- Email: [email protected]
Determination of the axial ratio of globular proteins in aqueous solution using viscometric measurements
Determination of the axial ratio of globular proteins in aqueous solution using viscometric measurements K. Monkos and B. Turczynski Department of Biophysics, Silesian Medical Academy, K. Marksa 19, 41-808 Zabrze 8, Poland (Received 15 November 1990; revised 11 April 1991) The paper presents the results of viscosity determinations on aqueous solutions of several globular proteins in a wide range of concentrations. On the basis of these measurements a general formula connecting the relative viscosity with concentration and axial ratio of the dissolved proteins was established. By applying the formula the axial ratios of bovine ),-globulin and horse albumin molecules were calculated. Keywords: Globular proteins; axial ratio; relative viscosity
Introduction Viscometric measurements still play an important role in investigations of biological macromolecules in solution ~-5. Like other hydrodynamic techniques, viscometry allows the study of molecular sizes and shapes. It should, however, be mentioned that the results obtained with different methods are not always in good agreement °'7. On the other hand, the correlation between the observed hydrodynamic behaviour and desired structural information can be established on the basis of theoretical calculations of hydrodynamic properties from the size and conformation of the macromolecule model. A comprehensive review of these problems, with regard to flexible and semirigid chain macromolecules, can be found in the monograph by Yamakawa s. Theories for rigid macromolecules of different shapes, including problems of translational and rotational motion, intrinsic viscosity, internal motions and flagellar motility, have been reviewed by Garcia de la Torre and Bloomfield9. It is worth noting that in recent years hydrodynamicists have been particularly interested in the representation of the gross conformation of macromolecules in terms of sophisticated bead models 9-12. For globular proteins, the assumption that in solutions they behave as solid particles, seems to be a good one 1°. Therefore, at the first approximation, it can be assumed that a globular protein is a rigid ellipsoid of revolution with one long axis (a) and two shorter axes (b). We propose a simple method of calculating the axial ratio p = a/b, based only on viscosity measurements. The method is applicable for globular proteins dissolved in water.
Experimental Materials The viscometric measurements were made for human, bovine and horse albumin, human and bovine ~-globulin, ovalbumin and human haemoglobin in a wide range of concentrations. Highly purified human and bovine albumins were obtained from Sigma and the other proteins from Polish Chemical Reagents factories. From
the crystalline form the material was dissolved in distilled water and then filtered by means of filter papers in order to remove possible undissolved fragments. The samples were stored at 4°C until just prior to viscometry measurements, when they were warmed to 25°C. The pH values of prepared samples were, except for ovalbumin, close to or below their isoelectric points and were as follows: 7.85 for ovalbumin, 4.75 for bovine albumin, 4.7 for human albumin, 7.7 for horse albumin, 5.7 for human and bovine v-globulin and 6.8 for human haemoglobin. These values changes insignificantly during the dilution of the solutions. Hsc ometr y We have measured the relative viscosity ~r = ~/~/o, where ~ and 7o are the viscosities of the solution and the solvent, respectively. The measurements were performed using an Ubbelohde (capillary) microviscometer placed in a waterbath controlled thermostatically at 25 + 0.1°C. The same viscometer was used for all measurements and was so mounted that it always occupied precisely the same position in the bath. Flow times were recorded to within 0.1 s. Solutions were temperature equilibrated and passed once through the capillary before any measurements were made. From five to 10 flow-time measurements were made on each concentration. Solution densities and protein concentrations were measured by weighing. For this purpose, 0.3 + 0.001 ml of a solution was weighed with the precision of + 0.1 rag. After water evaporation the residual proteins in a crystalline form were weighed once more. Thus obtained concentrations were checked for human haemoglobin by measuring the optical absorption at 2 = 540 nm. It was found that both methods gave a very good agreement with the measured concentrations.
Results and discussion We have measured the relative viscosity of the solutions for concentrations from several mg/ml up to about 330 mg/ml for human ~,-globulin, up to about 380 mg/ml for human and bovine albumin and up to about 440 mg/ml for ovalbumin and human haemoglobulin.
0141-8130/91/060341-04
© 1991 Butterworth-HeinemannLimited
Int. J. Biol. Macromol., 1991, Vol. 13, December
341
Determination of axial ratio of olobular proteins: K. Monkos and B. Turczynski
I
•
Ln ~1.~,
O
4
7O 60
50 2
40 30 20
0
10
1
I
I
I
I
5
10
15
20
25
•
p=Z
Figure 2 Plot of In rl, versus p2 for different protein concentrations 100
0
20O
30O
c ,:yml )
400
Figure 1 Plot of the relative viscosity qr versus concentration c for human haemoglobin ~,, ovalbumin O, bovine albumin A, human albumin × and human v-globulin • ; - - fitting obtained by using equation (4)
A 0.1
10
I
Io.o8 The results are shown in Fioure 1. The molecular weights and the axial ratios for the investigated proteins are presented in Table 1. As can be seen, the higher the axial ratio the higher the relative viscosity for a given concentration, regardless of the molecular weight. This is clear especially in the case of human haemoglobulin and human albumin, where both proteins have comparable molecular weight and differ substantially if their axial ratios are taken into account. While searching for the exact relation between the relative viscosity and the axial ratio of dissolved proteins we have found that the plot of In r/r versus p2 is linear for each fixed concentration (Fioure 2). So, the following relation could be written:
rlr = A ( c ) e x p [ B ( c ) p 2]
(1)
where A(c) and B(c) are coefficients which depend only on the concentration of the dissolved proteins. Using the standard method of curve linearization the coefficients were calculated up to the concentration of 380 mg/ml. The results, given in Fioure 3, show that both coefficients increase with an increase of concentration. Numerical analysis has shown that the following
6
.06
4
3.04 ,.02
0
'100
2 00 3 00 c ~g/ml )
4 00
Figure 3 Plot of the coefficients A(c) • and B(c) x versus concentration e; - and (3), respectively
fitting obtained by using equations (2)
functions give the best fit of the experimental values (see Appendix I and II):
A(c) = exp(~c
+
fl¢3/2)
(2)
Table 1 Values of molecular weight and axial ratio p for the studied proteins Protein Human haemoglobin Ovalbumin Bovine albumin Human albumin Human v-globulin Horse albumin Bovine ),-globulin
342
Molecular weight (M)
Axial ratio
68 000 43 800 65 000 69 000 156 000 70 000 160000
1.23 2.74 3 3.95 5.34 3.61 _ 0.11 5.38 ___0.04
Int. J. Biol. Macromol., 1991, Vol. 13, December
Reference
(p = a/b) Huisman 13 Young 6 Moser 14 Young6 Young6 Present data Present data
Determination of axial ratio of globular proteins: K. Monkos and B. Turczynski case, the coefficients in equation (4) are as follows:
where ~t = 5.2 x 10 -4 ml/mg
(2a)
• ' = 5.24 x 10-4ml/mg
(7a)
fl = 2.54 x 10 -4 mla/2/mg3/2
(2b)
f l ' = 2.53 x 10-4m13/2/mg 3/2
(7b)
)" = 1.19 x 10-4ml/mg
(7c)
6' = 7.33 x 10 -7 ml2/mg 2
(7d)
and B(c) = )'c + 6c 2
(3)
where ~ = 9.76 x 10 -2 ml/mg
(3a)
6 = 5.58 x 10 -7 ml2/mg 2
(3b)
Finally, by combining equations (1), (2) and (3), the following relation is obtained: qr = exp[(~t + ~p2)c + tic 3/2 + 3p:Zc2]
(4)
It can be seen that the relative viscosity depends only on concentration and axial ratio of the dissolved proteins. Thus, the above equation allows prediction of the values of relative viscosity for each concentration, when only p is known. For very low concentrations, when c tends to zero, expression (4) can be expanded in the power series of concentration. Limiting to the first order term, the following equation can be obtained: rlr = 1 + (o~ + )'p2)c
(5)
It is worth noting that this expression is qualitatively consistent with the relation obtained by Polsonl~: qr = 1 + (4 + 0.098 p2)NAVc/M
(6)
where N~,, V and M are the Avogadro's number, the volume of one dissolved protein and molecular weight, respectively. Poison measured the viscosity of solutions of various proteins, whose molecules had axial ratios varying from about 3 to 10. However, his measurements were limited only to very low concentrations. As has been shown in our earlier work 16, the linear Poison's approximation is valid only for concentrations up to several dozen of mg/ml. For example, for ),-globulin it gives good results for concentrations up to about 45mg/ml and for human albumin - - up to about 70 mg/ml. At this narrow range of concentrations the relative viscosity values obtained from Poison's formula and from relation (4) are in good agreement. This means that the values agree within the experimental errors of the measurements (which, in the whole experimental range of concentrations were less than 1.5%). It should also be mentioned that our limited relation (5) gives values of r/T lower than those predicted by the Polson's formula, and this discordance is outside of the experimental errors. How does hydration affect the coefficients in formula (4)? It is accepted by many authors that there are one or more layers of hydration around a protein 1°'17'18. The first water layer ought to be 3.5 A thick and it covers the immediate surface of the protein, making many H bond contacts ~8-z° If we assume that this layer uniformly coats the whole protein surface, then the axial ratio of such hydrated protein is different from that of nonhydrated one. We have repeated the whole procedure as in equations (1)-(4) assuming that proteins and their hydration shells form rigid structures with new axial ratios p' = (a + 3.5 A)/(b + 3.5 A). Functions A(c) and B(c) in equation (1) were found to depend on concentration in the same way as in formulae (2) and (3) but, in this
In some cases, however, the assumption that a hydration shell coats the whole surface of the protein is not a good one. In a study on the relationship between the structure of globular proteins and their translational friction coefficients Teller et al. 21 showed that the theoretical values were essentially in agreement with the experimental results when water 'beads' were placed only on the charged groups of the protein. Therefore, we suppose that within the frame of our scheme the experimental data enable only the calculation of the axial ratio of proteins without hydration shell using equation (4) with coefficients given by (2a), (2b), (3a) and (3b). At higher concentrations possible self-association effects may cause discordance between experimental values of relative viscosity and values predicted from relation (4). Because of these processes the proteins may create ensembles formed by 'end to end' or 'side by side' aggregations of two monomer molecules. These ensembles have the effective axial ratio different from that of monomer, e.g. in the case of 'end to end' aggregation the effective axial ratio increases and the experimental values of the relative viscosity should be higher than those predicted from relation (4) for monomers. In the case of 'side by side' aggregation the experimental values of the relative viscosity should be lower. Such deviations have been observed only with regard to bovine albumin for the concentrations about 400 mg/ml (the results are not given in Fioure 1), and they suggest that in this case the 'end to end' aggregation occurs. We suppose that relation (4) may be very useful for the determination of the axial ratio of dissolved proteins. We have tested it for aqueous solutions of horse albumin and bovine v-globulin. The viscosity measurements were made for concentrations up to 400mg/ml for horse albumin and up to 300 mg/ml for bovine v-globulin, and the axial ratios were calculated using relation (4). The average values of p are given in Table 1. It is interesting to note that relation (4) is obtained for the solutions out of their isoelectric points. Thus, the character of the relation is quite universal in the sense that it is not very 'sensitive' to the pH of the solution. The data presented in the present paper suggest that our method can be applied to all globular proteins (for which the axial ratio does not exceed the value of 10) dissolved in water. However, it is not clear whether the relation (4) can be applied to the other types of biological macromolecules such as fibrous proteins, polysaccharides or nucleic acids. Explanation of this problem will be the subject of further investigation.
References 1 2 3 4
Bothner, H., Waaler, T. and Wik, O. Int. J. Biol. Macromol. 1988, 10, 287 Axelos, M. A. V., Thibault, J. F. and Lefebvre, J. Int. J. Biol. Macromol. 1989, 11, 186 Gravanis, G., Milas, M., Rinando, M. and Clarke-Sturman, A. J. Int. J. Biol. Macromol. 1990, 12, 201 Dickinson, E., Rolfe, S. E. and Dalgleish, D. G. lnt. J. Biol. Macromol. 1990, 12, 189
Int. J. Biol. Macromol., 1991, Vol. 13, December
343
Determination of axial ratio o f 91obular proteins: K. Monkos and B. Turczynski 5
Freerksen, D. L., Shih, P. C. F., Vesta-Russell, J. F., Horlick, R. A. and Yau, W. W. Anal. Biochem. 1990, 189, 163 Young,E. G. in 'Comprehensive Biochemistry' (Eds M. Florkin and E. H. Stolz), Elsevier Pub. Co., Amsterdam, 1963 Marshall,A. G. 'BiophysicalChemistry', Wiley,New York, 1978 Yamakawa, H. 'Modern Theory of Polymer Solutions', Harper and Row, New York, 1971 Garcia de la Torre, J. and Bloomfield, V. A. Q. Rev. Biophys. 1981, 14, 81 Garcia, M. M. T., Rios, M. A. J. and Bernal, J. M. G. Int. J. Biol. Macromol. 1990, 12, 19 Freire,J.J.andGarciadelaTorre, J.Macromolecules1983,16,331 Garcia de la Torre, J., Lopez, M. C., Tirado, M. M. and Freire, J. J. Macromolecules 1983, 16, 1121 Huisman,T. H. J. Adv. Clin. Chem. 1963, 6, 236 Moser,P., Squire, P. G. and O'Konski, C. T. J. Phys. Chem. 1966, 70, 744 Polson, A. Kolloid - Z. 1939, 88, 51 Monkos, K., Monkos, J. and Turczynski, B. Post. Fiz. Med. 1988, 23, 215 Savage,H. and Wlodaver, A. Methods Enzymol. 1986, 127, 162 Savage,H. Water Sci. Rev. 1986, 2, 67 Kuntz, I. D. and Kauzmann, W. Adv. Prot. Chem. 1974, 28, 239 Hagler, A. T. and Moult, J. Nature 1978, 272, 222 Teller, D. C., Swanson, E. and de Haen, C. Adv. Enzymol. 1979, 61, 103
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
and
~ z,o,~,~~ ~,~- ~ ~,z,~ ~,~,~ fl=i=l
~
i=1 ¢
i=
i=x ~
-
~=x ~ ~
i=1
(I3) ~
i=1
Putting the experimental values of A~(c) into relations (I2) and (13) we have obtained numerical values of ~ and fl which are given in (2a) and (2b).
Appendix II We have found similarly that the square function of a concentration gives the best linearization for the B(c). Therefore, we have assumed that B(c) has the form given in the equation (3). To find the coefficients ~ and 6, we have minimized the square form Z2 :
~
(II1)
( B i -- ~c i -- 6 c / 2 ) 2
i=1
Appendix I While seeking the best fit of the coefficient A ( c ) to the concentration, we have found that the In A ( c ) with the function of c 3/2 gives the best linearization, so we have assumed that In A ( c ) = ~c + tic 3/2. To find the ~t and fl coefficients we have minimized the square form
with respect to the ~ and 6 which has resulted in the following expressions
i ~,~,-~ ~ o,~
r =
i=1
i=1
tii2)
~c~ i=l
Z1 =
~
(Zi -- 0~¢i -- fl¢/3/2)2
(I1)
i=1
where z~ = In At(c), with respect to the ~ and ft. A simple calculation shows that
~ c~z~-~~ c~,~ ~t=
i=1
i=1
~;#
2 .,~,~ 2 ~,~- 2 .,~, 2 ~,~ 6 =
~= ~
~=~
i= ~
i= ~
()
(II3)
,~ ~, ~ ~ - ,~i ~ (I2)
i~l
344
and
Int. J. Biol. Macromol., 1991, Vol. 13, December
The numerical values of these coefficients are given in (3a) and (3b).
Introduction Viscometric measurements still play an important role in investigations of biological macromolecules in solution ~-5. Like other hydrodynamic techniques, viscometry allows the study of molecular sizes and shapes. It should, however, be mentioned that the results obtained with different methods are not always in good agreement °'7. On the other hand, the correlation between the observed hydrodynamic behaviour and desired structural information can be established on the basis of theoretical calculations of hydrodynamic properties from the size and conformation of the macromolecule model. A comprehensive review of these problems, with regard to flexible and semirigid chain macromolecules, can be found in the monograph by Yamakawa s. Theories for rigid macromolecules of different shapes, including problems of translational and rotational motion, intrinsic viscosity, internal motions and flagellar motility, have been reviewed by Garcia de la Torre and Bloomfield9. It is worth noting that in recent years hydrodynamicists have been particularly interested in the representation of the gross conformation of macromolecules in terms of sophisticated bead models 9-12. For globular proteins, the assumption that in solutions they behave as solid particles, seems to be a good one 1°. Therefore, at the first approximation, it can be assumed that a globular protein is a rigid ellipsoid of revolution with one long axis (a) and two shorter axes (b). We propose a simple method of calculating the axial ratio p = a/b, based only on viscosity measurements. The method is applicable for globular proteins dissolved in water.
Experimental Materials The viscometric measurements were made for human, bovine and horse albumin, human and bovine ~-globulin, ovalbumin and human haemoglobin in a wide range of concentrations. Highly purified human and bovine albumins were obtained from Sigma and the other proteins from Polish Chemical Reagents factories. From
the crystalline form the material was dissolved in distilled water and then filtered by means of filter papers in order to remove possible undissolved fragments. The samples were stored at 4°C until just prior to viscometry measurements, when they were warmed to 25°C. The pH values of prepared samples were, except for ovalbumin, close to or below their isoelectric points and were as follows: 7.85 for ovalbumin, 4.75 for bovine albumin, 4.7 for human albumin, 7.7 for horse albumin, 5.7 for human and bovine v-globulin and 6.8 for human haemoglobin. These values changes insignificantly during the dilution of the solutions. Hsc ometr y We have measured the relative viscosity ~r = ~/~/o, where ~ and 7o are the viscosities of the solution and the solvent, respectively. The measurements were performed using an Ubbelohde (capillary) microviscometer placed in a waterbath controlled thermostatically at 25 + 0.1°C. The same viscometer was used for all measurements and was so mounted that it always occupied precisely the same position in the bath. Flow times were recorded to within 0.1 s. Solutions were temperature equilibrated and passed once through the capillary before any measurements were made. From five to 10 flow-time measurements were made on each concentration. Solution densities and protein concentrations were measured by weighing. For this purpose, 0.3 + 0.001 ml of a solution was weighed with the precision of + 0.1 rag. After water evaporation the residual proteins in a crystalline form were weighed once more. Thus obtained concentrations were checked for human haemoglobin by measuring the optical absorption at 2 = 540 nm. It was found that both methods gave a very good agreement with the measured concentrations.
Results and discussion We have measured the relative viscosity of the solutions for concentrations from several mg/ml up to about 330 mg/ml for human ~,-globulin, up to about 380 mg/ml for human and bovine albumin and up to about 440 mg/ml for ovalbumin and human haemoglobulin.
0141-8130/91/060341-04
© 1991 Butterworth-HeinemannLimited
Int. J. Biol. Macromol., 1991, Vol. 13, December
341
Determination of axial ratio of olobular proteins: K. Monkos and B. Turczynski
I
•
Ln ~1.~,
O
4
7O 60
50 2
40 30 20
0
10
1
I
I
I
I
5
10
15
20
25
•
p=Z
Figure 2 Plot of In rl, versus p2 for different protein concentrations 100
0
20O
30O
c ,:yml )
400
Figure 1 Plot of the relative viscosity qr versus concentration c for human haemoglobin ~,, ovalbumin O, bovine albumin A, human albumin × and human v-globulin • ; - - fitting obtained by using equation (4)
A 0.1
10
I
Io.o8 The results are shown in Fioure 1. The molecular weights and the axial ratios for the investigated proteins are presented in Table 1. As can be seen, the higher the axial ratio the higher the relative viscosity for a given concentration, regardless of the molecular weight. This is clear especially in the case of human haemoglobulin and human albumin, where both proteins have comparable molecular weight and differ substantially if their axial ratios are taken into account. While searching for the exact relation between the relative viscosity and the axial ratio of dissolved proteins we have found that the plot of In r/r versus p2 is linear for each fixed concentration (Fioure 2). So, the following relation could be written:
rlr = A ( c ) e x p [ B ( c ) p 2]
(1)
where A(c) and B(c) are coefficients which depend only on the concentration of the dissolved proteins. Using the standard method of curve linearization the coefficients were calculated up to the concentration of 380 mg/ml. The results, given in Fioure 3, show that both coefficients increase with an increase of concentration. Numerical analysis has shown that the following
6
.06
4
3.04 ,.02
0
'100
2 00 3 00 c ~g/ml )
4 00
Figure 3 Plot of the coefficients A(c) • and B(c) x versus concentration e; - and (3), respectively
fitting obtained by using equations (2)
functions give the best fit of the experimental values (see Appendix I and II):
A(c) = exp(~c
+
fl¢3/2)
(2)
Table 1 Values of molecular weight and axial ratio p for the studied proteins Protein Human haemoglobin Ovalbumin Bovine albumin Human albumin Human v-globulin Horse albumin Bovine ),-globulin
342
Molecular weight (M)
Axial ratio
68 000 43 800 65 000 69 000 156 000 70 000 160000
1.23 2.74 3 3.95 5.34 3.61 _ 0.11 5.38 ___0.04
Int. J. Biol. Macromol., 1991, Vol. 13, December
Reference
(p = a/b) Huisman 13 Young 6 Moser 14 Young6 Young6 Present data Present data
Determination of axial ratio of globular proteins: K. Monkos and B. Turczynski case, the coefficients in equation (4) are as follows:
where ~t = 5.2 x 10 -4 ml/mg
(2a)
• ' = 5.24 x 10-4ml/mg
(7a)
fl = 2.54 x 10 -4 mla/2/mg3/2
(2b)
f l ' = 2.53 x 10-4m13/2/mg 3/2
(7b)
)" = 1.19 x 10-4ml/mg
(7c)
6' = 7.33 x 10 -7 ml2/mg 2
(7d)
and B(c) = )'c + 6c 2
(3)
where ~ = 9.76 x 10 -2 ml/mg
(3a)
6 = 5.58 x 10 -7 ml2/mg 2
(3b)
Finally, by combining equations (1), (2) and (3), the following relation is obtained: qr = exp[(~t + ~p2)c + tic 3/2 + 3p:Zc2]
(4)
It can be seen that the relative viscosity depends only on concentration and axial ratio of the dissolved proteins. Thus, the above equation allows prediction of the values of relative viscosity for each concentration, when only p is known. For very low concentrations, when c tends to zero, expression (4) can be expanded in the power series of concentration. Limiting to the first order term, the following equation can be obtained: rlr = 1 + (o~ + )'p2)c
(5)
It is worth noting that this expression is qualitatively consistent with the relation obtained by Polsonl~: qr = 1 + (4 + 0.098 p2)NAVc/M
(6)
where N~,, V and M are the Avogadro's number, the volume of one dissolved protein and molecular weight, respectively. Poison measured the viscosity of solutions of various proteins, whose molecules had axial ratios varying from about 3 to 10. However, his measurements were limited only to very low concentrations. As has been shown in our earlier work 16, the linear Poison's approximation is valid only for concentrations up to several dozen of mg/ml. For example, for ),-globulin it gives good results for concentrations up to about 45mg/ml and for human albumin - - up to about 70 mg/ml. At this narrow range of concentrations the relative viscosity values obtained from Poison's formula and from relation (4) are in good agreement. This means that the values agree within the experimental errors of the measurements (which, in the whole experimental range of concentrations were less than 1.5%). It should also be mentioned that our limited relation (5) gives values of r/T lower than those predicted by the Polson's formula, and this discordance is outside of the experimental errors. How does hydration affect the coefficients in formula (4)? It is accepted by many authors that there are one or more layers of hydration around a protein 1°'17'18. The first water layer ought to be 3.5 A thick and it covers the immediate surface of the protein, making many H bond contacts ~8-z° If we assume that this layer uniformly coats the whole protein surface, then the axial ratio of such hydrated protein is different from that of nonhydrated one. We have repeated the whole procedure as in equations (1)-(4) assuming that proteins and their hydration shells form rigid structures with new axial ratios p' = (a + 3.5 A)/(b + 3.5 A). Functions A(c) and B(c) in equation (1) were found to depend on concentration in the same way as in formulae (2) and (3) but, in this
In some cases, however, the assumption that a hydration shell coats the whole surface of the protein is not a good one. In a study on the relationship between the structure of globular proteins and their translational friction coefficients Teller et al. 21 showed that the theoretical values were essentially in agreement with the experimental results when water 'beads' were placed only on the charged groups of the protein. Therefore, we suppose that within the frame of our scheme the experimental data enable only the calculation of the axial ratio of proteins without hydration shell using equation (4) with coefficients given by (2a), (2b), (3a) and (3b). At higher concentrations possible self-association effects may cause discordance between experimental values of relative viscosity and values predicted from relation (4). Because of these processes the proteins may create ensembles formed by 'end to end' or 'side by side' aggregations of two monomer molecules. These ensembles have the effective axial ratio different from that of monomer, e.g. in the case of 'end to end' aggregation the effective axial ratio increases and the experimental values of the relative viscosity should be higher than those predicted from relation (4) for monomers. In the case of 'side by side' aggregation the experimental values of the relative viscosity should be lower. Such deviations have been observed only with regard to bovine albumin for the concentrations about 400 mg/ml (the results are not given in Fioure 1), and they suggest that in this case the 'end to end' aggregation occurs. We suppose that relation (4) may be very useful for the determination of the axial ratio of dissolved proteins. We have tested it for aqueous solutions of horse albumin and bovine v-globulin. The viscosity measurements were made for concentrations up to 400mg/ml for horse albumin and up to 300 mg/ml for bovine v-globulin, and the axial ratios were calculated using relation (4). The average values of p are given in Table 1. It is interesting to note that relation (4) is obtained for the solutions out of their isoelectric points. Thus, the character of the relation is quite universal in the sense that it is not very 'sensitive' to the pH of the solution. The data presented in the present paper suggest that our method can be applied to all globular proteins (for which the axial ratio does not exceed the value of 10) dissolved in water. However, it is not clear whether the relation (4) can be applied to the other types of biological macromolecules such as fibrous proteins, polysaccharides or nucleic acids. Explanation of this problem will be the subject of further investigation.
References 1 2 3 4
Bothner, H., Waaler, T. and Wik, O. Int. J. Biol. Macromol. 1988, 10, 287 Axelos, M. A. V., Thibault, J. F. and Lefebvre, J. Int. J. Biol. Macromol. 1989, 11, 186 Gravanis, G., Milas, M., Rinando, M. and Clarke-Sturman, A. J. Int. J. Biol. Macromol. 1990, 12, 201 Dickinson, E., Rolfe, S. E. and Dalgleish, D. G. lnt. J. Biol. Macromol. 1990, 12, 189
Int. J. Biol. Macromol., 1991, Vol. 13, December
343
Determination of axial ratio o f 91obular proteins: K. Monkos and B. Turczynski 5
Freerksen, D. L., Shih, P. C. F., Vesta-Russell, J. F., Horlick, R. A. and Yau, W. W. Anal. Biochem. 1990, 189, 163 Young,E. G. in 'Comprehensive Biochemistry' (Eds M. Florkin and E. H. Stolz), Elsevier Pub. Co., Amsterdam, 1963 Marshall,A. G. 'BiophysicalChemistry', Wiley,New York, 1978 Yamakawa, H. 'Modern Theory of Polymer Solutions', Harper and Row, New York, 1971 Garcia de la Torre, J. and Bloomfield, V. A. Q. Rev. Biophys. 1981, 14, 81 Garcia, M. M. T., Rios, M. A. J. and Bernal, J. M. G. Int. J. Biol. Macromol. 1990, 12, 19 Freire,J.J.andGarciadelaTorre, J.Macromolecules1983,16,331 Garcia de la Torre, J., Lopez, M. C., Tirado, M. M. and Freire, J. J. Macromolecules 1983, 16, 1121 Huisman,T. H. J. Adv. Clin. Chem. 1963, 6, 236 Moser,P., Squire, P. G. and O'Konski, C. T. J. Phys. Chem. 1966, 70, 744 Polson, A. Kolloid - Z. 1939, 88, 51 Monkos, K., Monkos, J. and Turczynski, B. Post. Fiz. Med. 1988, 23, 215 Savage,H. and Wlodaver, A. Methods Enzymol. 1986, 127, 162 Savage,H. Water Sci. Rev. 1986, 2, 67 Kuntz, I. D. and Kauzmann, W. Adv. Prot. Chem. 1974, 28, 239 Hagler, A. T. and Moult, J. Nature 1978, 272, 222 Teller, D. C., Swanson, E. and de Haen, C. Adv. Enzymol. 1979, 61, 103
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
and
~ z,o,~,~~ ~,~- ~ ~,z,~ ~,~,~ fl=i=l
~
i=1 ¢
i=
i=x ~
-
~=x ~ ~
i=1
(I3) ~
i=1
Putting the experimental values of A~(c) into relations (I2) and (13) we have obtained numerical values of ~ and fl which are given in (2a) and (2b).
Appendix II We have found similarly that the square function of a concentration gives the best linearization for the B(c). Therefore, we have assumed that B(c) has the form given in the equation (3). To find the coefficients ~ and 6, we have minimized the square form Z2 :
~
(II1)
( B i -- ~c i -- 6 c / 2 ) 2
i=1
Appendix I While seeking the best fit of the coefficient A ( c ) to the concentration, we have found that the In A ( c ) with the function of c 3/2 gives the best linearization, so we have assumed that In A ( c ) = ~c + tic 3/2. To find the ~t and fl coefficients we have minimized the square form
with respect to the ~ and 6 which has resulted in the following expressions
i ~,~,-~ ~ o,~
r =
i=1
i=1
tii2)
~c~ i=l
Z1 =
~
(Zi -- 0~¢i -- fl¢/3/2)2
(I1)
i=1
where z~ = In At(c), with respect to the ~ and ft. A simple calculation shows that
~ c~z~-~~ c~,~ ~t=
i=1
i=1
~;#
2 .,~,~ 2 ~,~- 2 .,~, 2 ~,~ 6 =
~= ~
~=~
i= ~
i= ~
()
(II3)
,~ ~, ~ ~ - ,~i ~ (I2)
i~l
344
and
Int. J. Biol. Macromol., 1991, Vol. 13, December
The numerical values of these coefficients are given in (3a) and (3b).