X-ray diffraction studies indicate the existence of a novel doublestranded DNA helical conformation in which \(\Delta Z\) (the rise per base pair \()=0.32 \mathrm{nm}\) and \(P(\text { the pitch })=3.36 \mathrm{nm} .\) What are the other parameters of this novel helix: (a) the number of base pairs per turn, (b) \(\Delta \phi(\text { the mean rotation per base pair }),\) and (c) \(c(\) the true repeat)?

X-ray diffraction studies are a cornerstone method in molecular biology for determining the three-dimensional structure of molecules, especially crystalline solids. By directing X-rays at a crystalline sample and analyzing the pattern of rays that are diffracted, scientists can back-calculate the arrangement of atoms within the sample. In the context of DNA, X-ray diffraction studies have been instrumental in revealing the helical structure of the DNA molecule.

The pattern formed by the diffracted rays is called a diffraction pattern, which appears as a series of spots. These spots, when interpreted correctly, provide information about the spacing between layers of atoms and the angles at which these layers intersect. This information is crucial for constructing a detailed atomic model of the DNA helix. The pitch, rise, and other structural parameters of DNA can all be derived from carefully measuring diffraction patterns.

In a DNA helical conformation, 'base pairs per turn' refers to the number of nucleotide pairs that make up one complete revolution of the double helix. This number is a fundamental aspect of DNA structure as it affects the overall twist and compactness of the DNA molecule. For example, in the well-studied B-form of DNA, there are approximately 10.5 base pairs per turn.

The number of base pairs per turn is calculated by dividing the pitch, or the length of one complete helical turn, by the rise, which is the vertical distance between adjacent base pairs. As given in the example above, with a pitch (\( P \)) of 3.36 nm and a rise (\( \text{ΔZ} \) of 0.32 nm, we find approximately 10.5 base pairs per turn with the formula \( n = \frac{P}{\text{ΔZ}} \). Understanding this concept is critical for visualizing the spatial arrangement of the double helix and for explaining how DNA functions during replication and transcription.

The 'mean rotation per base pair' is another key parameter that describes DNA's helical conformation. It is the average angle by which the DNA helix rotates for every base pair that is added along the axis of the helix. This rotation, measured in degrees, helps in defining the twist of the DNA structure.

To calculate the mean rotation per base pair (\( \text{Δ}φ \) we divide 360 degrees (a full rotation) by the number of base pairs per turn. In the given example, by substituting approximately 10.5 base pairs per turn into the formula \( \text{Δ}φ = \frac{360}{n} \) we get a mean rotation of approximately 34.28 degrees per base pair. This indicates how tightly or loosely the DNA strands are wound around each other. A smaller number would suggest a tighter twist, while a larger number would imply a looser twist, potentially affecting the DNA's interactions with proteins and other molecules.