##### Project Model

### Abstract

In our modeling, we have studied three situations: formation of the Aga1 + Aga2/4D + Melanin complex, the kill switch module and the yeast's resistance to UV radiation with melanin. We have used Ordinary Differential Equations (ODE) for the varying quantities over time. With the objective of analyzing the behavior of the ODEs, we have used Euler’s method, implemented in Python 3, allowing us to visualize its evolution over time. This approach has allowed us to reveal crucial information for the experimental steps.

### Modeling

To tackle this problem, we have worked with a logic of temporal variation rates in which we considered the consuming and production factors to each temporal variable.

#### Melanin, Aga1 and Aga2/4D temporal variation

For the formation of the Aga1 + Aga2/4D + Melanin complex, a series of reactions occur as depict below:

Firstly, free Aga1 will bind to the cell’s exterior if is there surface available to its binding, forming the linked Aga1.

The second step is when free Aga2/4D binds to Aga1, constituting the linked Aga1+Aga2/4D.

The third step is related on the linkage of the free Melanin on the composite Aga1+Aga2/4D, forming the fully finished display Aga1+Aga2/4D+Melanin.

The reactions above are grouped in the reactions below:

Given the reactions above, we have that the varying quantities of each reagent and each product are mathematically represented by equations below. Each of them represents, respectively, the first, the second and the third reaction, as seen in Ref. [1].

The production rate of Aga1 and Aga2/4D are approximately constant, due to the nature of their genes’ promoters. Therefore, the temporal variation of the production of Aga1 and Aga2/4D can be given by a constant as seen in equations below:

Another important factor is the amount of surface area available for the binding of free Aga1. Therefore, the amount of Aga1 on the yeast's surface will be limited to a specific limit number, as shown by the inequation below:

This limit number is given by the ratio between the total available area on a cell and the area that Aga1 can occupy on its surface. To determine the total available area for each cell, we have considered the yeast to be a sphere with a radius given by Ref [2]. With these assumptions, we have calculated the superficial area of the sphere and also made an educated guess for the percentage of the available area (η) for Aga1 as being 30%

Replacing the values found in the literature and the estimated value, we have come to the conclusion that the limit number is given by the inequation previously mentioned.

The limit number (nl) is extremely significant for the formation of the complex and it represents the quantity of Aga1 that’ll be fixed on the surface, directly affecting the quantities of Aga2/4D and melanin. Thus, (nl) will be the initial value of (S) in our modeling. Through the equation above and by determining the final system of derivatives that represents the amount of Aga1, Aga2/4D and melanin overtime for the yeast, we have got the equation system in the equation below:

To consider the entire yeast’s population in the equation above, we needed to multiply the production rates of Aga1, Aga2/4D and the limit number by the number of individuals in the population (N), resulting in the equation below.

During the melanin coupling, the population considered in equation above can be approximated as a constant because of the small-time interval in which the binding process occurs. Thus, in this case, (N) is constant. On the other hand, to consider the population dynamics during the colony’s growth, in which the production and linkage of the display happen, we can use logistic equations, a non-Malthusian approach, to obtain a mathematical model able to describe its evolution over time, as seen in equation below.

With the obtained equations, we have used Euler’s method to visualize the dynamic system’s evolution over time. We have used Python 3 as a fundamental tool for this step. The code used for the simulation can be downloaded below.

Using the equation above, we were able to build the next code, in which we choose an interval to vary the parameters needed to simulate the behavior of one single cell of yeast. The code is available here

Now, using the equation above with a constant number of individuals in the population, we have a new code in Python as seen here. The output is shown in Figure 3.

Considering population dynamics and using the equation above, a new dynamical system is shown by the code here and the output is in Figure 4.

### Kill Switch

In a simplified approach, we have developed an intracellular model for the concentration of the mRNA, by DNA transcription, and the protein concentration, by mRNA translation, resulting in the following equations [1]:

In which (P) is the protein concentration, (M) is the mRNA concentration, (c) is the transcription rate (molecules per hour), (l) is the translation rate (per hour), (d) is the decay rates - or turnover rates - of mRNA and (σ) is the decay rates, or turnover rates, of protein. On the other hand, the transcription factors can inhibit the transcription of their own mRNA.

Using the Hill function, we had a new form for mRNA and protein variation [1].:

In which (c) is the maximum transcription rate, (l) remains as the translation rate and (h) is the saturation constant.

In our kill switch design, there are three ways to kill the cell: using the MET25 promoter, the ADH2-TDH3 promoter and the HXT6 promoter, for which, we will use the index 1, 2 and 3, respectively

The more protein there is, more individuals die. Then, the dynamical population is described by the following equation:

In which (α) is the proportionality parameter. Then:

In this section we have also used the Euler’s Method and generated the graphics in Python 3 using equations above. The code can be seen here, as well as the output in Figure 5.

### UV resistance and toxicity

The yeast population decays approximately exponentially when exposed to UV light and due to melanin toxicity. To mathematically explain this, we have proposed an ODE that describes the temporal behavior of the population when exposed to these factors (equation below). This equation is described in terms of the amount of melanin, since the greater the amount of melanin in the medium, the greater the toxicity and lower the death due to UV radiation.

It is possible to analytically solve this equation, considering that the amount of melanin is constant. Through the integral form, we have Equation 12.

Because pure melanin is very expensive, the experiments that we've led were made with the medium extract of an *Exophiala* sp. culture, which is mostly constituted by melanin.

Since we didn't have any information about the melanin concentration in the *Exophiala* medium extract, we have described the melanin in terms of the total volume used in the samples.

With curve fitting in experimental data, it was possible to determine the parameters in the equation above. We have used OriginPro8 to adjust the curve as shown in Figure 7.

In which (C) is the control yeast, where there is no presence of melanin, and (M) is null and constant (first equation below).

The (M1) curve describes the behavior of the sample that was prepared with a dissecated solution of *Saccharomyces cerevisiae* on a Teflon strip. Then, we deposited a layer of dissecated solution of *Exophiala* sp. extract (second equation below).

At last, the (M2) curve defines the behavior of a sample made up by a dissecated solution constituted of *S. cerevisiae* and the *Exophiala* sp. extract (third equation below).

M1 | M2 | |

κ[1/l] | -27061.3606 | -36483.1328 |

γtox[1/MIN 1/L] | 38399.2867 | 73652.6305 |

γuv[1/L] | 1.2379 | 1.2379 |

As seen in the first equation of this section, we had that the term that inhibits UV death and the term leading to toxicity death are dependent on the amount of melanin in the medium. The more melanin, the less the radiation-induced death and the more the population dies from toxicity.

To find the amount of melanin that stabilizes the number of individuals, we got the values in which the derivative from the first equation of this section is null.

Since the number of individuals is necessarily not zero (N), we know for a fact that the other factor multiplying it needs to be zero.

Thus, choosing an amount of melanin given by Equation 16, we had that the population remains stable.

For (M) to be a real number, the parameters must obey the following inequality:

And (M) must be a positive number.

Using the parameters from Table 2, we have that the ideal melanin is given by Table 3.

M1[uL] | M2[uL] | Concordance[%] | Average[uL] |

57.6262 | 39.1711 | 67.97 | 48.3986 |

Therefore, the modeling of our project has shown that there is an ideal amount of melanin that can be used until the toxicity effects start to overcome its protective properties. Consequently, in any case, if we want to protect our yeast colony from the harmful effects of the UV radiation, maximizing its survival rate, we must use the quantity of melanin that we've obtained.

### Glossary

### References

- [1] Prof Guy-Bart Stan. Modelling in biology. 2019.
- [2] Ian T. Cameron Paul Young G. M. Padilla Byron F. Johnson Anwar Nasim, Dennis E. Buetowand A. M. Zimmerman (Eds.).Molecular Biology of the Fission Yeast. Cell biology. AcademicPress, 1989.
- [3] Luiz Henrique Alves Monteiro. Sistemas Dinâmicos. Editora Livraria da Física, 4 edition, 2019.
- [4] Rob J. de Boer Kirsten ten Tusscher.Theoretical Biology. Utrecht University, 2019.