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 If we add two (or more) waves together a new wave results. If the original waves have overlapping crests and valleys (are in phase), the amplitude of the resultant wave will equal the sum of the amplitudes of the original waves, a phenomenon known as constructive interference. takes place.

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.

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.

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.

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.

The quality factor (Q factor) indicates how much energy is stored in a resonator (our cavity) versus how much is lost. is defined as the energy stored divided by the energy lost in each circle times \(2 \pi\). (Cory_and_Chaniotakis, 2006; Kao & Santosa, 2008). As its name indicates is a measure of the quality of a resonator. The method to find the quality factor for the biological laser is using by finding its eigenfrequencies, this was achieved by modeling the structure in COMSOL Multiphysics and investigating for modes.

Our models resulted 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 \(Q=1.3109\times10^3\). All the models and methods used can be found here