Biotechnol. Pmg. 1992, 8, 81-84
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Use of Various Measurements for Biomass Estimation T. Chattaway,t A. L. DemainJ and Gregory Stephanopoulos* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Electron, carbon, and nitrogen balances can be thought of as relationships among the time rates of change of the various compounds participating in a fermentation process. As such, they define the minimum number of necessary measurements from which the remaining rates can be determined through the use of the balances. All possibilities, however, are not equivalent, and some of them lead to singularities and solutions of high sensitivity. These possibilities are reviewed in this paper, and suggestions are offered in regard t o the combination of rate measurements from which robust biomass estimates can be produced.
Introduction Abundant literature exists on the application of mass balancing to fermentation data (1). Applications include yield and efficiency calculations (2,3),biomass estimation (4), and data reconciliation (5). Though sensitivity problems are known to occur in certain cases (61,no systematic study of the various combinations of measurements and balances has been made. Furthermore, although mass balancing has also been applied to fermentations on complex media (7), little is known about the validity of the approach in this case. In the case of biomass estimation, elemental balances are useful either for obtaining on-line estimates or for calculating biomass off-line, when a direct measurement is difficult, during fermentations on solid-containing media, for example. Since combining balances allows certain variables to be eliminated, it is necessary to investigate which combinations are likely to yield the best results. Though the number of such possibilities is rather large, it will be shown how this reduces to only a few usable solutions. For this, a systematic study of the various possibilities for estimating biomass is conducted and the practicalities are discussed. Sensitivity problems are classified, and solutions for alleviating them are proposed. The study theoretically is illustrated with data from a lysine fermentation on a complex medium, using an industrial strain. I t also provides insight into the possible sensitivity problems occurring when elemental balances are used for data reconciliation.
Theory All mass-balancing techniques are based on a description of the fermentation process by an equation of the following type, valid for any aerobic fermentation: a(substrates)
+ b o , + cNH, * d(biomass) + eCO, + fHzO + g(products) (1)
In eq 1, coefficients a-g are the unknowns. Substrates, including precursors, and products can be multiple (in which case coefficients a and g are the vectors a and g),
* To whom correspondence should be addressed.
+ Present address: IC1 Bioproducta and Fine Chemicals, Billingham, U.K. Present address: Department of Biology, MIT, Cambridge, MA
*
02139. 8756-7938/92/3008-0081$03.00/0
and the inorganic nitrogen source need not be ammonia. Under the assumption of constant and known compositions, the main elements (C, H, 0,and N) can be balanced over compounds occurring in significant quantities. After elimination off (since water can never be measured), this provides three independent equations. Quite equivalently, electron (or degree of reductance (311,carbon, and nitrogen balances can be written. The degree of reductance r of a compound of composition C,HaO,Na is defined as 1 r = -(4a + @ - 27 - 36) a
(2)
If the inorganic nitrogen source is not ammonia, a generalized degree of reductance is defined such that the inorganic nitrogen uptake does not contribute to the electron balance (3). Using this second set of equations allows the technique to be easily applied to complex media (7), where the composition of the substrates is ill-known but their degree of reductance can be obtained from their heat of combustion (2). Indeed, the two are almost proportional for biological compounds. Because of this regularity, the energy (or enthalpy) balance is almost proportional to the electron one, with the metabolic heat replacing oxygen uptake (8). The energy balance does not, therefore, introduce additional information, although it does allow metabolic heat loads to be used if they are measured. It will not be further considered, as its inclusion does not change any of the conclusions on the biomass estimation problem. In the case of complex media, substrates such as carbohydrates or organic nitrogen sources can be lumped (7), but the latter are most often impossible to measure accurately. It should be stressed that because the degrees of reductance and nitrogen contents of both the biomass and the organic nitrogen source are extremely close, it is impossible to estimate the two compounds simultaneously, using mass and energy balances. From now on, it is assumed that the main substrates contain no nitrogen and can be lumped and measured (as total reducing sugars for instance); other substrates are neglected. The validity of this last assumption will be tested on fermentation data. For simplicity, the case of a single product with no precursor is examined. By expressing all quantities in moles (moles of carbon for organic compounds) and introducing the degrees of reductance of substrate, biomass, and product (rB,Q, and rp,respectively (3)) and the nitrogen contents 6 and w of
0 1992 American Chemical Society and American Institute of Chemical Engineers
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biomass and product, respectively, the three available balances are written as -.
,
,.
-
-
- - PURl OUR
Table I. Ten Exact Solutions for Estimating Biomass from Measurements of SUR, OUR, NUR, CER, and PFR, Using Elemental BalancesP ~
~~
measurements not used
SUR
OUR
CER
PFR
NUR
rb - pa
1 C balance
rb balance
rb - rp
L
PFR
In eq 3, SUR through PFR stand for substrate uptake, oxygen uptake, inorganic nitrogen uptake, biomass production, carbon dioxide evolution, and product formation rates, respectively. Since the balance matrix is assumed constant, either instantaneous reaction rates, average ones, or differences in concentrations or quantities may be used with the appropriate units as best suits the problem at hand. The rates in eq 3 can usually be measured off-line (with the possible exception of biomass as stated). Gas rates are usually available on-line. The nitrogen uptake is sometimes available from alkali uptake such as in the case of fermentations on defined media using ammonium as the single nitrogen source and in the absence of acid production. Standard technology does not exist for measuring the other rates on-line, and biomass seems particularly difficult. In view of the need for biomass estimates during the course of a fermentation, the problem of biomass estimation (i.e., its calculation on- or off-line from other measurements) will now be considered. Since the matrix in eq 3 has rank three, measurement of three of the six rates suffices for determination of the others. If more measurements are available, either redundancy is introduced or certain balances can be dropped to arrive again at an exact solution. The solutions containing redundancy are briefly discussed first. If all measurements except biomass are available,biomass can be estimated from the three balances simultaneously, for instance by least squares. This is equivalent to a weighted combination of the balances with the weights chosen in such a way as to normalize all the balances with respect to their biomass coefficient. Alternatively, the weights can be derived from the measurement variances, which is the approach adopted for statistical data reconciliation (5). There are in fact an infinity of possibilities. This obvious disadvantage is a general feature of nonexact solutions. The other disadvantage is that they require more measurements, preferably free of systematic errors. If only four measurements are available, the fifth one (excluding biomass) can always be eliminated from two of the three balances. This leaves two equations to determine biomass, each of which would suffice on its own. A least-squares solution can again be used, with the remarks above also applying. In the case of the exact solutions, eq 3 can be thought of as a link between nine pieces of information (six rates and three balances), one of which (BPR) is unknown. An exact solution is obtained by selecting six known pieces of information out of the eight potentially available and performing the corresponding eliminations. There are, therefore, a priori, 28 (C:) such exact solutions. These can be classified into three groups: three solutions using only one balance (and five rates), 15 solutions using only two balances (and four rates), and 10 solutions using all three balances (and three rates). In the case of eq 3, the first two groups are in fact contained in the last one, as explained below. First, for any one of the three balances there are at least two measurements (neither of which is biomass) which do not appear in it. Hence, the inclusion of the first group in the last one. Second, for any balance
-s 1
rb-r,+ d
0
c
rb-r
-(rs - rp) w
CER OUR
-
-6
pw
N balance if w = 0-
8
6
N balance
N balance
8
8
N balance The solutions are classified by the measurements not used in the resulting equation, whose coefficient in biomass is also given.
there is a rate not appearing in either two of the remaining balances. If this rate is the one already discarded in a solution from the second group, there is redundancy in the calculations; if not, only three rates are effectively being used. Hence, all the solutions from the second group either are contained in the third or are not exact solutions. I t can thus be seen that all the exact solutions are covered by the third group where three rates are employed to estimate biomass and that there are at the most 10 such solutions. These solutions can be nicely tabulated on the basis of the two rates not used in the estimation of biomass: the BPR coefficients that result for each of the 10 cases after due elimination from eq 3 are shown in Table I. From Table I, it is clear that in fact the 10 solutions correspond to only eight different equations, further reducing to only five if the product contains no nitrogen. If the BPR coefficientis ill-known or close to zero, a sensitivity problem exists; this leads to biased estimates or to error amplification, respectively. This coefficient largely determines the quality of the biomass estimate. The remaining discussion further focuses on this point of sensitivity. The six cases where NUR is utilized are examined first (see Table I). Several of these are in fact the nitrogen balance, and all result in it when w = 0. This is undesirable for estimating biomass, because the solution relies heavily on the NUR measurement (which may not be very accurate, especially if it is obtained from alkali uptake and if acids are produced) and on 6 (which may very during a fermentation). Moreover, the nitrogen balance will not close if organic nitrogen sources are not negligible. When the product contains nitrogen (o # 0), out of the six solutions the three that do not reduce to the nitrogen balance usually have a coefficient in BPR close to zero (see Table I). This results from the fact that both pairs (rb,rp) and (6,w) are close in general and are especially so for amino acids or proteins as products. These solutions will not be further considered since they have limited practical use. The remaining four exact solutions, corresponding to the cases when neither the NUR measurement nor the nitrogen balance is used, are examined next (see Table I). Not using either OUR or CER (in addition to not using NUR) obviously leads to the carbon or electron balance, respectively. These do not present a sensitivity problem and are of course widely used. However, they both require SUR and PFR measurements, which may be difficult online. On the other hand, elimination of SUR or PFR leads to difficulties since r b is close to 4.4 (2). In particular, SUR should not be eliminated when substrates are
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Table 11. Quality of Biomass Estimates Obtained Using Various Balances, from Data of a Lysine Fermentation on a Complex Medium. ~~
criterion error norm, g/L max error, g/L final error, g/L
C 22.6 16.4 -4.1
E 25.3 17 2.52
N 31.7 -19 -19
S 195 113 108
S’ 62.4 37.2 32.7
P 118 -72 -71
P’ 48.4 -27 -16
LS 16.3 12.1 -6.3
The Euclidean norm of the vector of errors,the maximum error, and the final one are compared. C, E, and N designate the carbon,electron, and nitrogen balances, S and S’ (P and P’)result from elimination of substrate (product) between them, before and after reintroduction of the inorganic nitrogen measurement, respectively. LS corresponds to a weighted least-squares solution (see text).
carbohydrates (r, = 4) and PFR should not be eliminated for products such as many organic acids or amino acids and proteins. These sensitivity problems can also be evidenced by studying individual metabolic pathways (6). The rather surprising conclusion is that, in many cases, OUR and CER measurements should not both be used when an exact solution is required. However, the NUR measurement can be reintroduced in a different form to alleviate many of the above sensitivity problems. Since the nitrogen balance is of limited use and the valence of organic nitrogen is equal to -3, the nitrogen balance can be completely eliminated as an independent balance by adding it three times to the electron one, which yields a modified electron balance as follows (R is the vector of rates as in eq 3): [r, -4 3 -(rb -k 36) 0 -(rP + 3w)lR = 0 (4) This changes the effective degree of reductance of biomass and thereby reduces the sensitivity of its estimate. It should also be noted that knowledge of 6 is not needed. Eliminating SUR or PFR by using the carbon balance then gives respectively the following equations (R’ and R” correspond respectively to the rate vector after deletion of SUR or PFR in R): 1-4 3
-(rb
+ 36 - r,)
[ ( r ,- rp - 3w) -4 3
r, -(rp + 3w - r,)lR’ = 0 (5)
-(rb - rp
+ 3(6 - w ) )
(rp + 3w)I X R” = 0 (6)
These two options could be particularly useful for online use when NUR can be measured (or inferred from alkali addition). Using eq 5, biomass estimates are obtained from OUR, NUR, CER, and PFR. In certain cases, the PFR term may cancel (nil coefficient or no product). Indeed, eq 5 has been used with success for estimating yeast concentration during aerobic growth (9). In that instance, eliminating SUR from the carbon and electron balance (one of the cases in Table I) did not give usable estimates (unpublished results). Equation 6 is similar but with PFR being used rather than SUR. In reference to the earlier discussion, these equations actually correspond to one of the redundancy cases, with a particular weighting of the two remaining equations. The advantage of this weighting is that it cancels out the nitrogen content of biomass. When feasible, this approach provides a robust biomass estimator utilizing both OUR and CER (generally available simultaneously), NUR, and either SUR or PFR. Incorporating nitrogen measurements in this way can improve biomass estimates dramatically when SUR and PFR cannot both be measured.
Application to a Lysine Fermentation Data from a lysine fermentation on a complex medium conducted in our laboratory were provided by J. Vallino and R. Kiss. An industrial strain of Breuibacterium pauum was grown in a 15-L MBR fermentor (10-L working volume). Ten liters of preculture was grown first on a
very rich medium for about 20 h, after which half of the volume was bled off. The medium was then completed again with water and nutrients, among which were 200 g/L glucose, 15 g/L ammonium acetate, and 20 g/L yeast extract. The pH was corrected using a concentrated NH3 solution (approximately 17 N). Gas analysis and baseaddition measurements were collected on-line, and offline assays were performed for glucose and lysine (HPLC), ammonium concentration (IO), and biomass (dry cell weights). All quantities except biomass were converted to units of millimoles per liter, and the various combinations of balances described above were tried and compared with the measured biomass concentrations. In addition, a leastsquares solution using the three balances (eq 3) was calculated, with each balance normalized with respect to its coefficient in biomass to ensure comparable contributions. The coefficients in the balance matrix were as follows: r, = 4, rp = 4.67, w = 0.33; r b , 6, and the carbon content of Brevibacterium were taken from the literature (7) as 4.25,0.125, and 40mmol/g, respectively. The results are compared in Table 11, on the basis of the Euclidean norm of the vector of errors (differencebetween calculated and measured biomass), the maximum error, and the final one. Figure 1 shows results for the cases in which BPR was estimated from individual balances along with the least-squares solution. The modified electron balance gave very similar results to those of the original one. All the results showed high deviations during the initial part of the fermentation. The carbon and electron balances gave fairly good final estimates, whereas elimination of substrate or product from them gave completely unusable results. As shown in Table 11, reintroducing the inorganic nitrogen measurement (eqs 5 and 6) improved the predictions by a factor of 2-3. Moreover, including the nitrogen balance in the least-squares solution also gave a significant improvement, although when used alone it did not give good results (see Figure 1). The data therefore confirmed the conclusions of the theoretical analysis, regarding both the sensitivity problems and their improvement by including the inorganic nitrogen uptake. The deviations between calculated and measured biomass can be attributed to measurement errors or to unaccounted-for substances. Given the significant amount of yeast extract in the medium, the latter explanation is more likely. Considerable biases may nonetheless be present in the measurements, originating from fermentor volume and base titer uncertainties. Carbon and electron balances underestimate biomass during the first part of the fermentation, presumably when a lot of organic nitrogen is being consumed to form biomass. The maximum error is actually well within the bounds of the initial concentration of yeast extract. Later on, the amount of this substrate converted to biomass and product becomes negligible compared with that of the consumed glucose, which explains the estimate’s gradual improvement. If the errors are attributable to unmeasured substrate, the question remains why eliminating the SUR measurement
Biotechnol. Prog., 1992, Vol. 8, No. 1
04 45.0 ;
0
1
0
0
Table 111. Five Exact Solutions Likely To Lead to Robust Biomass Estimation from Other Measurements, Including Those Based on the Modified Electron Balance (See Text for Details)
measurements used 1 SUR, CER, PFR 8 SUR, OUR, PFR 6 NUR, PFR Q + 36 - rs OUR, NUR, CER, PFR eq 6 (text) Q - rp + 3(6 - w ) SUR, OUR, NUR, CER solution C balance t balance N balance eq 5 (text)
coefficients 1, -1, -1 rs, -4, -rp 1, -w -4,3, r,, -(rp 3w - r,) (rs- rp - 3w), -4,3, rp + 3w
+
i 0 v
m
-15 0
biomass coefficient
0
10
20
SC
VIS
3cl
es:imcte
so1utIon
LO
50
t i r e (b)
Figure 1. Measured and calculated biomass for a lysine fermentation on a complex medium. Calculations are based on individual carbon, electron, and nitrogen balances or a weighted least-squares combination of the three.
does not improve the prediction, once sensitivity problems have been alleviated. The answer probably lies in the nitrogen content of the cells and in inaccuracy of the base titer, and also in amore subtle fact. Indeed, in the modified electron balance of eq 4, the degree of reductance of the total substrate is quite different from that of glucose (since it contains some nitrogen), and hence even after elimination of SUR, the secondary substrate is still in fact being neglected in eq 5.
Conclusions As discussed, there are a priori 28 combinations for an exact determination of biomass from other measurements, using elemental balances. In fact, the particular structure of these equations leads to only eight different exact solutions at the most. The sensitivity study, BS well as the experimental data, showed that of these eight solutions the only ones of practical value are given by the carbon and the electron balance. These both necessitate substrate and product measurements, and neither takes advantage of OUR and CER measurements, both being generally available. The nitrogen balance relies on the exact determination of the biomass nitrogen content. Eliminating substrate or product leads, in most cases, to extreme sensitivity problems. These problems can be greatly alleviated (at least for fermentations on defined media) by incorporating the inorganic nitrogen uptake in a modified electron balance. In certain cases this measurement is available on-line through alkali-addition monitoring. When an exact solution is desired, the only five equations likely to lead to robust biomass estimates are summarized in Table 111. Depending on the availability and reliability of measurements, one of these equations should be chosen. If (and only if) extra measurements are available, a leastsquares solution can be elected, but the estimated biomass values depend on the weighting adopted for each equation used. In this discussion, the electron balance can be substituted for the energy one, replacing oxygen by metabolic heat measurements in so doing. For fermentations on complex media, the data show that the balances give rather poor results, except for final estimates. This is caused most likely from consumption of unmeasured substrate(& and elemental balances should
not be used for any application in this case. As stated, on the basis of component or energy balances, organicnitrogen sources and biomass can never be distinguished. If the latter were measured, balances could be used to calculate the former, and the same sensitivity considerations as for estimating biomass would apply. The sensitivity study also has significant implications for data reconciliation. In essence, it is not individual measurements, but rather groups of them, which can be designated as erroneous: substrate, biomass, and product form one group when their degrees of reductance are close. These groups can be narrowed down by using the nitrogen uptake as shown in eq 5 or 6. One possibility for solving this general sensitivity problem is the use of additional, independent balances, such as a proton balance, when available. Several difficulties need to be considered in this case. First, certain compounds, such as organic acids, though present in small quantities, have significant effects on the pH. Second, the reactions governing the pH are often complex and ill-understood, e.g., the change in pH resulting from protein degradation and assimilation.
Acknowledgment The contribution of Joseph J. Vallino and Robert D. Kiss in providing the experimental data is acknowledged. Financial support was provided from the National Science Foundation (PYIGrant CBT-8514729 and Grant EET871725). The MBR fermentor was donated by Sulzer Brothers Co.
Literature Cited (1) Acevedo, F. CRC Crit. Reu. Biotechnol. 1987, 6, 309. (2) Minkevich, I. G.; Eroshin, V. K. Folia Microbiol. 1973, 18, 376. (3) Roels, J. A. Biotechnol. Bioeng. 1980, 22, 2457. (4) Cooney, C.R.; Wang, H. Y.; Wang,D. I. C.Biotechnol.Bioeng. 1977, 19, 55. (5) Wang, N. S.; Stephanopoulos, G. Biotechnol. Bioeng. 1983, 25, 2177. (6) Grosz,R.;Stephanopoulos, G.;San,K. Y. Biotechnol. Bioeng. 1984,26, 1198. (7) Erickson, L. E.; Selga, S. E.; Viesturs, U. E. Biotechnol. Bioeng. 1978, 20, 1623. (8) Cooney, C. L.; Wang, D. I. C.; Mateles, R. I. Biotechnol. Bioeng. 1968, 11, 269. (9) Chattaway,T.; Stephanopoulos, G. BiotechnoLBioeng. 1989, 34, 647. (10) Weatherburn, N. W. Anal. Chem. 1967,39,971.
Accepted October 14, 1991. Registry No. Lysine, 56-87-1.
