Despite its complexity and the thousands of intracellular reactions involved, cell growth obeys the law of conservation of matter. All atoms of carbon, hydrogen, oxygen, nitrogen, and other elements consumed during growth are incorporated into new cells or excreted as products. Confining our attention to those compounds taken up or produced in signifi- cant quantity, if the only extracellular products formed are CO2 and H2O, we can write the following general equation for aerobic cell growth:
CwHxOyNz1aO21bHgOhNi!cCHαOβNδ1dCO21eH2O ð4:4ị InEq. (4.4):
• CwHxOyNzis the chemical formula for the carbon source or substrate (e.g., for glucose C6H12O6,w56,x512,y56, andz50). Once the identity of the substrate is known, CwHxOyNzis fully specified and contains no unknown variables.
• HgOhNiis the chemical formula for the nitrogen source (e.g., for ammonia NH3,g53, h50, andi51). Once the identity of the nitrogen source is known, HgOhNicontains no unknown variables.
• CHαOβNδis the chemical ‘formula’ for dry cells. The formula is a reflection of the dry biomass composition and is based on one C atom:α,β, andδare the numbers of H, O, and N atoms, respectively, present in the biomass per C atom. There is no fundamental
objection to having a molecular formula for cells even if it is not widely applied in biology. As shown inTable 4.10, microorganisms such asEscherichia colicontain a wide range of elements; however 90 to 95% of the biomass can be accounted for by four major elements: C, H, O, and N. Compositions of several microbial species in terms of these four elements are listed inTable 4.11. Bacteria tend to have slightly higher nitrogen contents (1114%) than fungi (6.39.0%)[4]. For a particular species, cell composition also depends on the substrate utilised and the culture conditions applied; hence the different entries inTable 4.11for the same organism. However, the results are remarkably similar for different cells and conditions; CH1.8O0.5N0.2can be used as a general formula for cell biomass when composition analysis is not available. The average ‘molecular weight’ of cells based on C, H, O, and N content is therefore 24.6, although 5 to 10% residual ash is often added to account for those elements not included in the formula.
• a,b,c,d, andeare stoichiometric coefficients. BecauseEq. (4.4)is written using a basis of one mole of substrate,amoles of O2are consumed anddmoles of CO2are formed, for example, per mole of substrate reacted. The total amount of biomass formed during growth is accounted for by the stoichiometric coefficientc.
As illustrated in Figure 4.11, Eq. (4.4) represents a macroscopic view of metabolism; it ignores the detailed structure of the system and considers only those components that have net interchange with the environment.Eq. (4.4)does not include a multitude of compounds such as ATP and NADH that are integral to metabolism and undergo exchange cycles in cells, but are not subject to net exchange with the environment. Components such as
TABLE 4.10 Elemental Composition ofEscherichia coliBacteria
Element % Dry weight
C 50
O 20
N 14
H 8
P 3
S 1
K 1
Na 1
Ca 0.5
Mg 0.5
Cl 0.5
Fe 0.2
All others 0.3
From R.Y. Stanier, J.L. Ingraham, M.L. Wheelis, and P.R. Painter, 1986,The Microbial World, 5th ed., Prentice Hall, Upper Saddle River, NJ.
vitamins and minerals taken up during metabolism could be included; however, since these materials are generally consumed in small quantities, we assume here that their contribution to the stoichiometry and energetics of reaction can be neglected. Other substrates and pro- ducts can be added easily if appropriate. Despite its simplicity, the macroscopic approach provides a powerful tool for thermodynamic analysis.
Equation (4.4) is not complete unless the stoichiometric coefficients a, b,c,d, ande are known. Once a formula for biomass is obtained, these coefficients can be evaluated using TABLE 4.11 Elemental Composition and Degree of Reduction for Selected Organisms
Organism Elemental formula Degree of reductionγ(relative to NH3) Bacteria
Aerobacter aerogenes CH1.83O0.55N0.25 3.98
Escherichia coli CH1.77O0.49N0.24 4.07
Klebsiella aerogenes CH1.75O0.43N0.22 4.23
Klebsiella aerogenes CH1.73O0.43N0.24 4.15
Klebsiella aerogenes CH1.75O0.47N0.17 4.30
Klebsiella aerogenes CH1.73O0.43N0.24 4.15
Paracoccus denitrificans CH1.81O0.51N0.20 4.19 Paracoccus denitrificans CH1.51O0.46N0.19 3.96
PseudomonasC12B CH2.00O0.52N0.23 4.27
Fungi
Candida utilis CH1.83O0.54N0.10 4.45
Candida utilis CH1.87O0.56N0.20 4.15
Candida utilis CH1.83O0.46N0.19 4.34
Candida utilis CH1.87O0.56N0.20 4.15
Saccharomyces cerevisiae CH1.64O0.52N0.16 4.12 Saccharomyces cerevisiae CH1.83O0.56N0.17 4.20 Saccharomyces cerevisiae CH1.81O0.51N0.17 4.28
Average CH1.79O0.50N0.20 4.19 (standard deviation53%) From J.A. Roels, 1980, Application of macroscopic principles to microbial metabolism,Biotechnol. Bioeng.22, 24572514.
Substrate CwHxOyNz
Cell b HgOhNi
Nitrogen source a O2
Biomass c CHαOβNδ d CO2 e H2O
FIGURE 4.11 Conversion of sub- strate, oxygen, and nitrogen for cell growth.
normal procedures for balancing equations, that is, elemental balances and solution of simultaneous equations.
C balance: w5c1d ð4:5ị
H balance: x1bg5cα12e ð4:6ị
O balance: y12a1bh5cβ12d1e ð4:7ị
N balance: z1bi5cδ ð4:8ị
Notice that we have five unknown coefficients (a, b, c, d, and e) but only four balance equations. This means that additional information is required before the equations can be solved. Usually this information is obtained from experiments. A useful measurable parameter is therespiratory quotient,RQ:
RQ5 moles CO2produced moles O2consumed 5d
a ð4:9ị
When an experimental value of RQis available, Eqs. (4.5) through (4.9) can be solved to determine the stoichiometric coefficients. The results, however, are sensitive to small errors in RQ, which must be measured very accurately. When Eq. (4.4) is completed, the quantities of substrate, nitrogen, and oxygen required for production of biomass can be determined directly.
E X A M P L E 4 . 7 S T O I C H I O M E T R I C C O E F F I C I E N T S F O R C E L L G R O W T H
Production of single-cell protein from hexadecane is described by the following reaction equation:
C16H341aO21bNH3!cCH1:66O0:27N0:201dCO21eH2O
where CH1.66O0.27N0.20 represents the biomass. If RQ50.43, determine the stoichiometric coefficients.
Solution
C balance: 165c1d ð1ị
H balance: 3413b51:66c12e ð2ị
O balance: 2a50:27c12d1e ð3ị
N balance: b50:20c ð4ị
RQ: 0:435d=a ð5ị
We must solve this set of simultaneous equations. Solution can be achieved in many different ways; usually it is a good idea to express each variable as a function of only one other variable.
bis already written simply as a function ofcin(4); let us try expressing the other variables solely in terms ofc. From(1):
d5162c ð6ị
From(5):
a5 d
0:4352:326d ð7ị
Combining(6) and (7)gives an expression forain terms ofconly:
a52:326ð162cị
a537:2222:326c ð8ị
Substituting(4)into(2)gives:
3413ð0:20cị51:66c12e 3451:06c12e
e51720:53c ð9ị
Substituting(8), (6), and (9)into(3)gives:
2ð37:2222:326cị50:27c12ð162cị1ð1720:53cị 25:4452:39c
c510:64 Using this result forcin(8), (4), (6), and (9)gives:
a512:48 b52:13 d55:37 e511:36
Check that these coefficient values satisfyEqs. (1) through (5).
The complete reaction equation is:
C16H34112:5 O212:13 NH3!10:6 CH1:66O0:27N0:2015:37 CO2111:4 H2O
Although elemental balances are useful, the presence of water inEq. (4.4) causes some problems in practical application. Because water is usually present in great excess and changes in water concentration are inconvenient to measure or experimentally verify, H and O balances can present difficulties. Instead, a useful principle is conservation of reduc- ing power or available electrons, which can be applied to determine quantitative relation- ships between substrates and products. An electron balance shows how available electrons from the substrate are distributed during reaction.