Types of Growth (Arithmetic and Geometric), Sigmoid Curve, Growth Rate, Factors Affecting Plant Growth, Practice Problems and FAQs

We all know that growth is the gradual development process that occurs in all living organisms. Can you tell me, which part of the body of an organism undergoes growth? Yes, Growth is defined as the irreversible increase in size of a cell or organ of an individual. So every part of the organism shows growth in one or another way. 

Every living organism exhibits growth in their life and hence, it is one of the characteristic features of living beings. Examples include growth of zygote into a plant. 

                   Fig: Seed germination

Starting from seed germination, to production of new seedlings and its development to a mature plant, it shows indefinite growth. Have you ever noticed why some of the plant organs grow constantly but some show slow growth? How does this take place? You are not getting the answer. To get answers for all these, let's understand more about the growth in plants in detail in this article.

Table of contents

• Types of growth on the basis of increase in cells
• Growth rate
• Factors affecting plant growth
• Practice Problems
• FAQs

Types of growth on the basis of increase in cells

Growth in plant is of two types as follows:

• Arithmetic growth
• Geometric growth

Arithmetic growth

In this type of growth, after mitotic division (a cell divides into two cells) one cell stays meristematic and continues division. But the other cell attains maturity and stops dividing. It is given in the below diagram. Here the cells increase from 1 to 2, 3, 4. 5 and so on. In this type of growth, the increase in the number of cells occurs in arithmetic progression. The rate of growth remains constant here. For example in roots, the elongation occurs at a constant rate.

Features of arithmetic growth

The following are the common features of arithmetic growth:

• Rate of growth is constant
• Growth is in arithmetic progression
• One daughter cell remains meristematic

                                    Fig: Arithmetic growth

Graphical representation

If plotted on a graph, arithmetic growth can be represented as a straight line where the X-axis represents the number of cells and Y-axis represents the time.

                  Fig: Graph of arithmetic growth

Mathematical expression 

Equation for a straight line can be represented as given below: 

y = mx + c

Similarly, slope of graph plotted for arithmetic growth is represented by the following equation:

                 Fig: Slope of linear graph of arithmetic growth

Gt = rt + G0

Where, 

Gt = Growth at time ‘t’ 

G₀ = Growth at time ‘zero’ 

r = Growth rate or it is the change in parameter or change in time

On plotting the length of the organ at different times, a linear curve is obtained. 

Lt = rt + L0

Where,

L_t = Length at time ‘t’ 

L₀ = Length at time ‘zero’ or at the beginning

r = Growth rate or elongation per unit time

Geometric growth

In this type of growth, a cell divides into two daughter cells and all the progeny cells retain the ability to divide. The rate of growth varies in different species and in different organs. For example, in certain species of plants like cacti, the rate of growth is slow. But in many plants, the growth rate is phenomenally rapid. For example, the young leaf sheath of banana grows for a time at the rate of almost three inches per hour. The rate of growth is not constant here.

                                                                   Fig: Geometric growth

Graphical representation

If plotted on a graph, geometric growth can be represented as an exponential growth curve where the X-axis represents the number of cells and Y-axis represents the time.

             Fig: Geometric growth curve

Mathematical expression

The equation for exponential growth can be represented as:

W₁ = W₀ert

where,

W₁ = Size at time t (weight, height, number etc.)

W₀ = Initial size

r = Growth rate

t = Time of growth 

e = Base of natural logarithms

Sigmoid Curve

Exponential growth can occur only in an ideal situation, that means the nutrients are unlimited. This is not possible in real life. In the real environment, nutrients are limited and a sigmoid growth curve is normally formed. The sigmoid growth curve is a S-shaped curve that depicts geometric growth. It contains three phases as follows:

• Lag phase
• Log phase or exponential phase
• Stationary phase

                              Fig: Sigmoid growth curve

Phases of sigmoid curve

The various phases of sigmoid curve are as follows:

Lag phase 

This is the initial phase of growth. The growth is very slow in this phase although the resources are available in abundance. This is due to the fact that only a few cells or individuals are available for growth in this phase. The cells take time to adjust to the new environment they are exposed here in this phase. 

                                                     Fig: Lag phase

Log phase or the exponential phase 

The growth rate is increased drastically in this phase, as the number of cells or individuals available for growth or division and resources are in abundance. After some time, resources start to decrease due to heavy consumption by the growing cells. By the end of this phase, there is a drastic decrease in resources. This means that not all individuals will have enough resources to grow or divide anymore. This signifies the end of the log phase.

                                                 Fig: Log phase 

Stationary phase 

In this phase, growth becomes saturated. This happens as the resources are limited and deposition of toxic waste materials occurs. The growth of the cells eventually slows down.

                                              Fig: Stationary phase

Arithmetic plus the geometric growth

The embryos of plants and animals exhibit geometric growth initially and later switch to arithmetic growth. Initially, there is rapid growth in the embryos of plants and animals. This decreases later when some cells differentiate and lose the ability to divide.

 
         Fig: Human embryo and plant embryo

Growth rate

The pace at which plant growth occurs is known as growth rate. 

Absolute growth rate

Absolute growth is considered the actual growth or the total growth per unit time. For the absolute growth rate, if we plot a graph then it appears bell shaped. The peak in this graph is formed when the growth is fastest. The first half of the curve indicates the period of increasing growth while the second half of the curve depicts the period of decreasing growth. It can be calculated using following formula:

AGR = Final parameter - Initial parameter/Time

Relative growth rate

It is defined as the growth per unit time per initial parameter. Growth that is quantified in comparison terms, such as percentage growth, is referred to as ‘relative growth’. It can be calculated using the following formula:

RGR = Final Parameter - Initial parameter/Time x Initial parameter

Comparison of absolute and relative growth rate

The comparison of absolute and relative growth rate can be done by the following experiment using two leaves. 

                        Fig: Comparison of absolute and relative growth rate

Here two leaves A and B of different sizes of area are taken into consideration. They show exact absolute increases in the area in a given time. But one of them shows a much higher relative growth rate. Here both A and B have grown by 5 cm2 in one day. But the initial size of A was 5 cm2 while that of the leaf B was 50 cm2. Here their absolute growth is the same, but the relative growth is faster in leaf A. 

                            Fig: Increase in the surface area by leaf A and leaf B

Factors affecting plant growth

Growth of an organism involves the processes of increased protoplasm synthesis, cell division, cell enlargement and cell differentiation. So, all the factors influencing the biosynthetic machinery like, the availability of O₂, H₂O, optimum light and temperature, minerals (both macro and micro) and absence of stress conditions, also affect growth of an organism. The major factors that affect the growth of plants are as follows:

Water

Water contributes to growth in many ways. It helps in cell enlargement and acts as a medium for enzymatic activities carried out during growth. Plants thrive when they have enough water. They react to water scarcity as well.

Oxygen

Oxygen acts as a medium for release of metabolic energy required for growth. 

Nutrients

Nutrients act as a source of energy for growth and aids the synthesis of protoplasm of new cells. Plants normally require enough nutrients to grow properly. Plant growth is influenced by the quantity and quality of nutrients provided. Few essential minerals required by plants for their growth and development are calcium, nitrogen, potassium, phosphorus, magnesium, sulphur, iron, etc.

The deficiencies of minerals like nitrogen, iron and sulphur result in stunted growth, insufficient chlorophyll in leaves, cell senescence etc.

Temperature

Temperature has a major role in the acceleration of growth and development. Certain seeds germinate when the temperature is favourable for germination. Enzymes can function effectively only in a certain range of temperatures. A temperature of 28℃ - 30℃ is optimum for the proper growth in most plants. Higher temperature above 45℃ hinders growth due to excessive transpiration, denaturation of enzymes and coagulation of protoplasm. Lower temperature inactivates enzymes and also increases the density of protoplasm.

Light

Light is one of the crucial requirements for photosynthesis in plants which ultimately results in providing energy to plants for growth and development. Many physiological processes in plants are influenced by the intensity, duration, and quality of light. Light is required for the processes like synthesis of photosynthetic pigments, tissue differentiation, and photosynthesis. Its absence results in etiolation. Light also influences certain stages of growth. The phenomenon is called photoperiodism. Direction of light is the main factor that determines the orientation in leafy shoots. 

Gravity

It determines the direction of shoot and root growth. 

                                                 Fig: Factors affecting plant growth

Practice Problems

Q 1. Which of the following statements is true regarding geometric growth?

a. Growth rate increases in the lag phase.
b.Growth is slow in the log phase. 
c. rowth slows down in the stationary phase.

1. I and II
2. II and III
3. III only
4. I only

Answer: The growth rate is not constant in geometric growth. In the lag phase in plants the growth rate is normally slow. Even in the presence of abundant resources, the growth rate does not increase as the number of cells available for multiplication is limited and they are getting adjusted to the environment. In the log phase, with abundant supply of resources, the growth rate increases exponentially. Due to continuous utilisation of resources in the log phase, there is a reduction in the resource availability. Now the growth of the cells slows down and they reach the stationary phase. Hence, option c is correct.

Q 2. Match the following.

a. A - II, B - III, C - IV, D - I
b. A - I, B - IV, C - III, D - II
c. A - I, B - III, C - IV, D - II
d. A - III, B - IV, C - II, D - I

Answer: Temperature influences the growth in plants by helping in germination of seeds, and effective action of enzymes in physiological processes of plants. Light acts as a major source in photosynthesis in plants, the end product of photosynthesis is food which provides energy to the plant body. Sufficient amount and quality of nutrients helps in synthesis of protoplasm in plant cells. Plants use water as a medium for most of the enzymatic reactions that occur during plant growth. Hence, option b is correct.

Q 3. Suppose a leaf’s initial area is 20 cm2 and it grows to the final leaf area of 30 cm2 in unit time. What will be the relative growth rate of the leaf in percentage?

Answer: Relative growth rate is defined as the growth per unit time per initial parameter. It is calculated by the formula: 

RGR = Final parameter - Initial parameter/Time x Initial parameter

Relative growth rate per unit time will be (Final parameter - Initial parameter) /Time x Initial parameter

= Final parameter - Initial parameter/ Initial parameter

(putting the value of time as 1 unit)

Substituting the values;

30 cm2 - 20 cm2/20 cm2

= 10 cm2/20 cm2

= 0.5

In percentage it will be

0.5 100=50 %

Thus, the relative growth rate of the leaf per unit time will be 50%.

Q 2. Mention the difference between arithmetic and geometric growth?

Answer: One of the daughter cells is meristematic and divides in arithmetic growth, while the other differentiates towards maturity and stops dividing. For example, arithmetic growth in roots is defined as the elongation of roots at a constant pace. Geometric growth is distinguished by moderate growth in the early phases and rapid growth in the later stages. In the lag phase, the growth in plants is slow. In the log phase, the resources or nutrients are surplus and the plant grows exponentially. Due to the decreased resources or nutrients, the growth slows down and eventually the cells go into stationary phase.

Q 4. Describe the sigmoid curve in plants?

Answer: Plants or plant parts do not usually grow at the same rate throughout their lives. It might be slow at times. A characteristic S-shaped curve emerges when the increase in cell number (growth rate) is plotted against time. This is known as a sigmoid growth curve. Sigmoid growth curve shows the geometric growth in plants. The three phases in the sigmoid curve are as follows:

• Lag phase - In this phase, the growth of plant cells is essentially slow as the number of cells or individuals available are low.
• Log phase- In this phase, the growth of the plant increases due to the surplus amount of resources available.
• Stationary phase- In this phase, the quantity of resources decreases due to consumption and eventually the growth in plant cells slows down.

                                            Fig: Sigmoid curve

FAQs

Q 1. What distinguishes an arithmetic growth rate from others?
Answer: The rate of growth is constant in arithmetic progression. Between the two progeny cells, only one cell is allowed to divide here. Hence one continues to divide, while the other is stopped in its tracks and begins to develop, differentiate, and mature.

Q 2. State a plant organ that shows both arithmetic and geometric growth?
Answer: Embryo develops inside the seed, initially exhibits geometric growth when the cells are actively dividing and growing. Later on, it switches to arithmetic growth as most of the cells differentiate and lose their ability to divide further.

Q 3. What effect does temperature have on plant growth?
Answer: Germination increases to a point due to increasing temperatures. Germination begins to drop once the seeds reach ideal temperatures, which vary depending on the plants. Most plant processes, such as photosynthesis, transpiration, respiration, germination, and blooming are influenced by temperature. All these processes increase as temperature rises up to a particular point. But beyond the optimum temperature, enzymes will get denatured and all the processes in plants will get affected. 

Q 4. What is considered as the meristematic phase of growth in plants?
Answer: The meristematic phase is considered as the formative phase. The tips of roots and shoots normally exhibit continuous growth and hence are meristematic. During this phase of growth, mitotic divisions of the pre-existing cells happen to form new cells.

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