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Already in the early stages of our project, we made use of modeling to guide us in designing the experimental part. Modeling provides a look into theoretical parts of the design and provides an evaluation of possible bottlenecks in the design. We used ordinary differential equation (ODE) and agent-based (AB) models, each best fit to the specific concerns. These models supported us in choosing the enzymes for melanin production and the design approaches to test in experiments and showed which results could be expected when different modifications are introduced.

Modeling supports the use of tyrosinase over laccase for melanin production

First, to save time and allow for faster screening, a double substrate model was used to analyze two potential candidates used for melanin production in Saccharomyces cerevisiae. Laccase and tyrosinase both have been identified as enzymes used by microbes to produce melanin from amino acid tyrosine or its metabolites (Tran-Ly et al., 2020). One drawback of laccase is that it requires L-Dopa as a precursor for melanin. This limitation could be avoided by using tyrosinase, in which case tyrosine, which in contrast to L-Dopa is abundant in yeast cells, could be used as the starting molecule for melanin synthesis (Eisenman & Casadevall, 2012). However, we wanted to evaluate if it was beneficial to introduce both enzymes. In this case, tyrosinase would synthesize L-Dopa, which is a substrate for laccase that converts it to dopaquinone. To see how effective this strategy could be, we decided to use a double substrate ODE model to model a single reaction of L-Dopa oxidation to dopaquinone when catalyzed by tyrosinase or laccase. (Figure 1).

Figure 1.

Double substrate ODE model shows higher reaction rate for tyrosinase

Plot shows the rate of L-Dopa oxidation to dopaquinone catalyzed either by tyrosinase or laccase.

In this model the two substrates, L-Dopa and oxygen, were the two limiting factors of the reaction. We kept the enzyme concentration constant in these models, because the aim was to compare the reaction rate when catalyzed by different enzymes.

The general equation for double substrate model of enzyme-catalyzed L-DOPA oxidation kinetics was introduced:

As the basis for building the model we used the constant of maximal reaction rate per mass of enzyme (Vm, U mg–1), Michaelis-Menten constants for L-DOPA and dissolved oxygen (Km ldopa and KmO2, mmol dm–3), constants that denote concentrations of L-DOPA (Cldopa, mmol dm–3), dissolved oxygen (CO2, mmol dm–3) and enzyme (Yenzyme, mg cm–3). The values for constants are presented in Supplementary Table 1.

For all our models we used the approximation of tyrosinase enzyme concentration taken from literature, because it is a realistic value which other organisms have reached (Tahany M. Abdel-Rahman, 2019).

The model revealed two significantly different concentration-rate dependencies of L-DOPA oxidation (Figure 1), which corresponded to the reaction being catalyzed by either laccase or tyrosinase. Tyrosinase has a substantial advantage in reaction velocity compared to laccase. We find that due to the marked difference in velocities of the two enzymes, the contribution of laccase in a two enzyme approach would be small, hence, we decided to continue with just tyrosinase as the main enzyme of the melanin pathway.

Modeling shows the effect of oxygen concentration on the reaction rate

As mentioned in our ENGINEERING page we tried different strategies in order to achieve the most efficient melanin production. After we carried out the EXPERIMENTS, we realized that extracellular melanin production is better than intracellular. We hypothesized that this could be because there is a more reducing environment inside the cells. We decided to modify the previous model to test how oxygen concentration affects the L-Dopa oxidation reaction rate.

Figure 2 shows three different states of the same model. Each represents L-DOPA oxidation reaction rate at the indicated dissolved oxygen concentration. On the figure from left to right parameters of dissolved oxygen concentration: 0.1 mmol dm–3, 0.3 mmol dm–3, 0.8 mmol dm–3

The detailed description of the model mathematics is the same as for Figure 1, described above. It is important to mention that the range of dissolved oxygen concentration value is set up according to the real estimated parameters (from 0 to 1 mmol dm-3).

The simulation shows a strong dependency of the reaction rate on oxygen concentration. A tendency of rapid increase in the reaction rate with the increase of dissolved oxygen concentration on each point of L-DOPA concentration can be observed (Figure 2). This model supports our hypothesis set from experimental data and encourages us to consider the redox state of the environment as a key parameter in melanin synthesis.

Modeling to evaluate the potency of melanin production in yeast

Next, we moved on to estimate the levels of melanin we could reasonably expect from the strain. For this, we proceeded to model the entire eumelanin biosynthesis pathway. This introduced certain difficulties, as the earlier steps of the pathway are enzymatically catalyzed, while later steps are independent of enzymes and their rates are difficult to predict (Eisenman & Casadevall, 2012). For this reason, we modeled the steps of the pathway up to dopaquinone using ODE’s. All reactions following dopaquinone were modeled using AB model. Due to the stochastic nature of AB models, we expect it to have a better chance to predict the random steps of the pathway.

We used the Michaelis-Menten equation for an ODE model to estimate the pathway steps from tyrosine to L-dopa and L-dopa to dopaquinone (Figure 3):

where for our model E is tyrosinase enzyme, S is tyrosine substrate, ES complex represents L-DOPA intermediate, P - dopaquinone.

We used SimBiology library in MATLAB to generate the model. We constructed the model object as sbiomodel and used it as a frame for defining the species, reactions and parameters. In the first part we introduced all species of the equation provided above. In addition, initial values of each of them were defined as concentration in mg/L. The two reactions with corresponding kinetic laws and parameters were defined as follows:

1) tyrosine + tyrosinase → l_dopa with “MassAction” kinetic law and reaction rate constants k1 (kon) and k1r (koff) expressed in 1/(mole*second) and 1/second respectively. We did not assign any value to k1r manually, because the toolkit calculates it automatically.
2) l_dopa → tyrosinase + dopaquinone with “MassAction” kinetic law and reaction rate constant k2 (kcat) expressed in 1/second

The values for these constants are shown in Supplementary Table 1.

After defining all necessary features for the model, we obtained such rate rules (ODEs) for each species:

Using sbiosimulate function we built the graph which represents the concentration of each species in time (Figure 3). The model indicated that L-DOPA approaches its maximal concentration of 16.2 mg/L in two minutes. It also estimates the amount of an important intermediate, dopaquinone, which is the last metabolite that is enzymatically derived. This is an input for our subsequent AB model and can, theoretically, be a good representation of melanin produced subsequently.

Figure 3.

For agent-based model the following pathway was introduced:

dopaquinone → cyclodopa → dopachrome → dhi → indolequinone → eumelanin

The AB model allows us to estimate product concentration using initial substrate concentration and reaction rates for each of the intermediate reactions.

The core principles and techniques for building AB models are uniform among biological modeling languages. We used Kappa language to build the AB model, because it has relatively simple logic and syntax.

To build the AB model we first introduced all agents (substrate with the states of dopaquinone, cyclodopa, dopachrome, dhi, indolequinone, eumelanin and agents that represent each reaction) and constants, which will be described later in this section. For initial concentration of dopaquinone substrate we used data from the ODE model (see the previous section). In the stochastic model we are using countable quantities of agents, so we need to convert values of agents’ molar concentrations to the numbers of molecules in the cell. For this we need to determine the cell volume. For our simulation we decided to model a fraction of the whole cell in order to save time and computing resources. A typical eukaryotic cell volume is about 2.5 x 10-12 L.

We used Avogadro's Number and the model volume to convert molar concentrations to number of molecules:

where n is the number of molecules of the agent a with molar concentration [a]. A represents Avogadro’s Number and V shows fractional cell volume that we used in our model.

There are several types of reaction modeling systems in Kappa language (Figure 4)

Figure 4.

For each reaction in the pathway we used several of these mechanisms, which are the limiting factors in our model. As these reactions are not catalyzed by enzymes, there is no timespace for binding/unbinding units to substrate, and the reactions are stochastic. We used three mechanism types: “binding”, “unbinding” and “modification” for all reactions, however, the constants for the binding and unbinding were set as the lowest possible number in order to denote their non-significance. The rate constants for “modification” mechanism were set according to the literature (Edge et al., 2006; Ito & Wakamatsu, 2008; Land et al., 2003; Odh et al., 1993).

After defining the rate rules and constants, the last step was to define initial conditions. For that we use stochastic conversion factor described earlier in this section. In this case AV quantity (Avogadro’s Number multiplied by fractional cell volume V) is needed to convert molar rate constants to stochastic rate constants.

Figure 5a.

Figure 5b.

Using this model we built a graph that illustrates the dependence of the number of synthesized eumelanin molecules from time (Fig 5A). For better visualization of the stochastic nature of the model, (Fig 5B) shows a selected part of the reaction time-course. Model predicts melanin production of around 14 000 molecules by one cell when production reaches its steady state. This is about 1.767e-5 mg/μL of melanin produced in a single cell. For comparison, a recent study showed production of around 0.420 g/L melanin in an optimized cell factory using black yeast (Elsayis et al., 2022). This modeling shows that while there is room for improvement in melanin production in S. cerevisiae compared to black yeast, we could expect reasonable production efficiency in our yeast strains.

Supplementary Table 1.

Edge, R., D’Ischia, M., Land, E. J., Napolitano, A., Navaratnam, S., Panzella, L., Pezzella, A., Ramsden, C. A., & Riley, P. A. (2006). Dopaquinone redox exchange with dihydroxyindole and dihydroxyindole carboxylic acid. Pigment Cell Research, 19(5), 443–450. https://doi.org/10.1111/J.1600-0749.2006.00327.X

Eisenman, H. C., & Casadevall, A. (2012). Synthesis and assembly of fungal melanin. Applied Microbiology and Biotechnology, 93(3), 931. https://doi.org/10.1007/S00253-011-3777-2

Elsayis, A., Hassan, S. W. M., Ghanem, K. M., & Khairy, H. (2022). Optimization of melanin pigment production from the halotolerant black yeast Hortaea werneckii AS1 isolated from solar salter in Alexandria. BMC Microbiology, 22(1), 1–16. https://doi.org/10.1186/S12866-022-02505-1/FIGURES/5

Ito, S., & Wakamatsu, K. (2008). Chemistry of mixed melanogenesis - Pivotal roles of dopaquinone. Photochemistry and Photobiology, 84(3), 582–592. https://doi.org/10.1111/J.1751-1097.2007.00238.X

Land, E. J., Ito, S., Wakamatsu, K., & Riley, P. A. (2003). Rate Constants for the First Two Chemical Steps of Eumelanogenesis. Pigment Cell Research, 16(5), 487–493. https://doi.org/10.1034/J.1600-0749.2003.00082.X

Odh, G., Hindemith, A., Rosengren, A. M., Rosengren, E., & Rorsman, H. (1993). Isolation of a new tautomerase monitored by the conversion of D-dopachrome to 5,6-dihydroxyindole. Biochemical and Biophysical Research Communications, 197(2), 619–624. https://doi.org/10.1006/BBRC.1993.2524

Tahany M. Abdel-Rahman, N. M. K. M. N. A. E.-G. E. Y. (2019). Purification, characterization and medicinal application of tyrosinase extracted from Saccharomyces cerevisiae. https://www.researchgate.net/publication/346738745_Purification_characterization_and_medicinal_application_of_tyrosinase_extracted_from_Saccharomyces_cerevisiae

Tran-Ly, A. N., Reyes, C., Schwarze, F. W. M. R., & Ribera, J. (2020). Microbial production of melanin and its various applications. World Journal of Microbiology and Biotechnology, 36(11), 1–9. https://doi.org/10.1007/S11274-020-02941-Z/FIGURES/3

Image references: https://www.digitalbiologist.com/blog/2018/7/stochastic-agent-based-model-of-a-cell-signaling-system

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