modelling Level 3

Photosynthesis and Light Response

Photosynthesis converts light energy into chemical energy (sugars). The rate increases with light intensity but saturates at high light—a classic rectangular hyperbola response. This model derives the light response curve, introduces quantum efficiency and saturation, and connects photosynthesis to gross primary productivity.

Prerequisites: rectangular hyperbola, saturation, quantum efficiency, michaelis menten

26 Feb 2026 Updated 12 Mar 2026 12 min read

1. The Question

How much carbon does a forest fix per day?

Photosynthesis is the process where plants use light energy to convert CO₂ and water into sugars and oxygen:

$$6\text{CO}2 + 6\text{H}_2\text{O} + \text{light} \to \text{C}_6\text{H}{12}\text{O}_6 + 6\text{O}_2$$

The rate depends on:

Light intensity (more light → more photosynthesis, up to a point)
CO₂ concentration (more CO₂ → more carbon fixed)
Temperature (warmer → faster enzymes, but too hot damages proteins)
Water availability (drought closes stomata, limiting CO₂ uptake)

The mathematical question: How do we model the relationship between light intensity and photosynthesis rate?

2. The Conceptual Model

The Light Response Curve

Plot photosynthesis rate ($A$, μmol CO₂/m²/s) vs. light intensity ($I$, μmol photons/m²/s):

Three regions:

Low light (< 200 μmol/m²/s): Photosynthesis increases linearly with light
- Light is limiting
- Slope = quantum efficiency ($\phi$)
Moderate light (200–1000 μmol/m²/s): Photosynthesis increases but sub-linearly
- Transition zone
- Enzyme capacity becoming limiting
High light (> 1000 μmol/m²/s): Photosynthesis saturates
- Asymptotes to maximum rate ($A_{\text{max}}$)
- Light no longer limiting; enzymes at maximum capacity

Shape: Rectangular hyperbola (or variants).

Key Parameters

Light compensation point ($I_c$):
Light intensity where photosynthesis equals respiration ($A = 0$).
Below this, plant loses more carbon through respiration than it gains through photosynthesis.

Quantum efficiency ($\phi$):
Initial slope of the light response curve (μmol CO₂ per μmol photons).
Typical value: $\phi \approx 0.05$ (20 photons needed per CO₂ fixed).

Light saturation point ($I_{\text{sat}}$):
Light intensity where photosynthesis reaches ~90% of maximum.
C3 plants: ~500–800 μmol/m²/s
C4 plants: ~1500–2000 μmol/m²/s

Maximum photosynthesis ($A_{\text{max}}$):
Asymptotic rate at very high light (limited by enzyme capacity, not light).
C3 crops: ~20–30 μmol/m²/s
C4 crops: ~40–60 μmol/m²/s

3. Building the Mathematical Model

Rectangular Hyperbola

The non-rectangular hyperbola model fits photosynthesis data well:

\[\theta A^2 - (\phi I + A_{\text{max}})A + \phi I A_{\text{max}} = 0\]

Where:

$A$ = photosynthesis rate (μmol CO₂/m²/s)
$I$ = light intensity (μmol photons/m²/s)
$\phi$ = quantum efficiency (dimensionless)
$A_{\text{max}}$ = maximum photosynthesis rate
$\theta$ = curvature parameter (0 ≤ θ ≤ 1)

Solving for $A$ (using quadratic formula):

\[A = \frac{(\phi I + A_{\text{max}}) - \sqrt{(\phi I + A_{\text{max}})^2 - 4\theta \phi I A_{\text{max}}}}{2\theta}\]

Special case: θ = 0 (rectangular hyperbola):

\[A = \frac{\phi I A_{\text{max}}}{\phi I + A_{\text{max}}}\]

This is the Michaelis-Menten form (common in enzyme kinetics).

Including Dark Respiration

Plants respire continuously (burning sugars for energy). Dark respiration ($R_d$) occurs day and night.

Net photosynthesis (observable rate):

\[A_{\text{net}} = A_{\text{gross}} - R_d\]

Light response with respiration:

\[A_{\text{net}}(I) = \frac{\phi I A_{\text{max}}}{\phi I + A_{\text{max}}} - R_d\]

At zero light ($I = 0$):

\[A_{\text{net}}(0) = -R_d\]

(Plant loses carbon through respiration)

Light compensation point ($I_c$): Where $A_{\text{net}} = 0$:

\[\frac{\phi I_c A_{\text{max}}}{\phi I_c + A_{\text{max}}} = R_d\]

Solve for $I_c$:

\[I_c = \frac{R_d A_{\text{max}}}{\phi(A_{\text{max}} - R_d)}\]

Typical value: $I_c \approx 20$–$50$ μmol/m²/s (dim indoor light).

From Leaf to Canopy

Leaf-level photosynthesis is measured in growth chambers under controlled light.

Canopy photosynthesis integrates over:

All leaves (sunlit and shaded)
Vertical light gradient (Beer’s law from Model 5)
Leaf area index (total leaf area per ground area)

Simplified canopy GPP:

\[\text{GPP} = \int_0^{LAI} A_{\text{leaf}}(I(L)) \, dL\]

Where $I(L) = I_0 e^{-k L}$ is light at cumulative leaf area $L$ (from Model 5).

4. Worked Example by Hand

Problem: A crop leaf has:

Quantum efficiency: $\phi = 0.05$
Maximum photosynthesis: $A_{\text{max}} = 30$ μmol/m²/s
Dark respiration: $R_d = 2$ μmol/m²/s

Calculate net photosynthesis at light intensities:
(a) $I = 100$ μmol/m²/s
(b) $I = 500$ μmol/m²/s
(c) $I = 2000$ μmol/m²/s

Solution

Using the rectangular hyperbola model:

\[A_{\text{gross}} = \frac{\phi I A_{\text{max}}}{\phi I + A_{\text{max}}}\] \[A_{\text{net}} = A_{\text{gross}} - R_d\]

(a) I = 100 μmol/m²/s

\[A_{\text{gross}} = \frac{0.05 \times 100 \times 30}{0.05 \times 100 + 30} = \frac{150}{5 + 30} = \frac{150}{35} = 4.29 \text{ μmol/m}^2\text{/s}\] \[A_{\text{net}} = 4.29 - 2 = 2.29 \text{ μmol/m}^2\text{/s}\]

(b) I = 500 μmol/m²/s

\[A_{\text{gross}} = \frac{0.05 \times 500 \times 30}{0.05 \times 500 + 30} = \frac{750}{25 + 30} = \frac{750}{55} = 13.6 \text{ μmol/m}^2\text{/s}\] \[A_{\text{net}} = 13.6 - 2 = 11.6 \text{ μmol/m}^2\text{/s}\]

(c) I = 2000 μmol/m²/s

\[A_{\text{gross}} = \frac{0.05 \times 2000 \times 30}{0.05 \times 2000 + 30} = \frac{3000}{100 + 30} = \frac{3000}{130} = 23.1 \text{ μmol/m}^2\text{/s}\] \[A_{\text{net}} = 23.1 - 2 = 21.1 \text{ μmol/m}^2\text{/s}\]

Light compensation point:

\[I_c = \frac{R_d A_{\text{max}}}{\phi(A_{\text{max}} - R_d)} = \frac{2 \times 30}{0.05 \times (30 - 2)} = \frac{60}{0.05 \times 28} = \frac{60}{1.4} = 42.9 \text{ μmol/m}^2\text{/s}\]

Interpretation:

At low light (100), photosynthesis is ~4× respiration → modest net gain
At moderate light (500), approaching half of maximum
At high light (2000), near saturation (~77% of max)

5. Computational Implementation

Below is an interactive photosynthesis simulator.

Plant type: Quantum efficiency (φ): 0.05 Max photosynthesis (A_max, μmol/m²/s): 30 Dark respiration (R_d, μmol/m²/s): 2.0 Current light (μmol/m²/s): 500

At Current Light Level:

Gross photosynthesis: μmol CO₂/m²/s

Net photosynthesis: μmol CO₂/m²/s

Light compensation: μmol/m²/s

Light saturation (90%): μmol/m²/s

Try this:

Switch to C4 crop: Higher $A_{\text{max}}$ and light saturation → more productive
Switch to shade plant: Lower $A_{\text{max}}$ but higher $\phi$ → efficient at low light
Increase respiration: Light compensation point rises → need more light to break even
Move current light slider: Watch how net photosynthesis changes
Low light (< 100): Linear increase (quantum-limited)
High light (> 1000): Saturates (enzyme-limited)

Key insight: The rectangular hyperbola captures the transition from light-limited to enzyme-limited photosynthesis.

6. Interpretation

C3 vs. C4 Photosynthesis

C3 plants (wheat, rice, trees):

Lower $A_{\text{max}}$ (~20–30 μmol/m²/s)
Saturate at moderate light (~500–800 μmol/m²/s)
Photorespiration at high temperature reduces efficiency

C4 plants (corn, sugarcane, millet):

Higher $A_{\text{max}}$ (~40–60 μmol/m²/s)
Saturate at high light (~1500–2000 μmol/m²/s)
CO₂ concentrating mechanism suppresses photorespiration

In full sunlight (2000 μmol/m²/s):

C3 plant: Photosynthesis ~25 μmol/m²/s (saturated)
C4 plant: Photosynthesis ~50 μmol/m²/s (still increasing)

C4 advantage in hot, bright environments (tropics, summer crops).

Shade Tolerance

Shade-tolerant plants (forest understory):

High $\phi$ (efficient at low light)
Low $A_{\text{max}}$ (don’t need high capacity)
Low $I_c$ (can survive deep shade)

Sun plants (crops, early successional):

Lower $\phi$ (less efficient per photon)
High $A_{\text{max}}$ (capitalize on full sun)
High $I_c$ (need more light to break even)

Gross Primary Productivity (GPP)

Daily GPP at canopy scale:

\[\text{GPP} = \int_{\text{day}} \sum_{\text{leaves}} A_{\text{net}}(I, T, ...) \, dt\]

Typical values:

Tropical rainforest: 6–8 g C/m²/day
Temperate forest: 3–5 g C/m²/day
Cropland: 4–10 g C/m²/day (depends on irrigation, fertilizer)
Grassland: 2–4 g C/m²/day
Desert: < 1 g C/m²/day

Annual global GPP: ~120 Pg C/year (120 billion tons of carbon fixed by photosynthesis annually).

7. What Could Go Wrong?

Assuming Light is Always Limiting

At low CO₂ or high temperature, photosynthesis can be limited by:

Rubisco capacity (CO₂ fixation enzyme)
RuBP regeneration (electron transport chain capacity)
Stomatal conductance (CO₂ diffusion into leaf)

Light saturation means light is no longer the limiting factor.

Ignoring Acclimation

Plants acclimate to their light environment:

Shade leaves: thin, high chlorophyll per mass → high $\phi$
Sun leaves: thick, more enzymes → high $A_{\text{max}}$

Same plant in different conditions has different parameters.

Forgetting Temperature Effects

Respiration increases exponentially with temperature (Q₁₀ ~ 2):

\[R_d(T) = R_d(20°\text{C}) \times 2^{(T-20)/10}\]

At high temperature:

Respiration increases faster than photosynthesis
Net productivity decreases
Heat stress can damage enzymes

Neglecting Water Stress

Drought → stomata close → CO₂ limited → photosynthesis drops even at high light.

Water-stressed plants have:

Lower $A_{\text{max}}$ (stomatal limitation)
Higher $I_c$ (respiration continues, photosynthesis suppressed)

8. Extension: Scaling to Ecosystems

Net Primary Productivity (NPP):

\[\text{NPP} = \text{GPP} - R_{\text{auto}}\]

Where $R_{\text{auto}}$ is autotrophic respiration (plant respiration from roots, stems, leaves at all times).

Net Ecosystem Productivity (NEP):

\[\text{NEP} = \text{NPP} - R_{\text{hetero}}\]

Where $R_{\text{hetero}}$ is heterotrophic respiration (decomposers breaking down dead organic matter).

Positive NEP: Ecosystem sequesters carbon (carbon sink)
Negative NEP: Ecosystem releases carbon (carbon source)

Next model will model soil moisture dynamics—how water availability controls transpiration and photosynthesis.

9. Math Refresher: The Rectangular Hyperbola

Form

\[y = \frac{ax}{b + x}\]

Asymptotes:

As $x \to 0$: $y \to 0$ (passes through origin)
As $x \to \infty$: $y \to a$ (horizontal asymptote)

Initial slope:

\[\frac{dy}{dx}\bigg|_{x=0} = \frac{a}{b}\]

Connection to Enzyme Kinetics

The Michaelis-Menten equation in biochemistry:

\[v = \frac{V_{\text{max}} [S]}{K_m + [S]}\]

Where:

$v$ = reaction rate
$V_{\text{max}}$ = maximum rate
$[S]$ = substrate concentration
$K_m$ = Michaelis constant (concentration at half-maximum rate)

Same mathematical form as photosynthesis light response!

Saturation

At $x = b$:

\[y = \frac{ab}{b + b} = \frac{a}{2}\]

Half-maximum rate occurs when $x = b$.

For photosynthesis: Half-saturation light $I_{1/2} = A_{\text{max}} / \phi$.

Summary

Photosynthesis converts light → chemical energy (CO₂ + H₂O → sugars + O₂)
Light response curve: Rectangular hyperbola shape
Low light: Linear increase (slope = quantum efficiency $\phi$)
High light: Saturates at $A_{\text{max}}$ (enzyme-limited)
Light compensation point ($I_c$): Where photosynthesis = respiration
Quantum efficiency: Typically $\phi \approx 0.05$ (20 photons per CO₂)
C4 plants have higher $A_{\text{max}}$ and light saturation than C3 plants
Gross Primary Productivity (GPP): Total carbon fixed by photosynthesis
Net Primary Productivity (NPP) = GPP - plant respiration