
MBW:Simulate a continuous operation photobioreactor utilizing the Photosynthetic Factory Model (PSF)From MathBioThe author of this wiki article is: Tobias Niederwieser (April 2016) The motivation for this wiki page was found in: Nauha, E. K., & Alopaeus, V. (2013). Modeling method for combining fluid dynamics and algal growth in a bubble column photobioreactor. Chemical Engineering Journal, 229, 559–568. doi:10.1016/j.cej.2013.06.065 ContentsExecutive SummaryLife support systems are a necessity to allow human life in the hostile environment of space. Currently physicochemical systems try to close the loop on air revitalization and water recycling. A complete closure on these systems has not been achieved and hence these systems require the continuous resupply of consumables. Additionally, physicochemical systems are incapable of producing food, making them unsuitable for long term exploration mission. Due to the historical approach of performing one function at a time, complexity, mass, and volume needs are high but at the same time redundancy is low. Algae have the potential to overcome these deficiencies by combining the critical life support system functions and basically simulating the human counterpart. Current research is conducted in smallscale bench top setups. In order to develop large scale facilities that are capable of supporting humans, this approach is not any more feasible due to cost and time constraints. Here is where models come into place that are taking the parameters of the smallscale bench top setups and can extrapolate the results to large scale facilities. Due to scaling effects, like shadowing and mixing, these models are fairly complex. This wiki article focuses on one basic model of algal growth, which is the Photosynthetic Factory (PSF). The model was taken from Nauha and Alopaeus (2013) and replicated in SimBiology. The wikiauthor was capable of using the presented equations and replicating the results presented in the paper. A steady state analysis was conducted and concluded that photobioreactors can only be used in batched mode. However, in order to continuously support humans, a continuous operation is needed. Instead of operating a high number of batched photobioreactors that potentially require crew time to load, this wikiarticle describes the modification of the model to allow a continuous operating photobioreactor. This is performed by implementing a continuous removal of resting algal cells. Via steady state analysis it is shown that the modification results in a stable continuously operating system resulting in steady life support in and outputs. IntroductionIn order to support human life in space, life support systems are necessary. They perform the critical functions of air revitalization, water provision, heat removal, waste removal and food provision. Currently, all these functions are performed with physicochemical life support systems that require consumables. This is necessary not just because they recycle rate of water and air is below 100 %, but mainly because those systems are incapable of producing food. In addition physicochemical systems, as utilized on the International Space Station, are far from being reliable as needed for long term exploration missions. As a solution bioregenerative life support systems are proposed that have potential of overcoming those deficiencies. One technology within bioregenerative life support systems is algae, which this article is focused on. Comparable to Earth, algae are capable of living in symbiosis with humans as seen in Figure 1. One reason for choosing algae is its fast doubling time as low as 9 hours ^{[1]}. This faster growth rate also correlates to a higher productivity of food and oxygen. In addition, algae are easier to handle due to its growth in solution ^{[2]}. The outputs of humans consisting of metabolic water, feces, urine and carbon dioxide gases can be used for inputs in an algal photobioreactor ^{[3]}. Under the additional input of light that can be freely gained from the sun, the algae then convert these inputs into oxygen, biomass, and clean water. These 3 outputs in turn can be used again as inputs for maintaining human life in space. As seen on the left side of Figure 1, this loop is not completely closed. As humans create heat while digesting an additional heat removal system to deep space is needed. Additionally, the ability of algae to deal with solid wastes and high amounts of trace contaminants is not yet investigated. Compared to the big mass flows of water, urine, feces and carbon dioxide, these mass flows are minimal and can be reused with other systems such as 3D printers. By combining the critical functions of historically separate life support systems into one technology could not just save complexity, mass, volume, and power, but can also increase the reliability of such a system. Current research is focusing on assessing the feasibility of photobioreactors for use in spaceflight applications ^{[4]}. For this purpose, small scale photobioreactors are investigated to characterize the productivity and design. Models are then used to scale this system up to full scale systems to assess robustness, design, and productivity of different configurations before experimentally verifying the final design ^{[5]}. Biological PhenomenonAlgae is a very broadly used word that is often intended for greenalgae, which are eukaryotic cells. Among other kinds, algae are sometimes also used for cyanobacteria, which have comparable functionality but are bacteria (prokaryotic cells). This article is focusing on green algae. Despite the naming convention, both are using photosynthesis to convert carbon dioxide and water to glucose and oxygen as seen in the equation below. This process is performed using light energy that is absorbed in the chloroplasts of algal cells. The entire carbon from CO2 is fixated in glucose, often also referred to as biomass. Half the oxygen from carbon dioxide is recycled towards breathing air whereas the other half is provided by water. Algal growth is dependent on several environmental factors for example temperature, light intensity, light spectrum, light cycle, carbon dioxide concentration, oxygen concentration, and nutrition to name just a few. One of the most influential factors is the light intensity. Commonly a light in the red and blue spectrum is utilized, which are the wavelengths that are absorbed by chloroplasts and hence only have minimal losses. In addition, most terrestrial photobioreactors are used outdoors, where the light cycle is usually 16/8. Lots of modeling approaches hence are based on the light intensity modeling as it allows a first order of magnitude estimation even though precise predictions are questionable. This model is justified as it was initially developed for terrestrial open algal photobioreactors. Here oxygen and carbon dioxide concentration is buffered by the atmosphere. Furthermore it is assumed that nutrition and temperature are in the photobioreactors sweet spot and constant. Mathematical ModelOne common approach for modeling the light intensity is the Photosynthetic factory (PSF) ^{[6]}. It is based on the assumption that algae can exist in 3 different states due to light intensity as seen in Figure 2. State 1 includes the resting algal cells that are receiving too little light intensity for photosynthesis but can be activated immediately. State 2 are the active algal cells that receive the suitable amount of light for growth. Once they perform cell division, they go back to Stage 1 before they can be light activated again. State 3 are the cells that have too much light and are inhibited in growth. Cells are able to recover but take time to readjust. The partial differential equations (PDE) for the model are presented below. The light intensity has to be modeled separately based on the biomass density. Equation 3 describes the decrease of light intensity based on distance to the light source and shadowing effect from the water and the biomass. Generally the deeper the photobioreactor, the lower the light intensity. The author of the paper decided to model the fractions of each state. The growth of the entire population was therefore modeled using Equation 4, taking into account the amount of active cells that are capable of dividing to form k new cells. In order to combine the equations presented above and yield the absolute cell counts the following equations were used by the author. Aside from modeling one single PSF, the paper also incorporated several of those PSF by compartmentalizing the photobioreactor using finite element flow models of the entire reactor volume. This approach results in higher fidelity models but also in higher complexity. Due to the fact that the finite element analysis was not presented in this paper and a one compartment approach was also modeled as a baseline, the wiki author made the simplification of focusing the PSF on one compartment. Results from the PaperThe paper plots fraction of PSF state over time in blue and green as seen in Figure 3. As the growth rate cannot be seen in these kind of plots it is plotted as additional line on the secondary axis. It can be seen that initially the active state becomes dominant and is filling up the resting state department, before it becomes light limited and decreases. During the entire operation of the photobioreactor, the inhibited state is negligible low and hence not shown. This behavior is expected as it can be explained by the initial optimal light intensity that, as the cell density increases, decreases with time. Replicated ResultsFor further analysis of the system, the author of the wiki page wanted to recreate these plots as a sanity check before generating new results. In order to build a system with SimBiology and to do analysis with mathematical tools the equations were simplified to one set of absolute values instead of several sets of equations using absolute and relative numbers. The resulting equations using absolute counts are seen below (Equation 6). By using SimBiology it was possible to recreate Figure 3 with own methods and the result is shown in Figure 4. The initial conditions such as equal fractions of X1 state and X2 state algal cells was assumed. The characteristic initial overshoot can be seen which then slowly results in a pure resting state algal solution.
For making the plots more intuitive the wiki author decided on a more suitable format of the plot by showing absolute instead of relative values, hence eliminating the need for the growth rate curve. Furthermore, the inhibited state was also chosen to be displayed as well as the average light intensity reaching the algal cells, mainly to verify the correct behavior of the model. The resulting plot is seen in Figure 5. It can be seen that the initial population grew to 1.2 times the population of biomass. Due to the continuous growth of the entire population, the initial characteristic spike is slightly disproportioned. As expected, the light concentration is continuously decreasing over time. This causes the overall rate of production to decrease resulting in a passive steady state at the end. It hast to be noted that the absolute values on the y axis do not have units as this model has not yet been experimentally verified to actual parameters. Extension of the ModelPhotobioreactors are usually operated in a batched mode, meaning that they are inoculated, grown to full capacity, and then harvested before the photobioreactor has to be reloaded. This approach is also shown in the presented paper, where one batch is lasting for about 100 hours. In order to reduce the crew time required to operate the photobioreactors in space habitats, a continuously operational photobioreactor is favorable, matching the in and output of human metabolism. On earth the atmosphere can buffer batches of algal growth, however due to the limited volume in space habitats, this option is not available. To investigate if the presented model is capable of producing such a steady state solution, steady states were calculated by setting the partial differential equations to 0 as shown below. Immediately one trivial steady state at (X1=0, X2=0, X3=0) can be derived which does not yield any practical use. The nontrivial steady state is more difficult as the resulting equation is not defined. States X2 and X3 can be immediately used to solve for the state as shown below: The equation for state X1 does not yield an immediate result as X1 cancels out. A result for this equation is not defined and therefore does not result in a steady state. This results is expected as batched photobioreactors will asymptotically decrease the light available for the algal cells due to shadowing but never completely stop the light intensity. In order to achieve a continuously operating photobioreactor, the model hence has to be modified. In order to make the first equation solvable, one solution is to add a process, that is constantly removing resting algal cells. This can be performed by strategically placing the inlet of a recirculating loop for example in the center of a photobioreactor. As the light intensity is decreasing with the depth of the photobioreactor and the cells with insufficient lighting are in the resting state, most of these cells should be found in the center of the photobioreactor. A state diagram of that model is comparatively shown in Figures 6 and 7. The new parameter for the removal of the resting cells was called ε. A new steady state calculation was performed shown below. Since only the state X1 was altered, X2 and X3 do not show any adaptation. The equation for state X1 again does not yield an immediate result as X1 cancels out as shown below. However, the equation for X1 is now solvable for I, which than in turn can be used to solve for X. Making the assumption that all parameters are nonzero and positive values results in 1 nontrivial and noncomplex steady state. Due to the multitude of variables it was decided not to show the solution in this wiki article, however it can be requested from the author. This state was then modeled in SimBiology and plotted as shown in Figure 9. The model diagrams and equations from SimBiology are shown in the Appendix. It can be seen that there is a constant absolute value of cells. Also the light intensity is shown to be at a constant level, compared to leveling out to 0 at the previous model. This allows a constant value of cells being in equilibrium between active and resting state. This is favorable as the constant growth results in a constant outcome of oxygen production. Initially it can be seen that the resting state is dominant, limiting the light intensity, but recovers towards the steady states. Overall the steady state is stable. ConclusionThe results of the paper by Nauha and Alopaeus (2013) were replicated by using SimBiology. A new representation was chosen to better show the interplays between light intensity and growth rate. By modifying the model to have a constant removal of algal cells it was shown that continuous operating photobioreactors are theoretically feasible. A mathematical model was used to perform a steady state analysis and the results where simulated in SimBiology. It has to be noted, that this model just focuses on the effects of light intensities. Other factors such as temperature, carbon dioxide concentration, oxygen concentration for example are assumed to be in their optimal range and constant. Further modeling work is needed to also include those parameters into the model to investigate the effects of the modification presented in this article. If those results are positive, it allows for experimental investigations of implementing such a mechanism in a photobioreactor. These experiments then can also be used to refine the model by adding absolute numbers to the processes. Further studies also have to focus on matching the productivity of the photobioreactor to the human metabolism. Some especially challenging aspects are the differences in metabolism that have to be buffered between high activities during extra vehicular activities (EVA), normal activity during the day and low activity during sleep. Appendix
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