MODELING AND COMPUTING OF DEFORMATION ...

SIAM J. APPL. MATH. Vol. 73, No. 5, pp. 1768–1792

c 2013 Society for Industrial and Applied Mathematics

MODELING AND COMPUTING OF DEFORMATION DYNAMICS OF INHOMOGENEOUS BIOLOGICAL SURFACES∗ M. MERCKER† , A. MARCINIAK-CZOCHRA‡ , T. RICHTER§ , AND D. HARTMANN¶ Abstract. Thin elastic surfaces frequently contain molecules influencing their mechanical properties. Such structures occur at different scales in biological systems. Prominent examples are bilayer membranes and cell tissues. We introduce a continuous dynamical model of deformation of lateral inhomogeneous surfaces, using the example of artificial membranes. In agreement with experimental observations, the membrane consists of different molecular species undergoing lateral phase separation and influencing the mechanical properties of the membrane. The model is based on minimization of free energy leading to a nonlinear PDE system of fourth order, related to the Willmore flow and the Cahn–Hilliard equation. The novelty of our work is in modeling and computing of time-dependent dynamics of the system, which requires local mass conservation. Parametric finite element simulations show how dynamics result in diverse equilibrium patterns. The presented simulations are in good agreement with recent theoretical and experimental observations. We investigate how each of the four elastic model parameters (line tension, spontaneous curvature, bending rigidity, and Gaussian rigidity) influences velocity and minimum shape of membrane budding. Key words. lipid bilayer, membrane, dynamics, continuous model, gradient flow, phase separation, budding AMS subject classifications. 74B20, 35Q92, 35B36, 74H15 DOI. 10.1137/120885553

1. Introduction. Elastic surfaces with large lateral dimensions but a relatively small thickness are one of the basic building blocks of a variety of structures in biology. One example is provided by cell membranes, which define mechanical boundaries of cells as well as of substructures inside cells. These structures usually have a size of up to a few micrometers, whereas the thickness of a membrane is a few nanometers. It has been proven that lateral inhomogeneity of molecules influencing mechanical properties of surfaces is essential for the genesis and maintenance of biological structures: In artificial membranes, such as liposomes, lateral phase separation of lipid molecules can lead to budding of vesicles [7]. It has been suggested that lateral organization is necessary for biogenesis and maintenance of cellular membrane systems [45]. Thin lateral homogeneous layers bend elastically and in the linear regime have been well described by the plate equation [14] as two-dimensional (2D) bent surfaces. This idea was further developed by Helfrich [28], describing the stable shape of a 2D surface by considering the minima of the classical bending energy ∗ Received by the editors July 23, 2012; accepted for publication (in revised form) May 24, 2013; published electronically September 4, 2013. We acknowledge ViroQuant Project (German FORSYS initiative) and WIN Project “Principles in the generation of the biological form” of the Heidelberg Academy of Sciences and Humanities. The research of the first and second authors was supported by the ERC Starting grant ‘Biostruct’ (210680) and the Emmy Noether Programme of DFG. http://www.siam.org/journals/siap/73-5/88555.html † BioQuant and Interdisciplinary Center of Scientific Computing (IWR), University of Heidelberg, INF 267, Heidelberg, Germany (mmercker [email protected]). ‡ Institute of Applied Mathematics, BioQuant and IWR, University of Heidelberg, INF 294, Heidelberg, Germany ([email protected]). § Institute of Applied Mathematics, University of Heidelberg, INF 294, Heidelberg, Germany ([email protected]). ¶ BioQuant and IWR, University of Heidelberg, INF 267, Heidelberg, Germany. Current address: Siemens AG, Corporate Technology, Munich, Germany ([email protected]).

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A DYNAMICAL MODEL OF INHOMOGENEOUS SURFACES

(1.1)

FHelfrich =

Γ

κ (H − H0 )2 ds + 2

1769

Γ

κG K ds,

without restrictions on the measures of curvature, such as the first metric tensor. Here, ds denotes the surface measure; H is the mean curvature and K the Gaussian curvature, both depending on the geometry of the surface Γ [17]. H0 , κ, κG are the mechanical moduli, which are constant if the surface is laterally homogeneous. H0 is the spontaneous curvature preferred in the relaxed state. Parameters κ and κG are the bending and Gaussian rigidity. In the case of κ = 1 and H0 = 0, the L2 gradient flow of (1.1) is known as Willmore flow [55]. Since most of the geometries in biological layers are intermediate between saddles, tubes, and spheres, both moduli may contribute to the energy penalty of curved surfaces. The Gauss–Bonnet theorem states that for closed surfaces Γ, the integral Γ K ds is constant, depending only on the topology of Γ [27]. Biological surfaces are typically inhomogeneous; for example, cell membranes are locally composed of many different molecules. We focus on modeling of artificially prepared vesicles (such as liposomes), which are composed of selected types of molecules. Membranes are built of different lipids, proteins, and other molecules with diverse functions [2]. In contrast to elastic bending, membrane molecules move freely in the lateral direction of the membrane, similarly to molecules in a 2D fluid as first described in the “fluid mosaic” model by Singer and Nicolson [50]. Lateral phase separation and clustering have been shown for lipids [7] and proteins [9] in living cells. The separation of an order parameter Φ is well described by a minimization of the Cahn–Hilliard energy [12] 2 ξ Γ 2 (∇ Φ) + f (Φ) ds, (1.2) FCahn–Hilliard = σ 2 Γ where σ is the (diffuse) line tension, ξ the transition length, ∇Γ the surface gradient, and f the double well potential. For already separated phases, the energy contribution of the lateral diffuse Cahn– Hilliard energy can be approximated by a sharp line energy acting at phase boundaries. Recent works show that corresponding sharp line tension models on curved surfaces emerge from the diffuse interface models as the thickness of the interface tends to zero [25]. However, considering the already separated phases is a strong restriction: The lateral sorting process determining the amount and location of interfaces is ignored. Even if the membrane geometry is fixed, it is a complex process [43, 47]. Moreover, as also shown in [43], these sorting processes influence the pattern selection process. Much effort has been invested in studying membrane behavior using lateral diffuse models based on minimization of free energy, consisting of different couplings of energy components (1.1) and (1.2). Some works used phenomenological coupling terms [3, 33, 52, 57], whereas other works derive coupling terms from first physical principles [25, 40, 53]. The techniques used to describe membrane surfaces range from graph and parametric representations [44, 52, 33, 24] of membranes to phase field descriptions [40, 53], where membranes of a finite thickness are embedded in fluid. We present a dynamic continuous parametric model of deforming inhomogeneous membranes consisting of two components. The mechanical model is coupled with a model of lateral phase separation of the components. The model is not restricted to small curvatures or symmetric geometries, and the coupling between the energies (1.1) and (1.2) is based on first physical principles. It takes into account the observation that different molecules vary in their shape and stiffness. In contrast to the recent works of Elliott and Stinner [24, 25], gradient flows considered in our paper

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MERCKER, MARCINIAK-CZOCHRA, RICHTER, AND HARTMANN

describe time-dependent dynamics and are not restricted to generating equilibrium configurations. That is, we consider lateral surface deformations advecting local mass combined with a local area constraint instead of a global constraint. Therefore, it is not sufficient to restrict variations of the energy F to the normal direction and to the order parameter as previously derived in [25]. Additionally, we have to take into account the tangential Fréchet derivatives of F . Steady state equations associated with our model describe a special case of the equations presented in [24, 25]. The explicit consideration of membrane dynamics is necessary to obtain biologically feasible equilibrium configurations. It appears that membrane model systems can be trapped in different local minimum patterns depending on the initial conditions (cf. our section 7.1 as well as [43]). Cellular membrane systems are arranged in a dynamic equilibrium rather than static configurations: Budding and fusion of membrane spheres occur with a flux maintaining the communication between subcellular membrane-bound systems [41]. Additionally, there exist chemical control mechanisms interacting with membranes during the deformation processes [39]. The dynamical model in this work provides a basis for describing those processes. To our knowledge, this is the first systematic derivation and computation of a three-dimensional (3D) parametric model of the dynamics of deforming lateral phase separating membranes with an incompressible lateral fluid-like behavior. Our study is restricted to biological membranes; nevertheless, the approach can be modified to describe tissues under the mechanical control of morphogens, e.g., following the idea of Cummings [16]. 2. Mathematical model. For any time t ∈ [0, T ), a bilayer is represented by a smoothly evolving family of smooth 2D hypersurfaces Γ(t) defined by a diffeomorphic : U ×[0, T ) → Γ(t) ⊂ R3 . In this work, we consider closed parametric representation X surfaces or surfaces with periodic boundary conditions. We consider a membrane composed of two different species such as two different lipids, or lipids and proteins. Concentration of two components A and B in Γ is described by the order parameter (phase field variable) Φ : Γ(t) → R, which is assumed to be a smooth function. If Φ ≈ 1, the membrane is locally composed of the species A, and if Φ ≈ −1, only the species B is present. See Figure 2.1. Hence, we consider the evolution (Γ(t), Φ(t))t≥0 with initial data Γ(0) = Γ0 , Φ(0) = Φ0 . Due to the parametric description we can 1 , u2 , t) with (u1 , u2 ) = identify material points on the evolving membrane by X(u u ∈ U . Thus, for each u ∈ U , we define the smooth function φ : U × [0, T ) → R u, t), t). This is motivated by our aim, which is to derive evolution by φ(u, t) = Φ(X( and φ in a Lagrangian setting, i.e., considering a parameter domain equations for X U = [0, 1]×[0, 1] and using derivatives with respect to u and t, e.g., ∂t , ∂u1 , ∂u2 . In the t) following, ∂t denotes a partial time derivative. If the material derivative of Φ(X, d is explicitly considered, we use the notation Φt = dt Φ(X(u, t), t) . Xt denotes the material velocity of particles. Square brackets are used for differential operators. Our model is based on minimization of the free energy containing the curvaturedependent part F1 + F2 and the Cahn–Hilliard energy F3 inducing lateral phase separation [12]: 1 F1 = κ(Φ)(H − H0 (Φ))2 ds, 2 Γ F2 = (2.1) κG (Φ)K ds, Γ 2 ξ Γ 2 F3 = σ (∇ Φ) + f (Φ) ds. 2 Γ


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Fig. 2.1. Continuous model of the bilayer membrane: The bilayer is represented by a 2D surface Γ and its lateral inhomogeneity by the order parameter Φ.

Since different components of the membrane may differ in their mechanical properties such as shape and stiffness, each macroscopic mechanical modulus h, h ∈ {κ, κG , H0 }, is a function of the order parameter Φ. Each function h is chosen so that h(1) = hA and h(−1) = hB , where hA and hB are the mechanical moduli of the pure components. Furthermore, σ is the (diffuse) line tension, ξ the transition length, ∇Γ the surface 9 gradient, and f = 32 (Φ2 − 1)2 the double well potential. The related sharp interface line tension value σsi is given by σsi = σξ [42]. Assuming local mass conservation, lateral dynamics are determined by the lateral continuity equation d [Φ] + ∇Γ · [j] = 0, dt where ∇Γ · is the surface divergence operator [17]. In the framework of the Lagrangian description used here, the material derivative and the partial derivative coincide, t). However, in the Eulerian approach it is necessary to t) = ∂t Φ(X, i.e., d/dtΦ(X, Γ evaluate Φt = ∇ [Φ] · Xt + ∂t [Φ]; i.e., the transport of Φ by the deforming surface would have to be considered. The flux j is determined by the lateral gradient of the chemical potential μ, with the dependence having the form j = ∇Γ [μ]. Here, following [5, 34, 52], we assume that μ is proportional to the variation of the free energy F with δ F . The mobility Lφ is assumed to be constant. It is respect to φ, and thus μ = Lφ δφ sufficient to consider variations with respect to φ, since we have adopted a Lagrangian framework; i.e., φ is defined in the parameter domain U and therefore independent of the actual configuration X. Hence, we obtain the following dynamical equation for φ on U : δ ∂t [φ] = Lφ ΔΓ F , δφ where ΔΓ is the Laplace–Beltrami operator [17]. For a fixed membrane, this equation provides conservation of the local mass of membrane components. Given a certain deformation, the membrane system evolves in the direction of the steepest descent of the free bilayer energy. Assuming overdamped motion due to dissipation of viscous forces (within not only the cell but also the cell membrane), which is a typical assumption for molecular systems, as well as lateral incompressibility [18], the dynamics in U × [0, T ) are given by the following gradient flow: of the deformation X

δ √ 2 (2.2) γ gd u , Xt = −LX F+ δX U √ with a local area constraint ∂t [ g] = 0 due to lateral incompressibility of the lipid bilayer, where LX is the kinetic coefficient, δδX [F ] denotes the variation of F with

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including normal as well as tangential components, γ is the local Larespect to X, √ grange multiplier modeling local membrane incompressibility, and g is the surface measure (cf. Appendix A). For details concerning the last equation and the realization of local incompressibility in the parametric form, we refer the reader to [26, 52]. The assumption of an incompressibility constraint and an overdamped motion is a significant simplification. Direct modeling of the fluid inside the vesicle and the fluidlike behavior of the membrane goes beyond the scope of this work. Comparisons with microscopic models resolving both aspects [43] underline the validity of our approach. X, t) ∈ R3 be the force acting on the membrane 3. Technical remarks. Let A( at X at time t. We assume that the force is determined by the variation F in the form of the following L2 -gradient flow: d A · ψ ds = −F , ψ = − F (X + εψ) (3.1) , dε Γ ε=0 ∈ C ∞ (Γ, R3 ) is an arbitrary test function. We consider a representation of ψ where ψ 0 ⊥ 1 2 in a local coordinate system, i.e., ψ = ψ n + ψ t1 + ψ t2 , where n is the local normal of the membrane and t1 , t2 are two nonparallel tangents (not necessarily normalized (i = 1, 2), we choose the following decomposition or orthogonal). Using ti = ∂i X

⊥ k ⊥ = ψ n + It holds that ψ n − u Au ∂u X. k ψ ∂k X and A = −A ds = − A⊥ ψ ds − ·ψ · ∂k Xψ k ds A Au ∂u X Γ

as well as

Γ

u,k

Γ

k F , ψ = F , ψn + F , ψ ∂k X

k d d k +ε + εψn) = F (X F X + ψ ∂k X dε dε ε=0 k

. ε=0

In the remainder of this paper, δ ⊥ and δ k denote the variations in the normal direction and the kth tangential

direction (k = 1, 2) with respect to Γ. Alternatively, we use the notation δ t = k δ k for tangential coordinates. 4. Statement of the main result. We assume vanishing boundary integrals, i.e., closed surfaces or periodic boundary conditions, and assume that the mechanical moduli and the function f (φ) are arbitrarily regular functions, i.e., κ, κG , H0 , f ∈ C ∞ ([−1, 1]). Theorem 4.1 (dynamic membrane equations). Parametric dynamic deformation of the lateral phase separating and incompressible two-component membrane is given by the equations

δ √ 2 ⊥ k (4.1) γ g d u = −LX A n + A ∂k X F+ Xt = −LX δX U k with the constraint (cf. section 2) (4.2)

∂t

√ g = 0,


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where Γ κG (φ) + Hγ − κ(φ)(H − H0 (φ))(H 2 − 2K) A⊥ = −ΔΓ κ(φ)(H − H0 (φ)) − Δ 2 2 ξ κ(φ) 2 2 ij Γ (H − H0 (φ)) H − ξ (∇ φ ) + f (φ) b ∂i φ ∂j φ + H + 2 2 i,j and 1 Ak = − ∂ k κ(φ) (H − H0 (φ))2 − ∂ k κG (φ) K − ∂ k γ 2

2 ξ +κ(φ)(H − H0 (φ))∂ k H0 (φ) + ξ 2 (∇Γ φ)2 + f (φ) . ∇l ∂ k φ∂ l φ − ∂ k 2 l

Furthermore, lateral dynamics of the order parameter φ are given by

δ 1 ∂t [φ] = Lφ ΔΓ F = Lφ ΔΓ κ (φ)(H − H0 (φ))2 (4.3) δφ 2

−κ(φ)(H − H0 (φ))H0 (φ) + κG (φ)K − ξ 2 ΔΓ φ + f (φ) .

The proof of the theorem is deferred to the following section, where each term of the Fréchet derivative of the energies is presented. Furthermore, in Appendix E it is shown that the free energy (2.1) is a Lyapunov functional for solutions of the evolutionary system (4.1)–(4.3). For notational convenience, we consider the case σ = 1. 5. Proof of the main result. We formulate energy (2.1) in terms of the parametric representation and calculate corresponding variations to derive the parametric and φ on U . To prove the main result, in the following we dynamic equations for X rely on variational expressions derived in Appendix B and on simple reformulations of geometric and tensor calculus results (cf. Appendix C). The first normal variation δ ⊥ of F and the variation with respect to the order parameter, δ φ , may be found in [25]. Since we consider time-dependent dynamics, tangential surface deformations have to be considered additionally. For derivations of variations with respect to normal deformations, δ ⊥ [F ], or with respect to the order parameter, δ φ [F ], we refer the reader to the original works cited below. Our proofs concentrate on tangential variations δ t , which so far have not been considered in the literature. √ Lemma 5.1 (normal variation of F + U γ g d2 u). δ ⊥ F + Γ γ ds = −ΔΓ κ(φ)(H − H0 (φ)) − κ(φ)(H − H0 (φ))(H 2 − 2K) δX κ(φ) Γ κG (φ) + Hγ + (H − H0 (φ))2 H − Δ 2 2 2 ξ 2 ij Γ −ξ (∇ φ ) + f (φ) . b ∂i φ ∂j φ + H 2 i,j Proofs of this lemma (or of its parts) are found in works using direct calculations of variations [1, 19, 26, 56, 59] or surface calculus [6, 17, 25]. They use product rule and Γ . The second Green’s identities for the first and the second surface gradient, ∇Γ and ∇

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surface gradient initially appeared in [46] and is based on the conjugate fundamental tensor, showing symmetric properties with respect to ∇Γ [58]. In contrast, the method of surface calculus is based on [20], using a variational curvature identity which is the = Hn. weak formulation of ΔΓ [X] Lemma 5.2 (tangential variation of F1 ). δ k F1 1 = − ∂ k κ(φ) (H − H0 (φ))2 + κ(φ)(H − H0 (φ))∂ k H0 (φ) . 2 δX Proof. Using the product rule we obtain

t 1 2√ 2 δ κ(φ)(H − H0 (φ)) g d u 2 √ √ 1 1 = κ(φ)δ t (H − H0 (φ))2 g d2 u + κ(φ)(H − H0 (φ))2 δ t g d2 u. 2 2 Lemmas B.4 and B.5 provide

√ 1 √ g ij ∇k bij ψ k g d2 u δt κ(φ)(H − H0 (φ))2 g d2 u = κ(φ)(H − H0 (φ)) 2 i,j,k √ 1 ∂ u κ(φ) (H − H0 (φ))2 guk ψ k g d2 u − 2 k,u √ κ(φ)(H − H0 (φ))∂ u H guk ψ k g d2 u − k,u

+

√ κ(φ)(H − H0 (φ))∂ u H0 (φ) guk ψ k g d2 u.

u,k

Due to Lemmas C.1 and C.2, and since the covariant derivatives and the first metric tensor commute, it follows that ij ij g ij ∇k bij = g ∇j bik = ∇j g bik i,j

i,j

=

∇j bjk

j

i

= ∂k H = guk ∂ u H . u

j

Since the first and third terms now cancel each other, we obtain the assertion. Lemma 5.3 (tangential variation of F2 ). δ k [F2 ] = −∂ k κG (φ) K. δX Proof. It holds that

√ 2 √ √ 2 t t ∂ κG (φ)K g d u = κG (φ)∂ K g d u + κG (φ)K∂ t g d2 u. Using Lemmas B.4 and B.6 and the product rule, we obtain

√ 2 √ t κG (φ)bi j ∇k bji Kψ k g d2 u ∂ κG (φ)K g d u = −

u,k

i,j,k

√ ∂ u κG (φ) Kguk ψ k g d2 u − k

√ κG (φ)∂k K ψ k g d2 u.


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Using the definition of the covariant derivative and the chain rule yields

√ 2 √ t l i i l i i ∂ b j bj Γkl − b j bl Γkj ψ k K g d2 u κG (φ)K g d u = κG (φ) −

i,j,k,l

i,j,k,l

u

√ ∂ κG (φ) Kguk ψ k g d2 u.

u,k

Since

l bi

lb j l

= δij , where δij is the Kronecker symbol, it follows that

√ 2 √ t ∂ ∂ u κG (φ) Kguk ψ k g d2 u, κG (φ)K g d u = − u,k

which is the claim. Our results are consistent with the Gauss–Bonnet theorem, since variations of F2 vanish in the case of a constant Gaussian rigidity κG . Lemma 5.4 (tangential variation of F3 ).

2 k δ k [F3 ] 2 u k ξ Γ 2 (∇ [φ]) + f (φ) . ∇u ∂ [φ]∂ [φ] − ∂ =ξ 2 δX u Proof. Using the chain rule, it follows that

2 ξ √ 2 (∇Γ φ )2 + f (φ) gd u δt 2 (5.1) 2 2 √ ξ ξ t Γ 2 √ 2 = δ (∇ φ ) (∇Γ φ )2 + f (φ) δ t g d2 u. gd u+ 2 2 Considering the first term of equation (5.1) results in 2 √ ξ t Γ 2 √ 2 ξ2 δ (∇ φ ) δt gd u = g ij ∂i φ ∂j [φ] g d2 u. 2 2 i,j From [37] it follows that δ t g ij = −∇i ψ j − ∇j ψ i . Thus, we obtain

√ ξ 2 t Γ 2 √ 2 2 δ (∇ φ ) g d u = −ξ g iu ∇u ψ j ∂i φ ∂j φ g d2 u. 2 i,j,u

Lemma C.4 yields 2 ξ t Γ 2 √ 2 · ∂u ∂k Xψ k √g d2 u. δ (∇ φ ) g d u = −ξ 2 ∂ u φ ∂ b φ ∂b X 2 u,b,k

Applying Green’s formula results in 2 ξ t Γ 2 √ 2 δ (∇ φ ) (5.2) gd u 2 · ∂k X∂ u [φ]√gψ k + ∂u √g∂ u [φ] gbk ∂ b [φ]ψ k d2 u. ∂u ∂ b [φ]∂b X = ξ2 u,b,k

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Applying Lemma C.4 to the first term on the right-hand side of (5.2), we obtain b √ 2 u k√ 2 2 ξ ∂u ∂ [φ]∂b X · ∂k X∂ [φ]ψ g d u = ξ ∇u ∂ b φ ∂ u [φ]gbk ψ k g d2 u. u,b,k

u,b,k

For the second term of (5.2) it holds that √ ξ2 ∂u g∂ u [φ] gbk ∂ b [φ]ψ k d2 u u,b,k

= ξ2

√ √ ∂u ∂ u [φ] g + ∂u g ∂ u [φ] ∂ b [φ]gbk ψ k d2 u. u,b,k

Applying the chain rule to the determinant results in b √ 2 u k√ 2 2 ξ ∂u ∂ [φ]∂b X · ∂k X∂ [φ]ψ g d u = ξ ∂u ∂ u φ ∂ b [φ]gbk ψ k g d2 u u,b,k

+ ξ2

u,b,k

· ∂j X∂ b [φ]gbk ψ k √g d2 u. g ij ∂u ∂i X

u,b,k,i,j

Using the alternative definition of the Christoffel symbol (see the proof of Lemma C.4), we obtain · ∂k X∂ u [φ]ψ k √g d2 u ξ2 ∂u ∂ b [φ]∂b X u,b,k

= ξ2

b

· ∂a ∂k X = Γz g zb ∂b X ak

√ √ ∂u ∂ u φ ∂ b [φ]gbk ψ k g d2 u + ξ 2 Γiui ∂ u [φ]∂ b [φ]gbk ψ k g d2 u.

u,b,k

u,b,k,i

Transposition of the indices u ↔ i, equality Γiui = Γiiu , and the definition of the covariant derivative lead to √ · ∂k X∂ u [φ]ψ k √g d2 u = ξ 2 ξ2 ∂u ∂ b [φ]∂b X ∇u ∂ u φ ∂ b [φ]gbk ψ k g d2 u. u,b,k

u,b,k

Reformulating the terms of (5.2), we obtain that 2 √ ξ t Γ 2 √ 2 ∂ ∇ [φ] g d u = ξ2 ∇u ∂ b [φ] ∂ u [φ]ψ k gbk g d2 u 2 u,b,k

(5.3) + ξ2

√ ∇u ∂ u [φ] ∂ b [φ]gbk ψ k g d2 u.

u,b,k

Substituting (5.3) in (5.1) and using the chain rule for covariant derivatives yield

2 ξ √ 2 √ (∇Γ φ )2 + f (φ) δt g d u = ξ2 ∇u ∂ b [φ]∂ u [φ] gbk ψ k g d2 u 2 u,b,k

2 ξ √ − (∇Γ φ )2 + f (φ) guk ψ k g d2 u. ∂u 2 u,k


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Lemma 5.5 (tangential variations of the Lagrange multiplier γ). √ δ k U γ g d2 u = −∂ k [γ]. δX This proof follows directly from Lemma B.5. Lemma 5.6 (variation of F with respect to the order parameter φ). δ F 1 = κ (φ)(H − H0 (φ))2 + κ(φ)(H − H0 (φ))H0 (φ) δφ 2 + κG (φ)K − ξ 2 ΔΓ φ + f (φ). This proof has been already presented in [25], using the calculus on surfaces. 6. Parametric finite element approximation. Surface finite element approaches have been applied to study lateral homogeneous membranes in [6, 17, 20, 48]. Recent works have adopted these approaches, considering phase separation on curved lateral inhomogeneous membranes. However, these studies either have been restricted to stationary geometries [21] or have not considered the local surface mass conserva of the tion [24]. Instead of surface finite elements, we choose a parametrization X 2 surface Γ over R with periodic boundary conditions and define all finite element spaces on a quadrangulation of the plane domain U . This parametric approach cannot represent closed surfaces; however, it simplifies the implementation, since the use of surface finite elements is avoided. The novelty of our numerical approach is in numerical simulations of a model of dynamics of lateral phase separating evolving membranes, maintaining effective local mass conservation (cf. section 6.5). Relying on a parametric model, a Lagrangian simulation approach is naturally chosen. In such an approach vertices of discretization follow physical particle paths of lipid molecules. Since physical motion of lipids is incompressible, adopting a Lagrangian framework ensures that the area of discretization elements is conserved. However, arbitrarily strong shearing of discretization elements is possible. This inevitably leads to difficulties in the underlying computational schemes. Thus, we adopt a different strategy: After each Lagrangian time step φm → φ˜m+1 of the physical model (section 6.4), we correct the grid with a nonphysical step φ˜m+1 → φm+1 (section 6.5 following ideas of [6]). Since induced motion of the correction step is tangential, the shape of the membrane and the global mass are conserved. The correction step might influence only the lateral distribution of the order parameter φ, i.e., the distribution of lipids. Using an additional transport equation, we ensure that the distribution of φ on the membrane in the laboratory coordinate system (Eulerian coordinate system) is unchanged. Thus, the scheme follows the concept of the classical arbitrary Lagrangian Eulerian (ALE) schemes [29]; i.e., vertices of the discretization grid do not follow lipid traces, but correction ensures realistic physical densities. 6.1. Mixed formulation: Introducing additional variables. Using the finite element library Gascoigne [8], we discretize the fourth order PDE model in a mixed formulation [11]. Variational formulations using H 1 spaces are approximated with H 1 conforming bilinear surface finite elements on quadrangulated surfaces using geometric identities [48] (6.1)

= Hn, ΔΓ [X]

(6.2)

|∇Γn|2 = H 2 − 2K,

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as well as introducing several additional unknown functions: the mean curvature H, the Gaussian curvature K [6], and also the chemical potential Y [23]. We consider Y = (6.3)

1 κ (φ)(H − H0 (φ))2 − κ(φ)(H − H0 (φ))H0 (φ) 2 + κG (φ)K − σ(ξ 2 ΔΓ φ − f (φ)).

The additional unknown function G, the Cahn–Hilliard energy density, is given by (6.4)

G=

ξ2 Γ (∇ [φ])2 + f (φ). 2

6.2. Analytical reformulation. The derivation of a weak formulation of the system (4.1)–(4.3) (cf. section 4) relies mainly on the chain rule and on integration by parts as well as on the identities presented below. The latter are based on reformulations of the parametric representation. (For details of the following proofs, we refer the reader to the Appendix D.) Using the definition of the covariant derivative yields u u ∇u ∂k [φ]∂ [φ] ψ ds = − ∂k [φ]∂ [φ]∂u [ψ] ds − Γluk ∂l [φ]∂ u [φ]ψ ds. U

U

l

U

Furthermore, for h ∈ {H0 , κ, κG } it holds that ∇Γ [h(φ)] =

i,j

= g ij ∂j [h(φ)]∂i X

= h (φ)∇Γ [φ] g ij h (φ)∂j [φ]∂i X

and

i,j

∂k [h(φ)] = h (φ)∂k [φ], using the chain rule. Thus, using Green’s formula (assuming vanishing boundary conditions, i.e., closed surfaces or periodic boundary conditions) it follows that Γ Δ [κ(φ)(H − H0 (φ))]ψ ds = − (H − H0 (φ))κ (φ)∇Γ [φ] · ∇Γ [ψ] ds U U Γ Γ − κ(φ)∇ [H] · ∇ [ψ] ds + κ(φ)H0 (φ)∇Γ [φ] · ∇Γ [ψ] ds U

U

as well as 1 − ∂k [κ(φ)](H − H0 (φ))2 + κ(φ)(H − H0 (φ))∂k [H0 (φ)] 2 1 = ∂k [φ](H − H0 (φ)) − κ (φ)(H − H0 (φ)) + κ(φ)H0 (φ) . 2 6.3. Finite element discretization. Our approach follows the approach of [6], but it is based on parametric finite elements in R2 on U = (0, 1) × (0, 1) with periodic boundary conditions instead of surface finite elements on the surfaces Γ itself. Let us assume that 0 = t0 < t1 < · · · < tM−1 < tM = T is a discretization of the time interval [0, T ] into time steps τm := tm+1 − tm , which are not necessarily equidistant. Further, let us assume that U m is a conforming quadrangulation of U = (0, 1) × (0, 1) σjm }Jj=1 . On U m , we denote by at time t = tm into open quadrangles U m = {ˆ Vˆ (U m ) := {ψˆ ∈ C(U m ; R), ψˆ|ˆσjm is bilinear for j = 1, . . . , J} ⊂ H 1 (U m , R)


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the usual finite element space of bilinear functions. Likewise, we introduce by Vˆ (U m ) ⊂ H 1 (U m , Rd ) the vector-valued finite element space. The approximation Γm of the de m ∈ Vˆ (U m ). formed surface Γ at time tm is given by means of the parametrization X m : σ Then, Γm = {σjm }Jj=1 is a quadrangulation into (curved) quadrangles X ˆjm → σjm . m m m m m By qk = X(u k ) we denote the vertices of Γ , where uk are the vertices of U . By m m m m m |σj | we denote the area of σj ∈ Γ . By qk1,j and qk2,j we denote the nearest m m qkm , qk1,j , qk2,j ∈ σm neighbor vertices of qkm in σ m j , i.e., j . Furthermore, let us define m m m hki,j := |uk − uki,j |. Since we work on U rather than on Γ we pursue a purely La m be the grangian approach; i.e., motion of lipids is directly traced by the grid. Let N j m unit normal on each quadrangle σj . We introduce the discrete vertex normals by ν m ( qkm ) =

(6.5)

and |αm k | :=

nm k , |nm k |

nm k :=

where

1 |αm k |

jm |σjm |N

m =∅ σjm ∩qk

|σjm | represents the total measure of the quadrangles coter uk , tm )]/|∂i [X( uk , tm )]| so that minous to qkm . Furthermore, we define tm i := ∂i [X( m m m 3 {ν , t1 , t2 } form a basis of R . m m k (u ) Next, we introduce the following finite difference quotient δi,j := X ki,j m m m k k is −X (uk ) /hki,j , where the index i and the sign of δi,j are chosen such that δi,j the differential quotient corresponding to ∂i [X(uk , tm )]. Using this finite difference quotient, the discrete vertex components of gxy can be computed as follows: m =∅ σjm ∩qk

m := g(x,y)k

(6.6)

1 |αm k |

k k |σjm |(δx,j · δy,j ).

m =∅ σjm ∩qk

In contrast to the approach of [6], which does not consider the Cahn–Hilliard part, we require a discrete approximation of the first fundamental tensor to approximate the resulting covariant derivatives. Discrete vertex Christoffel symbols are defined by m m 1 (i,l)m m i(m) m Γjz ( ∂z g(j,l)k + ∂j g(l,z)k (6.7) − ∂l g(j,z)k , qk ) := gk 2 l

(i,l)m

where gk

are the components of the inverse of the matrix gm , defined as m ( m (qkm )])i,l . gm = (∂i [X qkm )] · ∂l [X

To examine the quality of approximations (6.5)–(6.6), we have discretized a highly curved and nonsymmetric surface and compared approximations of the vertex norm )i,j using different spatial mals ν m and the first fundamental tensor gm := (g(i,j)k discretizations. The consistency check has shown the expected order O(h2 ), where h := maxj=1→J diam(σjm ). For scalar-valued, vector-valued, and matrix-valued func tions f, g ∈ L2 we introduce the L2 inner product ., . over U as f, g := U (f · g) ds. φ, 6.4. Discretized dynamical equations. We use the “natural functions” X, and γ, the auxiliary functions H, K, G, Y , the reformulations presented in section 6.2, and the discrete approximations (6.5)–(6.7). Additionally, we use functions B1 and B2 which appear in the weak approximation of (6.1) and play a role in tangential = grid control [6]. Splitting the test functions into normal and tangential parts ψ

k ψn + i ∂i Xψ for tangential movements Xt = −A ∂k X yields the weak equivalent

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t , ∂k Xψ = − i Ai gik , ψ , k = 1, 2. Hence, superscripts appearing in tangential X forces Ak of (4.1) are dropped in the weak formulation. The discrete approximation of (4.1)–(4.3) in its weak form containing only the first order derivatives reads as follows. m+1 , φ˜m+1 , γ m+1 , Discretization 1 (dynamic membrane equations). Find {X m+1 m+1 ˜ m+1 ˜ m+1 m+1 m+1 m 11 ˜ H ,K , B1 , B2 , G ,Y } ∈ V (U ) such that the following hold: (i) discrete approximation of (4.1), m+1 m ψν m X −X (6.8) , = −∇Γ [φm ], ∇Γ [ψ]κ (φm )(H m − H0 (φm )) τm LX m Γ ˜ m+1 Γ m m Γ m Γ −κ(φ )∇ [H ], ∇ [ψ] + κ(φ )H0 (φ )∇ [φ ], ∇ [ψ] κ(φm ) m m m Γ m 2 m 2 ˜ m+1 (H − H0 (φ ) H ,ψ +(H − H0 (φ ))|∇ [ν ]|m , ψ − 2 Γ [φm ], ∇Γ [ψ] + −κG (φm )∇ σξ 2 bij ∂i [φm ]∂j [φm ], ψ i,j

−H (G m

m+1

+γ

m+1

), ψ

∀ψ ∈ V (U m ),

(6.9) m+1 m ψ∂k [X m] X −X , τm LX κ (φm ) m m m m m m m = − ∂k [φ ](H − H0 (φ )) − (H − H0 (φ )) + κ(φ )H0 (φ ) , ψ 2 +κG (φm )K m+1 ∂k [φm ], ψ + σ ξ 2 ∂k [φm ]∂ u [φm ], ∂u [ψ] +σ

u

ξ 2 Γuk ∂l [φm ]∂ u [φm ], ψ + ∂k [Gm+1 + γ m+1 ], ψ l(m)

∀ψ ∈ V (U m ), k = 1, 2;

l,u

√ (ii) approximation of the lateral incompressibility equation (4.1) (∂t [ g] = ∇Γ · t ]) [26], [X (6.10)

m+1 m] ∇Γ · [X ] − ∇Γ · [X ,ψ = 0 τm

∀ψ ∈ V (U m );

(iii) discrete approximation of (6.1) (cf. [6]), m+1 m+1m m+1 m Γ Γ ˜ ˜ Bi (6.11) H ν + ti , ψ + ∇ X , ∇ [ψ] = 0

∈ V (U m ); ∀ψ

i

(iv) discrete approximation of (6.2), (6.12)

K m+1 , ψ =

1 2 Γ m tr (∇ ν ) − |∇Γ ν m |2m , ψ 2

(v) discrete approximation of (6.4), 2 ξ m+1 Γ m 2 m |∇ [φ ]|m + f (φ ), ψ (6.13) , ψ = σ G 2

∀ψ ∈ V (U m );

∀ψ ∈ V (U m );


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(vi) discrete approximations of the evolution of the components (4.3), (6.14)

˜m+1 φ − φm , ψ = −Lφ ∇Γ [Y m+1 ], ∇Γ [ψ] τm

∀ψ ∈ V (U m ),

and (6.3),

(6.15)

κ (φm ) m (H − H0 (φm ))2 , ψ 2 −κ(φm )(H m − H0 (φm ))H0 (φm ) + κG (φm )K m + σf (φm ), ψ + σξ 2 ∇Γ [φ˜m+1 ], ∇Γ [ψ] ∀ψ ∈ V (U m ).

Y m+1 , ψ =

m+1

˜ m+1 , B ˜ m+1 are intermediate and subject The solution components X , φ˜m+1 , H i to change by the second grid control step. The surface measure ds and the surface Γ depend on the detailed configuration of membrane X, which is gradients ∇Γ and ∇ unknown. Thus, gradients and other operators are functionals of the current deformation, leading to a highly nonlinear system. Here, we chose a dependence on the Γm . In this way, we m and hence use dsm , ∇Γm , and ∇ previous time step, i.e., on X avoid true nonlinear formulations to facilitate numerical analysis by linearizing the system. Additionally, we add nonphysical correction terms to ensure convergence to physically realistic solutions: Accumulation of numerical errors violating incompressibility of the membrane and, hence, mass conservation of φ˜m+1 , is globally corrected via an artificial global pressure. Furthermore, we correct the grid as outlined in the next section. We verify that the total area and mass at the min in all simulations ds and φ ds, differ from the initial values imum configurations, min min Γ Γ Γ0 ds0 and Γ0 φ ds0 by less than 2.5%. Nevertheless, simulations without global corrections still reveal good properties with respect to global mass and local area conservation. 6.5. Tangential grid control and effective local mass conservation. Numerical stability of the finite element scheme depends on a consistent and conforming distribution of vertices on the deformed surface Γm , which depends on the initial parametrization. Considering typical setups, undesirable tangential mesh distortions are observed in parametric membrane modeling. Although the area of discretization elements is conserved in the underlying model, arbitrary high shearing of discretization elements cannot be prevented due to the fluid character of the lipid bilayer. Different techniques have been developed to prevent such detrimental effects [10, 6, 22, 24]. In order to control the tangential motion of Γm , we follow the ideas of [6] by introducing a separate “mesh redistribution step” after each time step. To be more precise, it is not the mesh on the parametrizing surface U that is redistributed, but m : U m → Γm is the transformed mesh on the surface Γm ; i.e., the parametrization X adapted. Since the correction step involves only tangential corrections or movement on the surface, the surface does not evolve up to discretization errors (in the normal direction) and the curvatures and the overall area are not influenced by this correction step. However, local mass conservation is violated due to the induced transport or convection of the order parameter Φ. Since we adopt the Lagrangian point of view, d 1 , u2 , t), t] = ∂t [φ(u1 , u2 , t)] + X t · ∇Γ [φ] with X t = 0; i.e., φ is advected Φ[X(u Φt = dt with membrane during the correction movement. To compensate for this convection induced by the grid correction, we use the so-called ALE approaches [29]: During the correction step, spatial distribution of φ on evolving surface Γ(t) is advected or

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transported

kin the opposite direction at the rate −Xt . During the correction step, Xt = k A ∂k X. Using the definition of the first surface gradient, it follows that t · ∇Γ [φ] = Ak ∂k [φ]. The full grid control problem reads as X k m+1 , H m+1 , B m+1 , B m+1 , Discretization 2 (grid correction). Find {X 1 2 m+1 m 7 } ∈ V (U ) such that φ m+1 m+1 X −X m , ψν =0 τm

(6.16)

∀ψ ∈ V (U m )

holds and, for k = 1, 2, that the following holds:

(6.17)

m+1 m+1 X −X m m+1 , ψtm + cm = αm k k (δk Bk k ), ψ τm

(6.18)

∀ψ ∈ V (U m ),

φm+1 − φ˜m+1 m+1 ,ψ = − (δkm Bkm+1 + cm )∂ [φ ], ψ k k τm

∀ψ ∈ V (U m ),

k

m+1 m m m+1m =0 m+1 ], ∇Γ [ψ] ti , ψ + ∇Γ [X (6.19) ν + αi Bi H

∈ V (U m ). ∀ψ

i m m Here, 0 ≤ αm i , δi ∈ V (U ) are the coefficients influencing the strength of the ∈ V (U m ) are the forcing terms, determining the direction tangential correction and cm i of the tangential correction. A feasible choice for cm i is the tangential projection of the vector from each vertex qkm to the average zkm of its neighboring nodes, i.e., m ] [6]. Note that with the choice αm ≡ 0, (6.16)–(6.19) cm qkm ) ≡ τ1m (zkm − qkm ) · ∂i [X i ( i reduce to a system without any normal and tangential movement.

7. Simulations. Using the macroscopic modeling approach outlined above, we investigate the dynamics and minimal configurations of lateral sorting and deformation of membranes. We approximate numerically the fourth order system by discretized equations (6.8)–(6.15). Since, without lateral grid control, strong grid distortions during the evolution are observed, the grid is corrected after each physical time step. If not otherwise stated, we use stochastically perturbed initial conditions. The corresponding values at each discretization point are chosen using the standard random generator provided by C++. φ(t = 0) is uniformly distributed in an interval [b, c] with the average φ(t = 0) = Φ0 ≈ (b + c)/2. Simulations have been repeated three times to ensure that neither the dynamics nor the resulting shapes depend on stochastic perturbations. h(φ) ≡ hlin is chosen as a linear interpolation between the two values h(−1) = hA and h(+1) = hB ; unless otherwise stated, h ∈ {κ, κG , H0 }. 9 Furthermore, we use the double well potential f (φ) = 32 (1 − φ2 )2 , so that the sharp interface line tension value σsi is given by σsi = σξ [42]. All boundary conditions are periodic such that the mass and the order parameter are conserved. Simulations are stopped if a local minimum has been reached, i.e., the corresponding finite dif m , φm ])/τ m+1 is sufficiently small. We have m+1 , φm+1 ] − F [X ference quotient (F [X confirmed numerically the mass and area conservation as well as an experimental order of convergence of experimental order of convergence (EOC) of approx. 2 while refining the mesh.


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Fig. 7.1. Simulation of membrane dynamics (A)–(C) reveals budded minimum structures (C) starting from disordered initial condition (A). (D): Corresponding energy decay. In (A)–(C) the two phases correspond to locally high concentrations of species A and B, respectively.

Fig. 7.2. Lateral sorting depending on the functional relationship κ(φ). (A): Stochastic initial conditions. In (B)–(C) a linear function κ(φ) is chosen, contrary to (E)–(F), with κ(φ) having a tanh-form. (D): Corresponding decays of free energy (F) from unstable initial conditions. Different functions κ(φ) cause differences in early sorting (cf. circular marks), leading to different patterns.

7.1. Simulation of a membrane patch. To qualitatively compare the model with experimental data, we simulate membrane patches from slightly curved membranes and stochastically perturbed initial conditions uniformly distributed in [−0.57, −0.17] with the average Φ0 ≈ −0.37 (cf. Figure 7.1 A). We assume κ ≡ 1.5, κG ≡ −1.0, H0A = −10.0, H0B = 5.0, σ = 450.0, ξ = 0.03, LX = 0.0005, and Lφ = 0.05. Simulations show the transition from heterogeneous initial conditions to a single domain of one component with budded geometry, which is the minimal configuration (cf. Figure 7.1 A–C). The resulting patterns are visually comparable to those of stable structures observed in the experiments with real membranes [7]. Plotting the energy (2.1) over the simulation period reveals its decay in time (cf. Figure 7.1 D). 7.2. Influence of dynamics on minimum patterns. To investigate the dependence of dynamics and minimum patterns on the choice of the unknown function κ(φ), we perform simulations, starting again with a slightly curved membrane and stochastically perturbed and equally distributed initial conditions in the interval [−0.2, 0.2] with the average Φ0 ≈ 0 (see Figure 7.2 A). To ensure the comparability of different simulations, we use the same initial conditions. Corresponding results are

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presented in Figure 7.2. We compare the impact of the two different monotonous (2) functions κ(1) (φ) = a1 + b1 φ and κ0 (φ) = a2 + b2 tanh(−φ) on the dynamics and minimum patterns of lateral sorting. Here, ai and bi are chosen so that κ(i) (−1) = 3 and κ(i) (1) = 1 hold for i ∈ {1, 2}. Furthermore we set κG (φ) ≡ −1, H0 ≡ 0, σ = 450, ξ = 0.03, LX = 0.0005, and Lφ = 0.05. Linear approximations κ(1) (φ) have been frequently used in the past, whereas κ(2) (φ) has been recently determined from molecular dynamic studies [43]. We find that different choices of κ(φ) strongly influence the dynamics of the model as well as the minimum patterns. Depending on the choice of κ(i) , free energy decays at different times and at different rates from the unstable initial conditions (see Figure 7.2 D), resulting in two different equilibrium patterns (cf. Figure 7.2 C and F). This may be caused by the fact that differences d κ lead to differences in early sorting, influencing the development of the stable in dφ cross-connection between the phases (circular marks in Figure 7.2 B and E), and finally leading to different minimum patterns. Similar observations have been made in [43], investigating lateral sorting on fixed membranes. Thus, an explicit consideration of dynamics is essential to determine biologically relevant configurations. 7.3. Influence of elastic parameters on membrane budding. A biological cell is composed of a multitude of membrane systems. Each of the systems has a specific shape and composition of chemical components having highly specialized functions. Although most of the systems are physically separated, they are connected to each other through continued flow of membrane parts. This happens via small membrane spheres, called vesicles, constricting from a donor membrane in a process called budding: Budding plays a key role in healthy and pathogenic cellular processes such as sorting, transport, biogenesis, and infection [45, 31, 54]. Although details remain unknown, it is clear that mechanical properties and interactions of membrane molecules are important in such processes. We perform simulations to investigate the influence of different elastic parameters on dynamics and equilibrium configurations of membrane budding. In each experiment, we vary only one parameter h ∈ {σ, H0A , κA , κA G } related to the budded phase. Corresponding results are qualitatively and quantitatively compared to recent experimental and theoretical findings. In order to focus on membrane budding, we use the initial configuration supporting the development of a single bud in the middle of the 0 = (0, 0, a1 (1 + cos(a2 π))) and φ0 = −1 + a3 cos(a2 π), where membrane patch: X a1 , a2 , a3 are such that U ds = 1.3 and U φ ds = −0.85 (cf. Figure 7.3 A). In the case of a complete budding (e.g., Figure 7.3 C), the energy minimizing shape contains an infinitesimally narrow neck and, thus, it is beyond discrete approximations. Hence, we stop simulations just before the numerically computed solutions “blow up” due to geometric singularities, knowing that we are close to the energy minimum. Local expansion of the grid due to evolution of the membrane is avoided using local mesh refinement starting with J = 1024 vertices. If not otherwise stated, in the following we B set: κA = κB = 1.5, κA G = κG = −1 (ensuring the stability restriction 0 ≥ κG ≥ −2κ [49]), σ = 90, ξ = 0.03, LX = 5 · 10−4 , and Lφ = 0.05. Furthermore, for i ∈ {1, 2} we m choose αm i ≡ βi ≡ 0.1. Our simulations show that each of the four elastic parameters can induce or prevent membrane budding: For example, higher values of the line tension result in a more budded equilibrium shape (cf. Figure 7.4 A–B). After a critical value σ ∗ , we observe a qualitative change in the minimum configuration: The shape of an incomplete bud (cf. Figure 7.4 B) changes abruptly to that of a complete bud (cf. Figure 7.4 C). This observation fits well in the theoretical studies of [35, 36, 38]


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Fig. 7.3. Simulation snapshots of membrane budding. (A): Initial conditions at t = 0. (B): t = 0.2. (C): Budded configuration at t = 3.1, close to equilibrium.

Fig. 7.4. Choosing increasing values for the line tension σ results in a stronger budded geometry (A): σ = 90, (B): σ = 540, and finally in a complete bud (C): σ = 630. We observe the same effect by alternatively lowering the mechanical moduli H0A , κA , or κA G of the budded phase (results not shown). Shortening the domain size by an appropriate choice of initial conditions revokes the effect of budding transition (D). (E): time snapshots of a simulation with H0A = 5 which induces budding opposed to the initial local curvature of the membrane. (F)–(I): For each elastic parameter the budding time Tmin decays with increasing σ and decreasing mechanical moduli h ∈ {H0A , κA , κA G }, respectively.

∗ and experimental results of [51]. Calculating the critical sharp line tension value σsi obtained by [38] for equal domain sizes and transformation in the diffuse value [42] yields σ ∗ ≈ 711, which is close to the numerically estimated value σ ∗ ∈ [630, 700]. The same effect of budding transition is observed for the other three elastic parameters: Lowering the spontaneous curvature H0A of the budded phase leads to budding transition below a critical value, just as lowering the bending rigidity κA or the Gaussian rigidity κA G does. In all three cases, we have ensured that these numerically estimated critical values agree with the calculations of [36, 38]. Furthermore, in all cases, reducing the size of the budded domain by choosing appropriate initial conditions U φ ds = −0.95 results in the inhibition of the process (cf. Figure 7.4 D). It stays in agreement with the observations of [35]. To stress the influence of the spontaneous curvature on the budding direction, we further perform simulations in which the sign of H0A is opposed to the initial local curvature of the membrane (a1 , a2 , and a3 are chosen such that U ds = 1.08 and U φ ds = −0.55) choosing H0A = 5 (cf. Figure 7.4 E). All other elastic parameters remain constant: κ ≡ 10 ≡ −κG and σ = 900. We observe that, also in this case, the spontaneous curvature is capable of inducing a budding process opposite to the natural bending of the membrane. These results highlight that the sign of the spontaneous curvature might be an important

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mechanism to control the direction of budding. This might explain how budding in both directions of the same membrane could be achieved, similarly at the membrane of the endoplasmic reticulum. Our observations agree with the experimental results of [4] reporting a correlation between molecule structure and budding direction. To investigate the impact of the elastic parameters on the velocity and dynamics of the budding process, for each parameter we plot the budding time Tmin (h), h ∈ {σ, H0 , κ, κG } from the beginning of a simulation to the numerical beginning of the equilibrium configuration (cf. Figure 7.4 F–I). To ensure that the minimum geometry does not influence the results, we use only parameters in the regime of a complete membrane budding. Our results show that budding duration is exponentially shortened with increasing line tension σ, respectively decreasing values of hA , hA ∈ {H0A , κA , κA G }. A related effect has been previously described in dissipative particle dynamics studies, but solely for the line tension [30]. Appendix A. a a · b ab a ∂i [.] ∂i X ∂t . , dt . (gij )i,j (bij )i,j g ij bij αi aji ∇Γ [.] ΔΓ . Γ . ∇ Γ . Δ δα F u) δF/δ X( . . . ds H K Γijk P ∇k aij11...i ...jQ

Notation and definitions. covariant vector ai , (a)i = ai , or matrix, (a)ij = aij .

standard scalar product of vectors, (a · b) = i ai b i . standard matrix multiplication, (ab)ik = j aij bjk . inverse of a matrix, (a−1 )ij = (a)ij = aij . partial derivative with respect to ui . = ∂i X . basis vector of the tangential space, ∂i X partial and total time derivatives. · ∂j X; gij = gji . first fundamental tensor, gij = ∂i X second fundamental tensor, bij = −∂i X · ∂j n; bij = bji . component of the inverse first fundamental tensor.

bij = kl g ik g jl bkl . αi = j g ij αj , where (g ij )i,j is the inverse of the first fundamental tensor and α ∈ {a, aj , ∂, ∇}.

ui j ui mixed notation, aji = = u gju a u g aju and a i =

j j ju = u g ju aui . It holds that u g iu guj = gi = δi . u giu a first surface gradient ∇Γ f = i,j g ij ∂j f ∂i X.

√ ij 1 Γ first surface Laplacian Δ f = √g i,j ∂i gg ∂j [f ] . ij ij Γ f = ∇ )−1 . i,j b K∂j [f ]∂i X, where (b ) = (bij

√ Γ [f ] = √1 second surface Laplacian: Δ ∂i gbij K∂j [f ] . g

i,j

Fréchet derivative or variation with respect to α. u). strong formulation of δ X F in X( √ surface integral on a manifold, where ds = g d2 u, and g is the determinant of the first fundamental tensor. mean curvature, H = trace(bij ). Gaussian curvature, K = det(bij ). Christoffel symbol. covariant derivative of type-(P/Q)-tensor field in k direction.

Appendix B. Tangential variational expressions. Lemma B.1 (variation of the covariant first fundamental tensor). k · ∂j X + ∂j ∂k Xψ k · ∂i X. δ k gij = ∂i ∂k Xψ + ∂i X · ∂j δ k [X] · ∂j X . The claim of the Proof. It holds that δ k gij ] = ∂i δ k [X]

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k k. + ε∂k Xψ := d X lemma directly follows from δ k [X] = ∂k Xψ dε ε=0 Lemma B.2 (variation of the contravariant first fundamental tensor). (B.1) δ t g ij = −∇i ψ j − ∇j ψ i , where (gij ) is the first fundamental tensor, ∇i is the covariant derivative, and superscripts denote contravariant components. A proof of this lemma is given in [37]. Lemma B.3 (variation of the covariant second fundamental tensor; cf. [13]). k δ t bij = (B.2) ∇j ψ bik + ∇i ψ k bjk + ∇k bij ψ k . k

Lemma B.4 (variation of the mean curvature). ij k δt H = g ∇k bij ψ . i,j,k

i

Proof. It holds that δ t H = δ t bi = δ t g ij bij = δ t g ij bij + i i,j i,j

ij t ij i,j g δ bij , due to (B.2) and (B.3). Since (bij )i,j and (g )i,j are symmetric, it follows that ∇i ψ j bij + 2 g ij ∇i ψ k bjk + g ij ∇k bij ψ k . δ t H = −2 i,j

i,j,k

i,j,k

Furthermore, 2 i,j,k g ij ∇i ψ k bjk = 2 j,k ∇j ψ k bjk = 2 i,j ∇i ψ j bij ; i.e., the first two terms vanish, and the claim holds true. Lemma B.5 (variation of the surface measure). √ √ where η ∈ C 1 (U ). ηδ t [ g] d2 u = − ∂ u η guk ψ k g d2 u,

k,u

√ Proof. Applying the chain rule to the determinant yields ηδ k [ g] d2 u = √ ij k gg δ [gij ]η d2 u. Using Lemma B.1 and Green’s formula provides √ ij √ k ] · ∂j Xη d2 u ηδ k [ g] d2 u = gg ∂i [∂k Xψ

1 2

i,j

i,j

=−

√ · ∂k Xψ k d2 u. ∂i ηg ij g∂j X

i,j

√ · ∂k X = −√gΔΓ X · ∂k X = √gHn · ∂k X = 0 it holds that Since i,j ∂ g ij g∂j X √ · ∂k Xψ k √g d2 u ηδ k [ g] d2 u = − ∂i η g ij ∂j X i,j

=−

· ∂k Xψ k √g d2 u = − ∂ j η ∂j X

j

j

Since δ t = k δ k , the claim directly follows. Lemma B.6 (variation of the Gaussian curvature). i δt K = b j ∇k bij ψ k K. i,j,k

√ ∂ j η gjk ψ k g d2 u.

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Proof. It holds that δ t K = δ t det (bij ) = Kbij δ t bji =

Kbij δ t

i,j

i,j

g ki bjk = Kbij δ t g ki bjk + Kbij g ki δ t bjk .

k

i,j,k

i,j,k

Calculations using (C.2) and (B.3) yield δ t K = −2 Kbij ∇k ψ i bjk + 2 Kbij bjk ∇i ψ k + Kbij g ki ∇u bjk ψ u . i,j,k

i,j,k

i,j,k,u

Lemma C.3 yields δ t K = −2 gik K∇k ψ i + 2 Kgik ∇i ψ k + Kbij g ki ∇u bjk ψ u i,k

=

i,k

i,j,k,u

bij g ki ∇u bjk ψ u K.

i,j,k,u

Since the covariant derivatives and the first metric tensor commute, we obtain the claim. Appendix C. Results in differential geometry and tensor calculus. Lemma C.1 (Mainardi–Codazzi equation [15]). ∇c bab = ∇b bac . (C.1) Lemma C.2 (see [32]).

(C.2)

∇a bab = ∂b H .

a

Lemma C.3. bij bik = gjk ,

where bij are components of the inverse of the matrix (bij ).

i

Proof. It holds that it follows that

i

bij bik =

i

=

bij bik =

i

l

g li bjl bik , and since (AB)−1 = B −1 A−1 ,

g lj bil bik = glj bil bik

i

l

bil bik glj = δlk glj = gkj ,

l

i

l

i

l

as has been claimed. Lemma C.4. (C.3)

zb · ∂a ∂k Xψ k . ∇a ψ z = g ∂b X b,k

1789


Proof. It holds that zb · ∂a ∂k Xψ k = · ∂a ∂k X ψk + · ∂k X∂ a ψk g zb ∂b X g ∂b X g zb ∂b X

b,k

b,k

b,k

1 zb · ∂ a ∂k X + ∂k ∂b X · ∂a X − ∂ b ∂k X · ∂a X 2∂b X g 2 b,k · ∂k X − ∂b ∂a X · ∂k X ψk + + ∂a ∂b X g zb gbk ∂a ψ k

=

b,k

1 zb + ∂a ∂b X · ∂k X − ∂b ∂a X · ∂k X ψk + ∂k ∂b X · ∂a X g δkz ∂a ψ k 2 b,k k 1 g zb ∂k gba + ∂a gbk − ∂b gak ψ k + ∂a ψ z . 2

= =

k

b

Using the definition of the Christoffel symbol, we obtain that z k · ∂a ∂k Xψ k = ∂a ψ z + g zb ∂b X Γak ψ = ∇a ψ z , b,k

k

· ∂a ∂k X = Γz . which is the claim. In particular, we obtain b g zb ∂b X ak Lemma C.5. · ∂b ∂k Xψ k . (C.4) ∇b ψa = ∂a X k

Proof. The result follows directly by Lemma C.4. Appendix D. Analytical reformulations. Using the definition of the covariant derivative as well as integration by parts, it follows that √ √ ∇u ∂k [φ]∂ u [φ] ψ g d2 u = − ∂k [φ]∂ u [φ]∂u [ψ] ds − ∂k [φ]∂ u [φ]∂u [ g]ψ d2 u U U U u l l u + Γul ∂k [φ]∂ [φ] − Γuk ∂l [φ]∂ [φ] ψ ds. U

l

l

Applying the chain rule to the determinant and using

√the ldefinition of the Christoffel √ 1 rs gg ∂ [g ] = g Γul . This yields symbols lead to ∂u [ g] = r,s 2√ u rs l g u

U

=

√ ∇u ∂k [φ]∂ u [φ] ψ g d2 u

U

u

+ =−

∂k [φ]∂ [φ]∂u [ψ] ds −

−

u

l

u

l

Γuul ∂k [φ]∂ l [φ]ψ

U

∂k [φ]∂ u [φ]∂u [ψ] ds − U

ds − l,u

∂k [φ]∂ u [φ]Γlul ψ ds

U

l

Γluk ∂l [φ]∂ u [φ]ψ ds

U

Γluk ∂l [φ]∂ u [φ]ψ ds.

U

Appendix E. Lyapunov functional. We show that the total energy (2.1) is a Lyapunov functional for system (4.1)–(4.3). F is bounded from below on the set

1790


under consideration. Using the already calculated variations and the chain rule, we obtain δ √ 2 d δ = F (φ, X) F φt F Xt + gd u dt δφ δ X U 2 σξ 2 (∇φ)2 + f (φ) + κ(φ) H − H0 (φ) + κG (φ)K + 2 U d √ 2 [ g]d u. dt d √ t = −Lx δ F and φt = Lφ ΔΓ δ F , it holds [ g] = 0 and the dynamics X Due to dt δφ δX that

2 δ δ Γ δ d √ 2 − LX F + Lφ gd u. F (φ, X) = F Δ [F ] dt δφ δφ δX U Integration by parts, assuming vanishing boundary terms, leads to d = F (φ, X) dt

− LX

U

δ F δX

2

2

√ 2 Γ δ [F ] − Lφ ∇ gd u ≤ 0. δφ

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