The starting point for our agent-based stochastic model was work performed by Schützhold et al (2016). The authors offered an approach for computational modeling of yeast lipid metabolism. The agent-based stochastic mathematical model where each lipid-related reaction can be modified using a set of existing parameters was implemented. In this model only one TAG lipase, Tgl4, is present. The number of enzymes is reduced to simplify the model but save the dynamic approach and comprehensiveness. As was clarified by one of the authors of this research, Dr. Jens Hahn, Tgl4 was chosen due to the multifunctional nature of this enzyme in the yeast lipid metabolism: it efficiently hydrolyzes both TAGs and steryl esters (SE) lipid classes (Rajakumari & Daum, 2010). In our model, we assume that the lipase activity in our strains is absent. However, according to the commentary by Dr. Jens Hahn, the complete exclusion of the lipase activity from the model would affect the system more than the mere knockout of the Tgl4. Nonetheless, since in our wet lab experiments, we made triple deletion of the most active lipases (tgl3∆ tgl4∆ tgl5∆) , we assume that consequences for TAGs accumulation are comparable with the full loss of lipid-hydrolyzing enzymes (labeled as tglΔ), and remaining minor lipase activity can be ignored in the model (Fig 2).
The main unit of the model is an enzymatic reaction, which results in the production of a precursor for lipid synthesis or a lipid molecule itself. Other parameters such as the cell cycle length may be held constant. In the model, the duration of a single cell cycle is 7200 seconds. In every time point (time step equals 1 second) enzymatic reactions are executed according to their corresponding probabilities. The probability of the reaction to occur together with the maximum number of executions Nmax per time step can be changed. These parameters are based on the Michaelis-Menten equation for enzymatic reactions: Nmax corresponds to vmax, [S] corresponds to the substrate concentration, KM is a Michaelis-Menten constant and probability p is calculated as follows.
As a result, the production rates of different molecules can be adjusted depending on the genetic modifications. After every step of the simulation, probabilities of reactions are updated according to the substrate concentration in that step. The output of the simulation is 7200 data points, indicating the cellular content at each time step. We ran 10 simulations of the lipid accumulation in lipid droplets of both WT and the modified strain to consider the influence of lipase deletion (tgl∆) (Fig 1). In the case of deleted lipases, the number of TAG lipase reaction executions Nmax was set to 0, which corresponds to complete loss of the enzyme activity.
Figure 1. Lipid accumulation in lipid droplets in S. cerevisiae cell. The accumulation of the lipid molecules (TAGs and SEs) in the lipid droplets of a single wild-type cell (Wt) and a genetically modified cell with deleted lipases (tglΔ) during one cell cycle (7200 seconds).
The results show that at the beginning of the simulation (approx. during the first 30 min) both WT and tgl∆ strains accumulate lipids at similar rates. However, at later time points, lipid content continues to rise in the cells of the tgl∆ strain, while it gradually decreases in the WT cells (Fig 1). In the late G1 phase of the yeast cell cycle, buds emerge from the mother cells. This creates an increased need for lipids to build the growing membranes of the buds. TAGs are mobilized from lipid droplets due to lipase activity of Tgl4, which, in turn, is activated via phosphorylation by cyclin-dependent kinase (CDK1/CDC28) (Kurat et al., 2009). When the lipases are deleted, TAG and SE molecules are not utilized and their accumulation continues, resulting in an up to 2.25 fold increase in the number of lipid molecules (18000 molecules per cell compared to 7800 molecules per cell in the WT) according to the model (Fig 1).
To further increase the lipid accumulation, we aimed to model overexpression of two enzymes responsible for lipid production in S. cerevisiae: Phosphatidic acid phosphohydrolase 1, or PAP, and Diacylglycerol acyltransferase, or DGAT. PAP converts phosphatidic acid (PA) into DAG, and DGAT catalyzes the formation of TAG from DAG (Fig 2, see Description for more information). Consequently, the efficiency of DAG production could be a limitation in TAG assembly, and the enzymatic activity of the two mentioned enzymes could differently influence the accumulation of the final product. As it is described by Schützhold et al. (2016), altering different reactions’ parameters will variously affect the system. For instance, the sensitivity of the model to changes in TAG lipase kinetic parameters is drastically higher than the sensitivity to overexpression of TAG-producing enzymes. We made model simulations with an increased reaction rate (10 times) for both DGAT and PAP enzymes separately and together to observe the difference in the final lipid content (Fig 3, 4).
Figure 2. A concise scheme of the lipid metabolism in S. cerevisiae . Enzymes encoded by PAH1 and DGA1 genes catalyze reactions leading to TAGs formation, while lipases Tgl3, 4, and 5 degrade TAG molecules. FFA – free fatty acids. The figure is based on the data obtained by Ma et al., 2019.
First, we considered the effect of separate overexpression of DGAT and PAP. Although the decrease in lipid content after G1 phase does not occur when DGAT is overexpressed, unlike with PAP overexpression, the influence of these two enzymes on TAG accumulation in lipid droplets is equalized in the absence of the lipase activity (Fig 3). Secondly, we compared the accumulation of lipids with overexpressed DGAT and PAP genes separately and together (Fig 3, 4).
Figure 3. The effect of coupling each enzyme overexpression with the lipase deletions. The accumulation of the lipid molecules (TAGs and SEs) in the lipid droplets of a single wild-type cell (dark blue), a genetically modified cell with deleted lipases (tglΔ) (yellow), a cell with overexpressed DGAT (light blue), a cell with overexpressed and deleted lipases (light blue dashed), a cell with overexpressed PAP (green) and a cell with overexpressed PAP and deleted lipases (green dashed).
Higher yields of lipid molecules were observed in the cell with simultaneous overexpression of both PAP and DGAT enzymes and lipase deletions, while overexpression of the single enzyme in combination with tgl∆ resulted in lower TAG yield (Fig 4).
Finally, the number of lipid molecules in the modified yeast strain increased more than 4 times (Fig 4) compared to the wild-type strain (32000 molecules compared to 7800 molecules).
Figure 4. The effect of coupled PAP and DGAT overexpression in a cell with deleted lipases . The accumulation of the lipid molecules (TAGs and SEs) in the lipid droplets of a single wild-type cell (dark blue), a cell with overexpressed DGAT and PAP and deleted lipases (light blue), a cell with overexpressed DGAT and deleted lipases (green dashed) and a cell with overexpressed PAP a deleted lipases (yellow dashed).
The results of the agent-based model indicate that the combination of TAG lipase deletions together with overexpression of TAG-producing enzymes led to the highest amount of lipids in the cell. These modifications were later implemented in the lab.
Agent-based modeling helps to resolve the metabolic complexity and does not require many reaction equations and kinetic constants. However, the same benefits of this model can also become its limitations when it comes to modeling the expression of genes from complex synthetic constructs.
The agent-based model does not allow to simulate some types of experimental conditions. For example, changing the amounts of NADPH and acyl-CoA in the model did not affect the predicted lipid concentration significantly. Additionally, the model is limited to the duration of a single cell cycle, and thus does not allow for simulation duration that would resemble real production conditions.
In order to construct a more comprehensive model that would allow us to make predictions about the final TAG yield and thus evaluate the feasibility of our strain, we decided to use a set of deterministic ODEs. Unlike stochastic approaches to the modeling of biological systems, deterministic models such as ODE models are based solely on the kinetic characteristics and the initial conditions of a system (Palaniappan et al., 2019). For example, the following set of ODEs describes VP-EL222 mediated gene expression (Benzinger & Khammash, 2018).
On the molecular level, blue light illumination triggers the dimerization of VP-EL222 transcription factor (TF), which shifts the equilibrium towards its active form and subsequently increases the strength of its binding to DNA. This is reflected by an increase in the value of kon parameter. In (1), parameter I denote blue light intensity which acts as a scaling factor for kon. The model includes a total of 10 parameters whose description, values, and sources can be found in supplementary table 1.
The advantage of the VP-EL222 based system is its short response time and a low degree of leakiness (Benzinger & Khammash, 2018). To study such behavior, we modeled the protein expression levels of PAP (product ofPAH1) and DGAT (product of DGA1) under light intensities ranging from 0 μW cm-2 (no light) to 400 μW cm-2.
Use slider to adjust light intensity
I 0
Figure 5. The dependency of enzyme concentration on light intensity. Light intensity (I) is given in units μW cm-2. Blue line: PAH1 gene expression under the control of 5xBS-CYC180pr. Wild-type level of PAP (PAH1 gene product) is 2607 molecules per cell. Overexpression at light intensity 400 μW cm-2 is predicted to increase PAP abundance 4.5 times in comparison to wild-type strains. Yellow line: DGA1 expression under the control of 5xBS-GAL1pr. Wild-type level of DGAT (DGA1 gene product) is 1431 molecules/cell. Overexpression is predicted to increase DGAT abundance 6 times in comparison to wild-type strains. It can be seen that after I = 320 μW cm-2 increasing light intensity does not influence DGAT concentration significantly; for PAP, the same happens after I reaches 400 μW cm-2. The image is dynamic. Light intensity I is changed by pulling the slider left and right.
The results demonstrate that the system possesses fast response times and low basal expression, which allows us to fine-tune transcription factor activation to achieve the desired gene expression.
Clearly, increasing light intensity results in a higher gene expression. However, in practice, increasing light intensity will not necessarily lead to higher lipid production when the cells are coated with semiconductor nanoparticles. It has been shown that prolonged exposure to high light intensities can lead to photodamage (Prof. Guo, Fig 6). This is reflected by a decrease in quantum efficiency, which in our case is the ratio of the number of photogenerated electrons used to regenerate NADPH to the number of photons.
We contacted Prof. Guo regarding this, since we were interested in how this effect could be minimized or prevented. Prof. Guo informed us that it is possible that the excess of photogenerated electrons can lead to elevated levels of reactive oxygen species causing cellular stress. Therefore, in order to decrease phototoxicity, illumination power should be kept below the level at which induced cellular stress starts to adversely affect lipid production. Guo et al. (2017) have shown that after 5.60 mW cm-2 the yield started to decrease. This is applicable to our strain as well, since the species used in the study was the same.
Figure 6. External quantum yield of NADPH regeneration at different illumination times, adapted from Guo et al. (2017). This was estimated based on photochemical shikimic acid production. Error bars represent standard error of three independent replicates. External quantum yield describes the ratio of electrons participating in a reaction to the number of incident photons. There is a significant decrease in external quantum efficiency, possibly due to photodamage.
In order to assess the feasibility of our constructed yeast strain, we decided to model the production of TAGs and compare the final yield to the published values (Teixeira, David, Siewers & Nielsen, 2018). For that, we used the Michaelis-Menten model. Under the quasi-steady-state approximation, the rate of product formation can be expressed as Michaelis-Menten equation (Briggs & Haldane, 1925):
This equation allows us to find the rate of product formation, given the concentration of an enzyme and that of a substrate.
As was previously mentioned, in S. cerevisiae TAGs are synthesized from DAGs by DGAT (DGA1 gene product), and DAGs, in turn, are obtained through the conversion of phosphatidic acid (PA) by PAP (PAH1 gene product). Therefore, the following system of Michaelis-Menten equations describes the production of TAGs:
The currently accepted model of lipid biogenesis called budding model assumes that DGAT and other proteins involved in neutral lipid metabolism accumulate in certain regions of endoplasmic reticulum (ER). Indeed, protein localization studies have revealed that DGAT is a microsomal protein (Athenstaedt & Daum, 2006). This is in line with the fact that TAGs are nonpolar molecules which cluster in the hydrophobic region within the ER bilayer. Therefore, our model assumes the formation of DAGs and consequently TAGs from microsomal PA (i.e. PA found only in ER). In order to do that, we estimated the concentration of PA given 24.2 mg phospholipids/g CDW and the fraction of total phospholipids that is PA in microsomes which was found to be 0.2% (Zinser et al., 1991) (Fig 7, bottom panel). The kinetic constants found in both Michaelis-Menten equations are provided in Supplementary Table 1.
To account for the expression of endogenous PAH1 and DGA1, we use the median abundance of their products (2607 and 1431 molecules/cell, respectively (Ho et al., 2018)) and assume that it remains constant.
Figure 7. Accumulation of TAG and PA throughout the cultivation. Top panel: TAG production and accumulation dynamics according to the ODE model, wild type (WT), and a mutant with overexpressed PAP and DGAT enzymes. Final TAG concentration for the WT culture is predicted to be 13.9 mg/gCDW (CDW – cell dry weight); PAP and DGAT overexpression increases TAG content more than 6 times, raising the final concentration to 86.7 mg/gCDW. To model wild type accumulation, PAP and DGAT concentration were set corresponding to median cell abundance and multiplied by the number of cells (derived from the biomass). Light induction with the intensity of 420 μW cm-2 starts on the 25th hour of cultivation. Bottom panel: the model of biomass growth and glucose consumption of S. cerevisiae W303 strain (see our iGEM 2019 model for details and parameters) used as a basis for PA accumulation modeling.
Predicted TAG concentration reached the value 86.7 mg/gCDW at the end of the 72 h cultivation period (Fig 7, top panel). A similar study performed by Teixeira et al. reported yields of around 80 mg/gCDW of TAG for a strain with overexpressed DGA1, PAH1, deletions of TAG lipase genes TGL3/4/5, and the ACC1S659A,S1157A,S686 mutation which removes the feedback inhibition of ACC1 activity by SNF1 (Teixeira, David, Siewers & Nielsen, 2018).
Our mathematical model of TAG production in lipid droplets takes into account the overexpressed PAH1 and DGA1 genes as well as deleted TAG lipases TGL3/4/5 . The latter is reflected in the model by the fact that degradation of TAG is not considered. As a result, our model allows us to predict the final TAG yield at the end of a cultivation period of any given length and under various light conditions, which makes it a useful tool during the design of the production system.
Our model, however, does not consider the effect of electron generation by nanoparticles or the efficiency of our improved yeast autolysis method. Although these aspects will affect the final lipid yield, they are difficult to model mathematically using the available tools. The electron transfer from semiconductor to cytosol, and eventually to the enzyme, is a complex process which is not yet fully understood. There are some theories which explain the underlying mechanism; however it seems that a definitive consensus has not been reached yet (Guo et al., 2018). Therefore, modeling of electron generation and transfer will not be implemented in our dry lab this year.
In our dry lab, the agent-based model helped us to estimate the effect of each modification alone and their combinations. We found that a lipase knock-out is not efficient enough for reaching high theoretical yields, but when coupled with overexpression of PAH1 and DGA1 it can sufficiently increase the concentration of lipids. Our ODE model accuracy was successfully verified by comparing the simulation results to available values in literature. The results indicate that our cultivation approach with a 25-hour long growth phase and a subsequent production phase where PAP and DGAT are overexpressed increases theoretical yields by more than 6 times compared to the unmodified strain. In the literature higher TAG content of 254 mg/gCDW has been reported. Since our model does not consider other major modifications, such as Zwf1 deletion coupled with increased electron supply and PLIN3 overexpression, final theoretical yields of 86.7 mg/gCDW after 72 hours of cultivation of the strain with PAH1 and DGA1 overexpression, and lipase deletion seemed promising to us. We expect that the actual lipid yields in our final production strain will be higher because of the additional improvements. Overall, the results of our dry lab work indicate that our idea can be successfully implemented in the lab and we can move on.
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