Limiting Factors, Inverse Square Law & Greenhouse Economics (HT)

Why limiting factors interact

Photosynthesis depends on several conditions at the same time. The plant needs light energy and carbon dioxide, while its enzyme-controlled reactions require a suitable temperature. It must also have enough chlorophyll to absorb light. These factors do not act independently: the rate is restricted by whichever factor is preventing any further increase at that moment. This is the limiting factor.

Imagine that a leaf has plenty of carbon dioxide and a suitable temperature but receives very little light. Light is limiting, so increasing the light intensity increases the rate. Eventually, however, the plant may be receiving more light than it can use with the available carbon dioxide. At that point carbon dioxide becomes limiting. Adding still more light will not increase the rate.

The limiting factor can therefore change as conditions change. Improving one condition may remove one restriction only to reveal another. Temperature can also become limiting: at a low temperature, enzyme-controlled reactions proceed slowly, whereas temperatures above the optimum may reduce the rate because enzymes denature. A low amount of chlorophyll can limit how much light energy is absorbed even when the surroundings are bright.

Reading graphs with two factors

A graph of photosynthesis rate against one factor often rises and then reaches a plateau. While the curve is rising, the factor on the horizontal axis is limiting. At the plateau, increasing that factor has no further effect, so a different factor is limiting.

Two curves usually represent experiments in which the horizontal-axis factor changes while a second condition is kept at two different levels. Compare the curves at the same horizontal-axis value. If the curves separate, the condition named in their labels is affecting the rate.

Data for Interaction of carbon dioxide concentration and light intensity

In the graph, both curves rise at low carbon dioxide concentrations because carbon dioxide is limiting. At higher concentrations, the curve at 50 lux reaches a lower plateau: there is now enough carbon dioxide, but the lower light intensity limits the rate. At 100 lux, more light energy is available, so the rate can rise further before another factor becomes limiting.

The same reasoning applies to curves drawn for different temperatures. If a curve at a suitable higher temperature lies above one at a lower temperature, temperature was restricting the lower rate. However, do not assume that an even higher temperature must always produce a higher curve: above the optimum, denaturation can reduce the rate.

Deciding between three factors

A graph can combine three factors. For example, its horizontal axis might show carbon dioxide concentration, while different curves represent combinations of temperature and light intensity. Work systematically:

1. Use the rising part of each curve to decide whether the horizontal-axis factor is limiting.
2. At a plateau, compare curves whose labels differ in only one condition.
3. If increasing that condition produces a higher rate, it was limiting on the lower curve.
4. If changing it makes no difference, look for another condition that differs or could be limiting.

Suppose increasing carbon dioxide no longer raises the rate. Raising the temperature then produces a higher plateau, while light intensity is unchanged. This is evidence that temperature was limiting under the original conditions. If raising temperature makes no difference but increasing light produces a higher plateau, light was limiting instead. Conclusions must be based on the comparison shown, rather than on a rule that one particular factor is always limiting.

The inverse square law for light

For a lamp treated as a point source, its light spreads over an increasing area as distance increases. The area over which the light is spread is proportional to the square of the distance, so intensity follows the inverse square law:

$I \propto \frac{1}{d^2}$

For two positions, this can be written as:

$\frac{I_2}{I_1}=\left(\frac{d_1}{d_2}\right)^2$

If the distance from the lamp is doubled, the intensity becomes

$\left(\frac{1}{2}\right)^2=\frac{1}{4}$

of its original value. If the distance is trebled, it becomes $1/9$ of the original intensity. Conversely, halving the distance makes the intensity four times as great.

Be careful about the biological consequence. The inverse square law describes the change in light intensity. It does not guarantee that the photosynthesis rate changes by the same proportion. If light is limiting, reducing intensity is likely to reduce the rate. If carbon dioxide concentration or temperature is already limiting, a change in light may have little or no effect on the rate.

Manipulating greenhouse conditions

A greenhouse allows growers to control some limiting factors. Heating can maintain a more suitable temperature, artificial lamps can extend or strengthen illumination, and carbon dioxide can be added to the air. By removing a genuine limiting factor, a grower can increase photosynthesis, glucose production and plant growth. This may produce a larger or faster crop and increase sales income.

These changes cost money. Heaters and lamps require energy, equipment must be bought and maintained, and supplying carbon dioxide also has a cost. Once another factor becomes limiting, spending more on the factor already supplied sufficiently gives little or no extra yield. For example, brighter lamps are wasteful if carbon dioxide is now limiting.

The economically best conditions are therefore not necessarily those giving the absolute maximum possible rate of photosynthesis. Growers seek conditions where the value of the additional crop is greater than the additional production cost. Data on crop yield, energy prices and the effects of changing heat, light or carbon dioxide can be used to judge cost effectiveness. A small further rise in photosynthesis may not be worthwhile if achieving it costs more than the extra crop can earn.

Control also has to respond to changing conditions. Extra lighting brings no photosynthetic benefit during an already bright period if another factor limits the crop, while heating may be valuable on a cold day but unnecessary when the greenhouse is already warm. Effective management targets the factor currently limiting photosynthesis while avoiding overheating and unnecessary expenditure.

Interacting limiting factors

Light intensity, carbon dioxide concentration, temperature and the amount of chlorophyll interact. The factor currently restricting any further increase in photosynthesis is the limiting factor. Removing one limitation can cause a different factor to become limiting.

• Rising curve: the horizontal-axis factor is limiting.
• Plateau: that factor is no longer limiting; another factor restricts the rate.
• Higher comparison curve: the condition that differs between the curves affected the lower rate.
• Excessive temperature can make the rate fall because enzymes denature.

For graphs involving three factors, compare curves that differ in only one stated condition wherever possible.

Inverse square law

For light from a point source:

$I \propto \frac{1}{d^2}$