Modeling

To model the growth and lipid production of Chlorella vulgaris we used the Constraint-Based Modelling, being our principal tool the kit the COBRApy library offers ¹. The metabolic reconstruction we used was that of Chlorella vulgaris UTEX 395, published on July of this year ². This reconstruction is based on Chlamydomonas reinhardii, with experimental and bioinformatic data on the literature.

It includes 843 out of 7,100 annotated genes (around 12%), delineating 1,770 metabolites and 2,294 reactions, being the amino acid and lipid metabolisms the most accurately reconstructed ². Importantly the authors of this model report that the uptake rates were experimentally verified experimentally.

First, we analyzed which light wavelength was the best for the overall growth of the algae according to the metabolic model. When comparing different type of lights in the metabolic model, we couldn't find a notorious difference between the different light sources. It's important to emphasize that this only happens if we find and apply the optimal value in each of the conditions.

Spectral decomposition of solar light measured from outer space is the first studied condition.

In this case we find the solution that optimized best our model, more specifically the one that increases the overall production.

Solar Light (Optimized)

Spectral decomposition of growth room is the second studied condition. We repeated the previously described analysis.

With Growth room light search optimal intensity

Dynamics of the possible optimal values in each of the conditions.

With Growth room light search optimal intensity

With this, we can see that if you optimize the light conditions in a rank; you can obtain the same biomass production values, independently from your initial light source.

Then, we analyzed the resources and nutrients for the culture. We simulated more water and carbon sources in the medium. Also we tried the effect of oxygen, given that it is present in small quantities, because we were not able to get a sealed biorreactor. For the water and oxygen simulations, we used the photoautotroph model, while for the carbon source (acetate) we used the heterotroph one. The plots of these experiments are below.

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: dx=-fx(x-c) It is understood that f and c are parameters of positive values, when xc. On the other hand, f represent a modulation parameter that changes the maximum growth rate, that correspond to the maximum point in the parabola. As c corresponds to the maximum growth present in the medium, so it is a variable that depends on the amount of resources available for the seaweeds to use, it was consulted with the experimental section what variable would be more important when determining this carrying capacity it was advised that nitrogen was the principal limiting factor, so it is supposed that varying the nitrogen would change that carrying capacity. For our specific case (~0.29 M), the c value is approximated in 997.3 seaweeds per ml (the estimated carrying capacity in the previous part of the model), which might be extrapolated for any concentration of nitrogen present in the medium. For knowing the value of the parameter f, it might be seen that f must modulate the peak of the parabola over 301.5 (as it is reached in certain point of the parabola) but is would be needed to make more regular samples over the same period of time to know a more precise value for f.

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. The above assumptions derive the following model: dC/dt=μ_max (1-C/C_max )C dL/dt=aC-bL Where: C(t) -represents algal population. L(t) -represents the total amount of lipids, lipidic layer. μ_max - is the maximum rate of growth of the algae cells. C_max - is the carrying capacity, i.e., the maximum value of algal growth a - stands for rate at which lipids are produced in cells. d - is the removal factor of the lypids at the lipidic layer.

We note that (0,0) and (C*,L*)= (b/a , 〖a/b C〗_max) are the equilibrium points of the system. Evaluating in the Jacobean matrix in order to determine local analysis, we have:

〖A=J〗_((C*,L*)= ) (■(-μ_max&0@a&-b))

Which defines the eigenvalues: λ_1= -μ_max and λ_2= -b As tr(A)= -μ_max-b , det(A)= -μ_max b , Then 〖tr(A)〗^2-4det⁡(A)= 〖μ_max〗^2-2μ_max+b^2= 〖(μ_max-b)〗^2≥0

Now we have seen that equilibrium (b/a , 〖a/b C〗_max) , is a locally stable equilibrium . More precisely a stable spiral (whenever μ_max>b ).

In biological modeling it’s a good thing to see the presence of a stable equilibrium in the first quadrant of the phase plane. That warrants us to have solutions where the amount of lipids and algae cells are positive and therefore, exist for a determined amount of time. The presence of this behaviour has just told us that at non zero equilibrium (b/a , 〖a/b C〗_max), solutions of the system will tend to approach that equilibrium.

Some considerations about the above model:

We assumed that we removed the dead algae cells that could potentially form a layer of the same inside the bioreactor. If we would like to take that into consideration we would need to add an equation following the dynamics of a dead cell layer compartment.

As we managed to know from Wetlab, since nitrogen was assumed to be the principal limiting factor, we take μ_max, given by (~0.29 M); we take the carrying capacity C_max, which is a value approximated in 997.3 algal cells per ml. which might be extrapolated for any concentration of nitrogen present in the medium. For knowing the value of the parameter f, it might be seen that f must modulate the peak of the parabola over 301.5 (as it is reached in certain point of the parabola) but this would be needed to make more regular samples over the same period of time to know a more precise value for μ_max,.

Another differential equations system The advantage of the having a system described in such a way is that it is possible to incorporate all its parts in a single differential equations system, which consists on three part that were considered relevant on the lipid production; the seaweed population, the lipids present inside the seaweed's cell and the lipids that had been exocyted to the medium and are available for extraction. The equation system developed over this idea is shown in the next image:

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:

1. Ebrahim, A., Lerman, J. A., Palsson, B. O., & Hyduke, D. R. (2013). COBRApy: constraints-based reconstruction and analysis for python. BMC systems biology, 7(1), 74.

2. Zuñiga, C., Li, C. T., Huelsman, T., Levering, J., Zielinski, D. C., McConnell, B. O., ... & Betenbaugh, M. J. (2016). Genome-scale metabolic model for the green alga Chlorella vulgaris UTEX 395 accurately predicts phenotypes under autotrophic, heterotrophic, and mixotrophic growth conditions. Plant Physiology, pp-00593.

3. Blair, M. F., Kokabian, B., & Gude, V. G. (2014). Light and growth medium effect on Chlorella vulgaris biomass production. Journal of environmental chemical engineering, 2(1), 665-674.

4. JinShui Yang, Ehsan Rasa, Prapakorn Tantayotai, Kate M. Scow, HongLi Yuan, Krassimira R. Hristova (2011). Mathematical model of Chlorella minutissima UTEX2341 growth and lipid production under photoheterotrophic fermentation conditions. Bioresource 102(3):3077-82.

Statistic population aproximation The mathematical modeling section was focused in modeling the process of growth and production of the algae, as it is planed 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).

Fig 1. The experimental values of the population (left) against the population sizes provided by the model (right) it is important to notice that time is given in days and population size is in individuals per ml. 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.

Fig 2. Magnitude of the differential values (y axis) depending on the quantity of algae present in the medium (x axis) It is possible to obtain a simulation on the differential values of algae present in the medium using the following as a basis: dx=-fx(x-c) It is understood that f and c are parameters of positive values, when xc. On the other hand, f represent a modulation parameter that changes the maximum growth rate, that correspond to the maximum point in the parabola. As c corresponds to the maximum growth present in the medium, so it is a variable that depends on the amount of resources available for the seaweeds to use, it was consulted with the experimental section what variable would be more important when determining this carrying capacity it was advised that nitrogen was the principal limiting factor, so it is supposed that varying the nitrogen would change that carrying capacity. For our specific case (~0.29 M), the c value is approximated in 997.3 seaweeds per ml (the estimated carrying capacity in the previous part of the model), which might be extrapolated for any concentration of nitrogen present in the medium. For knowing the value of the parameter f, it might be seen that f must modulate the peak of the parabola over 301.5 (as it is reached in certain point of the parabola) but is would be needed to make more regular samples over the same period of time to know a more precise value for f. Differential equations system The advantage of the having a system described in such a way is that it is possible to incorporate all its parts in a single differential equations system, which consists on three part that were considered relevant on the lipid production; the seaweed population, the lipids present inside the seaweed's cell and the lipids that had been exocyted to the medium and are available for extraction. The equation system developed over this idea is shown in the next image:

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 and 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.