9.4 Marine Biotechnology

Step 1: Monod Growth Under Light Limitation

Begin with the specific growth rate $\mu$ as a function of average irradiance $\bar{I}$ inside the reactor, following a Monod (Michaelis-Menten-type) saturation curve:

$$\mu(\bar{I}) = \mu_{\max} \frac{\bar{I}}{\bar{I} + K_I}$$

Step 2: Beer-Lambert Light Attenuation

Light decays exponentially through the culture due to self-shading. The average irradiance across reactor path length $L$ is obtained by integrating Beer-Lambert law:

$$\bar{I} = \frac{I_0}{k_{\text{ext}} X L}\left(1 - e^{-k_{\text{ext}} X L}\right)$$

Step 3: Coupled Biomass-Nutrient Dynamics

Biomass $X$ grows at rate $\mu$ minus a maintenance/respiration loss $m$, while nutrient $N$ is consumed proportionally via yield coefficient $Y_{X/N}$:

$$\frac{dX}{dt} = (\mu - m)X, \quad \frac{dN}{dt} = -\frac{\mu X}{Y_{X/N}}$$

Step 4: Lipid Productivity and Biofuel Yield

The biodiesel yield depends on biomass concentration, lipid fraction $f_L$, and extraction efficiency $\eta$. Volumetric lipid productivity $P_L$ (g/L/day) is maximised at an intermediate biomass where light limitation and growth rate balance:

$$P_L = \mu(X) \cdot X \cdot f_L \cdot \eta, \quad \frac{dP_L}{dX} = 0 \;\Rightarrow\; X_{\text{opt}}$$

Step 5: Optimal Biomass Concentration

Setting $dP_L/dX = 0$ with the Beer-Lambert averaged growth rate yields a transcendental equation for $X_{\text{opt}}$. For the thin-culture limit ($k_{\text{ext}} X L \ll 1$), $\bar{I} \approx I_0$ and growth is nutrient-limited. For the dense-culture limit ($k_{\text{ext}} X L \gg 1$), $\bar{I} \approx I_0/(k_{\text{ext}} X L)$ and productivity scales as $\sim 1/X$, confirming a maximum at intermediate density.

Step 1: Elementary Reaction Mechanism

An enzyme $E$ binds substrate $S$ to form complex $ES$, which either dissociates back or converts to product $P$:

$$E + S \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} ES \overset{k_2}{\longrightarrow} E + P$$

Step 2: Quasi-Steady-State Assumption for [ES]

After a brief transient, the concentration of the enzyme-substrate complex reaches a quasi-steady state where its rate of formation equals its rate of breakdown:

$$\frac{d[ES]}{dt} = k_1[E][S] - k_{-1}[ES] - k_2[ES] \approx 0$$

Step 3: Solve for [ES] Using Conservation

Total enzyme is conserved: $[E_T] = [E] + [ES]$, so $[E] = [E_T] - [ES]$. Substituting into the steady-state equation and defining the Michaelis constant $K_M = (k_{-1} + k_2)/k_1$:

$$[ES] = \frac{[E_T][S]}{[S] + K_M}$$

Step 4: Derive the Michaelis-Menten Rate Law

The reaction rate (product formation rate) is $v = k_2[ES]$. Defining $V_{\max} = k_2[E_T]$ as the maximum rate when all enzyme is saturated:

$$v = \frac{V_{\max}[S]}{[S] + K_M}$$

Step 5: Temperature Dependence via Arrhenius

For marine extremophile enzymes, both $k_2$ (and hence $V_{\max}$) and $K_M$ depend on temperature through the Arrhenius equation. Cold-active enzymes achieve high $k_{\text{cat}}$ at low $T$ by having lower activation energies $E_a$, at the cost of reduced thermal stability:

$$k_2(T) = A \, e^{-E_a/(RT)}, \quad E_a^{\text{psychrophile}} < E_a^{\text{mesophile}} < E_a^{\text{thermophile}}$$