Team:TU Delft/Model

iGEM TU Delft

Modeling

Our innovative project changes the optical properties of bacteria in order to make them into biological lasers and lenses at length scales close to the wavelength of light itself. Modeling was required to explain and predict the possibilities within the physical limits. We modeled the various components of our system from scratch, using ray and wave optics as well as kinetic and dynamic models solved by analytical and numerical simulation techniques. Our models helped the lab-team by predicting the largest obstacles of turning E. coli into a lens or laser. We found that making lasing E. coli is physically impossible since either the cell size or the fluorophore concentration has to be increased by a factor 10. This result directed the project more towards biolens development.

A series of studies were performed to determine if there is any focusing effect from the polysilicate covered E. coli cells. Secondly, the optical properties of various shapes of the cell and various thicknesses of the polysilicate layer are characterized on how they influence the performance of the microlenses. Our models predict that microbial biolenses can focus light and scatter most of it forwards. Therefore, our biolenses are promising for microlens applications. Finally we concluded that the best shape is the round cells as it is orientation independent and advised the lab team to pursue this design. Therefore the labteam investigated the possibilities to making spherical E. coli cells.

Biolasers

We investigated the physics behind the possibilities of using E. coli as a laser cavity. To this end, we have modeled the various components of our system using ray- and wave optics, as well as kinetic and dynamic models, solved using analytical and numerical simulation techniques. Using our models, we predicted the physical limitations of using E. coli as a laser cavity. In order to understand what lasing cells are and how to achieve them, we need to get some background knowledge on conventional lasers first. We will give a short introduction to lasers below, you can find a more detailed description of lasers here.

‘Laser’ stands for Light Amplification by Stimulated Emission of Radiation. In conventional lasers light resonates in an optical cavityAn optical cavity is an arrangement of optical components which traps the light inside in a closed path where the light can resonate. For most conventional lasers this is accomplished by placing two mirrors directly opposite each other. The gain medium is located in between the mirrors. , which is a space between two mirrors filled with a gain medium (figure 1A). The molecules in the gain medium get excited (in a higher-energy state) by an excitation source, for example an electric pulse or another laser. When a light particle, a photon, collides with a molecule that is in the excited state, this molecule will release a copy of the incident photon. This process is called stimulated emission and results in light getting amplified every time it passes through the gain medium.

Figure 1: (A) Conventional laser. In a conventional laser an optical cavity is the space between two mirrors opposite to each other. The optical cavity is filled with a gain medium composed of molecules that can get excited by an external excitation source, such as light or electric current. (B) For biolasers we use a reflective surface instead of mirrors. As an excitation source we use a laser that can excite the gain medium. The gain medium we use is composed of fluorescent proteins.

To make biological lasers we constructed an optical cavity inside a bacterium. As explained above, to get a laser, an optical cavity spaced between mirrors and a gain medium is required. To create the optical cavity between mirrors in the biolaser, we engineered the bacteria to form a reflective layer. We investigated two options for making a reflective layer inside E. coli.

First we investigated the possibility of making an optical cavity from the entire cell. Therefore we encapsulated the cell in a reflective surface (figure 1B). To make a reflective surface, we engineered bacteria to grow a polysilicate or tin dioxide layer on its outer membrane using the enzyme silicatein (André et al., 2011, Müller et al.,, 2008). When silicic acid or tin dioxide monomers are present in the extracellular surroundings, silicatein can polymerize these monomers. Silicatein can then produce a layer on the outside membrane when it is expressed and transported to the outside membrane. Silicatein can produce such a layer form silicic acid as a substrate but also from tin dioxide. Using modeling we will investigate what the effects are of using these two substrates. The labteam was only able to use silicic acid as a substrate but since tin dioxide has a hihger refractive index than polysilicate, it is an interesting alternative for further research.

As an alternative, we investigated the possibility to make an optical cavity from part of the cell. To this end we would have to engineer E. coli to produce PHB granules, in which the light can resonate; the boundary of the granule then acts as a mirror when total internal reflection takes place.

The materials polysilica, tin dioxide and PHB can act as a reflective layer since they have a higher refractive index compared to the cytoplasm. To get amplification of photons (i.e. produce a gain medium inside the cell) we express the fluorescent proteins GFP, mVenus, and mCerulean, which we excite with an external (pumping) laser.

Below you can find several models we made to investigate the limitations and opportunities of turning E. coli into a laser. The first aim of our modeling work was to predict and explain how light can resonate in our biological laser cavities. The first question we addressed was what the minimal size of a cell is for light to resonate inside as in a biolaser (Q1). Then we computed the concentration of fluorophores (gain medium) we have in our cells and how this concentration changes over time (Q2). Based on this information we constructed a model where we take the mirror losses into account (Q3). From this model we can find the laser threshold concentration of fluorophores inside the cavity and the threshold size of the cavity. Furthermore we investigated what the quality of the laser cavity is. (Q4).

In order to get lasing, light has to resonate within an optical cavity, formed by mirrors or a reflective surface. The reflective surface produced by E. coli (e.g. silica and tin dioxide layer and PHB granules) is not a perfect mirror and therefore the light can only get reflected by total internal reflection When light approaches a (reflective) interface between two materials, at a large enough angle (with respect to the normal to the surface) and the refractive index on the other side is lower, all the light is reflected. This phenomenon is called total internal reflection. At which angle light gets totally reflected depends on the material characteristics (refractive indices) of both materials. when approaching the edge of the cavity at a large enough angle. When the light gets reflected by total internal reflection in a closed circular path inside a microcavity, this type of optical cavity is called a whispering gallery mode (WGM) micro-resonator (Humar et al., 2015)(figure 2). Whispering gallery modes are the phenomenon that waves are circulating in a spherical object in a closed path, that is the begin and end of one circulation is at the same point, as a result of total internal reflection at the surface (Humar et al., 2015, Wilson et al., 2012). A closed path of a an integer number of wavelengths is required so that constructive interference If we add two (or more) waves together, it results in a new wave. If the original waves have overlapping crests and valleys (are in phase), the amplitude of the resulting wave will equal the sum of the amplitudes of the original waves, a phenomenon known as constructive interference. takes place.

Figure 2: Circular resonation of light in a whispering gallery mode resonator.

For the first method described above, E. coli encapsulates itself by a layer of polysilicate or tin dioxide. Here we determine what the minimal size of the bacteria should be to fit light inside the encapsulated E. coli, using ray optics. We found that a polysilicate covered cell should have a minimal diameter of 1-1.3 µm depending on the fluorescent proteins used. For a cell covered with tin dioxide, the minimal diameter varies between 1.5-1.7 µm depending on the fluorophore used.

We investigate the possibility of using PHB granules inside the cells as an optical cavity, as an alternative to using the whole cell as an optical cavity. In a similar method as for the encapsulated E. coli, the minimal size of the PHB granules was determined by ray optics. We found that the minimal diameter of the PHB granules is about 1.7 µm, which is larger than the small axis of E. coli. Since the PHB granules cannot reach this size inside the cell, it will not be possible to use these as an optical cavity. Because getting lasing inside PHB granules was not possible, we decided to investigate lasing in a polysilicate covered cell.

The minimal sizes we calculated for an optical cavity in E. coli encapsulated by a reflective layer are comparable to the natural size of E. coli. However, this is the most optimistic model, because we computed the absolute minimum size to fit one wavelength of light between two times hitting the reflective surface. Our model thus shows that we can only trap light inside a cell in the most ideal case where we do not take losses into account.

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As we need enough fluorophores in the gain medium to obtain lasing it is important to know how the concentration of fluorophores evolves over time when using specific promoters. Therefore we made a kinetic model as in figure 3. To determine the growth rate we were able to fit the measured OD values of the fluorophore expressing bacteria to a growth equation. The growth rate was used in the kinetic model to determine the promoter strength by fitting experimentally obtained data. In our case we use constitutive promoters, however it appears that this model is only applicable to inducible promoters.

Figure 3: Flowchart of the fluorophore concentration model. Here we P is the promoter strength, µ, the growth rate, Kt the transcription rate, Km , the maturation rate of the fluorescent protein, \(\gamma_m\) and \(\gamma_{GFP}\) the degradation rate of mRNA and GFP, respectively.

To determine the intracellular concentration of the fluorophores, we made a calibration curve of EGFP. Although the fluorophores we have used in the experiments are GFPmut3b, mVenus and mCerulean. In order to determine their concentration, we used instead the calibration curve of EGFP and the relative brightness for each fluorophore compared to EGFP (Gambhir et al., 2010). The brightness of a fluorophore is the product of its quantum yield and extinction coefficient, and is proportional to the number of photons produced per molecule. Using the relative brightness we were able to determine the concentration of fluorescent molecules in the cells with the fluorophores mVenus and mCerulean. We were able to determine the concentration of both the fluorophores mVenus and mCerulean of about 20mM.

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In order to determine the minimal concentration of fluorophores required for lasing we made a model describing the kinetics of our biolaser. In this model we included spontaneous emission, stimulated emission and the cavity characteristics, such as mirror reflectivity and absorption by the medium. We determine the number of photons and the number of fluorophores in the excited state and solve this over time. We compared the results of our model with biolaser experiments from literature (Gather et al., 2011) and derived a threshold for lasing as a function of fluorophore concentration. We found that in a cavity the size of an E. coli cell, (1 µm in diameter), lasing can occur starting at fluorophore concentrations larger than 0.1 M. However, in our cells the maximum concentration we can achieve is about 20 mM. Since concentrations an order of magnitude higher is impossible to reach within the cell, lasing is physically not possible in our cells. Due to the small size of our cells, the mirror losses are much higher than in a large laser cavity since the photons will bounce more often to the mirrors while traveling the same distance. When we stick to the concentration of 20 mM we can determine the minimal cell size to get lasing. The minimal cell size found to achieve lasing at a fluorophore concentration of 20mM was 8 µm in diameter. We did not find a clear difference between using polysilicate and tin dioxide as a refractive layer. From our model we found that the there is no clear difference between using polysilicate or tin dioxide as a reflective surface around the cell. experiments

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The quality factor (Q factor) indicates how much energy is stored in a our cavity versus how much is lost,it is defined as the energy stored divided by the energy lost in each circle times \(2 \pi\). (Cory and Chaniotakis, 2006; Kao and Santosa, 2008). As its name indicates, the Q factor is a measure of the quality of a resonator. Using COMSOL Multiphysics, we modeled the structure of the system and determined its eigenfrequencies.

Our modeling focused on finding eigenfrequencies and eigenmodes around the green light frequency to see if we can have resonation of light in our cavities. Our models showed that there are no modes of interest for the polysilicate covered cell but there are two interesting modes for the tin dioxide covered cell. An example of a mode in the tin dioxide covered cell can be seen in figure 4. This mode has the eigenfrequency of \(\omega = 6.028\times10^{14} + i\times2.3 \times 10^{11} \) Hz and results in a Q factor of \(Q=1.3109\times10^3\).

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Figure 4: Eigenmode of the tin dioxide covered cell, omega is: \(\omega = 6.028\times10^{14} + i\times2.3 \times 10^{11} \).
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Biolenses

We modeled E. coli cells covered with a polysilicate layer to investigate whether they can act as a biological micro lenses. The interaction of the cells covered in polysilicate can be calculated from Gustav Mie’s solution of the Maxwell equations, usually referred to as the Mie Theory. In this project we used COMSOL Multiphysics and CST suit to model the electromagnetic field interacting with our structures. In both COMSOL and CST we are using the RF modules which solve the Maxwell equations and can be used for Mie scattering problems.

It is important here to note that the reason that Maxwell’s equations were used and not the simpler ray optics is that the size of the cells is around \(1\mu m\) which is in the same scale as the wavelength used (in the visible spectrum, usually 500 nm), which means that the wave nature of light has to be taken into account and not the particle nature used in the ray optics. Figure 1 demonstrates an example of a beam of light passing through our structure. This simulation was made using the CST Studio Suit, it is in the time domain, meaning that we have an electromagnetic field and we see how it propagates in time and how interacts with the cell All of this happens in the time span of femto seconds. This kind of simulations, even though they are impressive, are very computationally expensive and we used a server to run them. For this reason, all the other simulations shown in the studies below were performed in the frequency domain, meaning that we assumed steady state and modeled for one instance instead of a series of steps.

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Figure 1: Time domain simulation of a beam of light passing through a polysilicate covered cell that acts as a microlens.

To make the biological micro lenses we needed to cover the cell with a material with higher refractive index than the medium (water) or the cell itself. This material is a thin layer of polysilicate. The bacteria used in this part of the project are transformed with OmpA-silicatein construct, similar to the laser part of our project. Figure 2 demonstrates the concept behind creating the biological micro lenses.

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Figure 2: Concept of covering a cell with polysilicate to create a biological micro lens.

In order to know if the polysilicate covered cells can function as a lens, the first objective of modeling was to see if there is focusing of light, and what the difference is between the rod shaped cells and the spherical shaped cells in focusing the light (Q5) . Then we investigated how the light scatters far away from the cell and if we have much back scattering or broad scattering of the light (Q6) . Finally, after we determined the best shape for the lenses, we investigated how the thickness of the polysilicate layer affects the scattering far away from the structure and the focusing of light (Q7) .

We created several models to investigate how the light interacts with the bacterial lenses. In this question there were two main parts. Firstly, we investigated if the polysilicate covered cells are actually able to focus light. Secondly, we modeled how the rod shaped structure behaves compared to the more symmetrical spherical cells.

In order to determine how light scatters on the bacterial microlenses we created models that simulate the interaction of electromagnetic waves (light) with our structure. We concluded that focusing is possible both for the rod shaped and for the spherical cell. Figure 1 demonstrates a time domain investigation of the light scattering. It is important to note that there is not a traditional well defined focal point, as we see in the ray optics, but rather a broader focal area where the intensity of the field is higher. Even though this is not a focal point we can still clearly see focusing of the light which is useful for our application.

After we saw that indeed we can have focusing we wanted to see how the two different shapes behave. First was the simpler spherical model and second the better representation of a cell, the rod shaped model. The conclusion was that, although all types were able to focus light, the rod shaped models are orientation dependent and thus they are not the best option compared to the spherical ones that are orientation independent. This led the lab team to investigate ways to change the shape of the cells to spherical cells to improve their behavior.

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After investigating the focusing effect, and deciding on the best shape for our cells in Q5 we wanted to see how the light scatters far away from the cell. This model is important for our application because we can see if we have any backscattering of light, which is undesired because less light passes through the lenses, or if we have side scattering, dispersion, of light ,which is undesired as well.

As seen in figure 3, the model showed rather clearly that the light is scattered almost exclusively forward. There is hardly any back scattering or side scattering. This means that the lenses have a very directional far field scattering pattern and that they do not reflect light back. Those results were later verified from spectroscopy measurements of our cells.

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Figure 3: 3D representation of the far field scattering.
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The final part of the modelling is an investigation of the effect of the polysilicate layer’s thickness on the focusing and scattering properties of the microlenses. We investigated the impact of thickness on the far field scattering and on the focusing of light from a spherical cell.

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Figure 4: Linear relation between the thickness of the polysilicate layer and the maximum intensity at the focal area.

The conclusion was that the far field shape is not effected by different thicknesses, but the focusing area and the maximum field at that area increases as the thickness of the polysilicate increases. Figure 4 illustrates the linear increase of the focusing effects with the increase of thickness.

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References

  1. André, R., Tahir, M. N., Schröder, H. C. C., Müller, W. E., & Tremel, W. (2011). Enzymatic synthesis and surface deposition of tin dioxide using silicatein-α. Chemistry of Materials, 23(24), 5358-5365.
  2. Cory_and_Chaniotakis. (2006). Frequency response: Resonance, Bandwidth, Q factor Resonance. Tutorial, 1, 1–11. Retrieved from http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/lecture-notes/resonance_qfactr.pdf
  3. Gambhir, S. S., & Yaghoubi, S. S. (2010). Molecular Imaging with Reporter Genes: Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511730405
  4. Gather, M. C., & Yun, S. H. (2011). Single-cell biological lasers. Nature Photonics, 5(7), 406–410. https://doi.org/10.1038/nphoton.2011.99
  5. Humar, M., & Yun, S. H. (2015). Intracellular microlasers. Nature Photonics, 9(9), 572–576. https://doi.org/10.1038/nphoton.2015.129
  6. Kao, C. Y., & Santosa, F. (2008). Maximization of the quality factor of an optical resonator. Wave Motion, 45(4), 412–427. http://doi.org/10.1016/j.wavemoti.2007.07.012
  7. Müller, W. E., Engel, S., Wang, X., Wolf, S. E., Tremel, W., Thakur, N. L., Schröder, H. C. (2008). Bioencapsulation of living bacteria (Escherichia coli) with poly (silicate) after transformation with silicatein-α gene. Biomaterials, 29(7), 771-779.
  8. Wilson, K. A., & Vollmer, F. (2012). Whispering Gallery Mode Resonator Biosensors. Springer Reference, 1–14. https://doi.org/10.1007/978-90-481-9751-4_121
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