Thermodynamics of conformational transition and chain association of ι-carrageenan in aqueous solution: Calorimetric and chirooptical data

Thermodynamics of theConformational Transition ofBiopolyelectrolytes: The Caseof Specific Affinity ofCounterions

Sergio Paoletti1

Julio C. Benegas2

Sergio Pantano2

Amedeo Vetere1

1 Dipartimento di Biochimica,Biofisica e Chimica delle

Macromolecole,Universita di Trieste,

via L. Giorgieri 1,I-34127 Trieste, Italy

2 Departamento de Fisica—IMASL,

Universidad Nacionalde San Luis,

5700 San Luis, Argentina

Received 20 October 1998;accepted 15 June 1999

Abstract: A formal development of theCounterionCondensation theory (CC) of linear polyelec-trolytes has been performed to include specific (chemical) affinity of condensed counterions, forpolyelectrolyte charge density values larger than the critical value of condensation. It has beenconventionally assumed that each condensed counterion exhibits an affinity free-energy differencefor the polymer, (DGaff). Moreover, the model assumes that the enthalpic and entropic contributionsto DGaff, i.e., DHaff and DSaff, are both independent of temperature, ionic strength and polymerconcentration. Equations have been derived relative to the case of the thermally induced, ionicstrength dependent, conformational transition of a biopolyelectrolyte between two conformationsfor which chemical affinity is supposed to take place. The experimental data of the intramolecularconformational transition of the ionic polysaccharidek-carrageenan in dimethylsulfoxide (DMSO)have been successfully compared with the theoretical predictions. This novel approach provides theenthalpic and entropic affinity values for both conformations, together with the correspondingthermodynamic functions of nonpolyelectrolytic origin pertaining to the biopolymer backbonechange per se, i.e.,DHn.pol andDSn.pol, according to a treatment previously shown to be successfulfor lower values of the biopolyelectrolyte linear charge density. The ratio ofDHn.pol to DSn.pol wasfound to be remarkably constant independent of the value of the dielectric constant of the solvent,from formamide to water to DMSO, pointing to the identity of the underlying conformationalprocess. © 1999 John Wiley & Sons, Inc. Biopoly 50: 705–719, 1999

Keywords: conformational transition; enthalpy and entropy of transition; polyelectrolytes; spe-cific counterion affinity;k-carrageenan; polysaccharides, ionic

Correspondence to:S. Paoletti; email: [email protected]

Contract grant sponsor: MURST and University of TriesteBiopolymers, Vol. 50, 705–719 (1999)© 1999 John Wiley & Sons, Inc. CCC 0006-3525/99/070705-15

705

INTRODUCTION

Specific interactions between ionic biopolymers andcounterions, either metal ions or organic ions, mayoften be of the utmost importance to determine therelative stability of different conformations of abiopolymer, e.g., the ordered form (“helix”) or thedisordered one (“coil”). The free energy related tocounterion specificity is known to be crucial for themodulation of the corresponding conformational tran-sition, over and above the energetic and entropicconformational terms—“intrinsic” to the biopolymerbackbone—and the long-range “unspecific” terms ofpurely electrostatic origin. The problem of a detaileddescription of the counterion–polyion interactions hasbeen central in the polyelectrolyte literature for de-cades.1–5 The most widely diffused theories includedthe mean-field approaches of the Poisson–Boltzmannequation (PB)6–8and of theCounterionCondensationmodel (CC).9–13 Developments and applications ofboth theories,14–20 as well as their detailed compari-son,21–24 have been performed up to today, mostlypertaining to thermodynamic aspects. Among suchapplications, it is worth mentioning the cases of theconformational transitions of biopolymers16,25–27andthose of binding of ionic ligands to polyelectro-lytes.20,24,28,29However, in more recent times otherand more sophisticated approaches have proved to bemore accurate also regarding the dynamic aspects andthe fine details of the distribution of counterionsaround a charged macromolecule. Those newer meth-ods mainly use the discrete charge model28,30 orMonte Carlo treatments.31–33

Our interest has been mainly focused onto thethermodynamics of conformational transition(s) ofcharged biopolymers.14,25,34,35Albeit crude, the CCtheory was shown to be an effective tool in resolv-ing some controversial cases,25 and to lend itself tomodifications to take into account an importantfeature such as the biopolymer chain flexibili-ty.35–37 In previous works, a new method for theevaluation of the enthalpic and of the entropicterms of nonpolyelectrolytic origin accompanyingthe conformational transition has been developedand successfully applied both to aqueous38 and tononaqueous39 systems. The main underlying hy-pothesis is that for all the values of transition mid-point the free energy difference ofnonpolyelectro-lytic origin should be equal in value and opposite insign to the free energy difference of polyelectro-lytic origin. The latter value can be computed withreasonable accuracy by means of the CC theory,thereby affording a simple means of calculating theformer one. Both papers, however, were limited to

cases in which the dimensionless linear charge den-sity parameter of the CC theoryj for both confor-mations was lower than the critical value of 1, i.e.,in conditions of no counterion condensation nor, afortiori, of chemical (specific) affinity. The presenteffort is aimed at extending the previous formalismbased on the CC approach to include in a simpleway the proper state functions related to chemicalaffinity.

This paper is organized in the following way. Inthe first part, the complete derivation of the thermo-dynamic functions of polyelectrolytic origin will begiven for a transition between two conformations,both corresponding to the conditionj . 1. Next to it,the introduction of the enthalpic and of the entropicterm of counterion (specific) affinity will be made intothe general thermodynamic scheme of the CC theory.Experimental data are available for the intramolecularconformational transition of the ionic polysaccharidek-carrageenan in dimethylsulfoxide (DMSO).40 Sucha conformational transition is the same as that suc-cessfully analyzed in the preceding papers.38,39 InDMSO both the single-strandeddisorderedconforma-tion and the single-helicalorderedone share the con-dition of j . 1. At last, the application of the aboveformalisms to the real case of the transition ofk-car-rageenan in DMSO will turn out to be quite success-ful.

THEORY

To avoid a tedious repetition of the many terms usedin the CC theory, we resorted to collecting them all inAppendix 1, which the reader is referred to, as well asto the papers2,10–12,38,39that introduced and discussedthose formalisms.

The Evaluation of the ThermodynamicFunctions of the PolyelectrolyticContribution to ConformationalTransition

The content of this section is parallel to that containedin Ref. 38, under the same heading, the main differ-ence being related to the case ofj . 1 considered inthe present one.

The Gibbs free energy difference of polyelectro-lytic nature between the initial and the final confor-mational states is given by

~DGpol!tr 5 ~Gpol!f2(Gpol)i (1)

706 Paoletti et al.

In the case ofj . 1, in general,

Gpol 5 Gel 1 Gmix (2)

It holds

&el 5 2ne5Tj~zeff!2ln~1 2 e2kb!

5 2ne5Tj@zstr~1 2 r !#2ln~1 2 e2kb!

5 2ne5Tj~1 2 r !2ln~1 2 e2kb!

(3a)

and for the molar free energy,

Gel 5 &el/ne 5 25Tj~1 2 r !2ln~1 2 e2kb! (3b)

and for the reduced molar free energy:

;el 5 Gel/5T 5 2j~1 2 r !2ln~1 2 e2kb! (3c)

In the so-called chemical version of the CC theory ofG. S. Manning,11,12 an additional term ofpurely en-tropic nature, i.e., the “mixing” term, is added toaccount for the consequences of the “creation” of thevolume of the “condensed phase” on the variouschemical components of the system.

[It may be useful to recall that in the case ofj , 1there is no condensation; therefore,Gmix 5 0 and

Gpol 5 Gel 5 25Tjln~1 2 e2kb! (4)

which coincides with Equation (14) of Ref. 38]Gmix is made up by the following terms, each of

which stems from the declared component:

● the condensed counterions

&cond5 5Tne ln$@~nc1!/9#

/@~~~ns!1 1 ncp!1!/V#% (5a)

5 5Tner ln$r /@~Rs 1 1!~VpCp!#%

and for the molar free energy:

Gcond5 &cond/ne 5 5Tr

3 ln$r /@~Rs 1 1!~VpCp!#%(5b)


;cond5 Gcond/5T 5 r ln$r /@~Rs 1 1!~VpCp!#%

(5c)

Vp is the molar volume of the condensed phase;

● all the othercounterions(“ free”)

&free 5 5T@~ns!1 1 ~ncp 2 nc!#

3 ln$@~~~ns!1 1 ~ncp 2 nc!!1!

/~V 2 9!]/ @~~~ns!1 1 ncp!1!/V#} (6a)

5 5Tne@Rs 1 ~1 2 r !#ln$@Rs 1 ~1 2 r !#

/@~Rs 1 1!~1 2 VpCp!#}


Gfree 5 &free/ne 5 5T@Rs 1 ~1 2 r !#

3 ln$@Rs 1 ~1 2 r!#/@~Rs 1 1!~1 2 VpCp!#%(6b)


;free 5 Gfree/5T 5 @Rs 1 ~1 2 r !#

3 ln$@Rs 1 ~1 2 r!#/@~Rs 1 1!~1 2 VpCp!#%(6c)

● the anions (similions)

&sim 5 5T~ns!2ln$@~~ns!

21!

/~V 2 9!]/ @~~ns!21!/V#} (7a)

5 5TneRs ln$1/~1 2 VpCp!%


Gsim 5 &sim/ne 5 5TRs ln$1/~1 2 VpCp!% (7b)


;sim 5 Gsim/5T 5 Rs ln$1/~1 2 VpCp!% (7c)

● the solvent

&solv 5 5Tnc 5 5Tner (8a)


Gsolv 5 &solv/ne 5 5Tr (8b)

Transition of Biopolyelectrolytes 707


;solv 5 Gsolv/5T 5 r (8c)

The established procedure for the determination of thevalue of r requires minimization of all the terms ofEqs. (3c), (5c), (6c), (7c), and (8c), finding, in thelimit of infinite dilution of the polymer,

r 5 1 2 ~j21! (9)

The value ofVp corresponding to such a minimumcondition is also obtained at the end of the minimi-zation procedure. It can conveniently be expressed bythe dimensionless product:

~VpCp! 5 A/~A 1 1! (10)

where

A 5 @1 2 e2kb#2~1 2 j21!@Rs 1 j21#21

3 exp$1 1 @kb/~2j~ekb 2 1!~2Rs 1 j21!!#%(11)

In the often met condition of (VpCp) ! 1, Eq. (10)will reduce to (VpCp) 5 A, and it will then beidentical to Equation (11) of Ref. (14). All detailshave been reported elsewhere.11,12,14 The value of(DGpol)tr is then given by

~DGpol!tr 5 $~Gel!f 2 ~Gel! i% 1 $~Gcond!f 2 ~Gcond! i%

1 $~Gfree!f 2 ~Gfree! i% 1 $~Gsim!f 2 ~Gsim! i%

1 $~Gsolv!f 2 ~Gsolv! i%

(12)

in which each term has been defined above and withthe condition of Eq. (9).

The variablesr , j, Vp, k, andb will assume thesuffix “f” and “i” for the final and for the initialcondition, respectively.

The expression of Eq. (12) is completely general,inasmuch as it depends on the ionic strength and onthe temperature asseparately independentvariables,i.e., (DGpol)tr(I , T). For all transition midpoints, itwill be modified into (DGpol)tr(Im, Tm) and

kf 5 k i 5 km 5 $2lIm%1/2 (13)

The expressions of (DHpol)tr(I , T), and of (DSpol)tr(I ,T), are obtained using the pertaining expressions ofHpol and ofSpol, which in turn are obtained by takingthe proper derivatives ofGpol. The detailed expres-sions are reported in Appendix 2.

By using Eqs. (A1) and (A6) from Appendix 2 [thelatter equation in the form of2T(DSpol)tr], the en-thalpic and the entropic contributions to the respectivethermodynamic function of transition have been cal-culated at 298.15 K as a function of the ionic strengthI . The case considered is that of the intramoleculartransition ofk-carrageenan in DMSO40 for which bdis

5 10.3 Å andbord 5 8.2 Å, corresponding, at 298.15K, to jdis 5 1.162 andjord 5 1.459, respectively. Thevalues of (DGpol)tr have been calculated too. The(absolute) temperature dependence ofD has beentaken into account via Eq. (14)41:

D~T! 5 38.4761 0.16939T 2 0.4742T2 (14)

The results are graphically depicted in Figure 1. Sim-ilar to the findings forj , 1 of Ref. 38, an increase ofI swamps out the polyelectrolytic effect.

EXPERIMENTAL

Materials

The temperature of conformational transition midpoint of abiopolyelectrolyteTm is known to depend on the ionicstrength.42,43 For several biopolymers, linear plots (“Con-formational phase-diagrams”) have been found for the pairs[log(Im), (Tm)21], derived from the analysis of the exper-imental transition curves.Im is the value of the ionicstrength for which the temperature of transition midpoint isTm. The phase-diagram data of line A of Figure 2 of Ref. 40

FIGURE 1 Dependence on the logarithm of the ionicstrengthI of the polyelectrolytic thermodynamic functionsfor the conformational transition ofk-carrageenan inDMSO (T 5 298.15 K). (a)DHpol, (b) 2TDSpol, and (c)DGpol.

708 Paoletti et al.

were used in the present treatment. To obtain the value ofIm

at Tm from the corresponding value of the polymer concen-tration Cp, use of the following position was made:

Im~Tm! 5 ~1

2!Cp@~jhelix~Tm!!21 1 ~jcoil~Tm!!21# (15)

A linear regression of the log(Im) vs (1/Tm) plot has giventhe following parameters: intercept5 11.678; slope5 24633 K.

Fitting Procedures

The following six thermodynamic functions had to be cal-culated from the fitting procedures:DHn.pol and DSn.pol

(pertaining to the nonpolyelectrolytic component of confor-mational transition), (D*aff)coil and (D6aff)coil, (D*aff)helix,and (D6aff)helix (the four latter terms being the specificaffinity ones: see the next section), respectively.

The general calculation strategy used the following con-dition for all given values of (Im, Tm) [see later Discussionleading to Eq. (23)]:

@DHn.pol 2 TmDSn.pol# 1 r helix~Tm!@~D*aff!helix

2 Tm~D6aff!helix] 2 r coil~Tm!@~D*aff!coil (16)

2 Tm~D6aff!coil] 1 ~DGpol!~Im, Tm! 5 0

All symbols are defined in Appendix 1 and discussed in thefollowing section.

Two different approaches were followed.In the first one (method I), a system of six simultaneous

equations [of the general form of Eq. (16)] in the sixunknowns was solved for six different values of the pair(Im, Tm). The latter values were chosen to encompass thewhole (Im, Tm) range. The procedure was repeated fortwelve different sets of the six values of (Im, Tm), therebyaltogether analyzing seventy-two different data points. Sys-tems were solved using theFind subroutine of the Mathcad5.0 software program (MathSoft Inc., Cambridge, MA,USA). Convergence was reached without difficulty, inde-pendent of the initial guess of the values of the parameters.The only constraint used was

DHn.pol 1 r helix~D*aff!helix 2 r coil~D*aff!coil

, 26500 J mol21(17)

according to the experimental observation that the enthalpychange of transition is268806 300 J mol21.40

In the second approach (method II), a nonlinear regres-sion procedure was used to evaluate the six unknown pa-rameters. One hundred data pairs (Im, Tm), to be used aspredictors, were obtained using the coefficients of the linearregression of the log(Im) vs (1/Tm) plot (see above). TheMarquardt–Levenberg algorithm was used as a curve fitterto find the coefficients (parameters) of the independent

variable(s) giving the best fit between Eq. (20) and the (Im,Tm) data. (SigmaPlot software program, Jandel Corp., SanRafael, CA, USA). Also, in method II the above conditionof inequality—i.e., Eq. (17)—was applied. Convergencewas attained with no difficulty, and the values of the pa-rameters did not depend on those of their initial guesses.

RESULTS AND DISCUSSION

Evaluation of the Specific Affinity Term

The method proposed in Ref. (38) for the determina-tion of DHn.pol and ofDSn.pol is based on the assump-tion that, under the hypothesis that both the enthalpicand the entropic contributions of nonpolyelectrolyticorigin are independent of temperature and of ionicstrength, then atT 5 Tm

DGm 5 ~DGpol!~Im, Tm! 1 DGn.pol~Tm! 5 0 (18)

and then

~DGpol!~Im, Tm! 5 2DGn.pol~Tm! (19)

For all sets of pairs (Im, Tm) linear relationships wereshown to hold between the theoretical values of(DGpol)(Im, Tm) and Tm, or between [(DGpol)(Im,Tm)/Tm] and (Tm)21, respectively.38,39 For thepresent case, such plots are not linear (not reported).

In principle, the failure might be traced back to thelack of validity of the hypothesis underlying Eq. (19).However, given the encouraging results so far ob-tained,38,39it might very well be that additional termsare missing in the expression of the total free energydifference of transition.

The most relevant of them can be identified with afree energy difference of the (specific) affinity of thepolyion for the (condensed) counterionDGaff, as al-ready indicated.44 Such a working hypothesis is basedalso on the preliminary results45 obtained by explic-itly taking into account such a term.

The next step is based on the following assump-tions:

● For each condensed counterion, a free energydifference sets up between the reference stateand the condensed state.

D&aff 5 nc~D&8!aff 5 ner ~D&8!aff (20)

where (D&°)aff is the affinity free energy differenceper mol of condensed counterion.It follows that


DGaff 5 nc~D&8!aff/ne 5 r ~D&8!aff (21a)

● It is assumed that both the enthalpic and theentropic components of (D&°)aff, i.e., (D*°)aff

and (D6°)aff, are independent of temperature,ionic strength (namely, concentration ofall ionicspecies) and polymer concentration.

Moreover,

DHaff 5 nc~D*8!aff/ne 5 r ~D*8!aff (21b)

DSaff 5 nc~D68!aff/ne 5 r ~D68!aff (21c)

The total free energy of the system (per mole of fixedcharge) will then be given by the sum of the terms:

Gtot~I , T! 5 Gpol~I , T! 1 Gn.pol~T! 1 Gaff~T! (22)

and for the conformational transition:

DGtot~I , T! 5 DGpol 1 DGn.pol 1 DGaff 5 DGpol

1 DGn.pol 1 r f ~~D&8!aff!f 2 r i~~D&8!aff! i

5 DGpol~I , T! 1 @DHn.pol 2 TDSn.pol# (23)

1 $r f @~~D*8!aff!f 2 T~~D68!aff!f#%

2 $r i@~~D*8!aff! i 2 T~~D68!aff! i#%

[For mere simplicity, from this point onward the sym-bol “°” will be dropped from (D&°)aff, from (D*°)aff,and from (D6°)aff, without affecting their correct def-inition of standard thermodynamic entities.]

The first consequence of the introduction of theaffinity term is that the sum of all free-energy termscontaining the fraction “r ” of condensed counterionsmust include also that expressed by Eq. (18). Asalready demonstrated by G. S. Manning,42 this willNOT affect the limiting condensation condition, asgiven by Eq. (9), but will just add an additional termto Vp (and hence toVpCp):

@VpCp#aff 5 A9/~A9 1 1! (24a)

where

A9 5 A exp@~D*aff 2 TD6aff!/5T#

5 @1 2 e2kb#2~1 2 j21!@Rs 1 j21#21 (24b)

3 exp$1 1 @kb/~2j~ekb 2 1!~2Rs 1 j21!!#

1 ~D*aff 2 TD6aff!/5T}

However, the main bearing of the introduction of theaffinity term for our present purposes is that the va-lidity of Eq. (16) does not hold any longer. It has to besubstituted by

~DGpol!~Im, Tm! 1 r f~D&aff!f ~Tm!

2 r i~D&aff! i~Tm! 5 2DGn.pol~Tm!(25)

or, identically,

~DGpol!~Im, Tm! 1 r f~D&aff!f~Tm!

2 r i~D&aff! i~Tm! 1 DGn.pol~Tm! 5 0(26)

At this point, the possibility of using the dependenceof the left-hand side of Eq. (25) vsT (or of theleft-hand side divided byT vs T21) is vanished fortwo reasons:

1. There is no a priori analytical expression for thethermodynamic values of affinity (namely,D*aff andD6aff).

2. (DGpol)(Im, Tm) contains explicitly the term(VpCp)aff, which is dependent on the affinityterms as well.

Following the two different procedures outlined inExperimental, the values of the six unknowns [i.e.,DHn.pol, DSn.pol, (D*aff)helix, (D6aff)helix, (D*aff)coil,(D6aff)coil] were determined. The two sets of values arereported in column 2 of Table I for method I and incolumn 3 for method II, respectively. In the case ofmethod I, each value is given with the standard deviationof the mean of the twelve values for each unknown,whereas in the case of method II the standard error ofeach parameter is indicated as provided by the regressionroutine. The ratio (DH)n.pol/(DS)n.pol has also been indi-cated (see later). The following remarks can be made byinspection of Table I:

1. Both methods are successful in the parametricdescription of the system.

2. For each parameter, the values provided by thetwo approaches are extremely close, or evencoincident within the given accuracy.

3. The values of the parameters pertaining to thespecific affinity are the least accurate in bothmethods. However, in spite of a non-negligi-ble uncertainty on their absolute values, stillthe difference between the correspondingthermodynamic parameters between the “he-lix” and the “coil” forms are the same in bothmethods.

710 Paoletti et al.

4. Plotting the SUM of the polyelectrolytic freeenergy term and of the affinity one as a functionof T, or their sum divided byT vs T21, respec-tively, perfectly straight lines can be obtained,for both methods (not reported). Self-consis-tently, the linear regression coefficients, fromeither way of plotting, perfectlycoincidewiththe values ofDHn.pol andDSn.pol of Table I, asexpected from the model and already found inthe absence of specific counterion–polyion in-teractions.38,39

Method I is conceptually superior to method II inasmuchas it gives the exact (numerical) solution to any systemof six equations in the six unknowns. The uncertainty inthe values of the parameters stems only from the inac-curacy of the sampling procedure of the six pairs of datapoints—for each of the twelve sampled groups—fromthe [log(Im) vs (Tm)21] line, which in turn derives froma regression procedure. However, method I results to besuperior also as to the ability to reproduce the experi-mental data. In fact, from the last three lines of Table Iit can be seen that for the sample cases of the lowestvalue ofIm, the intermediate one and the highest one itgives a very good agreement between the calculated andthe experimental value ofTm, well within the experi-mental accuracy (about60.5 K) and much better than

that given by method II. Given such a superiority, onlythe values of the parameters given by method I will beconsidered further.

It can be of interest to analyze the relative contribu-tions of the thermodynamic functions of the specificaffinity interaction between the counterion and the dif-ferent polyion conformations. Table I reveals that theoverall contribution of counterion affinity in the courseof the conformational transition from the disordered(“coil”) to the ordered (“helix”) conformation is ener-getically unfavorable. It should be reminded that, onpassing from the “disordered” to the “ordered” form, thefraction of condensed counterionsr varies because of avariation of j. To separately assess the difference inaffinity between the “helix” (h) and the “coil” (c) form,it can be convenient to write for any thermodynamicfunction expressed per mole of repeating unit,Y(Y5 G,H, or S), as a function of the corresponding state func-tions expressed per mole of condensed counterion=(=5 &, *, or 6):

DYaff~T! 5 r h~T!~D=aff!helix 2 r c~T!~D=aff!coil

(27a)

r h~T!~D=aff!helix 2 r c~T!~D=aff!coil(27b)

5 r h~T!~D=aff!helix 2 r c~T!~D=aff!coil 1 r c~T!

Table I Parameters of the Conformational Transition of k-Carrageenan in DMSO

Property Method I Method II

(Dcoil3helixH)n.pol,J (mol repating unit)21

27903 6 103 27867 6 24

(Dcoil3helixS)n.pol,J (mol repating unit)21 K21

221.816 0.36 221.686 0.08

(DH)n.pol/(DS)n.pol, K 362.4 362.9[(D*)aff]coil,

kJ (mol condensed ion)215.256 1.04 5.4 6 4.5

[(D6)aff]coil,J (mol condensed ion)21 K21

27.5 6 4.1 23.2 6 15.0

[(D*)aff]helix,kJ (mol condensed ion)21

5.286 0.86 5.4 6 4.6

[(D6)aff]helix,J (mol condensed ion)21 K21

21.7 6 4.5 2.6 6 15.0

[(D*)aff]helix 2 [(D*)aff]coil,kJ (mol condensed ion)21

0.0 0.0

[(D6)aff]helix 2 [(D6)aff]coil,J (mol condensed ion)21 K21

5.8 5.8

u(Tm)calc 2 (Tm)expu/Kfor Im 5 0.0004M

0.05 0.10


0.09 0.19


0.21 1.9


~D=aff!helix 2 r c~T!~D=aff!helix

r h~T!~D=aff!helix 2 r c~T!~D=aff!coil

(27c)5 r c~T!@~D=aff!helix 2 ~D=aff!coil#

1 @r h~T! 2 r c~T!#~D=aff!helix

The first term at the right-hand side of Eq. (27c) is adirect measure of the difference of the thermodynamicfunction of affinity= to which the fraction ofalreadycondensed counterions (previously in the disorderedform) is subjected upon ordering. The second termtakes into account the component of function- due tothe “additional” amount of counterions that need tofurther condense in the ordered form, which has ahigher charge density. The values for changes of theGibbs free energy& for the enthalpy*, and for theentropy (as2TD6) for the particular case ofT5 298.15 K arereported in Table II.

The disorder-to-order conformational transition isa thermodynamically favored process for the ions thatare already condensed onto the polymer in the “coil”form (see the first row of Table II): in particular, it isthe entropy change that favors such a process. Thiscan be explained assuming that some molecules ofsolvation of the counterion (and possibly also fromthe polymer binding site) are released upon chainordering producing a positive change ofD6. How-ever, the additional fraction of counterions, i.e., [rh

2 rc], does not experience such a favourable entropychange (see the second row of Table II). The netbalance produces a positive change of thetotal freeenergy due to an overwhelming contribution of theenthalpic component over a favorable entropic one(see the last row of Table II). One can conclude thatfor the K1 form of k-carrageenan in DMSO specificcounterion/polyion affinity interactions per sedo notfavor the ordered conformation with respect to thedisordered one.

Evaluation of the Enthalpy ChangesAccompanying the ConformationalTransition

The various enthalpic contributions have been calcu-lated using the parameters determined by method I(Table I) and the expression of Eq. (27a) for theenthalpy change of affinity. The total enthalpy changeof transition has been evaluated according to the equa-tion:

~DHcalc!tot~I , T! 5 DHn.pol 1 DHpol~I , T! 1 DHaff~T!

(28)

The temperature dependence of the results forCp 5 6g L21 (corresponding toIm 5 0.011M, for which Tm

is 339.72 K) have been graphically depicted in Figure2A and reported in Table IIIa for the particular case ofT 5 Tm. It can be seen that the temperature depen-dence of bothDHaff [through ji ,f(T)] and DHpol isvery small. Moreover, the average absolute value ofthe former is about one order of magnitude smallerthan that ofDHn.pol, and that of the latter is about twoorders of magnitude smaller. As a consequence, theaverage value of (DHcalc)tot is practically independentof T and largely (i.e.,; 90%) determined by the valueof DHn.pol. This point will be addressed further in theDiscussion.

In a similar way, the entropic contributions havebeen calculated using the parameters determined bymethod I (Table I) and the expression of Eq. (27a) forthe entropy change of affinity. The total entropychange of transition has been evaluated according to

~DScalc!tot~I , T! 5 DSn.pol 1 DSpol~I , T! 1 DSaff~T!

(29)

The temperature dependence of the results still forCp

5 6 g L21 have been graphically depicted in Figure2B, asTDS values.

Table II Thermodynamic Parameters of Affinity at T 5 298.15 K

(D=aff)

(Dcoil3helix*)aff,J (mol repeating unit)21

2T(Dcoil3helix6)aff,J (mol repeating unit)21

(Dcoil3helix&)aff,J (mol repeating unit)21

rc[(D=aff)helix 2 (D=aff)coil] 4 2241 2237[ rh 2 rc](D=aff)helix 927 89 1016Total:rh(D=aff)helix 2 rc(D=aff)coil 931 2152 779

712 Paoletti et al.

For a better assessment of the results, it is conve-nient to focus on the particular value ofT 5 Tm,where

DGtruT5Tm5 DHtruT5Tm

2 TmDStruT5Tm5 0 (30)

and therefore,

~DHcalc!totuT5Tm5 TmDSn.pol

1 TmDSpoluT5Tm1 TmDSaffuT5Tm

(31)

The entropic components of the right-hand side of Eq.(31), as well as their sum have been reported in TableIIIb. The nonpolyelectrolytic contribution still largely

prevails, determining the sign and, largely, the abso-lute value of (DHcalc)totuT5Tm

; however, the polyelec-trolytic entropic term is one full order of magnitudelarger than the corresponding enthalpic one, stillkeeping the same sign. Also the entropic affinity termkeeps the same sign of the corresponding enthalpicone, but it is roughly reduced to a little less than onefifth of it. No surprise should come from noticing theperfect matching between (DHcalc)tot calculated fromEq. (28) (fourth column of Table IIIa) and the corre-sponding value obtained from Eq. (31) (fourth columnof Table IIIb), inasmuch as they derive (although in anot very simple way) from the same starting func-tions. Anyhow, the identity of the results obtained viacompletely independent calculation strategies shouldbe a convincing evidence at least of the internal self-consistency of the approach.

As to free energies, one should consider the SUMof the polyelectrolytic term and of the affinity one asan overall measure of the contribution of the long-range electrostatic interactions to the transition, thusreconciling the apparent inconsistency that the poly-electrolytic term is negative, contrary to expectations.It can be easily demonstrated that the latter resultsstems from the important role played by the term[VpCp] aff, which in turn is strongly influenced by the(D*aff) and (D6aff) terms. However, a detailed dis-cussion of this point will be given elsewhere.

Table III Calculated Contributions to theThermodynamics of Transition of k-Carrageenanin DMSO for Cp 5 6 g L21 at Tm 5 339.72 K[Values Expressed in J (mol repeating unit)21]

(a) From Eq. (28)

DHn.pol DHpol DHaff (DHcalc)tot

27903 19 935 26949

(b) From Eq. (31)

TDSn.pol TDSpol TDSaff (DHcalc)tot

27409 295 165 26949

(c) Difference Between the Corresponding Termsof (a) and (b)

DGn.pol DGpol DGaff (DG)tot

2494 2276 770 0

FIGURE 2 Dependence on temperature of the differentcalculated contributions to the thermodynamics of transitionof k-carrageenan in DMSO (Cp 5 6 g L21): A: (a) DHaff,(b) DHpol, (c) DHtot, and (d)DHn.pol. B: (a) TDSaff, (b)TDSpol, (c) TDStot, and (d)TDSn.pol.


Comparison of Theoretical Predictionswith Experiment

The experimental values of (DHexp)tot, for the ther-mally induced order–disorder conformational transi-tion of the K1 salt form ofk-carrageenan in DMSOare 6.76 0.6, 6.96 0.3, and 6.76 0.4 kJ mol21, forCp 5 3, 6, and 12 g L21, respectively.40 Thosevalues were determined by differential scanning cal-orimetry in a scanning cycle starting from 293.15 K(fully ordered conformation). The corresponding cal-culated values (DHcalc)tot are 26970, 26942, and26922 J mol21, for T 5 293.15 K. Forsake ofcuriosity, for T 5 323.15 K (DHcalc)tot are26962,26964, and26965 J mol21, respectively.

The agreement between the calculated and the ex-perimental data is really excellent and certainly wellwithin the experimental accuracy. This result is veryencouraging and it adds to the very good agreementalready demonstrated between the experimental val-ues of (Tm) and the calculated ones (see Table I,bottom lines).

Evaluation of the Enthalpy Change ofConformational Transition from theConformational Phase-Diagram

Possibly the most usual method to evaluate theoreti-cally the enthalpy change of conformational transitionof charged biopolymers,DHm(DHm 5 Hf 2 Hi), isbased on the analysis of the so-called conformationalphase-diagram (see Experimental).

Under the hypotheses that

1. both the enthalpy (DHn.pol) and the entropy(DSn.pol) changes ofnonpolyelectrolytic nature,“intrinsic” to the biopolymer backbone and es-sentially related to short-range interactions,donot depend neither on temperaturenor on ionicstrength;

2. the ionic strength dependent part of the enthalpychange of polyelectrolytic origin,DHpol(I , T),is much smaller than (DHn.pol);

3. the small temperature dependence ofj can beneglected;then theoretical derivations based on polyelec-trolyte theory indicate that a line is expected toconnect all such pairs. The following equationwas repeatedly demonstrated to hold25,42,43

~DHm!phase diag./@2d log~Im!/d~Tm!21#

5 ~12!2.3035@~jf!

21 2 ~j i!21#

(32)

for the process of “melting” (m) the ordered initial (i )conformation to give the disordered final one (f ).

The right-hand side term of Eq. (15) is a constantfor the given conformations; if the plot is linear, [2dlog(Im)/d(Tm)21] is a constant as well, and then(DHm)phase diag.should also be constant. In fact, in theabsence of specific affinity interactions, it holds that

~DHm!phase diag.5 DHn.pol 1 DHpol~I , T! (33a)

if the log(Im), (Tm)21 plot is linear, then the ionicstrength-dependent component ofDHpol(I , T) has tobe negligible with respect toDHn.pol. and it will be

~DHm!phase diag.> DHn.pol 1 ^DHpol&

> constant> ~DHexp!tot

(33b)

where^DHpol& is the average of the values of poly-electrolytic component of the transition enthalpychange, for the given range ofI andT, and (DHexp)tot

is the experimental value of the enthalpy change oftransition. For the case:ji , 1, jf , 1, such acomparison has repeatedly given excellent re-sults.38,39

It was independently shown,38,39at least forj , 1,that

~DHm!phase diag.> DHn.pol (34)

under the earlier hypothesis10,11 that

DHpol~I , T! ! DHn.pol (35)

In the present case, assuming average values (i.e.,temperature independent) forj i(^ji& 5 1.168) andjf(^jf& 5 1.457) and using the experimental value ofd log(Im)/d(Tm)21 (24633 K: see Experimental) oneobtains (DHm)phase diag.5 27513 J (mol repeatingunit)21. The calculated value ofDHn.pol is 27903 J(mol repeating unit)21. The difference between thetwo values is as low as about 5%. The good agree-ment found points to the reliability of the calculatedparameters, namelyDHn.pol, which are subjected toconditions only partly similar, but mostly differentfrom those of Eq. (32). Such a finding confirms theprevious ones,38,39which were however limited to therange ofj , 1 only. In our opinion, all that wouldlead to suggest that Eq. (34) may be considered, as afirst approximation, of general validity, forj valuesboth lower and higher than 1.

As a consequence, in the latter caseand in thepresence of specific affinity interactions one should

714 Paoletti et al.

expect a difference between (DHm)phase diag. and(DHexp)tot, corresponding to the neglected specificaffinity component,DHaff. In the present case theerror amounts to about 11% only. Such an error iscomparable to that (about 8%) deriving from the useof the linearized form of Eq. (3a) instead of thenonlinearized one, as already pointed out.25 At anyevent, additional observations will greatly help toassess the average extent of inaccuracy.

Equation (32) has been used to discriminate betweencompeting models for one of the conformations involvedin a conformational change.25,40 In those cases the dif-ference between the values ofj for the opposing modelswas of 100%, and the corresponding difference betweenthe calculated values of (DHm)phase diag.ranged from100% to 300%, depending on the conditions. The pres-ently found level of inaccuracy does not certainly impairthe ability of Eq. (32) to discriminate between confor-mations so grossly different in their physical properties.

CONCLUSIONS

From the whole of the preceding discussion, the fol-lowing conclusions can be drawn:

1. The starting idea put forth in the precedingpapers,38,39 which is presently reported in Eq.(19), can be still considered valid IF account isgiven for the presence of a priori unknowneffects of specific affinity between the counte-rions and the polyion.

2. The use of the simple plots of polyelectrolyticfree energy vs. temperature are useful diagnos-tics to reveal the presence of such specific af-finity interactions, detected as bare deviationsfrom linearity.

3. The determination of the enthalpy and entropychanges both of the “conformational” nonpoly-electrolytic component and of the specific af-finity term can be achieved without great diffi-culty by standard calculation procedures, once asufficiently large number of experimental datais available from the so-called conformationalphase-diagram. The reliability of the deter-mined values can be effectively tested by com-parison with easily accessible experimentaldata.

4. The Gibbs free-energy difference of specificaffinity may be totally unpredictable a priori, inthe sense that ions which are known to show a“phenomenological” positive affinity for thepolyelectrolyte, like K1 for k-carrageenan inwater, may show an overall “detrimental” con-

tribution toward binding to either conformationin a different solvent like DMSO. In the studiedcase, this seems particularly true for the corre-sponding enthalpy changes.

5. The value of the enthalpy of transition calculatedfrom the slope of the “conformational phase dia-gram” by use Eq. (32), (DHm)phase diag., has beenshown to be in good agreement with that of the“intrinsic” nonpolyelectrolytic part, DHn.pol

(namely, that derivable from molecular modelingcalculations).

In spite of the substantial success of coping withthe not simple problem of the specific affinity betweencounterion and polyion, one might be left with theapparent feeling that the calculated values be some-what phenomenological, and therefore devoid of theinterest which usually derives from theories based onfirst principles. However, this is not the case for the“intrinsic” enthalpy and entropy terms of nonpoly-electrolytic nature, which will hereafter be simplycalled “conformational.” The field of molecular mod-eling related to such conformational terms is muchmore active (and rapidly growing) than in any otherrelated to theoretical predictions of counterion speci-ficity or selectivity.

It might be of interest for those involved in mo-lecular modeling studies to notice that bothDHn.pol

andDSn.pol show a clear (inverse) dependence on thedielectric constantD. (see Table IV). A very goodlinear correlation exists among the two thermody-namic functions, which is confirmed by the very highvalue of the correlation coefficient, also reported inTable IV. The ratioDHn.pol/DSn.pol is practically con-stant for the three cases considered, which span overa rather wide range ofD values. This finding isstrongly indicative of the basic identity of the processunder the various dielectric conditions (Barclay–But-ler type of relationship).

The ratioDHn.pol/DSn.pol corresponds to the melt-ing temperature of the hypothetically uncharged ga-lactan backbone ofk-carrageenan. Not surprisingly,such a temperature (i.e., about 90°C) roughly corre-sponds to that experimentally observed for the melt-ing of pure agarose.46 Agarose is structurally verysimilar to k-carrageenan, but for the absence of sul-fate groups and for the presence of anL-galactoseresidue instead of aD-galactose in the repeating unit.Such a convergence lends further confidence to thepresently found values ofDHn.pol andDSn.pol.

As it was already pointed out,39 it is impossible onthe basis of thermodynamic data only to state what isthe absolute change of the initial and final conforma-tional state upon changingD. Still, these quantitative


findings may be of help for those conformationalanalysis studies increasingly concerned with the im-portance of the role of the solvent in the biopolymerstructure stabilization.

An additional conclusion deals with the nature ofthe conformation(s) ofk-carrageenan, the actual sam-ple case studied. The intramolecular nature of theconformational transition and the single-helical natureof the ordered conformation have recently receivedvery strong support from both theoretical47 and ex-perimental48,49 evidence. Those findings have beenconfirmed by this work, inasmuch as this is theonlyconformational hypothesis compatible with thepresenttheoretical polyelectrolytic treatmentand withthe relatedexperimentalcalorimetric data.

J. C. Benegas and S. Pantano are grateful to CONICET(Argentina) and to the Abdus Salam International Center forTheoretical Physics (Trieste, Italy) for the financial supportto their travel and stay in Trieste. This work was carried outwith the financial support of the Italian MURST and of theUniversity of Trieste. The scientific support of the POLY-bios Research Center, AREA Science Park (Trieste, Italy)and the technical support of Pierfrancesco Zuccato are alsogratefully acknowledged.

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APPENDIX 1

Definitions and Abbreviations

bi ,f Average distance between the projections ofthe polyelectrolyte charges onto the bio-polymer (ideal) axis, in theinitial andfinalconformations, respectively (Å)

bord,dis Average distance between the projections ofthe polyelectrolyte charges onto thebiopolymer (ideal) axis, in theorderedand thedisorderedconformations, respec-tively (Å)

Ce (Molar) concentration of polyelectrolytefixed charges (M)

Ccp (Molar) concentration of polyelectrolytecounterions [Ccp 5 ncp/(1V) 5 Ce] (M)

Cp (Molar) concentration of polyelectrolyte (ingeneral) (Cp 5 Ce 5 Ccp) (M)

CS (Molar) concentration of supporting 1:1 elec-trolyte [CS 5 (ns)

1/(1V) 5 (ns)2/

(1V)] (M)C2 (Molar) concentration of anions [C2

5 (ns)2/(1V)] (M)

C1 (Molar) concentration oftotal uncondensedcations {C1 5 [(ncp 2 nc) 1 (ns)

1]/(1V)} ( M)

cond (As a suffix) “of condensed counterions”D “Dielectric constant” (relative dielectric per-

mittivity) (dimensionless)e Value of the elementary electrical charge

(CGS units)e Base of natural logarithms (dimensionless)el (As a suffix) “of electrostatic origin” (De-

bye–Huckel)free (As a suffix) “ofnoncondensed (free) coun-

terions”& Gibbs free energy (Joule)G Molar Gibbs free energy (per mol of fixed

charge) (joule mol21); Reduced molar Gibbs free energy [; 5 G/

(5T)] (dimensionless)* Enthalpy (joule)H Molar enthalpy (per mol of fixed charge)

(joule mol21)' Reduced molar enthalpy [' 5 H/(5T)] (di-

mensionless)I Ionic strength {I 5 1

2¥ [C1( z1)2

1 C2( z2)2] 5 [CS 1 12

Ccp(1 2 r )]}(M)

Im Ionic strength at transition midpoint (M),B Bjerrum length [,B 5 e2/(D«0 kB z T) (for

DMSO at 298.15 K:,B 5 11.967 Å) (Å)kB Boltzmann’s constant (kB 5 5/1) (joule

K21)m (As a suffix) “of melting”mix (As a suffix) “of purely entropy of mixing

origin”nc Number of condensed counterions (dimen-

sionless)ncp Number of counterions from the polyelectro-

lyte (dimensionless)ne Number of (negative) polyelectrolytic fixed

charges (ne 5 ncp) (dimensionless)(ns)

1 Number of cations stemming from the sup-porting 1 : 1 electrolyte (dimensionless)


(ns)2 Number of anions stemming from the sup-

porting 1 : 1 electrolyte1 Avogadro’s number (dimensionless)n.pol (As a suffix) “ofnon-polyelectrolyte origin”pol (As a suffix) “of polyelectrolyte origin”r Fraction of condensed counterions [moles

of condensed counterions per mole offixed charge:r 5 1 2 (j21)] (dimension-less)

Rs Ratio of the 1 : 1added electrolyte to poly-electrolyte molar concentrations (Rs

5 CS/Cp) (dimensionless)5 Universal gas constant (5 5 8.314) (joule

mol21 K21)6 Entropy (joule K21)S Molar entropy (per mol of fixed charge)

(joule mol21 K21)s Reduced molar entropy (s 5 S/5) (dimen-

sionless)sim (As a suffix) “of similions (or “gegenions” or

“coions”)”T Absolute temperature (K)Tm Temperature at transition midpoint (K)tr (As a suffix) “of transition”Vp Volume of the condensed phase per mol of

fixed charge (Vp 5 9/ne) (L mol21)9 Volume of the condensed phase (L)w (As a suffix) “of mixing with the solvent”

(most frequently:water)V (Total) volume of the system (L)z1 Number of positive charges of the counter-

ion; in the present treatment,z1 5 11(relative number)

z2 Number of negative charges of the anion inthe present treatment,z2 5 21 (relativenumber)

zstr Number of negative charges of the fixedgroups on the polyelectrolyte; in thepresent treatment,zstr 5 21 (relativenumber)

zeff Effective number of negative charges on thefixed groups on the polyelectrolyte:zeff

5 zstr(1 2 r ) (relative number)«0 Vacuum dielectric permittivity (CGS units)k21 Debye length, (k21 5 {2[4p110227,B(Å)]

3 I } 21/ 2 5 {2 lI } 21/ 2) (Å)l l 5 0.091 for DMSO at 298.15 (M21 Å22)p pi (p 5 3.1415 . . . ) (dimensionless)j i ,f Linear charge density parameter of the CC

theory (j i ,f 5 ,B/bord,dis), in the initialand final conformations, respectively (di-mensionless)

jord,dis Linear charge density parameter of the CCtheory (jord,dis 5 ,B/bi ,f), in the orderedand thedisorderedconformations, respec-tively (dimensionless)

APPENDIX 2

As far as the calculation ofHpol is concerned, it is tobe recalled that all theGmix terms are defined asentropic terms, and thus for that purpose only theGel

term of Eq. (2) needs to be taken into account:

Hpol 5 ~Gel/T!/~1/T! 5 212~5T!j~1 2 r !2

3 ~1 1 d ln D/d ln T!@2 ln~1 2 e2kb! 1 kb

/~ekb 2 1!# 5 212~5T!j21~1 1 d ln D/d ln T!

3 @2 ln~1 2 e2kb! 1 kb/~ekb 2 1!#

(A1)

The polyelectrolytic component to the molar enthalpychange of transition is then given by

~DHtr!pol 5 21

2~5T!~1 1 d ln D/d ln T!

3 $~jf!21@2 ln~1 2 exp~2kf bf!! 1 kf bf

/~exp~kf bf! 2 1!# 2 $~ji!21

3 @2 ln~1 2 exp~2kibi!!

(A2)

1 k ibi/~exp~k ibi! 2 1!]}

For kfbf andkibi ! 1, Eq. (A2) reduces to

~DHtr!pol 5 21

2~5T!~1 1 d ln D/d ln T!

3 $~jf!21@2 ln~kfbf! 1 1!# (A3)

2 $~j i!21@2 ln~k ibi! 1 1!#}

As expected, Eqs. (A2) and (A3) bear a strong resem-blance to Eqs. (18) and (19) of Ref. (38), respectively,the only difference being the well-known change ofthe power law ofj from 11 to 21, on passing fromj , 1 to j . 1.11,12,38,39

For the polyelectrolytic molar entropy, besides theelectrostatic component, all the “mixing” term willcontribute

Spol 5 $2~Gel!/~T!% 1 $2~Gmix!/~T!% 5 Sel 1 Smix

Sel 5 25j~1 2 r !2$~d ln D/d ln T!ln~1 2 e2kb!

718 Paoletti et al.

112

~1 1 d ln D/d ln T!@kb/~ekb 2 1!#%

5 25j21$~d ln D/d ln T!ln~1 2 e2kb! (A4)

112

~1 1 d ln D/d ln T!@kb/~ekb 2 1!#}

Smix 5 25$r ln~r /@~Rs 1 1!~VpCp!#!

1 @Rs 1 ~1 2 r !#ln~@Rs 1 ~1 2 r !#

/@~Rs 1 1!~1 2 VpCp!#!(A5)

1 Rs ln~1/~1 2 VpCp!! 1 r }

Spol 5 Sel 1 Smix

5 25$j21$~d ln D/d ln T!ln~1 2 e2kb!

112

j21~1 1 d ln D/d ln T!@kb/~ekb 2 1!#%

1 ~1 2 j21!ln$~1 2 j21!/@~Rs 1 1!~VpCp!#%

1 @Rs 1 j21#ln~@RS 1 j21#

/@~Rs 1 1!~1 2 VpCp!#!

(A6)

1 Rs ln~1/~1 2 VpCp!! 1 ~1 2 j21!}

The polyelectrolytic component to the molar entropychange of transition is then given by

~DStr!pol 5 25$~d ln D/d ln T!$~jf!

21

3 ln~1 2 exp~2kf bf!! 2 ~ji!21 ln~1 2 exp~2kibi!!%

112~1 1 d ln D/d ln T!

3 $~jf!21@kf bf /~exp~kf bf! 2 1!#

2 ~ji!21@kibi/~exp~kibi! 2 1!#% 1 ~1 2 ~jf!

21!

3 ln$~1 2 ~jf!21!/@~Rs 1 1!~VpCp!#% 2 ~1 2 ~ji!

21!

3 ln$~1 2 ~ji!21!/@~Rs 1 1!~VpCp!#% 1 @Rs 1 ~jf!

21#

3ln$@Rs 1 ~jf!21#/@~Rs 1 1!~1 2 VpCp!#%

2 @Rs 1 ~ji!21#ln$@Rs 1 ~ji!

21#

/@~Rs 1 1!~1 2 VpCp!#% 1 ~ji!21 2 ~jf!

21%

(A7)

For kf bf andkibi ! 1, Eq. (A7) reduces to

~DStr!pol 5 25$~d ln D/d ln T!

3 @~jf!21 ln~kfbf! 2 ~j i!

21 ln~k ibi!#

112

~1 1 d ln D/d ln T!@~jf!21 2 ~j i!

21#

1 ~1 2 ~jf!21!ln$~1 2 ~jf!

21!

/@~Rs 1 1!~VpCp!#%

2 ~1 2 ~j i!21!ln$~1 2 ~j i!

21!

/@~Rs 1 1!~VpCp!#%

1 @Rs 1 ~jf!21#ln$@Rs 1 ~jf!

21#

/@~Rs 1 1!~1 2 VpCp!#%

2 @Rs 1 ~j i!21#ln$@Rs 1 ~j i!

21#

(A8)

/@~Rs 1 1!~1 2 VpCp!#}

1 ~j i!21 2 ~jf!

21}


Thermodynamics of conformational transition and chain association of ι-carrageenan in aqueous solution: Calorimetric and chirooptical data

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