Numerical Problems/ISC/Class 12
Created by Titas Mallick
Biology Teacher • M.Sc. Botany • B.Ed. • CTET (CBSE) • CISCE Examiner
BioNotesBioNotesCreated by Titas Mallick
Biology Teacher • M.Sc. Botany • B.Ed. • CTET (CBSE) • CISCE Examiner
Online
Numerical Problems - Biotechnology
Recombinant DNA (rDNA) technology and biotechnology rely heavily on precise quantitative measurements, including DNA concentration, restriction fragment lengths, molar ratios for ligation, and PCR amplification yields. This guide provides comprehensive, high-yield numerical problems to test deep conceptual understanding.
A researcher is working with two DNA molecules: a 6.5 kb circular plasmid (pDNA) and a 6.5 kb linear DNA fragment (lDNA). Both molecules contain exactly two recognition sites for the restriction enzyme EcoRI and one recognition site for BamHI. If both molecules are subjected to a complete double digest with EcoRI and BamHI, how many DNA fragments will be generated from each molecule?
Students often memorize "cuts = fragments" or "cuts + 1 = fragments" without considering the topology of the DNA. Always explicitly check if the problem states the DNA is circular (plasmids, bacterial chromosomes) or linear (eukaryotic chromosomes, PCR products).
A 10 kb linear DNA molecule is digested with restriction enzymes XhoI and HindIII, independently and together. The resulting fragment sizes are analyzed via agarose gel electrophoresis:
Construct a restriction map for this linear DNA fragment.
Restriction Map:
0 kb (Start) -------- 3 kb (XhoI cut) -------- 6 kb (HindIII cut) -------- 10 kb (End)
Note: The exact mirror image (HindIII at 4 kb, XhoI at 7 kb) is also a completely valid interpretation!
You are preparing a ligation reaction using a 4.5 kb pBR322 vector and a 1.5 kb gene of interest (insert). For optimal ligation efficiency, protocols often require a Vector:Insert molar ratio of 1:3. If you are using 50 ng of the prepared pBR322 vector, what mass of the insert DNA (in ng) must you add to the reaction mixture?
Molar ratios account for the number of molecules rather than their sheer weight. A smaller DNA fragment has more molecules per nanogram than a larger fragment.
Therefore, you need exactly 50 ng of the 1.5 kb insert DNA.
If the prompt asked for a 1:1 mass ratio, how many moles of insert would you have relative to the vector? Answer: Since the vector is 3 times larger (4.5 vs 1.5), equal masses mean you actually have 3 times as many insert molecules. A 1:1 mass ratio is inherently equivalent to a 1:3 molar ratio (Vector:Insert)!
A forensic scientist extracts a minute trace of DNA containing exactly 5 copies of a target locus. They subject this sample to a PCR run for 30 cycles.
The general formula for PCR amplification is: Final Copies = Initial Copies × (1 + Efficiency)^n Where Efficiency ranges from 0 to 1 (1 being 100%), and 'n' is the number of cycles.
Part 1: 100% Efficiency (E = 1)
Part 2: 85% Efficiency (E = 0.85)
Notice how a 15% drop in efficiency reduces the final yield by a massive factor of ~245x due to the exponential nature of PCR!
PCR Exponential Amplification:
You perform a heat-shock transformation of competent E. coli using 20 ng of a recombinant plasmid. After heat shock, you add 900 μL of SOC recovery medium to the 100 μL competent cell mixture, bringing the total volume to 1000 μL (1 mL). After 1 hour of incubation, you plate 50 μL of this suspension onto an LB-agar plate containing Ampicillin. The next day, you count 120 colonies on the plate. Calculate the Transformation Efficiency (TE) in terms of Colony Forming Units per microgram (CFU/μg) of DNA.
Transformation efficiency indicates how many cells successfully took up the plasmid per microgram of DNA used.
Determine the total number of transformants in the entire 1 mL volume:
Convert the mass of DNA used into micrograms (μg):
Calculate Transformation Efficiency (TE):
A frequent mistake is calculating TE based only on the colonies seen on the plate without adjusting for the dilution factor of the recovery medium! Always remember to scale up the colony count to the total recovery volume before dividing by the total DNA mass.