VOLUME 81, NUMBER 2

PHYSICAL REVIEW LETTERS

13 JULY 1998

High- and Low-Pitch Helical Structures of Tilted Chiral Lipid Bilayers
Shigeyuki Komura1 and Ou-Yang Zhong-can1,2
1Faculty of Computer Science and Systems Engineering, Kyushu Institute of Technology, Iizuka 820, Japan
2
Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing 100080, China(Received 13 November 1997)
Molecular packing in helical ribbons of tilted chiral lipid bilayers is investigated as cholesteric liquid
crystals. Two types of helices with high- and low-pitch angles observed in the pathway for cholesterol
crystallization (ChC) in native and model biles by Chung et al. [Proc. Natl. Acad. Sci. U.S.A. 90,
11 341 (1993)] correspond to parallel and antiparallel packing of molecules at the edges of the ribbon,
respectively. The structural change in the ChC is explained as a decreasing succession of free energyof the metastable intermediate structures. [S0031-9007(98)06588-0]
PACS numbers: 87.10. + e, 61.30. – v, 68.15. + e, 87.22.Bt

Tubular and helical structures are rather commonly
observed in nature [1]. For example, many authors foundvarious helical structures of chiral lipid bilayers (CLBs)
in recent experiments [2]. Among several theoretical
approaches, a fundamental basis which is consistent withthe experiments is that the molecular packing interaction
is chiral as in cholesteric liquid crystals (ChLCs) [3–5].
According to the early theories, the pitch angle of thehelix [f0 in Fig. 1(a)] has been predicted to be 45± whichis in good agreement with the previous experimental
observations [2].
Although the concept of molecular packing interaction in ChLCs has been experimentally confirmed [6]and theoretically extended [7,8], a recent challenge isto understand the new findings in cholesterol crystallization in native and model biles [9,10]: Micellar modelbiles composed of bile salt sodium taurocholate, lecithin,and cholesterol in a molar ratio of 97.5:0.8:1.7 were
initially prepared and contained both micelles and vesicles. Within 2–4 hours of dilution, filamentous structures were observed. A few days later, the filaments werebent to form high-pitch helices [ø54±, Fig. 2(a) in [10] ].
These helices grew laterally while maintaining the pitch
angle to form tubules [Fig. 2(b) in [10] ]. Within a few
weeks, high-pitch helices and tubules disappeared, while
new helices with low-pitch angle (ø11±) appeared andgrew to form new tubules [Figs. 2(c) and 2(d) in [10] ].
Eventually only platelike cholesterol monohydrate crystals (ChMs) remained [Figs. 2(e) and 2(f) in [10] ]. Oneof their interesting findings is the existence of helicalribbon structures with two distinctive pitch angles: high(54±) and low-pitch (11±) helices. To explain these structures, Chung et al. [10] used a previous theory [11]appropriate for anisotropic bilayers of tilted chiral amphiphiles case and derived a free energy per unit area as

where r0 and f0 are radius and pitch angle of the helicalribbon, respectively (see Fig. 1); k cc , k pp , k cp , and k p areelastic constants. For the equilibrium helix, they obtained
the relation tan f0 ­ sk cc yk pp d1y4 . Corresponding to
f0 ­ 54± and 11±, the calculated ratio is k cc yk pp ­3.4 and 0.0015, respectively. However, as commented
very recently by Selinger et al. [12], the ratio of 3.4
is reasonable, but that of 0.0015 is surprising. For thisreason, Selinger et al. assumed that the molecular tilt
direction (m in Fig. 1) varies linearly across the ribbon

g ­ s1y2r02 d fk cc cos4 f0 1 k pp sin4 f0

FIG. 1. Schematic illustration of the helical ribbon: (a) thegeometric parameters characterizing the ribbon; ( b) local coordinate system of TCLB; (c) helical ribbon with parallel packingof molecules (P-helix); (d) helical ribbon with antiparallel packing of molecules (A-helix).

1 sk cp y2d sin2 2f0 g 2 sk p y2r0 d sin 2f0 ,(1)0031-9007y98y81(2)y473(4)$15.00© 1998 The American Physical Society

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VOLUME 81, NUMBER 2

PHYSICAL REVIEW LETTERS

13 JULY 1998

∂
µ
k11
k22
k2 2
2
gLC ­
s= ? dd 1
d?=3d2
2
2
k22
k33
1
(2)
sd 3 = 3 dd2 ,2FIG. 2. (a) 2sgA 2 gP dykm2 versus q for several values oft ­ tan u0 and (b) qp and sin f0 versus t ­ tan u0 for 0 ,u0 , u0p stan u0p . 0.654d, respectively.

[12]. Although the observed low-pitch angle 11± has been
attributed to the difference between the helical direction
and the tilt direction in their paper, there is no evidence
for the linear ansatz and the theory has not yet reached
the conclusive explanation on the existence of two typesof helices. Thus, the general question can be posed asfollows: What is the nature of the existence of two typesof helices, and can we theoretically discuss the above
sequence of transition as well as each structure from theenergetic point of view?
In this Letter, we investigate the molecular packing
on a helical ribbon of CLBs as ChLCs by generalizing
our previous theory [4]. In Ref. [4], we employed onlygch ­ 2k2 d ? = 3 d as an energy, where d is a unit
vector parallel to the long axis of the molecules [seeFig. 1(b)] and k2 characterizes the chirality of ChLC.
With this simplified energy, the pitch angle was predicted
to be 45±. Here we give a calculation by using thecomplete free energy density per unit area of ChLC [13]474

where kii (i ­ 1, 2, and 3) are the splay, twist, and bending elastic constants, respectively, and = is the threedimensional gradient operator. To view (2) as an energy
per unit area, we regard kii and k2 as the corresponding elastic constants of ChLC multiplied by the thickness of the CLB, and H
thence the total free energy of the
CLB is given by F ­ gLC dA, where dA is the area element of the CLB surface. We allow the molecular tilt direction m (see below for the definition) to vary on thebilayer as proposed by Selinger et al. [5,12], but keep
the boundary condition that the tilt direction is alignedwith the helical direction at the ribbon edges. The latter condition has been also considered in the previous
theories [3,7,10] and confirmed by x-ray diffraction studies [14]. Especially, instead of assuming a simple linear ansatz for the tilt direction field [12], we obtain itexactly by solving the Euler-Lagrange equation derived
from gLC . With these refinements, we will reveal thatfor the high-pitch helices, molecules at the two edges ofbilayers tilt parallel with each other [Fig. 1(c)], whereas
for the low-pitch helices, they tilt antiparallel with eachother [Fig. 1(d)]. Helix formation takes place througha transition from a fluid CLB (chain-melted La ) phase
to a tilted CLB (TCLB, chain-frozen Lb p ) phase undercooling or dilution [6]. The observed sequence of structures from filaments to ChMs by way of two types ofhelices can be shown as a decreasing succession of freeenergy of the metastable intermediate structures and regarded as a quenchlike cooling process of the molecular tilt.
Mathematically, a surface of a helical ribbon woundaround a cylinder of radius r0 can be represented
by a vector function with two variables as Ys,, dd ­
where
v0 ­
sr0pcos v0 ,, r0 sin v0 ,, h0 v0 , 1 dd,
1y r 20 1 h20 [see Fig. 1(a)]. The pitch angle and thepitch of the helix are given by f0 ­ arctansh0 yr0 d
and 2pjh0 j, respectively. Moreover, , and d satisfy
0 # , # L and 2d0 y2 # d # d0 y2, respectively, where
L is the arc length along the edge of the ribbon, and d0
is the ribbon width along the z axis (the central axis of
the cylinder). We introduce the tilt of the director fieldby d ­ cos u0 n 1 sin u0 m, where n is the unit normal
vector of the ribbon surface, m is the unit vector parallelto the projection of d on the surface, and the tilt angle
u0 is the angle between n and d [see Fig. 1(b)] [11]. u0
characterizes the transition from the fluid (La ) phase withu0 ­ 0 to the TCLB (Lb p ) phase with a constant u0 . 0.
Generally, m can be written in terms of single angle
cs,, dd as m ­ cos cs,, ddz 1 sin cs,, dde, where z is
the unit vector along the z axis and e is the unit vector
along the circumference of the cylinder.

VOLUME 81, NUMBER 2

PHYSICAL REVIEW LETTERS

13 JULY 1998

For simplicity but without loss of generality, we consider the case where c is a function of only d, and use theone-constant approximation, i.e., k11 ­ k22 ­ k33 ­ k. Then (2) becomes [15]gLC ­ sky2r02 d fcos2 u0 1 sin2 u0 s4 sin2 c 1 v022 cd2 d 2 sin 2u0 sh0 cos c 1 r0 sin cdcd g
2 sk2 y2r0 d fsin 2u0 sh0 sin c 2 r0 cos cdcd 1 2 sin2 u0 sin 2cg 1 km2 y2 ,where cd ­ dcydd, and m ­ k2 yk is the inverse of
the pitch of ChLC [4,13]. For constant c, one canshow that (3) coincides with (1) when k pp ­ k cos2 u0 ,k cc ­ kscos2 u0 1 4 sin2 u0 d, k cp ­ k cos2 u0 , and k p ­2k2 sin2 u0 , except the constant term. Progress in thepresent work is to consider the case without the constraintof constant c. We therefore derive the Euler-Lagrange
equation from (3) as(4)
cdd ­ 2v02 sins2c 2 a0 dy cos a0 ,22
where cdd ­ d cydd and a0 ­ arctansmr0 d. The general form of the first integral of (4) ispp
cd ­ s2v0 y cos a0 d cosh2 C 2 cos2 sc 2 a0 y2d ,(5)where C is an integral constant.
Obviously, Eq. (4) has a constant solution of c ­c0 ­ a0 y2, and we let f0 ­ py2 2 c0 ­ py2 2 a0 y2
to satisfy the boundary condition such that the molecules
at the two edges of the ribbon tilt parallel with each other.
We call this configuration “parallel packing” (P-helix) as
shown in Fig. 1(c). It is necessary to check whether sucha helix is in mechanical equilibrium or not. We thendefine the curvature k0 and the torsion t0 of the helix by k0 ­ r0 v02 and t0 ­ h0 v02 , respectively. Thenthe elastic energy of the P-helix is expressed from (3) interms of k0 and t0 as gP ­ sky2d f4 sin2 u0 sk02 1 t02 d 1
cos2 u0 sk0 1 t02 yk0 d2 g 2 2k2 t0 sin2 u0 1 km2 y2. Minimizing this with respect to k0 and t0 , i.e., ≠gP y≠k0 ­≠gP y≠t0 ­ 0, we obtain the optimal geometry of the helical ribbon characterized by(6)
tan f0 ­ s1 1 4 tan2 u0 d1y4 ,2s1 1 4 tan2 u0 d1y4
, (7)s1 1 4 tan2 u0 d1y2 2 1
and the corresponding energy per unit area is gP ­km2 y2. These results reveal some important features of
the P-helix. First, we note that (7) is identical to theassumed boundary condition, f0 ­ py2 2 a0 y2 witha0 ­ arctansmr0 d, and hence the calculation is selfconsistent. This confirms the helix being in mechanical equilibrium since the minimization condition requires
mr0 ­ 2 tan 2f0 ­(3)the state of both free external force and moment. Second, the result of (6) essentially corresponds to that by
Chung et al. as mentioned before (note also the relations between k cc , k pp , and k), and predicts a high-pitch
angle of f0 . py4. We have also performed the minimizing calculation without using the one-constant approximation and obtained an equation as cos2 2f0 1 f2 1
sk11 yk33 d cot2 u0 g cos 2f0 1 1 ­ 0. Since this equation
possesses a negative solution for cos 2f0 , we can show
f0 . py4 as before, and it reduces to (6) when k11 ­k33 . Third, since the radius r0 is inversely proportional tothe chiral elastic constant k2 , we see that the origin of thecylindrical curvature is due to the chirality of molecules.
In fact, (7) reveals that the pitch (m21 ) should be proportional to the radius. This is one of the remarkable facts
observed in experiments (see Figs. 4–6 in Ref. [10]).
According to the experiments, a straight filament appears at the initial stage of the crystallization process inmodel biles and can be identified as r0 ­ 0 [9]. This
fact indicates that the initial stage of the TCLB helix
is quenched to the structure with f0 ­ py2 [notice (7)and f0 . py4]. From (6), one can then expect that thetilt angle takes the maximum value u0 ­ py2. After thequench, u0 will decrease from py2 to its stable value andthe straight filaments will bend and twist by decreasing
f0 and increasing r0 [9,10]. This is the expected mechanism underlying the transition from the straight filaments
to the high-pitch helices.
We now turn to consider the case where the tilt direction can vary on the bilayer across the ribbon. Since thetilt direction coincides with the helical direction at the ribbon edges, the allowed boundary condition for the nonuniform solution is cs2d0 y2d ­ py2 2 f0 and csd0 y2d ­cs2d0 y2d 1 p ­ 3py2 2 f0 . The molecular configuration subjected to this boundary condition is called “antiparallel packing” (A-helix) and is depicted in Fig. 1(d).
We will show below that the helical structure produced
from the antiparallel packing of molecules indeed resultsin the low-pitch angle. The integral constant C in (5)is now determined by the boundary condition as d0 ­p
cos a0 qKsqdyv0 with q ­ 1y cosh C, whereas the energy per unit area can be calculated from (3) as [15]pgA ­ skm2 y2d f1 1 s1 1 sin2 u0 dx 2 1 2 sin2 u0 x 1 1 x 2 s1 2 2yq2 dpp
1 8 sin2 u0 x 1 1 x 2 Es qdyq2 Ks qd 2 2 sin 2u0 x s1 1 x 2 d1y4 yqKs qdg ,

(8)

where
­ cot a0 ; Ksqd and Esqd Rare thep first and second complete elliptic integrals given by Ksqd ­Rpy2 x p
py2
dwy 1 2 q2 sin 2 w and Es qd ­ 0 dw 1 2 q2 sin2 w, respectively. To ensure that the helix is in mechanical
0475

VOLUME 81, NUMBER 2

PHYSICAL REVIEW LETTERS

equilibrium we must minimize gA . Because q is afunction of d0 , f0 , and a0 , minimization of gA overthese quantities leads simultaneously to ≠gA y≠q ­ 0.
From this and with the help of differential relations,dKs qdydq ­ fEs qdyqs1 2 q2 dg 2 Ks qdyq
and dEs qdydq ­ fEs qd 2 Ks qdgyq,p we can obtain
the relation between a0 and q as cos a0 y sin a0 ­q cot u0 y2Es qd. By substituting this into (8), gA thenbecomespgA ­ skm2 y2d f1 1 s1 1 sin2 u0 d s 1 1 4y 4 2 1dy2
1 2y 2 sin2 u0 2 cos2 u0 yE 2 s qdg ,(9)pwhere y ­ cos a0 y sin a0 [a0 ­ arctansmr0 d as
before].
In Fig. 2(a) we have plotted scaled gA 2 gP as afunction of q for several values of t ­ tan u0 . Thereexists a critical value of u0 (denoted as u0p ) such that for
0 , u0 , u0p there exists a critical value of q (denoted
as qp ) at which gP is equal to gA , whereas for u0p ,u0 , py2, gA is always smaller than gP predicting
the transition from P-helix to A-helix. In Fig. 2(b),qp is plotted against t ­ tan u0 for 0 , u0 , u0p . u0p
can be obtained by solving gP ­ gA at q ­ 1 andis tan u0p . 0.654. If we consider the case when thehelix closes to form a tube (d0 ­ 2ph0 ), then d0 ­s2pymd tan a0 tan f0 holds. With this relation, we cancalculate the pitch angle of the helix as
sin f0 ­Ks qd p 4
f q 1 4 tan4 u0 E 4 s qd
2p
2 2 tan2 u0 E 2 s qdg1y2 .

(10)

In Fig. 2(b), we have also plotted sin f0 versus t ­ tan u0
with q ­ qp . Except at the singular point u0 ­ u0p ,this graph generally shows f0 , 30± indicating that the
antiparallel packing results in the low-pitch angle. This is
one of the important results of this paper.
Another interesting feature is that for each curve in
Fig. 2(a), we see that q ­ 0 gives the minimum ofgA , and hence it is the unique stable state. The stateq ­ 0 leads to a0 ­ py2 which corresponds to r0 ! `since a0 ­ arctansmr0 d. In fact, this result confirms theobservation in experiments. At the final stage of thecrystallization process, the tubes fracture at their ends to
produce platelike ChMs (see Fig. 2 in Ref. [10]). In otherwords, P-helix or A-helix are metastable intermediates
although they are static solutions of the Euler-Lagrange
equation (4).
As a progress compared to the previous theories[10,12], Fig. 2(a) predicts an important fact that the transition from the low-pitch helices (A-helix) to the ChMs
can be continuous while that from the high-pitch helices(P-helix) to the ChMs is discontinuous. All the curves
for gA s qd are continuous and have their minima at q ­ 0.
This feature indicates that the helical tubes of the low476

13 JULY 1998

pitch angle transform continuously into platelike ChMs
as shown in Fig. 2(f) in Ref. [10]. For the high-pitch
helices, the dissolution process is necessary to transform
into the ChMs [10]. Since the energy for ChMs (q ­ 0)
is gChM ­ skm2 y2d f1 2 s2ypd2 cos2 u0 g, gP ­ km2 y2
is always larger than gChM except for u0 ­ py2. This
fact explains why the initial stage of the helical structuresof TCLB is quenched to u0 ­ py2 as mentioned before.
The experimental observations confirm very well theabove predictions [10].
This consideration now leads us to determine theoptimal high-pitch angle. For the transition from the highpitch helices to the ChMs, u0 must reduce to u0p in orderto satisfy the condition gP ­ gA from which the branchof low-pitch helix starts in Fig. 2(a). Hence the optimal
high-pitch angle is precisely obtained from (6) as [15]("f0 ­ arctan#1y4 )µ∂8
1
5
1. 52.1±,cos
arccos
1
3
3
32
3
(11)

which is in excellent agreement with the experimentallyobserved value f0 ­ 53.7 6 0.8± [10].
This work is supported by the Ministry of Education,
Science and Culture, Japan (Grant-in-Aid for Scientific
Research No. 09740318).

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[15] The details will appear in a full paper by the authors.