Growth Rate Calculators

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Growth rate is the rate at which a biological or chemical quantity increases over time. In microbiology and cell biology, the specific growth rate (μ, in h⁻¹ or day⁻¹) describes exponential growth: N(t) = N₀ × e^(μt). Doubling time (t_d) = ln(2)/μ = 0.693/μ. In ecology, r (intrinsic rate of increase) is the per-capita growth rate. Growth rate is calculated from the slope of a plot of ln(population size) vs. time during exponential phase. For bacteria, μ is determined from OD₆₀₀ measurements or CFU counts; for eukaryotic cells, from direct cell counting.

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Specific Growth Rate

μ = [ln(N₂) − ln(N₁)] / (t₂ − t₁)

Units: h⁻¹ or day⁻¹. From OD measurements: μ = slope of ln(OD₆₀₀) vs. time during exponential phase.

Doubling Time

t_d = ln(2) / μ = 0.693 / μ

Example: μ = 1.386 h⁻¹ → t_d = 0.693/1.386 = 0.5 h = 30 min. E. coli in LB at 37°C: t_d ≈ 20–30 min (μ ≈ 1.4–2.1 h⁻¹). Mammalian cells: t_d ≈ 18–48 h (μ ≈ 0.014–0.038 h⁻¹).

Population Growth Rate

r = intrinsic rate of increase (per capita, per time). r ≈ ln(R₀)/T. R₀ = net reproductive rate; T = generation time. N(t) = N₀ × e^(rt). λ = e^r = finite rate of increase (annual multiplier).

Relative Growth Rate (Plants)

RGR = (ln W₂ − ln W₁)/(t₂ − t₁) (g g⁻¹ day⁻¹). RGR = NAR × LAR. Fast-growing: 0.20–0.40; slow-growing: < 0.10 g g⁻¹ day⁻¹.

Glossary

Specific Growth Rate (μ)
μ = [ln(N₂)−ln(N₁)]/(t₂−t₁); per-capita rate of population or biomass increase per unit time; h⁻¹ for bacteria; doubling time = 0.693/μ; calculated from exponential phase.
Doubling Time (t_d)
Time for a population to double: t_d = 0.693/μ; E. coli ≈ 20 min; mammalian cells 18–48 h; lower t_d = faster growth; applies to exponential growth only.
Monod Equation
μ = μ_max × [S]/(Ks + [S]); relates specific growth rate to substrate concentration; Ks = half-saturation constant; at [S] >> Ks: μ ≈ μ_max; basis for fed-batch bioprocess control.

Frequently Asked Questions

Specific growth rate (μ) is the fractional rate of increase in population or biomass per unit time: μ = [ln(N₂) − ln(N₁)] / (t₂ − t₁). Calculated from the exponential (log) phase of growth only — where ln(N) vs. time is linear. Units: h⁻¹ for microbes; day⁻¹ for macroorganisms. From OD₆₀₀: μ = slope of ln(OD₆₀₀) vs. time plot during exponential phase. Doubling time: t_d = 0.693/μ.

Growth rate (μ): the instantaneous per-capita rate of increase; higher μ = faster growth; units h⁻¹ or day⁻¹. Doubling time (t_d): the time for a population to double in size; t_d = 0.693/μ; directly interpretable as a time period; lower t_d = faster growth. Example: E. coli μ = 2.08 h⁻¹ → t_d = 0.693/2.08 = 0.33 h = 20 min. Mammalian CHO cell μ = 0.030 h⁻¹ → t_d = 0.693/0.030 = 23 h. Population growth in ecology: r (intrinsic rate of increase) plays the same role as μ; λ = e^r = finite rate of increase (population multiplier per year).

Bacterial growth in batch culture: Lag phase: μ ≈ 0; cells adapting; no increase in OD or CFU. Exponential (log) phase: μ = constant maximum; OD doubles every t_d; ln(OD) increases linearly. Stationary phase: μ = death rate → net growth = 0; OD plateaus; nutrient depletion or waste accumulation. Death phase: death rate > μ → OD falls. μ is calculated only from exponential phase data. The maximum specific growth rate (μ_max) is a characteristic of the organism and culture conditions. In fed-batch bioreactors: feeding rate controls N by maintaining suboptimal [substrate] → prevents overflow metabolism.

Temperature: exponential increase in μ from minimum to optimum temperature (van't Hoff-Arrhenius); rapid decline above optimum (protein denaturation). Q₁₀ ≈ 2 for most biological reactions: each 10°C rise doubles rate up to the optimum. Substrate concentration: Monod equation: μ = μ_max × [S] / (Ks + [S]). At [S] >> Ks: μ ≈ μ_max. At [S] = Ks: μ = μ_max/2. pH: most bacteria: optimum pH 6.5–7.5; extremophiles outside this range. Oxygen: aerobes require O₂; sufficient aeration → prevents O₂ limitation → maintains μ; specific O₂ uptake rate = μ × Y_O₂.