**6.1 Disruption of microbial cells**

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).

The Release of Compounds from Microbial Cells 607

Fig. 3. Changes of rate constant determined for a linear model by method I (direct method based on computer-aided counting of cells under a microscope) and method II (indirect

method based on measurement of the quantity of released protein)

Fig. 4. The effect of suspension concentration on the cell disruption process

$$d\mathcal{N}\_d = k(\mathcal{N}\_\mathcal{o} - \mathcal{N}d)d\tau \tag{35}$$

Process rate constant k is described by Eq. (36).

$$k = \frac{F}{V\_a \cup} \mu \tag{36}$$

The sum of safe volumes *V*α<sup>∪</sup> for subsequent i-th material objects is given by Eq. (37).

$$V\_{\alpha \cup \downarrow} = \bigcup\_{i=1}^{n} V\_{\alpha i} \tag{37}$$

The regression line in Eq. (38) was obtained after integration of Eq. (35) by sides. The integration was preceded by the separation of variables. The regression line is often used to describe the disintegration kinetics of microbial cells in bead mills (Currie et al., 1974; Merffy & Kula, 1979; Limmon-Lason et al., 1976 Melendres et al., 1991; Garrido et al., 1994; Heim & Solecki, 1998; Solecki, 2007).

$$\ln\left(\frac{N\_0}{N\_0 - N\_d}\right) = k\tau\tag{38}$$

A correlation between variables in Eq. (38) was investigated. For linear regression describing the kinetics of yeast cell disruption the values of coefficient R were higher than 0.9835. Although advantageous values of the coefficient were obtained, the analysis of rate constant showed significant changes of its value which accompanied changes in the suspension concentration (Fig. 4). They are described by Eq. (39) for the suspension concentration S>0.

$$k = a\_1 S^2 + a\_2 \tag{39}$$

When estimating the coefficients in Eq. (39), data for the rate constants much deviating from a model line for very low and low concentrations of the suspension were neglected. Table 1 gives values of the determined coefficients and significant results of statistical analysis.

Changes in the rate constant of cell disruption accompanying a change of the suspension concentration are induced by deviation of the process kinetics from linearity. Their character depends on biomass concentration. In general, three concentration ranges of yeast suspension with similar cell disruption process can be distinguished. These concentrations are very low, up to about 0.01 g d.m./cm3, low – from about 0.01 to 0.10 g d.m./cm3 and medium and high – exceeding 0.10 g d.m./cm3. In Fig. 5, for selected biomass concentrations, the experimentally determined changes in the degree of yeast cell disruption are compared to the changes determined by the linear model (Eq. 38). At low concentrations of the suspension (S=0.08 g d.m./cm3) the process is linear. At the initial stage of the process, at very low suspension concentrations (S=0.002 g d.m./cm3), determined values of the rate constant are usually similar to the values obtained for low suspension concentrations. However, when the initial number of cells is small, the disruption rate decreases subsequently during the process. On the other hand, for medium and high concentrations (S=0.14 g d.m./cm3) the process of microbial cell disruption at the initial stage is much slower than the model process. At the next stage of the process the cell disruption rate increases so that the process is much faster than in the case of low concentrations.

τ

<sup>∪</sup> for subsequent i-th material objects is given by Eq. (37).

(35)

(36)

(38)

<sup>∪</sup> <sup>=</sup> ∪ (37)

1 2 *k aS a* = + (39)

( ) *o d <sup>d</sup> dN k N N d* = −

*<sup>F</sup> k u V*α

∪

1

*i V V* α

The regression line in Eq. (38) was obtained after integration of Eq. (35) by sides. The integration was preceded by the separation of variables. The regression line is often used to describe the disintegration kinetics of microbial cells in bead mills (Currie et al., 1974; Merffy & Kula, 1979; Limmon-Lason et al., 1976 Melendres et al., 1991; Garrido et al., 1994; Heim &

> 0 0

A correlation between variables in Eq. (38) was investigated. For linear regression describing the kinetics of yeast cell disruption the values of coefficient R were higher than 0.9835. Although advantageous values of the coefficient were obtained, the analysis of rate constant showed significant changes of its value which accompanied changes in the suspension concentration (Fig. 4). They are described by Eq. (39) for the suspension concentration S>0.

2

When estimating the coefficients in Eq. (39), data for the rate constants much deviating from a model line for very low and low concentrations of the suspension were neglected. Table 1 gives values of the determined coefficients and significant results of statistical analysis. Changes in the rate constant of cell disruption accompanying a change of the suspension concentration are induced by deviation of the process kinetics from linearity. Their character depends on biomass concentration. In general, three concentration ranges of yeast suspension with similar cell disruption process can be distinguished. These concentrations are very low, up to about 0.01 g d.m./cm3, low – from about 0.01 to 0.10 g d.m./cm3 and medium and high – exceeding 0.10 g d.m./cm3. In Fig. 5, for selected biomass concentrations, the experimentally determined changes in the degree of yeast cell disruption are compared to the changes determined by the linear model (Eq. 38). At low concentrations of the suspension (S=0.08 g d.m./cm3) the process is linear. At the initial stage of the process, at very low suspension concentrations (S=0.002 g d.m./cm3), determined values of the rate constant are usually similar to the values obtained for low suspension concentrations. However, when the initial number of cells is small, the disruption rate decreases subsequently during the process. On the other hand, for medium and high concentrations (S=0.14 g d.m./cm3) the process of microbial cell disruption at the initial stage is much slower than the model process. At the next stage of the process the cell disruption rate increases so that the process is much faster than in the case of low

⎛ ⎞ ⎜ ⎟ <sup>=</sup> <sup>−</sup> ⎝ ⎠

*N N*

*d <sup>N</sup> <sup>k</sup>*

τ

ln

=

*i*

α

*n*

=

Process rate constant k is described by Eq. (36).

α

The sum of safe volumes *V*

Solecki, 1998; Solecki, 2007).

concentrations.

Fig. 3. Changes of rate constant determined for a linear model by method I (direct method based on computer-aided counting of cells under a microscope) and method II (indirect method based on measurement of the quantity of released protein)

Fig. 4. The effect of suspension concentration on the cell disruption process

The Release of Compounds from Microbial Cells 609

diameter di are illustrated in Fig. 6. Values of the rate constant determined for the biggest cell fraction are over 10 times higher than those determined for the smallest cell fraction. Based on the size analysis of undisrupted cells it was found that at a very small initial number of microorganisms during the process three biggest yeast cell fractions disappear successively: 9.5-10 μm, 9-9.5 μm and 8.5-9 μm. The process analyzed for size fractions determined by the short axis of the ellipsoid did not reveal any effect of the cell size on microorganism disruption kinetics. This result is a consequence of a significant differentiation of cell sizes determined by the short axis of the ellipsoid. Cells in the given fraction can have the shape almost spherical to ellipsoidal with the 1 : 1.4 axis ratio (Table 2). The same percent deformation along axis c1i can be destructive for spherical cells but is still

Solecki (2011) considered the case of random transformation of objects from set *N* which differed by only one feature vr1 having an important effect on the process. Eq. (40) describes

( ) 0 0

 ζ

*d d*

( ) 0 0

 ζ

*d d*

 ζζ

*dN kN N d*

 ζζ

Eq. (41) describes the transformation of objects of the entire set taking into consideration the

*dN k N N d*

= −

0 0

1 1 0 0

*N N*

 ζ

ζ

*N N*

*m m*

= =

ζ

Fig. 6. The effect of cell size on the disintegration rate constant

ζ

ζ

*N N*

ζ

*N N*

0

ζ

 ζ

0

τ

 ζ

∑ ∑= <sup>−</sup> (41)

τ

∈ 1,*m* taking into account the

(40)

safe for much elongated cells.

existence of the whole set *N*.

division of set *N* into subsets.

the transformation of objects from an arbitrary interval


Table 1. Results of the estimation of parameters in Eq.(39) for the analytical methods applied

Studies on yeast cell morphology revealed a significant differentiation of cell sizes. Selected parameters are given in Table 2. Size distributions of axes c1i and c3i are illustrated in Fig. 5. The process of microbial cell disruption was analyzed taking into account different cell sizes for two cases: in the first one the cells were determined by the short axis of the ellipsoid c1i, while in the second one by mean diameter di.


Table 2. Chosen results of yeast morphology investigations

Fig. 5. Distribution of size of small and greater axis ellipsoid defining yeast cells

To model the disintegration process in relation to the size of yeast cells, for both cases mentioned above the set of cells was divided into m=12 size fractions. The effect of size fractions on the cell disruption kinetics was investigated using Eq. (35) to describe the disintegration of yeast cells from the given size fraction. This description allowed us to compare the process run in particular fractions with the process taken as whole. Results of the comparison of cell disruption kinetics for particular size fractions determined by mean

Table 1. Results of the estimation of parameters in Eq.(39) for the analytical methods applied Studies on yeast cell morphology revealed a significant differentiation of cell sizes. Selected parameters are given in Table 2. Size distributions of axes c1i and c3i are illustrated in Fig. 5. The process of microbial cell disruption was analyzed taking into account different cell sizes for two cases: in the first one the cells were determined by the short axis of the ellipsoid c1i,

ratio

Shape factor

Sphericity


Method a1 a2 n R

while in the second one by mean diameter di.

Parameter c1i c3i di Elongation Aspect

Table 2. Chosen results of yeast morphology investigations

I 0.070 0.00897 5 0,9948 II 0.045 0.0086 6 0,8812 III 0.125 0.0057 5 0,9786

unit μm μm μm – – – – min 3.314 4.020 3.643 1.004 1.010 0.846 0.503 max 8.007 9.433 8.686 1.432 1.439 1.126 0.992

Fig. 5. Distribution of size of small and greater axis ellipsoid defining yeast cells

To model the disintegration process in relation to the size of yeast cells, for both cases mentioned above the set of cells was divided into m=12 size fractions. The effect of size fractions on the cell disruption kinetics was investigated using Eq. (35) to describe the disintegration of yeast cells from the given size fraction. This description allowed us to compare the process run in particular fractions with the process taken as whole. Results of the comparison of cell disruption kinetics for particular size fractions determined by mean diameter di are illustrated in Fig. 6. Values of the rate constant determined for the biggest cell fraction are over 10 times higher than those determined for the smallest cell fraction. Based on the size analysis of undisrupted cells it was found that at a very small initial number of microorganisms during the process three biggest yeast cell fractions disappear successively: 9.5-10 μm, 9-9.5 μm and 8.5-9 μm. The process analyzed for size fractions determined by the short axis of the ellipsoid did not reveal any effect of the cell size on microorganism disruption kinetics. This result is a consequence of a significant differentiation of cell sizes determined by the short axis of the ellipsoid. Cells in the given fraction can have the shape almost spherical to ellipsoidal with the 1 : 1.4 axis ratio (Table 2). The same percent deformation along axis c1i can be destructive for spherical cells but is still safe for much elongated cells.

Solecki (2011) considered the case of random transformation of objects from set *N* which differed by only one feature vr1 having an important effect on the process. Eq. (40) describes the transformation of objects from an arbitrary interval ζ ∈ 1,*m* taking into account the existence of the whole set *N*.

$$\frac{N\_{0\zeta}}{N\_0}dN\_{d\zeta} = \frac{N\_{0\zeta}}{N\_0}k\_\zeta \left(N\_{0\zeta} - N\_{d\zeta}\right)d\tau\tag{40}$$

Eq. (41) describes the transformation of objects of the entire set taking into consideration the division of set *N* into subsets.

$$\sum\_{\zeta=1}^{m} \frac{N\_{0\zeta}}{N\_0} dN\_{d\zeta} = \sum\_{\zeta=1}^{m} \frac{N\_{0\zeta}}{N\_0} k\_{\zeta} \left( N\_{0\zeta} - N\_{d\zeta} \right) d\tau \tag{41}$$

Fig. 6. The effect of cell size on the disintegration rate constant

The Release of Compounds from Microbial Cells 611

Analysis of results of the process showed that at low concentrations of the suspension (0.05 g d.m./cm3) the process was very well described by linear relation in Eq. (38) (R=0.9938). The presence of the biggest yeast cells until reaching over 90% disintegration was confirmed using the direct method. In the case of yeast cell concentration equal to 0.05 g d.m./cm3, the initial number of cells is over 20 times bigger than in the case of cell concentration of 0.002 g d.m./cm3. Hence, it is more probable that bigger cells will survive longer in the process at

The disintegration of microorganisms in the bead mill is a random transformation of dispersed matter. If the experiments were carried out for a single cell, it might have appeared that the degree of disintegration obtained in the first experiment would be 100%, in the second one also 100%, and in the third one e.g. 0%. From the investigations carried out for the suspension concentration equal to 0.05 g d.m./cm3 it follows that if there are 2,355E9 cells dispersed at one time, after the process 0,2598E9 cells will remain alive on average. Hence, for an experiment carried out with a single cell, in 11 cases out of 100 experiments the result of disintegration at time τ=270 s will be 0% on average. It is probable, however, that in the process all cells will disintegrate. This probability decreases with an increase of the initial number of cells. Nonlinearity of the process caused by subsequent decay of objects from set *N* which are most susceptible to transformations can appear at every concentration of the suspension. This probability depends on the initial concentration of microorganisms in the suspension. For very low concentrations it is very big, while for high concentrations it is small. At a very small number of cells, a nonlinear process will be most probable, although it is not likely that the process will be linear. For a small and big number of cells, when no relations occur between the cells, a linear process is expected, but it may also happen that the process will be nonlinear. A result of this phenomenon are rate constants determined experimentally which are smaller than the values determined from

Fig. 8. Influence of suspension density on the distance between geometrical centres of adjacent

cells (distribution of yeasts in the nodes of compact hexagonal network was assumed)

higher concentrations of the suspension.

the model and given by Eq. (39).

After separating the variables in Eq. (41) and integration by sides, we get Eq. (42).

$$\sum\_{\zeta=1}^{m} \frac{N\_{0\zeta}}{N\_0} \ln \frac{N\_{0\zeta}}{N\_{0\zeta} - N\_{d\zeta}} = \sum\_{\zeta=1}^{m} \frac{N\_{0\zeta}}{N\_0} k\_{\zeta} \tau \tag{42}$$

Relations between rate constants in Eq. (35) and (40) are determined in Eq. (43).

$$\Phi \sum\_{\zeta=1}^{m} \frac{N\_{0\zeta}}{N\_0} k\_{\zeta} = k \tag{43}$$

Coefficient Φ is described by the relation given in Eq. (44).

$$\Phi = \ln \frac{N\_0}{N\_0 - N\_d} \left( \sum\_{\zeta=1}^{m} \frac{N\_{0\zeta}}{N\_0} \ln \frac{N\_{0\zeta}}{N\_{0\zeta} - N\_{d\zeta}} \right)^{-1} \tag{44}$$

The process of disintegration in the mill is described by Eq. (42) as a total effect of disintegration of particular cell size fractions. The disintegration kinetics of microorganisms in particular fractions is linear. Differences in cell sizes do not cause nonlinearity of the process. Deviations from linearity shown for very low concentrations of the suspension are caused by the decay of subsequent cell size fractions during the process. Nonlinearity of the process obtained at a very small initial number of microorganisms in the suspension N0 described by Eq. (42) is illustrated in Fig. 7.

Fig. 7. The course of microbial cell disruption taking into consideration the effect of size fraction on the disintegration rate

00 0 1 1 0 0 0

> 0 1 0 *<sup>m</sup> N*

*N* ζ

*NN N*

*NNN N*

*d*

 ζ

 ζ

*k k*

ζ

0 0 0 0 00 1 ln ln *m*

*N N N NN N N N* ζ

The process of disintegration in the mill is described by Eq. (42) as a total effect of disintegration of particular cell size fractions. The disintegration kinetics of microorganisms in particular fractions is linear. Differences in cell sizes do not cause nonlinearity of the process. Deviations from linearity shown for very low concentrations of the suspension are caused by the decay of subsequent cell size fractions during the process. Nonlinearity of the process obtained at a very small initial number of microorganisms in the suspension N0

ζ =

Fig. 7. The course of microbial cell disruption taking into consideration the effect of size

Φ = ⎜ ⎟ − − ⎝ ⎠

*d d*

⎛ ⎞

*k*

ζ

<sup>=</sup> <sup>−</sup> ∑ ∑ (42)

τ

Φ∑ <sup>=</sup> (43)

1

∑ (44)

−

 ζ

> ζ

> > ζ

ζ

After separating the variables in Eq. (41) and integration by sides, we get Eq. (42).

ζζ

Relations between rate constants in Eq. (35) and (40) are determined in Eq. (43).

ζ=

ζ

= =

ln *m m*

ζ

Coefficient Φ is described by the relation given in Eq. (44).

described by Eq. (42) is illustrated in Fig. 7.

fraction on the disintegration rate

Analysis of results of the process showed that at low concentrations of the suspension (0.05 g d.m./cm3) the process was very well described by linear relation in Eq. (38) (R=0.9938). The presence of the biggest yeast cells until reaching over 90% disintegration was confirmed using the direct method. In the case of yeast cell concentration equal to 0.05 g d.m./cm3, the initial number of cells is over 20 times bigger than in the case of cell concentration of 0.002 g d.m./cm3. Hence, it is more probable that bigger cells will survive longer in the process at higher concentrations of the suspension.

The disintegration of microorganisms in the bead mill is a random transformation of dispersed matter. If the experiments were carried out for a single cell, it might have appeared that the degree of disintegration obtained in the first experiment would be 100%, in the second one also 100%, and in the third one e.g. 0%. From the investigations carried out for the suspension concentration equal to 0.05 g d.m./cm3 it follows that if there are 2,355E9 cells dispersed at one time, after the process 0,2598E9 cells will remain alive on average. Hence, for an experiment carried out with a single cell, in 11 cases out of 100 experiments the result of disintegration at time τ=270 s will be 0% on average. It is probable, however, that in the process all cells will disintegrate. This probability decreases with an increase of the initial number of cells. Nonlinearity of the process caused by subsequent decay of objects from set *N* which are most susceptible to transformations can appear at every concentration of the suspension. This probability depends on the initial concentration of microorganisms in the suspension. For very low concentrations it is very big, while for high concentrations it is small. At a very small number of cells, a nonlinear process will be most probable, although it is not likely that the process will be linear. For a small and big number of cells, when no relations occur between the cells, a linear process is expected, but it may also happen that the process will be nonlinear. A result of this phenomenon are rate constants determined experimentally which are smaller than the values determined from the model and given by Eq. (39).

Fig. 8. Influence of suspension density on the distance between geometrical centres of adjacent cells (distribution of yeasts in the nodes of compact hexagonal network was assumed)

The Release of Compounds from Microbial Cells 613

*<sup>a</sup> <sup>N</sup> <sup>S</sup> k t N b* <sup>⎡</sup> <sup>⎤</sup> ⎛ ⎞ = + <sup>⎢</sup> <sup>⎥</sup> ⎜ ⎟ <sup>⎢</sup> ⎝ ⎠ <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

The process rate for high suspension concentrations increases only at the second stage of the process, after over 30 s. At the initial phase, the process is slower than the disruption of microorganisms at the concentration of 0.05 g d.m./cm3 (Fig. 4). Such a result was obtained at 0.11, 0.14 and 0.17 g d.m./cm3. The decreased process rate is caused by inefficient operation of the packing in the working chamber. If packing of the cells is so dense as that described above, the approaching surfaces cause deformation of the compressed cell perpendicular to the destruction axis. A result is forcing out of adjacent cells from the hazardous zone and consequently, a decreased efficiency of the packing. In this case, the relative rate of cell supply does not only result from cell concentration and the dislocation

In the case when cell disruption is equivalent to the release of intracellular compounds, the

( ) *dR k R R d* = − *II m*

The values of absorbance obtained at a given concentration of individual samples were calculated into the amount of pure nucleic acid RNA released during the process from the cells. Its concentration in the continuous phase of the suspension after time of the process is denoted as C. The calculations were made after assuming mean values for nucleotides defined by Benthin et al. (1991). Taking into account the method of analysis of disintegrated

( ) *dC k C C d* = − *III m*

A maximum concentration of protein Rm and nucleic acids Cm was determined in separate experiments based on the quantities of released intracellular compounds specified for cell disintegration level ranging from about 98% to over 99%. Experiments performer at the disintegration range 0-90% are represented according to the method applied by the

> ln *<sup>m</sup> II m*

ln *<sup>m</sup> III m*

When measuring method II was used, the values of coefficient R for data obtained in subsequent experiments were not smaller than 0.9605, while in the case of method III not smaller than 0.9826. The determined values of the rate constant of protein release are given in Fig. 3. Changes of this rate constant which proceed with concentration growth are in agreement with the dependence of yeast cell disruption rate constant on cell concentration. Rates constants of protein release at the highest concentration are about 1.5 times bigger

⎛ ⎞ ⎜ ⎟ <sup>=</sup> <sup>−</sup> ⎝ ⎠

*<sup>C</sup> <sup>k</sup> C C*

⎛ ⎞ ⎜ ⎟ <sup>=</sup> <sup>−</sup> ⎝ ⎠

*<sup>R</sup> <sup>k</sup> R R*

microorganisms, a mathematical description of the process has the form of Eq. (48).

τ

τ

τ

τ

<sup>0</sup> ln 1

rate of the suspension.

**6.2 The release of intracellular compounds** 

kinetics of protein release R from yeast cells is described by Eq. (47).

regression line given by Eq. (49) or Eq. (50), respectively.

5

(46)

(47)

(48)

(49)

(50)

An increase of the initial number of yeast cells in the suspension causes an increase of rate constant k (Fig. 3) according to Eq. (39). Analysis of the phenomenological model of the process shows that this can be due to an increase of volume *Vγi* and consequently the growth of volume *Vβi*. When microorganisms are concentrated, the interrelations between cells are intensified. That was confirmed by the investigations of the effect of biomass concentration and degree of disintegration on distances between geometric centers of adjacent yeast cells (Heim & Solecki 1999; Heim et al., 2007). It was assumed that they are placed in the nodes of the compact hexagonal network. Eq. (45) describes relationships of the distances between centers of the adjacent microorganisms b with biomass concentration S and disintegration degree X (Heim et al., 2007).

$$b = \frac{a\_4}{\left[S(1-X)\right]^{\frac{1}{3}}} + error \tag{45}$$

The analysis of yeast cell size distribution and distances between geometric centers of the cells indicated very significant intensification of relationships between the microorganisms, which follows an increase of the concentration of biomass. For instance, at the concentration 0.11 g d.m./cm3 about 30% of the cells show higher value of parameter c1i than distance b equal to about 6 μm. The increased intensity of cell interactions is confirmed also by the results of studies on rheological properties of the microbial suspension. It was proved that an increase of the initial concentration of microorganisms caused a decrease of apparent viscosity of the suspension. At the concentration 0.002 g d.m./cm3 its value is close to that obtained for water, at 0.08 g d.m./cm3 it is about 5 mPas, and at the concentration 0.17 g d.m./cm3 it reaches 20 mPas.

During the disintegration of yeast cells significantly changed the rheological properties of the suspension. Interesting is a rapid decrease of the apparent viscosity at the initial stage of the process at τ=0-30 s, related to the destruction of a large number of spatial structures of yeast cells. The cell disruption rate increases at a later stage of the process. According to the results shown in Fig. 5, the rate of disintegration increases in the process time ranging from 30 to around 120 s.

The presented analysis of relationships refers to the suspension of yeast cells placed in an ordinary reservoir. The concentration of microorganisms in space V is generally described by Eq. (29). After supplying the suspension into the mill working chamber, the relationships between cells are further intensified. Particular cells Ni fill up only volumes Vαi, while the continuous phase fills also volumes Vγji and Vβji. When the suspension is fed into the mill, the cells are filtered in volumes Vαi. With the moment of starting the process, the suspension of mean concentration Sα is introduced to volume Vγji according to Eq. (28).

The above analyses lead to a conclusion that an increase of volume *Vγi* which causes an increase of the cell disruption rate cannot be induced by yeast blocking by adjacent cells. Such an effect could be brought about by released intracellular compounds or some fragments of already disrupted cells. Nonlinearity of the process, which is revealed with an increase of suspension concentration, is described by Eq. (46) (Heim et al., 2007). It takes into account the concentration of the suspension of microorganisms and the change of distance between centers of adjacent cells which occurs during the process. A change of parameter b for a given concentration of the suspension is the measure of the number of disrupted cells and released intracellular compounds.

An increase of the initial number of yeast cells in the suspension causes an increase of rate constant k (Fig. 3) according to Eq. (39). Analysis of the phenomenological model of the process shows that this can be due to an increase of volume *Vγi* and consequently the growth of volume *Vβi*. When microorganisms are concentrated, the interrelations between cells are intensified. That was confirmed by the investigations of the effect of biomass concentration and degree of disintegration on distances between geometric centers of adjacent yeast cells (Heim & Solecki 1999; Heim et al., 2007). It was assumed that they are placed in the nodes of the compact hexagonal network. Eq. (45) describes relationships of the distances between centers of the adjacent microorganisms b with biomass concentration

> ( ) 4

The analysis of yeast cell size distribution and distances between geometric centers of the cells indicated very significant intensification of relationships between the microorganisms, which follows an increase of the concentration of biomass. For instance, at the concentration 0.11 g d.m./cm3 about 30% of the cells show higher value of parameter c1i than distance b equal to about 6 μm. The increased intensity of cell interactions is confirmed also by the results of studies on rheological properties of the microbial suspension. It was proved that an increase of the initial concentration of microorganisms caused a decrease of apparent viscosity of the suspension. At the concentration 0.002 g d.m./cm3 its value is close to that obtained for water, at 0.08 g d.m./cm3 it is about 5 mPas, and at the concentration 0.17 g

During the disintegration of yeast cells significantly changed the rheological properties of the suspension. Interesting is a rapid decrease of the apparent viscosity at the initial stage of the process at τ=0-30 s, related to the destruction of a large number of spatial structures of yeast cells. The cell disruption rate increases at a later stage of the process. According to the results shown in Fig. 5, the rate of disintegration increases in the process time ranging from

The presented analysis of relationships refers to the suspension of yeast cells placed in an ordinary reservoir. The concentration of microorganisms in space V is generally described by Eq. (29). After supplying the suspension into the mill working chamber, the relationships between cells are further intensified. Particular cells Ni fill up only volumes Vαi, while the continuous phase fills also volumes Vγji and Vβji. When the suspension is fed into the mill, the cells are filtered in volumes Vαi. With the moment of starting the process, the suspension

The above analyses lead to a conclusion that an increase of volume *Vγi* which causes an increase of the cell disruption rate cannot be induced by yeast blocking by adjacent cells. Such an effect could be brought about by released intracellular compounds or some fragments of already disrupted cells. Nonlinearity of the process, which is revealed with an increase of suspension concentration, is described by Eq. (46) (Heim et al., 2007). It takes into account the concentration of the suspension of microorganisms and the change of distance between centers of adjacent cells which occurs during the process. A change of parameter b for a given concentration of the suspension is the measure of the number of disrupted cells

of mean concentration Sα is introduced to volume Vγji according to Eq. (28).

*S X* = + ⎡ −⎤ ⎣ ⎦

1 1 <sup>3</sup> *<sup>a</sup> <sup>b</sup> error*

(45)

S and disintegration degree X (Heim et al., 2007).

d.m./cm3 it reaches 20 mPas.

and released intracellular compounds.

30 to around 120 s.

$$\ln \frac{N\_0}{N} = k \left[ 1 + \left( \frac{S}{b} \right)^{a\_5} \right] t \tag{46}$$

The process rate for high suspension concentrations increases only at the second stage of the process, after over 30 s. At the initial phase, the process is slower than the disruption of microorganisms at the concentration of 0.05 g d.m./cm3 (Fig. 4). Such a result was obtained at 0.11, 0.14 and 0.17 g d.m./cm3. The decreased process rate is caused by inefficient operation of the packing in the working chamber. If packing of the cells is so dense as that described above, the approaching surfaces cause deformation of the compressed cell perpendicular to the destruction axis. A result is forcing out of adjacent cells from the hazardous zone and consequently, a decreased efficiency of the packing. In this case, the relative rate of cell supply does not only result from cell concentration and the dislocation rate of the suspension.

#### **6.2 The release of intracellular compounds**

In the case when cell disruption is equivalent to the release of intracellular compounds, the kinetics of protein release R from yeast cells is described by Eq. (47).

$$dR = k\_{ll}(R\_m - R)d\tau\tag{47}$$

The values of absorbance obtained at a given concentration of individual samples were calculated into the amount of pure nucleic acid RNA released during the process from the cells. Its concentration in the continuous phase of the suspension after time of the process is denoted as C. The calculations were made after assuming mean values for nucleotides defined by Benthin et al. (1991). Taking into account the method of analysis of disintegrated microorganisms, a mathematical description of the process has the form of Eq. (48).

$$d\mathbf{C} = k\_{III}(\mathbf{C}\_m - \mathbf{C})d\tau \tag{48}$$

A maximum concentration of protein Rm and nucleic acids Cm was determined in separate experiments based on the quantities of released intracellular compounds specified for cell disintegration level ranging from about 98% to over 99%. Experiments performer at the disintegration range 0-90% are represented according to the method applied by the regression line given by Eq. (49) or Eq. (50), respectively.

$$\ln\left(\frac{R\_m}{R\_m - R}\right) = k\_H \tau \tag{49}$$

$$\text{lln}\left(\frac{\mathcal{C}\_m}{\mathcal{C}\_m - \mathcal{C}}\right) = k\_{III}\tau\tag{50}$$

When measuring method II was used, the values of coefficient R for data obtained in subsequent experiments were not smaller than 0.9605, while in the case of method III not smaller than 0.9826. The determined values of the rate constant of protein release are given in Fig. 3. Changes of this rate constant which proceed with concentration growth are in agreement with the dependence of yeast cell disruption rate constant on cell concentration. Rates constants of protein release at the highest concentration are about 1.5 times bigger

The Release of Compounds from Microbial Cells 615

Fig. 10. The effect of suspension concentration on the protein release process

Fig. 11. The effect of suspension concentration on the intracellular compounds release

process

than the values obtained for the lowest yeast cell concentration. A similar relation and character of changes were obtained when the release of intracellular compounds was studied on the basis of absorbance measurements. Results of these experiments are illustrated in Fig. 9. At suspension concentration S>0 changes of the process rate constant in methods II and III are described by Eq. (39) and results of estimation are collected in Table 1. Data obtained by Bradford's method for concentrations 0.02 and 0.05 were deleted from statistical analyses; in the case of absorbance measurements data obtained for the lowest concentration of the suspension were omitted.

Fig. 9. Changes of rate constant determined for a linear model – method III

Results shown in Fig. 3 refer to biological material taken from one fermentation process, while these illustrated in Fig. 9 were obtained for yeast from a different fermentation process. In general, the results show that protein release is slower than microbial cell disruption. These observations are in agreement with results obtained by Middelberg et al. (1991) for *Escherichia coli* disintegrated in a high-pressure homogenizer. For such bacteria whose cellular structure is different than yeast cellular structure, the authors showed that the process tested on the basis of determination of the number of living microorganisms and absorbance of light with the wavelength 260 nm. According to Middelberg et al. (1991), the process of protein release measured also by Bradford's method is slower. Dependence of the release rate of intracellular compounds on the place of their location in cells was published by Hetherington et al. (1971). The authors report that during disintegration of *E. coli* cells the periplasmic enzymes are released faster than total protein, all cytoplasmic enzymes are released at the same rate as total protein, whereas membrane-bound enzyme slower.

The release of intracellular compounds during disintegration of microorganisms in the bead mill is shown in Figs 10 and 11. In general, the results are concordant with the results obtained in the experiments with cell disruption process. A decreased rate of protein release at the initial stage of disintegration is observed already at low suspension concentrations; it is higher than the decrease of cell disruption rate at medium and high concentrations of the suspension. Character of the process investigated on the basis of absorbance measurement is

than the values obtained for the lowest yeast cell concentration. A similar relation and character of changes were obtained when the release of intracellular compounds was studied on the basis of absorbance measurements. Results of these experiments are illustrated in Fig. 9. At suspension concentration S>0 changes of the process rate constant in methods II and III are described by Eq. (39) and results of estimation are collected in Table 1. Data obtained by Bradford's method for concentrations 0.02 and 0.05 were deleted from statistical analyses; in the case of absorbance measurements data obtained for the lowest

Fig. 9. Changes of rate constant determined for a linear model – method III

Results shown in Fig. 3 refer to biological material taken from one fermentation process, while these illustrated in Fig. 9 were obtained for yeast from a different fermentation process. In general, the results show that protein release is slower than microbial cell disruption. These observations are in agreement with results obtained by Middelberg et al. (1991) for *Escherichia coli* disintegrated in a high-pressure homogenizer. For such bacteria whose cellular structure is different than yeast cellular structure, the authors showed that the process tested on the basis of determination of the number of living microorganisms and absorbance of light with the wavelength 260 nm. According to Middelberg et al. (1991), the process of protein release measured also by Bradford's method is slower. Dependence of the release rate of intracellular compounds on the place of their location in cells was published by Hetherington et al. (1971). The authors report that during disintegration of *E. coli* cells the periplasmic enzymes are released faster than total protein, all cytoplasmic enzymes are

released at the same rate as total protein, whereas membrane-bound enzyme slower.

The release of intracellular compounds during disintegration of microorganisms in the bead mill is shown in Figs 10 and 11. In general, the results are concordant with the results obtained in the experiments with cell disruption process. A decreased rate of protein release at the initial stage of disintegration is observed already at low suspension concentrations; it is higher than the decrease of cell disruption rate at medium and high concentrations of the suspension. Character of the process investigated on the basis of absorbance measurement is

concentration of the suspension were omitted.

Fig. 10. The effect of suspension concentration on the protein release process

Fig. 11. The effect of suspension concentration on the intracellular compounds release process

The Release of Compounds from Microbial Cells 617

microorganisms in the space which is safe for them. At high and very high concentrations, cells deformed due to compression can force out adjacent cells from the destruction zone causing a decrease of the process rate at its initial stage. An increase of the rates of cell disruption and release of intracellular compounds at medium and high concentration of microbial suspensions can be caused by mutual blocking of cells, blocking of cells by fragments of disintegrated microorganisms or the interaction of released intracellular compounds. Results of the discussed investigations confirmed the dependence of

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compound release rates on the position of these compounds in a cell.

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The study was carried out within the frames of the grant W-10/1/2011/Dz. St.

**8. Acknowledgments** 

39-42 , ISSN 0951-208X

ISSN 0958-6946

179X

**9. References** 

close to the one obtained on the basis of determination of the number of living cells. In the case of high suspension concentrations the decrease of the intracellular compound release rate was not so significant as in the case of results obtained by the direct measuring method. The sequence of events: cell wall disruption – release of intracellular compounds, was used by Melendres et al. (1993) to describe nonlinear release of enzymes. This theory can be used in the description of total protein release. Eq. (35) can be written in the form of Eq. (51).

$$N\_d = N\_0 \left(1 - e^{-kt}\right) \tag{51}$$

If protein release rate is proportional to that which can be released from disrupted cells, the increment of protein released in time dτ is described by Eq. (52).

$$dR = k\_{\rm II} (R\_D - R) d\tau \tag{52}$$

Protein can be released only from disrupted cells. The maximum amount of protein RD that can be released from cells disrupted in time τ is specified by Eq. (53).

$$R\_D = R\_m \left(\frac{N\_d}{N\_0}\right) \tag{53}$$

Upon substitution of Eq. (53) and Eq. (51) to Eq. (52) we obtain Eq. (54) for protein release in the process of microbial cell disruption.

$$dR = k\_{II} \left[ R\_m \left( 1 - e^{-k\tau} \right) - R \right] \tag{54}$$

Results illustrated in Fig. 4 show that differences in the process rates increase with an increase of the concentration of microorganism suspension. Character of changes of the rate constants determined on the basis of absorbance measurements is in agreement with that revealed on the basis of computer-aided analysis of microscopic images and Bradford's method (1976).
