Modelling1: Enzyme kinetics

Theory and derivation of enzyme kinetics :

We simplified our enzyme-EDCs system as the reaction with the process of the first and second order reactions:

The initial collision of E and S is a bimolecular reaction with the second-order rate constant k1. The ES complex can then undergo one of two possible reactions: k2 is the first-order rate constant for the conversion of ES to E and P, and k-1 is the first-order rate constant for the conversion of ES back to E and S.

We assume that the formation of product from ES (the step described by k2) does not occur in reverse.

The rate equation for product formation is:

However, measuring [ES] is more difficult because the concentration of the enzyme-substrate complex depends on its rate of formation from E and S and its rate of decomposition to E + S and E + P:

To simplify our analysis, we choose experimental conditions such that the substrate concentration is much greater than the enzyme concentration and ES remains constant until nearly all the substrate has been converted to product.

According to the steady-state assumption, the rate of ES formation must therefore balance the rate of ES consumption:

The total enzyme concentration, [E]T, is usually known:

This expression for [E] can be substituted into the rate equation to give:

Rearranging (by dividing both sides by [ES] and k1) gives an expression in which all three rate constants are together:

We defined KM and rearranged the equation:

We did some compute process for the equation and give:

Solving for [ES],

Finally, we can express the reaction velocity as:

The maximum reaction velocity, Vmax, can be expressed as:

And then we can get the equation,

In order to estimate the degradation capacity of the filter, we calculated horseradish peroxidase’s ability of degradation based on the equation of Michaelis-Menten kinetics :

We used several parameters according to the paper we found to calculate the initial velocity of different concentrations of BPA and NP. Also, their concentrations at different time after starting degradation.(Figure 1 and 2)

NP:

[H2O2]=10^(-5) M

Km=10.1*10^(-6) M

Vmax=0.056*10^(-6) M/s

S: 9.1*10^(-10)~9.1*10^(-6) M

BPA:

[H2O2]=0.02*10^(-3) M

Km=6*10^(-6) M

Vmax=2.22*10^(-9) M/s

S: 8.77*10^(-10)~8.77*10^(-6) M

From the results of the enzyme kinetics, we can find that when the concentration of EDCs increases, the degradation rate increases dramatically. As a result, our activated carbon in the system of our filter can help HRP to degrade EDCs in the more efficient way since its outstanding ability to capture EDCs in the water and accumulate more EDCs around HRP.

Modelling2: Concentration Test

Theory and derivation of the Freundlich Adsorption Isotherm from Kinetics

In general, the rate law can be written as:

If we imagine a small pore, adsorption sites near the open ends are different from at the center of the pore. This distinction occurs when the sites possess specific chemical properties. If a concentration is imposed at one end of the pore, solute must diffuse to find an open adsorption site first and it is similar as the desorption case.

In this point, the exponent n is a measure of the fractal dimension of the process and the rate constant for fractal reactions is also expected to be proportional to the molecular diffusion coefficient.

For both adsorption and desorption reactions but with distinct fractal dimensions (ni) and fractal rate constants (ki):

At equilibrium, the derivative is equal to zero and the equation can then be solved for S:

Finally, we can transform the equation to the Freundlich Adsorption equation,

Ce : the equilibrium concentration of the solute in the bulk solution (mg L−1)(the concentration of EDCs)

KF : Freundlich constant indicative of the relative adsorption capacity of the adsorbent (mg1−(1/n) L1/n g−1)

n : Freundlich constant indicative of the intensity of the adsorption

qe: the amount of solute adsorbed per unit weight of adsorbent at equilibrium (mg g−1)(the maximum of the adsorption)

We’ve conducted an experiment based on the kinetics of enzymes method described above to find out our filter’s maximum capability to capture EDCs. The experiment was designed as follow:

To measure the maximum capability, we need to know the EDCs concentration change between the input and output of running solution under different EDCs concentration solution. Thus, the fixed-volume (100mL) and fixed-flow (100mL/5 sec) solution with different EDCs concentration was used to pass through our filter.

Based on the EUR-Lex data for allowable BPA and NP concentration, we decided to conduct 4 different BPA/NP concentration (total 8 experiments) in this modeling experiment:

10-8M(maximum allowable concentration), 10-7M, 10-6M and 10-5M.

Additional information:

-An UV spectrum was used to detect the concentration of solution.

-Fixed amount of Activated Carbon (10g) was used in each experiment.