Closed, Two Dimensional Surface Dynamics

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 = rα. 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 ri 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 riTβk = @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 riSmn = 0, from where display projection of the velocity to the N;~ S~ ; S~ directions 1 2 follows S~m · riS~n = 0, meaning that S~m?riS~n are or- respectively. thogonal vectors and as so riS~n must be parallel to N~ the surface normal sufficient differentiability is achieved in both, space and time.

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