We created several modeling projects to support wet lab analysis efforts and hardware development. Our modeling projects focus on the ability of cyanobacteria to sequester carbon dioxide from the atmosphere through carbon fixation and photosynthesis, and their ability to generate electricity.
The main goals of our models:
The purpose of the model is to answer the following:
The growth of cyanobacteria is typically measured using optical densities (O.D.730). In the biophotovoltaic system, the cyanobacteria is printed on conductive ink and placed underneath a hydrogel soaked in BG-11 liquid media. While the cyanobacteria are still able to grow, analyzing their growth rate over time is difficult.
Typical Measures of Growth:
One of the most ubiquitous methods to measure growth of liquid cultures of bacteria is to analyze the turbidity, or O.D., of the sample. This is done using light scattering where measurements are made based on the light deflection of individual cells(1). Measurements of absorbance can be taken using a spectrophotometer. In cyanobacteria specifically, O.D. is taken at a wavelength of 730 nm.
Figure 1.1: Samples of Synechocystis sp. PCC 6803
The electrodes were printed using carbon based conductive ink, Nagase ChemteX CI-2042. This ink was used for the biophotovoltaic cell design as the experiments conducted by the hardware team showed this ink was the best for the viability of the cyanobacteria as well as having the least resistance of all of the conductive inks that were tested. The surface area of the anode was 1.5cm2 and the surface area of the cathode was 3cm2. Then, each cyanobacteria sample was screen printed in replicates of four onto the anode, with three layers being printed onto each. The final design of the printed voltaic cells are shown below in Figure 1.2. There is an overhang (0.25cm) of the cyanobacteria printed on the left side of each anode for the purpose of visualizing cyanobacteria on each sample. When printed onto the black ink, the cyanobacteria layer is not easily visible, which is necessary for the success of this calibration curve.
Figure 1.2: Set Up for Printed Cells
After all of the samples were printed, including a control that included CMC with no cyanobacteria, an image of the cells was taken using a Nikon D3200 camera. The photo was taken from a view directly above the samples. When taking the image it is imperative to minimize glare which can be done by diffusing light from above. Directly after imaging, a 1 inch by 1 inch square hydrogel soaked in 2% NaCl BG11 Media was placed on top of the electrodes (Figure 1.3). The hydrogel is used as a salt bridge to serve as a proton gradient in the biophotovoltaic cell. Additionally, the BG11 Media serves as nutrients for the cyanobacteria to grow, so soaking the hydrogels in this media keeps the printed cells alive. The cells were then imaged again using the camera.
Figure 1.3: Voltaic Cells with Hydrogels
This figure depicts what the cells looked like with the hydrogels placed on top.
An additional experiment was conducted to create a similar curve for the color versus a scale of O.D.730 of cyanobacteria in liquid culture. In this experiment we used a Kimtech Science Kimwipes box as a standard. This is ubiquitous in many labs and easy to acquire, so the wet lab and modeling teams decided it would be a practical standard to use. The purpose of the Kimwipe box is to serve as a standard so that another lab would be able to use this calibration curve as a measurement system for liquid culture growth. This is because the slope of the line of best fit will be the same for any conditions; however, the y-intercept will change based on conditions such as lighting or image quality. Therefore, by including the Kimwipe box as a standard they can adjust their y-intercept accordingly. In our standard curve, the O.D.730 of the Kimwipe box is 1.252. This is described more in depth in the measurement section of our Wiki. Additionally, six, 1mL samples of cyanobacteria at different O.D.730 in cuvettes were used. The concentration of each is noted in Table 1.1. This set of cuvettes was imaged using the Nikon D3200 camera against a white paper background, analyzed using ImageJ.
Table 1.1: O.D.s of Liquid Culture Calibration Curve
| Native Culture | 2X | 0.5X | 0.25X | 0.125X | 0.0625X | |
|---|---|---|---|---|---|---|
| O.D. | 0.939 | 1.321 | 0.560 | 0.298 | 0.162 | 0.078 |
The following are the images (Figure 1.4-1.6) from experimentation that were used for ImageJ analysis.
Figure 1.4: Printed cyanobacteria with no hydrogel
Figure 1.5: Printed cyanobacteria with hydrogel
Figure 1.6: Liquid cyanobacteria at different O.D.s compared to Kimwipe box
Analysis was performed in ImageJ, and the following results were collected. To analyze the data in ImageJ, the images were first converted to 16-bit–or black and white–images. Next, the measurements were set to analyze the mean gray value. The mean gray value measures the average grayness of the image. The overhang of each of the cells was selected and the measurements for this area were quantified. After each rectangle was quantified, the data was transferred to Excel. The average “mean gray value” was calculated for each replicate as well as the standard deviation. Additionally, Excel was used to calculate the best-fit linear regression lines for each of the calibration curves and the R2 value for each.
Graph 1.1: Printed cyanobacteria, no hydrogel
Table 1.2: Printed cyanobacteria, no hydrogel
| Trend Line | y=-0.4314x + 89.011 |
| R2 | 0.918 |
| r | -0.958 |
The high coefficient of determination of 0.918 indicates that 91.8% of the variance in the O.D.730 may be determined by the average measure of grayness of the samples. The high r value (correlation coefficient) of -0.958 shows that there is a strong, negative, linear correlation between the two variables and that average grayness is a good indicator of O.D.730. Additionally, the slope of the trend line indicates that as the average measure of grayness increases by 1, the O.D.730 of the sample decreases by 0.431.
Graph 1.2: Printed cyanobacteria, with hydrogel
Table 1.3: Printed cyanobacteria, no hydrogel
| Trend Line | y=-0.7456x + 104.78 |
| R2 | 0.55 |
| r | -0.7416 |
The results of the experiment with cyanobacteria printed onto the electrodes with the hydrogel placed on top show that the average measure of grayness is a moderately good predictor of O.D.730. The coefficient of determination (R2) is 0.55 which means 55% of the variance in O.D.730 can be explained by the average measure of grayness of the samples, which is not incredibly high. However, the correlation coefficient is -0.7416 which indicates that there is a moderately strong, negative, linear relationship between the average measure of grayness and the O.D.730 when the cells are under the hydrogel. On the other hand, the standard deviation bars show that there is a lot of variance in the average values for grayness, which decreases confidence in the prediction of this particular calibration curve. The trend line indicates that as the average measure of grayness increases by 1, the O.D. decreases by 0.7456.
Graph 1.3: Cyanobacteria liquid culture calibration curve
Table 1.4: Cyanobacteria liquid culture calibration curve
| Trend Line | y=-0.012x+1.748 |
| R2 | 0.9908 |
| r | -0.995 |
The calibration curve for the liquid cultures of cyanobacteria has a very high R2 value of 0.9908, which means that 99.08% of the variance in the O.D.730 can be explained by the average measure of grayness. In addition to this, the correlation coefficient of -0.995 indicates that there is a very strong negative, linear relationship between the average measure of grayness and O.D.730. Overall, these values show that the average measure of grayness using ImageJ is a good predictor of O.D.730. Also, the trend line of this relationship shows that as the average measure of grayness increases by 1, the O.D.730 decreases by 0.012.
The goal of this model was to create a measurement that would help to identify growth of the cyanobacteria once they were printed on paper and producing electricity. This was accomplished through the use of imaging and image analysis using ImageJ.
For the model of the printed cyanobacteria with no hydrogel there was a significantly strong correlation between the average measure of grayness found in ImageJ and the O.D.730, as the correlation coefficient for the linear regression was found to be -0.958. This indicates that the use of image analysis for printed cells is an accurate method to determine the O.D.730 of a sample if a spectrophotometer is not available. However, the standard deviations of the average grayness values were relatively high as shown by the standard deviation bars in the graph. To increase further confidence in this model, one should increase the number of replicates in order to decrease the standard deviation. This method may be more useful than a spectrophotometer because they typically can only reliably measure up to an O.D.730 of about 3(2), and this model shows a strong relationship with the average grayness measured and O.D.730 up to an O.D.730 of 40. The model’s ability to measure O.D.730 up to 40 is a necessity for the growth measurement with the printed cyanobacteria as our hardware is printing at very dense starting points (O.D.730 20). Therefore, to measure growth past this point we would need to use a measurement tool that can accurately measure at O.D.730 higher than 20. Another application of this particular model could be to measure the O.D.730 of solid cultures of cyanobacteria by taking a picture of a plate and analyzing the image to detect the average grayness and predict the O.D.730 of the sample.
The second model which uses the average grayness measurement to predict the O.D.730 of the cyanobacteria while the printed cells are under the hydrogel proved to be a moderately accurate measurement system. However, the correlation coefficient (-0.7416) was significantly lower than the correlation coefficient of the other models, and a higher correlation would be preferred as an accurate measurement of the O.D.730 is important to understand the impacts of electricity production on growth of cyanobacteria. Aside from the moderately strong correlation coefficient, the average grayness values for each sample had very high standard deviations which is not ideal and decreases confidence in the accuracy of the model.
TThe third model proposes a calibration curve to measure the O.D.730 of liquid cultures using average grayness measurements. This model may be utilized by many labs with limited resources to measure O.D.730 of cyanobacteria liquid samples. If a lab does not have access to a spectrophotometer, they would need an alternative method to detect growth of their cyanobacteria. This is an effective, efficient, and inexpensive alternative to other types of lab equipment. The high correlation coefficient of -0.995 indicates that the average grayness of a sample is a significant indicator of O.D.730.
This method of imaging the printed cells and completing image analysis using average measures of grayness provides a unique and effective model to predict O.D.730 of the printed cyanobacteria. Given the nature of the project and the necessity to measure growth of cyanobacteria while it is printed in the biophotovoltaic system and producing electricity, this measurement will prove to be incredibly useful. However, in order to accurately measure cyanobacteria growth within our hardware system a model with a higher correlation coefficient would be desirable. A way this may potentially be done would be by attempting to analyze the image in a different way in ImageJ. One way to do this may be using HSB analysis which controls for the hue, saturation, and brightness of the image. If HSB were to be used, all of the green in the image may be selected and the area of the “green” could be measured for all of the cells. When the O.D.730 is higher, it would be predicted that the area of green would be greater than when the cyanobacteria printed have a lower O.D.730. This method of analysis would allow us to select all of the area of the cyanobacteria that was printed underneath the hydrogel which is important because sometimes the bacteria spreads when the hydrogel is placed on top. One way the hardware team has tried alleviating the spread of the bacteria under the hydrogel is by increasing the concentration of CMC in the bio ink. This makes the ink thicker and typically, the bacteria will then stay in the printed area more effectively. However, due to the spread of the cyanobacteria under the hydrogel, HSB may be a better analysis method as for the grayscale analysis to be controlled, the same area must be selected for each cell, which would not be an accurate representation of the O.D.730 if the bacteria was outside of the bounds of the selected area.
Another potential future direction for this model would be to show the condition of the cultures using calibration curves as well. According to Schulze et al. (2011), one way to measure the health of a cyanobacteria culture is to use the chlorophyll concentration or the chlorophyll concentration divided by the O.D.730(3). Chlorophyll a concentration was measured at 450 nm and 670 nm. Given this information and the method determined through this model, new calibration curves could be developed to measure the health of the cyanobacteria when printed and in liquid cultures without the use of a spectrophotometer. This is useful for the analysis of the biophotovoltaic cell to evaluate whether or not the cyanobacteria begin to die when they are producing electricity because of added metabolic stress or other unknown reasons.
A final application for this model would be to predict cell count of the cyanobacteria based on the ImageJ analysis. This can be done as the conversion of O.D.730 to cellular concentration.(O.D.730 of 0.25 = 1x108 cells/mL). Using this conversion factor the model may be improved by converting the O.D.s to the cell count to additionally show cell growth.
Overall, this model utilizes a widely-used image analysis program, ImageJ, to create calibration curves that measure the growth of the cyanobacteria in printed conditions (both with and without a hydrogel) and while in a liquid culture. This is not only useful in the dimensions of this project which needs to measure the growth of cyanobacteria while in a biophotovoltaic system, but it is also a useful measurement tool for other labs that may not have access to a spectrophotometer, but need to measure cell growth of cyanobacteria in a simple, efficient, and cost effective way.
The purpose of this model is to develop an electrode pattern to optimize electricity production in the biophotovoltaic cell. The model aims to answer the following question:
Through testing variables such as length, width, and concentration difference of electrodes, and by using an analytical expression that describes the dynamics of microbial fuel cells, this model helps to describe the optimal design for electricity production.
Microbial Fuel Cells
Microbial fuel cells are bio-electrochemical devices with microorganisms on the electrodes that are used to generate and maintain an electrochemical gradient to produce electrical power (Figure 2.1). Unlike a battery, the microorganisms in microbial fuel cells can continue to produce electricity with the redox reactant in the environment that contains nutrition and organic substrate (ex. wastewater). In the general setup of the microbial fuel cells, the microorganisms are placed in the anaerobic chamber with an anode, and release electrons by breaking down substrates such as water waste. This releases electrons to the anode. The anode then transfers the electrons to the cathode, producing electricity. For the microorganisms to generate electricity, electrons that are produced intracellularly need to be transported to the electrode. There are several ways to achieve this process in microbial fuel cells. In a mediated microbial fuel cell, microorganisms can either generate the natural mediator or an artificial mediator will be added to achieve the transfer of electrons. In non-mediated microbial fuel cells, there are special microorganisms with an external membrane called conductive pili, which is a membrane that can directly transfer electrons out to the electrode(4)
Figure 2.1: Graphical representation of microbial fuel cells(5).
Biophotovoltaics
In biophotovoltaic cells, a subset of microbial fuel cells, photosynthetic organisms are the microorganisms used. These photosynthetic bacteria capture solar energy and convert it to usable energy. When the bacteria capture solar energy, the cell splits water molecules in photosystem II. This occurs in the thylakoid membrane of the cells. Some of these electrons are then transferred to the anode in which the electron flow is then powered by an external circuit to the cathode, where oxygen reduction occurs. The main benefits of this BPV system as compared to other photovoltaic devices are the lack of pollution and reduced energy consumption in the manufacturing process(6). The major pitfall associated with the BPV system is the low current output, which must be fine-tuned before large-scale integration.
Figure 2.2: Biophotovoltaics system(7)
This image depicts the function of biophotovoltaic cells and the process of water splitting through photosynthesis which provides electrons that go towards creating the “bioanode” and powering the chemical cell.
Hardware Design
An important part of our team's hardware design is based on a printed biophotovoltaic cell. The anode and cathode electrodes are printed parallel to each other with carbon conductive ink and the cyanobacteria are printed in a dense “bio ink” mixed with Carboxymethyl Cellulose on top of the anode. A hydrogel soaked in BG-11 liquid media with 2% NaCl is placed on top of the biophotovoltaic cell to serve as a salt bridge and to help sustain life for the cyanobacteria. The design is made up of several components of a typical biophotovoltaic cell, but also introduces a unique 2D design that is able to be printed. Sawa et al. serves as an example of this printed biophotovoltaic design, which is shown in Figure 2.3(8). When designing the electrodes for the biophotovoltaic cells, the article states the main focus was the dimension of the electrodes and the distance between them. The zigzag pattern was chosen to maintain a compact size while allowing for ion conduction between the anode and cathode. Sawa et al. (2017) also stated that the concentration of the cathode was made larger than that of the anode to increase the cathode’s exposure to oxygen and decrease potential catalytic limitations(8). The aim of this design is to create an efficient and cost effective process for producing carbon-negative energy using cyanobacteria. Printing several biophotovoltaic cells on paper in an efficient screen printing process can lead to greater current output, which is a downfall in the general design of biophotovoltaic cells. However, the 2D aspect of the electrodes poses a new challenge to maximizing the output of electricity compared to the traditional designs of biophotovoltaics.
Figure 2.3: Design of Printed BPV System from Sawa et al.(8)
1. Printed cyanobacteria; 2. Printed anode; 3. Printed cathode; 4. Paper substrate; The design in Sawa et. al consisted of an anode with a surface area of 1.36cm2 and a cathode with a surface area of 2.73cm2
To test for the optimal design of the printed anode and cathode, the parameters the modeling and hardware team chose to observe were length, distance between the electrodes, distance between the interactions, and concentration difference between anode and cathode.
Both subteams collaborated to test for the parameters used in the model. To conduct the experiment, the hardware team created multiple designs of electrodes, which are shown in Tables 2.1-2.4 below. For the electrode experiment which was testing optimal lengths, cells of different lengths ranging from 2cm to 6cm were printed in replicates of five measuring the resistance of the anode and cathode. The current of the circuit was also measured by inputting 5V into the system. Similar experiments were repeated for the distance between the cells (ranging from 0.25cm to 1.5cm), the difference in concentration of the cells (ranging from 1:1 to 3:1), and the distance from one interaction (spike of one zigzag) to the next (ranging from 0.25cm to 2cm).
The modeling team utilized the data collected from the hardware experiments to find the best-fit curves by testing different variables of the electrodes. The best-fit curves were then considered together,relating to the maximum electricity production. Next, the optimal design of the 2D electrode was generated.
Table 2.1: Experiment 1, Length of Electrodes
| Shapes | Length in cm | Surface Area: Anode in cm2 | Surface Area: Cathode in 2 | Average Resistance in Ohms (SD) | Average Current in mA (SD) | |
|---|---|---|---|---|---|---|
 |
2.0 | 1.0 | 1.0 | 53.204 (1.915) | 52.36 (6.830) | |
 |
2.5 | 1.25 | 1.25 | 80.908 (4.391) | 148.946 (11.948) | |
 |
3.0 | 1.5 | 1.5 | 81.64 (7.443) | 106.279 (12.916) | |
 |
3.5 | 1.75 | 1.75 | 112.198 (6.015) | 87.336 (13.217) | |
 |
4.0 | 2.0 | 2.0 | 102.168 (4.260) | 63.856 (7.493) | |
 |
4.5 | 2.25 | 2.25 | 113.83 (7.216) | 79.153 (9.397) | |
 |
5.0 | 2.5 | 2.5 | 112.884 (6.857) | 73.590 (12.885) | |
 |
6.0 | 3.0 | 3.0 | 158.989 (5.628) | 71.914 (9.118) | 6 |
Note: The width of anode and cathode are both 0.5cm.
Graph 2.1: The linear relationship between length and resistance built by Excel.
Graph 2.2: The linear relationship between length and Current built by Excel.
Table 2.2: Experiment 2, Distance between electrodes
| Shapes | Distance Between Electrodes (cm) | Surface Area: Anode | Surface Area: Cathode | Total Resistance in Ohms (SD) | Average Current in mA (SD) |
|---|---|---|---|---|---|
 |
0.250 | 1.5 | 3 | 80.498 (10.206) | 68.346 (6.838) |
 |
0.375 | 1.5 | 3 | 60.132 (6.139) | 90.846 (6.790) |
 |
0.500 | 1.5 | 3 | 36.6605 (6.523) | 94.905 (33.731) |
 |
0.625 | 1.5 | 3 | 70.57 (5.216) | 79.936 (8.405) |
 |
0.750 | 1.5 | 3 | 54.588 (3.605) | 101.464 (6.614) |
 |
0.825 | 1.5 | 3 | 89.918 (5.518) | 61.318 (2.686) |
 |
1.00 | 1.5 | 3 | 58.03 (5.738) | 98.874 (14.141) |
 |
1.50 | 1.5 | 3 | 86.788 (7.886) | 66.476 (4.988) |
Note: The Length of the anode and cathode is 3cm, the width of the anode is 0.5cm, and the width of the cathode is 1cm.
Graph 2.3: The linear relationship between the distance between electrode and resistance built by Excel.
Graph 2.4: The linear relationship between the distance between electrode and Current built by Excel.
Table 2.3: Experiment 3, Interactions between electrodes
| Shapes | Distance Between Interactions (cm) | Surface Area: Anode | Surface Area: Cathode | Total Resistance in Ohms (SD) | Average Current in mA (SD) |
|---|---|---|---|---|---|
 |
0.250 | 1.5 | 3 | 78.428 (9.501) | 83.27 (9.099) |
 |
0.500 | 1.5 | 3 | 87.768 (8.578) | 65.022 (10.638) |
 |
0.750 | 1.5 | 3 | 88.93 (10.546) | 67.852 (6.331) |
 |
1.00 | 1.5 | 3 | 76.55 (9.716) | 77.456 (6.921) |
 |
1.50 | 1.5 | 3 | 62.75 (4.578) | 87.574 (7.428) |
 |
2.00 | 1.5 | 3 | 59.36 (5.342) | 89.316 (7.252) |
The Length of the anode and cathode is 3cm, the width of the anode is 0.5cm, and the width of the cathode is 1cm.
Graph 2.5: The linear relationship between the distance between interaction and resistance built by Excel.
Graph 2.6: The linear relationship between the distance between interaction and Current built by Excel.
Table 2.4: Experiment 4, Difference in concentration between cathode and anode
| Shapes | Ratio of Cathode to Anode | Surface Area: Anode | Surface Area: Cathode | Average Resistance C:A in Ohms (SD) | Average Current in mA (SD) |
|---|---|---|---|---|---|
 |
1.00 | 1.5 | 1.50 | 67.477 (5.928) | 74.15 (5.620) |
 |
1.25 | 1.5 | 1.875 | 54.61 (7.696) | 92.265 (6.812) |
 |
1.50 | 1.5 | 2.25 | 43.9325 (1.551) | 107.0725 (3.180) |
 |
1.75 | 1.5 | 2.625 | 45.965 (4.692) | 107.16 (8.597) |
 |
2.00 | 1.5 | 3.00 | 46.92 (9.535) | 103.8025 (22.660) |
 |
2.50 | 1.5 | 3.75 | 61.4925 (10.045) | 84.45 (6.206) |
 |
3.00 | 1.5 | 4.50 | 42.785 (4.671) | 108.535 (7.024) |
Note: The Length of the anode and cathode is 3cm, and the width of the anode is 0.5cm.
Graph 2.7: The linear relationship between the surface area ratio of cathode to anode and resistance built by Excel.
Graph 2.8: The linear relationship between thethe surface area ratio of cathode to anode and Current built by Excel.
Parameters
To achieve our goal of modeling the best electrode design, we tested several different parameters:
Length = L (cm) → To test the changing of resistance, we measured the length of our electrode.
Distance between the electrodes = D (cm) → To test how distance affects the electricity output, we measure the distance between two close edges of the anode and cathode.
Distance between the interaction of electrodes = I_D (cm) → To test the interaction of electrodes, we designed a zigzag electrode and measured the distance between each peak of each anode and cathode.
The surface area ratio of anode and cathode = ratio (cm2/cm2) → To test how the concentration of anode and cathode affects the electricity output, we designed our electrodes at different surface areas and calculated the ratio between them.
Model 1 Set-up: Polynomial Regression
For our three independent variables, the distance between the electrode and anode (D), the distance between the interaction of electrode (I_D), and the surface area ratio of the cathode to the anode, we choose to build a regression model.
Set-up for model:
We constructed our model using R Studio, Excel, and Python. For each independent variable, we filtered the entire data set to focus on one variable individually. The average power (P) of 5 trials of electrodes was calculated based on Joule’s Law (Equation 2.1) and total resistance (RT) was calculated based on the Series Resistance formula (Equation 2.2). We use power as output to test different regression models to find the best fit for each variable.
Joule's Law P=I2RT (Equation 2.1)
P = power (W)
I = current (A)
RT = Resistance (Ω)
Series Resistance RT = Ra + Rc (Equation 2.2)
RT = total resistance (Ω)
Ra = resistance of anode (Ω)
Rc = resistance of cathode (Ω)
Distance between interaction (I_D):
We used Table 2.5 to test I_D as a variable, with the surface area ratio of the cathode to the anode (ratio) and distance (D) kept constant. This resulted in a linear regression with an R2 of 0.016, a logistic regression with an R2 of 0.009, and a polynomial regression with an R2 of 0.863, indicating that the best regression model is the 3rd-degree polynomial regression. This is due to the model having the greatest R2 value.
Table 2.5: The power (P) is the output, with the distance between interactions (I_D) as a variable, while the distance (D) and the surface area ratio of the cathode to the anode (ratio) remain constant. Each row in the table represents the average power output (P) from five identical trials under the same conditions.
Graph 2.10: 3rd degree Polynomial regression build by R studio, P = power (w), I_D = distance between interaction (cm)
Surface area ratio of the cathode to anode (C:A):
We used the same method and applied different regression methods using Table 2.6. Resulting in linear regression having an R2 of 0.446, logistic regression having an R2 of 0.660, and the 3rd-degree polynomial regression providing the best fit with an R2 value of 0.972.
Table Table 2.6: The power (P) is the output, with the surface area ratio of the cathode to the anode (ratio) as a variable, while the distance (D) and distance between interaction (I_D) remain constant. Each row in the table represents the average power output (P) from five identical trials under the same conditions.
Graph 2.9: 3rd-degree Polynomial regression built by R studio, P = power (w), ratio = the surface area ratio of the cathode to the anode
Distance (D):
We used Table 2.7 to test multiple regression including linear regression (R2 = 0.006), logistic regression (R2 = 0.010), 5th-degree polynomial regression (R2 = 0.286), splines regression (R2 = 0.180), and segmented regression (R2 = 0.122). We eventually found that when span = 0.5, LOESS (Locally Estimated Scatterplot Smoothing) regression is the best-fitting model. We used span = 0.5 and span = 0.7 to test the fitness of the model. Span is the proportion of the total data points used in each local regression. It controls the level of smoothing. When the span is equal to 0.5, the R2 is 1 and when the span is equal to 0.7 the R2 is 0.669. However, our data set provides 20 replicates when the distance is 0.5cm, therefore, we could not integrate the LOESS regression model with the other two polynomial regressions of the surface area ratio of the cathode to the anode and distance between interactions together. Therefore, we decided not to include distance as a parameter in our final 3D Polynomial regression model.
Table 2.7: The power (P) is the output, with the distance (D) as a variable, while the distance between interactions (I_D) and the surface area ratio of the cathode to the anode (ratio) remains constant. Each row in the table represents the average of power output (P) from five identical trials under the same conditions.
Graph 2.11: LOESS regression build by R Studio when span = 0.5, P = power (w), D = distance between cathode and anode
Graph 2.12: LOESS regression build by R Studio when span = 0.7, P = power (w), D = distance between cathode and anode
Since LOESS regression at span = 0.5 has better R2 than span = 0.7, we choose span = 0.5 as our final best-fit model for the distance between electrodes.
Model 2: Balance the Surface area and Resistance
Unlike the 3D electrode, the surface area of the 2D electrode is important because it determines how much cyanobacteria can be printed on the electrode, specifically the anode, to generate power. However, as the surface area increases, the resistance of the electrode will decrease which will cause the output power to decrease as well. Therefore, we want to find a balance between the surface area and resistance to maximize the power as well as minimize the resistance. This is shown in the following equation for the resistance of 3D electrodes (Equation 2.3). As the length increases, the resistance will also increase.
R = ρ/AL (Equation 2.3)
R = Resistance (Ω)
ρ = Resistivity (Ω⋅cm2)
A = Surface Area (cm2)
L = Length
We plotted the length versus total resistance which confirmed the prediction that the resistance would increase as the length increased. The total resistance was calculated based on series resistance (Equation 2.2).
Table 2.8: The average resistance is the dependent variable, and the length (L) is the independent variable for Table 2.8. Each row in the table represents the average resistance (Ohms) of the anode and cathode (the surface area and length of the anode and cathode are identical in this experiment) from five identical trials under the same conditions.
Graph 2.13: The linear trend of average resistance (Ohms) versus length (cm) was built by Excel. The linear trend line is y = 11.007x+9.001 and gives an R2 of 0.8869.
In order to achieve this goal, we generated a graph that has length on the x-axis with power and surface area on the y-axis. The average power (P) of five trials of electrodes was calculated based on Joule’s Law (Equation 2.1) and total resistance (RT) was calculated based on the Series Resistance formula (Equation 2.2).
Table 2.9: The power (P) and area (cm2) are the dependent variables, and the length (L) is the independent variable. Each row in the table represents the average of power output (P) from five identical trials under the same conditions. Graph 2.14: The linear relationship between length and surface area (red) as well as the power-law relationship between length and power output (blue) were built by Excel. The surface area follows a linear trend y1 = 0.5x, which gives an R2 of 1. The power output follows a power-law trend, y2 = 1.703x-0.755 (R2=0.7812).
Model 1: 3D Polynomial Surface Model
Graph Graph 2.15: The 3D Polynomial Surface Model built by Python, with the distance between the interaction (I_D) on the x-axis, the surface area ratio of the cathode to the anode (ratio) on the y-axis, and power on the z-axis.
In order to find the maximum power output, the surface area ratio of the cathode to the anode (ratio) and the distance between interactions (I_D) are used as variables. We integrate the two best-fit polynomial regressions that we found for these two independent variables and used Python to build a 3D Polynomial Surface Model. In this model, the distance between interactions (I_D) was on the x-axis, the surface area ratio of the cathode to the anode (ratio) was on the y-axis, and power (P) was on the z-axis. Based on this model, the maximum predicted power (P) is approximately 1.49 W, when the distance between interaction (I_D) = 0.65cm and the surface area ratio of cathode to the anode (ratio) = 3.00.
To find the best distance between the electrodes, we input the predicted max power output into our LOESS regression for distance. When the LOESS regression for D has span = 0.5, the predicted distance for a power of 1.49 W is 0.625 cm.
Model 2: Find the Best Length and Surface Area
Based on our second model, since the Area = Length*Width, we can generate a perfect linear equation that
y1 = 0.5x
y1 = Area, x = length (cm)
The trend line for the power-law graph of power versus length is
y2 = 1.703x-0.755 y2 = power (w), x = length (cm)
To find the best balance of power and length, we set these two equations together, which gives a final length of x=1.35cm.
1.703x-0.755 = 0.5x
x=1.35
The primary goal of this model is to find the best 2D electrode design to produce a maximum power output. The parameters we tested include the length of the electrode, the distance between electrodes, the distance between interactions of the electrode, and the surface area ratio of the cathode to the anode. The parameters were collected through experimentation that was completed by the hardware team. They tested several different designs of electrodes which varied in length, distance, interaction distance, and surface area difference. The data collected from these experiments was the total resistance of the cells and the current through the cells when 5V was input into the circuit.
The model determined that the maximum power output occurs when the surface area ratio of the cathode and anode is 3, the distance between the interactions is 0.65cm, the distance between the electrodes is 0.625 cm, and the length of the electrodes is 1.35cm. The finalized design of the electrode pattern is shown in Figure 2.4. The results of the model were used to choose the optimal experimental conditions for the electrodes in the biophotovoltaic cell design.
Figure 2.4: Final electrode design
This is the final design for the electrodes. The difference in concentration between the anode and cathode is 3, the distance between the interactions is 0.65cm, the distance between the electrodes is 0.625cm, and the length of the electrodes is 1.35cm. The design shows four cells printed side by side.
One limitation of the model is that we were not fully able to integrate the distance model with the other two parameters because we had 20 replicates when the distance was 0.5cm, but did not have 20 replicates for the other parameters. Additionally, this model only considers four parameters related to the power output, when there may be several other parameters that are necessary to consider to optimize the design such as electrode material or light intensity.
In our future work, we aim to experimentally apply the optimal design of electrodes that was determined by our model. This will help to test the reliability and accuracy of the predictions and ensure practical applicability. Additionally, to improve the model the team could collect more data and research other methods to effectively combine the results of the distance model with the polynomial regression model for the interaction distance and C:A surface area ratio. We also plan to explore a more in-depth model of the relationship between the surface area of the anode and power output. One way this could potentially be done is by integrating features of models for photovoltaic solar cells.
The main goal of this model is to describe the dynamics of cyanobacterial growth. This will allow the members of the wet lab team to observe the efficiency of carbon sequestration in both the wild-type and modified strains of cyanobacteria. It will also enable the wet lab team to explore how changing the rate of different variables, such as inorganic carbon or light intensity impacts the growth of the cells.
Through the use of a series of ordinary differential equations which assist in modeling the dynamics of cyanobacteria, these cellular processes may be explored.
This project creates a carbon negative energy source by using cyanobacteria and their photosynthetic properties to produce electricity. In order to do this it is necessary to understand the dynamics of cyanobacteria and the cellular processes of photosynthesis and the Calvin-Benson cycle. In cyanobacteria, the fixation of inorganic carbon through photosynthesis leads to increased biomass (9). Carbon transport is mediated through several transport proteins. There are three bicarbonate transporters in the cell: BCT1, BicA, and SbtA(9). There are also two CO2 transporters: NDH-I4 and NDH-I3(10). These transporters allow inorganic carbon to pass across the cell membrane. A mechanism called the CO2 concentrating mechanism is contained inside the carboxysome of cyanobacteria, which is a “bacterial microcompartment” that has a selectively permeable membrane shell(11). The purpose of the CO2 concentrating mechanism is to optimize CO2 fixation which is done through ensuring that there is more CO2 in the carboxysome than can be used by carbon-fixing enzymes(12). In optimizing the fixation of carbon dioxide in the cell, the process of photorespiration and oxygen binding to the active site of RuBisCo is inhibited(13).
Cyanobacteria are responsible for more than 25% of the world’s carbon fixation, which occurs through photosynthesis(14). In the process of photosynthesis cells use energy from light to convert water and carbon dioxide into organic compounds like sugars(15). The Calvin-Benson cycle is the main part of the carbon fixation process. In the Calvin-Benson cycle, carbon binds to the active site of RuBisCo and produces organic carbon, some of which goes towards the growth of the cells and some of which goes back towards the regeneration of ribulose 1,5-bisphosphate(16). This model explains the dynamics of cyanobacterial growth by including the transport of inorganic carbon, the conversion of carbon in the CO2 concentrating mechanism, and the fixation of CO2 through the Calvin-Benson cycle. The model also includes the dynamics of photosynthesis as it considers both light and inorganic carbon as potential limiting factors to the growth of the cell.
This model accurately depicts the dynamics of carbon sequestration and cellular growth of cyanobacteria. Our team developed a set of ordinary differential equations where each state variable represents a different species of carbon throughout the carbon sequestration process including the transport of inorganic carbon in the environment through bicarbonate and CO2 transporters, the conversion of inorganic carbon to CO2 ready to bind to RuBisCo through carbonic anhydrase, the binding of CO2 to RuBisCo, the regeneration of RuBisCo from the organic carbon produced in the Calvin-Benson cycle, and the production of biomass from the organic carbon produced in the Calvin-Benson cycle.
Figure 3.1: Network diagram of cyanobacterial dynamics
The following is the set of equations used to describe these cellular processes:
Several of the parameters were identified by finding the corresponding values in literature, which are shown in Table 3.1. In addition to these parameters, many more variables were developed based on the cyanobacterial dynamics in the paper, “Mathematical Modeling of Cyanobacterial Dynamics in a Chemostat” by Fadoua El Moustaid. This article explores how cyanobacteria use light and inorganic carbon as nutrients to grow. The following is a list of parameters that were derived from this source.
MIC - represents the Monod function which is used in the function for delta which is the growth rate when inorganic carbon is limiting.
MIC=[CinF]/K+CinF
delta- represents the cyanobacterial growth rate when inorganic carbon is a limiting factor
delta=rmax*MIC
v - the photons that are used to break up water in photosystem II which are used in the calculation for electrons.
v=alpha*vlight*A
elec - represents the electrons in the equation that are used to calculate e
elec = Yv*v
e - this function describes cyanobacterial growth when light is the growth limiting factor
e = elec/b[CinF]
r - describes the overall growth rate of cyanobacteria
r = min(e, delta)
Table 3.1: Parameters for CO2 Sequestration Model
Equation 3.1
dCin/dT=kin*[Cout]-dc1[Cin]-kc1[Cin]
The first term of this equation represents the transport of inorganic carbon from the environment into the cell. There are several transporters of cyanobacteria. Some of which transport bicarbonate into the cell and some of which transport CO2 into the cell. For the purpose of modeling the general dynamics of cyanobacteria, this model assumes a general transport rate of inorganic carbon. The parameter ‘kin’ represents the transport rate of inorganic carbon.
The second term of this equation represents the degradation of the inorganic carbon that was transported into the cell. This inorganic carbon is found in the cytoplasm before the carbon is transported into the carboxysome. The parameter ‘dc1’ represents the degradation rate of inorganic carbon in the cytoplasm.
The third term of this equation represents the rate at which inorganic carbon is being converted to CO2 that is ready to bind to RuBisCo. This is done with the enzyme carbonic anhydrase. Therefore, the parameter ‘kc1’ used in this equation is described by the rate of carbonic anhydrase that is found in literature.
Equation 3.2
dCin*/dT=kc1[Cin]-dc2[Cin*]-kc2[Cin*]+[CinF]*kc3
The first term of this equation describes the rate at which the inorganic carbon is being converted to CO2 ready to bind to RuBisCo.
The second term of this equation represents the degradation rate of Cin* which is the carbon that has been converted by carbonic anhydrase and is inside the carboxysome. The parameter ‘dc2’ represents this degradation rate.
The third term of the equation describes the process of the converted inorganic carbon binding to the active site of RuBisCo to produce organic carbon. The value for the parameter ‘kc2’ is the rate of the enzyme RuBisCo.
The fourth term of this equation represents the carbon produced in the Calvin-Benson cycle which goes back into the cycle to regenerate RuBisCo. The value of the parameter ‘kc3’ is the rate at which the regeneration of RuBisCo occurs.
Equation 3.3
dCinF/dT=kc2[Cin*]-dc3[CinF]-[CinF]*kc3-[CinF]*r*N*pf
The first term of the equation represents the binding of inorganic carbon to RuBisCo.
The second term of the equation represents the degradation of the carbon that is produced by the Calvin-Benson cycle, most of which is still inside of the carboxysome and is therefore assumed to have the same rate of degradation as the parameter dc2. This degradation is represented by the parameter ‘dc3’.
The third term of the equation describes the rate at which RuBP is regenerated with the carbon molecules that are produced through the Calvin-Benson cycle.
The last term describes the kinetics of cyanobacterial growth.
Equation 3.4
dN/dT=[CinF]*pf*r*N
This equation represents the growth of the cyanobacteria. The parameter ‘pf’ describes the proportion of carbon produced from the Calvin-Benson cycle that does not go back towards the cycle and the regeneration of RuBP. Figure 3.2 depicts the Calvin-Benson cycle and clearly represents the step in which one molecule of the G3P produced goes towards the production of biomass and the other 5 molecules go towards the regeneration of RuBP. The parameter ‘r’ describes the growth rate of cyanobacteria. The growth of cyanobacteria is limited by two factors: electrons due to light and carbon(21). Therefore, the parameter r is described by the minimum of these two variables.
Figure 3.2: Calvin-Benson Cycle
This figure represents the Calvin-Benson cycle which describes the steps of carbon fixation. The figure clearly shows the production of sugars which will ultimately go towards biomass, and the regeneration of RuBP by organic carbon molecules.
The parameters were calculated to be in congruent units and then the equations were input into Matlab to simulate the model. The following results were obtained from the simulation.
Figure 3.3: Carbon Sequestration Model Simulation
According to the model above the cyanobacterial growth increases at a linear rate until it reaches a concentration of about 102.5 Cmol/L, where it plateaus. During this time CinF decreases slowly until about 250s after which it decreases at a much faster rate until the cyanobacterial growth plateaus, and from there the concentration of CinF decreases again; however, at a much slower rate. The rate of Cin decreases at a constant rate until it plateaus at a concentration of about 10-9 Cmol/L. Finally, the concentration of Cin* increases momentarily, but then begins to decrease at a very slow rate over time.
These dynamics match what would be expected of the cyanobacterial model by observing the equations. The first equation representing dCin/dT shows that all of the concentration of Cin would be coming from the transport of the inorganic carbon outside of the cell. This concentration goes towards the conversion of inorganic carbon to CO2 that is ready to bind to RuBisCo, which happens at a faster rate than the transport of Cout. Therefore, it makes sense why all of the carbon that is coming into the cell is decreasing at a constant rate.
The second equation representing dCin*/dT shows that Cin* increases due to the conversion of inorganic carbon by carbonic anhydrase. The concentration of Cin* decreases due to the binding of carbon to the active site of RuBisCo; however, this occurs at a much slower rate than the conversion of inorganic carbon. Additionally, much of the carbon that binds to RuBisCo ends up going back into the cycle to produce RuBP. Therefore, the model simulation aligns well with what the equations would predict as the concentration of Cin* does not change much over time.
The third equation representing dCinF/dT shows that the concentration of CinF increases due to the binding of carbon to RuBisCo to produce organic carbon. However, the concentration of CinF decreases due to the carbon that goes back into the Calvin-Benson cycle to regenerate RuBP. In addition to this, the concentration of CinF decreases due to the organic carbon which goes towards the growth of the cell. The rate at which the concentration of CinF decreases due to the regeneration of RuBP happens much faster than the rate at which the concentration of CinF decreases due to growth. The simulation shows that the concentration decreases slowly and then decreases at a much faster rate for a short period of time, and then decreases at a much slower rate again. The slow decrease most likely represents the concentration of CinF going towards cell growth while the more steep decrease may represent the carbon going back towards the regeneration of RuBP.
The fourth equation representing dN/dT shows that the increase in final growth is due to the concentration of CinF. In the model, the growth is represented by either inorganic carbon being a limiting factor or light being a limiting factor. Given that light was not a limiting factor in this equation, it makes sense that the simulation depicts the growth increasing and then leveling off over time relative to the concentration of the other species of carbon within the cell.
In general the simulation of the model aligns well with the behavior which the equations are trying to predict. The next steps in the analysis of the model would be to change different parameters and analyze how the dynamics of the cells are impacted. Different parameters which may be changed include the transport rate of inorganic carbon (kin), the rate of carbonic anhydrase (kc1), the amount of inorganic carbon available outside of the cell (Cout), and the light intensity. Additionally, a sensitivity analysis should be completed in order to check the robustness of the model and the parameters which were assumed.
This model describes the dynamics within cyanobacteria, specifically in the photosynthetic process and Calvin-Benson Cycle. The benefit to the simplicity of this model is that it may be adapted by others who want to mutate some part of the cycle and observe the effects on cyanobacterial growth. The benefit in the contexts of this project is that it would be very easy to edit the equations and add parameters which will allow us to explore more in depth the dynamics of cyanobacterial growth and carbon sequestration for the modified strains of cyanobacteria. Currently, the model shows the steps of photosynthesis and carbon fixation, but adding additional parameters which show varied transport rates of inorganic carbon, or parameters which show the varying dynamics of different carbonic anhydrase enzymes would provide a more detailed explanation to the wet lab team of how the proposed biobricks (BBa_K5052310, BBa_K5052311, and BBa_K5052312) for the carbon sequestration module of the project, would impact the sequestration of carbon in cyanobacteria.
The general transport rate of inorganic carbon is described by the parameter ‘kin’. A future direction of the model would be to edit kin to describe the rate of all of the transporters in cyanobacteria. For example, the parameter could be the weighted average for the rate of all of the transporters. This would be calculated by taking the proportion of bicarbonate or CO2 being transported into the cell and the rate of each of the specific transporters. One of the goals of the wet lab team is to increase the expression of the bicarbonate transporters SbtA and BicA to increase the carbon fixation of cyanobacteria. SbtA is a high-affinity sodium dependent bicarbonate transporter(22), while BicA is a low-affinity sodium dependent bicarbonate transporter(23). The wet lab team plans to control the expression of these enzymes with the IPTG inducible promoter, cLac145 to avoid overexpression of transporters to a lethal degree. To adjust the model to observe the dynamics under these conditions, these new predicted transport rates would replace the value of the transport rates for these transporters in the wild type strain.
A second goal of the wet lab team is to improve the robustness of the cyanobacteria to intense light and temperatures. To do this, the team chose to modify the cyanobacteria with carbonic anhydrase from N.vulgaris because it is thermally robust(24). In doing so, the team predicts that cyanobacteria would have the ability to convert HCO3- to CO2 at a wider range of temperatures and therefore metabolize at more extreme conditions. To see how the dynamics of this modified species would change, a future direction for the modeling team would be to add a parameter which describes the effect of temperature on the degradation of the carbonic anhydrase enzyme. Under the conditions of the new species this parameter would then be changed to show the decreased degradation rate of the enzyme. In doing this, the wet lab team could then collect data on how changing the carbonic anhydrase enzyme to this more robust version affects the growth of the cyanobacteria, and therefore the carbon sequestration.
Furthermore, a major aspect of the project is the ability of the cyanobacteria to produce electricity. To predict the carbon that is sequestered and the growth of cyanobacteria while producing electricity in the biophotovoltaic system, the parameter ‘e’ may be edited to account for the electrons that go towards electricity production instead of cyanobacterial growth, which may be observed by looking more closely at the processes of photosystem II of photosynthesis.
The equations represent the processes responsible for cyanobacterial growth. Therefore, it would also be straightforward to look at conditions like limited inorganic carbon in the environment and decreased light intensity and see how this would impact the conditions for growth. This would be incredibly useful information for the wet lab team because it would help them determine what conditions would be most optimal to grow our cyanobacteria in, whether it be the mutated strains or the wild type strains. Due to this model’s simplicity, there is a flexibility to describe cyanobacterial dynamics under various conditions that enables diverse future applications for predictions of carbon sequestration and cell growth.