Nonpathogenic surrogates strains of B. cereus (CECT 495) and E. coli (CECT 102) were purchased to Colección Española de Cultivos Tipo (CECT, Universidad de Valencia, Spain). Both B. cereus and E. coli are common foodborne pathogens and were chosen by their differences at cell wall level, respectively Gram-positive and Gram-negative. Stock cultures were kept at -80°C in culture media supplemented with 25% (v/v) glycerol.

The culture media selected for the dose-response assays was meat-peptone broth (MPB) containing 5 gL⁻¹ meat extract (Scharlau SL, Barcelona, Spain), 10 gL⁻¹ neopeptone (BactoTM. BD Biosciences, Franklin Lakes, NJ, USA), and 5 gL⁻¹ NaCl (Emsure®, Merck KGaA, Darmstadt, Germany) in distilled water. DDAC (98 % purity; ABCR GmbH & Co KG, Karlsruhe, Germany) was added to the media to obtain the different desired concentrations for sub-MIC treatments. The pH was adjusted to 7.2 and the media was finally sterilized by autoclaving (121°C/15 min). Dose-response assays were performed in 300 mL Erlenmeyer flasks containing 180 mL of MPB. Flasks were inoculated with 900 μL from a 21 h culture and incubated in an orbital shaker (200 rpm) at 30°C ( B. cereus ) or 37°C ( E. coli ). A previous screening was carried out to know the range of DDAC concentrations with partial inhibitory effect under our specific experimental conditions. The resulting DDAC concentrations selected were 0.5, 0.75, 1.00, 1.50, 2.00, and 3.00 mg L-1 for E. coli and 0.05, 0.07, 0.25, 0.80, 1.00, and 1.50 mg L-1 for B. cereus . Each DDAC concentration was tested by duplicate. At predetermined incubation times (0, 2, 4, 8, 12, 16, 20, 24, 30, 36, and 48 h), samples from each flask were taken and properly diluted in peptone-buffered water for OD determination at 700 nm. For comparative purposes, 1 mL samples from each flask were several-fold diluted in peptone-buffered solution and 0.1 mL aliquots were plated by duplicate on agarized MPB (2% w/v agar), resulting in a total of four plates per DDAC concentration tested for each strain. Plates were incubated at previously indicated temperatures for each strain and manually counted after 24–48 h of incubation.

Different types of models have been studied in this study. First, we compared the performance of two types of models to reproduce OD growth and inhibition by DDAC: the classical logistic model with disinfectant inhibition following the Weibull functions and a dynamic model based on mechanisms of adaptation, growth, inhibition, and death. Second, we extend the mechanistic model to account for CFUs growth. Finally, we present the numerical methods used for the simulation, parameter estimation, and model analysis.

For the model simulation, calibration (estimation of unknown parameters), and analysis, we use numerical methods implemented in AMIGO2 (Advanced Model Identification using Global Optimization) software, a multi-platform toolbox implemented in Matlab (Balsa-Canto et al., 2016b). The code, with all the selected options for simulation and optimization, is freely available at https://doi.org/10.5281/zenodo.5167910.

For the model calibration, we have selected the maximization of the log-likelihood function, equivalent under common assumptions [refer to Balsa-Canto et al. (2016a) for details] to minimize the least-squares function (squares of the residuals) divided by the standard deviation. In this study, we calculate the standard deviation for each time-point from the experimental replicates (refer to calculated standard deviations for each time-point in Supplementary Figure S1 and calculate the mean for each bacteria ( B. cereus , E. coli ) and type of data (CFUs, OD). Therefore, the objective is to find the parameters that minimize:

where n_exp and n_t are the number of experiments (seven including the control without disinfectant for each bacterial strain) and the number of time data points, respectively, and being Y, σ, and Ŷ the data, standard deviation, and model predictions, respectively. Bounds considered for the parameters were [0, 15] with the exception of α∈[0, 1] and β∈[0, 100].

The confidence intervals for the parameters were estimated by ±tα/2γCii, where C_ii are the diagonal elements of the confidence matrix, tα/2γ is given by Student's t-distribution with γ the number of degrees of freedom and (1−α)100% selected to 95%. The Cramér-Rao inequality was employed to compute a bound for the confidence matrix using the Fisher information matrix (Balsa-Canto et al., 2016a).

To compare the performance among models, we employed the following well-known indexes: the adjusted R², the corrected Akaike information criterion (AICc), and the Bayesian information criterion (BIC). The three indexes are a function of the error existing between model and experimental data, penalizing the number of parameters to be estimated. Indexes with lower numbers imply better performance.

In this study, we compared a classical model (logistic growth with inhibition using the Weibull functions) with a model derived from mechanisms (inspired by biochemical reactions of adaptation, substrate-based growth with inhibition by cell density and death).

As explained in section 2.2.1, the classical model describes the effect of DDAC on bacterial growth based on the combination of the Weibull function as biocide-concentration model modifying the most relevant kinetic-parameters of the reparametrized logistic equation used for the bacterial growth description. We should mention that the derivative form of this model was used to allow its implementation in the AMIGO2 software, refer to https://doi.org/10.5281/zenodo.5167910 for details.

On the other hand, the model derived in this study assumes the following mechanisms (refer to section 2.2.2 for mathematical details of the model):

where the specific adaptation rate (or rate per capita) is a Hill-decreasing-function of the disinfectant concentration k_a(C), as described later, and the total rate follows the mass action law (K_a = k_aX).

where 1/Y_S is the yield coefficient, the specific growth rate is again a Hill-decreasing-function of disinfectant k_g(C) and k_i is the coefficient regulating the inhibition due to cell density (with low values for strong inhibition by large cell numbers). Note that this inhibition encodes different mechanisms like the inhibition for the production of waste in a cell cycle. This biochemical reaction describes both the exponential growth phase and the stationary phase.

Death occurs with a specific rate dependent on a Hill-increasing function on the concentration of disinfectant k_d(C). Note that death may be active during the whole dynamics and determines the death phase when this rate surpasses the growth rate (negative cell growth).

Under these assumptions, we derived a model consisting of a set of three ODEs plus an algebraic equation defining the measured variable OD (refer to section 2.2.2). We should mention that for its implementation in the AMIGO2 toolbox, the derivative equation of the OD function was implemented. Refer to code for details at https://doi.org/10.5281/zenodo.5167910.

Figure 1 depicts the performance of the logistic and mechanistic model to reproduce the growth of B. cereus and E. coli measured using OD. Both models reproduce the data trends, with the main difference being the ability of the mechanistic model to describe the smooth transition to the stationary phase in B. cereus (Figure 1B) and the initial death phase for E. coli (Figure 1D). Moreover, the mechanistic model starts from the experimental initial conditions, whereas the logistic model allows flexibility for this condition.

Table 3 shows three standard indexes to measure the model's ability to reproduce the data penalizing overparametrisation: the adjusted R², the corrected AICc, and the BIC. All criteria show a lower value (despite penalizing the models with many degrees of freedom) for the mechanistic model, where 13 parameters were estimated instead of the 12 estimation parameters for the logistic model.

The mechanistic model was selected for the rest of the study due to a better goodness-of-fit and its mechanistic insight. We want to stress that although both models show good performance, which was only slightly lower for the classical model, the insight from the mechanistic model was determinant to make the decision. Mechanistic models allow a deeper study of the process and (as shown in the next section) can be extended to account also for CFUs by adding only one extra parameter.

Both OD and CFUs are related standard methods to measure bacterial concentration dynamics but differ in their interpretation. OD is a fast method but is affected by many factors, including dead cells in the medium with intact cell wall (Stevenson et al., 2016), making estimations of alive cells unreliable, whereas CFUs provides reliable estimations of viable cells, but it is experimentally time-consuming. Estimations using only CFUs were tested with both logistic and mechanistic models (data not shown), with better estimations for the mechanistic model for including the death phase observed in the data.

We extended the mechanistic model derived in this study to explain both, OD and CFUs data. Let us recall here the OD variable:

being a function of latent, adapted, and dead cells. For the modeling of CFUs, we consider only alive cells (either latent or adapted):

with β10¹⁰ a scaling factor transforming OD to CFUs.

Figure 2 shows how assumed mechanisms are sufficient to reproduce OD and CFUs using the same adaptation, growth, and death parameters. The mechanistic model has 14 parameters: 12 common parameters plus α, defining contribution of dead cells to OD, and β10¹⁰, which represents the difference between OD (usually one order of magnitude) and CFUs (varying from 10 orders of magnitude for B. cereus and 12 for E. coli ).

Table 4 shows the model performance in terms of adjusted R², AICc, and BIC. Note that now the model has been challenged to reproduce both OD and CFUs with the same parameters and, although indexes are acceptable for the measured experimental errors, are worse than when reproducing only OD. To further analyze that the performance is satisfactory, we have included the confidence intervals of the estimated parameters, where the data uncertainty is also considered (refer to Supplementary Figure S1).

The graphical picture of how adaptation, growth, and death-specific rates change with DDAC is shown in Figure 3. Note that growth-specific rate is inhibited by limiting substrates and large cell density. To simplify the figure, we plotted k_g assuming that resources are not limiting (S is almost constant) and inhibition by cell density is not relevant, and therefore, K_g≈k_g.

In this study, we derived a new mechanistic model able to represent the different phases (adaptation, growth, stationary, and death) at sub-MIC concentrations of DDAC. The logistic model, the standard model used to quantify growth inhibition with bacteriostatic antimicrobials, was unable to reproduce the last death phase of the growth curves. This limitation was previously detected when modeling the inhibition of E. coli and Staphylococcus aureus due to silver nanoparticles (Chatterjee et al., 2015). As with sub-MIC DDAC concentrations, nanoparticles are mostly bacteriostatic but growth curves show a decline at long times.

The derived model was calibrated from and did reproduce the two most common measurements of bacterial population numbers, OD and CFUs, using the same parameters for adaptation, growth, and death rate. Results show that E. coli dead cells do not affect OD, probably due to severe cell lysis and their not appreciable effect in OD measurement (Stevenson et al., 2016). However, 32% of B. cereus dead cells contribute to OD. Two hypotheses can explain this estimation: either this contribution is due to viable but nonculturable cells or either some cells are non-lysed since B. cereus has more resistance to lysis than E. coli (Fykse et al., 2003).

The analysis of the developed model confirms the literature observation that DDAC is bacteriostatic and bactericidal depending on its concentration and the growth phase of the microbial population (Yoshimatsu and Hiyama, 2007; Kampf, 2018a). The growth rate is affected in both B. cereus and E. coli . In B. cereus the growth decreases sharply only by adding low concentrations of DDAC and remains almost constant afterward (γ_g close to zero), as can be seen in Figure 3, being, therefore, IC_{50, g} irrelevant and with large uncertainty. Although Hill functions are very uncertain, and not useful to infer exactly at which concentration is this sharp decrease, the growth rate should be at least 0.6 h⁻¹ when DDAC = 0 mg L-1and 0.4 h⁻¹ for DDAC = 0.5 mg L-1 (the DDAC concentrations experimentally tested) to fit the data. In E. coli , growth decreases smoothly when DDAC increases (γ_g = 0.078 and IC_{50, g} = 14). This can be also observed in Figures 3C,D for B. cereus and E. coli , respectively. On the other hand, the bactericidal effect is also detected for both B. cereus and E. coli despite being the estimated dying rates the most uncertain model kinetics (as evidenced by the parameter confidence intervals). Whereas for B. cereus the bactericidal effect increases smoothly with DDAC concentration (Figure 3C), this behavior is more abrupt for E. coli .

The model also allows the study of the DDAC changes in the lag phase, a mechanism that is not considered when simply classifying a disinfectant as bacteriostatic or bactericidal. Adaptation is fast in B. cereus (small lag phase) and slow in E. coli . The model captures this behavior using larger values of k_a for B. cereus (maximum allowed value ka0=15) than for E. coli (ka0=0.35). Note that large confidence intervals are reported for B. cereus adaptation rate as faster adaptation rates result in the same dynamics where adaptation is complete within minutes or seconds at c = 0. Despite both strains showing the different magnitude of adaptation, the change of adaptation with DDAC concentration (Figures 3A,B) shows a similar trend. Whereas for both adaptation is high without DDAC, when DDAC concentration increases there is a deep decrease of adaptation, therefore, showing longer lag phases at high DDAC concentrations.

The analysis of the lag phase with DDAC concentration is particularly relevant to understanding the implications of MIC values and how they depend on the growth phase (Yoshimatsu and Hiyama, 2007; Kampf, 2018a). Lack of growth after 24 h could be due to different mechanisms: either an extremely large lag phase or that the death rate is equal or larger than the growth rate [zero or negative net growth rate, also commented by Mouton and Vinks (2005) the stationary concentration]. Assuming no effect of the limiting substrate and MIC representing the concentration of zero growth rate (Coates et al., 2018), the model can be used to calculate the MIC using the intersection of growth and death rate (k_g and k_d). As shown in Figure 3C there is net growth of B. cereus for disinfectant concentrations below 1.39 mg/L and the bacteriostatic effect is greater at low concentrations of DDAC whereas the bactericidal effect is higher at larger concentrations. On the other hand Figure 3D shows a similar trend, but with less bacteriostatic effect and with a MIC of 2.86 mg/L.