Converting solar energy to chemical energy — the light response curve
2026-02-26
You should know
That plants use light to fix carbon, and that some processes rise quickly at first and then level off instead of increasing forever.
You will learn
How the photosynthetic light-response curve works, why it saturates, and how that shape connects light to carbon uptake.
Why this matters
This chapter links plant biology to geography at much larger scales, including forests, crop growth, and the global carbon cycle.
If this gets hard, focus on…
The key pattern is simple: more light helps a lot at low levels, but eventually the system hits a limit and extra light matters less.
A single mature spruce tree in a boreal forest fixes roughly 20–30 kg of carbon per year under good growing conditions. The entire boreal forest — 1.4 billion hectares of it across Canada, Russia, and Scandinavia — fixes around 2 billion tonnes of carbon annually, making it one of the largest carbon sinks on the planet. But that productivity is not constant: it rises from near zero at dawn, peaks around midday when light is bright, then declines. On an overcast day it might be half the clear-sky rate. In deep shade under the canopy it falls nearly to zero. All of this variation traces back to a single curve: the photosynthetic light response.
The relationship between light intensity and photosynthesis rate is not linear. At low light, photosynthesis increases steeply — each additional photon is efficiently used. At high light, the curve bends over and saturates: the biochemical machinery runs as fast as it can regardless of how much more light arrives. This saturation function — a rectangular hyperbola — is the same mathematical form that describes enzyme kinetics (Michaelis-Menten), uptake of nutrients by roots, and the response of many biological systems to a limiting resource. Understanding its shape, its parameters, and how they vary with temperature and species is essential for any model that connects light availability to carbon fixation and primary productivity.
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?
Plot photosynthesis rate (A, μmol CO₂/m²/s) vs. light intensity (I, μmol photons/m²/s):
Three regions:
Shape: Rectangular hyperbola (or variants).
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
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).
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).
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).
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
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)
Below is an interactive photosynthesis simulator.
<label>
Plant type:
<select id="plant-type">
<option value="c3-crop" selected>C3 Crop</option>
<option value="c4-crop">C4 Crop</option>
<option value="tree">Temperate Tree</option>
<option value="shade">Shade Plant</option>
</select>
</label>
<label>
Quantum efficiency (φ):
<input type="range" id="phi-slider" min="0.02" max="0.08" step="0.01" value="0.05">
<span id="phi-value">0.05</span>
</label>
<label>
Max photosynthesis (A<sub>max</sub>, μmol/m²/s):
<input type="range" id="amax-slider" min="10" max="60" step="5" value="30">
<span id="amax-value">30</span>
</label>
<label>
Dark respiration (R<sub>d</sub>, μmol/m²/s):
<input type="range" id="rd-slider" min="0.5" max="5" step="0.5" value="2">
<span id="rd-value">2.0</span>
</label>
<label>
Current light (μmol/m²/s):
<input type="range" id="light-current-slider" min="0" max="2000" step="50" value="500">
<span id="light-current-value">500</span>
</label><h4>At Current Light Level:</h4>
<p><strong>Gross photosynthesis:</strong> <span id="a-gross"></span> μmol CO₂/m²/s</p>
<p><strong>Net photosynthesis:</strong> <span id="a-net"></span> μmol CO₂/m²/s</p>
<p><strong>Light compensation:</strong> <span id="light-comp"></span> μmol/m²/s</p>
<p><strong>Light saturation (90%):</strong> <span id="light-sat"></span> μmol/m²/s</p>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.
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-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)
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).
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.
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.
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
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)
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.
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}
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!
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.