Multiple Measures of Bacterial Fitness from Growth Curves, Including Regrowth of Phage-Resistant Bacteria
by Stephen T. Abedon Ph.D.Steve Abedon, Ph.D., has taught well over 100 individual classes and ~10 individual courses over his 30+ years as a college professor. He has over 150 publications, including multiple monographs, and is ranked in the top 1 or 2 percent among researchers of all disciplines worldwide. His research focuses on bacteriophage biology and phage therapy. (abedon.1@osu.edu)
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bacfit.phage.org · Abedon’s Books
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Data are a table with time in the first column and one optical density (OD) column per well or culture, with a header row of names.
Comma-, tab-, or semicolon-separated text works, as do Excel files and plate-reader text exports whose times read as
h:mm:ss. Leading lines above the table, and a temperature column, are skipped.
Here are the five roles. Reference curves are the comparison, such as the phage-sensitive ancestor grown without phages. Test curves are compared with a reference over the whole growth curve. Regrowth curves are cultures exposed to phages that lysed and then grew back. Blank curves are medium alone and can be subtracted. Ignore leaves a column out. Roles are guessed from names and can be changed.
| Curve | Role | Compare with | First OD | Max OD | Last OD |
|---|---|---|---|---|---|
| No data loaded yet. | |||||
Each row compares one Test curve with its reference. Ratios are test ÷ reference, so values below 1 for rates, yield, area, and ln-fold change mean lower fitness by that measure. Differences in lag and in time to threshold are test − reference, so positive values mean the test curve is slower. When more than one reference is present, the italic rows compare references with each other, as a guide to how large a difference between repeats can be without meaning anything.
Load data and assign at least one Reference and one Test curve.
For each Regrowth curve the tool finds the lysis peak and the nadir, then fits the culture’s growth twice: once before lysis, when phage-sensitive bacteria dominate, and once after the nadir, when presumably phage-resistant bacteria do. The two rates are then compared with each other and with the phage-free reference.
Assign at least one curve the Regrowth role.
Enter CFU per mL per OD unit and the culture volume (Data tab, Settings) to convert the regrowth line into cell numbers.
All fitting is done on the natural logarithm of blank-subtracted OD, and only on readings at or above the fitting floor. A three-reading running median is used for peaks, nadirs, and threshold crossings, so that a single noisy reading cannot set a landmark. Here are the seven whole-curve measures, followed by the five regrowth measures.
The steepest slope of ln(OD) against time among the rolling windows before the curve’s maximum OD, in h−1, after each window’s slope is averaged with those of its two neighbors so that one noisy window cannot set it. The doubling time is ln 2 ÷ rmax. This is the Malthusian parameter in the sense used by Concepción-Acevedo et al. (2015).
Starting from the fastest window, neighboring windows are added for as long as they grow at least the set fraction (default 75%) of rmax; a single window below that cutoff is stepped over if the next one is back above it. One straight line is then fitted through every reading in that run. This rate rests on more readings than rmax and is less inflated by noise. The run’s length is reported as the duration of exponential growth.
The time at which the tangent through the fastest window, extended backward, reaches the starting OD (the median of the first three readings), after Zwietering et al. (1990). Fitness by this measure is shorter lag. Lag is reported only when the starting OD is at or above the fitting floor.
When the running median first reaches the threshold OD, interpolated on ln(OD). The difference between test and reference is how far behind the test curve runs at that density. It combines lag, growth rate, and inoculum size, which is a weakness as a measure of any single trait and a strength as a summary of how quickly a culture becomes dense.
The highest value of the running median. A lower maximum OD can mean less biomass per unit resource, a different cell size or shape, or earlier cell death, since OD measures light scattering rather than cell number.
The trapezoidal area under blank-subtracted OD from the first reading to the chosen end time, in OD·h. It integrates lag, rate, and yield into one number, as in Growthcurver (Sprouffske and Wagner, 2016), and so cannot say which of them differs.
ln[OD(t2) ÷ OD(t1)] for each curve, and the test-to-reference ratio of these. In a competition assay the same ratio, computed from each competitor’s own counts, is the relative fitness W of Lenski et al. (1991). Here the two cultures are grown separately, so the ratio is a monoculture analog of W, not W itself.
The peak is the reading followed by the largest fold decline, and the nadir is the lowest point after it. A curve that declines less than the set fold (default threefold) is reported as not lysed. The deviation of a phage-treated curve from its control, and other lysis landmarks, are the business of deviation.phage.org.
The maximum and exponential-phase rates (measures 1 and 2) are computed separately before the peak and after the nadir. Their ratio, regrowth ÷ pre-lysis, compares the regrowing population with the population that grew in the same well before lysis. The regrowth rate is also divided by the reference’s exponential-phase rate.
Where the regrowth line meets the plateau level after the nadir, taken as the median of the readings between the nadir and the start of the regrowth exponential phase, and never less than the fitting floor. Defined this way, onset is the intersection of two fits rather than the time an arbitrary OD is first exceeded. If the regrowth line meets the plateau before the nadir, as in a V-shaped curve with no plateau, onset is set to the nadir and flagged.
Two delays are reported. The delay relative to the reference is when the regrowth reaches the threshold OD minus when the phage-free reference did, as in the PHIDA approach of Martinez-Soto et al. (2021), who used a threshold of 0.2. The anchored delay is regrowth onset minus the time at which the pre-lysis growth line would have reached the plateau OD, which measures how far behind its own unperturbed trajectory the culture is when regrowth begins.
The highest OD reached after the nadir, flagged if still rising at the end of the run, in which case it is a lower bound.
In evolutionary terms, fitness is reproductive success relative to a competitor. For bacteria it is measured most directly by competing two strains in the same culture and following their ratio, which gives the relative fitness W of Lenski et al. (1991). Phage-resistant mutants were among the first bacteria to be assessed this way: Lenski (1988) found that Escherichia coli mutants resistant to phage T4 varied widely in competitive fitness, and that many paid a substantial cost for their resistance.
Competition assays need a way to tell the competitors apart and are laborious, so plate-reader growth curves are often used instead. The growth rate is the most common proxy, but it is only one component of fitness (Wiser and Lenski, 2015), and estimates made in plate readers can miss differences that competition detects (Concepción-Acevedo et al., 2015). Lag, yield, and area under the curve each capture other parts of the growth cycle (Zwietering et al., 1990; Sprouffske and Wagner, 2016), and growth curves can be combined to predict how strains will fare together (Ram et al., 2019). Reporting several measures side by side shows which part of the growth cycle a resistant strain loses ground in, which a single number cannot.
Resistance often comes from losing or altering the molecule a phage uses as its receptor, and that molecule usually has another job, which is why resistance is so often costly (Mangalea and Duerkop, 2020). These trade-offs can be put to use. Chan et al. (2016) isolated a phage that uses part of a multidrug efflux pump of Pseudomonas aeruginosa as its receptor, so that bacteria evolving resistance to the phage became more sensitive to antibiotics. Measuring fitness costs quickly, and for many resistant isolates at once, is part of judging which phages are likely to leave resistant bacteria at a disadvantage.
Turbidity curves of phage-treated cultures often fall and then rise again as phage-resistant bacteria grow back (Abedon, 2025). The time to regrowth has been used to compare phages (Martinez-Soto et al., 2021), but the shape of the regrowth carries more than its timing: its slope is the growth rate of the resistant population. Compared with the same culture’s growth before lysis, and with a phage-free reference, that slope becomes a fitness measure that requires no isolation of resistant clones. Read backward, the same slope also says when the resistant population was founded, which links fitness to the older question of how often resistance arises (Luria and Delbrück, 1943).
The view of regrowth trajectories as an automated route to the fitness of phage-resistant bacteria came from a conversation with Paul E. Turner (Yale University).
deviation.phage.org finds when a phage-treated curve departs from its control, along with other lysis landmarks. doublingtime.phage.org converts between growth rates and doubling times.