Suppleental Material for Correlation between Length and Tilt of Lipid Tails Ditry I. Kopelevich and John F. Nagle I. RESULTS FOR ALTERNATIVE DIRECTOR DEFINITIONS A. Alternative Director Definitions The goal of this section is to investigate the tilt-length correlations for the following alternative definitions of the tail director: A. Molecular directors, which characterize tilt of an entire olecule, rather than an individual tail. This approach to the director definition was taken in the earlier work [ 3]. Here, we investigate two different definitions of olecular directors (see Fig. Sa):. Mean tail director n, defined as the average of the end-to-end directors C N C of individual tails of a olecule. Specifically, n is defined as the vector pointing fro C () N to C(), C () i is the idpoint between atos C i of different tails of a lipid olecule.. Overall olecular director n P, which accounts for contributions of both the head- and tail-groups to olecular tilt. Following[3], wedefine n P asthe vectorpointing froc () N to the phosphorusatoin the lipid head-group. Siilarly to the individual tail directors, the length L of a olecular director is defined as the distance between its end-points. B. Gyration director n g defined as the principal direction of gyration of a lipid tail corresponding to its largest radius of gyration. In the calculations of the tensor of gyration S of a hydrocarbon chain, we neglect hydrogen atos belonging to the chain, i.e. S = N N (r k r C ) (r k r C ), () k= the suation is perfored over the chain carbon atos, r C is the center of ass of these atos, and r k is the position of the k-th ato. Furtherore, we require that the gyration director passes through r C. The length L g n g (a) (b) Figure S. (a) End-to-end directors of individual lipid tails (solid arrows) and the olecular directors n (blue dashed arrow) and n P (green dashed arrow). Spheres of different colors represent atos of different eleents: yellow = C, white = H, red = O, green = P, blue = N. (b) Gyration director of a hydrocarbon chain. For clarity, only carbon carbon atos of the chain are shown (yellow spheres); the black spheres represent projections of the positions of the carbon atos onto the gyration director. The blue sphere represents the chain center of ass.
of n g is defined as the end-to-end length of the chain fored by the projections of the carbon atos onto n g (see Fig. Sb), i.e. L g = ax k (r k r C ) ˆn g in k (r k r C ) ˆn g, () ˆn g = n g / n g is the noralized gyration director. Although radii of gyration are often used to characterize acroolecular shapes [4 6], the property of the tensor of gyration ost relevant to this work appears to be little known. Specifically, line l in, defined as having the sallest ean-square displaceent (MSD) fro a group of atos, contains the gyration director n g of this group. To prove this, let us consider a group of N atos with coordinates r k and obtain line l in with the sallest MSD fro these atos, = N N k= k in. (3) Here, k is the distance fro the k-th ato to the line. Define this line generally as being directed along unit vector û and passing through soe point with coordinates given by vector r. Then k = r k r [(r k r ) û]. (4) Let us now consider a syste of coordinates with the origin at the center of ass, k r k =. Then Eqs. (3), (4) yield: = N N r k + [ r (r û) ] û T Sû. (5) k= The first ter in (5) is independent of r and û. The ters contained in the square brackets represent the difference between the squared lengths of the vector r and the projection of this vector onto û. This difference is iniized if r is parallel to û, i.e. line l in passes through the origin (center of ass). Finally, to iniize, the unit vector û should axiize û T Sû, i.e. û is an eigenvector corresponding to the largest radius of gyration. Hence, line l in contains the gyration director. B. Analysis Results Dependence of the ean director length L() on tilt for all considered director definitions is shown in Fig. S. In all cases, increasing tilt leads to a slight decrease in L(). Moreover, L() for the end-to-end and gyration tail directors are nearly identical and the noralized lengths L()/ L() of both considered olecular directors, n and n P, exhibit a very siilar dependence on tilt. The latter observation suggests that the tilt-induced lipid shortening is doinated by shortening of tails, with head-groups aking a negligible contribution. The conditional distributions P(L ) of the directors n,n P, and n g are shown in Fig. S3. These distributions exhibit behavior siilar to that of the end-to-end tail directors C N C considered in the paper: as tilt increases, the distributions becoe slightly wider and their peak shifts towards saller L. II. -ROD MODEL A. Validation of the -rod Model for the DLPC bilayer In this section, we present evidence that the echanis of the tilt-induced tail shortening discussed in Section V of the paper is applicable to other saturated lipids. To this end, we repeated the analysis of Section V for a DLPC lipid bilayer. Results of this analysis are suarized in Figures S4 and S5. Qualitative siilarity between the DLPC and DPPC results indicate that the -rod odel holds for a wide class of saturated PC lipids. Moreover, since effect of the head-group in the lipid shortening is negligible (see Fig. S), it is likely that the conclusions of this paper hold for saturated lipids with different head-groups. In the reainder of this section, we report additional analysis in support of the -rod odel. Results of this analysis are the sae for the DPPC and DLPC bilayers. Hence, unless stated otherwise, only data for the DPPC bilayer are shown.
3.3 L()/ L().. Molecular director n Molecular director n P End-to-end tail director Gyration director n g IL odel IS odel PB odel.9.5.5 Figure S. Dependence of the average tail director length L on its tilt for different director definitions. In this plot, L() is noralized to its value at zero tilt: L() =.4 n for the olecular directors n, L() =.97 n for the olecular directors n P, and L() =.5 n for both the end-to-end and gyration directors of individual tails. The standard errors of the shown L() do not exceed. n. 3 4 P(L ) [.,.5] [.5,.] [.,.5] [.5,.] P(L ) [.,.5] [.5,.] [.,.5] [.5,.] P(L ) 3 [.,.5] [.5,.] [.,.5] [.5,.].6.8..4.6.8 L (a).5.5 L (b).6.8..4.6.8 L (c) Figure S3. Conditional probabilities P(L ) of the length L of (a) olecular directors n, (b) olecular directors n P, and (c) gyration directors n g. B. Gyration Director In this section we deonstrate that conclusions of Section V of the paper hold for the gyration tail director defined in Section I of SM. The hinge point of the -rod odel is defined here as the position of the chain carbon ato C h located farthest fro the gyration director. Gyration directors n and n of the chain segents C -C h and C h -C N are defined siilarly to the gyration director of the entire chain. As in the case of the end-to-end directors, the average angle between the gyration directors n and n of the chain segents decreases as the tilt of the overall tail director n increases, see Fig. S6. In addition, Fig. S7 indicates that dependence of the ean director lengths on the tilts and of the gyration directors n and n is qualitatively the sae as that for the end-to-end directors (see Fig. 6 of the paper). C. Additional Validation The two-rod odel is useful only if it provides a ore accurate representation of the tail shape than the single-rod odel assued in the existing theories. To copare accuracy of the single- and two-rod odels, we obtain the ean deviation of the chain carbon atos fro the overall chain director n and the directors n and n of the chain segents C -C h and C h -C N (C h is the hinge ato). Specifically, we copute the root ean square displaceent (RMSD) of carbon atos fro the directors, i i=i (Ò) = i (Ò) i i +. (6)
4.3 -.4 -.45 Pg.3.8 -.5 -.55 cos φ -.6.6.5.5 Figure S4. Sae as Fig. 4 of the paper but for the DLPC bulayer.. Noralized length.98.96.94 L() L ()+ L () L () L ( ) b () b ( ).9.5.5 Figure S5. Sae as Fig. 6 of the paper but for the DLPC bilayer. Here, the noralization values are L( = ) =. n, L ( = ) =.63 n, L ( = ) =.6 n, L ( = ) =.67 n, and b ( = ) = b ( = ) =. n..5.55 cos φ.6.65.5.5 Figure S6. Average relative orientation cosφ of the gyration directors n and n of the tail segents versus tilt of the overall tail director n.
5. Noralized length.98.96 L() L ()+ L () L ().94 L ( ) b () b ( ).9.5.5 Figure S7. Sae as Fig. 6 of the paper but for the gyration directors. Here, the noralization values are L( = ) =.5 n, L ( = ) =.85 n, L ( = ) =.8 n, L ( = ) =.9 n, and b ( = ) = b ( = ) =. n. Table S. Root ean square displaceents (Ò,) of atos fro the overall tail director n and the directors n and n of the tail segents C -C h and C h -C N at zero tail tilt. Data for both DPPC and DLPC bilayers are shown. Lipid Director type (n,) (n,) (n,) DPPC End-to-end.9 Å. Å. Å Gyration. Å.6 Å.6 Å DLPC End-to-end.5 Å.8 Å.8 Å Gyration.9 Å.5 Å.5 Å Here, Ò represents one of the directors (Ò = n, n, or n ), i (Ò) is the distance between ato C i and director Ò. For the end-to-end directors, the suation is perfored over all carbon atos in the chain (or the chain segent) excluding the chain (segent) end-points, i.e. i =, i = N for Ò = n, i =, i = h for Ò = n, and i = h+, i = N for Ò = n. For the gyration directors, the end-points are included in the suation, since these points do not necessarily lie on the directors. The RMSDs for individual chains are enseble averaged to obtain ean RMSDs, (Ò,), corresponding to different chain tilts. The RMSDs of chains with zero tilt are listed in Table S. Not surprisingly, the RMSDs are saller for the gyration directors. Nevertheless, for both considered types of the directors, the -rod odel yields a better approxiation to the chain shape, as evidenced by the substantial difference between the overall tail directors, (n,), and the directors of the individual rods, (n,) and (n,) of the -rod odel. If the tail shape were statistically a single cylinder, the RMSDs of the -rod odel would be only arginally saller than the single-rod RMSD. Effect of the chain tilt on the RMSDs is shown in Fig. S8. While all RMSDs growwith increasingtilt, the RMSDs of the -rod odel do not grow as fast as those of the single-rod odel, which indicates that the single-rod approxiation becoes even less accurate as the tail tilt increases. (7) D. Distribution of Gauche Rotaers in Lipid Tails The purpose of this section is to connect the -rod odel for lipid tails with distribution of gauche rotaers along the tails. To this end, we copute (i) the fraction P g,h of gauche rotaers in dihedral angles with the hinge ato C h in the central bond and (ii) the fractions P g, and P g, of gauche rotaers in the interior of segents C -C h and C h -C N. I.e., dihedral angle C i -C i+ -C i+ -C i+3 contributes to P g,h if i = h or i = h, to P g, if i < h, and to P g, if i > h.
6.5 (n,)/ (n,)..5..5 (n, ) (n,) (n,).95.5.5 Figure S8. Effect of the chain tilt on the RMSDs of carbon atos fro the overall tail director n and the rod directors n and n. RMSDs corresponding to end-to-end and gyration directors are shown by solid and dashed lines, respectively. The plotted RMSDs are noralized to their values at zero tilt..6 Pg.5.4 P g P g,h P g, P g,.3...5.5 Figure S9. Fraction of gauche rotaers versus tilt in the entire tail (P g), at the hinge ato (P g,h ), and in the segents C -C h and C h -C N (P g, and P g,, respectively). The dependence of P g,h, P g,, and P g, on the chain tilt is shown in Fig. S9. For coparison, the fraction P g () of the gauche rotaers in the entire chain is also shown. The data shown in Fig. S9 were obtained using the end-to-end tail director C N C to identify the hinge ato C h. The gyration director yields nearly identical results (data not shown). It is evident fro Fig. S9 that a dihedral angle containing the hinge ato C h is ore likely to be in gauche than trans conforation even at zero chain tilt. On the other hand, the fraction of the gauche conforations inside the segents C -C h and C h -C N is saller than the average fraction for the entire chain. This is in agreeent with the relative insensitivity of the segent length per ethylene group, b and b, to the chain tilt (see Fig. 6 in the paper and Fig. S7 in SM). III. TILT-LENGTH CORRELATION PREDICTED BY THE CONTINUUM MODEL FOR LIPID BILAYERS In section IV of the paper it is shown that the continuu odel for a single onolayer predicts zero correlation between the chain length and tilt. We now turn fro onolayers to bilayers and investigate the effect of onolayer coupling on the tilt-length correlation.
The total bilayer energy is the su of energies F W of individual onolayers [see Eq. () of the paper] and the energy F coupl of coupling between the onolayers. Assue haronic coupling, F coupl = k (z() z () ), (8) k is the coupling strength, the superscript (α) indicates the onolayer nuber (α =,; α = corresponds to the upper onolayer), and z (α) is the surface passing through end-points of lipid tails of the α-th onolayer. Up to the required accuracy, z (α) = z(α) +( ) α L (α), (9) z (α) (x,y) is the dividing surface of the α-th onolayer, i.e. surface separating the lipid head- and tail-groups. It is convenient to introduce syetric and antisyetric odes, 7 Then z + = z() +z () l + = l() +l (), z = z() z () L, (), l = l() l (). () F coupl = k (z L l + ). () The z-coponents of the tilt and director vectors do not contribute to the quadratic approxiation to the onolayer free energy [Eq. () of the paper] and it is thus sufficient to consider two-diensional vectors = ( x, y ) and n = (n x,n y ). Up to the required accuracy, these vectors are related by the following equation: It then follows that n (α) = (α) +( ) α+ z (α), (3) n + = + + z and n = + z +, (4) + = () + () n + = Ò() + Ò (), n = Ò() Ò () are the syetric and antisyetric coponents of the vectors and n. The total bilayer energy can then be written as, = () (), (5) F b = F W (n (), (),l () )+F W (n (), (),l () )+F coupl = F W (n,,l )+G(z, +,l + ), (8) G(z, +,l + ) = [ F W ( z + +, +,l + )+k (z L l + ) ]. (9) The variables (n,,l ) are decoupled fro (z, +,l + ). Furtherore, up to the factor of two, the free energy of (n,,l ) is the sae as that of a single onolayer. Hence, l is decoupled fro the tilt odes. However, l + ay be coupled with the tilt odes + because the onolayer-onolayerinteractions introduce coupling between l + and z and z is coupled with + through the ( n + ) ter in F W. Eq.(8) is a slight generalization of the expression for the free energy developed by Watson et al. [] it was assued that z () = z () = z, i.e. k. In this case, l + = z /L and l = ǫ/l, ǫ z z +. In the reainder of this section we assue that k =, i.e. G(z, + ) = F W ( z + +, +,z /L ). It is shown in Section IV that finite k yields siilar results. (6) (7)
Fourier transfors of the bilayer degrees of freedo are defined by g q = g(r)e iq r dr, g(r) = g q e iq r, () Ap Ap r = (x,y), q = (q x,q y ) is the wavevector, A p is the area of the ebrane projection onto the x y plane, and the relevant g(r) are z (r), (r), and l + (r). Here, (r) is the longitudinal (curl-free) coponent of the tilt vector field +. The Fourier transfor of (r) corresponds to the projection of + q on q. Following [] we oit the contribution of the transverse (divergence-free) coponent of + to the free energy since it is decoupled fro other degrees of freedo of the bilayer. Perforing Fourier transfor of G(z, + ) and integrating over the entire bilayer yields [] q 8 G = q f qb q f q, () f q = (z q, q), () the asterisk denotes adjoint atrix, and [ ] (κq B q = 4 q Ω/L +k A /L ) iq(κq Ω/L ) iq(κq Ω/L ) κq +κ θ (3) is the force atrix. The covariance between the longitudinal tilt coponent q and the chain length is (l + ) k B T q q = δ q,q (B q ) = δ q,q iqk B Tc(q ), (4) L c(q ) = The covariance between the chain tilt and length in real space is κl q / Ω/4 L κκ θq 4 +(κk A ΩL κ θ Ω /4)q +κ θ k A. (5) C + l (r) l+ (r ) + (r +r) = k BT iqc(q )e iq r = C + lφ (r), (6) A p C + lφ (r)isthecovariancebetweenthetaillengthl+ andthescalarpotentialφ + ofthetilt field + ( φ + = ), C + lφ (r) = k BT c(q )e iq r k BT A p 4π q q c(q )e iq r dq = k BT π c(q )J (rq)qdq. (7) Here, J (r) is the Bessel function. The last expression in Eq. (7) ephasizes that C + lφ has no aziuthal dependence and, hence, only the r-coponent of C + l is not zero and is equal to C + l (r) = C+ lφ (r) r k BT π qax c(q )J (rq)q dq, (8) q ax is the cut-off wavenuber. Eq. (8) indicates that coupling between the onolayers induces a long-range correlation between the chain tilt and length. However, since C + l () =, this coupling is unlikely to induce correlation between tilt and length of the sae chain. Nevertheless, one cannot copletely rule this out, since chains are not localized to one value of r and C + l (r) for sall r. Therefore, the tilt-length correlation should be considered over a range of values of r corresponding to a tilted chain. E.g., for a tilt angle of 3 and an effective chain length of Å the range for r is 6 Å. Such sall values of r are outside of the range of the continuu odel. Hence, even though Eq. (8) indicates that C + l (r) for this r, the specific value of C+ l (r) is likely different fro the one predicted by Eq. (8).
9 IV. TILT-LENGTH CORRELATION PREDICTED BY THE CONTINUOUS MODEL WITH A HARMONIC COUPLING BETWEEN MONOLAYERS In section III we investigated the tilt-length correlations assuing infinitely large value of the coupling constant k. Here, we apply a siilar analysis to the case of finite k. In this case, the free energy of the odes (z, +,l + ) can be written as G(z, +,l + ) = q f q Bq fq, (9) and B q = fq = (z q, q,l + q) (3) κq 4 +k iκq 3 Ωq / k L iκq 3 κq +κ θ iqω/. (3) Ωq / k L iqω/ k A +k L The covariance between the Fourier odes of the longitudinal coponent of and the chain length is c(q ) = (l + ) q q = δ k B T q,q ( B q ),3 = δ q,q iqk B T c(q ), (3) κl q / Ω/4 [L κκ θ +κ θ /k (κk A Ω /4)]q 4 +(κk A ΩL κ θ Ω /4)q +κ θ k A (33) Coparison of Eq. (33) with Eq. (5) reveals that the only effect of finite k is a correction to the coefficient in the q 4 ter in the denoinator. Therefore, finite k does not introduce any qualitatively new features to the covariance of the chain length and tilt. [] E. R. May, A. Narang, and D. I. Kopelevich, Phys. Rev. E. 76, 93 (7). [] M. C. Watson, E. S. Penev, P. M. Welch, and F. L. H. Brown, J. Che. Phys. 35, 447 (). [3] M. C. Watson, E. G. Brandt, P. M. Welch, and F. L. H. Brown, Phys. Rev. Lett. 9, 8 (). [4] D. N. Theodorou and U. W. Suter, Macroolecules 8, 6 (985). [5] V. Blavatska and W. Janke, J. Che. Phys. 33, 8493 (). [6] J. Vyětal and J. Vondrášek, J. Phys. Che. A 5, 455 ().