A model of HIV budding and self-assembly, role of cell membrane
aa r X i v : . [ q - b i o . S C ] M a r A model of HIV budding and self-assembly, role of cell membrane
Rui Zhang and Toan T. Nguyen
School of Physics, Georgia Institute of Technology, 837 State Street, Atlanta, Georgia 30332-0430 (Dated: November 1, 2018)Budding from the plasma membrane of the host cell is an indispensable step in the life cycleof the Human Immunodeficiency Virus (HIV), which belongs to a large family of enveloped RNAviruses, retroviruses. Unlike regular enveloped viruses, retrovirus budding happens concurrently with the self-assembly of retrovirus protein subunits (Gags) into spherical virus capsids on the cellmembrane. Led by this unique budding and assembly mechanism, we study the free energy profile ofretrovirus budding, taking into account of the Gag-Gag attraction energy and the membrane elasticenergy. We find that if the Gag-Gag attraction is strong, budding always proceeds to completion.During early stage of budding, the zenith angle of partial budded capsids, α , increases with timeas α ∝ t / . However, when Gag-Gag attraction is weak, a metastable state of partial buddingappears. The zenith angle of these partially spherical capsids is given by α ≃ ( τ /κσ ) / ina linear approximation, where κ and σ are the bending modulus and the surface tension of themembrane, and τ is a line tension of the capsid proportional to the strength of Gag-Gag attraction.Numerically, we find α < . π without any approximations. Using experimental parameters, weshow that HIV budding and assembly always proceed to completion in normal biological conditions.On the other hand, by changing Gag-Gag interaction strength or membrane rigidity, it is relativelyeasy to tune it back and forth between complete budding and partial budding. Our model agreesreasonably well with experiments observing partial budding of retroviruses including HIV. I. INTRODUCTION
The Human Immunodeficiency Virus (HIV) is famousfor its ability to induce Acquired Immunodeficiency Syn-drome (AIDS). It belongs to a large family of envelopedRNA viruses, retroviruses. Retroviruses are character-ized by the unique infection strategy of reverse transcrip-tion, in which the genetic information flows from RNAback to DNA (therefore the name “retro”) [1]. Buddingis an indispensable step in the retroviral life cycle [2, 3].After the major retroviral structural protein, Gags, aresynthesized inside the host cell, they are transported tothe cell membrane and self-assemble into spherical pro-tein shells called “capsids”, with viral RNA genome andother auxiliary viral proteins packaged inside. At thesame time, these capsids, enveloped by the cellular mem-brane, have to bud out of the membrane to target otherhost cells. In other words, budding and assembly ofretroviruses happen concurrently on the cell membrane.Despite a large body of experiments done within the last
FIG. 1: Electron microscopic image of partial budding ofHIV-1 viruses (reprinted from [2]). decade, the biological pathway and mechanism of retro-viral budding have still not been fully understood [2, 3].One important unexplained observation is that viral bud-ding can be inhibited partially or completely by chang-ing the cell environment or mutating the late domains ofthe Gag proteins. In these situations, capsids are onlypartially formed and stuck on the membrane (Fig. 1).Motivated directly by this partial budding phenomenon,in this paper, we propose a physical model to study HIV(and retroviruses in general) budding and assembly onthe elastic membrane. Physically, this situation is in-teresting because it provides a unique two dimensionalself-assembly mechanism in which the membrane elasticenergy plays an important role, since assembly is alwaysaccompanied with budding. Biologically, understandingthe physical mechanism of HIV budding and assemblyis certainly important toward understanding the HIVlife cycle. It is also important in the light of recent ef-fort from the virology community to develop assembly-oriented anti-viral therapy.Budding of regular enveloped viruses was studied the-oretically by several authors (TDGB) in Ref. [4, 5, 6].However, the viral budding pathway and the physicalmodel studied by TDGB is qualitatively different fromretroviral budding we study in this paper. For regularenveloped viruses, viral capsids are fully assembled in-side the cell [7, 8, 9]. After that, they are transported tothe cell membrane, bind to the viral spike proteins (em-bedded in the cell membrane) and then bud out throughthe cell membrane (see Fig. 2b). Therefore, capsids for-mation and budding out of the membrane are two sepa-rate processes. And the main driving force of budding isthe capsid-membrane attraction (mediated by embeddedspike proteins).Budding of retroviruses follows a completely different (a) h rR (b) G α α FIG. 2: (Color online) Schematic illustrations of two differenttypes of virus budding. (a): budding of retroviruses: cap-sid proteins (Gags) are first attracted to the membrane, thenself-assemble and bud on the membrane at the same time.This is the system studied in this paper. (b): budding ofregular enveloped viruses: capsid proteins first self-assembleinto complete capsids inside the cell, then bud on the mem-brane [4]. (a) also shows the cylindrical coordinate system( r, h, φ ) used in the model (the polar angle φ is not shown). pathway. Various TEM and X-ray tomography experi-ments suggest that retroviral capsids are assembled fromGag proteins on the cell membrane and bud out of thecell concurrently [2, 3]. Based on these experiments, westudy a different model for HIV (and retrovirus in gen-eral) budding and assembly shown in Fig. 2a. In thisnew model, we assume retroviral capsids are assembledfrom membrane-bound Gags only, neglecting the possi-bility that Gags from the interior of the cell may par-ticipate. In other words, the Gag-membrane attractionis strong such that Gags always bind to the membrane.This assumption is supported by various experimentalobservation where budding is completely inhibited (nocapsids are formed) but Gags are found in abundance atthe cell membrane [10] . In contrast to the TDGB model,the primary driving force of our retroviral budding is theshort range attraction between these membrane-boundGag proteins. This correlates well with experimental factthat point mutations changing Gag-Gag interactions af-fect the degree of viral budding. On the other hands,spike proteins or virus RNA seem not important forretroviral budding. In vitro, Gag proteins are directlyattracted to the membrane and they alone are usuallysufficient for the assembly and release of virus-like parti-cles [3, 11, 12]. We therefore neglect the contribution ofall other proteins or RNA components of retroviruses inour model.In this paper, for a given set of parameters (the mem- brane Gag concentration, the Gag-Gag interaction, andthe cell membrane bending and stretching rigidity), westudy the free energy profile of budded viral capsids.Two energies are considered explicitly: first, the elas-tic energy of the membrane including the bending andstretching energy; second, the Gag-Gag attraction en-ergy when a Gag makes contact to the other Gag (seeFig. 2a). Since the elastic energy scale is much largerthan k B T, for example, the bending rigidity of normalmembranes is about 20 k B T, thermal fluctuations arehigher order corrections and neglected in the theoreticaltreatment. Focusing on the budding process, we also as-sume that the Gag-Gag interaction is strong enough suchthat the entropic cost of bringing free Gags to the capsidcan be ignored. For simplicity, we assume the shape ofthe capsid together with the membrane attached to it is(partially) spherical with radius R (Fig. 2a). The sizeof a capsid is then characterized by the zenith angle α at its edge, the smallest being the angle of a single Gagprotein, α G (Fig. 2a). Since α G is very small ( α G = 0 . α G → α asa continuous variable. As budding proceeds to comple-tion, α increases from α G to π . When α = π , the capsidactually leaves the membrane through membrane fission.In this paper, we do not consider this fission process andthus, in our terminology, complete budding always means α → π . To simplify the calculation, we employ an scalingdescription where we neglect the variation in the degreeof viral budding and assume all capsids have the sameaverage zenith angle α . Π Α f Ž Τ>Τ c ΠΑ Α f Ž f Ž m f Ž Τ<Τ c FIG. 3: The schematic illustration of the total free energydensity as a function of the capsid size α . The left and rightprofiles correspond to strong or weak Gag-Gag attraction re-spectively. Here τ is the line tension of the rim of a partiallybudded capsid. τ is proportional to the strength of Gag-Gagattraction. τ c is the threshold line tension at which the localminimum at α appears. Our main result is shown in Fig. 3. The key parameteris the strength of Gag-Gag attraction which can be ad-justed experimentally by mutating of Gags, complexingof Gags with other molecules or by changing pH or salin-ity of the cell cytoplasm near the membrane [2, 13]. In apartially budded capsid, the line tension τ of the rim ofa capsid is directly proportional to this interaction Gag-gag interaction. When the Gag-Gag attraction is strong(or when τ is greater than a threshold value τ c ), as inthe normal biological conditions of HIV, budding alwaysproceeds to completion, i.e., α → π (the left panel ofFig. 3). At early stage of budding, the size of a partiallybudded capsid increases very slowly with time: α ( t ) ≈ ( t/τ diff ) / (1)where the time scale τ diff depends on the lateral mobilityof Gag, the radius of the capsid and the initial concen-tration Gag (see Eq. (51)). On the other hand, when theGag-Gag attraction is weak ( τ < τ c ), for example, aftermutation of the late domains of the Gag protein, partialbudding appears as a metastable state at the capsid size α (the right panel of Fig. 3). In this case, the free en-ergy barrier can be much larger than k B T and budding iskinetically trapped at α . Using a linear approximation,we find α ≃ r τ κσ , (2)where σ and κ are the surface tension and bending rigid-ity of the membrane.The energetics of HIV budding and assembly is studiedboth analytically and numerically in this paper. Analyt-ically, the complete scaling behaviors of the free energydensity profile in asymptotic limits of “soft” and ”stiff”membranes are calculated (the meaning of ”soft” and”stiff” membrane will be clear in later sections). In allcases, they agree with the numerical result well. How-ever, the numerical result gives a complete solution tothe problem including nonlinear regimes where the an-alytical result is normally not available. The inequility α < . π is found to always hold from the numericalcalculation without any approximations.It is worth to point out that budding in our model canbe considered as a consequence of the inhomogeneity ofthe membrane if one considers Gags as a part of the mem-brane. In this sense, our work is related to J¨ulicher andLipowsky’s works on domain-induced budding of vesi-cles [14, 15]. However, in their papers, the inhomogeneitywas introduced through two kinds of lipids which do notcarry a given curvature like our Gags. Domain-inducedbudding is a consequence of demixing of these differentmolecules. As a result, their budding happens in a muchlarger length scale (comparable to the size of the vesi-cle) where only two phases coexist, one budded out fromthe other. While in our case, we consider budding at amuch smaller length scale (a typical HIV-1 virus particleis about 130 nm in diameter, which is a hundred timessmaller than the size of a host cell) and actually havea multi-phase coexistence since there are more than onecapsid on the membrane.This paper is organized as follows. In Sec. II , we intro-duce the physical model of HIV budding and assembly. We then discuss the analytical solution to the elastic en-ergy of the membrane in Sec. III and to the total freeenergy density in Sec. IV. The numerical result is thenprovided and compared to the analytical results in Sec. V.After we get the complete theoretical result, we discussbudding kinetics and make connections to experiments inSec. VI. We finally conclude in Sec. VII. In this paper,the term “capsid” is used for both partial and completespherical shells of viral proteins. The meaning should beclear from the context. II. THE ELASTIC MODEL OF HIV CAPSIDBUDDING AND SELF-ASSEMBLY
Let us consider a membrane-capsids system in whichthe concentration of Gags on the membrane, c G , is fixed.We assume all capsids assembled by Gags have the sameaverage zenith angle α (see Fig. 2a), and an average con-centration, n . n is related with α by the conservation ofmass of Gags: n = c G A ( α G ) A ( α ) = c G − cos α G − cos α , (3)where A ( α ) = 2 πR Z α sin θdθ = 2 πR (1 − cos α ) (4)is the area of a capsid with zenith angle α , and α G is thezenith angle of a single Gag (see Fig. 2a). Within thisaverage description, it is convenient to think that thewhole membrane surface is divided into identical cells,each contains a single capsid. The average size of theseapproximately circular cells, d , is given by the condition π ( d/ n = 1 . (5)Generically, the free energy density of the membrane-capsid system can be written as f = nε = n ( ε m + ε c ) , (6)where ε is the free energy of one membrane cell. It in-cludes two parts: the elastic energy of the membrane, ε m , and the capsid energy ε c coming from the Gag-Gaginteraction and the Gag-membrane interaction.To calculate the elastic energy of the cell membrane,we use the standard Helfrich model [16, 17] where ε m isthe sum of two contributions from the bending energyand the stretching energy: ε m = Z dS h κ H − C ) + κ G K i + Z dSσ. (7)Here the integration with the area element dS is takenover the membrane surface. κ and κ G are the bend-ing rigidity and Gaussian bending rigidity, H and K are the mean and Gaussian curvatures, and C is thespontaneous curvature of the membrane surface. Usingthe Gauss-Bonnet theorem, one can show that the totalGaussian curvature of the membrane surface is propor-tional to the total area of capsids, in the generic casewhen κ G takes different values for the membrane at-tached to the capsid and the Gag-free membrane. Sincethe Gag concentration c G in our system is fixed, this termgives a constant in f and can be dropped from furtherconsideration [18]. For a given Gag concentration c G ,under our spherical capsid assumption, the shape andthe total area of all capsids are fixed. Therefore the to-tal elastic energy of the membrane attached to capsidsis also constant, and can also be dropped from consider-ation. As a result, the α -dependent contribution to ε m comes from the integration over the Gag-free membranesurface only. In this region, we take the spontaneouscurvature to be C = 0, corresponding to normal lipidbilayer membranes.In consideration of the single capsid energy ε c , since c G is constant, both the total Gag-membrane interactionenergy and the bulk part of the Gag-Gag interaction en-ergy are constant. The only α -dependent contribution to ε c comes from the rim energy of the capsid, due to thefact that the coordination number of Gags on the rim isnot as many as Gags inside the capsid. Since the perime-ter of the capsid rim with zenith angle α is 2 πR sin α , weset ε c = τ πR sin α. (8)The proportionality coefficient τ can be considered asthe “line tension” of the capsid. It is directly propor-tional to the strength of the Gag-Gag attraction and canbe changed experimentally by mutations of Gags or bychanging pH or salinity of the cell cytoplasm near themembrane.To proceed further, we take the “ideal capsids” ap-proximation when the distance between capsids is largeand the membrane mediated interaction between them isnegligible. Such an effective long-range interaction is pos- sible because the presence of the first capsid may changethe deformation of the membrane around the second cap-sid and provides an effective interacting energy betweenthe two. Qualitatively, this interaction is negligible whenthe capsid concentration n is small (the quantitative con-dition will be given in the next section). Under this non-interacting capsids approximation, ε m comes from themembrane deformation induced by a single capsid.The calculation procedure to find the free energy pro-file f ( α ) is as follows. We first minimize the membraneelastic energy ε m with respect of all possible membraneshapes for any given capsid size α . Here it is convenientto use a cylindrical coordinate system ( r, h, φ ) as shownin Fig. 2a (the azimuthal angle φ is not shown). Withour assumption of (partial) spherical capsids, the mem-brane profile is independent on φ . As a result, one canuse either the function h ( r ) or r ( h ) to parameterize themembrane. Correspondingly, the mean curvature andthe area element can be written as [19] H ( r ) = h ′ ( r ) + h ′ ( r ) + rh ′′ ( r )2 r [1 + h ′ ( r ) ] / , (9) dS = r p h ′ ( r ) drdφ ; (10)or H ( h ) = 1 + r ′ ( h ) − r ( h ) r ′′ ( h )2 r ( h )[1 + r ′ ( h ) ] / , (11) dS = r ( h ) p r ′ ( h ) dhdφ, (12)where h ′ ( r ) = dh/dr and h ′′ ( r ) = d h/dr are the firstand second derivatives of h with respect to r . Similarly, r ′ ( h ) = dr/dh and r ′′ ( h ) = d r/dh are the first and sec-ond derivative of r with respect to h . Functionally min-imizing the membrane energy ε m with respect to mem-brane shape r ( h ) or h ( r ), one obtains an elastic equationof the membrane shape, similar to the Euler-Lagrangeequation derived from the least action principle in theclassical mechanics. For the shape parametrization using r ( h ), δε m /δr = 0 leads to the equation: κ r [1 + r ′ ] / [ − r ′ − r ′ − r ′ − r ′ + rr ′′ − rr ′ r ′′ − rr ′ r ′′ + 2 r r ′′ − r r ′ r ′′ − r r ′ r ′′ − r r ′′ +30 r r ′ r ′′ + 4 r r ′ r (3) + 8 r r ′ r (3) + 4 r r ′ r (3) − r r ′ r ′′ r (3) − r r ′ r ′′ r (3) + 2 r r (4) + 4 r r ′ r (4) + 2 r r ′ r (4) ]+ σ r ′ − rr ′′ [1 + r ′ ] / = 0 , (13)where r (3) = d r/dh and r (4) = d r/dh are the thirdand forth derivatives of r with respect of h . This equationhas to be solved together with the boundary conditions.On the rim of the partial spherical capsid, the membraneitself and its slope must be continuous. We have h ( r ) | R sin α = R cos α, h ′ ( r ) | R sin α = − tan α, ; (14) or r ( h ) | R cos α = R sin α, r ′ ( h ) | R cos α = − cot α. (15)Far away from the capsid, the membrane becomes flat.we have h ′ ( r ) | ∞ = 0 (16)or r ′ ( h ) | ∞ = ∞ . (17)Solving the elastic equation (13) with the boundaryconditions, Eq. (15) and (17) (or Eq. (14) and Eq. (16) if h ( r ) is used), one obtains the membrane shape that min-imizes ε m . Substituting this shape into Eq. (7), one ob-tains the minimal ε m ( α ). Putting its value into Eq. (6),one gets the total free energy density profile f ( α ). Ingeneral, the elastic equation, Eq. (13), is highly non-linear and numerical calculations are needed to obtainthe exact membrane profile, as shown in Sec. V. How-ever, in certain asymptotic limits, analytical solutionscan be obtained which determine the scaling behavior ofthe system. This is done in the next two sections. III. ASYMPTOTIC SOLUTIONS OF THEMEMBRANE ELASTIC ENERGY
In calculating the free energy profile, the most nontriv-ial part is to find the minimal ε m ( α ), due to the nonlinearelastic equation involved. After the solution is found, it isstraightforward to add the other part of the energy ε c ( α )and get f ( α ). Therefore we focus on the solution of min-imal ε m in this section. Although not solvable in general,the problem do have analytical solutions in asymptoticlimits. To a large extent, they determine the analyticalbehavior of the system, especially the scaling behavior of ε m with the dimensionless parameter e σ = R r σκ , (18)which characterize the relative strength of the surfacetension to the bending rigidity. A. The small deformation solution
A typical approach to consider the elastic deformationof the membrane is to take the small deformation ap-proximation which assumes |∇ h | ≪ ∇ = ˆ r∂ r + ˆ φ r ∂ φ (19)in order to show similarity of the elastic equation to thelinearized Poisson-Boltzmann equation later. Expandingwith ∇ h and keeping terms of O ( ∇ h ) in δε m = 0, wereach a linearized elastic equation which can be writtenas H = 12 ∇ h, ∇ H − Hr s = 0 , (20) where we have introduced an important length scale inthe problem, r s = r κσ . (21)It is the length scale beyond which the stretching en-ergy becomes more important than the bending energy.Notice that Eq. (20) takes exactly the same form as alinearized Poisson-Boltzmann equation in electrolytes orplasma [21]. Therefore r s can be interpreted as an elasticscreening length, similar to the Debye-H¨uckel screeningradius. The local curvature H ( r ) induced by the capsiddecreases when r increases and becomes exponentiallysmall at projected distance larger than r s . This is a typ-ical linear solution of small deformation.Using boundary conditions Eqs. (14) and (16), the spe-cial solution to Eq. (20) is given by h ( r ) = R cos α + r s tan α K ( r/r s ) − K ( R sin α/r s ) K ( R sin α/r s ) , (22) h ′ ( r ) = − tan α K ( r/r s ) K ( R sin α/r s ) , (23) H ( r ) = tan α r s K ( r/r s ) K ( R sin α/r s ) , (24)where K and K are the zero and first order modifiedBessel function of the second kind. At r ≫ r s , both K ( r/r s ) and K ( r/r s ) decay like p r s /r exp( − r/r s ),and the deformation becomes exponentially small, as themeaning of r s suggested.Substituting this solution back to Eq. (7), we get theminimal elastic energy of the membrane ε m = πκ tan α R sin αr s K ( R sin α/r s ) K ( R sin α/r s ) . (25)Notice that this energy is proportional to the dimension-less parameter e σ = R p σ/κ = R/r s . Here the inverseproportion to r s is again a generic feature shared withthe theory of Debye-H¨uckel linear screening [21].The self-consistency of the small deformation approx-imation is warranted by | h ′ ( r ) | <
1, or, according toEq. (23), | tan α | <
1. Therefore this solution is appli-cable in the whole range of r for α < π/ al-ways serve as a “far-capsid” solution for the membraneshape, although the formula for ε m in Eq. (25) is not validin general. It describes the universal decaying behaviorof the deformation when the deformation itself becomessmall enough. We can formally define a characteristicdistance r c through | h ′ ( r c ) | = 1 , (26)beyond which the small deformation solution is valid. r c will be useful later when we discuss the complete solutionto the problem.With the small deformation solution in hand, we arenow ready to derive a quantitative condition for theideal capsid approximation introduced in the last sec-tion. Clearly, when the average projected distance be-tween capsids, d , is much larger than r s + 2 R , the mem-brane mediated interaction between capsids is negligible,since the deformations of the membrane by the capsidsat distance larger than r s are screened out. In this case,most of the membrane surface is flat, so d ≃ d (noticethat d is measured along the membrane surface which ingeneral is larger than d measured along r axis). Thusaccording to Eqs. (3) and (5), the ideal capsids approxi-mation is valid when d r s + 2 R = 2 sin( α/ r s + 2 R ) √ πc G sin( α G / ≫ . (27)In this work, we assume c G is small enough and this isalways the case. B. The catenoid solution
When the surface tension σ = 0, or, r s → ∞ , again ananalytical solution is available [22]. In this case, the sec-ond integral in Eq. (7) is zero. Our problem of finding theminimal ε m is reduced to a minimal surface problem indifferential geometry [19]. Namely, we look for the solu-tion to the equation H = 0 [23]. The only solution underthe rotational symmetry of our problem is the catenoidsolution, first discovered by Euler in 1740 [24].In this case, due to the possible multiple values of h at the same r , it is better to use the r ( h ) representation. H is then given by Eq. (11). Using boundary conditions(15) and (17), the special solution to H = 0 is r ( h ) = R sin α cosh h − R cos α − R sin α arcsinh(cot α ) R sin α . (28)The catenoid shapes for various α are depicted in Fig. 4.In this catenoid shape, the elastic energy ε m achieves itsabsolute minimum, zero.The catenoid solution is a solution to a nonlinear differ-ential equation. It involves large deformations which cannot be characterized by the linear solution discussed inthe last subsection. Although exact only when r s → ∞ ,this solution is still useful for large but finite r s [25]. Infact, since there are no other length scales in the elasticequation (13) ( R only shows up in the boundary con-ditions), a large r s actually means r s ≫ r . Thereforein the region of r ≪ r s , the catenoid solution shouldwork asymptotically. In this sense, this solution can al-ways serve as a “near-capsid” solution for the membraneshape, although ε m = 0 is not true in general. The char-acteristic length beyond which it fails is simply r s . Α= Π rh Α= Π rh Α= Π rh Α= Π rh FIG. 4: (Color online). In the limit of small surface tension,the optimal membrane shapes (thin line, blue online) aroundthe capsid (thick line, red online) are catenoids, as shown fordifferent capsid sizes.
In case of r s → ∞ and α ≪
1, both the catenoidsolution and the small deformation solution work in allrange of r . Indeed they become identical. C. Membrane elastic energy at two asymptoticlimits
The two solutions discussed in the last two subsectionsdetermine the analytical behavior of the system to a largeextent. When e σ ≪
1, they can be combined to get theanalytical expression of the minimal ε m . In general, theydetermine the scaling behavior of ε m with respect of e σ .We separate our discussion into two opposite limits ofsmall and large e σ , which can be called the soft membraneregime and the stiff membrane regime. Here “soft” meanseasy to stretch, “stiff” means the opposite.In the soft membrane regime, e σ ≪ R ≪ r s , thecatenoid solution is valid near the capsid when r ≪ r s .Calculating r c using Eqs. (28) and (26), we get r c = √ R sin α. (29)We see r c ≪ r s . Therefore the valid regions of the twoasymptotic solutions (one is r > r c , the other is r < r s )overlap largely and we can combine them to get a com-plete solution to the optimal membrane shape. Quanti-tatively, we artificially choose a projected distance some-where between r c and r s , say √ r c r s . For r < √ r c r s ,the catenoid solution is used. For r > √ r c r s , the smalldeformation solution is used. Notice that the special so-lution of the small deformation has to be calculated usingthe continuity conditions for h ( r ) and h ′ ( r ) at √ r c r s , de-rived from the catenoid solution. As a result, we have ananalytical expression for the optimal membrane shapecontinuously from the edge of the capsid to infinity. Thecorresponding ε m , keeping the leading order terms in thesmall parameter e σ , is given by ε m = πκ sin α R r s ln r s R . (30)When α ≪
1, this result agrees with the small deforma-tion solution in Eq. (25) in the same regime of small e σ .In this limit, we have ε ∝ e σ ln(1 / e σ ).In the stiff membrane regime, e σ ≫ R ≫ r s .Since r ≫ r s always, the “near capsid” region where thecatenoid solution holds disappears. On the other hand,for α < π/
4, the small deformation solution is valid inthe whole range of r . The membrane elastic energy isgiven by (25), which in this limit reads, ε m = πκ tan α sin α Rr s . (31)For α > π/ r c using Eq. (23)gives r c ≃ R sin α + r s ln | tan α | . (32)Since R ≫ r s , for most of α , we expect that the smalldeformation solution starts to work at places close to thecapsid. Probably because of this, the scaling behavior of ε m ∝ e σ is preserved even at large α , as shown by thenumerical result (see Sec. V). IV. ANALYTICAL RESULT OF THE TOTALFREE ENERGY DENSITY
After the information about the minimal ε m ( α ) isknown, we can add the line tension energy ε c ( α ) to it andconsider the total free energy density f ( α ). The presenceof ε c introduces the second dimensionless parameter tothe problem, e τ = Rτκ , (33)which characterize the relative strength of the line ten-sion on the capsid rim. In this section, we derive sev-eral simple scaling behaviors of the system, dependingon the two dimensionless parameters e σ and e τ . We againseparate our discussion into the soft and stiff membraneregimes corresponding to small and large e σ . A. The soft membrane regime
In the soft membrane regime, e σ = R/r s ≪
1. Substi-tuting Eqs. (8) and (25) to Eq. (6), we have f ≡ κc G (1 − cos α G ) e f = κc G (1 − cos α G ) π cot α e τ + e σ ln 1 e σ sin α ) , (34)where we have introduced the dimensionless free energydensity e f for convenience. e f ( α ) is plotted schematicallyin Fig. 3. When e τ is large, the only minimum of the freeenergy density is at α → π (the left panel of Fig. 3). Onthe other hand, when e τ < . e σ ln(1 / e σ ), a local minimumat the capsid size, α , appears ( the right panel of Fig. 3).Correspondingly, the threshold line tension at which thelocal minimum in the free energy density appears is τ c = 0 . Rσ ln 1 R p σ/κ . (35)Since transcendental equations are involved in mini-mization of e f , it is not easy to get the analytical expres-sion about this local minimum in general. However, α and the corresponding e f can be estimated in a linearapproximation. Assuming α is achieved at small α , wecan expand e f and keep only the leading order terms in α . We get e f = 4 π e τα + 2 π e σ ln 1 e σ α . (36)Taking ∂f /∂α = 0, we have α = s e τ e σ ln(1 / e σ ) = s τRσ ln( p κ/σ/R ) . (37)The fact that this is a minimum rather than a maximumis confirmed by ∂ e f /∂α | α >
0. For τ < τ c at which α shows up, this result is indeed much smaller than one,consistent with the initial assumption that α ≪
1. Inthe same limit, e f ≃ π pe τ e σ ln(1 / e σ ) = 4 π Rκ s τ σ ln p κ/σR . (38) B. The stiff membrane regime
In this case, we do not know the form of the membraneelastic energy ε m for large α . Still, in the same spirit oflinear analysis, we can assume that there is a minimumof f at small α , and use the small deformation solutionEq. (31) for ε m . Notice that the minimum found in thisway is only a local minimum, since we did not includethe information of large α .As a result, we have e f = π cot α e τ + e σ tan α ) ≃ π e τα + 2 π e σα. (39)Taking ∂f /∂α = 0, we get α = r e τ e σ = r τ κσ . (40)It is a minimum since ∂ e f /∂α | α >
0. For this result tobe meaningful, e τ ≪ e σ must hold, which will be checked incomparison with the numerical result. The correspond-ing free energy density is e f = 4 π √ e σ e τ = 4 πR r στ κ . (41) V. NUMERICAL RESULT AND DISCUSSION
In order to verify our analytical understanding and getthe complete solution to the problem, we solve the non-linear elastic equation derived from δε m = 0 numerically.Our computation procedure follows Refs. [6, 26]. Thisnumerical solution is then combined with ε c to give thetotal free energy density f . In this section, we show thenumerical result, compare it with the analytical formulas,and discuss the meaning of our results.
55 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 55 55 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5544 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4433 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 33 33 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3322 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 22 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2211 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Α ¶ Ž m ln J ΣŽ N ΣŽ ¶ Ž m ΣŽ FIG. 5: (Color online) Numerical result of the dimensionlessmembrane elastic energy e ε m = ε m /κ as a function of α . Theeleven sets of data points are at e σ = 10 − , − , − , ..., .They are labelled correspondingly as − , − , − , ...
5. The leftaxis e ε m / e σ is for all e σ ≥ e ε m / e σ ln(1 / e σ ) is for all e σ < . e σ ≪ The direct numerical result of ε m is plotted in Fig. 5,where for convenience we used the dimensionless elasticenergy e ε m = ε m /κ . The first important thing to notice is that the elastic energy profile always takes a “sanddune” shape, where two minimums, zeros, are achievedat α → , π , and a maximum shows up in the middle of α . Physically, this energy profile comes from the need ofmatching boundary conditions at the edge of the capsidand at infinity. The membrane deformed by the capsidedge at one end has to become flat far away from thecapsid. At α → α → π , the membrane is notdeformed at all, and the elastic energy is zero [27]. Whilefor α close to π/
2, the membrane is almost vertical at theedge of the capsid, and a large amount of elastic energyis needed to bend it flat.Secondly, we see clearly two kinds of asymptotic be-haviors of ε m depending on the parameter e σ = R/r s . Inthe stiff membrane regime, e σ ≫
1, the energy is propor-tional to e σ as shown by the collapse of the data pointsto a single curve with e σ varying from 10 to 10 . Themaximum of the energy is achieved at α m ≃ . π . α m isa nonlinear result and can not be calculated analytically.However, the proportionality of ε m to e σ is a small defor-mation result as shown in Eq. (31). In the soft membraneregime, e σ ≪
1, the energy is proportional to e σ ln(1 / e σ ),shown again by the collapse of the data points with e σ varying from 10 − to 10 − . Here the collapse is not aspronouncing as in the other regime mostly due to thelarger numerical error in dealing with smaller e ε m . Theabsolute value of e ε m in this regime is smaller at least infour order of magnitude than in the other regime. Themaximum of the energy here is arrived at α m = π/ α m . These fea-tures agree with our small e σ solution originating fromthe catenoid solution. In fact, Eq. (30) fits the numericaldata reasonably well, with an additional factor 1 . - - - - ΣŽΤŽ Α f ŽΑ f Ž FIG. 6: An effective “phase diagram” in the plane of two di-mensionless parameters, e σ = R/r s and e τ = Rτ /κ . In theupper-left part, the free energy density decreases monoton-ically with α , while in the lower-right part, it has a localminimum, as shown in the insets. The numerical data pointsmark the “phase boundary” at which the local minimum ap-pears. The dotdashed line and the solid lines fit the datapoints using e τ = 0 . e σ and e τ = 0 . e σ ln(1 / e σ ) respectively. The scaling of ε m with e σ suggests a simple way todo the numerical calculation to the free energy density.When e σ ≫ e f = e ε c + e ε m = 2 π cot α e τ + g ( α ) e σ = e σ (cid:20) π cot α e τ e σ + g ( α ) (cid:21) , (42)where e ε c = ε c /κ and g ( α ) is some function given by thenumerical computation. According to the last equality,up to an overall constant e σ , e f is completely determinedby only one parameter e τ / e σ . Similarly, when e σ ≪ e f = e σ ln 1 e σ (cid:20) π cot α e τ e σ ln(1 / e σ ) + g ( α ) (cid:21) , (43)where g ( α ) is again given by numerical computation, al-though we know it from our analytical result Eq. (34).In this regime, e f is determined by one parameter e τ / e σ ln(1 / e σ ). Below in studying the local minimum of e f , we therefore consider a single parameter dependence. ΤŽ (cid:144) ΣŽΑ ΣŽ >> ΤŽ (cid:144) ΣŽ ln H (cid:144) ΣŽ L Α ΣŽ << FIG. 7: The capsid size α as a free energy local minimum isshown as a function of e τ/ e σ and e τ / e σ ln(1 / e σ ) at two limits of e σ .The dots are numerical results taken at e σ = 10 , , , for the upper panel and e σ = 10 − , − , − , − for thelower panel. The curves are analytical results of Eq. (40)at e σ ≫ e σ ≪ . e τ plotted cor-responds to the lower-right “phase” in the “phase diagram”of Fig. 6. For all e σ and e τ , we get two different types of free en-ergy density profiles as shown in Fig. 3, consistent with the analytical result for small e σ . The global minimum ofthe free energy density is always at α → π . Physically,the line tension energy prefers the shortest length of thecapsid rim, which is zero for complete capsids ( α → π ).When e τ is very large, the line tension energy dominates,and the free energy density e f decreases with α monoton-ically to zero, as shown in the left panel of Fig. 3. Onthe other hand, when e τ is small, due to the maximumof the membrane elastic energy ε m , a local minimum atthe capsid size, α , shows up in the free energy den-sity, as shown in the right panel of Fig. 3. It is usefulto draw a “phase diagram” on the plane of e σ and e τ asFig. 6 to show this qualitative difference in the free energydensity profile. The lower right region of Fig. 6 corre-sponds to value of the parameters ( e σ , e τ ) where capsidbudding can be kinetically trapped. The two lines fit the“phase boundary” at large and small e σ with e τ = 0 . e σ and e τ = 0 . e σ ln(1 /σ ) respectively. As one can see,there is a very good agreement between numberical re-sults and our scaling formulas for e σ in two asymptoticlimits. According to the numerical fits, the threshold τ at which the local minimum in the free energy densityshows up are τ c = 0 . √ κσ (44)when e σ ≫
1, and τ c = 0 . Rσ ln 1 R p σ/κ (45)when e σ ≪
1. The later formula agrees with our analyticalresult Eq. (35) with a numerical factor 3 difference.The possible local minimum in e f (the right panel inFig. 3) suggests that budding may be trapped kineticallyat the capsid size α . Numerical and analytical results of α are shown in Fig 7. The analytical curves are drawnusing Eq. (40) at e σ ≫ e σ ≪
1, with addi-tional numerical factors of 2 and 1 . e τ . This is the parameter regime where thelinear approximation is no longer valid.The kinetic trapping becomes significant if the barrierin the free energy density is large. In Fig. 8, numericaland analytical results about this barrier are plotted. Forthe local minimum e f , up to a order one numerical fac-tor (1 . . σ ≫ σ ≪ e f m , whichis in the nonlinear regime. However, in the most impor-tant regime of small e τ and large barrier, Fig. 8 showsthat the main contribution to e f m comes from the mem-brane elastic energy ε m (the value of e f m at e τ = 0). Inthis regime, the additional contribution to e f m from theline tension energy ε c is negligible and e f m is almost aconstant. Combining the numerical result of e f m and theanalytical result of e f with proper numerical factors, we0 æ æ æ æ æ æ æ æ æ æ æà à à à à à à à à à àò ò ò ò ò ò ò ò ò ò ò ΤŽ (cid:144) ΣŽ f ŽΣŽ ΣŽ >> æ æ æ æ æ æ æ æ æ æ æ æ æà à à à à à à à à à à à àò ò ò ò ò ò ò ò ò ò ò ò ò ΤŽ (cid:144) ΣŽ ln H (cid:144) ΣŽ L f Ž ln I ΣŽ M ΣŽ ΣŽ << FIG. 8: (Color online). The local minimum e f , maximum e f m ,and barrier e f m − e f of the free energy density. The values of e σ and e τ plotted are the same as in Fig. 7. The circles (blueonline) are the numerical result of e f , fitted by the solid lines(blue online) using Eq. (41) at σ ≫ σ ≪ . . e f m ,marked by the dashed lines (green online) at their zero e τ values. The triangles (red online) are numerical result of thebarriers, e f m − e f , fitted by the dotdashed lines (red online)using Eqs. (46) and (47). get the asymptotic formulas for the barrier at e τ ≪ e f m − e f ≃ . e σ − . π √ e σ e τ = 7 . R r σκ − . πR r στ κ , ( e σ ≫ . (46) e f m − e f ≃ . e σ ln(1 / e σ ) − . π pe τ e σ ln(1 / e σ )= 4 . R σκ ln p κ/σR − . π Rκ s τ σ ln p κ/σR , ( e σ ≪ . (47)The largest barriers are achieved at e τ = 0 or e f = 0. VI. KINETICS OF BUDDING AND PARTIALBUDDING
As discussed in the previous sections, with a finite Gag-Gag attraction, budding always proceeds to completionthermodynamically. However, when this attraction isweak, or τ is small, a metastable state of partial bud-ding appears at a smaller capsid size α (see Fig. 3). It istherefore possible that the budding process is kineticallytrapped at α . In this section, we discuss this kinetic ef-fect and make connections of our theory to experiments.Let us first estimate the values of parameters. Anormal plasma membrane has κ ≃ − B T and σ ≃ . − / nm = 0 . − . B T / nm [28]. On theother hand, typical HIV have R ≃ − e σ = R p σ/κ ≃
10 and only the stiff membraneregime with e σ ≫ e σ ≪
1, one can show that the typi-cal energy scale ε m ∼ κ e σ ln(1 / e σ ) is comparable to k B Tand thus not important in the room temperature. In thissection, we therefore focus on the stiff membrane regimeonly.In order to see if budding can be kinetically trapped atthe local minimum α (see Fig. 3), we have to study thebudding kinetics and calculate the kinetic barrier. Forthis purpose, let us employ the standard kinetic pictureof the first order phase transition [29, 30], correspond-ing to the transition from a free-Gags phase to an aggre-gated Gags phase where Gags self-assemble into completeviral capsids. At the initial stage of aggregation, the con-centration of free Gags is large, Gags coagulate to formdimers. Dimers coagulate with free Gags or other dimersto form larger Gag clusters (small capsids). This initialcoagulation or nucleation is a fast process and is not arate limiting step in retroviral budding. Soon free Gagsare significantly depleted, and the main kinetic pathwayfor growth of capsids is for them to diffuse and mergewith each other. We will be concern with this later stageof coagulation. For simplicity, we work with the domi-nant capsid size, α ( t ) [with concentration n ( t )], assumingthese typical capsids carry all the mass of membrane-bound Gag proteins.Let us start with the case when α ( t ) is still small sothat the energy barrier for merging of capsids is smallerthan k B T. This is the regime of the well known diffu-sion limited aggregation [31]. The rate of the capsid area A ( α, t ) incretion is proportional to the probability thattwo capsids diffuse and merge with each other. The ki-netic rate equation reads dA ( α, t ) dt = 2 πR sin α ( t ) A ( t ) D ∇ n ( t ) | R sin α , (48)where D ≃ k B T /ηR sin α ( t ) is the lateral diffusion con-stant of a capsid on the membrane and ∇ n ( t ) | R sin α is thegradient of the concentration n ( t ) on the edge of the cap-sid. This gradient can be estimated assuming a steadystate in the diffusion and taking the adsorbing boundarycondition at the edge of the capsid and a given capsid1concentration [Eq. (3)] far away from the capsid. Solvingthe diffusion equation with these boundary conditions,we find ∇ n ( t ) | R sin α = c G A ( α G ) A ( α ) R sin α . (49)Substituting these relations and Eq. (4) into Eq. (48), weget α ( t )2 − sin 2 α ( t )4 = tτ diff + α G − sin 2 α G . (50)where τ diff = ηR /T c G A ( α G ) (51)is the time scale of diffusion proportional to the viscosity η of the membrane. In the small α regime correspondingto a small kinetic barrier, this equation can be written as α ( t ) = (3 t/τ diff + α G ) / , (52)which is a slow function of time.The regime of diffusion limited growth stops whenthe kinetic barrier between approaching partial capsid ismuch larger than k B T . At a later time, a different growthregime of Lifshitz-Slezov (LS) comes into play [29]. Inthis mechanism, the growth is no longer due to colli-sion and merging of partially budded capsids. Instead,smaller capsids shrink and release individual Gags. TheseGag are absorbed into larger capsids, leading to theirgrowth. This process of releasing and adsorbing of indi-vidual Gags (the so-called coalescence) has much smallerkinetic barrier than the barrier to capsid merging in thislater stage. The growth of capsid size in LS regime is thesame as that of diffusion limited growth [29]. However,the rate constant τ LS depends exponentially on the acti-vation energy to release individual Gag proteins from acapsid τ LS ∝ exp( −| ǫ | /k B T ) (53)where ǫ is the binding energy of Gag in a capsid, whichitself is also a function of the Gag-Gag interaction.The kinetic picture described above is good when τ >τ c and the free energy density decreases monotonicallywith increasing α (the left panel of Fig. 3). However,when τ < τ c and a local minimum α appears in thefree energy density (the right panel of Fig. 3), the abovepicture must be modified. For the cluster growth, eitherin the diffusion-limited regime or in the LS regime, thegrowth of the cluster size always reduces the free energyof the system. On the other hand, for the capsid growthof retroviral budding, after the capsid size reaches α ,the system free energy increases when the capsids growfurther. For α > α , the growth of capsids is determinedby the ability to tunnel through the kinetic barrier relatedto f m − f (see Fig. 3). The detail analysis of the rate ofcapsid growth for α > α is a very interesting problem by itself, requiring understanding of membrane energeticswhen a partially budded capsid absorbs other capsids ormany individual Gags to increase its size from α to α m .These calculations are beyond the scope of this paperand we will leave the detail treament of capsid growth inthis case to a future study. Nevertheless, one can expectthe rate of such process to inversely proportional to theexponential of the energy barrier τ tunnel ∝ exp[ − ( f m − f ) /nk B T ] , (54)where ( f m − f ) /n is the energy barrier of a membranecell with a single capsid in it. According to Eqs. (46)(see also the upper panel of Fig. 8), the maximum energybarrier is achieved at f = 0 or e τ = 0. Using Eqs. (3)and (34), it can be written as E m = f m n = κ (1 − cos α m ) e f m , (55)where e f m is given by Eqs. (46), and α m is the correspond-ing capsid size. A more useful expression of E m can begot if one recognizes that E m is nothing but the maxi-mum of ε m shown in Fig. 5. Using the numerical resultof that figure, we get E m = 11 . κ e σ = 11 . R √ κσ. (56)Clearly, E m ≫ k B T for e σ >
1. For example, for R = 70 nm , σ = 0 . B T / nm and κ = 20k B T, we get E m = 1765k B T. The true energy barrier is smaller than E m since e τ >
0. In the regime of small e τ , according toEq. (46), it is E ≃ κ (1 − cos α m )( e f m − e f )= E m − . r e τ e σ ! = E m − . r τ κσ ! . (57)In experiments, if one treats κ and σ as constants thenaccording to Eq. (56) the larger the retrovirus size R ,the bigger the kinetic barrier. On the other hand, theline tension τ is directly proportional to the strength ofthe Gag-Gag attraction and is experimentally adjustablethrough mutation of the late domain on the Gag pro-tein, binding of other molecules to Gags or changing thepH, salinity of water solution near the membrane [2, 13].As we know, the closest approach distance between twoGag proteins is about 10nm [1]. If Gags are denselypacked on the capsid, this gives τ ≃ . B T / nm ≃ τ < τ c (see Fig. 3). For a normal cell membrane with κ = 20k B T and σ = 0 . B T / nm , using Eq. (44), τ c ≃ . B T / nm = 1pN. Therefore for normal capsids, τ > τ c , and budding easily proceeds to completion (seethe left panel of Fig. 3). One the other hand, τ is biggerthan τ c only by a factor of 2. Therefore HIV buddingcan be fairly easily trapped at a partially budded state2with capsid size α by reducing the Gag-Gag interac-tion strength such as mutation of a single domain on theGag protein. The kinetic barrier E appeared at α = α can be much larger than k B T, and the time scale forcapsid growth beyond α , τ tunnel , is exponentially large.Qualitatively, this trend is consistent with experimentson mutation of the late domain of Gag proteins [2, 3].Numerically, we know that α < . ≃ . π (see theupper panel of Fig. 7). It agrees with experiments rea-sonably well, although there are many additional factorsthat we neglected in our treatment such as local varia-tion in membrane elasticity due to raft structures or thepresence of other proteins in in-vivo assembly and bud-ding. More controlled experiments are needed to to verifythe dependence on the membrane rigidities and Gag-Gagattraction of α given by Eq. (40). VII. CONCLUSION
In this paper, we developed an elastic model of HIV(and retroviruses in general) budding and self-assemblyon the elastic membrane. We studied the free energyprofile of the system as a function of the capsid size α . We showed that although always thermodynamicallyfavorable, complete budding and assembly may not beachieved if the Gag-Gag attraction is weak. In practice,for normal biological conditions, the Gag-Gag attractionis strong enough and HIV budding and assembly alwaysproceed to completion, as it should be. On the otherhand, it is fairly easy to trap HIV budding to a par-tially budded state with capsid size α by reducing theGag-Gag attraction. This can be done through the mu-tation of late domain on the Gag protein or binding ofother molecules to Gag. In principle, the trapping is also possible by increasing the membrane rigidities, althoughthis is not easy to do in vivo. Our theory agrees with rea-sonably well with experimental results. However, experi-ments with better controlled environments are needed toverify various aspect of the theory.The most interesting point of our model is probablythat it provides a unique self-assembly mechanism. Notlike self-assembly of other viruses or colloids, HIV assem-ble and bud concurrently on the membrane. Thereforethe membrane elastic energy plays an important role inthe assembly process. For example, the kinetic barrierwhich traps the HIV budding essentially comes from themembrane elastic energy around the capsids. In fact,our model developed for HIV budding and assembly canbe very well applied to other situations. For example,for a given concentration of membrane-bounded proteinswith a fixed spontaneous curvature, this kind of bud-ding and assembly phenomenon should also exist andcan be explained using our model. In this situation, itmay be easier to change the membrane properties andprotein-protein attraction in vitro to verify our theorymore quantitatively. Due to the interplay between themembrane elastic energy and the Gag-Gag attraction en-ergy, the kinetics of retrovirus budding is an interestingproblem by itself, as discussed in Sec. VI. We plan toaddress this question in more detail the near future. Acknowledgments
We wish to thank G. Bel, J. Mueller, B. I. Shklovskiiand T. A. Witten for useful discussions. T.T.N. acknowl-edges the junior faculty support from the Georgia Insti-tute of Technology. [1] J. M. Coffin, S. H. Hughes, and H. E. Varmus,
Retro-viruses (Cold Spring Harbor Laboratory Press, NewYork, 1997), 1st ed.[2] E. Morita and W. I. Sundquist, Annu. Rev. Cell Dev.Biol. , 395 (2004).[3] D. G. Demirov and E. O. Freed, Virus Research , 87(2004).[4] S. Tzlil, M. Deserno, W. M. Gelbart, and A. Ben-Shaul,Biophysical Journal , 2037 (2004).[5] M. Deserno and T. Bickel, Europhysics Letters , 767(2003).[6] M. Deserno, Physical Review E , 031903 (2004).[7] A. Zlotnick, Journal of Molecular Biology , 14 (2007).[8] M. F. Hagan and D. Chandler, Biophysical Journal ,42 (2006).[9] T. Hu and B. I. Shklovskii, Physical Review E , 051901(2007).[10] J. E. Dooher, B. L. Schneider, J. C. Reed, and J. R.Lingappa, Traffic , 195 (2007).[11] H. Garoff, R. Hewson, and D.-J. E. Opstelten, Microbi-ology and Molecular Biology Reviews , 1171 (1998). [12] S. Welsch, B. M¨uller, and H. Kra¨usslich, FEBS Lett. ,2089 (2007).[13] S. Campbell, R. J. Fisher, E. M. Towler, S. Fox, H. J.Issaq, T. Wolfe, L. R. Phillips, and A. Rein, Proc. Natl.Acad. Sci. USA , 10875 (2001).[14] F. J¨ulicher and R. Lipowsky, Physical Review Lettes ,2964 (1993).[15] F. J¨ulicher and R. Lipowsky, Physical Review E , 2670(1996).[16] P. Canham, Journal of Theoretical Biology , 61 (1970).[17] W. Helfrich, Z. Naturforsch. C , 693 (1973).[18] In Ref. [15], the Gaussian curvature term is importantsince the area of the budding region is not fixed.[19] Kreyszig, Differential Geometry (Dover, New York,1991).[20] Expansion in the opposite limit |∇ h | ≫ Statistical Physics, Part1 (Butterworth Heinemann, Oxford, 1980), 3rd ed.[22] In the opposite limit of κ = 0, although δε m = 0 still has a catenoid solution, it is only a stationary solutionbut does not correspond to an energy minimum. Actu-ally ε m can be arbitarily close to zero but not equal tozero, given a membrane shape arbitarilly close to the flatmembrane and only deformed a little bit at the capsidrim. This result is different from the minimal surface ofrevolution problem in the calculus of variation [32]. Thisis eccentially due to the fact that our boundary conditionrequires the membrane to be flat at infinity, but there isno confinment to its position there.[23] Since all energies we considered are positive definite, H =0 corresponds an absolute minimum for ε m . One can ofcourse still use δε m = 0 to get an elastic equation. It ismuch more complicated than H = 0 and the catenoidsolution indeed holds.[24] F. Morgan, Riemannian geometry: a beginner’s guide (Jones and Bartlett Publishers, Boston, London, 1993),1st ed.[25] The first order correction to this solution for large butfinite r s also involves a nonlinear differential equationand can not be solved analytically.[26] U. Seifert, K. Berndl, and R. Lipowsky, Physical Review A , 1182 (1991).[27] Intuitively, one may think that at α close to π , the “neck”of the membrane (see the lower-right panel of Fig. 4 foran illustration of the neck) cost a large elastic energy. Infact, this is not the case. In the soft membrane regime,the bending energy dominates. This neck can take acatenoid shape which has zero curvature energy. In thestiff membrane regime, the stretching energy dominates.The membrane can make a sharp turn to minimize thestretching energy again to almost zero.[28] C. E. Morris and U. Homann, The journal of membranebiology , 79 (2001).[29] E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics (Butterworth-Heinemann, Oxford, 1997).[30] T. T. Nguyen and B. I. Shklovskii, Physical Review E , 031409 (2002).[31] D. F. Evans and H. Wennerstr¨om, The Colloidal Do-main: Where Physics, Chemistry, Biology, and Technol-ogy Meet (Wiley-VCH, New York, 1999), 2nd ed.[32] B. van Brunt,