Question

The High Rate of Bacterial Metabolism Bacterial cells have a much higher rate of metabolism than animal cells. Under ideal conditions, some bacteria double in size and divide every \(20 \mathrm{~min}\), whereas most animal cells under rapid growth conditions require 24 hours. The high rate of bacterial metabolism requires a high ratio of surface area to cell volume. a. How does the surface-to-volume ratio affect the maximum rate of metabolism? b. Calculate the surface-to-volume ratio for the spherical bacterium Neisseria gonorrhoeae (diameter \(0.5 \mu \mathrm{m}\) ), responsible for the disease gonorrhea. The surface area of a sphere is \(4 \pi r^{2}\). c. How many times greater is the surface-to-volume ratio of Neisseria gonorrhoeae compared to that of a globular amoeba, a large eukaryotic cell (diameter 150 \(\mu \mathrm{m}\) )?

Short answer

a. Higher surface-to-volume ratios allow faster metabolism. b. The ratio for Neisseria gonorrhoeae is 12.00. c. Neisseria gonorrhoeae's ratio is 300 times greater than that of a globular amoeba.

Step-by-step solution

Understand the Concept

The surface-to-volume ratio is crucial in determining the rate of metabolism because it influences how efficiently a cell can exchange resources and waste with its environment. A higher ratio means more surface area relative to volume, allowing faster exchange processes, supporting higher metabolic rates.

Calculate Surface Area of Bacterium

Calculate the surface area of Neisseria gonorrhoeae using the formula for the surface area of a sphere: \[ \text{Surface area} = 4 \pi r^2 \] Given the diameter is 0.5 µm, the radius \( r = 0.25 \mu \text{m} \). Substitute this into the formula to get the surface area: \[ 4 \pi (0.25)^2 \approx 0.7854 \mu m^2 \].

Calculate Volume of Bacterium

Calculate the volume of Neisseria gonorrhoeae using the formula for the volume of a sphere: \[ \text{Volume} = \frac{4}{3} \pi r^3 \] Using the radius \( r = 0.25 \mu \text{m} \), the volume is: \[ \frac{4}{3} \pi (0.25)^3 \approx 0.06545 \mu m^3 \].

Calculate Surface-to-Volume Ratio for Bacterium

Use the calculated surface area and volume to determine the surface-to-volume ratio: \[ \text{Surface-to-volume ratio} = \frac{\text{Surface area}}{\text{Volume}} \] \[ \frac{0.7854 \mu m^2}{0.06545 \mu m^3} \approx 12.00 \].

Calculate Surface Area of Amoeba

For a globular amoeba with diameter 150 µm, first determine its radius: \( r = 75 \mu \text{m} \). Use the surface area formula for a sphere: \[ 4 \pi (75)^2 \approx 70785.74 \mu m^2 \].

Calculate Volume of Amoeba

Calculate the volume of the amoeba using its radius: \[ \frac{4}{3} \pi (75)^3 \approx 1,767,145.87 \mu m^3 \].

Calculate Surface-to-Volume Ratio for Amoeba

Determine the surface-to-volume ratio: \[ \frac{70785.74 \mu m^2}{1,767,145.87 \mu m^3} \approx 0.04 \].

Compare Ratios

Compute how many times greater the bacterium's ratio is compared to the amoeba's: \[ \frac{12.00}{0.04} = 300 \]. The surface-to-volume ratio of Neisseria gonorrhoeae is 300 times greater than that of the globular amoeba.

Key concepts

Surface-to-Volume Ratio

The surface-to-volume ratio is a key player in understanding the efficiency of bacterial metabolism. This ratio essentially measures how much surface area is available per unit volume within a cell. In simpler terms, it's about how much potential a cell has to get nutrients and expel waste in relation to its size.

• A higher surface-to-volume ratio means that for each unit of volume, there is more surface area available to interact with the environment.
• This higher ratio allows cells to exchange resources and waste more quickly, enhancing metabolic activities.
• Smaller cells, like bacteria, tend to have a larger surface-to-volume ratio. This enables them to have faster metabolic rates as compared to larger cells.

The quick growth and division potential in bacteria is closely linked to these efficient exchange processes. In the case of Neisseria gonorrhoeae, which has a very small diameter, the surface-to-volume ratio is significantly higher than in larger cells like amoebas.

Metabolic Rate

Metabolic rate refers to the speed at which organisms transform nutrients into energy and building blocks necessary for life.
Bacteria are well-known for their high metabolic rates, partly explained by their high surface-to-volume ratios.

• This rapid metabolism allows bacteria to double in size in a very short time under ideal conditions.
• High metabolic rates are advantageous for bacteria because they can utilize available resources quickly, outcompeting other organisms in the same environment.

However, it's important to note that higher metabolic rates require more efficient and larger exchange areas, which are represented by the surface area of the bacterial cells.
For example, the sphere shape of many bacteria like Neisseria gonorrhoeae is perfect for maximizing while still maintaining a compact size.

Cell Biology

In cell biology, understanding the function and structure of cells is crucial.
is heavily influenced by cellular structures such as the cell membrane, which facilitates the exchange of substances between the inside of the cell and its environment.

• Cell membranes are semipermeable, allowing selective substances to enter and exit based on the cell's needs.
• The efficiency of this exchange is greater in cells with a higher surface-to-volume ratio.

Bacteria’s, having simpler cellular structures compared to eukaryotic cells, optimize their cell biology to maintain a high metabolic rate.
Their small size and structure exploit this ratio for rapid growth and reproduction, which demonstrates the close relationship between form and function in the biological world.
This intricate balance allows bacteria to thrive in a variety of environments, showcasing the essential principles of cell biology.