Assume that a viral DNA molecule is a 50-µm-long circular strand with a uniform 20-Å diameter. If this molecule is contained in a viral head that is a 0.08-µm-diameter sphere, will the DNA molecule fit into the viral head, assuming complete flexibility of the molecule? Justify your answer mathematically.

Verified step by step guidance

1. Convert all given dimensions to consistent units, preferably micrometers (µm). For example, convert the DNA diameter from angstroms (Å) to micrometers using the conversion factor: 1 Å = 10\textsuperscript{-4} µm.
2. Calculate the volume of the viral DNA molecule by treating it as a cylinder with length equal to the DNA length and diameter equal to the DNA diameter. Use the formula for the volume of a cylinder: \(V_{DNA} = \pi \times \left(\frac{d}{2}\right)^2 \times L\), where \(d\) is the diameter and \(L\) is the length of the DNA.
3. Calculate the volume of the viral head, which is a sphere, using the formula: \(V_{head} = \frac{4}{3} \pi \times \left(\frac{D}{2}\right)^3\), where \(D\) is the diameter of the viral head.
4. Compare the volume of the DNA molecule (\(V_{DNA}\)) with the volume of the viral head (\(V_{head}\)). If \(V_{DNA} \leq V_{head}\), then the DNA can fit inside the viral head assuming complete flexibility; otherwise, it cannot.
5. Provide a conclusion based on the comparison, justifying mathematically whether the DNA molecule fits inside the viral head.

Key Concepts

DNA Molecular Dimensions and Volume

Understanding the physical size of the DNA molecule involves calculating its volume based on length and diameter. DNA is modeled as a cylinder, so volume is found using the formula π × (radius)^2 × length. Converting units consistently is essential to compare DNA volume with the viral head volume.

Volume of a Sphere

The viral head is a sphere, and its volume is calculated using the formula (4/3)πr³, where r is the radius. Knowing the viral head's volume allows comparison with the DNA volume to determine if the DNA can physically fit inside.

Concept of Molecular Flexibility and Packing

Assuming complete flexibility means the DNA can bend and coil without volume change, allowing it to occupy space efficiently. This assumption lets us focus on volume comparison rather than shape constraints, simplifying the problem to whether the DNA volume is less than or equal to the viral head volume.