386
`
`Macromolecules 1982,15, 386-395
`
`(16) Manning, G. S. J. Chem. Phys. 1969,51, 924.
`(17) Cleland. R. L. BioDolvmers 1979. 18. 2673.
`(18) Cleland, R. L., unpubiished results.
`(19) Cleland, R. L. Arch. Biochem. Biophys. 1977, 180, 57.
`(20) Strauss, U. P.; Helfgott, C.; Pink, H. J. Phys. Chem. 1967, 71,
`2550.
`
`(21) Strauss, U. P.; Ander, P. J. Am. Chem. SOC. 1958, 80, 6494.
`(22) Scatchard. G.: Batchelder. A. C.: Brown, A. J. Am. Chem. SOC.
`1946,68, 2320.
`(23) Cleland, R. L.; Wang, J. L. Biopolymers 1970, 9, 799.
`(24) Wik, K. 0. Doctoral Thesis, Faculty of Medicine, Uppsala
`University, Uppsala, Sweden, 1979.
`
`Polyelectrolyte Properties of Sodium Hyaluronate. 2.
`Potentiometric Titration of Hyaluronic Acid
`
`Robert L. Cleland,* John L. Wang,? and David M. Detweiler
`Department of Chemistry, Dartmouth College, Hanover, New Hampshire 03755.
`Received April 16, 1981
`
`ABSTRACT Titrations of 0.01 m glucuronic acid (GA) with NaOH were carried out in cells employing glass
`electrodes with a saturated calomel electrode (cell A) or a silver chloride electrode (cell B). The dissociation
`constant at zero ionic strength was given for either cell by pK = 3.23 (*0.02), consistent with previous
`determinations. Corrections for the liquid-junction potential (cell A) led to the expected behavior of apparent
`pK with concentration of added NaN03. Similar titrations of 0.0085 m hyaluronic acid (HA) from bovine
`vitreous humor gave essentially linear plots of apparent pK against degree of ionization a over the range a
`= 0.3-0.8 in the presence of added salt. Least-squares fits to these plots provided slopes which were fitted
`better as a function of concentration of added salt by the uniformly charged cylinder model than the infinite
`line charge model of a polyion. The cylinder radius required to obtain a good fit with the structural charge
`density is about 10 A, however, which is larger than the structural radius (4-5 A) of the charge sites. The
`discrepancy may be due in part to effects of charge discreteness and low dielectric constant of the polyion.
`The intrinsic dissociation constant for the polymer was estimated to be pK = 2.9 (*O.l), where the large error
`estimate reflects uncertainties in extrapolation to (Y = 0. The difference between polymer (HA) and monomer
`(GA) pK was attributed to effects of substitution at carbon 4 of the monomer. Although data for electrophoretic
`mobility at zero polymer concentration are limited for hyaluronate, agreement at one ionic strength (0.1 M)
`of the surface (f) potential calculated from this method and from potentiometric titration suggests that these
`techniques measure the same potential.
`
`Hyaluronic acid, which occurs in its ionized form in
`many connective tissues and fluids, is a linear poly-
`saccharide whose repeating disaccharide unit (see Figure
`1, paper 1') consists of D-glucuronic acid (@-linked at
`carbons 1 and 4) and N-acetyl-D-glucosamine (@-linked at
`carbons 1 and 3).2 At complete ionization of the COOH
`groups on the glucuronic acid residues, charges occur
`regularly on every second glucoside and the average charge
`spacing approximates the length of the disaccharide unit,
`about 1 nm.
`The thermodynamic properties of polyelectrolyte solu-
`tions have been extensively studied theoretically; in this
`work we investigate treatments which model the polyion
`as a smeared-out charge on an infinite line3 or rigid cyl-
`inder4ps or as a flexible polymer chain with discrete
`charges.6 In all of these treatments the chain may be
`characterized by a dimensionless charge density parameter
`€
`
`(1)
`[ e 2 Q / D k T L
`where e is the electron charge, Q is the number of (electron)
`charges in length L, D is the bulk solvent dielectric con-
`stant, k is the Boltzmann constant, and T i s the absolute
`temperature. For monovalent counterions the value 5 =
`1 has been sometimes regarded (see ref 3 and citations
`therein) as a critical value, above which counterion con-
`densation occurs in some fashion near the polyion and
`below which the latter may be regarded as fully ionized.
`
`+ Present address: Department of Biochemistry, Michigan State
`University, East Lansing, MI 48824.
`
`The value of 5 for the hyaluronate polyion has been taken
`previously' to be 0.70, which is below the critical value.
`The polymer can, therefore, be tentatively assumed to be
`fully ionized in the Debye-Huckel sense.
`Various experimental features of hyaluronate behavior
`agree rather well with this assumption in terms of the line
`charge t h e ~ r y . ~ These include data for the enthalpy of
`mixing of NaCl and hyaluronate' and for activity coeffi-
`cients in hyaluronate solution^.^^^ The assumption of
`complete ionization at 5 < 1 has been questioned in a
`recent discussion of activity coefficients in solutions of
`polyacrylate copolymers of low charge density,'O a point
`which will be dealt with in our discussion.
`In the present work the potentiometric titration behavior
`of hyaluronic acid is examined experimentally and the
`results are investigated in terms of the available theoretical
`models as applied to this experimental technique. The
`most extensive previous investigation by this method was
`that of Laurent."
`Experimental Section
`Materials. A crude sample of bovine vitreous humor hyal-
`uronate (K' form, Nutritional Biochemical Corp.) weighing 6.14
`g was dissolved in 500 cm3 of deionized water containing (as in
`all solutions of polymer) 1 mg
`of 5,7-dichloro-8-quinolinol
`(Eastman Kodak Co.) as preservative. The soluton, after dialysis
`for 48 h against deionized water, was treated with 5.0 g of kaolin
`(technical grade, washed and ignited, J. T. Baker Co.) for 30 min
`with stirring and 40 h unstirred, at room temperature. The
`suspension was clarified by centrifugation at ca. 104g for 15 min
`to give a supernatant containing about 0.08 mg cm-3 of protein
`contaminant. The supernatant (about 400 cm3) was dialyzed
`against approximately 5 kg of stirred deionized water for 5 days
`
`0024-9297/82/2215-0386$01.25/0 0 1982 American Chemical Societv
`
`ALL 2017
`PROLLENIUM V. ALLERGAN
`IPR2019-01505 et al.
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`

`

`Vol. 15, No. 2, March-April 1982
`
`with two changes of water. The dialyzed solution was passed
`through an ion-exchange column (Dowex 50W-X8 H+ form) to
`convert the hyaluronate salt to hyaluronic acid. The resulting
`solution contained 4.0 mg cm-3 of hyaluronate (as Na+ salt) and
`had a limiting viscosity number [7] = 354 cm3 g-l in 0.1 M HC1,
`corresponding to a weight-average molecular weight12 of 2.4 X 105
`(as Na+ salt).
`DGlucuronic acid was a commercial grade (Calbiochem, catalog
`no. 3473), which was found by titration with standard NaOH to
`have 0.954 equiv of acid per mole (194.1 g) weighed. Sodium
`hydroxide solutions were prepared from the 50% (w/w) reagent,
`which was centrifuged at 2000g for 10 min to remove suspended
`insoluble NaZCO3, and diluted with COz-free water prepared by
`boiling distilled water for 10 min and cooling in the absence of
`air. Solutions were stored in polyethylene bottles sealed with
`rubber stoppers through which were inserted drying tubes con-
`taining Ascarite (A. H. Thomas Co.) to prevent contamination
`with atmospheric COP The diluted solutions were standardized
`with weighed samples of potassium hydrogen phthalate (reagent
`grade, Fisher Scientific Co.) dried at 110 "C and assumed pure.
`Other analytical procedurea used in this work have been described
`previously13 and include our modifications of standard methods
`for protein and hexuronic acid analysis.
`Potentiometric Titrations. Measurements of pH or cell
`potential (in mV) were performed with a battery-operated po-
`tentiometric pH meter (Radiometer, Model PHM 4). Two kinds
`of electrochemical cells were used: cell A: glass electrode
`(G202C)ltest solutionlsaturated KCl(calome1 (K401); cell B: glass
`electrode (G202C)ltest solutionlAgCllAg (P501), where the Ra-
`diometer electrode numbers are given in parentheses. The test
`solutions were contained in a thermostated reaction vessel (Bolab,
`Inc.) with an outer jacket through which circulated water ther-
`mostated at 25.0 (iO.1) OC.
`The initial charge, usually about 5 cm3, of sample for titration
`or for standardizing electrodes, was pipetted (or weighed, when
`containing polymer) into the solution chamber. The electrodes
`were inserted, along with capillary tubes used for bubbling pre-
`purified nitrogen to stir the solution and maintain a C02-free
`atmosphere, the tip of a micrometer-driven microburet (Manostat
`Corp.), and a magnetic stirring bar to provide supplemental
`stirring. Spaces around the electrodes and glass tubes were packed
`with a moldable nondrying putty. As determined by the time
`required to reach a constant voltage (pH) reading, about 1 h was
`allowed for initial equilibration of the electrodes with solutions
`containing polymer or about 10 min with buffers or glucuronic
`acid. An additional 10 min was allowed after each addition of
`titrant (NaOH, NaN03, NaCl solution, or water). Before each
`titration electrodes were standardized with buffers at 25 "C as
`follows.
`Cell A. Standardization was carried out with 0.01 m HC1,0.09
`m KC1 (pH 2.098 on the National Bureau of Standards practical
`scale, hereafter called the NBS scale14). To correct for error in
`the change of voltage with pH due to nonideality of electrode
`behavior, a second buffer containing 0.025 m KH2P04, 0.025 m
`NazHP04 (pH 6.865 on the NBS scale at 25 "C) was used. The
`correction was performed in practice by use of the instrument's
`temperature compensation control to balance the potentiometer
`set at the latter pH value. Other buffers (such as 0.05 m potassium
`hydrogen phthalate, pH 4.008 at 25 "C on the NBS scale) were
`then checked and found to read correctly within 0.005 pH unit.
`All buffers were made with reagent grade compounds (J. T. Baker
`Co.) and distilled water.
`Cell B. Cell potential E was measured at 25 "C for 0.01 m HC1
`daily before each titration was performed. The standard cell
`potential E" (in mV) was redetermined from E for each run by
`use of the Nernst equation
`E" = E + b log UHCI
`(2)
`where log denotes base 10 logarithms throughout and uHCl is the
`HCI activity
`
`~ H C I = yHmwclmcl= y+zmHmcl
`@a)
`where mi is molality, yi is the activity coefficient on the molal
`concentration scale, and yt is the mean ionic activity coefficient.
`Values of b for a given electrode set were determined (and checked
`at intervals) from the slope of a calibration plot of E against log
`
`Titration of Hyaluronic Acid 387
`
`and 0.1 m.
`U H C ~ for a series of seven HCl solutions between
`Values of ya were taken from the literature.l6 Resulting values
`of b were within 1% of the ideal value b = RTfF = 59.16 mV,
`where R is the molar gas constant and F the molal Faraday
`constant.
`Treatment of Experimental Data
`The system considered is defined in terms of the fol-
`lowing components, having molality mi or molar concen-
`tration Ci: 1, the solvent, H20; 2, the acid to be titrated,
`composed of identical acidic sites HA, where HA repre-
`sents the ionizable COOH group of D-glucuronic acid (or
`its residue in the polymer); 3, the added electrolyte Nay,
`where Y- is C1- or NO;; 4, the titrant base, NaOH. The
`molality m2 (or molar concentration C,) refers to moles of
`monomer (disaccharide repeat units of the polymer) and
`hence moles of HA.
`The degree of neutralization v is defined by v = m4/m2.
`The degree of ionization a was calculated from the con-
`dition of electrical neutrality expressed in terms of mo-
`lalities mi of the various ionic species
`a 3 mA/m2 = (mNa + mH - my - mOH)/m2 =
`v + m ~ / m 2 (3)
`where A- is the carboxylate anion. The final equality
`results from stoichiometric relations mNa = m3 + vm2, when
`complete ionization of hyaluronate is assumed, and my =
`m3, with mOH = 0 to a sufficiently good approximation in
`the pH range of interest.
`The calculation of a thus requires estimates of mH based
`on experimental pH readings. For cell B this requires
`values of ya for HC1, according to eq 2 and 2a. For elec-
`trolyte solutions containing very dilute HC1 mixed with
`NaCl or buffers (such as glucuronic acid and its salt) and
`having total ionic strength I, Bates16 suggested that this
`coefficient could be approximated by yo, the limiting value
`at zero HC1 concentration of ya (for HCl) in an NaCl
`solution of ionic strength I. Bates used the DebyeHuckel
`formula for I < 0.1 M
`
`(4)
`where we take the constant A = 0.509 to calculate yo with
`po = 1.65 at 25 "C. In solutions containing glucuronic acid,
`I = c3 + acz.
`For solutions containing hyaluronic acid the effect of the
`ionized polymer on y+ of HC1 was assumed to be identical
`with its effect on y+ of NaC1. As shown in the Appendix,
`experimental dataa for NaCl in solutions of fully ionized
`hyaluronate can be fitted by the additivity rule
`
`(5)
`where gjs represents the effect of salt as given by eq 4 with
`an appropriate value of p., I = C3, and gjp represents the
`effect of the polyion in the form
`
`gjp = -log (7::)
`
`X = mA/m3 = am2/m3
`(7)
`T o allow for the dependence of &,' (defined in Appendix)
`on a, we adopt the approximate expression of the line
`charge theory3 4; = 1 - a512 and set 5 = 0.54 for this
`purpose from the empirical result (see Appendix) that &'
`= 0.73 at a = 1.
`For cell A, conversion of the experimental reading of pH
`pH(X) to pmH -log mH is more complicated due to the
`existence of a liquid-junction potential Ej(X) at the
`boundary test solutionlsaturated KC1. When standardi-
`
`

`

`388 Cleland, Wang, and Detweiler
`Table I
`Comparison of Calculated and Experimental Values of &
`-
`gH
`
`Macromolecules
`
`mHCl
`
`mxc1
`
`g H
`
`0.01
`0.01
`0.01
`0.01
`0.01
`
`0.03
`0.04
`0.09
`0.19
`0.49
`
`0.064
`0.068
`0.083
`0.097
`0.097
`
`AEj
`
`X = Na
`
`0.010
`
`0.000
`-0.011
`
`X = K
`p m H = 2
`
`-0.006
`-0.013
`-0.020
`-0.032
`pmH= 3
`
`X = Na
`calcd
`exptlC
`
`0.074
`
`0.083
`0.086
`
`0.078
`
`0.078
`0.078
`
`X = K
`exptld
`
`0.068 (0.065)
`0.093 (0.080)
`0.087 (0.081)
`0.084 (0.078)
`
`calcd
`
`0.062
`0.070
`0.077
`0.065
`
`0.004
`
`-0.006
`-0.017
`
`0.064
`0.039
`0.001
`0.058 (0.062)
`0.061
`-0.007
`0.068
`0.049
`0.001
`0.087 (0.076)
`0.067
`-0.016
`0.083
`0.099
`0.001
`0.082 (0.072)
`0.071
`-0.026
`0.097
`0.199
`0.001
`0.001
`0.071 (0.068)
`0.068
`-0.029
`0.097
`0.499
`The values of gH were calculated-from eq 4 with Q H = 2.96, as suggested by Kielland,l9 for I < 0.1 M; g H = 0.097 for
`I > 0.1 M (see text).
`Values of AE. were calculated as described in the text with the standardizing buffer taken as given
`ExpTdmental values from eq 11 of ref 17; standardizing buffer: 0.025 m KH,PO,, 0.025 m
`in footnotes c and d .
`Experimental values from mean values in Table I of ref 18; values
`Na,PO,; calculated value of Ej(S) = 0.032 (in pH unitsl,
`-
`-
`
`in parentheses are for KNO, in place of XCl; value of E i ( S ) taken as 0.050 (in pH units) within 0.01 pH unit of buffers a i
`and b of ref 18.
`zation is performed as in our experiments, the pH readings
`may be interpreted in terms of an operational coefficient
`& defined by17
`g~ E pH(X) - pmH = g H + A E j
`(8)
`where AEj E [Ej(X) - Ej(S)]F/(RT In 10) (in pH units) and
`Ej(S) is the liquid-junction potential for the standardizing
`buffer. Hedwig and Powell17 and McBrydels reported
`calibrations of cells like our cell A with solutions of known
`mH in NaC1" or K C P from which the experimental values
`of gH in Table I were obtained. While no reliable method
`exists for accurately calculating either g H or Qj,
`the fol-
`lowing procedure reproduces reasonably well these ex-
`perimental values. The single-ion activity coefficient g H
`has been estimated from eq 4 with the ionic diameter UH
`= 9 A suggested by Kielland,l9 so that pH = 0.329 X 9 =
`2.96 at 25 "C. At Z > 0.1 M this procedure overestimates
`gH, as judged by comparison of calculated and experi-
`mentalm values of yo (with Kielland's estimate of ya). In
`this range of I , g H was taken to be 0.097, as suggested by
`calculations from experimental yo at Z = 0.1,0.2, and 0.5
`M, with ycl taken equal to ya for NaCl solutions of equal
`Z (Bates-Guggenheim convention2'). The liquid-junction
`potentials may be estimated from the Henderson equation
`as outlined by Bates.22 The estimated values are listed for
`HC1-NaC1 and HC1-KC1 mixtures in Table I along with
`the calculated values of gH from eq 8. The agreement with
`the experimental values is within 0.01-0.02 pH unit over
`the pH range of interest and is comparable to the standard
`deviations in pH reported by McBryde for the calibra-
`tions.ls
`These calculations suggest that calibration of electrodes
`with solutions of known mH provides appropriate correc-
`tions for the effects of ionic strength of added salts on g H
`and the liquid-junction potential. Such calibration cannot
`be expected, however, to provide correct values of mH in
`solutions which contain polyions. Consider the analogous
`effect of the polyion on apparent yNa values in solutions
`of sodium salts of polyacids. For sodium hyaluronate, for
`which < 1, the sodium ions are assumed to be fully
`ionized, so that mNa is known. Measurements with glass
`electrodes specific to sodium ion on hyaluronate solutions
`without added salt have indicated values of gNa,p in the
`range 0.10-0.15 relative to calibrations with NaCl solutions
`of known m N 2 9 (see Appendix). While these values are
`
`0.068
`
`0.077
`0.080
`
`0.073
`
`0.073
`0.073
`
`uncorrected for effects of Qj, it seems unlikely that the
`latter can account for more than a small fraction of the
`observed cell potential change at a given mNa. In addition,
`the observed values of yNa are reasonably consistent with
`those expected from the line charge theory, as discussed
`later. In a mixture of "free" counterions, such as H+ and
`Na+, one would expect similar effects on the activity
`coefficient of each counterion in the polyion atmosphere,
`since there is no physical reason to distinguish between
`them, other than the binding capability of H+ due to the
`dissociation equilibrium. Although there are no experi-
`mental data available to justify the particular form of eq
`6 at the lower effective values of a due to the partial ion-
`ization in the titration region, inclusion of the correction
`gjp seemed preferable to ignoring the polyion effect on g H
`completely.
`All values of pmH were calculated from pH(X) readings
`(cell A) by use of eq 8, with g H obtained from eq 4 (gluc-
`uronic acid titrations) or eq 5 (hyaluronic acid titrations)
`and pH = 2.96, except for Z > 0.1 M, as noted above. The
`calculation of A,!?j was performed with equivalent mobilities
`ho from the literaturez3 and the value ho = 75.1 cm2 Q-'
`equiv-' for g l u ~ u r o n a t e . ~ ~ Although the mobility of hya-
`luronate is ionic-strength dependent in the presence of
`added salt (see below),
`is insensitive to the value
`chosen; for titrations without added salt, A was assumed
`to be given by the line charge theory result.25
`Substitution of a calculated from mH by eq 3 into
`pK' E pH + log [(I - C Y ) / f f ] = -log (aHmA/mHA)
`(9)
`then gives the apparent dissociation constant pK'for cell
`A. For cell B, as shown by Bates,lG experimental data are
`conveniently evaluated in terms of an approximate pH
`-log mH - log (YHYC1) = (E - E o ) / b + log mcl
`pwH
`(10)
`where the second equality is derived from eq 2. As men-
`tioned above, we assumed that log ( ~ H ~ c J was equal to 702
`and calculated the latter from eq 4 or, for hyaluronic acid
`solutions, with the extra term of eq 5. Values of a obtained
`from eq 3 with mH calculated from eq 10 were substituted
`into
`pK"5 PWH + log [(I - C r ) / f f ] = -log (aHmAyCl/mHA)
`(11)
`
`

`

`Vol. 15, No. 2, March-April 1982
`
`Titration of Hyaluronic Acid 389
`
`3 6'
`
`I
`
`I
`
`3'4__;
`
`\
`
`I
`
`i
`
`n *
`
`\ .
`
`I
`I
`
`3.4 c
`
`'"P
`
`I
`
`1
`I
`0.2
`0
`0 6
`I O
`0.8
`a
`O4
`Figure 2. Titration curves (cell A) for 0.0085 m hyaluronic acid
`titrated with NaOH in NaN03 solutions having the following
`values of C3* (NaN03 molarity in the corresponding dialysis
`equilibrium solvent): 0.01 M (v); 0.04 M (0); 0.10 M (0); 0.18
`M ( 0 ) ; 0.45 M (A). The straight linea represent linear least-squares
`fits to the linear sections of the data with the parameters and the
`range of a for the fitted points given in Table 11.
`I 1
`
`I
`
`I
`
`I
`
`4 5 I 4.0
`1
`
`3.5
`p K "
`
`2 . 4
`0
`
`0 2
`
`0 4
`
`0 6
`
`0.8
`
`Figure 1. Effect of ionic strength (as C3*'lz) on pK,', the value
`of pK'(cell A) at a = 0, for 0.01 m glucuronic acid (GA) and for
`0.0085 m hyaluronic acid (HA) at 25 "C. The effects of including
`corrections for the liquid-junction potential (see text) are shown
`as follows. Uncorrected points for GA (0) and HA (A, this work)
`are represented by the solid lines; corrected points for GA (V)
`and HA (e, this work; 0 , data of Laurent") are represented by
`the dashed lines. The half-filled circle gives the (uncorrected)
`value reported by Hirschm in 0.1 M KCl.
`Comparison of eq 9 and 11 shows that, to the extent our
`assumptions are valid,
`pK" -t log yci = pK'= PK i- log ( Y A / Y ~ ) (12)
`where K = 10-PK is the thermodynamic dissociation con-
`stant.
`Results
`Glucuronic Acid. As a check on experimental tech-
`nique and methods of data treatment, we carried out
`several titrations, with or without added salt, of D-gluc-
`uronic acid. The experimental data included duplicate
`titrations with NaOH at about 0.01 m in cell A and a single
`titration at 0.01 m in cell B. The values of pK'and pK"
`calculated from these titrations showed no significant
`dependence on CY (within f0.01 pK unit) and values were
`averaged to give pK" = 3.22 (kO.01) in 0.001 m NaCl and
`pK' = 3.23 (*0.01) at m3 = 0. In addition, experiments
`were performed in which neutral salts (NaN03 in cell A
`and NaCl in cell B) were added stepwise to partially
`neutralized samples of glucuronic acid to final salt con-
`centrations m3 of about 0.5. The values of pK"showed
`no dependence on NaCl concentration and averaged to 3.23
`(k0.02). The values of pK'showed a marked dependence
`on ionic strength and are plotted for 0.01 m glucuronic acid
`in Figure 1 against C31/2 to give a linear plot. The intercept
`was 3.23 (*O.Ol), in agreement with titrations at m3 e 0.
`The slope of the line, which is approximately -dgA/d11f2
`is -0.39. At I = 0, pK" = pK'- log yHA = 3.23 at m2 = 0.01.
`With the usual assumption that y = 1 for uncharged
`species in dilute solution, the result is pK = 3.23 (*0.02).
`Hyaluronic Acid. Data for titrations with NaOH were
`plotted as pK' (cell A) or pK" (cell B) as a function of CY.
`In Figure 2 are plotted data from a series of titrations with
`
`1
`
`Figure 3. Titration curvea (cell B) for hyaluronic acid with NaOH
`at C3* = 0.001 M. Points, corrected for dilution as described in
`the text, are shown for the following values of polymer equivalent
`molality mz: 0.0089 (V); 0.0045 (0); 0.0025 (0); 0.0013 (0); 0.0010
`(A). Fits to linear sections of the data are shown for 0.001-0.0025
`
`m (---) and for 0.0045-0.0089 m ( . a s ) . The solid curve is a fit
`to data calculated from tabulated solutions5 to the uniformly
`charged cylinder model appropriate for 0.001 M salt and radius
`a = 10 A.
`cell A at m2 = 0.0085 at concentrations of NaN03 between
`0.01 and 0.45 M. Data shown in Figure 3 represent a series
`of titrations with cell B at different m2 from 0.001 to 0.0087
`at C3* = 0.001 M NaC1, where C3* is the composition the
`polymer-free solvent (dialysate) would have at dialysis
`equilibrium. The actual points in both plots were corrected
`for dilution of salt and polymer during titration by the
`following procedure.
`The assumption was made, consistent with the Hill-
`Stigter model of polyion solutions,26 that C3* was the ap-
`propriate salt concentration for comparison with theo-
`retical models which neglect polyion interactions; extrap-
`
`

`

`.
`
`I
`
`3.6
`
`3.4
`
`I
`
`/ *
`
`I
`
`390 Cleland, Wang, and Detweiler
`
`m2
`
`0.0085
`0.0085
`0.0085
`0.0085
`0.0085
`0.0085
`0.0085
`0.0043
`0.0010
`0.0010
`
`0.000
`0.000
`0.010
`0.040
`0.100
`0.180
`0.450
`0.000
`0.000
`0.000
`
`2.92 (iO.O1)
`2.86 (iO.01)
`2.82 (20.01)
`2.78 (iO.O1)
`2.74 (i0.01)
`2.77 (iO.01)
`2.73 (iO.01)
`2.82 (i0.02)
`2.77 (iO.01)
`2.82 (i0.02)
`
`Macromolecules
`
`no. of points
`
`0.25-0.84
`0.39-0.93
`0.29-0.85
`0.33-0.87
`0.35-0.87
`0.36-0.84
`0.38-0.80
`0.34-0.58
`0.50-0.84
`0.45-0.57
`
`38
`22
`19
`19
`19
`19
`17
`7
`25
`8
`
`Table I1
`Parameters from Linear Least-Sauare Fits to Data for Titrations with NaOH
`oi range fitted
`(apK’/aalC3*
`PK,‘
`C,*“
`Cell A
`0.81 (20.02)
`0.84 (i0.02)
`0.77 (~0.01)
`0.55 (20.02)
`0.43 (iO.01)
`0.30 (iO.01)
`0.25 (iO.01)
`1.04 (k0.04)
`1.17 (iO.01)
`1.29 (i0.04)
`Cell B
`0.001
`5
`0.0089
`2.93 (20.01)
`1.21 (iO.01)
`0.35-0.81
`0.0045
`0.001
`1.13 (i0.02)
`2.97 (iO.01)
`11
`0.42-0.90
`1.30 (20.01)
`0.001
`13
`0.55-0.90
`2.93 (iO.01)
`0.0025
`0.001
`0.0023
`2.96 (kO.01)
`10
`0.49-0.90
`1.20 (iO.01)
`0.0010
`0.001
`2.96 (iO.01)
`20
`0.55-0.90
`1.29 (i0.02)
`a This molar concentration refers to the dialysis equilibrium solvent as explained in the text; the salt is NaNO, in cell A
`and NaCl in cell B.
`The intercept pK,‘ and the slope (apK’/acu)c,* are parameters from the least-squares fit over the range
`of 01 and with the number of points given. In cell B pK’ becomes pK”.
`Uncertainties given are standard deviations in the
`Three nearly identical titrations are combined for this fit.
`fitting procedure only.
`olation to infiite dilution of polymer should also be carried
`out. Values of C3* were calculated from eq 1-1 with the
`virial coefficients for salt distribution A2 = 0 and Al es-
`timated from a fit of eq 1-11 with Vi, the partial molal
`volume, taken to be 500 cm3 mol-l, appropriate to a
`cross-sectional chain radius of 5 A, as a reasonable fit to
`the data of Figure 1-3. Since the best empirical fit to the
`dependence of apparent pK values on salt concentration
`was given by the numerical solutions of Stigtel5 (see Figure
`5 and Discussion), an empirical correction to a single value
`of C3*
`(C3*)0 for each titration was made by use of a fit
`to the theoretical curve of Figure 5 marked ,310. Exper-
`imental values of pK’ (or pK’? were corrected by adding
`a term 6pK defined by
`6pK pK’(corrected) - pK’ =
`(apK’/d log C3*), log [C3*/(c3*)01 (13)
`A small correction for the effect of C3* on pK,-,’ (see eq 16),
`based on the slope in Figure 1, was also included. These
`corrections were negligible for titration at C3* > 0.01 M.
`Points shown in Figures 2 and 3 were corrected by this
`procedure.
`It is perhaps worth mentioning that the Manning-
`Holtzer formulation2’ could be used in a conceptually
`equivalent way. As the appropriate concentration variable,
`the latter authors used log (C3 + aC2/2), which is just log
`C3* when the Donnan equilibrium expression Al = -1/2 is
`used. The correction 6pK would then proceed through eq
`18 (see below) with an empirical fit of the data to obtain
`an effective 5.
`A series of titrations without added salt were also carried
`out with cell A at different values of m2 between 0.001 and
`0.0085. These titrations are not actually salt-free, since
`leakage from the saturated KC1 salt bridge was estimated
`at 0.0002 M/h, so that C, may have reached 0.00054.001
`M during a long titration. A correction procedure for
`polymer diultion similar to that described above was
`carried out by use of a fit to the Lifson-Katchalsky theory
`(curve LKlO in Figure 6), so that each curve refers to a
`single polymer concentration C2 in Figure 4, where the
`corrected points are plotted.
`Duplicate titrations were reproducible only to about
`0.01-0.02 pH unit; a discrepancy of 0.02 pH unit suffices
`to produce the differences in pK’ values for the two ti-
`trations at m2 = 0.001. A standard deviation in pK’values
`
`2.81
`I
`1
`I
`I
`0
`1.0
`0.2
`0.6
`0.8
`0.4
`a
`Figure 4. “Salt-free” titrations curves (cell A) for hyaluronic acid
`with NaOH at the following equivalent molar concentrations C2:
`0.0085 M (three titrations, 0 , 8 , and 0 ) ; 0.0043 M (0); 0.001 M
`(two titrations, A and A). Crosses represent corrections to the
`data at 0.0085 M to allow for the effect of the polymeric N-acetyl
`groups when the latter are assumed to have dissociation pK =
`0.8 (see text). The solid line represents a fit to the data at 0.0085
`M by the Lifson-Katchalsky theory with a taken to be 7.5 A.
`of about 0.02 pK unit for midrange values of a is estimated
`for titrations at m2 = 0.0085, at which most of our data
`were obtained. The effect of the correction term gjp in eq
`5 on calculated values of pK’ (and pK‘? was significant
`only for titrations in which m2 > m3, and then only in the
`initial stages of the titration., where the correction is of
`the order of a few hundredths (up to 0.05) pK unit.
`As discussed later, the correct procedure for extrapola-
`tion of the plots of apparent pK against a to obtain pK4,
`the intercept at a = 0, is not clear, and extrapolated values
`of pK,-,’ are therefore affected by some uncertainty. The
`plots in Figure 2 at C3* L 0.01 M are approximately linear,
`with positive curvature evident only at a > 0.9. Param-
`eters from least-squares linear fits to the data for a < 0.9
`are given in Table 11. At C3* = 0.001 M (Figure 3) and
`in the “salt-free” titrations of Figure 4, pronounced cur-
`
`

`

`Vol. 15, No. 2, March-April 1982
`
`Titration of Hyaluronic Acid 391
`
`vature appears also at values of a below about 0.3 in most
`titrations. In these cases we regard the fits to linear sec-
`tions, typically between a = 0.3 and 0.85, as reasonably
`accurate only for slopes at midrange values of a. Sec-
`typ-
`ond-degree fits including a term in a2 increase pK,'
`ically by 0.02-0.05 pK unit at C3* > 0.01 M and as much
`as 0.10 pK unit at 0.001 M, without significant effect on
`the slope at, say, CY = 0.6.
`The pK,' values obtained from the linear fits at C3* >
`0.01 M, as well as those obtained without correction for
`are plotted in Figure 1. As in the case of glucuronic
`acid, the intercept at C3* = 0, which we call pK, is some-
`what smaller; pK = 2.86 (h0.02) (where standard devia-
`tions refer only to fitting errors) when AI?j is included in
`the correction. While the absolute values of apparent pK
`are rather uncertain, these plots should indicate the ap-
`proximate dependence on C3*, which appears to be similar
`to that for the low-molecular-weight acid.
`Discussion
`Glucuronic Acid. The estimate pK = 3.23 (f0.02)
`found here for the acid dissociation constant of D-gluc-
`uronic acid may be compared with values reported pre-
`viously. In Figure 1 is plotted a value (half-filled circle)
`obtained in 0.01 M KC1 by Hirsch,% who also reported pK
`
`= 3.18 at 20 "C. Kohn and K o v ~ ~ E ~ ~ found pK' = 3.28
`(*0.01) at 20 O C for m2 = 0.003. The latter authors cited
`three other previous determinations of pK by potentiom-
`etry and one studyu employing both conductometric and
`potentiometric methods, which gave values of pK ranging
`from 3.20 to 3.24; one very early conductometric deter-
`mination30 gave pK = 3.33 (*0.02). The present deter-
`mination of pK thus confirms previous findings.
`The good agreement between results from cell A and cell
`B provides support for the correction procedure employed
`for cell A. While extrapolated values of pK are not very
`sensitive to the correction term hEj, the slope of apparent
`pK values with C3lI2 is significantly affected. The finding
`with cell B that no significant change of pK"occurs with
`change of C3 suggests, according to eq 12, that ye1
`YA.
`The slope of the plot for glucuronic acid in Figure 1 should
`therefore be similar to that expected for ycl. This ex-
`pectation may be compared with experiment by assump-
`tion of one of the single-ion conventions, such as the
`Bates-Guggenheim convention referred to above, for which
`yc1= yt,Nach The change of pK'with C2l2 shown for the
`correction with dj follows that predicted by the latter,
`consistent with the result for cell B.
`Hyaluronic Acid. A suitable starting point for the
`discussion of potentiometric titration of polyacids is the
`expression of Overbeek31 for the apparent dissociation
`constant of a polymer containing N2 identical sites of in-
`trinsic acid dissociation constant K
`pK' = PK + (0.434/N2kT)(aAel/d~)T
`(14)
`where Ael is the electrostatic free energy per polyion. As
`pointed out by Harris and Rice,6 this relation with pK'
`defined by eq 9 implies random mixing of charged and
`uncharged sites and neglect of specific counterion binding.
`Beside