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2 0 0 2 n Group analysis of the membrane shape a J 4 equation ] t f o ∗ G. De Matteis s . t a Dipartimento di Fisi a dell'Università di Le e, m Via Arnesano, CP. 193, I-73100 LECCE (Italy). - d n o 1st February 2008 c [ 1 v 4 Abstra t 4 0 A system of partial di(cid:27)erential equations des ribing bilayers am- 1 phiphili membranes is studied by the Lie group analysis. This algo- 0 2 rithmi approa h allows us to show all the symmetries of the system, 0 / to determine all possible symmetry redu tions, to re over the axisym- t a metri solutions, obtained by other euristi methods, and (cid:28)nally to m - address the question of new similarity solutions. d n o c 1 Introdu tion : v i X In the last de ades a great interest has been attra ted by the membraneous r a systems in biophysi s [1℄,[2℄. In parti ular the binary systems amphiphile- waterhasbeen onsidered[3℄. Inwatertheamphiphili mole ulesformsheets of ostant area, in the form of bilayers, be ause this minimizes the onta t of the hydro arbon tails with water. For a (cid:29)at pie e of membrane, the tails are still exposed to water onta t at the membranes edge. This leads to an e(cid:27)e tive line tension whi h, for large enough membranes, pulls the edges to- gether and loses the membrane, i.e. we get a vesi le. As reviewed in Se . 2, ∗ e-mail: dematt73yahoo.it,giovanni.dematteisle.infn.it 1 a detailed study of su h kind of membranes an be performed in a geometri al setting, following the Helfri h approa h. However most of the known analyti al solutions [4℄ an be obtained only under parti ular geometri al restri tions. So the aim of this arti le is to exploit a systemati group analysis (Se . 3), inorder to (cid:28)nd the symmetry properties of the system involving the shape equation and the Gauss-Codazzi-Mainardi equations. Then, in Se . 4 we use the above analysis in performingallthe possible symmetry redu tions and to determine lasses of parti ular solutions. Finally in Se . 5 we address some omments and (cid:28)nal remarks. 2 The Helfri h fun tional and the vesi les shape equations x(u1,u2) = Themembrane'sshapeisdes ribedbyathree- omponentve tor(cid:28)eld (x1(u1,u2),x2(u1,u2),x3(u1,u2)) depending on the two internal oordinates (u1,u2) of the membrane. To be self- onsistent, we introdu e a bit of the P geometri al on epts used below [6℄,[7℄,[8℄. The tangent plane to a point on the surfa e is spanned by two tangent ve tors t = ∂ x, α = 1,2 α α ∂ = ∂ where α ∂uα. The two tangent ve tors determine the metri tensor or the (cid:28)rst funda- Ω 1 mental form g (u1,u2) t t = ∂ x ∂ x, αβ α β α β ≡ · · (summation over repeated indi es is understood), by whi h we an express ds the in(cid:28)nitesimal element of distan e on the surfa e ds2 = g duαduβ. αβ dS The in(cid:28)nitesimal element of area is given by dS = detg du1du2. αβ p 2 g αβ The metri tensor hara terizes the intrinsi geometry of the surfa e. But for a omplete des ription of its embedding into eu lidean three- dimensional spa e we need the se ond fundamental form Ω = b duαduβ α,β = 1,2, 2 αβ where b (u1,u2) = (∂ ∂ x) n αβ α β · and t t n = 1 × 2 t t 1 2 k × k R3 is the unit normal ve tor on the surfa e. Embedding of the surfa e into is di tated by the Gauss-Weingarten equations (GWE) ∂ n = gβγb ∂ x α αγ β − (α,β,γ = 1,2), ( ∂α∂βx = bαβn+Γσαβ∂σx (1) gβγ = (g 1) Γσ where − βγ and αβ are the so- alled Christo(cid:27)el symbols 1 ∂g ∂g ∂g Γσ = gσδ αδ + βδ αβ . αβ 2 ∂uβ ∂uα − ∂uδ (cid:18) (cid:19) The two most important s alar quantities on the surfa e are the gaussian K H urvature and the mean urvature , given by the determinant and the bβ = gβγb : α αγ half-tra e of the matrix 1 K = det(bβ) = , α R R (2) 1 2 1 1 1 1 H = tr(bβ) = + , 2 α 2 R R (3) (cid:18) 1 2(cid:19) R R 1 2 where and are the prin ipal radii of urvature and they de(cid:28)ne the extrinsi properties of the surfa e. Inspired by the physi s of liquid rystals, Helfri h [5℄ onsidered the n (n1,n2,n3) u1 u2 derivatives of the unit ve tor ≡ with respe t to and as the urvature strains. The length and energy s ales are not the size of the 3 mole ular head group, or the mole ular energy involved in assembling the amphiphili sheet. Indeed the basi energy is that one needed to bend the sheet, i.e. an energy whi h arises mi ros opi ally from a hange in the lo al on entration of head and tail groups of the amphiphiles. Over short distan es, this energy will ensure that the membrane is rela- tively (cid:29)at. However, over larger distan es, the e(cid:27)e t of thermal (cid:29)u tuations is to hange appre iably the orientation of the membrane. The s ale of distan es at whi h an orientation orrelation o urs is alled the persisten e lenght. It an be al ulated from the free energy des ribing a system of sheets. We assume that the free energy whi h des ribes the deformations of the membrane depends on its shape only. The lo al free energy per unit area is written as an expansion in powers of lo al urvatures, and the integrated free energy depends parametri allyon theunknown oe(cid:30) ients inthis expansion. These oe(cid:30) ients arethe spontaneus, ornatural, urvature ofthe membrane, and its bending and saddle-splay moduli. Moreover, the free energy must be invariant under translations and rotations and under reparametrization transformations of the lo al oordinates, be ause of the (cid:29)uidity assumption. Thus, only the invariant linear and quadrati forms an enter into, namely ∂n1 ∂n2 ∂n1 ∂n2 2 ∂n1∂n2 ∂n1 ∂n2 + , + , . ∂u1 ∂u2 ∂u1 ∂u2 ∂u1 ∂u2 − ∂u2 ∂u1 (cid:18) (cid:19) Then, the elasti free energy takes the form k ∂n1 ∂n2 2 ∂n1 ∂n2 ∂n1∂n2 F = + c dA+k dA, c 4 ∂u1 ∂u2 − 0 ∂u1 ∂u2 − ∂u2 ∂u1 ZS(cid:18) (cid:19) ZS (cid:18) (cid:19) k k where the so- alled bending rigidity and the gaussian rigidity are the elasti moduli. Making use of the GWE (1) and of the expressions (2-3) for K H F c and , an be written as F = k (H H )2dS +k KdS c s − ZS ZS H = c0 where s − 2 is alled spontaneus urvature and takes a ount of the 4 asimmetry e(cid:27)e t of the membrane in its enviroment. k k T B Whenthebendingrigidity ≫ ,thepersisten elengthismu hlarger than the diameter of a vesi le. Thus we an negle t thermal (cid:29)u tuations and the shape of a vesi le is determined by minimizing the elasti energy. S Moreover, the area of the vesi le is (cid:28)xed for two reasons: a) there is very little ex hange of mole ules between the bilayer and the ambient on experimental time s ales; b) the elasti energy asso iated with displa ement within the membrane surfa e is negligible on the s ale of the bending elasti energy. Furthermore, we an assume that the variations of volume, that is of the water en losed in, are negligible. On the ontrary, a net transfer of water through the membrane would lead to a variation of the osmoti pressure, asso iated with an ex hange of the energy in the system, whi h is huge with respe t to the bending energy s ales. With these ostraints, the equilibrium shape of a vesi le is determined by the minimization of the Helfri h's shape F energy [1℄ given by F = k (H H )2dS +k KdS +λ dS +∆p dV s − (4) IS IS IS Z λ ∆p = p p e i where and − are treated as Lagrange multipliers or they an be pres ribed experimentally by volume or area reservoirs. The membrane shapes of equilibriumare de(cid:28)ned by the orresponding Euler-Lagrange equation (the shape equation) δ(1)F = 0, δ(1) where indi ates the (cid:28)rst variation of the fun tional under normal in- (cid:28)nitesimal distortion of the surfa e. The resulting equation is ∆p 2λH 2k(H H )2H +2k(H H )(2H2 K)+k∆H = 0, s s − − − − − (5) ∆ H where applied to represents the Lapla e-Beltrami operator on the surfa e 1 ∆ = ∂ gαβ detg ∂ . α ij β detg ij (cid:16) p (cid:17) p The integrability onditions of the GWE are known from the di(cid:27)erential 5 geometry [6℄ and they are (Γ2 ) (Γ2 ) +Γ1 Γ2 +Γ2 Γ2 Γ2 Γ2 Γ1 Γ2 = g K, 12 u1 − 11 u2 12 11 12 12 − 11 22 − 11 12 − 11 ∂b11 ∂b12 = b Γ1 +b (Γ1 Γ1 ) b Γ2 , ∂u2 − ∂u1 11 12 12 12 − 11 − 22 11 (6) ∂b12 ∂b22 = b Γ1 +b (Γ2 Γ1 ) b Γ2 . ∂u2 − ∂u1 11 22 12 22 − 12 − 22 12 g αβ These relations are ne essary and su(cid:30) ient onditions for the oe(cid:30) ients b αβ e to be the (cid:28)rst and the se ond fundamental forms of a surfa e immersed in the eu lidean spa e. For this reason they must be in luded in the set of equations, whi h des ribe the shape of a vesi le. So the equations (5) and R3 (6) des ribe the vesi le on(cid:28)gurations in . In order to simplifythe stru ture of the system, with no loss of generality, we hoose metri on the surfa e in the onformal form ds2 = e2α(x,y)(dx2 +dy2) (x,y) where are onformal lo al oordinates on the surfa e. Furthermore, in order to put in eviden e the role of the mean urvature, it is more usefull to introdu e the following variables 1 1 ϕ = = , 1e 2α(b +b ) H H H (7) 2 − 11 22 − s − s 1 H q = (b +b )e α seα. 11 22 − 4 − 2 (8) Thus, the onformal metri takes the form ds2 = 4q2ϕ2(dx2 +dy2). b b 11 12 For sake of larity, we rename the oe(cid:30) ients and of the se ond form as b ϑ, b ω, 11 12 ≡ ≡ 6 so the asso iated matrix is b b ϑ ω 11 12 = . b21 b22 ! ω 8q2ϕ(1+Hsϕ) ϑ ! − In on lusion, the system (6-5) takes the form q2(ϕ +ϕ )+2qϕ(q +q ) 2ϕ(q2 +q2)+q4(8ϕ+α ϕ2 +α ϕ3 +α ϕ4) = 0, xx yy xx yy − x y 2 3 4 ϑ ω = (8+ α2ϕ)q(ϕq +qϕ ),  y − x 3 y y  ω +ϑ = α2qϕ(ϕq +qϕ )+8qϕq ,  y x 3 x x x   4qϕ2(q +q )+4ϕq2(ϕ +ϕ ) 4ϕ2(q2 +q2) 4q2(ϕ2 +ϕ2) = xx yy xx yy − x y − x y = ω2 +ϑ2 (8+ α2ϕ)q2ϕϑ,   − 3    (9) α = 24H α = 8(2H2 λ), α = 4∆p 8H λ. where 2 s, 3 s − k 4 k − sk This is a system of nonlinear partial di(cid:27)erential equations in two di- mensions and in four dependent variables. Its integrability properties are unknown: only some parti ular solutions have been found on the basis of euristi onsiderations (for example by imposing axisymmetry) [20℄,[21℄,[4℄. 3 Group analysis 3.1 Symmetry algebra A spe i(cid:28) purpose of the present arti le is to study the Lie-point symmetry G group of the system (9). That is, we are looking for a lo al Lie group of pointtrasformationsonthespa eoftheindependentanddependent variables X U = (x,y,ϑ,ω,q,ϕ) : t.c. : q = 0,: ϕ = 0 R2 R4 × { 6 6 } ⊆ × g G into itself, su h that ea h trasformation ∈ a ts as (x,y,ϑ,ω,q,ϕ) = g (x,y,ϑ,ω,q,ϕ) = (Λ1, Λ2, Ω1, Ω2,..., Ω4), · g g g g g Λei e=e Λei(ex,ey,ϑ,ω,q,ϕ) Ωj = Ωj(x,y,ϑ,ω,q,ϕ) g g g g where , are smooth real r g fun tions, depending also on a suitable set of real parameters , onti- 7 Rr nously varying in a neighbourhood of the origin in . In this se tion we will follow the standard group analysis s heme developped in [11℄,[14℄,[15℄. For a symmetry group of a system of di(cid:27)erential equations we mean that g ea h element trasforms solutions of the system (9) into other solutions: (ϑ(x,y),ω(x,y),q(x,y),ϕ(x,y)) ((ϑ(x,y),ω(x,y),q(x,y),ϕ(x,y)) isasolutionwhenever iseone. Equivalently, we an say that the symmetry group leaves the system e e e e e e e e e e e invariant. Moreover we are interested in onne ted lo al Lie groups of symmetries, leaving aside the problem of the dis rete symmetries. Instead of onsidering (cid:28)nite trasformations, in the study of onne ted groups, we an restri t our attention to the asso iated algebra, that is to the in(cid:28)nitesimal g trasformations. The symmetry algebra of the system is realized by real ve tor (cid:28)elds, of the form b ∂ ∂ ∂ ∂ ∂ ∂ v = ξ1 +ξ2 +φ +φ +φ +φ , ∂x ∂y 1∂ϑ 2∂ω 3∂q 4∂ϕ ξi φ x,y,ϑ,ω,q,ϕ α where and are fun tions of . These fun tions have to be determined by the invarian e ondition of the di(cid:27)erential system (9), namely pr(2)v[∆ ] = 0 whenever ∆ = 0, for ν = 1,...,4, ν ν (10) ∆ = 0 ν where indi ates ea h of the equations in (9), and 4 4 4 4 4 ∂ ∂ ∂ ∂ ∂ pr(2)v = v+ φx + φy + φxx + φxy + φyy , α∂uα α∂uα α ∂uα α ∂uα α ∂uα α=1 x α=1 y α=1 xx α=1 xy α=1 yy X X X X X is the prolonged ve tor(cid:28)eld on the spa e ontaining the derivatives of the u1 = ϑ, u2 = ω, u3 = dependent variables. Here we have indi ated with q, u4 = ϕ φs φst and the omponents α and α an be omputed algorithmi ally [11℄. The equations (10) lead to a large number of linear partial di(cid:27)erential ξi φ α equations, alled determining equations, for the fun tions , . Solving them, we an distinguish two ases: I) (α ,α ,α ) = 2 3 4 if the phenomenologi al parameters are not vanishing 6 8 (0,0,0) L I , the symmetry algebra is spanned by the ve tor (cid:28)elds v(ξ1(x,y),ξ2(x,y)) = v(ξ(z))b= ξ∂ + z α q +ξ (ϑ iω)∂ (ω +iϑ 4iq2ϕ i 2q2ϕ2)∂ ∂ + z ϑ ω q − − − − − 6 − 2 +c.ch., i z = x + iy ξ = ξ1 + iξ2 ξ1,ξ2 where and , being arbitrary real harmoni fun tions satisfying the Cau hy-Riemann onditions ξ1 = ξ2, ξ1 = ξ2; y − x x y II) (α ,α ,α ) = 2 3 4 restri ting the phenomenologi al parameters to be (0,0,0) L II , the symmetry algebra is spanned by the ve tor (cid:28)elds v (ξ1(x,y),ξ2(x,y)) = v (ξ(zb)) = ξ∂ + 1 1 z q +ξ (ϑ iω)∂ (ω +iϑ 4iq2ϕ)∂ ∂ + z ϑ ω q − − − − − 2 +c.ch., i v = ϑ∂ +ω∂ +ϕ∂ , 2 ϑ ω ϕ with the same notation used above. The non trivial ommutation relations L I for are v(ξ(z)),v(ξ(z)) = v(ξξ ξξ ), z z b − (11) h i L e e e II and for b v (ξ(z)),v (ξ(z)) = v (ξξ ξξ ), 1 1 1 z z − (12) h i[v (ξ(z)),v ] = 0, 1 2 e e e (13) ξ(z) ξ(z) where and are arbitrary omplex analiti fun tions. In both ases the symmetry Liee algebras are in(cid:28)nite -dimensional. In parti ular, the alge- L I bra is a realization of the onformal transformations algebra in the plane L L L c II c [1b8℄. Moreover, is the dire t sum of and of the one-dimensional b b b 9 L v g 2 subalgebra generated by b L = L L . II c g ⊕ b Lb (Lb) I c A more expli it base for the algebra is obtainable by developing the ξ(z) fun tions in Laurent series, so thbat tbhe generi ve tor (cid:28)eld is L (α) = eiαzn+1∂ + n z α q + eiα(n+1)zn (ϑ iω)∂ (ω +iϑ 4iq2ϕ i 2q2ϕ2)∂ ∂ + ϑ ω q − − − − − 6 − 2 +c.c., h i α [0,2π[ where ∈ . Thus the ommutation relations be ome [L (α),L (β)] = (m n)L (α+β). n m n+m − (14) α = 0 L (0) n Inpassing,wenotethatfor thegenerators spana enterlessVira- L (0),L (π),L (0),L (π),L (0),L (π) soroalgebra[19℄. Thegenerators −1 −1 2 0 0 2 1 1 2 sl(2,C) (cid:8) (cid:9) span the algebra, whi h is the algebra of Möbius transformations in the omplex plane of lo al oordinates of the surfa e. Inordertogiveaninterpretationofthissymmetrystru ture ofthesystem (9), we onsider the e(cid:27)e t of (cid:28)nite transformations on the metri . The (cid:28)nite v v q ϕ z 1 transformations generated by ve tor (cid:28)elds and on the variables , , , z are given by z(ε,z) = Γ 1(ε+Γ(z)), z(ε,z) = Γ 1(ε+Γ(z)), − − (15) ε e e where is the group parameter and z ds Γ(z) = . ξ(s) Zz0 q ϕ While and transform as ξ(z) q(λ,q,z,z) = q(z,z) , ϕ(ϕ,z,z) = ϕ. ξ(z) (16) (cid:12) (cid:12) (cid:12) (cid:12) e (cid:12) (cid:12) e (cid:12) (cid:12) e 10