Bioprocess Biosyst Eng (2015) 38:199–205 DOI 10.1007/s00449-014-1256-8
SHORT COMMUNICATION
An autocatalytic kinetic model for describing microbial growth during fermentation Albert Ibarz • Pedro E. D. Augusto
Received: 22 April 2014 / Accepted: 7 July 2014 / Published online: 22 July 2014 Springer-Verlag Berlin Heidelberg 2014
Abstract The mathematical modelling of the behaviour of microbial growth is widely desired in order to control, predict and design food and bioproduct processing, stability and safety. This work develops and proposes a new semi-empirical mathematical model, based on an autocatalytic kinetic, to describe the microbial growth through its biomass concentration. The proposed model was successfully validated using 15 microbial growth patterns, covering the three most important types of microorganisms in food and biotechnological processing (bacteria, yeasts and moulds). Its main advantages and limitations are discussed, as well as the interpretation of its parameters. It is shown that the new model can be used to describe the behaviour of microbial growth. Keywords Biochemical engineering Bioprocessing Kinetics Modelling List of symbols a, b Parameters of model fit evaluation (Eq. 27) [different units] A, B Parameters for the partial fractions integration (Eq. 8) [different units]
A. Ibarz Department of Food Technology (DTA), School of Agricultural and Forestry Engineering (ETSEA), University of Lleida (UdL), Lleida, Spain e-mail:
[email protected] P. E. D. Augusto (&) Department of Agri-food Industry, Food and Nutrition (LAN), Luiz de Queiroz College of Agriculture (ESALQ), University of Sa˜o Paulo (USP), Piracicaba, SP, Brazil e-mail:
[email protected] k K1, K2, K3 M M0 M? MRSS Q S S0 t
Kinetic parameter of the reaction (Eq. 1) [g L-1 h-1] Parameters of the new model (Eq. 20) [g L-1, h-1, dimensionless, respectively] Microorganism biomass concentration [g L-1] Initial microorganism biomass concentration [g L-1] Maximum microorganism biomass concentration [g L-1] Residual sum-of-squares values (Eq. 26) [(g L-1)2] Fractional yield (Eqs. 3, 4) [dimensionless] Substrate concentration [g L-1] Initial substrate concentration [g L-1] Time [h]
Introduction A mathematical model of the behaviour of microbial growth is widely desired in order to control, predict and design food and bioproduct processing, stability and safety. The modelling of microbial growth through its biomass is especially interesting for fermentation processes. In fact, there are many mathematical models in the literature for describing microbial growth behaviour. The most widely applied models are those by Gompertz, generalized Verhulst (logistic) and Baranyi-Roberts, although many others are described, such as Richards, Stannard and Schnute [7, 11, 15]. Although those models fit the microbial growth data well, they are based on the microbial cell concentration count, which can be awkward in some
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fermentation processes. Furthermore, even though there are models based on the microbial biomass concentration, such as the Monod [2], the availability of new mathematical models is always desirable, as the real data does not follow specific models, and in general each particular data set can fit better to one specific model. Finally, whenever possible, it is desirable to use semi-empirical or constitutive models, based on the reaction kinetics. The aim of this work is to propose a new semi-empirical model to describe the behaviour of microbial growth based on biomass concentration. It is based on simplifying microbial growth through a reaction kinetic, which is explained and validated.
Model development and description The microbial growth process can be simplified as a reaction where microorganisms (M) consume substrate (S), forming products (P) and more microorganisms (M). Thus, it can be written as: k
S þ M ! M þ P
ð1Þ
concentrations, as well as the microbial concentration at that time (M): S ¼ ð S0 Q M 0 Þ Q M
ð5Þ
Substituting Eq. 5 in Eq. 2: dM ¼ k ½ ð S0 Q M 0 Þ Q M M dt
ð6Þ
To solve Eq. 6, after isolating the variables (Eq. 7), the expression on the left side must be integrated by the partial fraction method. Thus, this rational function can be rewritten as Eq. 8. dM ¼ k dt ½ ð S0 Q M 0 Þ Q M M
ð7Þ
1 ½ ð S0 Q M 0 Þ Q M M A B ¼ þ M ½ðS0 Q M0 Þ Q M
ð8Þ
The values of A and B (Eqs. 9, 10) are obtained by solving Eq. 8. Then, substituting it on Eq. 7, Eq. 11 is obtained:
As the reaction produces more microorganisms, its kinetic can be considered autocatalytic and described by Eq. 2:
A¼
1 S0 þ Q M 0
ð9Þ
dM ¼kSM dt
B¼
Q S0 þ Q M 0
ð10Þ
ð2Þ
To solve Eq. 2, it is necessary to express the substrate concentration (S) as a function of the microbial mass concentration (M). In ‘‘conventional’’ chemical reactions, the reactants and the product ratio are correlated by their stoichiometric coefficients. However, the stoichiometric coefficients are not easy to define in microbial growth. Therefore, rather than using the stoichiometric coefficients, the microbial growth can be described using the concept of fractional yields (Q). The fractional yield (Q) is defined as the amount of substrate consumed compared with the microbial mass formed, i.e.: Q¼
dðsubstrate consumedÞ dS ¼ dðmicroorganisms formedÞ dM
DS S0 S ¼ DM M M0
Equation 11 is thus integrated, with the following appropriated boundary conditions, resulting in Eq. 14: • when t = 0, M(t) = M0; • when t = 0, M(t) = M(t). ZM ZM dM dM þQ M ð S0 þ Q M 0 Þ Q M M0
M0
¼
ð4Þ
Thus, by rearranging Eq. 4, the substrate concentration at any time during fermentation (S) can be written as a function of the initial substrate (S0) and microbial (M0)
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ð11Þ
ð3Þ
Levenspiel [6] stated that the fractional yield (Q) can be considered constant in batch processes in mixed flow reactors for the period of exponential growth. If Q is considered constant for all compositions, it can be written as: Q¼
dM dM þQ ¼ ðS0 þ Q M0 Þ k dt M ð S0 þ Q M 0 Þ Q M
Zt
ðS0 þ Q M0 Þ k dt
0
ð12Þ
1 M ½lnððS0 þ Q M0 Þ Q M ÞM0 Q ¼ ðS0 þ Q M0 Þ k t ð13Þ M S0 ln ¼ ðS0 þ Q M0 Þ k t M0 ½ðS0 þ Q M0 Þ Q M ½lnðM ÞM M0 þQ
ð14Þ Finally, isolating M(t) in Eq. 14 and dividing Eq. 15 by (QM0), the proposed new model is obtained (Eq. 16):
Bioprocess Biosyst Eng (2015) 38:199–205
MðtÞ ¼
M0 ðS0 þ Q M0 Þ exp½kðS0 þ Q M0 Þ t S0 þ Q M0 exp½kðS0 þ Q M0 Þ t S0 þQM0 exp½kðS0 þ Q M0 Þ t Q S0 QM0
þ exp½kðS0 þ Q M0 Þ t
10
ð15Þ
K1 8
ð16Þ
Equation 16 can be simplified by defining three constants: K1 (Eq. 17), K2 (Eq. 18) and K3 (Eq. 19). S0 þ Q M 0 K1 ¼ ð17Þ Q
M (g / L)
MðtÞ ¼
201
6 4 2
(A)
ð18Þ
0
ð19Þ
10
Thus, the proposed new model can be written as Eq. 20:
8
MðtÞ ¼
K1 exp½K2 t K3 þ exp½K2 t
ð20Þ
The microbial growth curve has a typical sigmoidal shape, with three stages. In the first stage, the microorganism concentration (M(t)) is low and quasi-constant during a period. This is the lag phase, related to a period of adaptation. The growth rate at this period is close to zero. When the cells have adapted, they start to divide and then an exponential increase of M(t) is observed. This stage is called the exponential phase. Thus, after reaching a certain load, the growth rate starts to fall until it reaches zero, when the microorganism concentration reaches its maximum value (M?). This stage is called the stationary phase. Thus, at the beginning of the process (t = 0), the microorganism concentration is M0. Thus, Eq. 20 will give: Mðt ¼ 0Þ ¼ M0 ¼
K1 K3 þ 1
0
M (g / L)
S0 K3 ¼ Q M0
75
100
K2
4
(B) 0
0
25
50
75
100
10 8 6 4 2
For long enough times, the microorganism concentration when t ? ? is M?. Thus, Eq. 20 will give:
0
K3 0
25
50
(C) 75
100
t (h)
ð22Þ
The proposed model described in Eq. 20 is a sigmoidal mathematical function with three parameters, and can be evaluated to describe the microbial growth. A graphic description of the model and its parameters is shown in Fig. 1, which shows theoretical microbial growth patterns. It shows a curve with reference parameter values (K1 = 8 g L-1, K2 = 0.2 h-1 and K3 = 100, based on the values obtained in the following section). However, different curves appear with other values of K1, K2 and K3 and these can be compared to the initial one. The parameter K1 describes the maximum microbial load (Eqs. 22, 23). Thus, neither the growth curve shape (related to the kinetic parameter) nor the microbial lag
50
6
ð21Þ
Mðt ! 1Þ ¼ M1 ! K1
25
2
M (g / L)
K2 ¼ kðS0 þ Q M0 Þ
Fig. 1 Description of the parameters of the new model: theoretical microbial growth with reference parameter values (K1 = 8 g/L, K2 = 0.2 h-1 and K3 = 100; green dashed curve) and variations (blue continuous curves). a Different values of K1 (in g/L): 5, 6, 7, 8 and 9. b Different values of K2 (in h-1): 0.10, 0.20, 0.25, 0.30 and 0.40. c Different values of K3 (dimensionless): 100, 101, 102, 103 and 104
phase is changed by varying this value. Only the value of the maximum plateau (Fig. 1a) varies. K1 ¼ M1
ð23Þ
The parameter K2 is proportional to the reaction kinetic constant (k), and this is described by Eqs. 1, 2, 18 and 24.
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Thus, by varying this value, the time needed for the microbial load to go from its initial (M0) to its final (M?) values is changed. It can be seen that the growth curve changes its shape (Fig. 1b). The higher the K2 value, the higher the reaction kinetic constant (k), the microbial lag phase being shorter and the time of microbial development, shorter. K2 / k
ð24Þ
The parameter K3 describes the relative difference between the initial (M0) and final (M?) microorganism concentration values, for a specific initial load (as described in Eq. 25 after combining Eqs. 21 and 22). It can be seen as the relative growth. Thus, if K1 and K3 are constant, increasing K2 accelerates the microbial growth (Fig. 1b). On the other hand, if K1 and K2 are constant, a higher K3 describes a smaller initial load, or a smaller specific growth (Fig. 1c). Consequently, K3 can be indirectly related to the microbial lag phase. K3 ¼
M1 M0 M0
ð25Þ
The applicability of the new proposed model was then evaluated against 15 different microbial growth processes, covering data from bacteria, yeasts and moulds.
zero) and the coefficient of determination (R2; that must be as close as possible to unit). It is a simple and efficient approach to evaluating the model fit. n P
MRSS ¼ i¼1
ðMexperimental Mmodel Þ2i n
Mmodel ¼ a Mexperimental þ b
ð26Þ ð27Þ
The results obtained are shown in Table 1 and Fig. 2. It can clearly be seen that the proposed model describes the experimental data well, with high values of R2 (R2 [ 0.98; direct regression of Eq. 20 and evaluation by Eq. 27) and small values of a (|a-1| \ 0.03), b (|b| \ 0.16) and MRSS (MRSS \ 0.12). This can also be seen by evaluating the plotted model values in Fig. 2 (dashed curves), as well as the plot of experimental values versus those obtained by the proposed model (Fig. 3). Therefore, it was shown that the proposed model can be successfully used as an alternative for describing different microbial growth behaviours. Consequently, the proposed model is considered validated, as the mathematical expression here obtained well describes the microbial growth curve.
Advantages and limitations of the model Model validation The model evaluation was conducted considering 15 microorganism growth patterns under optimum and nonoptimum conditions, based on the previously published data for bacteria (Lactococcus lactis, Lactobacillus amylovorus, Rhodospirillum rubrum and Zymomonas mobilis [9, 12–14]), yeasts (Saccharomyces uvarum, Saccharomyces cerevisiae, Saccharomyces carlsbergensis, Yamadazyma stipites and Kluyveromyces marxianus [3–5, 10, 12] ) and moulds (Aspergillus oryzae and Humicola lanuginosa [1, 8] ). The data are presented on Table 1. The model parameters were obtained by non-linear regression using the CurveExpert Professional software (v.2.0.3, http://www.curveexpert.net/, USA) with a significant probability level of 95 %. The goodness of the obtained model was evaluated by the R2 regression value, the mean residual sum-of-squares values (MRSS, Eq. 26), and by plotting the microorganism concentration values obtained by the model (Mmodel) as a function of the experimental values (Mexperimental). The regression of those data to a linear function (Eq. 27) results in three parameters that can be used to evaluate the description of the experimental values by the model, i.e. the linear slope (a; that must be as close as possible to unit), the intercept (b; that must be as close as possible to
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The evaluated data covered the three most important types of microorganisms present in food and biotechnological processing (bacteria, yeasts and moulds), with different patterns of growth. Figure 2 and Table 1 show that the lag phase and specific growth vary with the type of microorganism. In all of these, the proposed model described the experimental values well. The proposed mathematical model is a semi-empirical model that describes the sigmoidal behaviour of microbial growth well. It is based on an autocatalytic kinetic reaction, successfully validated using the data from 15 different growth patterns. The main advantages of the model are related to its semi-empirical nature, as it is based on a described reaction kinetic (Eqs. 1–20), despite the mathematical functions being adjusted to the microbial growth curve, and its simplicity (Eq. 20). However, like every model, it has some limitations, which must be known in order for it to be employed adequately. Firstly, as in any mathematical modelling, an appropriate number of experimental data must be provided in order to obtain a suitable regression. Then, an appropriate amount of experimental data must be provided within the lag phase and after the microbial maximum load, in order
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Table 1 The 15 growth patterns evaluated by the proposed model #
Microorganism
Description
Reference
New model (Eq. 20) K1 (g L-1)
K2 (h-1)
Evaluation by Eq. 27
K3 (-)
R
2
MRSS
a
R2
b
1
Yeast
Saccharomyces uvarum, 25 C, fermentation broth
[5]
8.31
0.198
11.37
0.994
0.045
0.989
0.062
0.993
2
Mould
Humicola lanuginosa, 45 C, fermentation broth
[1]
9.50
0.230
40.76
0.998
0.036
0.979
0.151
0.998
3
Yeast
Saccharomyces cerevisiae, 30 C, fermentation broth
[3]
7.43
0.426
62.75
0.999
0.005
0.990
0.070
0.999
4
Yeast
Yamadazyma stipitis, 30 C, fermentation broth
[3]
5.99
0.236
54.27
0.998
0.013
1.013
0.010
0.998
5
Yeast
Kluyveromyces marxianus, 30 C, fermentation broth
[4]
8.67
0.293
19.98
0.993
0.071
0.981
0.112
0.994
6
Yeast
Kluyveromyces marxianus, 45 C, fermentation broth
[4]
3.02
0.485
27.46
0.997
0.004
0.998
0.010
0.997
7
Mould
Aspergillus oryzae, 30 C, fermentation broth
[8]
10.66
0.182
59.14
0.999
0.022
0.988
0.104
0.999
8
Yeast
Kluyveromyces marxianus, 30 C, whey ? lactose (166 g/L)
[10]
5.31
0.232
7.91
0.992
0.021
0.987
0.041
0.993
9
Yeast
Kluyveromyces marxianus, 30 C, whey ? lactose (127 g/L)
[10]
6.13
0.233
7.57
0.995
0.017
0.989
0.053
0.996
10
Yeast
Saccharomyces carlsbergensis, 30 C, fermentation broth
[12]
10.08
0.242
16.50
0.989
0.116
1.001
-0.038
0.990
11
Bacteria
Zymomonas mobilis, 30 C, fermentation broth
[12]
4.62
0.259
273.23
0.998
0.006
1.004
-0.034
0.998
12
Bacteria
Rhodospirillum rubrum, 35 C, fermentation broth 1
[13]
2.22
0.345
54.39
0.993
0.003
0.982
0.031
0.996
13
Bacteria
Rhodospirillum rubrum, 35 C, fermentation broth 2
[13]
1.80
0.211
54.65
0.995
0.002
0.975
0.030
0.997
14
Bacteria
Lactobacillus amylovorus, 30 C, fermentation broth
[14]
1.87
0.255
185.53
0.995
0.002
1.011
-0.030
0.997
15
Bacteria
Lactococcus lactis, 30 C, fermentation broth
[9]
2.53
1.735
1,786.05
0.990
0.009
1.005
-0.024
0.992
to estimate the model parameters (K1, K2 and K3) accurately. Nevertheless, this is not a specific limitation of this model, but common to all the existing microbial growth models. Secondly, the model obtained is based on the assumption that the fractional yield (Q) can be considered constant (Eqs. 3, 4). However, this is expected to be constant in most of the cases, as can be argued by the model fit to the different evaluated conditions. Even so, it may not be true at specific cases, but all the models have limitations related to non-fitting to specific data. Thirdly, the proposed model can only describe the lag, exponential and stationary phases, although the microbial growth curve can show a further mortality phase due to inhibitory mechanisms. However, this is not a limitation of the present model, but is common to almost all of the existing microbial growth models. Further, according to Peleg and Corradini [11], it makes good sense, as it is rare
for the mortality phase to be considered on food and bioproduct processing. Finally, most of the microbial growth models are written explicitly to show three properties: the duration of the lag phase, the maximum growth rate and the maximum microbial growth. The maximum microbial growth is shown by the parameter K1 in the present model. The maximum growth rate is the slope of the curve at the inflexion point. It is not directly shown in the present model, but the growth kinetic is described by the parameter K2. Furthermore, as described by Peleg and Corradini [11], the maximum growth rate alone is not a valuable information. Thus, the parameter K2 can be used to describe the microorganism ‘‘growth intensity’’. Although information about the lag phase is interesting, this does not have a clear duration, as the transition from the ‘‘no-growth’’ to the ‘‘exponential’’ phases is
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Bioprocess Biosyst Eng (2015) 38:199–205 10
12 6 5 4
Mmodel (g/L)
M (g / L)
8
3
6
2 1
4
8
4
2
0 0
0
0
10
20
30
40
50
4
8
12
Mexperimental (g/L)
12 10
Fig. 3 Experimental values versus those obtained by the proposed model (Eq. 20) for the 15 evaluated conditions. The dashed line represents a regression with a = 1 and b = 0 (Eq. 27)
9
M (g / L)
8
8
7
this being potentially useful for future studies on food and bioproduct processing and properties. 4
Conclusions 0
0
25
50
75
100
25
50
75
100
5 15 14 13 12 11
M (g / L)
4 3 2 1
A new semi-empirical mathematical model, based on an autocatalytic kinetic, was developed and proposed to describe the sigmoidal behaviour of microbial growth. The model is based on biomass concentration and was successfully validated using 15 microbial growth patterns, covering the 3 most important types of microorganisms present in food and biotechnological processing (bacteria, yeasts and moulds). Its main advantages and limitations are discussed, as well as the interpretation of its parameters. After evaluation, it can be concluded that the new model proposed here can be used to describe the microbial growth process.
0
0
t (h)
Fig. 2 Different growth patterns as biomass production. The dots are the experimental values; the curves are the new model (Eq. 20). See Table 1 for the microbial description (microorganism #1–15)
continuous. Consequently, this is determined in different ways in the literature [11]. Therefore, further than determine a specific value to its duration, it can be related to the parameter K3 (the higher K3, the longer the duration of the lag phase—Fig. 1c). Therefore, the three parameters of the proposed model can be used to describe the microbial growth behaviour. In conclusion, the new model proposed here can be used as an alternative to describe the microbial growth process,
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References 1. Bokhari SAI, Latif F, Rajoka MI (2008) Kinetics of high-level of b-glucosidase production by a 2-deoxyglucose resistant mutant of Humicola lanuginosa in submerged fermentation. Braz J Microbiol 39:724–733 2. Borzani, W. (1975). Cine´tica de processos fermentativos [Fermentation process kinetic]. In: Borzani, W.; Lima, U. A. L.; Aquarone, E. Biotecnologia: Engenharia Bioquı´mica [Biotechnology: Biochemical Engineering]. Sa˜o Paulo: Ed. Edgard Blu¨cher 3. Chmielewska J (2003) Selected biotechnological features of hybrids of Saccharomyces cerevisiae and Yamadazyma stipites. Electron J Polish Agric Univ 6:1 4. Hughes DB, Tudroszen NJ, Moye CJ (1984) The effect of temperature on the kinetic of ethanol production by a thermotolerant strain of Kluveromyces marxianus. Biotechnol Lett 6(1):1–6
Bioprocess Biosyst Eng (2015) 38:199–205 5. Lee JH, Williamson D, Rogers PL (1980) The effect of temperature on the kinetics of ethanol production by Saccharomyces uvarum. Biotechnol Lett 2(4):80–88 6. Levenspiel, O. (1986). El Omnilibro de los Reactores Quı´micos [The Chemical Reactor Omnibook]. Barcelona: Ed. Reverte´ 7. Martı´nez, A.; Rodrigo, M.; Rodrigo, D.; Ruiz, P.; Martı´nez, A.; Ocio, M. J. (2005). Predictive Microbiology and Role in Food Safety Systems. In: Barbosa-Ca´novas, G. V.; Tapia, M. S.; Cano, M. P. (Ed).. Martı´n-Belloso, O.; Martı´nez, A. (As.Ed.). Novel Food Processing Technologies. Boca Raton: CRC Press 8. Ottoni CA, Cuervo-Ferna´ndez R, Piccoli RM, Moreira R, Guilarte-Maresma B, Silva ES, Rodrigues MFA, Maiorano AE (2012) Media optimization for b-fructofuranosidase production by Aspergillus oryzae. Braz J Chem Eng 29(1):49–59 9. Parente E, Ricciardi A, Addario G (1994) Influence of pH on growth and bacteriocin production by Lactococcus lactis subsp, lactis 140NWC during batch fermentation. Appl Microbiol Biotechnol 41:388–394 10. Parrondo J, Garcı´a LA, Dı´az M (2009) Nutrient balance and metabolic analysis in a Kluyveromyces marxianus fermentation with lactose-added whey. Braz J Chem Eng 26(3):445–456
205 11. Peleg M, Corradini MG (2011) Microbial growth curves: what the models tell us and what they cannot. Crit Rev Food Sci Nutr 51(10):917–945 12. Rogers PL, Lee KJ, Tribe DE (1979) Kinetics of alcohol production by Zymomonas mobilis at high sugar concentrations. Biotechnol Lett 1(4):165–170 13. Tian Y, Yue T, Yuan Y, Soma PK, Williams PD, Machado PA, Lo YM (2010) Tobacco biomass hydrolysate enhances coenzyme Q10 production using photosynthetic Rhodospirillum rubrum. Bioresour Technol 101(20):7877–7881 14. Vuyst L, Callewaert R, Crabbe´ K (1996) Primary metabolite kinetics of bacteriocin biosynthesis by Lactobacillus amylovorus and evidence for stimulation of bacteriocin production under unfavourable growth conditions. Microbiology 142(4):817–827 15. Zwietering MH, Jongenburger I, Rombouts FM, van’t Riet K (1990) Modeling the bacterial growth curve. Appl Environ Microbiol 56(6):1875–1881
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