A coupled surface-cahn-hilliard bulk-diffusion system modeling lipid raft formation in cell membranes

Transcription

1 A couple surface-cahn-hilliar bulk-iffusion system moeling lipi raft formation in cell membranes Haral Garcke, Johannes Kampmann, Anreas Rätz, Matthias Röger Preprint September 2015 Fakultät für Mathematik Technische Universität Dortmun Vogelpothsweg Dortmun tu-ortmun.e/mathpreprints

2 A COUPLED SURFACE-CAHN HILLIARD BULK-DIFFUSION SYSTEM MODELING LIPID RAFT FORMATION IN CELL MEMBRANES HARALD GARCKE, JOHANNES KAMPMANN, ANDREAS RÄTZ, AND MATTHIAS RÖGER Abstract. We propose an investigate a moel for lipi raft formation an ynamics in biological membranes. The moel escribes the lipi composition of the membrane an an interaction with cholesterol. To account for cholesterol exchange between cytosol an cell membrane we couple a bulk-iffusion to an evolution equation on the membrane. The latter escribes a relaxation ynamics for an energy taking lipi-phase separation an lipi-cholesterol interaction energy into account. It takes the form of an (extene) Cahn Hilliar equation. Different laws for the exchange term represent equilibrium an non-equilibrium moels. We present a thermoynamic justification, analyze the respective qualitative behavior an erive asymptotic reuctions of the moel. In particular we present a formal asymptotic expansion near the sharp interface limit, where the membrane is separate into two pure phases of saturate an unsaturate lipis, respectively. Finally we perform numerical simulations an investigate the long-time behavior of the moel an its parameter epenence. Both the mathematical analysis an the numerical simulations show the emergence of raft-like structures in the non-equilibrium case whereas in the equilibrium case only macroomains survive in the long-time evolution. 1. Introuction Phase separation processes that lea to microomains of a well-efine length-scale below the system size arise in various physical an biological systems. A prominent example is the microphase separation in block-copolymers [7] or other soft materials, characterize by a fluilike isorer on molecular scales an high egree of orer at larger scales. Here micro-scale pattern arise by a competition between thermoynamic forces that rive (macro-)phase separation an entropic forces that limit phase separation. Different mathematical moels for microphase separation in i-block copolymers have been evelope, in particular built on self-consistent mean fiel theory (see for example [35] an the references therein) or ensity functional theory evelope in [29, 39, 6] that leas to the so-calle Ohta Kawasaki free energy. Diblock-copolymer type moels have also been stuie on spheres an more general curve surfaces [32, 51, 10, 31] an show a variety of ifferent stripe- or spot-like patterns. In contrast to materials science microphase separation on biological membranes is much less unerstoo, in particular in living cells. In this contribution we present an analyze a moel for so-calle lipi rafts, that represent microomains of specific lipi compositions. The outer (plasma) membrane of a biological cell consists of a bilayer forme by several sorts of lipi molecules an contains various other molecules like proteins an cholesterol. Besies being the physical bounary of the cell the plasma membrane also plays an active part in the functioning of the cell. Many stuies over the past ecaes have shown that the structure of the outer membrane is heterogeneous with several microomains of ifferent lipi an/or protein composition. Such omains have been clearly observe in artificial membranes such as giant unilamellar vesicles [48, 49]. Here a less fluiic (liqui-orere) phase of saturate lipis an cholesterol separates from a more fluiic (liqui-isorere) phase of unsaturate lipis. The nucleation of isperse microomains is followe by a classical coarsening process that leas to Date: September 1, Mathematics Subject Classification. 35K51, 35K71, 35Q92, 92C37, 35R37. Key wors an phrases. partial ifferential equations on surfaces, phase separation, Cahn Hilliar equation, Ohta Kawasaki energy, reaction-iffusion systems, singular limit. 1

3 2 H. GARCKE, J. KAMPMANN, A. RÄTZ, AND M. RÖGER coexisting omains with a length scale of the orer of the system size. The situation is much more complex an much less unerstoo for living cells. Their plasma membrane represents a heterogeneous structure with a complex an ynamic lipiic organization. The formation, maintenance an ynamics of intermeiate-size omains ( nm on cells of µm size [30]) calle lipi rafts are of prime interest. These are characterize as liqui-orere phases that consist of saturate lipis an are enriche of cholesterol an various proteins [44, 8]. These rafts contribute to various biological processes incluing signal transuction, membrane trafficking, an protein sorting [18]. It is therefore an interesting question to stuy the unerlying process, by which these rafts are generate an maintaine, an to unerstan the mechanism that allows for a ynamic istribution of intermeiate-size omains. Several phenomenological mesoscale moels have been propose, see for example the review [18]. One class of moels argues that raft formation is a result of a thermoynamic equilibrium process. Here one contribution is a phase separation energy that woul inuce a reuction of interfacial size between the raft an non-raft phases. The observation of nano-scale structures is then explaine by incluing ifferent aitional energy contributions. One proposal is that thermal fluctuations near the critical temperature for the phase separation are responsible for the intermeiate-size structures. Other explanations consier interactions between lipis an membrane proteins that coul act as a kin of surfactant or coul pin the interfaces ue to their immobility [54, 46, 53]. Finally, raft formation coul also be stabilize by inuce changes of the membrane geometry an elasticity effects. However, as argue in [18], all such moels are not able to reprouce key characteristics of raft ynamics; non-equilibrium effects essentially contribute to raft formation. Of particular importance are active transport processes of raft components that allow to maintain a non-equilibrium composition of the membrane. The competition between phase separation an recycling of raft components is argue to be of major importance for the ynamics an structure of lipi rafts. Foret [20] propose a simple mechanism of raft formation in a two-component flui membrane. This moel inclues a constant exchange of lipis between the membrane an a lipi reservoir as well as a typical phase separation energy. While relaxation of the latter tens to create large omains, the constant insertion an extraction of lipis in the membrane ensures inee the formation of rafts. The emerging microomains are static (in contrast to the lipi rafts on actual cell membranes) an the size istribution of these rafts is rather uniform, whereas in vivo cell membranes show a ynamic istribution of rafts of ifferent sizes (see the concluing remarks in [20] an the iscussion of Forets moel in [18]). Moreover, as alreay notice in [18], whereas Forets moel is motivate by incluing non-equilibrium effects his moel can be equivalently characterize as relaxation ynamics for an effective energy given by the Ohta Kawasaki energy of block-copolymers [39] that is well-known to generate phase-separation in intermeiate-size structures. A similar moel by Gómez, Sagués an Reigaa [22] consiers a ternary mixture of saturate an unsaturate lipis together with cholesterol an stuies the interplay of lipi phase separation an a continuous recycling of cholesterol. An energy that is etermine by the relative concentration φ of saturate lipis an the relative cholesterol concentration c is propose that in particular inclues a phase separation energy of Ginzburg Lanau type for the lipi phases an a preference for cholesterol saturate lipi interactions over cholesterol unsaturate lipi interactions. The ynamics inclue an exchange term for cholesterol that is given by an in-/outflux proportional to the ifference from a constant equilibrium concentration of cholesterol at the membrane. Our aim here is to propose an extene moel an to present both a mathematical analysis an numerical simulations. Similar as in [22] one ingreient of our moel are energetic contributions from lipi phase separation an lipi-cholesterol interaction. In aition, we inclue the ynamics of cholesterol insie the cytosol (the liqui matter insie the cell) an prescribe a etaile coupling between the processes in the cell an on the cell membrane. In particular, the outflow

4 SURFACE CAHN-HILLIARD MODELING RAFT FORMATION 3 of cholesterol from the cytosol appears as a source-term in the membrane-cholesterol ynamic an will be characterize by a constitutive relation. We will investigate ifferent choices of this relation an will illustrate the implications on the emergence of microomains A lipi raft moel incluing cholesterol exchange an cytosolic iffusion. To give a etaile escription of our moel let us fix an open boune set B R 3 with smooth bounary = B representing the cell volume an cell membrane, respectively. Let ϕ enote a rescale relative concentration of saturate lipi molecules on the membrane, with ϕ = 1 an ϕ = 1 representing the pure saturate-lipi an pure unsaturate-lipi phases, respectively. Moreover let v enote the relative concentration of membrane-boun cholesterol, where v = 1 inicates maximal saturation, an let u enote the relative concentration of cytosolic cholesterol. We then prescribe a phase-separation an interaction energy of the form ( ε (1.1) F(v, ϕ) = 2 ϕ 2 + ε 1 W (ϕ) + 1 2δ (2v 1 ϕ)2) H 2 with W a ouble-well potential that we choose as W (ϕ) = 1 4 (1 ϕ2 ) 2, constants ε, δ > 0, an enoting the surface graient. The first two terms represent a classical Ginzburg-Lanau phase separation energy, whereas the thir term moels a preferential bining of cholesterol to the lipi-saturate phase. We efine the chemical potentials (1.2) µ := δf δϕ = ε ϕ + ε 1 W (ϕ) δ 1 (2v 1 ϕ), (1.3) θ := δf δv = 2 (2v 1 ϕ), δ where enotes the Laplace Beltrami operator on, an prescribe for the ynamics of the concentrations over a time interval of observation (0, T ) the following system of equations (1.4) (1.5) (1.6) t u = D u in B (0, T ], D u ν = q on (0, T ], t ϕ = µ on (0, T ], (1.7) t v = θ + q = 4 δ v 2 δ ϕ + q on (0, T ], where ν enotes the outer unit-normal fiel of B on. The system is complemente with initial conitions u 0, ϕ 0, v 0 for u, ϕ an v, respectively. The first equation represents a simple iffusion equation for the cholesterol in the bulk, the secon equation characterizes the outflow of cholesterol. The thir equation is a Cahn Hilliar ynamics for the lipi concentration on the membrane, whereas the equation for the cholesterol on the membrane combines a mass-preserving relaxation of the interaction energy an an exchange with the bulk reservoir of cholesterol given by the flux from the cytosol. This combination yiels a iffusion equation with cross-iffusion contributions an a source term. To close the system it remains to characterize the exchange term q. We follow here two possibilities: First we prescribe a constitutive relation by consiering the membrane attachment as an elementary reaction between free sites on the membrane an cholesterol, an the etachment as proportional to the membrane cholesterol concentration, expresse by the choice (1.8) q = c 1 u(1 v) c 2 v. A similar coupling of bulk surface equations has been investigate in a reaction-iffusion moel for signaling networks [41, 42]. As a secon possibility we consier choices of q that allow for a global free energy inequality for the couple membrane/cytosol system. These two ifferent cases coul be consiere as a istinction between open an close systems an one important aspect of this work is to evaluate the consequences of these choices for the formation of complex phases.

5 4 H. GARCKE, J. KAMPMANN, A. RÄTZ, AND M. RÖGER We remark that the system conserves both the total cholesterol an the lipi concentrations since for arbitrary choices of q we obtain the relations ( ) (1.9) u x + v H 2 = 0, t B (1.10) ϕ H 2 = 0. t Let us contrast the above moel with the Ohta Kawasaki moel for phase separation in iblock copolymers mentione above. Let Ω R n enote a spatial omain, ϕ the relative concentration of one of the two polymers an let m := ffl ϕ be the prescribe average of ϕ over Ω. The mean fiel potential z is then given by z = ϕ m in Ω, z ν Ω = 0 on Ω, z = 0. Then a free energy is prescribe of the form ( ε F OK (ϕ) = 2 ϕ 2 + ε 1 W (ϕ)) x + σ (1.11) 2 Ω Ω Ω z 2 x, where σ > 0 is a fixe constant. Note that the last term can also be written as σ 2 ϕ ffl ϕ 2 H 1. A relaxation ynamics of Cahn Hilliar type is then typically consiere given as (1.12) t ϕ = µ, µ = δf OK δϕ = ε ϕ + ε 1 W (ϕ) + σz. The energies F an F OK both contain a Cahn Hilliar type energy contribution that favors macro-phase separation but are ifferent in the aitional terms. However we will see below that stationary patterns for our lipi raft moel with the choice (1.8) are for small δ > 0 closely relate to stationary points of F OK. If on the other han one consiers simple choices for q that lea to a global free energy inequality, we will observe a macro-scale separation of all saturate lipis in one connecte omain. This inicates that in fact non-equilibrium processes are responsible for lipi raft formation. Several asymptotic regimes are interesting in view of the ifferent parameters inclue in our moel. We will investigate the limit ε 0 that correspons to a strong segregation limit an leas to a moel where no mixing of the lipi phases is allowe an the omain splits into regions where ϕ = 1 an ϕ = 1 respectively. This limit correspons to the sharp interface limit in phase-fiel moels an connects to the analysis of the Ohta Kawasaki moel in [38]. Since the cytosolic iffusion in biological cells is known to be much faster than lateral iffusion on the cell membrane another natural reuction of the moel appears in the limit D that leas to a non-local moel efine solely on the cell membrane. Finally, assuming the effect of the lipi interaction with cholesterol to be large motivates to consier the asymptotic regime δ Outline of the paper an main results. In the next section we will erive the moel (1.4)-(1.7) from thermoynamic consierations. In particular we will show that for arbitrary choices of the exchange term q the surface equations an the bulk (cytosolic) equations are thermoynamically consistent when viewe as separate systems. Depening on the specific choice of q we may or may not have a global (that is with respect to the full moel) free energy inequality. We will present examples for both cases. In Section 3 we will first erive a reuce raft moel in the large cytosolic iffusion limit by formally taking D. We then analyze the qualitative behavior of the (reuce) system in terms of a characterization of stationary points an an investigation of their relation to stationary points of the Ohta Kawasaki moel. A formal asymptotic expansion for the sharp interface reuction ε 0 of our raft moel is presente in Section 4. Here we also briefly iscuss the resulting limit problem that takes the form of a free bounary problem of Mullins Sekerka type on the membrane with an aitional coupling to a iffusion process in the bulk an incluing

6 SURFACE CAHN-HILLIARD MODELING RAFT FORMATION 5 an interaction with the cholesterol concentration. In Section 5 we present numerical simulations of the full an reuce raft moel. In particular, we stuy spinoal ecomposition, coarsening scenarios an the possible appearance of raft-like structures as (almost) stationary states for ifferent choices of the exchange term q an for ifferent parameter regimes. Some conclusions are state in the final section. 2. Thermoynamic justification of the lipi raft moel In this section we will erive the governing equations for the lipi raft moel from basic thermoynamical conservation laws using a free energy inequality. Our arguments are similar to an approach use by Gurtin [24, 25], who erive the Cahn-Hilliar equation in the context of non-equilibrium thermoynamics. We first of all consier the equations which have to hol on the surface, will then consier the equations in the bulk an subsequently we will couple both systems. The basic quantities on the membrane surface are the rescale relative concentration of the saturate lipi molecules ϕ, the concentration v of the membrane-boun cholesterol, the mass flux J ϕ of the lipi molecules, the mass flux J v of the cholesterol, the mass supply of cholesterol q, the surface free energy ensity f an the chemical potential µ relate to the lipi molecules an the chemical potential θ relate to the surface cholesterol. The unerlying laws for any surface subomain Σ are the mass balance for the lipis (2.1) t Σ ϕ H 2 = the mass balance for the surface cholesterol (2.2) v H 2 = t Σ Σ Σ J ϕ n H 1, J v n H 1 + q H 2 Σ an the secon law of thermoynamics, which in the isothermal situation has the form (2.3) fh 2 (µj ϕ n ( t ϕf, ϕ n) + θj v n) H 1 + θq H 2 t Σ Σ where f enotes the surface free energy ensity an where n is the outer unit conormal to Σ in the tangent space of. In aition, we enote by H the integration with respect to the imensional surface measure an f, ϕ enotes the partial erivatives of f with respect to the variables relate to ϕ in a constitutive relation f = f(..., ϕ,...). Similarly we will enote with a subscript comma partial erivatives with respect to other variables. For a iscussion of these laws in cases in which source terms are present an at the same time f oes not epen on ϕ we refer to [24], [26, Chapter 62], an [40]. Thermoynamical moels of phase transitions with an orer parameter typically involve a free energy ensity f which oes epen on ϕ. In this case the free energy flux oes not only involve the classical terms µj ϕ an θj v but also a term t ϕf, ϕ. This is iscusse in [25, 5, 2]. Gurtin [25] introuces a microforce balance involving a microstress ξ in orer to erive the Cahn-Hilliar equation. Here, we o not iscuss the microforce balance an instea alreay use the form ξ = f, ϕ which coul be erive in our context in the same way as in [25]. However, in orer to shorten the presentation we o not state the etails. We hence obtain (2.3) as the relevant free energy inequality in cases where no external microforces are present. Since the above (in-)equalities (2.1) (2.3) hol for all Σ, we obtain with the help of the Gauß theorem on surfaces the local forms, compare [25], (2.4) (2.5) (2.6) t ϕ + iv J ϕ = 0 in (0, T ], t v + iv J v = q in (0, T ], t f + iv (µj ϕ t ϕf, ϕ + θj v ) θq in (0, T ]. Σ

7 6 H. GARCKE, J. KAMPMANN, A. RÄTZ, AND M. RÖGER With the constitutive relation f = f(v, ϕ, ϕ) we obtain from the local form of the free energy inequality f,ϕ t ϕ + f,v t v+ µ J ϕ + θ J v + Using the conservation laws (2.4) an (2.5) we obtain (iv J ϕ )µ + (iv J v )θ t ϕiv f, ϕ θq. (f,ϕ iv (f, ϕ) µ) t ϕ + (f,v θ) t v + µ J ϕ + θ J v 0. The fact that solutions of the conservation laws with arbitrary values for t ϕ an t v can appear is use in the theory of rational thermoynamics to show that the factors multiplying t ϕ an t v have to isappear as they o not epen on t ϕ an t v. We refer to Liu s metho of Lagrange multipliers [34] an to [26, 5] for a more precise iscussion on how the free energy inequality can be use to restrict possible constitutive relations. We now choose the following constitutive relations which guarantee that the free energy inequality is fulfille for arbitrary solutions of (2.4), (2.5). In fact, choosing (2.7) (2.8) (2.9) (2.10) µ = f,ϕ iv (f, ϕ), θ = f,v, J ϕ = D ϕ µ, J v = D v θ with D ϕ, D v 0 ensures that the free energy inequality is fulfille for all solutions of the conservation laws. More general moels, e.g. taking cross iffusion into account, are possible an we refer to [5] for an approach which can be use to obtain more general moels. We now consier the governing physical laws in the bulk. As variables we choose u which is the relative concentration of the cytosolic cholesterol, the bulk chemical potential µ u, the bulk free energy ensity f b (u), the bulk flux J u an the surface mass source term q u. We nee to fulfill the following mass balance equation in integral form which has to hol for all open U B: (2.11) t U u x = an the free energy inequality (2.12) f b (u) x t U ( U)\ ( U)\ J u ν H 2 + ( U) q u H 2 µ u J u ν H 2 + µ u q u H 2 ( U) which also has to hol for all open U B. Here we allowe for a source term q u on an in accorance to our iscussion above we introuce the free energy source µ u q u in (2.12). As on the interface the free energy source term is given classically as a prouct of the mass source term an the chemical potential. If we use the fact that we can choose an arbitrary open set U which is compactly supporte in B (which then implies ( U) = ) we obtain with the help of the Gauß theorem (2.13) (2.14) Choosing t u + ivj u = 0 in B (0, T ], t f b (u) + iv(µ u J u ) 0 in B (0, T ]. (2.15) µ u = f b (u), (2.16) J u = M(u) (f b (u)) makes sure that (2.14) is true for all solutions of (2.13). In the following we will often choose M(u) = Du an this will lea to the linear iffusion equation (1.4) (u) f b t u D u u = 0 in B (0, T ].

8 SURFACE CAHN-HILLIARD MODELING RAFT FORMATION 7 Choosing U such that U we obtain from (2.11) 0 = ( t u + ivj u ) x = (q u + J u ν)h 2 which gives, since U is arbitrary, U U q u = J u ν (= D u u ν) in (0, T ], an which yiels (1.5) for q = q u. We will now state global balance laws in the case where the mass supply for the interface stems from the bulk an vice versa. Lemma 2.1. We assume that the above state mass balance equations hol for the bulk an the surface an assume in aition that q u = q. Then it hols ( u x + v H 2) = 0 ϕ H 2 = 0. t t B If in aition, the free energy inequalities in the bulk an on the surface are true it also hols ( ) f(v, ϕ, ϕ) H 2 + f b (u) x q(θ µ u )H 2. t B Proof. The first two equations follow from (2.1), (2.2) with Σ = an (2.11) with U = B since U = an since =. The total free energy inequality follows similarly from (2.3) an (2.12). Remark 2.2. (i) In the above lemma we chose q u = q, that is the mass lost on the surface generates a source of mass for the bulk. (ii) Several constitutive laws for q make sense. It is possible to consier (2.17) q = c(θ µ u ), c 0 which leas to moel for which the total free energy ecreases. In this case we obtain ( ) f(v, ϕ, ϕ) H 2 + f b (u) x c(θ µ u ) 2 H 2 0. t (iii) We also consier the reaction type source term, compare (1.8), B q = c 1 u(1 v) + c 2 v. Also in this case we obtain a consistent moel which fulfills the bulk an surface free energy inequalities with source terms as state above. However, in this case the total free energy as the sum of the bulk an the surface free energy might increase which can be ue to the fact that we neglect energy contributions generate by the etachment an attachment process. (iv) One possible choice for the surface free energy ensity is, compare (1.1), f(v, ϕ, ϕ) = γε 2 ϕ 2 + γ ε W (ϕ) + 1 2δ (2v 1 ϕ)2, ε, δ, γ > 0. For f b (u) = 1 2 u2, q u = q an D u = D, γ = D ϕ = D v = 1 we obtain the system (1.2) (1.7). With the above quaratic choice for f b an arguing as in the erivation of the energy inequality one can prove the ientity (2.18) t ( ε 2 ϕ ε W (ϕ) + 1 2δ (2v 1 ϕ)2 H 2 1 ) + B 2 u2 x + ( µ 2 + θ 2 )H 2 + D u 2 = B q(θ u) where the last term is non-positive for the choice (2.17). Any other evolution that is base on a choice of q such that the right-han sie of (2.18) is always non-positive ecreases the

9 8 H. GARCKE, J. KAMPMANN, A. RÄTZ, AND M. RÖGER total free energy. This is in particular the case for choices of q such that (1.2)-(1.7) can be characterize as a graient flow. In the following we are mainly intereste in the epenence on the parameters ε, δ, D u an therefore as in (iv) above we always set D u = D, γ = D ϕ = D v = 1, in which case the above choices of free energy ensities an mass fluxes yiel the system (1.2) (1.7). 3. Qualitative behavior In this section we will investigate qualitative properties of the moel (1.2)-(1.7) an of the asymptotic reuction in the large cytosolic iffusion limit that we erive below. One key question here is whether or not our lipi raft moel supports the formation of mesoscale patterns. We will istinguish ifferent choices for the exchange term q an compare evolutions that reuce the total free energy with the evolution for the choice q given by the reaction-type law (1.8), which we consier as a prototype of a non-equilibrium moel. We remark that most of the arguments in this section are purely formal; a rigorous justification is out of the scope of the present paper. In particular, we assume the existence of smooth solutions, their convergence to stationary states as times tens to infinity, an that the long-time behavior of the full system asymptotically agrees with that of the reuce system evelope below. We first observe that uner an aitional growth assumption on the exchange term q we obtain energy bouns, even in the case that the system oes not satisfy a global energy inequality. Proposition 3.1. Assume that q has at most linear growth, that is there exists Λ > 0 such that (3.1) q(ϕ, u, v) Λ(1 + ϕ + u + v ) for all ϕ, u, v R. Then for all 0 < t < T an all D D 0 > 0, 0 < ε ε 0 any solution of (1.2)-(1.7) with initial ata ϕ 0, u 0, v 0 satisfies F(v(, t), ϕ(, t)) + 1 t u(, t) 2 D (3.2) u 2 C(δ, Λ, T, D 0, ε 0, v 0, ϕ 0, u 0 ). B 0 Proof. From (2.18) we euce that ( F(v(, t), ϕ(, t)) + 1 (3.3) u(, t) 2) t 2 B = D u 2 [ (, t) µ 2 (, t) + θ 2 (, t) (θ u)(, t)q(ϕ(, t), u(, t), v(, t)) ]. B For the last term we use the estimate ( ) (θ u)q(ϕ, u, v) θ 2 + u 2 + C Λ (1 + ϕ 2 + u 2 + v 2 ) (3.4) = θ 2 + C Λ (1 + ϕ 2 ) + (C Λ + 1) u 2 + C Λ For the secon term on the right-han sie we obtain (1 + ϕ 2 ) C ( (3.5) 1 + W (ϕ) ) C(ε 0 ) ( 1 + F(v(, t), ϕ(, t)) ). Using [28, Chapter 2, (2.25)] the thir term on the right-han sie of (3.4) can be estimate by bulk quantities, u 2 D u 2 + C(D) u 2 2 B B for a suitable constant C(D), which in particular yiels for all D D 0 u 2 D (3.6) u 2 + C(D 0 ) u 2. 2 B B B v 2.

10 SURFACE CAHN-HILLIARD MODELING RAFT FORMATION 9 To estimate the last integral on the right-han sie of (3.4) we first use Young s inequality an (1.3) to obtain an further euce that (3.7) 16 δ 2 v2 = θ δ θ(1 + ϕ) + 4 δ 2 (1 + ϕ)2 2θ δ 2 (1 + ϕ)2, v 2 δ2 8 θ2 + C(1 + W (ϕ)), ( v 2 (, t) C(δ, ε 0 ) 1 + F(v(, t), ϕ(, t)) ). We therefore obtain from (3.3)-(3.7) that ( F(v(, t), ϕ(, t)) + 1 u(, t) 2) + D ( ) u 2 (, t) + µ 2 (, t) + θ 2 (, t) t 2 B 2 B ( C(δ, Λ, D 0, ε 0 ) F(v(, t), ϕ(, t)) + 1 u(, t) 2) + C(δ, Λ). 2 By the Gronwall inequality we euce the claim. B Note that for the choice (1.8) of q assumption (3.1) is not satisfie. However, for any moification that coincies with that choice on a boune omain in the u, v plane an that has at most linear growth outsie the conclusion of Proposition 3.1 hols. For (2.17) or any other choice of q that implies a total free energy inequality we obtain an even better estimate, since now the right-han sie in (2.18) is non-positive A reuce moel in the limit of large cytosolic iffusion. Since in the application to cell biology the bulk (cytosolic) iffusion is much higher than the lateral membrane iffusion a reasonable reuction of the moel can be expecte in the limit D. In the case that the exchange term q satisfies assumption (3.1) we euce by Proposition 3.1 that T 0 B D u 2 is boune uniformly in D. The same conclusion hols in the case of any free-energy ecreasing evolution by (2.18). Therefore in the formal limit D we conclue that u is spatially constant an obtain the (non local) system of surface PDEs (3.8) (3.9) (3.10) t ϕ = µ on (0, T ], µ = ε ϕ + ε 1 W (ϕ) δ 1 (2v 1 ϕ) on (0, T ], t v = θ + q = 4 δ v 2 δ ϕ + q(ϕ, u, v) on (0, T ]. This system is complemente by initial conitions for ϕ an v. The cholesterol concentration u = u(t) is etermine by a mass conservation conition (3.11) u(t) + v(, t) = M, B where M > 0 is the total mass of cholesterol in the system. Note that the transformation of the couple bulk-surface system into a system only efine in the surface has the price of introucing a non-local term by the characterization of u through the mass constraint. The reuction (3.8)-(3.11) is similar to the reuction to a shaow system for 2 2 reaction-iffusion systems introuce by Keener [27], see also the iscussion in [37]. In the qualitative analysis below an the numerical simulations we will often restrict ourselves to the special choice q of the exchange function given in (1.8). For the reuce moel it is then possible to compute the evolution of the total mass of u an v, which are relate by (3.11). We

11 10 H. GARCKE, J. KAMPMANN, A. RÄTZ, AND M. RÖGER euce, cf. (2.11), (3.12) u(t) x = q(ϕ(, t), u(t), v(, t)) H 2 = c 1 u(t)(1 v)(, t) + c 2 v(, t) H 2 t B = (c 1 u(t) + c 2 )v(, t) H 2 c 1 u(t) x B B ( ) = (c 1 u(t) + c 2 ) M u(t) x c 1 u(t) x B B = c ( 2 1 u(t) x) + B B ( ) M c 1 c 2 u(t) x + c 2 M B B an therefore we see that u(t) remains nonnegative if it was initially nonnegative (which is the relevant case) an converges for t to u, which is the positive zero of thus (3.13) u = 1 2 p(z) = c 1 z 2 + ( c 1 M B (M B c 2 c 1 ) + 1(M 4 B B c 2 ) z + c 2 M B, c 2 c 1 ) 2 + c 2 M c 1 B. Since p(0) > 0 an p( M B ) < 0 we also obtain that v(t) remains in [0, M] for all times. We remark that if q is given as in (1.8), then assumption (3.1) oes not hol. Nevertheless we can obtain the reuce moel for this particular choice of q if we start with a moifie version: Replace first q by q(u, v) = c 1 u c 1 η(u)v c 2 v, where η : R R is any smooth, monotone increasing an uniformly boune function with η(r) = r for r M B 1. Now the above arguments apply an we obtain the non-local moel with exchange term q. By analogous computations as above we then euce u(t) x = c ( 1 u(t) x + c 1 η ( 1 u(t) x ) ) (M + c 2 u(t) x ), t B B B B B B which yiels 0 B u(t) x M for all t 0 if this property hols for the initial ata. But this implies q(u(t), v(, t)) = q(u(t), v(, t)) for all t 0. This justifies to consier in the following analysis an in the numerical simulations in Section 5 the exchange term q from (1.8) also for the non-local reuction Stationary points. We are in particular intereste in the long-time behavior of solutions an will therefore next investigate stationary points of our lipi raft system. For the full system (1.2)-(1.7) stationary points (ϕ, u, v ) are characterize by the equations (3.14) 0 = µ on, 0 = θ + q(ϕ, u, v ) on, u = 0 in B, D u ν = q(ϕ, u, v ) on. This implies that µ is constant an (3.15) (3.16) (3.17) ε ϕ + 1 ε W (ϕ ) = 1 δ (2v 1 ϕ ) + µ = θ 2 + µ, q(ϕ, u, v ) = 2 δ (2v 1 ϕ ), u = 0 in B, D u ν = q(ϕ, u, v ) on.

12 SURFACE CAHN-HILLIARD MODELING RAFT FORMATION 11 Alternatively, this system can be characterize by (3.18) ε ϕ + 1 ε W (ϕ ) = Q(ϕ ) + µ, where Q(ϕ ) = 1 2 θ is a nonlocal function of ϕ as u, v an hence θ are etermine by ϕ through (3.16), (3.17) The case of an energy-ecreasing evolution. Let us in the following first consier the case that q is chosen such that the right-han sie in (2.18) is nonpositive, hence the total free energy is ecreasing. For any stationary point u, v, ϕ the energy inequality (2.18) yiels that µ, θ an u are constant, in particular by (3.15) ε ϕ + 1 ε W (ϕ ) = θ 2 + µ R, q(ϕ, u, v ) = 0. In aition, we fix the total lipi an cholesterol masses as they are for any evolution etermine by the initial conitions. We therefore prescribe, for M, M 1 given, M = u + v = B u + v, B M 1 = ϕ. In particular, the stationary state ϕ coincies with a critical point of the Cahn Hilliar energy subject to a volume constraint. The conition q = 0 provies an aitional relation between ϕ, u an v. In the case of the exchange law (2.17) this etermines v an θ. We can elaborate the connection with stationary points of the Cahn Hilliar equation a bit more if we in aition assume that (u, v, ϕ ) is a local minimizer of the energy (1.1). We represent the latter as (3.19) F(v, ϕ) = F 1 (ϕ) + F 2 (v, ϕ), where ε F 1 (ϕ) := 2 ϕ 2 + ε 1 W (ϕ), F 2 (v, ϕ) := We also assume that both which hols in particular in case of the exchange law (2.17). For any v, ϕ with v = v an ϕ = ϕ we then have ( 2 (2v 1 ϕ) 2 = 2v 1 ϕ (2v 1 ϕ)) + 1 2δ (2v 1 ϕ)2. ϕ an v are fixe by the initial ata an the conition q = 0, ( ) 2 (2v 1 ϕ ) ( ) 2 (2v 1 ϕ) an euce that uner the respective mass constraints the minimizer of F 2 (v, ϕ) are given by those (v, ϕ) for which 2v 1 ϕ is constant, an that the minimum only epens on v an ϕ. Since θ an thus 2v 1 ϕ are constant, we see that u, v, ϕ minimizes F 2. Moreover, for any ϕ satisfying the mass constraint the minimum of F 2 is attaine by v = 1 2 (1 + ϕ + ffl (2v 1 ϕ)) an is inepenent of ϕ. Therefore ϕ as above nees to be a local minimizer of the Cahn Hilliar energy F 1 subject to a given mass constraint. In the case of the Cahn Hilliar ynamics in an open convex set in R n, is is known [45] that stable stationary points converge in the sharp interface limit ε 0 to configurations with one connecte phase bounary of constant curvature. In analogy, one therefore might expect that for ε > 0 sufficiently small the only local minimizer in our lipi raft moel for choices q that lea to an energy-ecreasing evolution are given by configurations with one lipi phase concentrate in a single geoesic ball

13 12 H. GARCKE, J. KAMPMANN, A. RÄTZ, AND M. RÖGER on. In particular, such local minimizer o not represent mesoscale pattern like lipi rafts. In Section we present a numerical simulation for energy ecreasing ynamics that confirms the expecte behavior The case of the exchange term (1.8). Let us next iscuss the choice of q as given in (1.8). The representation (3.18) shows some similarity to the equation for stationary points of the Ohta Kawasaki moel escribe above. We can make this more transparent in the case of the reuce system (3.8)-(3.11), at least if we presume that the long term behavior of the reuce systems captures the respective behavior of the full system (which means that the orer of limits D an t can be interchange). Since u is constant we obtain, writing c 1 = c 1 u for convenience, that (3.20) 0 = q(u, v ) = c 1 ( c 1 + c 2 ) v an further q(u, v ) = c 1 + c 2 2 = δ( c 1 + c 2 ) 4 = δ( c 1 + c 2 ) 4 = δ( c 1 + c 2 ) 4 (v 1 ϕ ) c 1 + c 2 (1 + ϕ ) + c 1 2 θ c 1 + c 2 (1 + ϕ ) + c 1 2 ) (θ θ δ( c 1 + c 2 ) 2 4 δ ) (θ θ c 1 + c 2 2 (v 1 ϕ ) c 1 + c 2 (1 + ϕ ) + c 1 2 (ϕ ϕ ), where we have use (3.20) in the thir line. This implies by (3.14) ( + δ( c 1 + c 2 ) ) ( ) θ θ = c ) 1 + c 2 (ϕ ϕ. 4 2 Since µ is constant we euce from equation (3.18) ε ϕ + ε 1 W (ϕ ) = µ + 1 θ + 1 ) (θ θ 2 2 = µ + 1 θ c 1 + c ( 2 + δ( c 1 + c 2 ) ) ) 1 (ϕ ϕ For δ 1 this can be approximate by (3.21) ε ϕ + ε 1 W (ϕ ) + c ( ) 1 + c 2 ( ) 1 ϕ ϕ 4 = µ The latter equation correspons to stationary points of the Ohta Kawasaki functional ( ε (3.22) F OK (ϕ) = 2 ϕ 2 + ε 1 W (ϕ)) + σ 2 ϕ ϕ 2 H 1 with (3.23) σ = c 1 + c 2, 4 where the constant on the right-han sie of (3.21) shoul be interprete as a Lagrange multiplier associate to a mass constraint for ϕ. The total lipi mass ϕ is given by the initial ata. We can also ientify c 1 + c 2 as a function of the ata. First its value is characterize by u an using (3.13) we euce c 1 + c 2 = 1 ( c 2 + M 2 B c 1 ) B c 1 ( 1 + c 2 + M 4 B c 1 ) 2 (3.24) B c 1 + c1 c 2 B. θ.

14 SURFACE CAHN-HILLIARD MODELING RAFT FORMATION 13 In Section we will present simulations that confirm that the long-time behavior of the reuce system is for δ 1 in fact very close to that of the Ohta Kawasaki ynamics. In particular, in contrast to any choice of q that inuces an energy-ecreasing evolution, in the case of the exchange term (1.8) we in fact see the occurrence of mesoscale patterns. Remark 3.2. Let us highlight one key ifference in the long-time behavior of our moel in the ifferent cases consiere above. For a free-energy ecreasing evolution stationary states are characterize by the properties that θ is constant an q = q(ϕ, u, v ) zero. For the choice (1.8) of the exchange term on the other han an the reuce system we have stationary states with non-vanishing q, which correspon to an persisting exchange of cholesterol between bulk an cell membrane. This process eventually allows for the formation of complex pattern on a mesoscopic scale. 4. Sharp interface limit ε 0 by formally matche asymptotic expansions In this section we formally erive the sharp interface limit of the iffuse interface moel (1.2) (1.7) as ε 0. We assume throughout this section that the tuple (u ε, ϕ ε, v ε, µ ε, θ ε ) solves (1.2) (1.7) an converges formally as ε 0 to a limit (u, ϕ, v, µ, θ). Furthermore, we suppose that the family of the zero level sets of the functions ϕ ε converges as ε 0 to a sharp interface. This interface, at times t [0, T ] is suppose to be given as a smooth curve γ(t) which separates the regions {ϕ(, t) = 1} an {ϕ(, t) = 1}. By a formal asymptotic analysis, we conclue that the limit functions (u, ϕ, v, µ, θ) can be characterize as the solutions of a free bounary problem on the surface, which is again couple to a iffusion equation in the bulk B. Other examples for this metho an more etails on formal asymptotic analysis can be foun in [2, 4, 9, 19, 21] which is by far not a comprehensive list of references. The obtaine limit problem escribes a time-epenent partition of the surface into ifferent phase regions (4.1) + (t) := {ϕ(, t) = 1} an (t) := {ϕ(, t) = 1} an the ynamic of the interface γ(t) between these two regions. We enote the corresponing separation of the surface-time omain (0, T ] by ± := {(x, t) (0, T ] : x ± (t)}. The limit problem then takes the following form. Sharp Interface Problem 4.1. The sharp interface moel obtaine from the formal asymptotic analysis is given by (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) ϕ = ±1 on ±, t u = D u in B (0, T ], D u ν = q := c 1 u(1 v) c 2 v on (0, T ], µ = 0 on ±, t v = θ + q on ±, θ = 2 δ (2v 1 1) on ±, 2µ + θ = c 0 κ g on γ, (4.9) [µ] + = 0 on γ, (4.10) [θ] + = 0 on γ, (4.11) 2V = [ µ] + ν γ on γ, (4.12) V = [ θ] + ν γ on γ,

15 14 H. GARCKE, J. KAMPMANN, A. RÄTZ, AND M. RÖGER where [ ] + is the jump across the interface γ an ν γ(x 0, t 0 ) T x0 enotes the unit normal to γ(t 0 ) in x 0 γ(t 0 ), pointing insie + (t 0 ). The geoesic curvature of γ(t) in is enote by κ g (, t) an V(x 0, t 0 ) enotes the normal velocity of γ(t 0 ) in x 0 γ(t 0 ) in irection of ν γ (x 0, t 0 ). For its precise efinition, let γ t : U γ(t), t (t 0 δ, t 0 + δ) be a smoothly evolving family of local parameterizations of the curves γ(t) by arc length over an open interval U R an let γ t0 (s 0 ) = x 0. Then the normal velocity in (x 0, t 0 ) is given by V(x 0, t 0 ) = γ t (s 0 ) ν γ (x 0, t 0 ), t t0 see also [11]. We first euce the existence of transition layers between the phase regions. By assuming that the functions (u ε, ϕ ε, v ε, µ ε, θ ε ) amit suitable expansions with respect to the parameter ε in the transitions layers an in the regions away from the interface respectively, we can then euce that the limit functions at least formally nee to fulfill (4.2) (4.12) Asymptotic analysis: Existence of transition layers an outer expansion. We start our analysis by expaning the solutions to the couple moel in the outer regions, where ϕ ε attains values away from zero. We assume that in these regions all functions in (1.2) (1.7) have expansions of the form f ε = ε k f k k=0 where f ε = ϕ ε, v ε,..., etc. Since we postulate the existence of ifferent phase regions characterize by the values of the limit function ϕ, we shoul first aress the existence of these phase regions. To this en, we collect all terms of orer ε 1 in (1.2) an obtain W (ϕ 0 ) = 0 an as a consequence we obtain ϕ 0 = ±1 as the only stable solutions. Since ϕ 0 is the ominant term in the assume expansion as ε 0, we euce ϕ ε ±1 as ε 0 an the existence of the claime transition layers. This justifies (4.1) an (4.2). The iscussion of equations (1.3) an (1.4) (1.7) is then straightforwar. Comparing the terms of orer O(1) in their corresponing equations allows us to euce (4.3) (4.7). Due to the (possibly steep) transition between the regions + an near the interface, the spatial erivatives occurring in the system might contribute terms which are not necessarily of orer O(1) (with respect to ε) in a neighborhoo of the interface. This motivates the nee for a more etaile analysis of the functions in the neighborhoo of the interface which we will aress in Section Coorinates for a neighborhoo of the interface. As state above, we suppose that the zero level sets of ϕ ε converge to some (smooth) curve γ(t) with inner (wrt. + (t)) unit normal fiel ν γ (, t). We then introuce on a small tubular neighborhoo N of γ(t) a new coorinate system which is more suitable for the analysis in the transition layer. We remark that the construction of these coorinates presente here is more complicate which is ue to the fact that N is a neighborhoo of γ(t) in the manifol. For a similar example, we refer the reaer to [17]. As above, let γ t : U γ(t) be a local parametrization of the curve γ(t) by arc length over an open interval U R. It is then possible to exten γ t to a local parametrization Ψ t of N by means of the exponential map from ifferential geometry. While etails on this map can be foun in the literature [12, 13], it is sufficient for our purpose to quickly recall its efinition. For a given point p an a vector a T p, there exists a unique geoesic curve c a such that c a (0) = p an c a (0) = a. The exponential map exp p in p is then efine for all a T p for which c a (1) exists an is given by exp p ( a) = c a (1). Note that for z [0, 1] one can easily check exp p (z a) = c a (z).

16 SURFACE CAHN-HILLIARD MODELING RAFT FORMATION 15 The istance between a point x an the interface γ(t) is efine as (x, t) := inf{l(c) c : I, c connects x an γ(t)} where l(c) enotes the length of the curve c. Setting Ψ t (s, r) = exp γt(s)(rν γ (γ t (s), t)), we obtain that Ψ t is a parametrization of a neighborhoo N(t) of γ(t). If we choose N(t) small enough, the properties of the exponential map imply that r is the signe istance between the point Ψ t (s, r) an γ(t), that is for (x, t) = Ψ t (s, r) N(t) we have { r = (x, (Ψ t (s, r), t) if r + (t) t) := (Ψ t (s, r), t) if r (t). For the asymptotic analysis, it is necessary to aapt the parametrization to the length scale of the transition layers. We therefore use the rescale parametrization (4.13) Λ(s, z, t) = Ψ t (s, εz) of N(t), where z = r ε. In particular, z can be written as a function of x an t. Let us remark that as γ(t) is the zero level set of the signe istance function, the tangent space of γ(t) is the subspace of the tangent space of which is orthogonal to the surface graient an thus the normal vector νγ of γ(t) is given by. Because of we can thus compute 0 = t (γ t (s), t) = (γt (s), t) t γ t (s) + t (γt (s), t) t γ t (s) ν = t (γ t (s), t) (γt (s), t) an therefore the time erivative t z on γ(t) fulfills t z = ε 1 V. Remark 4.2 (Graient, Divergence an Laplace Operator in the new coorinates). From the efinition of Λ in (4.13) we see (4.14) (4.15) (4.16) Λ(s, 0, t) = γ t (s), s Λ(s, 0, t) = s γ t (s) = γ t(s) an z Λ(s, 0, t) = εν γ (γ t (s)). Furthermore, the curve r Ψ t (s, r) is geoesic by efinition an hence we have P zz Λ(s, z, t) = 0 where P is the projection on the tangent space T Λ(s,z,t) on in Λ(s, z, t). These observations allow us to calculate z ( s Λ(s, z, t) z Λ(s, z, t)) = zs Λ(s, z, t) z Λ(s, z, t) + s Λ(s, z, t) zz Λ(s, z, t) = 1 2 s z Λ(s, z, t) 2 = 0 since z Λ(s, z, t) 2 = ε 2 by efinition. The equations (4.15) an (4.16) imply s Λ(s, 0, t) z Λ(s, 0, t) = 0, which yiels (4.17) s Λ(s, z, t) z Λ(s, z, t) = 0.

17 16 H. GARCKE, J. KAMPMANN, A. RÄTZ, AND M. RÖGER For simplification of the following calculations, we enote the variables s an z by s 1 an s 2 respectively. Given the arguments above, the metric tensor with respect to Λ is given by The matrix g 11 = g ss = s Λ s Λ, g 12 = g 21 = g sz = g zs = s Λ z Λ = 0 an g 22 = g zz = z Λ z Λ = ε 2. G := ( ) g11 g 12 g 21 g 22 is thus iagonal an as usual we enote the entries of its inverse G 1 by g ij. For a scalar function h(x, t) = ĥ(s(t, x), z(t, x), t) on N, the surface graient on N is hence expresse by (4.18) where γε (4.19) 2 h = g ij si ĥ sj Λ = g 11 s1 ĥ s1 Λ + 1 ε 2 s 2 ĥ s2 Λ = γε ĥ + 1 ε zĥ zλ i,j=1 is the surface graient on γ ε = {Λ(s, z, t) s U} for a fixe z [0, 1]. Similarly, a = γε â + 1 ε zâ z Λ for some vector value function a(x, t) = â(s(x, t), z(x, t), t). In analogy to the appenix in [2], we calculate the Laplace-Beltrami operator in the new coorinates. Due to the properties of the parametrization Λ(s, z, t) which alreay lea to the iagonal structure of the metric tensor above, we fin γε ĥ z Λ = 0 an γε ĥ zz Λ = 0. Hence ( z γε ĥ ) z Λ = ( z γε ĥ ) ) z Λ + γε ĥ zz Λ = z ( γε ĥ z Λ = 0 an substituting (4.18) in (4.19) we thus compute h = γε ĥ + 1 ( ) γε z ĥ z Λ + 1 ε ε zĥ γ ε z Λ + 1 ( ) zγ ε ε ĥ z Λ + 1 ε 2 zzĥ zλ + 1 ε 2 zĥ zzλ z Λ (4.20) = γε ĥ + 1 ε zĥ γ ε z Λ + 1 ε 2 zzĥ zλ where we have use the ientities above. Since ss Λ s Λ = 1 2 s ( s Λ s Λ) = 0, the curvature vector ss Λ of γ(t) (seen as a curve in R 3 ) is an element in span ( z Λ, ν ) where ν enotes the irection normal to the surface. The geoesic curvature κ g of γ(t) in is therefore given by As in [2], one can erive κ g = ss Λ z Λ = s ( s Λ z Λ) s Λ sz Λ = γε z Λ. γε ĥ = γ ĥ + R ε γε ĥ = γ ĥ + R ε γε ĥ = γ ĥ + R ε

18 SURFACE CAHN-HILLIARD MODELING RAFT FORMATION 17 where R ε is of higher orer in ε. Thus (4.20) reas h = γε ĥ 1 ε κ g z ĥ + 1 ε 2 zzĥ zλ (4.21) = γ ĥ 1 ε κ g z ĥ + 1 ε 2 zzĥ zλ + R ε Asymptotic analysis: Inner expansion. We assume now that the functions in (1.2) - (1.7) have inner expansions on N with respect to the new variables of the form f ε (x, t) = F (z, s, t; ε) = ε k F k (z, s, t) where again f ε = ϕ ε, v ε,... etc. Accoringly, the inner expansions for (v, φ, µ, θ) will be enote by (V, Φ, M, Θ). In orer for the inner an outer expansions to be consistent which each other, we prescribe the following matching conitions as z ± (again we use F as an placeholer for (V, Φ, M, Θ)) k=0 (4.22) F 0 (t, s, ± ) f 0 ± (x, t), (4.23) z F 0 (t, s, ± ) 0, (4.24) z F 1 (t, s, ± ) f ± 0 (x, t) ν γ where (x, t) = Λ(0, s, t) an f ± 0 (x, t) = lim δ 0 f 0 (exp x (±δν γ ), t). We refer the reaer to [9, 21] an [19] for a erivation of these matching conitions. We plug these asymptotic expansions in the equations (1.2)-(1.7) an use the results from Remark 4.2 where necessary. Again we collect all terms with the same orer in ε. As we assume that the limes ε 0 exists, we require that the terms associate with the leaing orers in ε cancel out. The terms of orer ε 1 in equation (1.2) hence yiel the ifferential equation zz Φ 0 + W (Φ 0 ) = 0 for Φ 0. Together with the matching conitions an the conition Φ 0 (0) = 0 we euce (4.25) Φ 0 (z) = tanh(z) an 1 2 zφ 0 2 (z) = W (Φ 0 )(z). Looking at the conitions impose by terms of orer ε 2, equation (1.6) yiels zz M 0 = 0 an integrating this equation from to z implies that M 0 is inepenent from z if we take the matching conition (4.23) into account. Thus M 0 (z = + ) = M 0 (z = ), which in turn gives (4.9). In a similar way, we obtain equation (4.10). The terms of orer ε 2 in (1.7) inee imply zz Θ 0 = 0 an integrating in z yiels (4.26) z Θ 0 = 0 by the matching conitions. Again we euce Θ 0 (z = + ) = Θ 0 (z = ) an (4.22) yiels (4.10). Let us observe for later use that [θ] + = 0 irectly implies (4.27) [v] + = 1 as we alreay saw that [ϕ] + = 2. A secon observation is motivate by the stuy of all terms of orer O(1) in (1.3). We can euce Θ 0 = 2 δ (2V 0 1 Φ 0 )

19 18 H. GARCKE, J. KAMPMANN, A. RÄTZ, AND M. RÖGER an since z Θ 0 = 0 by (4.26) we therefore obtain (4.28) 2 z V 0 (z, s, t) = z Φ 0 (z). Substituting the inner expansions in (1.6) also yiels terms of orer ε 1. The resulting equation reas V z Φ 0 = zz M 1. Equation (4.11) is then obtaine from the matching conitions by integrating in z. Next, we stuy the terms of orer O(1) in (1.2). Similar to the stuies on the relate Cahn- Hilliar equation we obtain M 0 = zz Φ 1 + κ g z Φ 0 + W (Φ 0 )Φ 1 1 δ (2V 0 1 Φ 0 ) = zz Φ 1 + κ g z Φ 0 + W (Φ 0 )Φ Θ 0 an apply the following solvability conition for Φ 1 erive in [4, Lemma 2.2]. Lemma 4.3 ([4]). Let A(z) be a boune function on < z <. Then the problem has a solution if an only if zz φ + W (F 0 (z))φ = A(z) z R, φ(0) = 0 R A(z)F 0(z) z = 0. φ L (R), It is inee easy to see that this conition is necessary if one multiplies the equation by F 0 an uses integration by parts. The assertion that the conition is also sufficient can be erive from the metho of variation of constants, etails are given in [4]. Lemma 4.3 irectly yiels where c 0 is given by 2M 0 = c 0 κ g 1 2 c 0 := Θ 0 z Φ 0 z z Φ 0 2 z. Given the fact that Θ 0 is inepenent of z (see (4.26)) we euce We thus conclue Θ 0 z Φ 0 z = Θ 0 z Φ 0 z = 2Θ 0. 2µ + θ = c 0 κ g as we claime in (4.8). In orer to conclue our analysis, we collect all terms of orer ε 1 from equation (1.7). We thus have V z V 0 = zz Θ 1 an another integration with respect to z allows us to euce V [v] + = [ θ] + ν γ from the matching conitions. Our observation (4.27) on [v] + above hence implies (4.12).

20 SURFACE CAHN-HILLIARD MODELING RAFT FORMATION Free energy inequalities an conservation properties. The iscussion of the thermoynamical backgroun in Section 2 can be extene to the sharp interface moel (4.2) (4.12) erive above if one consiers the surface free energy F(ϕ, v, γ) := c 0 1 H (4.29) (2v 1 ϕ) 2 H 2 γ 2δ = c 0 1 H [ ] (2v) 2 H 2 + (2v 2) 2 H 2 γ 2δ + an the bulk free energy (4.30) F b (u) := 1 2 B u 2 x. In particular, choosing these energies the energy inequality erive in Lemma 2.1 for the iffuse interface moel from the secon law of thermoynamics has a counterpart in the sharp interface moel, i.e. t [F(ϕ, v, γ) + F b(u)] q(θ u) H 2. As the following calculations show, this is mostly ue to the relations specifie in the equations (4.8) (4.12) on γ. They come into play since (4.31) t γ 1 H 1 = γ κ g V H 1. We refer the reaer to Section 5.4 in [12] for a erivation of this ientity. Let us now calculate the time erivative of (4.29) in orer to erive the esire energy inequality. Reynols transport theorem an (4.31) imply (4.32) t F(ϕ, v, γ) = c 0 κ g V H ( ) δ 2 γ 2δ t 2 θ H 2 = c 0 κ g V H 1 + δ ( ) θ 2 H 2 + θ 2 H 2 γ 8 t + = c 0 κ g V H 1 + δ θ t θ H 2 + δ θ t θ H 2. γ We have use equation (4.10) to erive the last equality. Because of (4.8), the first integral c 0 γ κ gv H 1 coincies with γ (2µ + θ) V H1. Using (4.11) an (4.12) yiels c 0 κ g V H 1 = µ [ µ] + ν γ H 1 θ [ θ] + ν γ H 1 γ γ γ (4.33) = µ µ + ν γ H 1 + µ µ ν γ H 1. γ γ θ θ + ν γ H 1 + θ θ ν γ H 1. γ By the ivergence theorem for tangential vector fiels on manifols, we see that all integrals equate to integrals over + an respectively. That is, we fin µ µ + ν γ H 1 = iv (µ µ) H 2 γ + = µ 2 H 2 µ µ H = µ 2 H 2. + γ

21 20 H. GARCKE, J. KAMPMANN, A. RÄTZ, AND M. RÖGER The last equation hols because of (4.5). Keeping in min that the orientation of γ seen as the bounary of + iffers from the orientation of γ seen as the bounary of, similar calculations lea to µ µ ν γ H 1 = µ 2 H 2. γ For the other integrals in (4.33), we use (4.6) an (4.7) to calculate θ θ + ν γ H 1 = θ 2 H 2 θ θ H 2 γ + + = θ 2 H 2 θ ( t v q) H = θ 2 H 2 δ θ t θ H 2 + θq H an the analogue result θ θ ν γ H 1 = θ 2 H 2 δ θ t θ H 2 + θq H 2. γ 4 Plugging these finings in (4.33) yiels c 0 κ g V H 1 = µ 2 H 2 γ an by (4.32) we euce F(ϕ, v, γ) = t θ 2 H 2 δ 4 θ t θ H 2 + θq H 2 µ 2 H 2 θ 2 H 2 + θq H 2 θq H 2. Similarly to the erivation of Lemma 2.1, we also fin t F b(u) uq H 2. This leas to the free energy inequality for the sharp interface moel state in 4.1, (4.34) t [F(ϕ, v, γ) + F b(u)] q(θ u) H 2. Furthermore, equation (4.2) an Reynols transport theorem allow us to euce ϕ H 2 = 1 H 2 (4.35) 1 H 2 t t + (t) t (t) = V ν γ H 2 V ν γ H 2 = 0. γ(t) We remark that the signs in the above equation epen on the values of ϕ in + (t) an (t) respectively as well as the fact that γ(t) is seen as the bounary of + (t) for the first integral an as the bounary of (t) for the secon integral. For energy ensities chosen accoring to (4.29) an (4.30), equations (4.3) an (4.4) correspon to the mass balance equation for cytosolic cholesterol (2.11), while the mass balance equation (2.2) for the membrane boun cholesterol correspons to (4.6). We use these equations to euce (4.36) t [ B u x + γ(t) v H 2 ] = 0. Equations (4.35) an (4.36) show that the conservation laws erive in Lemma 2.1 hol for the sharp interface moel as well. Equation (4.34) is the analogue to the general energy inequality in

22 SURFACE CAHN-HILLIARD MODELING RAFT FORMATION 21 Lemma 2.1 for the sharp interface moel, if one chooses the constitutive relations in such a way that the resulting free energy ensities lea to (4.29) an (4.30). 5. Numerical simulations In this section we present numerical results for the reuce non-local system (3.8) (3.11) an for the fully couple bulk surface moel (1.2) (1.8). For the first, we propose a iscretization which is semi-implicit in time an which is base on surface finite elements in space [14, 15]. The etails of the iscretization are shown in Section 5.1. We valiate the numerical metho in Section 5.2 with two benchmark tests an subsequently, in Section 5.3, we present simulation results showing properties of the moel. Moreover, in Section 5.4, we present a numerical simulation for a iffuse interface approximation of the fully couple bulk surface moel (1.2) (1.8). For a large bulk iffusion coefficient, the results show a goo agreement with those obtaine for the reuce moel in Section 5.3. This further justifies to use numerical simulations for the reuce moel in orer to stuy the properties of the original moel. Cahn Hilliar systems on stationary surfaces have been numerically investigate by parametric finite elements [43], level set techniques [23] an with iffuse interface methos [43] mostly applie for the simulation of spinoal ecomposition an coarsening scenarios. For corresponing results on moving surfaces we refer to [16, 36] (sharp interface approach) an [52] (iffuse interface approach), respectively. Recently, a Cahn Hilliar like system has been numerically stuie for the simulation of the influence of membrane proteins on phase separation an coarsening on cell membranes [53] Surface FEM Discretization. For the iscretization of the reuce system (3.8) (3.11) with q given by (1.8) we evelop a scheme similar to the one in [41], where a relate non-local reaction iffusion system has been numerically treate. To iscretize in time, we introuce steps τ m, m = 1,..., M T. Denoting by ϕ (m), µ (m), v (m), u (m) the time iscrete solution at time t m = m τ l, the time iscrete weak formulation then reas (5.1) (5.2) (5.3) l=1 ϕ (m+1) ψ + µ (m+1) ϕ (m) ψ = ψ, τ m+1 τ m+1 µ (m+1) η + ε ϕ (m+1) η + 1 W (ϕ (m) )ϕ (m+1) η 2 ε δ + 1 ϕ (m+1) η = 1 δ ε (W (ϕ (m) )ϕ (m) W (ϕ (m) ))η 1 η, δ v (m+1) ζ + 4 v (m+1) ζ 2 ϕ (m+1) ζ τ m+1 δ δ + (c 1 u (m) + c 2 )v (m+1) v (m) ζ = ζ + c 1 u (m) ζ τ m+1 for all ψ, η, ζ H 1 (), where the non-local term (5.4) u (m) = 1 B ( M v (m) ) v (m+1) η is treate fully explicitly. For the spatial iscretization, we introuce a triangulation h of the surface, an we use linear surface finite elements to obtain from (5.1) (5.4) a linear system of equations for the coefficients of the iscrete solutions µ (m+1) h, ϕ (m+1) h, v (m+1) h with respect to the stanar Lagrange basis. The resulting linear system is solve by a irect solver or a stabilize bi-conjugate graient (BiCGStab) metho epening on the number of unknowns. Furthermore, we apply a simple time aaptation

23 22 H. GARCKE, J. KAMPMANN, A. RÄTZ, AND M. RÖGER strategy, where time steps τ m+1 are chosen (within bouns) inversely proportional to ϕ (m) h (x) ϕ(m 1) h (x) max, x h τ m see e.g. [43]. The above numerical scheme is implemente in the aaptive FEM toolbox AMDiS [50] Benchmark tests. In this section we consier a triangulation h = Sh 2 of the unit sphere S 2, an we use the following set of parameters: (5.5) c 1 = c 2 = 500, ε = 0.02, M = 5 B = 20π 3, δ = Analytic solution for a prescribe right han sie. In orer to verify the valiity of the above numerical scheme, we efine right han sies for (3.8) an (3.10) such that an analytic solution for the resulting system can be given. To be more precise, we let ( ) ϑ(x) + β t (5.6) ϕ(x, t) := tanh 2ε for x S 2 an t [0, T ). Moreover, β is an angle relate to the initial value ϕ(x, 0), an by ϑ = ϑ(x) we enote the angle given by ϑ(x) := arccos(x 3 ) [0, π] for x = (x 1, x 2, x 3 ) S 2. Furthermore, we let v(x, t) := 1 2 (1 + ϕ(x, t)) such that θ = 0 hols. Then we efine F 1, F 2 : S 2 [0, T ] R such that (5.7) (5.8) (5.9) hol with u = u(t) given by t ϕ µ = F 1 on S 2 (0, T ], µ = ε ϕ + ε 1 W (ϕ) on S 2 (0, T ], t v q(u, v) = F 2 on S 2 (0, T ] (5.10) u = 3 4π ( ) M v. S 2 In this way we obtain expressions for F 1 an F 2, which are incorporate in the numerical scheme escribe in Section 5.1. Finally, we use the aopte scheme to fin approximate numerical solutions ϕ h, µ h, v h to (5.7) (5.10) for initial conitions for ϕ h, v h given by corresponing values etermine by (5.6) an v = 1 2 (1 + ϕ). Note that the expression for F 2 involves an integral of ϕ which is numerically approximate in polar coorinates in every time step. With the usual Lagrange interpolation operator I h, we efine the relative errors e (t) := I hϕ(, t) ϕ h (, t) L (Sh 2), ϕ(, t) L (S 2 ) e 1 (t) := I hϕ(, t) ϕ h (, t) H 1 (S 2 h ) I h ϕ(, t) H 1 (S 2 h ) of ϕ h (, t) in the L (Sh 2)-, H1 (Sh 2 )-norms, respectively, an we consier their values for simulations with T = π 4, β = π 4 an ifferent spatial resolutions in Table 1. The results inicate a linear epenence of the L (Sh 2)- an H1 (Sh 2 )-errors, respectively, on the gri-size, as expecte for elliptic problems. As a further illustration of this benchmark, in Figure 1, we see contour plots of ϕ(, 0) an ϕ(, T ), respectively. We remark that in the following we use a gri with vertices for all simulations with spherical geometry.

24 SURFACE CAHN-HILLIARD MODELING RAFT FORMATION 23 number of vertices e (T ) e 1 (T ) Table 1. Relative errors for simulations of (5.7) (5.10) with ifferent resolutions. ϕ(t = 0) ϕ(t π 4 ) Figure 1. Benchmark simulation with prescribe right han sie in (5.7) (5.9). Contour plots of initial values of analytic solution (5.6) an of values at time T Valiation of v. We return to (3.8) (3.11) an remark that one easily verifies from (3.12) that z := v fulfills the ODE (5.11) ż = c 1 M B We compare solutions of the above ODE with ( ) c 1 B + c M 2 + c 1 z + c 1 B B z2. obtaine by numerical solution of (5.1) (5.4), where we have chosen the initial conitions S 2 h v h ϕ(, 0) = R, v(, 0) = Thereby, R : h [ 0.001, 0.001] provies a perturbation given by an (irregular an nonperioic) oscillation aroun zero. In Figure 2, z h = v h obtaine by numerical simulation of (5.1) (5.4) is compare with a numerical solution of the ODE (5.11) with initial conition z(0) = v(, 0) S 2 S 2 h obtaine with Maple TM [1]. The excellent agreement further justifies the chosen numerical scheme Simulation results Variation of parameters. We use the basic set of parameters given in (5.5). Furthermore, we use the initial conitions ϕ(, 0) = ϕ+r, v(, 0) = 1 4, where ϕ = 0.5 an R : h [ 0.001, 0.001] again enotes a perturbation by an (irregular an nonperioic) oscillation aroun zero. In Figure 3, we see corresponing results, where contour plots of the iscrete solutions ϕ h (, t) (first row) an v h (, t) (secon row) are isplaye for various times t. The evolution shows a spinoal ecomposition with subsequent interrupte coarsening scenario. The geometric shape of the particles can rastically change, if one changes the values of ϕ(, 0). In a secon example, we have use ϕ(, 0) = 0 + R, while all remaining parameters have not been change. The corresponing numerical results can be seen in Figure 4.

25 24 H. GARCKE, J. KAMPMANN, A. RA TZ, AND M. RO GER z reuce system ODE for z t Figure 2. Comparison of a numerical solution of the ODE (5.11) with 2 Sh vh obtaine by simulation of (3.8) (3.11). ϕh (, t = 0) ϕh (, t ) ϕh (, t ) ϕh (, t ) ϕh (, t ) vh (, t = 0) vh (, t ) vh (, t ) vh (, t ) vh (, t ) Figure 3. Numerical results for ϕ(, 0) = R. First row: contour plots of ϕ(, t) for several choices of times t; secon row: corresponing contour plots of v(, t). In the following, we compare almost stationary states for simulations with varying parameters, in orer to illustrate the properties of the moel an its solutions. We use the basic set of parameters for the results shown in Figure 3 an moify one parameter. Let ϕ := ϕ(, 0) = ϕ. Varying ϕ. We change ϕ by changing the initial conition for ϕ, or rather ϕ. In Figure 5 one observes the almost stationary states from the previous two examples with ϕ = 0 an ϕ = 0.5 an examples with intermeiate values, where one can see a crossover from circular lipi rafts to stripe like patterns. For ϕ = 0.75 the system allows for a constant stationary state with orer parameter away from the classical phases ϕ { 1, 1}. Varying δ. For ϕ = 0.5 we consier almost stationary iscrete orer parameter ϕh for ifferent values for the parameter δ. In Figure 6, for large δ the almost stationary ϕh has one lipi raft, as one woul expect for classical Cahn Hilliar system on the sphere. For ecreasing but still positive values of δ we expect to approach stationary solution of Ohta Kawasaki base ynamics, as shown in Section 3.4. For a more etaile comparison with almost stationary states of Ohta Kawasaki base ynamics we refer to the subsequent Section

26 SURFACE CAHN-HILLIARD MODELING RAFT FORMATION 25 ϕh (, t = 0) ϕh (, t ) ϕh (, t ) ϕh (, t ) ϕh (, t ) vh (, t = 0) vh (, t ) vh (, t ) vh (, t ) vh (, t ) Figure 4. Numerical results for ϕ(, 0) = 0 + R. First row: contour plots of ϕ(, t) for several choices of times t; secon row: corresponing contour plots of v(, t). ϕ = 0.1 ϕ = 0 ϕ = 0.25 ϕ = 0.5 ϕ = 0.75 Figure 5. Almost stationary iscrete solutions ϕh for ifferent values of ϕ. δ = 0.3 δ = 0.1 δ = 0.02 δ = Figure 6. Almost stationary iscrete solutions ϕh for ifferent values of δ. Varying c1. Returning to the previous stanar parameters ϕ = 0.5 an δ = 0.02 we investigate the influence of ifferent values of c1 on the almost stationary iscrete states. Increasing c1 correspons to increasing the number of lipi rafts as shown in Figure 7. Varying c2. With our stanar choice c1 = 500 we obtain for variation of values for c2 similar results as for the previous examples (varying c1 ). In Figure 8, one observes increasing number of rafts of ecreasing sizes for increasing c2. From the analysis in Section 3.4 we conclue that increasing c1 or c2 correspons to increasing the non-local energy contribution in the Ohta Kawasaki functional (3.22), see also (3.23), which is expecte to escribe the ynamics in the limit δ 0. This fact provies an explanation of the increasing number of particles for increasing c1, c2.

27 26 H. GARCKE, J. KAMPMANN, A. RA TZ, AND M. RO GER c1 = 5 c1 = 100 c1 = 500 c1 = 2000 Figure 7. Almost stationary iscrete solutions ϕh for ifferent values of c1. c2 = 5 c2 = 100 c2 = 500 c2 = 2000 Figure 8. Almost stationary iscrete solutions ϕh for ifferent values of c Comparison with Ohta Kawasaki moel. For the result in Figure 9 we have investigate the following scenario. We run a simulation of (3.8) (3.11) with parameter values as in (5.5) except δ = towars an almost stationary state (see Figure 9, left). With the resulting iscrete orer parameter ϕh we continue a simulation with Ohta Kawasaki-base ynamics with parameters accoring to (3.23) an (3.24) again towars an almost stationary state (see Figure 9, mile). The ifference between these two stationary results for ϕh is isplaye in Figure 9 (right). These results confirm the analytic results of Section 3.4. Figure 9. Almost stationary ϕh obtaine from a simulation of the reuce system (left) an from subsequent Ohta Kawasaki-base ynamics (mile), ifference between the previous numerical solutions (right) Non spherical membrane shape. In a further example, we show in Figure 10 results with the stanar parameter set (5.5) for the case that is not a sphere. One obtains similar patterns as for the sphere. However, this configuration appears to be more stable than the previous spherical ones, where sometimes very slow arrangements coul take place. This behavior has not been observe for this geometry. We remark that for this simulation, we have chosen a similar resolution as for the previous examples with spherical geometry Comparison with energy ecreasing ynamics. Here, we stuy a choice of q leaing to an energy ecreasing evolution. From the analysis in Section 3.3, we expect that solutions of the system (3.8) (3.11) generically converge to configurations with one lipi phase concentrate in a single geoesic ball. For this purpose, we choose q = c(θ u)

28 SURFACE CAHN-HILLIARD MODELING RAFT FORMATION ϕh (, t = 0) ϕh (, t ) ϕh (, t ) 27 ϕh (, t ) Figure 10. Numerical results for a non spherical surface an ϕ(, 0) = R. Contour plots of ϕ(, t) for several choices of times t. with c = 500, corresponing to (2.17) an fb (u) = 21 u2 in (2.15). The results in this case are isplaye in Figure 11 an illustrate the evolution towars an almost stationary state with a single spherical raft particle. ϕh (, t = 0) ϕh (, t ) ϕh (, t ) ϕh (, t ) ϕh (, t ) Figure 11. Numerical results for a choice of q leaing to an energy ecreasing evolution an ϕ(, 0) = R. Contour plots of ϕ(, t) for several choices of times t Simulation of the full moel. For the simulation of the couple bulk surface moel (1.2) (1.8), we propose a iffuse interface approximation base on the iffuse approaches for the treatment of PDE s on an insie close surfaces in [43] an [33], respectively. For a relate phasefiel escription of a couple bulk surface reaction iffusion system, we refer to [42]. Moreover, a further iffuse interface escription of a couple bulk surface system has been given in [47], an in [3] a iffuse interface approach of a linear couple elliptic PDE system has been analyze with respect well-poseness of the iffuse system an convergence towars its sharp interface counterpart. We choose a omain Ω R3 containing B an provie a phase-fiel approximation of (1.2) (1.7) by (5.12) (5.13) ψ t u = D (ψ u) ε 1 g b(ψ)q(u, v) in Ω (0, T ], b(ψ) t ϕ = (b(ψ) µ) in Ω (0, T ], b(ψ)µ = ε (b(ψ) ϕ) + ε b(ψ)w (ϕ) δ b(ψ)(2v 1 ϕ) in Ω (0, T ], 4 2 (5.15) b(ψ) t v = (b(ψ) v) (b(ψ) ϕ) + b(ψ)q(u, v) in Ω (0, T ], δ δ where the phase-fiel function ψ : Ω R escribing the geometry, that is B an, is given by 1 3r(x) (5.16) ψ(x) := 1 tanh 2 εg (5.14) for a (small) real number εg > 0 an a signe istance r to being negative in B. Note that ψ is obtaine by smearing out the characteristic function χb of B on a length proportional to εg. Moreover, b = b(ψ) := 36ψ 2 (1 ψ)2 is small outsie the iffuse interface region. We use a semi-implicit aaptive FEM iscretization (see [42]) an choose Ω = ( 2, 2)3 with all solutions assume Ω-perioic, εg = 81, D = 100. All otherwise egenerate mobility functions appearing in secon orer terms in (5.12) (5.15) are regularize by aition of a parameter

29 28 H. GARCKE, J. KAMPMANN, A. RÄTZ, AND M. RÖGER δ r = The resulting linear system is solve by a stabilize bi-conjugate graient (BiCGStab) solver. The numerical scheme is implemente in AMDiS [50]. Moreover, we apply the parameters in (5.5) an initial conitions as in Section 5.3 an aitionally u(, 0) = 4.25 corresponing to M = 20π 3 in (5.5). In Figure 12 we see plots of ϕ h (, t) {ψh = 1 2 } for various times t, where ψ h is obtaine by replacing r by a iscrete signe istance r h in (5.16). The results are in goo qualitative agreement with the results in Section 5.3 (see Figure 3) an hence further justify the reuce moel formally obtaine in the limit D. ϕ h (, t = 0) ϕ h (, t ) ϕ h (, t ) ϕ h (, t ) ϕ h (, t ) Figure 12. Numerical results for iffuse interface approximation of the full moel (1.2) (1.8). Contour plots of ϕ(, t) {ψh = 1 } for several choices of times t Conclusions We have presente a moel for lipi raft formation that extens in particular the moel by Gómez, Sagués an Reigaa [22]. The key new aspect in our work is an explicit account for the cholesterol ynamics in the cytosol, which leas to a complex system of partial ifferential equations both in the bulk an on the cell membrane. These are couple by an outflow conition for the cytosolic cholesterol an a source term in the surface membrane equation, etermine by a constitutive relation. These consierations lea to an interesting an complex mathematical moel. The surface equations combine a Cahn Hilliar type evolution equation for an orer parameter an an equation for the surface cholesterol. The latter is a iffusion equation incluing a cross-iffusion term an an aitional (nonlocal) term. In the bulk we have a iffusion equation with a Robin-type bounary conition that couples both systems. We have shown that our moel can be erive from thermoynamical conservation laws an free energy inequalities for bulk an surface processes. Depening on the specific choice of the exchange term we obtain a total free energy ecrease or not, which can be seen as a istinction between equilibrium (close system) an non-equilibrium (open system) type moels. The analysis of both classes reveals striking ifferences in the behavior: whereas equilibrium-type moels only support the formation of macroomains an connecte phases, a prototypical choice of a non-equilibrium moel leas to the formation of raft-like structures. These finings are the result of both a (formal) qualitative mathematical analysis an careful numerical simulations. More precisely, in a certain parameter regime (large interaction strength between lipi an cholesterol) we have emonstrate that stationary states of the prototypical non-equilibrium moel are close to stationary states of the Otha Kawasaki moel, whereas in the case of equilibrium moels stationary states coincie with stationary states for a surface Cahn Hilliar equation. For a better unerstaning of the (complex) moel asymptotic reuctions are instrumental. We in particular justify the sharp interface limit of the moel, that is the reuction when the parameter ε (relate to the with of transition layers between the istinct lipi phases) tens to zero. Moreover, we have erive a simplifie moel by taking the cytosolic iffusion (typically much larger than the lateral membrane iffusion) to infinity. This reuction leas to a non-local moel that only inclues variables on the surface membrane.