VOL. XVIII (1976)
BIOTECHNOLOGY AND BIOENGINEERING
A Model for Bacterial Growth on Methanol With the growing interest in biological processes there has been an increasing need to study and understand the growth kinetics of microorganisms. Current theories are primarily based on the relationship between substrate concentration and specific growth rate, as originally proposed by Monod ( p = pmaxS/[K S ] ) . Although the Monod equation has provided a reasonable description for growth of many different species of microorganisms, it is inadequate for those substrates which are inhibitory to the microbial growth a t high concentrations. In recent years numerous growth kinetic models have been proposed to describe the inhibitory effect of substrate on the specific growth rate. Most of these models are based on either a strict empirical fit or a substrate inhibition model of enzyme kinetics. Wayman and Tsengl recently proposed the following empirical model
+
P
+S) PmaxS/(K + S ) - i (S - SB)
= Pmax8/(K
P =
S
< Se
S
> SB
(1)
where i represents an inhibition constant and SOrepresents the threshold concentration that separates the concentration range of substrate-enhanced growth from that of substrate-inhibited growth. While the two-equation model seems to fit the experimental data they considered, it is nevertheless empirical. In addition, because of the two regions used in the model, a Monod region and a linearly decreasing growth region, the model is incapable of fitting growth rates that deviate from this functional form. For example, the data of Pilat and Prokopl show a plateau of constant growth rate over a wide range of substrate concentration. Substrate inhibition models based on enzyme kinetics have also been proposed by A n d r e ~ s Yano , ~ et al.,4 and Edwards.6 However, the models based on substrate inhibition mechanisms predict specific growth rates that approach zero value asymptotically and therefore, are incapable of predicting zero growth rate a t a finite substrate concentration, a situation commonly observed with microorganisms growing on alcohols. I n addition, since the specific growth rate is taken to be proportional to the substrate consumption rate in these models, a constant yield is implied. This raises some questions as to the validity of models that are solely based on enzyme kinetics since variable yield has frequently been reported. Hence, an alternate approach on the modeling of specific growth rate as a function of limiting substrate concentration is suggested here. Specific growth rates a t various methanol concentrations of a methanol utilizing bacteria were determined by shake flask batch experiments. A simple, mineralsalt medium containing methanol as the sole carbon and energy source was used. Nitrogen, magnesium, and phosphate were provided in the medium by NH,OH, MgS04 . i’H20, and NaHzPOl . H 2 0 . Tap water was used to provide any other possible trace element requirements. The p H was adjusted to 7.0 using HC1 and the temperature was maintained a t 30°C for the experiments. As shown in Figure 1 the specific growth rate increases rapidly with methanol concentration up to about 0.1% w/v, remains practically constant over the concentration range 1629 @ 1976 by John Wiley & Sons, Inc.
1630 BIOTECHNOLOGY AN11 BIOENGINEERING VOL. XVIII (1976)
=\?
T = 30°C pH
3
7.0
I
0
Fig. 1. (Y
pz
+ +
I
1
1
10 20 30 I N I T I A L M E T H A N O L CONCENTRATION ( W / V
4 .O
YO)
Comparison of specific growth rate models with experimental data. pz = (0.445 S - 0.0963 S2)/(0.00712 S - 0.0598 82); [0.4.55S/(0.00712 S)][(l- 0.216 A S ) / ( ~ - 0.0598 S)]. (- - -) Model 2, = 0.488 S / ( O . O l O S 0.461 S2); (- - - -) Model 3, pz = 0.412 S/(0.00522 S - 0.157 S 2 0.208 S3). ( - - - ) Model 4, pz = 0.477 Se-0.310S/(0.00940 S ) ; (0) experimental data; pz has units of hr-l, S has units of yo w/v. ) Model 1,
+
+ + +
+
of 0.1 to 0.3% w/v, and then decreases thereafter, reaching the critical concentration of 4.60/, w/v, a t which point no growth takes place. Using four of the models proposed by Edwards the experimental data were fitted by a nonlinear least-square fit technique which makes use of Marquardt’s method. Comparisons of the models with the experimental data are shown in Figure 1. Models 2, 3, and 4 all give poor fits especially at the critical methanol concentrat,ion (4.6% w/v). Furthermore, one of the parameters in Model 3 is negative and therefore cannot be accepted in the context of the proposed mechanism. On the other hand, Model 1, which is based on a substrate inhibition model of enzyme involving two active sites per molecule, fits the experimental data almost, perfect,ly, although it. also suffers from the same shortcoming in that two of the kinetic paramet.ers in the model are negative. Additional models, which were construct,ed on the basis of combinations of passive or active transport through the wall/membrane complex with various enzyme kinetics within the cell, also suffered from the same shortcomings in that the fit was extremely poor a t high substrat,e concent,rations, especially a t the critical concentration. In fact it is not difficult to realize that any model based on enzyme kinetics (for example, a subst,rat.e inhibition model) with or without mass transfer phenomena, is incapable of predicting zero growth at a nonzero finite substrate concentration. Therefore, a different approach is taken here to model the specific growth rate. We assume that the substrate consumption rat.e either follows the usual enzyme kinetics (Michaelis-Menten kinetics or substrat,e inhibition kinetics) or is limited
COMMUNICATIONS TO T H E EDITOR
1631
by active transport (Michaelis-Menten form). We also know that only a fraction of the substrate consumed is transformed into biomass (cells), i.e., pz = Y ( - p a ) where pz is the specific cell growth rate, - p s is the specific substrate consumption rate, and Y is the yield factor. Since the yield factor, Y , often depends heavily on the substrate concentration, we allow it to be an arbitrary function of substrate, Y ( s ) . In other words we allow the specific substrate consumption rate, - p a , to take on any functional form of enzyme kinetics or active transport and account for zero pr a t the critical concentration by the yield function, Y ( s ) . Thus, the specific growth rate expressions are not restricted to the models that are based on enzyme kinetics having positive parameter constraints. Instead it can take on any functional form due to the arbitrariness of the yield function, Y ( s ) . Since most enzyme kinetic expressions are ratios of polynomials in sub( k B k2S2)/(K S K,S2), strate concentrations (for example, k S / ( K etc.), i t is convenient to assume that the yield function, Y ( s ) ,which is allowed to be arbitrary, can also be modeled by a ratio of polynomials in substrate concentration, S. Therefore, ratios of polynomials in S are good candidates for modeling the specific growth rates. Model 1, which fits the data well, is already in this form p, = (0.4.5-5S - 0.0963 S2)/(0.00712 S - 0.0398 S 2 ) (2)
+
+ s),
+ +
+
which can be rearranged to pz
= (0.453 S/(0.00712
+ S ) ][ ( l - 0.216 S)/(l - 0.0598 S ) ]
S 5 4.6 (3)
which may be interpreted as Pz =
--IYDP.l [Y(s)/l-ol
(4)
In other words, if we assume that the substrate consumption rate is of MichaelisMenten type, then the implied yield function is
I.(S) = l-o(l - 0.216 S)/(l- 0.05988)
S 5 4.6
(5)
where may be interpreted as the maximum yield factor a t zero substrate concentration. This interpretation, the usual Michaelis-Menten type substrate consumption with a decreasing yield function, is yet to be verified. Nevertheless, it represents a plausible interpretation of a model which fits the data extremely well. A natural question to ask a t this point is then, would a ratio of higherorder polynomials lead to a different interpretation? We took a model which is second-order in the numerator and third-order in the denominator and fitted it to the data, pz =
+ 0.0309 S3) ZS [0.434 S/(0.00644 + S ) ][ ( l - 0.208 S)/(1 - 0.125 S + 0.0309 S 2 ) ](6) (0.434 S - 0.0901 S2)/(0.00644
+S
- 0.125 S2
and obtained a good fit as shown in Figure 2. Equation (6) is subject to the same interpretation, Michaelis-Menten type substrate consumption and an almost linearly decreasing yield. I t is simple to show that due to the negativity of the second-order coefficient in the denominator polynomial, eq. (6) cannot be rearranged into a combination of the usual substrate inhibition form for ps with an arbitrary yield function in the form of ratios of polynomials. Apparently these two models, eqs. (3) and (6), not only fit the experimental data very well (Figs. 1 and 2), but also lead to the same interpretation. They both suggest Michaelis-
1632 BIOTECHNOLOGY AND BIOENGINEERING VOL. XVIII (1976)
I N I T I A L METHANOL CONCENTRATION ( W / V 7 0 )
Fig. 2. A specific growth rate model. pz = (0.434 S - 0.0901 S2)/(0.00644 - 0.125 Sz 0.0309 S 3 ) 1: [0.434 S/(0.00644 S ) ][ ( l - 0.208 S)/(l -0.125 S 0.0309 S2)]; pz has units of h r F , S has units of % w/v.
+S
+
+
+
Menten type substrate consumption rate and monotonically decreasing yield functions. Although the above two models do seem to provide a plausible explanation for the experimental data obtained, there are many questions that need t.o be answered in addition to experimental verification of the proposed interpretation. Among them are the questions concerning the number of models (the order of polynomials) that need to be tested and the discriminatory selection of the most adequate model from those that fit the data well. However, this is not a proper place to go into these questions as the validity of the proposed interpretation must be checked out experimentally first. Suffice it to say that between the two models proposed, eqs. (3) and (6), eq. (3) is adequate since it fits the experimental data as well as eq. (6) even though it has one less parameter. Although no experimental data are available at present to confirm that the yield function decreases with the substrate concentration, such trends have been indicated by Pilat and Prokop,* Reuss e t a1.,6 and Tseng,' for yeasts growing on alcohols. While the approach taken here is also empirical in the sense that a ratio of polynomials in substrate concentration is used to fit the specific growth rate, the resulting model can lead to certain physical interpretations. For example, the model may predict a substrate consumption rate (p,) of the Monod or substrate inhibition type with a variable yield function. Thus, with the approach demonstrated here one can conceivably obtain both pz and p, without an a priori assumption on the mechanism of substrate consumption and the assumption of a constant yield factor.
COMMUNICATIONS TO THE EDITOR
1633
References 1 . M. Wcyman and M. C. Tseng, Biotechnol. Bioeng., 18, 383 (1976). 2. P. Pilat and A. Prokop, Biotechnol. Rioeng., 17, 1717 (1975). 3. John F. Andrews, Riolechnol. Bioeng., 10, 707 (1968). 4. T. Yano, T. Nakahara, S. Kamiyama, and K. Yamada, Agr. Riol. Chem., 30, 42 (1966). 5. V. H . Edwards, Riotechnol. Rioeng., 12, 679 (1970). 6. M. Reuss, H. Sahm, and F. Wagner, Chemi Zngenieur Technik, Nr. 16,669 (1974). 7. M. M. C. Tseng, Ph.1). Thesis, University of Toronto, Ontario, Canada, 1974.
BILLJ. CHEN HENRYC. LIM GEORGE T.Tsao School of Chemical Engineering Purdue University West Lafayette, Indiana 47907 Accepted for Publication May 28, 1976
BIOTECHNOLOGY AND BIOENGINEERING
A Model for Bacterial Growth on Methanol With the growing interest in biological processes there has been an increasing need to study and understand the growth kinetics of microorganisms. Current theories are primarily based on the relationship between substrate concentration and specific growth rate, as originally proposed by Monod ( p = pmaxS/[K S ] ) . Although the Monod equation has provided a reasonable description for growth of many different species of microorganisms, it is inadequate for those substrates which are inhibitory to the microbial growth a t high concentrations. In recent years numerous growth kinetic models have been proposed to describe the inhibitory effect of substrate on the specific growth rate. Most of these models are based on either a strict empirical fit or a substrate inhibition model of enzyme kinetics. Wayman and Tsengl recently proposed the following empirical model
+
P
+S) PmaxS/(K + S ) - i (S - SB)
= Pmax8/(K
P =
S
< Se
S
> SB
(1)
where i represents an inhibition constant and SOrepresents the threshold concentration that separates the concentration range of substrate-enhanced growth from that of substrate-inhibited growth. While the two-equation model seems to fit the experimental data they considered, it is nevertheless empirical. In addition, because of the two regions used in the model, a Monod region and a linearly decreasing growth region, the model is incapable of fitting growth rates that deviate from this functional form. For example, the data of Pilat and Prokopl show a plateau of constant growth rate over a wide range of substrate concentration. Substrate inhibition models based on enzyme kinetics have also been proposed by A n d r e ~ s Yano , ~ et al.,4 and Edwards.6 However, the models based on substrate inhibition mechanisms predict specific growth rates that approach zero value asymptotically and therefore, are incapable of predicting zero growth rate a t a finite substrate concentration, a situation commonly observed with microorganisms growing on alcohols. I n addition, since the specific growth rate is taken to be proportional to the substrate consumption rate in these models, a constant yield is implied. This raises some questions as to the validity of models that are solely based on enzyme kinetics since variable yield has frequently been reported. Hence, an alternate approach on the modeling of specific growth rate as a function of limiting substrate concentration is suggested here. Specific growth rates a t various methanol concentrations of a methanol utilizing bacteria were determined by shake flask batch experiments. A simple, mineralsalt medium containing methanol as the sole carbon and energy source was used. Nitrogen, magnesium, and phosphate were provided in the medium by NH,OH, MgS04 . i’H20, and NaHzPOl . H 2 0 . Tap water was used to provide any other possible trace element requirements. The p H was adjusted to 7.0 using HC1 and the temperature was maintained a t 30°C for the experiments. As shown in Figure 1 the specific growth rate increases rapidly with methanol concentration up to about 0.1% w/v, remains practically constant over the concentration range 1629 @ 1976 by John Wiley & Sons, Inc.
1630 BIOTECHNOLOGY AN11 BIOENGINEERING VOL. XVIII (1976)
=\?
T = 30°C pH
3
7.0
I
0
Fig. 1. (Y
pz
+ +
I
1
1
10 20 30 I N I T I A L M E T H A N O L CONCENTRATION ( W / V
4 .O
YO)
Comparison of specific growth rate models with experimental data. pz = (0.445 S - 0.0963 S2)/(0.00712 S - 0.0598 82); [0.4.55S/(0.00712 S)][(l- 0.216 A S ) / ( ~ - 0.0598 S)]. (- - -) Model 2, = 0.488 S / ( O . O l O S 0.461 S2); (- - - -) Model 3, pz = 0.412 S/(0.00522 S - 0.157 S 2 0.208 S3). ( - - - ) Model 4, pz = 0.477 Se-0.310S/(0.00940 S ) ; (0) experimental data; pz has units of hr-l, S has units of yo w/v. ) Model 1,
+
+ + +
+
of 0.1 to 0.3% w/v, and then decreases thereafter, reaching the critical concentration of 4.60/, w/v, a t which point no growth takes place. Using four of the models proposed by Edwards the experimental data were fitted by a nonlinear least-square fit technique which makes use of Marquardt’s method. Comparisons of the models with the experimental data are shown in Figure 1. Models 2, 3, and 4 all give poor fits especially at the critical methanol concentrat,ion (4.6% w/v). Furthermore, one of the parameters in Model 3 is negative and therefore cannot be accepted in the context of the proposed mechanism. On the other hand, Model 1, which is based on a substrate inhibition model of enzyme involving two active sites per molecule, fits the experimental data almost, perfect,ly, although it. also suffers from the same shortcoming in that two of the kinetic paramet.ers in the model are negative. Additional models, which were construct,ed on the basis of combinations of passive or active transport through the wall/membrane complex with various enzyme kinetics within the cell, also suffered from the same shortcomings in that the fit was extremely poor a t high substrat,e concent,rations, especially a t the critical concentration. In fact it is not difficult to realize that any model based on enzyme kinetics (for example, a subst,rat.e inhibition model) with or without mass transfer phenomena, is incapable of predicting zero growth at a nonzero finite substrate concentration. Therefore, a different approach is taken here to model the specific growth rate. We assume that the substrate consumption rat.e either follows the usual enzyme kinetics (Michaelis-Menten kinetics or substrat,e inhibition kinetics) or is limited
COMMUNICATIONS TO T H E EDITOR
1631
by active transport (Michaelis-Menten form). We also know that only a fraction of the substrate consumed is transformed into biomass (cells), i.e., pz = Y ( - p a ) where pz is the specific cell growth rate, - p s is the specific substrate consumption rate, and Y is the yield factor. Since the yield factor, Y , often depends heavily on the substrate concentration, we allow it to be an arbitrary function of substrate, Y ( s ) . In other words we allow the specific substrate consumption rate, - p a , to take on any functional form of enzyme kinetics or active transport and account for zero pr a t the critical concentration by the yield function, Y ( s ) . Thus, the specific growth rate expressions are not restricted to the models that are based on enzyme kinetics having positive parameter constraints. Instead it can take on any functional form due to the arbitrariness of the yield function, Y ( s ) . Since most enzyme kinetic expressions are ratios of polynomials in sub( k B k2S2)/(K S K,S2), strate concentrations (for example, k S / ( K etc.), i t is convenient to assume that the yield function, Y ( s ) ,which is allowed to be arbitrary, can also be modeled by a ratio of polynomials in substrate concentration, S. Therefore, ratios of polynomials in S are good candidates for modeling the specific growth rates. Model 1, which fits the data well, is already in this form p, = (0.4.5-5S - 0.0963 S2)/(0.00712 S - 0.0398 S 2 ) (2)
+
+ s),
+ +
+
which can be rearranged to pz
= (0.453 S/(0.00712
+ S ) ][ ( l - 0.216 S)/(l - 0.0598 S ) ]
S 5 4.6 (3)
which may be interpreted as Pz =
--IYDP.l [Y(s)/l-ol
(4)
In other words, if we assume that the substrate consumption rate is of MichaelisMenten type, then the implied yield function is
I.(S) = l-o(l - 0.216 S)/(l- 0.05988)
S 5 4.6
(5)
where may be interpreted as the maximum yield factor a t zero substrate concentration. This interpretation, the usual Michaelis-Menten type substrate consumption with a decreasing yield function, is yet to be verified. Nevertheless, it represents a plausible interpretation of a model which fits the data extremely well. A natural question to ask a t this point is then, would a ratio of higherorder polynomials lead to a different interpretation? We took a model which is second-order in the numerator and third-order in the denominator and fitted it to the data, pz =
+ 0.0309 S3) ZS [0.434 S/(0.00644 + S ) ][ ( l - 0.208 S)/(1 - 0.125 S + 0.0309 S 2 ) ](6) (0.434 S - 0.0901 S2)/(0.00644
+S
- 0.125 S2
and obtained a good fit as shown in Figure 2. Equation (6) is subject to the same interpretation, Michaelis-Menten type substrate consumption and an almost linearly decreasing yield. I t is simple to show that due to the negativity of the second-order coefficient in the denominator polynomial, eq. (6) cannot be rearranged into a combination of the usual substrate inhibition form for ps with an arbitrary yield function in the form of ratios of polynomials. Apparently these two models, eqs. (3) and (6), not only fit the experimental data very well (Figs. 1 and 2), but also lead to the same interpretation. They both suggest Michaelis-
1632 BIOTECHNOLOGY AND BIOENGINEERING VOL. XVIII (1976)
I N I T I A L METHANOL CONCENTRATION ( W / V 7 0 )
Fig. 2. A specific growth rate model. pz = (0.434 S - 0.0901 S2)/(0.00644 - 0.125 Sz 0.0309 S 3 ) 1: [0.434 S/(0.00644 S ) ][ ( l - 0.208 S)/(l -0.125 S 0.0309 S2)]; pz has units of h r F , S has units of % w/v.
+S
+
+
+
Menten type substrate consumption rate and monotonically decreasing yield functions. Although the above two models do seem to provide a plausible explanation for the experimental data obtained, there are many questions that need t.o be answered in addition to experimental verification of the proposed interpretation. Among them are the questions concerning the number of models (the order of polynomials) that need to be tested and the discriminatory selection of the most adequate model from those that fit the data well. However, this is not a proper place to go into these questions as the validity of the proposed interpretation must be checked out experimentally first. Suffice it to say that between the two models proposed, eqs. (3) and (6), eq. (3) is adequate since it fits the experimental data as well as eq. (6) even though it has one less parameter. Although no experimental data are available at present to confirm that the yield function decreases with the substrate concentration, such trends have been indicated by Pilat and Prokop,* Reuss e t a1.,6 and Tseng,' for yeasts growing on alcohols. While the approach taken here is also empirical in the sense that a ratio of polynomials in substrate concentration is used to fit the specific growth rate, the resulting model can lead to certain physical interpretations. For example, the model may predict a substrate consumption rate (p,) of the Monod or substrate inhibition type with a variable yield function. Thus, with the approach demonstrated here one can conceivably obtain both pz and p, without an a priori assumption on the mechanism of substrate consumption and the assumption of a constant yield factor.
COMMUNICATIONS TO THE EDITOR
1633
References 1 . M. Wcyman and M. C. Tseng, Biotechnol. Bioeng., 18, 383 (1976). 2. P. Pilat and A. Prokop, Biotechnol. Rioeng., 17, 1717 (1975). 3. John F. Andrews, Riolechnol. Bioeng., 10, 707 (1968). 4. T. Yano, T. Nakahara, S. Kamiyama, and K. Yamada, Agr. Riol. Chem., 30, 42 (1966). 5. V. H . Edwards, Riotechnol. Rioeng., 12, 679 (1970). 6. M. Reuss, H. Sahm, and F. Wagner, Chemi Zngenieur Technik, Nr. 16,669 (1974). 7. M. M. C. Tseng, Ph.1). Thesis, University of Toronto, Ontario, Canada, 1974.
BILLJ. CHEN HENRYC. LIM GEORGE T.Tsao School of Chemical Engineering Purdue University West Lafayette, Indiana 47907 Accepted for Publication May 28, 1976
