Modeling
Statistic population approximation The mathematical modeling section was focused in modeling the process of growth and production of the algae, as it is planned to be used as a way of large-scale bio fuel production, it is fundamental to know how the algae is growing, how every seaweed is going to produce such a material and how the population is growing. Based on the population size that was registered by the experimental section, it was possible to estimate a continuous model, that could predict which population size would be reached by a certain population after any amount of time. Using the logistic growth model as a basis, it was possible to extrapolate some parameters to describe a system that approximates the experimental values using the limits when t->0 and t->infinite as a basis;
It is important to notice that, as a logistic equation, it must reach a maximum given enough time, which would be generally associated with P(t)=a/b as e^(-t) would be nearer to zero, when considering this, it is good to point out that a/b=115.61/0.12=963.42, which is quite similar to the 997.3 that was registered as the amount of algae reached by day 10 (the difference might be interpreted as lost when rounding).
The problem with the model previously described is that it is only valid when the conditions in which the seaweed is growing are the exact same conditions as those described previously. Differential equation development Nevertheless, it is possible to study how the algae is growing according to the already present algae, which would mean, generate a differential equation model. For such a result, it was thought that when studying each difference between two values divided between the time lapse, it is possible to have an approximation of the differential value, based on the definition of derivate and associating each change with the average population size between the two used to calculate the difference.
It is possible to obtain a simulation on the differential values of algae present in the medium using the following as a basis:
About our two compartment model We used a nonlinear system of ordinary differential equations to model the dynamics between Chlorella vulgaris´s cells growing inside a bioreactor and the quantity of lipids that they generate. For that, we used the idea that (once modified the algae cells), a vast amount of lipids are exocyted without cells being killed. With that in mind, the exocyted lipids will tend to form a lipid layer at the top of bioreactor or at least lipids will remain out of algae cells. The formation of the layer holds because of the differences of densities with surrounding water on the bioreactor. Thus, we can use a model in compartments which allows us to ignore how the lipids are excreted and accumulated. We are interested in what the population of cells is going to be at an amount of time, and how much lipids we can get out of them.
Further modelling includes studying the dynamics of the cells, the lipids inside cells, and the cells dying inside bioreactor i.e. cells which should be removed from bioreactor. That would allow us to determine whether it is more convenient to stop extracting the lipids exocyted and sacrifice the whole population of old or dying cells, or to just remove the amount of dying-dead cells out of bioreactor. The hypothesis used to deduce the model are: Chlorella´s cells C, grow following a Logistic growth. The lipids are generated and accumulated inside the algae cells but exocyted at a proportional rate to its concentration inside cells. (That defines the “aC” term in 2nd equation which adds size to the Lipid compartment). Lipidic layer is removed at a proportional rate to the amount of lipids produced. This is the amount of lipids used for creating biofuel. The segregation of cells and lipids is caused by the difference of densities between lipids and water. Lipids are supposed to form a layer at surface of bioreactor.
For easier understanding, it is important to follow the next relation between the symbols and their meaning: p-Seaweed population x-Lipids inside the seaweed's cell y-Lipids outside the seaweed's cell f-Modulator of the maximum growth rate c-Carrying capacity modulator a-Lipid production, which should be measured as production per seaweed b-Lipid excretion rate The differential equation for dx might be understood as a gradient between the lipids produced, which depend on the individual production, and the amount of seaweeds, and the lipids that are exocyted of the cell, which depend on the population size and the lipids that are available for exocytosis. The last equation, which corresponds to dy, represents the amount of lipids outside the cell and, as such, is just the positive version of the exocytosis part in the last equation. It is interesting to analyze the equilibrium points for the system, it has been already pointed that dp reaches equilibrium (dp=0) whenever p=0,c , in which c corresponds to a stable equilibrium and 0 is unstable, which is also congruent with the general suggestion of literature on population models. Also, we might observe that dx reaches equilibrium when ap=bxp, as a and b are parameter of a constant value, it might be concluded that such an equilibrium happens when p=0 or x=a/b. On the other hand, dy never reaches a complete equilibrium, unless p=0 (which might help us to identify a trivial equilibrium for all three variables when p=0), it might be considered unimportant as y functions as a measurement of the lipid already produced and available for extraction. Now, if we analyze the seaweed population size and the amount of lipids inside the cell, it might be seen that there is an equilibrium for both variables when (p,x)=(c,a/b), implying an equilibrium for both equations when such a point is reached. Interestingly, we might remark that it happens to be an dynamic equilibrium as the production and exocytosis of lipids is still going and new seaweed is born while other is dying, implying that the lipid production is not stopped at this point, even reaching its maximum as that is the point where the variables associated with y are at its maximum.
References:
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