A theory of ordering of elongated and curved proteins on membranes driven by density and curvature
Caterina Tozzi, Nikhil Walani, Anabel-Lise Le Roux, Pere Roca-Cusachs, Marino Arroyo
AA theory of ordering of elongated and curved proteins on membranes drivenby density and curvature
Caterina Tozzi, a † Nikhil Walani, a † , Anabel-Lise Le Roux b , Pere Roca-Cusachs bc , and Marino Arroyo (cid:63) abd Cell membranes interact with a myriad of curvature-active proteins that control membrane morphology and areresponsible for mechanosensation and mechanotransduction. Some of these proteins, such as those containing BARdomains, are curved and elongated, and hence may adopt different states of orientational order, from isotropic tomaximize entropy to nematic as a result of crowding or to adapt to the curvature of the underlying membrane.Here, extending the work of [Nascimento et. al,
Phys. Rev. E , 2017, 96, 022704], we develop a mean-fielddensity functional theory to predict the orientational order and evaluate the free-energy of ensembles of elongatedand curved objects on curved membranes. This theory depends on the microscopic properties of the particles andexplains how a density-dependent isotropic-to-nematic transition is modified by anisotropic curvature. We alsoexamine the coexistence of isotropic and nematic phases. This theory lays the ground to understand the interplaybetween membrane reshaping by BAR proteins and molecular order, examined in [Le Roux et. al,
Submitted , 2020].
Many cellular processes rely on the ability of cell membranes tochange their shape, including area and tension regulation , orthe transport of cargo within the cells in membrane bound vesi-cles . Membrane can change shape in response to cytoskele-tal dynamics , changes in pH of the surrounding medium , orthe recruitment of proteins that are either curved or bulky anddisordered . In this study we focus on elongated curved pro-teins such as those containing the BAR domain (amphiphysin,endophilin, F-CHO) and others like dynamin, EHD2, etc . Whenthese proteins lack positional order but tend to point in a givendirection, at least locally, they are in a so-called nematic state.These proteins are “banana shaped” and can impinge anisotropiccurvatures on the membranes upon binding through a scaffoldingeffect , which allows them to tubulate liposomes , stabilize tubu-lar necks in Caveolae or bind to necks of budding vesicles anddrive endocytic transport . The generation of anisotropic curva-ture has been associated with a nematic ordering of the elongatedproteins along the high-curvature direction at very high cover-age . Besides anisotropic curvature, elongated proteins canalso create isotropically curved (spherical) domains, as F-BARs inthe initial stages of assembly of clarthin coats or during fastendocytosis by endophilin . This suggests a multi-functionalityof curved and elongated proteins and a correlation between cur-vature anisotropy, density, and nematic order. Controlled in-vitroexperiments capturing the dynamics of this interplay have beenelusive. Giant Unilamellar Vesicles (GUVs) exposed to curved andelongated proteins exhibit no change in membrane shape below atension-dependent protein coverage threshold, above which verythin protein-rich tubules are violently shed by the vesicle , andin GUVs-tether systems, the high membrane tension strongly re- a Universitat Politècnica de Catalunya-BarcelonaTech, 08034 Barcelona, Spain. Email:[email protected] b Institute for Bioengineering of Catalonia (IBEC), The Barcelona Institute for Scienceand Technology (BIST), 08028 Barcelona, Spain. c Universitat de Barcelona, 08036 Barcelona, Spain. d Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), 08034 Barcelona,Spain. † These authors contributed equally to this work. duces the ability of the membrane to change shape .Theoretical continuum models coupling membrane’s elasticitywith orthotropic proteins have been restricted to a prescribedconfiguration (density and orientation) of proteins that are al-ways in the nematic phase. Coarse grained MD and Monte-Carlosimulations on the other hand have been successfully used to un-derstand molecular aspects of proteins in shaping of membranesuch as scaffolding as compared to helical insertions , arclength , lateral interactions or chirality and the generationof topological defects in closed vesicles transforming into tubu-lar liposomes . However, these simulations are limited to shorttime-scales and small length-scales. While continuum modelshave shown promise in describing the dynamics of membraneprotein interactions at time and length scales relevant to manybiophysical processes , they are phenomenological and dis-connected to the relevant microscopic details. Thus, there is aneed for the development of effective field theories capturing themicroscopic details of protein interactions on lipid membranes.To this end, we develop a mean field density functional the-ory for the free energy of the proteins accounting for protein areacoverage, orientational order and membrane curvature that canprovide a basis for upscaling. In the present work, membranecurvature is taken as given. In a companion paper, we couplethe model presented here with one for a deformable membraneto study the mechano-chemistry of membrane reshaping by BARproteins . In Section 1, the entropy of elongated proteins onthe membrane is obtained by adapting to 2D a recent theory byNascimento et. al for hard ellipsoidal particles, which correctsOnsager’s classical theory of isotropic-to-nematic transitions fornon-spherical particles to provide quantitative prediction at highdensities and moderate particle aspect ratio. The crucial differ-ence with Onsager’s work is the enforcement of a compact sup-port of the orientational probability distribution beyond a certainspatial density. In Section 2, we generalize the theory to accountfor the elastic curvature energy of the proteins depending on theirorientation and on the second fundamental form of the under-lying surface. We also examine the effect of curvature on theisotropic-to-nematic transition and on the orientational probabil- a r X i v : . [ c ond - m a t . s o f t ] S e p ty distribution. Finally, in Section 3, we study the coexistence ofisotropic and nematic phases. Before accounting for the bending energy of proteins, we adaptand extend the theory presented by to two-dimensions to ap-ply it to proteins on a membrane. We consider proteins to be 2Delliptical particles such that the length of their major and minoraxes are given by a and b as shown in Fig. 1. The state of pro-tein i , qqq i , is given by its position on a surface Γ , rrr i and orientationover the unit circle S , given by the angle γ i of the long axis of theellipse relative to a fixed direction on the surface. We assume thatproteins are rigid, non-overlapping but otherwise non-interacting.Thus, for two particles with states qqq and qqq , their interaction po-tential U is purely repulsive and can be defined as U ( qqq , qqq ) = (cid:40) ∞ if particles overlap, otherwise . (1)The configurational free energy for N identical proteins up to ad-ditive constant is given by F e = − β ln 1 N ! (cid:90) Ω N e − β Σ ≤ i < j ≤ N U i , j dqqq ... dqqq N , (2)where β = / k B T and Ω is the domain in phase space for eachof the proteins, accounting for the translational and orientationaldegrees of freedom. Since all the proteins are equivalent, invok-ing a mean field approximation and a passage to the continuumlimit (see for further details), the free energy can be written as F e = β (cid:90) Ω ρ ( qqq ) ln ( ρ ( qqq )) dqqq − β (cid:90) Ω ρ ( qqq ) ln [ − W ( qqq )] dqqq (3)where ρ ( qqq ) is the number density of particles with state qqq and W ( qqq ) is the average fraction of excluded area for a given particle,i.e. the fraction of phase space inaccessible to a particle due topresence of other particles. This quantity is postulated to take theform W ( qqq ) = λ (cid:90) Ω ρ ( qqq ) (cid:104) − e − β U ( qqq , qqq ) (cid:105) dqqq , (4)where λ is an adjustable parameter discussed in Fig. 1 accountingfor the high-density packing. We note that in the dilute limit, thefirst term in Eq. (3) is dominant, and thus in this limit λ does notplay an important role. We further express the number density ofproteins in terms of positional and orientational contributions ρ ( qqq ) = φ ( rrr ) f ( rrr , γ ) , (5)so that (cid:82) S f ( rrr , γ ) d γ = and (cid:82) Γ φ ( rrr ) drrr = N . Substituting the aboverelation into Eq. (4), we obtain W ( qqq ) = λ (cid:90) S (cid:90) Γ φ ( rrr ) f ( rrr , γ ) (cid:104) − e − β U ( rrr , γ , rrr , γ ) (cid:105) drrr d γ , (6)where we note that drrr should be interpreted as the element ofarea on the surface Γ . Note that the term between square brack-ets is zero unless rrr and rrr are within a small distance commen- surate to the particle size. Thus, assuming that φ varies slowly inthis length-scale, it is reasonable to approximate φ ( rrr ) f ( rrr , γ ) ≈ φ ( rrr ) f ( rrr , γ ) in the equation above, finding W ( qqq ) = λ φ ( rrr ) (cid:90) S f ( rrr , γ ) A e ( γ , γ ) d γ , (7)where A e ( γ , γ ) is given by A e ( γ , γ ) = (cid:90) Γ (cid:104) − e − β U ( rrr , γ , rrr , γ ) (cid:105) drrr . (8)The integrand in this expression is 0 unless the two particles over-lap, in which case it is 1. It thus contains purely geometric infor-mation and can be interpreted as the excluded area per particlefor two particles oriented along the angles γ and γ . Note that, bytranslational invariance, it is independent of rrr , and by rotationalinvariance is should depend on γ and γ through their difference.Introducing Eqs. (5,7) into Eq. (3), we can write the configura-tional free energy of the system as F e = β (cid:90) Γ φ ( rrr ) ln φ ( rrr ) drrr + β (cid:90) Γ φ ( rrr ) (cid:26) (cid:90) S f ( rrr , γ )[ ln f ( rrr , γ ) − ln g ( rrr , γ )] d γ (cid:27) drrr , (9)where we have defined g ( rrr , γ ) = − λ φ ( rrr ) (cid:90) S f ( rrr , γ ) A e ( γ , γ ) d γ . (10)We note that for circular particles, this free energy reduces to thatof a Van der Waals gas. We also note that, even though A e is scaledependent (it has units of area), φ is also scale dependent in sucha way that g is dimensionless and scale-independent. To evaluate the free-energy in Eq. (9), we need to evaluate A e ( γ , γ ) . For this, we note that the excluded area between twoellipses can be computed in terms of the distance of closest ap-proach R , see Fig. 1(a,b), as A e ( γ , γ ) = (cid:90) S R ( ω , ξ , a , b ) d ξ . (11)The calculation of R ( ω , ξ , a , b ) is algebraically complex . It canbe shown that R solves the equation = ( f − f )( f − f ) − ( − f f ) , (12)where f α = + G − (cid:18) Ra sin θ α (cid:19) − (cid:18) Rb cos θ α (cid:19) , for α = , , (13)with θ = ξ , θ = ξ − ω , and G = + (cid:18) ab − ba (cid:19) sin ω . (14)The above equations allow us to compute A e ( γ , γ ) numerically,see Fig. 1(c), which shows how the dependence of the excludedarea on particle alignment depends on the aspect ratio. c N o r m a li z ed () -1 -0.5 0.50 1 -0.5)-18+47(cos( ) -0.5)(cos( ) -0.5)-44+63 b d a/b =5.3 a/b =3 a/b =1 Fig. 1 (a) Contact configuration of two ellipses in the plane. The modulus of the vector joining the centers, R = | RRR | , is the so-called distance ofclosest approach for two ellipses whose major axis forms an angle ω = γ − γ , where γ α is the orientation of each of the ellipses with respect to afixed direction. This distance obviously depends on ω , on the length of the major and minor axes, a and b , but also on the location of the contactpoint, parametrized by the angle ξ . (b) Illustration of the calculation of the excluded area (shaded in green) for a given ω . (c) Excluded area perparticle for different ellipses of fixed area as a function of cos ω . The excluded area is normalized by the area of an ellipse π ab and all the ellipses withdifferent aspect ratios a / b have the same area. Elongated ellipses need to align to reduce the excluded area per ellipse. Dots represent the excludedarea computed numerically with high accuracy while the solid line corresponds to a least-squares fit with the second-order polynomial approximationin Eq. (15) . (d) Schematic view for the choice of λ in Eq. (4) estimated through the relation a eff = λ A e , where a eff is the average area effectivelyoccupied by one particle in the rectangle and given by a ef f = A rect / N particles = ab and A e is the excluded area for a pair of particles with parallel longaxis, A e = π ab . The average fraction of excluded area for a given particle W ( qqq ) is only relevant at high packing but we are estimating it usingthe excluded area between two particles, A e . To reconcile thesetwo quantities through the parameter λ , we follow . In a densepacking limit, such as in Fig. 1(d), W ( qqq ) should approach 1, whichwe can express as the number density of particles φ = N particles / A tot times an effective area per particle a eff in such a dense arrange-ment. We thus obtain a eff = ab . Examining Eq. (7), in this high-packing limit ≈ λ ( N particles / A tot ) A e , where A e is the excludedarea between two ellipsoidal particles with parallel long axis, i.e. A e = π ab . We thus conclude that λ = / π ≈ / .Since A e ( γ , γ ) depends on the orientations of the particlesthrough ω = γ − γ , it can be approximated by an expansion ofLegendre polynomials depending on cos ω as A e ( γ , γ ) = B + B P ( cos ω ) + ··· (15)where B and B are constants depending on a and b , and P ( x ) = x − / . Note that by symmetry arguments, only evenpolynomials appear in the expansion. The expansion can be ex-tended to higher order but the second-order approximation al-ready provides a good approximation, see Fig. 1(c). Interestingly,the second order expansion allows us to express A e ( γ , γ ) in termsof the symmetric and traceless tensor σσσ ( γ ) = [ (cid:96)(cid:96)(cid:96) ( γ ) ⊗ (cid:96)(cid:96)(cid:96) ( γ ) − III ] , (16)where III is the surface identity and (cid:96)(cid:96)(cid:96) ( γ ) is the local orientation ofproteins. The latter can be expressed in an arbitrary orthonormalframe of the tangent plane to the surface { λλλ , µµµ } as (cid:96)(cid:96)(cid:96) = cos γ λλλ + sin γ µµµ , see Fig. 1. This tensor describes the local (or microscopic)second moment of the orientation of proteins, and as shown later,it leads to a theory where orientational order appears through theclassical nematic tensor QQQ . By noting that (cid:96)(cid:96)(cid:96) ( γ ) · (cid:96)(cid:96)(cid:96) ( γ ) = cos ω , a direct calculation showsfrom Eq. (15) that λ A e ( γ , γ ) = c − d σσσ ( γ ) : σσσ ( γ ) , (17)where, c = λ B and d = − λ B depend on a and b and can be com-puted by fitting a second order polynomial in cos γ to A e ( γ , γ ) . Given a density field φ , we can find the optimal angular distri-bution at each point in space rrr by minimizing the free energy inEq. (9) with respect to f subject to the normalization constraint.To minimize the free energy and account for this constraint, weintroduce the Lagrangian functional L [ f , µ ] = (cid:90) S f ( ln f − ln g ) d γ + µ (cid:18) (cid:90) S f d γ − (cid:19) , (18)where µ is a Lagrange multiplier field. Since we perform thisminimization point-wise, we drop for notational simplicity thedependence on rrr of f , g , µ , φ and all quantities depending onthese fields.Recalling Eq. (17), we have g ( γ ) = − φ ( c − d σσσ ( γ ) : QQQ ) , (19)where we have introduced the nematic tensor describing the av-erage particle orientation QQQ = (cid:90) S f ( γ ) σσσ ( γ ) d γ = (cid:104) σσσ ( γ ) (cid:105) . (20)Note that QQQ inherits from σσσ the properties of being symmetricand traceless. We further introduce the auxiliary tensor ψψψ = d φ (cid:90) S f ( γ ) g ( γ ) σσσ ( γ ) d γ , (21) hich is also symmetric and traceless. The stationarity conditioncan then be written as = δ f L = (cid:90) S ( ln f − ln g − ψψψ : σσσ + µ ) δ f d γ , (22)for all admissible variation δ f , and thus the term between paren-theses must vanish. It is clear that when g → , then necessar-ily f → . We can thus define the support of f ( γ ) as S + = { γ ∈ ( − π , π ) such that g ( γ ) > } . Determining µ though the normal-ization of f , we find an expression for the angular probabilitydensity function f ( γ ) = g ( γ ) e σσσ ( γ ) : ψψψ (cid:82) S + g ( γ (cid:48) ) e σσσ ( γ (cid:48) ) : ψψψ d γ (cid:48) if γ ∈ S + otherwise. (23)We note that this expression is far from being explicit, since g depends on QQQ , which in turn depends on f , and ψψψ also dependson f . However, as developed below, it allows us to evaluate thefree energy. We also note that the probability density function f vanishes in a region of the orientational space as the areal numberdensity of proteins φ increases, and thus g in Eq. (19) becomesnegative. As discussed in , this is critical to quantitativelypredict density based ordering for moderately elongated particles.Finally, due to the symmetry of particles with respect to rotationsby π , it follows that f ( γ ) = f ( γ + π ) .Since QQQ is symmetric and traceless, it has two real eigenvaluesof opposite sign and it is diagonal in an orthonormal eigenframe.We let { λλλ , µµµ } be this eigenframe. Thus, the nematic tensor canbe expressed as QQQ = S ( λλλ ⊗ λλλ − µµµ ⊗ µµµ ) , (24)where we call S the order parameter, which contracting the aboverelation and Eq. (20) with λλλ ⊗ λλλ can be expressed as S = (cid:68) ( (cid:96)(cid:96)(cid:96) · λλλ ) − (cid:69) = (cid:104) P ( cos γ ) (cid:105) , (25)where γ is the angle between the nematic direction λλλ and thedirection of a microscopic particle, (cid:96)(cid:96)(cid:96) . Combining Eqs. (16) and(24), we also find that σσσ : QQQ = SP ( cos γ ) , and thus g ( γ ) = − φ [ c − dSP ( cos γ )] . (26)A traceless symmetric tensor such as ψψψ can be expressed in theeigenframe of QQQ as ψψψ = ψ ( λλλ ⊗ λλλ − µµµ ⊗ µµµ ) + ¯ ψ ( λλλ ⊗ µµµ + µµµ ⊗ λλλ ) . (27)We show next that in fact, { λλλ , µµµ } is also an eigenfame of ψψψ , andthus ¯ ψ = .With the above representation of ψψψ , we find that σσσ : ψψψ = ψ P ( cos γ ) + ¯ ψ sin γ cos γ . (28)The condition that { λλλ , µµµ } is an eigenframe of QQQ implies that = λλλ · QQQ · µµµ and hence = (cid:90) S + f ( γ ) λλλ · σσσ ( γ ) · µµµ d γ = (cid:90) S + f ( γ ) sin γ cos γ d γ . (29) Noting that S + is symmetric about γ = since g ( γ ) is an evenfunction, Eq. (26), and using the symmetry f ( γ ) = f ( γ + π ) , theabove relation implies that = (cid:90) S + ∩ ( − π / , π / ) g ( γ ) e σσσ ( γ ) : ψψψ sin γ cos γ d γ = (cid:90) S + ∩ ( − π / , π / ) g ( γ ) sin γ cos γ e ψ P ( cos γ ) e ¯ ψ sin γ cos γ d γ = (cid:90) S + ∩ ( , π / ) g ( γ ) sin γ cos γ e ψ P ( cos γ ) (cid:16) e ¯ ψ sin γ cos γ − e − ¯ ψ sin γ cos γ (cid:17) d γ , where in the last step we have used the fact that g ( γ ) sin γ cos γ e ψ P ( cos γ ) is an odd function of γ . Since in theintegration domain S + ∩ ( , π / ) the function g ( γ ) , sin γ and cos γ are strictly positive, it follows that the integral above is strictlypositive if ¯ ψ > and strictly negative if ¯ ψ < , and we thusconclude that ¯ ψ = , that QQQ and ψψψ have the same eigenframe,and that σσσ : ψψψ = ψ P ( cos γ ) . iiiiii iiiiii inaccessiblespace E ne r g y den s i t y () O r de r ( S ) Protein coverage ( )
Fig. 2 Free-energy density landscape as a function of protein coverage,expressed as the area fraction a p φ with a p the area of a protein, and ofnematic order S . The white region is inaccessible due to crowding. Thereis a discontinuous isotropic-to-nematic transition for an area fraction ofabout 0.5. We consider ellipses with aspect ratio a / b = on a flat mem-brane. Dots represent minima (red) and maxima (white) of the energyprofile for fixed φ . The diagrams on top illustrate states i, ii and iii. Combining this last expression with Eqs. (23,25,26), we findthat S = (cid:82) S + P ( cos γ ) { − φ [ c − dSP ( cos γ )] } e ψ P ( cos γ ) d γ (cid:82) S + { − φ [ c − dSP ( cos γ )] } e ψ P ( cos γ ) d γ . (30)Importantly, the above relation provides an implicit relation forthe auxiliary variable ψ ( φ , S ) given the particle number density φ and the order parameter S . With these expressions, the configu-rational free-energy in Eq. (9) can be rewritten as F e [ φ , S ] = β (cid:90) Γ φ (cid:26) ln φ + S ψ − ln (cid:90) S + { − φ [ c − dSP ( cos γ )] } e ψ P ( cos γ ) d γ (cid:27) drrr , (31)in terms of the fields φ ( rrr ) and S ( rrr ) . We note that, given the lack f a preferred orientation, this effective free-energy depends onthe nematic tensor QQQ only through S , and whenever S (cid:54) = , thenematic direction is arbitrary. Figure 2 shows the landscape of the energy density, the integrandin Eq. (31), as a function of density and order parameter. Inthe figure, we express density or coverage as the area fraction a p φ , where a p is the area of a protein. The figure shows that,as area fraction becomes large, the energy grows rapidly irre-spective of S and blows up at finite density, defining a region ofinaccessible states where g ( γ ) , see Eq. (26), becomes negative.Given the density φ , we can minimize the energy profile with re-spect to S to determine the degree of order in equilibrium as afunction of φ , defining the equilibrium path shown by red dotsin the figure. For low area-fraction, proteins maximize their en-tropy by being randomly oriented and hence S = is the onlysolution branch. As density increases beyond a threshold, we ob-serve the emergence of another stable branch characterized byhigh protein order. There is a range of densities where both thedisordered and the ordered branches coexist and are separatedby unstable equilibrium points marked in the figure with whitedots. Density-based ordering for such elliptical molecules thusproceeds through a discontinuous phase transition. The proce-dure described here, which adapts that in to 2D systems withelliptical particles, predicts how the energy landscape dependson the particle aspect ratio. As shown in Fig. 3, increasing it de-creases the size of the region of accessible states, decreases thethreshold density of the phase transition, and increases the maxi-mum packing limit. Having described the phase transition of ellipses on flat surfaces,we now consider curved proteins adhered to a curved lipid mem-brane approximated as a surface. In doing so, we ignore the effectof curvature in the calculation of the excluded area but accountfor the bending energy of the proteins. Since binding of BAR pro-teins occurs due to electrostatic interaction with the lipids, theyadhere along their charged faces . We assume that adsorbedproteins sample the curvature of the underlying membrane alongtheir long axis (cid:96)(cid:96)(cid:96) , Fig. 4(b), i.e. the surface normal curvature alongthis direction, which can be computed as k (cid:96)(cid:96)(cid:96) = (cid:96)(cid:96)(cid:96) · kkk · (cid:96)(cid:96)(cid:96) where kkk isthe second fundamental form of the membrane surface character-izing its local curvature.Since both the nematic tensor QQQ , see Eq. (20), and the secondfundamental form of the surface are symmetric tensors, they pos-sess respective tangential orthonormal eigenframes, { λλλ , µµµ } for QQQ and one given by the principal curvature directions { vvv , vvv } for kkk , Fig. 4(a). The principal curvatures of the surface are the cor-responding eigenvalues kkk · vvv i = k i vvv i , i = , . In general, these twoeigenframes are different and are rotated by an angle θ , Fig. 4(a),which can be assumed to lie in the interval θ ∈ ( − π / , π / ] sinceeigenvectors can be flipped. We can express one frame in terms of the other as λλλ = cos θ vvv − sin θ vvv and µµµ = sin θ vvv + cos θ vvv . Fromthe definitions of angles γ and θ , the angle between the princi-pal curvature direction vvv and a microscopic particle direction (cid:96)(cid:96)(cid:96) is γ − θ , and hence k (cid:96)(cid:96)(cid:96) = k cos ( γ − θ ) + k sin ( γ − θ ) . (32)Denoting the bending rigidity (with units of energy) of a pro-tein by κ p , its preferred curvature along the long axis by ¯ C and itsarea by a p , we can write its elastic bending energy as U b ( kkk , γ ) = κ p a p ( k (cid:96)(cid:96)(cid:96) − ¯ C ) . (33)We note that this energy depends on position (through the princi-pal curvatures k and k ) and on the relative orientation betweenthe particle direction and the principal curvature direction. Com-bining the bending energy with the interaction energy discussedin Section 1 for N proteins, Eq. (1), we obtain the free energy F = − β ln 1 N ! (cid:90) Ω N e − β ( Σ ≤ i < j ≤ N U i , j + Σ ≤ i ≤ N U bi ) dqqq ... dqqq N . (34)Following a similar mean field approximation and a passage tothe continuum limit as in Section 1 and noting that the bendingenergy of a protein molecule does not depend on the state of otherproteins, we arrive at the following expression for the free energyof the system (see Appendix A) F [ φ , f ] = β (cid:90) Γ φ ln φ drrr + β (cid:90) Γ φ (cid:26) (cid:90) S f [ ln f − ln g ] d γ (cid:27) drrr + (cid:90) Γ φ (cid:90) S fU b d γ drrr . (35)As before, we find the optimal particle angle distribution f byminimizing the free-energy, resulting in an effective energy thatwill depend on S as before, but now also on θ and hence on thefull nematic tensor QQQ . Analogously to before, we introduce theLagangian L = (cid:90) S f (cid:104) ln f − ln g + U b (cid:105) d γ + µ (cid:18) (cid:90) S f d γ − (cid:19) . (36)Minimization with respect to f requires that = δ f L = (cid:90) S (cid:104) ln f − ln g − ψψψ : σσσ + U b + µ (cid:105) δ f d γ , (37)where the auxiliary symmetric and traceless tensor ψψψ was de-fined in Eq. (21), and hence, with the same argument leading toEq. (23), we find that f ( γ ) = g ( γ ) e σσσ ( γ ) : ψψψ e − U b (cid:82) S + g ( γ (cid:48) ) e σσσ ( γ (cid:48) ) : ψψψ e − U b d γ (cid:48) (38)if γ ∈ S + and 0 otherwise.As before, we express ψψψ in the eigenframe of QQQ as ψψψ = ψ ( λλλ ⊗ λλλ − µµµ ⊗ µµµ ) + ¯ ψ ( λλλ ⊗ µµµ + µµµ ⊗ λλλ ) . (39)However, we cannot make the same argument as before to con-clude that ¯ ψ = because, unless θ = or θ = π / , U b is not an O r de r ( S ) a/b=3 Protein coverage ( ) O r de r ( S ) a/b=5.3 Protein coverage ( ) O r de r ( S ) a/b=8.3 Protein coverage ( ) E ne r g y den s i t y () E ne r g y den s i t y () -1.8 0 3.6 1.8 E ne r g y den s i t y () Fig. 3 Energy density landscape for ellipses with varying aspect ratio on a flat membrane. λμℓ v v γθ ℓ v γ − θ a b direction of maximum curvature d i r ec t i o n o f m i n i m u m c u r v a t u r e N e m a t i c d i r ec t i o n I n d i v i d u a l p r o t e i n d i r ec t i o n Fig. 4 (a) Illustration of the eigenframe { λλλ , µµµ } of the nematic tensor QQQ ,where λλλ is the nematic direction, and of the eigenframe { vvv , vvv } of thesecond fundamental form of the surface kkk , where these vectors determinedirections of maximum and minimum curvature of the surface. We alsoillustrate a microscopic direction (cid:96)(cid:96)(cid:96) , the angle γ between the nematicdirection λλλ and (cid:96)(cid:96)(cid:96) , and the angle θ between the two eigenframes. (b)An adsorbed protein along vector (cid:96)(cid:96)(cid:96) samples the normal curvature of thesurface in this direction. even function of γ , see Eqs. (32,33). Recalling Eq. (28), we canwrite the angular probability distribution f ( γ ) = [ − φ ( c − dSP )] e ψ P e ¯ ψ sin γ cos γ e − U b (cid:82) S + [ − φ ( c − dSP )] e ψ P e ¯ ψ sin γ (cid:48) cos γ (cid:48) e − U b d γ (cid:48) , (40)where P stands for P ( cos γ ) . In Section 1, we used Eq. (25) todetermine ψ . Now, however, we need two equations since we alsoneed to determine ¯ ψ . For this, we recall that = λλλ · QQQ · µµµ leadingto Eq. (29). Thus, we have two conditions = (cid:90) S + P [ − φ ( c − dSP )] e ψ P e ¯ ψ sin γ cos γ e − U b d γ − S (cid:90) S + [ − φ ( c − dSP )] e ψ P e ¯ ψ sin γ cos γ e − U b d γ , (41) = (cid:90) S + sin γ cos γ [ − φ ( c − dSP )] e ψ P e ¯ ψ sin γ cos γ e − U b d γ , (42)the second of which was trivially satisfied previously as the inte-grand is an odd function of γ for ¯ ψ = and U b = . Now, how-ever, these two relations provide a system of nonlinear equationsto solve for ψ and ¯ ψ .Examining the above equations, it is clear that f ( γ ) , ψ and ¯ ψ depend on φ and S , but also on θ , k and k through U b . PluggingEq. (40) into Eq. (35), we obtain a computable expression of the free energy accounting for the curvature energy of the proteins F [ φ , S , θ , k , k ] = β (cid:90) Γ φ (cid:26) ln φ + S ψ (43) − ln (cid:90) S + { − φ [ c − dSP ( cos γ )] } e ψ P ( cos γ ) e ¯ ψ sin γ cos γ e − U b d γ (cid:27) drrr . It is interesting to note that, as mentioned earlier, in the spe-cial case that the nematic direction λλλ is aligned with one of theprincipal directions, θ = or θ = π / , then ¯ ψ = , U b becomes aneven function of γ , and hence f ( γ ) is symmetric with respect tothe nematic direction. For a general nematic orientation relativeto the principal curvatures, however, f is not symmetric about thenematic direction. θ The free energy in Eq. (43) can then be minimized with respectto θ to yield an effective energy ˆ F [ φ , S , k , k ] = min θ ∈ ( − π / , π / ) F [ φ , S , θ , k , k ] . (44)This process identifies the energetically optimal nematic orienta-tion relative to the curvature of the surface. To do that, we make L stationary with respect to θ to find = (cid:90) S + g ( γ ) e σσσ : ψψψ e − U b ∂ U b ∂ θ d γ . (45)Expanding the last term in the integral, we find = (cid:90) S + g ( γ ) e σσσ : ψψψ e − U b (cid:104) k cos ( γ − θ ) + k sin ( γ − θ ) − ¯ C (cid:105) (46) ( k − k ) cos ( γ − θ ) sin ( γ − θ ) d γ . This equation, together with Eqs. (41,42), provides a system ofthree nonlinear equations for three unknowns, ψ , ¯ ψ and θ . Fora sphere, k = k , this equation is an identity showing that anydirection is equally possible. Suppose that ¯ ψ = and θ = . Inthis case, U b is an even function of γ , Eqs. (42,46) are identicallysatisfied, and Eq. (41) provides an equation for ψ . Thus, there isalways a solution with ¯ ψ = and θ = but in general there maybe others and their relative stability must be examined to selectthe ground state. b c O r de r ( S ) O r de r ( S ) O r de r ( S ) Protein coverage ( )
Protein coverage ( )
Protein coverage ( ) ij Decreasing cylinder radiusIncreasing protein coverage E ne r g y den s i t y () e g inaccessiblespace O r de r ( S ) Cylinder f Protein coverage ( ) low coverage ( )high coverage ( )
Cylinder dh A ng l e () O r de r ( S ) /4/8/10/6 Protein coverage ( )
Cylinder inaccessiblespace O r de r ( S ) Protein coverage ( ) E ne r g y den s i t y () CylinderCylinder or high coverage ( ) low coverage ( ) O r de r ( S ) Cylinder E ne r g y den s i t y () -0.4 0 -0.2 O r de r ( S ) -0.6 Cylinder E ne r g y den s i t y () Cylinder0.70.81 O r de r ( S ) E ne r g y den s i t y () Cylinder0.70.81 O r de r ( S ) E ne r g y den s i t y () Cylinder0.70.81 O r de r ( S ) E ne r g y den s i t y () O r de r ( S ) E ne r g y den s i t y () Cylinder ororor
Fig. 5 (a)-(d) Energy density contours as a function of density and order on spherical and cylindrical surfaces of different radii, R = , , and 10 nm, with ¯ C = nm. Red dots denote stable states, which minimize the free-energy for a given protein coverage. (e,g) Angular probabilitydistribution f ( γ ) plotted against γ − θ , i.e. against the angle of a particle relative to the direction of maximum curvature of the cylinder vvv , Fig. 4.The inset pictorially illustrates the state of the system, where the double-ended arrow indicates the nematic direction. (f,h) Protein net orientationexpressed as the angle θ between the nematic direction and vvv as a function of density and order for the cylindrical surfaces in (b) and (d). In (f), θ = everywhere. (i) Energy landscapes in the ( θ − S ) plane for high protein coverage ( φ a p = . ) and cylinders of decreasing radius. (j) Energylandscape in the ( θ − S ) plane for a thin cylinder ( ¯ CR = / ) and varying coverage. .3 Free-energy landscapes on curved surfaces Energy density landscapes exhibiting the isotropic-to-nematictransition for proteins on spherical and cylindrical surfaces areshown in Fig. 5 (a-d) using the expression given by Eq. (43) andminimizing the energy with respecto to θ (the angle between thenet orientation of proteins and the maximum curvature direction)as described in the previous section. We depict stable equilibriumstates minimizing the free energy for a given protein coveragewith respect to S and θ with red dots. For a sphere, Fig. 5(a), theisotropic curvature does not bias alignment along any specific di-rection and hence the phase transition is solely driven by entropicinteractions. In fact, examining Eq. (35), it is clear that since U b does not depend on orientation, the last bending term in the free-energy density is simply linear in φ and hence does not alter thepath of minimizers marked by red dots. As a result, the systemshows the same discontinuous transition upon crowding as in theplanar case.On anisotropically curved surfaces such as cylinders with ra-dius R , proteins are biased to orient along specific directions tofavorably adapt their curvature to that of the underlying surface,Fig. 5(b-d). This creates a competition between a curvature-dependent bias and the entropic part of the free-energy, whichleads to partial order (finite S ) even in the dilute limit. Further-more, this curvature bias changes the character of the isotropic-to-nematic transition, which now becomes continuous. Ourmodel not only provides the free-energy landscape as a functionof φ and S but also the nematic orientation relative to the direc-tion of maximum curvature of the cylinder vvv (Fig. 4) given by θ and represented in Fig. 5(f,h), and the distribution of proteinorientations f ( γ ) , which we represent relative to vvv , i.e. against γ − θ , Fig. 5(e,g). Figure 5(f) illustrates the observation that for ¯ CR ≥ the optimal nematic orientation is always that of max-imum curvature of the cylinder, θ = . Figure 5(e) shows theangular distribution for two values of protein coverage markedin (b). Both distributions are unimodal and symmetric about thedirection given by vvv but as coverage increases, order increases aswell and the distribution becomes more localized and compactlysupported.For cylinders with higher curvature than that of proteins, ¯ CR < , the situation is more complex since now proteins aligned with vvv will be bent beyond their spontaneous curvature whereas pro-teins forming an angle α with vvv given by cos α = ¯ CR , see Eq.32, will store no elastic energy. The isotropic-to-nematic transi-tion becomes discontinuous again, Fig. 5(d), and the nematic di-rection is along vvv for low S but θ becomes different from zero forlarger order, Fig. 5(h). Interestingly, at low densities when θ = ,the angular distribution is bimodal, broad, and symmetric about vvv , indicating a state where proteins are disordered but preferen-tially adopt orientations forming a finite angle with the directionof maximum curvature, which is too curved compared to the pro-tein curvature. This detailed information is lost if the nematicstate is described in terms of a moment of f such as the nematictensor QQQ or equivalently S and θ alone, rather than in terms ofthe full distribution. For high density, we find that the systemadopts a non-symmetric, very narrow and compactly supported distribution (or a symmetry-related distribution), indicative of anematic state with nematic direction forming a finite angle with vvv , consistent with the fact that F-BAR proteins at high coveragesadopt increasingly helical arrangements on increasingly thinnertubes .The symmetry-breaking transition for thin tubes at high cover-ages can be nicely examined through free-energy maps at givencoverage and radius as a function of θ (the angle between ne-matic direction and vvv ) and S . On the one hand, we can ob-serve how at fixed high-coverage and as the cylinder radius de-creases, a single minimum given by θ = splits into two minimagiven by θ = ± θ when ¯ CR < , Fig. 5(i). Similarly, given a high-curvature cylinder, as coverage increases the optimal nematic di-rection switches from being aligned with vvv to adopting eitherone of two symmetry-related orientations, Fig. 5(j).We finally note that the model proposed here can be used toquantify the free-energy of curved elongated particles on othersurfaces, such as those of negative curvature. Figure 6 shows theisotropic-to-nematic transition on different regions of a catenoid,characterized by having zero mean curvature and negative Gaus-sian curvature. The results are similar to those on cylinders, albeitwith a larger bias towards nematic states, compare Fig. 5(b,c) andFig. 6(b,c). Coarse-grained simulations suggest the possibility of coexistencebetween isotropic and nematic phases . To examine such coex-istence using our theory, we first consider the situation of a flatmembrane. Figure 7(a) shows the landscape of minimum free en-ergy as a function of protein coverage, which as discussed earlierand shown here with the color representing order, has an isotropicand a nematic branch. The slope of the energy density as a func-tion of protein coverage is precisely the chemical potential of pro-teins in a given state. For coexistence of isotropic and nematicphases in equilibrium, the chemical potentials of the two phasesshould be equal. If the number of proteins populating these twophases is fixed with average density ¯ φ , then coexistence addi-tionally requires the double tangency condition, see Fig. 7(a,b)and the line with slope µ coex following the Maxwell construction.Thus, under these conditions, coexistence is possible only when φ I < ¯ φ < φ N . When the membrane can exchange proteins witha bulk solution with chemical potential µ b , the double tangencycondition is no longer required and coexistence requires simplythat µ b = µ I = µ N , see Fig. 7(c). This slightly relaxes the possibil-ity of coexistence but the figure shows that it can only occur in arather narrow range of densities.Since as discussed earlier the energy landscape on spheres isthat of a planar surface with a tilt proportional to φ , the condi-tions for coexistence are similar. On tubes, however, there ex-ists essentially no isotropic phase, and as illustrated in Fig. 8the energy is convex in φ , leaving no room for coexistence. Yet,as suggested by experiments where thin tubes are pulled off gi-ant vesicles and exposed to a solution with BAR proteins , itis reasonable to expect isotropic-nematic coexistence in cylinder-sphere systems in equilibrium. Indeed, for moderate coverages,spheres adopt an isotropic state whereas thin-enough tubes are in ba O r de r ( S ) Protein coverage ( ) iii O r de r ( S ) Protein coverage ( ) iii O r de r ( S ) Protein coverage ( ) iiiiii iiiiii E ne r g y den s i t y () Fig. 6 (a)-(c) Energy density contours as a function of density and order on negatively curved surfaces of zero mean curvature, where the principalcurvatures are k , = ± / R , for different values of R . By way of illustration, we identify these curvatures with different positions on a catenoid surface. µ nem =2.32 µ iso =0.13=1.55 µ coex O r de r ( S ) µ nem =1.55 µ iso = Chemical equilibriumand double tangent: µ nem =1.45 µ iso = Chemical equilibrium: M a x w e ll li n e Zoom in Zoom in E ne r g y den s i t y () E ne r g y den s i t y () E ne r g y den s i t y () cba Fig. 7 (a) Lowest energy density as a function of protein coverage for ellipses on a flat membrane, i.e. the energy along the minimum-energy pathmarked with red dots in Fig. 2. The color code represents order, highlighting the parts of the energy landscape corresponding to isotropic and tonematic phases. The slopes of the tangent lines represent the rate of change of energy density with respect to protein coverage, i.e. the chemicalpotential. The red line is doubly tangent (a Maxwell line) to the isotropic and nematic branches and represents a situation of coexistence in whichprotein number is fixed, see zoom in (b). If proteins can be exchanged with a bulk solution where they have a given chemical potential, then coexistenceof isotropic and nematic phases does not require the double tangency constraint (c). µ cylinder =0.02 µ sphere = µ cylinder =0.39 µ sphere = µ cylinder =1.49 µ sphere = O r de r ( S ) Chemical equilibrium: E ne r g y den s i t y () Protein coverage ( )
TubeSphere
Fig. 8 Energy landscape for a sphere with radius R s = / ¯ C and atube with radius R t = / ¯ C . Three different isotropic-nematic states ofcoexistence in an ensemble in which proteins can be exchanged with abulk solution are highlighted by pairs of tangents with the same slope. a significantly nematic state. It is thus possible to find infinitelymany equilibrium states of coexistence over a broad range of bulkchemical potentials, Fig. 8, in all of which area coverage and or-der are higher on the tube. Curved proteins on membranes are responsible for many biolog-ical functions, which rely on the mechanisms of curvature sens-ing and generation. When these proteins are elongated, such asthose containing BAR domains, their physics crucially depend ontheir orientation. Here, extending the work of , we have devel-oped a mean-field density functional theory connecting physicsfrom a micro- to a mesoscale to evaluate the free-energy of elon-gated and curved proteins on a curved membrane applicable tolarge protein coverage. The free-energy landscape is expressed interms of the net orientation of proteins relative to the principalcurvature directions ( θ ), of the classical order parameter S , andof the number density φ , and it depends on the aspect ratio andintrinsic curvature of these proteins, on their bending rigidity, andon the second fundamental form of the membrane. In addition tothe free-energy landscape, the theory provides the orientationalprobability distribution of proteins.We have shown that, while on planar surfaces and spheres thesystem exhibits a density-dependent discontinuous isotropic-to-nematic transition, this transition is continuous on surfaces withanisotropic curvature such as cylinders or catenoids. We haveshown that anisotropic curvature biases the system towards aslightly nematic state even at low protein concentrations. Whenthe curvature of cylindrical membranes is higher than that of theproteins, then the orientational distribution becomes bimodal atlow densities and asymmetric with respect to the principal direc-tion of curvatures at high densities. Our theory has also allowedus to examine the coexistence of isotropic and nematic phasesunder different conditions.Our theory provides physical rules to understand the statecurved and elongated proteins on surfaces of given curvature.However, it does not say anything about how the proteins, with a given density or orientational distributions, affect the shape ofthe underlying membrane. This situation is examined experimen-tally and computationally elsewhere , where the present modelis coupled with one of membrane dynamics. Conflicts of interest
There are no conflicts to declare.
Acknowledgements
This work was supported by the Spanish Ministry of Economyand Competitiveness/FEDER (BES-2016-078220 to C.T.), the Eu-ropean Commission (H2020-FETPROACT-01-2016-669 731957),the European Research Council (CoG-681434 to M.A.), the Gen-eralitat de Catalunya (2017-SGR-1602 to P.R-C., 2017-SGR-1278to M.A.), the prize “ICREA Academia” for excellence in research toP.R-C. and to M.A., and Obra Social “La Caixa”. IBEC and CIMNEare recipients of a Severo Ochoa Award of Excellence from theMINECO.
A Mean field free-energy of bendable proteins on acurved membrane
The free-energy of N proteins interacting with the pairwise poten-tial as mentioned in Eq. (1) and also with the underlying curvedsurface as given by Eq. (33) is F ∗ = − β ln 1 N ! (cid:90) Ω N e − β ( Σ ≤ i < j ≤ N U i , j + Σ ≤ i ≤ N U bi ) dqqq ... dqqq N . (47)As shown in , the above free energy can be approximated bythe mean field energy F ∗ ≈ F = − β ln 1 N ! (cid:28) (cid:90) Ω e − β ∑ ≤ j ≤ N U , j e − β U b dqqq (cid:29) N (48) = − β ln 1 N ! (cid:18) (cid:90) Ω (cid:68) e − β ∑ ≤ j ≤ N U , j e − β U b (cid:69) dqqq (cid:19) N , (49)where in the second line we have changed the order of integrationand where the ensemble average (cid:104) f (cid:105) = (cid:90) Ω N − f p ( qqq ,..., qqq N ) dqqq ... dqqq N , (50)is with respect to the probability distribution of N − particlesgiven by p ( qqq ,..., qqq N ) = e − β ( Σ ≤ i < j ≤ N U i , j + Σ ≤ i ≤ N U bi ) (cid:82) Ω N − e − β ( Σ ≤ i < j ≤ N U i , j + Σ ≤ i ≤ N U bi ) dqqq ... dqqq N . (51)The mean free approximation is an upper bound to the exact freeenergy.Unlike the hard-core repulsion energy, which depends on theconfiguration of other proteins, the bending energy of a proteinmolecule does not depend on the configuration of other proteins.
10 | 1–12 e can thus write F = − β ln 1 N ! (cid:18) (cid:90) Ω (cid:68) e − β ∑ ≤ j ≤ N U , j (cid:69) e − β U b dqqq (cid:19) N = − β ln 1 N ! (cid:18) (cid:90) Ω [ − W ( qqq )] e − β U b dqqq (cid:19) N (52)or equivalently F = − β ln 1 N ! (cid:18) (cid:90) Ω [ − W ( qqq )] e − β U b ( qqq ) dqqq (cid:19) N (53)with W ( qqq ) as defined before in Eq. (4). Following , we discretizethe phase space in subdomains Ω = ∪ i Ω i , each with N i particles,and obtain free energy within this domain F i after using Stirling’sapproximation as F i = − β ln (cid:18) N i (cid:90) Ω i [ − W ( qqq i )] e − β U b ( qqq i ) dqqq i (cid:19) N i . (54)Thus, the total free energy is given by F = ∑ i F i = − β ln ∏ i (cid:18) N i (cid:90) Ω i [ − W ( qqq i )] e − β U b ( qqq i ) dqqq i (cid:19) N i . (55)Assuming N i ≈ ρ ( qqq i ) ∆ qqq i and passing onto the continuum limit, weobtain F = β (cid:90) Ω ρ ( qqq ) ln ρ ( qqq ) dqqq − β (cid:90) Ω ρ ( qqq ) ln [ − W ( qqq )] dqqq + (cid:90) Ω ρ ( qqq ) U b ( qqq ) dqqq . (56)Further, separating the particle density ρ into spatial and orienta-tional components as mentioned in Eq. (5), we obtain the expres-sion for free energy in Eq. (35) Notes and references
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