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Instantaneous and Overall Indicators for Determination of Bottleneck Ranking in Metabolic Reaction Networks Kansuporn Sriyudthsak and Fumihide Shiraishi* Department of Bio-System Design, Bio-Architecture Center, Kyushu UniVersity, Hakozaki, Fukuoka 812-8581, Japan
A dynamic logarithmic gain expresses the percentage change in a dependent variable or metabolite concentration in response to an infinitesimal percentage change in an independent variable or enzyme activity. This article discusses the usefulness of several bottleneck ranking (BR) indicators that are functions of the logarithmic gain, to determine the most likely bottleneck enzyme in a dynamic metabolic reaction system. Mathematical models for penicillin V and ethanol fermentations are considered as case studies. The calculated results reveal that the instantaneous BR indicator, which is a product of the dynamic logarithmic gain and the metabolite concentration, is an effective measure for instantaneous bottleneck enzyme ranking changed during the fermentation period, whereas the overall BR indicator, which is a time-averaged value of the instantaneous BR indicator, is an effective measure for bottleneck enzyme ranking throughout the entire fermentation period. cillin V fermentation6 and ethanol fermentation13 are used as case studies for discussion.
Introduction A number of research studies have focused on improving productivities in fermentation processes.1-3 One approach to improving the productivity of a process is to identify the most likely bottleneck in metabolic pathways and then enhance the activity of the relevant enzyme through genetic manipulation.4,5 Sensitivity analysis is an effective means of pinpointing a bottleneck enzyme.6 In general, metabolite concentrations in a fermentation process change with time because of both the multiplication of microbial cells and changes in substrate feed rate during the fermentation period. For this reason, dynamic sensitivity analysis is a useful tool for identifying bottlenecks at different stages of the processes being studied. In a previous work, Shiraishi and Suzuki7 performed a complete calculation of dynamic logarithmic gains over the entire period of a penicillin V fermentation process6,8-11 and found that the ranking of the logarithmic gains changed with time, meaning that the ranking of bottleneck enzymes also changed with time. Therefore, the dynamic logarithmic gains of the penicillin V concentration in response to an infinitesimal percentage change in each enzyme activity were integrated over the entire fermentation period. This integrated index was successfully used for the determination of the most likely bottleneck enzyme. However, it is not always effective to use the time-averaged dynamic logarithmic gains, as they are percentages normalized by the metabolite concentration at each time point and can be large in the region of low metabolite concentration. This study therefore investigates the usefulness of several alternative bottleneck ranking indicators including the previously employed method, namely, the time-averaged logarithmic gain,12 in order to determine a practical indicator that is applicable to various fermentation processes. Mathematical models for peni* To whom correspondence should be addressed. Tel.: +81-92-6427603. Fax: +81-92-642-7603. E-mail:
[email protected].
Theory Dynamic Logarithmic Gain and Its Time-Averaged Value. The dynamic logarithmic gain expresses the percentage change in a dependent variable at time t in response to an infinitesimal percentage change in an independent variable at time zero and is written as Li,j ) L[Xi(t), Yj] )
∂ ln Xi(t) ∂Xi(t) Yj ) ∂ ln Yj ∂Yj Xi(t) (i ) 1, 2, ..., n;j ) 1, 2, ..., m)
(1)
where Xi and Yj are the dependent and independent variables, respectively, and t is the time. The dynamic logarithmic gain, or dynamic sensitivity, can be used as an instantaneous bottleneck ranking indicator.6,12 Because the dynamic logarithmic gain changes throughout the fermentation period, it can be difficult to judge which enzyme is the most sensitive. Thus, it is reasonable to rank the time-averaged dynamic logarithmic gain, defined as j ij ) 1 L tf
∫
tf
0
Lij dt
(2)
where tf is the end of the fermentation time. This is one of the overall bottleneck ranking indicators that was used to identify the bottleneck in a fed-batch penicillin V fermentation process.12 Alternative Bottleneck Ranking Indicators. The logarithmic gain has a normalized form where an infinitesimal change in the dependent variable is divided by the dependent variable. This means that, when the dependent variable is very small, the response of the dependent variable cannot express a substantial increase in the dependent variable. To overcome this problem, we introduce the following instantaneous bottleneck ranking indicator Li,jXi ) L[Xi(t), Yj] Xi(t) )
10.1021/ie901531d 2010 American Chemical Society Published on Web 01/22/2010
∂Xi(t) Yj ∂Xi(t) X (t) ) Y ∂Yj Xi(t) i ∂Yj j (i ) 1, 2, ..., n;j ) 1, 2, ..., m)
(3)
Ind. Eng. Chem. Res., Vol. 49, No. 5, 2010
The integration of this indicator value provides the overall bottleneck ranking indicator, given by LijXi )
1 tf
∫
tf
(i ) 1, 2, ..., n;j ) 1, 2, ..., m)
LijXi dt
0
(4) We also introduce the following overall bottleneck ranking indicator
c ij ) L
LijXi ¯ X i
∫ L X dt 1 ) ∫ ∫ X dt t X¯i tf
)
ij i
0
tf
tf
0
f
i
0
LijXi dt
(i ) 1, 2, ..., n;j ) 1, 2, ..., m)
(5)
where Xi is the time-averaged product concentration, defined as ¯ X i)
1 tf
∫
tf
0
Xi dt
(i ) 1, 2, ..., n)
(6)
The performances of the instantaneous bottleneck ranking indicators, eqs 1 and 3, and overall bottleneck ranking indicators, eqs 2, 4, and 5, will be compared in the Results and Discussion to identify the bottlenecks in the metabolic networks of penicillin V and ethanol fermentations. The largest value of a bottleneck ranking indicator indicates that a desired product concentration is probably the highest with a change in the relevant enzyme activity, in other words, the relevant enzyme is a bottleneck enzyme. It should be noted that all of our bottleneck ranking indicators are based on L(X,Y), but not on L(V,Y), which expresses a logarithmic gain of flux with respect to a change in an enzyme activity. This is mainly for the following two reasons: First, the determination of a bottleneck enzyme is to identify a parameter that has the most significant impact on an increase in the productivity of a desired product. In some cases, there can be plural influxes producing the desired product. Although it might not be difficult to judge which is the main flux and it might be possible to confirm which flux certainly has a significant impact on productivity, the desired product is more or less produced through not only the identified flux but also other fluxes. If one uses L(X,Y), therefore, it is possible to identify exactly which enzyme has the most significant impact on an increase in the desired product concentration. Second, if possible, one should experimentally validate the bottleneck enzyme determined using a mathematical model. This can be done more easily if L(X,Y) is used, because the experimental measurement of metabolite concentrations is easier than that of fluxes. Computational Method for Calculation of Dynamic Sensitivities. The software SoftCADS (Software for the Calculation of Dynamic Sensitivities), developed in our laboratory,14,15 was used to calculate the instantaneous and overall BR indicators. The differential equations for metabolite concentrations are given by dXi ) dt
dSi,j ) dt
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n
∂fi(X, Y) ∂fi(X, Y) Sk,j + ∂Xk ∂Yj k)1 (i ) 1, 2, ..., n;j ) 1, 2, ..., m)
∑
where Si,j (i ) 1, 2, ..., n; j ) 1, 2, ..., m) represents the gains or local sensitivities, defined by Si,j )
∂Xi ∂Yj
i
(i ) 1, 2, ..., n)
(7)
i)1
and the differential equations for their sensitivities are given by
(9)
Equation 8 must be derived by partial differentiation of eq 7. This mathematical operation is laborious and can lead to mistakes, especially in large-scale systems. SoftCADS automatically calculates the values of differentiation terms in eq 8 by a highly accurate numerical differentiation method14 and solve eqs 7 and 8 simultaneously by a Taylor-series method16 to obtain metabolite concentrations and local sensitivities at each time point. These time-course values are then used to calculate the instantaneous BR indicators (i.e., the dynamic logarithmic gains) expressed by eq 1 and the products of dynamic logarithmic gains and metabolite concentrations expressed by eq 3. Finally, the overall BR indicators, Lj ij, LijXi, and Lc ij (i ) 1, 2, ..., n; j ) 1, 2, ..., m), are calculated by an appropriate numerical integration method17,18 Mathematical Models. Penicillin V Fermentation Model. Figure 1 illustrates a simplified metabolic pathway model of penicillin V fermentation using Penicillium chrysogenum. The original mathematical model for this system was first constructed by Nielsen et al.8-11 This model was then extended and used to optimize a fed-batch fermentation process by Conejeros and Vassiliadis.6 The system contains both intracellular and extracellular metabolites. The differential mass-balance equations and kinetic expressions in Michaelis-Menten form are presented in Table 1, and the kinetic parameter values are provided in Table 2. Ethanol Fermentation Model. Figure 2 shows a simplified pathway model of ethanol fermentation using Saccharomyces cereVisiae. The original mathematical model for this system was first constructed by Galazzo and Bailey.19,20 This model was then transformed to an S-system model within the framework of BST (biochemical systems theory) by Curto
q
∑ f (X, Y)
(8)
Figure 1. Metabolic pathways of a penicillin V fermentation system.
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Table 1. Model Equations of a Penicillin V Fermentation Mass-Balance Equations dX1 dt
dX3 dt
dX5 dt
dX7 dt
dX9 dt
dX11 dt
dX13 dt
) V1 - V2 - V9 - V12X1
(10)
) (V3 - V4 - V6 + V8)FcX13
(12)
) V9FcX13
(14)
) V6
(16)
)-
1 PACVS
(18)
1 PAT
(20)
)-
) V12X13 -
X13F X12
{
dX2 dt
dX4 dt
dX6 dt
dX8 dt
dX10 dt
dX12 dt
dX14
(22)
dt
) V2 - V3 - V5 - V10 - V12X2
(11)
) (V4 + V5 - V8)FcX13
(13)
) V10FcX13
(15)
) V7FcX13
(17)
)-
1 PIPNS
(19)
)F
)
(21)
X14F FsF - V12V13X13 X12 X12
Kinetic Equations Y1X9 Y2X1X10[O2] , ν2 ) KCYS KVAL [GSH] KAAA X1 X1 + Km + + 1+ 1+ Ki [AAA] [CYS] [VAL] KACV Y3X2X11 Y4X3X11[PoaCoa] ν3 ) , ν4 ) X2 + Km,IPN X3[PoaCoa] + Km,6APA-Poa[PoaCoa] + Km,PoaX3 Y5X2X11[PoaCoa] ν5 ) , ν6 ) Y6X3 X2[PoaCoa] + Km,IPN-Poa[PoaCoa] + Km,PoaX2 Y7X4X11 ν7 ) ν3, ν8 ) Km,PenV + X4
ν1 )
)(
(
V9 ) Y8(X1 - X5),
ν11 ) m )
mMX14 , X14 + Ksm
ν12 ) µ )
)
(
V10 ) Y9(X2 - X6)
µMX14 , KsX13 + X14
ν13 )
)
(23)
}
ν11 1 1 ) + YX/s Y* ν12 X/s
Table 2. Parameter Values in a Mathematical Model for a Fed-Batch Penicillin V Fermentation parameter
value
parameter
value
KmIPN-Poa KmPoa KmIPN Km6APA-Poa Ki Km KAAA KCYS KVAL KACV KPenV PACVS PIPNS PAT [PoaCoa] [GSH] [O2] [AAA]
0.023 mM 0.006 mM 4.0 mM 0.0093 mM 8.9 mM 0.13 mM 0.63 mM 0.12 mM 0.3 mM 12.5 mM 2.0 mM 615 mM-1h 330 mM-1h 420 mM-1h 0.006 mM 3.0 mM 0.1157 mM 1.2 mM
[CYS] [VAL] Ksm mM Ks µM Y0 Fc SF Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9
0.25 mM 2.2 mM 0.0001 g L-1 0.02 g gcell-1 0.006 g g-1 0.06 h-1 0.13 gcell g-1 0.0024 L mL gcell-1 450 g L-1 17.6 h-1 74.5 h-1 4.03 h-1 1.95 h-1 13.74 h-1 0.2 h-1 0.4 h-1 0.05 h-1 0.03 h-1
Figure 2. Metabolic pathways of an ethanol fermentation system.
(24)
(25)
(26)
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Table 3. Rankings of Enzyme Activities in a Penicillin V Fermentation Using Three Overall Bottleneck Ranking Indicators Yj Lj 4j ranking L4jX4 ranking Lc 4j ranking Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9
Figure 3. Time course of penicillin V concentration in a penicillin V fermentation process.
et al.13 When cell growth and death are taken into consideration, a complete set of the differential equations is obtained as follows dX1 ) 0.8122X2-0.2344Y1 - 2.8632X10.7464X50.0243Y2 dt
(27)
dX2 ) 2.8632X10.7464X50.0243Y2 dt 0.5239X20.739X5-0.394Y30.999Y60.001
(28)
dX3 ) 0.5232X20.7318X5-0.3941Y3 dt 0.0148X30.584X40.03X50.119Y40.944Y70.056Y90.575
(29)
dX4 ) 0.022X30.6159X50.1308Y4Y9-0.6088 dt 0.0945X30.05X40.533X5-0.0822Y5
(30)
dX5 ) 0.0913X30.333X40.266X50.024Y40.5Y50.5Y9-0.304 dt 3.2097X10.198X20.196X50.372Y20.265Y30.265Y60.0002Y80.47 (31) dX6 ) 0.002X30.05X40.533X5-0.0822Y5X7 dt
(32)
dX7 ) 0.04Y1X6-0.4X7 - 0.012X60.7X7 dt
(33)
The initial values of the dependent variables were chosen as X1 ) 0.0345, X2 ) 1.011, X3 ) 9.144, X4 ) 0.0095, X5 ) 1.1278, X6 ) 0.01, and X7 ) 0.0001, and the independent variables were set as Y1 ) 19.7, Y2 ) 68.5, Y3 ) 31.7, Y4 ) 49.9, Y5 ) 3440, Y6 ) 14.31, Y7 ) 203, Y8 ) 25.1, and Y9 ) 0.042. All of these parameter values were based on experiments at extracellular and intracellular pH values of 4.5 and 6.94, respectively.13,19,20 Results and Discussion Penicillin Fermentation Model. Figure 3 shows the time course of the penicillin V concentration in fed-batch penicillin V fermentation. The penicillin V concentration increases almost linearly from the beginning without exhibiting a significant lag phase. Table 3 compares the overall bottleneck ranking indicac 4j (j ) 1, 2, ..., 9), calculated from the j 4j, L4jX4, and L tors, L
0.241 0.4061 -0.00082 0.0450 0.339 -0.00114 -0.0166 -0.00239 -0.00144
3 1 9 4 2 8 5 6 7
9.32 22.2 -0.0895 1.90 13.0 -0.0711 -0.744 -0.110 -0.0586
3 1 7 4 2 8 5 6 9
0.206 0.491 -0.00198 0.0421 0.288 -0.00157 -0.0165 -0.00243 -0.0013
3 1 7 4 2 8 5 6 9
time courses of the logarithmic gains, L4j (j ) 1, 2, ..., 9), and the metabolite concentration X4 in the same fermentation. The enzyme activities, Yj (j ) 1, 2, ..., 9), were ranked according to magnitudes of the overall BR indicator values. The rankings of c 4j (j ) 1, 2, ..., 9) are theoretically the Yj based on L4jX4 and L same. Because L4jX4 is not a normalized value, the value of this overall indicator is much larger than the values of the other c 4j is the value of two indicators. On the other hand, because L L4jX4 divided by an integrated value of X4, the magnitude of its j 4j. Therefore, the rankings of Yj must value is on the order of L c 4j (or L4jX4). It is clear that the j 4j and L be compared between L rankings of Yj for j ) 1-6 are completely the same. The sensitivity for Y2 is the highest, followed by those for Y5, Y1, j 4j and Y4. The rankings of Yj (j ) 7-9) are different between L c 4j (or L4jX4). However, because L j 4j, L4jX4, and L c 4j (j ) and L 7-9) are so small that the relevant enzymes cannot significantly influence the penicillin V production, one can neglect this difference in the rankings. According to the rankings of Yj for the overall BR indicators, it is possible to identify Y2 as the most likely bottleneck enzyme of this fermentation system. Figure 4 compares the time courses of the instantaneous bottleneck ranking indicators, L4j and L4jX4 (j ) 2 and 5), for Y2 and Y5 which are ranked in first and second positions, respectively. Each value of L4j tends to become large in the initial stage where the penicillin concentration is low. However, its period is short. In addition, the tendency for an increase or decrease in its value is fundamentally similar to that in the value of L4jX4. Consequently, the overall BR indicators obtained from the integrations of L4j and L4jX4 are less influenced by the high sensitivity in the initial stage. As indicated later using the calculated results for ethanol j ij and L c ij (or L j ijXi) in the fermentation, the rankings of Yj for L penicillin V fermentation might be theoretically different. However, the calculated result for this fermentation provided almost the same rankings. This can be explained as follows: The penicillin V concentration has a short lag phase and increases almost linearly in the initial stage (Figure 3). Therefore, throughout the entire fermentation period, Lij is always calculated using a sufficiently large value of the penicillin V concentration, j ij so that the calculated value varies in the neighborhood of L without deviating much from its time-averaged value (Figure c ij. j ij are almost on the order of L 4). As a result, the values of L Ethanol Fermentation Model. Figure 5 shows the time course of ethanol concentration in the investigated ethanol fermentation process. The ethanol concentration is very low until 3 h, at which time the cell concentration starts increasing quickly. Table 4 lists the overall bottleneck ranking indicators, c 6j (j ) 1, 2, ..., 9), calculated from the time j 6j, L6jX6, and L L courses of the logarithmic gains, L6j (j ) 1, 2, ..., 9), and the metabolite concentration, X6, in the same fermentation. The enzyme activities, Yj (j ) 1, 2, ..., 9), were ranked according to the magnitudes of the overall indicator values. As for the calculated results of the penicillin V fermentation, the rankings
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Figure 4. Time courses of instantaneous bottleneck ranking indicators L4j (left column) and L4jX4 (right column) for j ) 2 (upper line) and 5 (lower line) in a penicillin V fermentation process.
Figure 5. Time course of ethanol concentration in an ethanol fermentation process. Table 4. Rankings of Enzyme Activities in an Ethanol Fermentation Using Three Overall Bottleneck Ranking Indicators Y Lj 6j ranking L6jX6 ranking Lc 6j ranking Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9
2.24 -6.97 × 10-5 0.0200 -0.0177 0.004123 -5.51 × 10-5 -0.004119 0.0104 0.0108
1 8 2 3 6 9 7 5 4
84.8 -0.00202 0.275 -0.529 0.0584 -0.000795 -0.05835 0.185 0.322
1 8 4 2 6 9 7 5 3
1.68 -4.01 × 10-5 0.00545 -0.0105 0.00116 -1.57 × 10-5 -0.00116 0.00367 0.00639
1 8 4 2 6 9 7 5 3
c 6j (j ) 1, 2, ..., 9) are the same. Therefore, of Yj for L6jX6 and L j 6j and L c 6j (or the rankings of Yj must be compared between L L6jX6).
j 6j and L c 6j (or L6jX6) values rank Y1 in the first Both the L position, indicating that this enzyme is the most likely bottleneck. It should be note that Y1 is the activity of the enzyme associated with the flux to supply the substrate from the outside to the inside of the cells through the cell membrane, and this type of sensitivity is always high.21-23 The ranking based on j 6j provides the order Y3, Y4, Y9, and Y8 after Y1, whereas that L c 6j (or L6jX6) gives Y4, Y9, Y3, and Y8. However, Y8 based on L and Y9 (ATPase and NAD+, respectively) are excluded from being bottleneck candidates because these are the factors affecting plural parts. j 6j and L c 6j (or L6jX6) The reason for the different rankings of L j is that L6j is the integrated value of the changed quantity of X6 c 6j (or L6jX6) is the divided by X6 at each time point, whereas L integrated value of the changed quantity of X6. This is clear from Figure 6, where the time courses of L6j and L6jX6 (j ) 3 and 4) for Y3 and Y4 are compared. As described above, the ethanol concentration is very low until 3 h and then rapidly increases (Figure 5). However, both L63 and L64 present their respective maxima in the first 3 h of the fermentation (Figure 6). This is because, when the system is in its lag phase, the ethanol concentration, X6, is very low, and the use of this value to divide the changed value of X6 gives j 63 and a large logarithmic gain. Clearly, the integrated values, L jL64, from the time courses of logarithmic gains do not reflect any substantial change in X6 as a result of perturbation in the region of high ethanol concentration. On the other hand, L63X6 and L64X6 are small in the lag phase and exhibit their respective maxima at around 12 h, where the ethanol concentration has c 63 and L c 64 (or L63X6 and been sufficiently increased. Because L L64X6, respectively) are calculated using the time-transient values
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Figure 6. Time courses of instantaneous bottleneck ranking indicators L6j (left column) and L6jX6 (right column) for j ) 3 (upper line) and 4 (lower line) in an ethanol fermentation process.
of L63X6 and L64X6, they can reflect a substantial change in the ethanol concentration as a result of perturbation in the region c 6j (or L6jX6) gives a of high ethanol concentration. Therefore, L j better ranking indicator than L6j. On the other hand, the penicillin V fermentation has a short lag phase and a sufficiently high product concentration throughout the entire fermentation period. c ij. j ij or L Consequently, it becomes possible to use either L As in the penicillin V fermentation system, the values of L6jX6 in the ethanol fermentation system (Table 4) are not normalized and can result in large values in comparison with the values of c 6j is the value of L6jX6 the other two overall indicators. Because L divided by the integrated value of X6, the magnitude of the j 6j. resulting value is reduced to the order of the magnitude of L c ij is a useful indicator for the overall It is thus clear that L bottleneck ranking when Yj is ranked. Alternatively, when the bottleneck ranking is considered with respect to time, LijXi is a useful indicator for the instantaneous bottleneck ranking. Validation of Bottleneck Ranking Indicators. Although the bottleneck ranking indicators can be successfully employed in metabolic bottleneck predictions, it is important to test the validity of their predictions by a finite change in each enzyme activity. To successfully perform this test, we changed the enzyme activities by the same finite percentages and then compared the final product concentrations. In this case, the enzyme showing the highest product concentration was identified as a bottleneck enzyme. Table 5 indicates the final penicillin V concentrations and their rankings for 1% and 50% finite changes in enzyme activities. The highest penicillin V concentration is given with an increase in Y2. With a change of more than 1%, Y1 comes second instead of Y5. This can be explained by the time course of the IPN concentration, X2, found in our previous work.7 The
Table 5. Penicillin V Concentrations and Their Rankings for 1% and 50% Finite Changes in Enzyme Activities Y
50% change
ranking
1% change
ranking
normal Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9
93.958 95.98 99.30 93.86 94.16 94.44 94.04 94.14 94.09 94.00
2 1 9 4 3 7 5 6 8
93.958 94.11 94.52 93.96 93.98 94.08 93.96 93.96 93.96 93.95
2 1 8 4 3 7 5 6 9
concentration of the substrate IPN in the Y5-catalyzed reaction decreases with the fermentation time. This means that the efflux is larger than the influx at the X2 pool. Under this condition, therefore, the increase in Y5 accelerates a decline in X2, which does not contribute to the penicillin V production. However, this is a special case because the system is strongly affected by the deactivation of enzymes. Table 6 shows the ethanol concentrations and their rankings for 1% and 50% finite changes in enzyme activities. The results confirm that the rankings of the enzymes for the finite changes are very similar to the ranking of Lc 6j (or L6jX6) in Table 4. Only the 6 and 7 rankings are different between the infinitesimal and finite changes, but they can be regarded as the same because the indicator values have little difference. On the other hand, the ranking of Lj 6j obviously differs from those of the other two. The results obtained here strongly c ij (or LijXi) work support that the rankings of enzymes using L well in practical applications. The results suggest that there is a minor limitation to the determination of a bottleneck enzyme using only the bottleneck
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Table 6. Ethanol Concentrations and Their Rankings for 1% and 50% Finite Changes in Enzyme Activities Y
50% change
ranking
1% change
ranking
normal Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9
107.588 164.650 107.590 107.638 109.816 107.599 107.588 107.607 107.634 107.799
1 8 4 2 7 9 6 5 3
107.588 108.819 107.88802 107.5891 107.592 107.588262 107.58801 107.588265 107.5890 107.590
1 8 4 2 7 9 6 5 3
ranking indicators. It is clear, however, that the indicators can effectively give good predictions that agree to a considerable extent with the ranking of final production concentrations for finite changes in enzyme activities. Conclusions The present work investigated the usefulness of bottleneck ranking indicators for the identification of the most likely bottleneck in a metabolic reaction network. As a result, the following conclusions were obtained: (1) The ranking of the bottleneck enzymes changed during the fermentation period and can be successfully observed using the instantaneous bottleneck ranking indicator given by Li,jXi ) L[Xi(t), Yj]Xi(t)
(i ) 1, 2, ..., n;j ) 1, 2, ..., m)
(2) The ranking of the bottleneck enzymes over the entire fermentation period can be successfully determined using the overall bottleneck ranking indicator given by
∫ L X dt ∫ X dt tf
cij ) L
ij i
0
tf
0
(i ) 1, 2, ..., n;j ) 1, 2, ..., m)
i
Nomenclature Li,j ) logarithmic gain or normalized sensitivity for a response of a metabolite concentration with respect to an infinitesimal change in an enzyme activity at time t defined by eq 1 Lj i,j ) time-averaged dynamic logarithmic gain defined by eq 2 LijXi ) instantaneous bottleneck ranking indicator at time t defined by eq 3 LijXi ) overall bottleneck ranking indicator defined by eq 4 cij ) overall bottleneck ranking indicator defined by eq 5 L Sij ) gain or local sensitivity for a response of a metabolite concentration with respect to an infinitesimal change in an enzyme activity at time t (mM or mM h) Ethanol Fermentation Model X1 ) intracellular glucose concentration (mM) X2 ) G6P concentration (mM) X3 ) FDP concentration (mM) X4 ) PEP concentration (mM) X5 ) ATP concentration (mM) X6 ) ethanol concentration (mM) X7 ) cell concentration (mM) Y1 ) rate constant for extracellular glucose concentration (h-1) Y2 ) rate constant for the reaction Vhk catalyzed by HK (h-1) Y3 ) rate constant for the reaction Vpfk catalyzed by PFK (h-1) Y4 ) rate constant for the reaction Vgapd catalyzed by GAPD (h-1) Y5 ) rate constant for the reaction Vpk catalyzed by PK (h-1) Y6 ) rate constant for the reaction Vpol catalyzed by POL (h-1) Y7 ) rate constant for the reaction Vgol catalyzed by GOL (h-1)
Y8 ) rate constant for the reaction VATPase catalyzed by ATPase (h-1) Y9 ) rate constant for inhibition of NAD+ catalyzed by NAD+ (h-1) AbbreViations ADP ) adenosine diphosphate ATP ) adenosine triphosphate FDP ) fructose-1,6-phosphate G6P ) glucose-6-phosphate GAPD ) glyceraldehyde 3-phosphate dehydrogenase (E.C. 1.2.1.12) D-glyceraldehyde-3-phosphate:NAD+ oxidoreductase GOL ) glycerol production HK ) hexokinase (E.C. 2.7.1.1) ATP:D-hexose-6-phosphotransferase NAD+ ) nicotinamide adenine dinucleotide PEP ) phosphoenolpyruvate PFK ) phosphofructokinase (E.C. 2.7.1.11) ATP:D-fructose-6phosphate 1-phosphotransferase PK ) pyruvate kinase (E.C. 2.7.1.40) ATP:pyruvate O2-phosphotransferase POL ) polysaccharide production (glycogen + trehalose) Penicillin V Fermentation Model F ) feed flow rate (L h-1) KAAA ) saturation constant of ACVS for L-R-aminoadipate (mM) KACV ) inhibition constant of ACVS for ACV (mM) KCYS ) saturation constant of ACVS for L-cysteine (mM) Ki ) inhibition constant of IPNS for glutathione (mM) Km ) saturation constant of IPNS for ACV (mM) Km,6APA-Poa ) saturation constant of AAT for 6-APA with phenoxyacetyl CoA (mM) Km,IPN ) saturation constant of IAH for IPN (mM) Km, IPN-Poa ) saturation constant of IAT for IPN with phenoxyacetyl CoA (mM) Km,PenV ) saturation constant of PA for penicillin V (mM) Km,Poa ) saturation constant of AAT and IAT for phenoxyacetyl CoA (mM) Ks ) saturation constant for the specific growth rate with substrate (g L-1) Ksm ) saturation constant for the maintenance coefficient expression (g L-1) KVAL ) saturation constant of ACVS for L-valine (mM) kACV ) secretion constant of intracellular ACV (h-1) kIPN ) secretion constant of intracellular IPN (h-1) k1 ) rate constant for the reaction 1 catalyzed by ACVS (h-1) k2 ) rate constant for the reaction 2 catalyzed by IPNS (h-1) k3 ) rate constant for the reaction 3 catalyzed by IAH (h-1) k4 ) rate constant for the reaction 4 catalyzed by AAT (h-1) k5 ) rate constant for the reaction 5 catalyzed by IAT (h-1) k6 ) rate constant for the reaction 6, carboxylation of 6-APA into 8-HPA in extracellular medium (h-1) k7 ) rate constant for the reaction 7 catalyzed by PA (h-1) M ) number of independent variables m ) maintenance coefficient (g g-1 h-1) mM ) maximum maintenance coefficient (g g-1 h-1) N ) number of dependent variables PACVS ) linear decay constant for ACVS activity (mM-1 h-1) PAT ) linear decay constant for acyltransferase activity (mM-1 h-1) PIPNS ) linear decay constant for IPNS activity (mM-1 h-1) s ) substrate, extracellular glucose concentration (g L-1) SF ) substrate concentration in the feed stream (g L-1) t ) time (h) tf ) time for the entire fermentation period (h) Vi ) local flux of the ith pathway (mM h-1) XACVS ) ACVS concentration (mM)
Ind. Eng. Chem. Res., Vol. 49, No. 5, 2010 XAT ) AT concentration (mM) XF ) concentration of a desired metabolite (mM) XIPNS ) IPNS concentration (mM) X1 ) ACV concentration (mM) X2 ) IPN concentration (mM) X3 ) 6APA concentration (mM) X4 ) PenV concentration (mM) X5 ) extracellular concentration of ACV (mM) X6 ) extracellular concentration of IPN (mM) X7 ) 8HPA concentration (mM) X8 ) OPC concentration (mM) X9 ) ACVS concentration (mM) X10 ) IPNS concentration (mM) X11 ) AT concentration (mM) X12 ) volume (L) X13 ) cell concentration (g L-1) X14 ) extracellular glucose concentration (g L-1) x ) cell concentration (g L-1) YX/S ) growth yield (gcell g-1) Y*X/S ) true growth yield (gcell g-1) Y1 ) rate constant for the reaction 1 catalyzed by ACVS (h-1) Y2 ) rate constant for the reaction 2 catalyzed by IPNS (h-1) Y3 ) rate constant for the reaction 3 catalyzed by IAH (h-1) Y4 ) rate constant for the reaction 4 catalyzed by AAT (h-1) Y5 ) rate constant for the reaction 5 catalyzed by IAT (h-1) Y6 ) rate constant for the reaction 6, carboxylation of 6-APA into 8-HPA in extracellular medium (h-1) Y7 ) rate constant for the reaction 7 catalyzed by PA (h-1) Y8 ) secretion constant of intracellular ACV (h-1) Y9 ) secretion constant of intracellular IPN (h-1) Greek Symbols µ ) specific growth rate (h-1) µM ) maximum specific growth rate (h-1) Fc ) cell specific volume (mL gcell-1) AbbreViations AAA ) R-aminoadipic acid ACV ) L-R-aminoadipyl-L-cysteinyl-D-valine ACVout ) extracellular L-R-aminoadipyl-L-cysteinyl-D-valine ACVS ) L-R-aminoadipyl-L-cysteinyl-D-valine synthetase AT ) acyltransferase CYS ) L-cysteine GSH ) glutathione HPA ) 8-hydroxypenillic acid IAH ) isopenicillin N aminohydrolase IAT ) acetyl CaA to isopenicillin N acyltransferase IPN ) isopenicillin N IPNout ) extracellular isopenicillin N IPNS ) isopenicillin N synthetase O2 ) dissolved oxygen OPC ) 6-oxopiperidine-2-carboxylic acid PA ) penicillin amidase PenV ) penicillin V PoaCoa ) phenoxyacetyl CoA VAL ) L-valine 6APA ) 6-aminopenicillanic acid 8HPA ) 8-hydroxypenillic acid [ ] ) metabolite concentration (mM)
Literature Cited (1) Hilge-Rotmann, B.; Rehm, H.-J. Relationship between fermentation capability and fatty acid composition of free and immobilized Saccharomyces cereVisiae. Appl. Microbiol. Biotechnol. 1991, 34, 502–508.
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(2) Ingram, L. O.; Gomez, P. F.; Lai, X.; Moniruzzaman, M.; Wood, B. E.; Yomano, L. P.; York, S. W. Metabolic engineering of bacteria for ethanol production. Biotechnol. Bioeng. 1997, 58 (2&3), 204–214. (3) Lei, F.; Olsson, L.; Jorgensen, S. B. Experimental investigations of multiple steady-states in aerobic continuous cultivations of Saccharomyces cereVisiae. Biotechnol. Bioeng. 2003, 87 (7), 766–776. (4) Nissen, T. L.; Kielland-Brandt, M. C.; Nielsen, J.; Villadsen, J. Optimization of ethanol production in Saccharomyces cereVisiae by metabolic engineering of the ammonium assimilation. Metab. Eng. 2000, 2, 69–77. (5) Westerhoff, H. V.; Kell, D. B. Matrix method for determining steps most rate-limiting to metabolic fluxes in biotechnological processes. Biotechnol. Bioeng. 1987, 30, 101–107. (6) Conejeros, R.; Vassiliadis, V. S. Analysis and optimization of biochemical process reaction pathways. 1. Pathway sensitivities and identification of limiting steps. Ind. Eng. Chem. Res. 1998, 37, 4699–4708. (7) Shiraishi, F.; Suzuki, Y. Method for determination of the main bottleneck enzyme in a metabolic reaction network by dynamic sensitivity analysis. Ind. Eng. Chem. Res. 2009, 48, 415–423. (8) Jorgensen, H.; Nielsen, J.; Villadsen, J.; Mollgaad, H. Analysis of penicillin V biosynthesis during fed-batch cultivations with a high-yielding strain of Penicillium chrysogenum. Appl. Microbiol. Biotechnol. 1995, 43, 123–130. (9) Nielsen, J.; Jorgensen, H. S. Metabolic control analysis of the penicillin biosynthetic pathway in a high-yielding strain of Penicillium chrysogenum. Biotechnol. Prog. 1995, 11, 299–305. (10) Nielsen, J.; Jorgensen, H. S. A kinetic model for the penicillin biosynthetic pathway in Penicillium chrysogenum. Control Eng. Pract. 1996, 4 (6), 765–771. (11) de Noronha Pissara, P.; Nielsen, J.; Bazin, M. J. Pathway kinetics and metabolic control analysis of a high-yielding strain of Penicillium chrysogenum during fed-batch cultivations. Biotechnol. Bioeng. 1996, 51, 168–176. (12) Shiraishi, F.; Suzuki, Y. Method for Determination of the Main Bottleneck Enzyme in a Metabolic Reaction Network by Dynamic Sensitivity Analysis. Ind. Eng. Chem. Res. 2009, 48, 415–423. (13) Curto, R.; Sorribas, A.; Cascante, M. Comparative characterization of the fermentation pathway of Saccharomyces cereVisiae using biochemical systems theory and metabolic control analysis: Model definition and nomenclature. Math. Biosci. 1995, 130, 25–50. (14) Shiraishi, F.; Hatoh, Y.; Irie, T. An efficient method for calculation of dynamic logarithmic gains in biochemical systems theory. J. Theor. Biol. 2005, 234, 79–85. (15) Shiraishi, F.; Tomita, T.; Iwata, M.; Berrada, A. A. A reliable Taylor series-based computational method for the calculation of dynamic sensitivities in large-scale metabolic reaction systems: Algorithmic and software evaluation. Math. Biosci. 2009, 222, 73–85. (16) Shiraishi, F.; Tomita, T.; Hirayama, H. A rapid and highly-reliable method for calculation of dynamic sensitivities in a large-scale metabolic reaction system. In Proceedings of the 9th International Conference on Molecular Systems Biology (ICMSB-2006); Munchen, 2006. (17) Atkinson, K. Elementary Numerical Analysis, 2nd ed.; John Wiley & Sons, Inc.: New York, 1993; p 425. (18) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes: The Art of Scientific Computing, 3rd ed.; Cambridge University Press: New York, 2007; p 1235. (19) Galazzo, J. L.; Bailey, J. E. Fermentation pathway kinetics and metabolic flux control in suspended and immobilized Saccharomyces cereVisiae. Enzyme Microb. Technol. 1990, 12, 162–172. (20) Galazzo, J. L.; Bailey, J. E. Fermentation pathway kinetics and metabolic flux control in suspended and immobilized Saccharomyces cereVisiae, Errata. Enzyme Microb. Technol. 1991, 13, 363. (21) Cascante, M.; Curto, R.; Sorribas, A. Comparative characterization of the fermentation pathway of Saccharomyces cereVisiae using Biochemical Systems Theory and Metabolic Control Analysis: Steady-state analysis. Math. Biosci. 1995, 130, 51–69. (22) Torres, N. V.; Voit, E. O. Pathway Analysis and Optimization in Metabolic Engineering; Cambridge University Press: Cambridge, U.K., 2002. (23) Voit, E. O. Computational Analysis of Biochemical Systems: A Practical Guide for Biochemists and Molecular Biologists; Cambridge University Press: Cambridge, U.K., 2000; p 531.
ReceiVed for reView September 29, 2009 ReVised manuscript receiVed November 24, 2009 Accepted January 6, 2010 IE901531D
