Team:Exeter/Model
Modelling
We have produced this informative model influenced by experimental data that provides knowledge into the scalability of our filtration system and incorporates it within a user-friendly interface on our wiki to inform the future implementation of our research. The model can guide future-users on the most efficient flowrate and volume of metal binding reactor (MBR) to use for their purpose.
This model can be used to resolve the issue of knowing when the metal binding reactor needs to be changed as it informs when the pili on the E. coli in the metal binding reactor are too saturated to still be efficient. This can be used in conjunction with the 'backwash' method suggested by representatives at South West Water. For the 'backwash' method we would require two or more filters in series such that one can be taken offline without leading to a complete shutdown. Then we would pass water through the offline filter to be tested. The model can then help inform how often the water should be tested and can be rescaled for more accuracy based on these results.
User Interface
We have designed and developed a user interface for the model to make it easily accessible to future users. Once the users values are inputted and the "run model" button is clicked, the outputs for their specific inputs will be produced along with a graph of output concentration against time to provide a visual representation.
This model outputs a graph of the concentration of ions which make it through the filter over time. The red line is the concentration of ions which the user will allow to pass through the filter. The point where the plot crosses the red line is the point at which the filter needs to be changed because it can no longer sufficiently reduce the concentration of ions in the water.
Inputs:
Metalsⓘ
Select the metal you are extracting.
Conciⓘ
Initial Concentration: the metal concentration of the water flowing into the filter.
Concdⓘ
Desired Concentration: the metal concentration you want the filtration system to reduce the concentration down to.
Flowrateⓘ
Type here the rate of flow of water going into the filtration system.
MBR Volumeⓘ
Type here the volume of the part of the filtration system that houses the genetically modified E. coli.
Outputs:
Run Timeⓘ
The time the filtration system should be run for to reduce the concentration down to the desired concentration.
Meⓘ
Extracted Metal Mass: The total mass of all the metal that has been removed from the water in the run time.
Errors:
Errors will be displayed here after the model is run.
The output graph will be displayed here.
How the Model Works
Given a specific metal that the user wishes to filter from the water, they can input the concentration of metal ions in the water which they are filtering and the amount they wish to reduce the concentration down to. From this they have the ability to alter the flowrates and volume of the reactor vessel in order to find out the most suitable conditions for their purpose.
Our filtration system involves genetically modified E. coli that have a finite number of pili, therefore, a limited number of binding sites. This method of extraction leads to the inefficiency that as ions are bound by the pili there will be a reduction in the number of unbound pili remaining able to bind to the rest of the ions flowing through. This model incorporates this to find the most efficient time to change the filter, which helps to avoid water flowing through without enough pili to bind to the ions. It includes proportions based on the initial concentration of the metal ions in the water, the volume of the reactor chamber, and the flowrate of the system to help provide potential users with information to help implement this unique filtration system effectively.
Output concentration is the ion concentration which leaves the reactor.
Example Graphs and their Physical Explanations
Example 1
Using a small metal binding reactor, like the one in our prototype (which has a volume of 1.5L and a pump which allows for a flow rate of 0.07L/s) we will find that if we pump through a zinc solution of 10g/L with the intention of reducing the concentration to below 9g/L, then an inverse exponential decay curve is produced for concentration over time. This is because initially there are lots of unbound pili able to bind to the zinc ions and so a higher proportion of zinc ions is taken out of solution. As more pili become bound to the zinc ions, fewer collisions occur between the zinc ions and the unbound pili resulting in the zinc becoming bound less frequently. As such, the filter becomes less efficient and a lower proportion of zinc ions is taken out of solution until eventually the filter is not taking enough zinc out of the water to meet the desired concentration. A similar shape graph was obtained experimentally using a solution of mannose and unmodified MG-1655 E. coli.
Example 2
If, however, we had access to a metal binding reactor just ten times larger with a volume of 15 litres, then the model predicts that this graph would be obtained. The larger reactor needs replacing significantly less often. Initially the graph is almost horizontal with an output concentration of approximately 0. This is because there is such an abundance of pili in the reactor that the zinc ions are very likely to have a successful collision with an unbound pilus in the time it takes to get through the reactor. Thus, the vast majority of zinc ions entering the reactor become bound to pili. The exponential curve is also more pronounced. This is because the greater size of the filter and so the greater abundance of pili, means that smaller proportion of the pili need to be free in order to continue taking a sufficient number of zinc ions out of solution. As the proportion of free pili reduces, the number of successful collisions between pili and zinc ions decreases and so the number of free pili reduces far more slowly. This allows for an extended period of time when the filter is operating at a lower, but sufficient efficiency.
Example 3
On the contrary, if the volume of the metal binding reactor were reduced to just 1 litre, then the graph looks almost linear. This is because of the relative abundance of the zinc ions in comparison to the number of pili, means that each pilus is experiencing many more collisions per unit time and so the pili are all bound very quickly and there is no extended period of time where the reactor can operate at a lower efficiency, as in the other scenarios.
Assumptions
We made an assumption for the "number of pili per E. coli". The decision made was based on information from literature (Group, 2017) (Christof K. Biebricher, 1984). We used sensitivity analysis to check the effect that an inaccuracy in this value would have on the output of the model and we concluded that it would not have a large effect on the output. Additionally, there may be a range of number of pili per E. coli but due to the large number of E. coli considered in this model we can assume an average for this. Another assumption made was that the metal ions will have the same binding rate to our modified pili as mannose have to the pili on the MG-1655 E. coli. With further time, however, it would be possible to alter the model to take this into account by comparing it to relevant experimental data. Additionally, this model implicitly assumes that the metal binding reactor is a long, narrow pipe. As such, if this reactor was in the shape of a wide vat then the results of our model may be less accurate.
The Model's Evolution
The Interface
After receiving feedback on the user interface, changes were made to increase its user-friendliness. The first was enabling the user interface to produce a graph to provide the user with a visual representation of the models process and outputs. A further improvement made was including example values in the input boxes to provide the users with feasible values to input as, for example, typical concentrations of metal ions may not be known. Another alteration was enabling the interface to produce error messages to explain why certain input values were not feasible so that the user could alter the inputs accordingly.
The Outputs
Initially, the model produced the outputs ‘concentration left’ and ‘efficiency’ in addition to the current outputs of the model. It was decided that these were unnecessary as the concentration left would either be the same as desired concentration, if the volume of the reactor is large enough, or the desired concentration would never be reached therefore the error message would be displayed. Therefore, post this change in adding the error messages, this output became redundant. Furthermore, the efficiency of the system is merely based on the ratio of the input concentration to desired concentration, therefore, it was decided that this output was also not necessary for the user.
Curve Fitting
In order to increase the accuracy of the model, we performed experiments using a solution of mannose and unmodified MG-1655 E. coli in our metal binding reactor, similar to those detailed on the metal binding reactor page. By using a known initial concentration, recording the flowrate of the water and calculating the volume of the metal binding reactor, we were able to input the same values into our model.
We collected the outflow water into 100ml samples, then measured the concentration of mannose in each sample using High-Performance Liquid Chromatography. Using the known flowrate, we calculated the time intervals from the volumes. Graphs were then plotted of output concentration against time for both the models output concentration values at the timestep intervals and the experimental data.
This enabled us to calculate a scaling factor which takes into account the frequency of collisions and binding constants. The scaling factor is dependent on the flowrate and volume of reactor as we became aware of the fact that the collision rate is affected by the time that the ions will be inside the reactor, which depend on both of these factors. Comparing to the experimental data also made us aware that the number of E. coli per unit area calculated from the scanning electron microscope was out by a large factor. This is potentially due to the E. coli aggregating making it difficult to count the number of them in the image.
With this correction to our model, Figure 4 shows that the output from our model now agrees with 6 of the 8 data points obtained through experimental measurements.
Error Handling
Our model works by calculating how the ion concentration changes over discrete time steps, or iterations. During the development of this model it became apparent that, while running online, if the number of iterations exceeds a maximum value then an error occurs. As such, we have made some changes such that if the number of iterations exceeds 150,000 then the model starts again using larger time steps in order to reduce the number of iterations. This process repeats several times and allows for the model to predict outcomes for reactors which can run for up to 174 days.
Stakeholder Engagement
Veolia Water
We contacted Veolia Water Treatment Plant, a current treatment plant that removes metal ions from contaminated mine water, to obtain feedback on the model, particularly to find out whether there were other outputs that it would be useful for the model to produce. Their response was that we had covered similar bases to those required at their treatment plant.
In order to inform the future implementation of our research we ran the model with values provided by a representative from Veolia water treatment plant to see how our metal binding reactor system would work at an industrial scale. We were provided with data for the minewater intial concentrations, final effluent permit limit (desired concentration), the volume of their reactor chambers (equivalent to our MBR) and the flowrates of the system. The model was ran for the values provided for Zinc and Iron. The values for the other 4 metals were not used for the following reasons: the inital concentration of Nickel and Copper were already below the final effluent permit limit, Magnesium did not have a final permit effluent limit and no data was provided for Cobalt.
The input values for Zinc were: Initial Concentration 19.95 mg/L, Desired Concentration 0.02 mg/L. The input values for Iron were: Initial Concentration 82.20 mg/L, Desired Concentration 3 mg/L. The Flowrate and MBR Volume were the same for both metals, with the Flowrate input 600 L/s and MBR Volume input 1340000 L. Below is the graphs of output concentration against time for Zinc and Iron.
The graph indicates that our metal binding reactor system would be effective in extracting the metals down to their required concentrations from the minewater metal concentrations at an industrial scale.
Centre for Biomedical Modelling and Analysis
They suggested the idea of creating a user interface for the model to display on our presentation at the iGEM Jamboree. We built on this idea and integrated the model within a graphical user interface on our group website, as seen above.
Parameter Finding
| Parameter | Value | Source |
|---|---|---|
| Number of Pili per E. coli | 250 | Literature search (Group, 2017) (Christof K. Biebricher, 1984) |
| Number of E. coli per Unit Area | 9.22x1016 (m-2) | Scanning Electron Microscope imaging and comparison of model to experimental data. |
| Surface area to volume ratio of substrate | 2412.4 (m-1) | Measurements |
Sensitivity Analysis
We performed sensitivity analyses on all of the parameters in the above table and the duration of each time step in order to identify the parameters requiring further investigation. Using MATLAB, we ran the model varying one parameter at a time, keeping all others constant. The run time of the reactor was plotted against the relevant parameter to see the effect that the change had. The most sensitive parameter was the duration of each time step shown in Figure 6. At low time step durations, there is a comparatively high variation in the run time of the reactor and so is considered a sensitive region. Since we are using the smallest possible time steps in our model for the greatest degree of accuracy, this means that the time step which we are using falls in this sensitive region and requires attention. We used the experimental results described previously in order to make our model more accurate. One way we fitted the model to the experimental results was to tweak how the flow of ions interacts with the pili during each time step and as such, we are confident that the change during each time step is as precise as possible.
The following graphs show the other parameters which we tested. These graphs are very linear with no particularly sensitive regions, as such while we have been as accurate as possible with determining the values of the parameters, there are no regions with cause for concern.
References
Group, G., 2017. ETH Zurich. [Online]
Christof K. Biebricher, E.-M. D., 1984. F and Type 1 Piliation of Escherichia coli. [Online]
