The bacterial growth curve

Concept

The bacterial growth curve

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Statement

Lag, exponential, stationary and death phases.

Why it matters

A bacterial population's size is not a fixed property of a species but the outcome of a dynamic process, and that process has a characteristic, reproducible shape whenever a fixed volume of medium is inoculated with cells and left undisturbed. antibiotic-resistance depends directly on this shape: selection for a resistant mutant acts most powerfully while the population is expanding exponentially, since a small reproductive advantage compounds every generation and is amplified enormously by the sheer number of divisions occurring in that phase. The same curve underlies industrial fermentation, food microbiology, and the design of continuous culture systems such as the chemostat, all of which either exploit or deliberately avoid one particular phase of the curve.

The curve also gives microbiology its basic quantitative vocabulary — generation (doubling) time, specific growth rate, carrying capacity of a batch culture — terms that recur throughout this unit and, in modified form, throughout ecology's treatment of population growth more broadly (logistic-population-growth).

Hypotheses

The culture is grown in a fixed, finite volume of medium with no addition or removal of nutrients or waste (a closed batch culture).This is what forces the curve through all four phases in sequence: growth cannot continue exponentially forever, because the finite nutrient supply must eventually be exhausted and metabolic waste must eventually accumulate. A continuously fed system (a chemostat) removes this constraint and can hold a population in the exponential phase indefinitely (Corollaries). Every cell divides by binary fission into two genetically identical daughter cells, and division events are frequent and asynchronous enough for the population to be treated as a smooth, continuous variable.Individual cells divide in discrete steps, but with an initial population of many thousands of asynchronously dividing cells, division events are so densely and randomly spread in time that the population count \(N(t)\) is well approximated as a smooth exponential function rather than a jagged staircase. The medium is well mixed and physically uniform, so every cell experiences the same nutrient concentration at a given time. In structured environments (biofilms, gradients) growth rate can vary substantially with position, and the simple curve derived here no longer applies uniformly across the population.

Proof

\frac{dN}{dt} = \mu N

During unrestricted growth, the rate at which new cells appear is proportional to the number of cells already present, since each existing cell divides at the same per-capita (specific) rate \(\mu\); this is the defining assumption of exponential growth. A

N(t) = N_0\,e^{\mu t}

Separating variables and integrating Step 1 from an initial population \(N_0\) at \(t=0\) gives this closed-form solution directly; a population growing at a constant per-capita rate grows exponentially in absolute size, even though the per-capita rate itself never changes. A

g = \frac{\ln 2}{\mu}\ \ \Rightarrow\ \ N(t) = N_0\cdot 2^{\,t/g}

Define the generation (doubling) time \(g\) as the time for \(N\) to double; setting \(N(g)=2N_0\) in Step 2 and solving for \(g\) gives \(g=\ln2/\mu\), and substituting back expresses the same exponential law in the equivalent, more intuitive base-2 form used throughout microbiology. A

\ln N(t) = \ln N_0 + \mu t

Taking logarithms of Step 2 linearises the exponential relationship: a plot of \(\ln N\) (or, equivalently, \(\log_{10}N\)) against time is a straight line of slope \(\mu\) precisely during, and only during, the exponential phase — the standard diagnostic used to identify that phase experimentally and to measure \(\mu\) or \(g\) from real data (Worked examples). A

\text{Lag: } \mu\approx 0;\ \ \text{Exponential: } \mu=\mu_{\max}>0;\ \ \text{Stationary: } \mu\approx 0 \text{ (birth} \approx \text{death)};\ \ \text{Death: } \mu<0

The full batch-culture curve is this same specific growth rate \(\mu\) taking different values over time as nutrients are progressively depleted and waste accumulates (Hypotheses): newly inoculated cells first synthesise enzymes and adjust metabolically before dividing (lag), then divide at a constant maximal rate while resources remain abundant (exponential, Steps 1–4), then reach a dynamic balance where division and death rates are equal rather than division ceasing outright (stationary), and finally decline as death outpaces division once resources are exhausted (death). B

Result

N(t) = N_0\cdot 2^{\,t/g}, \qquad g = \frac{\ln 2}{\mu}

Reading. A bacterial population growing without restriction doubles every fixed interval \(g\), so \(N\) grows exponentially in absolute terms even though each individual cell's own division rate never changes; over the full batch-culture cycle, this exponential law holds only transiently, sandwiched between an initial lag and an eventual stationary plateau.

Scope. The exponential relation \(N=N_0\cdot2^{t/g}\) applies strictly only within the exponential phase itself (Hypotheses); \(g\) is not a fixed property of a species but depends on temperature, medium richness, and other environmental conditions, and can differ by orders of magnitude between organisms and even for the same organism under different conditions.

Corollaries & converses

A chemostat continuously replaces spent medium with fresh medium at a fixed dilution rate, removing the closed-culture assumption (Hypotheses) entirely and allowing \(\mu\) to be held constant indefinitely at any chosen sub-maximal value — the standard technique for studying cells specifically in steady-state exponential growth rather than only transiently.
Because reproductive advantage compounds every generation during exponential growth, antibiotic-resistance mutations that arise (by chance, independent of the antibiotic itself) are amplified fastest, and fixed most quickly in a population, when selection is imposed during this phase rather than during lag or stationary phase.
Converse: from a series of population counts taken during unrestricted growth, plotting \(\ln N\) against \(t\) (Step 4) and reading off the slope gives \(\mu\), and hence \(g\), without needing to know either quantity in advance — the standard method by which generation times are actually measured.

Fails without

Drop the closed-batch assumption (e.g. a chemostat, with continuous nutrient inflow and waste outflow): the population can be held indefinitely in a steady, exponential-like state rather than passing through lag, exponential, stationary and death phases in sequence — exactly why continuous culture is treated as a distinct technique from the batch growth curve (Discussion).
Drop the desynchronisation assumption and instead treat division as one discrete, synchronous event across the whole population: a real growth curve would show stepwise jumps in cell number rather than the smooth exponential curve the model predicts; the smooth continuous approximation only holds because, in an actual culture, individual cells' division timing is spread out across the population rather than perfectly aligned.

Common errors

Assuming the entire batch-culture curve is exponential, rather than recognising that Steps 1–4's exponential law describes only one of four phases (Step 5).
Believing stationary phase means cells have stopped dividing altogether; in fact division continues, but is matched by an equal rate of cell death, so net population size is roughly constant (Discussion).
Plotting raw population count \(N\) against time and expecting a straight line, rather than plotting \(\ln N\) (or \(\log_{10}N\)) as required by Step 4 to obtain a linear exponential-phase segment.
Confusing generation time \(g\) (time for the population to double) with generation number (how many doublings have occurred) — these are related but distinct quantities, related by \(N=N_0\cdot2^{\,\text{generation number}}\).

Discussion

The four-phase batch-growth curve, and the chemostat as a device for holding a culture in steady-state exponential growth indefinitely, are most closely associated with Jacques Monod's kinetic studies of bacterial growth in the 1940s, which also established the now-standard relationship between growth rate and limiting-nutrient concentration used throughout microbial physiology.

Real growth curves deviate from the idealised four-phase picture in well-documented ways: diauxic growth, in which a population presented with two different usable nutrients exhausts the preferred one, passes through a brief second lag phase while re-adjusting its metabolism, and only then resumes exponential growth on the second nutrient, producing a visibly two-humped curve rather than the single smooth exponential segment of Step 2.

Common misconception: that the death phase simply mirrors the exponential phase, with population size decaying at the same rate it once grew. Death-phase kinetics are generally slower and less regular than exponential-phase growth, since cells vary considerably in their resistance to starvation and accumulated waste, and some organisms enter a distinct low-metabolism survival state rather than dying outright.

Worked examples

N_0 = 100\text{ cells},\quad g = 20\text{ min (}\textit{E. coli}\text{ under optimal conditions)},\quad t = 2\text{ h} = 120\text{ min}

Number of generations elapsed: \(t/g = 120/20 = 6\). Applying the Result directly with \(t/g\) as an exact integer count of doublings. A

N(120) = 100 \times 2^{6} = 100 \times 64 = 6400\text{ cells}

Six successive doublings multiply the initial population by \(2^6=64\); note how rapidly this compounds — a further hour (three more doublings) would multiply the population by a further \(2^3=8\), to over \(51{,}000\) cells, provided the exponential phase has not yet ended. A

N(120\text{ min}) = 6400\text{ cells}

Reading. A modest twenty-minute generation time produces rapid, dramatic population growth once compounded over multiple generations — the same compounding logic (Corollaries) that makes exponential phase the decisive window for the fixation of any newly arisen advantageous mutation.

Scope. Valid only while the culture remains genuinely within exponential phase; extrapolating this formula indefinitely would predict physically impossible population sizes within days, precisely because real cultures leave exponential phase once nutrients are depleted (Result, Scope).

Problems

A culture is inoculated with \(500\) cells and grows exponentially with generation time \(g=30\) minutes. How many cells are present after \(3\) hours?
Solution
\(t/g = 180/30 = 6\) generations. \(N = 500\times2^{6} = 500\times64 = 32{,}000\) cells.
A student plots \(\ln N\) against time for a growing culture and measures a slope of \(0.0347\ \text{min}^{-1}\) over the linear portion of the graph. Find the generation time.
Solution
The slope of \(\ln N\) vs. \(t\) is \(\mu\) (Step 4), so \(\mu = 0.0347\ \text{min}^{-1}\). By Step 3, \(g=\ln2/\mu = 0.693/0.0347 \approx 20\) minutes.
Explain, using the Result and Corollaries, why an antibiotic that kills only actively dividing cells is typically far less effective against a culture already in stationary phase than against the same culture in exponential phase.
Solution
Many antibiotics target processes specific to active cell division (e.g. cell-wall synthesis during elongation). In exponential phase \(\mu=\mu_{\max}\) and nearly every cell is dividing, so nearly every cell is vulnerable. In stationary phase (Step 5), \(\mu\approx0\) because birth and death rates are balanced, not because division has ceased for every individual cell, but a much smaller fraction of the population is actively dividing at any instant — leaving fewer vulnerable targets and making the same antibiotic considerably less effective, a phenomenon closely related to (though distinct from) genetically encoded antibiotic-resistance.