Verhulst-Pearl growth model versus Malthusian growth model for in vitro evaluation of lead removal in wastewater by Photobacterium sp

Introduction

It is well-known that some genera of bacteria, such as Pseudomonas, Xanthomonas, Ferroxidans, Ralstonia, Acidobacillus^1–3, have very interesting capacities as heavy metal remediating agents, since they are capable of extracting metals from solid substrates or transform them into harmless forms. Furthermore, bacteria can be used as bio absorbents for the recovery of metals and for the treatment of industrial effluents^4,5. For this reason, the application of these technologies and research for continuous improvements is necessary.

In a previous study, the application of Photobacterium sp to remove lead from wastewater was shown to be possible⁶. However, understanding the behavior of this strain within the experimental conditions was not studied. Particularly, computational modeling of the growth kinetics that aims to extract information about interactions between bacteria, nutrients and in this case the pollutant (lead; Pb) could be useful. For this reason, the present study intends to analyze growth of Photobacterium sp using two well-known mathematical models, the Verhulst-Pearl growth model and the Malthusian growth model⁷. These models have a long history of application in the fields of ecology, environmental remediation and industrial fermentation⁸.

Methods

The study uses kinetics data collected from the previous study in Quito-Ecuador, which studied the application of Photobacterium sp to remove lead from wastewater⁶. Briefly in the previous experiments, Photobacterium sp was exposed to two concentrations of Pb (20–100 ppm) in a general growth broth. The experiments were performed in times ranging between 0 and 86 hours, under a controlled temperature (25°C) and unsupervised at room temperature, ranging 5–23°C⁶ at Quito-Ecuador.

Results

Fitting a bacteria growth model aims to understand its kinetics. In Table 1 it is evident that Verhulst-Pearl model has a good performance since a R-squared > 0.90, which is considered as acceptable to model bacteria kinetics curves^5,9. In these fitted curves, it is noticeable that room temperature parameters differed drastically from 25°C in carrying capacity (A) and middle time (to) to achieve middle biomass concentration. Therefore, these parameters suggest that 25°C controlled temperature increases the performance in both Pb concentrations. Room temperature seems to reduce the growth considering that Quito-Ecuador is located at 2800 m.a.s.l. Moreover, this model explains the reduction in maximum growth ratio due to the increase of Pb concentration. This effect is present at both temperature conditions (Figure 1).

Figure 1. Modeling data with Verhulst-Pearl growth model.

(A) 25°C, 20ppm lead; (B) Room temperature, 20ppm lead; (C) 25°C, 100ppm lead; and (D) Room temperature, 100ppm lead.

Table 2 shows that the Malthusian growth model had an unacceptable goodness of fit, R-squared < 0.90 for all fitted curves. Therefore, this is considered as an unacceptable model for the bacteria kinetics curves tested. In these fitted curves, it is difficult see differences in temperature and Pb concentration. However, this model provides information about variations in growth ratio (Figure 2). It would be difficult to extend analysis since this model does not fit as well as the Verhulst-Pearl growth model.

Figure 2. Modeling data with Malthusian growth model.

(A) 25°C, 20ppm lead; (B) Room temperature, 20ppm lead; (C) 25°C, 100ppm lead; and (D) Room temperature, 100ppm lead.

Conclusions

Analysing the goodness of fit reveals that the Verhulst-Pearl growth model is the best option to model the kinetics of Photobacterium sp instead of the Malthusian growth model, at least in this particular case. Photobacterium sp is also suitable to remove lead from water as shown in Ramirez-Cando et al.⁶. The Malthusian model has no manner to estimate Carrying capacity since parameters in the model are developed to explain only the exponential growth phase. Moreover, parameters determined by Verhulst-Pearl growth model are very important in design further research and scaling to preindustrial process in microbiology as well as Monods model¹⁰.

Data availability

Dataset 1: Kinetics data obtained in Ramirez-Cando et al.⁶. Uploaded with permission of all the authors.¹¹