25. At the transcription start site of a gene, any of the four nucleotides can occur with equal probability p. The Shannon Entropy S, given by S = −Σi=14 pi ln pi, for this start site is ___________.
Given data: ln(2) = 0.69
Shannon Entropy Calculation at Transcription Start Site with Equal Nucleotide Probability
Shannon entropy quantifies uncertainty at a gene’s transcription start site when each of the four nucleotides (A, T, G, C) appears with equal probability. The calculated value is 1.386, representing maximum nucleotide variability.
Formula Breakdown
The Shannon entropy formula is:
S = −∑(pi ln pi)
For four equally likely nucleotides:
p = p₁ = p₂ = p₃ = p₄ = 1/4 = 0.25
Substituting gives:
S = −4 × (0.25 ln 0.25)
Step-by-Step Calculation
ln(0.25) can be rewritten using exponent rules:
0.25 = 2−2 → ln(0.25) = ln(2−2) = −2 ln 2
Given ln(2) = 0.69:
ln(0.25) = −2 × 0.69 = −1.38
Now compute one term:
0.25 × (−1.38) = −0.345
Multiply by 4:
S = −4 × (−0.345) = 1.386
Why Equal Probability Maximizes Entropy
Entropy is highest when all outcomes are equally likely because uncertainty is maximal.
| Scenario | Probabilities | Entropy |
|---|---|---|
| Equal | [0.25, 0.25, 0.25, 0.25] | 1.386 |
| Biased | [0.7, 0.1, 0.1, 0.1] | < 1.0 |
Biological Relevance
In genomic regions such as promoters:
- High Shannon entropy → flexible transcription initiation (housekeeping genes)
- Low entropy → stronger consensus sites (tissue-specific genes)
Understanding entropy is valuable for bioinformatics, promoter analysis, and GATE Biotechnology preparation.
Final Result
Shannon entropy S = 1.386 (using ln values rounded appropriately)
