**2. Theory**

A basis for the theory of random transformations of dispersed matter and principles of constructing general phenomenological and mathematical models were presented by Solecki (2011). In the case of disintegration of microorganisms in bead mills, a material medium V(τ) is a suspension of microorganisms which fills the working mill chamber. The concept of material objects belonging to set *N* covers microbial cells dispersed in the liquid. A random transformation consists in cell disruption between circulating elements of the packing and release of intracellular compounds.

Basic conditions prevailing in space V of the mill chamber are safe for microbial cells. The process of random transformation of objects belonging to set *N* proceeds as follows.

In any instant τ (τ ≥ 0), p families of transformation volumes Vγj dispersed in space V are generated at random, where p is the natural number. During the process the number of generated families of volumes Vγj is big and can change in time according to Eq. (1).

$$p = f\_3(\tau) \tag{1}$$

Volume Vγji is the transformation space of the i-th cell of dispersed microorganisms which belongs to the j-th family. A set of the conditions in which a transformation proceeds at the intensity no less than γti, occurring in this volume, will ensure the transformation of the i-th object from set *N*. The transformation consists in the change ς of cell properties (ς>0) belonging to the set of identified properties of microorganisms *Pr*. After the transformation an object from set *N* does not belong to set *N* any longer. In the case of disintegration of microbial cells in a bead mill, the set of systemic transformation conditions is limited mainly to the transforming action of mechanical factors. According to Eq. (2), space V is incomparably bigger than any i-th volume Vγji.

$$V\_{\gamma\vec{\mu}} << V \tag{2}$$

The range of changes in transformation volumes Vγji generated in space V is described by Eq. (3) in reference to the volume, and by Eq. (4) in view of the intensity of transformation conditions.

$$\left(V\_{\gamma j i}\right)\_{\text{min}} \le V\_{\gamma j i} \le \left(V\_{\gamma j i}\right)\_{\text{max}}\tag{3}$$

$$V\_{\gamma j \bar{i} (\gamma\_t = \gamma\_{\bar{u}})} \le V\_{\gamma j \bar{i} (\gamma\_t)} \le V \gamma\_{\gamma j \bar{i} (\gamma\_t = \gamma\_{t \text{max}})} \tag{4}$$

Symbol γtmax denotes the highest intensity of transformation conditions which can be generated in space V.

As a result of relative movement, the i-th cell is introduced to appropriate transformation volume Vγji. This volume is limited by surface F<sup>γ</sup>αji belonging to it according to Eq. (5).

$$F\_{\gamma\alpha\;\,\dot{\mu}} \in V\_{\gamma\;\,\dot{\mu}}\tag{5}$$

develop, basing on this theory, a phenomenological model of microorganism disintegration in a bead mill and to present a mathematical description of the process which would include

A basis for the theory of random transformations of dispersed matter and principles of constructing general phenomenological and mathematical models were presented by Solecki (2011). In the case of disintegration of microorganisms in bead mills, a material medium V(τ) is a suspension of microorganisms which fills the working mill chamber. The concept of material objects belonging to set *N* covers microbial cells dispersed in the liquid. A random transformation consists in cell disruption between circulating elements of the

Basic conditions prevailing in space V of the mill chamber are safe for microbial cells. The

In any instant τ (τ ≥ 0), p families of transformation volumes Vγj dispersed in space V are generated at random, where p is the natural number. During the process the number of

> <sup>3</sup> *p f* = ( ) τ

Volume Vγji is the transformation space of the i-th cell of dispersed microorganisms which belongs to the j-th family. A set of the conditions in which a transformation proceeds at the intensity no less than γti, occurring in this volume, will ensure the transformation of the i-th object from set *N*. The transformation consists in the change ς of cell properties (ς>0) belonging to the set of identified properties of microorganisms *Pr*. After the transformation an object from set *N* does not belong to set *N* any longer. In the case of disintegration of microbial cells in a bead mill, the set of systemic transformation conditions is limited mainly to the transforming action of mechanical factors. According to Eq. (2), space V is

> *V V* γ

The range of changes in transformation volumes Vγji generated in space V is described by Eq. (3) in reference to the volume, and by Eq. (4) in view of the intensity of transformation

( ) ( ) min max *V VV*

( ) () ( ) *t ti <sup>t</sup> t t*max

 γ γγ

γ

 γγ

*ji ji ji*

Symbol γtmax denotes the highest intensity of transformation conditions which can be

As a result of relative movement, the i-th cell is introduced to appropriate transformation volume Vγji. This volume is limited by surface F<sup>γ</sup>αji belonging to it according to Eq. (5).

> *F V* γα

 γ

γ

γ γγ

*V VV*

 γ γ (1)

*ji* << (2)

*ji* ≤ ≤ *ji ji* (3)

<sup>=</sup> <sup>=</sup> ≤ ≤ (4)

*ji* ∈ *ji* (5)

process of random transformation of objects belonging to set *N* proceeds as follows.

generated families of volumes Vγj is big and can change in time according to Eq. (1).

the effect of cell size on the rate of cell disintegration.

packing and release of intracellular compounds.

incomparably bigger than any i-th volume Vγji.

**2. Theory** 

conditions.

generated in space V.

On the other hand, volume Vγji can be limited by surface Fγβji when it is formed. This surface belongs to the transformation volume according to Eq. (6).

$$F\_{\gamma\beta\,\,\dot{\mu}} \in V\_{\gamma\,\,\dot{\mu}} \tag{6}$$

The i-th object is transformed immediately after translocation of all its points to the transformation volume Vγji. If volume Vγji is bigger than the volume of the i-th object, then volume Vβji is formed in it. It is limited by surface F<sup>γ</sup>αji which does not belong to it according to Eq. (7)

$$F\_{\rho\alpha ji} \not\ll V\_{\beta ji} \tag{7}$$

Space Vβji cannot include total unconverted material object belonging to set *N* for which an appropriate transformation volume is Vγji. Before being translocated to Vβji, every i-th object must be first transformed in volume Vγji.

Volume Vαi is safe for the unconverted i-th object from set *N*. We assume that intensive stirring takes place in it. Its aim is to make the concentration of unconverted objects from set *N* uniform within appropriate volumes which are safe for them. Volume Vαi is composed of two parts: Vαit which is part of volume Vαi whose subsets can be transformed to volumes Vγ<sup>i</sup> or Vβi; and Vαic which is part of volume Vαi whose subsets are never transformed to other volumes. In the case of the bead mill, volumes Vαic occur near all surfaces of the working chamber. These volumes are distant from the mill surface by the size of a cell, and the thickness of their layer is slightly smaller than the radius of the smallest bead in the packing. In volume Vαic the i-th cell is fully safe. The quotient of volumes Vαic to Vαi, the quotient of the sum of volumes Vγi and Vβi to Vαit and the rate of relative movement of the i-th cell to Vγ<sup>i</sup> are the factors that determine efficiency of the system of transformation of the i-th cell in a given technical device. An increase of the first factor causes a decrease of the transformation efficiency. An opposite effect is caused by an increase of other factors.

For instance, studies were carried out to increase differences in the velocity of points on the surfaces of adjacent beads of the packing which circulates in the mill (Solecki, 2007). Such an effect was to be induced by the presence of immobile baffles between stirring disks of a classical mill. In many cases of geometric solution of the mill interior, because of introduction of the baffles the efficiency of microorganism disintegration was deteriorated (Solecki, 2007). The development of a narrow-clearance construction did not bring about elimination of classical mills equipped with multi-disk impellers from the market. Rate constants determined for the process of disintegration in the mill with a bell-shaped impeller were often lower than those determined for a classical mill in comparable process conditions (Solecki, 2007). Optimum operating conditions of narrow-clearance mills are obtained for smaller values of packing degree. In the case of a classical mill it is about 90%, while for the mill with a bell-shaped impeller 60%, and with a cylindrical stirrer only 40%.

Part of space V is composed of volume Vδ which is safe for microorganisms from set *N* (Solecki, 2011). No stirring takes place in it and neither it nor its subsets are transformed to other volumes. Between volumes Vα and Vδ microbial cells can migrate freely. For a correctly constructed mill chamber, volumes Vδ (these can be slots in the place where two elements meet) are negligibly small and insignificant from the technology point of view, particularly when the device is sterilized between subsequent processes. In further studies it was assumed that Vδ=0.

The Release of Compounds from Microbial Cells 599

surfaces approach each other, the cell is deformed. The contact area of the cell with solid bodies described by diameter dδi increases. Walls of the cell with axis c1i are disrupted when compression surfaces approach each other at the third stage to destruction distance lδi min

> *i i* min 1 *l c* δ

From the point of view of random transformation, the first two destruction stages have no significance, providing naturally that the number of load cycles has no effect on the cell strength. Hence, it can be assumed that starting with instant τ=0, all cells present in the suspension are so deformed that an arbitrarily small increase of deformation in the transformation volume causes their disruption. The distance of destruction is measured along the destruction axis. It was assumed that the cell-destroying surface contact area in the

Disruption of cells dispersed in the liquid occurs once the cells have been translocated to destruction volume Vγji which is in the place of formation of a single destruction system. The continuity of grinding surfaces was assumed for the set of points belonging to circular environment of diameter δi max around the point of destruction. At least one grinding surface has a spherical shape. Thus, in the mill two destruction elements can generate a single destruction axis or a set of axes which constitute a cylindrical surface. At the present stage of studies we make a simplified assumption that elements destroying microbial cells are

We consider the case of batch mill operation, when during the process the supply of microbial suspension is constant. Its volume is equal to V. The process is carried out for the known initial biomass concentration. The initial number of microorganisms is N0. During the process no microbial cells are added from the outside (it is assumed that the disintegration process is carried out in sterile conditions) and no cells are removed to the outside. The time of disintegration is very short as compared to the life time of microorganisms and the time of formation of new cells in the process conditions. Hence, it can be assumed that function f13 which describes cell growth and function f14 describing elimination of cells different than that being a result of the transformation, satisfy equations

> *f N* 13 0 ( , 0 τ

> *f N* 14 0 ( , 0 τ

In the case of disintegration of microorganisms in the bead mills the transformation consists in the disruption of microbial cell walls. In the process carried out in the bead mill, high levels of packing of the mill chamber and big rotational speed of the stirrer are applied. Because of a high level of packing the mill with beads and high intensity of their circulation resulting from high rotational speed of the stirrer, it was assumed that during the process:

1

*i*

=

*ji*

*V const* γ

*n*

1. the sum of all volumes Vγi is constant (Eq. (16)),

moment of cell wall disruption was represented by a spherical cap of diameter δi max.

< (13)

) = (14)

) = (15)

∪ <sup>=</sup> (16)

described by the relation given in Eq. (13).

perfectly rigid.

(14) and (15).

An ellipsoidal shape of microorganisms and their different sizes were assumed. The size of the i-th cell is described by the length of three mutually perpendicular axes c1i, c2i and c3i, which satisfy relation (8).

$$\mathcal{L}\_{1i} = \mathcal{c}\_{2i} \le \mathcal{c}\_{3i} \tag{8}$$

Microorganisms were assumed to be much smaller than packing elements in the mill dk according to Eq. (9).

$$c\_{3i} \ll d\_k \tag{9}$$

For the whole population of microorganisms subjected to disintegration it was assumed initially that cell walls had the same mechanical strength and that the compressibility of fluids protected by them was identical.

Microorganisms in the working mill chamber are disintegrated between two surfaces belonging to different solids. One of them is a single element of the packing, and the other one is another element of the packing or an element of the mill working chamber. This can be the external surface of the reservoir of diameter Dz, the surface of an impeller with disk diameter dt or of its shaft of diameter dw. Due to the relations in Eq. (9) through (12), it was assumed that all inner surfaces of the mill chamber and stirrer were planes.

$$d\_k \ll D\_z \tag{10}$$

$$d\_k \ll d\_w \tag{11}$$

$$d\_k \ll d\_t \tag{12}$$

It was assumed that microorganisms were destroyed as a consequence of disruption of cell walls caused mainly by compressive load which acted along the shorter cell axis c1i. This assumption was made after analysis of the results of many experiments which included disruption of single yeast cells using micro-tools controlled by micromanipulators. Results of these experiments have not been published yet. Non-homogeneity of the structure, thickness and rigidity of cell walls causes that microorganisms compressed along the long axis are deformed asymmetrically and due to accumulated energy catapult from the hazardous zone. This property formed by the evolution guarantees a better survival rate of microorganisms in their environment. In the case of actions characterized by much higher dynamics than the typical one occurring in nature, the probability of cell destruction along the long axis can be significant. In such a case the above assumption simplifies the problem.

It is assumed that cell walls are disrupted when compounds inside the cell can flow out of it. This state is equivalent to cell disintegration – transformation of the object. Three stages are distinguished in the destruction of microorganisms (Solecki, 2009):


At the first stage the cell of axis c1i contacts two destroying surfaces. The cell is not deformed. Points of contact of solid bodies with the i-th cell are called the points of destruction. They form a single destruction system. The segment which connects destruction points of cell c1i is defined as the destruction axis. At the second stage, when destroying

An ellipsoidal shape of microorganisms and their different sizes were assumed. The size of the i-th cell is described by the length of three mutually perpendicular axes c1i, c2i and c3i,

Microorganisms were assumed to be much smaller than packing elements in the mill dk

For the whole population of microorganisms subjected to disintegration it was assumed initially that cell walls had the same mechanical strength and that the compressibility of

Microorganisms in the working mill chamber are disintegrated between two surfaces belonging to different solids. One of them is a single element of the packing, and the other one is another element of the packing or an element of the mill working chamber. This can be the external surface of the reservoir of diameter Dz, the surface of an impeller with disk diameter dt or of its shaft of diameter dw. Due to the relations in Eq. (9) through (12), it was

It was assumed that microorganisms were destroyed as a consequence of disruption of cell walls caused mainly by compressive load which acted along the shorter cell axis c1i. This assumption was made after analysis of the results of many experiments which included disruption of single yeast cells using micro-tools controlled by micromanipulators. Results of these experiments have not been published yet. Non-homogeneity of the structure, thickness and rigidity of cell walls causes that microorganisms compressed along the long axis are deformed asymmetrically and due to accumulated energy catapult from the hazardous zone. This property formed by the evolution guarantees a better survival rate of microorganisms in their environment. In the case of actions characterized by much higher dynamics than the typical one occurring in nature, the probability of cell destruction along the long axis can be significant. In such a case the above assumption simplifies the problem. It is assumed that cell walls are disrupted when compounds inside the cell can flow out of it. This state is equivalent to cell disintegration – transformation of the object. Three stages are

At the first stage the cell of axis c1i contacts two destroying surfaces. The cell is not deformed. Points of contact of solid bodies with the i-th cell are called the points of destruction. They form a single destruction system. The segment which connects destruction points of cell c1i is defined as the destruction axis. At the second stage, when destroying

assumed that all inner surfaces of the mill chamber and stirrer were planes.

distinguished in the destruction of microorganisms (Solecki, 2009):

<sup>123</sup> *iii ccc* = ≤ (8)

<sup>3</sup>*i k c d* << (9)

*k z d D* << (10)

*k w d d* << (11)

*k t d d* << (12)

which satisfy relation (8).

fluids protected by them was identical.

according to Eq. (9).

1. contact,

2. cell deformation, 3. disruption of cell walls. surfaces approach each other, the cell is deformed. The contact area of the cell with solid bodies described by diameter dδi increases. Walls of the cell with axis c1i are disrupted when compression surfaces approach each other at the third stage to destruction distance lδi min described by the relation given in Eq. (13).

$$d\_{\delta i \min} < c\_{1i} \tag{13}$$

From the point of view of random transformation, the first two destruction stages have no significance, providing naturally that the number of load cycles has no effect on the cell strength. Hence, it can be assumed that starting with instant τ=0, all cells present in the suspension are so deformed that an arbitrarily small increase of deformation in the transformation volume causes their disruption. The distance of destruction is measured along the destruction axis. It was assumed that the cell-destroying surface contact area in the moment of cell wall disruption was represented by a spherical cap of diameter δi max.

Disruption of cells dispersed in the liquid occurs once the cells have been translocated to destruction volume Vγji which is in the place of formation of a single destruction system. The continuity of grinding surfaces was assumed for the set of points belonging to circular environment of diameter δi max around the point of destruction. At least one grinding surface has a spherical shape. Thus, in the mill two destruction elements can generate a single destruction axis or a set of axes which constitute a cylindrical surface. At the present stage of studies we make a simplified assumption that elements destroying microbial cells are perfectly rigid.

We consider the case of batch mill operation, when during the process the supply of microbial suspension is constant. Its volume is equal to V. The process is carried out for the known initial biomass concentration. The initial number of microorganisms is N0. During the process no microbial cells are added from the outside (it is assumed that the disintegration process is carried out in sterile conditions) and no cells are removed to the outside. The time of disintegration is very short as compared to the life time of microorganisms and the time of formation of new cells in the process conditions. Hence, it can be assumed that function f13 which describes cell growth and function f14 describing elimination of cells different than that being a result of the transformation, satisfy equations (14) and (15).

$$f\_{13}\left(N\_{0'}\tau\right) = 0\tag{14}$$

$$f\_{14} \left( N\_{0'} \tau \right) = 0 \tag{15}$$

In the case of disintegration of microorganisms in the bead mills the transformation consists in the disruption of microbial cell walls. In the process carried out in the bead mill, high levels of packing of the mill chamber and big rotational speed of the stirrer are applied. Because of a high level of packing the mill with beads and high intensity of their circulation resulting from high rotational speed of the stirrer, it was assumed that during the process: 1. the sum of all volumes Vγi is constant (Eq. (16)),

$$\bigcup\_{i=1}^{n} V\_{\gamma ji} = const \tag{16}$$

The Release of Compounds from Microbial Cells 601

process is carried out and according to a technical device selected, it is necessary to know the morphology of packing in the case of vibrating mixers and bead mill, the distribution of shearing forces and velocity of liquid flow through the valve in the case of high-pressure homogenizers, and finally the size of cavitation bubbles in the case of ultrasound

Fig. 1. The set of possibilities to generate a single family of transformation volumes and

In the case of cell disintegration in the bead mill packed with beads of the same diameter, ps families of destruction volumes are formed by the spherical surface-spherical surface transformation system and pp families by the spherical surface-flat surface transformation system. Numbers ps and pp are the natural numbers and their sum is equal to the number of transformation families p formed in space V. The phenomenological model consists of ps layers of the first type shown in Fig. 1 and pp layers of the second type specific of the spherical surface-flat surface system. The range of space divisions in subsequent layers encompasses only the divisions which were formed for one specified family of transformation volumes. If, for instance, two contacting spherical surfaces generate a transformation family, then the layer encompasses space divisions for all vertical segments from point H' to point H". On the other hand, if the distance between two spherical surfaces is such that the biggest axially compressed cell which is most susceptible to disintegration is disrupted, then in the component layer there will be a space division described (dotted lines) by the division of vertical segment passing through point G". In a general case, for the i-th cell from set *N* the component layer will contain divisions marked on vertical segments from point 3' to point 3". The sum of all volumes contained in p layers is equal to V. By

related volumes for microorganisms from set *N* dispersed in space V

homogenizers.

2. the sum of all volumes Vβi is constant (Eq. (17)).

$$\sum V\_{\beta i} = \text{const} \tag{17}$$

We also assume that the process of microbial disintegration is carried out in steady-state conditions during the whole process duration. The mill start-up is neglected.

#### **3. A phenomenological model**

Figure 1 shows a theoretical set of possibilities to generate the families of transformation volumes between two spherical surfaces. It is prepared on the basis of a description of the general phenomenological model presented by Solecki (2011).

Single microbial cells are reduced to material points which have, among the others, such properties as spatial shape, volume, mass, etc. On each vertical segment between *AC* and *BD* , division of volume V(τ) is marked. This division is a result of forming transformation volume Vγji for the i-th cell. We consider an arbitrary vertical segment 15 . It is divided by dashed lines *RS* , *GH* and *IJ* into four main parts. Volume Vαci which is safe for cell Ni occupies interval 〈1,2) . In this volume intensive stirring takes place. Neither the whole volume Vαci nor any of its subsets can be transformed to other volumes. Interval 〈2,4) occupies volume Vαti. It is safe for cell Ni and intensive stirring is observed in it. Subsets of volume Vαti can be transformed to volumes Vγji and Vβji. Vγji is the transformation volume of object Ni and in Fig. 1 it occupies interval 〈3,4〉 . It includes surface F<sup>γ</sup>αji which separates it from volume Vαti. Volume Vγji is also limited by surface Fγβji belonging to it, in case it is formed. This occurs when Vγji has the size big enough in relation to the object transformed in it. Volume Vβji occupies interval (3,5〉 and is inaccessible for non-transformed object Ni. Surface F<sup>γ</sup>αji which separates it from volume Vα does not belong to volume Vβji. It was assumed that object Ni is the material point that has feature (Vγi)min among the others. This is the smallest transformation volume characteristic of a given object. Components of space V in Fig. 1 are defined by two parameters: the length of relevant vertical segments and color intensity which determines a functional relation. The family of volumes Pj is formed of a group of transformation volumes for various material objects from set *N* defined by segment 3'3". For objects from set *N* of the smallest characteristic transformation volume (Vγs)min, the division into Vαci, Vαti, Vγji and Vβji is marked by continuous lines *R S*' ' , *G H*' ' and *I J*' ' . For objects with the biggest characteristic transformation volume (Vγb)min, the division into volumes with different properties is marked by dotted lines *R S*" ", *G H*" " and *I J*" " .

Like for the cell-disrupting spherical surface-spherical surface system, also for the spherical surface-plane surface system we can determine a map to generate volumes related to the cell transformation process.

To determine the set of possibilities of generating transformation volumes, it is essential to know results of morphological studies and strength tests of microorganisms dispersed in space V. The current development of computer techniques for microscopic image analysis causes that morphological studies are not a problem now. Studies on the strength of microbial cells were carried out by Mashmoushy et al. (1998), Shiu et al. (1999) and Svaldo-Lanero et al. (2007). In the case of the bead mill it is especially important to know destroying transformations in which cells are disrupted (Smith et al., 2000, Stenson et al., 2010). Mechanical stresses in *Saccharomyces cerevisiae* yeast cells caused by high hydrostatic pressure were tested by Hartman et al. (2006). Depending on the method by which the

∑*V const* β

We also assume that the process of microbial disintegration is carried out in steady-state

Figure 1 shows a theoretical set of possibilities to generate the families of transformation volumes between two spherical surfaces. It is prepared on the basis of a description of the

Single microbial cells are reduced to material points which have, among the others, such properties as spatial shape, volume, mass, etc. On each vertical segment between *AC* and *BD* , division of volume V(τ) is marked. This division is a result of forming transformation volume Vγji for the i-th cell. We consider an arbitrary vertical segment 15 . It is divided by dashed lines *RS* , *GH* and *IJ* into four main parts. Volume Vαci which is safe for cell Ni occupies interval 〈1,2) . In this volume intensive stirring takes place. Neither the whole volume Vαci nor any of its subsets can be transformed to other volumes. Interval 〈2,4) occupies volume Vαti. It is safe for cell Ni and intensive stirring is observed in it. Subsets of volume Vαti can be transformed to volumes Vγji and Vβji. Vγji is the transformation volume of object Ni and in Fig. 1 it occupies interval 〈3,4〉 . It includes surface F<sup>γ</sup>αji which separates it from volume Vαti. Volume Vγji is also limited by surface Fγβji belonging to it, in case it is formed. This occurs when Vγji has the size big enough in relation to the object transformed in it. Volume Vβji occupies interval (3,5〉 and is inaccessible for non-transformed object Ni. Surface F<sup>γ</sup>αji which separates it from volume Vα does not belong to volume Vβji. It was assumed that object Ni is the material point that has feature (Vγi)min among the others. This is the smallest transformation volume characteristic of a given object. Components of space V in Fig. 1 are defined by two parameters: the length of relevant vertical segments and color intensity which determines a functional relation. The family of volumes Pj is formed of a group of transformation volumes for various material objects from set *N* defined by segment 3'3". For objects from set *N* of the smallest characteristic transformation volume (Vγs)min, the division into Vαci, Vαti, Vγji and Vβji is marked by continuous lines *R S*' ' , *G H*' ' and *I J*' ' . For objects with the biggest characteristic transformation volume (Vγb)min, the division into

volumes with different properties is marked by dotted lines *R S*" ", *G H*" " and *I J*" " .

Like for the cell-disrupting spherical surface-spherical surface system, also for the spherical surface-plane surface system we can determine a map to generate volumes related to the cell

To determine the set of possibilities of generating transformation volumes, it is essential to know results of morphological studies and strength tests of microorganisms dispersed in space V. The current development of computer techniques for microscopic image analysis causes that morphological studies are not a problem now. Studies on the strength of microbial cells were carried out by Mashmoushy et al. (1998), Shiu et al. (1999) and Svaldo-Lanero et al. (2007). In the case of the bead mill it is especially important to know destroying transformations in which cells are disrupted (Smith et al., 2000, Stenson et al., 2010). Mechanical stresses in *Saccharomyces cerevisiae* yeast cells caused by high hydrostatic pressure were tested by Hartman et al. (2006). Depending on the method by which the

conditions during the whole process duration. The mill start-up is neglected.

general phenomenological model presented by Solecki (2011).

*<sup>i</sup>* = (17)

2. the sum of all volumes Vβi is constant (Eq. (17)).

**3. A phenomenological model** 

transformation process.

process is carried out and according to a technical device selected, it is necessary to know the morphology of packing in the case of vibrating mixers and bead mill, the distribution of shearing forces and velocity of liquid flow through the valve in the case of high-pressure homogenizers, and finally the size of cavitation bubbles in the case of ultrasound homogenizers.

Fig. 1. The set of possibilities to generate a single family of transformation volumes and related volumes for microorganisms from set *N* dispersed in space V

In the case of cell disintegration in the bead mill packed with beads of the same diameter, ps families of destruction volumes are formed by the spherical surface-spherical surface transformation system and pp families by the spherical surface-flat surface transformation system. Numbers ps and pp are the natural numbers and their sum is equal to the number of transformation families p formed in space V. The phenomenological model consists of ps layers of the first type shown in Fig. 1 and pp layers of the second type specific of the spherical surface-flat surface system. The range of space divisions in subsequent layers encompasses only the divisions which were formed for one specified family of transformation volumes. If, for instance, two contacting spherical surfaces generate a transformation family, then the layer encompasses space divisions for all vertical segments from point H' to point H". On the other hand, if the distance between two spherical surfaces is such that the biggest axially compressed cell which is most susceptible to disintegration is disrupted, then in the component layer there will be a space division described (dotted lines) by the division of vertical segment passing through point G". In a general case, for the i-th cell from set *N* the component layer will contain divisions marked on vertical segments from point 3' to point 3". The sum of all volumes contained in p layers is equal to V. By

The Release of Compounds from Microbial Cells 603

*<sup>N</sup>*<sup>0</sup> *<sup>S</sup>*

For process duration τ=0 volumes Vγji, Vβji and Vαi are formed in space V. Unconverted objects can be only in volume Vαi appropriate for them. Between safe volumes for particular

*VVV*

*V V VV*

*V V VV*

 *s b sb* = ⇒= γ

 *s b bs* < ⇒< γ

In the case illustrated by Eq. (20) identical cells of microorganisms belonging to set *N* are

given in Eq. (21) follows the inequality of dispersion of the objects from set *N* in volume

they contain. The process of ideal mixing ensures uniformity of the dispersion of objects from set *N* only within the volume limited by the surface of type Fγβ. In the case presented in Eq. (21), owing to relations given in Eq. (2) and (22) and uniform dispersion of elements of set *N* and volume Vγji in space V, we can use mean concentration of dispersed material objects.

> ( )min *V V* γ

Hence, after starting the process, appropriate objects Ni of set *N* will be introduced at random to the formed volumes Vγji. Mean concentration of the transformed objects in the

> *<sup>N</sup>*<sup>0</sup> *<sup>S</sup> V* α

In instant τ=0 of the process duration the number of unconverted objects N(τ) present in

The conversion rate of microorganisms X(τ) defined by the ratio of the number of disrupted

0 *Nd <sup>X</sup>*

The number of cells Nd(τ) which were disrupted in instant τ=0 is given by Eq. (25).

cells Nd to the initial number of cells N0 according to Eq. (26) is equal to 0.

α∪

*<sup>i</sup>* is equal to the initial number of objects N0, as described by Eq. (24).

α

*<sup>i</sup>* . Every additional surface Fγβi introduced in space V additionally separates

For the relation given in Eq. (19) equivalence can occur between Eq. (20) and Eq. (21).

*bis* ⊂⊂ ⊂⊂ … … α

 α

 αα

 αα

*<sup>i</sup>* into two parts. They differ in the concentration of microorganisms which

cells of microorganisms from set *N* there is a relationship described by Eq. (19).

( ) ( ) min min

( ) ( ) min min

α

γ

γ

uniformly dispersed in the whole volume ∪*V*

volume occupied by them is described by Eq. (23).

∪*V*α

volume ∪*V*

volume ∪*V*

α

α

*<sup>V</sup>* <sup>=</sup> (18)

(19)

(20)

(21)

*<sup>i</sup>* . On the other hand, from the equivalence

*ji* << (22)

= (23)

*N N*= 0 (24)

0 *Nd* = (25)

*N*= (26)

analogy, phenomenological models can be constructed for other methods of microorganism disintegration: physical, chemical and biological.

In the phenomenological model of microorganism disintegration it was assumed that space V consisted of volumes Vα, Vγ and Vβ. Additionally, volume Vα was divided into Vαt and Vαc. Destruction volumes Vγji are formed by circulating beads of the packing. The general and specific schematics of cell disruption between spherical surfaces were discussed in an earlier study (Solecki, 2011). The general scheme of cell disruption between the spherical and flat surface is shown in Fig. 2a. It covers all possible cases contained between segments *AC* and *BD* in Fig. 1, referring to the division of the volume of microbial suspensions into Vαji, Vγji and Vβji. After limiting deformation of the i-th cell, its walls are disrupted (Fig. 2a). The formed transformation volume Vγji is limited by surface F<sup>γ</sup>αji (orange dashed line), surface Fγβji (red dashed line), the spherical surface of packing element and the flat surface of the mill chamber. Volume Vγji with the axis of symmetry OU includes limiting surfaces F<sup>γ</sup>αji and Fγβji. Surface Fγβji limits volume Vαji, but it does not belong to this volume. The volume inaccessible for the i-th living cell Vβji is limited by the spherical plane and surface F<sup>γ</sup>αji. The limiting surfaces do not belong to volume Vβji. The axis of symmetry of volume Vβji is straight line OU.

Fig. 2. The model of cell disruption during hitting with spherical elements: a) a general case – a non-axial impact, b) a particular case – an axial impact

In the specific case shown in Fig. 2b, the line of division of space V is in the AC position (Fig. 1) and volume (Vγji)min is formed. It is limited by the spherical surface, flat surface and surface F<sup>γ</sup>αji (orange dashed line). The axis of symmetry of volume (Vγji)min passes through points O and U.

It follows from the presented phenomenological model that it can encompass not only the working chamber of the mill but even the entire system of technical devices used to disintegrate microorganisms during continuous mill operation. Then volume Vαc is composed of suspension volume in the whole system connected to the working chamber including inlet and outlet reservoirs.

#### **4. A mathematical model**

The concentration of microorganisms introduced into space V is determined by the initial number of non-transformed cells N0 in space V according to Eq. (18).

analogy, phenomenological models can be constructed for other methods of microorganism

In the phenomenological model of microorganism disintegration it was assumed that space V consisted of volumes Vα, Vγ and Vβ. Additionally, volume Vα was divided into Vαt and Vαc. Destruction volumes Vγji are formed by circulating beads of the packing. The general and specific schematics of cell disruption between spherical surfaces were discussed in an earlier study (Solecki, 2011). The general scheme of cell disruption between the spherical and flat surface is shown in Fig. 2a. It covers all possible cases contained between segments *AC* and *BD* in Fig. 1, referring to the division of the volume of microbial suspensions into Vαji, Vγji and Vβji. After limiting deformation of the i-th cell, its walls are disrupted (Fig. 2a). The formed transformation volume Vγji is limited by surface F<sup>γ</sup>αji (orange dashed line), surface Fγβji (red dashed line), the spherical surface of packing element and the flat surface of the mill chamber. Volume Vγji with the axis of symmetry OU includes limiting surfaces F<sup>γ</sup>αji and Fγβji. Surface Fγβji limits volume Vαji, but it does not belong to this volume. The volume inaccessible for the i-th living cell Vβji is limited by the spherical plane and surface F<sup>γ</sup>αji. The limiting surfaces do not belong to volume Vβji. The axis of symmetry of volume Vβji is

Fig. 2. The model of cell disruption during hitting with spherical elements: a) a general case

In the specific case shown in Fig. 2b, the line of division of space V is in the AC position (Fig. 1) and volume (Vγji)min is formed. It is limited by the spherical surface, flat surface and surface F<sup>γ</sup>αji (orange dashed line). The axis of symmetry of volume (Vγji)min passes through

It follows from the presented phenomenological model that it can encompass not only the working chamber of the mill but even the entire system of technical devices used to disintegrate microorganisms during continuous mill operation. Then volume Vαc is composed of suspension volume in the whole system connected to the working chamber

The concentration of microorganisms introduced into space V is determined by the initial

number of non-transformed cells N0 in space V according to Eq. (18).

– a non-axial impact, b) a particular case – an axial impact

disintegration: physical, chemical and biological.

straight line OU.

points O and U.

including inlet and outlet reservoirs.

**4. A mathematical model** 

$$S = \frac{N\_0}{V} \tag{18}$$

For process duration τ=0 volumes Vγji, Vβji and Vαi are formed in space V. Unconverted objects can be only in volume Vαi appropriate for them. Between safe volumes for particular cells of microorganisms from set *N* there is a relationship described by Eq. (19).

$$V\_{ab} \subset \dots \subset V\_{ai} \subset \dots \subset V\_{as} \tag{19}$$

For the relation given in Eq. (19) equivalence can occur between Eq. (20) and Eq. (21).

$$\left(V\_{\gamma s}\right)\_{\text{min}} = \left(V\_{\gamma b}\right)\_{\text{min}} \implies V\_{\alpha s} = V\_{\alpha b} \tag{20}$$

$$\left(\left(V\_{\gamma s}\right)\_{\text{min}} < \left(V\_{\gamma b}\right)\_{\text{min}} \implies V\_{ab} < V\_{\alpha s} \tag{21}$$

In the case illustrated by Eq. (20) identical cells of microorganisms belonging to set *N* are uniformly dispersed in the whole volume ∪*V*α*<sup>i</sup>* . On the other hand, from the equivalence given in Eq. (21) follows the inequality of dispersion of the objects from set *N* in volume ∪*V*α*<sup>i</sup>* . Every additional surface Fγβi introduced in space V additionally separates volume ∪*V*α*<sup>i</sup>* into two parts. They differ in the concentration of microorganisms which they contain. The process of ideal mixing ensures uniformity of the dispersion of objects from set *N* only within the volume limited by the surface of type Fγβ. In the case presented in Eq. (21), owing to relations given in Eq. (2) and (22) and uniform dispersion of elements of set *N* and volume Vγji in space V, we can use mean concentration of dispersed material objects.

$$\left(V\_{\gamma^{\bar{\mu}}}\right)\_{\text{min}} << V \tag{22}$$

Hence, after starting the process, appropriate objects Ni of set *N* will be introduced at random to the formed volumes Vγji. Mean concentration of the transformed objects in the volume occupied by them is described by Eq. (23).

$$S\_{\alpha} = \frac{N\_0}{V\_{\alpha \cup}} \tag{23}$$

In instant τ=0 of the process duration the number of unconverted objects N(τ) present in volume ∪*V*α*<sup>i</sup>* is equal to the initial number of objects N0, as described by Eq. (24).

$$N = N\_0 \tag{24}$$

The number of cells Nd(τ) which were disrupted in instant τ=0 is given by Eq. (25).

$$N\_d = 0\tag{25}$$

The conversion rate of microorganisms X(τ) defined by the ratio of the number of disrupted cells Nd to the initial number of cells N0 according to Eq. (26) is equal to 0.

$$X = \frac{N\_d}{N\_0} \tag{26}$$

The Release of Compounds from Microbial Cells 605

Microorganisms were disintegrated in a horizontal bead mill with a multi-disk impeller. The working chamber about 1 dm3 in volume had the diameter of 80 mm. The impeller was equipped with six round disks 66 mm in diameter. They were mounted centrally on the shaft at a distance of 30 mm from each other. The mill was filled in 80% with ballotini of the diameter ranging from 0.8 to 1.0 mm. They were made of lead-free glass of specific density around 2500 kg/m3. 50% water solution of ethylene glycol at the temperature 275 K was supplied to the cooling jacket of the mill. Experiments were carried out in the mill at periodic operating conditions (constant feed). Rotational speed of the impeller was 261.8 rad/s. The experiments were performed for commercial baker's yeast *S. cerevisiae* produced by Lesaffre Bio-corporation (Wołczyn, Poland). The concentration of yeast suspension ranged from about 0.002 to over 0.17 g d.m./cm3. Microorganisms were dispersed in the water

The kinetics of cell disruption was determined on the basis of the count of living microorganisms present in the suspension samples. A computer-aided analysis of microscopic images (method I) was used. Cells were counted under the Olympus BX51 microscope (Olympus Optical Co.). Photographs were taken using a CCD digital camera of resolution 2576×1932×24 bit (Color View III, Soft Imaging System). Preparations were stained with methylene blue. Thom neu chamber (Paul Marienfeld & Co.) was used to count cells. Photographs were analyzed by means of a specialist software (analySIS 5, Soft Imaging

The amount of protein *R* dissolved in the continuous phase was determined by Bradford's method (1976) (method II). The supernatant was obtained after 20 min centrifugation at centrifugal force 34000 g. Measurements were made in a spectrophotometer at the wavelength 595 nm (Lambda 11, Perkin Elmer). A standard protein concentration curve prepared

The degree of release of intracellular compounds was analyzed also on the basis of light absorbance A in the supernatant (method III). The measurements were made using a Lambda 11 spectrophotometer (Perkin Elmer) at the wavelength λ = 260 nm (Middelberg et al., 1991; Heim & Solecki, 1998, 1999). Near the applied wavelength, spectral characteristics of RNA and DNA nucleic acids reach maximum values. The supernatant was obtained after centrifugation of the suspension in a 3K30 B centrifuge (Braun Biotech International) for 20 min at centrifugal acceleration of 34 000 g. The inside of the centrifuge was cooled down to

In rheological investigations a RC 20 rotational rheometer (RheoTec) operating in a two-slot cylindrical tank – bell-shaped stirrer system was used. Measurements for the suspension of yeast cells and supernatant were made at the temperature 4°C. The degree of disintegration of microbial cells was changed from 0 to nearly 100%. Supernatant was obtained after 20

In the case monogeneity of yeast cells the set of variable properties *Vr* of objects *N* is the empty set (Solecki, 2011). At such an assumption the loss of microorganisms dNd in time interval dτ will be expressed by the right-hand side of Eq. (34) with an opposite sign. Kinetics of the disruption of microorganisms in holistic approach is described in Eq. (35).

**5. Materials and methods** 

System).

4°C.

**6. Results and discussion 6.1 Disruption of microbial cells** 

solution containing 0.15 M NaCl and 4 mM K2HPO4.

for bovine albumin (Albumin A 9647, Sigma) was applied.

minute centrifugation of the suspension at centrifugal force 40 000 *g*.

At any time of the process duration τ the number of transformed microorganisms is Nd. The transformed cells can be in any place in space V. The number of living microorganisms present only in volume ∪*V*α*<sup>i</sup>* is determined by Eq. (27).

$$N = N\_0 - N\_d \tag{27}$$

According to Eq. (28), after time τ of the process, microbial cells at mean concentration S<sup>α</sup> determined by number N of unconverted cells in volume ∪*V*α*<sup>i</sup>* are introduced to all volumes Vγji.

$$S\_{\alpha} = \frac{N}{\bigcup\_{i=1}^{n} V\_{\alpha i}} \tag{28}$$

Naturally, the overall concentration of microorganisms in space V will be determined by Eq. (29). The concentration will be recorded, for instance, in the suspension samples taken from the mill and in the inlet or outlet reservoir.

$$S = \frac{N}{V} \tag{29}$$

The rate of conversion of microorganisms after time τ of the process is given by Eq. (26). The increase of the number of transformed objects dNd in all volumes Vγji after arbitrarily short time interval dτ is specified by Eq. (30).

$$d\mathcal{N}d = \mathcal{S}\_a dV\tag{30}$$

Volume dV displaced from Vαji to Vγji in time increment dτ depends on the size of limiting surface F through which dV is displaced and on the displacement rate u. This is described by Eq. (31).

$$dV = \mathfrak{u}F d\mathfrak{\tau} \tag{31}$$

Upon substitution of Eq. (28) and (31) to Eq. (30) we obtain Eq. (32) which describes the increase of objects transformed in volumes Vγji.

$$d\mathcal{N}\_d = k\mathcal{N}d\tau \tag{32}$$

The process rate constant k of the transformation of microbial cells is described by Eq. (33).

$$k = \frac{F}{\bigcup\_{i=1}^{n} V\_{ai}} \mu \tag{33}$$

Surface F is the sum of these parts of surface F<sup>γ</sup>αji through which microbial cells pass to volume Vγji.

Based on Eq. (33) the loss of non-transformed objects can be represented by Eq. (34).

$$dN = -k(N\_0 - N\_d)d\tau\tag{34}$$
