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 the length of the DNA molecule from micrometers to angstroms. Since 1 µm = 10,000 Å, multiply 50 µm by 10,000 Å/µm.
2. Calculate the volume of the DNA molecule. Use the formula for the volume of a cylinder: V = πr^2h, where r is the radius (half of the diameter) and h is the length of the DNA.
3. Convert the diameter of the viral head from micrometers to angstroms. Since 1 µm = 10,000 Å, multiply 0.08 µm by 10,000 Å/µm.
4. Calculate the volume of the viral head. Use the formula for the volume of a sphere: V = (4/3)πr^3, where r is the radius (half of the diameter).
5. Compare the volume of the DNA molecule to the volume of the viral head to determine if the DNA can fit inside.

Key Concepts

Molecular Dimensions

Understanding the dimensions of both the viral DNA and the viral head is crucial. The DNA is a circular strand measuring 50 µm in length and 20 Å in diameter, while the viral head is a sphere with a diameter of 0.08 µm. Converting these measurements into consistent units is essential for accurate comparison.

Geometric Fitting

Geometric fitting involves determining whether one shape can fit within another based on their dimensions. In this case, we need to assess if the circular DNA strand can be accommodated within the spherical viral head, considering the DNA's flexibility and the head's volume.

Mathematical Justification

Mathematical justification requires applying formulas to calculate the volume of the viral head and the effective space occupied by the DNA. By comparing these values, we can determine if the DNA can fit inside the viral head, thus providing a quantitative basis for the conclusion.