Derivation of a mathematical model for the mechanism of casein micelle coagulation by rennet

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.

Application of polymer cross-linking theory to rennet-induced milk gels

A general theory of gel formation by cross-linking of polymer molecules, orginally developed for the study of rubber vulcanization, is proposed as a model for milk gel formation. The model divides gel formation and development into 4 stages, an enzymic and a non-enzymic stage preceding gelation, then a stage of material transition from the sol to the gelled state and a stage of increase in gel strength due to internal cross-link formations. The 2 post-gelation stages overlap to a variable degree depending upon particular circumstances. Gel formation and properties are interpreted in terms of the number of initial units which combine to form the gel and the number of cross-links eventually formed by each unit. When combined with a kinetic expression for the rate of cross-link formation the time course of gel shear modulus development or of the incorporation of material into the gel may be predicted. The classical asymmetric sigmoid shape of gel shear modulus development curves results from the third stage of material transition into the gel, which in turn depends upon the number of cross-links eventually formed by each initial unit. The modulus of mature gels is found to depend upon the square of material concentration, and gel breaking strength is found to be proportional to the modulus, in agreement with the predictions of the theory.

Coagulation of homogenized milk particles by rennet

The action of rennet on homogenized milk was studied using turbidimetric and light scattering techniques, and compared with results obtained previously for skim milk. The time required for the onset of coagulation was shorter for homogenized milk than for skim milk. The rate of coagulation of fully renneted particles increased with increasing temperature, and with increasing Ca2+ concentration, but was only slightly influenced by changes in ionic strength. The von Smoluchowski rate constant for the coagulation reaction was two orders of magnitude smaller for homogenized milk than for skim milk. Results suggest that coagulation of homogenized milk is controlled in general by the same factors as skim milk, and that the reaction may be inhibited owing to a reduction in the amount of casein available for mutual interaction, rather than to disruption of the micelles on homogenization.

Effects of cations and anions on the rate of the acidic coagulation of casein micelles: the possible roles of different forces

Raw skim milk was diluted 1000-fold using distilled water or various salt solutions as specified. Smooth, hyperbolic profiles of coagulation ratev.pH for casein were calculated from recordings of turbidity (400 nm) with time. The effects of pH, cation type, anion type and cleavage ofk−casein by chymosin (EC 3.4.23.4) were determined. The maximum of pH-coagulation rate profiles decreased by 63, 85 and 94% when the skim milk diluent was changed from water to salt solutions of NaCl (100 mM), CaCl2(50 mM) or MgCl2(50 mM). The maximum of the pH–coagulation rate profile was 15 times greater when the Ca salt was changed from CaCl2to Ca(SCN)2(50 mM). The highest pH at which casein coagulation occurred increased from 4·45 to > 6·0 when Cu2+(1 mM) was included with casein micelles dispersed in CaCl2solution (50 mM). The addition of chymosin to casein micelles suspended in CaCl2solution (70 mM) eliminated the inhibition of casein coagulation by Ca2+at pH 4·5. It is proposed that ions such as Mg2+, Ca2+, and Na+, which generally associate with casein phosphate and carboxylate groups, increased the H+concentration required to initiate the coagulation of casein, because H+must displace bound Ca2+, Mg2+or Na+to reduce repulsive hydration forces between casein micelles, allowing attractive hydration forces (e.g. hydrophobic phenomena) to cause casein coagulation. Furthermore, it is proposed that ions such as Cl−, Br−,and SCN−bind to lysine, arginine and histidine groups and thereby decrease repulsive hydration forces between cationic 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.

Use of Platelet Aggregometer to monitor the chymosin-initiated coagulation of casein micelles

The chymosin-initiated coagulation of casein micelles was followed by monitoring light transmission using a Platelet Aggregometer. The release of macropeptide by chymosin was monitored using fluorescamine. The lag period in the clotting reaction was proportional to clotting time and the reciprocal of enzyme concentration. The average rate of coagulation, which was approximately equal to the reciprocal of clotting time (Tc), increased in proportion to enzyme concentration at low enzyme concentrations and reached a limiting value at high enzyme concentrations. The percentage hydrolysis at the Tc was 47 ± 5% in the presence of 20 mM-CaCl2 and it was calculated that a 5-fold decrease in the speed of the enzyme-catalysed reaction would decrease this value at the Tc to 43 ± 5%. The possible uses and limitations of the Platelet Aggregometer for determining the influence of the chemical environment on the velocity of the chymosin-catalysed reaction and para-casein micelle aggregatability are discussed.

A mathematical model for the description of chymosin action on casein micelles

A mathematical model for chymosin action on casein micelles is presented in a two-stage equation which results in a single curve demonstrating the lag time from enzyme addition to the end of coagulum firming. The model uses the Michaelis–Menten enzyme kinetics equation for the first reaction followed by an nth order reaction for the casein micelles agglomeration stage. The computer output using these equations shows that lag time is elongated as enzyme concentration is lowered. Regression analysis of time of gelation against l/E0 shows good correlation. Viscosity of the milk drops at the beginning of the κ-casein hydrolysis and increases thereafter, when the coagulum is being formed.

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.

Characterization of Enzymatically Induced Aggregation of Casein Micelles in Natural Concentration by in Situ Static Light Scattering and Ultra Low Shear Viscosimetry

The aggregation of casein micelles in undiluted skim milk after the addition of chymosin was studied by static light scattering and ultra low shear viscometry. The static light scattering measurements were made with two different sample thicknesses, 72 and 16 µm. The scattering data were analyzed by indirect Fourier transformation and by the polydispersity inversion technique which led to pair distance distribution functions and size distribution function, respectively. The minimum scattering angle was 1 degrees, which allows for the determination of particle sizes up to a maximum diameter of 12 µm. The fractal dimension determined from double logarithmic plots of intensity versus scattering vector resulted in values between 1.9 and 2.0. The influence of multiple scattering was determined by comparison of the measurements with the different sample thicknesses. The measurements show no significant influence of multiple scattering when the transmission is above 0.85. Due to the very complex and porous structure of the casein aggregates the Rayleigh-Debye-Gans scattering theory has been used in the data analysis. Measurements with a new instrument using ultra low shear showed good agreement with theory. Copyright 1999 Academic Press.

Structures and functionalities of milk proteins

During the last decade, marked progress has been made in the study of the fine details of the structures of milk proteins such as caseins, beta-lactoglobulin, alpha-lactalbumin, and lactotransferrin. Many of the functional properties of the individual milk proteins, as well as the milk protein products, may be described at the molecular level. This article is an attempt to thoroughly review the three-dimensional structures of major milk proteins, and to correlate them with the functional aspects of these proteins as food ingredients.