Envisioning the co-culture in silico
Co-cultures are a synthetic mimicking of nature's relationships between cells that are able to make biological processes more efficient, faster and even cheaper. They serve as a powerful tool as they leverage the ecological niches and specialties inherent from the evolutionary process. Hence, in order to make the production of our Nisin and PHB containing bacterial cellulose our team more efficient our team decided to harness the power of a co-culture. This co-culture involves K. xylinus producing bacterial cellulose and two different recombinant E. coli producing Nisin or PHB.
Though the general concept of co-cultures is becoming an increasingly attractive field of bioprocessing, in the process of designing our co-culture experiments we discovered that there existed a lack of available literature describing this system. Due to the novelty and complexity of this system we had to validate that it would be possible while gaining an understanding of the kinetics of what goes on in the co-culture. Hence it is for this reason that we developed a model capable of describing the intracellular and extracellular dynamics of this co-culture system. The objectives of this model were as follows:
Figure 1. Schematic representation of the Co-culture
In order to determine if the co-culture of E. coli and K. xylinus would be feasible through modelling we started off by describing their growth given environmental circumstances and the growth dynamics of both chassis. Thus we implemented version 1 of our model which described how both chassis would grow just based on substrate concentration which was glucose. In order to base off growth on substrate consumption we also had to model the change in the substrate. To do this we implemented a system of ordinary differential equations that could mathematically give us an understanding of the happenings in this system. To model the growth we made use of the growth rate μ as described by Monod's equation (1). Ultimately enabling us to derive the following equations
Equations that describes the biomass growth of K. xylinus and E. coli
The yield coefficient is a constant that can be used to relate cell growth to the rate of substrate consumption. Thus, the yield coefficients of both K. xylinus and E. coli was used to generate an equation to describe the utilization of their substrate, glucose, in the co-culture:
Equation that describes glucose consumption during microbial growth
Throughout our modelling process we derived several series of systems of differential equations. In order to solve the system of equations numerically we implemented these equations in python and made use of the odeint and solve_ivp method from the scipy package. The solutions to these equations were then plotted to give us a better visualization of the changes in these modelled parameters. An assumption made in this simulation involved that only the substrate amount affects biomass growth. Here are the graphs generated for the biomass growth and substrate consumption.
Figure 2. Simulation showing the Model’s Results for the consumption of glucose
Figure 3. Simulation showing the Model’s Results for the biomass growth of both K. xylinus and E. coli
All parameters used in the simulation run can be found in Table #
Since our two bacterial species are grown together there is a possibility that extracellular secretions from each have the possibility of affecting the growth and proper functioning of the other. Hence it was important that we verified if there existed such substances and if so what the effects of one on the other was. It was discovered from literature that K. xylinus produces by-products of gluconic acid and acetic acid (2). Since these are acidic in nature they then have the capacity to reduce the pH when the two bacteria are growing together. K. xylinus grows in an optimal pH of 5 but E. coli has a wider pH range. As both bacteria have an optimal pH to be grown in, our aim was to look into how the production of these substances affect pH and how this could affect growth. We derived a system of equations describing the production of gluconic and acetic acid using equations (3).
Equations that describes extracellular acid production of K. xylinus
The equations were coupled with the equations describing biomass growth and substrate consumption and were then solved using python’s ‘odeint’ method from the scipy package.
However, we then had to find a way to connect their concentrations to pH. Hence we derived this equation
Equations that describes the pH of the co-culture
Figure 4. Simulation showing the Model’s Results for the Production of Gluconic and Acetic acid
Figure 5. Simulation showing the Model’s Results for the change in the co-culture's pH
From our simulation run, we realised that Gluconic acid is produced at a higher rate than that of Acetic acid. Since it has a higher dissociation constant Ka, it must have resulted in significant dip in pH as the production of acid occurs. Hence, a strategy has to be implemented to ensure that pH changes do not affect the optimal growth of our bacteria.
After several attempts to understand the dynamics of E. coli and K. xylinus we decided to make an attempt to understand the dynamics of just K. xylinus’s growth. The reason for this is because the wet lab began their experiments to understand the best medium to grow k.ylinus in before conducting a co-culture. This is to allow for an optimal production of bacterial cellulose. We therefore implemented a model to help them further understand it.
We used the same equations in previous iterations of the model to describe the factors such as growth and extracellular substance (gluconic acid and acetic acid) production. However we made a slight change to that of substrate consumption as it was now only dependent on K. xylinus.
Here are our system of ordinary differential equations used to model the monoculture
Figure 6. Simulation showing the Model’s Results for the growth of a K. xylinus monoculture
Figure 7. Simulation showing the Model’s Results for the growth of a K. xylinus monoculture
Figure 8. Simulation showing the production of Bacterial Cellulose
Due to inconsistencies and difficulties in measuring the amount of K. xylinus the wet lab decided to base the biomass of K. xylinus on the amount of BC produced and thus shifted their experiments to reflect that change. Wethen shifted our model to show that so as to determine which means is a more accurate method of measuring Biomass. Because biomass was based on the BC we had to base our BC production on another parameter. From literature review and observations from the wet lab we determined that the amount of BC produced can be correlated to the amount of glucose which is the major substrate of K. xylinus that is used in the production of BC.
The biggest change to this version of the model was the modification of the BC production equation. The equation was based on this paper. We used initial value concentrations from the wet lab’s experiment. Hence, we were able to derive this equation
Figure 9. Simulation showing Model's results for substrate consumption
Figure 10. Simulation showing Model's results for Bacterial Cellulose Production in the monoculture
Having implemented a model that bases K. xylinus growth of BC growth we then adapted it to a co-culture version of the model. The simulation resulted in the following
Figure 11. Simulation showing Model's results for substrate consumption for the co-culture
Figure 12. Simulation showing Model's results for Bacterial Cellulose and E. coli's concentrations in the co-culture
The python files for all implementations of the model can be found here
Having described biomass growth, substrate consumption, bacterial cellulose production, and the production of other extracellular substances, it was essential that we also have an understanding of how Nisin, our recombinant peptide responsible for antimicrobial activity, would be produced in this co-culture. Our protein production model was taken directly from Calleja et al. 2016 (4). A detailed look at the model can be found in the Nisin Modelling page. After modyfying the system of equations to fit our premise, a simulation of the model resulted in the following:
Figure 13. Change in Substrate Consumption forthe Co-culture during the simulation run
Figure 14. Change in Bacterial Cellulose in a Co-culture (note that bacterial cellulose amount is used to quantify the amount of K. xylinus)
Figure 15. Change in E. coli biomass in a co-culture
Figure 16. Change in Acetic acid production by K. xylinus in a co-culture
Figure 17. Change in Gluconic acid production by K. xylinus in a co-culture
Figure 18. Protein production in the co-culture during the simulation run
Note here that the blue line signifies the uninduced portion of the protein produciton, while the red signifies the induced (IPTG induction) portion.
As with all models, there exist shortcomings due to assumptions and the limitations of the mathematical concept used. One such limitation of this model is that it has a minimum simulation time of 4 hours. This is so because the model predicts that all the substrate would be consumed after those four hours. This is not so; the co-culture is expected to be grown for about five days (120 hours). It behaves this way due to literature values implemented in the model's parameters, which are not representative of the actual experiments. Hence, conducting experiments to get values more representative of the lab’s results should annul this behaviour.
Another limitation of this model is that when modelling the pH change, there exists a massive drop in the pH. We hypothesize this is so because the equations are not holistic representations of all the factors affecting the system's pH.
We would be implementing several strategies to combat these limitations to improve the model. One includes using the wet lab’s data to improve the model by fitting it into our model. Another method is exploring changes to our equations to represent other parameters that could affect the particular factor we are studying.
Having shaped our wet lab experiments based on our first round of modelling and results we sought to improve our model based on the data collected from their experiments. Thus we collected BC values over a period of time and will use it to fit into our model using non-linear regression. This would enable us to get an optimized model capable of helping us understand dour co-culture and make better predictions. Not only does this shape our wet lab experiments it enables us to understand and predict our system behavior when taken to an industrial scale. This is important in helping us as we seek to understand the entrepreneurial framework of our project.
The lab uses an intermittent feeding strategy that aims to increase the production of BC by increasing the amount of media given to the bacteria. The most useful one is the 24 hour feeding strategy. In order to make the model reflective of the wet lab’s experiments we worked on modifying the model to reflect this change.
In order to do this we first had to find a function that can describe the change in the change in volume also known as the feeding flow rate. From literature we did not find an accurate function that was able to describe this. Hence, we looked into deriving our own function. The function that most accurately showed the pattern in this was the square wave function.
We were able to change certain parameters in this equation to model this feeding strategy. Such was the frequency, and duty. We used the python SciPy signal.square wave function to implement the function.
We were unable to include this function in the solve.ivp system of equations as the function would miss time periods and not accurately integrate the function.
Due to this we had to use another function to integrate the feeding flow rate to show the change in volume. For this we made use of the python SciPy cumulative.trapeziod function. Which was able to accurately integrate the function. However, the challenge was how to incorporate this into the rest of the system of equations. We were not able to implement this, so as a future iteration of this algorithm we would be utilizing methods that can allow for the intermittent feeding strategy to be added onto the rest of the model. The aim is that this periodic feeding strategy will counteract the limitation of the model having a short growth period of the co-culture.
Implementation of this algorithm can be found here
Figure 19. Schematic representation of Intermittent Feeding algorithm
Figure 20. Simulation Results of Periodic Feeding over a period of 72 hours as calculated by the intermittent feeding algorithm
Figure 21. Simulation Results of Periodic Feeding over a period of 72 hours as calculated by the intermittent feeding algorithm
The graphs created by the model revealed that E. coli would grow at a rate about five times as fast as K. xylinus when in HS media, non-optimal growth conditions, and extracellular secretions from K. xylinus. To counteract this, we need to add one part of E. coli culture for every five parts of HS media in the co-culture. With these results, we can better control the rate of E. coli growth to prevent E. coli from outcompeting K. xylinus when in a co-culture.
Another key result highlighted from the model is the effect of the production of acetic acid and gluconic acid on the system's pH. This could cause the co-culture to grow in non-optimal conditions that can affect the production of bacterial cellulose, Nisn and PHB. However, our wet lab experiments discovered that using the intermittent feeding strategy allowed the pH to remain fairly constant. Hence, this validates another benefit of using the intermittent feeding strategy in our co-culture.
The model also revealed that using bacterial cellulose to quantify K. xylinus resulted in a more accurate model representative of the wet lab’s experiments.
In conclusion, in partnership with wet lab experiments, the model shows that it is possible to have a co-culture of K. xylinus and E. coli to produce bacterial cellulose, Nisin and PHB. This is as long we can ensure starting concentrations of K. xylinus are in rations higher than E. coli as this would cause competition. Also, ensuring that other environmental factors are regulated in a way that ensures the possible optimal growth of the co-culture.
Looking into future iterations of the co-culture, we would work on modifying the model to represent the entire growth period of K. xylinus which is about five days compared to the 3 hours the model predicted. This design could also be modified to model industrial constructs to help us determine the productivity of this co-culture when taken to this stage. Lastly, we understand that more than pH, biomass concentration, and substrate amount, there exist other factors that would be responsible for shaping how the co-culture would behave. These could include a surface area for growth, oxygen and even the amount of other nutrients such as proteins. Implementing these in future iterations of the model would be most beneficial to represent the co-culture holistically.