Mean field kinetics of the enzyme-triggered gelation of casein micelles

A new calculation of the kinetics of the renneting reaction

A detailed calculation of the growth of molecular weight during the renneting of milk is given, based on a first-order breakdown of κ-casein followed by development of instability caused either by a decrease in the intermicellar repulsive potential or by the formation of holes in the stabilizing surface layer of the micelles. Unlike most of the models which have been described, this model allows a complete analytical solution. The solution is, however, complex and difficult to use simply, although it is shown that the calculations are in accord with experimental observations of the dependence of the coagulation process upon the enzyme concentration and the concentration of the milk. The calculations are also compared with those from other models of the reaction.

Precision conductometry in milk renneting

On renneting, the electrical conductivity of milk decreased as viscosity increased. The sigmoidal time course of the decrease resembled the time course of shear modulus, but was more rapid. The total amount of change was independent of the amount of rennet and proportional to milk conductivity and its casein content. The conductivity change was interpreted as a change in the way casein micelles obstructed the path of the charge-carrying ions.

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.