4.2 Phytoplankton

Step 1: Start from enzyme kinetics (Michaelis-Menten)

Nutrient uptake by a phytoplankton cell involves binding of the substrate $S$ to membrane transport proteins $E$, forming a complex $ES$ that is internalized:

$$E + S \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} ES \xrightarrow{k_2} E + S_{\text{internal}}$$

Step 2: Apply the steady-state approximation

At steady state, the rate of $ES$ formation equals its rate of breakdown. Setting $d[ES]/dt = 0$:

$$k_1 [E][S] = (k_{-1} + k_2)[ES]$$

Step 3: Define the half-saturation constant

Let $E_T = [E] + [ES]$ be the total transporter concentration, and define $K_s = (k_{-1} + k_2)/k_1$. Solving for $[ES]$:

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

Step 4: Derive the uptake rate

The uptake rate is $V = k_2 [ES]$. Defining $V_{\max} = k_2 E_T$ gives the Michaelis-Menten uptake equation:

$$V = V_{\max} \cdot \frac{S}{K_s + S}$$

Step 5: Connect uptake to growth (Monod model)

Monod (1949) proposed that the specific growth rate $\mu$ follows the same functional form as uptake, assuming growth is directly proportional to nutrient uptake and the internal cell quota is at steady state:

$$\mu = \mu_{\max} \cdot \frac{S}{K_s + S}$$

Step 6: Ecological interpretation of $K_s$

When $S = K_s$, the growth rate is exactly half of $\mu_{\max}$. The half-saturation constant determines competitive ability: species with low $K_s$ (e.g., Prochlorococcus, $K_s \approx 5$ nmol/L for NH₄⁺) dominate in oligotrophic waters, while species with high $\mu_{\max}$ but high $K_s$ (e.g., diatoms) win in nutrient-rich conditions. At low $S$:

$$\mu \approx \frac{\mu_{\max}}{K_s} \cdot S \quad \text{(linear, affinity-limited regime)}$$

The ratio $\alpha_{\text{nutrient}} = \mu_{\max}/K_s$ is the "nutrient affinity," analogous to the initial slope of the P-I curve.

Step 1: Define the photosynthetic apparatus model

Consider a photosystem with $n$ reaction centers. Each center can be in an open (ready) state $O$ or closed (processing) state $C$. Light drives transitions from open to closed at rate $\sigma I$ (where $\sigma$ is the effective absorption cross-section), and turnover returns closed centers to open at rate $\tau^{-1}$:

$$O \xrightarrow{\sigma I} C \xrightarrow{\tau^{-1}} O$$

Step 2: Steady-state fraction of open centers

At steady state, the fraction of open reaction centers $f_O$ satisfies:

$$\sigma I \cdot f_O = \tau^{-1}(1 - f_O) \quad \Longrightarrow \quad f_O = \frac{1}{1 + \sigma I \tau}$$

Step 3: Express photosynthetic rate

The photosynthetic rate is proportional to the rate of light absorption by open centers. The rate of electron transport per reaction center is $\sigma I \cdot f_O$, so the total rate (normalized to chlorophyll) is:

$$P = n \cdot \sigma I \cdot f_O = \frac{n \sigma I}{1 + \sigma I \tau}$$

Step 4: Identify P-I parameters

Rewriting in terms of conventional P-I parameters: $P_{\max} = n/\tau$ (maximum rate when all centers cycle at maximum turnover) and $\alpha = n\sigma$ (initial slope = quantum yield times absorption):

$$P = \frac{\alpha I}{1 + \alpha I / P_{\max}} = P_{\max} \cdot \frac{\alpha I / P_{\max}}{1 + \alpha I / P_{\max}}$$

Step 5: Approximate with the tanh model

The rectangular hyperbola above can be closely approximated by the smoother $\tanh$ formulation (Jassby & Platt, 1976), which has the same initial slope $\alpha$ and asymptote $P_{\max}$:

$$P = P_{\max} \cdot \tanh\left(\frac{\alpha I}{P_{\max}}\right)$$

Step 6: The light saturation index

The light saturation parameter $I_k$ is defined as the irradiance at the intersection of the initial slope with the saturated rate:

$$I_k = \frac{P_{\max}}{\alpha} = \frac{1}{\sigma \tau}$$

Below $I_k$, photosynthesis is light-limited ($P \approx \alpha I$); above $I_k$, it is enzyme-limited ($P \approx P_{\max}$). Shade-adapted species have low $I_k$ (large antenna, slow turnover), while high-light-adapted species have high $I_k$ (small antenna, fast turnover).

Step 7: Adding photoinhibition (Platt et al., 1980)

At very high irradiances, photodamage reduces the photosynthetic rate. Platt et al. added an inhibition term with coefficient $\beta$:

$$P = P_s \left(1 - e^{-\alpha I / P_s}\right) e^{-\beta I / P_s}$$

where $P_s$ is the hypothetical maximum without inhibition. The realized maximum occurs at $I_{\max} = (P_s/\alpha)\ln((\alpha+\beta)/\beta)$.