Question 4 of 5: Product Kinetics, Washout, Maintenance Coefficient & Nutritional Requirements

Question 4: Product Kinetics, Washout, Maintenance Coefficient & Nutritional Requirements (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(i) Growth-associated and non-growth-associated products

The Luedeking–Piret model partitions the rate of product formation into a term proportional to the instantaneous growth rate and a term proportional to biomass concentration alone: $$\frac{dP}{dt}=\alpha\frac{dX}{dt}+\beta X$$ A growth-associated product is one for which α≫0 and β≈0 — its formation rate tracks the specific growth rate directly, so production is fastest during exponential growth and essentially stops once growth stops. Primary metabolites that are intermediates or direct outputs of energy metabolism (e.g. ethanol from yeast fermentation, or the biomass itself) are typically growth-associated. A non-growth-associated product is one for which β≫0 and α≈0 — its formation rate is essentially independent of the instantaneous growth rate and continues (or even peaks) after growth has slowed or stopped, because it draws on cellular machinery/energy not tied to biosynthesis of new cells. Many secondary metabolites (e.g. penicillin, produced predominantly in the stationary/idiophase rather than the growth/trophophase) are non-growth-associated. Many real fermentations are mixed, showing both terms simultaneously.

(ii) Why D must not exceed μ_max in a chemostat

At steady state, the biomass balance on a chemostat (feed sterile, F in = F out = F, volume V constant) gives X(μ−D)=0 with D≡F/V, so any non-washed-out steady state requires μ=D — the culture's specific growth rate exactly matches the rate at which cells are diluted out of the vessel. But μ itself is bounded by Monod kinetics, $$\mu=\frac{\mu_{max}S}{K_s+S}\;<\;\mu_{max}\quad\text{(approached only as }S\to\infty\text{)}$$ so μ can never reach, let alone exceed, μ_max for any finite substrate concentration. If the operator sets D>μ_max, there is no value of S for which the culture can grow fast enough to satisfy μ=D — cells are removed from the vessel faster than they can reproduce at any concentration, biomass declines monotonically, S rises back toward the feed value S_in (less is being consumed as X falls), and the culture is progressively washed out (X→0). D must therefore be kept strictly below μ_max for a viable, non-washed-out steady state to exist at all. In practice the limit is slightly tighter: with sterile feed at finite S_in the tank concentration can never exceed S_in, so washout actually sets in at D_crit=μ_maxS_in/(K_s+S_in) (just below μ_max when S_in≫K_s), and biomass productivity DX peaks somewhat below that.

(iii) Cell maintenance coefficient and the Pirt equation

Cells consume substrate not only to build new biomass (growth) but also for "maintenance" functions that consume energy/substrate even at zero net growth — maintaining transmembrane ion gradients and membrane potential, osmoregulation, motility, and turnover/repair of macromolecules. The maintenance coefficient m_s (units: g substrate·g biomass⁻¹·h⁻¹) is the specific rate of substrate consumption attributable to maintenance alone, independent of growth rate. The Pirt equation partitions the total specific substrate consumption rate q_s into a growth term and this maintenance term: $$q_s=\frac{\mu}{Y_G}+m_s$$ where Y_G is the "true" (maximal) growth yield — biomass formed per substrate consumed for growth alone, excluding maintenance. Because the observed yield is Y_obs=μ/q_s, dividing through by μ and inverting gives the equivalent, often-quoted form $$\frac{1}{Y_{obs}}=\frac{1}{Y_G}+\frac{m_s}{\mu}$$ which shows that the observed yield always falls short of the true yield, and falls further short as μ decreases — at low growth rate (as in a chemostat run at low D) a larger fraction of every substrate unit consumed is diverted to maintenance rather than new biomass, since maintenance draws at a roughly constant specific rate m_s regardless of how slowly the culture is growing.

(iv) Major nutritional requirements for microbial cultivation