Composition variation and underdamped mechanics near membrane proteins and coats
CComposition variation and underdamped mechanicsnear membrane proteins and coats
S. Alex Rautu, George Rowlands, and Matthew S. Turner
1, 2 Department of Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom Centre for Complexity Science, University of Warwick, Coventry CV4 7AL, United Kingdom (Dated: October 5, 2018)We study the effect of transmembrane proteins on the shape, composition and thermodynamicstability of the surrounding membrane. When the coupling between membrane composition andcurvature is strong enough the nearby membrane composition and shape both undergo a transitionfrom over-damped to under-damped spatial variation, well before the membrane becomes unstablein the bulk. This transition is associated with a change in the sign of the thermodynamic energyand hence favors the early stages of coat assembly necessary for vesiculation (budding) and maysuppress the activity of mechanosensitive membrane channels and transporters. Our results suggestan approach to obtain physical parameters of the membrane that are otherwise difficult to measure.
PACS numbers: 87.14.ep 87.15.kt 87.16.D-
Biological membranes are crucial to the structure andfunction of living cells [1]. Transmembrane proteins es-sential for transport, adhesion and signalling are embed-ded in membranes [2, 3] consisting of a mixture of lipidsand other amphipathic components. The interactionwith the adjacent lipid molecules is known to regulatethe function of membrane proteins [4–7]. Here, we areprimarily interested in the non-specific lipid-protein in-teractions that arise from the coupling of their hydropho-bic regions [8–14], although we can also allow for selectiveenrichment of membrane component(s) near the protein.We employ a continuum theory in which small deforma-tions of the lipid environment near a rigid inclusion canbe described by a number of local field variables, such asthe profile of the mid-plane of the bilayer, its composi-tion and membrane thickness [15–36]. Furthermore, thefree-energy cost associated with thickness deformation iscompletely decoupled at lowest order [21], and it can beindependently analyzed although we do not do so here.We allow for selective enrichment/depletion of curva-ture sensitive inclusions in the vicinity of a membraneprotein or, equivalently, lipid asymmetry between leafletsthat is characterized by a local spontaneous curvature,the preferred mean curvature in the absence of any me-chanical stresses on the membrane [33–41]. This localvariation may be relatively large near a membrane pro-tein if its geometry is such that it bends or deforms thesurrounding membrane (see Fig 1). Our approach leadsto a real-space description of the membrane around aninclusion of arbitrary symmetry.We consider a two-component membrane in which thelocal compositional asymmetry between the different lay-ers and/or the density of curvature-sensitive inclusions isphenomenologically coupled to the local mean curvatureof the membrane [33, 34]. When the compositional vari-ation is weak and the membrane displacement is small,the free-energy can be written as a Landau-Ginzburg ex-
FIG. 1. (color on-line) Sketch of a membrane inclusion show-ing the mid-plane of the bilayer (blue line) at height u ( r, θ ).The surface variation of the rigid inclusion in the ˆz direc-tion is coarse-grained out so that the geometry is defined byits radius r and two functions describing the height U ( θ )and contact angle U (cid:48) ( θ ) of the hydrophobic belt. These pa-rameterize the protein-membrane interface (red line), where u ( r , θ ) = U ( θ ) and ˆn · ∇ u ( r , θ ) = U (cid:48) ( θ ), with ˆn as the in-ward unit normal vector. We require both the normal forceand the torque on the inclusion to vanish. The latter can leadto an equilibrium tilt angle ψ about the axis labeled by ε . pansion [33–35, 42–44], F ϕ = 12 (cid:90) M (cid:2) a ϕ + b ( ∇ ϕ ) + 2 c ϕ (cid:0) ∇ u (cid:1)(cid:3) d r , (1)where only the lowest-order terms are retained and a , b and c are phenomenological constants. The scalar fields ϕ ( r ) and u ( r ) are the local composition difference (as an a r X i v : . [ c ond - m a t . s o f t ] F e b area fraction) and bilayer mid-plane height, respectively,see Fig 1. Both deformation fields are described within aMonge representation, which allows us to write the free-energy associated with mid-plane deformation as F u = 12 (cid:90) M (cid:2) σ ( ∇ u ) + κ ( ∇ u ) (cid:3) d r , (2)where σ and κ are the surface tension and bending rigid-ity of the membrane, respectively [45].We now seek the ground state of the membrane andneglect fluctuations throughout. The membrane shape u ( r ) and its compositional field ϕ ( r ) can then be com-puted exactly by minimizing the free-energy functional, F = F u + F ϕ , leading to the Euler-Lagrange equations: ∇ u = ( ∇ − β ) φ, (3) ∇ ( ∇ − α ) u + γ ∇ φ = 0 , (4)where φ ( r ) = ( b/c ) ϕ ( r ) and the coefficients α = (cid:112) σ/κ , β = (cid:112) a/b and γ = c / √ κb represent the relevant inverselength scales of the model [46]. By combining (3) and (4),a single equation for φ ( r ) can be obtained [47]:( ∇ − k )( ∇ − k − ) φ = 0 , (5)where k ± is given by k ± = 12 (cid:104)(cid:112) ( α + β ) − γ ± (cid:112) ( α − β ) − γ (cid:105) . (6)By separation of variables, a solution to equation (5) thatvanishes in the far-field limit can be found to be φ ( r, θ ) = φ + ( r, θ ) + φ − ( r, θ ) , (7)where r and θ are the usual polar coordinates, as illus-trated in Fig 1, and φ ± is defined by φ ± ( r, θ ) = k ± k ± − β ∞ (cid:88) n =0 V ± n ( θ ) K n ( k ± r ) , (8)where K n are the modified Bessel functions of the secondkind of order n , and V ± n ( θ ) = A ± n cos( nθ ) + B ± n sin( nθ ),with A ± n and B ± n arbitrary constants. From this we ob-tain the membrane shape through Eq (3), which yields u ( r, θ ) = u + ( r, θ ) + u − ( r, θ ) + u h ( r, θ ) , (9)where the solutions that diverge at infinity are excluded.Here, u h ( r, θ ) is the homogeneous solution of (3), namely u h ( r, θ ) = ∞ (cid:88) n =0 W n ( θ ) r − n , (10)where W n ( θ ) = X n cos( nθ ) + Y n sin( nθ ), with X n and Y n some constants. The remaining two terms in (9) are theinhomogeneous solutions, which are found to be u ± ( r, θ ) = ∞ (cid:88) n =0 V ± n ( θ ) K n ( k ± r ) . (11) FIG. 2. (color on-line) Membrane profiles induced by anasymmetrical inclusion, where the contact angle is given by U (cid:48) ( θ ) = 15 ◦ if | θ | < w/ θ measuredfrom the x − axis (see text). The membrane parameters arehere α r = 0 . β r = 1 . γ r = 0 .
5, with r the radiusof the inclusion (not depicted). The compositional asymme-try φ ( r ) is shown as the color-map of the surface plots. The angular functions V ± n ( θ ) and W n ( θ ) are deter-mined by the boundary conditions at the interface ∂ M ,located at a distance r from the symmetry axis. Theseare specified by the height U ( θ ) and contact angle U (cid:48) ( θ )at which the mid-plane of the bilayer meets the inclu-sion (see Fig 1). This choice is motivated by assuming astrong coupling between the transmembrane domain ofthe inclusion and the membrane hydrophobic core. Also,the normal derivative of φ is chosen to vanish on ∂ M ,which is used to obtain an unique solution [48].This methodology allows us to compute exactly thelowest order estimates to the membrane profile, its localphase behavior, and the total deformation energy, givenan arbitrary model for the shape of the inclusion, through U ( θ ) and U (cid:48) ( θ ), i.e. a general solution to the problem.This makes contact with experiments that might mea-sure membrane shape (cryo TEM [49] or perhaps TIRFmicroscopy) and composition (NMR [50] or FRET [51]).First, we consider a simple illustrative example in whichthe height, U ( θ ), is chosen to be a constant z , while thecontact angle has a non-zero value only within an angularinterval w , see Fig 2. This corresponds to a rigid inclusionthat induces a local mid-plane deformation only withina specific region along its hydrophobic belt, with the re-maining part preferring a flat membrane. The Connollysurface of a leucine transporter, LeuT, exhibits similarfeatures [52, 53]. The height z is not entirely arbitrary,being set by the overall balance of normal forces. Sim-ilarly, the condition of torque balance leads to a tilt ofthe inclusion (see [47] for details), as illustrated in Fig 1.Typical solutions due to such an asymmetrically-shapedinclusion that exerts no net torque are shown in Fig 2for physiologically reasonable values of α , β , and γ [54].The induced φ ( r ) shows a rich variation as the angle w isvaried between 0 and 2 π , which correspond to inclusionswith a cylindrical and a conical shape, respectively.To better understand the role of the coupling constant γ , we consider symmetric conical inclusions ( w = 2 π )in what follows, noting that the transition from over- tounder-damped variation also appears for rigid inclusionswith other (or no) symmetry. For values of γ less than γ d = | α − β | , the solutions are found to be monotonicallydecaying, see Fig 3(b). However, as γ is increased abovethis point, the solutions show an underdamped behav-ior, with the membrane displacement decaying to zerofor large distances. The magnitude of this amplitude be-comes large as γ approaches γ c = α + β , suggesting thepresence of an instability. In fact γ > γ c , where k ± < γ = γ d instead corresponds to a crit-ically damped system, separating the real and complexdomain of k ± . The solutions are thermodynamically sta-ble on either side of this boundary. When γ d < γ < γ c ,the decay rate λ of the membrane undulations and itswave number ω can be determined by approximating K n ( ρ ) ≈ e − ρ ( πρ / − / for ρ (cid:29) n in Eq (9) [55], i.e. u ( r ) ∼ e − λ ( r − r ) (cid:112) r/r cos [ ω ( r − r ) + ϑ ] , (12)where ϑ is a phase angle that only depends on α , β and γ . Here, λ and ω are given by the real and imagi-nary parts of (6), respectively. Thus, we find that thewavelength of the pre-critical undulations diverges aswe approach γ = γ d , and the decay length divergesfor γ = γ c , which signals the presence of a bulk mem-brane instability. Physically and mathematically distinctunderdamped solutions have also been found in studiesof membrane thickness mismatch without any composi-tional field that couples to mid-plane curvature [17].While α can be measured through various experimen-tal techniques [56–59], the parameters β and γ are moreelusive [48]. Our analysis suggests a possible method tomeasure them, e.g. by tuning the system to lie near theinstability threshold γ (cid:46) γ c . Here, the amplitude of theundulations are large and long-ranged, and γ and β canbe inferred by comparison with (9) or (12). This tun-ing might be achieved by controlling the surface tension,e.g. using a micropipette aspiration technique [56], so as FIG. 3. (color on-line) (a) Plot of k ± from Eq (6) against thecoupling constant γ , with α r = 0 . β r = 1 . r theinclusion radius. This illustrates that both k + (red line) and k − (blue line) are real for γ < | α − β | , and purely imaginarywhen γ > α + β . The grey shaded area illustrates the regionwhere k ± are complex, while the green line shows the realpart only. The domain given by γ > α + β corresponds toLeibler’s unstable regime [33, 34]. (b) Radial profiles of thebilayer mid-plane u ( r ) and the compositional asymmetry φ ( r )induced by a conical inclusion, with a modest contact angleof 15 ◦ , for different values of the coupling constant γ , where α and β have the same values as those in panel (a). to approach the critical tension σ c = κ ( γ − β ) , althoughthe presence of thermal fluctuations may mean that someaveraging will be required to resolve the ground state,particularly far away from the membrane inclusion. Thisillustrates the predictive power of our model.Mechanosensitive membrane channels have beenwidely studied and reveal the interplay between the bi-ological function of transmembrane proteins and the ad-jacent membrane structure and composition. Throughconformational changes from a closed to an open statethat allows the passage of solvent through the membrane,they can equilibrate an osmotic imbalance between theinterior and exterior of cells [62–64]. Although many ex-amples of these channels are found in nature, the bacte-rial mechanosensitive channels of large (MscL) and smallconductance (MscS) are prototypes of such proteins. Ex-perimental studies have have shown that the channelopening probability is related to the membrane tensionand the size of the open pore [60–67]. One possibility FIG. 4. (color on-line) (a) The top sketches, with the bi-layer membrane represented by a thick green line, show twoidealized schemes for channel gating: the gating-by-tilt model(left); and the dilational gating model (right). The figures be-low this show the angle for gating-by tilt that would accountfor the absolute value of the entire conformational energychange F measured for MscL and MscS [60, 61]. The dashedline indicates F = 0 separating two domains where the mem-brane acts to open ( F > F < ◦ , which are probably unphysical andwhere our perturbative approach anyway breaks down. (b)The top sketch shows a membrane deformed by the assemblyof a protein coat such as clathrin or a viral coat protein. Thegraphs below this show the radial compositional field φ ( r )when the coat is of size r = 10 nm (left) and the changein membrane energy-per-area of coat monomers ∆ f c due tocoupling to the composition field against the coat radius r (right). In both cases we assume a typical intrinsic coat cur-vature with R c = 50 nm , β = 1 . nm − and γ = 1 . nm − .In the underdamped regime the energy change ∆ f c can ex-hibit an initial decrease, which may drive coat assembly. Inboth figures (a) and (b) we use κ = 20 k B T . is that the channel simply dilates open at high tensionbut the transition between the closed and open statesmight also involve, e.g. a change in slope at the protein-membrane interface (here, δ ) [68]. In a two-componentmembrane such a change in boundary conditions betweenthe closed and open states couples to both the shape andasymmetry field in the nearby membrane and hence con- tributes to a change in the free energy of the channel-membrane system. Here, for simplicity, we consider theangle at the channel wall δ to be non-zero in a conicalclosed state and δ = 0 in the open state. We explorethe thermodynamic effect of this gating-by-tilt by com-paring the deformation energy F of the membrane to theexperimental estimate of the energy required to open thechannels at zero tension, inferred by assuming purely di-lational opening. Fig 4(a) shows that the even modestchanges in the boundary angle at the face of the chan-nel could give rise to a significant thermodynamic energyunder gating-by-tilt. Moreover, a regime is identified inwhich the membrane can act to close , rather than open,the channel, characterized by a negative total energy F relative to the open state, although a similar result waspreviously identified in a model that neglects spatial vari-ation of coupling to curvature [23, 24]. Our results in-dicate that lipid composition variation, and its couplingto mean curvature, could play a role in regulating thefunction of mechanosensitive membrane channels.Finally, the presence of a negative deformation energyin the underdamped regime motivated us to study thethermodynamics of protein coat formation on a mem-brane. Such coats are important in regulation, e.g. mem-brane trafficking using clathrin coats, or in infection,where viral coats assemble at the plasma membrane [1].Fig 4(b) shows both the compositional field φ around aprotein coat of size r = 10 nm and the variation with r of ∆ f c , i.e. the change in membrane energy due tocoupling to φ only, scaled by the coat area. Thus, ∆ f c renormalizes the chemical potential for binding of earlycoat monomers to the membrane, and it is computedby adding both the contribution from the membrane in-side (under) and outside the coat [47]. Two striking fea-tures are observed in the underdamped regime. Firstly,this free-energy change can support an initial decrease with coat size. In this case the deformation of the mem-brane (with its associated composition) is energeticallyfavorable. This is true (even) for membranes that re-main thermodynamically stable, i.e. in the absence ofbulk instability. This may represent a new mechanismfor driving (controlling) coat formation in cells. Thismight be tested by tracking coat assembly at different α (tension), e.g. controlled by micropipette aspiration [56]:We predict a dramatic increase in the rate of assemblynear the critical tension σ c . Secondly, the existence of aminimum in ∆ f c corresponds to a characteristic coat sizewith metastable character; we note that partially formedcoats are often observed [69].We acknowledge the stimulating discussions withDr P. Sens (Paris) and Profs M. Freissmuth and H.Sitte (Vienna) and funding from EPSRC under grantEP/E501311/1 (a Leadership Fellowship to MST). [1] B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts,and P. Walter, Molecular Biology of the Cell , 5th ed.(Garland Science, 2008).[2] S. J. Singer and G. L. Nicolson, Science , 720 (1972).[3] D. M. Engelman, Nature , 578 (2005).[4] T. M. Suchyna, S. E. Tape, R. E. Koeppe, O. S. An-dersen, F. Sachs, and P. A. Gottlieb, Nature , 235(2004).[5] P. Moe and P. Blount, Biochemistry , 12239 (2005).[6] D. Krepkiy, M. Mihailescu, J. A. Freites, E. V. Schow,D. L. Worcester, K. Gawrisch, D. J. Tobias, S. H. White,and K. J. Swartz, Nature , 473 (2009).[7] M. Milescu, F. Bosmans, S. Lee, A. A. Alabi, J. I. Kim,and K. J. Swartz, Nature Struct. Biol. , 1080 (2009).[8] A. G. Lee, Biochim. Biophys. Acta , 62 (2004).[9] K. Mitra, I. Ubarretxena-Belandia, T. Taguchi, G. War-ren, and D. M. Engelman, Proc. Natl. Acad. Sci. USA , 4083 (2004).[10] W. Dowhan, E. Mileykovskaya, and M. Bogdanov,Biochim. Biophys. Acta , 19 (2004).[11] T. K. M. Nyholm, S. Ozdirekcan, and J. A. Killian,Biochemistry , 1457 (2007).[12] O. S. Andersen and R. E. Koeppe, Annu. Rev. Biophys.Biomol. Struct. , 107 (2007).[13] R. Phillips, T. Ursell, P. Wiggins, and P. Sens, Nature , 379 (2009).[14] J. A. Lundbaek, S. A. Collingwood, H. I. Ing´olfsson,R. Kapoor, and O. S. Andersen, J. R. Soc. Interface , 373 (2010).[15] P. B. Canham, J. Theor. Biol. , 61 (1970).[16] H. W. Huang, Biophys. J. , 1061 (1986).[17] H. Aranda-Espinoza, A. Berman, N. Dan, P. Pincus, andS. A. Safran, Biophys. J. , 648 (1996).[18] C. Nielsen, M. Goulian, and O. S. Andersen, Biophys.J. , 1966 (1998).[19] S. Mondal, G. Khelashvili, and H. Weinstein, Biophys.J. , 2305 (2014).[20] P. Sens and S. Safran, Eur. Phys. J. E , 237 (2000).[21] J.-B. Fournier, Eur. Phys. J. B , 261 (1999).[22] T. Weikl, M. Kozlov, and W. Helfrich, Phys. Rev. E ,6988 (1998).[23] P. Wiggins and R. Phillips, Proc. Natl. Acad. Sci. USA , 4071 (2004).[24] P. Wiggins and R. Phillips, Biophys. J. , 880 (2005).[25] N. Dan, P. Pincus, and S. A. Safran, Langmuir , 2768(1993).[26] M. Goulian, R. Bruinsma, and P. Pincus, Europhys.Lett. , 145 (1993).[27] K. S. Kim, J. Neu, and G. Oster, Biophys. J. , 2274(1998).[28] M. S. Turner and P. Sens, Biophys. J. , 564 (1999).[29] V. S. Markin and F. Sachs, Phys. Biol. , 110 (2004).[30] T. Ursell, K. C. Huang, E. Peterson, and R. Phillips,PLoS Comput. Biol. , e81 (2007).[31] C. A. Haselwandter and R. Phillips, Europhys. Lett. ,68002 (2013).[32] C. A. Haselwandter and R. Phillips, PLoS Comput. Biol. , e1003055 (2013).[33] S. Leibler, J. Phys. France , 507 (1986).[34] S. Leibler and D. Andelman, J. Phys. France , 2013(1987). [35] D. Andelman, T. Kawakatsu, and K. Kawasaki, Euro-phys. Lett. , 57 (1992).[36] W. Helfrich, Z. Naturforsch C. , 693 (1973).[37] U. Seifert, Adv. Phys. , 13 (1997).[38] J. N. Israelachvili, D. J. Mitchell, and B. W. Ninham, J.Chem. Soc., Faraday Trans. , 1525 (1976).[39] S. M. Gruner, J. Phys. Chem. , 7562 (1989).[40] H. T. McMahon and J. L. Gallop, Nature , 590(2005).[41] A. Callan-Jones, B. Sorre, and P. Bassereau, Cold SpringHarb Perspect Biol. (2011).[42] P. B. Sunil Kumar, G. Gompper, and R. Lipowsky, Phys.Rev. E , 4610 (1999).[43] M. Schick, Phys. Rev. E , 1 (2012).[44] U. Seifert, Phys. Rev. Lett. , 1335 (1993).[45] D. Nelson, T. Piran, and S. Weinberg, Statistical Me-chanics of Membranes and Surfaces , 2nd ed. (World Sci-entific Publishing Company, 2004).[46] The sign choice of γ is simply a convention. The fields u and φ are invariant under a sign change in γ , itself relatedto the convention of a direction for “up” and whether onerefers to the enrichment of a component that couples topositive curvature (depending on one’s choice for “up”)or the depletion of one that couples to negative curvature.[47] See Supplemental Material, which includes Refs. [70–72].[48] Although Dirichlet boundary conditions could be usedas well, our choice gives the ground state solution in theabsence of any constraints on the composition asymmetryfield φ at the boundary; see Ref. [47] for more details.[49] A. Shimada, H. Niwa, K. Tsujita, S. Suetsugu,K. Nitta, K. Hanawa-Suetsugu, R. Akasaka, Y. Nishino,M. Toyama, L. Chen, Z.-J. Liu, B.-C. Wang, M. Ya-mamoto, T. Terada, A. Miyazawa, A. Tanaka, S. Sug-ano, M. Shirouzu, K. Nagayama, T. Takenawa, andS. Yokoyama, Cell , 761 (2007).[50] P. J. Judge and A. Watts, Curr. Opin. Chem. Biol. ,690 (2011).[51] L. M. S. Loura and M. Prieto, Front. Psychol. , 82(2011).[52] A. Yamashita, S. K. Singh, T. Kawate, Y. Jin, andE. Gouaux, Nature , 215 (2005).[53] S. K. Singh, C. L. Piscitelli, A. Yamashita, andE. Gouaux, Science , 1655 (2008).[54] Typically the membrane correlation length might be α − ∼ r [73]. Eq. (1) can be associated with a typi-cal interfacial width for a strongly segregated system ofa few nm, say (cid:112) b/ | a | ∼ r , and a line tension of about apN ( (cid:112) b | a | ∼ pN) [74], which combine to give b ∼ k B T .Sorting of strongly curvature-coupling components intomembrane tubes gives an upper bound of c ∼ κ/ (2 . r )and hence a value for γ ∼ /r [75], indicating that the allregimes discussed in the main text are accessible. Whilethe use of a phase separated system to estimate the pa-rameters a and b is questionable, given that they aremotivated within a model in which the system remainsessentially one phase, with φ very small, we are reassuredthat the primary limit on the accessibility of the under-damped regime is that β is not too large. Given thatspontaneous phase separation is often seen on biologicalmembranes there would seem to be no lower limit on areasonable magnitude for a and hence β .[55] M. Abramowitz and I. A. Stegun, Handbook of Mathe-matical Functions (Dover Publications Inc., 1965). [56] E. Evans and W. Rawicz, Phys. Rev. Lett. , 2094(1990).[57] L. Bo and R. Waugh, Biophys. J. , 509 (1989).[58] M. Kummrow and W. Helfrich, Phys. Rev. A , 8356(1991).[59] J. P´ecr´eaux, H.-G. D¨obereiner, J. Prost, J.-F. Joanny,and P. Bassereau, Eur. Phys. J. E , 277 (2004).[60] C.-S. Chiang, A. Anishkin, and S. Sukharev, Biophys.J. , 2846 (2004).[61] S. Sukharev, Biophys. J. , 290 (2002).[62] E. Perozo, Nat. Rev. Mol. Cell Biol. , 109 (2006).[63] I. R. Booth, M. D. Edwards, S. Black, U. Schumann, andS. Miller, Nature Rev. Microbiol. , 431 (2007).[64] E. S. Haswell, R. Phillips, and D. C. Rees, Structure ,1356 (2011).[65] S. I. Sukharev, P. Blount, B. Martinac, F. R. Blattner,and C. Kung, Nature , 265 (1994).[66] C. Kung, B. Martinac, and S. Sukharev, Annu. Rev.Microbiol. , 313 (2010).[67] E. Perozo, A. Kloda, D. M. Cortes, and B. Martinac, Nature Struct. Biol. , 696 (2002).[68] M. S. Turner and P. Sens, Phys. Rev. Lett. , 118103(2004).[69] J. Heuser, J. Cell. Biol. , 560 (1980).[70] K. F. Riley, M. P. Hobson, and S. J. Bence, Mathemati-cal Methods for Physics and Engineering , 3rd ed. (Cam-bridge University Press, 2006).[71] F. W. Byron and R. W. Fuller,
The Mathematics ofClassical and Quantum Physics (Dover Publications Inc.,1992).[72] I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, and D. Zwill-inger,
Table of Integrals, Series, and Products , 6th ed.(Academic Press, 2000).[73] M. P. Sheetz and J. Dai, Trends Cell Biol. , 85 (1996).[74] A. Tian, C. Johnson, W. Wang, and T. Baumgart, Phys.Rev. Lett. , 208102 (2007).[75] S. Aimon, A. Callan-Jones, A. Berthaud, M. Pinot,G. E. S. Toombes, and P. Bassereau, Dev. Cell28