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Enzyme-induced coagulation of casein micelles: a number of different kinetic models

Published online by Cambridge University Press:  01 June 2009

Douglas B. Hyslop
Douglas B. Hyslop
Affiliation:
Department of Food Science, University of Wisconsin, Madison, WI 53706, USA
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Summary

Several mathematical models are presented in an attempt to describe the kinetics of the enzyme-induced coagulation of casein micelles. In each model the primary phase of the clotting reaction is assumed to follow first order kinetics. The only differences amongst the various models centre on the definition of the flocculation rate constant, which is defined in seven different ways. The rate constants are defined and discussed in terms of activation energy and functionality theory. The first model is such that the number of functional sites is two. The second is such that the number is much larger. The third and fourth are such that there is an exponential energy barrier, one which has a magnitude proportional to the extent of proteolysis caused by the clotting enzyme. These two definitions differ only in the pre-exponent. In one case the pre-exponent is a constant, whereas in the other it is dependent on the size of clotting particles. The fifth and sixth definitions are also energy barrier rate constants, but the energy barrier changes in an arbitrary fashion with respect to time during proteolysis. The seventh definition assumes a large number of functional sites, but such that the number increases with extent of proteolysis. In the Payens nomenclature (Payens, 1989), all models could be considered to be ‘source’ models, and all are derived using the Drake moment equation (Drake, 1972). Only the first model has a truly constant flocculation rate parameter, and only this model has a relatively simple analytical solution. All other models yield analytical solutions only by way of infinite series expansions. Thus, all models are presented in terms of power series expansions, and only through the first five time-dependent coefficients. This confines all models to the early stages of coagulation. In all cases the first three coefficients are virtually the same. The first two coefficients involve only proteolysis, and the third includes initial flocculation information. Time-dependent changes in the flocculation rate constant begin to take effect in the fourth coefficient. When the fourth coefficients of the third and seventh models are compared, a simple relationship is suggested between free energy barrier removal and functional site generation, but only assuming that the number of functionalities is large.

Type
Original Articles
Information
Journal of Dairy Research , Volume 60 , Issue 4 , November 1993 , pp. 517 - 533
Copyright
Copyright © Proprietors of Journal of Dairy Research 1993

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References

REFERENCES

Brown, R. J. & Ernstrom, C. A. 1988 Milk-clotting enzymes. In Fundamentals of Dairy Chemistry, 3rd edn pp. 609–633 (Ed. Wong, N. P.). New York: Van Nostrand ReinholdGoogle Scholar
Carlson, A., Hill, C. G. & Olson, N. F. 1987 Kinetics of milk coagulation. II. Kinetics of the secondary phase: micelle flocculation. Biotechnology and Bioengineering 29 590–600Google Scholar
Cohen, R. J. & Benedek, G. B. 1982 Equilibrium and kinetic theory of polymerization and the sol-gel transition. Journal of Physical Chemistry 86 3696–3714CrossRefGoogle Scholar
Dalgleish, D. G. 1980 A mechanism for the chymosin-induced flocculation of casein micelles. Biophysical Chemistry 11 147–155Google Scholar
Dalgleish, D. G. 1988 A new calculation of the kinetics of the renneting reaction. Journal of Dairy Research 55 521–528CrossRefGoogle Scholar
Dalgleish, D. G., Brinkhuis, J. & Payens, T. A. J. 1981 The coagulation of differently sized casein micelles by rennet. European Journal of Biochemistry 119 257–261CrossRefGoogle ScholarPubMed
Dalgleish, D. G. & Holt, C. 1988 A geometric model to describe the initial aggregation of partly rennetted Casein micelles. Journal of Colloid and Interface Science 123 80–84CrossRefGoogle Scholar
Darling, D. F. & Van Hooydonk, A. C. M. 1981 Derivation of a mathematical model for the mechanism of casein micelle coagulation by rennet. Journal of Dairy Research 48 189–200CrossRefGoogle Scholar
Drake, R. L. 1972 The scalar transport equation of coalescence theory: moments and kernels. Journal of the Atmospheric Sciences 29 537–547Google Scholar
Ernst, M. H., Ziff, R. M. & Hendriks, E. M. 1983 Coagulation processes with a phase transition. Journal of Colloid and Interface Science 97 266–277Google Scholar
Green, M. L., Hobbs, D. G., Morant, S. V. & Hill, V. A. 1978 Inter-micellar relationships in rennet-treated separated milk. Journal of Dairy Research 45 413–422CrossRefGoogle Scholar
Holt, C. 1975 The stability of bovine casein micelles. In International Conference on Colloid and. Surface Science, pp. 641–644 (Ed. Wolfram, E.). Budapest: Akademiai KiadoGoogle Scholar
Holt, C. 1985 The size distribution of bovine casein micelles. A review. Food Microstructure 4 1–10Google Scholar
Hyslop, D. B. 1989 Enzymatically initiated coagulation of casein micelles: a kinetic model. Netherlands Milk and Dairy Journal 43 163–170Google Scholar
Johnson, J. 1991 The Effects of Sodium, Calcium and Magnesium Chloride Salts on the Chymosin-induced Coagulation of Skim Milk. MS Thesis, University of Wisconsin-MadisonGoogle Scholar
Lushnikov, A. A., Tokar, Ya. I. & Tsitshishivili, M. S. 1981 [Two precisely resolvable models of coagulating systems with a particle source.] Doklady Akademii Nauk SSSR 256 1155–1158Google Scholar
Moore, J. W. & Pearson, R. G. 1981 Kinetics and Mechanism, 3rd edn.New York: John Wiley.Google Scholar
Overbeek, J. Th. G. 1952 Kinetics of flocculation. In Colloid Science, vol. 1, Irreversible Systems, pp. 278–301 (Ed. Kruyt, H. R.). Amsterdam: ElsevierGoogle Scholar
Payens, T. A. J. 1976 On the enzyme-triggered clotting of casein; a preliminary account. Netherlands Milk and Dairy Journal 30 55–59Google Scholar
Payens, T. A. J. 1977 On enzymatic clotting processes. II. The colloidal instability of chymosin-treated casein micelles. Biophysical Chemistry 6 263–270CrossRefGoogle ScholarPubMed
Payens, T. A. J. 1989 The enzyme-triggered coagulation of casein micelles. Advances in Colloid and Interface Science 30 31–69CrossRefGoogle Scholar
Payens, T. A. J. & Brinkhuis, J. 1986 Mean field kinetics of the enzyme-triggered gelation of casein micelles. Colloids and Surfaces 20 37–50CrossRefGoogle Scholar
Stockmayer, W. H. 1943 Theory of molecular size distribution and gel formation in branched-chain polymers. Journal of Chemical Physics 11 45–55Google Scholar
Van Hooydonk, A. C. M., Boerrigter, I. J. & Hagedoorn, H. G. 1986 pH induced physico-chemical changes of casein micelles in milk and their effect on renneting. 2. Effect of pH on renneting of milk. Netherlands Milk and Dairy Journal 40 297–313Google Scholar
Van Hooydonk, A. C. M., Olieman, C. & Hagedoorn, H. G. 1984 Kinetics of the chymosin-catalysed proteolysis of κ-casein in milk. Netherlands Milk and Dairy Journal 38 207–222Google Scholar
Van Hooydonk, A. C. M. & Walstra, P. 1987 a Interpretation of the kinetics of the renneting reaction in milk. Netherlands Milk and Dairy Journal 41 19–47Google Scholar
Van Hooydonk, A. C. M. & Walstra, P. 1987 b A consistent model for the clotting of milk by rennet; a reply to a comment. Netherlands Milk and Dairy Journal 41 293–295Google Scholar
Verwey, E. J. W. & Overbeek, J. Th. G. 1948 Theory of the Stability of Lyophobic Colloids: the interaction of sol particles having an electric double layer. Amsterdam: Elsevier.Google Scholar
Von Smoluchowski, M. 1917 [Research on a mathematical theory of the kinetics of coagulation of colloidal suspensions.] Zeitschrift für Physikalische Chemie 92 129–168Google Scholar
Walstra, P. 1979 The voluminosity of bovine casein micelles and some of its implications. Journal of Dairy Research 46 317–323CrossRefGoogle ScholarPubMed