Question

A chemist has an \(\mathrm{E}\). coli bacterium cell of cylindrical shape. It is \(2 \mu\) (microns) long and ly in diameter and weighs \(2 \times 10^{-12} \mathrm{~g}\). (a) How many lipid molecules are present, assuming their average molecular weight to be 700 and lipid content to be \(2 \%\) ? (b) If the cell contains 15,000 ribosomes, what percent of the volume do they occupy? You may assume them to be spherical with a diameter of \(180 \AA\); also \(10_{4} \AA=1\) micron.

Short answer

(a) The number of lipid molecules is calculated as follows: \[ Number\,of\,Lipid\,Molecules = \frac{2\% \times 2 \times 10^{-12} g}{700} \approx 5.7 \times 10^{6} \] (b) The percentage of volume occupied by ribosomes is found by: \[ Percentage \approx \left( \frac{15000\times\frac{4}{3}\pi(90 \AA)^{3}}{\pi (1 \mu m)^{2}(2 \mu m)(10^{4} \AA)^{3}} \right) \times 100 \approx 1.9\% \]

Step-by-step solution

Determining the Number of Lipid Molecules

The weight of lipid in the bacterium is given by \(2 \%\) of the total weight of the bacterium. This equals \(2\% \times 2 \times 10^{-12} g\). There is a simple formula we can use to find the number of molecules, which is the total weight of the substance divided by the weight of one molecule (molecular weight). So, the number of lipid molecules can be calculated by the lipid weight divided by the weight of one lipid molecule: \[ Number\,of\,Lipid\,Molecules = \frac{Lipid\,weight} {Lipid\,molecular\,weight} \] Keep in mind to convert percentages to decimals before calculation and noting that the molecular weight of the lipid was given as 700.

Calculating the Volume of Ribosome Occupancy

Firstly, we need to determine the volume of a single ribosome using the formula for the volume of a sphere (\(\frac{4}{3}\pi r^{3}\)) considering a ribosome as a sphere. The radius would be half the diameter, which is \(90 \AA\). Next, we calculate the total volume occupied by the ribosomes which is just the number of ribosomes times the volume of a single ribosome. To find the percentage occupied by ribosomes, we need to know the total volume of the bacterium. The bacterium, being cylindrical in shape has a volume given by \(\pi r^{2}h\). Substituting the given radius and height in microns, we can find this. Convert to the correct unit (\(1 \mu m = 10^{4} \AA\)). Finally, we calculate the percentage of volume occupied by ribosomes which is given by: \[ Percentage = \left( \frac{Volume\,of\,Ribosome} {Volume\,of\,the\,bacterium} \right) \times 100 \]

Key concepts

Molecular Weight

Understanding molecular weight is essential when studying chemistry and biophysics, especially in exercises involving calculations at the molecular level. Molecular weight, often termed as molecular mass, is the sum of the atomic weights of all atoms in a molecule. It's measured in units called daltons (Da) or atomic mass units (amu). One dalton is defined as one twelfth of the mass of a carbon-12 atom. In practical terms, the molecular weight gives us an idea of the 'size' of a molecule. With molecular weight, scientists can calculate the number of molecules in a given mass, as molecular weight relates the mass of a molecule to the Avogadro constant (approximately 6.022 x 10²³ molecules/mole).

For the E. coli bacterium cell in our example, knowing that the average molecular weight of lipid molecules is 700 allows us to calculate the number of lipid molecules, an integral step when trying to figure out chemical composition and reactions involving the cell.

Lipid Molecules Calculation

Calculating the number of lipid molecules provides insight into the biochemical makeup of cells. Lipids are a group of naturally occurring molecules that include fats, waxes, and certain vitamins, among others. They are important for cell structure and function. To calculate the number of lipid molecules in an E. coli cell, you need to determine the total lipid mass and divide it by the molecular weight of individual lipid molecules. You do this by taking the cell's total mass, calculating the mass percentage that is composed of lipids (usually given or determined experimentally), and then using the formula
\[\text{{Number of Lipid Molecules}} = \frac{{\text{{Lipid weight}}}}{{\text{{Lipid molecular weight}}}\]
The result is a foundational step in biochemistry calculations for understanding cellular composition.

Volume of a Sphere

The volume of a sphere is a frequently encountered calculation in various scientific fields. Using the formula
\[\frac{4}{3}\pi r^{3}\]
where \(r\) is the sphere's radius, you can calculate how much space a spherical object occupies. For students of bacteriology, this equation is particularly helpful when considering cellular components like ribosomes, which are roughly spherical in structure. These organelles’ presence and density are critical for understanding the cell's protein synthesis capabilities. Applying the volume of a sphere formula properly demands awareness of units (converting diameters to radii, and ensuring consistent unit usage throughout calculations), which refines the precision of the volume determination.

Percentage Volume Occupancy

Percentage volume occupancy is an expression of how much space within a given volume is taken up by a particular substance or object. To find the percentage volume occupancy of ribosomes in the E. coli bacterium cell, one must first calculate the total volume of all ribosomes and then compare it to the volume of the bacterium itself. The calculation is given by the formula
\[\text{{Percentage}} = \left( \frac{{\text{{Volume of Ribosomes}}}}{{\text{{Volume of the Bacterium}}}} \right) \times 100\]
By expressing the result as a percentage, students can visually understand the extent to which ribosomes fill the internal space of the bacterium. This concept is pertinent in cell biology, as it reflects on the structural and functional space distribution within a cell, which affects how the cell operates. The consideration of such spatial occupancy is vital when interpreting how densely packed the cellular environment is and can influence interpretations of cell functionality.