Team:Exeter/Model
Modelling
We have produced this informative model, that provides knowledge into the scalability of our filtration system and incorporated 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 fluidised media reactor to use for their purpose.
It can be used to resolve the issue of knowing when the Fluidised Media Reactor is saturated as it informs when the E. coli in the Fluidised Media Reactor are too saturated to still be efficient, therefore, the filter needs changing. 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 turned 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.
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 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 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.
User Interface
We have a 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
Inputs:
Metals:
Select the metal you are extracting.
Initial Concentration:
Input the metal concentration of the water flowing into the filter.
Desired Concentration:
Input 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.
FMRVolume:
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.
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.
Example Graphs and their Physical Explanations
Using a small fluidised media 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 a typical logarithmic 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
If, however, we had access to a fluidised media 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 logarithmic 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.
On the contrary, if the volume of the fluidised media 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 much 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.
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.
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.
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. The other assumption made initially with the model was instantaneous binding. However, we compared the model with experimental data to take this into account along with other extraneous variables.
Parameter Finding
| Parameter | Value | Source |
|---|---|---|
| Number of Pili per E. coli | 250 | Literature search |
| Number of E. coli per Unit Area | 9.22x1016 (m-2) | Scanning Electron Microscope imaging and comparison of model to experimental data. |
| Surface Area of Each Torus | 0.36186 (m2) | Measurements |
| Volume of each Torus | 150x10-6 (m3) | Measurements |
Senstivity Analysis
We performed sensitivity analysis, on all of the parameters in the above table and the timestep that the model depends on, in order to identify the parameters requiring further investigation. Using MATLAB, we ran the model varying one parameter at a time over a range of values, keeping the other parameters constants. The output values were stored then plotted against the parameter to see the effect that a change in that parameter had on the output. See the graphs below which show the effect that a change the parameter ' E. coli per Unit Area' have on efficiency and output concentration.
Areas of high perturbation indicated the range at which parameters are sensitive because any change in parameters value has a larger effect on the output when in this range. In this graph we can see it occurs around 3X1011 and below. Originally the value calculated for 'Number of E. coli per Unit Area' was 5X1011 which is close to this region. This indicated that a small error in calculating this could have a large effect on the output. To deal with we sent off samples for imaging again and recalculated the parameter value to check for any errors, the recalculated value for this parameter is 9.22x1010.
Below are the graphs for the parameters ‘Surface Area of Torus’, ‘Volume of Torus’ and ‘Average Number of Pili per E. coli’ and ‘Timestep’.
We can see from the above graphs that the parameter values calculated for ‘Surface Area of Torus’, ‘Volume of Torus’ and ‘Average Number of Pili per E. coli’ are not in particularly sensitive regions. However, ‘timestep’ appears to affect the output value with greater perturbation as it increases in value. Therefore, it is important to keep the timestep value as low as possible to gain the most accurate value from the model, this is as expected.
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 fluidised media reactor, similar to those detailed on the Fluidised Media Reactor page. By using a known initial concentration, recording the flowrate of the water and calculating the volume of the Fluidised Media 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.
References
Group, G., 2017. ETH Zurich. [Online]
Christof K. Biebricher, E.-M. D., 1984. F and Type 1 Piliation of Escherichia coli. [Online]
