Problem 41 - Calculus for Biology and Medicine

Question

The diameter of a typical bacterium is about \(0.5\) to \(1 \mu \mathrm{m}\). An exception is the bacterium Epulopiscium fishelsoni, which is about \(600 \mu \mathrm{m}\) long and \(80 \mu \mathrm{m}\) wide. The volume of \(E\). fishelsoni is about order(s) of magnitude larger than that of a typical bacterium. (Hint: Approximate the shape of a typical bacterium by a sphere and the shape of \(E\). fishelsoni by a cylinder.)

Step-by-Step Solution

Answer

Epulopiscium fishelsoni is about 7 to 8 orders of magnitude larger in volume than a typical bacterium.

Step 1: Volume of a Typical Bacterium

Assume the shape of a typical bacterium is spherical with a diameter of \(0.5\) to \(1 \, \mu \mathrm{m}\). The volume \(V_s\) of a sphere is calculated using the formula: \[ V_s = \frac{4}{3} \pi r^3 \] where \(r\) is the radius. For a diameter of \(0.5 \, \mu \mathrm{m}\), the radius \(r\) is \(0.25 \, \mu \mathrm{m}\), and for a diameter of \(1 \, \mu \mathrm{m}\), the radius \(r\) is \(0.5 \, \mu \mathrm{m}\). This results in the volumes being approximately:\[ V_s = \frac{4}{3} \pi (0.25)^3 \approx 0.065 \mu \mathrm{m}^3 \]\[ V_s = \frac{4}{3} \pi (0.5)^3 \approx 0.524 \mu \mathrm{m}^3 \] Thus, the volume of a typical bacterium ranges from about \(0.065\) to \(0.524 \, \mu \mathrm{m}^3\).

Step 2: Volume of Epulopiscium fishelsoni

Approximating the shape of \(E. fishelsoni\) as a cylinder, we use the formula for the volume \(V_c\) of a cylinder:\[ V_c = \pi r^2 h \] where \(r\) is the radius and \(h\) is the height (or length). The diameter is \(80 \, \mu \mathrm{m}\), so the radius is \(40 \, \mu \mathrm{m}\), and the length is \(600 \, \mu \mathrm{m}\). Plugging these values in:\[ V_c = \pi (40)^2 (600) \] Calculating, the volume is approximately:\[ V_c = \pi \times 1600 \times 600 \approx 3,019,200 \mu \mathrm{m}^3 \] Therefore, the volume of \(E. fishelsoni\) is about \(3,019,200 \mu \mathrm{m}^3\).

Step 3: Comparing Orders of Magnitude

To find the difference in orders of magnitude, compare the volume of \(E. fishelsoni\) with the range for typical bacteria:- Typical bacterium ranges from \(0.065\) to \(0.524 \, \mu \mathrm{m}^3\).- \(E. fishelsoni\) is \(3,019,200 \, \mu \mathrm{m}^3\).To determine orders of magnitude, divide volumes and take logarithms:- From smallest: \( \frac{3,019,200}{0.065} \approx 46,449,231\).- From largest: \( \frac{3,019,200}{0.524} \approx 5,762,977\).Taking the base 10 logarithm gives approximately an increase of 7 to 8 orders of magnitude.

Key Concepts

Bacterial Shapes

Bacteria can come in various shapes and sizes, making them fascinating to study. Most commonly, bacteria have three main shapes:

• **Cocci**: Round or spherical.
• **Bacilli**: Rod-shaped, resembling cylinders.
• **Spirilla**: Spiral or corkscrew shaped.

These simple forms can combine in various ways to create chains or clusters. Understanding the shape is essential for categorizing bacteria and determining how best to analyze or measure them. For instance, when calculating volumes, approximating a bacterium's shape accurately is crucial to ensuring that our measurements are realistic. Simpler shapes such as spheres and cylinders can help make these calculations more straightforward.

Cylindrical Volume Calculation

A cylindrical shape is quite common in both nature and technology. Calculating the volume of a cylinder is a straightforward process and can be applied in various scenarios, including estimating the volume of certain bacteria. The formula to find the volume of a cylinder is:\[V_c = \pi r^2 h\]Here, \(r\) is the radius, and \(h\) is the height or length of the cylinder. By plugging in values for \(r\) and \(h\), you can easily determine the volume. For example, for the bacterium _Epulopiscium fishelsoni_, which resembles a long cylinder, identifying these measurements accurately can offer insights into its enormous volume compared to other bacteria.

Spherical Volume Calculation

The volume of a sphere is distinctively different from a cylindrical shape, offering a different approach to calculations. The formula for calculating the volume of a sphere is:\[V_s = \frac{4}{3} \pi r^3\]In this equation, \(r\) represents the radius of the sphere. The cubic nature of this equation means that even small changes in the radius significantly affect the calculation of a sphere's volume. When considering typical bacteria like cocci, we often approximate them as small spheres, making this formula highly useful in microbiology to estimate sizes and compare different bacterial species' volumes.

Orders of Magnitude

Orders of magnitude help compare quantities that differ vastly in size. It essentially indicates how much larger or smaller one number is compared to another, often using powers of ten. When comparing the volumes of typical bacteria to that of _Epulopiscium fishelsoni_, this concept comes into play.

• A typical bacterium might have a volume around 0.065 to 0.524 \(\mu m^3\),
• While _Epulopiscium fishelsoni_ around 3,019,200 \(\mu m^3\).

By taking the ratio of these values and applying logarithms, we can calculate the orders of magnitude. In this case, we find that _Epulopiscium fishelsoni_ is about 7 to 8 orders of magnitude larger than a typical bacterium, illustrating its incredibly large size in the microscopic world.

Epulopiscium fishelsoni

_Epulopiscium fishelsoni_ is an extraordinary bacterium due to its immense size. Unlike typical bacteria, which are usually microscopic, this bacterium can be as large as a grain of salt, measuring approximately 600 \(\mu m\). Found in the intestines of specific surgeonfish, it is cylindrical in shape, with its diameter approximately 80 \(\mu m\) long.This gigantic size poses unique biological questions and adaptations, as larger bacteria may face different environmental challenges and life processes compared to their smaller peers. The fact that this bacterium requires a size similar to tiny multicellular organisms hints at fascinating evolutionary traits and adaptive strategies in its marine environment.