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 towards biolens development. Our models predict that microbial biolenses scatter most of the light forwards and focus it therefore our biolenses are promising for microlens applications. From the models we find that the E. coli focuses the light about \(0.5-1 \mu m\) after the cell.

Biolasers

Using modeling we investigated the physics behind the possibilities of using E. coli as a laser cavity or lenses. Therefore 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, but we also made a more detailed description of lasers, which can be found here.

‘Laser’ stands for Light Amplification by Stimulated Emission of Radiation. In conventional lasers light resonates in an An 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 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 (higher-energy 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 use a reflective surface instead of mirrors. As an excitation source we use a laser which can excite the gain medium. The gain medium we use are fluorophores.

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 biosilica or tin dioxide layer on its outer membrane using the enzyme silicatein. Silicatein can produce a such a layer when expressed and transported to the outside membrane. When silicic acid or tin dioxide monomers are present in the extracellular surrounding, silicatein can polymerize the monomers.

Alternatively, we investigated the possibility to make an optical cavity from part of the cell. Therefore 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, the inside of the cell. To get amplification of photons (i.e., produce a gain medium inside the cell) we express the fluorescent molecules 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 making 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 these pieces of 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 factor of the cavity is (Q4).

In order to get lasing, light has to resonance within an optical cavity formed by mirrors or a reflective surface. In bacteria the light has to be reflected in a spherical cavity since the E. coli is rod shaped. The reflective surface produced by E. coli (e.g. silica and tin dioxide layer and PHB granules) are not perfect mirrors and therefore the light can only get reflected by total internal reflection 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 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 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 biosilica or tin dioxide. Here we determine what the minimal size of the bacteria should be to fit light inside the E. coli encapsulated with biosilica or tin dioxide using ray optics. We found that a biosilica covered cell should have minimal diameter of \(1.3 \mu m\) when GFPmut3b or mVenus is used as a gain a diameter of \(1\mu m\) when mCerulean is used as a gain medium. When a cell is covered with tin dioxide, the minimal diameter is \(1.6 \mu m\) when GFPmut3b is used as a gain medium, a diameter of \(1.7 \mu m\) and \(1.5\mu m\) when mVenus and mCerulean are used as a gain medium, respectively.

The alternative to using the whole cell as an optical cavity we investigated the possibility of using PHB granules inside the cells as an optical cavity. In a similar method as for the encapsulated E. coli the minimal size of the PHB granules is determined by ray- optics. We found that the minimal diameter of the PHB granules is \(d\approx 1.7\mu m\) which is larger than the small axis of E. coli. Therefore we expected that the granules will not grow to this size and therefore we won’t be able to fit light inside the PHB cavity. Thus lasing won’t be physically possible in PHB granules in E. coli since the light does not fit within the cell to resonate.

The sizes we found 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 where 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 if it functions as a perfect laser cavity.

The complete model ca be found here.

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Because we need enough fluorophores in the gain medium to get lasing we wanted to know how the concentration of fluorophores evolves over time when we use specific promoters. Therefore we made a kinetic model which takes the promoter strength- (P), growth- (\(\mu\)), transcription rate- (\(K_t\)), degradation of mRNA- (\(\gamma_m\)) and maturation (\(K_m\)) of fluorescent protein, and the degradation of non-fluorescent protein (\(\gamma_{GFP}\)) and fluorescent protein (\(\gamma_{GFP}\)) into account as in figure 3. The full description of this model can be found here. To determine the growth rate we were able to fit the measured OD values of the bacteria to a growth equation. The growth rate can then be used in the kinetic model to determine the promoter strength. This model can be used to predict the protein concentration for inducible promoters. It appears that this model cannot be used when using a constitutive promoter which we use in our fluorophore constructs.

Figure 3: Flowchart of the fluorophore concentration model

To determine the intracellular concentration of the fluorophores we make a calibration curve of EGFP. The fluorophores we are using in the experiments are however GFPmut3b, mVenus and mCerulean. In order to determine their concentration we can use 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 are able to determine the concentration of fluorescent molecules in the cells with the fluorophores mVenus and mCerulean. The concentration of both fluorophores is about 20mM. How we exactly determine these concentrations and their results can be found here.

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In order to determine the limit concentration of fluorophores required for lasing we made a model that described 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. The complete model can be found here. 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, (\(1\mu m\) diameter), lasing can occur starting at fluorophore concentrations larger than 0.1 M. However, in our cells the maximum concentration we can achieve is in the nM to µM range and therefore lasing is physically not possible in our cells. To get lasing, we would either need much larger cells or a much higher concentration of fluorophores. The minimal size to get lasing with a concentration of 20 mM is around \(8\mu m\). We did not find a clear difference between using polysilica and tin dioxide as a refractive layer. The lab team did not test the tin dioxide covered cells but only the polysilica covered cells which should give comparable results.

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Biolenses

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

It is important here to note that the reason that Maxwell equations were used and not the simpler ray optics is that the size of the cells is around \(1\mu m\) that is very comparable with the wavelength used (in the visible spectrum, usually 500 nm), this means that the wave nature of light had to be taken into account and not the particle nature used in the ray optics. Figure 3 demonstrates an example of a beam of light passing through our structure. This simulation was made using the CST modeling suit, it is in time domain meaning that we have an electromagnetic field and see how it propagates in time a and how interacts with the cell, see the time in femptoseconds in the left down corner. Those kind of simulations even though they are impressive they are very computationally expensive and we used a server to run it. All the other simulation shown in the studies below were performed in the frequency domain meaning that we waited until steady state reached and modeled for one instance instead of a series of steps.>

Electromagnetic_spectrum
Figure 2: 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. To create this thin layer of polysilicate we engineered bacteria to grow a layer on its outer membrane using the enzyme silicatein. Silicatein can produce a such a layer when expressed and transported to the outside membrane. When silicic acid or tin dioxide monomers are present in the extracellular surrounding, silicatein can polymerize the monomers. This part is the same as the one used in the aforementioned lasers investigation. Figure 3 demonstrates the concept behind creating the biological micro lenses.

Electromagnetic_spectrum
Figure 3: Concept of covering a cell with polysilicate to create a biological micro lens.

Below you can find all the models we created to investigate the possibility of the E.coli cells covered with polysilicate layer to act as biological microlenses. The first objective was to see if we have any focusing effect and what is the difference between the rod shaped cells and the spherical shapes 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 effects the scattering in the far field and the focusing of light (Q7) .

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. The models were created in 3D in the software ‘COMSOL Multiphysics’ and we obtained two results: the fact that there is focusing of light after the lens and that the fact that there is no back scattering of light after meeting the lenses. Models of the rod shape, which is the regular shape of E. coli, and spherical shape (not orientation related) were made. More details about those models can be found here.

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References

  1. Humar, M., & Yun, S. H. (2015). Intracellular microlasers. Nature Photonics, 9(9), 572–576. https://doi.org/10.1038/nphoton.2015.129
  2. 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
  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
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