Cell Stoichiometry

Key points

1. 01

   The general growth equation has five unknown coefficients but only four elemental balances (C, H, O, N) — a fifth relation (measured $RQ$, measured yield, or an electron balance) is always needed to solve it.

2. 02

   If a nitrogen-containing carbon source supplies more nitrogen than the biomass formed needs, the nitrogen-source coefficient $b$ comes out negative — meaning $NH_3$ is a product, not a reactant.

3. 03

   Respiratory quotient $RQ = \dfrac{\text{mol } CO_2}{\text{mol } O_2}$: $\approx 1$ for carbohydrates, $< 1$ for reduced substrates (fats, hydrocarbons), $> 1$ for already-oxidized substrates (organic acids, amino acids).

4. 04

   Available electrons = the sum of (valency × atom count) for every element in a formula; $C=+4$, $H=+1$, $O=-2$, $P=+5$, $S=+6$ are fixed, but $N$'s valency depends on the compound it's found in.

5. 05

   Degree of reduction $\gamma$ = available electrons ÷ number of carbon atoms — it measures how reduced a substance is per carbon, and is what lets an electron balance replace an elemental balance when solving the general growth equation. $CO_2$ is the zero-point ($\gamma=0$); $CH_4$ is the highest single-carbon value ($\gamma=8$).

6. 06

   One mole of $O_2$ = $4$ electron equivalents — the conversion used to turn "electrons left over" into moles of $O_2$ in any electron balance, and it follows directly from $O_2 + 4H^+ + 4e^- \rightarrow 2H_2O$.

7. 07

   $\gamma_X$ (biomass) doesn't depend on the nitrogen source — it's fixed by biomass's elemental composition, using amine-nitrogen ($-3$) valency, because that's the level nitrogen sits at once it's inside the cell either way. Feeding $NO_3^-$ instead of $NH_3$ changes the electron balance instead (an extra $8$ electrons per N atom to reduce nitrate down to the amine level), which shows up as lower $O_2$ demand, not a lower $\gamma_X$.

8. 08

   A biomass elemental formula is written per C-mole (one carbon atom per formula unit), so mass-percent composition is just each element's (subscript × atomic weight) over the formula's total weight.

9. 09

   Product formation adds a sixth term to the general equation ($f\,C_jH_kO_lN_m$) and a matching term to the electron balance ($\gamma_P j f$) — the product's electron total needs the $\times j$ because, unlike biomass, it isn't written on a C-mole basis.

10. 10

    Maximum theoretical yield ($Y_{X/S}^{max}$, $Y_{P/S}^{max}$) is the electron balance with $O_2$ zeroed out — all substrate electrons land in biomass or product instead. Real aerobic cultures always fall below it, since some electrons are always spent on $O_2$.

11. 11

    Theoretical oxygen demand (ThOD) is the same electron balance run the other way: $c = f = 0$, every electron goes to $O_2$ — the $O_2$ needed to fully combust the substance.