Bacterial Growth Curve and Generation Time

Bacterial growth is one of the few places in a microbiology course where you are handed a formula and a number to compute. It is also a place where the vocabulary and the math have to work together: you need to know what each phase of the growth curve means before you know when the doubling formula applies. This guide covers both, following the cards in Encodr's free Microbiology deck.

How a bacterium divides

Bacteria reproduce by binary fission. DNA replication begins at a fixed spot on the circular chromosome, the origin of replication, and proceeds in both directions around the loop. A protein called FtsZ assembles into a ring, the Z ring, at the future division site. That ring helps build a divisome, and the divisome synthesizes new peptidoglycan to form an inward-growing septum that finally splits the cell into two daughter cells.

One cell becomes two, two become four, four become eight. That doubling is the whole reason bacterial populations can grow so fast, and it is the idea behind every formula below.

The four phases of the growth curve

When you put a small number of bacteria into fresh medium and count them over time, the population follows a characteristic curve with four phases.

Two things trip students up. First, lag phase is not a pause in activity, only in numbers: the cells are busy getting ready. Second, stationary phase does not mean nothing is happening. Division and death are both going on and they cancel out. The course ties the shift into stationary phase to nutrients becoming limited and/or waste products accumulating.

The growth formulas

The doubling formula is:

Nn = N0 x 2^n

N0 is the starting number of cells, Nn is the final number, and n is the number of generations. Because each generation doubles the population, growth is exponential, not linear. A population that added the same fixed number of cells each generation would be growing linearly, and the 2^n term is what rules that out.

The generation time, g, is the time for one complete round of binary fission. In prokaryotes, "generation time" and "doubling time" are synonyms. Since n generations take a total time t:

g = t / n

Often you are given the counts and the time and must find n first. Rearranging the first formula with logarithms gives:

n = log10(Nn / N0) / log10(2)

Dividing by log10(2) is the same as multiplying by about 3.3, which is why you may see the formula written with a 3.3 in it.

Worked examples

Example 1: how many cells after a day? Start from one cell (N0 = 1) that divides every 30 minutes for 24 hours. That is 24 x 60 = 1,440 minutes, so n = 1,440 / 30 = 48 generations. Then Nn = 1 x 2^48 = 281,474,976,710,656, or about 2.8 x 10^14 cells. The number is huge because 48 doublings compound.

Example 2: find the generation time. A culture grows from 1.0 x 10^6 cells to 8.0 x 10^6 cells over 120 minutes. The fold increase is 8, so n = log10(8) / log10(2) = 3 generations. Then g = t / n = 120 / 3 = 40 minutes.

Example 3: a smaller culture. Start with 1,000 cells and a generation time of 30 minutes. After 240 minutes, n = 240 / 30 = 8 generations, so Nn = 1,000 x 2^8 = 256,000 cells.

You can check any of these with the bacterial growth calculator, which shows the working for the final population, the number of generations or the generation time.

When the formula applies, and when it does not

The doubling formula describes exponential growth, so it applies only during log phase. Run it across a whole growth curve and it will over-predict, because the curve flattens in stationary phase and falls in death phase. Exam questions usually signal this by giving you a time window of steady growth. If a question hands you counts from different phases, treat that as a prompt to think about the curve rather than to plug in numbers.

How the counts are made

The N values in these formulas come from counting. The course covers several methods:

• Direct microscopic count. Cells counted divided by the volume of the counting-chamber square gives cells/mL. The named chamber is a Petroff-Hausser chamber. This method counts every visible cell, whether alive or dead.
• Viable plate count. Colonies counted on agar, scaled by the dilution, give colony-forming units (CFU) per mL. This counts only cells able to form a colony. The full method is in serial dilutions and CFU calculations.
• Spectrophotometry (turbidity). An indirect estimate of cell density that needs a calibration curve, built by plotting known absorbance values against cell density measured independently, for example by plate count.
• Most probable number (MPN). A pattern of positive tubes is read off a standard statistical table to estimate bacteria per 100 mL.

What else shapes growth

Beyond time, a culture's growth depends on its environment. The course sorts organisms into five oxygen classes (obligate aerobes, obligate anaerobes, facultative anaerobes, aerotolerant anaerobes and microaerophiles), three pH classes (neutrophiles, acidophiles and alkaliphiles) and five temperature classes (psychrophiles, psychrotrophs, mesophiles, thermophiles and hyperthermophiles). Mesophiles have an optimum near 37 degrees C.

Bacteria also do not always grow as free-floating cells. A biofilm is a cluster of microbes in a self-produced matrix of extracellular polymeric substances (EPS), which make up roughly 50 to 90 percent of its dry mass. Cells in a biofilm coordinate through quorum sensing, detecting local density through signaling molecules called autoinducers.

Study tips for the math

The formulas are short, which makes them easy to feel you know and then fumble under time pressure. Practicing them is better than rereading them: write the formula from memory, then solve a fresh problem. Active recall beats passive review for exactly this reason. To see what else the course covers, read what's in microbiology, and for a unit-by-unit plan see how to study for microbiology.