## Y

Application of the Mathematical Stoichiometry Relations

The obtained stoichiometric relations (equations 9a-9e) can now easily be applied. In this section their use is demonstrated with the following subjects:

• Calculation of the complete growth stoichiometry

• Calculation of maintenance coefficients and maximal growth yields

• Calculation of the limit to growth yield posed by the second law

• Calculation of COD-based growth yields

• Calculation of the relation between heat production and Gibbs energy dissipation

• Calculation of maximal product yields in anaerobic metabolism

Calculation of the Complete Growth Stoichiometry. If for a given growth system the C source, the electron donor (DGed to decide on RET using equation 7c), the temperature, and the growth rate are known, then equations 2 and 3 allow a direct calculation of the required Gibbs energy 1/YSX to produce 1 C-mol of biomass at high growth rates l. Knowing the electron donor couple and acceptor couple, and using Table 3 and equations 5a-5c, the values of DGed, DGea, DHed, DHea, yD, and yA can be calculated and from this ( —DGcat) and ( —DHCAT) using equations 7a and b. Using equations 9b-9e subsequently allows the complete stoichiometric calculation where H + , N source, H2O, and HCO— must be calculated using the conservation equations of electric charge, N, O or H, and carbon.

EXAMPLE 9a

### Calculation of stoichiometry using equations 9b-9e

Consider Example 4, where a microorganism is grown anaerobi-cally on methanol, producing acetate. Assume first that the growth rate is high, such that maintenance can be neglected. Equation 3a then shows that l/Ycx — 698 kJ Gibbs energy per C-mol biomass. Methanol is the C-source, methanol/HCO- is the electron donor, and HCO-/acetate is the electron acceptor. From Table 3 and using equations 5a-5c, we can then find that DGed — 36.032 kJ/e-mol, DHed — -23kJ/e-mol; DGEA — 26.801 kJ/e-mol; DHea — -33.5 kJ/e-mol. Also, yD — 6, yA — -4, and yx — 4.2.

This provides that -DGCAT — 6(36.032 - 26.801) — 55.386 and -DHcat — -23 - (-33.5) — 63 kJ/C-mol methanol. Using equations 9b-9e, we obtain the maximal growth yields (maintenance neglected).

I/Yax — 18.9 C-mol acetate/C-mol biomass — 9.45 mol acetate/C-mol biomass

YSA — 1.42 C-mol acetate/C-mol methanol — 0.71 mol acetate/mol methanol

Using the C- balance 1/YjAx is calculated as 6.6 mol HCO-consumed/C-mol biomass produced. This overall stoichiometric result is very close to the exact solution obtained in Example 4.

The small deviation arises from the assumption that D GEX ^ D GED (as discussed before).

In general it can be shown that the simple set of equations 9b-9e seldom deviates more than 5% from the exact solution.

Maintenance Coefficients and Maximal Yield Coefficients.

As indicated previously, relations between the maintenance coefficients follow from the catabolic reaction. The following relations can now be written to link the various maintenance coefficients to the maintenance Gibbs energy mG using — DGCAT (see also Example 9b). It is noted that mG follows from the correlation (equation 2) and that DGcat is calculated for the catabolism of 1 (C)-mol of electron donor.

mD = mG/( — DGCAT) m a = (Cd/ — CA)mG/( — DGcat) ( — DHCAT)

Using equations 10a-10d, the value of the maintenance Gibbs energy requirement mG can be calculated from either measured maintenance coefficients (for electron donor mD, electron acceptor mA, heat production mq, or carbon dioxide production mC). Furthermore, it can easily be understood that the maximal biomass yields for electron donor, acceptor, and heat are found from equations 9b-9e by substitution of l/Y^X (instead of 1/YGX) because the maintenance contribution is then neglected.

### EXAMPLE 9b

These values can also be obtained directly from equations 9b-9e by substituting the complete Gibbs energy of growth and maintenance according to equation 1c:

mG 13

= 1348 kJ Gibbs energy/C-mol X

Clearly, comparing Examples 9a and 9b, one observes that the yield of biomass YDX drops from 0.077 to 0.04 due to maintenance, but the acetate/methanol yield YDA increases from 1.42 to 1.46 C-mol acetate/mol methanol.

Second Law Limit of Growth Yield. As for any chemical reaction, the microbial growth yield is also limited by the second law of thermodynamics. This limit is achieved if 1/Ygx = 0, because this defines equilibrium. Equations 9c-9f then show that for the thermodynamic limits we can write [see also Ref. 4] the following:

Thermodynamic limits for growth yields

Clearly, the more reduced electron donors (cD higher) have a higher YDX limit. This limit has already been determined (1).

COD-Based Yields. In wastewater treatment, the biomass yield is calculated on COD basis. YCOD is the gram biomass COD over gram-substrate COD. Based on the COD definition we can write

### Effect of maintenance on stoichiometry

In Example 9a the maintenance contribution was neglected. Assume that the microorganism is growing at 37 °C. Equation 2 then leads to mG = 13 kJ/C-mol biomass h. Using equations 10a-10c and using D GCAT and DHCAT (Example 9) we obtain

This allows the following relation for YCOD from equation 9b mD = 0.2347 mol methanol/C-mol biomass/h mA = 0.3521 C-mol acetate/C-mol biomass/h mQ = 14.787 kJ/C-mol biomass/h

Further assume that the growth rate i = 0.02 h—1. Using equation 1b, the YX values obtained in Example 9a, and the m( values obtained here, one obtains for the stoichiometry: