Parth Ajmera

General equation of aerobic growth

C_wH_xO_yN_z + aO_2 + bH_gO_hN_i \rightarrow cCH_\alpha O_\beta N_\delta + dCO_2 + eH_2O

$w, x, y, z$ — subscripts of the carbon source's molecular formula
$a$ — stoichiometric coefficient of oxygen (mol)
$b$ — stoichiometric coefficient of the nitrogen source (mol)
$g, h, i$ — subscripts of the nitrogen source's molecular formula
$c$ — stoichiometric coefficient of biomass (mol)
$\alpha, \beta, \delta$ — subscripts of the biomass formula, on a C-mole basis (one carbon atom per formula unit)
$d$ — stoichiometric coefficient of carbon dioxide (mol)
$e$ — stoichiometric coefficient of water (mol)

RQ = \frac{\text{no. of moles of } CO_2 \text{ produced}}{\text{no. of moles of } O_2 \text{ consumed}}

$RQ$ — respiratory quotient (ratio of CO $_2$ produced to O $_2$ consumed — indicates how reduced or oxidized the substrate is: a **reduced** substrate is rich in hydrogen relative to oxygen (fats, hydrocarbons — still holds a lot of energy/available electrons, so needs a lot of $O_2$ to fully combust), while an **oxidized** substrate already has more oxygen bonded in and fewer available electrons left to give up (organic acids, and $CO_2$ at the fully-oxidized extreme) — see degree of reduction, below, for the precise measure of this), dimensionless

\gamma = \frac{\text{no. of available electrons}}{\text{no. of carbon atoms}}

$\gamma$ — degree of reduction (available electrons per carbon atom — measures how reduced/energy-rich a substance is; higher $\gamma$ needs more $O_2$ to fully oxidize), dimensionless

\eta_X = \frac{\gamma_X c}{\gamma_S w} \qquad\qquad \eta_{O_2} = \frac{4a}{\gamma_S w}

$\eta_X$ — fraction of substrate electrons transferred to biomass, dimensionless
$\eta_{O_2}$ — fraction of substrate electrons transferred to $O_2$ , dimensionless
$\gamma_S w$ — total available electrons entering with the substrate (denominator for every $\eta$ )

C_wH_xO_yN_z + aO_2 + bH_gO_hN_i \rightarrow cCH_\alpha O_\beta N_\delta + fC_jH_kO_lN_m + dCO_2 + eH_2O

$f$ — stoichiometric coefficient of the product (mol)
$j, k, l, m$ — subscripts of the product's molecular formula — written as the real molecule (e.g. $C_2H_6O$ for ethanol), not on a C-mole basis the way biomass is

Y_{X/S} = \frac{\Delta X}{\Delta S} \qquad\qquad Y_{P/S} = \frac{\Delta P}{\Delta S}

$Y_{X/S}$ — biomass yield on substrate (fraction of substrate mass converted into biomass), mass basis, in grams of biomass per gram of substrate consumed ( $\text{g/g}$ )
$Y_{P/S}$ — product yield on substrate (fraction of substrate mass converted into product), mass basis, in grams of product per gram of substrate consumed ( $\text{g/g}$ )
$\Delta X$ , $\Delta P$ , $\Delta S$ — biomass formed, product formed, and substrate consumed, in grams ( $\text{g}$ )

\text{ThOD} = \frac{\text{available electrons per mole}}{4}

$\text{ThOD}$ — theoretical oxygen demand ( $O_2$ needed to fully combust the substance — the ceiling on $O_2$ consumption, reached when no electrons go to biomass or product), in moles of $O_2$ per mole of substance ( $\text{mol } O_2/\text{mol}$ ); multiply by $32\ \text{g/mol}$ and divide by the molar mass for a mass basis ( $\text{g } O_2/\text{g}$ )

\%\,\text{element}_i = \frac{n_i M_i}{\sum_j n_j M_j} \times 100

$\%\,\text{element}_i$ — mass contribution of element $i$ to dry biomass (what fraction of the cell's dry weight that element accounts for), percent ( $\%$ )
$n_i$ — subscript (atom count) of element $i$ in the biomass formula, dimensionless
$M_i$ — atomic weight of element $i$ , in grams per mole ( $\text{g/mol}$ )
$\sum_j n_j M_j$ — sum of (subscript × atomic weight) over every element in the formula, in grams per mole ( $\text{g/mol}$ ) — the total C-mole formula weight