Selection of Best Indicators for Ranking and Determination of

Sep 17, 2010 - at a given time is the best instantaneous BR indicator, whereas its value at the end time of the fermentation process is the best overa...
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Ind. Eng. Chem. Res. 2010, 49, 9738–9742

Selection of Best Indicators for Ranking and Determination of Bottleneck Enzymes in Metabolic Reaction Systems Kansuporn Sriyudthsak and Fumihide Shiraishi* Department of Bio-System Design, Bio-Architecture Center, Kyushu UniVersity, Hakozaki, Fukuoka 812-8581, Japan

Dynamic logarithmic gain and its modifications can be theoretically used as a bottleneck ranking (BR) indicator in a metabolic reaction system. However, it is not sufficiently explicit whether they can be successfully used in practical applications. The present work therefore focuses on the selection of best BR indicators in both instantaneous and overall cases. A modified ethanol fermentation model is used as a case study. The results indicate that the mathematical product of dynamic logarithmic gain and desired product concentration at a given time is the best instantaneous BR indicator, whereas its value at the end time of the fermentation process is the best overall BR indicator. The former is useful for observing the time course of the desired product concentration and determining a process time to be terminated. The latter is for ranking bottleneck enzymes to determine which enzyme activity should be changed to attain a higher desired product concentration. Moreover, discussion is made on the utilization of the overall BR indicators to predict how much the final desired product concentration is increased. Introduction A recent development of high-performance analytical instruments has made it possible to comprehensively determine timevarying metabolite concentrations inside cells,1-3 although the accuracy of measurement still needs to be successively improved in the future. The time-varying concentration data, in turn, enable us to construct a more accurate mathematical model for an in vivo enzyme reaction network system. These mathematical models can be used to efficiently identify bottleneck enzymes in a metabolic reaction network and then optimize a fermentation process to increase the productivity of a desired metabolite. We have discussed the performances of several indicators for ranking enzymes in order to identify bottleneck enzymes in metabolic reaction networks.4-6 If a system is in steady state, dynamic logarithmic gains, namely, normalized sensitivities, can be directly used as a bottleneck ranking (BR) indicator. On the other hand, when the reaction condition is varied with time, like fed-batch fermentations, the bottleneck enzyme must be identified from the metabolite concentrations changed throughout the entire fermentation period, because the ranking of bottleneck enzymes changes with time.5-7 In this case, it is important to take into consideration both instantaneous BR indicators that rank enzymes at a certain time and overall BR indicators that rank enzymes over the entire period of time. We previously ranked and identified bottleneck enzymes in the penicillin V and ethanol fermentation systems5,6 using dynamic logarithmic gain and its modifications. Here, the dynamic logarithmic gain Lij expresses the percentage change in the metabolite concentration Xi at time t in response to an infinitesimal percentage change in the enzyme activity Yj at time zero. Also, its modification LijXi represents the product of dynamic logarithmic gain and desired product concentration and is equivalent to the time-varying semirelative sensitivity. The results revealed that two instantaneous BR indicators, including the dynamic logarithmic gain Lij and the mathematical product of dynamic logarithmic gain and desired product concentration LijXi, were effective in the penicillin V fermentation system. In the ethanol fermentation system with a long lag phase, on the * To whom correspondence should be addressed. Tel.: +81-92-6427603. Fax: +81-92-642-7603. E-mail: [email protected].

other hand, Lij could not correctly rank bottleneck enzymes but c ij, which is the LijXi was effective. The overall BR indicator L integrated value of LijXi (denoted as LijXi) divided by an integrated value of the desired product concentration Xi (denoted j i), was successfully used in both systems.6 as X The main objectives of the present work are to determine the best instantaneous and overall BR indicators from the BR indicator candidates including other new overall BR indicators, which are Lij and LijXi at the end time of the fermentation process, and to investigate the prediction of the desired product concentration increased by a finite change in an enzyme activity by use of the values of Lij and LijXi at the end time. The ethanol fermentation model5,8-10 will be modified to use as a case study, because this model has a long lag phase where a low desired product concentration may give misleading information in the determination of bottleneck enzymes using the BR indicators. Consequently, the modified model will be used to examine the performances of the BR indicators. Theory Instantaneous and Overall BR Indicators. The dynamic logarithmic gain, i.e., normalized sensitivity, expresses the percentage change in a dependent variable at any time t in response to an infinitesimal percentage change in an independent variable at time zero,11-13 written as Li,j ) L(Xi(t), Yj) )

∂ln Xi(t) ∂Xi(t) Yj (i ) 1, 2,...,n; ) ∂ln Yj ∂Yj Xi(t) j ) 1, 2,...,m)

(1)

where Xi and Yj are the dependent and independent variables, respectively. The instantaneous BR indicators include Lij and LijXi, whereas the overall BR indicators include the time-averaged values of Lij and LijXi over the entire fermentation period of time (denoted j ij and L X respectively) and L X divided by the timeas L ij i ij i j i (denoted average value of the desired product concentration X 5,6 c as Lij). In addition, the present work introduces the logarithmic gains at the end time of the fermentation process Lij|t)end and its modification LijXi|t)end as alternative overall BR indicators.

10.1021/ie100911h  2010 American Chemical Society Published on Web 09/17/2010

Ind. Eng. Chem. Res., Vol. 49, No. 20, 2010

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Table 1. Instantaneous and Overall Bottleneck Ranking Indicators instantaneous BR indicators ∂Xi(t) Yj ∂Yj Xi(t)

1

Lij )

2

LijXi )

∂Xi(t) Yj X (t) ∂Yj Xi(t) i

overall BR indicators j ij ) 1 L tf

∫L

LijXi )

1 tf

tf

ij

0

dt

∫ L X dt tf

0

ij i

∫ L X dt ) ∫ X dt tf

3

cij ) L

LijXi ¯ X i

ij i

0

tf

0

4

Lij | t)end )

5

LijXi | t)end )

i

∂Xi(t) Yj ∂Yj Xi(t) t)end ∂Xi(t) Yj X (t) ∂Yj Xi(t) i t)end

Figure 1. Metabolic pathways in an ethanol fermentation model. Table 2. Initial Values and Modified Parametersa

Because these two indicators theoretically provide the same enzyme rankings, the following discussion will consider only LijXi|t)end. All of these BR indicators are presented in Table 1. Prediction of Final Concentration of Desired Metabolite. Calculation of the overall BR indicator is based on the infinitesimal change in an enzyme activity. In practice, however, it is impossible to infinitesimally change an enzyme activity. One therefore needs to confirm whether the enzyme ranking based on the overall BR indicators is identical to that based on finite changes in enzyme activities. If these are identical, one needs to clarify to what extent of the finite change in the enzyme activity the final desired product concentration calculated using the overall BR indicator is identical to that calculated from finite changes in enzyme activities. As stated later, the use of either Lij|t)end or LijXi|t)end is recommended as the overall BR indicator. The equations for prediction of the final desired product concentration using the two BR indicators are given as Xfinal ) X*i + L · X*i P i 100 Xfinal ) X*i + LijXi | t)end i

P 100

X1 ) 0.0345 X2 X3 X4 X5 X6 X7

) ) ) ) ) )

1.011 9.144 0.0095 1.1278 0.01 0.0001

Y1 ) 19.7 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9

) ) ) ) ) ) ) )

68.5 31.7 49.9 3440.0 14.31 230.0 25.1 0.042

R1 ) 0.01354

β1 ) 0.04772

R2 R3 R4 R5 R6 R7

β2 ) 0.00873 β3 ) 0.00025 β4 ) 0.00158 β5 ) 0.05350 β7 ) 0.012 g76 ) -0.7 h76 ) 0.6 g77 ) 1.0

) ) ) ) ) )

0.04772 0.00872 0.00037 0.00152 0.002 0.045

a The symbols have the following meanings: Xi, the metabolite concentration [mmol/L-cell]; X6, the cell concentration [g/L-liq]; X7, the ethanol concentration [g/L-liq]; Yj, the enzyme activity [mmol/min]; Ri and βi, the rate constants for overall incoming and outgoing fluxes, respectively; and gij and hij, the kinetic orders for overall incoming and outgoing fluxes, respectively.

(2)

(3)

is the final desired product concentration, X*is the where Xfinal i i desired product concentration under a normal condition without any change in the enzyme activity, L is the overall BR indicator c ij, or L j ij), and P is the percentage of change in an (Lij|t)end, L calculated from Lij|t)end in enzyme activity. The value of Xfinal i eq 2 is theoretically the same as that calculated from LijXi|t)end in eq 3. Modified Ethanol Fermentation Model. Figure 1 illustrates the metabolic pathways in the simplified ethanol fermentation model.5,6 The relevant differential equations and parameter values are given elsewhere.5,6 The present work uses a partially modified set of their parameter values to facilitate meaningful discussion. The modified parameter values and initial values for the metabolite concentrations are listed in Table 2. In the following, the ethanol fermentation model with modified parameter values is called the modified ethanol fermentation model. Figure 2 shows the time course of ethanol concentration in the modified ethanol fermentation model under a normal

Figure 2. Time course of ethanol concentrations in an ethanol fermentation model with modified parameters under a normal condition.

condition. The ethanol concentration increases very slowly during the lag phase and then starts rising rapidly. Results and Discussion Instantaneous BR Indicators. Of the two kinds of instantaneous BR indicators presented in Table 1, the logarithmic gain Lij can be successfully used to rank enzymes at a specified time point, but it cannot be used to make a judgment of when the desired product concentration is most affected by a change in each enzyme activity at t ) 0. This is because Lij may give a

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Figure 3. Time courses of two kinds of instantaneous BR indicators. c ij, L j ij, and L X Table 3. Values and Rankings of Lij|t)end, L ij i c j j ijXi Yi LijXi|t)end Lij Lij L Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9

84.83584 0.26667 4.19445 -6.99927 0.52379 -0.00602 -0.58500 -1.56614 4.26555

1 8 4 2 7 9 6 5 3

1.16988 0.00343 0.03927 -0.01576 0.00490 -0.00006 -0.00573 -0.03876 0.00966

1 8 2 4 7 9 6 3 5

1.36173 0.00289 0.02864 0.02606 0.00376 -0.00005 -0.00456 -0.04216 -0.01580

1 8 3 4 7 9 6 2 5

57.82530 0.16967 1.94108 -0.77910 0.24199 -0.00284 -0.28327 -1.91575 0.47743

1 8 2 4 7 9 6 3 5

large value when the desired product concentration is low. For instance, Lij was able to successfully rank bottleneck enzymes in the penicillin V fermentation model4,5 where the desired product concentration monotonically increased from the beginning. On the other hand, it gave a partially incorrect enzyme ranking in the ethanol fermentation model6 where several enzyme activities were sensitive to the ethanol concentration during the long lag phase as seen in Figure 2. In contrast, LijXi does not have such a difficulty because it is not dimensionless with respect to the metabolite concentration. Although this fact may prompt us to use the local sensitivity Sij ) ∂Xi/∂Yj as an instantaneous indicator, the use of this indicator is not recommended because the local sensitivity is not normalized with respect to the enzyme activity. We thus consider that LijXi is the best instantaneous BR indicator,6 which correctly gives information on when the desired product concentration is most affected by a change in each enzyme activity at t ) 0. Overall BR Indicators. The previous publications5,6 indicated that if the instantaneous BR indicator LijXi changes only c ij can correctly rank in the positive region or negative region, L bottleneck enzymes. When LijXi changes in both the positive and negative regions, on the other hand, it is unclear whether c ij can still give a correct ranking. We will therefore consider L this issue in more detail by introducing alternative overall BR indicators, Lij|t)end and LijXi|t)end. These represent the values of Lij and LijXi at the end time of the fermentation process. To facilitate meaningful discussion, we will use the modified ethanol fermentation model where a part of the parameters in the original ethanol fermentation model have been changed to magnify the degree of variation in LijXi in both the positive and negative regions. Figure 3 illustrates the time-transient behavior of L6j and L6jX6 (j ) 1, ..., 9) in the fermentation period of 0-24 h. Table 3 c 6j, L j 6j, and compares the values and ranking of L6jX6|t)24h, L L6jX6 obtained from Figure 3. The calculated values are given with five digits in order to indicate the presence of differences and to observe values of ranked BR indicators including j 6, it is c 6j is a function of L X and X insignificant ones. Since L 6j 6

c 6j. understandable that the ranking of L6jX6 is equal to that of L It is noticeable that the ranking of L6jX6|t)24h is partially different j 6j. Two reasons explain this c 6j, L X and L from those of L 6j 6 result. One is that several values of L6j become large in the lag phase where the desired product concentration is low (see the left figure in Figure 3), which mistakenly gives higher ranks to relevant enzymes. The other is that LijXi changing in both the positive and negative regions gives almost the same integrated values (see the right figure in Figure 3). For example, L64X6 is positive in the period of 0-13.71 h and then becomes negative after t ) 13.71 h. Similarly, L69X6 is negative in the period of 0-13.83 h and then becomes positive after t ) 13.83 h. In these cases, the integrated values of L64X6 and L69X6 in the positive region are not significantly different from their respective values c 6j obtained in the negative region. As a result, the value of L from the integrated values in the positive and negative regions can no longer rank bottleneck enzymes correctly. It is certain that, when LijXi changes in both the positive and c ij may give a wrong ranking. As demonstrated negative regions, L in the analyses of the penicillin V fermentation model5 and the original ethanol fermentation model,6 however, it seems that the integrated values of LijXi in the positive and negative regions c ij to are usually quite different, which makes it possible to use L rank bottleneck enzymes. In any case, the enzyme ranking based on the values of LijXi|t)end is theoretically correct. Consequently, it is considered that LijXi|t)end is the best overall BR indicator. Furthermore, as far as the enzyme ranking at a specified time point is concerned, Lij|t)end can also be used as the overall BR indicator. Applications of Instantaneous and Overall BR Indicators. The instantaneous BR indicator can be used to investigate which enzyme can greatly contribute to an increase in the desired product concentration at a certain time during the fermentation period. It can also be used to determine the termination time of the fermentation process to obtain a maximum desired product concentration. On the other hand, the overall BR indicator can be used to rank bottleneck enzymes at a specified time point, for example, at the end time of the fermentation process. As stated earlier, the best instantaneous and overall BR indicators are LijXi and LijXi|t)end, respectively. The use of these indicators enables us to determine which enzyme activity should be changed and when the fermentation process should be terminated to maximize the desired product concentration Xk. The identification procedure is as follows. First, calculate LkjXk (j ) 1, 2, ...) from t ) 0 to a certain point and plot the calculated values against the time. Second, select the highest value LkmXk through the entire fermentation period. Thus, LkmXk enables us to elucidate the termination time

Ind. Eng. Chem. Res., Vol. 49, No. 20, 2010 a

Table 4. Normalized Variations of Final Ethanol Concentrations for Changes of 1-75% in Yi Yi

1%

5%

10%

25%

50%

75%

Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9

1.00983 1.00003 1.00048 1.00081 1.00006 1.00000 1.00007 1.00018 1.00049

1.04896 1.00015 1.00235 1.00390 1.00030 1.00000 1.00035 1.00096 1.00233

1.09738 1.00027 1.00455 1.00747 1.00059 1.00001 1.00071 1.00204 1.00438

1.23960 1.00058 1.01034 1.01599 1.00139 1.00002 1.00194 1.00602 1.00926

1.46751 1.00092 1.01797 1.02168 1.00255 1.00005 1.00463 1.01526 1.01452

1.68558 1.00114 1.02387 1.01152 1.00355 1.00010 1.00909 1.02750 1.01764

a The final ethanol concentration takes a value of 86.16360 g/L without any changes in enzyme activities. Each value was calculated by dividing the final ethanol concentration after a finite change in an enzyme activity by the value 86.16360.

of the fermentation process and the enzyme activity Ym that should be changed to maximize Xk. The identification of a bottleneck enzyme using BR indicators may be one of the optimization methods because the BR indicators can reasonably predict which enzyme activity should be changed to increase the productivity of the desired product. The use of BR indicators enables us to quickly identify the bottleneck enzyme in a straightforward manner. Unlike the general optimization methods, however, our method is not based on simultaneous changes in plural enzyme activities. Therefore, the method may have to be extended in the future. Enzyme Rankings and Final Desired Product Concentrations with Infinitesimal and Finite Changes in Enzyme Activities. Once the end time of a given fermentation process is determined, the ranking of bottleneck enzymes based on LijXi|t)end or Lij|t)end elucidates which enzyme activity should be

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changed to maximize the final desired product concentration. However, it should be noted that this ranking is based on increases in the desired product concentration in response to infinitesimal changes in enzyme activities, although the activity of the enzyme identified as a bottleneck can be changed only finitely. It is thus meaningful to confirm to what extent the ranking of the final desired product concentrations based on the values of LijXi|t)end or Lij|t)end is identical to that of the final desired product concentrations calculated by finite changes in the corresponding enzyme activities. Table 4 lists the normalized variation of final ethanol concentrations calculated from finite changes of 1, 5, 10, 25, 50, and 75% in Yi (i ) 1, ..., 9). In the present ethanol fermentation model, changes in several of the enzyme activities hardly have an impact on the final ethanol concentration. Again, the calculated values in the list are therefore given with five digits in order to indicate the presence of differences. The final ethanol concentration fundamentally becomes larger as the enzyme activity changes more highly. It is clear that there is no change in the ranking of bottleneck enzymes even when Yi is changed at least until 50%, and this ranking is equal to that based on LijXi|t)end (see Table 3). The result clearly shows that, if the result of the enzyme ranking based on LijXi|t)end is used, the bottleneck enzyme can be determined without finitely changing the enzyme activities to check the validity of the ranking. When applying the values of Lij|t)end and LijXi|t)end to eqs 2 and 3, respectively, one can approximately calculate the final desired product concentration for a finite change of P % in each enzyme activity. It is important to investigate to what extent of the changes in the enzyme activities the final desired product

Figure 4. Comparison between final ethanol concentrations approximately calculated from eq 3 using LijXi|t)end and exactly calculated from a mathematical model with finite percentage changes in Y1, Y4, Y9, and Y3.

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concentration calculated from eq 2 or 3 is identical to that obtained from the mathematical model with finite changes in enzyme activities. In experiments, it is difficult to investigate the effects of finite changes in all enzyme activities on the final product concentration. As a result of investigation, we have confirmed that the identification of a bottleneck enzyme based on the ranking of BR indicators is an appropriate means. Also, we have clarified already that the ranking of final product concentrations based on the BR indicators is identical to that based on finite changes in enzyme activities in a certain range. Figure 4 compares the final desired metabolite concentrations calculated from eq 3 using the values of LijXi|t)end with those calculated from the mathematical model for the finite changes of 0-75% in the top four ranking enzyme activities (application of Lij|t)end to eq 2 also provides the same results). When Y1 is changed, the desired product concentration calculated from the mathematical model increases linearly with the increase in P. This calculated value is in good agreement with the value approximately calculated by eq 3 even when Y1 is changed up to 75%. When Y3, Y4, and Y9 are changed, on the other hand, the desired product concentrations calculated from the mathematical model slightly decrease the percentage of their increase for changes of >25% in P. For this reason, the calculated values by eq 3, increasing linearly with the increase in P, deviate from the values calculated from the mathematical model, although their differences are not remarkable because the extent of change in the final desired metabolite concentration is substantially small. As a result, it is concluded that, in the present ethanol fermentation model, eq 3 can perfectly predict the final ethanol concentration until 10% change in the enzyme activity and can be effectively used until 25% change. Conclusions The best instantaneous and overall BR indicators are LijXi and LijXi|t)end, respectively. The instantaneous BR indicator LijXi can be effectively used to observe the behavior of a system and make a judgment of where to terminate a given fermentation process. The overall BR indicator LijXi|t)end can be successfully used to rank bottleneck enzymes. Also, the desired product concentration in the present ethanol fermentation model can be perfectly predicted using the values of LijXi|t)end up to a change of 10% in the enzyme activity. From the observation of the time course of LijXi, one can determine the termination time of the fermentation process and which enzyme activity should be changed to maximize the desired metabolite concentration. 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 in Table 1

LijXi ) instantaneous bottleneck ranking indicator at time t, defined in Table 1 Lj i,j ) time-averaged dynamic logarithmic gain, defined in Table 1 LijXi ) overall bottleneck ranking indicator, defined in Table 1 c ij ) overall bottleneck ranking indicator, defined in Table 1 L Lij|t)end ) overall bottleneck ranking indicator, defined in Table 1 LijXi|t)end ) overall bottleneck ranking indicator, defined in Table 1 Xifinal ) predicted final desired product concentration X*i ) product concentration at normal condition P ) percentage of change in an enzyme activity

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ReceiVed for reView April 19, 2010 ReVised manuscript receiVed July 26, 2010 Accepted August 31, 2010 IE100911H