GATE 2020 BT Q.43: solution and answer

Official answer: 4096Source: Official answer key, GATE 2020 BT (IIT Delhi)

1. EcoRI's recognition site, GAATTC, is a specific, defined hexanucleotide (6-nucleotide) sequence.

2. In a random DNA sequence with all four bases (A, T, G, C) occurring in equal proportion, the probability of any one specific base occurring at a given position is $1/4$.

3. Because the sequence is random, base identities at successive positions are independent. The probability that a specific 6-position window reads exactly GAATTC is therefore $$p={\left(\frac{1}{4}\right)}^6=\frac{1}{4096}$$

4. Treating each nucleotide position along the DNA as a trial for "does an EcoRI site start here," the occurrences of the site follow a geometric distribution with success probability $p$ per position. The expected (average) number of nucleotides between two successive occurrences of such an event equals the reciprocal of its per-position probability, $1/p$.

5. Average spacing between successive EcoRI sites: $$\frac{1}{p}=\frac{1}{1/4096}=4096\text{ nucleotides}$$

6. This result depends only on the recognition-site length and base composition, not on the total genome size; the 1 Mb length only tells us that the site is expected to recur about $1,000,000/4096\approx 244$ times, enough occurrences for the average spacing to be a meaningful, stable figure.