Describe bacterial growth curves

Overview

When bacteria are introduced into a nutrient-rich medium, their population doesn't grow uniformly. Instead, they follow a predictable S-shaped pattern called the bacterial growth curve, which reveals how microorganisms respond to their environment, resources, and waste accumulation.

[!intuition] Why Growth Isn't Constant

Think about moving to a new city with unlimited pizza. At first, you're figuring out where everything is (lag). Then you eat pizza like crazy (exponential). Eventually, you're full and the pizza runs out (stationary). Finally, you get sick from old pizza (death). Bacteria do the same with nutrients.

The growth curve exists because:

• Initial adaptation takes time – bacteria must synthesize enzymes for the new environment
• Resources are finite – exponential growth can't continue forever
• Waste accumulates – metabolic byproducts become toxic
• Space is limited – physical crowding inhibits division

[!definition] The Four Phases

1. Lag Phase

The lag phase is the period of zero or minimal population growth immediately after inoculation.

What's happening:

• Bacteria synthesize ribosomes, enzymes, and cofactors needed for the new medium
• Cells repair damage from previous stationary phase or storage
• Gene expression adjusts to available nutrients
• Individual cells grow in size but don't divide yet

Duration: Hours to days, depending on:

• Inoculum age (old cultures lag longer)
• Medium similarity (less lag if similar to previous environment)
• Species metabolic flexibility

2. Exponential (Log) Phase

The exponential phase (or log phase) is the period of maximum, constant growth rate.

What's happening:

• Every cell divides at regular intervals (generation time or doubling time)
• Nutrients are abundant, waste is minimal
• All cells are metabolically active
• Population size increases geometrically: $N = N_0 \times 2^n$

Why exponential? Each division creates two cells that themselves divide. If generation time $g = 20$ minutes:

• Start: 1 cell
• After 20 min: 2 cells
• After 40 min: 4 cells
• After 60 min: 8 cells
• After $t$ minutes: $2^{t/g}$ cells

[!formula] Mathematical Description of Exponential Growth

Starting from first principles:

Rate of change = rate of production $\frac{dN}{dt} = \mu N$

Where:

• $N$ = number of cells at time $t$
• $\mu$ = specific growth rate (divisions per unit time)
• $\frac{dN}{dt}$ = instantaneous rate of population increase

Why this form? The rate of increase is proportional to the current population because every cell contributes to growth.

Solving by separation of variables: $\frac{dN}{N} = \mu \, dt$

Integrate both sides: $\int_{N_0}^{N} \frac{dN}{N} = \int_{0}^{t} \mu \, dt$

$\ln N \bigg|_{N_0}^{N} = \mu t \bigg|_{0}^{t}$

$\ln N - \ln N_0 = \mu t$

$\ln\left(\frac{N}{N_0}\right) = \mu t$

Exponentiate: $N = N_0 e^{\mu t}$

In terms of doublings: If cells divide every $g$ hours (generation time), then after $n$ generations: $N = N_0 \times 2^n$

Where $n = \frac{t}{g}$, so: $N = N_0 \times 2^{t/g}$

Connection between $\mu$ and $g$: Since $N_0 e^{\mu t} = N_0 \times 2^{t/g}$, equating exponents when $t = g$: $e^{\mu g} = 2$ $\mu = \frac{\ln 2}{g} \approx \frac{0.693}{g}$

[!example] Worked Example 1: Calculating Final Population

Problem: An E. coli culture starts with $10^6$ cells. The generation time is 20 minutes. How many cells after 3 hours?

Solution:

Step 1: Convert time to generations. $n = \frac{t}{g} = \frac{180 \text{ min}}{20 \text{ min}} = 9 \text{ generations}$

Why? Each generation is one doubling, so we need to count how many doublings occur.

Step 2: Apply the exponential formula. $N = N_0 \times 2^n = 10^6 \times 2^9$

Why $2^n$? Each generation multiplies the population by 2.

Step 3: Calculate. $2^9 = 512$ $N = 10^6 \times 512 = 5.12 \times 10^8 \text{ cells}$

Answer: $5.12 \times 10^8$ cells (512 million)

[!example] Worked Example 2: Finding Generation Time

Problem: A bacterial culture increases from $5 \times 10^4$ to $1.28 \times 10^7$ cells in 4 hours. What is the generation time?

Solution:

Step 1: Find the number of generations. $N = N_0 \times 2^n$ $\frac{N}{N_0} = 2^n$ $2^n = \frac{1.28 \times 10^7}{5 \times 10^4} = \frac{1.28 \times 10^7}{0.5 \times 10^5} = 256$

Why this ratio? It tells us the fold-increase in population.

Step 2: Solve for $n$. $2^n = 256 = 2^8$ $n = 8 \text{ generations}$

Why logarithms? We could also use $n = \frac{\ln(N/N_0)}{\ln 2}$.

Step 3: Calculate generation time. $g = \frac{t}{n} = \frac{4 \text{ hours}}{8} = 0.5 \text{ hours} = 30 \text{ minutes}$

Why divide? Generation time is the time per generation.

Answer: 30 minutes per generation

3. Stationary Phase

The stationary phase is the period where growth rate equals death rate, so net population change is zero.

What's happening:

• Nutrient depletion (especially carbon, nitrogen sources)
• Oxygen limitation (in aerobic cultures)
• Waste accumulation (organic acids, ammonia lower pH)
• Space constraints in agar plates
• Cells enter stress response: some produce spores, secondary metabolites, biofilms

Population dynamics: $\frac{dN}{dt} = \mu N - d N = 0$

Where $d$ = death rate. Growth $\mu N$ balances death $dN$.

Why important?

• Many antibiotics and secondary metabolites (penicillin, streptomycin) are produced in stationary phase
• Cells are most resistant to stress (thicker cell walls, DNA repair upregulated)

4. Death (Decline) Phase

The death phase is the period of exponential decrease in viable cell count.

What's happening:

• Nutrients exhausted
• Toxic waste levels lethal
• Autolysis (self-digestion by released enzymes)
• Loss of cell membrane integrity

Death is also exponential: $N = N_{\text{max}} e^{-d t}$

Where $d$ = death rate constant.

Why exponential death? Toxic conditions affect all cells similarly, so a constant fraction dies per unit time—the reverse of exponential growth.

[!example] Worked Example 3: Optical Density Conversion

Problem: You measure $\text{OD}_{600} = 0.8$ for an E. coli culture. The calibration is $\text{OD}_{600} = 0.1$ corresponds to $10^8$ cells/mL. Estimate the cell concentration.

Solution:

Step 1: Assume linear relationship in the valid range. $\frac{\text{cells/mL}}{\text{OD}_{600}} = \frac{10^8}{0.1} = 10^9 \text{ cells/mL per OD unit}$

Why linear? Light scattering is proportional to cell concentration for OD < 1.0.

Step 2: Calculate concentration. $\text{cells/mL} = 0.8 \times 10^9 = 8 \times 10^8 \text{ cells/mL}$

Answer: $8 \times 10^8$ cells/mL

[!mistake] Common Errors and Misconceptions

Mistake 1: "Stationary phase means all growth stops"

Why it feels right: The population number is constant, so it looks like nothing is happening.

Why it's wrong: Individual cells are still dividing, but the birth rate equals the death rate. The population is dynamic, not frozen. Some cells divide while others die.

The fix: Stationary phase is a steady state, not stasis. Think of a bathtub with the faucet on and drain open—water level is constant, but water is flowing.

Mistake 2: "Generation time is constant across all phases"

Why it feels right: We calculate a single value for $g$ in textbook problems.

Why it's wrong: Generation time is only constant during exponential phase. In lag phase, cells don't divide at all. In stationary phase, only some cells divide. In death phase, more cells die than are born.

The fix: When you calculate or report $g$, always specify it's for exponential growth. $g$ is a characteristic of log phase only.

Mistake 3: "More starting cells = faster growth"

Why it feels right: More cells should make more cells faster, right?

Why it's wrong: The growth rate $\mu$ (doublings per hour) is an intrinsic property of the species and conditions—it doesn't change with $N_0$. More starting cells give you a higher final number, but the exponential rate is the same.

The fix: Distinguish between:

• $\mu$ or $g$ (intrinsic growth rate) – doesn't depend on $N_0$
• Absolute cell production – does depend on $N_0$

If Culture A starts with $10^4$ cells and Culture B with $10^6$ cells, both with $g = 20$ min, after 1 hour:

• Culture A: $10^4 \times 2^3 = 8 \times 10^4$
• Culture B: $10^6 \times 2^3 = 8 \times 10^6$

Both doubled 3 times at the same rate, but B has more cells because it started with more.

[!mnemonic] LEGS Mnemonic

Lag – Learning the environment
Exponential – Explosive reproduction
Going nowhere (stationary) – Growth equals death
Sayonara (death) – Starving and dying

[!recall]- Explain to a 12-Year-Old

Imagine you're a bacteria dropped into a swimming pool full of food (sugar water).

Lag phase: You just got there. You're looking around, figuring out what kind of food is here, and getting your body ready to eat. You're not making baby bacteria yet—you're just getting prepared. This is like the first day at a new school.

Exponential phase: NOW you're ready! You eat, grow, and split into two bacteria. Each of those splits into two more. Then those four become eight, eight become sixteen—it's crazy fast because everyone is making copies of themselves. It's like if every person in class could make a clone every 20 minutes. The room fills up FAST.

Stationary phase: Uh-oh. The pool is getting crowded, the food is running low, and there's bacteria poop everywhere making the water gross. Some bacteria are still being born, but just as many are dying from the bad conditions. The total number stays the same—it's like a classroom where some kids leave and new kids arrive, but the class size doesn't change.

Death phase: The food is gone, the water is toxic, and bacteria start dying faster than new ones are born. Everyone's getting sick. The population shrinks. Sad times.

The whole thing makes an S-shaped curve when you graph it: slow start, fast middle, flat top, then downward.

Experimental Considerations

Measuring Growth

1. Turbidity (Optical Density)

   • Spectrophotometer at 600 nm
   • Fast, non-destructive
   • Measures total cells (living + dead)
   • Linear only for OD < 1.0

2. Viable Count (CFU)

   • Serial dilution + plate counting
   • Measures only living cells
   • Time-consuming (24-48 hr incubation)
   • Gold standard for viability

3. Direct Count (Hemocytometer)

   • Microscopic counting in chamber
   • Total cells (can't distinguish viable)
   • Fast but tedious

Factors Affecting Growth Rate

• Temperature: Optimal vs. too hot/cold affects enzyme activity
• pH: Most bacteria prefer pH 6.5-7.5; extremes denature proteins
• Oxygen: Aerobes need O₂; anaerobes are poisoned by it; facultative adapt
• Nutrient quality: Complex media (yeast extract) vs. defined minimal media
• Inoculum age: Old stationary-phase cells lag longer

Connections

• Bacterial Reproduction and Binary Fission – how individual cells divide
• Bacterial Metabolism – why nutrient type affects growth rate
• Antibiotics and Bacterial Growth – most effective in log phase when cells are actively synthesizing
• Continuous Culture Systems – chemostats maintain perpetual exponential growth
• Biofilm Formation – often triggered in stationary phase
• Spore Formation – survival strategy in death phase for Bacillus, Clostridium
• Exponential Functions in Biology – same math applies to viral replication, cancer cell growth

Flashcards

Concept Map

Hinglish (regional understanding)