Minimum Unique DNA Sequence Length Calculation

This solution determines the minimum length of a DNA sequence segment that is statistically unique within a large genome (3 billion base pairs).

Understanding DNA Composition

DNA uses 4 bases (A, T, C, G). A sequence of length $L$ can have $4^L$ different combinations.

Probability of Sequence Occurrence

In a genome of size $N = 3 \times 10^9$ base pairs, the expected number of times a specific sequence of length $L$ appears is approximately $N / 4^L$. For a sequence to be considered unique, this value should ideally be less than 1.

Setting the condition for uniqueness:

$ \frac{N}{4^L} < 1 $

$ 4^L > N $

Calculating Required Length (L)

We need to find the smallest integer $L$ such that $4^L > 3 \times 10^9$.

Taking logarithm base 4:

$ L > \log_4(3 \times 10^9) $

Using base 10 logarithms for calculation:

$ L > \frac{\log_{10}(3 \times 10^9)}{\log_{10}(4)} $

$ L > \frac{\log_{10}(3) + 9}{\log_{10}(4)} $

Using approximations $\log_{10}(3) \approx 0.477$ and $\log_{10}(4) \approx 0.602$:

$ L > \frac{0.477 + 9}{0.602} \approx \frac{9.477}{0.602} \approx 15.74 $

Minimum Integer Length

Since $L$ must be an integer and $L > 15.74$, the smallest integer value for $L$ is 16.

Let's verify:

• For $L=15$: $4^{15} \approx 1.07 \times 10^9$ (less than $3 \times 10^9$, so not guaranteed unique)
• For $L=16$: $4^{16} \approx 4.30 \times 10^9$ (greater than $3 \times 10^9$, so likely unique)

Therefore, the minimum unique sequence length is 16 base pairs.