# New Insights into Modelling Bacterial Growth with Reference to the Fish Pathogen Flavobacterium psychrophilum

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

_{0})] against time and attempting to ascribe increased biological meaning to some of the parameters. However, due to these modifications the resulting growth functions are not simple transformations of the original mechanistically-derived Gompertz and logistic equations and are better referred to as modified Gompertz and logistic equations [22]. As a result, mechanistic interpretation is no longer straightforward, and these modified models tend to give empirical descriptions of the sigmoidal pattern of microbial growth. Using a more detailed mechanistic approach, Baranyi and Roberts developed a six-parameter model [23], again plotting log-transformed data [ln(x)] against time. However, fitting this particular model can be challenging when using standard nonlinear regression programs [24,25]. Following re-parameterization and incorporation of a number of additional assumptions, a simplified four-parameter model was proposed which is the most widely applied of the Baranyi equations [2,7,15,24].

## 2. Materials and Methods

#### 2.1. Datasets

#### 2.2. Mathematical Considerations

#### 2.2.1. Potential Growth

#### 2.2.2. Actual Growth

_{0}, with Equation (14) yielding:

#### Rectangular Hyperbola

#### Simple Exponential

_{m}and T are as described above. In the Baranyi model, parameter µ

_{max}is defined as the theoretical maximum relative rate, with µ

_{max}also representing maximum relative rate in the modified logistic.

#### 2.3. Model Fitting

_{0})]. The BAR and MLOG models were derived and modified, respectively, to describe logarithmic transformed microbial growth data and therefore these models were fitted to the logarithmic transformed data. In contrast, the LOG, log × hyp and log × exp models were derived to describe absolute growth data and thus fitted to the untransformed data. All models were fitted by means of nonlinear regression using the Newton algorithm in the NLIN procedure of SAS [38]. A range of starting values for each parameter was determined through visual inspection of the growth curve. For parameter values that were difficult to estimate using visual inspection, x

_{m}and x

_{0}were fixed in the first attempt at model fitting, providing an estimated starting value for the remaining parameters for future iterations whereby all parameters of the model were estimated. Using the range of starting values, PROC NLIN forms a grid and evaluates the model at each point on the grid. The values on the grid that yield the smallest objective function are used as initial parameter estimates for the first iteration of the fitting processes [38]. The range of starting values was the same between models that contained common parameters.

#### 2.4. Statistical Analysis

_{u}) and lower (d

_{l}) critical values were calculated according to Draper and Smith [44]. When d is less than the lower critical value d

_{l}, evidence of positive autocorrelation occurs, when the d value is greater than the upper critical value d

_{u}, evidence of negative autocorrelation occurs. The test is inconclusive when d falls between the upper and lower critical values. Positive serial autocorrelation in the residuals suggests the model has the tendency to systemically over- or under-estimate projected values [44]. The quantitative bias factor (BF) was calculated according to Ross [42]. Perfect agreement between predictions and observations will result in a BF of one. Values of BF greater than one occur when a model’s predictions are on average greater than observed values, while a value of less than unity occurs when a models’ predictions are on average less than observed values. When calculating the AF and BF statistics, both observed and predicted values underwent a ’x + 1’ transformation. This transformation allowed for the inclusion of data points whereby x = 0 when calculating these statistics. Likewise, data points that approached zero, and due to the logarithmic nature of the calculation result in nonsensical AF and BF values, were able to be included in the calculations following ‘x + 1’ transformation.

#### 2.5. Model Validation

## 3. Results

_{0})] of the datasets as a means of affirmation. All five models examined in this study were evaluated based upon fitting behavior, examination of residuals, measures of goodness-of-fit and cross-validation.

#### 3.1. Paramater Estimates

_{max}and T for BAR and MLOG) are presented in Table 2. Unlike MLOG and BAR, lag times for log × hyp and log × exp are not explicitly represented as a parameter in the equation and must be calculated using Equation (15). Likewise, maximum growth rate for log × hyp and log × exp is not explicitly represented as an equation parameter and therefore was calculated using Equation (12), while scaled maximum growth rate was calculated by dividing maximum growth rate by microbial biomass at the point of inflexion (Equation (13)). Estimates of s, s/x* and T were influenced by dataset. In Study 1, using OD data, the LOG, log × hyp and log × exp estimates of s, s/x* and T were in close agreement with one another. These three models determined the longest lag time and slowest maximum growth rate to occur when F. psychrophilum was cultured on the Cy7 medium (Table 2—Dataset 4). All three models were in agreement that fastest maximum growth rate and highest scaled maximum growth rate occurred when grown on the modified Cytophaga medium (Table 2—Dataset 3), with the shortest lag time and lowest scaled maximum growth occurring with the Shieh medium (Table 2—Dataset 2). When applied to logarithmic transformed OD data in Study 1, close agreement was found between the µ

_{max}and T of the BAR and MLOG. In agreement with the log × hyp, log × exp and LOG, the models BAR and MLOG determined the longest lag time when F. psychrophilum was cultured on Cy7 (Table 2—Dataset 4). Both BAR and MLOG were in agreement that the shortest lag time occurred with the TYES medium (Table 2—Dataset 1). Highest relative growth rate with BAR and MLOG occurred on Shieh (Table 2—Dataset 2), with lowest relative growth rate occurring on Cy7 (Table 2—Dataset 4). It is important to note that µ

_{max}of the Baranyi represents a maximum relative rate (units of per unit time, h

^{−1}), whereas the s of log × hyp and log × exp is a maximum rate (units of microbial biomass or its surrogate per unit time) and s/x* is a scaled maximum rate (units of per time, h

^{−1}).

#### 3.2. Growth Prediction

#### 3.3. Model Evaluation

#### 3.4. Model Validation

## 4. Discussion

^{−1}), LOG, log × hyp and log × exp were in close agreement with values of 0.17, 0.18 and 0.17 when grown on the TYES liquid medium. In comparison, when MLOG and BAR were fitted to the logarithmic transformation of this same dataset, lag times were determined to be 4.6 and 3.7 h, respectively. Maximum relative growth rates (h

^{−1}) of MLOG and BAR were determined to be 0.314 and 0.292, respectively. Due to the logarithmic transformation effecting the overall shape of a growth curve, it is clear that lag times are not comparable between models applied to absolute OD data and logarithmic transformed data, viz. LOG, log × hyp and log × exp versus MLOG and BAR, respectively. However, BAR and MLOG estimate lag times that are very much shorter than lag times when using the absolute untransformed OD data. Lag time estimates of MLOG and BAR are much shorter than the time at which OD starts to rise over the baseline OD

_{0}using the actual untransformed data; therefore these two models appear to underestimate the real lag time. This can be seen through comparison of lag time values in Table 2 and visualization of Figure 1, in addition to the corresponding figures (Figures S1–S6 in the Supplementary Materials). The extent of underestimation of lag time for BAR and MLOG is more apparent in Study 1 compared to Study 2, whereby Study 1 is characterized by lower x

_{0}values and a more prolonged lag phase.

^{−1}), while LOG, log × hyp and log × exp represent maximum growth in an absolute sense (units of microbial biomass per h) in addition to being able to represent maximum growth on a relative basis through scaled maximum growth rate (h

^{−1}). Very little information exists concerning parameters estimates resulting from models fitted to bacterial growth data in the field of aquaculture, with even less being available concerning F. psychrophilum. Stenholm et al. [50] reported maximum growth rates between 0.062 and 0.098 h

^{−1}for 26 strains of F. psychrophilum. It is unclear however if these values were generated using a model, with the authors describing growth rates as being determined from the exponential increase in OD at 525 nm. Growth rates of two pathogenic bacteria Aeromonas hydrophila and Vibrio alginolyticus, obtained by fitting modified forms of the logistic and Gompertz, have been published [5]. However, the authors omitted the units when describing parameter estimates. Clearly, many factors must be standardized before meaningful comparisons of model-derived growth parameter estimates can be made. These factors include but are not limited to: method by which bacterial growth is measured, transformation applied to the dependent variable, and actual definition of growth rate [7].

^{6}–10

^{7}CFU/mL detection thresholds of OD measurement devices [4]. Therefore, this technique is best suited in circumstances whereby high cell densities are reached. In the field of predictive food microbiology spoilage resulting from bacterial load is a concern at high levels, making the absorbance technique practical in this field of study [4]. Likewise, high levels of the pathogenic bacteria used in this current study, F. psychrophilum, are commonly incubated to elicit typical clinical signs of rainbow trout fry syndrome. In a challenge test, Aoki et al. [51] inoculated ~10

^{6}CFU/mL of F. psychrophilum on a modified Cytophaga liquid medium, the same media as in Dataset 3 of this study. Bacteria reached the exponential phase of growth, at CFU/mL of ~10

^{7}, and subsequently experimental fish were challenged with this load resulting in clinical signs of disease. Therefore, it is it feasible that the relatively high level of F. psychrophilum used in studies permits the use of the absorbance technique to determine lag and rate parameter estimates when studying this bacterium.

_{m}for the logistic [14]. The remaining equations display flexible points of inflexion that result in a more robust and versatile growth curve. The point of inflexion of these growth functions can all be calculated by equating the second derivative of the function to zero and solving for x* (the microbial biomass at the point of inflexion). However, log × hyp is the only equation out of log × hyp, log × exp and BAR whereby t* (time at which the point of inflexion occurs) and x* can be solved for analytically, with log × exp requiring numerical methods to solve for t* and the second derivative of the BAR unable to be solved in such a way that it is explicit in t, the independent variable [7]. Therefore, without the ability to solve for t* for the Baranyi, it is not possible to calculate maximum rate of actual growth using $\mathrm{d}x/\mathrm{d}t={\mu}_{\mathrm{max}}\alpha \left({t}^{*}\right)u\left({x}^{*}\right){x}^{*}.$ The ability of a growth function to be easily manipulated mathematically has important ramifications for their application and is an example of the advantage of log × hyp over the other functions examined.

## 5. Conclusions

## Supplementary Materials

**Table S1**: Observed growth data of Flavobacterium psychrophilum on eight liquid mediums, (1) TYES, (2) Shieh, (3) modified Cytophaga, (4) Cy7, (5) FLPB, (6) TYESB, (7) CBCM and (8) MAOB;

**Table S2**: Resulting parameter estimates of log × hyp fit to Flavobacterium psychrophilum grown on eight liquid mediums: (1) TYES, (2) Shieh, (3) modified Cytophaga, (4) Cy7, (5) FLPB, (6) TYESB, (7) CBCM and (8) MAOB;

**Table S3**: Resulting parameter estimates of log × exp fit to Flavobacterium psychrophilum grown on eight liquid mediums: (1) TYES, (2) Shieh, (3) modified Cytophaga, (4) Cy7, (5) FLPB, (6) TYESB, (7) CBCM and (8) MAOB;

**Figure S1**: Optical density growth data and model predictions for Flavobacterium psychrophilum. Growth predictions using (a) LOG, log × hyp, log ×exp (b) MLOG and BAR grown on a Sheih liquid medium;

**Figure S2:**Optical density growth data and model predictions for Flavobacterium psychrophilum. Growth predictions using (a) LOG, log × hyp, log × exp (b) MLOG and BAR grown on a modified Cytophaga liquid medium;

**Figure S3**: Optical density growth data and model predictions for Flavobacterium psychrophilum. Growth predictions using (a) LOG, log × hyp, log × exp (b) MLOG and BAR grown on a Cy7 liquid medium;

**Figure S4**: Optical density growth data and model predictions for Flavobacterium psychrophilum. Growth predictions using (a) LOG, log × hyp, log × exp (b) MLOG and BAR grown on a TYESB liquid medium;

**Figure S5**: Optical density growth data and model predictions for Flavobacterium psychrophilum. Growth predictions using (a) LOG, log × hyp, log × exp (b) MLOG and BAR grown on a CBCM liquid medium;

**Figure S6**: Optical density growth data and model predictions for Flavobacterium psychrophilum. Growth predictions using (a) LOG, log × hyp, log × exp (b) MLOG and BAR grown on a MAOB liquid medium.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Optical density growth data and model predictions for Flavobacterium psychrophilum. Growth predictions using (

**a**) LOG, log × hyp, log × exp and (

**b**) MLOG and BAR, grown on a TYES liquid medium, and (

**c**) LOG, log × hyp, log × exp and (

**d**) MLOG and BAR, grown on a FLBP liquid medium. In panels (

**a**) and (

**c**), models are fitted to untransformed optical density measurements (OD) while (

**b**) and (

**d**) models are fitted to [ln(OD/OD

_{0})].

**Figure 2.**The effect of dampening potential growth rate, resulting actual growth, and actual rate of growth (AGR) of Flavobacterium psychrophilum. Dampening effect is represented by: a rectangular hyperbola (

**a**), a simple exponential (

**b**) when grown on a TYES liquid medium and a rectangular hyperbola (

**c**), a simple exponential (

**d**) when grown on FLBP liquid medium.

**Figure 3.**Comparison of actual growth rate (AGR), predicted growth and maximum actual growth rate (s) from liquid media resulting in highest and lowest maximum growth rates (s) from Study 1 and Study 2 when fitting log × hyp and log × exp. Panels (

**a**) and (

**b**) are taken from Study 1, and panels (

**c**) and (

**d**) from Study 2.

Functional Form | |
---|---|

Simple logistic (LOG) | $x=\frac{{x}_{0}{x}_{m}}{{x}_{0}+\left({x}_{m}-{x}_{0}\right){\mathrm{e}}^{-\mu t}}$ |

Modified logistic (MLOG) | $x=\frac{{x}_{m}}{\left\{1+\mathrm{exp}\left[\frac{4{\mu}_{\mathrm{max}}}{{x}_{m}}\left(T-t\right)+2\right]\right\}}$ |

Baranyi four parameter (BAR) | $x={x}_{0}+{\mu}_{\mathrm{max}}A\left(t\right)-\mathrm{ln}\left[1+\frac{{\mathrm{e}}^{{\mu}_{\mathrm{max}}A\left(t\right)}-1}{{\mathrm{e}}^{\left({x}_{m}-{x}_{0}\right)}}\right]$ $A\left(t\right)=t+\frac{1}{{\mu}_{\mathrm{max}}}\mathrm{ln}\left[{\mathrm{e}}^{-{\mu}_{\mathrm{max}}t}+{\mathrm{e}}^{-{h}_{0}}-{\mathrm{e}}^{-{\mu}_{\mathrm{max}}t-{h}_{0}}\right]$ where ${h}_{0}=$µ _{max}T = ln[1 +$\frac{1}{{q}_{0}}]$ |

log × hyp | $x=\frac{{x}_{0}{x}_{m}}{{x}_{0}+\left({x}_{m}-{x}_{0}\right){\left(\frac{\lambda}{\lambda +t}\right)}^{\lambda \mu}}$ |

log × exp | $x=\frac{{x}_{0}{x}_{m}}{{x}_{0}+\left({x}_{m}-{x}_{0}\right)exp\left[-\frac{\mu}{\lambda}\left(1-{\mathrm{e}}^{-\lambda t}\right)\right]}$ |

**Table 2.**Comparison of maximum growth rate (s, units of microbial mass per h), scaled maximum growth rate (s/x*, h

^{−1}), lag time (T, h) and maximum relative growth rate (µ

_{max}, h

^{−1}), generated by the standard logistic (LOG), logistic × hyperbola (log × hyp) and logistic × exponential (log × exp) fitted to optical density (OD) growth data and the modified logistic (MLOG) and four-parameter Baranyi (BAR) fitted to logarithmic transformed growth data [ln(OD/OD

_{0})].

LOG | log × hyp | log × exp | MLOG | BAR | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Dataset | s | s/x* | T | s | s/x* | T | s | s/x* | T | µ_{max} | T | µ_{max} | T |

1 | 0.122 | 0.168 | 18.8 | 0.122 | 0.182 | 18.7 | 0.122 | 0.170 | 18.3 | 0.314 | 4.6 | 0.292 | 3.7 |

2 | 0.096 | 0.138 | 16.8 | 0.095 | 0.151 | 16.6 | 0.097 | 0.163 | 17.1 | 0.362 | 5.6 | 0.348 | 5.3 |

3 | 0.122 | 0.191 | 19.8 | 0.124 | 0.207 | 19.8 | 0.123 | 0.197 | 19.5 | 0.302 | 5.5 | 0.291 | 5.1 |

4 | 0.052 | 0.165 | 24.5 | 0.053 | 0.181 | 24.7 | 0.052 | 0.168 | 24.2 | 0.219 | 7.8 | 0.221 | 7.7 |

5 | 0.111 | 0.103 | 3.1 | 0.124 | 0.174 | 1.7 | 0.137 | 0.161 | 1.6 | 0.285 | 1.7 | 0.334 | 1.8 |

6 | 0.133 | 0.161 | 3.7 | 0.145 | 0.225 | 3.5 | 0.148 | 0.237 | 3.6 | 0.309 | 2.9 | 0.506 | 4.4 |

7 | 0.120 | 0.206 | 4.1 | 0.119 | 0.217 | 4.0 | 0.107 | 0.279 | 4.8 | 0.847 | 7.6 | 1.656 | 8.3 |

8 | 0.031 | 0.130 | 1.4 | 0.031 | 0.146 | 1.2 | 0.030 | 0.130 | 0.7 | 0.457 | 7.7 | 1.452 | 8.7 |

**Table 3.**Goodness-of- fit: Akaike information criterion (AIC), mean square prediction error (MSPE) concordance correlation coefficient (CCC) and accuracy factor (AF) values obtained when fitting models to microbial growth data from Study 1 and 2.

Measure of Goodness-of-Fit | LOG | log × hyp | log × exp | MLOG | BAR |
---|---|---|---|---|---|

AIC | |||||

Study 1-Average (±SE) | −117.1 (11.5) | −110.0 (11.5) | −111.6 (11.3) | −58.8 (3.1) | −67.5 (5.5) |

SStudy 2-Average (±SE) | −44.6 (3.9) | −43.9 (2.7) | −45.0 (2.9) | −44.4 (3.6) | −44.0 (2.9) |

SOverall-Average (±SE) | −80.8 (15.1) | −77.0 (14.0) | −78.3 (14.0) | −51.6 (3.7) | −55.7 (5.6) |

MSPE | |||||

SStudy 1-Average (±SE) | 0.001 (0.0002) | 0.001 (0.0003) | 0.001 (0.0002) | 0.024 (0.003) | 0.014 (0.004) |

SStudy 2-Average (±SE) | 0.005 (0.002) | 0.004 (0.001) | 0.003 (0.001) | 0.005 (0.002) | 0.004 (0.001) |

SOverall-Average (±SE) | 0.003 (0.001) | 0.002 (0.001) | 0.002 (0.001) | 0.013 (0.004) | 0.009 (0.003) |

CCC | |||||

SStudy 1-Average (±SE) | 0.998 (0.001) | 0.998 (0.001) | 0.998 (0.001) | 0.997 (0.001) | 0.998 (0.001) |

SStudy 2-Average (±SE) | 0.987 (0.004) | 0.988 (0.005) | 0.989 (0.005) | 0.997 (0.001) | 0.998 (0.001) |

SOverall-Average (±SE) | 0.992 (0.003) | 0.993 (0.003) | 0.993 (0.003) | 0.997 (0.001) | 0.998 (0.001) |

AF | |||||

SStudy 1-Average (±SE) | 1.013 (0.002) | 1.016 (0.003) | 1.015 (0.002) | 1.039 (0.005) | 1.024 (0.005) |

SStudy 2-Average (±SE) | 1.033 (0.005) | 1.030 (0.003) | 1.027 (0.002) | 1.022 (0.007) | 1.016 (0.003) |

SOverall-Average (±SE) | 1.023 (0.005) | 1.023 (0.003) | 1.021 (0.002) | 1.031 (0.005) | 1.020 (0.003) |

**Table 4.**Examination of residuals using the runs test, Durbin–Watson (DW) and averaged bias factor (BF) statistics obtained when fitting models to growth data from Study 1 and 2.

Test for Examination of Residuals | LOG | log × hyp | log × exp | MLOG | BAR |
---|---|---|---|---|---|

Runs test | |||||

Runs were random | 6 | 5 | 6 | 8 | 8 |

Too few runs | 2 | 3 | 2 | 0 | 0 |

No. curves exhibiting serial correlation determined by DW statistic (α = 0.01) | |||||

No serial correlation | 7 | 7 | 7 | 8 | 8 |

Positive Correlation | 1 | 1 | 1 | 0 | 1 |

BF | |||||

SStudy 1-Average (±SE) | 0.996 (0.002) | 0.993 (0.001) | 0.994 (0.000) | 1.017 (0.001) | 1.003 (0.001) |

SStudy 2-Average (±SE) | 1.001 (0.003) | 0.994 (0.002) | 0.994 (0.003) | 1.006 (0.003) | 1.001 (0.000) |

SOverall-Average (±SE) | 0.999 (0.002) | 0.994 (0.001) | 0.994 (0.001) | 1.011 (0.003) | 1.002 (0.001) |

**Table 5.**Predicted residual error sum of squares (PRESS) and concordance correlation coefficient (CCC) resulting from cross-validation of Datasets 1-2 from Study 1 and Datasets 5-6 from Study 2.

Cross-Validation Test | Model | ||
---|---|---|---|

LOG | log × hyp | log × exp | |

PRESS | |||

Dataset 1 | 0.0717 | 0.0718 | 0.0599 * |

Dataset 2 | 0.0110 * | 0.0125 | 0.0134 |

Dataset 5 | 0.2640 | 0.2135 | 0.1858 * |

Dataset 6 | 0.1184 | 0.1153 | 0.1149 * |

CCC | |||

Dataset 1 | 0.9945 | 0.9945 | 0.9953 * |

Dataset 2 | 0.9990 * | 0.9989 | 0.9988 |

Dataset 5 | 0.9726 | 0.9782 | 0.9814 * |

Dataset 6 | 0.9806 | 0.9807 | 0.9808 * |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Powell, C.D.; López, S.; France, J.
New Insights into Modelling Bacterial Growth with Reference to the Fish Pathogen *Flavobacterium psychrophilum*. *Animals* **2020**, *10*, 435.
https://doi.org/10.3390/ani10030435

**AMA Style**

Powell CD, López S, France J.
New Insights into Modelling Bacterial Growth with Reference to the Fish Pathogen *Flavobacterium psychrophilum*. *Animals*. 2020; 10(3):435.
https://doi.org/10.3390/ani10030435

**Chicago/Turabian Style**

Powell, Christopher D., Secundino López, and James France.
2020. "New Insights into Modelling Bacterial Growth with Reference to the Fish Pathogen *Flavobacterium psychrophilum*" *Animals* 10, no. 3: 435.
https://doi.org/10.3390/ani10030435