Model

Overview

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A mathematical model is a simulated data-generating rule for an observed object described using mathematical methods. In particular, in the natural sciences, they are often described in the form of differential equations. This is because the derivative is the rate of “change” and is very compatible with the natural sciences, which observe natural phenomena and look at “change. The mathematical models in iGEM are often related to gene expression and protein function, and help in the design and understanding of projects. the DBTL cycle is one of the most important concepts in iGEM, through which projects are refined.

Our project will create a biological system that converts the energy of radiation, which is harmful to living organisms, into useful energy. The project has four key words

• Melanin
• Radiation and Ultraviolet radiation
• Space
• Kill switch

From these four perspectives, we have developed six models for a deeper understanding of the project.

1. Tyrosinase production
2. Melanin production
3. UV tolerance and melanin toxicity
4. Radiation resistance
5. Cosmic radiation
6. Kill switch

Each model and keyword corresponds to the following.
Melanin: model1, model2, model3, model4
Radiation and Ultraviolet radiation: model4, model5
Space: model5
Kill switch: model6

The above models were developed from different perspectives to provide a better understanding of the project. An overview of each of the models is given below.

The results from our wet experiments revealed that melanin production takes longer than expected. Dry experiments suggested that this delay might be caused by insufficient IPTG concentration. We decided to examine how the expression of tyrosinase, the enzyme responsible for melanin production, varies with different IPTG concentrations. The model results indicated that increasing IPTG concentration up to 5.0 mM could improve melanin production. However, since enzymatic reactions like those involving tyrosinase are typically very efficient, even a small amount of the enzyme may be sufficient for the reaction. Therefore, we proceeded to investigate the optimal tyrosinase concentration needed for rapid melanin production in the next model.

Based on the results of Model 1, we further explored melanin production at various tyrosinase concentrations. As the IPTG concentration increased, so did melanin production efficiency. However, the rate of production reached a saturation point, and from a cytotoxicity perspective, an IPTG concentration between 0.5 to 1.0 mM was found to be optimal. Through additional parameter changes and simulations, we discovered that structural changes in tyrosinase could potentially enhance melanin production efficiency.

We predicted the survival rate of Shewanella cells expressing melanin when exposed to ultraviolet or radiation. While melanin is known to protect cells from radiation, it can also introduce cytotoxicity, leading to cell death. Considering both factors, the model showed the progression of cell population over time and calculated the optimal melanin concentration. Although the optimal concentration varies depending on the radiation wavelength, melanin concentrations above 0.8 g/L lead to stronger cytotoxic effects, reducing growth rates despite increased radiation shielding.

In our project, we use melanin's radiation protection capabilities to shield microorganisms from radiation. To test this, we developed a simple model simulating melanin’s radiation shielding ability. The results showed that a melanin concentration of approximately 0.2 g/L in the periplasm is effective for radiation protection. However, when considering high-energy radiation, particle radiation, and radiation permeability, a higher melanin concentration would likely be required. Future simulations will focus on constructing models under more realistic conditions to evaluate the system's feasibility in practice.

ENEducer is intended to be cultivated inside spacecraft, where strong cosmic radiation is present and could impact cell viability. In this model, we calculated the radiation dose absorbed by cells using absorbed dose rates. Focusing on secondary radiation generated when primary radiation (protons and electrons) from space collides with the spacecraft walls, we estimated the dose cells might receive. Simulations were conducted by varying the distance from Earth and the position of the cells within the spacecraft.

Our designed kill switch operates in a copper ion concentration-dependent manner. In the growth medium, where copper ions are present at 10^-6 M, the kill switch remains off. However, when the bacteria leave the medium and the copper ion concentration around the bacteria drops below 10^-10 M, the kill switch clearly turns on. This ensures controlled cell death when cells stray from their designated environment, enhancing safety in applications.

Model-1

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Check the change of Tyrosinase expression level by IPTG concentration. Wet experiment showed that Tyrosinase expression level is low and Melanin production is low (takes time to produce). In Dry, we will examine how Tyrosinase expression level changes with IPTG concentration, thinking that IPTG concentration may be the cause.

A schematic diagram of the reaction pathway is shown below. Based on this, we constructed an ODE model in MATLAB and performed simulations using MATLAB R2024a. Our preliminary research led us to a paper that constructs a model based on homodimerization and binding/dissociation reactions of each molecule. The parameter values were taken from this paper and the model was used as is.

Why did we use this model?

One of the most important factors in modeling is assumptions. No matter how accurate the model is, it is useless if it is outside of the assumptions we set. We decided to use the model in this paper because it assumed the same situation we wanted to simulate.

The lactose operon model is used. A repressor expressed from the lacI gene on the genome of E. coli DH5α forms a dimer and binds to the operator (lacO) on the plasmid to repress melA (Tyrosinase) gene expression. IPTG added to the medium forms a dimer and binds to the free lacI dimer, thereby inhibiting the binding of lacO to the repressor. Furthermore, the IPTG dimer dissociates the lacI dimer from the lacI dimer-lacO complex to form the lacI dimer-IPTG dimer complex. This removes the physical barrier of lacO and allows RNA polymerase to bind to the Promoter, thereby promoting transcription. Of the lacI dimer-IPTG dimer complex generated by the above process, only lacI is degraded and two molecules of IPTG are released and reused. In addition, the model takes into account that even if the lacI dimer is bound to lacO, there is leakage that allows transcription to proceed. The following differential equations were developed.

Several assumptions are made in this model: first, the IPTG concentration in the medium (IPTG concentration added to the medium) is assumed to be equal to the intracellular IPTG concentration. This is made because the diffusion rate of the substance is sufficiently fast that IPTG is taken up into the cell and the final concentrations are equal.

• Intracellular IPTG concentration and medium IPTG concentration are equal, no change in IPTG concentration
• Plasmid copy number is 25
• E. coli volume is 8 x 10^-16 L

The list of parameters and initial values for this model are as follows.

Parameters

Parameters	Description	value	unit	source
ksMR	lacI transcription rate	0.23	nM/min	[1]
ksR	lacI translation rate constant	15	/min	[1]
dmRNA	mRNA degradation rate constant	0.462	/min	[1]
d1	protein degradation rate constant	0.2	/min	[1]
k2R	lacI dimerization rate constant	50	/nM・min	[1]
k-2R	lacI dimer dissociation rate constant	10-3	/min	[1]
kr	association rate constant for repression	960	/nM・min	[1]
k-r	disassociation rate constant for repression	2.4	/min	[1]
kdr1	association rate constant of IPTG and lacI	3.0×10-7	/nM-2・min	[1]
k-dr1	dissociation rate constant of IPTG2:lacI2	12	/min	[1]
kdr2	association rate constant of IPTG and lacI2:lacO	3.0×10-7	/nM-2・min	[1]
k-dr2	dissociation rate constant of IPTG2:lacI2 (association rate constant of lacO and IPTG2:lacO)	4.8×103	/nM・min	[1]
ks1Mm	melA transcription rate constant from lactose operon	0.5	/min	[1]
ks0Mm	leak melA transcription rate constant	0.01	/min	[1]
ksTyr	Tyrosinae translation rate constant	20	/min	[1]

Initial Value

Abbreviation	Description	initial value	unit	source
[genome]	Concentration of E.colli genome	2.1	nM	calculated
[lacO]	oncentration of Plasmid (pHSG398)	50	nM	estimated
[IPTG]	Concentration of IPTG	0, 0.1, 0.5, 1.0, 10, 100	mM	depends on conditions

In the wet experiment, the IPTG concentration in the doubling medium is 0.5 mM. Therefore, we first examined how Tyrosinase expression changes over time at an IPTG concentration of 0.5 mM.

Tyrosinase expression increased with increasing IPTG concentration, indicating that at low IPTG concentrations, repressor-mediated transcriptional repression is active, whereas at high IPTG concentrations, the repression is removed and transcription is promoted. The IPTG concentration of 0.5 mM in the Wet experiment is approximately the middle level of Tyrosinase expression. This may be due to leakage, where transcription occurs even if the repressor is bound to the operator.

Tyrosinase expression increased with increasing IPTG concentration, indicating that at low IPTG concentrations, repressor-mediated transcriptional repression is active, whereas at high IPTG concentrations, the repression is removed and transcription is promoted. The IPTG concentration of 0.5 mM in the Wet experiment is approximately the middle level of Tyrosinase expression. This may be due to leakage, where transcription occurs even if the repressor is bound to the operator.

To better understand the mechanism of IPTG-induced derepression, we not only examined changes in Tyrosinase expression as a result of changes in IPTG concentration, but also changes in other substances.

First, we examined changes in IPTG concentrations of lacO without the repressor bound and lacI2:lacO complex with the repressor bound.

Next, the mRNA for melA, the gene encoding Tyrosinase, was examined.

The behavior of mRNAmelA is consistent with that of Tyrosinase. However, the concentration of mRNA is negligible compared to the concentration of Tyrosinase. This indicates that protein synthesis from mRNA proceeds with high efficiency, and even a small amount of leakage can result in the production of a significant amount of protein. In melanin production, a slight leakage is not a problem, but in cases where leakage is critical, such as in kill switches or substance detection, caution must be taken when using the lactose repressor for transcriptional repression.

The results suggest that the IPTG concentration should be increased to a maximum level of 5.0 mM. However, reactions with enzymes such as Tyrosinase generally proceed so efficiently that a smaller amount of enzyme compared to substrate may be sufficient in terms of reaction rate. Therefore, we investigated the following model to determine what concentration of Tyrosinase is required to produce melanin fast enough.

1. Stamatakis, M. (2009). Comparison of deterministic and stochastic models of the lac operon genetic network. Biophys J, 96(3), 887-906. https://doi.org/10.1016/j.bpj.2008.10.028

Model-2

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From model (1), it was found that Tyrosinase expression changes with IPTG concentration. Therefore, we next examined how melanin production changes with each Tyrosinase concentration.

Melanin is produced from tyrosine in vivo via multiple reactions. A schematic diagram of the reaction pathway is shown below. The model used the Michaelis-Menten equation, which represents a general enzymatic reaction, and a first-order chemical reaction.

The reaction tyrosine→dopa→dopaquinone is catalyzed by Tyrosinase. The process of melanin formation from dopaquinone through multiple reaction intermediates is a spontaneous oxidation reaction. At first, we attempted to construct and analyze a model with differential equations for all of the above reactions. However, we simplified the model as follows because the model would become complicated due to the large number of parameters and variables, and because there is essentially no problem in treating the enzymatic reaction and the spontaneous oxidation reaction together.

The enzyme-substrate reaction of tyrosine with tyrosinase produces dopaquinone. Subsequently, melanin is produced from dopaquinone by an oxidation reaction. The following differential equation was formulated.

Several assumptions are made in this model.

• Tyrosine → dopaquinone is a single enzymatic reaction, and dopaquinone → melanin is a single oxidation reaction.
• The amount of tyrosinase is constant.
• Melanin degradation is not considered.

The list of parameters and initial values for this model are as follows.

Parameters	Description	value	unit	source
k1	Tyrosine binding rate constant	11.443	L/g・h	[1]
k2	Tyrosine dissociation rate constant	6.049	/h	[1]
k3	Tyrosinase reaction rate constant	0.209	/h	[1]
k4	Melanin puroducing reaction rate constant	0.030	/h	[1]
-	Tyrosinase molecular weight	66330	-	calculated

Parameters	Description	initial value	unit	source
[Tyrosine]	Tyrosine concentration	0.6	g/L	Wet Lab
[Tyrosinase]	Tyrosinase concentration	0.02, 0.166, 0.279, 0.358	g/L	depends on conditions

The Tyrosinase concentration was simulated at 2500 nM, or 0.166 g/L. Despite the rapid decrease in Tyrosine and the rapid increase in Dopaquinone, the Melanin concentration takes about 140 hours to reach steady state. This result indicates that the rate-limiting step in the reaction of melanin production is the spontaneous oxidation reaction of Dopaquinone→Melanin.

Next, we examined how Melanin production changes when the amount of Tyrosinase is varied. From the results of Model 1, the Tyrosinase concentration at each IPTG concentration was inferred.

Melanin production was found to be faster as the amount of Tyrosinase (IPTG) increased. However, there was a limit to the increase in speed, and not much difference was seen when IPTG was above 1.0 mM. This suggests that the optimal IPTG concentration is 1.0 mM. These results were reported to the Wet Lab, and the IPTG concentration in the medium is currently under discussion.

The Wet Lab is experimenting with a melA mutant in the gene melA encoding Tyrosinase for rapid melanin production. This melA mutant has been shown to speed up melanin production, but the detailed factors are not clear. We therefore investigated the increase in Melanin production efficiency by varying several parameters (k₁, k₂, k₃).

Melanin production efficiency increased with increasing k₁ and k₃ and decreasing k₂. The time required to produce 0.3 g/L of melanin, half of the maximum, was approximately 45 hours before and 30 hours after changing the parameters. The larger k₁ and smaller k₂ indicate stronger binding of Tyrosine to Tyrosinase. In melAmut, where melanin production is more efficient, mutations in Tyrosinase amino acids may alter the structure of Tyrosinase. Tyrosinase structure may be altered by mutation of Tyrosinase amino acids. The results suggest that such conformational changes may result in increased Tyrosine binding strength and increased oxygen binding rate, which may increase the rate of Melanin production.

The model can simulate different situations by changing the values of the parameters. Here, k₁, k₂, k₃, and k₄ are varied independently to see if the Melanin production rate increases.

Varying k1

The figure shows that as k₁ increases, the rate of melanin production increases. k₁ represents the binding rate between Tyrosine and Tyrosinase, so a larger k₁ represents a stronger binding.

Varying k₂

The figure shows that the rate of melanin production increases as k₂ decreases. k₂ represents the dissociation rate of Tyrosine and Tyrosinase, so a smaller k₂ represents a stronger binding.

Varying k₃

The figure shows that as k₃ increases, the rate of melanin production increases. k₃ represents the enzymatic reaction rate of Tyrosinase, so its increase represents an increase in its rate.

Varying k₄

The figure shows that as k₄ increases, the rate of melanin production rises. Since k₄ represents the reaction rate of the spontaneous oxidation from dopaquinone to melanin, its increase indicates a faster reaction rate. The increase in melanin production rate due to k₄ was more significant than that caused by other parameters (k₁ to k₃). This result aligns with the discussion in Figure 2-3, suggesting that the rate-limiting step in melanin production is the spontaneous oxidation reaction from dopaquinone to melanin.

Using this model, we investigated how melanin production varies with IPTG concentration: the greater the IPTG concentration, the more efficient the melanin production. However, there is a limit to the production rate increase, and we concluded that the optimal range of IPTG concentration is 0.5~1.0 mM in terms of cytotoxicity.

Simulations with different parameters showed that the melanin production efficiency may increase due to the conformational change of Tyrosinase.

In the future, we would like to perform fitting using the data obtained in the Wet Lab to build a more accurate model.

1. iGEM 2022 NCKU_Tainan https://2022.igem.wiki/ncku-tainan/model

Model-3

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To predict the number of S.oneidensis cells expressing melanin during UV irradiation. Melanin acts to protect cells from UV light and radiation, and we will investigate how melanin concentration affects survival.

The temporal changes in the number of cells in the normal state can be expressed using a logistic function.

In melanin-expressing cells, survival may be reduced due to melanin's cytotoxicity. Those models are shown below.

Solving these differential equations in variable-separated form yields the following equations.

The model was created with the following assumptions

• Temperature is set at 22°C (growth rate and carrying capacity vary with temperature)
• Melanin expression does not reduce carrying capacity
• The initial number of cells (N0) is set to 0.1 g/L

Parameters were set as follows with reference to [1] N0 = 0.1, which was considered sufficient to meet the time required for cells to enter the logarithmic growth phase in [1].

Parameters	Description	value	unit	source
r	Specific growth rate	0.2451	/h	[1]
rtox	Toxicity death rate	0.2	L/g・h	Estimated
M	melanin concentration	0.6	g/L	Model2
K	Maximum cell mass	1.93	g/L	[1]

Changes in cell number over time were examined in the absence of melanin expression (control) and in the presence of melanin expression (melanin expressed).

Melanin expression was shown to have a slower growth rate.

Cell number decreases exponentially during UV irradiation. The number of cells can be mathematically estimated by using ordinary differential equations for temporal behavior. The more melanin expressed, the less UV-induced death, but the more death due to cytotoxicity. This model was created based on the model done by the igem team iGEM 2019 Sao_Carlos-Brazil[2]. However, we considered that when UV tolerance increased, the number of cells may not decrease, but rather trend upward. To achieve this, we varied the parameter of the reduction coefficient depending on the UV intensity and introduced a logistic function to account for the possibility of an increase in cell number. The model under UV irradiation is shown below.

When cells increase, the number of cells increases slowly up to the carrying capacity, and when they decrease, the number of cells decreases exponentially without being affected by the carrying capacity. Solving this differential equation yields the following equation.

In this model, the change in the number of cells in the radiation environment of the International Space Station (ISS) is examined. It is known that the radiation resistance of S.oneidensis is lower compared to other bacteria. The absorbed dose for S.oneidensis's D10 (the dose that reduces the cell count to 10%) is 0.07 kGy [3]. Using the fact that the radiation dose at ISS is 1 mGy/day, we found that it takes 0.81hour to reach 10% cell count at ISS. r_uv=-2.843 was calculated by substituting N=0.1, t=0.81h into N= N0e^-r_uvt.

Following the above assumptions, we can calculate that 34.7hour is required for the number of melanin transduced S.oneidensis cells to reach 10% in ISS, r_uv = -0.0663. Other parameters λ, κ, and c were predicted from r_uv for melanin 0 g/L and 0.6 g/L.

Parameters	Description	value	unit	source
r	Specific growth rate	0.2451	/h	[1]
rtox	Toxicity death rate	0.2	L/g・h	Estimated
K	Maximum cell mass	1.93	g/L	[1]
λ	Radiation intensity rate	0.7477	/eV・h	Aforementioned
ϰ	Radiation protection rate	25.22	L/g	Aforementioned

We examined how r_uv (rate of increase of cells) changes with melanin concentration.

At low melanin concentrations, r_uv is small due to the strong influence of UV radiation. It increases as melanin concentration increases, but once it reaches the peak, it decreases due to the cytotoxic effect of melanin. Cell viability is high when the UV energy is low, and at 500 nm there is a melanin concentration at which r_uv is positive. At all wavelengths, cell numbers were not maximal at concentrations above 0.8 g/L.

Next, the number of cells at different melanin concentrations was examined at 300 nm, 400 nm, and 500 nm.

At 300 nm and 400 nm, cell number decreased exponentially over time at all melanin concentrations. At low melanin concentrations, the cell number decreased due to UV radiation, but at too high melanin concentrations, the survival rate decreased due to toxicity, which can reflect cytotoxicity. At 500 nm, r_uv was positive at the optimum melanin concentration and increased slowly toward the carrying capacity.

From fig3-2, it was found that there is a melanin concentration at which survival is maximized when the effects of UV light and cytotoxicity are balanced. By finding this optimal melanin concentration, the number of cells can be maximized.

At melanin concentrations where the derivative of r_uv is zero, r_uv is maximal.

In order for M to be a real number, the inside of √ must be positive, but since all parameters are positive, the condition is satisfied. In addition, the condition for the melanin concentration to be positive is as follows.

Using the parameters in Table 2, we find that E_uv≥0.01. Therefore, the equation is valid for UV and radiation with energies above 0.01 eV.

Using the parameters in Table 2, the optimal melanin at each UV wavelength is shown in the table below. The shorter the UV wavelength, the higher the optimum melanin concentration.

	300nm	400nm	500nm
Optimal Melanin	0.74g/L	0.64g/L	0.57g/L

The model examined the number of cells surviving in the ISS radiation environment and inferred the effects of UV energy and melanin concentration on the increase or decrease in cell numbers. The optimum amount of melanin was also indicated, taking into account UV-protection capacity and cytotoxicity. In reality, there should be a limit to the amount of melanin that can be expressed by cells, so the survival rate can be maximized with the optimal amount of melanin or a smaller amount.

Model-4

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Our project uses the radioprotective potential of melanin to protect micro-organisms from radiation. We will use models to understand Compton scattering and melanin ghosting, which are considered important for radioprotection by melanin.

Compton scattering

As explained in the description, when melanin is irradiated, photons are scattered by Compton scattering and recoil electrons are produced at the same time. First, this Compton scattering is understood in detail. A schematic diagram of the Compton scattering reaction is shown below.

Why Compton scattering?
Interactions between photons (electromagnetic waves) such as γ-rays and X-rays and matter include the photoelectric effect, Compton scattering and electron pair production. Which interaction is likely to occur depends on the energy of the photon and the atomic number in the substance. C, H and O atoms, which are abundant in melanin, and photons tend to induce Compton scattering over a wide range of photon energies. Therefore, we focused on Compton scattering. In addition, the photons are more likely to induce the photoelectric effect at lower photon energies.

The top panel shows the likelihood of carbon-C interactions, with Compton scattering being the most common in the energy range covered by the project.

From the schematic diagram of Compton scattering(Fig.4-1), the energy and momentum conservation laws are formulated and solved to obtain the energy of scattered photons and recoiled electrons produced by Compton scattering.

Memo
If the mass of the photon is mp, E=mpc^2 follows from the equivalence of mass and energy. Therefore, the momentum of the photon is expressed as mpc=E/c. The law of conservation of momentum decomposed each momentum vector into vertical and horizontal components, which were then compared between the vertical and horizontal components.

The above equations can be organised as follows.

Here, the electron rest energy m_ec² = 511 (keV) is used.

Once the energy E0 of the incident photon is determined, the energy Esc of the scattered photon is found to be determined only by the scattering angle θ. The smaller the absolute value of θ, the larger the value of Esc, and the larger the absolute value of θ, the smaller Esc. The angle θ is randomly determined and is called forward scattering when it is between -90 and 90 degrees, and backscattering otherwise.

Scattered photons can further cause Compton scattering with another electron or cause other interactions. Such reactions, called electromagnetic cascades, are speculated to occur in melanin and have been suggested to be important in radiation protection. A schematic diagram of the reaction is shown below.

Eventually, the recoil electrons return to melanin again and only the energy of the radiation is considered to decrease.

We first developed a simple model of radiation protection by melanin and melanin concentration.

To confirm the basic principle, we used Cryptococcus neoformans, which achieves radiation protection by melanin, as a model.

Melanin ghosting

Cryptococcus neoformans has a structure called the melanin ghost. Melanin ghosts are structures in which melanin is arranged along the cell wall to cover the cell, and it has been suggested that this structure may be important for radiation protection. Melanin exists in the form of melanin grains, and the irregular arrangement of melanin grains forms the melanin ghost.

For the sake of simplicity, we assumed that melanin grains are melanin molecules and further assumed that they are regularly arranged. In the following model, we focused our calculations on micro-regions within the melanin ghosts and finally calculated how much melanin is required for these micro-regions to cover the cells. In addition, simple simulations were performed using the calculated values to gain a better understanding of radiation protection by melanin.

The first model investigated the relationship between scattering probability and melanin concentration. Greater melanin concentration leads to greater Compton scattering probability, which makes it easier to protect cells from radiation. On the other hand, it is important to know the optimum melanin concentration, as excessive melanin production is toxic for the cells. In this model, the following assumptions are made for simplicity

Assumptions

• All melanin molecules are circular with radius r
• The distance between the centres of one melanin molecule and the nearest melanin molecule is l
• Melanin molecules do not overlap each other (l is greater than or equal to 2r)
• Compton scattering occurs when a photon strikes the circumference of a melanin, but the reaction occurs at the centre of the melanin molecule
• The angle θ of Compton scattering is determined according to the following rules

0-90 degrees when E0 is above 300 keV
90-180 degrees when E0 is less than 300 keV

• When Esc is less than 50 keV, photoelectric effect occurs instead of Compton scattering (termination of reaction)

The hypothetical situation can be represented in the diagram as follows.

What we want to find is the probability of Compton scattering. If Compton scattering occurs at the centre of the melanin in the bottom centre of the diagram (let M-1), then Compton scattering is considered to occur next when scattered photons are scattered within the angle θ formed by the two tangents drawn from the centre of M-1 to the next melanin circle. Therefore, the probability of scattering occurring can be calculated by dividing the sum of the angles at which the scattered photons hit the next melanin by 180 degrees (π).

From the diagram above, the total angle at which scattered photons strike from one melanin to the next is 3θ. In other words, the probability to find is 3θ/π.

Why do we consider 180 degrees (π)?
In the hypothetical situation, it is symmetrical vertically, so there is no difference between thinking up to 180 degrees and 360 degrees. If we were to consider 360 degrees, the probability we seek would be 6θ/2π, which is ultimately no different from considering up to 180 degrees. Also, if backscattering occurs, the photon cannot escape from the circle of melanin molecules. This means that only forward scattering, 180 degrees (forward), should be considered.

Next, the value of θ is determined. In the present model, we want to investigate the relationship between the scattering probability and the size of melanin and the distance between melanin, so we aim to express θ in terms of l and r.

If a straight line of length l is drawn between the centres from one melanin to the next as shown in the diagram above, the angle between the line and the tangent line is θ/2. From the definition of the trigonometric ratio sin, θ can be expressed as using l and r.

Next, we calculate this to determine the value of θ.

What is cos^-1 (inverse trigonometric function)?
Inverse trigonometric functions (sin^-1, cos^-1, tan^-1) are the inverse functions of trigonometric functions (sin, cos, tan), where "x and y are exchanged". For example, sin(π/6) = 1/2, so sin^-1(1/2) = π/6. It is also sometimes denoted as Arcsin instead of sin^-1, and cos^-1 and tan^-1 are Arccos and Arctan respectively. In our model, the expression for cosθ was used to calculate the value of θ by substituting cos^-1. θ = cos^-1(cosθ)

Since the probability we are seeking is 3θ/π,

Here, the probability was expressed using n as l = nr. This is because, in this model, we wanted to investigate the relationship between scattering probability and melanin size or inter-melanin distance, so we thought that n, the ratio of the inter-melanin distance l to the melanin radius r, could be used to represent the relationship with probability. It also has the advantage that it is easier to understand as there are fewer variables, and that it is easier to model by using n, which is dimensionless.

Since l is greater than or equal to 2r from the assumption, l≥2r and nr≥2r, n is a real number greater than or equal to 2.

If the probability of finding

Therefore, the following can be drawn graphically for probability and n.

When n = 2, i.e. when the melanins are in contact with each other, the probability is 1. As n increases, i.e. as the inter-melanin distance increases, the probability decreases exponentially. It is counter-intuitive that as the value of n, the ratio of the inter-melanin distance to the radius of the melanin circle, increases, the probability of a scattered photon hitting one melanin to another decreases.

Let us now recheck the hypothetical situation. Then we can see that it is not sufficient to consider only the nearest melanin (melanin only l away). See figure below.

As shown in the figure, the previous model only considered melanin l away from a melanin, but as melanin is multilayered in melanin ghosts, the next closest melanin (melanin √3l away) must also be considered. Therefore, a slightly improved version of this model was created and tested.

Why not consider more distant melanin?
When you read the explanation above, you probably wondered why we don't consider more distant melanin, such as the third nearest melanin (2l away) or the fourth nearest melanin. Of course, we considered this possibility as well. In our model, we are looking for a value of n at which the balance between scattering probability and melanin concentration is just right: as n increases, the scattering probability decreases, which is not our goal. In other words, we are considering the case when n is somewhat small. Since more distant melanin affects the scattering probability when n is large, we considered that for the range of small values of n, such as the one we are considering, up to melanin √3l away would be sufficient.

As mentioned above, melanin √3l away also needs to be considered. Their effect on the probability can be obtained by replacing l → √3l; adding the probability of a scattered photon hitting a melanin l away and the probability of a scattered photon hitting a melanin √3l away gives a probability that takes both into account.

the following can be drawn graphically for probability and n.

In the range n=2~about 3, the probability exceeded 1. This is due to the fact that in the range where n is small, there is melanin covering at distance l and melanin covering at distance √3l. In other words, the probability was greater than 1 because of overlapping calculations.

We then tried to find a value of n for which the probability would be exactly one, to see if the overlapping areas could be expressed in a mathematical formula.

As shown in the figure, the previous model only considered melanin l away from a melanin, but as melanin is multilayered in melanin ghosts, the next closest melanin (melanin √3l away) must also be considered. Therefore, a slightly improved version of this model was created and tested.

As shown in the diagram above, the angle between the tangent line of the melanin at distance l and the tangent line of the melanin at distance √3l overlaps, as shown in the area indicated in red. Because of the calculation of this coverage, the probability exceeded 1. As n increases, the area indicated in red becomes smaller and eventually the tangents just overlap each other. At that point, the probability should be exactly one. To find such a value of n, the following situation was set up and calculated.

Suppose that there is a circle of radius r centred at the origin O of the xy-plane and that scattering occurs in this circle = melanin. As in the previous models, the distance between neighbouring melanins is l. Let C1 and C2 be the tangents to the melanin at distance l and to the melanin at distance √3l from the origin, respectively, and let θ1 and θ2 be the angles that each tangent makes with the x-axis, respectively. Let the circle with contact point C1 be circle 1 and the circle with contact point C2 be circle 2.

The angle between the covered areas (shown in red) in the previous diagram can be expressed as θ1-θ2. There are six of these same ones within 180 degrees, so the probability of this overlapping area is as follows.

The equation of the tangent line at point C1 on circle 1 can be expressed as ①. Similarly, the equation of the tangent line at point C2 on circle 2 can be expressed as ②. Also, since C1 and C2 are points on circles 1 and 2 respectively, ③ and ④ are satisfied. Since the tangent line passes through the origin, substitute x=0 and y=0 for ① and ② respectively, and obtain x1 and y1 from equations ① and ③, and x2 and y2 from equations ② and ④ respectively.

Now we can express the coordinates of C1 and C2 using l and r. We want to find θ1 and θ2, so we use the trigonometric ratio tan. The tangent line we are considering passes through the origin, so the slope of the tangent line is the value of tan. The slope of the tangent line is y1/x1 and y2/x2, so tanθ1 = y1/x1 and tanθ2 = y2/x2. This is calculated as,

The θ1-θ2 to be found can be expressed using the inverse trigonometric function (tan^-1) as follows. (see note above on inverse trigonometric functions)

We now want to find a value of n such that θ1 = θ2. In this case, the scattering probability is 1 without just covering the melanin, which is the situation we want to find, where the melanin concentration is as small as possible (large inter-melanin distance) and the probability is large; since 0 ≤ θ1, θ2 ≤ π/2, the angle and tan values correspond one to one.

The calculation gives n = √(28/3), which is approximately 3.055.

The graph for probability and n is shown below.

The red line represents the probability of correction except for cover, and the blue line represents the probability of no correction; it can be seen that the red and blue lines intersect exactly where θ1 = θ2, i.e. at n = √(28/3), when no cover correction is required. The green line represents the probability of the correction term only, which also has probability 0 at n = √(28/3), indicating that no correction is required in the larger n range.

The respective equations and ranges of n for the red, blue and green lines are as follows.

To ascertain the radioprotective capacity when n = √(28/3), we designed the following model.

In our project, melanin ghosting is simulated by expressing melanin in the periplasm and in micro-organisms without it. The size of the periplasm is then assumed to be as follows to ascertain the concentration of melanin and the number of melanin layers.

Assumptions

• The volume of the bacteria is 8 x 10^-16 L
• The bacteria have a spherical shape with a radius of 2 µm and a periplasm of 23.5 nm thickness
• The periplasm is rectangular with a length of 23.5 nm and a width of 4πμm
• There is one melanin per rectangular area of √3l/2nm in length and lnm in width.
• As in the model above, for simplicity, a two-dimensional plane is considered.

The hypothetical situation is illustrated in Fig.

First, the relationship between melanin concentration and n is ascertained by considering how many melanin can be contained within the periplasm.

On the other hand, if the concentration of melanin in the periplasm is m(g/L), the melanin concentration is m/318(M), since the molecular weight of melanin is 318.

The volume ratio of the periplasm to the assumed total bacterial volume is now determined. The volume of periplasm is determined by subtracting the volume of a sphere of radius (2-0.0235) µm from the volume of a sphere of radius 2 µm. Thus, the ratio of periplasm to the total is 0.2787/8.

Multiplying this by the volume of the whole bacterium, 8 x 10^-16 L, gives the volume of periplasm as follows.

Given that the melanin concentration is m/318(M), the number of melanin in the periplasm is

Formula ★ and Formula ♦︎ both express the number of melanin in the periplasm, from which the relationship between m and l is derived.

Therefore, m = 6.478/l².

Assuming that r = 2 nm, l = 2√(28/3) when n = √(28/3). Therefore, the concentration of m can be calculated as follows.

When r = 2 nm, the optimum melanin concentration is 0.17 g/L.

The length of the periplasm is 23.5 nm. In this case, it is ascertained how many layers of melanin enter the periplasm.

As shown in the figure, the longitudinal length from one melanin layer to the next is √3l/2nm. Therefore, the integer part of the quotient of 23.5 divided by √3l/2 + 1 represents the number of melanin layers at 23.5 nm.

Substituting r = 2 nm and n = √(28/3),

There are five melanin layers. Therefore, the number of forward scattering times is at most 5.

Finally, a simple simulation of radiation protection by melanin is performed. According to the Compton scattering equation, Ere and Esc are calculated for randomly determined angles θ. The calculations were performed using Python. From the previous calculations, it was found that with a scattering probability of 1 and a melanin radius of 2 nm, periplasm has five layers of melanin, so the number of scattering times was calculated as five.

In addition to the assumptions used in the model above, the following assumptions were made.

The angle θ of Compton scattering is determined according to the following rules.

0-90 degrees when E0 is above 300 keV.

90-180 degrees when E0 is below 300 keV.

The simulated (n=500) results are shown below.

The scatter diagram on the left plots the scattering angle θ versus the energy of the incident photon (blue) and the energy of the recoil electron (orange). As assumed, it can be seen that only forward scattering (0°≤θ≤90°) occurs when the incident photon energy is above 300 keV, while only backward scattering (90°≤θ≤180°) is simulated when the incident photon energy is below 300 keV. It can be seen that for both types of scattering, the recoil electron energy increases as the angle increases. When backscattering predominates, the incident photon and recoil electron energies are roughly divided into two groups. The scatter plot on the top right shows the energy of the scattered photons and the number of scattering attempts, where 0 represents the first incident photon of 662 keV. Due to the large number of trials and the large scatter, the energies of the 1st to 5th scattered photons are shown below as a box-and-whisker diagram. Furthermore, only the energy of the fifth scattered photon is shown as a histogram.

At 662 keV, five times seems sufficient.

The reason is that the majority of the scattered photons are below 300 keV, where backscattering predominates.

In the present model, it was assumed that melanin and radiation always cause Compton scattering. In reality, however, radiation and melanin do not react 100% of the time. This assumption was made because the aim of the present model was to gain a better understanding of the radioprotection mechanism by melanin.

The difference between the model 3 and the model 2 is the difference in sensitivity to radiation and UV. In general, UV radiation reacts better with bacteria, so it is likely that the melanin concentration required for protection was higher in model 3.

Radioprotection by melanin, which plays an important role in the project, has been better understood using a simple model. Melanin concentrations in periplasm do not have to be that high to be useful for radiation protection. However, higher melanin concentrations are required for high energy radiation and particle radiation, as well as for radiation permeability. In the future, we would like to simulate the implementation of this biological system by constructing a model under more realistic conditions.

1. Nishiyama, H. (2011). 電磁波の周波数、波長、エネルギー　物質の大きさとの相対比較. https://www5.dent.niigata-u.ac.jp/~nisiyama/ElectroMagneticRadiation.pdf
2. Yamazaki, H. １. 放射線と物質の相互作用. http://accwww2.kek.jp/oho/textbook/from2022/2023/text/1_Yamazaki.pdf
3. Particle Data Group. (2022). Review of Praticle Physics. Progress of Theoretical and Experimental Physics, Volume 2022. https://doi.org/10.1093/ptep/ptac097
4. Dadachova, E. (2008). The radioprotective properties of fungal melanin are a function of its chemical composition, stable radical presence and spatial arrangement. Pigment Cell and Melanoma Research, 21(2), 192-199. https://doi.org/10.1111/j.1755-148X.2007.00430.x
5. Meredith, P. (2006). The physical and chemical properties of eumelanin. Pigment Cell Research, 19(6), 572-594. https://doi.org/10.1111/j.1600-0749.2006.00345.x
6. Muroya, Y. (2017). 5 水と水溶液の放射線化学. Radioisotopes, 66(10), 425-435. https://www.jstage.jst.go.jp/article/radioisotopes/66/10/66_661006/_pdf
7. Kobayashi, K. (1984). Pulse Radiolysis Studies in Biological Systems. The Institute of Scientific and Industrial Research Osaka University. 24(6). https://www.jstage.jst.go.jp/article/biophys1961/24/6/24_6_277/_pdf
8. Charles E. Turick. (2011). Gamma radiation interacts with melanin to alter its oxidation–reduction potential and results in electric current production. Bioelectrochemistry, 82(1), 69-73. https://doi.org/10.1016/j.bioelechem.2011.04.009
9. Vasileiou T, Summerer L (2020) A biomimetic approach to shielding from ionizing radiation: The case of melanized fungi. PLoS ONE 15(4): e0229921. https://doi.org/10.1371/journal.pone.0229921
10. Dadachova E, Bryan RA, Huang X, Moadel T, Schweitzer AD, Aisen P, et al. (2007) Ionizing Radiation Changes the Electronic Properties of Melanin and Enhances the Growth of Melanized Fungi. PLoS ONE 2(5): e457. https://doi.org/10.1371/journal.pone.0000457

Model-5

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This model estimates how much radiation a cell cultured in a spacecraft would be affected by. This would help predict the amount of energy available for cell survival and metabolism due to the effects of space radiation.

There are three types of cosmic radiation near the earth: galactic cosmic rays, solar particle rays, and particles trapped by the earth's magnetic field. Galactic cosmic rays and solar particle rays are mostly composed of protons, but the number of particles near the Earth is small because the Earth is protected by a magnetic field. The number of particles varies depending on the location of the spacecraft and solar activity, but is omitted near the Earth because it is unlikely that microorganisms can utilize these high-energy particles.

The main influence is from particles trapped in the Earth's magnetic field. These are cosmic radiation that falls to the earth and is trapped by the earth's magnetic field, and are radiation belts with high intensity of high-energy protons and electrons.

To avoid complications in the model, the following assumptions were made.

• Assume the cell is a mass of water
• Mainly consider particles trapped by the earth's magnetic field. In this model, the range of altitude from 0 km to 60,000 km is investigated, and the cosmic radiation in this range is mainly accounted for by protons and electrons.
• Only electromagnetic waves of secondary radiation are considered.

Primary radiation in the environment penetrates through the walls of the ISS and rocket, and secondary radiation is produced by collisions with the walls. The model assumes that protons and electrons are primary radiation and that secondary radiation, which is electromagnetic radiation, is diffused into the ship and used by microorganisms.

Specifically, the amount of energy that a cell receives from secondary radiation on board is expressed using the “absorbed dose rate”. The absorbed dose rate is the energy absorbed by a substance per unit mass per unit time, and since it takes into account the energy of photons and the ease with which a substance absorbs energy, a value closer to practical conditions can be calculated. In preparing this model, we refer to the commentary [1].

Since the absorbed dose rate ϵ_abs (J/kg∙h) of a cell represents the energy absorbed by the substance, we can imagine that it is proportional to the energy E(J) of the photon and the number I(/m²∙h) (velocity) of photons passing through the substance. Now we need to consider how the photon energizes the cell. Consider a photon, which is secondary radiation emitted from the wall of a plane, reaching a cell and passing through the cell. In this case, the photon reacts and interacts with atoms in the cell, and is scattered or absorbed. From this principle, not all of the energy possessed by the photon will react with the cell, and the probability that the photon interacts with molecules in the cell must be introduced. The mass energy absorption coefficient ζ_en^water (m²/kg) is used here. The mass energy absorption coefficient is a constant that depends on the energy of the photon and the type of substance, and in this model, the substance is assumed to be water. In summary, the absorbed dose rate ϵ_abs is expressed as follows

Next, we investigate the relationship between the number of photons passing through the cell and the secondary radiation emitted from the wall. In this case, we assume that the source of the secondary radiation is at the wall, where nuclear decay occurs, although this is not the actual principle. The produced secondary radiation flies in all directions from the source (Fig.5-1). The surface area of a sphere of radius l centered on the source is 4πl², and during time t, ηP photons are produced. The number of photons N on the microscopic surface A can be expressed as follows.

Since the number of photons passing through the microplane can also be expressed as N=IAt, the flow velocity I is

which is the same as the number of photons passing through the microscopic surface. Therefore, substituting ② into ①, the absorbed dose rate ϵ_abs of a cell at a distance l from the source of intensity P can be expressed as follows

From this point on, we set simplified conditions on board (Fig.5-2,3). Primary radiation flying through space collides with the walls of the ship, producing secondary radiation that diffuses into the ship. As indicated earlier, secondary radiation flies in all directions with the walls as the source. The range of photons reaching the cell is assumed to be the source in the area indicated by the circle of radius r in Fig.5-2, and photons from further away are ignored. Assume that the cell is located at a vertical distance h from the wall and that the wall is uniformly distributed with sources of the same intensity. If the source has an intensity density p per microplane ∆S, the overall intensity is P = p∆S. Therefore, the absorbed dose rate ϵ_abs from the microplane ∆S can be expressed as

Then, to obtain the absorbed dose rate over the entire area, all the micro planes ΔS are added together. In Fig.4-4, the microplane ΔS is the area of the gray area,

From Fig.4-2, if the angle formed by the perpendicular distance h between the cell and the wall and the distance √(r²+h² ) farthest from the cell is θ, then tanθ=r/h. Substituting this, we can express the following.

Equation ③ shows that the absorbed dose rate ϵ_abs is independent of the distance h from the source, and although the ISS is filled with air, which causes radiation to decay to a lower energy, radiation from far away can be excluded by setting the distance R at which the contribution to cells disappears due to decay. We will now use equation ③ to investigate the absorbed dose rate at an altitude of around 400 km, which is the ISS orbit. For this purpose, we thought it important to deepen our understanding of each constant.

• Photon energy E

The magnitude of energy possessed by a single photon of secondary radiation. The energy of a photon of cosmic radiation is commonly expressed as E' (eV), and the unit conversion is performed using the electric elementary quantity e=1.602×10^-19), which represents the amount of energy that a 1 eV particle has.


• Mass energy absorption coefficient

This is a constant that expresses the probability of a photon interacting with an atom in a cell and the rate at which the energy of the interacting photon is absorbed by the cell. It depends on the type of material and the magnitude of the photon energy.

• Number of photons passing through a cell

The number of photons that emit from a source with a small area and pass through a cell, considering that the flux of photons in the ISS (/m²∙h) is known, so we assign the flux to the number of photons, ηp. The flux is the number of photons per unit area and unit time.

3-1 : Change in mass energy absorption coefficient at each photon energy

The change in the mass energy absorption coefficient of photons at each photon energy was investigated. The mass energy absorption coefficient of water is evident up to 10 MeV [2]. This data was plotted. Since we may now consider photons above 10 MeV, we need values for large photon energies; since there is a constant decreasing trend from 1.0 MeV to 10 MeV, we used an approximate curve of those data to estimate the mass energy absorption coefficient above 10 MeV.

It was found that the mass-energy absorption coefficient decreases with increasing photon energy, and since the degree of decrease is small after 0.1 MeV, the absorbed dose rate is considered to increase in dependence on photon energy.

The product of photon energy and mass-energy absorption coefficient was then determined. As expected earlier, the effect of the mass-energy coefficient becomes smaller as the energy increases.

3-2 : Absorbed dose rate depending on the source range

The absorbed dose rate as a function of the variation of the angle θ, which determines the area of the source, was considered useful for estimating the size of the source to be considered and the appropriate wall-to-cell distance. the larger θ, the larger the area of the source and therefore the larger the absorbed dose rate should be. the absorbed dose rate was varied by varying θ in the range 0~85° and the The magnitude of the absorbed dose rate was investigated by varying θ in the range 0~85° (Fig.5-7). Other conditions are described in the table below.

	E(MeV)	ζ(m2/kg)	ηp(/cm2)	θ
photon	0.1	2.52×10-3	107	0~85

3-3 : Absorbed dose rate on the ISS

The absorbed dose rate when cells are cultured in the ISS at an altitude of 400 km was investigated, assuming that the ISS is not large enough to have an unaffected area for the cells, it was decided to use the entire wall as the source of radiation (Fig.5-8). It was also assumed that the cells were in the centre of the ship and that the secondary radiation was flying equally from all six walls, and the absorbed energy was expected to be six times higher. The θ was then 45°.

The values in the table below were used in the calculations.

	E(MeV)	ζ(m2/kg)	ηp(/cm2・day)	θ	εabs(J/kg・h)
photon	0.1	2.52×10-3	107	45	1.8×10-7

The calculation result was ϵ_abs=1.75×10^-7 (J/kg∙h). The dose inside the ISS is generally said to be 0.5~1 mGy per day; unit conversion of 1 mGy gives ϵ_abs=4.17×10^-5 (J/kg∙h), which is about 200 times larger than the calculation result in the model. As the model calculations focused only on photons, this is thought to be due to the lack of energy given by the protons, neutrons and electrons passing through the ISS wall.

In addition, the model considered that at an altitude of 400 km, in addition to the protons trapped in the Van Allen zone, the ISS is also affected by galactic cosmic rays and solar particle rays, and therefore these particle rays also have an impact. The calculations were performed with reference to the values.

	E(MeV)	ζ(m2/kg)	ηp(/cm2・day)	θ	εabs(J/kg・h)
photon	0.1	2.52×10-3	107	45	1.8×10-7
electron	0.3	3.20×10-3	104	45	6.7×10-10
Proton(capture)	100	7.57×10-4	104	45	5.2×10-8
Neutrons(capture)	1	3.11×10-3	104	45	2.2×10-9
Protons (galaxies)	1000	3.68×10-4	10	45	2.6×10-10
Helium(galaxies)	200	6.09×10-4	10	45	8.4×10-12
Proton(flare)	100	7.57×10-4	105	45	5.2×10-7
Neutron(flare)	1	3,20×10-3	105	45	2.2×10-8

The possible effects of space radiation were calculated, but even when all were added up, the dose was not equal to 1 mGy/day. As described in 3-2, the dose varied depending on the measurement location on board, which was considered to be the reason for the difference from the actual measured values on board the ISS. Assuming that the number of photons indicated by the flux (/sq-cm・h) passes directly through the cells, the photon flux was substituted into equation ①.

	E(MeV)	ζ(m2/kg)	ηp(/cm2・day)	εabs(J/kg・h)
photon	0.1	2.52×10-3	107	1.7×10-7

The calculated result was ϵ_abs=1.68×10^-7 (J/kg∙h), which was almost identical to the result calculated using equation ④.

Next, the absorbed dose rate in the Van Allen zone is estimated. The Van Allen belts are radiation belts thought to be formed by particles originating from the solar wind that are trapped by the Earth's magnetic field. The Van Allen zone is divided into an inner zone at altitudes between 1000 and 20 000 km and an outer zone between 20 000 and 60 000 km. Protons are mainly trapped in the inner zone and electrons in the outer zone.

4-1 : Photon flux in the Van Allen zone

Simulations of the proton flux at various altitudes in the inner Van Allen zone have already been done and the data are used [4]. However, as the data are for protons, they need to be converted to photon data. The approximate curve derived from the plot in [4] was used to calculate the flux at an altitude of 400 km. The value was 6.66×10⁸ (/m2∙h), which is 6.26 times higher than the value calculated for photons in 3-3 at 400 km. It was therefore decided to multiply the proton flux by a factor of 6.26 to convert it to the photon flux on board. As the electrons in the inner zone have enough energy to be shielded, only the secondary radiation produced by proton collisions needs to be considered, and it can be assumed that the photon flux depends on the proton flux. The results of the conversion of protons in the inner zone into photon flux are shown below.

The flux of electrons in the outer band was also done in [4] and, as in the inner band, the flux at 400 km altitude was calculated using an approximate curve derived from the plot. The value is 1.56×10¹¹ (/m²∙h), which is 37.4 times larger than the photon flux of 4.17×10⁹ (/m²∙h) performed in 3-3. Therefore, the electron flux was converted to photon flux by multiplying the electron flux by 1/37.4. The results of the conversion of the secondary radiation of the electrons in the outer band as photons are shown below.

4-2 : Absorbed dose in the van Allen zone

The absorbed dose rate is obtained using the proton and photon fluxes plotted in 4-1. In the inner zone, protons account for most of the impact. Electrons in the inner zone have low energy and can be shielded by the walls of the aircraft, so electrons at altitudes below 1000 km are neglected. It is assumed that the secondary radiation produced by protons is uniformly 10 MeV and that due to electrons is uniformly 1 MeV. This is based on the assumption that the energy of the protons in the inner zone is around 100 MeV and that secondary radiation with an energy of 10% is emitted when they undergo a nuclear reaction with the atoms in the ship's wall. It is assumed that the electrons have an energy of 1 MeV and that the excitation and bremsstrahlung of the ship wall atoms results in secondary radiation reduced to 10%. In the absence of data, an approximate curve of about four nearby points was used to predict the flux. Equation ④ was used in the calculations.

The estimated absorbed dose rate when the rocket is located at each altitude is shown below. The inner and outer bands take into account secondary radiation due to protons only and electrons only, respectively.

For the Van Allen belt as a whole, the absorbed dose rate was found to be around 10^-6~10^-3 (J/kg∙h). The absorbed dose rate also varied with the photon flux, but in reality, the energy of the protons and electrons at different altitudes should be a factor in the variation of the absorbed dose rate.

It has been reported that the dose to a satellite crossing the Van Allen zone was 0.22 Gy/h for the inner zone crossing and 0.054 Gy/h for the outer zone crossing [5]. The calculations in this model were calculated to be at most 1.36 x 10^-3 J/kg∙h for the inner zone and 9.59 x 10^-5 J/kg∙h for the outer zone, which is smaller than the measured values. This is thought to be due to the fact that only the effects of electromagnetic radiation, which became secondary radiation, were estimated, and the doses given by the direct action of protons and electrons were not taken into account. Furthermore, galactic cosmic rays and solar particle rays should in fact also contribute to the dose rate.

The absorbed dose rate of cells in the Earth's magnetosphere, starting from the ISS orbit, could be estimated. Although the model calculation results are small compared to the measured values, we believe that we have calculated the dose that the cells can use as energy. It is difficult to calculate the exact absorbed dose rate because the flux of particles, which significantly affects the dose, varies with the thickness of the shielding, the angle of the aircraft and solar activity. We would like to use the results of this calculation to predict how much electrons can be obtained from the reaction of photons with melanin expressed in bacteria.

Model-6

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Our project involves the use of genetically modified organisms in space. The organisms could be released into the spacecraft or space environment, e.g. due to an unforeseen accident. As a biosafety measure, we have designed a Kill Switch that works in a concentration-dependent manner with copper ions in the culture medium.

In this model, we analyse the kinetics of the designed kill switch and confirm that this kill switch works.

These can be divided into three steps.

Step.1 - Input: copper ion concentration-dependent regulation of transcription

Step.2 - Inversion: reversing the sign

Step.3 - Output: cell death is induced by toxin

A schematic representation of the Step.1 reaction is shown below.

Some enzymes cannot work by themselves and require an auxiliary factor. Such enzymes alone are called apoenzymes, while enzymes that are bound to and activated by an auxiliary factor are called holoenzymes. Although CueR is not an enzyme, it requires copper(I) ions as an auxiliary factor to act as a transcription factor. Therefore, CueR without copper ions bound is referred to as apoprotein (Apo) and CueR with copper bound as holoprotein (Holo). First, when CueR is transcribed and translated, Apo is produced, which binds to intracellular copper ions to form Holo. The dissociation constant for copper ions is 10^-21 M, which is very small and once bound by copper ions, they rarely dissociate from the protein: Apo binds near the copA promoter upstream of the TetR gene used in Step 2 with a rate of k₁,k_-1 and Holo with a rate of k₂,k_-2 Transcription is repressed (off) when Apo binds to DNA and activated (on) when Holo binds to DNA. This transcription factor is thought to have two very unique transcriptional regulatory mechanisms: a substitution reaction (k_2a), in which Holo bound to DNA is replaced by Apo, and a dissociation-assisted reaction (k_2b), in which Holo bound to DNA is pulled apart by Apo. These mechanisms are thought to switch transcription on and off in a copper ion concentration-dependent manner.

A schematic representation of the reactions in Steps.2 and 3 is shown below.

In Step.1, when Holo binds to DNA, transcription is on, and when Apo binds to DNA, transcription is off; in Step.2, this sign is reversed so that the final output (transcription) is off when Holo binds to DNA and the output (transcription) is on. For this purpose, a tetracycline repressor (TetR) was used in Step.2. TetR represses transcription of downstream genes by binding to tetracycline operators. In other words, transcription of the toxin gene is repressed when the TetR is present (binding of Holo to DNA) and the transcriptional repression of the toxin gene is released when the TetR is absent (binding of Apo to DNA). Finally, in Step.3, the output part of the KILL SWITCH works, where toxin is produced and kills the bacteria.

The following differential equations were formulated.

In this model, several assumptions are made.

• The speed of transcription and translation is equal regardless of promoter and RBS
• Plasmid copy number is constant at 25
• The volume of E. coli is 8 x 10^-16 L
• Cu2+ in the culture medium is reduced to Cu+ and the Cu+ concentration is constant

The parameters and initial values used in the model are as follows

Parameters

Parameters	Description	value	unit	source
k1	Papo binding rate constant	6×106	/M・s	[1]
k-1	Papo unbinding rate constant	0.4	/s	[1]
k2	Pholo binding rate constant	1.1	/s	[1]
k-2	Pholo unbinding rate constant	9×106	/M・s	[1]
k2a	Pholo substitution rate constant	134×106	/M・s	[1]
k2b	Papo assisted dissociation rate constant	54×106	/M・s	[1]
kc	Copper ion binding rate constant	107	/s	estimated
km	Transcription rate constant	0.04	/s	calculated from [2]
kp	Translation rate constant	0.0077	/s	calculated from model1-[1]
dm	mRNA degradation rate constant	0.25	/s	calculated from model1-[1]
dp	protein degradation rate constant	0.0033	/s	calculated from model1-[1]
Ktet	TetR dissociation constant	5×10-8	M	[3],[4]
n	Hill coefficient	3	-	[5]

Initial Value

Abbreviation	Description	initial value	unit	source
[DNA]	Plasmid Concentration	5×10-8	M	estimated
[lacO]	oncentration of Plasmid (pHSG398)	50	nM	estimated
[Cu+]	Copper ion concentration	10-5, 10-6, 10-7, 10-8, 10-9, 10-10	M	depends on conditions

The time variation of Holin concentration at copper ion concentrations of 10^-5, 10^-6, 10^-7, 10^-8, 10^-9 and 10^-10 M was investigated. The results are shown below.

It can be seen that the Holin concentration (blue) increases with decreasing copper ion concentration.

The threshold concentration of Holin was set at 5×10^-7 M (more detail iGEM 2013 TU-Delft). At Holin concentrations greater than this, cell death is induced. The steady state is greater than this concentration when the copper ion concentration is below 10^-10 M. The kill switch is switched on after about 10 min when the copper ion concentration in the environment around the cell drops below 10^-10 M.

At 10^-9M, about 80% of the threshold level of holin is produced. At this concentration, the bacteria remain stressed and the number of individuals induced to cell death is expected to increase over time. As the concentration of copper ions in the medium is 10^-5~10^-6 M, cell death induced at 10^-9 M is not considered to be a problem.

For k _c values.

From previous studies, it is off at [Cu+]=10^-6 M and on at [Cu+]=10^-10 M.

The binding of copper ions to apoproteins is sufficiently faster than the dissociation of copper ions from holoproteins that the dissociation reaction is negligible.

We want to guess the value of k_c. k_c is varied at 10⁵, 10⁷ and 10⁹ s^-1. The most appropriate value of k_c would be such that the switch is off at [Cu+]=10^-6M and on at [Cu+]=10^-10M. The figure shows that the optimum value of k_c is 10⁷. We have therefore used this value in our simulations.

Our designed kill switch can switch the switch on and off in a copper ion concentration-dependent manner. The kill switch is off when the doubling medium contains copper ions at a concentration of 10^-6M. However, when the bacteria leave the medium and the copper ion concentration around the bacteria falls below 10^-10 M, the kill switch is clearly switched on; Holin is produced in sufficient quantities for cell death.

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