Calculate cell proliferation and doubling time for exponential growth phases.
Hours
Minutes
Days
Specific Growth Rate (μ):–
Doubling Time (Td):–
Number of Generations:–
function calculateGrowthRate() {
// Get Elements
var initialInput = document.getElementById('sgr_initial');
var finalInput = document.getElementById('sgr_final');
var timeInput = document.getElementById('sgr_time');
var unitSelect = document.getElementById('sgr_time_unit');
var errorDiv = document.getElementById('sgr_error');
var resultDiv = document.getElementById('sgr_results');
// Get Values
var n0 = parseFloat(initialInput.value);
var nt = parseFloat(finalInput.value);
var t = parseFloat(timeInput.value);
var timeUnit = unitSelect.value;
// Reset Display
errorDiv.style.display = 'none';
resultDiv.classList.remove('visible');
// Validation
if (isNaN(n0) || isNaN(nt) || isNaN(t)) {
errorDiv.innerHTML = "Please enter valid numeric values for all fields.";
errorDiv.style.display = 'block';
return;
}
if (n0 <= 0 || nt <= 0) {
errorDiv.innerHTML = "Cell concentration must be greater than zero.";
errorDiv.style.display = 'block';
return;
}
if (t <= 0) {
errorDiv.innerHTML = "Time elapsed must be greater than zero.";
errorDiv.style.display = 'block';
return;
}
// Logic: Specific Growth Rate (µ) = (ln(Nt) – ln(N0)) / t
// Logic: Doubling Time (Td) = ln(2) / µ
// Logic: Generations = (log10(Nt) – log10(N0)) / log10(2) OR t / Td
var logNt = Math.log(nt);
var logN0 = Math.log(n0);
var mu = (logNt – logN0) / t;
// Handle negative growth (decay) or static
var doublingTime = 0;
if (mu !== 0) {
doublingTime = Math.log(2) / mu;
} else {
doublingTime = Infinity;
}
// Generations n = (log(Nt) – log(N0)) / 0.301
var generations = (Math.log10(nt) – Math.log10(n0)) / Math.log10(2);
// Update DOM
var timeUnitShort = "";
if(timeUnit === "Hours") timeUnitShort = "h⁻¹";
else if(timeUnit === "Minutes") timeUnitShort = "min⁻¹";
else timeUnitShort = "d⁻¹";
document.getElementById('res_mu').innerHTML = mu.toFixed(4) + " " + timeUnitShort;
if (doublingTime < 0) {
document.getElementById('res_doubling').innerHTML = "N/A (Decay)";
} else if (doublingTime === Infinity) {
document.getElementById('res_doubling').innerHTML = "No Growth";
} else {
document.getElementById('res_doubling').innerHTML = doublingTime.toFixed(2) + " " + timeUnit.toLowerCase();
}
document.getElementById('res_generations').innerHTML = generations.toFixed(2);
// Show Results
resultDiv.classList.add('visible');
}
Understanding Specific Growth Rate in Microbiology
The specific growth rate ($\mu$) is a fundamental parameter in microbiology, cell biology, and biotechnology. It quantifies how fast a microbial population is growing per unit of time during the exponential (log) phase. Unlike a simple linear rate, specific growth rate accounts for the fact that cells multiply based on the current population size—a characteristic of exponential growth.
The Formula
Specific growth rate is calculated using natural logarithms (ln) of the biomass or cell count over a specific time interval. The standard formula used by this calculator is:
μ = (ln(Nₜ) – ln(N₀)) / t
Where:
μ (mu): Specific growth rate (usually expressed in inverse hours, $h^{-1}$).
Nₜ: Final cell concentration or biomass at time $t$.
N₀: Initial cell concentration or biomass at time $0$.
t: Time interval between the two measurements.
What is Doubling Time?
Doubling time ($T_d$) is the amount of time it takes for the population to double in size. It is inversely proportional to the specific growth rate. A higher growth rate means a shorter doubling time.
The relationship is defined as: $T_d = \ln(2) / \mu \approx 0.693 / \mu$.
Example Calculation
Imagine you are cultivating E. coli in a bioreactor.
At 0 hours, your optical density (OD) or cell count is 0.1.
This metric is critical for optimizing fermentation processes, determining the potency of antibiotics (via growth inhibition), and modeling population dynamics in ecology. By knowing the $\mu$, bioprocess engineers can determine the optimal feed rates for nutrient media to maintain steady-state growth in chemostats.
Why use Natural Log (ln) vs Log Base 10?
In biology, growth is a continuous process best described by the exponential function $e^{x}$. Therefore, the natural logarithm ($\ln$) is the mathematical standard for calculating rates. If you use $\log_{10}$, you are calculating the number of orders of magnitude change, which requires a conversion factor ($2.303$) to get the true specific growth rate.