How to Calculate Specific Growth Rate from Growth Curve

Calculate microbial or cell culture kinetics from exponential growth data

Units (e.g., CFU/mL, OD600, or g/L)

Must be higher than N₁

Time units (e.g., hours or days)

Calculation Results:

Specific Growth Rate (μ):

Doubling Time (g or t_d):

Number of Generations (n):

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How to Calculate Specific Growth Rate from a Growth Curve

In microbiology and fermentation technology, the specific growth rate (μ) is a fundamental parameter that describes the increase in biomass per unit of time. It is most accurately measured during the exponential (log) phase of growth, where the population increases at a constant rate relative to the existing population.

The Specific Growth Rate Formula

The relationship between biomass and time during the log phase is defined by the following mathematical formula:

μ = [ln(N₂) – ln(N₁)] / (t₂ – t₁)

Where:

• μ = Specific growth rate (expressed in time^-1, e.g., hr^-1)
• N₁ = Population or biomass at the start of the observation period
• N₂ = Population or biomass at the end of the observation period
• t₁ = Starting time
• t₂ = Ending time
• ln = Natural logarithm (base e)

Doubling Time and Generations

Once the specific growth rate is known, you can calculate the Doubling Time (t_d), which is the time required for the population to double in size. The formula is:

t_d = ln(2) / μ ≈ 0.693 / μ

Practical Example Calculation

Suppose you are monitoring a bacterial culture. At hour 2 (t₁), the optical density (OD600) is 0.15 (N₁). At hour 6 (t₂), the optical density has increased to 1.20 (N₂).

1. Calculate the time interval: 6 – 2 = 4 hours.
2. Calculate the log difference: ln(1.20) – ln(0.15) = 0.182 – (-1.897) = 2.079.
3. Divide the log difference by time: 2.079 / 4 = 0.519 hr^-1 (μ).
4. Calculate doubling time: 0.693 / 0.519 = 1.33 hours.

Why This Matters in Bioprocessing

Calculating the specific growth rate allows researchers to compare the fitness of different strains, optimize culture media, and determine the best time to harvest cells or induce protein expression. A high specific growth rate generally indicates favorable environmental conditions and efficient nutrient utilization.

function calculateGrowthRate() { var n1 = parseFloat(document.getElementById("initialPop").value); var n2 = parseFloat(document.getElementById("finalPop").value); var t1 = parseFloat(document.getElementById("startTime").value); var t2 = parseFloat(document.getElementById("endTime").value); var resultBox = document.getElementById("growthResultBox"); var errorBox = document.getElementById("growthError"); // Reset display resultBox.style.display = "none"; errorBox.style.display = "none"; // Validation if (isNaN(n1) || isNaN(n2) || isNaN(t1) || isNaN(t2)) { errorBox.innerText = "Please fill in all fields with valid numbers."; errorBox.style.display = "block"; return; } if (n1 <= 0 || n2 <= 0) { errorBox.innerText = "Population values must be greater than zero."; errorBox.style.display = "block"; return; } if (n2 <= n1) { errorBox.innerText = "Final population (N2) must be greater than Initial population (N1) for growth."; errorBox.style.display = "block"; return; } if (t2 <= t1) { errorBox.innerText = "End time (t2) must be greater than start time (t1)."; errorBox.style.display = "block"; return; } // Math var deltaTime = t2 – t1; var lnN1 = Math.log(n1); var lnN2 = Math.log(n2); // Specific Growth Rate (mu) var mu = (lnN2 – lnN1) / deltaTime; // Doubling Time (td) var td = Math.log(2) / mu; // Number of Generations (n) var n = (lnN2 – lnN1) / Math.log(2); // Display document.getElementById("resMu").innerText = mu.toFixed(4) + " (time⁻¹)"; document.getElementById("resTd").innerText = td.toFixed(2) + " (time units)"; document.getElementById("resGen").innerText = n.toFixed(2); resultBox.style.display = "block"; }