Hairy-Root Culture Kinetics — Specific Growth Rate, Sugar Uptake and Yield: 04-BS-13, December 2013, Q7

Question 7: Hairy-Root Culture Kinetics — Specific Growth Rate, Sugar Uptake and Yield (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.

Given. The 12-point biomass (g/L) and residual-sugar (g/L) time course above, $t=0$ to 55 d.

Find. $\mu(t)$ and $q_S(t)$ over each 5-day interval; identify any interval of constant $\mu$; the overall observed $Y_{XS}$.

Approach. Both specific rates are computed interval-by-interval from the discrete data: $\mu\approx\ln(X_{i+1}/X_i)/\Delta t$ (log-mean growth rate over the interval) and $q_S\approx-(\Delta S/\Delta t)/\bar X$ (sugar consumed per unit time per unit average biomass), each plotted against the interval midpoint. The overall yield is simply total biomass formed divided by total sugar consumed across the whole run.

(a) Specific growth rate per interval. $$\mu_i=\frac{\ln(X_{i+1}/X_i)}{t_{i+1}-t_i}.$$ Evaluated for all 11 intervals (full table plotted below): $\mu$ falls from $0.223\ \text{d}^{-1}$ (0–5 d) to $0.154\ \text{d}^{-1}$ (5–10 d) to $0.055\ \text{d}^{-1}$ (10–15 d), and continues declining, with minor noise, down to $0.003\ \text{d}^{-1}$ by 50–55 d.
(b) Specific sugar-uptake rate per interval. $$q_{S,i}=\frac{S_i-S_{i+1}}{(t_{i+1}-t_i)\,\bar X_i},\qquad \bar X_i=\tfrac12(X_i+X_{i+1}).$$ $q_S$ tracks $\mu$ closely in shape, falling from $0.40\ \text{g sugar g}^{-1}\text{d}^{-1}$ (0–5 d) to $0.009\ \text{g g}^{-1}\text{d}^{-1}$ by 50–55 d.
(a) When is $\mu$ constant? Strictly, it is not: $\mu$ is highest in the very first interval and declines monotonically (with minor scatter) over the whole 55-day run — there is no window in which successive $\mu_i$ values are equal within the data's noise. This is a genuine biological result, not a data-reading failure: differentiated hairy-root tissue grows as a branching, increasingly diffusion-limited mat (nutrient and O₂ penetration into the root interior becomes progressively harder as biomass accumulates), so hairy-root cultures characteristically show a continuously decaying specific growth rate rather than the sharp, constant-$\mu$ exponential phase seen in well-mixed single-cell suspension cultures. The closest approach to "constant" is the initial 0–10 d window, where $\mu$ is highest and changing most slowly in relative terms.
(c) Overall observed yield. $$Y_{XS}=\frac{X_f-X_0}{S_0-S_f}=\frac{13.7-0.64}{30-5.1}=\frac{13.06}{24.9}=\boxed{0.525\ \text{g biomass/g sugar}}.$$

Specific growth rate μ(t) and specific sugar-uptake rate qₛ(t), computed interval-by-interval from the batch data. Both decline continuously from the start — there is no window of constant μ in this hairy-root culture.

Quantity	Result
$\mu$, 0–5 d (highest)	0.223 d⁻¹
$\mu$, 50–55 d (lowest)	0.003 d⁻¹
Constant-$\mu$ window	none — continuously declining (diffusion-limited root mat)
$q_S$, 0–5 d (highest)	0.40 g g⁻¹d⁻¹
Overall observed $Y_{XS}$	0.525 g biomass/g sugar

Concept & Theory

Topic. Discrete-data estimation of specific growth and substrate-uptake rates for a differentiated (non-single-cell) plant-tissue culture.

Key relations. $\mu=(1/X)(dX/dt)\approx\ln(X_{i+1}/X_i)/\Delta t$; $q_S=-(1/X)(dS/dt)\approx-(\Delta S/\Delta t)/\bar X$; overall $Y_{XS}=\Delta X/(-\Delta S)$.

Why this works. The log-mean form for $\mu$ is the exact finite-difference estimator of the instantaneous specific rate under the local assumption of exponential growth between sample points, and using the interval-average biomass $\bar X$ (not just the endpoint) for $q_S$ avoids biasing the rate toward either end of an interval over which $X$ itself is changing.

Common pitfall. Forcing an "exponential phase" onto the data because textbook batch-culture curves always show one — differentiated tissue cultures (hairy roots, plant cell aggregates, mycelial fungi) frequently do not exhibit a true constant-$\mu$ window because growth becomes diffusion-limited by the tissue's own increasing mass/geometry well before any nutrient in the bulk medium is depleted; reporting a false "exponential phase" from noisy early-time data is a more serious error than reporting that none exists.

Source. Shuler & Kargi, Bioprocess Engineering: Basic Concepts, Ch. 6 (Batch growth kinetics, specific rate estimation from discrete data).

Question 7 of 10: Hairy-Root Culture Kinetics — Specific Growth Rate, Sugar Uptake and Yield

Question 7: Hairy-Root Culture Kinetics — Specific Growth Rate, Sugar Uptake and Yield (20 marks)