Difference between revisions of "Team:TU Delft/Model"

Revision as of 14:23, 19 October 2016

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 to 1 µm after the cell.

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

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 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 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).

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 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) does not act as 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 biosilica 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 biosilica covered cell should have 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.

The alternative to using the whole cell as an optical cavity is using a part of the cell as optical cavity. Therefore, we investigated the possibility of using PHB granules inside the cells. 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 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.

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 in the most ideal case where we do not take losses into account.

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Q2. How does the fluorophore concentration in the gain medium evolve over time?

Because 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, K_t the transcription rate, K_m , 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 are using in the experiments are 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.

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Q3. Can we determine the laser threshold fluorophore concentration and cell size when taking into account the kinetics and dynamics of photons inside a biolaser cavity?

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 was found to be 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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Q4. What is the quality factor of the cavity?

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

Electromagnetic_spectrum — **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 biosilica 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 3 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 cellAll of this happens in the time span of fempto seconds. This kind of simulations even though they are impressive they 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.

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. This part is the same as the one used in the aforementioned lasers investigation but without expressing the fluorescent proteins in the cells. Figure 3 demonstrates the concept behind creating the biological micro lenses.

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 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 affects the scattering far away from the structure and the focusing of light (Q7) .

Q5. How does the polysilicate covered cell focus the light?

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 doesthe 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 2 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 usefull 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 round to improve their behavior.

More detailed explanation about the models describing how the polysilicate covered cells interact with the light can be found here

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Q6. How does the polysilicate layer covered cell scatters the light?

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 4, 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. More details about the model can be found here

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Q7. How does the thickness of the polysilicate layer influences the performance of the spherical microlenses?

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.

The conclusion was that the far field shape is not effected bydifferent thicknesses, but the focusing area and the maximum field at that area increases as the thickness of the polysilicate increases. Figure 5 illustrates the linear increase of the focusing effects with the increase of thickness. More detailed description of the models used and the results can be found here.

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Humar, M., & Yun, S. H. (2015). Intracellular microlasers. Nature Photonics, 9(9), 572–576. https://doi.org/10.1038/nphoton.2015.129
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
Gambhir, S. S., & Yaghoubi, S. S. (2010). Molecular Imaging with Reporter Genes: Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511730405
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
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
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

[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

[5] 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

[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

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 <p>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 <i>E. coli</i> 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.</p>
-<p>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.</p>
+<p>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.</p>
 <p>Below you can find several models we made to investigate the limitations and opportunities of making <i>E. coli</i> 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 <b><a href="#Qs">(Q1)</a></b>. Then we computed the concentration of fluorophores (gain medium) we have in our cells and how this concentration changes over time <b><a href="#Qs">(Q2)</a></b>. Based on this information we constructed a model where we take the mirror losses into account <b><a href="#Qs">(Q3)</a></b>. 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.   <b><a href="#Qs">(Q4)</a></b>.</p>
@@ Line 78: / Line 78: @@
 <p> The alternative to using the whole cell as an optical cavity is using a part of the cell as optical cavity. Therefore, we investigated the possibility of using PHB granules inside the cells. In a similar method as for the encapsulated <i>E. coli</i>, the minimal size of the PHB granules is 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 <i>E. coli</i>. Since the PHB granules cannot reach this size inside the cell, it will not be possible to use these as an optical cavity. </p>
 <p>The sizes we found for an optical cavity in <i>E. coli</i> encapsulated by a reflective layer are comparable to the natural size of <i>E. coli</i>.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 in the most ideal case where we do not take losses into account. </p>
-<p>The complete model can be found <b><a href="/Team:TU_Delft/Model/Q1" target ="_blank"> here</a></b>.</p>
 <a href="https://2016.igem.org/Team:TU_Delft/Model/Q1" class="btn btn-info" role="button" style="text-decoration:none; color:#f3f4f4; float:right;">Complete model</a>
 <a href="#Qs" class="btn btn-info" role="button" onclick="$('html,body').animate({scrollTop: $('#Qs').offset().top}, 'slow');" style="text-decoration:none; color:#f3f4f4; float:right;">Back to Top</a>
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-<p> 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 <b><a href="https://2016.igem.org/Team:TU_Delft/Model/Q2" target ="_blank"> here</a></b>. 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.</p>
+<p> Because 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. </p>
 <figure>
 <img src="https://static.igem.org/mediawiki/2016/a/a8/T--TU_Delft--modeling3.png" alt="">
-<figcaption>Figure 3: Flowchart of the fluorophore concentration model</figcaption>
+<figcaption>Figure 3: Flowchart of the fluorophore concentration model. Here we P is the promoter strength, µ, the growth rate, K<sub>t</sub>  the transcription rate, K<sub>m</sub> , the maturation rate of the fluorescent protein, \(\gamma_m\) and \(\gamma_{GFP}\) the degradation rate of mRNA and GFP, respectively.  </figcaption>
 </figure>
-<p> 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 <a href="#references">(Gambhir <i>et al.</i>, 2010)</a>. 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 <b><a href="https://2016.igem.org/Team:TU_Delft/Model/Q2" target="_blank"> here</a></b>.</p>
+<p> To determine the intracellular concentration of the fluorophores we made a calibration curve of EGFP. Although the fluorophores we are using in the experiments are 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 <a href="#references">(Gambhir <i>et al.</i>, 2010)</a>. 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. </p>
+<a href=" https://2016.igem.org/Team:TU_Delft/Model/Q2" class="btn btn-info" role="button" style="text-decoration:none; color:#f3f4f4; float:right;">Complete model</a >
 <a href="#Qs" class="btn btn-info" role="button" onclick="$('html,body').animate({scrollTop: $('#Qs').offset().top}, 'slow');" style="text-decoration:none; color:#f3f4f4; float:right;">Back to Top</a>
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-<p>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 <b><a href="https://2016.igem.org/Team:TU_Delft/Model/Q3" target="_blank">here</a></b>. We compared the results of our model with biolaser experiments from literature <a href="#references">(Gather  <i>et al.</i>,  2011)</a> and derived a threshold for lasing as a function of fluorophore concentration. We found that in a cavity the size of an <i>E. coli</i> 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. </p>
+<p>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 <a href="#references">(Gather  <i>et al.</i>,  2011)</a> and derived a threshold for lasing as a function of fluorophore concentration. We found that in a cavity the size of an <i>E. coli</i> 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 was found to be 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</p>
+<a href=" https://2016.igem.org/Team:TU_Delft/Model/Q3" class="btn btn-info" role="button" style="text-decoration:none; color:#f3f4f4; float:right;">Complete model</a >
 <a href="#Qs" class="btn btn-info" role="button" onclick="$('html,body').animate({scrollTop: $('#Qs').offset().top}, 'slow');" style="text-decoration:none; color:#f3f4f4; float:right;">Back to Top</a>
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Difference between revisions of "Team:TU Delft/Model"

Revision as of 14:23, 19 October 2016

Modeling

Biolasers

Q1. How can light resonate in the cell and what is the minimal size of the cavity in order to fit light inside it?

Q2. How does the fluorophore concentration in the gain medium evolve over time?

Q3. Can we determine the laser threshold fluorophore concentration and cell size when taking into account the kinetics and dynamics of photons inside a biolaser cavity?

Q4. What is the quality factor of the cavity?

Biolenses

Q5. How does the polysilicate covered cell focus the light?

Q6. How does the polysilicate layer covered cell scatters the light?

Q7. How does the thickness of the polysilicate layer influences the performance of the spherical microlenses?

References