Computational Modelling

Biofilm Growth Model

Biofilms are an integral part of attached-growth wastewater treatment processes like trickling filters and rotating biological contactors. The nutrients in wastewater allow biofilms to grow and thrive¹. Our biofilms will be grown in the lab, instead of allowed to form naturally, so we need to estimate the incubation period required for sufficient biofilm growth. This should increase the conversion efficiency of the chosen contaminants.

We used two similar empirical mathematical models to predict the rate of biofilm growth. The Logistic and Gompertz models can accurately describe this phenomenon using only a few parameters ². The use of a controlled setting for incubation also simplified this analysis by allowing us to neglect the presence of any additional agents, inhibitory or otherwise ². The simplified equations (taken from Verotta et al. ²) are shown below:

Where:
B(t) is the amount of biofilm present at time t
B₀ is the initial amount of biofilm present (at t=0)
K_b is the growth rate of the biofilm
B_max is the maximum amount of biofilm that is produced

The units for quantifying biofilms depend on the method of measurement used. Frequently-used methods include dry weight, optical density, and heterotrophic plate count (HPC) ¹. Had we been able to reach this phase in the lab, we would have most likely chosen dry weight as our method of measurement. Initial measurements at the start of growth would provide the value of B₀, and final measurements after growth has ceased would provided the value for B_max. Periodic measurements throughout the growth period would provide a series of points to form a curve. Using curve-fitting software such as MATLAB, we would able to solve for the optimal value of k_b to fit the empirical model to the data. As seen in the plot below (which uses arbitrary values also provided by Verotta et al. ²) the Logistic and Gompertz growth models follow the shape of an S-curve, indicating that biofilm growth will plateau after a certain length of time. The objective of these models is to predict the incubation period required to maximize biofilm growth. Knowing this will also save time and reduce costs by allowing us to use the minimum appropriate growth time. The MATLAB file for this model can be found at the bottom of this page.

Figure 1: Logistic and Gompertz models showing biofilm growth over time.

Contaminant Conversion Models

Simple empirical models were also developed to predict the rate of conversion of the selected emerging contaminants by laccase. These models predict enzymatic activity with the objective of estimating the reaction time needed for adequate conversion of contaminants. Both equations are derived from fundamental kinetic principles. Kinetic data specific to laccase from Trametes Versicolor with the selected contaminants as substrate were taken from relevant literature. The activity of cutinase was not modelled as cutinase was mainly used as a positive control; cutinase has previously been shown to be exported using the type III flagellar secretion system³. It does not react with estradiol or diclofenac. These models are developed from research on wild-type laccase. Discrepancies between model-predicted activity and the observed behaviour may indicate the relative efficiency of our fusion proteins compared to the wild-type enzyme. Both models are linked at the bottom of this page.

Conversion of 17-β estradiol

Rangelov & Nicell ⁴ provide a simple mathematical model (shown below) to predict the concentration of a substrate as a function of time and concentration of laccase in its oxidized state. This model was shown to be effective in predicting the conversion of 17-β estradiol ⁴.

where:
α = dimensionless exponent of [E*] describing rate of substrate oxidation (0<α≤1)
β = dimensionless exponent of [S] describing rate of substrate oxidation (0<β≤1)
[E*] = Concentration of laccase in the oxidized state
[S] = concentration of substrate (17β-estradiol)

The graph shows the rate of change of the substrate concentration with an arbitrary value used for [E*]. It was assumed that [E*] would remain constant for the purposes of modelling. Had we progressed further in the lab, measurements of the substrate concentration, [S] at various points in time throughout the reaction would have produced an empirical curve. Using curve fitting tools such as MATLAB, we would be able to determine an equation representing [E*].

Conversion of diclofenac

The mathematical model for conversion of diclofenac was developed using Michaelis-Menten kinetics. Lonappan et al. ⁵ provided the following kinetic data for wild-type laccase with diclofenac as substrate. These parameters are used in the Michaelis-Menten equation, also seen below.

k_m = 9.032646 x 10^-3 mM
V_max = 0.0806 mg/L/h

where:
k_m is the Michaelis-Menten constant
V_max is the maximum speed of the reaction
[S] is the concentration of the substrate (diclofenac)

The Michaelis-Menten equation is plotted to show the rate of change of the concentration of the reaction product. This would also indicate the rate of conversion of diclofenac. To predict the length of time needed for sufficient conversion of a given concentration of diclofenac, the equation would be integrated with respect to time.

Figure 2: Rate of reaction as a function of diclofenac concentration.

References

1. Sehar, S. & Naz, I. Role of the Biofilms in Wastewater Treatment. in Microbial Biofilms - Importance and Applications (IntechOpen, 2016). doi:10.5772/63499

2. Verotta, D., Haagensen, J., Spormann, A. M. & Yang, K. Mathematical Modeling of Biofilm Structures Using COMSTAT Data. Comput. Math. Methods Med. 2017, 1–11 (2017).

3. Green, C. A. et al. Engineering the flagellar type III secretion system: improving capacity for secretion of recombinant protein. Microb. Cell Fact. 18, 1 (2019).

4. Rangelov, S. & Nicell, J. A. Modelling the transient kinetics of laccase-catalyzed oxidation of four aqueous phenolic substrates at low concentrations. Biochem. Eng. J. 132, 233–243 (2018).

5. Lonappan, L. et al. Agro-industrial-Produced Laccase for Degradation of Diclofenac and Identification of Transformation Products. ACS Sustain. Chem. Eng. 5, 5772–5781 (2017).

Files

To download the biofilm growth model, click here.
To download the estradiol conversion model, click here.
To download the estradiol conversion model, click here.