Question 4 of 8: Theoretical Yield Coefficients — ATP Method and Recombinant Protein Production
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2018 — 04-BS-13, Biology. Three-hour, closed-book exam (one double-sided aid sheet permitted, approved Casio/Sharp calculator allowed). Format: Part I offers 5 questions (any 3 constitute a complete answer, 20 marks each) and Part II offers 3 questions (any 2 constitute a complete answer, 20 marks each) — a full paper is 5 questions. All 8 numbered questions are solved below for completeness (renumbered Q1–Q8 continuously: Q1–Q5 = Part I, Q6–Q8 = Part II). Q1, Q2, Q3, and Q4 are calculation/stoichiometry questions; Q5, Q6, Q7, and Q8 are essay questions.
Reference texts: Shuler & Kargi, Bioprocess Engineering: Basic Concepts (2nd ed., Prentice Hall) — elemental/electron balances, yield coefficients, maintenance-associated product formation, fermenter mass and energy balances; Madigan et al., Brock Biology of Microorganisms (15th ed., Pearson) — bacterial nutrition, transport mechanisms, cell-wall structure, pure-culture technique, sterilization methods; Toledo, Fundamentals of Food Process Engineering (3rd ed., Springer) — plant/animal tissue rheology and gross structure.
Question 4: Theoretical Yield Coefficients — ATP Method and Recombinant Protein Production (20 marks)
Find. (a) $Y_{X/S}$, $Y_{P/S}$ (theoretical); (b) $Y_{X/S}$, $Y_{P/S}$ (true, stoichiometric) and $Y'_{X/S}$, $Y'_{P/S}$ (observed, with maintenance).
Approach. (a) Convert the ATP yield to a biomass mass yield via $Y_{X/ATP}$, and read the ethanol mass yield directly off the 2:1 fermentation stoichiometry. (b) Get the "true" (maintenance-free) yields directly from the given stoichiometric coefficients, then apply Pirt's substrate balance ($q_S=\mu/Y_{X/S}+m_S$) and a Luedeking–Piret-type product balance ($q_P=\mu\,Y_{P/S}+m_P$) to get the total specific substrate-uptake and product-formation rates, and finally the maintenance-corrected ("observed") yields as their ratios to $q_S$.
(a) Theoretical growth yield via the ATP method. Biomass formed per mole of glucose $=Y_{X/ATP}\times(\text{ATP yield})=10.5(2)=21$ g biomass/mol glucose:
$$Y_{X/S}=\frac{21}{180}=\boxed{0.1167\ \text{g biomass/g glucose}}.$$
Theoretical product (ethanol) yield reads directly off the given stoichiometry (2 mol ethanol/mol glucose):
$$Y_{P/S}=\frac{2(46)}{180}=\boxed{0.5111\ \text{g ethanol/g glucose}}$$
(this is the same thermodynamic-maximum figure used as the comparison benchmark in Q1(b)(ii)).
(b) True (stoichiometric) yields. Directly from the given reaction coefficients, per mole of glucose:
$$Y_{X/S}=\frac{3.46(25)}{180}=\boxed{0.4806\ \text{g cell/g glucose}},\qquad Y_{P/S}=\frac{0.79(22.03)}{180}=\boxed{0.0967\ \text{g protein/g glucose}}.$$
(b) Observed (maintenance-corrected) yields. Total specific glucose-uptake rate (Pirt's equation, growth-associated $+$ maintenance):
$$q_S=\frac{\mu}{Y_{X/S}}+m_S=\frac{0.4}{0.4806}+0.2=0.8324+0.2=\boxed{1.032\ \text{h}^{-1}}.$$
Observed growth yield:
$$Y'_{X/S}=\frac{\mu}{q_S}=\frac{0.4}{1.032}=\boxed{0.3875\ \text{g cell/g glucose}}.$$
Total specific protein-formation rate (growth-associated $+$ non-growth-associated, Luedeking–Piret form):
$$q_P=\mu\,Y_{P/S}+m_P=0.4(0.0967)+0.1=0.0387+0.1=\boxed{0.1387\ \text{h}^{-1}}.$$
Observed product yield (per total substrate consumed, i.e. $q_P$ divided by the same total uptake rate $q_S$):
$$Y'_{P/S}=\frac{q_P}{q_S}=\frac{0.1387}{1.032}=\boxed{0.1343\ \text{g protein/g glucose}}.$$
Quantity
Result
(a) $Y_{X/S}$ theoretical
0.1167 g/g
(a) $Y_{P/S}$ theoretical
0.5111 g/g
(b) $Y_{X/S}$ true (stoichiometric)
0.4806 g/g
(b) $Y_{P/S}$ true (stoichiometric)
0.0967 g/g
(b) $Y'_{X/S}$ observed
0.3875 g/g
(b) $Y'_{P/S}$ observed
0.1343 g/g
Check: part (b) gives $m_S$ and $m_P$ in units of h-1, which is unusual for a maintenance coefficient (normally g substrate or g product per g cell per hour). They are treated here, per the standard Pirt/Luedeking–Piret framework, as specific maintenance-associated rate constants with those implied mass-per-mass-per-time units so that $q_S$ and $q_P$ come out dimensionally consistent with $\mu$ (h-1) — the only interpretation that makes the requested g/g yield ratios computable from the given numbers.
