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Chapter 1: Problem 10

To get a feeling for the size of cells (and to practice the use of the metric system), consider the following: the human brain weighs about \(1 \mathrm{kg}\) and contains about \(10^{12}\) cells. Calculate the average size of a brain cell (although we know that their sizes vary widely), assuming that each cell is entirely filled with water \(\left(1 \mathrm{cm}^{3} \text { of water weighs } 1 \mathrm{g}\) ). What \right. would be the length of one side of this average- sized brain cell if it were a simple cube? If the cells were spread out as a thin layer that is only a single cell thick, how many pages of this book would this layer cover?

Short Answer

Expert verified
The average side length of a brain cell cube is approximately 1 碌m, and it would cover about 1608 pages.

Step by step solution

01

Determine Volume of Brain Cells

Since the human brain weighs about 1 kg, and 1 cm鲁 of water weighs 1 g, the volume of the brain is approximately 1000 cm鲁. The entire brain is assumed to be filled with cells, so the combined volume of these brain cells is also 1000 cm鲁.
02

Calculate Average Volume per Cell

Given that there are \(10^{12}\) cells in the brain, we find the average volume of a single cell by dividing the total volume of the brain by the number of cells. Therefore, the average volume per cell is \(\frac{1000 \text{ cm}^3}{10^{12}} = 10^{-9} \text{ cm}^3\).
03

Determine Side Length of Cube-Shaped Cell

Assuming each cell is a cube, the volume (\(V\)) of a cube is given by \(V = a^3\), where \(a\) is the side length. Hence, \(a^3 = 10^{-9} \text{ cm}^3\). Solving for \(a\), we have \(a = (10^{-9})^{1/3} \text{ cm} \approx 10^{-3} \text{ cm} = 1 \mu m\).
04

Calculate Area Covered by a Single Cell Layer

If the cells are spread out in a single layer, the area covered by each cell (being a square) is \(a^2 = (10^{-3} \text{ cm})^2 = 10^{-6} \text{ cm}^2\). With \(10^{12}\) cells, the total area covered by a single cell thick layer is \(10^{12} \times 10^{-6} \text{ cm}^2 = 10^6 \text{ cm}^2\).
05

Convert Area to Number of Pages

Assuming each book page is approximately 21 cm x 29.7 cm, the area of one page is 621.7 cm虏. Therefore, the total number of pages covered is \(\frac{10^6 \text{ cm}^2}{621.7 \text{ cm}^2} \approx 1608\) pages.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Metric System
The metric system is a decimal-based system of measurement. It is used worldwide, especially in scientific contexts, because of its simplicity and precision. Here, the metric system helps us understand the size of cells by using units like kilograms for mass and cubic centimeters and micrometers for volume and length.
  • Mass: 1 kilogram (kg) is defined as 1000 grams (g).
  • Volume: 1 cubic centimeter (cm鲁) is the volume that one gram of water occupies at 4掳C.
  • Length: 1 centimeter (cm) is 10 millimeters (mm), and 1 micrometer (碌m) is 1/1000 of a millimeter.
In this exercise, understanding the metric system allows us to express brain mass, the volume of the brain, and the dimensions of brain cells consistently and accurately. For example, the brain weighs about 1 kg, which corresponds to 1000 cm鲁 of water, and the dimensions of a small cell can be expressed in micrometers.
Brain Cell Volume
To estimate brain cell volume, we assume that each brain cell is filled with water. This simplification is reasonable since brain cells, like all cells, are mostly composed of water. Given that the entire human brain, weighing about 1 kg, is assumed to be filled with these cells, the total volume attributed to these cells is about 1000 cm鲁, equivalent to the volume of 1 L of water.
We then calculate the average volume for a single brain cell.
  • Given there are 1012 cells in the brain, we find the average volume of one cell as 1000 cm鲁 divided by 1012.
  • This results in a very tiny volume per cell: 10-9 cm鲁. Such a small volume reflects the minuscule size of each cell.
Although this is an average, real brain cells vary widely in size and shape depending on their type and function.
Cube-Shaped Cell
Conceptualizing a brain cell as a cube simplifies calculations of its dimensions from its volume. In geometry, for a cube, the volume is the cube of its side length. This means if a cell has a volume of 10-9 cm鲁, each side of the cube (denoted as \( a \)) must satisfy \( a^3 = 10^{-9} \ \text{cm}^3 \).
Solving this using cube roots, we have:
  • \( a = (10^{-9})^{1/3} \text{ cm} \)
  • \( a \approx 10^{-3} \text{ cm} = 1 \ \mu m \)
So, each side of the average brain cell is approximately 1 碌m long. Micrometers are especially useful in cell biology because they provide a more intuitive understanding of cell sizes, which are not visible to the naked eye.
Surface Area Calculation
Calculating surface area for a geometric shape like a cube gives insight into how much physical space a cell occupies when spread out. For our cube-shaped brain cell with a side length of 1 碌m, its surface area is calculated by squaring the side and multiplying by six (since a cube has six faces).
  • The area of one face of the cell is: \( (10^{-3} \text{ cm})^2 = 10^{-6} \text{ cm}^2 \).
  • Thus, the full cube has a surface area of: \( 6 \times 10^{-6} \text{ cm}^2 \).
When considering the entire single cell layer formed by 1012 such cells, the total area becomes 106 cm虏. This is a large area and helps illustrate how a vast number of minuscule cells can collectively cover significant space, equivalent to almost 1600 pages.

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Most popular questions from this chapter

"Life" is easy to recognize but difficult to define. According to one popular biology text, living things: 1\. Are highly organized compared to natural inanimate objects. 2\. Display homeostasis, maintaining a relatively constant internal environment. 3\. Reproduce themselves. 4\. Grow and develop from simple beginnings. 5\. Take energy and matter from the environment and transform it. 6\. Respond to stimuli. 7\. Show adaptation to their environment. Score a person, a vacuum cleaner, and a potato with respect to these characteristics.

Draw to scale the outline of two spherical cells, one a bacterium with a diameter of \(1 \mu \mathrm{m}\), the other an animal cell with a diameter of \(15 \mu \mathrm{m}\). Calculate the volume, surface area, and surface-to-volume ratio for each cell. How would the latter ratio change if you included the internal membranes of the cell in the calculation of surface area (assume internal membranes have 15 times the area of the plasma membrane)? (The volume of a sphere is given by \(4 \pi r^{3} / 3\) and its surface by \(4 \pi r^{2},\) where \(r\) is its radius.) Discuss the following hypothesis: "Internal membranes allowed bigger cells to evolve."

Suggest a reason why it would be advantageous for eukaryotic cells to evolve elaborate internal membrane systems that allow them to import substances from the outside, as shown in Figure \(1-24.\)

When bacteria are grown under adverse conditions, i.e., in the presence of a poison such as an antibiotic, most cells grow and proliferate slowly. But it is not uncommon that the growth rate of a bacterial culture kept in the presence of the poison is restored after a few days to that observed in its absence. Suggest why this may be the case.

Mutations are mistakes in the DNA that change the genetic plan from the previous generation. Imagine a shoe factory. Would you expect mistakes (i.e., unintentional changes) in copying the shoe design to lead to improvements in the shoes produced? Explain your answer.
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