The goal of our project is to use genetically modified phototrophs to desalinate seawater as a solution to the worldwide freshwater shortage. We particularly studied the cyanobacteria *Synechocystis sp. PCC 6803*. Parallel to the experiments in the lab, we worked on describing our desalination system mathematically.

Mathematical models provide the possibility to recreate a scenario and change the different parameters without having to conduct countless experiments. This way, our models would eliminate the need for multiple wet-lab experiments and bring important insights which we can use to upscale our project to an industrial scale.

Our modeling is split into three parts like the experimental procedure – bacterial growth stage, desalination stage, and separation stage. In the bacterial growth stage, we aimed to model the growth curve of *Synechocystis sp. PCC 6803* under different conditions. This enables us to predict the growth under a combination of different factors, in our case pH and salinity values.
The desalination stage is especially important for upscaling our project to an industrial scale. Here, we aimed to estimate the number of bacteria that we need in order to desalinate 1 liter of seawater. In the separation stage, we aimed to increase the efficiency of the filtration of the organism from water using a carbohydrate binding domain, CBD, displayed on the surface of the organism. Different environmental conditions can affect the interactions between CBD_{CipA} and its ligand. Therefore, a mathematical model of the binding affinity of CBD to cellulose in different conditions could be constructed to optimize the process. Some useful parameters to look at could be pH, temperature, and salinity levels.

For all three parts, we worked according to the design - build - model - test - repeat principle. On the one hand, this means that our model has a direct influence on adjustments of experimental settings. On the other hand, this also means we need some experimental data to build the model in the beginning.

We used Matlab for the modeling. You can have a look at our code here.

One of the disadvantages of using cyanobacteria in research is their relatively slow growth rate, with a doubling time of 4-8 hours [1-3]. Investigating the influence of different factors on bacterial growth in the lab would take a long time as well as a large number of experiments. For our desalination process, it is important to know how the growth of phototrophic organisms is influenced by environmental factors, for example, the pH value and the salinity of the saltwater. This information is also important in order to find ideal growth conditions for an industrial scale and to see if bacterial growth is affected by the modification of the organism. Therefore, we wanted to create a model which could predict bacterial growth under variable conditions. We used the growth model from the iGEM team NCTU_Formosa 2018 as a starting point and adjusted it to describe the growth of phototrophic organisms like the cyanobacteria *Synechocystis sp. PCC 6803* [7].

To investigate the effect of pH and salinity on the cyanobacteria growth, we started by fitting their model using our experimental data and in a final step calculating the relationship between both factors. Parameters and variables used for this can be found in table 1. In the end, this should enable us to evaluate further changes in pH values or salinity on the growth without having received new experimental data.

First, the following experiments were performed to establish optimal growth conditions for cyanobacteria *Synechocystis sp. PCC 6803*. The cyanobacteria were cultured for 26 days using a) 3 different pH and a constant salinity as well as using b) 3 different salt concentrations and a constant pH. Originally, we intended to include different temperatures as well but it was not possible as we had to keep the incubator at 37°C in order to not disturb cultures from other experiments. In order to evaluate bacterial growth, absorbance/Optical Density (OD) was measured every 24 hours at a wavelength of 544 nm. Day 0 started with a base OD between 0.02 - 0.04 after subtraction of the blank. The growth was evaluated respectively.

We started the modeling process using a general equation that describes the bacterial growth rate. This is the logistic regression function (1).

(1)

We fitted the values C and B using the data obtained in the growth curve experiments. In the equation, C sets the limit for the maximum number of bacteria. We can determine the value for A by using the time point 0 to solve the equation. Since the value B reflects the maximum bacteria growth rate under certain conditions, we need a special equation depending on the influence factor.

First, we looked at the growth rate depending on the pH value. Therefore, we used the cardinal pH equation (2). We used the following parameters for the equation: pH_{min}=5 pH_{max}=11 pH_{opt}=7.5

(2)

We cultured the cyanobacteria *Synechocystis sp. PCC 6803* in three different pH conditions
( pH= 6, 7, 8) and used the obtained data to solve the equation system for c, d, and g (see figure 1A). This gave us the following values: c = 0.2070, d = 1.00 and g = 0.

With this, we were able to calculate the bacterial growth rate under different pH. Additionally, we were also able to model the bacterial growth curve depending on the pH (3). Figure 1 B shows the data gained experimentally compared to the model for the pH value 7.

(3)

**Figure 1** (A) experimental growth curve of cyanobacteria *Synechocystis sp. PCC 6803* under different pH conditions. (B) Simulated growth curve of cyanobacteria *Synechocystis sp. PCC 6803* at pH 7 in comparison to the experimental growth curve.

Next, we looked at the growth rate depending on the salinity. For this, we used the equation stated by the iGEM team NCTU_Formosa 2018 (4)[4].

(4)

By culturing the cyanobacteria *Synechocystis sp. PCC 6803* in three different salinities, we were able to use the experimental data to solve the equation system for h, i, and j (see figure 2 B). This gave us the following values: h = 0.3581, i = -0.3324 and j = 0.2527.

(5)

With this, it is again possible to calculate the bacterial growth rate under different conditions – In this case, we looked at varying salinity (5). Figure 2 B shows the data gained experimentally compared to the model for the NaCl concentration of 0.16%.

**Figure 2** (A) Experimental growth curve of cyanobacteria *Synechocystis sp. PCC 6803* under different salinity conditions. (B) Simulated growth curve of cyanobacteria *Synechocystis sp. PCC 6803* at a salinity of 0.16 % in comparison to the experimental growth curve.

After accounting for the two factors on their own, we need to find out how much each factor influences the growth in the merged equation (6).

(6)

After accounting for the two factors on their own, we need to find out how much each factor influences the growth in the merged equation.

We used two sets of different pH and salinity (sal) values to solve the system for α and β. The first set of factors was pH= 6 and sal= 0.16 %, the second set was pH=7 and sal= 3.5% (see figure 3 A). This gave us the following values: α = 1.2800 and β = - 0.2395.

Again, for validation, we compared our model to actual experimental data (see figure 3 B).

**Figure 3** (A) experimental growth curve of cyanobacteria *Synechocystis sp. PCC 6803* under different salinity and pH conditions. (B) Simulated growth curve of cyanobacteria *Synechocystis sp. PCC 6803* at a salinity of 3.5 % and a pH of 7 in comparison to the experimental growth curve.

Finally, our combined model can predict the bacterial growth for a range of pH and salinity values. For demonstration we modeled the growth under the following two conditions: pH= 8, sal= 0.3 and pH=7, sal=0.5

**Figure 4** Simulated growth curve of *Synechocystis sp. PCC 6803* at a Salinity of 0.3 and a pH of 8 and a Salinity of 0.5 and a pH of 7.

Although we did not use experimental data from transformed cyanobacteria to fit the model, it is only a small step to apply and adjust our growth model to new data. Thus, it will be fairly easy to model the growth curve of transformed bacteria in a short time without unnecessarily conducting a large number of new experiments.

**Table 1** Variables and parameters used for the growth modeling

Symbol | Unit | Description |
---|---|---|

pH | none | The input pH value |

pHmin | none | Minimum pH value resistance |

pHmax | none | Maximum pH value resistance |

pHopt | none | Optimized pH value |

sal | [mM] | Input salinity |

t | [days] | Time |

A | [O.D.544/day] | initial bacterial growth rate |

B | [O.D.544/day] | Maximum bacterial growth rate |

C | [O.D.544/day] | Bacterial growth rate under different pH |

RpH (pH) | [O.D.544/day] | Maximum amount of bacteria under certain conditions |

Rsal (sal) | [O.D.544/day] | Bacterial growth rate in different salt concentrations |

Our project would strongly benefit from modeling the desalination process on an industrial scale, since it will be important to know how much salt one phototrophic organism can absorb. This in turn helps us find the amount of bacteria needed to desalinate a certain amount of sea water.

Using the approach by the iGEM SJTU BioX-Shanghai 2015 Team as a starting point, our plan was to first create a mathematical model of the working stage of the bacterium where C^{l−} are transported against the concentration gradient of Cl^{−} via halorhodopsin inside the cell [5]. This means that the ions move from a low concentration outside the cell to a higher concentration inside the cell. In a second step, we model the influx of Na^{+} into the cell via channelrhodopsin. The execution of this plan is a two step process.

Firstly, we conduct wet-lab experiments in different conditions to get the following parameters:

On the one hand, the desalination process is complex. It is very important to understand what is happening during this process. On the other hand, it is also important to simplify the problem to a degree that can be addressed in a simple model based on the law of conservation.

Therefore, as a second step in our plan, we build the simplified model based on the following assumptions:

Due to the lack of actual experimental data, we could only develop a theoretical plan and could not complete the steps. However, this can be used as a boilerplate for future work.

After the completion of the desalination process, cyanobacteria, *Synechocystis sp. PCC 6803*, needs to be removed from the water. This could be achieved by utilizing the natural abilities of the carbohydrate-binding domain, also known as CBD.

CBDs are proteins derived from cellulose-degrading microorganisms. Cellulase enzyme is associated with the presence of CBD, which greatly takes part in the attachment of the microorganism to the cellulose via glucose moieties and diverse residues. This complex of CBD with carbohydrate-degrading enzymes is usually referred to as an anchor-enzyme complex. There are various CBD families depending on their structure and function. For example, CBD_{CipA} belongs to the CBD family 3 and is derived from Clostridium Thermocellum. CBD_{CipA} creates polar and non-polar regions with a calcium-binding site at a slight distance from either region, which is believed to stabilize the complex. CBD_{CipA} anchor-enzyme complexes have a high affinity for cellulose, and this is why the cellulose-based filter could be used as a possible separation method of bacteria expressing this CBD on their surface[6-10].

Different environmental conditions can affect the interactions between CBD_{CipA }and its ligand. Therefore, a mathematical model of the binding affinity of CBD to cellulose in different conditions could be constructed to optimize the process. Some useful parameters to look at could be pH, temperature, and salinity levels. This model would eliminate the need for multiple wet-lab experiments and would, in the end, hopefully increase the efficiency of the filtration of the molecule/organism the CBD is bound to [7|.

Due to time constraints and focusing on other aspects of our project, we sadly did not have the time to fully complete this model with complementary input data. However, our team laid some of the fundamental groundwork needed for future models to be made.

Using iGEM Linköping 2019and iGEM Imperial 2014 as inspiration, our plan was to first create a mathematical model of the binding of CBD to cellulose. Firstly, we would do wet-lab experiments in different conditions to get needed parameters, which in our case, would determine the Kd value.

In the planned wet-lab setup, as shown in figure 5, we aimed to use two columns; one containing a liquid with bacteria expressing CBD and fluorescent protein, and another with the cellulose filter. The fluorescence of the 7C liquid is measured, and after that, the bacterial liquid is poured through the column containing the filter. The fluorescence of the filtered liquid is measured, and the Kd is obtained. This would be done at different pH, salinity levels, and temperatures. With the acquired parameters, the model could be calibrated to be even more precise at predicting the affinity at different conditions, and thereafter the experimental setup could be adjusted with more accuracy.

In the future, the model analyzing the effect of other parameters than the ones mentioned above (pH, temperature, salinity) on the Kd could be included such as heavy metal ions in polluted seawater.

**Figure 5** Our planned experimental setup. The top column contains bacteria expressing CBD and fluorescent protein, and the bottom column contains cellulose (cotton). The bacterial liquid is poured through the bottom column, and the filtrate (which is bacteria-free desalinated water) is collected in a beaker. The fluorescence is measured before and after filtration, and the Kd is obtained.

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