### Failing to understand the difference between linear and exponential time can undermine your research.

by Dr. Barry Hall

Over the last decade high-throughput measurements of bacterial growth rates by using microtiter plate readers has become increasingly common. Indeed, some plate readers, including the BioTek LogPhase 600 and the EnzyScreen GrowthProfiler 960, are designed specifically for that purpose. However, the documentation and various application notes for those and several other plate readers, including Tecan’s Infinite200 Pro and SoftMax Pro 7.0 software for use with Molecular Devices’ instruments, contain serious and misleading errors.

All of that documentation (references 1-4) includes one or more plots of bacterial growth showing OD vs time. In each case the error is that the plots show OD on a linear scale. That scale should be ln OD vs time. The linear plot reflects a fundamental misunderstanding of bacterial growth kinetics. That misunderstanding may arise from years of experience of monitoring enzyme kinetics, which is linear in time. It leads to incorrect expression of bacterial growth rates as OD per minute.

Enzyme reactions are generally linear in time; i.e. the change in product concentration is a linear function of time. The amount of product released (or substrate consumed) per minute is a constant and is properly expressed as µmol/minute.

That is **not** the case for bacterial growth kinetics. Bacterial growth is a first-order process. The absolute increase in cell number per unit time is a function of the number of cells already present. Formally, the change in cell number (N) with respect to time is the number of cells present, times some constant µ.

µ is the first-order growth rate.

rearranging we obtain:

When that equation is integrated from *t* = 0 to *t* = *t*

the relationship becomes:

The left side is a dimensionless number, so the right side must also be dimensionless. *t* is in units of time so µ must be in units of reciprocal time.

OD is used as a measure of the number of cells. When ln OD is plotted vs time, during the exponential phase of growth the points fall along a straight line. Indeed, that is where the terms “exponential phase” and “exponential growth” come from. It is particularly ironic that the Application Note for BioTek’s LogPhase 600 instrument (reference 2) makes that same error. Figure 1 in that Application Note shows a bacterial growth curve with lag phase, log phase, and stationary phases clearly marked. The scale, however, is linear and the axis label is OD, not ln OD. Figures 3 through 6 show max growth rates under various conditions but the rates are always expressed as OD/min, which is incorrect. The correct unit for growth rates, or for any first-order process, is simply min-1.

When such a fundamental error is included in the documentation one wonders about the rates that are calculated by their software. Are the values reported as erroneous as are the units?

The figure below shows bacterial growth data [ExampleData1 from the GrowthRates program __http://www.bellinghamresearch.com/__ well A2]. OD values are corrected for the OD of a blank well, an essential step in correctly calculating growth rates. Graph A shows a plot of OD vs time, while Graph B shows a plot of ln OD vs time.

Based on Graph A, the maximum growth rate would be 0.067 OD per minute, but that is misleading. In fact, that rate increases from 0.002 per minute starting at 80 minutes and increases to the maximum rate at 230 minutes. There is no log phase in this linear plot.

Based on Graph B the exponential growth rate is 0.037 per minute from 120 through 170 minutes, with a lag time of only 100 minutes before entering the log phase. This is the correct plot and the growth rate is correctly expressed a reciprocal time. The fitted equation is ln OD = 0.037 x minutes -7.91.

A legitimate alternative to plotting ln OD vs time is a *semi-log* plot of OD vs time, in which the Y-axis (OD) is scaled logarithmically.

A legitimate alternative to plotting ln OD vs time is a *semi-log* plot of OD vs time, in which the Y-axis (OD) is scaled logarithmically.

Notice the log scale for OD. Note that the shape of this curve is identical to that of the Exponential plot above. An exponential fit of points 120 through 170 fts the equation OD = 0.0003 x exp(0.037 x minutes). The power 0.037 is identical to the slope from the exponential plot above.

**Whenever a plot shows a linear scale for OD, it is incorrect and misleading. Either use a linear fit ofln OD vs time or an exponential fit of a semi-log plot.**

Fortunately, it is not necessary to use the manufacturer’s software to calculate growth rates. All of those instruments output the OD readings for each well in the microtitre plates at each time point. The output is usually available in the form of an Excel file.

The program ** GrowthRates**, distributed by the Bellingham Research Institute

__www.bellinghamresearch.com__, uses that Excel output file to correctly calculate growth rates and writes files of those rates. For each well,

**writes the growth rate both in min-1 and hour-1, and the standard error of that rate. Because many microbiologists are more comfortable with describing growth rates in terms of doubling times,**

*GrowthRates***also reports the doubling time in minutes. Both the lag time and the maximum OD are reported. The correlation coefficient R is reported as a measure of the accuracy of the reported rate.**

*GrowthRates*For a 384 well microtiter plate, the total time to accomplish all of those calculations using ** GrowthRates** is less than 30 seconds. That can be compared with the 32 hours that would be required to manually calculate those rates for all 384 wells using Excel. No programming skills in Matlab, R, or Python are required to use

**.**

*GrowthRates*** GrowthRates** automatically corrects OD reading for background, then automatically identifies the optimal time points to use to calculate the maximum growth rate.

Different instruments output their data in Excel using different formats.** GrowthRates** automatically translates those output files to a specific format it requires.

**currently supports 16 different plate readers, and support for additional instruments can be provided upon request. For more details see the GrowthRates Technical Brief (reference 5).**

*GrowthRates*We encourage you to visit __http://www.bellinghamresearch.com/__ to explore the ** GrowthRates** features, and find other articles and helpful details about analyzing bacterial growth rates.

### References

- “Advanced kinetic analysis of a bacterial growth assay”. Application note by Molecular Devices. https://www.moleculardevices.com/en/assets/app-note/br/advanced-kinetic-analysis-of-bacterial-growth-assay#gref
- “Monitoring Bacterial Growth under Different Environmental Conditions?: Using the Agilent BioTek LogPhase 600 to provide high quality kinetic data: Application Note.
__https://www.biotek.com/resources/application-notes/m-onitoring-bacterial-growth-under-different-environmental-conditions/__ - “Automated solution for monitoring growth of Staphylococcus aureus: optimizing bacterial growth studies on the Infinite® 200 PRO multimode reader platform using permanent shaking and heating” Tecan Application Note.
__https://lifesciences.tecan.com/plate_readers/fluorescence_absorbance_luminescence?p=tab–3__. Click Infinite 200 Pro, then under Literature click List of all downloads, then see Infinite 200 Pro Application Notes. - GrowthProfiler 960.
__https://www.enzyscreen.com/growth_profiler.htm__ - GrowthRates Technical Brief, Bellingham Research Institute, https://drive.google.com/file/d/1pBPh8eyf117uGgFjkclM-Dp0kkmViA5b/view

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