Team:TU Delft/Model
Modeling
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 optical cavity, 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.
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 need to engineer 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 we let the bacterium 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.
These materials 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, facilitating the lab team. The first aim of our modeling work was to predict and explain how light can resonate in our biological laser cavities. For this project we addressed several questions. 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 limit concentration of fluorophores inside the cavity and the minimal size of the cavity. Furthermore we investigated what the quality factor of the cavity is (Q4).
Q1. How can light resonate in the cell and what is the minimal size of the cavity in order to fit light inside it?
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.
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 \(0.9 \mu m\) and a cell covered with thin dioxide should have a minimal diameter of \(1.3 \mu m\) as can be found here.
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 geometrical optics. We found that the minimal diameter of the PHB granules is \(d\approx 1.5\mu m\) 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.
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 for hitting the reflective surface two times. Our model thus shows that we can only trap light inside a cell if it functions as a perfect laser cavity.
Back to TopQ2. How does the fluorophore concentration in the gain medium evolve over time?
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(\(\gamma_m\)) , non-fluorescent protein (\(\gamma_{GFP}\)), fluorescent protein (\(\gamma_{GFP}\)) and maturation (\(K_m\)) of fluorescent protein 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.
