Team:TU Delft/Model/Q3

iGEM TU Delft

Modeling

Q3: Can we determine the limit fluorophore concentration and limit size when taking into account the kinetics and dynamics of photons inside a biolaser cavity?

Introduction

In order to get lasing cells we encapsulate E.coli in polysilica or tin dioxide and we use fluorophores as a gain medium. In Q1 we already determined the minimal size in order to fit one wavelength of light inside the cells when the light resonates in whispering gallery modes. In this section we determine the lasing threshold. Therefore we take losses due to the mirrors and absorption and the gain due to stimulated emission into account. We will determine here what the minimal size and the minimal concentration of fluorophores is to get lasing.

Laser Kinetics and Dynamics

To achieve lasing the gain of photons in the system through stimulated emission has to be larger than the loss through absorption and escape. The rate of stimulated emission therefore has to be higher than a critical value, which we call the lasing threshold. In this section we determine this threshold and whether it will be reached in our system. To do so we describe the dynamics of the number of photons with the emission wavelength (N1), the number of GFP molecules in the excited state ( GFP1) and the number of GFP molecules in the ground state (GFP0).

Figure 1: Energy diagram spontaneous emission (A) and stimulated emission (B). (Super-Resolution Imaging Center)

When light is absorbed by an atom (or molecule), the energy of that atom increases with the energy of the photon that is adsorbed, a higher energy level (S1) (figure 1a) (Lakowicz, 2007). The atom will not stay in this higher energy level, but through a series of rapid non-radiative transitions its energy is lowered to a metastable state (S1,0). From the metastable state the atom will be able to release its energy in the form of an emitted photon. For spontaneous emission the emission spectrum is often broad and overlaps with the absorption spectrum. The release of energy does not necessarily go by emission of a photon as also non-radiative relaxation can take place. In that case the energy is released by heat.

Next to spontaneous emission, the metastable state of the atom can also be relaxed through stimulated emission, which is the basic principle that allows lasers to work (Hecht, 2002, Svelto et al,2010). In stimulated emission a photon hits the metastable particle and forces it to release its energy as another photon (figure 1b). This new photon has the same characteristics as the incident photon, meaning that they are in phase, with the same polarization, same direction, and same wavelength (Hecht, 2002, Svelto et al,2010). Stimulated emission can only take place when enough atoms are in the excited state, otherwise normal absorption is much more likely to occur. Furthermore the photons have to be trapped inside a cavity so that they will pass the excited molecules many times.

In a conventional laser system the gain medium (the fluorophores) are placed between 2 mirrors. Between these mirrors photons can oscillate to create a burst of photons emitted by stimulated emission. However the mirrors are never perfect and therefore some photons are lost from the system. Furthermore the medium between the mirrors (which in our case is the cytoplasm) does absorb photons, so we also should take absorption losses into account.

We can describe the dynamics of the fluorophores in excited state (GFP1) and the photons N1 in our system by equations (1,2). $$ \frac{\partial GFP_1}{\partial t} = Nonradiative Relaxation + (Spontaneous +Stimulated) Emission$$ $$ \frac{\partial N_1}{\partial t} = (Spontaneous +Stimulated) Emission - Mirror losses- Absorption losses$$

Because the total GFP concentration (GFPt) is fixed the dynamics of the ground state GFP molecules follows directly from equation (1,3).

$$ GFP_T=GFP_0+GFP_1$$

In lasing experiments done by other groups (Humar et al, 2015, Gather et al, 2011) . With fluorophores as a gain medium, a pulsing laser is used to excite the fluorophores in the system. A pulsing laser is used to create a population inversion while minimizing photo-bleaching. In our model we will look at the photon dynamics directly after a pulse, and therefore population inversion has taken place. We will assume that initially all the GFP molecules are in the excited state. Furthermore we assume that initially there are no photons at emission wavelength (N1) in the system. Therefore the initial conditions are given by equation (4,5).

$$GFP_1(0)=GFP_t$$ $$ N_1(0)=0$$

The non-radiative emission and spontaneous emission in equation (1) are both dependent on the relaxation time of GFP. Therefore the spontaneous relaxation rate of GFP1 can be given by equation (6) where ksp is the spontaneous emission rate and \(\tau_{sp}\) the spontaneous emission time (Svelto et al, 2010).

$$k_{sp} = \frac{1}{\tau_{sp}}$$

Because not all the spontaneous relaxation transitions result in emission of a photon, we should take the efficiency of the fluorophores into account. The efficiency of fluorophores can be described by the quantum yield (QY), the number of photons emitted per photon absorbed by a GFP molecule (Lakowicz, 2007). Therefore the spontaneous emission term in equation (1,2) becomes \(QY\cdot k_{sp}\).

The stimulated emission rate kstim (equation 7) is proportional to the stimulated emission cross section (\(\sigma_e\)) of the fluorophore and the volume of the system(Svelto et al, 2010). Here V is the volume of the optical cavity.

$$ k_{stim} = \frac{\sigma_ec}{V}$$

The stimulated emission cross-section is comparable to the absorption cross section (\(\sigma_a\)) (Mit et al, 2009), therefore we take \(\sigma_e=\sigma_a\) as literature values of \(\sigma_a\) can be easily found.

The losses due to absorption can be approximated by the law of Lambert-Beer. The time dependent version of Lambert-Beer follows from the differential Lambert-Beer law (Parnis et al. , 2013) and the speed of the photons, c (equation 8). In equation 8, x is the distance traveled, I0 is the intensity of the light, \(\alpha/ ) the absorption coefficient and N1 the number of photons with the emission wavelength.

$$ \left.\begin{matrix} \frac{\partial I_0}{\partial x} = -\alpha I_0 \\ \frac{\partial x}{\partial t} = c\\ I_0\propto N_1 \end{matrix}\right\} \frac{\partial N_{1,absorption}}{\partial t} = -\alpha \cdot c \cdot N_1 = K_{abs} \cdot N_1 $$

The absorption constant \(\alpha\) is determined experimentally by measuring the absorption of light at emission wavelength. The total absorption A is given by\(A=\alpha l=\log_{10}\frac{I_0}{I}\). The plate we used has a path length of 10.2 mm. We obtained a value of \(\alpha\) of \(15.2 m^{-1}\).

The losses due to the mirrors depend on the number of times a photon hits a mirror and the reflectivity of each mirror (Wilmsen et al, 2001, Svelto et al, 2010). The mirror loss rate (kmirror) is given by equation 9 where c is the speed of light and l the length of the optical cavity; its value is negative because the reflectivity of the mirrors R is always smaller than 1: \(R_1, R_2 <1\). We will determine the reflectivity below.

$$ K_{mirror} = \frac{c}{2l}\ln(R_1R_2)$$

Combining all contributions the dynamics of the system can be described by the following set of differential equations:

$$\frac{dGFP_1}{dt} = (k_{sp} + k_{stim}\cdot N_1)GFP_1 = \Big(\frac{1}{\tau_sp} +\frac{\sigma_a c}{V} N_1\Big)GFP_1$$ $$\begin{split} \frac{dN_1}{dt} = (QY\cdot k_{sp}+&k_{stim}\cdot N_1)GFP_1+(k_{mirror}+k_{abs})N_1\\ & = \Big(\frac{QY}{\tau_{sp}}+\frac{\sigma_a c}{V}N_1\Big)GFP_1 + \Big(\frac{c}{2l}\ln(R_1R_2)-\alpha\cdot c\Big)N_1 \end{split}$$

References

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