Exponential Growth of Biomass

The simplest statement in microbiology: cells make more cells, so the rate of increase is proportional to how many there already are. Write that as $dX/dt = \mu X$, integrate, and you get $X = X_0 e^{\mu t}$. Take logarithms and you get a straight line, which is why every growth curve in the literature is plotted with a logarithmic biomass axis — on linear axes the exponential is an uninformative hockey stick, and on log axes the phases of a batch culture separate cleanly.

Those phases are worth naming, because this equation describes exactly one of them. Lag phase comes first: the inoculum is rebuilding the enzymes and ribosomes it needs for the new medium, and $\mu$ is effectively zero. Its length depends on how different the new medium is from the one the seed grew in and on the physiological state of the seed, and no equation predicts it — which is precisely why a seed train exists, transferring cultures in mid-exponential phase into medium of the same composition so that the lag stays near zero. Exponential (or log) phase is this equation's territory, and it is the only stretch where a single $\mu$ applies. Stationary phase arrives when something runs out or accumulates: the limiting substrate falls through $K_s$, or oxygen transfer caps respiration, or a product reaches an inhibitory concentration. Death phase follows, and is itself often exponential, with a negative rate.

Measuring $\mu$ properly means plotting $\ln X$ against time across the whole run and taking the slope of the STRAIGHT stretch, discarding the lag at the front and the tail-off at the back. Two points chosen carelessly — one in the lag and one in exponential phase — average a real growth rate with a period of no growth and return a number that describes neither. This is also the commonest way $\mu$ and $\mu_{max}$ get confused: the slope of a clean exponential in genuine substrate excess is $\mu_{max}$, and the slope of anything else is just $\mu$.

A word on what "biomass" means here, because the measurement is less obvious than the equation. Cell dry weight is the reference method and the slowest. Optical density at 600 nm is what everybody actually uses, and it is a proxy: it measures light scattering, which depends on cell size and shape as well as number, so the OD-to-dry-weight factor drifts as a culture moves from exponential into stationary phase and the cells change morphology. Viable plate counts measure something different again — colony-forming units, which miss cells that are alive but not culturable and count a clump as one. None of the three is wrong; they answer different questions, and a growth curve should say which one it plotted.

The logarithm has a consequence worth internalising for planning. Because time enters logarithmically, doubling the biomass you are aiming for adds ONE doubling time — not double the hours. Aiming ten times higher adds only 3.3 doublings. The corollary is more useful in practice: a bigger inoculum saves the same fixed number of hours whatever the target, which is why seed-train volume is one of the cheapest levers available on a batch cycle time, and why the standard 5 to 10 percent inoculum is a compromise between the cost of growing the seed and the cost of an idle production vessel.

Finally, resist extrapolating far past your data. The exponential is real and it is real for a few hours. Continued indefinitely, a single bacterium would exceed the mass of the Earth in a couple of days, which is the standard classroom illustration and also a serious engineering warning: a fermentation planned on an extrapolated exponential can be planned for a biomass the vessel could never physically hold, aerate or mix.