# Rennet-Induced Casein Micelle Aggregation Models: A Review

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{2+}, the micellar aggregation process [6]. Hardening is typically considered an extension of aggregation, but, indeed, it seems not to be the case, as the study of this step actually requires a different approach and instrumentation compared to aggregation. In fact, gel hardening is the most complex and currently least understood aspect of enzymatic milk coagulation. Various instruments are available to study gel rheological properties that are critical for cheese manufacturing control, such as the rheologically determined gelation time or the appropriate elastic modulus for cutting [7]. As a result, there is particular interest in developing methods that can be used to study/quantify gelation in the cheese vat [8], as control of the gel hardness is critical for the objective selection of the optimal curd cutting time that affects both cheese quality and yield. Figure 1 outlines the coagulation process of milk.

_{max}); and (B) the period corresponding to the non-enzymatic phase of coagulation which consists of two phases: (a) the period corresponding to the aggregation phase of the destabilized micelles that goes from t

_{max}until the moment in which the gel begins to offer mechanical resistance (e.g., the point that could be identified, approximately, as the time at which a value of G′ = 1 Pa is reached, determined by a rheometer [10,11], related to t

_{2min2}[12]) and (b) the asymptotic period corresponding to the hardening phase of the casein gel that continues until the optimum curd cutting point is reached. Harboe et al. [13] and Fox et al. [8] also describe differences between the aggregation phase and the hardening phase, but from the point of view of changes in viscosity and rheological parameters. Horne and Lucey [14] carried out the last review of the aggregation models but, as mentioned above, without differentiating the aggregation phase from the hardening phase. As seen above, the controversy raised by the separation of the non-enzymatic phase of milk coagulation into phases of aggregation and hardening will serve as the basis for studying, in a first approach, the different models that describe the phenomenon of micellar aggregation.

## 2. Micellar Aggregation

^{2+}, as the van der Waals attraction is not sufficient. The need of a sufficient amount of Ca

^{2+}in the serum phase of milk clearly shows its role. Presumably, the effect of Ca

^{2+}ions is double. First, Ca

^{2+}decreases electrostatic repulsion by neutralizing negative charges on the micelles [21], being more efficient than H

^{+}ions in the pH range under consideration. Second, Ca

^{2+}ions can bridge between negative sites on paracasein micelles by means of salt bridges. It is very well-known that lowering the pH of milk considerably increases Ca

^{2+}activity [22] by shifting the existing ionic balance between the soluble and micellar phases, releasing H

^{+}from the micelle.

_{max}(Figure 2) was proposed as an approximate target value of aggregation start time [28,29].

## 3. Aggregation Models of Hydrolyzed Casein Micelles

#### 3.1. Holter Model

_{c}is the coagulation time, E is the enzyme concentration, and k is the constant.

_{h}) and the aggregation time (t

_{a}), defining the coagulation time as t

_{c}= t

_{h}+ t

_{a}. Subsequently, Foltmann [32], assuming Holter’s proposal, proposed the following equation for t

_{c}:

_{a}is the aggregation time, t

_{c}is the coagulation time, [E] is the enzyme concentration, and C is the constant.

_{a}, would come to represent the minimum duration of coagulation when the enzyme concentration tends to infinity (i.e., hydrolysis would be instantaneous, not limiting the aggregation rate). As can be seen, this relationship is purely empirical, but it is a relevant relationship that must be satisfied as a starting point for any more descriptive mechanistic models. Figure 5 shows the representation of Equation (2). The equation assumes that there is no overlap between proteolysis and the aggregation phase, and that the degree of proteolysis is always the same at the time of coagulation [33].

#### 3.2. Payens Model

_{c}is the coagulation time, k

_{s}is the aggregation rate constant, and V

_{max}is the maximum rate of hydrolysis of κ-casein.

_{s}) of the hydrolyzed micelles was very low compared to the constants found experimentally. This has been attributed to the fact that k

_{s}is an average of all values, from very low values at the beginning and high values at the end of the aggregation reaction [3,40,41,42].

#### 3.3. Step Function Model

_{c}is the coagulation time, t

_{h}is the hydrolysis time, t

_{a}is the aggregation time, k

_{M}is the Michaelis–Menten constant (it approximates the dissociation constant for the enzyme-substrate complex), V

_{max}is the maximum rate of proteolysis at infinite substrate concentration, α

_{c}is the extent of κ-casein hydrolysis, S

_{0}is the initial concentration of κ-casein, k

_{s}: is the aggregation rate constant, C

_{0}is the concentration of aggregating material, M

_{crit}is the average molecular weight at coagulation time (≈10 micellar units), and M

_{0}is the average molecular weight at t = 0.

#### 3.4. The Energy Barrier Model

_{t}is the stability factor at time t for casein micelles, W

_{0}is the initial stability factor for casein micelles, C

_{m}is the constant that relates the stability of the casein micelle to the concentration of κ-casein, V is the enzymatic reaction rate, and t is the time.

_{c}:

_{c}is the coagulation time, S

_{0}is the initial concentration of κ-casein, k

_{s}is the aggregation rate constant, n

_{c}is the concentration of casein aggregates at time t

_{c}, and n

_{0}is the initial concentration of casein micelles.

_{c}) as a function of temperature, rennet concentration, and total protein concentration. Additionally, it demonstrated the existence of a delay time in the start of aggregation equivalent to 60% clotting time [45]. This energy barrier model has been tested extensively [4]. The difficulty with the model lies in its simplification of Michaelis kinetics to allow calculations [39].

#### 3.5. The Functionality Model

_{ij,}should be a function of the number of free reactive sites on the aggregation particles, which met the required conditions for a polyfunctional model of the Flory–Stockmayer type [48]:

#### 3.6. Adhesive Hard-Sphere Model

_{r}, of skim milk is described as:

#### 3.7. Fractal Aggregation Model

_{p}, is obtained as follows:

_{A}, is given by:

_{A}will decrease with increasing values of R as the group grows [63]. At a certain radius (R

_{g}), to which the aggregate is known as the fractal ‘blob’, φ

_{A}will be equal to the volume fraction of particles in the system φ and all aggregates will touch, forming a continuous gel network [5]. Bremer et al. [55] defined the gelation point base on this event, which infers that all the particles present in the system are incorporated into the aggregates. The real question is whether this corresponds to the experimentally measured rheological gel point (crossover point—when the elastic modulus, G′, becomes higher that the viscous modulus, G″, which is sometimes generalized empirically as G′ = 1 Pa); this seems to be the assumption made [4]. Furthermore, assuming that the fractal blob coincides with the moment in which the rheometer yields a measurable rheological response, the obvious question is how the increase in the elastic modulus of the structure takes places, according to the fractal model [64].

#### 3.8. Light Scattering Model

_{a}is the diffuse reflectance ratio attributed to the aggregation period; R

_{∞A}is the diffuse reflectance ratio at t = ∞, during the aggregation period; R

_{0A}is the diffuse reflectance ratio at t

_{max}; k

_{2}is the second order reaction rate constant; t′ is the time after enzyme addition; and t

_{max}is the starting time for the aggregation reaction.

## 4. Summary

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Coagulation curves of pasteurized whole milk. R is the light scattering ratio, R’ and R″ are the first and second derivatives of R, respectively. t

_{max}is the time to the maximum of the first derivative of R. t

_{2min2}is the time to the second minimum of the second derivative of R. A is the enzymatic period. B is the nonenzymatic period. a is the aggregation period. b is the hardening period. Reproduced with permission from [Payne, F.A.; Castillo, M.], [Light Backscatter Sensor Applications in Milk Coagulation. In Encyclopedia of Agricultural, Food, and Biological Engineering]; published by [Taylor & Francis], [2007].

**Figure 3.**Release of macropeptides, expressed as fraction (●). changes in relative viscosity (■) during the course of rennet coagulation. On top of the figure, casein micelle aggregation progress is graphically represented. Reproduced with permission from [Fox, P.F.; Guinee, T.P.; Cogan, T.M.; McSweeney, P.L.H], [Fundamentals of Cheese Science, 2nd ed.]; published by [Springer], [2017].

**Figure 4.**Release of casein macropeptide during coagulation of milk at different pH values. Reproduced with permission from [Corredig, M.; Salvatore, E.], [Enzymatic Coagulation of Milk. Advanced Dairy Chemistry: Volume 1B: Proteins: Applied Aspects]; published by [Springer], [2016].

**Figure 5.**Relationship between enzyme concentration (E), aggregation time (t

_{a}), and coagulation time (t

_{c}). Reproduced with permission from [Dalgleish, D.G.], [A New Calculation of the Kinetics of the Renneting Reaction.]; published by [J. Dairy Res.], [1988] [34].

**Figure 6.**Two interacting casein micelles. The hard-core repulsion is preceded by an attractive interaction modeled as a “square well” potential. The depth of the well depends on the stability of the κ-casein brush. Reproduced with permission from [De Kruif, C.G.], [Casein Micelle Interactions]; published by [Int. Dairy J.], [1999].

**Figure 7.**Relative viscosity of skim milk during the coagulation process. The drawn line represents the theoretical calculation in which the micelles are considered to be sticky hard spheres. Reproduced with permission from [De Kruif, C.G.], [Casein Micelle Interactions]; published by [Int. Dairy J.], [1999].

**Figure 8.**Formation of a weak gel in which most of the particles are weakly bound to the network. Larger circles represent the fractal ‘blob’ concept. Reproduced with permission from [Horne, D.S.], [Banks, J. Rennet-induced Coagulation of Milk. In Cheese Chemistry, Physics and Microbiology]; published by [Elsevier], [2004].

**Figure 9.**Short-term rearrangement. (

**a**) Particle rearrangement. (

**b**) Fractal group in two dimensions, where a short-term rearrangement has occurred. Reproduced with permission from [Walstra, P.], [Physical Chemistry of Foods]; published by [Marcel Dekker], [2003].

**Figure 10.**Effect of temperature and protein concentration on apparition of a shoulder in the curve of diffuse reflectance ratio versus time. (

**a**) Data corresponded to skim milk adjusted to 30 g protein/kg and coagulated at 20, 25, 30, 35, and 40 °C and at constant calcium chloride and enzyme concentration. (

**b**) Data corresponded to skim milk adjusted to 30, 50, and 70 g protein/kg and coagulated at 20 °C and at constant calcium chloride and enzyme concentration. Reproduced with permission from [Castillo, M.Z.; Payne, F.A.; Hicks, C.L.; Laencina, J.S.; López, M.-B.M.], [Modelling Casein Aggregation and Curd Firming in Goats’ Milk from Backscatter of Infrared Light.]; published by [J. Dairy Res.], [2003].

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**MDPI and ACS Style**

Salvador, D.; Acosta, Y.; Zamora, A.; Castillo, M.
Rennet-Induced Casein Micelle Aggregation Models: A Review. *Foods* **2022**, *11*, 1243.
https://doi.org/10.3390/foods11091243

**AMA Style**

Salvador D, Acosta Y, Zamora A, Castillo M.
Rennet-Induced Casein Micelle Aggregation Models: A Review. *Foods*. 2022; 11(9):1243.
https://doi.org/10.3390/foods11091243

**Chicago/Turabian Style**

Salvador, Daniel, Yoseli Acosta, Anna Zamora, and Manuel Castillo.
2022. "Rennet-Induced Casein Micelle Aggregation Models: A Review" *Foods* 11, no. 9: 1243.
https://doi.org/10.3390/foods11091243