2.1 Growth associated

Growth linked products are formed by growing cells and hence primary metabolites. Figure 1 clearly shows that product is formed simultaneously with growth of cells. That is product concentration increases with cell concentration. The formation of growth associated product may be described by Eq. (1);

$$\frac{dP}{dt} = r\_p = q\_p X \tag{1}$$

where P = concentration of product qp = specific rate of product formation X = biomass concentration.

### 2.2 Non-growth associated

They are formed by cells which are not metabolically active and hence are called secondary metabolites. Figure 2 clearly shows that product formation is unrelated

Figure 1. Growth associated.

Figure 2. Non-growth associated.

to growth rate but is a function of cell concentration. The formation of Non-growth associated product may be described by Eq. (2);

$$q\_p = \beta = \text{constant} \tag{2}$$

where μ is the specific growth rate (h-1), S is substrate concentration (g/L) and KS and μmax are the Monod constant (g/L) and maximum specific growth rate, (h-1)

At the end of the lag phase, the growth of microorganisms is well acclimatized for its contemporary environment. Then the cells were multiplied hastily. The major active part of the cell growth curve which is called as the exponential (log) phase is used for the adjudication of kinetic parameters. The period of balanced growth that is the log phase, in which all components of a cell grow at the equivalent

In Contois model, Michaelis constant is directly proportional to cell concentration and specific growth rate is inversely proportional to cell concentration which is described by Eq. (5). The Monod equation was also modified with the maintenance

<sup>μ</sup> <sup>¼</sup> <sup>μ</sup>maxS

where X is cell mass concentration (g/L) and t is time (h). Separation of variables and integrating Eq. (4) yields Eq. (5). The above equations were used to enumerate the cell growth and product accumulation during the batch experiments [17]. The relationship between cell growth and product formation were identified by

Leudeking-Piret model (Eq. (7)) was used for kinetic analysis of cell production.

where α and β are the associated and non-associated growth factor respectively.

S Ks þ S 

Ks<sup>X</sup> <sup>þ</sup> <sup>S</sup> (5)

dt <sup>þ</sup> <sup>β</sup><sup>x</sup> (7)

� m (6)

rate. Malthus model was also used for the cell growth behavior.

term which was incorporated in the Herbert model (Eq. (6)).

μ ¼ ð Þ μmax þ m

dp dt <sup>¼</sup> <sup>α</sup> dx

x and p show the concentration of dry cell weight (DCW) and product

respectively.

Mixed growth associated.

Kinetic Studies on Cell Growth

DOI: http://dx.doi.org/10.5772/intechopen.84353

Figure 3.

Leudeking-Piret kinetics.

17

#### 2.3 Mixed-growth associated

The product formation from the microorganism depends on both growth and Non-growth associated. It takes place during growth and stationary phases. In Figure 3, product formation is a combination of growth rate and cell concentration. The formation of Mixed-growth associated product may be described by Eq. (3);

$$q\_p = a\_\mu + \beta \tag{3}$$

#### 2.4 Production kinetics

Microbial growth kinetics, i.e., the relationship between the specific growth rate (μ) of a microbial population and the substrate concentration (s), is an indispensable tool in all fields of microbiology, be it physiology, genetics, ecology, or biotechnology, and therefore it is an important part of the basic teaching of microbiology [16]. Unfortunately, the principles and definitions of growth kinetics are frequently presented as if they were firmly established in the 1940s and during the following "golden age" in the 1950s and 1960s the key publications are those of Monod. Monod, logistic, modified logistic model, and Leudeking-Piret models were used to describe the batch growth kinetics of cell. The Monod kinetic model is given as Eq. (4):

$$
\mu = \frac{\mu\_{\text{max}} \text{S}}{K\_t + \text{S}} \tag{4}
$$

to growth rate but is a function of cell concentration. The formation of Non-growth

The product formation from the microorganism depends on both growth and Non-growth associated. It takes place during growth and stationary phases. In Figure 3, product formation is a combination of growth rate and cell concentration. The formation of Mixed-growth associated product may be described by Eq. (3);

Microbial growth kinetics, i.e., the relationship between the specific growth rate (μ) of a microbial population and the substrate concentration (s), is an indispensable tool in all fields of microbiology, be it physiology, genetics, ecology, or biotechnology, and therefore it is an important part of the basic teaching of microbiology [16]. Unfortunately, the principles and definitions of growth kinetics are frequently presented as if they were firmly established in the 1940s and during the following "golden age" in the 1950s and 1960s the key publications are those of Monod. Monod, logistic, modified logistic model, and Leudeking-Piret models were used to describe the batch growth kinetics of cell. The Monod kinetic model

<sup>μ</sup> <sup>¼</sup> <sup>μ</sup>maxS

qp ¼ β ¼ constant (2)

qp ¼ αμ þ β (3)

Ks <sup>þ</sup> <sup>S</sup> (4)

associated product may be described by Eq. (2);

2.3 Mixed-growth associated

Figure 2.

Cell Growth

Non-growth associated.

2.4 Production kinetics

is given as Eq. (4):

16

where μ is the specific growth rate (h-1), S is substrate concentration (g/L) and KS and μmax are the Monod constant (g/L) and maximum specific growth rate, (h-1) respectively.

At the end of the lag phase, the growth of microorganisms is well acclimatized for its contemporary environment. Then the cells were multiplied hastily. The major active part of the cell growth curve which is called as the exponential (log) phase is used for the adjudication of kinetic parameters. The period of balanced growth that is the log phase, in which all components of a cell grow at the equivalent rate. Malthus model was also used for the cell growth behavior.

In Contois model, Michaelis constant is directly proportional to cell concentration and specific growth rate is inversely proportional to cell concentration which is described by Eq. (5). The Monod equation was also modified with the maintenance term which was incorporated in the Herbert model (Eq. (6)).

$$
\mu = \frac{\mu\_{\text{max}} \mathbf{S}}{K\_\text{s} \mathbf{X} + \mathbf{S}} \tag{5}
$$

$$
\mu = (\mu\_{\text{max}} + m) \left(\frac{\text{S}}{\text{K}\_{\text{s}} + \text{S}}\right) - m \tag{6}
$$

where X is cell mass concentration (g/L) and t is time (h). Separation of variables and integrating Eq. (4) yields Eq. (5). The above equations were used to enumerate the cell growth and product accumulation during the batch experiments [17]. The relationship between cell growth and product formation were identified by Leudeking-Piret kinetics.

Leudeking-Piret model (Eq. (7)) was used for kinetic analysis of cell production.

$$\frac{dp}{dt} = a\frac{d\mathbf{x}}{dt} + \beta\mathbf{x} \tag{7}$$

where α and β are the associated and non-associated growth factor respectively. x and p show the concentration of dry cell weight (DCW) and product

concentration. The Logistic equation was used to analyze the exponential growth phase kinetics while Malthus kinetics was used to express the death phase kinetics (Eqs. (8) and (9)) [16, 18].

$$\frac{d\boldsymbol{\mathfrak{x}}}{dt} = \mu\_m \left(\mathbf{1} - \frac{\boldsymbol{\mathfrak{x}}}{\boldsymbol{\mathfrak{x}}\_m}\right) \mathbf{x} \tag{8}$$

$$\frac{d\mathbf{x}}{dt} = \boldsymbol{\mu} \cdot \mathbf{x} \tag{9}$$

$$\mathcal{X}(t) = \frac{\varkappa\_0 \exp\left(\mu\_m \cdot t\right)}{\left[1 - \left(\frac{\chi\_0}{\chi\_m}\right) (1 - \exp\left(\mu\_m \cdot t\right))\right]} \tag{10}$$

$$\mathbf{1}n\left(\frac{\boldsymbol{\omega}}{\boldsymbol{\omega}\_0}\right)\boldsymbol{\mu}\cdot\boldsymbol{t}\tag{11}$$

The value of xm can be obtained from the experimental growth kinetic data and

<sup>μ</sup> exp ð Þ <sup>μ</sup> � <sup>t</sup> (16)

the value of parameter α was obtained from the slope of the linear plot of

The kinetic modeling of product production by Leudeking-Piret model using carbon source.

p tðÞ¼ <sup>p</sup><sup>0</sup> <sup>þ</sup> <sup>α</sup>x<sup>0</sup> exp ð Þþ <sup>μ</sup> � <sup>t</sup> <sup>β</sup> <sup>x</sup><sup>0</sup>

¼ p<sup>0</sup> þ αA tðÞþ βB tð Þ

by Leudeking-Piret model are shown in Figure 4. It is the combination of kinetic models for better agreement between experimental data and model

conversions are criteria with the main attention toward productivity.

exponential growth phase and death phase, respectively.

Eqs. (13) and (16) show the kinetic model of product production in the

The resulting graph obtained from kinetic modeling of product production

predictions which are employed in cell growth and Product production. The product accumulation mostly adhered to growth-associated kinetic pattern. Matlab ver. 7.12 computer software was used to define the interpretation of growth kinetic

One of the very important practical applications of this model is the evaluation of the product formation kinetics. Mathematical models facilitate data analysis and provide a strategy for solving problems encountered in fermentations. Information on fermentation process kinetics is potentially valuable for the improvement of batch process performance. Finally, the product yields and substrate

This is an original work of the authors and it has not been submitted to any other open access publishers previously. Here we have declared that there is no conflict

p tðÞ� p<sup>0</sup> � βB against A(t).

Kinetic Studies on Cell Growth

DOI: http://dx.doi.org/10.5772/intechopen.84353

Figure 4.

parameters.

3. Conclusion

Conflict of interest

of interest.

19

where xm, x<sup>0</sup> and μ<sup>m</sup> are the initial DCW or biomass concentration, maximum biomass concentration and maximum specific growth rate of the microorganism, respectively. Also, tm is the required time (seed age) for maximum product concentration by the microorganism. According to Eq. (10), in order to estimate the value of the μm, a plot of 1n <sup>x</sup> xm�<sup>x</sup> against t will yield a straight line that the value of its the slope corresponds to μm and the intercept equals to 1n xm <sup>x</sup><sup>0</sup> � 1 � �: The substrate and product inhibitory effect on cell growth has been presented by Eq. (11), where x is biomass concentration with respect to time and x<sup>0</sup> is the initial biomass concentration.

$$1n\frac{\boldsymbol{\infty}}{\boldsymbol{\infty}\_m - \boldsymbol{\infty}} = \mu\_m \cdot \boldsymbol{t} - \mathbf{1}n \left(\frac{\boldsymbol{\infty}\_m}{\boldsymbol{\infty}\_0} - \mathbf{1}\right) \tag{12}$$

The growth pattern of micro-organism followed the modified Logistic model. Maximum cell concentration was obtained for sugarcane bagasse incubated for 48 h when compared to glucose as carbon source. The experimental values deviate slightly towards the end of stationary phase because the modified logistic equation used does not distinguish the decrease in cell density that normally occurs at the end of stationary phase [19]. Substituting Eqs. (8) and (10) into Eq. (7) and integrating, will yield Eq. (13).

$$\begin{split} p(t) &= p\_0 + a \mathbf{x}\_0 \left\{ \frac{\exp\left(\mu\_m \cdot t\right)}{\left[\mathbf{1} - \left(\frac{\mathbf{x}\_0}{\mathbf{x}\_m}\right) (\mathbf{1} - \exp\left(\mu\_m \cdot t\right))\right]} - \mathbf{1} \right\} \\ &+ \beta \frac{\mathbf{x}\_m}{\mu\_m} \mathbf{1} n \left[\mathbf{1} - \left(\frac{\mathbf{x}\_0}{\mathbf{x}\_m}\right) (\mathbf{1} - \exp\left(\mu\_m \cdot t\right))\right] \end{split} \tag{13}$$

Eq. (13) can be rewritten as Eq. (14)

$$p(t) = p\_0 + aA(t) + \beta B(t) \tag{14}$$

The value of dx/dt is equal to zero and x = xm in the stationary phase. Using Eqs. (7) and (13), one can obtain:

$$\beta = \frac{\frac{dp}{dt}(st \cdot phase)}{\chi\_m} \tag{15}$$

concentration. The Logistic equation was used to analyze the exponential growth phase kinetics while Malthus kinetics was used to express the death phase kinetics

dt <sup>¼</sup> <sup>μ</sup><sup>m</sup> <sup>1</sup> � <sup>x</sup>

dx

x tðÞ¼ <sup>x</sup><sup>0</sup> exp ð Þ <sup>μ</sup><sup>m</sup> � <sup>t</sup> <sup>1</sup> � <sup>x</sup><sup>0</sup> xm � �

> 1n x x0 � �

and product inhibitory effect on cell growth has been presented by Eq. (11), where x is biomass concentration with respect to time and x<sup>0</sup> is the initial biomass

xm � <sup>x</sup> <sup>¼</sup> <sup>μ</sup><sup>m</sup> � <sup>t</sup> � <sup>1</sup><sup>n</sup>

The growth pattern of micro-organism followed the modified Logistic model. Maximum cell concentration was obtained for sugarcane bagasse incubated for 48 h when compared to glucose as carbon source. The experimental values deviate slightly towards the end of stationary phase because the modified logistic equation used does not distinguish the decrease in cell density that normally occurs at the end of stationary phase [19]. Substituting Eqs. (8) and (10) into Eq. (7) and integrating,

> <sup>1</sup> � <sup>x</sup><sup>0</sup> xm � �

> > xm � �

The value of dx/dt is equal to zero and x = xm in the stationary phase. Using

dt ð Þ st � phase xm

β ¼ dp

<sup>1</sup><sup>n</sup> <sup>1</sup> � <sup>x</sup><sup>0</sup>

8 < :

its the slope corresponds to μm and the intercept equals to 1n xm

x

1n

p tðÞ¼ p<sup>0</sup> þ αx<sup>0</sup>

Eq. (13) can be rewritten as Eq. (14)

Eqs. (7) and (13), one can obtain:

<sup>þ</sup> <sup>β</sup> xm μm

where xm, x<sup>0</sup> and μ<sup>m</sup> are the initial DCW or biomass concentration, maximum biomass concentration and maximum specific growth rate of the microorganism, respectively. Also, tm is the required time (seed age) for maximum product concentration by the microorganism. According to Eq. (10), in order to estimate the

xm � �

ð Þ 1 � exp ð Þ μ<sup>m</sup> � t

x (8)

dt <sup>¼</sup> <sup>μ</sup> � <sup>x</sup> (9)

μ � t (11)

<sup>x</sup><sup>0</sup> � 1 � �

> 9 = ;

: The substrate

(12)

(13)

(15)

h i (10)

xm�<sup>x</sup> against t will yield a straight line that the value of

xm x0 � 1 � �

exp ð Þ μ<sup>m</sup> � t

� �

ð Þ 1 � exp ð Þ μ<sup>m</sup> � t h i � <sup>1</sup>

ð Þ 1 � exp ð Þ μ<sup>m</sup> � t

p tðÞ¼ p<sup>0</sup> þ αA tðÞþ βB tð Þ (14)

dx

(Eqs. (8) and (9)) [16, 18].

Cell Growth

value of the μm, a plot of 1n <sup>x</sup>

concentration.

will yield Eq. (13).

18

Figure 4. The kinetic modeling of product production by Leudeking-Piret model using carbon source.

The value of xm can be obtained from the experimental growth kinetic data and the value of parameter α was obtained from the slope of the linear plot of p tðÞ� p<sup>0</sup> � βB against A(t).

Eqs. (13) and (16) show the kinetic model of product production in the exponential growth phase and death phase, respectively.

$$p(t) = p\_0 + a\mathbf{x}\_0 \exp\left(\mu \cdot t\right) + \beta \frac{\varkappa\_0}{\mu} \exp\left(\mu \cdot t\right) \tag{16}$$

$$= p\_0 + aA(t) + \beta B(t)$$

The resulting graph obtained from kinetic modeling of product production by Leudeking-Piret model are shown in Figure 4. It is the combination of kinetic models for better agreement between experimental data and model predictions which are employed in cell growth and Product production. The product accumulation mostly adhered to growth-associated kinetic pattern. Matlab ver. 7.12 computer software was used to define the interpretation of growth kinetic parameters.

## 3. Conclusion

One of the very important practical applications of this model is the evaluation of the product formation kinetics. Mathematical models facilitate data analysis and provide a strategy for solving problems encountered in fermentations. Information on fermentation process kinetics is potentially valuable for the improvement of batch process performance. Finally, the product yields and substrate conversions are criteria with the main attention toward productivity.

### Conflict of interest

This is an original work of the authors and it has not been submitted to any other open access publishers previously. Here we have declared that there is no conflict of interest.
