Question

The Size of Cells and Their Components A typical eukaryotic cell has a cellular diameter of \(50 \mu \mathrm{m}\). a. If you used an electron microscope to magnify this cell 10,000-fold, how big would the cell appear? b. If this cell were a liver cell (hepatocyte) with the same cellular diameter, how many mitochondria could the cell contain? Assume the cell is spherical; that the cell contains no other cellular components; and that each mitochondrion is spherical, with a diameter of \(1.5\) \(\mu \mathrm{m}\). (The volume of a sphere is \(4 / 3 \pi r^{3}\).) c. Glucose is the major energy-yielding nutrient for most cells. Assuming a cellular concentration of \(1 \mathrm{~mm}\)

Short answer

a) The magnified size is 500,000 \( \mu m \). b) The cell could contain approximately 36,977 mitochondria.

Step-by-step solution

Calculate the Magnified Size

The given cell diameter is 50 \( \mu m \). With a magnification of 10,000 times, the apparent size of the cell can be calculated as:\[ \text{Magnified size} = \text{original size} \times \text{magnification} = 50 \mu m \times 10,000 = 500,000 \mu m\]So, the cell would appear to be 500,000 \( \mu m \) in diameter.

Calculate the Volume of the Cell

Assuming the cell is spherical, use the formula for the volume of a sphere, \( \frac{4}{3} \pi r^3 \), where the radius \( r = \frac{50}{2} = 25 \mu m \).\[\text{Volume of the cell} = \frac{4}{3} \pi (25)^3\]Calculate this to find:\[\text{Volume of the cell} \approx 65,450 \mu m^3\]

Calculate the Volume of a Mitochondrion

Each mitochondrion is also spherical, with a diameter of 1.5 \( \mu m \), so its radius \( r = \frac{1.5}{2} = 0.75 \mu m \).\[\text{Volume of a mitochondrion} = \frac{4}{3} \pi (0.75)^3\]Calculate this to find:\[\text{Volume of a mitochondrion} \approx 1.77 \mu m^3\]

Calculate the Maximum Number of Mitochondria

To find the number of mitochondria that could fit in the cell, divide the volume of the cell by the volume of a mitochondrion, assuming the cell contains no other components:\[\text{Number of mitochondria} = \frac{65,450}{1.77} \approx 36,977\]Therefore, about 36,977 mitochondria could fit in the cell.

Key concepts

Eukaryotic Cells

Eukaryotic cells are complex structures that make up multicellular organisms like plants, animals, and fungi. Unlike prokaryotic cells, which lack a nucleus, eukaryotic cells have their genetic material enclosed within a nuclear membrane. This distinct nucleus is a hallmark of eukaryotic cells, allowing for greater organization and regulation of genetic information.

These cells are larger and more intricate than their prokaryotic counterparts. They contain numerous organelles, such as mitochondria for energy production, endoplasmic reticulum for protein and lipid synthesis, and Golgi apparatus for packaging and sorting molecules. This compartmentalization allows eukaryotic cells to perform specialized functions efficiently.

• Contain a nucleus and membrane-bound organelles
• Found in multicellular organisms
• Enable complex and regulated biological processes

Cellular Structures

Within eukaryotic cells, cellular structures play critical roles in maintaining cellular function and integrity. These structures include organelles like mitochondria, ribosomes, and the cytoskeleton.

Each organelle has a unique role that contributes to the overall operation of the cell. Mitochondria, for example, are known as the powerhouses of the cell due to their role in converting nutrients into energy-rich molecules like ATP. The cytoskeleton provides structural support and facilitates cell movement.

• Mitochondria: energy conversion
• Ribosomes: protein synthesis
• Cytoskeleton: structural support and movement

Exploring these structures helps us understand how cells operate at a microscopic level, balancing the needs for energy, growth, and reproduction.

Electron Microscopy

Electron microscopy is a powerful technique used to visualize the detailed structure of cells at very high magnifications. Unlike traditional light microscopes, electron microscopes use beams of electrons, which have much shorter wavelengths, allowing for higher resolution images. This makes it possible to see even the smallest cellular components, such as the intricate architecture of organelles and molecules.

There are two main types of electron microscopes: transmission electron microscopes (TEM) and scanning electron microscopes (SEM). TEM provides detailed internal images of thin cell sections, revealing internal structures with high clarity. SEM, on the other hand, is used for viewing surface details, providing three-dimensional images.

• TEM: detailed internal cell structure
• SEM: surface structure and 3D imaging

This technology is invaluable in research and diagnostics, allowing scientists to study cell biology at the molecular level.

Cellular Dimensions

Understanding cellular dimensions is essential for grasping the scale and functionality of cells. In the case of eukaryotic cells, which are typically much larger than prokaryotic cells, their size can range from 10 to 100 micrometers in diameter.

These dimensions have significant implications for the cell's ability to house various organelles. Larger cell volumes provide space for numerous mitochondria, as seen in the exercise, which is crucial for energy-intensive cells like liver cells. Calculating cellular volumes and the volume of organelles, such as mitochondria, helps in understanding how cells are structured to meet their metabolic needs. With the formula for the volume of a sphere, \ \( \frac{4}{3} \pi r^3 \ \), scientists can determine how many organelles can fit within a cell.

• Larger cell size means more organelles
• Essential for energy production and cellular processes
• Volume calculations help understand cell capacity

These calculations and insights are critical for fields like cell biology and bioengineering, where cellular design impacts functionality.