Team:KU ISTANBUL/Results
Results
We have created computational models of our biological resonators and performed finite difference time domain simulations using Lumerical FDTD. Details of our models can be found in the modelling page.
We have determined the minimum sizes of cells for each model geometry in an experiment where we would cover the cell with proteins with a high-refractive index. We have taken the experimentally achievable refractive index limit as 1.7 since there are some experimental results in the literature in which researchers can measure the refractive index of reflectins in different media [1-4 ].
First of all, as a general remark we can see that we need higher refractive index materials to build a resonator as the resonator size gets smaller. This correlation becomes even more drastic in the size limit of below 5 µm for structures that have spherical symmetry.
When we analyze the spherical models, we can see that one can obtain whispering gallery modes from the resonators with diameters 10 µm, 9 µm and 8 µm if these resonators were built from a material with refractive index around 1.7. Resonance peaks of these cavities are very clear as can be seen in Figures 1-4, 8-11, 15-18. Also heatmaps belonging to these modes are given in Figures 5-7, 12-14, 19-21.
If we look at Figures 22-25, 26-28, we can see that the refractive index limit for clear resonance peaks rose up, from 1.7 to 1.8 or 1.9; as we are decreasing the diameter of the cavity, the cavity can no longer sustain the whispering gallery modes. They start appearing at higher refractive indices, and they are again very sharp at higher indices. Heatmaps belonging to these modes are given in Figures 26-28, 33-35.
We can realize that we can no longer see sharp resonance peaks in a cavity with 1.9 refractive index membrane for 4-5 µm diameters as can be seen in Figures 36-37, 41-42. Although the resonance occurs for higher refractive index around 1.9-2.0, we can see these resonance modes in Figures 38-40, 43-44.
As we are analyzing even smaller cavities of diameters 1-3 µm, it is very clear that we can not observe whispering gallery modes in these cavities if they were built with a material of refractive index smaller than 2.0. We have performed simulations with higher refractive index materials just for these small cavities, and we have found that one can observe resonance modes if these cavities are built with higher refractive index materials around 2.25-2.5. Resonance peaks obtained for these modes can be seen in Figure 45-48.
Now let's discuss disk-shaped models. Actually, we have found very similar results for disk-shaped models to spherical models. Initial guess for this similarity would be that it is because we are running simulations on a 2-dimensional plane. If you did not take into account the z-axis of the structures in the model, a sphere and a disk would be very similar although there are minor differences. Nevertheless, we have found resonance modes in different places and at different frequencies which points out the real differences between these two geometries of cavities.
For disk-shaped cavities of diameters between 10-8 µm, there is enough evidence for the proposal that one can obtain whispering gallery modes in a cavity which is built from a material with refractive index 1.7. The resonance peaks are clearly visible as they were in spherical models (see Figure 49-51, 55-57, 61-64). Also, heatmaps for these resonance modes can be seen in Figures 52-54, 58-60, 65-67.
If the disk-shaped cavity has a diameter between 7-6 µm, the refractive index of the membrane should be around 1.8 to properly observe resonance modes. The sharp peaks belonging to these cavities can be seen in Figures 68-70, 73-75 whereas the heatmaps can be found in Figures 71-72, 76-78.
The same pattern exists when the diameters get smaller. The membrane should have a refractive index of minimum 1.9 if the disk has a diameter of 4-5 µm. Resonance peaks of these cavities can be seen in Figures 79-81, 84-85 along with their corresponding heatmaps in Figures 82-83, 86.
For a very small disk-shaped cavity, diameters smaller than 3 µm, our results reveal that it is not possible to obtain resonance modes in these cavities if the membrane has refractive index smaller than 2.0. Resonance peaks obtained with higher refractive indices for these cavities can be found in Figure 87-89.
We have designed toroidal models for red blood cells in mind although these models can represent some inhomogeneity in biological cells inside our computational models. So we find them helpful for designing lasing experiments.
Actually, our results revealed that these toroidal models are not so different from the spherical and disk-shaped models in perspective for the correlation between refractive index, size of the cavity and resonance modes. Again we have obtained a stair like relation between the minimum refractive index required for resonance and the size of the cavity. As the size of a toroidal cavity gets smaller towards 4 µm, the required refractive index for resonances is getting higher and higher by 0.1 steps in the refractive index with 2 µm steps in diameter of the cavity.
For toroidal cavities with diameters between 8-10 µm, refractive index should be around 1.7-1.8 as can be seen in Figures 90-93, 97-100, 104-107 with corresponding heatmaps in Figures 94-96, 101-103, 108-110.
If the diameters of toroids are between 5-7 µm, the refractive index should be around 1.8-2.0 as can be seen in Figures 111-113, 117-119, 123-125 with corresponding heatmaps in Figures 114-116, 120-122, 126-128.
When the diameters of toroids are smaller than 4 µm, the refractive index of the membrane should be much higher than the 2.0 limit as can be seen in Figures 129-131. We have achieved resonance in this region, using refractive indices of around 2.5.
Overall results from our computational models is that we can obtain whispering gallery modes from cavities with diameters 5 µm or bigger using special proteins as a coating material for the membrane of the cell. From this result and the recommendations of the previous iGEM team TU_Delft 2016, we have included S. cerevisiae (baker’s yeast) into our experimental plan since they are much larger than most bacteria and have nearly spherical shapes before they start budding. We are confident about achieving lasing from resonators built from yeast cells by covering them reflectins and silicatein. Our computational results are satisfactory for our purposes even if they are still lacking some details of real biological cells and their environment.
Note: When we use the term “refractive index” in all of our documents, we are referring to the value of this parameter around the visible spectrum. Since most cellular materials show very small differences in refractive index, we have found it appropriate to use the term everywhere as it is easier to understand for a novice.
Note 2: Figure numbers appearing in this page are continuation of the figures presented in the supplementary materials. So, any figure mentioned on this page is either here or in the supplementary materials.
[1] Crookes, W. J.; Ding, L.-L.; Huang, Q. L.; Kimbell, J. R.; Horwitz, J.; McFall-Ngai, M. J. Reflectins: The Unusual Proteins of Squid Reflective Tissues. Science (80-. ). 2004, 303 (5655), 235 LP – 238. https://doi.org/10.1126/science.1091288
[2] Ghoshal, A.; DeMartini, D. G.; Eck, E.; Morse, D. E. Experimental Determination of Refractive Index of Condensed Reflectin in Squid Iridocytes. J. R. Soc. Interface 2014. https://doi.org/10.1098/rsif.2014.0106
[3] Ogawa, J.; Iwata, Y.; Tonnu, N.; Gopinath, C.; Huang, L.; Itoh, S.; Ando, R.; Miyawaki, A.; Verma, I.; Pao, G. Genetic Manipulation of the Optical Refractive Index in Living Cells. bioRxiv 2020, 2020.07.09.196436. https://doi.org/10.1101/2020.07.09.196436
[4] Chatterjee, A.; Cerna Sanchez, J. A.; Yamauchi, T.; Taupin, V.; Couvrette, J.; Gorodetsky, A. A. Cephalopod-Inspired Optical Engineering of Human Cells. Nat. Commun. 2020, 11 (1), 2708. https://doi.org/10.1038/s41467-020-16151-6
A. Spherical Models
We have created spherical models with varying diameters and refractive indices of the membrane. Here, we will be explaining the effects of every combination of these two major parameters onto the whispering gallery modes of our model resonator.
1. For 10 µm diameter spherical cavity:
Figure 1: Power spectra of cavity with 10 µm diameter. Refractive index of the membrane in this model is 2.0.
Figure 2: Power spectra of cavity with 10 µm diameter. Refractive index of the membrane in this model is 1.9.
Figure 3: Power spectra of cavity with 10 µm diameter. Refractive index of the membrane in this model is 1.8.
Figure 4: Power spectra of cavity with 10 µm diameter. Refractive index of the membrane in this model is 1.7.
Figure 5: Heatmap of the resonance mode obtained from a spherical cavity of 10 µm diameter. The resonance mode is at wavelength 443nm.
Figure 6: Heatmap of the resonance mode obtained from a spherical cavity of 10 µm diameter. The resonance mode is at wavelength 458nm.
Figure 7: Heatmap of the resonance mode obtained from a spherical cavity of 10 µm diameter. The resonance mode is at wavelength 474nm.
B. Disk-Shaped Models
We have created disk-shaped models with varying diameters and refractive indices of the membrane. Here, we will be explaining the effects of every combination of these two major parameters onto the whispering gallery modes of our model resonator.
1. For 10 µm diameter disk-shaped cavity:
Figure 49: Power spectra of cavity with 10 µm diameter. Refractive index of the membrane in this model is 2.0.
Figure 50: Power spectra of cavity with 10 µm diameter. Refractive index of the membrane in this model is 1.9.
Figure 51: Power spectra of cavity with 10 µm diameter. Refractive index of the membrane in this model is 1.8.
Figure 52: Heatmap of the resonance mode obtained from a disk-shaped cavity of 10 µm diameter. The resonance mode is at wavelength 443 nm.
Figure 53: Heatmap of the resonance mode obtained from a disk-shaped cavity of 10 µm diameter. The resonance mode is at wavelength 457 nm.
Figure 54: Heatmap of the resonance mode obtained from a disk-shaped cavity of 10 µm diameter. The resonance mode is at wavelength 473 nm.
C. Toroidal Models
We have created toroidal models with varying radii and refractive indices of the membrane. Here, we will be explaining the effects of every combination of these two major parameters onto the whispering gallery modes of our model resonator.
1. For 10 µm diameter toroidal cavity:
Figure 90: Power spectra of cavity with 10 µm diameter. Refractive index of the membrane in this model is 2.0.
Figure 91: Power spectra of cavity with 10 µm diameter. Refractive index of the membrane in this model is 1.9.
Figure 92: Power spectra of cavity with 10 µm diameter. Refractive index of the membrane in this model is 1.8.
Figure 93: Power spectra of cavity with 10 µm diameter. Refractive index of the membrane in this model is 1.7.
Figure 94: Heatmap of the resonance mode obtained from a toroidal cavity of 10 µm diameter. The resonance mode is at wavelength 429 nm.
Figure 95: Heatmap of the resonance mode obtained from a toroidal cavity of 10 µm diameter. The resonance mode is at wavelength 443 nm.
Figure 96: Heatmap of the resonance mode obtained from a toroidal cavity of 10 µm diameter. The resonance mode is at wavelength 457 nm.