Maximum likelihood estimators for colony-forming units

Abstract

Measuring the abundance of microbes in a sample is a common procedure with a long history, but best practices are not well-conserved across microbiological fields. Serial dilution methods are commonly used to dilute bacterial cultures to produce countable numbers of colonies, and from these counts, to infer bacterial concentrations measured in colony-forming units (CFUs). The most common methods to generate data for CFU point estimates involve plating bacteria on (or in) a solid growth medium and counting their resulting colonies or counting the number of tubes at a given dilution that have growth. Traditionally, these types of data have been analyzed separately using different analytic methods. Here, we build a direct correspondence between these approaches, which allows one to extend the use of the most probable number method from the liquid tubes experiments, for which it was developed, to the growth plates by viewing colony-sized patches of a plate as equivalent to individual tubes. We also discuss how to combine measurements taken at different dilutions, and we review several ways of analyzing colony counts, including the Poisson and truncated Poisson methods. We test all point estimate methods computationally using simulated data. For all methods, we discuss their relevant error bounds, assumptions, strengths, and weaknesses. We provide an online calculator for these estimators.Estimation of the number of microbes in a sample is an important problem with a long history. Yet common practices, such as combining results from different measurements, remain sub-optimal. We provide a comparison of methods for estimating abundance of microbes and detail a mapping between different methods, which allows to extend their range of applicability. This mapping enables higher precision estimates of colony-forming units (CFUs) using the same data already collected for traditional CFU estimation methods. Furthermore, we provide recommendations for how to combine measurements of colony counts taken across dilutions, correcting several misconceptions in the literature.

Keywords: CFU; MPN; bacterial counts; colony count estimation; dilution experiments; dilution plating; maximum likelihood estimator; most probable number.

Fig 1

Visual equivalence between plate and tube-based assays. The left panel is a cartoon of a typical plate containing colonies, where the growing colonies are shown as dark disks. In the middle panel, the plate is divided into N (here 16) approximately colony-sized regions. If a region contains one or more colony centers (black dots), this region can be mapped to a positive (dark) tube as shown in the right panel. Similarly, regions containing no colony centers are mapped to negative (light) tubes. This demonstrates that plating is equivalent to a massive parallel version of a tube-based assay with $N\approx \frac{A_{plate}}{A_{colony}}.$ Furthermore, it demonstrates that the MPN method can be used for plate data.

Fig 2

Bias and under-estimation of the true bacterial concentration as a function of crowding, illustrated using the Poisson estimator. We illustrate this by plotting the ratio of the estimated concentration (with the error bands denoting ±1 SEM at N = 5,000) to the true concentration. Here, crowding is measured by the ratio of the average number of colonies to the maximum number of colonies that can fit within a plate ${\displaystyle f=\frac{⟨n⟩}{N}}$. At low crowding values (encompassing the conventionally recommended 25–250 colonies per plate), the naive estimator has low bias, but large uncertainty. At a crowding value of 0.2 (∼1,000 colonies on a 10 cm plate—an ambitious task, and not recommended), the naive Poisson estimator under-estimates the true concentration by about 10%, and many-fold under-estimation is possible as crowding approaches 1.

Fig 3

The probability distributions of estimated CFU concentrations from different estimators generated from 1,000 independent numerical experiments with dilutions 0.1, 0.1, 0.01, 0.01, 0.001, 0.001, r = 100,000, V = 0.2, N = 5,000. Here, the segmented-plate average (one-quarter of the plate is counted), naive Poisson, “pick the best,” traditional average, Poisson with cutoff, and MPN methods are compared. The MPN method demonstrates the best combination of high precision and accuracy.