Problem 23 - Biochemistry: A Short Course

Question

Uniqueness. The human genome contains 3 billion base pairs arranged in a vast array of sequences. What is the minimum length of a DNA sequence that will, in all probability, appear only once in the human genome? You need consider only one strand and may assume that all four nucleotides have the same probability of appearance.

Step-by-Step Solution

Answer

The minimum length is 16 base pairs.

Step 1: Understanding the Probability

We want a sequence that is unique in the genome, which means it should appear only once in 3 billion base pairs. If any random sequence of a certain length can be present in the genome at any location, we need to consider the probability of it appearing.

Step 2: Estimate Total Possible Sequences

A DNA sequence can be made up of any of the four nucleotides: adenine (A), thymine (T), guanine (G), or cytosine (C). For a sequence of length \( n \), there are \( 4^n \) possible different sequences. This number must be larger than the number of base pairs (3 billion) for it to likely appear only once by chance.

Step 3: Setting Up the Equation

We set \( 4^n \ge 3 \text{ billion} \) as our inequality to ensure a sequence of length \( n \) appears uniquely. To be more clear, we rewrite 3 billion as \( 3 \times 10^9 \).

Step 4: Solve for n

To find \( n \), we need \( n \ge rac{ ext{log}_{10}(3 \times 10^9)}{ ext{log}_{10} 4} \). Calculating, we find \( n \ge rac{9.477}{0.602} \approx 15.741 \, \). Since \( n \) must be an integer, we round up to \( n = 16 \).

Step 5: Final Conclusion

The shortest sequence length \( n \) for which \( 4^n > 3 \times 10^9 \) is 16.

Key Concepts

DNA Sequence

DNA sequences are the specific order of nucleotides that make up a DNA molecule. These sequences carry the genetic instructions used in the development and functioning of all known living organisms and many viruses. DNA consists of four types of nucleotides, each identified by a letter: adenine (A), thymine (T), guanine (G), and cytosine (C). Together, these nucleotides form a structure known as a double helix, resembling a twisted ladder. The order in which these nucleotides appear determines the genetic information available for building and maintaining an organism. Each particular sequence can code for a specific function or trait, making the DNA sequence incredibly important in genetics. In the human genome, these sequences are part of 3 billion base pairs that constitute the complete set of genetic information.

Nucleotide Probability

Nucleotide probability refers to the likelihood of each type of nucleotide—A, T, G, and C—appearing at any position in a DNA sequence. In scenarios where nucleotides are assumed to occur with equal probability, each one has a 25% chance of being selected for each position in the sequence. This assumption simplifies calculations and predictions about the likelihood of certain sequences appearing within a given length of DNA. When calculating the probability of a specific DNA sequence, it is important to consider the sequence length. For each position in the sequence, the probability of any one nucleotide being present is \( \frac{1}{4} \) . Therefore, the probability of any specific sequence of length \( n \) occurring is \( \left( \frac{1}{4} \right)^n \) . This formula helps in understanding how likely it is for a specific sequence to appear by chance in a genome.

Unique Sequence Calculation

A unique sequence calculation involves determining the minimum length required for a DNA sequence to appear only once in a given genome. For the human genome, which contains roughly 3 billion base pairs, this calculation requires understanding the number of possible sequences of a given length. Each position in the sequence can be occupied by any of the four nucleotides, so for a sequence of length \( n \), there are \( 4^n \) possible sequences.To ensure that a sequence is unique, the number of possible sequences must exceed the total number of base pairs in the genome. For the human genome:- Set the inequality \( 4^n \geq 3 \times 10^9 \)- Solve for \( n \) to find that \( n \geq \frac{\text{log}_{10}(3 \times 10^9)}{\text{log}_{10} 4} \)- Upon solving, it results in a minimum integer length of \( n = 16 \). This means that a sequence of 16 base pairs is statistically likely to be unique in the human genome.

Base Pairs

Base pairs are the pairs of nucleotides that make up the rungs of the DNA double helix. In human DNA, adenine (A) pairs with thymine (T) and guanine (G) pairs with cytosine (C) through hydrogen bonding. These specific pairings ensure the DNA's stable structure and accurate replication. Base pairs form the framework for DNA's genetic encoding, with three billion base pairs making up the human genome. Each pair of nucleotides forms a "rung" in the DNA helix, and the sequence of these rungs makes up genes. Since the human genome is made up of approximately 3 billion of these base pairs, identifying and understanding them is crucial in studying genetic functions and variations. Understanding base pairs helps in exploring concepts such as genetic uniqueness, replication, and the overall architecture of genetic material in living organisms.