Polyelectrolyte properties of proteoglycan monomersXiao Li and Wayne F. Reed Citation: J. Chem. Phys. 94, 4568 (1991); doi: 10.1063/1.460585 View online: http://dx.doi.org/10.1063/1.460585 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v94/i6 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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Polyelectrolyte properties of proteoglycan monomers Xiao Li and Wayne F. Reeda)

Physics Department, Tulane University, New Orleans, Louisiana, 70118

(Received 23 November 1990; accepted 12 December 1990)

Light scattering measurements were made on proteoglycan monomers (PGM) over a wide range of ionic strengths Cs ' and proteoglycan concentrations [PG]. At low Cs there were clear peaks in the angular scattering intensity curve I(q), which moved towards higher scattering wave numbers q, as [PG] 1/3. This differs from the square root dependence of scattering peaks found by neutron scattering from more concentrated polyelectrolyte solutions. The peaks remained roughly fixed as Cs increased, but diminished in height, and superposed I(q) curves yielded a sort of isosbestic point. Under certain assumptions the static structure factor S( q) could be extracted from the measured I(q), and was found to retain a peak. A simple hypothesis concerning coexisting disordered and liquidlike correlated states is presented, which qualitatively accounts for the most salient features of the peaks. There was evidence of a double component scattering autocorrelation decay at low C" which, when resolved into two apparent diffusion coefficients, gave the appearance of simultaneous "ordinary" and "extraordinary" phases. The extraordinary phase was "removable," however, by filtering. At higher Cs the proteoglycans appear to behave as random nonfree draining polyelectrolyte coils, with a near constant ratio of 0.67 between hydrodynamic radius and radius of gyration. The apparent persistence length varied as roughly the - 0.50 power of ionic strength, similar to various linear synthetic and biological polyelectrolytes. Electrostatic excluded volume theory accounted well for the dependence of A2 on Cs •

INTRODUCTION

Proteoglycan monomers (PGM), consisting chiefly of sulfated glycosaminoglycans (GAGs) linked to a protein backbone, I are major constituents of extracellular matrices. In cartilage, for example, it is thought that aggregates of these structures, formed by noncovalent attachment of the PG monomers to a hyaluronic acid backbone,2-4 playa fun­damental role in maintaining swelling pressure and provid­ing tissue resilience and compressibility. 5.6 It seems prob­able, in turn, that these pressure related properties are strongly linked to the polyelectrolyte nature of the proteog­Iycans.

The current picture of proteoglycan monomers, such as are extracted from bovine nasal cartilage, is that of a linear protein backbone on the order of 4000 A in contour length and of molecular weight around 300 000, with sulfated GAGs of varying molecular weights up to 20000 daltons and contour lengths up to around 400 A covalently attached at serine-glycine residue pairs.7-9 In this picture for the PGM the distribution of GAGs (chiefly chondroitin and keratan sulfate and smaller oligosaccharide fragments) is not thought to be simple or regular. There are roughly 150 GAG chains along the core protein. The core protein itself contains subregions, including a link protein for binding to hyaluronic acid in forming the proteoglycan aggregate. 10.1 1 The total weight of the PG monomer ranges from 1 to 4 X 106 daltons. 12 Proteoglycan polydispersity is thought to be due chiefly to proteolytic degradation of the core protein. 8

This type of simply branched polymer is generally referred to as a "graft copolymer."

a) To whom correspondence should be addressed.

The aim of the current work is to determine how pro­teoglycan monomers behave as individual polyelectrolytes and how they interact with each other at low concentrations. Comparison will then be made with polyelectrolyte theories and other experimental results.

In the picture of polyelectrolyte behavior evolving over the last few decades, there have emerged certain general con­cepts and experimental properties most directly applicable to linear polyelectrolytes. The Poisson-Boltzmann equation at various levels of approximation and in different geome­tries has been used to predict electrostatic persistence lengthsl3

•14 and to predict counterion condensation. 15.16

Viscosity and light scattering data have been related to the concepts of apparent intrinsic and electrostatic persistence lengths via the wormlike chain model. The problem of ex­cluded volume becomes especially difficult with addition of the electrostatic factors, and has led to a number of different approximate theories. 17-19 The hydrodynamic friction fac­tor has also been treated in the wormlike chain model in a manner similar to neutral polymers.20-22 Significant diver­gences from such theories, however, have been found. 23

-25

At low ionic strengths, neutron scattering peaks have been found for polyelectrolytes in solutions free of added electro-I t 26-28 d' . k r . y e, an, m one case, scattenng pea s were lound m the visible for extremely dilute linear polyelectrolytes.29 Oddly, a so-called "extraordinary" hydrodynamic phase observed by dynamic light scattering at low ionic strengths has been found for some polyelectrolytes, such as poly-L-lysine,30 DNA,31 polyvinyl pyrrolidone,32 polystyrene sulfonate,33 but not for others. 23.34

The current work combines dynamic and static light scattering data on bovine nasal PGM over a wide range of ionic strengths and proteoglycan concentrations [PG]. At no added salt there are very well defined scattering peaks for

4568 J. Chern. Phys. 94 (6),15 March 1991 0021-9606/91/064568-13$03.00 © 1991 American Institute of PhysiCS

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X. Li and W. F. Reed: Proteoglycan monomers 4569

I( q), whose position shifts as [PO] 1/3. As such solutions are titrated with small amounts of salt the peaks change in a fashion reminiscent of an isosbestic point. These results are discussed in terms of a simple hypothesis concerning mixed "locally correlated liquidlike" and "disordered" states. Dy­namic scattering behavior at low ionic strength is marked by a two component autocorrelation decay, which is reminis­cent of the extraordinary and "ordinary" phases seen for other polyelectrolytes in low ionic strength conditions. The extraordinary phase, however, seems largely removable by filtering. At ionic strengths above about 0.5 mM the predic­tion of a ratio of 0.67 for random coil hydrodynamic radius to radius of gyration in the nonfree draining limit is closely obeyed. The apparent persistence length follows an inverse square root dependence on ionic strength, which is not well fit by electrostatic persistence length and excluded volume theories. The latter theories, however, give a good account of the dependence of A2 on Cs •

MATERIALS AND METHODS

Proteoglycan monomers were a gift from Dr. Anna Plaas. According to the basic method of Heinegard and Sommarin,35 they were extracted from bovine nasal cartilage in guanidine hydrochloride (GnHCl) and purified by two cesium chloride density gradient ultracentifugations under "associative conditions" (i.e., at moderate ionic strength, so that the proteoglycan monomers remain associated in aggre­gates around a hyaluronate backbone). Final purification involved a cesium chloride density gradient ultracentrifuga­tion under "dissociative conditions" (i.e., at high ionic strength, 4 M OnHCI, to dissociate the porteoglycan mon­omers from the hyaluronate backbone). The high buoyant density fraction (p> 1.6 g/cm3), often referred to as "AIAID}" PG monomer, was collected and dialyzed against 50 mM sodium acetate at pH 6.8. The samples at 7 mg/ml, were then stored at - 20°C until use.

To prepare samples for light scattering, they were dia­lyzed exhaustively against pure, deionized water (conduc­tivity <} pS) for about 66 h, with four changes of water. The conductivity of the water external to the dialysis sack was practically equal to that of the pure water by the end of the dialysis period. The final external water was filtered through 0.22 pm Nylon 77 membranes and centrifuged} h at 10 000 g to eliminate dust and particles. This water was then used in diluting the salt-free proteoglycans solutions to desired concentrations. The pH of these final solutions was around 3.8-4.0. It usually then sufficed to filter these solu­tions through a 0.22 pm membrane just prior to light scatter­ing experiments. In some cases solutions were filtered through 0.1 pm polypropylene membranes. Data from such solutions are specifically indicated.

For determining [PG] after dialysis and other handling procedures, a calibration was made by careful dilution of the stock 7 mg/ml PG in 50 mM sodium acetate solution to lower concentrations (0.1-2.5 mg/ml) and measuring the absorption shoulder at 210 nm. The concentration of the original 7 mg/ml stock was determined by a dimethyl meth­ylene blue sulfated glycosaminoglycan binding assay.36 The absQrption measurements were made in a 1 mm path length

cell on a Hewlett-Packard 8451A diode array spectropho­tometer, and gave an extinction coefficient of 3.856 mg/ml X cm. This factor was subsequently used for all deter­minations of [PG].

Investigation of ionic strength effects was done with NaCI solutions. Throughout the text Cs will be used to desig­nate in mM, the amount of added NaCI in a solution. For purposes of estimation, the ionic strength due to the PGM counterion can be taken as 1.8 mM/( (mg/ml ofPG).

Light scattering measurements were made with a verti­cally polarized 5 W argon ion laser (A = 4880 A) as the source, a custom built goniometer with a Thorn EMI head­on photon counting photomultiplier tube, and a BI 2030 autocorrelator. Details of the apparatus, including calibra­tion procedures, were reported previously.37 Static scatter­ing data is usually expressed as Ke/ I, where K = [4~n2(dn/de)2/(A 4NA )], e is the solute concentra­tion in g/cm3, dn/de is the differential refractive index (measured on a Bryce-Phoenix differential refractometer), NA is Avogadro's number, and I is the absolute Rayleigh ratio. All measurements were carried out at 25°C. The angu­lar range of static and dynamic scattering experiments was typically 20°_140° with steps every 2°_5°. Dynamic light scat­tering data were generally analyzed by the standard cumu­lants procedure38 and the ratio of the second cumulant to first cumulant value squared Q, was used as a rough indica­tor ofpolydispersity. As no detailed knowledge of the PGM population molecular mass distribution was available, no at­tempt at polydispersity corrections was made in interpreting the data.

RESULTS

Figure 1 shows a typical Zimm plot for PO in 1 M NaCl. A value of 0.147 was measured for dn/de. The weight aver­age molecular weight M w is 1.16 ± 0.3 million, the root mean square z-average radius of gyration (R ~ > z is 600 ± 50 A, and the second virial coefficient is 1.7 ( ± 0.2) X 10 - 4

cm3 mol/g. Other Zimm plot determinations gave M w clos­er to 1.5 million. As the precise M w is not critical to the results or their interpretation the value of 1.16 million will be used consistently in subsequent calculations. At any rate, these M w, Az, (R; > z and dn/ de values are comparable to those obtained by other workers. 12,39 The straightness of the

3.0 ,----------------------,

Ke/l

( X106)

2.0

1.0

0.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

sin2 (eI2) + [PG]

FlO. I. Zimm plot for PO in 1000 mM C" pH unadjusted.

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4570 X. Li and W. F. Reed: Proteoglycan monomers

lines over this range of q2 (R ;) makes it probable that the PGM are random-type structures without any significant spheroidal character. We do not, however, attempt to make deductions concerning the branched nature of the PG mon­omers on the basis of the scattering data. Kitchen and Cle­land39 made such fits and concluded that they could not assert a branched structure solely on the basis of light scat­tering data.

Figure 2 shows the scattering intensity, expressed as 1/ Ke, plotted as a function of sin (e /2) for different [PG], with no added salt. The scattering vector q = (41Tn/ A) sin (e /2), e being the scattering observation angle, is equal to 0.003 42 A - I at sin (e /2) = 1 for all such graphs. Except for 0.08 mg/ml [PG] and lower concentra­tions, the I(q) peaks were fairly symmetric. Intensity peaks move toward higher q values as [PG] increases and finally moves off the axis above about 0.32 mg/ml. No secondary peaks at higher q were discernible. These peaks appear simi­lar in nature to those found by Drifford et al.29 with visible light for sodium polystyrene sulfonate (NaPSS) at very low concentrations, and to those generally found by neutron scattering for polyelectrolytes at higher concentrations in salt-free solutions.26-28 The inset in Fig. 2 shows log(qp) vs 10g[PG], where qp refers to the wavelength at the peak of I( q). The linear fit to these points indicates that qp increases as [PG] 1/3. This is similar to the qI/3 dependence for tRNA (Ref. 28) but differs markedly from most other works where qp increased as C!12 (Cp is the polyelectrolyte concentration).26.29,32,4O Those reports were based on fairly long, linear polyelectrolytes, and a sort of cylindrical pack­ing of nearly extended rods accounted for the C !/2 observa­tions. Since the PGM are short, dense structures, a simple cubic "arrangement" of isotropic objects, at least for deter­mining average interparticle spacing dj , may be taken as the simplest model in this case. Then the locations of the Bragg peaks qp' are easily calculated by

qp(cm- I) = 21T[e(g/cm3)XNA/M] 1/3. (1)

Using M = 1.16 X 106 the calculated line is shown on the Fig. 2 inset and is remarkably close to the actual qp found experimentally. Without pursuing polydispersity consider-

12 ~ ,."

I/Kc (,0.-'1

(Xl0- 5)

,~,

,~

6

O~ ____ ~ ____ ~ ____ ~ ____ ~ ____ ~ 0.0 02 0.4 0.6 0.8 1.0

5;n(812)

FIG. 2. I(q)IKc vs sin (012) for Cp = ... 0.08,.0.12,.0.15, A 0.2, 0 0.253,.0.32, and V 0.6 mglml. Inset to Fig. 2 is log(qp) vs loge Cp ); 0 are experimental qp' Cp points, dotted line are the predicted Bragg peaks qp' assuming M w = 1.16 million and a simple cubic arrangement of scatterers is also shown.

ations here we note that with M = 1.6 million, the calculated and experimental points in the Fig. 2 inset almost perfectly overlap.

It is well known that to a first approximation the scat­tered intensity should be proportional to the reciprocal ofthe diffusion coefficient.41 Figure 3 shows the reciprocal diffu­sion coefficient 1/ D(q) plotted vs q2, together with the cor­responding I(q). Both D(q) and I(q) in the figure were from a solution of [PG] = 0.1 mg/ml. l/D(q) follows the peak of I( q), at least qualitatively. Refinements ofthe theory include a hydrodynamic interaction term H(q), such that I(q) = [I + H(q) ]!D(q).42 We do not pursue here the ex­traction of H(q) from the data. Such has been done, for example, on latex spheres in benzene.43

If the peak locations were related to correlation dis­tances controlled by electrostatic potentials, then the peak may be expected to shift before it disappears as ionic strength increases. Figure 4 shows the behavior of a peak for 0.253 mg/ml PG at various ionic strengths. Interestingly, the peak does not shift in any obvious fashion, but rather just dimin­ishes in height, and finally disappears by about 0.3 mM Cs •

The peak in S( q) in Ref. 26 also did not shift with increasing C" although Cp in that case was much higher (29 mg/ml deuterated polystyrene sulfonate of M w = 72 000). Also re­markable is that in the superposition of the curves there is one value of q for which I(q) remains almost constant. This is reminiscent of an isosbestic point, which results when spectrophotometrically observing the conversion of one chemical species to another. This suggests the coexistence of two "states," each of which produces a particular I(q). This will be pursued in the Discussion. (Similar behavior of the I(q) peaks as Cs increases is observed in Fig. 10, which is presented later.)

Figure 5 (a) shows curves of Ke/ I vs q2 for 1.0 mg/ml PG at various Cs • The slope is initially negative, but as Cs

increases it becomes less negative, then zero, then grows in the positive sense, then decreases. The Ke/ I lines are ex­trapolated to higher q2 for purposes to be made clear in the Discussion. Figure 5(b) shows Ke/I vs i curves for 0.05 mg/ml PG at various Cs • The slope starts at a positive value and, after a slight increase from Cs = 0 to about Cs = 0.3 mM, falls smoothly with Cs to its final small positive value at high Cs ' Thus, as expected, the correlating influence of the

4.0 8.0

0- 1 (X107) I/Kc

(5/c m2 ) (X 10-5)

20 4.0

0.0 ~ ____ ~ ____ ~ ____ ~ ____ ~ ____ ----' 0.0

0.0 0.2 0.4 0.6 0.8 1~

5;n(812)

FIG. 3. I(q)IKc and I/D(q) vs sin(O!2) for Cp = 0.1 mglml: 0 I(q),. I/D(q).

J. Chem. Phys., Vol. 94, No.6, 15 March 1991

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X. Li and W. F. Reed: Proteoglycan monomers 4571

7.0.--________________________________ -.

I/Ke

(x 1 0- 5)

3.0

. . . , .

00 L-____ -'-___ ~ __ ___'_ ___ ~ __ ___'

00 0.2 0.4 0.6 0.8 1.0 sin(e/2)

FIG. 4. I(q)IKcYssin(8!2) for Cp = 0.253mglml, for different C,(mM): 00, • 0.0466, !:::. 0.0995,00.205, • 0.297.

electrostatic interactions is effectively screened out at low Cs

forlow Cp ' and data such as from Fig. 5(b) allows estimates of (R i)z to be made.

Using the well-known Zimm relation

Ke/! = (1 + q2(R i>z/3)/Mw + 2A 2Q(q)e (2)

yields

(R 2) = 3M d(Ke/!) g z W dq2 ' (3)

Ca)

10

~-- .... ,. ... ..... ~ '1- ~ .... ,. U·-'-"-----

OOL-__ ~ __ ~ __ ---'-___ ~_~ __ ~ __ ---'-~~ 00 02 0 4 06 08 1.0 1.2 1.4 1.6

sin 2C8/2)

4.0 .--_____________________ ---,

(b)

2,0

0.0 L-____ --'-___ ...L-___ ~ _____ ~ ____ ----'

0.0 0.2 0.4 0.6 0.8 1,0

sin 2C8/2)

FIG. 5.(a) Kcl!I(q) yssin2 (8/2) for 1.0mg/mi PGat C, (mM) = 00,. 0.25,60.5,.1,01.5, ... 5.04,.20.85, V 1003. (b) KclI(q) yssin2 (8/2) for 0.05 mg/ml PG at C, = 6. 0.53,.4.65, \( 25.28, ... 149.65.

providing that Q(q) is a constant, as is seen to be true, e.g., in Fig. 1. Using M W = 1.16 X 106 the values of (R ;)!12 shown in Fig. 6 are obtained. The data is plotted vs C s- 0.25 because of the linearity of the plot. Henceforth (R ; > z will be simply denoted by R g •

WithQ(q) constant,andsetequalto 1 inEq. (2),A2 can be calculated as a function of Cs using the q = 0 intercepts of Ke/!, such as from Fig. 5(b). The values of A2 calculated this way are shown in Fig. 7. The solid line represents values of A2 according to electrostatic excluded volume theory, and is discussed below.

Figure 8 shows the second cumulant values of the z­average translational diffusion coefficient D vs [PO] at dif­ferent values of Cs for () = 90·. Strangely, at values of Cs

below 0.5 mM, D is a decreasing function of [PO]. Assum­ing, at least for Cs > 0.5 mM, that D at Cp = 0 represents the isolated hydrodynamic behavior of the PGM, we use the Stokes-Einstein spherical approximation for the hydrody­namic diameter Dh (Dh = kT /31f'1]D, where 1] is the solu­tion viscosity) to plot the values of D near [PO] = 0 as the equivalent hydrodynamic diameter D h' in Fig. 6. Except be­low 0.5 mM Cs ' Dh follows Rg, and in fact Dh/Rg is about 1.4, which resembles the behavior expected for a non-free­draining polymer coil. This seems to be the simplest inter­pretation for the dependence of Don Cs ' and no other effects, such as electrolyte friction,44 are invoked.

The non-free-draining nature of the PG monomers is clear, since Dh is independent of Rg in the free-draining case. The non-free-draining character is also significant in that, whereas non-free-draining is often simply assumed to hold for linear polyelectrolytes, recent reports on very long linear polyelectrolytes (i.e., containing many Kuhn segment lengths) show that they are far from the non-free-draining coillimit.23

-25 In fact, within error bars, the Dh ofhyaluron­

ate and variably ionized sodium polyacrylate (PA) were found to be independent of Rg over a wide ionic strength regime. The fact that the POM are branched and much denser structures than these linear molecules makes the non­free-draining character of the PGM plausible.

From the data an approximate expression for (R ; > vs Cs (in mM) is

(4)

This shows a considerably smaller effect on R g than reported

1500 r---------------------------------, 1500

o

1000 1000

500 500

OL------~----~----~---~O

0.0 0.5 1.0 1.5 2.0 [Cs(mM)]-O 25

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4572 X. Li and W. F. Reed: Proteoglycan monomers

1.6 ~-----------------------------------,

A2

( X103)

0.8 I. I I

• -----'0-------------0--_ . 50 100 150 200

FIG. 7. A2 vsC,. Determined by + Zimm plot, intercept of Kc/I(q) via Eq. (2) for 0 I mg/ml and. 2.5 mg/ml [PG]. The solid line represents the excluded volume correction according to Eq. (24). The dotted line is the hard sphere A2 values from Eq. (27).

by Pasternak et al.45 from light scattering studies of PG monomers; Rg - 590 A in 4 M guanidine hydrochloride and Rg -1590 A in 50 mM GnHCl. An absence of a noticeable dependence on es was reported by Stivala et al. from small angle x-ray scattering (SAXS).46 Because the x-ray mea­surements required higher [PG] (3-15 mg/ml), however, the authors argue that the contribution to ionic shielding from the PGM's own counterions may be significant. By the factor of 1.785 mM/ (mg/ml of PG) given above, in pure water the equivalent total ionic strength would be - 5-15 mM. From 5 mM to 1 M es Rg varies by roughly 20%. The high concentration SAXS measurements were probably made over the overlap threshold, however, so that the Rg reported may have another interpretation.

Taking the overlap concentration e * to be

c*~M /4(R !>3/2 (5)

gives c* - 0.80 mg/ml at 1 mM es and es - 2.8 mg/ml for 1 Mes '

The positive slopes of D vs [PG] at es > 2 mM in Fig. 8 agree qualitatively with Harper et al.,47 who found positive slopes using a boundary relaxation technique for bovine na-

80

D(X109)

(cm2/s) 60

40

20

. ..... ~.

~

0

o 0

o 0.0

· 0 . ·

· . . 0

0

0

0.5

. . . =

--"-. ~ . . . . . . 0 . . .

0 0 0

0 0 0 0

1.0 1.5 20 2.5 3.0 [PG] (mg/ml)

FIg. 8. D (from second cumulant) vs Cp for C, 00,.0.495,6, 1.98, ... 11.86,049.77,. 153, 'V 503.5.

sal PG monomers. Using light scattering Reihanian et al.48

and Reed et al.37 found negative slopes of D vs polymer con­centration for PG monomers and hyaluronate, respectively. These negative slopes were probably due to polyion aggre­gates which dominated the light scattering intensity. This phenomenon is discussed elsewhere.23

•25

The linear limit of the hydrodynamic coupled mode the­ory30 for charged spheres can be used to attempt analysis of the ep dependence of the apparent (measured) diffusion co­efficient D,

(6)

where ep is the molarity of polymer, Dp is the ep = 0 limit of D (and is assumed independent of es ' which is clearly not true for the PGM in light of Fig. 6), D; is the small ion diffusion coefficient and is much larger than Dp' and Zp is the net charge on the polymer. Values of Zp ranged from 23 to 148 in going from es = 2 to 500 mM, which is against the expected trend of decreasing Zp with increasing es , and much smaller than the expected titration charge of around 3000. Such underestimates of Zp using the above theory are typical. 29

Figure 9(a) shows the second order cumulant values of D vs es for several different polymer concentrations. The positive slope, except above about 5 mM for 2.5 mg/ml [PG], is in contradiction to Eq. (3), but the abrupt rise in D

100

D(Xl09 ) (cm2/s)

75

50

25

0 0.1

2.0

D(Xl07)

(cm2/s)

1.0

0.0 0.1

(a)

1.0

10 Cs(mM)

10.0

Cs(mM)

(b)

100 1000

100.0 1000.0

FIG. 9. (a) D (from second cumulants) vs C, for Cp = 00.05,'" 1,02.5 (all 0.22 flm filtered), and. 1 mg/ml (0.1 flm filtered). (b) Large and small D components (obtained by separate single exponential fits to each halfofthecorrelator) vs C, for 6, ... 2.5 mg/ml and 0.0.6 mg/ml [PG].

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X. Li and W. F. Reed: Proteoglycan monomers 4573

vs Cs ' especially at higher Cp is reminiscent of the extraor­dinary diffusional phase reported for several different polye­lectrolytes.3

0-33 As Cs increased the polydispersity index Q

( Q = f.J./r2, where f.J. and r are the first and second moments of the series expansion of the logarithm of the scattered elec­tric field autocorrelation function) decreased from around 0.5 to 0.15. This suggested mUltiple relaxation times in the autocorrelation decay curve.

To investigate the multiple decays in the 0.22 f.J.m fil­tered solutions the correlator channels were divided into two halves (the "multitau" option of the BI 2030 correlator), the first half having a sample time of 10 f.J.s/channel and the second half having 160 f.J.s/channel. Figure 9(b) shows the () = 90· for D vs Cs at [PG] = 0.6 and 2.5 mg/ml, corre­sponding to the short and long decay components, obtained by single exponential fits to the first and second halves, re­spectively, of the autocorrelator. These curves look similar to those in the reports on simultaneously observed ordinary and extraordinary phases,31-33,49 including the existence of a Cs at which point the two phases "split." The Cs of the split­ting point increases with increasing Cp ' although the data is not extensive or accurate enough to quantify the trend. Drif­ford and Dalbiez33 reported a linear relationship between the Cs of splitting and Cp • For the 2.5 mg/ml [PG] the contin­ued drop of the large D vs Cs is qualitatively in agreement with Eq. (6) from the coupled mode theory. The large D for 1 mg/ml drops and then rises slightly vs Cs ' which does not agree with Eq. (6). The rise of the small D vs Cs in Fig. 9 (b) for both 1.0 and 2.5 mg/ml [PG] does not follow coupled mode theory but is suggestive ofthe dissolution of aggregates by salt. Such an increase in D vs Cs can be observed when a solution known to contain polyelectrolyte aggregates is ti­trated with salt.

In fact, recent data23•25 suggest that the long autocorre­

lation decay, or extraordinary phase (EP), may be simply associated with a small population of polyelectrolyte aggre­gates stable for long periods at low ionic strength. Because D vs Cp is negative for such aggregates, their size increases with concentration, which may bear on the fact that the Cs for the EP splitting depends on Cpo These aggregates and hence the EP may be "removable" by proper filtration, even though centrifugation may be ineffective. Furthermore, under cer­tain precipitation and drying conditions for obtaining the polymer powder, there is never an aggregate phase. Thus, for example, hyaluronate and variably ionized PA showed re­movable EPs, whereas some heparin samples from different sources showed an EP, and others had no EP at all. 50 There was a strong relationship between the absence of an EP and clear, visible crystallinity of the dry heparin, as seen under a microscope.

Accordingly, because the PG monomers are smaller than P A and HA, a series of experiments was performed by filtering the salt-free PG solutions through 0.1 f.J.m mem­branes instead of the customary 0.22 f.J.m membranes. For 1.0 mg/ml [PG], for example, this eliminated the large EP jump in D at low Cs ' as seen in Fig. 9(a), where data are shown for both the 0.22 and 0.1 f.J.m filtered samples. Kc/I(q) performed on the same sample, however, continued to give the negative slope of Fig. 5(a) at low Cs • Filtering a

0.2 mg/ml PG solution with the 0.1 f.J.m membrane yielded the J / Kc curve of Fig. 10. Also shown on Fig. 10 is D( q), which is broadly peaked at low Cs and then becomes fairly flat by 5 mM Cs • It thus seems that the unusual negative slope behavior of Kc/J(q) , as well as the J(q) peaks at low ionic strength, are independent of the long autocorrelation time, since such J(q) behavior persists even after the long autocorrelation decay component is almost entirely re­moved by filtration.

We have confidence in the D vs Cs data at high Cs in Fig. 9(a) and in the higher value Dcurves in Fig. 9(b). We con­sider the issue of negative and positive slopes in terms of coil expansion in the Discussion.

DISCUSSION

Scattering peaks at low C.

Probably the most salient feature of the scattering re­sults is the existence of the well-defined scattering peaks in Fig. 2. We first propose a very simple hypothesis and carry out a corresponding data analysis which qualitatively fol­lows the major trends in the data.

Figures 4 and 10 are striking in that there is a point on J(q) which remains nearly constant as the PG solution is titrated with tiny amounts of Nae!. This point is reminiscent of the "isosbestic" point encountered in photochemistry, i.e., the wavelength at which the absorbance or emission in­tensity remains constant as one chemical species is converted to another. The existence of such a point follows from the simple fact that if for two functions g(x) and hex) there exists an x' such that g(x) = hex') = A, then a*g(x') + b *h(x') = A for all a and b such that a + b = 1.

We propose that the behavior in Fig. 4 results from the progressive loss of polymer in a "locally correlated" state to polymer in a disordered state as the solution is titrated with salt. We use the term locally correlated state as the weakest term for a scattering state which can give scattering peaks. In this "mixed state" hypothesis we assume that the intensity scattered from each state is independent, and that the net observed intensity is the sum of these

6.0 6.0 D-l

I/Ke (XlO- 7) (X 10-5) (s/em2)

.3.0 .35

• •• ----........~

0.0 1.0 0.0 0.2 0.4 0.6 0.8 1.0

s;n(8/2)

FIG. 10.0.1 Jim filtered 0.2 mg/ml: I(q)/Kc result for C, (mM): 0 0, 6. 0.0428,00.114, \7 0.185, 0 4.97 and l/D(q) result for C,(mM):. 0,. 4.97.

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4574 X. Li and W. F. Reed: Proteoglycan monomers

I(q) = II (q) + I 2(q)

= KM [CI (Cs )PI (q,Cs )SI (q,C.,Cp )

+ C2 ( Cs )P2 (q,Cs )S2(q,C"Cp)]' (7)

where CI and C2 are the concentrations of polyion in the locally correlated and disordered states, respectively, subject to CI + C2 = Cp ' where Cp is the total polymer concentra­tion, and the functions P and S refer to the form and struc­ture factors, respectively, in each phase. The possible depen­dence of these functions on each of the main variables q, Cs '

and C is indicated explicitly. p ••

If the disordered state were truly in theta condItions (perfect disorder) then there would be no correlation among independent polyions, so that S2(q,Cs ,Cp ) would equal uni­ty. In the experiments on the PGM theta conditions were never achieved under any Cs or [PG] (T = 25 'C), but since data at high enough Cs is fit well by Eq. (2), we use this equation for I 2 (q), which makes use of the approximation

P2-I(q)~(1 +q2(R;)J3). (8)

It is now hypothesized that the nature of the locally correlated state does not change upon titration with small amounts of salt, (possibly because the added salt ions are absent from the region of the PGM participating in the local­ly correlated state) but that only the percentage of the po­lyion popUlation involved in that state diminishes. Thus SI (q,Cs'Cp ) = SI (q,Cp ).It is further assumed that theac~u­al shape of the polyion is practically independent of WhICh state it is in, so that PI (q,Cs) = P2 (q,Cs) = P(q,Cs)' Fur­thermore, since the dramatic changes in Figs. 4 and 10 occur over such a narrow range of added salt, 0--0.3 mM, it is rea­sonable to approximate P(q,Cs) = P(q,O) over this narrow range. Thus

I(q,Cs'Cp ) ~KM{CI (Cs )P(q,O)SI (q,Cp )

+ C2(Cs )/[ 1/P(q,O) + 2A2C2(C.)M J}. (9)

Equation (9) predicts that for low total Cp ' such that 2A2C2M~ 1/P(q) , there will be an approximate or "quasi­isosbestic" point at a certain q if there exists a point q; where SI (q;) is near unity. A perfect isosbestic point would exist if theta conditions prevailed in the disordered state (A 2 = 0) for S(q;) = 1.

We can now analyze the data as follows: Since CI (Cs )

= 0 for low salt concentrations (e.g., > - 1 mM for 0.05 mg/ml [PG]), the experimental values of Kc/I(q) vs q2 can be used to extrapolate to Cs = 0 and Cp = 0 to obtain P(q,O). Performing this extrapolation on the data yields a form factor for polyions at Cs = 0 with all influence of SI (q) from the locally correlated state eliminated;

P(q,O) = [1 +2.76sin2 (B/2)] -I. (10)

We note parenthetically that use of this form factor implies that the I(q) peaks cannot be due wholly to an intramolecu­lar effect. An intramolecular effect seems extremely unlike­ly, both on theoretical grounds, and on the linearity of P- I (q) even at C. = 0 for very low [PG] [e.g., Fig. 5(b)].

Now, at no added salt we assume that CI (0) = Cp (all the polyions are in the locally correlated state), so that we

can extract SI (q) by

SI(q,Cp ) =I(q,O,Cp)/[KCpMP(q,O)], (11)

where Mis the molecular weight of the PG monomers. Some representative curves of S\ (q,Cp ) are shown in Fig. II. First of all, for 0.08 mg/ml, [PG] ,0.32 mg/ml there isa peak to SI (q,Cp ), so that the total intensity peaks seen in Fig. 2 can­not be due, for those concentrations, to the product of a monotonic increasing Seq) and a monotonic decreasing P( q), which has been proposed, for example, to explain neu­tron scattering peaks in polymer melts51 (the "correlation hole" argument). This correlation hole argument may be valid, however, for [PG] ,0.08 mg/ml, for which the I(q) peaks are less symmetrical and well pronounced than for the higher concentrations.

Since theS\ (q,Cp ) curves of Fig. 11 all have a point near unity, the quasi-isosbestic point must exist. Using the SI (q,Cp ) and P(q,O) curves obtained from the data via Eqs. ( 10) and ( 11 ), the total intensity is calculated by Eq. (9). A value of A2 = 1.55 X 10 - 3 from the manual extrapolation of the experimental A2 to C. = a in Fig. 7 is used. The results for [PG] = 0.253 mg/ml are shown in Fig. I2(a) for differ­ent values of the fraction of PGM in the locally correlated state CI/C ,and are to be compared with Fig. 4. The qualita­tive agree~ent is remarkable. The quasi-isosbestic point seen in Fig. 4, as well as the trend for I( q) at low q to rise with C., for I(qp) to diminish, and for the peak to finally disap­pear are reproduced in Fig. 12(a) by Eq. (9).

The mixed state hypothesis also gives qualitative agree­ment with the scattering for higher [PG], where the peak was off the high end of the q axis, and KC /1 vs i appeared as negative-slope lines at low Cs' Figure 12(b) shows theKC /I vsq2 curves calculated according to Eq. (9) for [PG], = 1.0 mg/ml, which can be compared with Fig. 5(a). The isosbes­tic point is even more smeared out and harder to identify than in Fig. I2(a) (as in ray optics of astigmatism it can be located by the vertical "circle ofleast confusion"), which is likewise predicted by Eq. (9) as an effect of the 2A 2C2M term becoming larger with respect to 1/ P( q) as C2 increases. The Kc/I linesforhigherCs values in Figs. 4 and I2(b) are not extrapolated to higher i, as P(q) changes significantly

2,0,-----------------

S(q)

1.5

1.0

0,5

0,0 L-=~~~=='==::::;::::::=_~_~ 0,0 0.2 0.4 0,6 0,8 1,0

sin(e/2)

FIG. 11. SI (q,Cp ) obtained from I(q,O,Cp ) curves (as in Fig. 2) via Eq. (11) for Cp = ... 0.08,00.12,. 0.15,0 0.2,.1 mg/ml.

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X. Li and W. F. Reed: Proteoglycan monomers 4575

70r---------------------------------~

(a)

30 ••• 0 0

.' 0 •••••• 0° •••••••••••••••••••• 0°0

000°0

OOL-____ ~ ______ ~ ____ ~ ______ ~ ____ ~

00 0.2 0.4 0.6 0.8 1.0 sin(8/2)

KC/I

(Xl00) (b)

20

0.0 '------'------'----~----~--~~--~-~ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

sin 2(8/2)

FIG. 12. Calculated I(q,C"Cp)IKc vs sin(812) from Eq. (9) using the values of SI (q) and P(q,O) obtained from the data via Eqs. (10) and (11), for (a) 0.253 mglml; for fraction of polymer in locally correlated state, C,ICp = 00.2, ~ 0.4, L. 0.5, .0.8, Oland (b) KcII(q) vs sin2 (8 /2) for 1.0 mglml; for C/Cp = 0 0.6, ~ 0.7, L. 0.8,.0.9,00.96, .0.98.

and cannot be taken as in Eq. (10), andA 2 also changes (Fig. 7).

Although this phenomenological mixed state hypothe­sis accounts for the most prominent features of the scattering peaks, it does not elucidate the nature of the locally correlat­ed state or the quantitative dependence of C\ and C2 on Cs •

There are several theories on the similar peaks found in light and neutron scattering. These generally start with the ap­proximation, also used above, that l(q) = S(q)P(q). A form for the radial distribution function g(r) is then usually assumed or modeled, which depends on particle shape, inter­particle potential energy and concentration functions. Seq) is then calculated from g(r) by

Seq) = 1 + 41Tn LX> [g(r) - 1]

X [sin(qR)/qR ]R 2dR,

where n is the number density of polyions.

(12)

It is not our intent to make detailed comparisons between our results and these many different theories, but rather to summarize relevant features of these theories to get an idea of the simplest criteria for the existence of the ob­served peaks.

The simplest g(r) is for a hard sphere of radius R, for whichg(r) = 0 for r<R andg(r) = constant for r> R. Via Eq. (12) this leads to an Seq) oftheform52

Sex) = (1- 24CJ\(x)/x] , (13)

where x = 2qR, C u ( = 41TR 3Cp /3) is the solute volume fraction and J\ (x) is the first order spherical Bessel function, which manifests damped oscillations. Thus even for a system of spheres interacting only through a hard core potential, an oscillatory Seq) will build in as Cu increases. At low Cu Seq) merely increases monotonically over our range of q, so that, again, the correlation hole argument may apply for [PG] <0.08 mg/ml. Undulatory x-ray patterns from liquids often resemble those expected for hard spheres of high vol­ume fraction. In Fig. 2, however, [PG] runs from 0.06 to 0.32 mg/ml, which would give a rough range of volume frac­tions from 0.005 to 0.03, assuming R = 350 A (using R equal to the hydrodynamic radius R h , at Cp = 0 and low salt). Clearly a simple hard sphere potential for the PGM, with R ~Rh could not give the kind of high contrast peaks [contrast defined as the ratio of S(qp )/S(O) seen for Seq) in Fig. 11]. Even if there were a high volume fraction of PG acting as hard spheres with R~Rh' then the peaks would not totally disappear as Cs increased, but rather their con­trast would simply diminish due to a decreased Cu as Rh shrank.

A simple approximation for treating the screened Cou­lomb potential between macroions is to continue to use a squareg(r) as above but to replace R with an effective radius Relf , which depends on the ionic strength. Benmouna et al.53

examined this case, and, defining r = Relf/R, obtained Eq. (13) where 24Cur1J\(xr)/xr replaces the second term in brackets. Thus with substantially lower Cu ' high contrast peaks in Seq) should be observed when r> 1. This square g(r) with ionic strength dependent Relf may thus be the sim­plest possible model to account for the existence of a high contrast Seq) peak at Cs = O.

The correlated state, even at low actual Cu , can then be said to resemble liquid-like order, due to the electrostatically enhanced effective Cu' If r is varied in the S(x) of Eq. (13), using the electrostatically enhanced Relf mentioned in the last paragraph, however, the effective square well model does not even qualitatively predict the behavior of the ob­served l(q). Notably, using parameters appropriate to the experiment, an isosbestic point is not generated. Note that in testing such models for the range 0.08 mg/ml < [PG ] <0.32 mg/ml, Seq) must go from a peaked form at no added salt to a horizontal line at high Cs ' in order to agree with experi­ment. A refinement to the effective square well potential in Ref. 53 uses a linearized screened potential to obtain g(r), and thence Seq). In using experimentally appropriate pa­rameters this model is actually worse, in that it predicts only a monotonically increasing Seq), and never a peaked form, even at Cs = O.

A perhaps more realistic potential is that of a hard sphere at r < R, with a screened potential, exp( - Kr)/r, for t> R. K is the Debye-Hiickel screening parameter. Seq) cal­culated in this case54 shows qualitatively similar oscillatory behavior at high Cu between pure hard spheres and those with the screened potential. At low Cv (e.g., ..... 0.05) there

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4576 X. Li and W. F. Reed: Proteoglycan monomers

are high contrast peaks with strong Coulombic potentials which quickly diminish to the broad low contrast peak for Seq) as the potential is reduced.

As R goes to a (approaching a point ion) in this latter model, Seq) reduces to

(14)

where (7 is a screening length dependent on the macroion charge. This Seq), while able to lead to a peak in l(q) when multiplied by an appropriate monotonically decreasing P( q) (correlation hole-type argument, although P(q) = I for a point scatterer) has itself no peak and hence cannot be used to model the Seq) curves for PGM, at least above 0.08 mg/ml.

Thus, qualitatively, the square g(r) with Relf and the square g(R) with screened potential are adequate to account qualitatively for the existence of the l(qp) at Cs = 0, al­though the effects of increasing Cs are not accounted for by it.

The simplest picture for the mixed state model can now be stated: At no added salt there is a correlation between PG monomers, at least between nearest neighbors, whose aver­age distance from each other is very close to that predicted by a simple cubic packing of point masses (inset of Fig. 2). Without an explicit calculation of energy changes compared to kT for excursions away from the average interparticle spacing, it can be surmised that the effective volume frac­tio?, in the sense of Relf due to the Coulombic potential, is qUite large compared to the hard sphere (actual) volume (R~Rh)' so that high contrast peaks at low actual C can occur. Since no secondary peaks were observed we c;nnot assert, nor is it necessary to assert, that correlations persist significantly beyond nearest neighbors [JI (x)/x peaks rap­idly decrease in height, anyway]. This model hence does not require long range order, but does not exclude its possibility, either.

Now, as salt is added, the mixed state model asserts that correlations between a certain fraction of the particles are destroyed to the extent that a q-independent A2 is sufficient to describe the residual interactions for the disordered popu­lation. Where the interparticle Coulomb potential between neighboring particles is not weakened (perhaps because of "gaps" around some particles in the very low concentration added salt ions), those particles will remain correlated at around the same average interparticle spacing do with the same S(q), their contribution to the total scattering still be­ing given by the first term in Eq. (10). As is clear in Figs. 4 and 10, very small values of Cs quickly lead to the loss of the correlated state (e.g., a nearly flat l(q) results by C';C ~ 1 in Figs. 4 and 10). At present we do not attempt to finlhow the fraction of polymer in each state C I and C2 , depends on CJCp and possibly other parameters.

Thus this mixed state hypothesis is to be carefully distin­guished, say, from the "two state" model of Ise et 01.,40 who proposed a coexistence of disordered and ordered macroion states. The mixed state hypothesis here envisions ( 1 ) a popu: lation of PG monomers disordered by local salt ions, whose interactions are adequately described by a q-independent A 2,

and (2) a population of PG monomers which are pairwise

correlated via an electrostatic potential unscreened by salt ions, which leads to a liquidlike order.

Why two states should coexist is not clear to us, so that the two state model remains phenomenological, supported by the agreement of its predictions with experiment. It is not clear, for example, ifit is possible for added salt to be distrib­uted in such a heterogeneous fashion as to produce the two states (e.g., gaps in the concentration of added salt ions maintaining the correlated state), rather than just to be dis­tributed uniformly, thus gradually diminishing the effective radii of the entire population. At high enough ionic strength, at any rate, all PGM are screened and only the disordered state exists. Since the existence of a quasistable lattice of any type is not necessary, PGM can pass between correlated and uncorrelated states rather rapidly, this being dependent mainly on the instantaneous distribution of small ions. The fact that the long autocorrelation decay curve at low ionic strengths is largely removable by filtration, and hence prob­ably represents stable, entangled PG aggregates rather than some type of organized structure, suggests that there is no inherent large difference in diffusion rates between PGs in the two states.

Having established that Seq) in the correlated state has a genuine peak, it is tempting to interpret the peak width in terms of an average mean square deviation from the Bragg spacing d;. Without attempting to extract g(R) from Eq. (12) using the experimental Seq), we simply resort to the disorder parameter of Guinier (Ref. 55), a1, whichcharac­terizes a loss of long range order. In such a treatment the integral peak width of Seq) is related to the mean square deviation of neighboring particles from the Bragg distance corresponding to the peak. In fact, Podgornik et 01.56 adapt­ed Guinier's treatment to interpreting x-ray peak widths from hexagonally organized DNA rod like arrays under high osmotic stress. Using similar reasoning for a simple average cubic PG arrangement we get for the mean square deviation a1 of nearest neighbors

a1=aqd1/ffl, (15)

where aq is found from integration of the experimental Seq) by

aq = Sl(q)dq . l(qp)

(16)

We obviously cannot consider correlations between more distant neighbors in this fashion without experimentally measured secondary scattering peaks. From 0.06 to 0.32 mg/ml [PG] there is no detectable systematic change in aq, but rather averages aq = 1.43 ± 0.22 X 105 cm - I • This cor­responds t<? root mean square values of aB of from about 935-2150 A over the range 0.32 to 0.06 mg/ml [PG].

The ratio (a 1 ) 1/2/ d B thus varies from around 0.52-0.68 in this picture, so that even the locally correlated PGMs are far from being in a neat lattice arrangement.

Apparent persistence length and excluded volume considerations

The decreases in Rg and Rh with increasing C seen in Fig. 6 show that PG monomers behave as flexible ;olyelec-

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X. Li and W. F. Reed: Proteoglycan monomers 4577

trolytes in terms of the sensitivity of their dimensions to ionic strength. The phenomenon in this case is probably due to two related effects; as the ionic strength increases the short GAG chains forming the branches of the PG monomer be­come shielded and shorten as their persistence lengths de­crease, which decreases R g • Second, and probably more pre­dominantly, the shielding of the individual GAG chains from one another at their irregular root spacings allows the PG to relax and shorten along its long protein backbone axis. Development of a theory to account for these effects and predict the electrostatic contribution to the total persistence length is beyond the scope of this article. It is instructive, nonetheless, to analyze the ionic strength dependence of the Rg in terms of existing theories for linear polyelectrolytes.

Since there is no evidence in the P( q) data of any spheri­city to the GAG monomers (Stivala et al.,46 however, claim a spherically symmetric mass distribution based on compari­son of experimental and theoretical particle distance distri­bution functions) it seems a more random model, such as a wormlike chain type is more appropriate. A slightly better model for the PGM according to current structural models would be a "wormlike cylinder" with finite backbone radius around the root mean square end-to-end length of the pen­dant GAG chains. If we recall that for a rigid cylinder of length L and radius R the radius of gyration is R; = L 2/12 + R 2/2, then considering that a 20000 dalton GAG would have a maximum extension of around 400 A, and the protein backbone has a contour length of around 4000 0 2

A, then the L /12 term contributes around 94% of the total R;. Stivala et al. report PG cross sectional R values between 25-100 A, compared to full R ~500 A,gso that there is experimental evidence that it is th~ dimensions of the core protein which dominate the total (R ;). Thus it may be reasonable to relate the mean square unperturbed R to the total persistence length LT and the contour length L by the usual formula

(R ;)0 = LLT/3 - L} + 2L }/L - 2(L j./L 2)

X[1-exp(-L/L T )]. (17)

Since light scattering measures the perturbed (R ;), an ap­parent total persistence length L T can then be extracted from the measured (R ;), at each value of ionic strength assuming a total contour length L for the protein backbone. The experimental fit to Rg given by Eq. (4) was used to calculate the L T by Eq. (17). Using an algorithm that searches for the exponent r which gives the maximum linear correlation coefficient r, in the relation L T = L 0 + bC r, a value of r around - 0.49 was found for the range of C = 1-1000 mM, assuming L = 4000 A. Assuming higher ~alues for L led to progressively smaller values of r. The L T assum­ing a contour length of L = 4000 A for the protein backbone are shown plotted against C ,- 0.5 in Fig. 13. Since the number of Kuhn statistical segments varies from about 7.8 at Cs

= 1000 mM to about 2 at Cs = 1 mM, the PG M are far from being in either the rod or random coil limits. Considering L T to be composed of apparent intrinsic and electrostatic persis­tence lengths

(18)

1000,------------------------------------

(A)

500

o+----------------+----------------~ 0.0 1.0

FIG. 13. L ~ vs C ,- 0.5, where the L ~ are calculated from measured (R 2 > viaEq.(17). 8

the intrinsic portion is found from the y intercept (infinite ionic strength limit) to be around 240 A. This apparent val­ue is considerably higher than the 50 A for the true intrinsic persistence lengths reported in Ref. 46 by small angle x-ray scattering (SAXS) measurements and in Ref. 39 based on a "branched block copolymer" model in conjunction with light scattering data, and may indicate a considerable ex­cluded volume effect.

A model for the electrostatic persistence length which might fit the branched PG monomer is not known to the authors. Using the well-known Odijk 13 and Skolnick/Fix­man 14 theories of electrostatic persistence lengths for infi­nitely thin linear polyelectrolytes, in the limit where LK~ 1 the unperturbed Le is approximated by

Le = 52/4~Q, (19)

where 5 is the Manning condensation parameter (number of elementary charg~s per Bjerrum length) and Q is the Bjer­rum length (7.16 A m water at 25 °C). This equation, which takes no account of excluded volume effects predicts that Le~Cs-I.

Since two of the basic assumptions of this model for L in the case ofPG monomers are clearly violated, namely tha~ the polyelectrolyte be infinitely thin and that there be a uni­formly smeared out linear charge density, the discrepant rough experimel!tal powerlaw dependence of r ~ - 0.5 (for L around 4000 A) may not seem surprising, and the linear model may be simply dismissed as inappropriate for this case. What is interesting, however, is the fact that a power law of L ; = C s- 0.5 has now also been found for a wide var­iety oflinear polyelectrolytes. These include polystyrene sul­fonate,57 hyaluronic acid23 and variably ionized polyacrylic acid.25 One possibility given57 was that taking account of the chain diameter and solving the full nonlinear Poisson-Boltz­mann equation, as did LeBret58 numerically, can lead to a rough r = - 0.5 power law at ionic strengths above around O.lmM.

Another possibility was that since Eq. (17) gives the unperturbed (R ;) 0' and light scattering measurements give the perturbed (R ;), electrostatic excluded volume theories might be used to obtain an approximate value of the expan­sion factor a ( (R ;) = a2 (R ;) 0)' This is done in Ref. 23 by

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4578 X. Li and W. F. Reed: Proteoglycan monomers

using the Gupta-Forsman equation for a

a5 - a 3 = (134/105) (1 - 0.885N k- 0.462)Z, (20)

where Nk is the number of Kuhn segment lengths in the polymer and z is the standard excluded volume parameter from perturbation theory

Z= (3/2rr/1)312{3NJ/2, (21)

where lk is the Kuhn segment length. The Skolnick-Fixman form for the binary cluster integral p, is used;

p~8L ~K-1R(w), (22)

where w = 2rrs 2Q -IK-I exp( - Kd), d being the cylinder diameter, and

('T12 rWlsin (J

R(w)=Jo sin2BJo x-1e-xdxdB. (23)

In the coil limit of hyaluronate and partially ionized PA this correction was qualitatively good. Recent Monte Carlo sim­ulations using the Debye-Hiickel approximation for an infi­nitely thin chain in a threefold rotational isomeric state mod­el59 also show that a rough Le - C ,- 0.5 power law arises, regardless of how smoothly or irregularly the charge is dis­tributed. Excluded volume corrections of the type outlined above brought the theoretical (R; > into good agreement with the Monte Carlo values. In the case ofPGM it is hard to define a linear charge density. Taking there to be roughly one million daltons of sulfated GAGs, condensed to S = 1, i.e., with roughly one elementary charge per 350 daltons of GAG, yields around 2900 elementary charges distributed over the 4000 A contour length of the protein backbone. The value of sin win Eqs. (22) and (23) is thus taken as around 5.2. Applying the same corrections to PG monomers, how­ever, gives a small difference between the perturbed and un­perturbed Rg's (i.e., a< 1.1). Kitchen and Cleland39 also calculated a very small a ( ~ 1.06) for bovine nasal PG in 4 M GdHCI. In the absence of appropriate theories for the peculiar graft polymer nature of the PGM it remains puz­zling why L ;aC;- 0.5.

Surprisingly, however, excluded volume calculation based on slight modification ofYamakawa's21 Eq. (21.5) for A2

(24)

give a remarkably good fit to the A2 vs Cs data shown in Fig. 7. The solid line in Fig. 7 represents A2 as calculated by Eq. (24), where

(25)

and

hoCz) = [1- (1 + 3.903z) -04683]11.828z, (26)

where a, p, and z are as given by the above expressions. A2,hs

represents high Cs limit of A2 and was found experimentally to be around 1.7 X 10 - 4. The excluded volume correction was somewhat sensitive to the cylinder diameter d and so a value of 50 A (twice the cross sectional Rg for PGM from Ref. 46) was used. It was quite insensitive to intrinsic persis­tence length (50 and 240 A gave virtually the same fit). It should be noted that the A2 in Fig. 7 were determined at Cs

high enough to consider that C1 in Eq. (9) (concentration in

locally correlated state) is negligible, so that [PG] ~ C2 and Eq. (2) with Q(O) = 1 is valid and represents the total scat­tering intensity. This assumption may be responsible for the small but systematic difference in A2 values obtained at [PG] = 1.0 and 2.5 mg/ml.

A hard sphere calculation of A2 with R = 300 A ( = Rh at high Cs ) and M = 1.16 million gives A2 = 2.02 X 10 - 4 . Although this value is in reasonable agreement withA 2,hs the inability of the purely hard sphere intera~tion to account for A2 vs Cs is shown by the dotted line in Fig. 7, calculated from

A2 = (16rrNA R 3)/(3M2), (27)

where R is taken as 0.57 times the value of (R ; > 1/2 given in Eq. (4), which guarantees convergence of A2 to A2,hs at high Cs • The hard sphere curve is seen to underestimate A2 at low Cs·

Poly ion expansibility correction to the coupled mode Eq. (6)

The linearized coupled-mode result, Eq. (6), assumes that Dp is independent of Cs' Since Dp is an experimentally known function of Cs (Fig. 6), Eq. (6) can be rewritten simply as

D~Dp.hs(1 + 500CpZ~/Cs)/(1 + 1.65C s-O

.2S

), (28)

where Dp,hS is the diffusion coefficient at high salt and Cs is now expressed in mM to match Eq. (4). When substituting Cp values into Eq. (28) with typical Zp values (20-150) the qualitative behavior of Figs. 9(a) (above 5 mM) and 9(b) (all Cs, higher value D curves) is recovered, i.e., a negative slope for high Cp and an initially negative then positive slope forintermediate Cp (e.g., 0.6 mg/ml). Thus even though the Zp values are low, the coupled mode theory with polyion expansivity included, continues to agree qualitatively with the experimental trends.

Removability of the extraordinary phase

As seen in Figs. 9(a) and 9(b) there appears to be a low value of D, originating in a "slow mode" of the scattering autocorrelation decay function, which is reminiscent of the so-called extraordinary phase reported for many polyelec­trolytes in low ionic strength conditions. Also seen in Fig. 9(a), however, is evidence of the removability of the EP by filtration of the PGM solution through 0.1 f.lm membranes. In that case there is a gradual, smooth increase of D vs Cs

rather than ajump. Also, Q was -0.2 even at no added salt and no "slow phase" could be observed when dividing the correlator into two halves as was done in Fig. 9(b). In fact, the difference in Fig. 9(a) between the D for the 1 mg/ml [PG] filtered through 0.22 and 0.1 f.lm membranes is strik­ingly similar to Fig. 4 in a study by Zero and Ware60 on the EP of poly-L-Iysine (mol. wt - 90000). In that figure the abrupt rise in D [similar to the data for 0.22 f.lm filtered 1 mg/ml PGM in Fig. 9(a)] was obtained by dynamic light scattering, whereas the smoothly rising data [similar to the 0.1 f.lm filtered 1 mg/ml PGM in Fig. 9(a)] were obtained by the fluorescence recovery after photobleaching technique (FRAP). Zero and Ware point out that dynamic light scat­tering is the only known method which reveals the EP and its

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X. Li and W. F. Reed: Proteoglycan monomers 4579

transition to the ordinary phase. In light of the fact that light scattering is extremely sen­

sitive to the presence of even small amounts of aggregates, whereas techniques such as FRAP are not, and considering that the apparent EP for PGM is largely removable by filter­ing, we conclude that the EP is due simply to the presence of a tiny amount of entangled polyelectrolyte aggregates. The pore size of the filter with respect to the dimensions of the polymer is clearly of critical importance in removing the aggregates. These aggregates appear fairly stable at very low ionic strengths, but become untangled as Cs increases. There are a number of ways that these aggregates can form, includ­ing lyophilization prior to dispersal in solvent, and no im­purities need be involved. We defer conjectures as to the mechanisms of the aggregates' stability at low Cs to a future work, but note that the trend seems fairly general for other polyelectrolytes.23.25.50 Significantly, Fig. 10 shows that the existence of the I(q) peaks at low Cs are not affected by the absence ofthe EP, i.e., the peak persists even after removal of aggregates by filtering. The aggregates are not "temporal" or fleeting in the sense that, at least on a time scale of hours, they do not reappear after removal. That this temporal be­havior may not be general for all polyelectrolytes, however, is evidenced in a recent work on short (150 base pair) DNA fragments, which reported the gradual reappearance, after filtration, of what those authors termed "loose aggregates or localordering."6I,62

SUMMARY

Bovine nasal PG monomers behave as non-free-drain­ing polyelectrolytes whose apparent electrostatic persistence length varies roughly as C s- 0.5. At very low ionic strengths there are well-defined light scattering peaks in the range q = 0.001 to 0.003 A - I for [PG] = 0.062--0.3 mg/ml. A simple mixed state hypothesis (coexisting disordered and liquidlike locally correlated states) is presented which qual­itatively explains the behavior of I(q) at different Cs and Cpo Interpretation of the integral peak-width t::.q of the Seq) peaks in terms of Bragg center disorder t::.~ shows that the local correlations are probably quite weak. The slow or ex­traordinary diffusional phase apparent at low Cs appears to be removable by filtration through 0.1 pm membranes. Thus its origin is thought to involve PG aggregates. The persis­tence of the locally correlated phase static scattering behav­ior even after the removal of the extraordinary phase strong­ly suggests the two phenomena [peaks in I(q) and a long term decay in autocorrelation] are not related. The rough C s- 0.5 dependence of the apparent electrostatic persistence length L; is hard to understand in the light of persistence length and electrostatic excluded volume theories for linear polyelectrolytes, although the latter theories account quite well for the dependence of A2 on Cs •

ACKNOWLEDGMENTS Support of this work from National Science Foundation

Grant No. DMB-8803760 is gratefully acknowledged. We are grateful to Dr. Anna Plaas for providing the purified, characterized proteoglycans and for extensive orientation in their handling and properties.

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