Microbial products formulas

X is biomass concentration (g/L). In exponential growth with constant $\mu$, $X=X_0{e}^{\mu t}$.

Valid only in the exponential phase, where $\mu$ is constant. $\mu$ is in h${}^{-1}$, so $t_d$ is in hours.

S is the concentration of the growth-limiting substrate (g/L). $K_s$, the saturation constant, is the value of S at which $\mu ={\mu }_{\max }/2$. When $S\gg K_s$, $\mu \approx {\mu }_{\max }$; when $S\ll K_s$, $\mu \approx ({\mu }_{\max }/K_s)S$. Linearised (Lineweaver-Burk form): $1/\mu =1/{\mu }_{\max }+(K_s/{\mu }_{\max })(1/S)$. Applies when one substrate limits growth.

Grams of biomass formed per gram of substrate consumed, taken as constant over the culture. Similarly $Y_{P/S}=\Delta P/(-\Delta S)$ for product and $Y_{P/X}=\Delta P/\Delta X$.

$\alpha$ (g product per g biomass) is the growth-associated coefficient and $\beta$ (g product per g biomass per h) the non-growth-associated coefficient. A purely growth-associated product has $\beta =0$; a purely non-growth-associated product, such as a typical secondary metabolite, has $\alpha \approx 0$. Mixed products such as lactic acid need both terms.

F is the feed flow rate (L/h), V the culture volume (L), D the dilution rate (h${}^{-1}$) and $S_0$ the limiting-substrate concentration in the feed. Assumes a sterile feed, Monod kinetics, constant yield, and no death or maintenance. Washout occurs when D reaches ${\mu }_{\max }{S}_0/(K_s+S_0)$, the highest growth rate the feed can support.

Theoretical ethanol yield is $2\times 46/180=0.511$ g ethanol per g glucose, and CO${}_2$ is $2\times 44/180=0.489$ g per g glucose. Real yields are lower because some sugar goes to cells and by-products.

Three moles of methanol per mole of triglyceride, usually with excess methanol to push the equilibrium forward and an alkali catalyst; lipases can also catalyse it.

Assumes first-order death at a constant temperature. N is the number of viable cells or spores, $k_d$ the specific death rate (min${}^{-1}$), and D the time at that temperature that cuts the viable count tenfold (one log). The z value is the temperature rise that cuts D tenfold.