Microbial Growth Curves

The math of how bacteria grow — and why it determines everything in your vat

The Lab MAT 172 BIO 111
The log phase isn't just a biology concept — it's exponential mathematics happening in real time in your fermentation vat. When S. thermophilus is doubling every 20 minutes at 42°C, you're watching y = ae^(kt) play out in yogurt.
Lag phase
Cells adjusting to environment. No growth yet. Synthesizing enzymes and ATP.
Log phase
Exponential growth. Cells dividing at maximum rate. This is y = ae^(kt).
Stationary phase
Growth = death rate. Nutrients depleted, waste accumulating. Population plateaus.
Death phase
Death exceeds growth. Nutrients gone, toxic waste builds. Population collapses.
Y-axis scale:

Toggle between linear and log scale — watch how the log phase becomes a straight line on a log plot

📈
Why the log phase is a straight line on a log plot
▼
During log phase, population doubles every generation time: N = N₀ × 2^(t/g) where g is generation time. Taking the log of both sides: log(N) = log(N₀) + (t/g) × log(2). That's a linear equation — y = b + mx. On a linear scale the growth looks like a J-curve that rockets upward. On a log scale it becomes a straight line, which is why microbiologists always plot CFU data on a log scale. The slope of that line IS the growth rate.
💡 This is the same math as Blogarithmic Functions — taking the log of exponential data transforms a curve into a line. Every Petrifilm count you've ever done assumed log-scale thinking.
⏱️
What happens in each phase biologically
▼
Lag phase: No division yet. Cells are metabolically active — synthesizing RNA, enzymes, and other molecules needed to divide. Length depends on inoculum size, age of cells, and how different the new environment is from the previous one. A healthy active culture has a shorter lag than a stressed or cold culture.

Log phase: Maximum growth rate for those conditions. Every cell is dividing at the minimum possible generation time. This is when metabolic byproducts (like lactic acid) accumulate fastest. The culture is at peak activity.

Stationary phase: Nutrient depletion and waste accumulation bring growth and death to equilibrium. For LAB in yogurt, this is when pH has dropped enough to start inhibiting the organisms that produced the acid. Self-limiting by design.

Death phase: Nutrients exhausted, pH or other waste products at lethal levels. Cell death exceeds any remaining growth. Relevant for understanding pasteurization — you're forcing pathogens into rapid death phase by heat.
Exponential growth equation: N(t) = N₀ × e^(kt) — or equivalently N₀ × 2^(t/g). The log phase is literally named for the mathematics happening inside it.
Interactive growth calculator
10³ CFU/mL
25 min
3 hrs
Doublings
7.2
Final count
1.5×10⁵
Log increase
+2.2
Fold increase
151×
S. thermophilus at 42°C with 25 min generation time: after 3 hours you have 7.2 doublings — about 151× more cells than you started with.
🧮
Generation time, doubling time, and growth rate
▼
Generation time (g): Time for one doubling. S. thermophilus at 42°C ≈ 20–30 min. E. coli at 37°C ≈ 20 min. Listeria at 37°C ≈ 40–60 min. At refrigeration temperatures (4°C), generation times extend to hours — which is why refrigeration works.

Growth rate constant (k): k = ln(2) / g = 0.693 / g. At g = 25 min, k = 0.0277 per minute or 1.66 per hour.

Number of doublings: n = t / g. In 3 hours (180 min) at g = 25 min: n = 180/25 = 7.2 doublings.

Final population: N = N₀ × 2^n. Starting at 10³ CFU/mL, after 7.2 doublings: 10³ × 2^7.2 ≈ 150,000 CFU/mL.
📊
Why we plot on log scale — the MAT 172 connection
▼
Exponential data spans enormous ranges — from 10¹ to 10⁹ CFU/mL in a single growth curve. Plotting that on a linear axis makes everything below 10⁷ invisible. A log scale compresses the range into something readable. This is the same reason decibels and pH exist — the data is logarithmic in nature and deserves a logarithmic axis.

On a log scale: log(N) = log(N₀) + (t × log(2)/g). This is y = b + mx — a straight line. The slope tells you the growth rate. A steeper slope = faster growth = shorter generation time. This is how you compare growth rates between different organisms or conditions.
💡 Your Petrifilm counts, your serial dilutions, your log reductions — all of this assumes log-scale thinking. When you count 45 colonies on a 10⁻³ dilution plate, you're doing a log-scale calculation to get back to original concentration.
🔢
Log reductions revisited — death phase math
▼
Death phase follows the same mathematics as log phase, just in reverse. Pasteurization forces pathogens into accelerated death phase by applying lethal heat. The D-value (decimal reduction time) is the time at a given temperature to achieve a 1-log (90%) reduction — exactly one step back on the growth curve.

A 5-log reduction means the population went through 5 decades of decline on the log scale — from 10⁵ to 10⁰, or 100,000 cells to 1 cell. Same math as growth, opposite direction.

💡 The growth curve and the death curve are mirror images mathematically. What takes bacteria hours to build up, pasteurization destroys in seconds — because the death rate at 72°C is orders of magnitude faster than the growth rate at 42°C.
Your yogurt vat is a controlled growth curve experiment. The incubation window, the inoculation rate, the temperature — all of it is managing where on the curve the culture is when you pull the product.
S. thermophilus growth at 42°C — yogurt fermentation timeline

The target pH window corresponds to the late log / early stationary transition — pull too early and you don't have enough acid, too late and you overshoot

🥛
The yogurt fermentation curve — what's actually happening
▼
When you inoculate warm milk at 42°C with S. thermophilus and L. bulgaricus, the lag phase is short — typically 30–60 minutes — because the cultures are thermophilic and the environment is optimal. Then log phase begins: cells doubling every 20–30 minutes, producing lactic acid as a metabolic byproduct. As lactic acid accumulates, pH drops. At pH ~5.4, casein proteins start to aggregate (approach the isoelectric point). By pH ~4.6, the gel sets. S. thermophilus begins to slow as the environment becomes inhibitory — the organism is self-limiting. That's stationary phase setting in and it's your signal to cool.
💡 Slow acidification = something is suppressing the log phase. Temperature too low, inoculation rate too low, phage attack, inhibitory substances. The growth curve explains every fermentation problem.
🦠
Phage attack on the growth curve
▼
A bacteriophage attack doesn't look like the culture stops — it looks like the log phase slope flattens. Instead of doubling every 25 minutes, growth slows or reverses as phage lyse cells faster than they replicate. On a pH monitoring chart you see the acidification rate slow — the curve bends when it should be straight. That's the signature. By the time the pH drop has obviously stalled, the culture is already compromised. The earlier you catch the flattening slope, the more options you have.
💡 This is why titratable acidity monitoring during fermentation is more sensitive than pH endpoint checks — you catch the rate change before the endpoint tells you something went wrong.
❄️
Refrigeration — pushing organisms back to lag
▼
Refrigeration (4°C) doesn't kill spoilage organisms — it extends their generation time from minutes to hours, pushing them effectively into a prolonged lag phase. L. monocytogenes, which is psychrotrophic, can still grow at refrigeration temperatures — just slowly, with a generation time of 12–24 hours at 4°C vs 40–60 minutes at 37°C. This is why shelf life exists and why temperature abuse matters so much — even a few hours at elevated temperature shifts the organism from extended lag back into active log phase, and the population growth that follows is exponential.
The shape of the growth curve is not fixed. Temperature, pH, nutrients, inoculum size, and competing organisms all shift the curve — changing the lag length, the log phase slope, and where stationary hits.
Compare growth conditions
42°C
10³ CFU/mL
At 42°C — optimal for S. thermophilus. Generation time ~25 min. Log phase is steep.
🌡️
Temperature — the most powerful variable
▼
Temperature affects every phase of the growth curve. Near the optimal temperature, generation time is minimized and log phase is steepest. As temperature drops below optimal, generation time increases dramatically — the Arrhenius equation describes this relationship and it's exponential. As temperature rises above optimal, proteins denature and growth slows then stops. For S. thermophilus: optimal ~42°C, minimum ~20°C, maximum ~52°C. For Listeria: grows from 0°C to 45°C, with optimal around 30–37°C — which is why it's so dangerous in a dairy environment.
🧪
Inoculum size — why starter culture rate matters
▼
A larger inoculum shortens the apparent lag phase and gets you into log phase faster — but the slope of log phase (growth rate) is the same. Doubling the inoculum doesn't double the growth rate; it just means you started higher on the curve. In yogurt production, starter culture rate (typically 2–3%) is optimized to hit the pH target within a defined incubation window. Too low and you're waiting hours longer for acidification. Too high and you overshoot pH before you can respond.
💡 This is also why a phage-compromised culture doesn't recover by adding more starter — you're increasing N₀ but the phage are attacking the organisms as fast as they divide. The slope of the curve is compromised, not the starting point.
⚔️
Competitive exclusion on the growth curve
▼
In a yogurt vat, S. thermophilus and L. bulgaricus have faster generation times at 42°C than most spoilage organisms. Their log phase gets them to inhibitory pH levels before competitors can establish significant populations — this is competitive exclusion in action on the growth curve. The LAB win the race and change the environment to one their competitors can't survive. Yeasts and molds are more acid-tolerant, which is why elevated YM counts are the concern even in properly acidified yogurt — they can still grow where bacteria cannot.

Quick reference

Key equations
N = N₀ × 2^(t/g)
n = t / g
k = ln(2) / g
log(N) = log(N₀) + nt
S. thermophilus at 42°C:
Generation time: ~20–30 min
Lag phase: ~30–60 min
Target pH: 4.5–4.6
Incubation: ~4–6 hours

Key vocabulary

Generation time
Time for one complete cell division (one doubling). Shorter = faster growth. S. thermophilus at 42°C: ~25 min.
Log phase
Exponential growth phase. Named because plotting count on a log scale produces a straight line. Maximum metabolic activity.
Growth rate constant (k)
k = ln(2)/g. The slope of the growth curve on a log scale. Larger k = faster growth.
Lag phase
Adaptation period before growth begins. Cells synthesizing enzymes and adjusting to environment. Not dormant — metabolically active.
Stationary phase
Growth rate equals death rate. Nutrient depletion and waste accumulation bring net growth to zero.
D-value
Decimal reduction time — time at a given temperature to achieve 1-log (90%) reduction. Death phase math applied to pasteurization.
Psychrotrophic
Capable of growing at refrigeration temperatures. Listeria is psychrotrophic — grows slowly at 4°C. Most pathogens are not.

Exam reminders

Log phase is exponential growth — N = N₀ × 2^(t/g). Doubling every generation time.
On a log scale, exponential growth becomes a straight line. Slope = growth rate.
Lag phase ≠ no activity. Cells are actively preparing to divide — synthesizing enzymes and ATP.
Refrigeration extends generation time, it doesn't stop growth. Listeria still grows at 4°C.
Slow acidification = suppressed log phase. Temperature, phage, inhibitory substances, low inoculum.
D-value is death phase math — the same log relationship as growth, running backwards.

The Lab connection

📝 Log reductions → Blogarithmic Functions
🦠 Phage flattening the curve → Trophic Systems
⚔️ Competitive exclusion → BIO 111 Ecology
📉 Death phase = D-value = Food Safety Math