Closed, 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 for some 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 dimensional, homogeneous surface with time invariable surface energy density adopts constant mean curvature shape when it comes in equilibrium with environment. As an example, we apply the formalism to analyze equilibrium shapes of micelles and explain why they adopt spherical, lamellar and cylindrical shapes. We show that theoretical calculation for micellar optimal radius is in good agreement with all atom simulations and experiments.

I. INTRODUCTION balance equations [11, 12]. Specific dynamical equations accounting for bending as well as electrodynamic effects Biological systems exhibit a variety of morphologies have also been reported [13–15]. Furthermore, active and experience large shape deformations during a mo- membrane theories have extended our understanding of tion. Such ’choreography’ of shape motility is charac- passive membranes. Active membrane theories include teristic not only for all living organisms and cells [1] external forces [16–19] and provide a framework for the but also for proteins, nucleic acids and to all biomacro- study of active biological or chemical processes at sur- molecules in general. Shape motility, which is a motion faces, such as the cell cortex, the mechanics of epithelial of two-dimensional surfaces, may be a result of active tissues, or reconstituted active systems on surfaces [16]. (by consuming energy) or passive (without consuming Among the remarkable aspects of fluid lipid mem- energy) processes. The time scale for shape dynamics branes deduced from the large body of theoretical work may vary from slow (nanometer per nanoseconds) to very [8–10], is that the physical behavior of a membrane on the fast (nanometer per femtosecond) [2,3]. Slowly moving length scale not much bigger than its own thickness, can surfaces are considered as over-damped systems. An ex- be described with high accuracy by a purely geometric ample is cell motility. In that case one may use well Hamiltonian [4, 20, 21]. Associated Euler-Lagrange equa- developed the Helfrich formalism to describe the motion. tions [22, 23], so called shape equations, are fourth order This is a coarse-grained description of membranes with partial nonlinear differential equations, and finding a gen- an expansion of the free energy in powers of the curvature eral analytical solution is typically difficult, even though tensor [4]. However, while the formalism [4] are applica- it has been analytically [24] and numerically solved for ble to slowly moving surfaces they are not applicable to some specific [25–32] and general cases [33, 34]. fast moving surfaces, where biomolecules maybe fitted. In fluid dynamics, material particles can be treated as Surface dynamics for proteins or DNA [2,3] may reach a vertex of geometric figure and virtual layers as sur- nm/fs range. So that surfaces may be represented as faces and equations of motion for such surfaces can be virtual three dimensional pseudo Riemannian manifolds. searched. We refer to the formalism as differentially vari- We derived fully generic equations of motions for three ational surfaces (DVS) (or DVS formalism) [5]. manifolds [5], but purposefully omitted lengthy discus- In this paper, we propose different approach to the sion about motion of two-dimensional surfaces, which is ’shape choreography’ problem. We use DVS formal- a topic for this paper. ism, tensor calculus of moving surfaces and the first law Currently, significant progress on fluidic models of of thermodynamics to derive the final equation for the membrane dynamics has already been made. The role closed 2D surface dynamics (later on referred as surface) of geometric constraints in self-assembly have been elu- and to solve it analytically for the equilibrium case. In cidated by linking together thermodynamics, interaction other words, we derive generic equations of motions for free energies and geometry [6,7]. The Helfrich formal- closed two-dimensional surfaces and without any a pri- arXiv:1802.07166v1 [physics.bio-ph] 18 Feb 2018 ism provides the foundation for a purely differential ge- ori symmetric assumptions, we show that constant mean ometric approach whereby the membrane surface poten- curvature shapes are equilibrium solutions. In contrast tial energy density is considered as a functional of the to the Young-Laplace law these solutions, are univer- static curvature [4], see also review papers[8–10]. The sally correct descriptions of capillary surfaces as well as model has been improved by adding force and torque molecular surfaces. In addition, our equations of motions (20-25) are generic and exact. It advances our understanding of fluid dynamics because generalizes ideal magneto-hydrodynamic and Naiver-Stokes equations [5] ∗ Present address: Max Plank Institute for the Physics of Complex and in contrast to Navier-Stokes, as we demonstrate in System, 01187 Dresden, Germany. this paper, are trivially solvable for equilibrium shapes. [email protected], [email protected] To demonstrate the validity of these equations and their DAVID V. SVINTRADZE analytical solutions we apply them to micelles. Within written as Xα = Xα(t, Si). Let the position vector R~ be our formalism it becomes simple task to show micelles expressed in coordinates as lamellar, cylindrical, spherical shapes and assert their optimal spherical radius. R~ = R~(Xα) = R~(t, Si) (1) For clarity, we shall give brief description of micelles and their structures. A micelle consists of monolayer Latin letters in indexes indicate surface related tensors. of lipid molecules containing hydrophilic head and hy- Greek letters in indexes show tensors related to Euclidean drophobic tail. These amphiphilic molecules, in aqueous ambient space. All equations are fully tensorial and fol- environment, aggregate spontaneously into a monomolec- low the Einstein summation convention. Covariant bases for the ambient space are introduced as X~ α = ∂αR~, where ular layer held together due to a hydrophobic effect α [35, 36] (see also [5, 37–40]) by weak non-covalent forces ∂α = ∂/∂X . The covariant metric tensor is the dot [41]. They form flexible surfaces that show variety of product of covariant bases shapes of different topology, but remarkably in thermo- X = X~ X~ (2) dynamic equilibrium conditions they are spherical, lamel- αβ α β lar (plane) or cylindrical in shape. The contravariant metric tensor is defined as the matrix αβ inverse of the covariant metric tensor, so that X Xβγ = α α δγ , where δγ is the Kronecker delta. As far as the ambi- II. METHODS ent space is set to be Euclidean, the covariant bases are linearly independent, so that the square root of the met- In the section we provide basics of tensor calculus ric tensor determinant is unit. Furthermore, the Christof- α ~ α ~ for moving surfaces and summarize the theorems we fel symbols given by Γβγ = X · ∂βXγ vanish and set used directly or indirectly to derive equations for two- the equality between partial and curvilinear derivatives dimensional surface dynamics. Differential geometry pre- ∂α = ∇α. liminaries we used here are available in tensor calculus Now let’s discuss tensors on the embedded surface with textbook [42] and in our work [5]. arbitrary coordinates Si. Latin indexes throughout the text are used exclusively for curved surfaces and curvilinear derivative ∇i is no longer the same as the partial i A. Basics of differential geometry. derivative ∂i = ∂/∂S . Similar to the bases of ambient space, covariant bases of an embedded manifold are de- i Suppose that S (i = 1, 2) are the surface coordinates fined as S~i = ∂iR~ and the covariant surface metric tensor of the moving manifold (or the surface) S and the ambi- is the dot product of the covariant surface bases: ent Euclidean space is referred to coordinates Xα (Fig- ~ ~ ure1). Coordinates Si,Xα are arbitrarily chosen so that Sij = Si · Sj (3) The contravariant metric tensor is the matrix inverse of the covariant one. The matrix inverse nature of covariant-contravariant metrics gives possibilities to raise and lower indexes of tensors defined on the manifold. The i ~i ~ surface Christoffel symbols are given by Γjk = S · ∂jSk and along with Christoffel symbols of the ambient space provide all the necessary tools for covariant derivatives to be defined at tensors with mixed space/surface indexes:

αj αj γ α νj γ µ αj ∇iTβk = ∂iTβk + Xi Γγν Tβk − Xi ΓγβTµk + j αm m αj ΓimTβk − ΓikTβm (4)

γ where Xi is the shift tensor which reciprocally shifts space bases to surface bases, as well as space metric FIG. 1. Graphical illustration of the arbitrary sur- α to surface metric; for instance, S~i = X X~ α and Sij = face and its local tangent plane. S~1, S~2, N~ are local i ~ ~ α ~ β ~ α β tangent plane base vectors and local surface normal respec- Si · Sj = Xi XαXj Xβ = Xi Xj Xαβ. Note that in (4) tively. X~ 1, X~ 2, X~ 3 are arbitrary base vectors of the ambient Christoffel symbols with Greek indexes are zeros. Euclidean space and R~ = R~(X) = R~(t, S) is radius vector Using (2,4), one may directly prove metrilinic prop- ~ of the point. V is arbitrary surface velocity and C,V1,V2 erty of the surface metric tensor ∇iSmn = 0, from where display projection of the velocity to the N,~ S~ , S~ directions 1 2 follows S~m · ∇iS~n = 0, meaning that S~m⊥∇iS~n are or- respectively. thogonal vectors and as so ∇iS~n must be parallel to N~ the surface normal sufficient differentiability is achieved in both, space and time. Surface equation in ambient coordinates can be ∇iS~j = NB~ ij (5)

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α N~ is a surface normal vector with unit length and Bij velocity V , interface velocity C and tangential velocity is the tensorial coefficient of the relationship and is gen- V i are given on Figure1. There is a clear geometric inter- erally referred as the symmetric curvature tensor. The pretation of the interface velocity [42]. Let the surfaces trace of the curvature tensor with upper and lower in- at two nearby moments of time t and t+∆t be St, St+∆t dexes is the mean curvature and its determinant is the correspondingly. Suppose that A ∈ St (point on St) and Gaussian curvature. It is well known that a surface with the corresponding point B ∈ St+∆t, B has the same sur- constant Gaussian curvature is a sphere, consequently a face coordinates as A (Figure2), then AB~ ≈ V~ ∆t. Let sphere can be expressed as: P be the point, where the unit normal N~ ∈ St intersect the surface S , then for small enough ∆t, the angle Bi = λ (6) t+∆t i ~ ~ ∠AP B → π/2 and AP → V · N∆t, therefore, C can be where λ is some non-zero constant. According to (5,6), defined as finding the curvature tensor defines the way of finding AP covariant derivatives of surface base vectors and as so, C = lim (11) defines the way of finding surface base vectors which in- ∆t→0 ∆t directly leads to the identification of the surface. and can be interpreted as the instantaneous velocity of the interface in the normal direction. It is worth of men- B. Basics of tensor calculus for moving surfaces. tioning that the sign of the interface velocity depends on the choice of the normal. Although C is a scalar, it is called interface velocity because the normal direction is All Equations written above are generally true for mov- implied. ing surfaces. We now turn to a brief review of definitions of coordinate velocity V α, interface velocity C (which is the same as normal velocity), tangent velocity V i (Figure C. Invariant time differentiation. 1), time ∇˙ -derivative of surface tensors and time differentiation of the surface integrals. The original definitions of time derivatives for moving surfaces were given in [43] Among the key definitions in calculus for moving surfaces, perhaps one of the most important is the invariant and recently extended in tensor calculus textbook [42]. ˙ Let’s start from the definition of coordinate velocity time derivative ∇. As we have already stated, invariant V α and show that the coordinate velocity is α component time derivative is already well defined in the literature of the surface velocity. Indeed, by the definition [42, 43]. In this paragraph, we just give geometrically intuitive definition. ∂Xα Suppose that invariant field F is defined on the surface V α = (7) ∂t at all time. The idea behind the invariant time derivative is to capture the rate of change of F in the normal direc- ~ On the other hand the position vector R given by (1) is tion. Physical explanation of why the deformations along i tracking the coordinate particle S . Taking into account the normal direction are so important, we give below in partial time differential of (1) and definition of ambient integration section. This is similar to how C measures base vectors, we find the rate of deformation in the normal direction. Let for ~ i ~ α i a given point A ∈ St, find the points B ∈ St+∆t and P ∂R(t, S ) ∂R ∂X (t, S ) α V~ = = = V X~ (8) the intersection of St+∆t and the straight line orthogo- ∂t ∂Xα ∂t α nal to St (Figure2). Then, the geometrically intuitive Therefore, V α is ambient component of the surface veloc- definition dictates that ity V~ . Taking into account (8), normal component of the F (P ) − F (A) surface velocity is dot product with the surface normal, ∇˙ F = lim (12) so that ∆t→0 ∆t

α β β α α As far as (12) is entirely geometric, it must be an in- C = V~ · N~ = VαX~ N X~ β = VαN δ = VαN (9) β variant (free from choice of a reference frame). From the It is easy to show that the normal component C of the co- geometric construction one can estimate value of F in ordinate velocity, generally referred as interface velocity, point B, so that is invariant in contrast with coordinate velocity V α and ∂F its sign depends on a choice of the normal. The projec- F (B) ≈ F (A) + ∆t (13) tion of the surface velocity on the tangent space (Figure ∂t 1) is tangential velocity and can be expressed as On the other hand, F (B) is related to F (P ) because i α i V = V Xα (10) B,P ∈ St+∆t and are nearby points on the surface St+∆t, then according to deﬁnition of covariant derivative Taking (9,10) into account one may write surface velocity i i as V~ = CN~ + V S~i. Graphical illustrations of coordinate F (B) ≈ F (P ) + ∆tV ∇iF (14)

3 DAVID V. SVINTRADZE

D. Time diﬀerentiation of integrals.

The remarkable usefulness of the calculus of moving surfaces becomes evident from two fundamental formulas for integrations that govern the rates of change of volume and surface integrals due to the deformation of the domain [42]. For instance, in evaluation of the least action principle of the Lagrangian there is a central role for 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 Euclidean domain Ω with boundary S evolving with the interface velocity C, the evolution of the space integral and surface integral for closed surfaces are given by the formulas

d Z Z ∂F Z F dΩ = dΩ + CF dS (17) dt Ω Ω ∂t S Z Z Z d ˙ i F dS = ∇F dS − CFBi dS (18) dt S S S FIG. 2. Geometric interpretation of the interface velocity C and of the curvilinear time derivative ∇˙ ap- The first term in the integral represents the rate of change plied to invariant field F . A is arbitrary chosen point so of the tensor field, while the second term shows changes in that it lays on F (St) ∈ St curve and B is its’ corresponding the geometry. Of course there are rigorous mathematical point on the St+∆t surface. P is the point where St surface proofs of these formulas in the tensor calculus textbooks. normal, applied on the point A, intersects the surface St+∆t. We are not going to reproduce proof of these theorems By the geometric construction, for small enough ∆t → 0, here, but instead we give less rigorous but completely ~ ~ ~ ~ ∠AP B → π/2, AB ≈ V ∆t and AP ≈ V N∆t. On other intuitive explanation of why only interface velocity has hand, by the same geometric construction the field F in the to be count. Rigorous mathematical proof follows from point B can be estimated as F (B) ≈ F (A) + ∆t∂F/∂t, while fundamental theorem of calculus from viewpoint of the St+∆t surface the F (B) value can be estimated as F (P )+∆tV i∇ F , where ∇ F shows rate of change i i Z b(t) Z b(t) in F along the surface St+∆t and along the directed distance d ∂F (t, x) 0 i F (t, x)dx = dx + b (t)F (t, b(t)) BP ≈ ∆tV . dt a a ∂t (19) In the case of volume integral or surface integral it can since ∇ F shows rate of change in F along the surface and i be shown that b0(t) is replaced by interface velocity C. ∆t · V i captures the directed distance BP . Determining F (A),F (P ) values from (13,14) and putting it in (12), Intuitive explanation is pretty simple. Propose there gives is no interface velocity then closed surface velocity only has tangent component. For each given time tangent ve- ∂F ∇˙ F = − V i∇ F (15) locity (if there is no interface velocity) translates each ∂t i point to its neighboring point and therefore, does not Extension of the definition (15) for any arbitrary tensors add new area to the closed surface (or new volume to with mixed space and surface indexes is given by the the closed space, or new length to the closed curve). As formula so, tangential velocity just induces rotational movement (or uniform translational motion) of the object and can ∂T αi αi βj k αi γ α µi γ µ αi be excluded from additive terms in the integration. Per- ∇˙ T = − V ∇kT + V Γ T − V Γ T βj ∂t βj γµ βj γβ µj haps, it is easier to understand this statement for one ˙ i αk ˙ k αi +ΓkTβj − Γj Tβk dimensional motion. Let’s assume that material point (16) is moving along some trajectory (some closed curve or loop), then, in each point, the velocity of the material The derivative commutes with contraction, satisfies sum, point is tangential to the curve. Now one can translate product and chain rules, is metrinilic with respect to the this motion into the motion of the closed curve where ambient metrics and does not commute with the surface the loop has only tangential velocity. In this aspect, the derivative [42]. Also from (12) it is clear that the invari- embedded loop only rotates (uniformly translates in the ant time derivative applied to time independent scalar plane) without changing the length locally, therefore tan- ˙ i vanishes. Christoffel symbol Γj for moving surfaces is gential velocity of the curve does not add new length to ˙ i i i defined by the formula Γj = ∇jV − CBj. the curve (Same is true for open curve with fixed ends).

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III. GENERAL EQUATIONS OF SURFACE MOTIONS

Fully non-restrained and exact equations for moving three-dimensional surfaces in electromagnetic ﬁeld, when the interaction with an ambient environment is ignored, reads

˙ i i ∇ρ + ∇i(ρV ) =ρCBi (20)

α i i j α 1 µν µ ∂α(ρV (∇˙ C + 2V ∇iC + V V Bij) − V ( Fµν F + AµJ )) =fa (21) 4µ0 Z Z ˙ i j i i j i i ρVi(∇V + V ∇jV − C∇ C − CV Bj)dS = f aidΩ (22) S Ω

where ρ is the surface mass density, V α, V i are coor- in our work [5], we don’t reproduce derivation of this dinate and tangential components of the surface veloc- set in this paper, rather just mention that ﬁrst one is ity, C is interface velocity, α = 0, 1, 2, 3 for Minkowski the consequence of mass conservation, second and third four-dimensional space-time ambient space, i = 0, 1, 2 equations come from minimum action principle of a La- for pseudo-Riemannian manifold (surface), Bij is the grangian and imply motion in normal direction (21) and surface curvature tensor, F µν is electromagnetic tensor, in tangent direction (22). α α βα α α F = J − ∂βF , J is α component of J~ = (J ) For two dimensional surface dynamics, Minkowskian four current, f, f i are normal and tangential components space becomes Euclidean, so that α = 1, 2, 3 and the sur- α of F~ = (F ), a, ai are the normal and tangential com- face is two-dimensional Riemannian manifold i = 1, 2. So ponents of the partial time derivative of the four vector that, after modeling the potential energy as a negative volume integral of the internal pressure and inclusion in- potential A~ = (Aα), S, Ω stand for surface and space integrals respectively. Exact derivation of (20-22) is given teraction with an environment, (20-22) further simpliﬁes as

˙ i i ∇ρ + ∇i(ρV ) =ρCBi (23) α i i j α + α + ∂α(ρV (∇˙ C + 2V ∇iC + V V Bij) + V (P + Π)) = − V ∂α(P + Π) (24) ˙ i j i i j i ρVi(∇V + V ∇jV − C∇ C − CV Bj) =0 (25)

where P +, Π are internal hydrodynamic and osmotic equation of motion in normal direction (39) as we get pressures, respectively. Derivation of (20-22) can be from using the first law of thermodynamics (see below). found in [5]. We derive (23-25) in appendix section. It α i + α + α is noteworthy that from the last equations set only the ∂α(σV Bi + (P + Π)V ) = −(∂αP + ∂αΠ)V (27) second equation (24) differs from the dynamic fluid film It is worth of mentioning that (23-25) also follows from equations [42, 44] (20-22) if one applies same formalism as it is given in (30- 32). Indeed, for relatively slowly moving surfaces space ρ(∇˙ C + 2V i∇ C + V iV jB ) = σBi (26) i ij i is three dimensional Euclidean so that α = 1, 2, 3, the where σ is surface tension. (26) is only valid when the surface is two-dimensional Riemannian (i = 1, 2) and the surface can be described with time invariable surface ten- potential energy becomes sion [42, 44], meaning that the surface is homogeneous Z 1 µν µ and the surface tension is constant, while (24) does not U = ( Fµν F + AµJ ) 4µ have that restriction. Using (26) in (24) and taking into Ω 0 Z 1 account that in equilibrium processes internal pressure is = (− 0 E2 + B2 − qϕ + A~J~)dΩ (28) the same as external pressure, one gets exactly the same Ω 2 µ0

5 DAVID V. SVINTRADZE where E,~ B~ are electric and magnetic fields and q, ϕ, A,~ J~ law of thermodynamic, as far as there is no dissipated or are charge density, electric potential, magnetic vector po- absorbed heat, the change of the internal energy of the tential and current density vector respectively. Using surface must be (30-32) formalism into account, we find dE = δW (30) 1 dU = −(P + + Π)dΩ = (− 0 E2 + B2 − qϕ + A~J~)dΩ where δW is infinitesimal work done on the subsystem 2 µ0 and dE is infinitesimal change of the internal energy. (29) Because the temperature of the system is constant, the Taking into account (29) and that the pressure comes differential of the subsystems’ internal energy can be re- from the normal force applied to the surface, we find α + i modeled as fa = −V ∂α(P + Π) and in tangent direction f ai = 0, then (20-22) becomes (23-25). Electromagnetic potential dE = dU (31) energy can be generalized if one takes into account environment, which enters in energy terms as bound and where U is the total potential energy of the surface. By free charges and electric/magnetic fields are replaced by the definition the elementary work done on the subsystem polarization and magnetization vectors [5]. is δW = (P − + Π)dΩ (32)

IV. RESULTS AND DISCUSSION where , P −, Π are external hydrodynamic and osmotic pressures applied on the surface by the surroundings cor- A. General assumptions. respondingly and Ω is the volume that surface encloses with boundary of S surface area. Let’s propose that the surface is homogeneous (i.e material particles are homo- In this section we apply basics of thermodynamics and geneously distributed on the surface) so that the total fundamental theorems of calculus of moving surfaces to potential energy is integration of the potential energy demonstrate shortest derivation of the equation, describ- per unit area over the surface, then ing motion of homogeneous, closed two dimensional surface with time invariable surface tension at normal direc- dU = σdS (33) tion (27). We consider the system consisted of aqueous media with the formed closed surface in it (Figure3). where σ is the potential energy per unit area and is called The system is isolated with constant temperature and surface tension in the paper. As far as we discuss sim- there is no absorbed or dissipated heat on the surface; in plest case of the system consisted of aqueous medium and other words, a process is adiabatic. According to the first single closed surface, we can suggest that the surface tension is not time variable. Using (30-33) after few lines of algebra, we fined Z Z σdS = (P − + Π)dΩ (34) S Ω Assuming the surface is moving so that (34) stays valid for any time variations, then time differentiation of the left side must be equal to time differentiation of right integral. As far as on the right hand side we have space integral, time differentiation can be taken into the integral, using general theorems for differentiation of space and surface integrals (17-18), so that integration theorem for space integral holds and the convective and advective terms due to volume motion are considered Z Z α d − − ∂X (P + Π)dΩ = (∂αP + ∂αΠ) dΩ dt Ω Ω ∂t Z + C(P − + Π)dS (35) S To calculate time derivative of the surface integral we have to take into account the theorem about time differ- FIG. 3. Graphical illustration of the isolated system containing aqueous solution. Water molecules are repre- entiation of the surface integral (18), from which follows sented as red and white sticks. The system boundary is shown that for time invariable surface tension as white faces with black edges. The subsystem-micelle is Z Z d i closed surface, blue blob in the center of the system. σdS = −σCBi dS (36) dt S S

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α Where C = V Nα is interface velocity, Nα is α com- derivation explains all the shapes surfaces can adopt in ponent of the surface normal and V = ∂Xα/∂t is coor- aqueous solution at equilibrium conditions.1 If the com- α i dinate velocity, X is general coordinate and Bi is the pactness condition is relaxed then (40) predicts that in trace of the mixed curvature tensor generally known as addition to cylinder and plane all other CMC surfaces mean curvature. After few lines of algebra putting (34- are also equilibrium shapes for moving surfaces. Taking 36) together, we find into account that the surface tension in general can be a function of many variables, such as Gaussian curva- Z Z i − − α ture, bending rigidity, spontaneous curvature, molecules (σCBi + C(P + Π))dS = − (∂αP + ∂αΠ)V dΩ S Ω concentration, geometry of surfactant molecules and etc., (37) then (39) may predict possible deformations of differently Generalized Gauss theorem converts the surface integral shaped surfaces and their wide range of static shapes. In of the left hand side of (37) into space integral, so that fact, if considered that surface tension, which is defined Z Z as potential energy per unit area, can be a function of α i − α i i NαV (σB + P + Π)dS = ∂α(σV B mean curvature σ = σ(Bi ), then Taylor expansion of i i i S Ω σ(Bi ) naturally rises all additional terms. These gener- +(P − + Π)V α)dΩ (38) alizations and temperature fluctuations can be included in the equations, but it is not scope of this paper and Combination of (37) and (38) immediately gives equation should be addressed separately. One may even propose of motion for surface in normal direction σ as time independent the Helfrich Hamiltonian and then (39, 40) will become equation of static shapes for homo- α i − α − α ∂α(σV Bi + (P + Π)V ) = −(∂αP + ∂αΠ)V (39) geneous surfaces with time invariable surface tension. For equilibrium processes internal and external pressures − + are identical P = P , so that (39) becomes identical B. Physical application, micelle. to the equation of motion in normal direction observed from master equations (23-27). Also, we should note that We can put equation (39) and its solution (40) under (39) is only valid for motion of the homogeneous surfaces the test for homogeneous micellar surface equilibrated with time invariable surface tension at normal direction, with the aqueous solution. Based on (40) we can calcu- therefore, it does not display any deformation in tangent late minimal value of a micelle radius. The value of the directions. (39) further simplifies when the surface comes trace of the mixed curvature tensor for a sphere is in equilibrium with the solvent where divergence of the surface velocity ∂ V α (stationary interface) along with 2 α Bi = − (42) ∂P/∂t (where P = P − + Π) vanishes, then the solution i R to (39), taking into account the condition (35), becomes where R is radius. P Bi = − (40) i σ The result (40) shows that the solution is constant mean curvatures (CMC) surfaces. Such CMC are rare and can be many if one relaxes the condition we restricted to the system. We consider isolated system where the surface is closed subsystem, these two preconditions mathemat- ically mean that the surface we discuss is compact embedded surface in R3. According to A. D. Alexandrov uniqueness theorem for surfaces, a compact embedded surface in R3 with constant non-zero mean curvature is a sphere [45]. Correspondingly the solution (40) is a sphere FIG. 4. Simulated three dimensional coordinates (as far as we have compact two-manifold in the Euclidean of the micelle in aqueous solution display sphere space). When with diameter 38.5A˚. (Left) dihexanoylphosphatidylcholine (DHPC phospholipids) are modeled as orange balls. (Right) P 6= 0 (41) Gaussian mapping at contour resolution 8A˚ of the micelle σ shows spherical structure. the surface is spheroid (or a cylinder if one relaxes compactness restriction making the cylinder infinitely long) and becomes plane (again when compactness argument is 1 Even though we set environment as aqueous, it enters into equa- relaxed) or other zero mean curvature shape when com- tions as osmotic pressure term, which due to a generality of ar- pactness argument is not relaxed but contour of the sur- guments can be anything. Therefore, as a medium one may pick face remains fixed. This surprisingly simple and elegant any liquid or gas.

7 DAVID V. SVINTRADZE

3 Let’s calculate value of the surface pressure when the 894A˚ . Using this value, one gets micelle still can exist. Lipids in a micelle are confined in 1.99 · 10−20 the surface by hydrophobic interactions with average en- P ≈ ≈ 2.22 · 107N/m2 (45) ergy in the range of hydrogen bonding. As far as values of 0.894 · 10−27 hydrogen bonding energy are somewhat uncertain in the On other hand, using the same surface tension of a fluid literature, by the first approximation we take average en- monolayer at the optimal packing of the lipids, one gets ergy for the hydrogen bonding energy interval and assign R = 27±0.1A˚. All atom simulation also generates spher- it to the lipid molecule. Low boundary of the interval ical structure with diameter 54 ± 0.1A˚ (Figure5). There (minimum energy) for XH ··· Y hydrogen bond is about is still some uncertainty in this estimation because we as- 1 kJ per mol (CH ···C unit) and high boundary is about signed 1 kJ/mol energy per −CH2− unit and we based 161 kJ per mol (FH···F unit), the low and high values are on references data [46, 47], while in other literature it is taken according to references [46, 47]. Therefore, average mentioned that the hydrophobic interactions are about 4 energy is about (1 + 161)/2 = 81kJ/mol ≈ 13 · 10−20J. kJ/mol per −CH2− unit [55]. In our opinion, this dis- To estimate hydrogen bonding energy per molecule with crepancy can be resolved if one calculates hydrophobic the undefined shape (lipid molecule) in the first approxi- energy based on the potential energy mation is to assign average energy to it and consider the spherical shape with the gyration radius. Of course it is Z U = − 0 E2 dΩ (46) CH2 low level approximation, but even such rough calculations Ω 2 produce reasonable results. After all these rough estima- ~ tions the pressure to move one lipid from the surface, in where ECH2 is electric field per −CH2−, 0 is dielectric order to induce critical deformations of the surface, is constant in the vacuum and Ω stands for the volume of µν about average energy per the average volume of the lipid the lipid molecules. (46) directly emerges from Fµν F molecule term written in the equations of motion (20-22 and 28- 29). For electrostatics 3 · 13 · 10−20 P ≈ ≈ 3.1 · 107N/m2 (43) 4πr3 Z 1 Z G U = F F µν dΩ = − 0 E2 dΩ (47) µν CH2 3 Ω 4µ0 Ω 2 where 4πrG/3 is the estimated volume of a lipid molecule considered as sphere with the gyration radius rG ≈ 1nm. so one should go to the scrutiny of calculating electric On the other hand, surface tension of a fluid monolayer field for each −CH2− units, then take a sum of the elec- at optimal packing of the lipids is about σ ≈ 3·10−2N/m tric field and square it (we are not going to do it in this [7, 48, 49], using these and (42,43) in (40) the estimated paper). Also, one may ask why the hydrophilic inter- micelle radius is action energy is not taken into account in these calculations. Hydrophilic head of the lipid molecule is in contact 2 · 3 · 10−2 ˚ with water molecules so there is no work needed to drag it R ≈ 7 = 19.3 ± 0.1A (44) 3.1 · 10 in aqueous solution from the lipids layer. Therefore, hy- These calculations put the minimum radius in nanome- drophilic interaction energy can be neglected. The most ter scale and is in very good agreement with experimen- work goes on overcoming hydrophobic interactions be- tal as well as computational frameworks [50, 51]. To tween lipid tails. further validate the (44) result, we ran a CHARMM based Micelle Builder simulation [52, 53] for 100 phospholipid molecules (DHPC lipids). The simulation result V. CONCLUSION (Figure4) generated a spherical micelle with diameter 38.5±0.1A˚. These calculations indeed indicate that even We have presented a framework for the analysis of two such rough estimations produce reasonable accuracy. dimensional surface dynamics (identified as micelle in the To get more convincing estimations it is necessary text) using first law of thermodynamics and calculus of to take into account that neither lipids are spherical moving surfaces. In final equations of normal motion nor hydrophobic interactions per lipid are average en- (39,27) we assume that a surface is homogeneous and ergy of single hydrogen bond. In the second approxi- has time invariable surface tension. However, the gen- mation lipids are no longer undefined spheres, but have eral equations (23-25) don not have these constrains and well defined surfactant geometry. The Hydrophobic en- indicate arbitrary motion along normal deformation, as ergy is no longer average energy of single hydrogen well as into tangent directions, but are analytically more bond, but is 1 kJ per mol per −CH2− unit. In all complex. The solution to the normal equations of mo- atom simulations we used dihexanoylphosphatidylcholine tion in equilibrium conditions are surprisingly simple and (DHPC) lipid molecule having 12 − CH2− units (Figure display all possible equilibrium shapes. We applied the 5) per hydrophobic tail, so hydrophobic energy is about formalism to estimate micelle optimal radius and com- 12kJ/mol ≈ 1.99 · 10−20J. Accurate calculation of the pared estimations to all atom simulations. Even for low- lipid molecule volume using cavity, channel and cleft vol- level approximations, we found remarkable agreement be- ume calculator [54], gives the volume estimation of about tween theoretically calculated radius and one obtained

8 MOVING MANIFOLDS ...

FIG. 5. All atom simulation of DHPC micelle. (A) The figure shows a geometry of the DHPC surfactant molecule used in simulation and gives parametric description of volume, surface area, sphericity and effective radius. (B) Indicates atomistic simulation result contoured by Gaussian map and the diameter of the micelle, measured by PyMol. The diameter of the simulated micelle appears to be 54.0A˚ with the uncertainty of the measurement 0.1A˚. from atomistic simulations and from experiments. One in it’s current form was initiated at Aspen Center for can readily apply the theory to any closed surfaces; such Physics, which is supported by National Science Founda- are vesicles, membranes, water droplets or soap films. tion grant PHY-1607611 and was partially supported by As a final remark, even though the analytic solution a grant from the Simons Foundation. (40) looks like generalized Young-Laplace law, the differ- i ence is obvious. Bi is a trace of mixed curvature tensor, known as mean curvature, and when the mean curvature is constant, it defines whole class of constant mean curvature (CMC) surfaces. Generalized the Young-Laplace law is a priori formulated for spherical morphologies and therefore in some particular cases can be obtained from (40) constant mean curvature shapes. The condition for holding the particular case is a compactness. However, the compactness argument can be relaxed in our derivation if the considered system is set to be much larger than APPENDIX: DERIVATION OF EQUATIONS OF the subsystem. Therefore, the solution (40) effectively MOTIONS FOR CLOSED, TWO DIMENSIONAL SURFACES predicts formation of all CMC surfaces while Young- Laplace law is correct for spherical structures alone. Also in derivation of Young-Laplace relation one of corner- stone idea is suggestion of spherical symmetries, while Now we turn to the derivation of (23-25) without using our derivation is free of symmetries and explains why any information from (20-22) (though derivation of (23- CMC surfaces are such abundant shapes in nature, ob- 25) from (20-22) is strightforwad and trivial if one sets servable even on molecular levels. In fact, according to V 0 = 0 in (20-22 equations [5]). To deduce the equations the results, any homogeneous closed surface with time of motion we derive the simplest one from the set (23) invariable surface tension adopts CMC shape when it first. It is direct consequence of generalization of conser- comes in equilibrium with environment. vation of mass law. Variation of the surface mass density R must be so that dm/dt = 0, where m = S ρdS is surface mass with ρ surface density. Since the surface is closed, ACKNOWLEDGMENTS i at the boundary condition v = niV = 0, a pass integral along any curve γ across the surface must vanish (ni is a We thank Dr. Frank Julicher (MPIPKS) and Dr. Er- normal of the curve and lays in the tangent space). Us- win Frey (LMU), for stimulating discussions. The paper ing Gauss theorem, conservation of mass and integration

9 DAVID V. SVINTRADZE formula (18), we find lead Z 2 Z 2 2 Z Z Z δ ρV ˙ ρV i ρV i i dS = (∇ − CBi )dS 0 = vρdγ = niV ρdγ = ∇i(ρV )dS δt S 2 S 2 2 γ γ S Z 2 2 2 Z ˙ V ˙ V i ρV i i i = (∇ρ + ρ∇ − CBi )dS = (∇i(ρV ) − ρCBi + ρCBi )dS S 2 2 2 S Z V 2 V 2 ρV 2 Z Z d Z i i ˙ i i i ˙ = ((ρCBi − ∇i(ρV )) + ρ∇ − CBi )dS = (∇i(ρV ) − ρCB )dS + ∇ρdS − ρdS 2 2 2 i dt S S S S Z 2 2 Z i V V i i = (−∇i(ρV ) + ρ∇˙ )dS = (∇˙ ρ + ∇i(ρV ) − ρCB )dS (48) i S 2 2 S Z 2 2 2 i V i V V = (−∇i(ρV ) + ρV ∇i + ρ∇˙ )dS Since last integral must be identical to zero for any inte- S 2 2 2 grand, one immediately finds first equation from the set Z V 2 = (−∇ (ρV i ) + ρV~ (V i∇ V~ + ∇˙ V~ ))dS (51) (23). To deduce second and third equations, we take a i 2 i Lagrangian S At the end point of variations the surface reaches station- Z 2 Z ary point and therefore by Gauss theorem (as we used it ρV + L = dS + (P + Π)dΩ (49) already in (48)), we find S 2 Ω Z 2 Z 2 i V i V and set minimum action principle requesting that −∇i(ρV )dS = − ρV ni dγ = 0 (52) 2 2 δL/δt = 0. Evaluation of space integral is simple and S γ straightforward, using integration theorem for space in- γ is stationary contour of the surface and ni is the normal tegral where the convective and advective terms due to to the contour, therefore interface velocity for contour volume motion is properly taken into account (17), we i v = niV = 0 and the integral (52) vanishes, correspond- find ingly

δ Z Z Z Z 2 Z (P ++Π)dΩ = ∂ (P ++Π)V αdΩ+ C(P ++Π)dS δ ρV ~ i ~ ˙ ~ δt α dS = ρV (V ∇iV + ∇V )dS (53) Ω Ω S δt S 2 S (50) Derivation for kinetic part is a bit tricky and challenging To decompose dot product in the integral by normal and that is why we do it last. Straightforward, brute math- tangential components and, therefore, deduce ﬁnal equa- ematical manipulations, using ﬁrst equation from (23), tions, we do following algebraic manipulations

˙ ~ i ~ ˙ ~ i ~ i j ~ i j ~ ˙ ~ i ~ i j α ~ i j ~ ∇V + V ∇iV = ∇V + V ∇iV + CV Bi Sj − CV Bi Sj = ∇V + V ∇iV + CV Bi Xj Xα − CV Bi Sj (54)

α j α ~ ~ α ~ Now using Weingartens formula Xj Bi = −∇iN , ∇iXα = 0, definition of surface normal N = N Xα metrinilic property of Euclidian space base vectors and taking into account definition of surface velocity i V~ = CN~ + V S~i and its derivatives, we find

˙ ~ i ~ i j α ~ i j ~ ˙ ~ i ~ i ~ α i j ~ ∇V + V ∇iV + CV Bi Xj Xα − CV Bi Sj = ∇V + V ∇iV − CV Xα∇iN − CV Bi Sj ˙ ~ i ~ i α ~ i j ~ = ∇V + V ∇iV − CV ∇i(N Xα) − CV Bi Sj ˙ ~ i ~ i ~ i j ~ = ∇V + V ∇iV − CV ∇iN − CV Bi Sj ˙ ~ i ~ i j ~ i ~ i j ~ = ∇V + V ∇i(CN) + V ∇i(V Sj) − CV ∇iN − CV Bi Sj ˙ ~ i ~ i j ~ i j ~ = ∇V + V N∇iC + V ∇i(V Sj) − CV Bi Sj ˙ ~ ˙ j ~ i ~ i j ~ i j ~ = ∇(CN) + ∇(V Sj) + V N∇iC + V ∇i(V Sj) − CV Bi Sj (55)

10 MOVING MANIFOLDS ...

Continuing algebraic manipulations using Thomas for- of interface velocity N~ ∇iC = ∇˙ S~i and the deﬁnition of i mula ∇˙ N~ = −∇ CS~i, the formula for surface derivative curvature tensor (5) yield

˙ ~ ˙ j ~ i ~ i j ~ i j ~ ∇(CN) + ∇(V Sj) + V N∇iC + V ∇i(V Sj) − CV Bi Sj j i i j j = ∇˙ (CN~ ) + C∇ CS~j + 2V N~ ∇iC + V V BijN~ + ∇˙ (V S~j) i ~ i j ~ i j ~ j ~ i j ~ − V N∇iC + V ∇i(V Sj) − V V BijN − C∇ CSj − CV Bi Sj i i j j j = ∇˙ (CN~ ) − C∇˙ N~ + 2V N~ ∇iC + V V BijN~ + ∇˙ (V S~j) − V ∇˙ S~j i j ~ i j ~ j ~ i j ~ + V ∇i(V Sj) − V V ∇iSj − C∇ CSj − CV Bi Sj ˙ i i j ~ ˙ j i j j i j ~ = (∇C + 2V ∇iC + V V Bij)N + (∇V + V ∇iV − C∇ C − CV Bi )Sj (56)

Taking dot product of (56) on V~ and combining it with (53) last derivation finally reveals variation of kinetic energy, so that we finally find

Z 2 Z δ ρV ˙ i i j ˙ i j i i j i dS = (ρC(∇C + 2V ∇iC + V V Bij) + ρVi(∇V + V ∇jV − C∇ C − CV Bj))dS (57) δt S 2 S

Combining (48-50) and (57) together and taking into ac- surface normal, we immediately ﬁnd ﬁrst (23) and the count that the pressure acts on the surface along the last equation (25) of the set. To clarify second equation (24), we have

Z Z Z i i j + α + ρC(∇˙ C + 2V ∇iC + V V Bij)dS = −∂α(P + Π)V dΩ − C(P + Π)dS S Ω S Z Z i i j + + α C(ρ(∇˙ C + 2V ∇iC + V V Bij) + P + Π)dS = −∂α(P + Π)V dΩ (58) S Ω

After applying Gauss theorem to the second equation (58), the surface integral is converted to space integral so that we ﬁnally ﬁnd

α i i j + α + α ∂α(ρV (∇˙ C + 2V ∇iC + V V Bij) + (P + Π)V ) = −∂α(P + Π)V (59)

and, therefore, all three equations (23-25) are rigorously clariﬁed.

[1] Michael D. Cahalan and Ian Parker, “Choreography of of Immunology 26, 585–626 (2008), pMID: 18173372, cell motility and interaction dynamics imaged by two- https://doi.org/10.1146/annurev.immunol.24.021605.090620. photon microscopy in lymphoid organs,” Annual Review [2] Samir Kumar Pal, Jorge Peon, Biman Bagchi, and

11 DAVID V. SVINTRADZE

Ahmed H. Zewail, “Biological water:âfemtosecond dy- ity and deformation of a spherical vesicle by pressure,” namics of macromolecular hydration,” The Journal Physical review letters 59, 2486 (1987). of Physical Chemistry B 106, 12376–12395 (2002), [23] Ou-Yang Zhong-Can and Wolfgang Helfrich, “Bending http://dx.doi.org/10.1021/jp0213506. energy of vesicle membranes: General expressions for the [3] Chaozhi Wan, Torsten Fiebig, Olav Schiemann, Jacque- first, second, and third variation of the shape energy and line K. Barton, and Ahmed H. Zewail, “Femtosecond applications to spheres and cylinders,” Physical Review direct observation of charge transfer between bases in A 39, 5280 (1989). dna,” Proceedings of the National Academy of Sciences [24] IvaAlo˜ M. Mladenov, Peter A. Djondjorov, Mariana Ts. of the United States of America 97, 14052–14055 (2000). Hadzhilazova, and Vassil M. Vassilev, “Equilibrium con- [4] Wolfgang Helfrich, “Elastic properties of lipid bilayers: figurations of lipid bilayer membranes and carbon nanos- theory and possible experiments,” Zeitschrift fürNatur- tructures,” Communications in Theoretical Physics 59, forschung C 28, 693–703 (1973). 213 (2013). [5] David V. Svintradze, “Moving manifolds in electromag- [25] SaˇsaSvetina and Boˇstjan Zekˇs,“Membraneˇ bending en- netic fields,” Frontiers in Physics 5, 37 (2017). ergy and shape determination of phospholipid vesicles [6] Jacob N. Israelachvili, D. John Mitchell, and Barry W. and red blood cells,” European biophysics journal 17, Ninham, “Theory of self-assembly of hydrocarbon am- 101–111 (1989). phiphiles into micelles and bilayers,” J. Chem. Soc., Fara- [26] Udo Seifert and Reinhard Lipowsky, “Adhesion of vesi- day Trans. 2 72, 1525–1568 (1976). cles,” Physical Review A 42, 4768 (1990). [7] Jacob N. Israelachvili, D.John Mitchell, and Barry W. [27] Reinhard Lipowsky, “The conformation of membranes,” Ninham, “Theory of self-assembly of lipid bilayers Nature 349, 475–481 (1991). and vesicles,” Biochimica et Biophysica Acta (BBA) - [28] Udo Seifert, Karin Berndl, and Reinhard Lipowsky, Biomembranes 470, 185 – 201 (1977). “Shape transformations of vesicles: Phase diagram for [8] Markus Deserno, “Fluid lipid membranes: From differ- spontaneous- curvature and bilayer-coupling models,” ential geometry to curvature stresses,” Chemistry and Phys. Rev. A 44, 1182–1202 (1991). physics of lipids 185, 11–45 (2015). [29] Frank Jülicher and Reinhard Lipowsky, “Domain- [9] R. Lipowsky and E. Sackmann, Structure and Dynamics induced budding of vesicles,” Physical review letters 70, of Membranes (Elsevier, Amsterdam, 1995). 2964 (1993). [10] Udo Seifert, “Configurations of fluid membranes and vesi- [30] Frank Jülicher and Reinhard Lipowsky, “Shape transfor- cles,” Advances in Physics 46, 13–137 (1997). mations of vesicles with intramembrane domains,” Phys- [11] R Capovilla and J Guven, “Stresses in lipid membranes,” ical Review E 53, 2670 (1996). Journal of Physics A: Mathematical and General 35, [31] Frank Jülicher and Udo Seifert, “Shape equations for ax- 6233 (2002). isymmetric vesicles: a clarification,” Physical Review E [12] Jean-Baptiste Fournier, “On the stress and torque ten- 49, 4728 (1994). sors in fluid membranes,” Soft Matter 3, 883–888 (2007). [32] Ling Miao, Udo Seifert, Michael Wortis, and Hans- [13] L.E. Scriven, “Dynamics of a fluid interface equation of Günther Döbereiner, “Budding transitions of fluid- motion for newtonian surface fluids,” Chemical Engineer- bilayer vesicles: the effect of area-difference elasticity,” ing Science 12, 98 – 108 (1960). Physical Review E 49, 5389 (1994). [14] Marino Arroyo and Antonio DeSimone, “Relaxation dy- [33] Volkmar Heinrich, Saˇsa Svetina, and Boˇstjan Zekˇs,ˇ namics of fluid membranes,” Phys. Rev. E 79, 031915 “Nonaxisymmetric vesicle shapes in a generalized bilayer- (2009). couple model and the transition between oblate and pro- [15] Ling-Tian Gao, Xi-Qiao Feng, Ya-Jun Yin, and Huajian late axisymmetric shapes,” Physical Review E 48, 3112 Gao, “An electromechanical liquid crystal model of vesi- (1993). cles,” Journal of the Mechanics and Physics of Solids 56, [34] Vera Kralj-Igliˇc,SaˇsaSvetina, and Boˇstjan Zekˇs,“Theˇ 2844 – 2862 (2008). existence of non-axisymmetric bilayer vesicle shapes pre- [16] Guillaume Salbreux and Frank Jülicher, “Mechanics of dicted by the bilayer couple model,” European biophysics active surfaces,” Phys. Rev. E 96, 032404 (2017). journal 22, 97–103 (1993). [17] Sriram Ramaswamy, John Toner, and Jacques Prost, [35] David Chandler, “Interfaces and the driving force of hy- “Nonequilibrium fluctuations, traveling waves, and in- drophobic assembly,” Nature 437, 640–647 (2005). stabilities in active membranes,” Phys. Rev. Lett. 84, [36] Sergey Leikin, V Adrian Parsegian, Donald C Rau, and 3494–3497 (2000). R Peter Rand, “Hydration forces,” Annual Review of [18] Hsuan-Yi Chen, “Internal states of active inclusions and Physical Chemistry 44, 369–395 (1993). the dynamics of an active membrane,” Phys. Rev. Lett. [37] David V Svintradze, “Hydrophobic and hydrophilic in- 92, 168101 (2004). teractions,” Biophysical Journal 98, 43a–44a (2010). [19] N. Gov, “Membrane undulations driven by force fluctu- [38] David V Svintradze, “Moving macromolecular sur- ations of active proteins,” Phys. Rev. Lett. 93, 268104 faces under hydrophobic/hydrophilic stress,” Biophysical (2004). Journal 108, 512a (2015). [20] Peter B Canham, “The minimum energy of bending as a [39] David V Svintradze, “Cell motility and growth factors possible explanation of the biconcave shape of the human according to differentially variational surfaces,” Biophys- red blood cell,” Journal of theoretical biology 26, 61–81 ical Journal 110, 623a (2016). (1970). [40] David V Svintradze, “Geometric diversity of living organ- [21] Evan A Evans, “Bending resistance and chemically in- isms and viruses,” Biophysical Journal 112, 309a (2017). duced moments in membrane bilayers,” Biophysical jour- [41] C Tanford, The Hydrophobic Effect Formation of Mi- nal 14, 923 (1974). celles and Biological Membranes (Wiley-Interscience, [22] Ou-Yang Zhong-Can and Wolfgang Helfrich, “Instabil-

12 MOVING MANIFOLDS ...

New York, 1973). [50] Scott E Feller, Yuhong Zhang, and Richard W Pas- [42] Pavel Grinfeld, Introduction to tensor analysis and the tor, “Computer simulation of liquid/liquid interfaces. ii. calculus of moving surfaces (Springer, New York, 2010). surface tension-area dependence of a bilayer and mono- [43] J. Hadamard, Mmoire sur le problme danalyse relatif layer,” The Journal of chemical physics 103, 10267– lquilibre des plaques elastiques encastres (Oeuvres, Her- 10276 (1995). mann, Tome 2, 1968). [51] Egbert Egberts, Siewert-Jan Marrink, and Herman JC [44] Pavel Grinfeld, “Exact nonlinear equations for fluid films Berendsen, “Molecular dynamics simulation of a phos- and proper adaptations of conservation theorems from pholipid membrane,” European biophysics journal 22, classical hydrodynamics,” J. Geom. Symm. Phys 16, 1– 423–436 (1994). 21 (2009). [52] Sunhwan Jo, Taehoon Kim, Vidyashankara G Iyer, and [45] A. D. Alexandrov, “Uniqueness theorem for surfaces in Wonpil Im, “Charmm-gui: a web-based graphical user in- the large,” Leningrad Univ. 13, 19 (1958), 58, Amer. terface for charmm,” Journal of computational chemistry Math. Soc. Trans. (Series 2) 21, 412–416 (1958). 29, 1859–1865 (2008). [46] JW Larson and TB McMahon, “Gas-phase bihalide and [53] Xi Cheng, Sunhwan Jo, Hui Sun Lee, Jeffery B pseudobihalide ions. an ion cyclotron resonance determi- Klauda, and Wonpil Im, “Charmm-gui micelle builder nation of hydrogen bond energies in xhy-species (x, y= f, for pure/mixed micelle and protein/micelle complex sys- cl, br, cn),” Inorganic Chemistry 23, 2029–2033 (1984). tems,” Journal of chemical information and modeling 53, [47] J Emsley, “Very strong hydrogen bonding,” Chemical So- 2171–2180 (2013). ciety Reviews 9, 91–124 (1980). [54] Neil R Voss and Mark Gerstein, “3v: cavity, channel and [48] F Jähnig,“What is the surface tension of a lipid bilayer cleft volume calculator and extractor,” Nucleic acids re- membrane?” Biophysical journal 71, 1348–1349 (1996). search , gkq395 (2010). [49] Jacob N Israelachvili, Intermolecular and surface forces: [55] A. Leitmannova Liu, Advances in Planar Lipids Bilayers revised third edition (Academic Press, 2011). and Liposomes, Vol. 4 (Academic Press, Elsevier, 2011).