function calculateGrowthRate() {
// 1. Get DOM elements
var t1Input = document.getElementById('time1');
var x1Input = document.getElementById('biomass1');
var t2Input = document.getElementById('time2');
var x2Input = document.getElementById('biomass2');
var unitSelect = document.getElementById('timeUnit');
var errorDiv = document.getElementById('error-display');
var resultsDiv = document.getElementById('results');
var muDisplay = document.getElementById('muResult');
var doublingDisplay = document.getElementById('doublingResult');
var kDisplay = document.getElementById('kResult');
// 2. Parse values
var t1 = parseFloat(t1Input.value);
var x1 = parseFloat(x1Input.value);
var t2 = parseFloat(t2Input.value);
var x2 = parseFloat(x2Input.value);
var unit = unitSelect.value;
var unitLabel = "";
// Set unit abbreviation
if(unit === 'hours') unitLabel = "h⁻¹";
else if(unit === 'minutes') unitLabel = "min⁻¹";
else unitLabel = "d⁻¹";
// 3. Validation
errorDiv.style.display = 'none';
resultsDiv.style.display = 'none';
if (isNaN(t1) || isNaN(x1) || isNaN(t2) || isNaN(x2)) {
errorDiv.innerText = "Please fill in all fields with valid numbers.";
errorDiv.style.display = 'block';
return;
}
if (x1 <= 0 || x2 = t2) {
errorDiv.innerText = "Time Point 2 must be greater than Time Point 1.";
errorDiv.style.display = 'block';
return;
}
// 4. Calculation Logic
// Formula: mu = (ln(x2) – ln(x1)) / (t2 – t1)
var lnX2 = Math.log(x2);
var lnX1 = Math.log(x1);
var timeDiff = t2 – t1;
var mu = (lnX2 – lnX1) / timeDiff;
// Formula: Doubling Time = ln(2) / mu
var doublingTime = Math.log(2) / mu;
// Formula: Multiplication rate (generations per unit time) k = mu / ln(2)
// Usually k = 1 / doublingTime
var k = mu / Math.log(2);
// 5. Output Formatting
if (mu < 0) {
errorDiv.innerText = "Warning: The value decreased over time. This indicates decay, not growth.";
errorDiv.style.display = 'block';
}
muDisplay.innerText = mu.toFixed(4) + " " + unitLabel;
doublingDisplay.innerText = doublingTime.toFixed(2) + " " + unit;
kDisplay.innerText = k.toFixed(4) + " gen/" + unit;
resultsDiv.style.display = 'block';
}
How to Calculate Specific Growth Rate from a Graph
The specific growth rate ($\mu$) is a fundamental metric in microbiology, biotechnology, and population ecology. It represents the rate at which a population (such as bacteria, yeast, or cells) increases per unit of time during the exponential phase. To calculate this value accurately, researchers often rely on semi-logarithmic graphs derived from experimental data.
Understanding the Graph
When growing a culture, you typically measure the biomass, optical density (OD), or cell count over time. If you plot the Logarithm of Biomass ($\ln X$) on the Y-axis against Time ($t$) on the X-axis, the exponential growth phase appears as a straight line.
The slope of this straight line is equal to the specific growth rate ($\mu$).
The Formula
To calculate the specific growth rate, you need two points from the linear portion of your semi-log graph:
$\mu = \frac{\ln(X_2) – \ln(X_1)}{t_2 – t_1}$
Where:
$\mu$ (Mu): Specific growth rate (units: $time^{-1}$, e.g., $h^{-1}$).
$X_1$: Biomass, absorbance (OD), or cell number at time $t_1$.
$X_2$: Biomass, absorbance (OD), or cell number at time $t_2$.
$t_1, t_2$: The time points corresponding to your measurements.
Note: Ensure you are using the natural logarithm ($\ln$), not the base-10 logarithm ($\log$), unless you adjust the formula by multiplying by 2.303.
Step-by-Step Calculation Guide
Plot your data: Create a graph with Time on the X-axis and the natural log ($\ln$) of your cell count or OD on the Y-axis.
Identify the Exponential Phase: Look for the section of the graph that forms a straight line. Do not include the lag phase (flat start) or stationary phase (flat end).
Select Two Points: Pick two points on this straight line. It is often better to pick points on the "best fit" line rather than raw data points to reduce experimental error. Let's call them ($t_1, X_1$) and ($t_2, X_2$).
Input Data: Enter these values into the calculator above.
Interpret Results: The calculator provides $\mu$, which tells you the proportional increase of the population per unit of time.
Doubling Time ($T_d$)
A related and highly useful metric is the Doubling Time (also known as generation time). This is the time it takes for the population to double in size. It is inversely proportional to the specific growth rate: