(C) PLOS One
This story was originally published by PLOS One and is unaltered.
. . . . . . . . . .



Inferring fungal growth rates from optical density data [1]

['Tara Hameed', 'Department Of Bioengineering', 'Imperial College London', 'London', 'United Kingdom', 'Natasha Motsi', 'Elaine Bignell', 'Medical Research Council Centre For Medical Mycology', 'University Of Exeter', 'Exeter']

Date: 2024-05

Abstract Quantifying fungal growth underpins our ability to effectively treat severe fungal infections. Current methods quantify fungal growth rates from time-course morphology-specific data, such as hyphal length data. However, automated large-scale collection of such data lies beyond the scope of most clinical microbiology laboratories. In this paper, we propose a mathematical model of fungal growth to estimate morphology-specific growth rates from easy-to-collect, but indirect, optical density (OD 600 ) data of Aspergillus fumigatus growth (filamentous fungus). Our method accounts for OD 600 being an indirect measure by explicitly including the relationship between the indirect OD 600 measurements and the calibrating true fungal growth in the model. Therefore, the method does not require de novo generation of calibration data. Our model outperformed reference models at fitting to and predicting OD 600 growth curves and overcame observed discrepancies between morphology-specific rates inferred from OD 600 versus directly measured data in reference models that did not include calibration.

Author summary Quantifying fungal growth is essential for antifungal drug discovery and monitoring antifungal resistance. As fungal growth is complex, with fungal morphology (shape) dynamically changing over time, previous studies have quantified fungal growth by estimating growth rates during specific fungal morphologies (morphology-specific growth rates) or by mathematically modelling fungal growth. However, collecting time-series data that captures the morphological information required for mathematical model fitting or estimating morphology-specific growth rates is prohibitively time consuming for large-scale drug testing in most microbiology laboratories. Alternatively, fungal growth can be quickly, although indirectly, quantified by measuring the optical density (OD) of a broth culture. However, changes in OD are not always reflective of true changes in fungal growth because OD is an indirect measure. This paper proposes a method to model fungal growth and estimate a morphology-specific growth rate from indirect OD 600 measurements of the major mould pathogen, Aspergillus fumigatus. We explicitly model the relationship between measured indirect OD 600 data and true fungal growth (calibration). The presented work serves as the much-needed foundation for estimating and comparing morphology-specific fungal growth rates in varying antifungal drug concentrations using only OD 600 data.

Citation: Hameed T, Motsi N, Bignell E, Tanaka RJ (2024) Inferring fungal growth rates from optical density data. PLoS Comput Biol 20(5): e1012105. https://doi.org/10.1371/journal.pcbi.1012105 Editor: Attila Csikász-Nagy, Pázmány Péter Catholic University: Pazmany Peter Katolikus Egyetem, HUNGARY Received: November 22, 2023; Accepted: April 24, 2024; Published: May 16, 2024 Copyright: © 2024 Hameed et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability: All code used for model fitting and plotting is available on a GitHub repository at https://github.com/tah17/Inferring-fungal-growth-rates-from-OD-data. All data used in the paper are included as Supporting Information files. Funding: This research was funded in whole, or in part, by the Wellcome Trust [Grant number 215358/Z/19/Z to T.H.], https://wellcome.org/. For the purpose of open access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission. This project was partly funded by National Centre for the Replacement, Refinement and Reduction of Animals in Research (NC3Rs) Studentship (NC/P00217X/1), https://nc3rs.org.uk/. E.B acknowledges support from the MRC Centre for Medical Mycology at the University of Exeter (MR/N006364/2 and MR/V033417/1). Work in the laboratory of E.B. was funded by MRC project grants MR/M02010X/1, MR/S001824/1, and MR/L000822/1 and by a Biotechnology and Biological Sciences Research Council (BBSRC) project grant BB/V017004/1. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist.

Introduction Fungal infections affect over a billion people worldwide [1] and remain a global threat to human health [2]. Combatting fungal infections is essential for treating patients with life-threatening invasive infections [1]. In the clinic, the invasive fungal infection, invasive aspergillosis, is widely prevalent [1] and associated with unacceptably high mortality rates [2] of around 29% [3], even with treatment. Without appropriate treatment, invasive aspergillosis can result in 100% mortality, as seen with COVID-associated-pulmonary-aspergillosis [4]. Whilst most fungal infections are treated with antifungal drugs [5,6], the issue of antifungal resistance [7], spanning to multi-drug resistance, may shrink an already small pool of treatment options for patients [8]. To avoid patients being left with no treatment options, there is a real need to locate new antifungal drugs and for increased antifungal stewardship [7]. Antifungal stewardship requires detecting and quantifying fungal growth in different conditions to evaluate potential resistance or drug efficacy. Multiple fungal pathogens have recently been identified as critical threats to public health by the world health organisation (WHO) [9]. These fungi can exhibit complex growth dynamics, such as filamentation, due to morphogenesis [10,11], where morphogenesis is also promoted by exposure to antifungal drugs [12]. The current standards for antifungal susceptibility testing for filamentous fungi, set by the Clinical and Laboratory Standards Institute (CLSI) and the European Committee for Antimicrobial Susceptibility Testing (EUCAST), involve evaluating minimum inhibitory/fungicidal/effective concentrations (MIC/MFC/MEC) in vitro [13–15], which is the minimum drug concentration required to reduce or alter the morphology of fungal biomass within a certain time [15–18]. However, further development of antifungal susceptibility testing for filamentous fungi is required for the following two reasons. First, the current CLSI and EUCAST standards include subjective measurements that depend upon visual inspection of changes in growth [19]. Second, the current guidelines cannot capture time-dependent antifungal activity or antifungal activity specific to a particular fungal morphology (morphology-specific antifungal activity). The MIC/MFC/MEC do not describe temporal dynamics of fungal growth because they are based upon a single end-point measurement. These metrics also cannot capture morphology-specific antifungal activity while fungal morphology evolves over time. Popular antifungal drugs, such as azoles, have morphology-specific mechanisms of action [20] and different fungal morphologies have been noted to be future antifungal targets [21]. We need a more quantitative, temporal, and morphology-specific evaluation of fungal growth to further mechanistic understanding of both differential growth of filamentous fungi and a drug’s antifungal activity. Several previous studies have calculated fungal growth rates to quantitatively evaluate temporal changes in fungal growth in different experimental conditions [22–25]. Growth rates specific to a fungal morphology (morphology-specific growth rates), such as germination rates of fungal spores or proliferation rates of hyphae, can be estimated from morphology-specific growth data to evaluate morphology-specific changes in fungal growth over time. For example, hyphal extension rates have been derived from hyphal microscopy data to investigate hyphae-specific growth defects in mutant strains of Aspergillus fumigatus [22]. Such morphology-specific growth rates can also be estimated from mechanistic models of fungal growth fit to directly measured time-course data. For example, hyphal growth rates have been estimated using a fungal growth model fit to hyphal area image data [26]. Mechanistic models that include morphology-specific growth mechanisms [23,26,27] can not only estimate morphology-specific growth rates, but also further understanding of morphological dynamics during growth [26]. However, to the best of our knowledge, estimating morphology-specific growth rates and developing mechanistic models of dynamic filamentous fungal growth are not routine in clinical microbiology laboratories potentially because it is too difficult to quickly collect the required time-course data that encapsulates morphological information or directly measures dynamic fungal growth. The acquisition and analysis of directly measured time-course morphology data requires specialist technology and analyses that lie beyond the scope of clinical microbiology laboratories. To collect morphology time-course data, fungal growth is directly measured over time using techniques such as enumerating colony forming units (CFUs), microscopy, or histology. These direct data collection methods are only feasible for large-scale experimentation in specialist research microbiology laboratories because of the high cost of technologies and skill-level required to perform automated direct measurement of fungal growth. Technological developments have been made to aide processing of fungal image data [25], such as FIJI [28] or fungal feature tracker (FFT) [29]. But they are either not fully automated, relying on point-and-click methods, or are still not used in clinical microbiology labs due to difficulty of use or implementation. Instead of directly measuring fungal growth to collect time-course fungal growth data, indirect measures of fungal growth can be obtained for growth rate estimation or mechanistic modelling by measuring optical density (OD). Collecting OD data is an automated, quick, and popular method to obtain time-dense fungal growth data that is widely used for estimating microbial growth rates [30] and is amenable to high-throughput phenotyping and antimicrobial susceptibility studies [31]. For fungi, the use of OD data to directly and empirically infer fungal growth rates is commonplace. Moreover, simple mechanistic models of fungal growth, such as logistic [32] and Gompertz [33] population growth models, have been directly fit to OD data to estimate fungal growth rates [23]. These simple mechanistic models could be extended to include morphology-specific growth mechanisms to estimate morphology-specific growth rates, as has been done with mechanistic models fit to directly measured data [26]. However, all these approaches to estimate growth rates from OD, both empirically or using modelling, harbour intrinsic inaccuracies because OD is an indirect measure of growth whereby changes in OD measurements may not even be reflective of true fungal growth and changes in fungal morphology are generally overlooked [11]. To collect data that has a higher fidelity to true fungal growth, OD data can be converted to a direct measure of growth using OD values at known fungal concentrations [34,35] (calibration). However, it is difficult for clinical microbiology laboratories to routinely collect time-dense direct measures of fungal growth required for robust calibration because automated collection of such data is not yet standard. In this paper, we aim to model filamentous fungal growth and estimate morphology-specific fungal growth rates from OD measured at a wavelength of 600nm (OD 600 ) without collecting calibration data but still taking into consideration that OD is an indirect measure of growth. We proposed a model of fungal growth that treats measured OD 600 as an observed variable distributed around a linear transform of the true unobserved fungal concentration, thereby explicitly including calibration in the model structure. We demonstrate the model’s ability to fit to and predict OD 600 using time-course OD 600 data for the filamentous fungus, A. fumigatus, and evaluated the model’s estimated morphology-specific growth rate using directly measured data of A. fumigatus growth.

Discussion The need for surveillance of antifungal drug susceptibility is a pressing problem in medical mycology. However, investigating time- and morphology-specific antifungal modes of action using mechanistic modelling or estimating growth rates is currently hampered in clinical microbiology laboratories by arduous methodologies reliant upon sophisticated technologies. A potential solution is to use indirect measures of fungal growth, such as OD, for model fitting and fungal growth rate estimation. We found a logistic model could not be directly fit to OD 600 data for low density fungal cultures and the estimated growth rate was distinct from reference rates estimated from directly measured data (Fig 2). To overcome these issues, we proposed a mathematical model that described the dynamic changes in fungal growth and incorporated calibration in the model. The model captured observed dynamics in OD 600 growth data (Fig 3) and outperformed population growth models previously used in OD modelling in terms of RMSE and relative LPD when predicting held-out test data during CV (Fig 4). We then used the model to infer a hyphal growth rate from indirect OD 600 data and a hyphal extension rate and growth rate from direct (HL, NC) measures (Fig 5). The fungal growth rate inferred from the OD 600 data using our model was closer to those inferred from the direct measures of fungal growth compared to a reference logistic model that did not include calibration. Our model was able to infer growth rates with similar medians from all the collected data sets (OD 600 , HL, NC), indicating our estimates were robust to the data collection method. Two similar attempts for modelling calibration have been previously made. A statistical model [43] was proposed to model a calibration process required for estimating unknown concentrations of any samples. Its success was demonstrated using cockroach allergens [43]. However, this model has not been used in conjunction with a population growth model to infer microbial growth rates to the best of our knowledge. Another study [38] proposed predicting microbial growth curves from OD by fitting population growth models to data that details the first time the OD reader can detect growth in wells initialised with different initial inocula. However, the method requires that the well’s maximum carrying capacity is known along with the microbial number at the first time the OD reader can detect growth or a pre-chosen OD value [38]. For filamentous fungi, we do not know the maximum carrying capacity of a well or the fungal concentration at the first OD detection time or any pre-chosen OD without the collection of calibration data. In the wider microbiology community, calculating growth rates to characterise and compare growth dynamics of microbes, such as bacteria or yeast, in different conditions is common [38,40,41,44–52]. Growth rates have been estimated directly from OD using empirical methods with application to antimicrobial susceptibility testing [50,51] or using population growth models fit to OD data [48]. In addition, various methods to improve bacterial growth rate estimation from OD have been proposed, such as the use of GP models, where growth rates are calculated empirically post-inference [40,41,52]. However, none of these models [40,41,48,52] can be reused to estimate growth rates of filamentous fungi from OD because all the models either require OD data that has been pre-processed using collected calibration data or the models are directly fit to OD, with one study even highlighting OD calibration as a problem when estimating growth rates [44]. Calibration data that are available for microbes such as bacteria cannot be reused for filamentous fungi since OD values are sensitive to the size and shape of the microbe measured [34,35]. Moreover, we found that directly fitting models of true fungal growth to OD 600 data resulted in model misfit and distinctly lower growth rates compared to growth rates estimated from direct measures for filamentous fungi. For bacteria, non-parametric models have been preferentially used to estimate growth rates from OD because it has previously been recommended to calculate growth rates empirically instead of using population growth models [53]. In this study, we used non-parametric GP models (GP-OD and GP-OD-calibration) as reference models. The GP-OD-calibration model could successfully estimate an empirical growth rate that had CIs that included the CIs of reference rates estimated from direct measures of fungal growth, without the need for calibration data. However, we argue that our parametric Logistic-OD-calibration model is preferred for fungal growth modelling and growth rate estimation for the following three reasons. Firstly, we demonstrated that our model outperformed the non-parametric GP-based models both at fitting to and predicting fungal OD 600 curves during CV (Fig 4). Secondly, our model benefits from explicit inclusion of a growth rate, as opposed to GP-based models that provide an empirical post-hoc estimation of the growth rate, because it increases biological interpretability. Lack of an explicit growth rate parameter in a model makes it more difficult to detect potential bias in the estimated growth rate arising from unknown external factors not accounted for in the model. Thirdly and finally, parametric models more readily allow for known fungal growth mechanisms to be included in the model than non-parametric models, for example including different initial fungal inocula of an experiment in the model. Previously, a non-mechanistic piecewise-linear model was fit to OD 600 data to detect the number of change-points in the growth curves [54] that does not consider OD being an indirect measure of growth. We did not include it as a reference model in this study because it was not targeted at estimating growth rates or mechanistically modelling fungal growth. Fungal growth rates would need to be calculated empirically and subsequently compared between different experimental conditions. Different conditions may result in different numbers of estimated change-points, and difficult-to-collect calibration data would still be needed to estimate these growth rates. In this paper, we assumed that the true latent growth follows a deterministic logistic growth model and that measured OD 600 is a linear transform of true fungal growth (a linear calibration curve). Future research could investigate whether incorporating the natural stochasticity of fungal growth in our latent model would improve model fit. Our chosen logistic function for true fungal growth could also be expanded to investigate more complex morphology-specific growth mechanisms or to include antifungal activity. Methods have already been proposed to accelerate Bayesian inference of large-scale models whose linear transform is the only observed output [55], which would facilitate estimation of fungal growth rates should the expansion of our model dramatically increase computation time. Another interesting avenue for future research would be to explore more complex relationships between measured OD and fungal growth, since a linear calibration curve is not necessarily reflective of the true relationship between measured OD and fungal growth for large OD [34]. However, we anticipate that modelling a nonlinear calibration curve whilst retaining a parametric model would make key parameters of interest (e.g. fungal growth rates) difficult to infer because large differences in the parameter values may no longer correspond to measurable differences in observed OD. In summary, this study proposed a mathematical model to infer a biologically interpretable and morphology-specific fungal growth rate from indirect OD 600 data without the need for collecting calibration data and, hence, is accessible to all microbiology laboratories that collect OD 600 data, rather than just specialist laboratories. The results shown here serve as the much-needed groundwork for investigating time- and morphology-specific changes to filamentous fungal growth using mathematical models with OD 600 data. Future mechanistic models can use our presented model as a foundation to investigate mechanisms of antifungal action using only OD 600 data by inferring and subsequently comparing growth rates of fungi in varying antifungal concentrations.

Methods In vitro OD 600 data of fungal growth We obtained OD 600 data from an experiment with wells containing different initial inocula of A1160P+ [56] A. fumigatus (0, 2×102, 2×103, 2×104 and 2×105 spores) suspended in 200μl of fRPMI (0, 1, 101, 102 and 103 [N/μl]) in three dimensional (3D) wells. The OD 600 was recorded every hour for 24–25 hours for each well, for nine (three technical times three biological) replicates at wavelength 600nm by a BioTek OD reader. The wells with 0 spores were used to determine the background OD 600 level. In vitro hyphal length data of fungal growth We obtained in vitro HL data of fungal growth for A1160P+ [56] A. fumigatus. A 50μl sample from a spore suspension of 2×104 spores/ml was added to 950μl of fRPMI in 3D wells to give an initial inoculum of 1 [N/μl] and images of the wells were captured every hour from 4 to 25 hours. The fRPMI growth media was chosen to match the growth media used in the OD 600 experiment. A hypha was chosen at random in the images for each replicate (n = 3) and the hyphal length was measured (μm) using ImageJ [36] until the single hypha could not be tracked anymore (18 hours) due to overcrowding or the hypha growing in a plane that is not in the field of vision of the image (e.g., upwards towards the camera). In vitro nuclear count data of fungal growth A 100μl sample from a 105 spores/ml suspension of fluorescent A. fumigatus spores (CEA10 strain (FGSC#1163) with the histone H1 tagged with GFP (PGpdA-H1-sGFP) [57]) was grown in 3D wells containing 900μl of fRPMI without phenol red (equivalent to an initial inoculum per well of 101 [N/μl]) and images were taken every hour for 16 hours. A1160P+ and the histone tagged CE10 strain of A. fumigatus all derive from the CE10 strain [56,57]. fRPMI without phenol red was used in this experiment to avoid autofluorescence of the fRPMI interfering with counting fluorescent nuclei during data collection and we do not expect the exclusion of phenol red in the fRPMI media to alter A. fumigatus growth dynamics. Hyphae (n = 7) were followed and their number of nuclei per hypha were counted in a point-and-click fashion using the particle counting tool provided by ImageJ [36] until hyphae could no longer be followed (13 hours). Logistic-OD-calibration model We describe the dynamics of true (latent) fungal concentration, f(t) [N/μl], at time t by a logistic function that begins growth at t = τ hours, where β is the hyphal growth rate, K is the carrying capacity and τ represents the time delay for the fungal growth to begin after inoculation. The initial condition, f(0), is the corresponding initial fungal inoculum used in the OD 600 experiment. We model the measured OD 600 , y t , at time t, as distributed around a linear transform of the latent fungal growth with multiplicative noise that is proportional to the OD 600 value (as we expect the noise to increase for high OD 600 values without exceeding measured OD 600 for small OD 600 values), where δ (>1) is a proportionality constant that represents the proportion of light absorbance, σ is the scale of the measurement noise and B is the mean basal OD 600 reading corresponding to the mean OD 600 of the background fRPMI media (background correction). Priors were chosen for each of the parameters (B, δ, σ, β, K, τ) (details in S1 Text) and a prior sensitivity analysis was conducted for the growth rate β (Fig H in S1 Text), where Bayesian inference for the Logistic-HL model with the less informative prior was conducted with adapt_delta set to 0.99 to ensure there were no signs of non-convergence. The model is reduced to a logistic function with multiplicative noise, when we fit the model to direct measurements of fungal growth (HL data, y H,t , and the NC data, y N,t ), where a different noise scale (σ . ) is inferred for the HL and the NC data, σ H and σ N , respectively. Models considered We considered the following fifteen reference models in addition to our Logistic-OD-calibration model (Table 1). PPT PowerPoint slide

PNG larger image

TIFF original image Download: Table 1. Our model and the reference models considered in this study. https://doi.org/10.1371/journal.pcbi.1012105.t001 The models have three defining aspects: the fungal growth model, f(t), (logistic, Gompertz, GP, exponential or logistic with no delay), the data they are fit to (OD 600 , HL or NC) and whether calibration is included in the model (modelled, not modelled or not needed). We considered the following four underlying fungal growth models in conjunction with three ways to model the measured OD 600 , depending on whether calibration is modelled, not modelled, or not needed. 1. Fungal growth models: Logistic for t>τ, with parameters K, β and f(0) being the carrying capacity, the hyphal growth rate and a parameter for the initial OD 600 value or the known initial fungal inoculum when calibration is included in the model. For the logistic with no delay, τ = 0. Gompertz for t>τ, with K, c, and f(0) being the carrying capacity, the initial growth rate and a parameter for the initial OD 600 , respectively. When calibration is included in the model the known initial fungal inoculums of the OD 600 experiment are used instead of a parameter for the initial OD 600 value. GP log-transformed fungal growth, g(t) = log f(t), is modelled using a GP [58] equipped with a zero mean GP prior and an exponentiated quadratic kernel, K(t, t), which is an n by n matrix whose (i, j)-th element is [58], fit to the log-transformed OD 600 data. We assumed Gaussian noise on the log OD 600 centred around the GP, g(t), that had a scale of σ, . As described in Swain et al. [40], the growth rate was calculated by taking the maximum value of the time derivative of the fitted GP, which is analogous to the maximum time derivative of the logarithm of the growth curve. We did not apply a media correction to the data before fitting to ensure strictly positive data for calculating logarithms. This does not alter the inferred growth rate as derivatives are invariant to translations. Exponential for t>τ, where β is the growth rate and f(0) specifies the parameter for the initial fungal inoculum, which is known when calibration is included in the model. 2. Modelling calibration: When calibration is modelled, OD 600 is assumed to be distributed around a linear transform of the (latent) true fungal growth model with multiplicative noise that has a scale of , where B represents the OD 600 of the background fRPMI media and is the proportionality constant. To avoid negative values for the true latent fungal growth when using the GP fungal growth model (GP-OD-calibration model), we model g(t) (= log f(t)) as a GP for the log-transformed latent fungal growth, . The calculated growth rate is again the maximum value of the time derivative of the fitted GP, which is analogous to taking the maximum value of the time derivative of the logarithm of the latent growth curve. Hence, the growth rate estimated from the GP-OD-calibration model is comparable to growth rate estimated using GP-OD where calibration is not modelled. In the mixed logistic-OD-calibration model, we assume random effects on the background media parameter B. We model a background media OD 600 value, Bj, for the j-th replicate , and assume that the Bj are distributed around the sample mean, , of the observed blanks (wells that had an inoculum size of 0 [N/μl]), . The sample mean is calculated by , where denotes the measured OD 600 values of the blanks and N b is the total number of data points of the blanks. If the calibration is not modelled, either OD 600 is assumed to be distributed around the fungal growth model with multiplicative noise with a scale of σ, or the fungal growth model will specify its own model for the OD 600 data (as in the GP fungal growth model). Including calibration in the model is not needed if a model is fit to the directly measured (HL and NC) data. Model inference For all models considered, we sampled posterior distributions using the Hamiltonian Monte Carlo (HMC)-based No-U-Turn sampler (NUTs) [59] provided by RStan [60]. The models were run for 2000 iterations using 50% for warm-up for four chains. The Logistic-OD model was run with the control parameter adapt_delta set to 0.99 during model fitting to ensure there were no signs of non-convergence. The GP-OD-calibration model was run for 4000 iterations with adapt_delta set to 0.99 during model fitting to ensure there were no signs of non-convergence. The Logistic-OD-calibration model was also run for 4000 iterations with adapt_delta at 0.99 for sampling only during prior predictive checks only to ensure no signs of non-convergence. The convergence was monitored using the Gelman-Rubin metric [61], where was used as a heuristic to diagnose a lack of convergence. All models were developed and checked using a full Bayesian Workflow through prior, fake data and posterior predictive checks [62]. Prior predictive and fake data checks were conducted to confirm that priors reflected expert knowledge about fungal OD 600 values (Fig I in S1 Text) and that all the parameters could be estimated using the model with these priors from fake data (Fig J in S1 Text). Posterior predictive checks were conducted to assess the models’ potential ability to explain the observed data. We fit each of the models to the OD 600 fungal growth data ( where I is the total number of replicates and T is the final time the OD 600 is measured (= 25 hours)) and sampled replicates, , from its posterior predictive distributions, . We then visually checked if demonstrate and a sufficiently close resemblance to . Predictive performance We assessed the performance of the models on predicting entire held-out time-series of replicates using 5-fold CV. The training-testing data split was stratified by the replicates. Model fit or predictive performance was assessed using the following two metrics: Relative log (pointwise) predictive density (LPD): m,k is the mean LPD per replicate averaged over the testing replicates, k test , in the k-th fold. The offset (+1) is added to enable taking logarithms (Fig 4). The mean LPD is the LPD averaged over all the points j (j = 1,…,T i ) in the i-th replicate in 63], where k train and k test are the training and testing replicate indices, respectively, within the k-th fold. The mean LPD is calculated by 63], where s is the index of a sample from the posterior, test . The root mean squared error (RMSE): RMSE m,k is calculated between the model’s predictions, The RMSE of an intercept model (mean of all OD 600 values) was included during model comparison as a baseline (Fig 4).

Acknowledgments We thank the anonymous reviewers for their insightful comments during the peer review process.

[END]
---
[1] Url: https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1012105

Published and (C) by PLOS One
Content appears here under this condition or license: Creative Commons - Attribution BY 4.0.

via Magical.Fish Gopher News Feeds:
gopher://magical.fish/1/feeds/news/plosone/

You are viewing proxied material from magical.fish. The copyright of proxied material belongs to its original authors. Any comments or complaints in relation to proxied material should be directed to the original authors of the content concerned. Please see the disclaimer for more details.