Indian Journal of ChemistryVol. 35A, July 1996, pp. ~36-545

Papers

Microscopic origin of the chirality driven morphologies of the amphiphilicmonolayers and bilayers

Nilashis Nandi & Biman Bagchi'"

Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560 012, India

Received 10 January 1996

It is widely known that the compressed monolayers and bilayers of chiral lipids or fatty acidsform helical morphologies, while the corresponding racemic modification gives only flat platelets with-out twist. No molecular explanation of this phenomenon is yet available, although subtle interactions atthe chiral centers have' often been proposed as the driving force behind the morphology of the aggreg-ate to form a particular shape. In the present study, the morphologies of the chiral amphiphilic assem-blies have been predicted on the basis of an effective pair potential between the molecules, which de-pends on the relative sizes of the groups attached to the chiral centers, the orientation of the amphi-philic molecules and also on the distance between them. It is shown that for a pair of same kind of en-antiomers, the minimum energy conformation favours a twist angle between them. This twist betweenthe neighbouring molecules gives rise to the helicity of the aggregate. The present theory also showsfrom the molecular considerations that for a pair of mirror-image isomers (i.e. the racemic modifica-tion) the minimum energy conformation corresponds to the zero angle between the molecules, thus giv-ing rise to flat platelets as observed in experiments. Another fascinating aspect of such chirality drivenhelical structures is that the sense (or the handedness) of the helix is highly specific about the chiralityof the monomer concerned. The molecular theory shows, for the first time, that the sense of the helicalstructures in many cases is determined by the sizes of the groups attached to the chiral centers and theeffective potential between them. The predicted senses of the helical structures are in complete agree-ment with the experimental results.

I. IntroductionIt is widely known that the compressed mono-

layers and bilayers of npids or fatty acids havingat least one chiral carbon atom form many inter-esting supramolecular morphologies like helicalribbons, tubules, fibers, strands and flat crystals 1 •

In the gel state, the mono layers and bilayers formhelical structures, provided one kind of enantiom-er (either D- or L-) is present in excess within theaggregate'. The corresponding racemic modifica-tion, on the other hand,.. gives only flat plateletswithout twist'. The driving force for the formationof such helical morphologies from the chiral am-phiphilic molecules is known to come partly from .the interactions at the chiral centers of the amphi-philes+'. Another fascinating aspect of such chi-rality driven helicity is that the sense (or the han-dedness) of the helical structure is highly specificabout the chirality of the monomer concerned.More explicitly, if right handed helix has been ob-served to be formed from the aggregate of the D-

'Also at: Jawaharlal Nehru Center for Advanced ScientificResearch, Bangalore;

enantiomers of a particular -chiral amphiphile, thecorresponding aggregate of the L-enantiomersshould gives rise to the helix of left handednessand vice-versa, in the compressed gel statev",

Customarily, in studying the aggregate of thechiral amphiphilic moleculesv" (he individualmolecules are represented by a pseudovector(usually termed as a director) whose directiongives the orientation of the long hydrophobic tailof the amphiphile and it lacks the symmetry ofreflection (i.e. not superimposable with its mirrorimage). Many of such continuum theories startwith an assumption of the presence of a intrinsicbending force due to chirality+'>", It is believedthat within the assembly of one kind of enantiom-ers, any two neighbouring directors are tilted rela-tive to each other and this tilt from neighbour toneighbour gives rise to the helicity of the aggreg-ate. In theoretical formulation one usually em-ploys a Ginzburg-Landau type free energy func-tional which is common in the study of the liquidcrystals. This free energy provides only a coarsegrained description where long wavelength pro-perties are explicitly considered. Consequently, in

NANDI et al.: ORIGIN OF CHIRALITY DRIVEN MORPHOLOGIES OF MONO- & BILAYERS 537

the above description, the detailed microscopicfeatures of the molecules are completely averagedout and one cannot expect to understand the mic-roscopic length scale interactions driving the as-sembly to a particular morphology, such as thesense of helix.

However, a fully microscopic description of theHamiltonian of such a system is enormously diffi-cult due to the complexities of the interactions be-tween the molecules. One can, of course, describethe system in a mesoscopic scale in which the es-sential microscopic features of the effective chiralinteraction are explicitly retained but other inter-actions like the hydrophobic effect or the electri-cal head group interaction, which are not differentfor the optical isomers. are averaged out.

It has been pointed out recently that it might bepossible to predict and understand the structureformation from an effective intermolecular pair.potential between the chiral centers of the aggreg-ate!", because, in these structures the subtle stere-ogenicity at the chiral centers is surely responsiblefor driving the aggregate to a particular morphol-ogy. Such reduced effective potentials may bederived after summing up many other detailed in-teractions between the groups attached to the chi-ral centers of a pair of chiral molecules and mini-mally this potential should depend only on thedistance and orientation between a pair of chiralmolecules. These minimal energy conformationscan be studied by changing the orientation be-tween the groups and the distance between thechiral centers. In this approach it has been as-sumed that the pair potential between the adja-cent groups of the chiral centers has a Lennard-Jones (6-12) form. Thus, in principle, such mic-roscopic theories are expected to be able to pre-dict the tilt between the two neighbouring amphi-philes from the effective potential by studying theminimum energy configuration between a pair ofthe same kind of enantiomers. Thus, by knowingthe sizes and the orientations of the groups att-ached to the two chiral centers, one can know therelative tilt of a pair of amphiphilic molecules.Consequently, the arrangement of an array ofmolecules in space can be predicted from thepresent molecular considerations. The sense ofthe array is thus predetermined from the relativesizes of the neighbouring groups of the chiral cen-ters.

The present study is devoted to achieve such agoal. Here, we have calculated the tilts of thegroups attached to the two chiral centers andhave attempted to predict the sense of the aggreg-ate from the tilts. Other parameters characterizing

the helical structure like pitch, pitch angle, etc.,are also expected to be predicted from the molec-ular theory. As indicated earlier, the flat morphol-ogy of the racemic modification is another prob-lem left to be solved from molecular theory. Wehave also tried to understand the morphology ofthe racemic modification from the effective inter-action between a pair of mirror image isomers.

The organization of the rest of the paper is asfollows. In Section II we have described the calcu-lation of the effective interaction potential be-tween a pair of chiral amphiphiles. In Section IIIwe have described the calculation of the tilt be-tween a pair of same kind of enantiomers fromthe effective potential as calculated in Section II.In Section IV, we have described the systematicstudy of the sense of the helicity from the tiltangles calculated from the theory. In Section V,we have described the parameters characterizingthe helicity like the pitch, pitch to diameter ratio,etc. In Section VI we have calculated the mini-mum energy configuration for a pair of mirror im-age isomers. Section VII includes discussionswhich is followed by brief concluding remarks.

II. Theoretical formulation for the effective in-teraction potential

In this section we briefly recapitulate the theor-etical formulation for calculating the effective in-teraction between the two chiral molecules, eachhaving one chiral center (designated by C1 andC2, respectively). Now, consider the plane of aand b groups as shown in Fig. 1. For a pair ofmolecules this plane contains four groups, twofrom each chiral center. The relative arrange-ments of these four groups depend on whetherthe molecules belong to D-D or D-L pair. General-ly, we can represent these groups by four groupsin a plane, m, n, 0 and p, respectively (see Fig. 2).The m and n groups are attached with the C1 chi-ral center and 0 and p groups are attached withthe C2 chiral center, respectively. We assume thatthe C1 carbon atom is situated at the center of anarbitrary frame of reference (designated by a setof axes, X and Y). a is the orientation of the linejoining the two chiral centers and r is the distancebetween them. X' and Yare the set of axes situat-ed at C2 and are parallel to X and Y respectively.'1 and '2 are the orientations of m and 0 groupswith respect to the X and X' respectively. Pmn andPOP are the angle between m and n and that be-tween the 0 and p groups, respectively.

We represent the effective sizes of the groupsattached to the chiral centers by urn' o., Uo andUP' respectively. These U values correspond to the

538 INDIAN J CHEM, SEe. A, JULY 19~o

(41

(bl

Fig. 1- Three-dimensional models for (a) a pair of same kind ofenantiomers (D-D) and (b) a pair of mirror image isomers (D-L). Inboth the figures lis the group which includes the largest part of thehydrocarbon chain, b' includes the hydrophilic head group. a andb are the two other groups attached to the chiral center and theirrelative arrangement is different in the D-D (Fig. (a)) and D-L pairs(Fig. (b)). The bgroup is largerthan the agroup. Note that rmay be

either larger or smaller than b'.

effective diameters of the corresponding groupsadded with the effective radius of the chiral car-bon atom. It is well known that the effective sizesof the alkyl groups increase linearly with the in-crease in the length of the corresponding carbonchain 11. We have calculated the effective diametersof the groups using the empirical correlations pro-vided by Ben-Arnotz and Herschbach" and thegroup increments tabulated by Bondi 12. The em-pirical relations are as follows 11,

Vhs = 1.086 {V.- 9.94),

a= 1.244 ( Vhs)1/3,

... (1)

... (2)

where V. is the "space-filling" volume, which canbe computed by summing the increments for thevarious atoms and the functional groups tabulatedby Bondi'". These values are expected to be re-markably accurate, as indicated in the literature 1I.

Also, these values are insensitive to substantialdeviation in the shape of the group from spheric-ity. In order to facilitate connection with real ex-perimental situation, we have given the sizes ofthe groups attached to the chiral centers of com-mon amphipbiles (forming helical morphologies)in Table 1.

The Lenn ••rd-Jones energy parameters of thegroups, m, n, 0 and p, have been represented byEm' En' Eo and Ep' respectively. From the values ofthe Eu of several classes of compounds like al-kanes, alcohols, haloalkanes, etc., tabulated by

Ben-Arnotz and Herschbach 11, a linear depend-ence of the ~u on the sizes of the groups is ob-served. For 1 A increment in the effective diame-ter of a group, the EUlk8 values of the group (k8is the Boltzmann constant) increases by -1000 K.In the present study we have taken the Eu valuesof the groups as proportional to their effective di-ameters.

As already discussed, the interaction betweenthe chiral centers has been calculated by assuminga Lennard-Jones (6-12) form of the potential+' be-tween the adjacent groups. If we assume that theadjacent groups belonging to the neighbouringchiral centers interact through pair potentials VIand V2, respectively, then the total interactionpotential between the two chiral centers is givenby

Table 1- The effective diameters of the groups (in A) attached tothe chiral center of the amphiphiles (A - E) 1-5forming helical mor-phology. The groups are designated as I, b', a and b, respectively.The tilt angles (in degrees) in the plane of I, b' (represented by OM)and that in the plane of a and b(represented by ¢M)in the minimal

energy configuration are also shown

Amphiphile I • b' OM a b ¢MAi 8.19 6.51 11 1.4 7.81 45B2 7.81 8.00 1 1.4 8.24 45C3 7.81 8.81 6 1.4 2.99 31D4 8.18 8.00 0.1 1.4 7.29 44P 5.53 7.35 14 1.4 1.92 15

1Amphiphile '1\ is the double chain ammonium am phi phi Ie citedin the ref. 5 with n= 12 and m= 2, where n-l and rn- I are thenumber of the methylene groups attached to the carboxylic groupin the hydrophobic tails and the N+ atom of the head group, re-spectively. Here, I is -(CH2h'COO'(CHi)II'(CH3) and b' is-NH·CO·(CH2)·N(CH3);. The a and b groups are -H andCOO'(CH2)II'CH3, respectively .2Amphiphile 'B' is the double chain ammonium amphiphile citedin the ref. 5 with n= 12 and m= 11,where n-l and rn-l are thenumber of the methylene groups attached to the carboxylic groupin the hydrophobic tails and the N + atom of the head group, re-spectively. Here, I is -(CH2)'COO'(CH2)1I'(CH3) and b' isCOO'(CH2)II'CH3' The a and b groups are - H andNH'CO'(CH2)IO'N(CH3);, respectively.3Amphiphile 'C' is the single chain ammonium amphiphile cited .in: Kunitake T; Yamada ill, J chem Soc Chern Commun; 1986,655-656. Here, I is - COO'(CH2)11'(CH3) and b' isNH·CO·(C6H5h'o·(CH214·N(CH3h. The aand bgroupsare - Hand - CH3 , respectively.4Amphiphile 'D' is the double chain phospholipid with a nucleo-tide head group as cited in the ref. 1. Here, I is-(CH2)'COO'(CH2)1I'(CH3) and b'is -COO'(CH2)12'CH3'The aand b groups are - H and a phosphonucleotide head group(see figure 11 ofreference 1), respectively.5Amphiphile 'E' is 12-hydroxystearic acid, cited in ref. 4. The mo-lecular projection formula is shown in Fig. 4. Here, t is- COO'(CH2)1I and b'is (CH2)4CH3' The aand bgroups are - Hand OH respectively,

NANDI et al.: ORIGIN OF CHIRALITY DRIVEN MORPHOLOGIES OF MONO- & BILAYERS 539

V VI V2-=-+-kT kT n: , .. (3)

Here, VI gives the interaction between the groupsm and 0 and the V2 gives the interaction betweenthe groups n and p respectively. These interac-tions themselves are given by the following ex-pressions

... (4)

... (5)

where, gl is the median distance between m and 0

and g2 is the median distance between n and pgroups, respectively. al and a2 are given by

... (6)

... (7)

and E I and E2 are given by the Berthelot rule':'

... (8)

... (9)

From Fig. 2 it is clear that the orientations ofthe groups attached to one chiral center (say C I)relative to those attached to the other cbiral cen-ter (C 2) and also the separation between the twocbiral centers (r) influence the median distancebetween the groups. Consequently, the effectivepair potential depends on the orientation and theseparation of the groups. In order to calculate theeffective interaction potential between the cbiralcenters, we need to consider explicitly the orien-tation and the distance between the cbiral centersin the case of 0-0 and D-L pairs. In the followingtwo subsections we describe the' explicit expres-sions of the pair potential of the D-D and D-L

pairs.In the foregoing formulation the interaction

potential in the plane of the a and b groups hasbeen explicitly considered. The minimal energyconfiguration of the t and b' groups attached tothe CI and C2 cbiral centers (Fig. 1) can also beunderstood in an analogous way.

y

~"'n~~~------.-----------_. x

Fig. 2 - Relative arrangement of the two pairs of groups attachedto the two chiral centers C 1and C 2' m and nare attached to the C 1

and oand pare attached to the Cj. C 1 is situated at the center of thearbitrary frame of reference XY. X' and Y' are parallel to the X andY, respectively. r is the line joining C 1 and C2 and a is the orienta-tion of r. 01 and O2 are the orientations of the m and 0 groups, re-spectively. {J mn and {J op are the angle between the m and n and thatbetween the oand p, respectively. These groups are the represen-

tatives ofthe a, bor t,b' groups (see Fig. 1)in the two dimension.

III. Calculation of the tilt between a pair ofsame kind of enantiomers (e.g. 0-0) fromthe effective pair potential

It is again to be noted that when we are consid-ering the interaction in the plane of a and bgroups of the two chiral centers of a 0-0 pair (ora i-i. pair), m is equivalent to a, 0 is equivalent toa group, n is equivalent to b group and p is equiv-alent to b group, respectively. The explicit expres-sion for the effective pair potential in this case isthen given by

V (4) jEa Ea[( r 1-= - --- -+-COS('2- a)kT 1', k k o; 2

-~Sin('2- a)cot('I- a)fl2

(r 1 1

- - + -2 cos ('2 - a) - - sin ('2 - a)aa 2

xcotlk af'j+ (~)h~[(;,-~x cOS(fJ-'1 + a) +.!. sin(fJ-'1 + a)

2

)-12 ( 1

x cot(fJ- '2) -.!... - - cOS(fJ-'1 + a)ab 2

1 ) -6]+"2 Sin(fJ-'1 + a)cot(fJ- '2) ... (to)

540 INDIAN J CHEM, SEe. A, JULY 1996

As indicated earlier, it is expected that a pair ofsame kind of enantiomers should prefer to beoriented with respect to each other at the mini-mum energy conformation. This orientations canbe studied by minimizing equation to. It is to benoted that there should be a tilt in the plane ofthe a and b groups (let us designate this angle byOM) arising out of the interaction between the aand b groups of the neighbouring molecules. Syn-chronously, there should be another tilt in theplane of the t and b' groups due to the interactionbetween the t and b' groups (designated by OM).The helical structure formed from these tilts isschematically shown in the Fig. 3.

IV. Prediction of the sense using the tilts calcu-lated from the effective intermolecular potential

From the foregoing molecular considerations, itis seen that there are two tilts in the plane of tand b' and in the plane of a and b groups. Thesetwo synchronous tilts in the two almost perpendi-cular planes give rise to a twist between a pair ofchiral amphiphilic molecules in the compressedstate. Thus, when same kind of chiral enantiomers(only D- or only L-) are stacked together, due tothe twist between each pair of molecules, the chi-ral centers' of all the molecules cannot lie on astraight line. Instead, they should follow a helicalpath. It is well known that twist between the twosuccessive neighbours can lead to helicity, but it isfor the first time that the handedness of the helic-ity can be predicted from the sizes and the effec-tive potentials of the groups attached to the chiralcenters. To predict the sense, we shall follow thefollowing stepwise systematic approach.

(1) First, consider a chiral amphiphilic mole-cule having only one chiral center. The Fisherprojection formula has been drawn for the mole-cule [see Fig. 4{a)]. The longest carbon chainshould be vertical and the most highly oxidizedend of the molecule should be at the top.

(2) The tetrahedral structure of the amphiphileis drawn from the projection formula. Accordingto the standard conventions 14, all the verticalbonds in the projection formula point backwardsin the corresponding tetrahedral configuration[see Fig. 4{b)].

(3) To construct an array of amphiphilic mole-cules, we shall place the successive molecules insuch a way that the second molecule is placed be-hind the first, third behind the second and so on.Note that it is customary to observe the handed-ness of the helix by placing it in such a way thatthe helix propagates away from the eye. From theconsideration of the tilt (see Section Ill) we can

Fig. 3-Schematic diagram of a circular helix formed from the ar-rangement of the chiral centers of an array of D-D or L-L molecules.The angles 8M and ¢1Mare the tilt angles developed at the minimumof the pair potential, in the plane of the land b and tr.l'tin the planeof aand bgroups, respectively and TM is the separatiun between the

two adjacent chiral centers.

see how the array of the molecules propagates inthe space. It is clearly seen that depending on theeffective interaction potential of the groups, thehelical assembly would turn either in a left-'handed way or in a right-handed way [see Fig.4{c) and 4{d) for a left-handed helix formed byD-12 hydroxystearic acid].

Thus, it is clearly seen that once we know theabsolute configuration of the chiral molecule andthe effective interaction potential of the groupsattached to the chiral center, we can easily predictthe sense of the helical aggregate formed from thechiral monomers.

Next, we have attempted to predict the sensesof the helical structures of some amphiphi)ic as-semblies and compared the theoretical predictionwith the experimentally determined senses.

(i) Prediction ofthesenseoft» 12 hydroxysteancacid:From the Fisher projection formula, the abso-

lute configuration is drawn as shown in Fig. 4{a,b). In the figures, the dotted bonds point back-wards to the plain of the paper and the filled inbonds point towards the top of the plain of thepaper. The sizes of the different groups are givenin Table 1. Among the four groups, the size de-creases as follows: {CH2)11 - COOH > (CH2)4-

CH» -OH> -H.

NANDI eta!.: ORIGIN OF CHIRALITY DRIVEN MORPHOLOGIES OF MONO- & BILAYERS 541

(rZk-COOH

H,-OH

(CHz). CH,

(a)

(~HZ)"-COOHI

II

II

I--(

H > \ """'OH\\\\

(CHz).CH,(b)

(d)

Fig. 4-(a) ~ishe~projectionformuIaofD-12-hydroxystearic acid;(b) three dimensional structure of the o-12-hydroxystearic acidderi:-ed ~rom the proj~tion formula. (c) A pair of o-12-hydroxys-teanc acid molecules In the closed packed state and (d) a section ofthe helical aggregate formed by an array of D-12-hydroxystearic

acid molecules.

Thus, as shown in Fig. 4(c), when we place thechiral center of the second amphiphile (designatedby C2) behind the crural center of the first amphi-phile (designated by C1), the second molecule istilted with respect to the first and the magnitudesof the tilts in the top and bottom planes and theleft and right planes (see Fig. 1) are given inTable 1 as predicted from the theory outlined insection III of the present paper. In constructing ahelix from an array of amphiphiles we have toplace molecules one behind another in successionbecause the sense of helix is observed as the turnof the helix goes away from the eye. It is clearlyseen from Fig. 4( c) that in the plane drawn bydotted lines, the two - OH groups are away andthe two - H atoms attached to the C1 and C2 chi-ral centers are closer. Similarly, in the perpendi-cular plane, the two (CH2)1I -COOH groups areaway relative to each other and the two(CH;!)4- CH3 groups are closer compared tothem: If we construct an array of molecules thenit is easily seen from Figs 4{c) and 4{d) that theassembly should have a left-handed twist.

(ii) Prediction of the sense of L-12-hydroxystearicacid

The second helical aggregate we have consid-ered is the L-hydroxystearic acid. From the Fisherprojection formula, the absolute configuration isdrawn as shown in Fig. 5(a). The sizes of the dif-ferent groups are given in Table 1. Then, follow-ing the steps outlined earlier to determine thesense of n-isomer, we can easily find that thesense of the assembly should be right handed. Wehave not depicted the assembly of i-isomers be-cause it can be easily visualized from the corre-sponding D-isomers.

(iii) Prediction of the sense of2-CI2-L-Glu-Cll-N +

amphiphileThe third helical aggregate we have considered

is the above mentioned t-ammonium amphiphile.The molecular formula is given in Fig. 6. TheFisher projection formula and the absolute confi-guration are shown in the Fig. 6(a,b). The size ofthe group decreases as follows: CH3(CH2)uCOO-(CH2lz > COO(CH2)1I - CH3 > NH·CO·(CH2)1O-

N+ -(CH3)3> - H.

The sizes of the different groups are given inTable 1. In the plane containing (indicated by thedashed lines in Fig. 6c) the CH3(CH2)IICOO(CH2}.zand the COO(CH2l11 -CH3 groups, the formerwith greater effective interaction potential andlarger effective size should be more apart than the

542 INDIAN J CHEM, SEC. A, JULY 1996

latter. In the plane containing NH'CO'(CH2hoN-(CH3h and - H, the former being much largerthan the latter, the two - H groups should becloser than the two NH'CO'(CHZ)lON+(CH3hgroups. From Fig. 6(d) we can easily find that thesense of the assembly should be right-handed.The steps are outlined in the Fig. 6(c,d).

(iv) Prediction of the sense o/2-C12-rrGlu-Cu-N+amphiphile

The fourth and the last example we have con-sidered is the n-isomer of the ammonium amphi-phile considered just before. The molecular for-mula is given in Fig. 5(b). From the Fisher: projec-tion formula, the absolute configuration is drawnas shown in Fig. 5(b). The sizes of the differentgroups are given in Table 1. Then, following thesteps outlined earlier to determine the sense, wecan easily find that the sense of the assemblyshould be left-handed.

V. Prediction of the parameters characterizingthe helicity from the calculated tilts

In order to facilitate. comparison with experi-ments, it is necessary to calculate the characteris-tic parameters of the helix that would be predict-ed by the molecular theory. In the following, wecalculate the pitch of the helix formed by the ag-gregate of one kind of enantiomer. If the adjacentchiral centers (of D-J) or L-L pair) follow a circularhelix due to the twist between each .molecule,then the pitch of the helix, P, is given by the fol-lowing expression

2.1l . [ -1 ( tan tPM l]P=- SIn tan rM,OM J2(1 - cos OM)

... (11)

'1""-,<00,

~1-H(CHz). CHI

(0)

oIC-O-(CHz),ICH)

I CHIIe

H--C -- NH-i-lCHzho-tF-CH)

I 0 JH)(CHzlz-!-O-(CHz>'1 CH)

(b)

Fig. 5 -.(a) Fisher projection formula of L-12-hydroxystearic acidand (b) Fisher projection formula of 2-C Iz-D-Glu-C IIN + .

o~-O-(C"')1ratJI •

I

/I

I--4

(CH2~tCOOH(C~)n COOHr---f+------7

/ I I I

/ ~'I /1I 1C2 OH I

/ I I I/ H~ I

/ H ••••••...Cti ~OH 1I I I IL ..J_L I

I (CH2'),CH3(CH2),CH 3

(e)

CnH COOH CnH22COOH22 ~- •••

('X " "I, ~' \ ,/",..:r~ ::.IOH/ ~1C2dH I

/ \ ,I : I /;' , ,,-4....:~/ ICH) CH-L-"-"'2I,CH3/ 2, 3 ",

I ",I /I /, II II I, \, "\ '\ .•............, ,, ', ', ',

" ;'..•.. '"..•.•. -.-'(d)

Fig. 6-(a) Fisher projection formula of 2-C12-L-Glu-CIIN+; (b)three dimensional structure of the 2-C12-L-Glu-CIIN+ moleculederived from the projection formula. (c) A pair of 2-CI2-L-Glu-C'I N + molecules in the closed packed state and (d) a section of thehelical aggregate formed by an array of 2-CI2-L-Glu-CIIN + mole-

cules.

NANDI et III.: ORIGIN OF CHIRALITY DRIVEN MORPHOLOGIES OF MONO- & BILAYERS 5·n

where the angles OM and ~'''I are the angles corre-sponding to the minimum of the pair potentialand rM is the corresponding straight line distancebetween the two chiral centers. From the expres-sion 11 it is seen that P increases rapidly as OMbecomes smaller. For ~M= 45° and r.\l = 2.4A, thecalculated value of the pitch corresponding to the0M=0.75° is 1105 A and for OM=0.165°, it is5002 A. Pitches of comparable order are ob-served in the superhelical strands I) (P= 1100 A)and in the duplex) (P= 5000 A). The angle OM isin the plane containing the t and b' groups. Thesegroups constitutes the major parts of the hydro-phobic tail and the head group. The sizes of thegroups attached to the chiral centers of commonamphiphiles are given in Table 1.

The forces responsible for the aggregation ofthe amphiphile act in this plane along the hydro-phobic chain and do not favour the tilts of the Iand the b' groups. Thus, the splay of these groupsis expected to be much small. Note that the mainchiral interaction is expressed by the twistthrough the angle ~M(in the plane of the a and bgroups).

However, it should be pointed out that if we as-sume that the chiral interaction is the only drivingforce for a helical structure and that there is noforce to resist the twist due to the chirality, the si-tuation is unrealistic. This will certainly lead to aprediction of the twist which will be much higherthan that observed in a real. situation. The reasonis, when the chirality is the only driving force forhelical morphology, the surface is elastically re-laxed and the net curvature becomes equal to thespontaneous curvature/. In bilayers, due to theelastic properties of the system, the constituentmolecules try to minimise the OM angle. The chi-ral interaction, on the other hand, helps the ag-gregate to shift the potential to a minimal state bydeveloping a twist through the angle ~M' In fact,deGennes has pointed out long ago that the natu-ral twist or the microscopic twist is always smallin the molecular scale":

It has been observed in. some recent experi-ments 17 that the ratios of the pitch and the diame-ter (d) of the helices are higher than the valuepredicted by Helfrich's theory [(Pld) = .7l~l.Fromthe present molecular consideration, we find thatthe ratio of the pitch to the diameter is given bythe following, somewhat modified, expressionP tan~M-=.7l-- ... (12)d OM

The above expression indicates that, when OM issmall, the ratio Pld should be significantly higher

than n. This ratio should also depend upon thevalue of ~M' In general, OM should be small and~M> > OM' Thus, the present theory indicates thatthe ratio Pld can be higher than the value of .7l

and should, in fact, depend upon the sizes of thegroups attached to the chiral center of the specificamphiphile under consideration. Experimentalevidences seem to support this conclusion 17.

VI. Minimum energy conformation for a pair ofmirror-image isomers (racemic modjfication,D-L form) from the effective pair potential

In this Section we consider the effective interac-tion between a pair of mirror image isomers. Itmust be mentioned that when we are consideringthe interaction between the two chiral centers of aD-L pair, in the plane of the a and the b groups,then m is equivalent to a, 0 is equivalent to bgroup, n is equivalent to b group and p is equiva-lent to a group, respectively. The median distancebetween the groups can be easily obtained fromthe median of the tallest isosceles trapezoid thatcan be drawn between C1 m, C20 and that be-tween C1 nand C2p (Fig. 2), respectively. Theother parameters have been explained earlier. Theexplicit expression for the distance and the orien-tation-dependent effective pair potential is thengiven by

~=(~) ~[(~-~Sin(~2-a)kT t: Vkk aa+ ab aa+ ab

)

-12

XCOt(~I- a)+~cos(~2- a)aa+ ab

- (~-~Sin(~2- a)cot(~I- a)aa+ ab aa+ ab

... (13)

544 INDIAN J CHEM, SEe. A, JULY 1996

The parameters used In the equation have beenexplained in Fig. 2.

VII. DiscussionThe pair potential profiles of the pure (L-L or

D-D) and racemic modifications (D-L pair) are de-picted in Figs 7 and 8,. respectively. We have pre-sented here the plots with a = 0° only. However,plots with other a values have the samefeatures 10. It is clear from Figs 7 and 8 that theeffective pair potential profile of the D-D pair(Fig. 7) is strikingly different from that of the D-Lpair (Fig. 8). Unlike the single minimum observedin the case of D-L pair, there are two minima inthe case of D-D pair. One minimum is at( ¢2 - ¢ I ) = 0° and at a small separation, while theother minimum is at (¢z - ¢!) = - 45°. The globalminimum is the latter one in which the groups areoriented at a certain angle and the separation be-tween the chiral centers is much less than that inthe former, thus favouring a more closed packedstate. The reason for the difference in behaviourof D-D and D-L pairs is that as we increase theangle (¢2 - ¢I)' the 0 group is tilted towards the mgroup and the p goes away from the n group (seeFig. 2). The molecules being in a compressed orgel state, should try to pack as efficiently as possi-ble. This can be achieved by minimizing the se-paration between the chiral centers. The lattercan, of course, be achieved without invoking thehindrance between the group.s of the adjacent chi-ral centers. For a D-L pair (racemic modification),as the a group attached to the Cz chiral centerorients more towards the b group of the C1 chiralcenter, the other two a and b groups (on theother side of the X and X' axes) move equallyaway from each other (see Fig. 2). So, the orient-ations of the groups towards each other do notlead the packing arrangement to a more favour-able state and parallel arrangement is thereby fa-voured. From Fig. 7 it is seen that as the value of(¢2 - ¢I) increases, the pair potential becomes in-creasingly unfavourable.

From the present study it is clear that the senseof helix should be predetermined by the effectivepair potential. The reason for this is that the pot-ential and hence the relative tilt between the twochiral molecules depend on the sizes of thegroups attached to the dural center. Thus, it isexpected that a complete knowledge of the abso-lute conformation of the monomer, sizes arrd theeffective intermolecular pair potential, should en-able one to predict the sense of the helical shapedaggregate formed from the chiral monomers inthe closed packed state. The D-12-hydroxystearic

o

~.1

::J -2

-3

-4

Fig. 7 - Effective pair potential profile for a pair of mirror-imageisomers with the variation in (i>2-¢1 as well as the separation be-tween the chiral centers from equation 10.The aand bgroups havediameters 1.5 A and 4.5 A respectively.

Ejk= 150 and E/k= 450./1= 1l0oanda=0°.

o

I- -1<II~::J -2

-3

-4

Fig. 8-Pair potential profile for a pair of same kind of en-antiomers with the variation in ¢2-¢1 as well as the separationbetween the chiral centers from equation 11. The a and bgroups have diameters 1.5 A and 4.5 A respectively. Ejl

k: = 150 and E/k= 450, /1= 110° and a = 0°.

acid gives left-handed helix and the L-12-hydroxy-stearic acid gives right-handed helix". The i-isomer ofthe ammonium amphiphile (2-C 12-L-Glu-CIlN+)gives right-handed helix, whereas the o-isomer of theammonium amphiphile (2-C1Z-D-Glu-CIlN+) givesthe left-handed helix. We have calculated the tiltangles ()and ¢for all the molecules from the effectivepair potential between the molecules following thetheoretical framework' as given in Section ill. The tiltshave been listed in Table 1. In Fig. 4(a-d) and in Fig.6(a-d) we have schematically outlined the steps fol-lowed to determine the senses of the helical aggre-gates of the D-and t-hydroxystearic acids and the D-and i-ammonium amphiphiles. The complete agree-ment between the theoretically predicted senses andthose observed experimentally (Table 2) suggests thatthe effective intermolecular pair potential, at leastpartially, governs the chirality driven helix formation.

NANDI et af.: ORIGIN OF CHIRALITY DRIVEN MORPHOLOGIES OF MONO- & BIL\I\YERS 545

Table 2- The theoretically predicted senses of the amphiphilicaggregate forming helical morphology calculated from the effec-

tive intermolecular pair potential

Amphiphile Theoretically Experimentallypredicted observed sensesense

D-12-hydroxystearic acid Left-handed Left-handed (ref. 4)

L-12-hydroxystearic acid Right-handed Right-handed (ref. 4)

o-ammoniurn amphiphile Left-handed Left-handed (ref. 5)

t-arnmonium amphiphile Right-banded Right-handed (ref. 5)

It has been shown earlier that the present microscopicapproach can successfully predict and help in under-standing the experimentally observed pitch and pitchto width ratio of several helical aggregates. The agree-ment between the theoretical and experimentalsenses observed in the present study further substan-tiates the view that the intermolecular pair potentialbetween a pair of chiral molecules has a major role indictating the morphology of the aggregate.

As indicated in the Introduction, that the amphi-philic molecules have an erect conformation in thehelical state" and approximately have an area permolecule, close to the cross-sectional area of a - CH2group of an alkyl chain 19.In such a closed packed statethe preferred minimum is the one which is at shorterseparation and at a finite twist angle. However, in thepair potential profile ofthe 0-0 pair (Fig. 4) a secondminimum is observed at nearly zero twist angle at re-latively large intermolecular separation. At an elevat-ed temperature, the constituent molecules may thusbe trapped into this second minimum at nearly zeroangle relative to each other. Consequently, the helic-ity would not be observed. Theoretical and experi-mental studies indicate that at high temperatures,where the ordered state of the lipid bilayer isunstable,the system cannot express the chirality even if it ispresent at the molecular level/". The observed mor-phology is thus dependent on the temperature and theconcentration of the amphiphile. Experimental stud-ies on tubule formation also indicate that the aggreg-ate morphology is indeed dependent on the lipid con-centration".

VIII. SummaryIn this work we have presented a scheme to calcul-

ate the effective pair potential between the two chiralamphiphile molecules in order to understand the rel-ative orientation among them. We find that the rela-tive arrangement between a pair of molecules of onekind of enantiomers (0-0 or L-L) favours a twist be-tween them. Fr.om the calculated tilts the sense of thehelix has been predicted. Complete agreement be-tween the theoretically predicted senses and the ex-

perimental result has been observed in all the cases.The theory also explains the experimentally observedcharacteristics of the helical aggregate, such as thepitch and the sense, and is found to be dependent onthe sizes of the groups attached to the chiral centerand also upon the concentration of amphiphile mole-cules. The theory also indicates that for a pair of mir-ror-image isomers (O-Lpair), the minimized effectivepotential favours parallel alignment giving rise to aflat state. The surprising success of the molecular ap-proach strongly indicates that the chirality driven he-lix formation is governed by the subtle stereochemi-cal interactions at the chiral centers, which in turn arecontrolled by the effective pair potential between thegroups attached to the chiral centers of the pair of am-phiphiles.

AcknowledgementIt is a pleasure to thank Prof. Santanu Bhattacharya

for suggesting the problem, for pointing out import-ant references, helpful discussions and also for a criti-cal reading of the manuscript. Authors are thankful toProf. P.Balaram for pointing out some important ref-erences and helpful discussions. Discussions withProf. Manju Bansal are thankfully acknowledged.This work is supported by the financial assistancefrom the Department of Science and Technology.

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