Factors affecting photosynthesis (light, CO2, temperature)
The rate of photosynthesis depends on multiple environmental factors, but at any given moment
one factor often has the strongest limiting effect. A practical way to model this is to treat
light, CO2, and temperature as three separate response functions, then combine them
into a single predicted net rate.
Concept: limiting factors
A limiting factor is the variable that is farthest from its “saturating” or optimal range.
If light is very low, increasing CO2 may not help much, because the light reactions cannot supply
enough ATP and NADPH. Likewise, if temperature is far from the enzyme optimum, carbon fixation slows
even if light and CO2 are high.
Model used in this calculator
The calculator uses a simple multiplicative model for gross photosynthesis and then subtracts
a constant respiration offset Rd to estimate net photosynthesis.
The outputs are best interpreted as relative rates unless Pmax and Rd are supplied in real units.
\[
\begin{aligned}
P_{\mathrm{gross}} &= P_{\max}\cdot f_L(I)\cdot f_C(C)\cdot f_T(T) \\
P_{\mathrm{net}} &= P_{\mathrm{gross}} - R_d
\end{aligned}
\]
The multiplicative form is a common teaching model: each factor acts like a “fraction of the maximum”
that reduces the achievable rate. Real plants are more complex, but this is very useful for exploration.
1) Light response curve (saturating)
Photosynthesis rises with light at low intensity because more photons drive the light reactions.
At higher intensities the response saturates as electron transport and carbon fixation become limiting.
A simple saturating function is:
\[
\begin{aligned}
f_L(I) &= \frac{I}{I + K_I}
\end{aligned}
\]
- I = light intensity (µmol photons m−2 s−1 or a relative scale)
- KI = half-saturation constant (the value of I where fL = 0.5)
A useful reference point is the “90% saturation” level. Solving \(f_L=0.9\) gives:
\[
\begin{aligned}
0.9 &= \frac{I}{I + K_I}
\quad\Rightarrow\quad
I \approx 9K_I
\end{aligned}
\]
2) CO2 response curve (saturating)
Increasing CO2 generally increases carboxylation and reduces photorespiration in C3 plants,
so the net rate rises and then saturates when other processes become limiting.
The calculator uses:
\[
\begin{aligned}
f_C(C) &= \frac{C}{C + K_C}
\end{aligned}
\]
- C = CO2 concentration (ppm in this calculator)
- KC = half-saturation constant (C where fC = 0.5)
Similarly, “90% saturation” for CO2 is:
\[
\begin{aligned}
C \approx 9K_C
\end{aligned}
\]
3) Temperature response curve (bell-shaped)
Photosynthesis has an optimal temperature range because enzyme kinetics speed up with temperature
to a point, but at higher temperatures proteins become less efficient, membranes change, and
photorespiration often increases in C3 plants. A simple bell-shaped function is a Gaussian peak:
\[
\begin{aligned}
f_T(T) &= \exp\!\left(
-\frac{1}{2}\left(\frac{T - T_{\mathrm{opt}}}{\sigma_T}\right)^2
\right)
\end{aligned}
\]
- T = temperature (°C)
- Topt = optimum temperature for the peak
- σT = “width” parameter controlling how quickly the response falls away from the optimum
The calculator also reports an approximate “near-optimal” band where \(f_T \ge 0.9\).
From the Gaussian:
\[
\begin{aligned}
f_T(T)\ge 0.9
&\Rightarrow
\exp\!\left(-\frac{1}{2}\left(\frac{T - T_{\mathrm{opt}}}{\sigma_T}\right)^2\right)\ge 0.9 \\
&\Rightarrow
\left|T - T_{\mathrm{opt}}\right|
\le
\sigma_T\sqrt{2\ln\!\left(\frac{1}{0.9}\right)}
\end{aligned}
\]
Net photosynthesis and Rd
Even in the light, leaves respire. This is represented here by a constant offset Rd:
\[
\begin{aligned}
P_{\mathrm{net}} &= P_{\max}\cdot f_L\cdot f_C\cdot f_T - R_d
\end{aligned}
\]
If \(P_{\mathrm{net}}<0\), the model predicts that respiration exceeds gross photosynthesis under those conditions.
Compensation points (simple interpretation)
The calculator estimates “compensation points” for light and CO2 by solving \(P_{\mathrm{net}}=0\)
while holding the other factors fixed at your current values. For light:
\[
\begin{aligned}
0 &= P_{\max}\cdot f_C(C)\cdot f_T(T)\cdot \frac{I}{I + K_I} - R_d \\
\Rightarrow\quad
\frac{I}{I + K_I} &= \frac{R_d}{P_{\max} f_C f_T}
\end{aligned}
\]
If the right-hand side is between 0 and 1, the light compensation point is:
\[
\begin{aligned}
I_{\mathrm{comp}} &= \frac{A K_I}{1 - A}
\quad\text{where}\quad
A=\frac{R_d}{P_{\max} f_C f_T}
\end{aligned}
\]
The CO2 compensation point is analogous:
\[
\begin{aligned}
C_{\mathrm{comp}} &= \frac{B K_C}{1 - B}
\quad\text{where}\quad
B=\frac{R_d}{P_{\max} f_L f_T}
\end{aligned}
\]
These are simplified “model compensation points” (not full physiological CO2 compensation points that include
photorespiration biochemistry explicitly). They are still useful for understanding thresholds.
How the calculator identifies the limiting factor
Each factor \(f_L, f_C, f_T\) ranges from 0 to 1. The calculator labels the limiting factor as the smallest
of the three values:
\[
\begin{aligned}
\text{limiter}=\arg\min\{f_L, f_C, f_T\}
\end{aligned}
\]
This is a clear, practical rule for learning: the “shortest bar” indicates the factor farthest from its best case.
Reading the graphs
-
Rate vs Light: shows the predicted net rate as light changes (CO2 and temperature fixed at your values).
The highlighted point is your current setting.
-
Rate vs CO2: shows how raising CO2 changes net rate (light and temperature fixed).
-
Rate vs Temperature: shows the bell-shaped temperature dependence (light and CO2 fixed).
-
Limiter dashboard: three bars show \(f_L\), \(f_C\), and \(f_T\). The smallest bar corresponds to the limiter label.
-
Heatmap: explores net rate across a grid of light and CO2 values at your chosen temperature.
Limitations
-
Real photosynthesis depends on many additional factors: humidity, leaf age, nitrogen status, stomatal conductance,
water potential, and acclimation history.
-
The temperature response of photosynthesis can be asymmetric and may shift with species and growth conditions.
The Gaussian peak is a helpful teaching approximation.
-
CO2 and light responses can also change with temperature and water stress; the multiplicative form assumes
separable effects for simplicity.
