^{*}

Conceived and designed the experiments: LV JS. Performed the experiments: VM. Analyzed the data: LV MG. Contributed reagents/materials/analysis tools: HPK MG JS. Wrote the paper: LV MG JS HPK.

LV, JS, HPK, VM, and MG are employees of Deutsche Sammlung von Mikroorganismen und Zellkulturen GmbH (DSMZ). The DSMZ is an independent, non-profit organisation. There are no patents, products in development or marketed products to declare. The authors adhere to all the PLoS ONE policies on sharing data and materials.

The Phenotype MicroArray (OmniLog® PM) system is able to simultaneously capture a large number of phenotypes by recording an organism's respiration over time on distinct substrates. This technique targets the object of natural selection itself, the phenotype, whereas previously addressed ‘-omics’ techniques merely study components that finally contribute to it. The recording of respiration over time, however, adds a longitudinal dimension to the data. To optimally exploit this information, it must be extracted from the shapes of the recorded curves and displayed in analogy to conventional growth curves.

The free software environment R was explored for both visualizing and fitting of PM respiration curves. Approaches using either a model fit (and commonly applied growth models) or a smoothing spline were evaluated. Their reliability in inferring curve parameters and confidence intervals was compared to the native OmniLog® PM analysis software. We consider the post-processing of the estimated parameters, the optimal classification of curve shapes and the detection of significant differences between them, as well as practically relevant questions such as detecting the impact of cultivation times and the minimum required number of experimental repeats.

We provide a comprehensive framework for data visualization and parameter estimation according to user choices. A flexible graphical representation strategy for displaying the results is proposed, including 95% confidence intervals for the estimated parameters. The spline approach is less prone to irregular curve shapes than fitting any of the considered models or using the native PM software for calculating both point estimates and confidence intervals. These can serve as a starting point for the automated post-processing of PM data, providing much more information than the strict dichotomization into positive and negative reactions. Our results form the basis for a freely available R package for the analysis of PM data.

The so called ‘-omics’ techniques yielded tremendous insights in the biology of cellular organisms. They address different steps in the information transfer from coding DNA (genomics)

A major biological feature, the phenotype, was until recently not accessible with high-throughput techniques. This is unfortunate, as it is the phenotype which is the object of selection and, hence, is the level at which evolutionary directions are governed

In microbiology, a simple way to assess the phenotype is to characterize an organism's replication behavior under specific conditions

The Phenotype MicroArray (PM) system appears to close the gap of capturing a large number of phenotypes in high-throughput systems. In this approach, a physiological reaction producing NADH engenders a redox potential and flow of electrons to reduce a tetrazolium dye

In common ‘-omics’ techniques, the recorded value is a mostly qualitative information on the difference between two experiments, usually obtained from measurements at a single time point, which is often an endpoint

This wealth of data was till now hardly exploited, as the kinetics were usually only qualitatively assessed

The native OmniLog® PM software

The development of statistical methods for the analysis of longitudinal data started with the pioneering work of Laird and Ware

Most empirical equations such as the logistic law

Here we explored the free software environment R

Our results enable us to propose software solutions for exploiting multiple respiration kinetics from automated systems such as PM. Since we consider mainly biologically focused users, we believe that the introduction and availability of convenient and reliable data exploration techniques

The first dataset comprised two strains of two species of bacteria (^{T},

To additionally investigate the impact of the age of cultures on the technical and biological reproducibility, the third dataset focused on a single strain only, _{1}), 18.00 h (t_{2}), 19.33 (t_{3}), 20.50 (t_{4}), 21.92 (t_{5}), 23.25 h (t_{6}), 24.5 h (t_{7}), 25.58 h (t_{8}) or 40.33 h (t_{9}), respectively, and subsequently measured on GEN III MicroPlates™ in the PM modus over 91 hours. For each growth duration age four technical replicates were performed except for t_{9}, which was repeated only twice. Dataset 3 thus comprised one strain × eight growth durations × four technical replications × 96 substrates plus (t_{9}) one strain × one growth duration × two technical replicates × 96 substrates, hence 3072+192 = 3264 individual curves.

All raw measurements are included in

As the functionality of the native OmniLog® PM software

Plots of all respiration curves are included in

For the description of functional dependencies of two measured variables a mathematical function can be fitted onto the data. In general, such a fit aims at minimizing the distances between the raw data points and the values predicted by the function. The choice of a type of function is usually motivated by some basic assumption about the underlying system. The selection of a function is an interpreting activity and a crucial step in the analysis

The parametric analysis method of the native OmniLog® PM software ^{th} percentile highest value among all values over all time points, and the minimum height (“MinHeight”) is calculated as the 12^{th} smallest value among the first 48 reads over all time points. The length of the lag phase is calculated from the raw data using the formula “MidTime - (MidHeight – MinHeight)/Slope”

In contrast, the basic part of R's add-on package _{μ} and the time point t_{μ} of its occurrence. A tangent at this point has the form y(t) = μ(t-λ) and thus yields the length of the lag-phase λ via y_{μ} = μ(t_{μ}-λ) (Kschischo, pers. comm.).

In addition to the point estimates for the parameters from both model and spline, confidence limits can be calculated

We assessed in detail in how many (and which) cases a model fit was impossible using one of the default models: (i) logistic growth, (ii) Gompertz growth, (iii) modified Gompertz growth and (iv) Richards growth

The accuracy with which the parameters (estimated using the three different approaches) fitted to the raw data was investigated for all types of observed curve shapes (see above) and visualized using a set of individual curves representative for each shape. These exemplars could also be used to illustrate the difference in parameter estimation between model and spline fit and thus for the identification and explanation of the effect of difficult-to-fit curve shapes on parameter estimates. Moreover, they were used to determine the most useful way of displaying parameters estimated together with their CIs. The proposed methods here intentionally resign any multiplicity adjustment, because the analyses are expected to detect all interesting phenomena while it would be worse to miss some of them.

Because there is no restriction on the type of sample to be analyzed, the PM technique is capable of dealing with a rather unlimited amount of distinct experimental questions. That is, not only isolated strains or well-defined mutants are manageable, but also mixed or environmental samples are feasible

Since up to now the

Parameter estimates from all respiration curves including CI limits are available in

To explicitly infer the optimal number of distinct shape classes for classifying the curves, we applied the R package ^{st} and 2^{nd} dataset. Archetypes are characteristic extreme types of combinations of multivariate observations found by minimizing a convex residual sum of squares (RSS) criterion; the implementation ensures that the real measurements can be represented by convex combinations of the archetypes. The algorithm alternates between finding the best set of coefficients for the given archetypes and finding the best archetypes for a given set of coefficients. The overall RSS is reduced successively because in each step several convex least-squares problems are solved. While the number of archetypes is predefined in each run, the algorithm can be restarted for a series of numbers of archetypes (we tested 1 to 10) and, according to the “elbow criterion”, the optimal number is the one that resulted in the largest step towards a lower RSS compared to subsequent improvements. We used the

Using a subset of dataset 1 the visualization of PM curves as heat maps

Each respiration curve is displayed as a thin horizontal line, in which the curve height as measured in OmniLog® units is represented by color intensity (darker parts indicate higher values). The x-axes correspond to the measurement time in hours. The upper part shows an overview of two plates. Here, the descriptions of the y axes (only visible if enlarged, but see below) list the names of the wells; the descriptions of the x axes list the measurement time in hours. The boxes below represent magnified parts of the upper panel to illustrate the color changes in the case of decreasing color intensities (regions surrounded by black ellipses) or technical problems such as short-term intermediate peaks (positions marked by eight-pointed stars).

In

PM curves from a representative technical repetition from the first dataset were arranged according to the original 8×12 wells plate layout. The respective curves from all four strains are superimposed; the affiliation to each strain is indicated by color as follows: black, ^{T}; green,

We felt that data quality and integrity could be checked faster and more comprehensively using the second, curve-based visualization approach. The curve display gave a more intuitive and straightforward overview of the data, while simultaneously facilitating the development of an overall assessment of an organism's behavior in the experiment. Moreover, color codes for results from distinct organisms, replications and experiments enabled informative superimposed displays (

Left, enlarged view of the curves from wells G11 and H11 as depicted in ^{T}; green,

We estimated the parameters length of the lag phase λ, respiration rate μ (slope), maximum cell respiration A and the area under the curve (AUC) using both the model fitting and model-free spline fitting approach from the basic part of R's add-on package

Value | Dataset 1 | Dataset 2 | Dataset 3 | |

curves | # total | 3840 | 3840 | 3264 |

# without fittable models | 244 (6.35%) | 64 (1.66%) | 44 (1.35%) | |

# without fittable splines | 0 (0%) | 0 (0%) | 0 (0%) | |

experimental groups | # total | 768 | 768 | 864 |

# without fittable models | 12 (1.56%) | 1 (0.13%) | 2 (0.23%) | |

# without fittable splines | 0 (0%) | 0 (0%) | 0 (0%) | |

# model parameter λ<0 | 281 (36.59%) | 229 (29.82%) | 205 (23.73%) | |

# spline parameter λ<0 | 221 (28.78%) | 154 (20.05%) | 124 (14.35%) | |

# model parameter A>400 | 10 (1.3%) | 3 (0.39%) | 12 (1.39%) | |

# spline parameter A>400 | 0 (0%) | 0 (0%) | 0 (0%) | |

# model parameter A<100 | 282 (36.72%) | 245 (31.9%) | 162 (18.75%) | |

# spline parameter A<100 | 291 (37.89%) | 236 (30.73%) | 146 (16.9%) | |

spline fits | # A<100 | 1464 | 1196 | 583 |

# A<100 and λ<0 | 884 (60.38%) | 733 (61.29%) | 485 (83.19%) | |

mean λ if A<100 and λ<0 | −207.6±2871.0 | −6.8±8.1 | −12.1±12.4 | |

# A>100 | 2376 | 2664 | 2681 | |

# A>100 and λ<0 | 106 (4.46%) | 106 (3.98%) | 130 (4.85%) | |

mean λ if A>100 and λ<0 | −3.3±4.4 | −3.3±4.3 | −3.0±3.5 |

Depending on the specific dataset, between 1.4% and 6.4% of the curves could not be fitted by the modeling approach. Hence, for some experimental groups no parameter estimation was possible at all, resulting in one to twelve groups without parameter estimation depending on the dataset. In contrast, the spline resulted for all datasets, yielding parameter estimates for every group (

In contrast, the spline fit yielded negative λ in only 14.4% to 28.8% of the groups; hence around 10% fewer groups with unreasonable λ point estimators. Not a single group was found with an estimate for A exceeding 400 (

Kendall and Spearman correlations between the parameters describing the curves are listed in

Model | Spline | ||||||||

λ | μ | A | AUC | λ | μ | A | AUC | ||

Model | λ | 0.428 | 0.367 | 0.294 | 0.641 | 0.360 | 0.371 | 0.295 | |

μ | 0.248 | 0.536 | 0.732 | 0.296 | 0.846 | 0.717 | 0.734 | ||

A | 0.362 | 0.320 | 0.7 | 0.324 | 0.538 | 0.748 | 0.700 | ||

AUC | 0.372 | 0.712 | 0.522 | 0.205 | 0.721 | 0.843 | 0.998 | ||

Spline | λ | 0.571 | 0.131 | 0.225 | 0.211 | 0.299 | 0.285 | 0.203 | |

μ | 0.940 | 0.837 | 0.272 | 0.620 | 0.025 | 0.723 | 0.736 | ||

A | 0.437 | 0.658 | 0.571 | 0.963 | 0.051 | 0.584 | 0.854 | ||

AUC | 0.372 | 0.713 | 0.523 | 0.999 | 0.046 | 0.621 | 0.963 |

In

On substrate H11 (

On substrate G11 (

The green curve describes an intrinsically negative reaction (no respiration curve), but instead a slight and linear increase in color development. This probable noise was apparently sufficient as a data basis on which model fit was possible, but some of the resulting parameters, especially the negative length of the lag phase λ, were not biologically sensible. In contrast, the corresponding spline fit yielded a positive value for λ with a broad CI. Again in contrast, the OmniLog® software yielded a value near zero for this parameter. The estimate for μ was slightly positive only from the spline fit, whereas the other two methods yielded zero values. The other parameters were rather similarly estimated by the different methods.

To further explore the causes of the respective differences and characteristics of parameter estimation in certain data situations, the best-fitting model and the spline fit for the red and green curves from well G11 are shown in

The raw colour intensities (black circles), measured over time (x-axis, hours) were fitted by both a parametric model (thick blue line) and a model-free spline (thick red line). The thin dashed lines indicate the maximum slope of each approach (thin dashed black line corresponds to the model fitting approach, the thin dashed red line to the spline, respectively). In the left panel the irregularity is better customized by the spline fit, whereas the model straightens it with the consequence of underestimating the maximum height (A). In case of (almost) no respiration (right panel), the fitted model apparently yielded biologically reasonable parameter estimates for µ but not for λ. In contrast, the spline approach exhibited overfitting and yielded overestimated µ and also overestimated, but biologically meaningful λ. Note the particularly broad CIs for these parameter estimates in

The example corresponding to (almost) no respiration (

In

Left, a plot of the raw respiration data illustrates their courses individually for each of the ten repetitions. The red curve (D12/4) was used as an exemplar for demonstrating the detection of significant differences via CIs. In the right panel, point estimates and 95% CIs for each of the four parameters from the spline approach are given for each replication. The blue lines highlight the position of the upper and lower limit of CIs from D12/4's parameters. A non-overlap of the CIs of different curves indicates a difference of a statistically detectable amount, and the distance between two intervals provides information about the expected minimum difference.

We used the red-colored curve (D12/4) as an exemplar for demonstrating the detection of significant differences

The results from the time series approach in the third dataset are shown in _{5}) was chosen as an example and the corresponding CI limits highlighted. For both λ and μ, all other CIs overlapped with that from curve 20, indicating no detectable differences between the curves. Considering the maximal respiration A and the integral AUC, several CIs did not overlap with that from curve 20, but the effect size for the maximal respiration is at most 4 OmniLog® units ( = 1.5%) for A and 978 units ( = 5%) for AUC. Again, the user was now free to decide whether these differences should be regarded as biologically relevant.

The upper panels show the plot of the respiration curves of _{1}), 18 h (t_{2}), 19.33 (t_{3}), 20.5 (t_{4}), 21.92 (t_{5}), 23.25 h (t_{6}), 24.5 h (t_{7}), 25.58 h (t_{8}) or 40.33 h (t_{9}), respectively, and subsequently measured on GEN III microplates™ in the PM modus over 91 h. The lower panels shows point estimates and 95% CIs for each of the four parameters from the spline approach. The blue lines highlight the position of the upper and lower limits of the CIs from repetition no. 4 at t_{5}. A non-overlap of the CIs of different curves indicates a difference of a statistically detectable amount, and the distance between two intervals provides information about the expected minimum difference.

Regarding the comparison of group means,

The upper panels show the results from the preliminary calculation, a simple calculation of group means of confidence limits and point estimators. The groups, here the distinct pretreatments (cultivation times t1 to t9), are given on the y-axis. For orientation, the blue lines highlight the position of upper and lower limit of CIs from repetition no. 4 at t_{5}, in analogy to

The multiple-mean comparison testing procedure also provides 95% CIs, but for the differences between the group means (here: the considered parameter estimators), thus yielding precise information about the significance of the differences between the groups regarding the considered parameter(s).

To examine whether it is valid to subtract the negative controls (A01) from the measurements from all other wells before estimating curve parameters, we compared the parameter values for maximum height (A) from the A01 with that from selected wells with a negative reaction. Our findings suggest that the negative control might display a reproducible, strain-specific growth-like behavior, and even though these curves are shallower than unambiguously positive reactions, their maximum height can well be larger than that of typical negative reactions on the same plate. This makes it impossible to regard it as an approximation of an error term to be subtracted from the measurements from each other well. These findings are described in detail in

Analysis of archetypes (

The outer figure is a scree plot in which the residual sums of squares (RSS, y-axis) are plotted against the corresponding predefined numbers of archetypes (x-axis). Apparently either four or five archetypes are optimal according to the “elbow criterion”. The insert (upper right) is a parallel coordinates plot showing the original measurements (gray lines) as well as the optimal archetypes (green, black, blue, violet and red lines) obtained if five archetypes are requested. On the x-axis, the names of the curve parameters are indicated. The minima and maxima of the four y-axes are also indicated. For an interpretation of the archetypes, see the main text.

When facing huge and complicatedly structured datasets such as the PM ones discussed here or that commonly occurring in other -omics analyses, the only way to get a comprehensive insight into the experimental results is a suitable graphical raw data representation. Such exploratory graphics have to be comprehensible in short time but also be highly informative

In this study we explored an open-source solution for these specifications and showed that curve kinetics offer a more powerful data visualization than level plots

The information content of the longitudinal PM raw data is a multiple of what an endpoint measurement could ever provide. A suitable analysis strategy thus has to be able to summarize this information and eliminate noise. These requirements can be met by model-fitting and spline-fitting approaches aiming on both dimension reduction and noise reduction

Although the OmniLog® PM software

Our results indicate, however, that the parameter estimation procedures perform best if applied to curves that follow the typical sigmoidal shape. But the parameters A (maximum height) and AUC (area under the curve) are less influenced by possibly uncommon shapes than the lag phase (λ) and the slope (μ). When comparing the two main approaches for curve description, it turned out that the spline smoother is flexible enough to follow even extreme curve shapes and is therefore superior for general parameter estimation, while the model-fitting approach appeared to be more constrained by the underlying model equations and straightened the curves to much. While 14% to 28% of the estimates for λ were biologically unreasonable in a strict sense (negative), most of these were only slightly negative and could safely be regarded as minor mis-estimates for 0.0. Also, the interpretability of the other parameters was not affected in these cases, and extremely negative λ can still be qualitatively interpreted as indicative of overall negative reactions.

The high amount of negative estimates for λ suggests that there is still space for algorithmic improvement. In this study, the default parameters for the smoothing spline and the number of knots were used, since the evaluation of best-performing parameters was beyond the scope of this study. However, the selection of these two kinds of parameters is the critical step in this method

Compared to both model and spline methods, the slope estimates from the OmniLog® PM software

However, the spline-based approach exhibits overfitting behavior in the case of certain curves that strongly deviate from a sigmoidal curve shape. This appears to occur especially when almost no reaction takes place, as shown in

It is well known that phenomena such as autocorrelation (which is usual for growth curves) and non-homoscedasticity of the residuals violate the underlying assumptions of model- and spline-fitting

To enable the user to extract all necessary information, we provided a feasible graphical solution displaying the point estimator together with its CI limits. The function

With two exemplars (

It may often be of interest not to compare single curves but distinct groups of curves. In

Using the more sophisticated simultaneous calculation of differences of user-defined means in combination with the visualization of their CIs, the experimenter is empowered to investigate the data set more specifically regarding the biological hypotheses. The advantages of simultaneous CIs for drawing testing decisions are that the significance, relevance, and direction (increase or decrease) of the effect of interest, as well as the uncertainty concerning the estimates, can be interpreted in a scale close to that of the measured variable, which is often easier than interpreting p values in the scale of probability

Since the width of a CI is a critical measure in the interpretation of testing decisions, a common spline fit for a bundle of repetitions in combination with the above mentioned improvements of spline fitting itself would be of interest for further method developments. On the other hand, the estimation methods for the curve descriptive parameters should be regarded as an interesting point for improvement. As mentioned above, μ and λ are sensitive to uncommon curve shapes, at least partly because of their estimation procedure.

Beyond the here proposed strategies for testing local hypotheses, global-hypothesis frameworks, as they are known, e.g., from the already well explored gene-expression microarray analyses, should be considered. For example, comparisons between complete plates measured from distinct strains or treatments could be managed by a difference-of-means approach. To get the results from the distinct wells comparable to each other, they would need to be normalized by, e.g., dividing by the well-specific means calculated over all plates. The thus normalized parameter estimates could regarded as one sample per plate or groups of plates and accordingly compared against each other.

The conducted archetype analysis

With the here presented approach to OmniLog® PM data analysis highly structured graphics can easily be produced while retaining the flexibility of systematically decoupling the various elements of a display. Itemization by substrate, tested strain or even repetition number is quite simple and no constraints about the number of displayed curves or the position of the subpanel are imposed at all.

The smoothing-spline method for dimension and noise reduction appears to prevalently result in more meaningful parameter estimates than parametric model fitting when applied to PM data.

In conjunction with the proposed parameter estimation using models or preferably splines, the experimenter is free to define limits within which statistically detectable differences are not considered biologically relevant, which makes this method easily adaptable and more powerful than conventional mean comparison procedures

Although the strategy introduced here depends on CIs calculated on the basis of single curves, the approach could be easily extended to include the calculation of mean curves and corresponding intervals

As demonstrated here, for a comprehensive comparison of the curves several parameters have to be considered to come to a meaningful decision. This is connected to fundamental ideas from multivariate data analysis

Even though the analysis of the biological causes for the respiration behaviors of the here tested strains is beyond the scope of the study, a few remarks on the study design and implementation of controls should be placed. On the GEN III plates the A01 well is defined as the control well, containing no substrate. By construction no reaction should occur on this well unless some kind of artifact was involved. The vendor's recommendation is, understandably, to adjust the experimental procedure until this point is met ^{T}, if pooled over all replicates, the values of A in well A01 turned out to be significantly larger than those in, e.g., well D03 (see

In our assessments of the PM technique we observed a series of other experimental sources of errors. One of them is a false-positive color development due to some chemical conversion of the redox dye, actually not caused by respiration (data not shown). Especially pentose sugars such as L-arabinose or xylose might be susceptible to these reactions (B. Bochner, pers. comm.). Thus, we recommend to measure one plate inoculated with only the inoculation fluid but no cell material and to check if such false-positive reactions occur. Wells with such reactions should be excluded from further analysis, since their color development cannot safely be attributed to a physiological reaction.

To conclude, we believe to have demonstrated that tools provided in the free statistical software environment R can be successfully applied to PM data. These tools allow the user to visualize the kinetics in several meaningful respects, to conduct parameter estimation and, hence, dimension and noise reduction with the measurements and to detect statistically significant differences between the curves. All of these techniques can be conducted after selecting and rearranging the data in a sensible way depending on the respective scientific questions of interest. The outcome can even be used to improve the experimental setup itself as, e.g., by determining the necessary minimum number of replications. Additional work is necessary, however, to optimize details of the parameter estimation procedure (see above), and particularly to integrate all mentioned tools together with data input and output, and addition of metadata, in an easy-to-use R package. We recently released the first version of such a package, ‘opm’, by making it available at the comprehensive R archive network CRAN (

The application of more sophisticated spline estimation methods might solve the remaining difficulties. One main issue is the selection of a suitable degree of smoothness. Approaches such as cross-validation

Alternatively or additionally the methods for parameter extraction from the spline could be critically revised. A more sophisticated estimation method for the length of the maximum slope μ and the lag-phase λ, possibly making use of information from the second spline derivative, would probably be able to deal with the so far frequently suboptimal spline fits for these parameters in the case of intrinsically negative reactions.

With the here established strategies for data processing and analysis, the results from Phenotype Microarray experiments are commuted in a framework similar to that for the thoroughly acquainted gene-expression microarray analysis. Thus, the next steps leading to functional data analysis would be to test the applicability of statistical analysis tools such as global multiple testing procedures

Raw measurements from dataset 1 in CSV format.

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Raw measurements from dataset 2 in CSV format.

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Raw measurements from dataset 3 in CSV format.

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Plots of all respiration curves from dataset 1 in PDF format.

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Plots of all respiration curves from dataset 2 in PDF format.

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Plots of all respiration curves from dataset 3 in PDF format.

(PDF)

Parameter estimates from all respiration curves and their CI limits in CSV format.

(CSV)

Parameter estimates from all respiration curves and their CI limits as R code.

(TXT)

Behavior of the negative controls compared to negative reactions in other wells.

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All-against-all correlation plots of the parameter estimates.

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The authors are grateful to Barry Bochner (BiOLOG Inc.) and John Kirkish (BiOLOG Inc.) for helpful advice. LV sincerely thanks Benjamin Hofner (IMBE, University of Erlangen-Nürnberg) for many helpful comments and fruitful discussions on the issues presented here.