# Modelling and Predicting the Growth of Escherichia coli and Staphylococcus aureus in Co-Culture with Geotrichum candidum and Lactic Acid Bacteria in Milk

^{1}

^{2}

^{*}

## Abstract

**:**

^{6}CFU/mL, LAB not only induced an early stationary phase of E. coli (two isolates BR and PS2) and S. aureus (isolates 2064 and 14733) but also affected their death phase. In co-cultures with LAB and G. candidum, the numbers of E. coli and S. aureus increased in 2 logs and 1 log, respectively, reaching maximum population densities (MPDs) of less than 5 and 4 logs, respectively. After that, the populations of both bacteria represented with two isolates decreased in more than 2 logs and 3 logs within 2 days compared to their MPDs, respectively. G. candidum was found to be the subject of interactions with LAB within a given temperature range only partially. To develop a tertiary model for the growth curves of the populations, a one-step approach was used, combining the Huang-Gimenez and Dalgaard primary model with secondary square-root models for growth rate and lag time. Furthermore, the reparametrized Gompertz-inspired function with the Bigelow secondary model was used to describe the death phase of the E. coli and S. aureus isolates. The prediction ability of the growth of the H-GD tertiary model for co-cultures was cross-validated within the isolates and datasets in milk and milk medium with 1% NaCl. The study can be used as knowledge support for the hygiene guidelines of short-ripened raw milk cheeses, as was our case in Slovakia.

## 1. Introduction

## 2. Material and Methods

#### 2.1. Microorganisms and Culture Conditions

^{®}Fresco

^{®}1000NG and isolate J of G. candidum were used during all co-cultivation experiments. Mesophilic starter culture consisting of Lactococcus lactis subsp. lactis, L. lactis subsp. cremoris, and Streptococcus salivarius subsp. thermophilus was kept frozen at −45 °C. G. candidum was refrigerated at 5 °C on plate count skim milk agar slants (SMA; Merck, Darmstadt, Germany) and periodically sub-cultured in diluted SMA agar. There were four different series of E. coli and S. aureus co-cultivations using 2 isolates of each, BR, PS2 for E. coli; and 2064, 14733 for S. aureus. All bacterial cultures were maintained in Brain Hearth Infusion (BHI) broth (Sigma-Aldrich, St. Louis, MO, USA) at 5 ± 0.5 °C before analysis.

#### 2.2. Preparation of Inoculum and Experiments

^{6}CFU/mL, of G. candidum at approximately 10

^{2}CFU/mL, and E. coli and S. aureus isolates at approximately 10

^{3}CFU/mL.

_{w}-values were estimated by the LabMaster-aw (Novasina, Lachen, Switzerland).

#### 2.3. Quantification of Microorganisms

#### 2.4. Mathematical Models

#### 2.4.1. Modelling the Microbial Interaction in Co-Cultures

- A.
- H-GD model with the competition coefficients

_{t}〉

_{t}〉

_{t}, t〉

_{max}

^{i=Lab,P,Gc}are the maximum specific growth rates of the LAB, E. coli or S. aureus and G. candidum (Gc), respectively, t

_{λ}

^{i}are the lag times of microorganisms, α is the lag phase transition coefficient, taking a value of 4 [31]. Concentrations x, which include x

_{i}= log N

_{i}, x

_{0,i}= log N

_{0,i,}x

_{max,i}= log N

_{max,i,}x

_{res,i}= log N

_{res,i}represent the real, initial, maximum and residual (or tail) cell density, N

_{i}, N

_{0,i}A

_{max, i}and N

_{res},

_{i}. I

_{LP}, I

_{PL}are the competition coefficients representing the effects of LAB (Fresco) on E. coli or S. aureus and E. coli or S. aureus on LAB (Fresco), respectively, in the H-GD model type R. k

_{max,P}is the maximum death rate of E. coli or S. aureus, t

_{λs}

^{i=Ec, Sa}is the survival curve shoulder, t

_{t}is the transitioning breakpoint time from stationary to survival phase for E. coli/S. aureus that is determined so that the time t

_{λ}is equal to zero, k

_{Gc}is the reduction coefficient for the G. candidum growth rate.

_{max}

^{i}and t

_{λ}

^{i}are a function of temperature, the following secondary square root models were used to incorporate the effect of temperature on growth parameters [32]:

_{T}(h

^{−1}·°C

^{−1}) is the slope and depends on additional growth conditions and the microorganism involved, T (°C) is the temperature, and T

_{min}is its theoretical minimum for growth.

_{λ}) and T was used in the H-GD models according to [33]:

_{λ}

_{,I}is the regression coefficient.

_{i}parameter was used to help in the theoretical description of the influence of temperature and other factors acting in this phase such as the pH drop and addition of NaCl.

_{max}on temperature as follows:

_{efi}

^{t}is k

_{max}

^{i}at a reference temperature (T

_{ref}) and z represents a temperature required for a 10-fold reduction of E. coli or S. aureus numbers.

#### 2.4.2. Parameter Determination and Evaluation of Model Performance

_{max}, competition coefficients I

_{LP}, I

_{PL}; and reduction coefficient k

_{Gc}in the H-GD model type R and reduction coefficient k

_{Gc}(Equations (1)–(4)) and parameters b

_{T}, T

_{min}, b

_{λ}from the square-root models (Equations (5)–(7)). The parameters N

_{res}, k

_{ref}, and z are derived from the reparametrized Gompertz-inspired survival model (Equation (3)), as well as the Bigelow secondary model (8), respectively.

_{H-GD}is the vector of parameters of H-GD models for the simultaneous competitive growth of the co-culture series.

_{E}): the average maximum density counts of microorganisms (x

_{max,Lab}, x

_{max,P}, x

_{max,G}), competition coefficients (I

_{LP}, I

_{PL}), reduction coefficient k

_{Gc}, regression coefficient, b

_{λ}

_{,Gc}, maximum declination rates at a reference temperature T

_{ref}(k

_{ref,Ec}, k

_{ref,Sa}), and z-values (z

_{Ec}, z

_{Sa}). Regression coefficients b

_{λ}

_{,Ec}for the isolate E. coli PS2 were also evaluated by using one-step kinetic data analysis. The remaining parameters in Equation (8), which were previously optimised by nonlinear regression analysis for single cultures [19] or taken from the following scientific articles [24,36], were fixed as constants. This approach has the advantage that some parameters for co-culture growth prediction in milk could be estimated from the growth of individual species.

^{2}) to evaluate its suitability to fit the whole set of observation points according to Equations (1)–(8).

_{f}) and accuracy (A

_{f}) factors [37] on the datasets of different E. coli (BR and PS2) and S. aureus (2064 and 14733) isolates within the temperature range of 15 to 21 °C for the cases without and with 1% NaCl addition

_{f}) and accuracy (A

_{f}), were calculated using Microsoft Excel (Microsoft, Redmond, WA, USA).

## 3. Results and Discussion

#### 3.1. One-Step Analysis of Competitive Growth

_{0}) of 3.2 ± 0.3 log CFU/mL in all co-culture trials in the shortest time. These results aligned with our previous work [19] and those reported for the co-culture growth of Lactiplantibacillus plantarum with S. aureus by Rodríguez-Sánchez et al. [48].

_{EL}(Equation (2)) in the H-GD model showed a similar growth reduction (<60%) for both isolates of E. coli compared with their original capacity as individual species in milk [19]. On the other hand, the competitive effect of LAB on S. aureus isolates was stronger and strain-dependent. The coefficients I

_{SL}in Table 3 were significantly lower and different for isolates 2064 and 14733. Thus, they confirmed the higher sensitivity to non-specific inhibition caused by LAB that was found for isolate 14733.

^{6}CFU/mL) of a starter culture favoured the growth of LAB in milk (the competition coefficient I

_{LP}is greater than one in almost all cases and was able not only to induce an early stationary state in E. coli (isolates BR and PS2) and S. aureus (isolates 2064 and 14733) for cases without and with 1% NaCl addition but also subsequently reduced their population. LAB growth of the LAB slightly suppressed the growth rate of G. candidum of its original ability as a single species in milk. The reduction coefficients of the growth rate of G. candidum k

_{Gc}were within the region 0.710–0.995. Naturally, their values were lower for the cases with NaCl addition (Table 2 and Table 3).

#### 3.2. Model Validation

#### 3.2.1. E. coli Isolates in Co-Cultures

_{f}) and accuracy (A

_{f}) factors [37] for each microorganism in the E. coli co-cultures. For E. coli, the HG-D model data of isolate BR were validated with the experimental data of isolate PS2. The calculated B

_{f}values for the growth of E. coli isolates were within 0.993–1.387 and the A

_{f}values ranged from 1.283 to 1.704. With high probability, the prediction of E. coli growth was affected by the growth ability and sensitivity of the PS2 isolate to the addition of NaCl.

_{f}values were between 0.996 to 1.003, while the A

_{f}values ranged from 1.026 to 1.035 showing that, on average, the predicted value was 2.6 to 3.5% different (either smaller or greater) from the observed values. Accurate prediction of LAB growth also confirmed the fact that their growth was minimally affected by other co-culture populations such as G. candidum or E. coli as the values of the interaction coefficient I

_{LE}≅ 1.0 indicated before (Table 2).

#### 3.2.2. S. aureus Isolates in Co-Cultures

_{f}and A

_{f}), the fast-growing isolate 2064 model data and the experimental data of isolate 14733 were used. While the calculated B

_{f}values for the growth of S. aureus isolates were 1.184 and 1.284, the A

_{f}values were 1.371 and 1.500 in milk and 1% NaCl in milk, respectively.

_{f}values were within 0.980 to 1.016 and the A

_{f}values ranged from 1.023 to 1.035. Furthermore, the values of the interaction coefficient I

_{LS}varied between 0.91 and 1.26 (Table 3).

_{f}when used for model performance evaluations involving pathogens, three categories were recommended by [55]. B

_{f}in the range of 0.90 to 1.05 can be considered good; 0.70 to 0.90 or 1.06 to 1.15 can be considered acceptable and less than 0.70 or greater than 1.15 should be considered unacceptable. In almost all cases with two isolates of each E. coli and S. aureus contaminant in our study, the values of the B

_{f}factors were in the range of 0.90–1.05, which means that the H-GD model can be considered as suitable also for growth prediction in co-cultures with three different microbial populations.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Co-culture growth of LAB Fresco, E. coli BR, and G. candidum in milk at 15, 18, and 21 °C (without and with 1% NaCl, respectively). The continuous lines represent the growth predicted values by the H-GD model and the dots represent the experimental values (▪ LAB, ▲ Ec, ● Gc).

**Figure 2.**Co-culture growth of LAB Fresco, S. aureus 14733, and G. candidum in milk at 15, 18, and 21 °C (without and with 1% NaCl, respectively). The continuous lines represent the growth predicted values by the H-GD model and the dots represent the experimental values (▪ LAB, ▲ Sa, ● Gc).

Microorganism | Isolate | Origin |
---|---|---|

DVS^{®} Fresco^{®} 1000NG | - | commercial LAB culture; Christian Hansen, Hoersholm, Denmark |

G. candidum | J | Slovakian traditional cheese “Bryndza” |

E. coli | Br | Slovakian traditional cheese “Bryndza” |

PS2 | laboratory-produced pasta-filata cheese from raw cows’ milk | |

S. aureus | 2064 | Slovakian ewes’ lump cheese |

14733 | milk vending machine biofilm |

**Table 2.**Parameters of the H-GD model with 95% highest posterior density interval (Bayesian estimation) for growth of E. coli (isolates BR and PS2) in co-cultures with G. candidum and LAB in milk.

Parameters | E. coli (Isolate BR) | E. coli (Isolate PS2) | ||
---|---|---|---|---|

In Milk | 1% NaCl in Milk | In Milk | 1% NaCl in Milk | |

x_{max,Lab} | 9.34 ± 0.04 | 9.32 ± 0.03 | 9.36 ± 0.04 | 9.33 ± 0.04 |

x_{max,Ec} | 4.17 ± 0.16 | 3.95 ± 0.10 | 5.14 ± 0.17 | 5.14 ± 0.10 |

x_{max,Gc} | 5.96 ± 0.08 | 6.09 ± 0.10 | 5.72 ± 0.08 | 6.04 ± 0.17 |

I_{LE} | 1.158 ± 0.093 | 1.254 ± 0.100 | 0.957 ± 0.059 | 0.951 ± 0.054 |

I_{EL} | 0.526 ± 0.045 | 0.536 ± 0.049 | 0.588 ± 0.042 | 0.513 ± 0.035 |

k_{Gc} | 0.850 ± 0.038 | 0.710 ± 0.025 | 0.931 ± 0.048 | 0.749 ± 0.046 |

k_{ref} | 0.101 ± 0.006 | 0.101 ± 0.006 | 0.133 ± 0.006 | 0.081 ± 0.006 |

x_{res,Ec} | 0.4 ^{a} | 0.42 ± 0.16 | 1.20 ± 0.29 | 0.5 ^{d} |

z_{Ec} | 30.67 ± 5.68 | 32.25 ^{d} | 6.38 ± 0.70 | 28.21 ± 5.76 |

b_{λ}_{,Gc} | 0.0109 ± 0.0003 | 0.0101 ± 0.0003 | 0.0096 ± 0.0002 | 0.0085 ± 0.0002 |

b_{T,Gc} ^{b} | 0.0228 ^{b} | 0.0228 ^{b} | 0.0228 ^{a} | 0.0228 ^{a} |

T_{min,Gc} ^{b} | 0.00 ^{b} | 0.00 ^{b} | 0.00 ^{a} | 0.00 ^{a} |

b_{λ,Lab} ^{c} | 0.0343 ^{c} | 0.0343 ^{c} | 0.0343 ^{b} | 0.0343 ^{b} |

b_{T,Lab} ^{c} | 0.0384 ^{c} | 0.0384 ^{c} | 0.0384 ^{b} | 0.0384 ^{b} |

T_{min,Lab} | 1.11 ^{c} | 1.11 ^{c} | 1.11 ^{b} | 1.11 ^{b} |

b_{λ}_{,Ec} | 0.0493 ^{c} | 0.0493 ^{c} | 0.0365 ± 0.0045 | 0.0366 ± 0.0044 |

b_{T}_{,Ec} | 0.0421 ^{c} | 0.0421 ^{c} | 0.052 ^{c} | 0.052 ^{c} |

T_{min,Ec} | 4.16 ^{c} | 4.16 ^{c} | 4.80 ^{c} | 4.80 ^{c} |

**Table 3.**Parameters of H-GD model with 95% highest posterior density interval (Bayesian estimation) for growth of S. aureus (isolates 2064 and 14733) in co-cultures with G. candidum and LAB in milk.

Parameters | S. aureus (Isolate 2064) | S. aureus (Isolate 14733) | ||
---|---|---|---|---|

In Milk | 1% NaCl in Milk | In Milk | 1% NaCl in Milk | |

x_{max,Lab} | 9.43 ± 0.03 | 9.40 ± 0.05 | 9.34 ± 0.03 | 9.25 ± 0.03 |

x_{max,Sa} | 3.83 ± 0.15 | 4.17 ± 0.11 | 4.43 ± 0.12 | 4.43 ± 0.16 |

x_{max,Gc} | 5.65 ± 0.12 | 5.82 ± 0.17 | 5.85 ± 0.11 | 6.04 ± 0.15 |

I_{LS} | 1.262 ± 0.056 | 1.083 ± 0.057 | 1.064 ± 0.044 | 0.912 ± 0.043 |

I_{SL} | 0.308 ± 0.144 | 0.174 ± 0.089 | 0.705 ± 0.079 | 0.526 ± 0.054 |

c_{LS} | - | - | - | - |

c_{SL} | - | - | - | - |

k_{Gc} | 0.995 ± 0.067 | 0.778 ± 0.058 | 0.906 ± 0.048 | 0.850 ± 0.055 |

k_{ref} | 0.133 ± 0.022 | 0.102 ± 0.007 | 0.107 ± 0.007 | 0.094 ± 0.006 |

x_{res,Sa} | 1.47 ± 0.13 | 0.3 ^{c} | 0.5 ^{c} | 0.5 ^{c} |

z_{Sa} | 9.46 ± 1.21 | 10.44 ± 0.51 | 11.49 ± 1.18 | 13.79 ± 1.67 |

b_{λ}_{,Gc} | 0.0092 ± 0.0002 | 0.0086 ± 0.0003 | 0.0104 ± 0.0003 | 0.0086 ± 0.0002 |

b_{T}_{,Gc} | 0.0228 ^{a} | 0.0228 ^{a} | 0.0228 ^{a} | 0.0228 ^{a} |

T_{min,Gc} | 0.00 ^{a} | 0.00 ^{a} | 0.00 ^{a} | 0.00 ^{a} |

b_{λ}_{,Lab} | 0.0343 ^{b} | 0.0343 ^{b} | 0.0384 ^{b} | 0.0384 ^{b} |

b_{T}_{,Lab} | 0.0384 ^{b} | 0.0384 ^{b} | 1.11 ^{b} | 1.11 ^{b} |

T_{min,Lab} | 1.11 ^{b} | 1.11 ^{b} | 0.0302 ^{b} | 0.0302 ^{b} |

b_{λ}_{,Sa} | 0.0302 ^{b} | 0.0302 ^{b} | 0.0409 ^{b} | 0.0409 ^{b} |

b_{T}_{,Sa} | 0.0409 ^{b} | 0.0409 ^{b} | 5.02 ^{b} | 5.02 ^{b} |

T_{min,Sa} | 5.02 ^{b} | 5.02 ^{b} |

**Table 4.**Goodness-of-fit indices and models comparison of H-GD model for the E. coli and S. aureus isolates in co-culture with G. candidum and LAB Fresco in milk.

Indices | E. coli BR | E. coli PS2 | S. aureus 2064 | S. aureus 14733 | ||||
---|---|---|---|---|---|---|---|---|

in Milk | 1% NaCl in Milk | in Milk | 1% NaCl in Milk | in Milk | 1% NaCl in Milk | in Milk | 1% NaCl in Milk | |

SSE | 14.719 | 16.080 | 19.450 | 25.719 | 10.625 | 11.725 | 15.184 | 17.592 |

R^{2} | 0.992 | 0.991 | 0.987 | 0.986 | 0.991 | 0.991 | 0.989 | 0.988 |

p | 10 | 10 | 11 | 11 | 10 | 10 | 10 | 10 |

RMSE | 0.251 | 0.254 | 0.289 | 0.324 | 0.280 | 0.284 | 0.270 | 0.282 |

_{EC BR}(milk) = 244; n

_{EC BR}(milk + 1% NaCl) = 254; n

_{EC PS2}(milk) = 244; n

_{EC PS2}(milk + 1% NaCl) = 256; n

_{Sa 2064}(milk) = 146; n

_{Sa 2064}(milk + 1% NaCl) = 155; n

_{Sa 14733}(milk) = 218; n

_{Sa 14733}(milk + 1% NaCl) = 232.

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## Share and Cite

**MDPI and ACS Style**

Ačai, P.; Koňuchová, M.; Valík, Ľ.
Modelling and Predicting the Growth of *Escherichia coli* and *Staphylococcus aureus* in Co-Culture with *Geotrichum candidum* and Lactic Acid Bacteria in Milk. *Appl. Sci.* **2023**, *13*, 8713.
https://doi.org/10.3390/app13158713

**AMA Style**

Ačai P, Koňuchová M, Valík Ľ.
Modelling and Predicting the Growth of *Escherichia coli* and *Staphylococcus aureus* in Co-Culture with *Geotrichum candidum* and Lactic Acid Bacteria in Milk. *Applied Sciences*. 2023; 13(15):8713.
https://doi.org/10.3390/app13158713

**Chicago/Turabian Style**

Ačai, Pavel, Martina Koňuchová, and Ľubomír Valík.
2023. "Modelling and Predicting the Growth of *Escherichia coli* and *Staphylococcus aureus* in Co-Culture with *Geotrichum candidum* and Lactic Acid Bacteria in Milk" *Applied Sciences* 13, no. 15: 8713.
https://doi.org/10.3390/app13158713