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A Mathematical Model Describing Tannin–Protein Association
ANALYTICAL BIOCHEMISTRY ARTICLE NO.
263, 46 –50 (1998)
AB982817
A Mathematical Model Describing Tannin±Protein Association Moris L. Silber,* Bruce B. Davitt,* Rafail F. Khairutdinov,†,1 and James K. Hurst*,† *Department of Natural Resource Sciences, and †Department of Chemistry, Washington State University, Pullman, Washington 99164-6410
Received May 6, 1998
A quantitative model is derived for tannin–protein binding and protein precipitation in solution. This model is based on the assumption that precipitation occurs when the number of tannin molecules associated with one protein molecule attains a critical value. Precipitation occurs at this point because tannin crosslinking causes formation of large protein aggregates. Analytical expressions were derived for the dependence of protein precipitate yields upon the concentrations of the protein and tannin in solution. This expression fits reasonably well the experimentally observed bell-shaped dependencies of bovine serum albumin or gelatin precipitation upon total protein and tannin concentrations. © 1998 Academic Press
The ability of small molecules, such as dyes or other ligands, to bind with protein molecules is widely used in biochemical and clinical analyses for determination of small amounts of proteins in solution (1, 2). Coomassie brilliant blue G-250 (CBB)2 is one of the most popular dyes in such analyses (3– 6). The strongly bluecolored anionic form of this dye exists at very low concentration at the pH of the assay reaction mixture, but increases markedly upon binding to proteins (6, 7). Macromolecule–ligand binding is studied by variety of techniques, such as equilibrium dialysis (8), absorption spectroscopy (5, 6, 9) and calorimetry (10). Ordinarily, the number of moles of ligands bound per mole of protein is obtained as a function of unbound ligand concentration, and the data are treated by using a mathematical model in which the protein is assumed to have a fixed number of equivalent binding sites. Other ligands, like polyphenols and simple phenols, can also 1 To whom correspondence should be addressed. Fax: (509) 3358867. E-mail: [email protected]. 2 Abbreviations used: CBB, Coomassie brilliant blue; BSA, bovine serum albumin; TA, tannic acid.
46
bind proteins, most commonly through hydrogen bonding and hydrophobic interactions (10, 11). This has given rise to a separate group of analytical methods, which measure natural polyphenols (tannins) based on their ability to precipitate proteins (12–18). It has been found that the number of tannin molecules bound per protein molecule in the precipitate formed with phenols depends markedly on the protein concentration; generally, the lower the protein concentration, the greater the tannin/protein ratio (19, 20). It has also been found that precipitation is followed by resolubilization of the tannin–protein complex upon further addition of protein (21, 22). In this paper a mathematical model is developed that describes quantitatively the dependence of tannin–protein complex precipitation upon the protein concentration. This model is used to estimate the average number of tannic acid molecules bound to protein that cause precipitation. MATERIALS AND METHODS
Tannic acid, Coomassie brilliant blue G-250, and bovine serum albumin were obtained from Sigma. All other chemicals were the best available grade and were used without further purification. Nanopure water was used in all experiments. All absorbance measurements were carried out on IBM 9430 UV/ VIS or Hewlett–Packard 8452A diode array spectrophotometers interfaced to a ChemStation data acquisition/analysis system. A Coomassie brilliant blue– bovine serum albumin complex (CBB–BSA) was prepared using previously developed protocols (22). A stock solution of tannic acid in 50% aqueous methanol was brought to a working solution by diluting with an equal amount of 0.2 M sodium acetate, pH 5.0. Aliquots of the working solution were transferred to 16 polyethylene test tubes and sufficient CBB–BSA solution was added to give a linear range of concentrations varying from 1.4 3 1026 to 1.0 3 1024 M. The final concentration of tannic acid in all test tubes was equal to 1.4 3 1024 M. The resulting 0003-2697/98 $25.00 Copyright © 1998 by Academic Press All rights of reproduction in any form reserved.
PROTEIN PRECIPITATION BY TANNINS
47
FIG. 1. (a) The dependence of CBB–BSA precipitate yields on the total concentration of CBB–BSA in the reaction mixture. Solid circles, experimental data; open circles, best fit by Eq. [3] with l 5 14 and m # calculated from the amount of tannic acid and CBB–BSA in precipitate; solid line, best fit by Eqs. [3] and [5] with K 5 2 3 10 5 M21, n 5 50, l 5 14. The total concentration of tannic acid in these experiments was 1.4 3 1024 M. (b) The dependence of gelatin precipitate yields on the total concentration of gelatin in the reaction mixture. Solid circles, experimental data from ref. 21; solid line, best fit by Eqs. [3] and [5] with K 5 3.4 3 10 5 M21, n 5 200, l 5 130. The total concentration of tannic acid in these experiments was 1.2 3 1024 M.
mixtures were briefly vortexed, refrigerated for 3 h, and sedimented for 30 min at 5000 rpm in a Sorvall RC-5B centrifuge using a SM-24 rotor. The supernatants were collected, and the concentration of free (unbound) tannic acid in the supernatants was determined from the intensities of their characteristic absorption bands (e 5 7.3 3 104 M21 cm21 at 272 nm). The free fraction of tannic acid was isolated via an Amicon Bioseparation technique utilizing 2.0 mL Centricon-30 centrifugal microconcentrators that provided a 30,000 molecular weight cutoff. Ultrafiltration was carried out in a Sorvall RC-5B centrifuge (Rotor Model SS-34) at 3000 rpm for 2 min. Under these conditions, an unbound tannic acid fraction comprising less than 5% of the original supernatant volume could be consistently collected in the retentate vial. The absorbance of tannic acid followed the Lambert–Beer law over the range of concentrations used in this study. The unwashed precipitates were dissolved in 2.0 mL of a solution containing 1% (v/v) sodium dodecyl sulfate (SDS) and 5% (v/v) triethanolamine by stirring intermittently for 10 min. The dissolved precipitates were diluted with twice the volume of buffer and measured spectrophotometrically at 600 nm, corresponding to the maximum absorbance of the CBB–BSA complex (e 5 5.8 3 104 M21 cm21). All measurements were carried out at ambient temperature (23.0 6 2.0°C). Linear least-squares methods were applied to all data analyses. RESULTS AND DISCUSSION
Figure 1 shows the amount of CBB–BSA in precipitates from tannic acid (Fig. 1a) as a function of the total
amount of CBB–BSA. A bell-shaped dependence is observed. According to a previously proposed model (9), tannic acid molecules engage in interprotein binding, effectively crosslinking them; precipitation occurs at high tannic acid concentration due to formation of large protein aggregates. At low CBB–BSA concentrations, when the number of tannic acid molecules bound to each protein molecule is large, practically all of the CBB–BSA complex precipitates. Precipitation continues so long as the average number of sites per CBB– BSA molecules occupied by tannic acid molecules (m #) exceeds the critical number (l ), i.e., m # . l. Beyond this point, the number of tannic acid bonds between CBB–BSA complexes falls below the level necessary for large aggregate formation, i.e., m # , l; further addition of protein molecules to the solution reduces the average number of tannic acid bonds to each protein, causing the precipitate to redissolve. This phenomenon is schematically illustrated in Fig. 2. A mathematical model describing this behavior is developed below, based upon the binding equilibrium between CBB–BSA and tannic acid (TA) molecules in solution: TA 1 CBB–BSA º TA µ CBB–BSA
[1]
Binding of the tannic acid molecules to the CBB–BSA complexes is assumed to occur randomly, so that, at a given total tannin/protein ratio, individual proteins will contain differing numbers of bound tannins. Assuming a Poisson type distribution ( J(k)) of tannin molecules bound to CBB–BSA, i.e.,
48
SILBER ET AL.
FIG. 2. Stylized drawing of CBB–BSA aggregation at different average numbers of tannic acid molecules per protein molecule.
J~k! 5 e 2m#
~m # !k , k!
[2]
the amount of CBB–BSA complexes in the precipitate (P p) is given by Pp 5 Pt 3
O
k5l11
J~k! 5 P t 3
O
k5l11
e
2m #
~m # !k , k!
[3]
where P t is the total concentration of the protein. The sum in the right side of Eq. [3] is the portion of CBB– BSA complexes for which the number of occupied tannin binding sites exceeds l. If each protein molecule has n identical noninteracting binding sites for tannic acid molecules, one has from the mass action law (see Appendix): m # 5n
Equations [4] and [5] predict that the extent of protein precipitation will exhibit a bell-shaped dependence on the total protein concentration when the total tannin is held constant. As an example, Fig. 3 shows results of model calculations for the effect of protein concentration on the amount of the protein in the precipitate at three different concentrations of tannic acid. Numerical analysis of Eqs. [3]–[5] also shows that when n 3 Pt @ r 3 Tt, K 2 1, the average number of tannic acid molecules per each protein molecule at the point of maximum precipitation is related to l by the relation r 3 Tt/Pt < 1.25 3 l. This simple relation gives a lower limit of the value of l; when r 5 5 (see below), one obtains from the data in Fig. 1, l $ 11. Equations [4] and [5] treat tannic acid binding to protein in terms of the simple mass action law, for which all binding sites are assumed equivalent and independent. With this assumption, the standard Scatchard analysis (23) can be used to evaluate the number (n) of binding sites for tannin molecules on one CBB– BSA complex and the association constant K. For this determination, Eq. [4] is rewritten in the form
F
G
1 Pt r 3 Tt 1 1 5 , Y n3K 12Y n
[6]
where 1/(1 2 Y). The constants n and K can be evaluated from the dependence of P t/Y on Y 5 (T t 2 T f)/T t, provided that a linear relationship is obtained. In the case where various binding sites on the protein have different binding affinities (19, 21), or are interacting, one expects a nonlinear dependence of P t/Y on 1/(1 2 Y). Our analysis shows that the exper-
# 3 P t! K 3 ~r 3 T t 2 m K 3 r 3 Tf 5n , 1 1 K 3 r 3 Tf 1 1 K 3 ~r 3 T t 2 m # 3 P t! [4]
where K is the equilibrium binding constant, r is the number of binding sites on the protein molecule that can be occupied by one tannic acid molecule, T f and T t are the concentrations of free (unbound) and of the total (bound and unbound) tannic acid, and P t is the total concentration of protein. From Eq. [4] we derive n 3 Pt 1 r 3 Tt 1 K21 2 Î~n 3 Pt 1 r 3 Tt 1 K21!2 2 4n 3 r 3 Pt 3 Tt m # 5 . 2Pt [5] Note that when n 3 Pt @ K 2 1, a condition, which corresponds to strong binding, and n 3 Pt @ r 3 Tt, a condition which corresponds to low occupancy of available binding sites, Eq. [5] reduces to m # 5 r 3 Tt/Pt.
FIG. 3. Graphical presentation of the theoretical dependencies of the amount of precipitated protein (P p) on the initial concentration of the protein in solution (P t) at three different concentrations of tannic acid. Parameters used in the calculations are n 5 150, K 5 10 3 M21, l 5 33. The tannic acid concentration is 1.5 3 1024 M (curve 1), 1.1 3 1024 M (curve 2), and 8 3 1025 M (curve 3).
49
PROTEIN PRECIPITATION BY TANNINS
imental dependence of P t/Y on 1/(1 2 Y) for CBB–BSA precipitation by tannic acid did not fit Eq. [6]. Specifically, increasing the tannic acid concentration resulted in an apparent increase in the value of n and a decrease in the value of K. Therefore, the data fits in Fig. 1 made according to Eq. [3] were based upon values of m # calculated directly from the amounts of tannic acid and CBB–BSA in the precipitates. For these calculations, we assumed a pentagalloylglucose structure for tannic acid. For these structures, all five galloyl residues can participate in hydrogen bonding to one peptide chain without introducing bond angle strain (21). Consequently, the value of r was taken to be 5. The best fit of experimental data presented in Fig. 1 by Eq. [3] corresponds to l 5 14. This means that, on average, L 5 l/r 5 2.8 tannic acid molecules per CBB– BSA were required to induce protein precipitation. This number for l is also very close to the strong binding limit, i.e., l $ 11. 3 We also fitted the bell-shaped precipitation curve assuming that the protein-binding sites were identical and noninteracting (Eqs. [3] and [5]). In this procedure K, n, and l were treated as fitting parameters. The solid line in Fig. 1a shows the best fit to the data, which was obtained by using K 5 2 3 10 5 M21, n 5 50, l 5 14. Equations [3] and [5] were also used to fit experimental data for gelatin precipitation by tannic acid (21). The best fit of this dependence (Fig. 1b) corresponded to K 5 3.4 3 10 5 M21, n 5 200, l 5 130. Thus, binding of ;26 tannic acid molecules was required to precipitate one gelatin molecule from solution. In this analysis, the molecular weight of gelatin was assumed 285,000. The best fit values of K and n fell between previous estimates of the upper and lower limits for K and n (21). The value of l for gelatin is about nine times larger than that for CBB–BSA. This difference may be due to a larger size of gelatin molecule.
acid concentrations. For bovine serum albumin and gelatin, this expression fits reasonably well the experimental data. APPENDIX
Assuming that each tannic acid molecule has r residues that can participate independently in bonding to one peptide chain, each protein molecule has n identical noninteracting binding sites, and m # is the average number of sites per CBB–BSA molecule occupied by tannic acid molecules, one has from the mass action law m # 3 Pt 5 K 3 n 3 Pf 3 r 3 Tf ,
[A1]
where T f and P f, are concentrations of free tannic acid and protein, respectively, and P t is the total protein concentration. Replacing in Eq. [A1] n 3 P f by n 3 P f 5 (n 2 m # ) 3 P t one has m # 3 P t 5 K 3 ~n 2 m # ! 3 Pt 3 r 3 Tf .
[A2]
Or, after algebraic transformation, m # 5n
K 3 r 3 Tf 1 1 K 3 r 3 Tf
[A3]
# Substitution in Eq. [A3] with r 3 T f 5 r 3 T t 2 m 3 P t results in Eq. [A4]: m # 5n
# 3 P t! K 3 ~r 3 T t 2 m 1 1 K 3 ~r 3 T t 2 m # 3 P t!
[A4]
Equations [A3] and [A4] are exactly Eq. [4] used above in the analyses of experimental data of tannic acid binding. REFERENCES
CONCLUSION
A quantitative model for tannic acid-induced protein precipitation in solution has been developed based upon the assumption that precipitation occurs when the number of tannic acid molecules associated with one protein molecule attains a critical value necessary to form tannin-crosslinked protein aggregates. Analytical expressions were derived for the dependence of the protein precipitate yields on the protein and tannic 3 The value of l defined from Eq. [3] does not depend on r, i.e., on the assumption about a pentagalloylglucose structure for tannic acid. The number of binding sites on the protein molecule that can be occupied by one tannic acid molecule, for other than r 5 5, results only in the different number of tannic acid molecules per CBB–BSA required to induce protein precipitation.
1. Sedmak, J. J., and Grossberg, S. E. (1977) Anal. Biochem. 79, 544 –552. 2. Chial, H. J., and Splittgerber, A. G. (1993) Anal. Biochem. 213, 362–369. 3. Bradford, M. M. (1976) Anal. Biochem. 72, 248 –254. 4. Splittgerber, A. G., and Sohl, J. L. (1988) Anal. Biochem. 179, 503. 5. Atherton, B. A., Cunningham, E. L., and Splittgerber, A. G. (1996) Anal. Biochem. 233, 160 –168. 6. Matejovicova, M., Mubagva, K., and Flameng, W. (1997) Anal. Biochem. 245, 252–254. 7. Compton, S. J., and Jones, C. G. (1985) Anal. Biochem. 151, 369 –374. 8. Dryer, R. L., and Lota, G. F. (1989) Experimental Biochemistry, pp. 103–104, Oxford Univ. Press, New York. 9. Hussain, A., and Chitre, A. V. (1997) J. Ind. Chem. Soc. 74, 50 –51.
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10. Haslam, E., and Lilley, T. H. (1986) in Plant Flavonoids in Biology and Medicine (Cody, V., Middleton, E., Jr., and Harborne, J. B., Eds.), pp. 53– 65, A. R. Liss, New York. 11. Hagerman, A. E., and Klucher, K. M. (1986) in Plant Flavonoids in Biology and Medicine (Cody, V., Middleton, E., Jr., and Harborne, J. B., Eds.), pp. 67–76, A. R. Liss, Inc, New York. 12. Goldstein, J. L., and Swain, T. (1965) Phytochemistry 4, 185–192. 13. Bate-Smith, E. C. (1973) Phytochemistry 12, 907. 14. Hagerman, A. E., and Butler, L. G. (1978) J. Agric. Food Chem. 26, 809 – 812. 15. Hagerman, A. E., and Butler, L. G. (1980) J. Agric. Food Chem. 28, 944 –947. 16. Hagerman, and A. E., Robbins C. T. (1987) J. Chem. Ecology 13, 1243–1259.
17. Martin, J. S., and Martin, M. M. (1982) Oecologia 54, 205– 211. 18. Makkar, H. P. S., Dawra, R. K., and Singh, B. (1987) Anal. Biochem. 166, 435– 439. 19. McManus, J. P., Davis, K. G., Lilley, T. H., and Haslam, E. (1981) J. Chem. Soc. Chem. 309 –311. 20. Hagerman, A. E. (1992) Polyphenolic Compounds in Food and their Effects on Health: Tannin–Protein Interactions, pp. 237– 247, ACS. 21. Van Buren, J. P., and Robinson, W. B. (1969) J. Agr. Food Chem. 17, 772–777. 22. Silber, M. L., and Davitt, B. B. (1997) Patent Application Ref. 940091, 402P1. 23. Scatchard, G. (1949) Ann. N.Y. Acad. Sci. 51, 660 – 672.
263, 46 –50 (1998)
AB982817
A Mathematical Model Describing Tannin±Protein Association Moris L. Silber,* Bruce B. Davitt,* Rafail F. Khairutdinov,†,1 and James K. Hurst*,† *Department of Natural Resource Sciences, and †Department of Chemistry, Washington State University, Pullman, Washington 99164-6410
Received May 6, 1998
A quantitative model is derived for tannin–protein binding and protein precipitation in solution. This model is based on the assumption that precipitation occurs when the number of tannin molecules associated with one protein molecule attains a critical value. Precipitation occurs at this point because tannin crosslinking causes formation of large protein aggregates. Analytical expressions were derived for the dependence of protein precipitate yields upon the concentrations of the protein and tannin in solution. This expression fits reasonably well the experimentally observed bell-shaped dependencies of bovine serum albumin or gelatin precipitation upon total protein and tannin concentrations. © 1998 Academic Press
The ability of small molecules, such as dyes or other ligands, to bind with protein molecules is widely used in biochemical and clinical analyses for determination of small amounts of proteins in solution (1, 2). Coomassie brilliant blue G-250 (CBB)2 is one of the most popular dyes in such analyses (3– 6). The strongly bluecolored anionic form of this dye exists at very low concentration at the pH of the assay reaction mixture, but increases markedly upon binding to proteins (6, 7). Macromolecule–ligand binding is studied by variety of techniques, such as equilibrium dialysis (8), absorption spectroscopy (5, 6, 9) and calorimetry (10). Ordinarily, the number of moles of ligands bound per mole of protein is obtained as a function of unbound ligand concentration, and the data are treated by using a mathematical model in which the protein is assumed to have a fixed number of equivalent binding sites. Other ligands, like polyphenols and simple phenols, can also 1 To whom correspondence should be addressed. Fax: (509) 3358867. E-mail: [email protected]. 2 Abbreviations used: CBB, Coomassie brilliant blue; BSA, bovine serum albumin; TA, tannic acid.
46
bind proteins, most commonly through hydrogen bonding and hydrophobic interactions (10, 11). This has given rise to a separate group of analytical methods, which measure natural polyphenols (tannins) based on their ability to precipitate proteins (12–18). It has been found that the number of tannin molecules bound per protein molecule in the precipitate formed with phenols depends markedly on the protein concentration; generally, the lower the protein concentration, the greater the tannin/protein ratio (19, 20). It has also been found that precipitation is followed by resolubilization of the tannin–protein complex upon further addition of protein (21, 22). In this paper a mathematical model is developed that describes quantitatively the dependence of tannin–protein complex precipitation upon the protein concentration. This model is used to estimate the average number of tannic acid molecules bound to protein that cause precipitation. MATERIALS AND METHODS
Tannic acid, Coomassie brilliant blue G-250, and bovine serum albumin were obtained from Sigma. All other chemicals were the best available grade and were used without further purification. Nanopure water was used in all experiments. All absorbance measurements were carried out on IBM 9430 UV/ VIS or Hewlett–Packard 8452A diode array spectrophotometers interfaced to a ChemStation data acquisition/analysis system. A Coomassie brilliant blue– bovine serum albumin complex (CBB–BSA) was prepared using previously developed protocols (22). A stock solution of tannic acid in 50% aqueous methanol was brought to a working solution by diluting with an equal amount of 0.2 M sodium acetate, pH 5.0. Aliquots of the working solution were transferred to 16 polyethylene test tubes and sufficient CBB–BSA solution was added to give a linear range of concentrations varying from 1.4 3 1026 to 1.0 3 1024 M. The final concentration of tannic acid in all test tubes was equal to 1.4 3 1024 M. The resulting 0003-2697/98 $25.00 Copyright © 1998 by Academic Press All rights of reproduction in any form reserved.
PROTEIN PRECIPITATION BY TANNINS
47
FIG. 1. (a) The dependence of CBB–BSA precipitate yields on the total concentration of CBB–BSA in the reaction mixture. Solid circles, experimental data; open circles, best fit by Eq. [3] with l 5 14 and m # calculated from the amount of tannic acid and CBB–BSA in precipitate; solid line, best fit by Eqs. [3] and [5] with K 5 2 3 10 5 M21, n 5 50, l 5 14. The total concentration of tannic acid in these experiments was 1.4 3 1024 M. (b) The dependence of gelatin precipitate yields on the total concentration of gelatin in the reaction mixture. Solid circles, experimental data from ref. 21; solid line, best fit by Eqs. [3] and [5] with K 5 3.4 3 10 5 M21, n 5 200, l 5 130. The total concentration of tannic acid in these experiments was 1.2 3 1024 M.
mixtures were briefly vortexed, refrigerated for 3 h, and sedimented for 30 min at 5000 rpm in a Sorvall RC-5B centrifuge using a SM-24 rotor. The supernatants were collected, and the concentration of free (unbound) tannic acid in the supernatants was determined from the intensities of their characteristic absorption bands (e 5 7.3 3 104 M21 cm21 at 272 nm). The free fraction of tannic acid was isolated via an Amicon Bioseparation technique utilizing 2.0 mL Centricon-30 centrifugal microconcentrators that provided a 30,000 molecular weight cutoff. Ultrafiltration was carried out in a Sorvall RC-5B centrifuge (Rotor Model SS-34) at 3000 rpm for 2 min. Under these conditions, an unbound tannic acid fraction comprising less than 5% of the original supernatant volume could be consistently collected in the retentate vial. The absorbance of tannic acid followed the Lambert–Beer law over the range of concentrations used in this study. The unwashed precipitates were dissolved in 2.0 mL of a solution containing 1% (v/v) sodium dodecyl sulfate (SDS) and 5% (v/v) triethanolamine by stirring intermittently for 10 min. The dissolved precipitates were diluted with twice the volume of buffer and measured spectrophotometrically at 600 nm, corresponding to the maximum absorbance of the CBB–BSA complex (e 5 5.8 3 104 M21 cm21). All measurements were carried out at ambient temperature (23.0 6 2.0°C). Linear least-squares methods were applied to all data analyses. RESULTS AND DISCUSSION
Figure 1 shows the amount of CBB–BSA in precipitates from tannic acid (Fig. 1a) as a function of the total
amount of CBB–BSA. A bell-shaped dependence is observed. According to a previously proposed model (9), tannic acid molecules engage in interprotein binding, effectively crosslinking them; precipitation occurs at high tannic acid concentration due to formation of large protein aggregates. At low CBB–BSA concentrations, when the number of tannic acid molecules bound to each protein molecule is large, practically all of the CBB–BSA complex precipitates. Precipitation continues so long as the average number of sites per CBB– BSA molecules occupied by tannic acid molecules (m #) exceeds the critical number (l ), i.e., m # . l. Beyond this point, the number of tannic acid bonds between CBB–BSA complexes falls below the level necessary for large aggregate formation, i.e., m # , l; further addition of protein molecules to the solution reduces the average number of tannic acid bonds to each protein, causing the precipitate to redissolve. This phenomenon is schematically illustrated in Fig. 2. A mathematical model describing this behavior is developed below, based upon the binding equilibrium between CBB–BSA and tannic acid (TA) molecules in solution: TA 1 CBB–BSA º TA µ CBB–BSA
[1]
Binding of the tannic acid molecules to the CBB–BSA complexes is assumed to occur randomly, so that, at a given total tannin/protein ratio, individual proteins will contain differing numbers of bound tannins. Assuming a Poisson type distribution ( J(k)) of tannin molecules bound to CBB–BSA, i.e.,
48
SILBER ET AL.
FIG. 2. Stylized drawing of CBB–BSA aggregation at different average numbers of tannic acid molecules per protein molecule.
J~k! 5 e 2m#
~m # !k , k!
[2]
the amount of CBB–BSA complexes in the precipitate (P p) is given by Pp 5 Pt 3
O
k5l11
J~k! 5 P t 3
O
k5l11
e
2m #
~m # !k , k!
[3]
where P t is the total concentration of the protein. The sum in the right side of Eq. [3] is the portion of CBB– BSA complexes for which the number of occupied tannin binding sites exceeds l. If each protein molecule has n identical noninteracting binding sites for tannic acid molecules, one has from the mass action law (see Appendix): m # 5n
Equations [4] and [5] predict that the extent of protein precipitation will exhibit a bell-shaped dependence on the total protein concentration when the total tannin is held constant. As an example, Fig. 3 shows results of model calculations for the effect of protein concentration on the amount of the protein in the precipitate at three different concentrations of tannic acid. Numerical analysis of Eqs. [3]–[5] also shows that when n 3 Pt @ r 3 Tt, K 2 1, the average number of tannic acid molecules per each protein molecule at the point of maximum precipitation is related to l by the relation r 3 Tt/Pt < 1.25 3 l. This simple relation gives a lower limit of the value of l; when r 5 5 (see below), one obtains from the data in Fig. 1, l $ 11. Equations [4] and [5] treat tannic acid binding to protein in terms of the simple mass action law, for which all binding sites are assumed equivalent and independent. With this assumption, the standard Scatchard analysis (23) can be used to evaluate the number (n) of binding sites for tannin molecules on one CBB– BSA complex and the association constant K. For this determination, Eq. [4] is rewritten in the form
F
G
1 Pt r 3 Tt 1 1 5 , Y n3K 12Y n
[6]
where 1/(1 2 Y). The constants n and K can be evaluated from the dependence of P t/Y on Y 5 (T t 2 T f)/T t, provided that a linear relationship is obtained. In the case where various binding sites on the protein have different binding affinities (19, 21), or are interacting, one expects a nonlinear dependence of P t/Y on 1/(1 2 Y). Our analysis shows that the exper-
# 3 P t! K 3 ~r 3 T t 2 m K 3 r 3 Tf 5n , 1 1 K 3 r 3 Tf 1 1 K 3 ~r 3 T t 2 m # 3 P t! [4]
where K is the equilibrium binding constant, r is the number of binding sites on the protein molecule that can be occupied by one tannic acid molecule, T f and T t are the concentrations of free (unbound) and of the total (bound and unbound) tannic acid, and P t is the total concentration of protein. From Eq. [4] we derive n 3 Pt 1 r 3 Tt 1 K21 2 Î~n 3 Pt 1 r 3 Tt 1 K21!2 2 4n 3 r 3 Pt 3 Tt m # 5 . 2Pt [5] Note that when n 3 Pt @ K 2 1, a condition, which corresponds to strong binding, and n 3 Pt @ r 3 Tt, a condition which corresponds to low occupancy of available binding sites, Eq. [5] reduces to m # 5 r 3 Tt/Pt.
FIG. 3. Graphical presentation of the theoretical dependencies of the amount of precipitated protein (P p) on the initial concentration of the protein in solution (P t) at three different concentrations of tannic acid. Parameters used in the calculations are n 5 150, K 5 10 3 M21, l 5 33. The tannic acid concentration is 1.5 3 1024 M (curve 1), 1.1 3 1024 M (curve 2), and 8 3 1025 M (curve 3).
49
PROTEIN PRECIPITATION BY TANNINS
imental dependence of P t/Y on 1/(1 2 Y) for CBB–BSA precipitation by tannic acid did not fit Eq. [6]. Specifically, increasing the tannic acid concentration resulted in an apparent increase in the value of n and a decrease in the value of K. Therefore, the data fits in Fig. 1 made according to Eq. [3] were based upon values of m # calculated directly from the amounts of tannic acid and CBB–BSA in the precipitates. For these calculations, we assumed a pentagalloylglucose structure for tannic acid. For these structures, all five galloyl residues can participate in hydrogen bonding to one peptide chain without introducing bond angle strain (21). Consequently, the value of r was taken to be 5. The best fit of experimental data presented in Fig. 1 by Eq. [3] corresponds to l 5 14. This means that, on average, L 5 l/r 5 2.8 tannic acid molecules per CBB– BSA were required to induce protein precipitation. This number for l is also very close to the strong binding limit, i.e., l $ 11. 3 We also fitted the bell-shaped precipitation curve assuming that the protein-binding sites were identical and noninteracting (Eqs. [3] and [5]). In this procedure K, n, and l were treated as fitting parameters. The solid line in Fig. 1a shows the best fit to the data, which was obtained by using K 5 2 3 10 5 M21, n 5 50, l 5 14. Equations [3] and [5] were also used to fit experimental data for gelatin precipitation by tannic acid (21). The best fit of this dependence (Fig. 1b) corresponded to K 5 3.4 3 10 5 M21, n 5 200, l 5 130. Thus, binding of ;26 tannic acid molecules was required to precipitate one gelatin molecule from solution. In this analysis, the molecular weight of gelatin was assumed 285,000. The best fit values of K and n fell between previous estimates of the upper and lower limits for K and n (21). The value of l for gelatin is about nine times larger than that for CBB–BSA. This difference may be due to a larger size of gelatin molecule.
acid concentrations. For bovine serum albumin and gelatin, this expression fits reasonably well the experimental data. APPENDIX
Assuming that each tannic acid molecule has r residues that can participate independently in bonding to one peptide chain, each protein molecule has n identical noninteracting binding sites, and m # is the average number of sites per CBB–BSA molecule occupied by tannic acid molecules, one has from the mass action law m # 3 Pt 5 K 3 n 3 Pf 3 r 3 Tf ,
[A1]
where T f and P f, are concentrations of free tannic acid and protein, respectively, and P t is the total protein concentration. Replacing in Eq. [A1] n 3 P f by n 3 P f 5 (n 2 m # ) 3 P t one has m # 3 P t 5 K 3 ~n 2 m # ! 3 Pt 3 r 3 Tf .
[A2]
Or, after algebraic transformation, m # 5n
K 3 r 3 Tf 1 1 K 3 r 3 Tf
[A3]
# Substitution in Eq. [A3] with r 3 T f 5 r 3 T t 2 m 3 P t results in Eq. [A4]: m # 5n
# 3 P t! K 3 ~r 3 T t 2 m 1 1 K 3 ~r 3 T t 2 m # 3 P t!
[A4]
Equations [A3] and [A4] are exactly Eq. [4] used above in the analyses of experimental data of tannic acid binding. REFERENCES
CONCLUSION
A quantitative model for tannic acid-induced protein precipitation in solution has been developed based upon the assumption that precipitation occurs when the number of tannic acid molecules associated with one protein molecule attains a critical value necessary to form tannin-crosslinked protein aggregates. Analytical expressions were derived for the dependence of the protein precipitate yields on the protein and tannic 3 The value of l defined from Eq. [3] does not depend on r, i.e., on the assumption about a pentagalloylglucose structure for tannic acid. The number of binding sites on the protein molecule that can be occupied by one tannic acid molecule, for other than r 5 5, results only in the different number of tannic acid molecules per CBB–BSA required to induce protein precipitation.
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