Overview

The genetic approach to microcystin (MC) degradation is a novel and promising topic. Compared to traditional methods such as activated carbon and ozonation, bioremediation is not as energy intensive but just as effective. However, several questions remained unsolved, including finding the optimal conditions for MC degradation or characterizing the process. While recent experimental contributions yielded appreciable results, few theoretical studies have been published. At the earlier stage of microcystin degradation system development, there is a strong need to investigate its qualities and identify potential issues in use. Therefore, we believe that an improved predictive understanding of this system is required, and a mathematical model should be necessary.

We present an initial model of the MC metabolism in our bacteria population. It can help us understand the degree of enzyme-MC binding interaction and the growth characteristics of our recombinant bacteria. Several relevant parameters can be predicted or estimated by fitting the model to our experimental data. From them, we are more likely to foresee and address issues when implementing MC degradation systems in bioremediation practices.

MlrA function in Hill kinetics

Common models for enzyme-substrate interaction include Michaelis-Menten equations and Hill equations, depending on the enzyme and reaction of interest. More specifically, it involves the binding of substrate molecules to the active sites on the enzyme to capacity, described by “binding affinity”, the relative strength of attraction between enzyme and substrate, and “interactions”, the degree of inhibition or facilitation between binding sites on a given enzyme.

Assumptions underlying these models are:

1.There is only 1 intermediate complex in the reaction, and it can be characterized as:

(For Michaelis-Menten equation, n=1)

2.The 1st step reaction is instantaneously at chemical equilibrium, so the concentration of the intermediate complex does not change on the timescale of product formation.

3.The 2nd step is the rate-limiting reaction, and it is not high enough to disrupt the equilibrium of the 1st step.

4.The 2nd step is irreversible. This can be fulfilled when either of the following is true:

a)[S] >> [P].

b)The energy released during the reaction is very large, soΔG << 0 and the forward reaction can naturally proceed.

In our case, (b) is true and the assumption stands.

5.The enzyme is not consumed during the reactions.

Through previous experiment evidence in the enzymatic activity of MlrA, we recognized that Hill kinetics may better describe the linearization reaction as compared to Michaelis-Menten kinetics. In the study “Heterologous expression and characterization of microcystinase” by Dariusz Dziga et al, the Hill Coefficient was estimated to be 1.57 (unitless), indicating unfitness for the Michaelis-Menten model. Moreover, the Hill Constant suggests that interaction between enzyme binding sites and MCs are positively cooperative in nature: the binding potential of MC molecules to the enzyme increases as the number of previously bound molecules increases.

We therefore selected Hill equation to model our enzyme kinetics. The concentration of cyclic MCs was measured by High Performance Liquid Chromatography (HPLC) in our experiments each hour, and we estimated the degradation rate and concentration in the middle of each time interval. We then used Wolfram Mathematica to fit the given equation to our data.

The fitted model provided us with several constants:

When compared with our experimental data, the Hill Fit demonstrated a good match with R Squared = 0.893. Our model thus effectively captures the qualities of the MlrA enzyme within our surface display system.

Bacteria & MC interaction ODE

When compared with our experimental data, the Hill Fit demonstrated a good match with R Squared = 0.893. Our model thus effectively captures the qualities of the MlrA enzyme within our surface display system.

In addition to the previous assumptions for Hill equation, we make the following assumptions to approach the reaction in a simpler unstructured kinetics model:

1.The growth of our recombinant bacteria is only limited by a single substrate. In this case, MC is the sole factor influencing the bacteria population.

2.The cell composition and physiological state remain constant during the time scope of interest.

3.The environmental indexes, i.e., pH level and temperature, remain constant during the time scope of interest.

4.The bacteria growth is uniform and can be characterized using its biomass concentration as the only parameter.

5.There is only one limiting substrate in the culture medium. Other components do not interfere with the bacteria growth.

6.The cell yield coefficient remains constant in our scope of interest.

Considering the positively cooperative nature of MlrA-MC binding, we derive the model involving the Hill Constant.

At equilibrium, it can be derived for the 1st step:

With the assumption that the 2nd step is the rate-limiting step, we consider .

The dissociation of the enzyme-substrate complex can thus be written from EQNS (1):

Conservation of the enzyme can be expressed as:

Using EQNS. (2) (3):

The rate of substrate depletion and product formation is:

This expression can further be simplified as:

where:

As the initial enzyme concentration is proportional to the present cell concentration, and the cell growth rate is proportional to the substrate depletion rate:

where Y is the cell yield coefficient.

From EQNS. (5) (7) (8):

Considering the endogenous decay of bacteria, we simplify EQN. (9) to:

where:

From its mathematical form, EQNS (12) can be seen as a modification to the classic Monod model.

EQNS (10) (11) (12) are our final equations for substrate-coupled microbial growth. They are summarized to the following:

We now obtain a set of ODEs that can be solved for S[t]. X[t]. μ[t], respectively referring to the concentration of MC, biomass concentration of bacteria, and the growth rate of bacteria population that is correlated to MCs.

We found that the 3 ODEs do not yield analytical solutions when we input our experiment parameters. So, we used the NDSolve function in Mathematica for a graph of the numerical solutions to be compared with the experimental curve.

Results indicated quite a good fit, with R Squared = 0.963.

For data from 1 to 7 hours, all predicted concentrations of MC fall into the error interval (full range error) of data points.

At the end of the degradation curve (8-9 hours), the predicted concentrations are higher than the measured data. It was theoretically calculated that degradation would be insufficient at 9 hours, while experimental evidence suggests that circular MCs have been fully removed before 9 hours.

A possible explanation in that our constructed degradation system is not as sensitive to the concentration of MCs as previously expected, especially at lower concentrations. This discovery advocates our engineered bacteria for practical implementations, considering that MC concentrations are usually as high as several milligrams per litre, and degradation capabilities at lower concentrations would be helpful.

Most of the parameters in our model were estimated from known experimental conditions, and were further validated when compared with parameters estimated from previous research. There was good match in value of some, and order of magnitude for all.

In conclusion, our two models succeed in reflecting our system, and provide us with a more thorough understanding on implementing the MC degrader bacteria to real life use. Its application may be further optimized using our obtained parameters as well.