GATE Biotechnology (BT) Calculators
Calculators for the numerical questions in GATE Biotechnology: enzyme and microbial kinetics, bioreactors and sterilisation, and the molecular-biology bench calculations, each showing the working.
29 calculators · 3 converters · common conversions
Engineering Mathematics
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Eigenvalue Calculator
det(A − λI) = 0
Σλ = trace A, Πλ = det A
Example: The matrix [[4, 1], [2, 3]]: Eigenvalues: λ₁ = 5, λ₂ = 2.
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Normal Distribution Calculator
z = (x − μ)/σ, P(X < x) = Φ(z)
Φ standard normal cumulative distribution
Example: X normal with μ = 50 and σ = 10, between 40 and 65: P(40 < X < 65) = 0.7745375448.
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- Poisson Distribution Calculator – Mean rate λ and count k → exact, cumulative and tail Poisson probabilities.
- Binomial Distribution Calculator – Trials n, probability p and successes k → exact, cumulative and tail probabilities, mean and variance.
- Correlation & Regression Calculator – Paste x and y values → correlation r, R², slope, intercept and regression equation.
- Bayes’ Theorem Calculator – Priors and likelihoods → total probability and posterior probabilities, with a Bayes table.
- Numerical Integration Calculator – f(x), limits and intervals → trapezoidal, Simpson 1/3 and 3/8 results with a reference value.
- Euler Method Calculator – An ODE, initial value, step and end point → y by Euler, Heun, RK2 or RK4 with every step.
General Biology
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Michaelis-Menten Calculator
v = Vmax[S] / (Km + [S])
competitive: Km × (1 + [I]/Kᵢ); uncompetitive: Km and Vmax ÷ (1 + [I]/Kᵢ); non-competitive: Vmax ÷ (1 + [I]/Kᵢ)
Example: An enzyme with Vmax = 100 and Km = 2 at [S] = 5: Reaction rate v = 71.4286.
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Microbial Growth Calculator
N = N₀ e^(μt), t_d = ln 2 / μ
generations n = log₂(N/N₀)
Example: A culture growing from 1000 to 64,000 cells in 6 hours: Specific growth rate μ = 0.693147 per h.
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Nernst Equation Calculator
E = (RT/zF) ln([C]ₒᵤₜ/[C]ᵢₙ)
R = 8.314 J/(mol·K), F = 96,485 C/mol, z the ion’s valence; 61.5 mV per decade at 37 °C for z = 1
Example: Potassium with 5 mM outside and 140 mM inside the cell at 37 °C: Equilibrium potential E = -89.0587 mV.
Genetics and Molecular Biology
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Hardy-Weinberg Calculator
p² + 2pq + q² = 1, p + q = 1
p = (2·AA + Aa)/(2N); χ² = Σ(O − E)²/E with 1 degree of freedom
Example: A sample of 360 AA, 480 Aa and 160 aa individuals: Allele frequency p (A) = 0.6.
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Primer Tm Calculator
Tm = 64.9 + 41(G + C − 16.4)/N
Wallace (under 14 nt): Tm = 2(A + T) + 4(G + C); salt-adjusted 81.5 + 16.6 log[Na⁺] + 0.41(%GC) − 675/N
Example: The M13 reverse primer AGCGGATAACAATTTCACACAGGA: Melting temperature Tm = 53.9667 °C.
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- PCR Amplification Calculator – Starting copies, cycles and efficiency → copies, fold amplification and cycles needed.
- DNA Concentration Calculator – A260, dilution and length → concentration, purity, copy number and molarity.
Bioreaction Engineering
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Monod Equation Calculator
μ = μmax S / (Ks + S)
Ks the substrate concentration at μ = μmax/2; doubling time ln 2/μ
Example: A culture with μmax = 0.5 per hour and Ks = 0.2 g/L at S = 1 g/L: Specific growth rate μ = 0.416667 per h.
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Chemostat Calculator
D = F/V = μ, S = Ks D/(μmax − D), X = Y(S₀ − S)
washout above D_crit = μmax S₀/(Ks + S₀); productivity DX
Example: A 2 L chemostat fed 0.5 L/h with 10 g/L substrate, μmax 0.5 per h, Ks 0.2 g/L, Y = 0.5: Steady-state biomass X = 4.9 g/L.
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- Thiele Modulus Calculator – Particle size, rate constant and effective diffusivity → Thiele modulus and effectiveness factor.
- CSTR and PFR Reactor Calculator – Rate constant and detention time (or target removal) → outlet concentration and reactor sizes.
- Sterilization Time Calculator – Spore load, acceptable survivors, temperature and D/z (or A, Eₐ) → holding time and Del factor.
- Second-Order System Calculator – Damping ratio and natural frequency → overshoot, peak time, rise time, settling time and poles.
Upstream and Downstream Processing
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Centrifuge RCF Calculator
RCF = 1.118 × 10⁻⁵ × r × rpm²
r rotor radius in cm (centre to the tube bottom); rpm = √(RCF/(1.118 × 10⁻⁵ r))
Example: A microcentrifuge at 12,000 rpm with a 10 cm rotor radius: RCF = 16099.5 × g.
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Stokes' Law Settling Calculator
vₛ = g(ρₚ − ρ_f)d² / (18μ)
valid for particle Reynolds number below about 1 (laminar settling)
Example: A 50 µm sand grain (2650 kg/m³) in water at 20 °C: Settling velocity vₛ = 2.24532 mm/s.
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- Dilution Calculator – Stock and target concentrations and volume → how much stock and solvent to mix.
- Molarity Calculator – Mass, molar mass and volume → molarity; or the mass to weigh out for a target molarity.
Analytical Tools
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Beer–Lambert Calculator
A = ε l c = −log₁₀ T
ε absorptivity, l path length (cm), c concentration, T transmittance
Example: A dye solution with absorbance 0.6 in a 1 cm cell, ε = 12 000 L/(mol·cm): Concentration c = 80.3859 µmol/L.
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Hemocytometer Cell Count Calculator
Cells/mL = average per large square × dilution × 10⁴
each large square holds 0.1 µL (1 mm × 1 mm × 0.1 mm); viability = live ÷ total
Example: Counts of 45, 52, 48 and 50 cells in four squares after a 1:1 trypan blue dilution: Cell density = 975000 cells/mL.
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- pH Calculator – [H⁺], [OH⁻], pH or pOH → the other three and whether the solution is acidic or basic.
- Molar Mass Calculator – A chemical formula → molar mass and the mass percent of each element.
Unit Converters for GATE Biotechnology (BT)
- Weight and Mass Converter – kg, g, mg, lb, oz, st, t, ton, oz t, ct
- Volume Converter – L, mL, m³, ft³, in³, US gal, imp gal
- Temperature Converter – °C, °F, K, °R
Common GATE Biotechnology (BT) Conversions
How to use these GATE Biotechnology calculators
GATE Biotechnology is mostly conceptual biology, but its numerical questions concentrate on a few areas: enzyme kinetics and inhibition, microbial growth and the Monod and chemostat models, diffusion limits in immobilised enzymes, thermal sterilisation, and bench calculations such as primer melting temperature, PCR amplification, DNA quantification, Hardy–Weinberg frequencies, Beer–Lambert absorbance, centrifuge speeds and cell counts.
The calculators show each formula with your numbers substituted, so you can follow the reasoning as well as the answer. Fitting tools such as the Lineweaver–Burk regression take raw data pairs, which is how practical questions present them.
Work the question first, then check it. Watch the units: rates per hour versus per minute, µg/mL versus ng/µL (the same number), and base-e versus base-10 logarithms in growth and death kinetics.