BIOPOLYMERS
VOL. 14, 409-417 (1975)
Infinite-Dilution Viscoelastic Properties of Tobacco Mosaic Virus NOR10 NEMOTO, JOHN L. SCHRAG,andJOHN D. FERRY, Department of Chemistry; and ROBERT W. FULTON, Department of Plant Pathology, University of Wisconsin, Madison, Wisconsin 53706
Synopsis The storage and loss shear moduli, G‘ and G”, have been measured for dilute solutions of unaggregated and aggregated tobacco mosaic virus samples in glycerol-water mixtures, by the Birnboim-Schrag mu1tiple-lumped resonator modified for use with aqueous solvents. The frequency range was 100-5800 Hz, the concentration range 0.6-2.1 X 10-3 g/ml, and the temperatures 25.0” and 37.8”C. The number-average and weightaverage molecular weights of the aggregated sample were estimated as 1.4 and 2.0 X 108, respectively, from electron microscopy. The extrapolated intrinsic moduli [G’] and [G”] were compared with the predictions of the Kirkwood-Auer theory for rigid rodlike molecules. For the unaggregated sample, the frequency dependence of [G’] and [G”] agreed well with the theory assuming the intrinsic viscosity to be 27 ml/g, though the asymptotic limit of [G’]M/RTa t higher frequencies was slightly larger than the theoretical value of 3/5. For the aggregated sample, the data agreed with the theory for rigid rods as modified to account for molecular-weight distribution.
INTRODUCTION The viscoelastic properties of macromolecules a t infinite dilution are very sensitive to their conformations as well as their shapes. For several years, extensive m e as~ remen tsl-~ have been made on dilute solutions of synthetic polymers in organic solvents by the multiple-lumped resonator (MLR) with a computerized data acquisition and processing system.* The results were successfully extrapolated to infinite dilution and interpreted in terms of several molecular as recently reviewed.’ Similar studies on biological polymers with the MLR were impossible until very recently, because the low-loss aluminum alloy from which the resonators were fabricated was very susceptible to corrosion in aqueous solvents. After several attempts t o correct this deficiency, a titanium alloy (Ti-6A1-4V) with special heat treatment was found to be highly resistant to corrosion and also to have the required low mechanical loss (high &). I n this paper we describe the first viscoelastic measurements on a biological polymer in a n aqueous solvent by the MLR with a titanium resonator. According to several molecular theories,8--’2the frequency dependence of the storage and loss shear moduli, G’ and G”, of a very dilute solution of a 409
@ 1975 by John Wiley & Sons, Inc.
YEMOTO ET AL.
410
rigid rodlike macromolecule is characterized by a single relaxation time associated with end-over-end rotat,ion. I n earlier m e a s u r e r n e n t ~on ~ ~solutions of paramyosin in glycerol-water mixtures, the frequency dependence of G‘ and GI‘ was found to be close to the theory for rigid prolate ellipsoidsl0-l2 a t low frequencies; however, a t high frequencies did not approach a constant value as the thcory predicts, but increascd with frequency. Since the earlier measurements were made a t concentrations too high to permit extrapolation to infinite dilution, this deviation could be attributed t.o either intermolecular interaction or some degree of int,ramolecular flexibility, or both. With thc RILR, it is possible to extrapolate measurements of viscoelastic properties to infinite dilution to eliminate any cffccts of intermolecular interaction. Since tobacco mosaic virus (TMV) appears to be the most rigid rodlike macromolecule known, it was chosen for thc present study, which provides the first demonstration of agreement wit,h molecular t.heory for rigid rods.
THEORY The predictions of molecular theory for rigid rods8t9and prolate ellipsoids1°-12 have been summarized in a previous we use the same notation here. It is convenient to express the viscoelastic properties in terms of the reduced intrinsic moduli [(if], and [ G ” ] R , defined as follows: [c’]R =
( M / R T ) lim C’/c c-0
[G”],
=
( M / R T )lim (GI‘
- wqS)/c
(2)
c+o
where w is radian frequency, M molecular weight, R the gas constant, T absolute tempcrature, c concentration (g/ml), and qs the solvent viscosity. The predicted frequency dependences are givcn by
[G’]R = mlw2rO2(1
[G”]R=
WTo 70
+ w2r02)-1
[“Zi(l $-
=
W2T02)-1
(3)
+
(4)
.In21
m[aI?l,M/RT
(5)
where T O is the relaxation time (inversely proportional to the rotary diffusion coefficient) and [ q ] is the intrinsic viscosity; m = l/(ml m2). For the Kirkwood-Auer theory, which we shall use here, corresponding t o a rigid rod with a very large ratio of length to the rcpeat distance between points of frictional resistance, ml = 3/5 and m2 = 1/5. Since our experimental data have been obtained a t frequencies higher than r O - l ,we emphasize the limiting behavior of [G’IR and [G”]R a t high frequencies; as w ---t m ,
+
[G’IR
=
[e“]R=
ml
(6)
m2r0w.
(7)
VISCOELASTICITY OF TOBACCO MOSAIC VIRUS
41 1
From Eq. (7), the ratio of the high-frequency limiting intrinsic viscosity [q’], to [71 ism2l(m1 m2). When the effect of molecular-weight distribution is taken into account in the theories, the intrinsic moduli [G’] and [G”] should be expressed as
+
[G‘] = lim G’/c c-0
=
RT T
cnj 1?n+lw ro3 RT c njwroj Mn
firn
[G”] = lim (G” - wq,)/c c-0
u2roj2
=
+ m2)
T
j
(9)
m h I,q3f,IRT. (10) Here is the number-average molecular weight, n3 and ro3 are the mole fraction and the characteristic relaxation time of molecules with the molecular weight If,. Comparison of Eq. (8) with Eq. (3) shows that the numberaverage molecular weight is a suitable substitute for A4 to define the dimensionless moduli [(;’IR and [G”IR in the presence of the molecularweight distribution. Equation (6) is then applicable to a rigid polymer with any molecular-weight distribution. Equation (7) may be written as 705 =
an
nn
[C”],
=
[G”]dd,/RT
=
m2w~n3r0,
(11)
I
The ratio of [q’],/[ql0 has the same value m2/(ml molecular-weight distribution.
+ m2),unaffected by
EXPERIMENTAL Materials Tobacco mosaic virus was purified from extracts of infected tobacco (Nicotina tabacom L. var. Havana 38) by ammonium sulfate precipitation and isoelectric precipitation, followed, after storage, by one cycle of highand low-speed centrifugation. After purification, the solutions of TAW in 0.001 M EDTA were stored in a refrigerator. The viscoelastic measurements wcre made on two samples with different storage periods; TMV-RI for two weeks and TAN-A for three years. Centrifugations in sucrose density gradients showed that the former consisted mainly of monomers with a small amount of dimers, whereas the latter was highly aggregated by end-to-end association. For the highly aggregated sample, the number-average and weight-average molecular weights were calculated from measurements on electron micrographs (magnification 18,lOO), with counts of numbcrs of species corresponding to j monomeric lengths, j ranging from 1 t o 15. The mole fraction n3 of j-mers is given in Table I, and the calculated averages in Table 11. The molecular weight of sample TAN-M was taken as that of a monomer,14neglecting the presence of dimers. Mixtures of glycerol and water containing 0.001 A f EDTA were used as solvents; Table I1 also gives the weight percents of glycerol, densities ( p ) , and viscosities.
NEMOTO ET AL.
412
TABLE I Distribution of Lengths in Sample TMV-A j ?L j
j nj
1 0.133 7 0.034
2 0.200 8 0.021
3 0.239 9 0.006
4 0.1.56
6 0.081 12 0.008
)
0.098 11 0.014
10
0.007
1.i 0.002
TABLE I1 Polymer-Solvent Systems Solvent
BW X
TMV-M TMV-A
A?,
X
3.9 20.
lo-' 3.9
14.
Wt. % Glycerol
Temp. ("C)
7)s
P
(poise)
(g/ml)
,510.7
25.0 37.8 25.0
0.0316 0.0343 0.127
1.113 I . 112 1 . I55
65.4
Solutions were made up by adding glycerol, containing 0.001 A4 EDTA, to solutions of TMV in 0.001 M EDTA; the polymer concentration c in g/ml was calculated assuming additivity of volumes, taking the density of the polymer as 1.37 g/ml. The most concentrated solutions were measured first and then sequentially diluted several times. The concentration range was from 1.6 to 0.G X lov3g/ml for TMV-M and from 2.1 to 1.0 X g/ml for TMV-A.
Method The storage and loss moduli G' and G" of the solutions were measured with the Birnboim-Schrag multiple-lumped resonator (MLR) with a computerized data acquisition and processing ~ y s t e m . ~One titanium alloy resonator, with five working resonance frequencies from 100 to 5800 Hz, was used. The resonator housing was constructed of gold-plated aluminum, which was sufficiently corrosion-resistant for these solutions, through stainless-steel housings will be used in the future. For the unaggregated sample, measurements were made a t two temperatures, 25.0" and 373°C; for the aggregated sample, a t 25.0"C only. Temperature control during measurements was better than *O.Ol"C. As deduced from Eqs. (1) and ( 3 ) , the magnitude of the storage modulus G', a t the same value of W T ~ is , mainly determined by the number of polymer molecules per unit volume, that is, proportional to c / M . Since the molecular weight of TMV is very high, the values of G' were very small and subject to relatively large experimental errors. The small intrinsic viscosity value for unaggregated TMV, originating from its small axial ratio, also gave considerable experimental uncertainty to the determination of G" W7S.
VISCOELASTICITY O F TOBACCO MOSAIC VIRUS
413
RESULTS The quantities (G’/c)‘’‘ and (GI’ - wq,)/c were plotted against c and extrapolated to zero c, as described in previous to give the corresponding intrinsic moduli [GI]and [G”]. An example is shown in Figure 1. Owing to the scattering of the data, all quantities except [G”] of TMVA were determined within considerable uncertainty, which will be shown with error bars. The initial concentration dependences may be described by the ratios y’ and y”, which are, respectively, the limiting value at low concentrations of the ratio (G’/c) -l d(G‘/c)/dc and the corresponding expression with G” - wqs substituted for G’. For TMV-34, they were almost independent of frequency. The value of the ratio y = y‘/y’’ a t higher frequencies was about 0.1, which is sufficiently close to zero, the limiting value a t high frequencies predicted by theories for rigid macromolecules. For TAN-A, 7’ increased with increasing frequency and y “ had a constant value; the ratio y, a t the highest frequencies measured, was somewhat uncertain but appeared to be surprisingly large (at least 10). The extrapolated values [GI]and [G“] were reduced to the corresponding dimensionless quantities by Eqs. (1) and (a),and are plotted logarithmically against the reduced frequency wv,M/RT in Figures 2 and 3. For TMV-A, was substituted for M . the number-average molecular weight
cx
lo4, g./rnl.
c x lo4, q . / m ~ .
Fig. 1. Plots of ( G ’ / C ) ’ /and ~ (G’’-cqS)/c against c for TMV-M in 50.7% glycerolwater a t 25.0°C, each a t five frequencies (Hz) as follows, bottom to top: 103, 393, 984, 2370, and 5810.
NEMOTO ET AL.
414 I
I
I
/ I
I
-2
-I
I
0
log wq,M/RT
Fig. 2 . Logarithmic plots of reduced intrinsic shear moduli [G']Rand [G"]R against reduced frequency q , M / R T for TMV-M. Open circles: at 37.S°C, slotted circles a t 23.0"C. Solid lines, K-,4 theory: dashed lines, C-S-S theory.
DISCUSSION TMV-M (Unaggregated) To test agreement between data and theory, the value of the intrinsic viscosity [g] must be known to evaluate T~ by Eq. ( 3 ) . Since our measurements did not cover the terminal zone, [g] cannot be obtained from these data. Instead, from several values of [ T ] in the l i t e r a t ~ r e , ' ~ -ranging '~ from 27 to 37 ml/g, 27 was chosen for the unaggregated virus. The theories of Kirkwood-Auer (I
VOL. 14, 409-417 (1975)
Infinite-Dilution Viscoelastic Properties of Tobacco Mosaic Virus NOR10 NEMOTO, JOHN L. SCHRAG,andJOHN D. FERRY, Department of Chemistry; and ROBERT W. FULTON, Department of Plant Pathology, University of Wisconsin, Madison, Wisconsin 53706
Synopsis The storage and loss shear moduli, G‘ and G”, have been measured for dilute solutions of unaggregated and aggregated tobacco mosaic virus samples in glycerol-water mixtures, by the Birnboim-Schrag mu1tiple-lumped resonator modified for use with aqueous solvents. The frequency range was 100-5800 Hz, the concentration range 0.6-2.1 X 10-3 g/ml, and the temperatures 25.0” and 37.8”C. The number-average and weightaverage molecular weights of the aggregated sample were estimated as 1.4 and 2.0 X 108, respectively, from electron microscopy. The extrapolated intrinsic moduli [G’] and [G”] were compared with the predictions of the Kirkwood-Auer theory for rigid rodlike molecules. For the unaggregated sample, the frequency dependence of [G’] and [G”] agreed well with the theory assuming the intrinsic viscosity to be 27 ml/g, though the asymptotic limit of [G’]M/RTa t higher frequencies was slightly larger than the theoretical value of 3/5. For the aggregated sample, the data agreed with the theory for rigid rods as modified to account for molecular-weight distribution.
INTRODUCTION The viscoelastic properties of macromolecules a t infinite dilution are very sensitive to their conformations as well as their shapes. For several years, extensive m e as~ remen tsl-~ have been made on dilute solutions of synthetic polymers in organic solvents by the multiple-lumped resonator (MLR) with a computerized data acquisition and processing system.* The results were successfully extrapolated to infinite dilution and interpreted in terms of several molecular as recently reviewed.’ Similar studies on biological polymers with the MLR were impossible until very recently, because the low-loss aluminum alloy from which the resonators were fabricated was very susceptible to corrosion in aqueous solvents. After several attempts t o correct this deficiency, a titanium alloy (Ti-6A1-4V) with special heat treatment was found to be highly resistant to corrosion and also to have the required low mechanical loss (high &). I n this paper we describe the first viscoelastic measurements on a biological polymer in a n aqueous solvent by the MLR with a titanium resonator. According to several molecular theories,8--’2the frequency dependence of the storage and loss shear moduli, G’ and G”, of a very dilute solution of a 409
@ 1975 by John Wiley & Sons, Inc.
YEMOTO ET AL.
410
rigid rodlike macromolecule is characterized by a single relaxation time associated with end-over-end rotat,ion. I n earlier m e a s u r e r n e n t ~on ~ ~solutions of paramyosin in glycerol-water mixtures, the frequency dependence of G‘ and GI‘ was found to be close to the theory for rigid prolate ellipsoidsl0-l2 a t low frequencies; however, a t high frequencies did not approach a constant value as the thcory predicts, but increascd with frequency. Since the earlier measurements were made a t concentrations too high to permit extrapolation to infinite dilution, this deviation could be attributed t.o either intermolecular interaction or some degree of int,ramolecular flexibility, or both. With thc RILR, it is possible to extrapolate measurements of viscoelastic properties to infinite dilution to eliminate any cffccts of intermolecular interaction. Since tobacco mosaic virus (TMV) appears to be the most rigid rodlike macromolecule known, it was chosen for thc present study, which provides the first demonstration of agreement wit,h molecular t.heory for rigid rods.
THEORY The predictions of molecular theory for rigid rods8t9and prolate ellipsoids1°-12 have been summarized in a previous we use the same notation here. It is convenient to express the viscoelastic properties in terms of the reduced intrinsic moduli [(if], and [ G ” ] R , defined as follows: [c’]R =
( M / R T ) lim C’/c c-0
[G”],
=
( M / R T )lim (GI‘
- wqS)/c
(2)
c+o
where w is radian frequency, M molecular weight, R the gas constant, T absolute tempcrature, c concentration (g/ml), and qs the solvent viscosity. The predicted frequency dependences are givcn by
[G’]R = mlw2rO2(1
[G”]R=
WTo 70
+ w2r02)-1
[“Zi(l $-
=
W2T02)-1
(3)
+
(4)
.In21
m[aI?l,M/RT
(5)
where T O is the relaxation time (inversely proportional to the rotary diffusion coefficient) and [ q ] is the intrinsic viscosity; m = l/(ml m2). For the Kirkwood-Auer theory, which we shall use here, corresponding t o a rigid rod with a very large ratio of length to the rcpeat distance between points of frictional resistance, ml = 3/5 and m2 = 1/5. Since our experimental data have been obtained a t frequencies higher than r O - l ,we emphasize the limiting behavior of [G’IR and [G”]R a t high frequencies; as w ---t m ,
+
[G’IR
=
[e“]R=
ml
(6)
m2r0w.
(7)
VISCOELASTICITY OF TOBACCO MOSAIC VIRUS
41 1
From Eq. (7), the ratio of the high-frequency limiting intrinsic viscosity [q’], to [71 ism2l(m1 m2). When the effect of molecular-weight distribution is taken into account in the theories, the intrinsic moduli [G’] and [G”] should be expressed as
+
[G‘] = lim G’/c c-0
=
RT T
cnj 1?n+lw ro3 RT c njwroj Mn
firn
[G”] = lim (G” - wq,)/c c-0
u2roj2
=
+ m2)
T
j
(9)
m h I,q3f,IRT. (10) Here is the number-average molecular weight, n3 and ro3 are the mole fraction and the characteristic relaxation time of molecules with the molecular weight If,. Comparison of Eq. (8) with Eq. (3) shows that the numberaverage molecular weight is a suitable substitute for A4 to define the dimensionless moduli [(;’IR and [G”IR in the presence of the molecularweight distribution. Equation (6) is then applicable to a rigid polymer with any molecular-weight distribution. Equation (7) may be written as 705 =
an
nn
[C”],
=
[G”]dd,/RT
=
m2w~n3r0,
(11)
I
The ratio of [q’],/[ql0 has the same value m2/(ml molecular-weight distribution.
+ m2),unaffected by
EXPERIMENTAL Materials Tobacco mosaic virus was purified from extracts of infected tobacco (Nicotina tabacom L. var. Havana 38) by ammonium sulfate precipitation and isoelectric precipitation, followed, after storage, by one cycle of highand low-speed centrifugation. After purification, the solutions of TAW in 0.001 M EDTA were stored in a refrigerator. The viscoelastic measurements wcre made on two samples with different storage periods; TMV-RI for two weeks and TAN-A for three years. Centrifugations in sucrose density gradients showed that the former consisted mainly of monomers with a small amount of dimers, whereas the latter was highly aggregated by end-to-end association. For the highly aggregated sample, the number-average and weight-average molecular weights were calculated from measurements on electron micrographs (magnification 18,lOO), with counts of numbcrs of species corresponding to j monomeric lengths, j ranging from 1 t o 15. The mole fraction n3 of j-mers is given in Table I, and the calculated averages in Table 11. The molecular weight of sample TAN-M was taken as that of a monomer,14neglecting the presence of dimers. Mixtures of glycerol and water containing 0.001 A f EDTA were used as solvents; Table I1 also gives the weight percents of glycerol, densities ( p ) , and viscosities.
NEMOTO ET AL.
412
TABLE I Distribution of Lengths in Sample TMV-A j ?L j
j nj
1 0.133 7 0.034
2 0.200 8 0.021
3 0.239 9 0.006
4 0.1.56
6 0.081 12 0.008
)
0.098 11 0.014
10
0.007
1.i 0.002
TABLE I1 Polymer-Solvent Systems Solvent
BW X
TMV-M TMV-A
A?,
X
3.9 20.
lo-' 3.9
14.
Wt. % Glycerol
Temp. ("C)
7)s
P
(poise)
(g/ml)
,510.7
25.0 37.8 25.0
0.0316 0.0343 0.127
1.113 I . 112 1 . I55
65.4
Solutions were made up by adding glycerol, containing 0.001 A4 EDTA, to solutions of TMV in 0.001 M EDTA; the polymer concentration c in g/ml was calculated assuming additivity of volumes, taking the density of the polymer as 1.37 g/ml. The most concentrated solutions were measured first and then sequentially diluted several times. The concentration range was from 1.6 to 0.G X lov3g/ml for TMV-M and from 2.1 to 1.0 X g/ml for TMV-A.
Method The storage and loss moduli G' and G" of the solutions were measured with the Birnboim-Schrag multiple-lumped resonator (MLR) with a computerized data acquisition and processing ~ y s t e m . ~One titanium alloy resonator, with five working resonance frequencies from 100 to 5800 Hz, was used. The resonator housing was constructed of gold-plated aluminum, which was sufficiently corrosion-resistant for these solutions, through stainless-steel housings will be used in the future. For the unaggregated sample, measurements were made a t two temperatures, 25.0" and 373°C; for the aggregated sample, a t 25.0"C only. Temperature control during measurements was better than *O.Ol"C. As deduced from Eqs. (1) and ( 3 ) , the magnitude of the storage modulus G', a t the same value of W T ~ is , mainly determined by the number of polymer molecules per unit volume, that is, proportional to c / M . Since the molecular weight of TMV is very high, the values of G' were very small and subject to relatively large experimental errors. The small intrinsic viscosity value for unaggregated TMV, originating from its small axial ratio, also gave considerable experimental uncertainty to the determination of G" W7S.
VISCOELASTICITY O F TOBACCO MOSAIC VIRUS
413
RESULTS The quantities (G’/c)‘’‘ and (GI’ - wq,)/c were plotted against c and extrapolated to zero c, as described in previous to give the corresponding intrinsic moduli [GI]and [G”]. An example is shown in Figure 1. Owing to the scattering of the data, all quantities except [G”] of TMVA were determined within considerable uncertainty, which will be shown with error bars. The initial concentration dependences may be described by the ratios y’ and y”, which are, respectively, the limiting value at low concentrations of the ratio (G’/c) -l d(G‘/c)/dc and the corresponding expression with G” - wqs substituted for G’. For TMV-34, they were almost independent of frequency. The value of the ratio y = y‘/y’’ a t higher frequencies was about 0.1, which is sufficiently close to zero, the limiting value a t high frequencies predicted by theories for rigid macromolecules. For TAN-A, 7’ increased with increasing frequency and y “ had a constant value; the ratio y, a t the highest frequencies measured, was somewhat uncertain but appeared to be surprisingly large (at least 10). The extrapolated values [GI]and [G“] were reduced to the corresponding dimensionless quantities by Eqs. (1) and (a),and are plotted logarithmically against the reduced frequency wv,M/RT in Figures 2 and 3. For TMV-A, was substituted for M . the number-average molecular weight
cx
lo4, g./rnl.
c x lo4, q . / m ~ .
Fig. 1. Plots of ( G ’ / C ) ’ /and ~ (G’’-cqS)/c against c for TMV-M in 50.7% glycerolwater a t 25.0°C, each a t five frequencies (Hz) as follows, bottom to top: 103, 393, 984, 2370, and 5810.
NEMOTO ET AL.
414 I
I
I
/ I
I
-2
-I
I
0
log wq,M/RT
Fig. 2 . Logarithmic plots of reduced intrinsic shear moduli [G']Rand [G"]R against reduced frequency q , M / R T for TMV-M. Open circles: at 37.S°C, slotted circles a t 23.0"C. Solid lines, K-,4 theory: dashed lines, C-S-S theory.
DISCUSSION TMV-M (Unaggregated) To test agreement between data and theory, the value of the intrinsic viscosity [g] must be known to evaluate T~ by Eq. ( 3 ) . Since our measurements did not cover the terminal zone, [g] cannot be obtained from these data. Instead, from several values of [ T ] in the l i t e r a t ~ r e , ' ~ -ranging '~ from 27 to 37 ml/g, 27 was chosen for the unaggregated virus. The theories of Kirkwood-Auer (I
