Order-Disorder Conformation Change of Xanthan in 0.01M Aqueous Sodium Chloride:

Dimensional Behavior

WE1 LIU and TAKASHI NORISUYE, Department of Macromolecular Science, Osaka University, Toyonuka,

Osaka 560, Japan

Synopsis

Weight-average molecular weights M,, second virial coefficients, and z-average radii of gyra- tion (S2):/2 were determined by light scattering as a function of temperature T for four sodium salt samples of xanthan in 0.01M aqueous NaCI, in which the polysaccharide undergoes an order-disorder conformation change with increasing T. The data for (S2):/' and 44, at 25 and 80"C, the lowest and highest temperatures studied, confirmed the previous conclusion that the predominant conformation at the former T, i.e., in the ordered state, is a double helix, while that a t the latter T, i.e., in the disordered state, is a dimerized coil expanded by electrostatic repulsions between charged groups of the polymer. As T was increased from 25 to W C , (SZ):/' sigmoidally decreased or increased depending on the dimer's molecular weight. This temperature dependence of (S2):/2 and that determined elsewhere for a high molecular weight sample were found to be described almost quantitatively by a simple dimer model in which the double helix melts from both ends, when the double-helical fraction in the dimer at a given T estimated previously from optical rotation data was used.

INTRODUCTION

Xanthan, an ionic polysaccharide produced by the bacterium Xanthomonas campestris, undergoes an order-disorder conformation change when its aque- ous salt solution of low ionic strength is heated.'-3 Although much experimen- tal work'-12 has been done on this conformation change, no reasonable molecular explanation for it is as yet given. The present paper reports a light-scattering study made in an attempt to clarify the conformation change.

In such an attempt, the most basic is the determination of the ordered and disordered conformations of the polysaccharide. Re~ently, '~ we explored these conformations of sodium salt xanthan (degree of pyruvation of about 1/3) in 0.01M aqueous sodium chloride by light scattering, viscometry, and polarime- try, and drew the following conclusion: the ordered conformation at 25°C is a double helix, as proposed by Holzwarth and Pre~tridge'~. l5 and substantiated later by us16-18 and other group^,^^,^^ at a higher NaCl concentration of O.lM, while the disordered conformation at 80°C is a dimerized extended coil containing a small number of double-helical residues. Thus, the order-dis- order conformation change of Na xanthan in 0.01M aqueous NaCl may be considered the process of partial melting of the double helix.

Biopolymers, Vol. 27, 1641-1654 (1988) 0 1988 John Wiley & Sons, Inc. CCC 0006-3525/88/101641-14$04.00

1642 LIU AND NORISWE

In a more recent study," we found that as T was increased from 25 to 80"C, the intrinsic viscosity [ q ] of Na xanthan in the aqueous salt decreased or increased depending on the dimer's molecular weight, whereas the specific rotation [a] 300 a t a 300-nm wavelength sigmoidally increased for any samples studied. We deemed this striking temperature dependence of [ 171 very infor- mative for the purpose of finding the molecular mechanism of the order-dis- order change, and tried to interpret it in terms of a plausible model. It was found that the observed changes in [ q ] with T are very similar to those in the mean-square radius of gyration ( S 2 ) calculated, as a property analogous to [q], for a simple dimer model in which the double helix melts only from both ends.

This similarity was encouraging, and led us to check directly by light scattering whether the model indeed describes the actual dimensional behav- ior of Na xanthan in 0.01M aqueous NaC1. Thus, in this work, z-average mean-square radii of gyration ( S 2 ) , were determined for this system as functions of weight-average molecular weight M, and T in the range from 25 to 80°C, in which the substantial change in occurs.

EXPERIMENTAL

The previously characterized ~ a m p l e s ' ~ ~ ~ ~ of Na xanthan, X14-2-3, X12-2-3, and X13-3-3, were used for the present light-scattering study. One more Na salt sample X10-4, whose radii of gyration in 0.01M aqueous NaCl had been determined previ~usly'~ a t a few T, was also investigated to complete the curve of (SZ)>1I2 vs T.

Test solutions of these four samples were optically clarified first by filtra- tion into centrifuge tubes through three Millipore filters connected in series (MILLEX-HV, 0.22-0.45-pm pore size) and then by centrifugation in a Sorvall RC2-B centrifuge at about 40000 gravities and 25-30°C for 4 h. A central portion of the supernatant in each centrifuge tube was sucked into a pipette and directly transferred into a light-scattering cell. The tube, pipette, and cell had been rinsed with refluxing acetone for about 10 h.

Intensities of light scattered from xanthan solutions were measured on a Fica 50 light-scattering photometer in an angular range from 22.5 to 150", using vertically polarized incident light of a 436- or 546-nm wavelength. The temperature was successively raised, but in most cases, fresh solutions of different polymer concentrations were used at each T.

Pure benzene at 25°C was used to calibrate the photometer. Its Rayleigh ratio was taken to be 46.5 x l o p 6 and 16.1 x cm-' for 436 and 546 nm, re~pectively,'~ and its depolarization ratio was determined to be 0.41, a t 436 nm and 0.40, a t 546 nm by the method of Rubingh and Y u . ~ ~ Specific refractive index increments a t fixed chemical potentials of diffusible compo- nents for Na xanthan in 0.01M aqueous NaCl a t desired T were obtained by interpolating or extrapolating the previously determined valued3 a t 25, 38,50, 60, and 70°C.

The intensity data obtained were analyzed with the use of Zimm's plot. No correction for optical anisotropy was made; we note that according to Sat0 et a1.,16 anisotropy effects on M,, light-scattering second virial coefficient A,, and (Sz)>t/2 are negligible for xanthan samples with M, higher than lo5.

ORDER-DISORDER CONFORMATION CHANGE

11

9

c I

5 7 E ul

\

e \

a

W s 5

z 3

1 0 0.5 1

sin2(e/2)+ kc

1643

1

01 I I I I I I 1 1 1 I I 1

sin 2 ( e m + kc

0 0.5 1 .o

Fig. 1. Zimm plots for Na xanthan sample X14-2-3 in 0.01M aqueous NaCl at 25°C (a) and 80°C (b). Wavelength of incident light, 546 nm; k, lo2 cm3 g-'.

RESULTS

Molecular Weight and Second Virial Coefficient

Figures l(a and b) illustrate the Zimm plots for Na xanthan sample X14-2-3 in 0.01M aqueous NaCl at 25 and 80°C. Here, K is the optical constant, c the polymer mass concentration, and R , the reduced excess scattering intensity a t scattering angle 8.

Figure 2 shows that no discernible change in M , with T occurs a t least up to 80°C when 0.01M NaCl solutions of samples X14-2-3, X12-2-3, X10-4, and

1644 LIU AND NORISUYE

90

70

"0 40

% z 7

20

0

X 14-2-3

P

X 12-2-3

h

I I I I I I I

10 30 50 70 90 T/OC

Fig. 2. Weight-average molecular weights of indicated xanthan samples in 0.01M aqueous NaCl at different 2'.

~

I 20 40 60 eo

T / O C

0

0

0

0

Fig. 3. Temperature dependence of A, for indicated xanthan samples in 0.01M aqueous NaC1.

ORDER-DISORDER CONFORMATION CHANGE 1645

X13-3-3 are heated. The values of M, are approximately twice the viscosity- average molecular ~ e i g h t s ' ~ ~ ~ ~ or MA7 of the respective samples (40.5 X lo4 for X14-2-3, 18 x lo4 for X12-2-3,ll x lo4 for X-10-4, and 5.3 X lo4 for X13-3-3) in cadoxen [Tris(ethylene diamine) cadmium dihydroxide] at 25"C, a single-coil solvent for Na xanthan.16 Thus, they confirm the previous con~lusion'~ that the dimerized form of Na xanthan in 0.01M aqueous NaCl is maintained up to 80°C.

The temperature dependence of A, for the four samples in the aqueous salt is depicted in Fig. 3. The curves fitting the data points except ones at 80°C rise sigmoidally with increasing T. Interestingly, the substantial rises are almost complete in a range of T roughly from 40 to 70°C in which at infinite dilution was previously found to increase sigmoidally (see Fig. 5 of Ref. 21). Similar correspondence between the increases in A , and specific rotation was recently found by Hacche et a1.l' at a lower NaCl concentration of 0.005M.

The apparently small A , values at 80°C in Fig. 3 can hardly be explained unless, at this temperature, aggregates of dimers are present in 0.01M aqueous NaCl solutions of finite c and become less as c is lowered. Hacche et a1.l' suggested the presence of aggregates at 0.005M on the basis of their M , data, but we did not pursue possible aggregation at finite c and 80°C. Our M, data in Fig. 2 show that in .0.01M aqueous NaC1, only dimers are present at the infinite dilution with which we are concerned in the present work.

Radius of Gyration

Figure 4 illustrates the temperature dependence of (S2)>yz for the four samples, along with that reported previously13 for samples X9-3 ( M , = 96 x lo4) and X10-4. With increasing T, the (Sz)>yz for the three higher-molecular weight samples sigmoidally decrease, while those for the two lower molecular weight samples stay almost constant or increase at T - 50°C. These features are very similar to what was observed for [q] previously.21

The open circles in Fig. 5 show (Sz)>y2 plotted double-logarithmically against M , for Na xanthan in 0.01M aqueous NaCl at 25 and 80°C (see Ref. 13 for the data other than those given in Fig. 4). For comparison, the (Sz)>y2 data of Sat0 et al.l6?" for the double helix in 0.1M aqueous NaCl at 25°C are also shown by solid circles; we note that their data are described accurately by Benoit and Doty's equation25 [see Eq. (S)] for (S ' ) of an unperturbed wormlike chain with a persistence length qh of 120 (*lo) nm and a linear mass density ML, of 1940 ( f 100) nm-', and that the ML, value of 1940 nm-' corresponds to the established pitch h of 0.47 nm16> per main-chain glucose residue of the xanthan double helix. The agreement between the solid and open circles at 25°C may be taken to substantiate that the ordered conformation of Na xanthan in 0.01M aqueous NaCl is essentially the same double helix as that in 0.1M aqueous NaC1, but two remarks should be made.

First, the open circles at 25°C for M , < 2 X lo5 appear appreciably above the solid circles when compared at the same M,. Since in this M, region (S2)>y2 in 0.01M aqueous NaCl is larger for disordered xanthan at 80°C than for ordered xanthan at 25"C, the double helices of the two lowest molecular weight samples ( M , = 10.6 X lo4 and 5.85 X lo4) are likely to be partially

1646 LIU AND NORISUYE

15c

12a

90

60

50

E 45 \

(Y

>N

( Y ~ 40

34

v) V

32

30

20

15

10

XQ-3

I

X 1 2-2-3

X10-4

.-

x 13-3-3

.--%/---

I I I I I L , 20 40 60 80

T / "C

Fig. 4. Temperature dependence of (S2):/' for indicated xanthan samples in 0.01M aqueous NaC1. (6), previous data.13

broken in the aqueous salt at 25"C, as discussed elsewhere21 on the basis of viscosity and optical rotation data.

Second, when analyzed by use of the Benoit-Doty equation, our ( S 2 ) , data for M , > 20 x lo4 in 0.01M aqueous NaCl at 25°C give qh = 125 ( f 20) nm and ML, = 1760 (t- 100) m-'. The former is comparable to or slightly larger than 120 nm in 0.1M aqueous NaC1, and consistent with the intrinisically high rigidity of the xanthan double helix a t infinite ionic strength, i.e., a negligibly small contribution from electrostatic repulsions to qh.27 On the other hand, the latter yields an h of 0.52 ( f 0.3) nm, which is larger than the established value of 0.47 nm; the molar mass per main-chain residue of our Na xanthan is 460 g mol- '. The discrepancy in h results from the fact that for M , > 20 X l o4 our (S2)>tI2 data a t 25°C appear slightly above the solid circles. However, since in this M , region ordered xanthan has a larger statistical radius than does disordered xanthan, the slightly larger pitch is probably immaterial. In fact, the corresponding [ 113 data21 at 25°C were indistinguishable from those of

ORDER-DISORDER CONFORMATION CHANGE 1647

A cv s

4 10 100 400

Fig. 5. Molecular-weight dependence of (S2)i/2 for Na xanthan in 0.01M aqueous NaCl at 25 and 80°C (see Ref. 13 for the data other than those shown in Fig. 4). Solid circles, data of Sato et al.16,18 in 0.1M aqueous NaCl at 25°C.

Sato et al.17 in 0.1M aqueous NaC1, and gave ML, = 1940 ( f 100) nm-' and qh = 100 (k 20) nm, i.e., substantially the same r e ~ u 1 t . s ~ ~ ~ ~ ~ as those of these authors, when analyzed by the theory of Yamakawa et al.28929 for wormlike cylinders, with the diameter assumed to be 2.2 (k 0.2)

The open circles for disordered xanthan at 80°C in Fig. 5 are fitted by a smooth curve whose slope decreases from about 0.7 to 0.6 with increasing M,. The slope in this range is consistent with the previous conclusion21 from [q] that the overall conformation of the disordered xanthan dimer is a coil expanded by electrostatic repulsion of charged groups. This coil should be stiff to some extent, since the cellulosic backbone is semiflexible.za*

DISCUSSION

Model and Assumptions

The previously proposed modelz1 for the xanthan dimer in 0.01M aqueous NaCl at a given T consists of one wormlike double helix whose ends are linked by pairs of wormlike coils separated from paired helical residues (see Fig. 6 of Ref. 21). Namely, the model contains no loop. This was suggested by the fact31 that the melting behavior of short double-stranded DNA ( M , < lo6) can be explained by a model in which no loop formation is allowed. The other assumptions underlying our model are as follows:

(1) Any dimers at a given T have an average double-helical length; no distribution of double-helical length is considered.

(2) The double-helical section can occupy any possible positions on the dimer with equal probability. This is an approximation for the xanthan dimer

1648 LIU AND NORISUYE

whose order-disorder change is not a truly reversible process, as detected by optical rotation', 12, l3 and vis~ometry.".~~

(3) The two pairs of wormlike coils are unperturbed; this approximation was suggested by Tricot's analysis,33 which showed that the molecular weight dependence of [q] for several polyelectrolytes can be described by the theory of Yamakawa et al.28,29 for unperturbed wormlike cylinders.

(4) The intercoil repulsion is significant only near the helix ends. It is taken into account by introducing a parameter +, where 2+ equals the angle between the initial tangent vectors of either pair of coils (at the helix end) separated from paired helical residues (see Ref. 21 for more details); when + is zero, each pair of wormlike coils are connected tangentially to the wormlike double helix.

It was found that the theoretically calculated (S2) of the model dimer21 is relatively insensitive to + in the range from 0 to 64" (the supplementary bond angle for the j3-1,4-~-glucosidic oxygen; see, for example, Ref. 34) unless the molecular weight is too low. As in our previous work,21 we here assume c$ to be 45O, for which the theoretical expression21 of ( S 2 ) is simplified to be

( S 2 ) = 4[ f 2 d 4 + (1 - f ) 2 q , 2 ( 1 2 + 1 3 ) + f ( 1 - f >q34] (1)

with

1 5 1 1 +(:)2[2 - 2 + -(I 45 - e-2{)

where

ORDER-DISORDER CONFORMATION CHANGE 1649

In these equations, f is the weight fraction of double-helical residues in the dimer, M the dimer’s molecular weight, and q, and ML,c the persistence length and linear mass density of each wormlike coil, respectively. In the limits of f = 1 and 0, Eq. (1 ) reduces respectively to the Benoit-Doty equationz5 for unperturbed wormlike chains and the e~pression’~ (with cp = 45’) for a disordered dimer with a negligibly short double-helical section, i.e.,

with

and

L, 3 1 3 4 a 8Lc 1 6 ~ :

( 1 - Ci-4 ( f - 0) (10)

with

Model Parameters

If [a]3m is the sum of contributions from the double-helical and coiled conformations, it is related to f in Eq. ( 1 ) by

Here, [a]!& and [a]&, denote the values of [aim for the intact double helix and the disordered dimer containing virtually no double-helical residues, respectively, and all the [aim refer to infinite dilution. In the analysis given below, we assume Eq. (12) and use the values of f estimated previouslyz1 as a function of M , and T according to the equation. In connection with this assumption, two points should be noted:

(1) Although the change in optical rotation of xanthan arises predomi- nantly from that in the conformation of the side chains,2 there is evidence35 that breaking of the xanthan double helix in water-cadoxen mixtures occurs concomitantly with the change in side-chain conformation, i.e., that in optical rotation.

( 2 ) However, the order-disorder conformation change in the present system may differ from the cadoxen-induced d i s s o ~ i a t i o n ~ ~ , ~ ~ of the double helix, since though irreversible, it cannot be a simple kinetic process toward the complete breaking and dissociation of the helix; when T is increased to a given value, a certain local “equilibrium” should be attained in the xanthan dimer. In this case, some backward reaction is likely to occur in addition to partial breaking and reformation of the double-helical conformation, and may lead some coil residues to a state different in local conformation (for example, in side-chain conformation) from the coiled state. Any effect of such a likely but unknown backward reaction on [a]3m is ignored in Eq. (12) .

1650 LIU AND NORISUYE

Equation (1) contains q h ) ML,h) q,, and ML,, in addition to f as the parameters. Both ML, and ML, may be considered independent of T in a good approximation, while q h and q, generally depend on T. If the theory of Odij k37 or that of Skolnick and F i ~ m a n ~ ~ for the electrostatic persistence length is applied to the double-helical and coiled portions in the dimer, and if the bending force constants39 for these portions are taken as constants independent of T, both q h and q, can be shown to vary almost in proportion to 1/T ( T = absolute temperature).21 Accepting this T dependence of the persistence lengths and the approximation to the linear mass densities, we can obtain q h ) ML,h) q,, and ML,, at any T from the values determined at a single temperature. In this work, the values of q h and ML, a t desired T were evaluated from q h = 125 nm and ML,h = 1760 nm-' at 25°C obtained from the (S2), data in Fig. 5.

The values of q, and ML, , at 80°C were estimated previously13 to be 21 (k 3) nm and 750 ( k 100) nm-', respectively, from limited (S2), data on the assumption that f at the temperature is about zero for any samples. The q, value of 21 nm was close to the persistence length of 18 nm reported for Na carboxymethyl cellulose33 in 0.01M aqueous NaC1, while the ML of 750 nm-' was comparable to the value of 840 nm-' evaluated with the virtual bond length of 0.547 nm for p-1,4-D-gluco~e~~ and the molar mass of 460 per main-chain residue of our Na xanthan. However, the values of f at 80°C actually evaluated according to Eq. (12) were 0.15 for M , above 20 X lo4 and about 0 for M, below 11 X 104.21 To obtain q, and ML, consistent with this evaluation, we therefore reanalyzed the ( S 2 ) , data a t 80°C in Fig. 5, using Eq. (1) with these f values and the ML, and qh values a t 80°C. We found that a q, of 25 nm and an ML,c of 760 nm-', values not different from our previous estimates, lead to the closest agreement between the calculated and experi- mental (S2),. This is shown in Fig. 6, in which the interrupted solid line represents the calculated values.

The dashed line in Fig. 6 represents (S2)ll2 calculated from Eq. (10) with q, = 25 nm and M L , , = 760 nm-'. I t also fits the data points fairly closely, which indicates that theoretical ( S 2 ) for M between 2 X lo5 and lo6 happens to be insensitive to whether the model dimer contains helical residues of 15% or virtually none. This theoretical fact may have something to do with our finding in Fig. 4 that (S2)i/2 of low molecular weight xanthan is almost independent of T and hence of f , or increases with increasing T, i.e., with a decrease in f .

Temperature Dependence of Radius of Gyration

With qh = 125 nm (at 25°C)) ML, h = 1760 nm-', qc = 25 nm (at 8OoC), and ML, , = 760 nm estimated in this work, along with the previously estimated f values, we computed (S2)1/2 as functions of M and T from Eq. (1). The results are compared with the experimental data in Fig. 7. Although the calculated values appear to change with T somewhat more sharply than do the experimental (S2)>y2 values in a range of T from 50 to 65°C) the agreement is generally good. In particular, those for the two lower molecular weight samples X10-4 ( M , = 20.8 x lo4) and X13-3-3 ( M , = 10.6 x lo4) well reproduce the characteristic behavior of Na xanthan that (S2)>s/2 for the

ORDER-DISORDER CONFORMATION CHANGE 1651

M,,,/ 104 Fig. 6. Comparison between the calculated and measured (S2):/2 for disordered xanthan in

0.01M aqueous NaCl at 80°C. Interrupted solid line, Eq. (1) with ML, = 1760 nm-', qh = 106 nm (at 80OC), ML,c = 760 nm-', qe = 25 nm, and f = 0.15 (for M, > 2 X lo5) or 0 (for M, < 1.1 X lo5); dashed line, Eq. (10) with ML,c = 760 nm-' and qc = 25 nm.

former is almost independent of T, while that for the latter increases at T - 50°C with increasing T and levels off. Qualitatively, the increase in (S2)1/2 of the model dimer can be explained by increasing contributions from the distances between the paired coil portions,13 and the subsequent leveling off by the balance between these contributions and the negative contribution from decreasing qc.

In sum, our dimer model describes almost quantitatively the dimensional behavior of Na xanthan in 0.01M aqueous NaCl over the range of tempera- tures from 25 to 80°C. Thus, it may be concluded that the model takes account of an essential part of the changes in global properties that occur when an aqueous salt solution of the double-helical polysaccharide is heated.

CONCLUDING REMARKS

The slightly steeper changes in theoretical (S2)ll2 observed in Fig. 7 may have something to do with our approximation in which xanthan dimers with different f in a given solution are replaced by one with an average f . Consideration of some statistical distribution of f at a given T is desirable, but we do not know how to introduce such a distribution to the present system, which attains no true thermodynamic equilibrium and hence cannot be treated by equilibrium theory.

In this connection, it is interesting to ask why virtually no single chains were detected even at f - 0, i.e., in solutions of samples with M, lower than 11 X lo4 at 80°C. Our tentative answer to this question is that complete separation of dimers in 0.01M aqueous NaCl at the high T requires a long time. A recent attempt4' made by viscometry (as a function of time) in hope

1652 LIU AND NORISUYE

1 4 0 1 Mw=96X104

E c I 70

hl tN v) '7 1, M,=80X104

v Mw =2O.8X1O4

30 -0 0 0 a - - I I 1 I 1

20 40 60 80 T/ O C

Fig. 7. Comparison between the calculated and measured (S2):/' for Na xanthan samples in 0.01M aqueous NaCl. Solid line, Eq. (1) (see the text for the parameter values used).

of detecting dimer separation with use of a high molecular weight sample at 80°C ended up with overwhelmingly observing degradation of the poly- saccharide, and thus in the present work, we did not pursue such a kinetic process, focusing on the dimensional behavior of undegraded xanthan. Further kinetic work on dimer separation seems almost imperative, and is now in progress in our laboratory, using a low molecular weight sample ( M , = 9 x lo4) and a method different from that employed previou~ly.~~ We wish to add here that preliminary M, data on this sample evidenced a very slow sep- aration of xanthan dimers at high T. The results will be reported in a future paper.

Finally, we comment on a very recent paper by Jones et al.,41 who investi- gated the thermally induced order-disorder conformation change of potas- sium salt xanthan in aqueous KBr or salt-free solution by conductivity and optical rotation. Their data analysis based on Manning's polyelectrolyte theory42-44 yielded for ordered xanthan a charge spacing b of about 0.58 nm, which would be expected if xanthan assumed a single-stranded form. Jones et al. concluded from this finding that their conductivity and optical rotation data are incompatible with the double-helical structure of the polysaccharide. However, this conclusion does not always follow from the b value and must be accepted with reservation. The reason is as follows:

Salt-free solutions of Jones et al.41 were dialyzed against deionized water for 48 h. Although the temperature of the solutions in the dialysis is not given

ORDER-DISORDER CONFORMATION CHANGE 1653

in Ref. 41, the double-helical structure of the polysaccharide is likely to be broken in water by strong electrostatic repulsions between charged groups; in order to prepare solutions of double-stranded xanthan, salt-free solutions of purified high molecular weight sample (Na salt) need to be carefully treated below 15°C (preferably below 10°C),16,32 probably because separated single- chain portions in dimers tend to aggregate randomly instead of complete pairing when the solvent condition is changed to that favorable for the double helix. Figure 5 of Ref. 41 shows that with decreasing T, the optical rotation in aqueous KBr levels off at a value larger for a lower salt concentration. This behavior is similar to what was observed for Na samples1,’2’13 disordered once a t high T, indicating that the “ordered form” of Jones et al. at low salt concentrations is largely disordered and differs from the one we consider the ordered form. In this case, the estimated b is likely to be concerned, to a large extent, with single-chain portions in dimers (or in their aggregates), and does not seem to be incompatible with our dimer model. Partial ordering of single-chain portions at higher KBr concentrations may not significantly alter b, since the charge spacing should be insensitive to whether single-chain portions are in the disordered or chirooptically ordered state.

We acknowledge with thanks a Grant-in-aid for Scientific Research from the Ministry of Education, Science and Culture, Japan. W. Liu expresses his appreciation to the National Education Commission, the People’s Republic of China, which allowed him to study at Osaka University. Thanks are extended to a referee for informing us of Ref. 41.

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1654 LIU AND NORISUYE

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chap. 2.

185-191.

Received February 22, 1988 Accepted April 29, 1988