Team:KUAS Korea/Model
Model
Modeling of Our Project
Abstract
Many people fell in love with synthetic biology dreams modifying creatures from DNA scale. They have images at first a tree with metal-involved fruit for resource, a cow giving its meet anticancer contained and a human with "super-powered" abilities from other creatures. These dreams are in some senses quite far from reality and in other hands gradually coming true.
In the future when we can finally make such products mentioned above, bioluminescence will be very important. It can show if the system works well as designed like power on/off button. And it can show whether any wanted state is reached like detection. Bioluminescence is a very simple interface and straightforward.
So far, no biochemical pathway for luciferin has been fully described but one found only in bacteria, prokaryotes. Knowing the whole pathway provides several applications in diverse area. Prokaryotic bioluminescence systems were rarely applied to eukaryotes because of its low efficiency adopted to eukaryotes. Recently December 11th 2018, a full pathway of fungal bioluminescence was unveiled opening the door to application of bioluminescence system to eukaryotes.[1]
BTS(Bioluminescence tattoo system) chose fungal bioluminescence. In modeling section, we will see how many luciferins of fungal bioluminescence are needed to be visible so that our system operates well. Especially, cells are loaded using encapsulation for biocompatibility so this will be also considered.
So Let's see how many luciferins are needed to make a visible light and how much plausible the system would be.
Background
1) Brightness : why lux is eligible
The light emitted from BTS must be bright enough to be seen. We had been searching how to define ‘brightness’ in a strict physical term and quantify it. Finally, it was figured out that there is no strict definition of brightness but frequently using luminance. [2]
However, Luminance is about luminous flux and is a value of the light source not a light seen. How much far we see the light doesn't change the value of luminance. Intuitively, if we see a light in a distance, it should be darker. Luminance is not an eligible physical quantity that we can adopt to BTS.
Lux is the quantity that represents brightness in the condition of ours. Luminance divided by area. If we see a light in a distance, Lux of the light goes down proportionally to inversed squared distance. This follows our intuition and we chose Lux represents the brightness.
Then how much Lux should we make? It is relative. Brightness changes by background lights, dark adaptation and intrinsic difference person-by-person. KAIST 2011 iGEM chose \(0.1 lux\) the minimum intensity of light that cone cells in human can perceive. [3] This is plausible because see the table below. The light with \(0.1 lux\) will be as bright as full moon. The brightness of full moon is properly bright enough to be seen in the dark condition like covering below a hand that we supposed.
[table 1. values of lux in diverse conditions.][4]
Analysis
1) Chemiluminescence yield rate
First, let’s consider luciferins to be visible. Chemiluminescence quantum yield is \( 0.6 \% \) at \( \mathrm{pH} 7 \). [5] \(\mathrm{pH}\) is supposed to be \(7\). The value can change as we fix the strain of BTS. If the number of luciferins \(N\) is given, we can calculate how many photons come out.
\[N \times 0.6 \% \]
2) Watt to lux conversion
The wavelength of the luciferin luminescence is \( 520nm \) [6]. The energy of the photon is as below.
\[E=\frac{hc}{\lambda}\]
where \(h=6.63\times10^{-34}m^2 kg/s\), \( c=2.99 \times 10^8 m/s \). So \( E=3.83 \times 10^{-19}J \) comes up.
Here, we assume the time of one luciferin is reacted as \(10^{-5}s\) and \( 100 \% \) of luciferins are reacted. These are based on no scientific background, so the result can be out of reality. If the luciferase, Luz, is characterized, the values can be measured specifically. Then the modeling will be more plausible.
Assuming the reaction time of one luciferin, we can get power of one luciferin.
\[W=\frac{E}{t}\]
where \(E=3.83 \times10^{-19} J\), \(t=10^{-5} s\). So \(W=3.83 \times 10^{-14}Watt\) comes up.
This can be converted to unit of lumen, using watt to lumen conversion table [7].
\[W\times(\frac{lm}{W})=LM\]
where \(W=3.83 \times 10^{-14}Watt\), \(\frac{lm}{W}=484.930\). So \(LM=1.86 \times 10^{-11}lm\)
And we assume that we see the light from 1 meter distance and in an absolutely dark room. Because this is the modeling for BTS. BTS will be seen covering hand over the region that it emits light. If BTS emits large amount of light that it can be easily seen, it will be a big burden to the cells carrying BTS genes. So the interface changes for human to see it covering hand. Then we can assume the condition is no background light. Human arm is assumed to be 1 meter long. So the condition that BTS is seen is 1 meter distance and no background light.
Then lumen can be converted to \( lux \) as below.
\[\frac{LM}{r^2}=LUX\]
where \(LM=1.86 \times10^{-11}lm\), \(r=1m\). So \(LUX=1.86 \times10^{-11} lux \).
If there are more luciferins, the value of lux goes up linearly. In order to make brightness of luciferins more than \(0.1 lux \), the number of luciferins needed is as below.
\[LUX \times \mathrm{number\; of\; luciferin} \times \mathrm{chemiluminescence\; quantum\; yield} = 1.86 \times10^{-11} lux \times N \times 0.006 \gt 0.1 lux \]
Then \(N=8.96 \times10^11\). It is a very large number so converted to \(mol\). Then mol of luciferin \(n=1.49 \times10^{-12}mol\).
3) The amounts of cells in one capsule
Here, we will see how many luciferins are needed considering the size of the capsule and cells and the number of cells in the capsule.
First, we assumed the size of the cell that we use to make BTS is \(20 \mu m\). This is grounded on rare reasons. The size can vary as see the figure below
[figure 1. the relative sizes of various cells and cellular components] [8]
The encapsulated cell types would change depending on the environments of injected creatures and means of the system. The cells might be animal cell because that is the most durable to live inside animal body’s environment compared to plant cells or prokaryotes.
We took human cell to be an example of animal cell to be considered here. Many histology and molecular biology textbooks state that there are \(∼200\) types of cells in the adult human body. [9] About \(200\) types of human cells vary in shape and size. The result of this section will be carefully considered because it can be dramatically different change the cell type. The size should be taken into account when choosing the cell type.
[table 2. Characteristic average volumes of various human cells] [10]
We assume the diameter of capsule is about \(300 \mu m\) and cell is about \(20 \mu m\). The capsule size can be different from very small[11] to very large[12]. It can change flexibly on means of the system.
[figure 2. size of the capsule and the cell]
If the capsule and the cell is a sphere shape, we can get the volume from \(V= \frac{4}{3} \pi r^3\) (\(r\) is diameter).
The volume of the capsule is \(V= \frac{4}{3}\pi(150)^3 \approx 1.4 \times 10^7 \mu m^3\) .
The volume of the cell is \(V= \frac{4}{3}\pi(10)^3 \approx 4.2 \times 10^3 \mu m^3 \) .
So we could get maximum cell number could be encapsulated from following formula.
\[\mathrm{Maximum\; Cell\; number\; could\; be\; encapsulated}=\frac{\mathrm{volume\; of\; the\; capsule}}{\mathrm{volume\; of\; the\; cell}}=\frac{1.4 \times 10^7}{4.2 \times 10^3 }\approx 3.33 \times 10^3\]
Therefore about \(3.33 \times 10^3\) cells could be encapsulated in the capsule.
If there are \(3.33 \times 10^3\) cells in the capsule, we can calculate how many luciferins are needed to make \(1.49 \times10^{-12}mol \) all luciferins of each cell combined. \(1.49 \times10^{-12}\) mol is the number of luciferins that we need to make the light visible as we modeled above. The mol of luciferin needed to make a visible light in one cell is as below.
\[\frac{\mathrm{Minimum\; number\; of\; luciferins\; to\; make\; a\; visible\; light}}{\mathrm{amount\; of\; cells\; in\; one\; capsule}}=\frac{1.49\times10^{-12}mol}{3.33\times10^3 cells}= 4.48\times10^{-16} mol/cell\]
4) The amounts of capsules in one tattoo
Let’s assume that the tattoo that we make is 5cm long in radius. As we assumed above, the capsule is 150 micro meter long in radius. Then we can derive the amount of cells in one capsule. The tattoo area is divided by the capsules area.
\[\frac{\pi (5\times10^{-2} )^{2}}{\pi (3\times10^{-4} )^{2}}=2.77\times10^{4}capsules/tattoo\]
The amount of cells contained in one tattoo is \[3.33 \times 10^{3}cells/capsule\times2.77\times10^{4}capsules/tattoo = 9.22\times10^{7}cells/tattoo\]. Then the mol/g·cell value above can be reduced. The mol of luciferins needed to make a visible light in one cell is as below
\[\frac{\mathrm{Minimum\; number\; of\; luciferins\; to\; make\; a\; visible\; light}}{\mathrm{amount\; of\; cells\; in\; one\; tattoo}}=\frac{1.49\times10^{-12}mol}{9.22\times10^7 cells}= 1.62\times10^{-20} mol/cell\]
Cells in the tattoo need to have \(1.62\times10^{-20} mol\) in each. If the mass of the cell is assumed to be \(10^{-9} g\).
\[\frac{1.62\times10^{-20} mol/cell}{10^{-9} g}=16.2\times10^{-12} mol/g·cell\]
\(25 \sim 1000 \times10^{-12} mol/g·cell\) of luciferins is known to exist in the fruiting bodies of Mycena chlorophos, Omphalotus japonicus, and Neonothopanus gardneri which intrinsically have hispidin as a luciferin. [13]
Conclusion
\(1.49 \times10^{-12}mol\) of luciferins are needed to make a visible light. And we looked up how many luciferins are needed considering various condition, luciferins only, cells in capsule, capsules in tattoo. The insight derived from the result suggests that luciferins needed in the cells to make a visible light are a little bit less than that in the original cells. The value can change manipulating the size of capsules and cells. Or the result of characterization of luciferase can change the value. So making the system is quite plausible.
What matters forward is luciferin reaction rate and luciferin reaction time. Both of them can be discovered if Luz luciferase is characterized. If characterization of the luciferase is done, the result may suggest that more luciferins are needed. Even though the luciferin reaction time is discovered to be less than \(10^{-5}\) s that we assumed, the assumed luciferin reaction rate \(100 \% \) is so much ideal.
But seeing the wide range of luciferin concentrations in the original cells. Even though the characterization is done and the reaction rate is discovered to be 1% which is 1/100 ratio from the assumption, the luciferin concentration is not so much far from the range. So again, the system is plausible.
Reference
[1] Alexey A. Kotlobay et al., Genetically encodable bioluminescent system from fungi, PNAS, 2018.
[2] Human eye sensitivity and photometric quantities, Link
[3] 2011 iGEM KAIST Team, Link
[4] Recommended Light Levels, Link
[5] Zinaida M. Kaskova et al., Mechanism and color modulation of fungal bioluminescence, Science, 2017.
[6] Zinaida M. Kaskova et al., Mechanism and color modulation of fungal bioluminescence, Science, 2017.
[7] Luminous Efficacy Tables, Link.
[8] Fowler, S., Roush, R., & Wise, J. (2018). Concepts of Biology: OpenStax.
[9] Hatano, A. et al., CELLPEDIA: a repository for human cell information for cell studies and differentiation analyses, Database Oxford, 2011.
[10] Milo, R., & Phillips, R., Cell biology by the numbers. Garland Science, 2015.
[11] Sakai S. et al., Biocompatibility of subsieve-size capsules versus conventional-size microcapsules, J Biomed Mater Res A. 2006.
[12] Sugiura, S. et al., Size control of calcium alginate beads containing living cells using micro-nozzle array. Biomaterials, 2005.
[13] Yuichi Oba et al., Identification of hispidin as a bioluminescent active compound and its recycling biosynthesis in the luminous fungal fruiting body, Photochem. Photobiol. Sci., 2017.