Question

Given two complementary strands of DNA containing 100 base pairs each, calculate the ratio of two separate strands to hydrogen-bonded double helix in solution at \(300 \mathrm{~K}\). (Hint: The formula for calculating this ratio is \(e^{-\Delta E / R T}\), where \(\Delta E\) is the energy difference between hydrogen-bonded double-strand DNAs and single-strand DNAs and \(R\) is the gas constant.) Assume the energy of hydrogen bonds per base pair to be \(10 \mathrm{~kJ} / \mathrm{mol}\).

Short answer

The ratio of individual strands to double helix is effectively 0, meaning nearly all DNA is double-stranded.

Step-by-step solution

Identify the Given Values

The problem states that each strand consists of 100 base pairs. The energy of hydrogen bonds per base pair is given as 10 kJ/mol. The temperature is 300 K, and the gas constant \( R \) is 8.314 J/(mol·K) or 0.008314 kJ/(mol·K).

Calculate Total Energy Difference \( \Delta E \)

Since there are 100 base pairs, and each contributes 10 kJ/mol of energy for a hydrogen bond, the total energy difference \( \Delta E \) for the double strand is:\[ \Delta E = 100 \text{ base pairs} \times 10 \text{ kJ/mol} = 1000 \text{ kJ/mol} \]

Insert Values Into the Formula

The formula to calculate the ratio of the separate strands to the double helix is:\[ e^{-\Delta E / (RT)} \]Substituting the values, we have \( \Delta E = 1000 \text{ kJ/mol} \), \( R = 0.008314 \text{ kJ/(mol·K)} \), and \( T = 300 \text{ K} \).

Solve for the Ratio

Calculate the exponent:\[ -\frac{\Delta E}{RT} = -\frac{1000}{0.008314 \times 300} \approx -400.48 \]Thus the ratio becomes:\[ e^{-400.48} \approx 0 \]

Consider the Physical Meaning

The value of the exponential function \( e^{-400.48} \) is effectively zero, indicating that nearly all DNA strands are in the double-stranded form at equilibrium.

Key concepts

Hydrogen Bonding in DNA

Hydrogen bonds are essential for the formation of the DNA double helix. These weak attractions occur between the hydrogen atom of one molecule and a highly electronegative atom, such as nitrogen or oxygen, of another molecule. In DNA, hydrogen bonds specifically form between nitrogenous bases of two complementary strands. Adenine (A) pairs with thymine (T) forming two hydrogen bonds, while guanine (G) pairs with cytosine (C), forming three hydrogen bonds.

These bonds may be weak individually, but collectively, they provide significant stability to the double helix structure. For a 100 base pair DNA sequence, the cumulative effect of these bonds sustains the structural integrity, making the DNA stable and resistant to unwinding at physiological temperatures.

Energy Calculation in DNA Stability

The energy involved in hydrogen bonding for DNA stabilization can be calculated when given key parameters such as the number of base pairs and energy per bond. In the problem, each base pair contributes 10 kJ/mol of hydrogen bonding energy. Thus, for a DNA strand with 100 base pairs, the total energy difference (\( \Delta E \)) contributing to the DNA's double helical formation is:

• \( \Delta E = 100 \text{ base pairs} \times 10 \text{ kJ/mol} = 1000 \text{ kJ/mol} \)

This calculation indicates the energetic cost associated with breaking all these bonds, effectively delineating the energy landscape between the single-stranded and double-stranded states of DNA.

Thermodynamics of Hydrogen Bonding

In thermodynamics, reactions are often explored through parameters such as energy differences and temperature, which affect the likelihood of formation or dissociation of molecular structures. The formula \( e^{-\Delta E / RT} \) allows us to calculate the ratio between single-stranded and double-stranded DNA. Here, \( \Delta E \) signifies the total energy needed to disrupt the hydrogen bonds, \( R \) is the gas constant (0.008314 kJ/(mol·K)), and \( T \) is the absolute temperature in Kelvin (300 K in this exercise).

This thermodynamic expression provides insight into the relative stability of two states: higher values of \( \Delta E \) indicate a stable double-stranded form since more energy is required to break it down, and a large negative exponent results in a near-zero ratio, suggesting a preference for the double-stranded state at equilibrium.

The Role of Base Pairs in DNA Structure

The base pairs in DNA play a critical role in maintaining the integrity and function of the DNA molecule. Each base pair consists of two complementary nucleotides bound together by hydrogen bonds. The pairing rule (A with T, and G with C) ensures uniformity in DNA's diameter and is vital for accurate replication and transcription processes.

The arrangement and number of base pairs in a DNA molecule determine its length and genetic information it can encode. With 100 base pairs, as stipulated in this exercise, a segment of DNA can encode about 33 amino acids, assuming each triplet codon corresponds to one amino acid. This highlights the significance not only of the chemical bonding between bases but also the genetic coding potential embedded within the sequence of base pairs.