Growth rates and conditions for growth

How fast does a plant grow?

A bamboo shoot can grow tens of centimetres in a day. A tree in a pot may grow only a few centimetres in a year. To compare them, we need a number. The number is the growth rate, the increase in growth per unit of time.

A plant or an organ can make new cells in two ways. So there are two kinds of growth, arithmetic and geometric.

Arithmetic growth

In arithmetic growth, a cell divides by mitosis, and then only one of the two daughter cells goes on dividing. The other one differentiates and matures.

The simple example is a root that lengthens at a steady rate. It adds the same length in every hour. If you plot its length against time, you get a straight line.

• $L_t$ is the length at time $t$.
• $L_0$ is the length at time zero, the start.
• $r$ is the growth rate, the length added in one unit of time.

Try it. A root is $4\ \text{cm}$ long and lengthens by $0.5\ \text{cm}$ a day. After $10$ days it is $4 + 0.5 \times 10 = 9\ \text{cm}$ long.

Geometric growth

In geometric growth, a cell divides, and both daughter cells go on dividing. The number of dividing cells doubles at each step, so the growth speeds up.

This does not go on for ever. The growth has three stages.

1. Lag phase. The start is slow. The cells are getting ready.
2. Log phase, also called the exponential phase. Growth rises fast, because there are more and more dividing cells.
3. Stationary phase. Nutrients run short, so growth slows and then stops.

If you plot size against time, you get a typical sigmoid curve, a curve in the shape of the letter S.

The sigmoid curve describes cells in a culture, a seedling, a single organ such as a leaf or a fruit, and an annual plant over its whole life.

The growth of the log phase is written like this.

• $W_1$ is the final size. It may be weight, height or the number of cells.
• $W_0$ is the initial size, at the start of the time.
• $r$ is the growth rate, called the relative growth rate.
• $t$ is the time of growth.
• $e$ is the base of natural logarithms.

Look at what the equation says. The final size $W_1$ depends on the initial size $W_0$. A big plant makes more new matter than a small one, because it has more material to start with. The value of $r$ is also called the efficiency index. It tells how well the plant makes new plant material.

You should find that the two curves start together and then part. The red curve bends upward and keeps climbing. The blue line stays straight. A larger rate $r$ makes the parting faster.

Absolute and relative growth rate

There are two ways of comparing the growth of two living things.

• The absolute growth rate is the total growth per unit of time. We measure the gain and compare it.
• The relative growth rate is the growth per unit of time, shown on a common basis. For example, the gain per unit of the starting size.

The two do not always agree. Here is an example with two leaves.

Both leaves gain $5\ \text{cm}^2$ in a week, so their absolute growth rate is the same. But leaf A has doubled its area, a relative growth of $\dfrac{5}{5} \times 100 = 100\%$ a week. Leaf B has grown by only $\dfrac{5}{20} \times 100 = 25\%$ a week. Leaf A is growing faster for its size.

A useful number: a plant part that doubles in a fixed time has $W_1/W_0 = 2$. Then $rt = \ln 2 = 0.693$, so the relative growth rate is $r = \dfrac{0.693}{t}$. A part that doubles in $4$ days has $r = 0.173$ per day.

Conditions for growth

Growth does not happen on its own. Look at what a seedling needs.

• Water. Cells enlarge when they take in water and become turgid, which means swollen and firm. Water is also the medium in which the enzymes of growth work. A seed that is kept dry does not grow.
• Oxygen. It is needed to release metabolic energy from food by respiration. This energy drives the growth activities.
• Nutrients. The macro and micro elements are used to make protoplasm, and some of them also act as a source of energy.
• Temperature. Every plant grows best in a certain range of temperature, its optimum. Too cold or too hot, and the enzymes work badly, and growth slows or stops.
• Light and gravity. These are signals from the surroundings. They affect certain phases of growth, for example the direction in which a shoot or a root grows.

Try it in Python

The program grows a plant part by both rules, and prints the two columns side by side.

Practise with new numbers

Common mistakes

• Mixing arithmetic and geometric growth. In arithmetic growth only one daughter cell divides again, and the graph is a straight line. In geometric growth both do.
• Thinking the S-curve is the same as the straight line. The S-curve has a lag, a log and a stationary phase.
• Saying that the stationary phase has no growth because the cells die. Growth slows because nutrients or space run short.
• Saying that equal absolute growth means equal relative growth. Compare the gain with the starting size.
• Leaving out the starting size. In $W_1 = W_0 e^{rt}$ the final size depends on $W_0$.

Remember these

• Growth rate: the increase in growth in a unit of time.
• Arithmetic growth: one daughter cell divides again, a straight line, $L_t = L_0 + rt$.
• Geometric growth: both daughter cells divide again, a sigmoid curve with a lag, a log and a stationary phase, $W_1 = W_0 e^{rt}$.
• $r$ is the relative growth rate, also called the efficiency index.
• Absolute growth rate: the total gain in a unit of time. Relative growth rate: the gain compared with the starting size.
• Growth needs water, oxygen, nutrients and a suitable temperature. Light and gravity are signals.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. What happens to the two daughter cells in arithmetic growth?

2. Name the three phases of the sigmoid growth curve, in order.

3. What does each symbol mean in $W_1 = W_0 e^{rt}$?

4. A leaf grows from $10\ \text{cm}^2$ to $16\ \text{cm}^2$ in $3$ days. What is its absolute growth rate, and what is its relative growth rate per day, as a per cent of its starting area?

5. Give the reason why oxygen and water are both needed for growth.