To find the k_on value for our PR bacteria, we fitted our model to our experimental data (Click here to learn how we fit our model using RMSE!). 1.9×10-7 µL cells^-1 hr^-1 was the k_on 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 4-3 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, k_off, into account, we now needed to determine this value.

Binding and Dissociation Model (k_on and k_off)

For the first hour, 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, k_off, is 0 from t = 0 to t = 1. Based on this assumption, we found a k_on value that fitted our model to the first two points (the first hour) of our experimental data. As the yellow curve (step 2, figure 4-4 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 k_off value that would fit our model to the asymptotic behavior of our experimental data. 0.32 hr^-1 was the k_off value that yielded the lowest RMSE. (Click here to learn how we found k_off using RMSE!). The green curve (step 3 figure 4-4 below) includes this k_off value, and fits our experimental data much better with the lowest RMSE. Thus, our final Binding and Dissociation model includes both k_on and k_off.

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.

Initial NP Concentration (micromolar)
Target NP Concentration (micromolar)
Retention Time (the amount of time water stays in the tank, hours)
Calculate!
Initial Bacteria Concentration Needed (# of bacteria/microliter):

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.

Initial NP Concentration (micromolar)
Initial Bacteria Concentration (# of cells/microliter)
Amount of time that can be used for the process (hours)
Calculate!
Resulting NP Concentration (micromolar):

Example Application of Completed Model

We used our Binding and Dissociation model (in figure 4-5 below) to determine the trapping of CC-NP concentration over time by our PR bacteria using the k_on and k_off 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).

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

We used Euler’s Method to create our FDM model, and our formulas for the Binding-Only Model are shown below (table 4-1). As the table shows, by plugging in values for L and R at t = 0, a fixed value for k_on, and the time interval Δt, dC/dt = Y₁ at t = 0 can be calculated. In other words, we can find the trapping rate at t = 0. As per FDM methods, we assume that the trapping rate is Y₁ for t = [0, t₀ + Δt] = [0, t₁], so = [NP] trapped during the time interval t = [0, t₁]. Therefore, to find the new L at t = t₁, we subtract (Δt)Y₁ from L₀ to get L₁. The same process is done for R to find R₁ because the decrease in [NP] is directly proportional to the decrease in unsaturated bacteria. Then, using the new values L₁ and R₁ and the same values of k_on and Δt, a new trapping rate Y₂ for t = [0, t₁ + Δt] = [0, t₂] is calculated. This process is repeated for each subsequent value of t.

Table 4-2 shows a sample of our calculations for the Binding-Only Model using the formulas above, initial L and R conditions that match our experimental trial, the value of k_on determined from experimental data, and Δt = 0.1 up to 1 hour.

At t = 0, no NPs have been trapped and all PR bacteria are available, so we can set L = initial [NP] and R = initial [bacteria], and use these initial conditions to calculate subsequent values.

For our Binding and Dissociation Model, the formulas were modified to include the effect of k_off, which is shown below (table 4-3). This shows one of the advantages of Euler’s Method and other FDM models ‒ the form of dC/dt is adaptable to further variables or complications, and the model can still provide results for complex equations.

Table 4-4 shows a sample of our calculations for the binding and dissociation model using the formulas above, initial L and R conditions that match our experimental trial, the values of k_on and k_off determined from fitting to experimental data, and Δt = 0.1 up to 1 hour.

When using FDM, it is important to keep in mind that values after the set initial conditions are numerical approximations. This means that the larger the value of Δt, the more inaccurate the final result, as shown in figure 4-6.

For example, Figure 4-7 below shows how the trapping curve changes with different values of Δt. Larger values of Δt result in a final [NP] that is too low. This is because trapping rate is fastest at the beginning (when there are more unbound NPs and unsaturated bacteria), and large values of Δt assume this faster trapping rate for a larger interval of time, thus resulting in a faster decrease in [NP] than if smaller values of Δt were used. To account for error caused by FDM, we ran our model with decreasing values of Δt until there was a negligible difference in final [NP], which was at Δt = 0.1. Since the error due to FDM is negligible for Δt = 0.1, our model used time intervals of Δt = 0.1 hours.

Minimizing Root Mean Square Error (RMSE) to find k_on and k_off

To determine the best fit of our model to our experimental data, we calculated the Root Mean Square Error (RMSE). We did this by squaring the difference between each experimental value and its respective modeling value, averaging these values, and finding the square root. The lower this RMSE value, the better the fit of our model to our experimental data.

For our binding-only model (the blue curve in Figure 4-4), the RMSE was calculated for all of the experimental data points, and we found an initial k_on value that gave the lowest RMSE. For our binding and dissociation model, we first found a new k_on value that fit the first two experimental data points, then found a k_off value that fit the rest of the data points. To find the k_on value, the RMSE was calculated for the first two points. To find the k_off value, the RMSE was calculated for all the points.

BIOFILM TRAPPING MODEL

Free-floating PR bacteria and biofilm have different characteristics, so we have to consider different variables in our biofilm trapping model. Our goal here is to use mathematical modeling to 1) understand how biofilms interact with NPs, and 2) use this information to predict how many biofilm-coated biocarriers should be used in WWTP sedimentation tanks.

Based on experiments described below, we first determined that biofilm surface area (SA) is the main factor affecting NP trapping rate. This led us to develop a biofilm trapping model which treats NPs as solutes in wastewater that get trapped as they are carried into the biofilm surface by the movement of water (as shown in Figure 4-8).

We modeled this interaction using a differential equation (click here to see how we derived this equation):

which relates the change in NP concentration to an effective trapping rate (k_trap), the speed of the water carrying the NP into the biofilm (S_water), the total surface area of biofilm (SA_biofilm), and the total volume of the treatment container (V_{water total}).

As with the PR bacteria model, we used FDM to solve this equation numerically. We ran experiments to find k_trap based on experimentally controlled values of the other variables.

Finally, we developed two calculators to help WWTPs use our biofilm to clean up NPs.

Determining Key Parameters of Biofilm-NP Interaction

To model our biofilm’s NP trapping rate, we first needed to determine the significance of various factors that would affect biofilm-NP interaction. Initially, we assumed that the surface area (SA) and volume of the biofilm would be the two major factors that affect trapping rate.

Biofilm Volume vs NP Trapping

In our experiments, we observed that NPs were trapped in a matrix of biofilm (see SEM images on Experimental Summary), but we did not know how deep the NPs would travel into the biofilm layer. When we tested 4 different volumes of biofilm for their NP trapping abilities over a period of 5 hours (to mimic WWTP sedimentation tank conditions), we found that volumes greater than 1-1.5 mL—which covered the entire bottom of the cylindrical container—did not significantly improve NP trapping (see our experiment). This led us to conclude that NPs are not likely to travel through the entire biofilm layer.

Biofilm Depth vs NP Trapping

Wanner et al. (2006) previously reported that biofilms could only trap particles up to a depth of 200 μm. To see if our own biofilm constructs have a similar trapping depth, we analyzed our volume trial results by converting the biofilm volumes to depths. Since our biofilm is different from the ones used by Wanner et al., it possibly has a different maximum depth for trapping NPs.

In our experiments, biofilm was added directly to the cylindrical containers, where volume can be found by the equation v=r²h. We tested 4 different biofilm volumes, so the height (h) or depth of the biofilm was found by plugging in known values for volume (v) and radius (r).

We found that more AuNPs were trapped when the biofilm increased from 500 to 1000 μm, but when the depth exceeded 1000 μm, the amount of AuNPs trapped by our biofilm remained similar (figure 4-9). Therefore, our experimental results agree with the general conclusion of Wanner et al. (2006) that when biofilm depth exceeds a certain limit, depth no longer impacts NP trapping. In future experiments, we made sure to use biofilm with a thickness over 1000 μm (1 mm) to ensure that the depth is not a factor affecting NP trapping.

Determining Equations of Biofilm’s NP-trapping interaction

Surface Area

Example Application

Initial nanoparticle concentration (micromolar)
Velocity of water in contact with biofilm (cm/sec)
Total Volume of Container (L)
Time for biofilm to be in contact with NP solution (seconds)
Surface Area of biofilm used
Calculate!
Resulting Nanoparticle Concentration (micromolar):

Initial NP concentration (micromolar)
Target NP Concentration (micromolar)
Velocity of Water in contact with Biofilm (cm/s)
Total Volume of tank (L)
Total time biofilm is in contact with NP solution (s)
Calculate!
Surface Area of Biofilm Needed(cm2):

REFERENCES

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