of the curvature tensor with upper and lower in-dexes is the mean curvature and its determinant is theGaussian curvature. It is well known that a surface withconstant Gaussian curvature is a sphere, consequently asphere can be expressed as: B ii = λ (6)where λ is some non-zero constant. According to (5,6),finding the curvature tensor defines the way of findingcovariant derivatives of surface base vectors and as so,defines the way of finding surface base vectors which in-directly leads to the identification of the surface. B. Basics of tensor calculus for moving surfaces.
All Equations written above are generally true for mov-ing surfaces. We now turn to a brief review of definitionsof coordinate velocity V α , interface velocity C (which isthe same as normal velocity), tangent velocity V i (Figure1), time ˙ ∇ -derivative of surface tensors and time differ-entiation of the surface integrals. The original definitionsof time derivatives for moving surfaces were given in [43]and recently extended in tensor calculus textbook [42].Let’s start from the definition of coordinate velocity V α and show that the coordinate velocity is α componentof the surface velocity. Indeed, by the definition V α = ∂X α ∂t (7)On the other hand the position vector (cid:126)R given by (1) istracking the coordinate particle S i . Taking into accountpartial time differential of (1) and definition of ambientbase vectors, we find (cid:126)V = ∂ (cid:126)R ( t, S i ) ∂t = ∂ (cid:126)R∂X α ∂X α ( t, S i ) ∂t = V α (cid:126)X α (8)Therefore, V α is ambient component of the surface veloc-ity (cid:126)V . Taking into account (8), normal component of thesurface velocity is dot product with the surface normal,so that C = (cid:126)V · (cid:126)N = V α (cid:126)X α N β (cid:126)X β = V α N β δ αβ = V α N α (9)It is easy to show that the normal component C of the co-ordinate velocity, generally referred as interface velocity,is invariant in contrast with coordinate velocity V α andits sign depends on a choice of the normal. The projec-tion of the surface velocity on the tangent space (Figure1) is tangential velocity and can be expressed as V i = V α X iα (10)Taking (9,10) into account one may write surface velocityas (cid:126)V = C (cid:126)N + V i (cid:126)S i . Graphical illustrations of coordinate velocity V α , interface velocity C and tangential velocity V i are given on Figure 1. There is a clear geometric inter-pretation of the interface velocity [42]. Let the surfacesat two nearby moments of time t and t + ∆ t be S t , S t +∆ t correspondingly. Suppose that A ∈ S t (point on S t ) andthe corresponding point B ∈ S t +∆ t , B has the same sur-face coordinates as A (Figure 2), then (cid:126)AB ≈ (cid:126)V ∆ t . Let P be the point, where the unit normal (cid:126)N ∈ S t intersectthe surface S t +∆ t , then for small enough ∆ t , the angle ∠ AP B → π/ AP → (cid:126)V · (cid:126)N ∆ t , therefore, C can bedefined as C = lim ∆ t → AP ∆ t (11)and can be interpreted as the instantaneous velocity ofthe interface in the normal direction. It is worth of men-tioning that the sign of the interface velocity depends onthe choice of the normal. Although C is a scalar, it iscalled interface velocity because the normal direction isimplied. C. Invariant time differentiation.
Among the key definitions in calculus for moving sur-faces, perhaps one of the most important is the invarianttime derivative ˙ ∇ . As we have already stated, invarianttime derivative is already well defined in the literature[42, 43]. In this paragraph, we just give geometricallyintuitive definition.Suppose that invariant field F is defined on the surfaceat all time. The idea behind the invariant time derivativeis to capture the rate of change of F in the normal direc-tion. Physical explanation of why the deformations alongthe normal direction are so important, we give below inintegration section. This is similar to how C measuresthe rate of deformation in the normal direction. Let fora given point A ∈ S t , find the points B ∈ S t +∆ t and P the intersection of S t +∆ t and the straight line orthogo-nal to S t (Figure 2). Then, the geometrically intuitivedefinition dictates that˙ ∇ F = lim ∆ t → F ( P ) − F ( A )∆ t (12)As far as (12) is entirely geometric, it must be an in-variant (free from choice of a reference frame). From thegeometric construction one can estimate value of F inpoint B , so that F ( B ) ≈ F ( A ) + ∆ t ∂F∂t (13)On the other hand, F ( B ) is related to F ( P ) because B, P ∈ S t +∆ t and are nearby points on the surface S t +∆ t ,then according to definition of covariant derivative F ( B ) ≈ F ( P ) + ∆ tV i ∇ i F (14)3AVID V. SVINTRADZE FIG. 2.
Geometric interpretation of the interface ve-locity C and of the curvilinear time derivative ˙ ∇ ap-plied to invariant field F . A is arbitrary chosen point sothat it lays on F ( S t ) ∈ S t curve and B is its’ correspondingpoint on the S t +∆ t surface. P is the point where S t surfacenormal, applied on the point A , intersects the surface S t +∆ t .By the geometric construction, for small enough ∆ t → ∠ AP B → π/ (cid:126)AB ≈ (cid:126)V ∆ t and AP ≈ (cid:126)V (cid:126)N ∆ t . On otherhand, by the same geometric construction the field F in thepoint B can be estimated as F ( B ) ≈ F ( A ) + ∆ t∂F/∂t , whilefrom viewpoint of the S t +∆ t surface the F ( B ) value can be es-timated as F ( P )+∆ tV i ∇ i F , where ∇ i F shows rate of changein F along the surface S t +∆ t and along the directed distance BP ≈ ∆ tV i . since ∇ i F shows rate of change in F along the surface and∆ t · V i captures the directed distance BP . Determining F ( A ) , F ( P ) values from (13,14) and putting it in (12),gives ˙ ∇ F = ∂F∂t − V i ∇ i F (15)Extension of the definition (15) for any arbitrary tensorswith mixed space and surface indexes is given by theformula˙ ∇ T αiβj = ∂T αiβj ∂t − V k ∇ k T αiβj + V γ Γ αγµ T µiβj − V γ Γ µγβ T αiµj + ˙Γ ik T αkβj − ˙Γ kj T αiβk (16)The derivative commutes with contraction, satisfies sum,product and chain rules, is metrinilic with respect to theambient metrics and does not commute with the surfacederivative [42]. Also from (12) it is clear that the invari-ant time derivative applied to time independent scalarvanishes. Christoffel symbol ˙Γ ij for moving surfaces isdefined by the formula ˙Γ ij = ∇ j V i − CB ij . D. Time differentiation of integrals.
The remarkable usefulness of the calculus of movingsurfaces becomes evident from two fundamental formu-las for integrations that govern the rates of change ofvolume and surface integrals due to the deformation ofthe domain [42]. For instance, in evaluation of the leastaction principle of the Lagrangian there is a central rolefor time differentiation of the surface and space integrals,from where the geometry dependence is rigorously clari-fied.For any scalar field F = F ( t, S ) defined on a Euclideandomain Ω with boundary S evolving with the interfacevelocity C , the evolution of the space integral and surfaceintegral for closed surfaces are given by the formulas ddt (cid:90) Ω F d
Ω = (cid:90) Ω ∂F∂t d Ω + (cid:90) S CF dS (17) ddt (cid:90) S F dS = (cid:90) S ˙ ∇ F dS − (cid:90) S CF B ii dS (18)The first term in the integral represents the rate of changeof the tensor field, while the second term shows changes inthe geometry. Of course there are rigorous mathematicalproofs of these formulas in the tensor calculus textbooks.We are not going to reproduce proof of these theoremshere, but instead we give less rigorous but completelyintuitive explanation of why only interface velocity hasto be count. Rigorous mathematical proof follows fromfundamental theorem of calculus ddt (cid:90) b ( t ) a F ( t, x ) dx = (cid:90) b ( t ) a ∂F ( t, x ) ∂t dx + b (cid:48) ( t ) F ( t, b ( t ))(19)In the case of volume integral or surface integral it canbe shown that b (cid:48) ( t ) is replaced by interface velocity C .Intuitive explanation is pretty simple. Propose thereis no interface velocity then closed surface velocity onlyhas tangent component. For each given time tangent ve-locity (if there is no interface velocity) translates eachpoint to its neighboring point and therefore, does notadd new area to the closed surface (or new volume tothe closed space, or new length to the closed curve). Asso, tangential velocity just induces rotational movement(or uniform translational motion) of the object and canbe excluded from additive terms in the integration. Per-haps, it is easier to understand this statement for onedimensional motion. Let’s assume that material pointis moving along some trajectory (some closed curve orloop), then, in each point, the velocity of the materialpoint is tangential to the curve. Now one can translatethis motion into the motion of the closed curve wherethe loop has only tangential velocity. In this aspect, theembedded loop only rotates (uniformly translates in theplane) without changing the length locally, therefore tan-gential velocity of the curve does not add new length tothe curve (Same is true for open curve with fixed ends).4OVING MANIFOLDS ... III. GENERAL EQUATIONS OF SURFACEMOTIONS
Fully non-restrained and exact equations for movingthree-dimensional surfaces in electromagnetic field, whenthe interaction with an ambient environment is ignored,reads ˙ ∇ ρ + ∇ i ( ρV i ) = ρCB ii (20) ∂ α ( ρV α ( ˙ ∇ C + 2 V i ∇ i C + V i V j B ij ) − V α ( 14 µ F µν F µν + A µ J µ )) = f a (21) (cid:90) S ρV i ( ˙ ∇ V i + V j ∇ j V i − C ∇ i C − CV j B ij ) dS = (cid:90) Ω f i a i d Ω (22)where ρ is the surface mass density, V α , V i are coor-dinate and tangential components of the surface veloc-ity, C is interface velocity, α = 0 , , , i = 0 , , B ij is thesurface curvature tensor, F µν is electromagnetic tensor, F α = J α − ∂ β F βα , J α is α component of (cid:126)J = ( J α )four current, f, f i are normal and tangential componentsof (cid:126)F = ( F α ), a , a i are the normal and tangential com-ponents of the partial time derivative of the four vectorpotential (cid:126)A = ( A α ), S , Ω stand for surface and space in-tegrals respectively. Exact derivation of (20-22) is given in our work [5], we don’t reproduce derivation of thisset in this paper, rather just mention that first one isthe consequence of mass conservation, second and thirdequations come from minimum action principle of a La-grangian and imply motion in normal direction (21) andin tangent direction (22).For two dimensional surface dynamics, Minkowskianspace becomes Euclidean, so that α = 1 , , i = 1 ,
2. Sothat, after modeling the potential energy as a negativevolume integral of the internal pressure and inclusion in-teraction with an environment, (20-22) further simplifiesas˙ ∇ ρ + ∇ i ( ρV i ) = ρCB ii (23) ∂ α ( ρV α ( ˙ ∇ C + 2 V i ∇ i C + V i V j B ij ) + V α ( P + + Π)) = − V α ∂ α ( P + + Π) (24) ρV i ( ˙ ∇ V i + V j ∇ j V i − C ∇ i C − CV j B ij ) =0 (25)where P + , Π are internal hydrodynamic and osmoticpressures, respectively. Derivation of (20-22) can befound in [5]. We derive (23-25) in appendix section. Itis noteworthy that from the last equations set only thesecond equation (24) differs from the dynamic fluid filmequations [42, 44] ρ ( ˙ ∇ C + 2 V i ∇ i C + V i V j B ij ) = σB ii (26)where σ is surface tension. (26) is only valid when thesurface can be described with time invariable surface ten-sion [42, 44], meaning that the surface is homogeneousand the surface tension is constant, while (24) does nothave that restriction. Using (26) in (24) and taking intoaccount that in equilibrium processes internal pressure isthe same as external pressure, one gets exactly the same equation of motion in normal direction (39) as we getfrom using the first law of thermodynamics (see below). ∂ α ( σV α B ii + ( P + + Π) V α ) = − ( ∂ α P + + ∂ α Π) V α (27)It is worth of mentioning that (23-25) also follows from(20-22) if one applies same formalism as it is given in (30-32). Indeed, for relatively slowly moving surfaces spaceis three dimensional Euclidean so that α = 1 , ,
3, thesurface is two-dimensional Riemannian ( i = 1 ,
2) and thepotential energy becomes U = (cid:90) Ω ( 14 µ F µν F µν + A µ J µ )= (cid:90) Ω ( − (cid:15) E + 1 µ B − qϕ + (cid:126)A (cid:126)J ) d Ω (28)5AVID V. SVINTRADZEwhere (cid:126)E, (cid:126)B are electric and magnetic fields and q, ϕ, (cid:126)A, (cid:126)J are charge density, electric potential, magnetic vector po-tential and current density vector respectively. Using(30-32) formalism into account, we find dU = − ( P + + Π) d Ω = ( − (cid:15) E + 1 µ B − qϕ + (cid:126)A (cid:126)J ) d Ω(29)Taking into account (29) and that the pressure comesfrom the normal force applied to the surface, we find f a = − V α ∂ α ( P + + Π) and in tangent direction f i a i = 0,then (20-22) becomes (23-25). Electromagnetic potentialenergy can be generalized if one takes into account en-vironment, which enters in energy terms as bound andfree charges and electric/magnetic fields are replaced bypolarization and magnetization vectors [5]. IV. RESULTS AND DISCUSSIONA. General assumptions.
In this section we apply basics of thermodynamics andfundamental theorems of calculus of moving surfaces todemonstrate shortest derivation of the equation, describ-ing motion of homogeneous, closed two dimensional sur-face with time invariable surface tension at normal direc-tion (27). We consider the system consisted of aqueousmedia with the formed closed surface in it (Figure 3).The system is isolated with constant temperature andthere is no absorbed or dissipated heat on the surface; inother words, a process is adiabatic. According to the first
FIG. 3.
Graphical illustration of the isolated systemcontaining aqueous solution.
Water molecules are repre-sented as red and white sticks. The system boundary is shownas white faces with black edges. The subsystem-micelle isclosed surface, blue blob in the center of the system. law of thermodynamic, as far as there is no dissipated orabsorbed heat, the change of the internal energy of thesurface must be dE = δW (30)where δW is infinitesimal work done on the subsystemand dE is infinitesimal change of the internal energy.Because the temperature of the system is constant, thedifferential of the subsystems’ internal energy can be re-modeled as dE = dU (31)where U is the total potential energy of the surface. Bythe definition the elementary work done on the subsystemis δW = ( P − + Π) d Ω (32)where , P − , Π are external hydrodynamic and osmoticpressures applied on the surface by the surroundings cor-respondingly and Ω is the volume that surface encloseswith boundary of S surface area. Let’s propose that thesurface is homogeneous (i.e material particles are homo-geneously distributed on the surface) so that the totalpotential energy is integration of the potential energyper unit area over the surface, then dU = σdS (33)where σ is the potential energy per unit area and is calledsurface tension in the paper. As far as we discuss sim-plest case of the system consisted of aqueous medium andsingle closed surface, we can suggest that the surface ten-sion is not time variable. Using (30-33) after few lines ofalgebra, we fined (cid:90) S σdS = (cid:90) Ω ( P − + Π) d Ω (34)Assuming the surface is moving so that (34) stays validfor any time variations, then time differentiation of theleft side must be equal to time differentiation of right in-tegral. As far as on the right hand side we have spaceintegral, time differentiation can be taken into the inte-gral, using general theorems for differentiation of spaceand surface integrals (17-18), so that integration theoremfor space integral holds and the convective and advectiveterms due to volume motion are considered ddt (cid:90) Ω ( P − + Π) d Ω = (cid:90) Ω ( ∂ α P − + ∂ α Π) ∂X α ∂t d Ω+ (cid:90) S C ( P − + Π) dS (35)To calculate time derivative of the surface integral wehave to take into account the theorem about time differ-entiation of the surface integral (18), from which followsthat for time invariable surface tension ddt (cid:90) S σdS = (cid:90) S − σCB ii dS (36)6OVING MANIFOLDS ...Where C = V α N α is interface velocity, N α is α com-ponent of the surface normal and V = ∂X α /∂t is coor-dinate velocity, X α is general coordinate and B ii is thetrace of the mixed curvature tensor generally known asmean curvature. After few lines of algebra putting (34-36) together, we find (cid:90) S ( σCB ii + C ( P − + Π)) dS = − (cid:90) Ω ( ∂ α P − + ∂ α Π) V α d Ω(37)Generalized Gauss theorem converts the surface integralof the left hand side of (37) into space integral, so that (cid:90) S N α V α ( σB ii + P − + Π) dS = (cid:90) Ω ∂ α ( σV α B ii +( P − + Π) V α ) d Ω (38)Combination of (37) and (38) immediately gives equationof motion for surface in normal direction ∂ α ( σV α B ii + ( P − + Π) V α ) = − ( ∂ α P − + ∂ α Π) V α (39)For equilibrium processes internal and external pressuresare identical P − = P + , so that (39) becomes identicalto the equation of motion in normal direction observedfrom master equations (23-27). Also, we should note that(39) is only valid for motion of the homogeneous surfaceswith time invariable surface tension at normal direction,therefore, it does not display any deformation in tangentdirections. (39) further simplifies when the surface comesin equilibrium with the solvent where divergence of thesurface velocity ∂ α V α (stationary interface) along with ∂P/∂t (where P = P − + Π) vanishes, then the solutionto (39), taking into account the condition (35), becomes B ii = − Pσ (40)The result (40) shows that the solution is constant meancurvatures (CMC) surfaces. Such CMC are rare and canbe many if one relaxes the condition we restricted to thesystem. We consider isolated system where the surfaceis closed subsystem, these two preconditions mathemat-ically mean that the surface we discuss is compact em-bedded surface in R . According to A. D. Alexandrovuniqueness theorem for surfaces, a compact embeddedsurface in R with constant non-zero mean curvature is asphere [45]. Correspondingly the solution (40) is a sphere(as far as we have compact two-manifold in the Euclideanspace). When Pσ (cid:54) = 0 (41)the surface is spheroid (or a cylinder if one relaxes com-pactness restriction making the cylinder infinitely long)and becomes plane (again when compactness argument isrelaxed) or other zero mean curvature shape when com-pactness argument is not relaxed but contour of the sur-face remains fixed. This surprisingly simple and elegant derivation explains all the shapes surfaces can adopt inaqueous solution at equilibrium conditions. If the com-pactness condition is relaxed then (40) predicts that inaddition to cylinder and plane all other CMC surfacesare also equilibrium shapes for moving surfaces. Takinginto account that the surface tension in general can bea function of many variables, such as Gaussian curva-ture, bending rigidity, spontaneous curvature, moleculesconcentration, geometry of surfactant molecules and etc.,then (39) may predict possible deformations of differentlyshaped surfaces and their wide range of static shapes. Infact, if considered that surface tension, which is definedas potential energy per unit area, can be a function ofmean curvature σ = σ ( B ii ), then Taylor expansion of σ ( B ii ) naturally rises all additional terms. These gener-alizations and temperature fluctuations can be includedin the equations, but it is not scope of this paper andshould be addressed separately. One may even propose σ as time independent the Helfrich Hamiltonian and then(39, 40) will become equation of static shapes for homo-geneous surfaces with time invariable surface tension. B. Physical application, micelle.
We can put equation (39) and its solution (40) underthe test for homogeneous micellar surface equilibratedwith the aqueous solution. Based on (40) we can calcu-late minimal value of a micelle radius. The value of thetrace of the mixed curvature tensor for a sphere is B ii = − R (42)where R is radius. FIG. 4.
Simulated three dimensional coordinatesof the micelle in aqueous solution display spherewith diameter . ˚ A . (Left) dihexanoylphosphatidylcholine(DHPC phospholipids) are modeled as orange balls. (Right)Gaussian mapping at contour resolution 8˚ A of the micelleshows spherical structure. Even though we set environment as aqueous, it enters into equa-tions as osmotic pressure term, which due to a generality of ar-guments can be anything. Therefore, as a medium one may pickany liquid or gas. XH · · · Y hydrogen bond is about1 kJ per mol ( CH · · · C unit) and high boundary is about161 kJ per mol ( F H ··· F unit), the low and high values aretaken according to references [46, 47]. Therefore, averageenergy is about (1 + 161) / kJ/mol ≈ · − J .To estimate hydrogen bonding energy per molecule withthe undefined shape (lipid molecule) in the first approxi-mation is to assign average energy to it and consider thespherical shape with the gyration radius. Of course it islow level approximation, but even such rough calculationsproduce reasonable results. After all these rough estima-tions the pressure to move one lipid from the surface, inorder to induce critical deformations of the surface, isabout average energy per the average volume of the lipidmolecule P ≈ · · − πr G ≈ . · N/m (43)where 4 πr G / r G ≈ nm .On the other hand, surface tension of a fluid monolayerat optimal packing of the lipids is about σ ≈ · − N/m [7, 48, 49], using these and (42,43) in (40) the estimatedmicelle radius is R ≈ · · − . · = 19 . ± . A (44)These calculations put the minimum radius in nanome-ter scale and is in very good agreement with experimen-tal as well as computational frameworks [50, 51]. Tofurther validate the (44) result, we ran a CHARMMbased Micelle Builder simulation [52, 53] for 100 phos-pholipid molecules (DHPC lipids). The simulation result(Figure 4) generated a spherical micelle with diameter38 . ± . A . These calculations indeed indicate that evensuch rough estimations produce reasonable accuracy.To get more convincing estimations it is necessaryto take into account that neither lipids are sphericalnor hydrophobic interactions per lipid are average en-ergy of single hydrogen bond. In the second approxi-mation lipids are no longer undefined spheres, but havewell defined surfactant geometry. The Hydrophobic en-ergy is no longer average energy of single hydrogenbond, but is 1 kJ per mol per − CH − unit. In allatom simulations we used dihexanoylphosphatidylcholine(DHPC) lipid molecule having 12 − CH − units (Figure5) per hydrophobic tail, so hydrophobic energy is about12 kJ/mol ≈ . · − J . Accurate calculation of thelipid molecule volume using cavity, channel and cleft vol-ume calculator [54], gives the volume estimation of about 894˚ A . Using this value, one gets P ≈ . · − . · − ≈ . · N/m (45)On other hand, using the same surface tension of a fluidmonolayer at the optimal packing of the lipids, one gets R = 27 ± . A . All atom simulation also generates spher-ical structure with diameter 54 ± . A (Figure 5). Thereis still some uncertainty in this estimation because we as-signed 1 kJ/mol energy per − CH − unit and we basedon references data [46, 47], while in other literature it ismentioned that the hydrophobic interactions are about 4kJ/mol per − CH − unit [55]. In our opinion, this dis-crepancy can be resolved if one calculates hydrophobicenergy based on the potential energy U = − (cid:90) Ω (cid:15) E CH d Ω (46)where (cid:126)E CH is electric field per − CH − , (cid:15) is dielectricconstant in the vacuum and Ω stands for the volume ofthe lipid molecules. (46) directly emerges from F µν F µν term written in the equations of motion (20-22 and 28-29). For electrostatics U = (cid:90) Ω µ F µν F µν d Ω = − (cid:90) Ω (cid:15) E CH d Ω (47)so one should go to the scrutiny of calculating electricfield for each − CH − units, then take a sum of the elec-tric field and square it (we are not going to do it in thispaper). Also, one may ask why the hydrophilic inter-action energy is not taken into account in these calcula-tions. Hydrophilic head of the lipid molecule is in contactwith water molecules so there is no work needed to drag itin aqueous solution from the lipids layer. Therefore, hy-drophilic interaction energy can be neglected. The mostwork goes on overcoming hydrophobic interactions be-tween lipid tails. V. CONCLUSION
We have presented a framework for the analysis of twodimensional surface dynamics (identified as micelle in thetext) using first law of thermodynamics and calculus ofmoving surfaces. In final equations of normal motion(39,27) we assume that a surface is homogeneous andhas time invariable surface tension. However, the gen-eral equations (23-25) don not have these constrains andindicate arbitrary motion along normal deformation, aswell as into tangent directions, but are analytically morecomplex. The solution to the normal equations of mo-tion in equilibrium conditions are surprisingly simple anddisplay all possible equilibrium shapes. We applied theformalism to estimate micelle optimal radius and com-pared estimations to all atom simulations. Even for low-level approximations, we found remarkable agreement be-tween theoretically calculated radius and one obtained8OVING MANIFOLDS ...
FIG. 5.
All atom simulation of DHPC micelle. (A) The figure shows a geometry of the DHPC surfactant molecule usedin simulation and gives parametric description of volume, surface area, sphericity and effective radius. (B) Indicates atomisticsimulation result contoured by Gaussian map and the diameter of the micelle, measured by PyMol. The diameter of thesimulated micelle appears to be 54 . A with the uncertainty of the measurement 0 . A . from atomistic simulations and from experiments. Onecan readily apply the theory to any closed surfaces; suchare vesicles, membranes, water droplets or soap films.As a final remark, even though the analytic solution(40) looks like generalized Young-Laplace law, the differ-ence is obvious. B ii is a trace of mixed curvature tensor,known as mean curvature, and when the mean curvatureis constant, it defines whole class of constant mean cur-vature (CMC) surfaces. Generalized the Young-Laplacelaw is a priori formulated for spherical morphologies andtherefore in some particular cases can be obtained from(40) constant mean curvature shapes. The condition forholding the particular case is a compactness. However,the compactness argument can be relaxed in our deriva-tion if the considered system is set to be much larger thanthe subsystem. Therefore, the solution (40) effectivelypredicts formation of all CMC surfaces while Young-Laplace law is correct for spherical structures alone. Alsoin derivation of Young-Laplace relation one of corner-stone idea is suggestion of spherical symmetries, whileour derivation is free of symmetries and explains whyCMC surfaces are such abundant shapes in nature, ob-servable even on molecular levels. In fact, according tothe results, any homogeneous closed surface with timeinvariable surface tension adopts CMC shape when itcomes in equilibrium with environment. ACKNOWLEDGMENTS
We thank Dr. Frank Julicher (MPIPKS) and Dr. Er-win Frey (LMU), for stimulating discussions. The paper in it’s current form was initiated at Aspen Center forPhysics, which is supported by National Science Founda-tion grant PHY-1607611 and was partially supported bya grant from the Simons Foundation.
APPENDIX: DERIVATION OF EQUATIONS OFMOTIONS FOR CLOSED, TWO DIMENSIONALSURFACES
Now we turn to the derivation of (23-25) without usingany information from (20-22) (though derivation of (23-25) from (20-22) is strightforwad and trivial if one sets V = 0 in (20-22 equations [5]). To deduce the equationsof motion we derive the simplest one from the set (23)first. It is direct consequence of generalization of conser-vation of mass law. Variation of the surface mass densitymust be so that dm/dt = 0, where m = (cid:82) S ρdS is surfacemass with ρ surface density. Since the surface is closed,at the boundary condition v = n i V i = 0, a pass integralalong any curve γ across the surface must vanish ( n i is anormal of the curve and lays in the tangent space). Us-ing Gauss theorem, conservation of mass and integration9AVID V. SVINTRADZEformula (18), we find0 = (cid:90) γ vρdγ = (cid:90) γ n i V i ρdγ = (cid:90) S ∇ i ( ρV i ) dS = (cid:90) S ( ∇ i ( ρV i ) − ρCB ii + ρCB ii ) dS = (cid:90) S ( ∇ i ( ρV i ) − ρCB ii ) dS + (cid:90) S ˙ ∇ ρdS − ddt (cid:90) S ρdS = (cid:90) S ( ˙ ∇ ρ + ∇ i ( ρV i ) − ρCB ii ) dS (48)Since last integral must be identical to zero for any inte-grand, one immediately finds first equation from the set(23). To deduce second and third equations, we take aLagrangian L = (cid:90) S ρV dS + (cid:90) Ω ( P + + Π) d Ω (49)and set minimum action principle requesting that δL/δt = 0. Evaluation of space integral is simple andstraightforward, using integration theorem for space in-tegral where the convective and advective terms due tovolume motion is properly taken into account (17), wefind δδt (cid:90) Ω ( P + +Π) d Ω = (cid:90) Ω ∂ α ( P + +Π) V α d Ω+ (cid:90) S C ( P + +Π) dS (50)Derivation for kinetic part is a bit tricky and challengingthat is why we do it last. Straightforward, brute math-ematical manipulations, using first equation from (23), lead δδt (cid:90) S ρV dS = (cid:90) S ( ˙ ∇ ρV − CB ii ρV dS = (cid:90) S ( ˙ ∇ ρ V ρ ˙ ∇ V − CB ii ρV dS = (cid:90) S (( ρCB ii − ∇ i ( ρV i )) V ρ ˙ ∇ V − CB ii ρV dS = (cid:90) S ( −∇ i ( ρV i ) V ρ ˙ ∇ V dS = (cid:90) S ( −∇ i ( ρV i V ρV i ∇ i V ρ ˙ ∇ V dS = (cid:90) S ( −∇ i ( ρV i V ρ(cid:126)V ( V i ∇ i (cid:126)V + ˙ ∇ (cid:126)V )) dS (51)At the end point of variations the surface reaches station-ary point and therefore by Gauss theorem (as we used italready in (48)), we find (cid:90) S −∇ i ( ρV i V dS = − (cid:90) γ ρV i n i V dγ = 0 (52) γ is stationary contour of the surface and n i is the normalto the contour, therefore interface velocity for contour v = n i V i = 0 and the integral (52) vanishes, correspond-ingly δδt (cid:90) S ρV dS = (cid:90) S ρ(cid:126)V ( V i ∇ i (cid:126)V + ˙ ∇ (cid:126)V ) dS (53)To decompose dot product in the integral by normal andtangential components and, therefore, deduce final equa-tions, we do following algebraic manipulations˙ ∇ (cid:126)V + V i ∇ i (cid:126)V = ˙ ∇ (cid:126)V + V i ∇ i (cid:126)V + CV i B ji (cid:126)S j − CV i B ji (cid:126)S j = ˙ ∇ (cid:126)V + V i ∇ i (cid:126)V + CV i B ji X αj (cid:126)X α − CV i B ji (cid:126)S j (54)Now using Weingartens formula X αj B ji = −∇ i N α ,metrinilic property of Euclidian space base vectors ∇ i (cid:126)X α = 0, definition of surface normal (cid:126)N = N α (cid:126)X α and taking into account definition of surface velocity (cid:126)V = C (cid:126)N + V i (cid:126)S i and its derivatives, we find˙ ∇ (cid:126)V + V i ∇ i (cid:126)V + CV i B ji X αj (cid:126)X α − CV i B ji (cid:126)S j = ˙ ∇ (cid:126)V + V i ∇ i (cid:126)V − CV i (cid:126)X α ∇ i N α − CV i B ji (cid:126)S j = ˙ ∇ (cid:126)V + V i ∇ i (cid:126)V − CV i ∇ i ( N α (cid:126)X α ) − CV i B ji (cid:126)S j = ˙ ∇ (cid:126)V + V i ∇ i (cid:126)V − CV i ∇ i (cid:126)N − CV i B ji (cid:126)S j = ˙ ∇ (cid:126)V + V i ∇ i ( C (cid:126)N ) + V i ∇ i ( V j (cid:126)S j ) − CV i ∇ i (cid:126)N − CV i B ji (cid:126)S j = ˙ ∇ (cid:126)V + V i (cid:126)N ∇ i C + V i ∇ i ( V j (cid:126)S j ) − CV i B ji (cid:126)S j = ˙ ∇ ( C (cid:126)N ) + ˙ ∇ ( V j (cid:126)S j ) + V i (cid:126)N ∇ i C + V i ∇ i ( V j (cid:126)S j ) − CV i B ji (cid:126)S j (55)10OVING MANIFOLDS ...Continuing algebraic manipulations using Thomas for-mula ˙ ∇ (cid:126)N = −∇ i C (cid:126)S i , the formula for surface derivative of interface velocity (cid:126)N ∇ i C = ˙ ∇ (cid:126)S i and the definition ofcurvature tensor (5) yield˙ ∇ ( C (cid:126)N ) + ˙ ∇ ( V j (cid:126)S j ) + V i (cid:126)N ∇ i C + V i ∇ i ( V j (cid:126)S j ) − CV i B ji (cid:126)S j = ˙ ∇ ( C (cid:126)N ) + C ∇ j C (cid:126)S j + 2 V i (cid:126)N ∇ i C + V i V j B ij (cid:126)N + ˙ ∇ ( V j (cid:126)S j ) − V i (cid:126)N ∇ i C + V i ∇ i ( V j (cid:126)S j ) − V i V j B ij (cid:126)N − C ∇ j C (cid:126)S j − CV i B ji (cid:126)S j = ˙ ∇ ( C (cid:126)N ) − C ˙ ∇ (cid:126)N + 2 V i (cid:126)N ∇ i C + V i V j B ij (cid:126)N + ˙ ∇ ( V j (cid:126)S j ) − V j ˙ ∇ (cid:126)S j + V i ∇ i ( V j (cid:126)S j ) − V i V j ∇ i (cid:126)S j − C ∇ j C (cid:126)S j − CV i B ji (cid:126)S j = ( ˙ ∇ C + 2 V i ∇ i C + V i V j B ij ) (cid:126)N + ( ˙ ∇ V j + V i ∇ i V j − C ∇ j C − CV i B ji ) (cid:126)S j (56)Taking dot product of (56) on (cid:126)V and combining it with (53) last derivation finally reveals variation of kinetic en-ergy, so that we finally find δδt (cid:90) S ρV dS = (cid:90) S ( ρC ( ˙ ∇ C + 2 V i ∇ i C + V i V j B ij ) + ρV i ( ˙ ∇ V i + V j ∇ j V i − C ∇ i C − CV j B ij )) dS (57)Combining (48-50) and (57) together and taking into ac-count that the pressure acts on the surface along the surface normal, we immediately find first (23) and thelast equation (25) of the set. To clarify second equation(24), we have (cid:90) S ρC ( ˙ ∇ C + 2 V i ∇ i C + V i V j B ij ) dS = (cid:90) Ω − ∂ α ( P + + Π) V α d Ω − (cid:90) S C ( P + + Π) dS (cid:90) S C ( ρ ( ˙ ∇ C + 2 V i ∇ i C + V i V j B ij ) + P + + Π) dS = (cid:90) Ω − ∂ α ( P + + Π) V α d Ω (58)After applying Gauss theorem to the second equation (58), the surface integral is converted to space integral sothat we finally find ∂ α ( ρV α ( ˙ ∇ C + 2 V i ∇ i C + V i V j B ij ) + ( P + + Π) V α ) = − ∂ α ( P + + Π) V α (59)and, therefore, all three equations (23-25) are rigorously clarified. [1] Michael D. Cahalan and Ian Parker, “Choreography ofcell motility and interaction dynamics imaged by two-photon microscopy in lymphoid organs,” Annual Review of Immunology , 585–626 (2008), pMID: 18173372,https://doi.org/10.1146/annurev.immunol.24.021605.090620.[2] Samir Kumar Pal, Jorge Peon, Biman Bagchi, and Ahmed H. Zewail, “Biological water:ˆa femtosecond dy-namics of macromolecular hydration,” The Journalof Physical Chemistry B , 12376–12395 (2002),http://dx.doi.org/10.1021/jp0213506.[3] Chaozhi Wan, Torsten Fiebig, Olav Schiemann, Jacque-line K. Barton, and Ahmed H. Zewail, “Femtoseconddirect observation of charge transfer between bases indna,” Proceedings of the National Academy of Sciencesof the United States of America , 14052–14055 (2000).[4] Wolfgang Helfrich, “Elastic properties of lipid bilayers:theory and possible experiments,” Zeitschrift f¨ur Natur-forschung C , 693–703 (1973).[5] David V. Svintradze, “Moving manifolds in electromag-netic fields,” Frontiers in Physics , 37 (2017).[6] Jacob N. Israelachvili, D. John Mitchell, and Barry W.Ninham, “Theory of self-assembly of hydrocarbon am-phiphiles into micelles and bilayers,” J. Chem. Soc., Fara-day Trans. 2 , 1525–1568 (1976).[7] Jacob N. Israelachvili, D.John Mitchell, and Barry W.Ninham, “Theory of self-assembly of lipid bilayersand vesicles,” Biochimica et Biophysica Acta (BBA) -Biomembranes , 185 – 201 (1977).[8] Markus Deserno, “Fluid lipid membranes: From differ-ential geometry to curvature stresses,” Chemistry andphysics of lipids , 11–45 (2015).[9] R. Lipowsky and E. Sackmann, Structure and Dynamicsof Membranes (Elsevier, Amsterdam, 1995).[10] Udo Seifert, “Configurations of fluid membranes and vesi-cles,” Advances in Physics , 13–137 (1997).[11] R Capovilla and J Guven, “Stresses in lipid membranes,”Journal of Physics A: Mathematical and General ,6233 (2002).[12] Jean-Baptiste Fournier, “On the stress and torque ten-sors in fluid membranes,” Soft Matter , 883–888 (2007).[13] L.E. Scriven, “Dynamics of a fluid interface equation ofmotion for newtonian surface fluids,” Chemical Engineer-ing Science , 98 – 108 (1960).[14] Marino Arroyo and Antonio DeSimone, “Relaxation dy-namics of fluid membranes,” Phys. Rev. E , 031915(2009).[15] Ling-Tian Gao, Xi-Qiao Feng, Ya-Jun Yin, and HuajianGao, “An electromechanical liquid crystal model of vesi-cles,” Journal of the Mechanics and Physics of Solids ,2844 – 2862 (2008).[16] Guillaume Salbreux and Frank J¨ulicher, “Mechanics ofactive surfaces,” Phys. Rev. E , 032404 (2017).[17] Sriram Ramaswamy, John Toner, and Jacques Prost,“Nonequilibrium fluctuations, traveling waves, and in-stabilities in active membranes,” Phys. Rev. Lett. ,3494–3497 (2000).[18] Hsuan-Yi Chen, “Internal states of active inclusions andthe dynamics of an active membrane,” Phys. Rev. Lett. , 168101 (2004).[19] N. Gov, “Membrane undulations driven by force fluctu-ations of active proteins,” Phys. Rev. Lett. , 268104(2004).[20] Peter B Canham, “The minimum energy of bending as apossible explanation of the biconcave shape of the humanred blood cell,” Journal of theoretical biology , 61–81(1970).[21] Evan A Evans, “Bending resistance and chemically in-duced moments in membrane bilayers,” Biophysical jour-nal , 923 (1974).[22] Ou-Yang Zhong-Can and Wolfgang Helfrich, “Instabil- ity and deformation of a spherical vesicle by pressure,”Physical review letters , 2486 (1987).[23] Ou-Yang Zhong-Can and Wolfgang Helfrich, “Bendingenergy of vesicle membranes: General expressions for thefirst, second, and third variation of the shape energy andapplications to spheres and cylinders,” Physical ReviewA , 5280 (1989).[24] Iva ˜Alo M. Mladenov, Peter A. Djondjorov, Mariana Ts.Hadzhilazova, and Vassil M. Vassilev, “Equilibrium con-figurations of lipid bilayer membranes and carbon nanos-tructures,” Communications in Theoretical Physics ,213 (2013).[25] Saˇsa Svetina and Boˇstjan ˇZekˇs, “Membrane bending en-ergy and shape determination of phospholipid vesiclesand red blood cells,” European biophysics journal ,101–111 (1989).[26] Udo Seifert and Reinhard Lipowsky, “Adhesion of vesi-cles,” Physical Review A , 4768 (1990).[27] Reinhard Lipowsky, “The conformation of membranes,”Nature , 475–481 (1991).[28] Udo Seifert, Karin Berndl, and Reinhard Lipowsky,“Shape transformations of vesicles: Phase diagram forspontaneous- curvature and bilayer-coupling models,”Phys. Rev. A , 1182–1202 (1991).[29] Frank J¨ulicher and Reinhard Lipowsky, “Domain-induced budding of vesicles,” Physical review letters ,2964 (1993).[30] Frank J¨ulicher and Reinhard Lipowsky, “Shape transfor-mations of vesicles with intramembrane domains,” Phys-ical Review E , 2670 (1996).[31] Frank J¨ulicher and Udo Seifert, “Shape equations for ax-isymmetric vesicles: a clarification,” Physical Review E , 4728 (1994).[32] Ling Miao, Udo Seifert, Michael Wortis, and Hans-G¨unther D¨obereiner, “Budding transitions of fluid-bilayer vesicles: the effect of area-difference elasticity,”Physical Review E , 5389 (1994).[33] Volkmar Heinrich, Saˇsa Svetina, and Boˇstjan ˇZekˇs,“Nonaxisymmetric vesicle shapes in a generalized bilayer-couple model and the transition between oblate and pro-late axisymmetric shapes,” Physical Review E , 3112(1993).[34] Vera Kralj-Igliˇc, Saˇsa Svetina, and Boˇstjan ˇZekˇs, “Theexistence of non-axisymmetric bilayer vesicle shapes pre-dicted by the bilayer couple model,” European biophysicsjournal , 97–103 (1993).[35] David Chandler, “Interfaces and the driving force of hy-drophobic assembly,” Nature , 640–647 (2005).[36] Sergey Leikin, V Adrian Parsegian, Donald C Rau, andR Peter Rand, “Hydration forces,” Annual Review ofPhysical Chemistry , 369–395 (1993).[37] David V Svintradze, “Hydrophobic and hydrophilic in-teractions,” Biophysical Journal , 43a–44a (2010).[38] David V Svintradze, “Moving macromolecular sur-faces under hydrophobic/hydrophilic stress,” BiophysicalJournal , 512a (2015).[39] David V Svintradze, “Cell motility and growth factorsaccording to differentially variational surfaces,” Biophys-ical Journal , 623a (2016).[40] David V Svintradze, “Geometric diversity of living organ-isms and viruses,” Biophysical Journal , 309a (2017).[41] C Tanford, The Hydrophobic Effect Formation of Mi-celles and Biological Membranes (Wiley-Interscience,
New York, 1973).[42] Pavel Grinfeld,
Introduction to tensor analysis and thecalculus of moving surfaces (Springer, New York, 2010).[43] J. Hadamard,
Mmoire sur le problme danalyse relatiflquilibre des plaques elastiques encastres (Oeuvres, Her-mann, Tome 2, 1968).[44] Pavel Grinfeld, “Exact nonlinear equations for fluid filmsand proper adaptations of conservation theorems fromclassical hydrodynamics,” J. Geom. Symm. Phys , 1–21 (2009).[45] A. D. Alexandrov, “Uniqueness theorem for surfaces inthe large,” Leningrad Univ. 13, 19 (1958), 58, Amer.Math. Soc. Trans. (Series 2) , 412–416 (1958).[46] JW Larson and TB McMahon, “Gas-phase bihalide andpseudobihalide ions. an ion cyclotron resonance determi-nation of hydrogen bond energies in xhy-species (x, y= f,cl, br, cn),” Inorganic Chemistry , 2029–2033 (1984).[47] J Emsley, “Very strong hydrogen bonding,” Chemical So-ciety Reviews , 91–124 (1980).[48] F J¨ahnig, “What is the surface tension of a lipid bilayermembrane?” Biophysical journal , 1348–1349 (1996).[49] Jacob N Israelachvili, Intermolecular and surface forces:revised third edition (Academic Press, 2011). [50] Scott E Feller, Yuhong Zhang, and Richard W Pas-tor, “Computer simulation of liquid/liquid interfaces. ii.surface tension-area dependence of a bilayer and mono-layer,” The Journal of chemical physics , 10267–10276 (1995).[51] Egbert Egberts, Siewert-Jan Marrink, and Herman JCBerendsen, “Molecular dynamics simulation of a phos-pholipid membrane,” European biophysics journal ,423–436 (1994).[52] Sunhwan Jo, Taehoon Kim, Vidyashankara G Iyer, andWonpil Im, “Charmm-gui: a web-based graphical user in-terface for charmm,” Journal of computational chemistry , 1859–1865 (2008).[53] Xi Cheng, Sunhwan Jo, Hui Sun Lee, Jeffery BKlauda, and Wonpil Im, “Charmm-gui micelle builderfor pure/mixed micelle and protein/micelle complex sys-tems,” Journal of chemical information and modeling ,2171–2180 (2013).[54] Neil R Voss and Mark Gerstein, “3v: cavity, channel andcleft volume calculator and extractor,” Nucleic acids re-search , gkq395 (2010).[55] A. Leitmannova Liu, Advances in Planar Lipids Bilayersand Liposomes , Vol. 4 (Academic Press, Elsevier, 2011)., Vol. 4 (Academic Press, Elsevier, 2011).