A new calculation of the kinetics of the renneting reaction

Rennet-Induced Casein Micelle Aggregation Models: A Review

Two phases are generally recognized in the enzymatic coagulation of milk: hydrolysis and aggregation, although nowadays more and more researchers consider the non-enzymatic phase to actually be a stage of gel formation made up of two sub-stages: micellar aggregation and hardening of the three-dimensional network of para-κ-casein. To evaluate this controversy, the main descriptive models have been reviewed. Most of them can only model micellar aggregation, without modeling the hardening stage. Some are not generalizable enough. However, more recent models have been proposed, applicable to a wide range of conditions, which could differentiate both substages. Manufacturing quality enzymatic cheeses in a cost-effective and consistent manner requires effective control of coagulation, which implies studying the non-enzymatic sub-stages of coagulation separately, as numerous studies require specific measurement methods for each of them. Some authors have recently reviewed the micellar aggregation models, but without differentiating it from hardening. Therefore, a review of the proposed models is necessary, as coagulation cannot be controlled without knowing its mechanisms and the stages that constitute it.

Kinetics of pepsin-induced hydrolysis and the coagulation of milk proteins

Hydrolysis-induced coagulation of casein micelles by pepsin occurs during the digestion of milk. In this study, the effect of pH (6.7-5.3) and pepsin concentration (0.110-2.75 U/mL) on the hydrolysis of κ-casein and the coagulation of the casein micelles in bovine skim milk was investigated at 37°C using reverse-phase HPLC, oscillatory rheology, and confocal laser scanning microscopy. The hydrolysis of κ-casein followed a combined kinetic model of first-order hydrolysis and putative pepsin denaturation. The hydrolysis rate increased with increasing pepsin concentration at a given pH, was pH dependent, and reached a maximum at pH ∼6.0. Both the increase in pepsin concentration and decrease in pH resulted in a shorter coagulation time. The extent of κ-casein hydrolysis required for coagulation was independent of the pepsin concentration at a given pH and, because of the lower electrostatic repulsion between para-casein micelles at lower pH, decreased markedly from ∼73% to ∼33% when pH decreased from 6.3 to 5.3. In addition, the rheological properties and the microstructures of the coagulum were markedly affected by the pH and the pepsin concentration. The knowledge obtained from this study provides further understanding on the mechanism of milk coagulation, occurring at the initial stage of transiting into gastric conditions with high pH and low pepsin concentration.

Application of numerical analysis to a number of models for chymosin-induced coagulation of casein micelles

Four models describing renneting kinetics are evaluated for their ability to describe well documented attributes of the coagulation of casein micelles. The first model is based on a constant flocculation rate parameter. In the second the flocculation rate constant is proportional to the product of the sizes of the aggregating particles. Both models fail to predict proper dependence of rennet coagulation time on enzyme concentration. The third model is based on an energy barrier being reduced in linear proportion to the degree of proteolysis. The enzyme dependency of this model only works when the initial energy barrier is larger than ∼ 50 kBT (where kB is Boltzmann's constant and T the absolute temperature), which does not seem feasible. The fourth model, based on functionality theory, is able to predict proper dependence of rennet coagulation time on enzyme concentration when functionality is ∼ 2.

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

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.