INTRODUCTION
We aim to implement our NP trapping systems in different steps of the wastewater treatment process. There are several factors that will affect the NP trapping efficiency for the proteorhodopsin (PR) bacteria and biofilm models. Our PR bacteria would be added to aeration tanks, where water movement is fast and turbulent, while our biofilm (attached to biocarriers), would be placed in the clarifier or sedimentation tanks, where water movement is calmer to prevent biofilm detachment.
Due to the lack of literature on our proposed NP-trapping techniques using PR and biofilm, experimental trials and our prototype design were integral to the modeling process. Experimental trapping rates from our prototype were used to fit our model to the current trapping abilities of our PR construct (BBa_K2229400) and our biofilm construct (BBa_K2229300). After experimentally determining the rate constants for our PR bacteria and biofilm constructs, the mathematical models can be used to determine two objectives, given an initial NP concentration and a final target NP concentration.
Objective 1: What PR bacteria concentration is needed in the aeration tanks?
Objective 2: How many biofilm-coated biocarriers are needed in the secondary sedimentation tank?
PROTEORHODOPSIN TRAPPING MODEL
Proteorhodopsin and citrate binding modeled as a ligand-receptor interaction
To model the binding of PR bacteria to CC-NPs, we used a coarse-grained model for ligand-receptor interaction (Ruiz-Herrero et al. 2013). The model is based on the chemical interaction between a freely diffusing ligand L (nanoparticle), and a cell membrane receptor R (PR bacteria), which combine to form a complex C (nanoparticle-loaded PR bacteria) in the following reaction scheme:
Figure 3-1 Reaction scheme for PR bacteria and nanoparticles. Our bacteria (L) and nanoparticles (R) bind with the affinity rate k(on) to form the complex C. Conversely, starting with the complex C, nanoparticles fall off bacteria with the dissociation rate k(off). Figure: Justin Y.
where kon is the binding rate constant of our PR bacteria to CC-NPs and koff is the rate constant of NPs dissociating from PR bacteria. L, R, and C are all functions of time because our PR bacteria binds to NPs over time, which decreases the concentration of free NPs and available bacteria while increasing the concentration of NP-loaded PR bacteria. Therefore, we can use the following differential equation to model the progression of NP trapping over time by our PR bacteria:
Equation 1: Binding and Dissociation Model
To determine the kon and koff rate constants, we fit our model to match experiments (figures 3-3 and 3-4) that show a decrease in CC-AgNPs when mixed with a known concentration of our PR bacteria (Bba_K2229400). Determining kon and koff rate constants will enable us to use equation 1 and inform WWTPs what concentration of PR bacteria is needed to treat their tanks given a starting NP concentration and desired final NP concentration. Below, we will explain how we obtained both the kon and koff rate constants.
Determining NP binding rate (kon) and dissociation rate (koff) constants using experimental data
Binding-Only Model (kon only)
We first determined kon while assuming a best-case scenario where NPs do not fall off of PR once they bind, which means that kon is zero. Thus, our initial model can be described with the following equation:
Equation 2: Binding-Only Model
Since the time-dependent functions [L] and [R] for binding between PR bacteria and citrate NP are unknown in literature, we used finite-difference methods (FDM) to model this equation. (Click here to learn more about how we used FDM!)
We obtained kon, the binding rate constant, from experimental data where CC-AgNPs were mixed with a known concentration of our PR bacteria (BBa_K2229400) (Figure 3-2 and 3-3 below).
Figure 3-2 The binding rate k(on) for our PR bacteria was determined experimentally. A known concentration of our PR bacteria (BBa_K2229400) was mixed with a known concentration of citrate-capped AgNPs, and the decrease in nanoparticle concentration over 5 hours was measured. By inputting the known bacteria and nanoparticle concentrations, we fitted our model to our experimental data to find k(on). Figure: Justin Y.
Figure 3-3 Experiment: Justin Y
To find the kon value for our PR bacteria, we fit our model to our experimental data (Click here to learn how we fit our model!). 1.9×10-7 µL cells-1 hr-1 was the kon value for which the Root Mean Squared Error (RMSE), a measure of the error between the model and the experiment, was the lowest. However, as Figure 3-3 shows, the Binding-Only Model does not reflect how the experimental data reaches an asymptote towards the end of 5 hours. One of the things that may account for this discrepancy between our Binding-Only Model and our experimental data is NPs falling off of the PR bacteria. As we show below, our revised model does a better job of describing the experimental data after accounting for this effect. Since our Binding-Only Model does not take the NP dissociation rate constant, koff, into account, we now needed to determine this value.
Binding and Dissociation Model (kon and koff)
For the first hours, few NPs are bound by the bacteria and there are many PR receptor sites open, so we assumed that the number of NPs falling off is negligible in the first hour. Thus, we assume that the dissociation rate, koff, is 0. Based on this assumption, we found a kon value that fit our model to the first two points (the first hour) of our experimental data. As the yellow curve (step 2) in Figure 3-4 below shows, the model fits the first two data points but falls below the experimental data as time goes on. We assumed that this discrepancy is due to NPs falling off the bacteria. Thus, our next step was finding a koff value that would fit our model to the rest of the experimental data. (Click here to learn how we found koff). The green curve in Figure 3-4 below includes this koff value, and fits our experimental data much better with the lowest RMSE. Thus, our final binding and dissociation model includes both kon and koff.
Figure 3-4
CC-NP Trapping by PR Bacteria Calculator
We developed two calculators to help WWTPs use our PR bacteria to clean up CC-NPs.
Calculator 1 allows WWTPs to input their NP concentration, their target NP concentration, and the time water spends in the tank to determine the initial PR bacteria concentration they need to add.
Calculator 2 allows WWTPs to input their NP concentration, PR bacteria concentration they plan to add, and the time water spends in the tank to determine the final NP concentration of the water leaving the tank.
Example Application of Completed Model
In Figure 3-5 below, we used our Binding and Dissociation model to determine the trapping of CC-NP concentration over time by our PR bacteria using the kon and koff values above (3.5×10-7 µL cells-1 hr-1 and 0.32 hr-1 respectively) and time intervals of 0.1 hours (click here to learn why).
Figure 3-5
In this example, the initial conditions of L and R were set to the same values as our experimental trial, which means that [NP] = 1.078 µM and [PR bacteria] = 569600 cells/µL. Under these conditions, our model predicts that NP concentration after 5 hours is 0.693 µM (the percent difference of our modeled value from our experimental value (0.708 µM) is 2.19%). WWTPs can obtain a graph like this one by inputting the variables specific to their treatment plant, such as initial NP concentration and how much time water spends in the tank.
Calculation Explanations for FDM and RMSE
Finite-Difference Method (FDM) Explanation
Minimizing Root Mean Square Error (RMSE) to find kon and koff
BIOFILM TRAPPING MODEL
Evaluating the trapping rate through the change in substrate concentration and volumetric flow rate
Determining the significance of different factors
Surface Area
Example Application
REFERENCES
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