Q.42 A cell has five molecules of a rare mRNA.
Each cell contains 4 × 105 mRNA molecules.
How many clones one will need to screen to have 99% probability
of finding at least one recombinant cDNA of the rare mRNA,
after making cDNA library from such cell?
Options:
(A) 4.50 × 105
(B) 3.50 × 105
(C) 4.20 × 105
(D) 4.05 × 105
Calculation of Number of Clones Required to Identify a Rare cDNA with 99% Probability
Screening cDNA libraries to identify rare transcripts is a common task in molecular
biology. This numerical problem explains how probability theory is used to calculate
the number of clones required to detect at least one rare cDNA with high confidence.
Question Overview
- Rare mRNA molecules per cell = 5
- Total mRNA molecules per cell = 4 × 105
- Desired probability of detection = 99%
Key Concept
The probability of finding at least one desired clone is given by:
P = 1 − (1 − f)N
Where:
- P = desired probability
- f = fraction of rare mRNA
- N = number of clones screened
Step 1: Calculate Fraction of Rare mRNA
f = (Number of rare mRNA molecules) / (Total mRNA molecules)
f = 5 / (4 × 105)
f = 1.25 × 10−5
Step 2: Apply Probability Equation
0.99 = 1 − (1 − f)N
(1 − f)N = 0.01
Step 3: Logarithmic Approximation
Taking natural logarithm:
N ln(1 − f) = ln(0.01)
For small f, ln(1 − f) ≈ −f
N (−1.25 × 10−5) = −4.605
Step 4: Calculate Number of Clones
N = 4.605 / (1.25 × 10−5)
N ≈ 3.68 × 105
Approximation for exams:
N ≈ 3.5 × 105
Correct Answer
Option (B): 3.50 × 105
Important Exam Tips
- Always calculate the fraction of rare mRNA first.
- Use ln(1 − f) ≈ −f for rare transcripts.
- Remember: ln(0.01) ≈ −4.6.
Conclusion
To achieve a 99% probability of identifying at least one recombinant cDNA corresponding
to a rare mRNA, approximately 3.5 × 105 clones must be screened.
Hence, Option (B) is the correct answer.
