Team:TU Delft/Model/Q2

iGEM TU Delft

Modeling

Question 2:

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

Introduction

Kinetic Model

In order to predict the fluorophore concentration in our cells, we model their kinetics. The kinetics of the fluorescent protein can be described by the flow diagram in figure 1. In our model transcription and translation are summarized into one variable: the promoter strength P. Furthermore we take the copy number (cn) into account to determine the mRNA concentration (m). When GFP is synthesized at the rate (\(K_t\)) it has to go through some extra folding (maturation) before it can become fluorescent (\(K_m\)). Therefore n(t) is the concentration of immature GFP molecules and f(t) is the concentration of mature GFP molecules. (\(\gamma_{GFP}\)) is the degradation rate for GFP assuming first order degradation and (\(\gamma_m \)) is the degradation rate of mRNA. (\(\mu \)) is the growth rate.The dynamics can be described by differential equations 1, 2 and 3.

Figure 1: Flowchart fluorophore kinetics
$$\frac{\partial m}{\partial t} = c_n P-(\mu(t)+\gamma_m) m(t)$$ $$ \frac{\partial n}{\partial t} = K_t m(t) - (K_m + \mu(t) +\gamma_{GFP} ) n(t)$$ $$ \frac{\partial f}{\partial t} = K_m n(t) -(\mu(t) +\gamma_{GFP}) f(t)$$

The growth rate can be found by fitting the optical density (OD600 values) to the growth curve in equation 4 (Koseki et al., 2014).

$$OD = \frac{N_mN_0(q\cdot e^{\mu_{max}t}+1)}{N_0q(e^{\mu_{max}t}-1)+N_m(q+1)}$$

Here Nm is the maximal population density, N0 is the population density at \(t=1\), q is a dimensionless quantity related to the physiological state of the cell and \(\mu_{max}\) is the maximal growth rate. The initial guesses used were \(\mu_{max} = 0.96, q=0.2, N_{max} = 1, N_0 = N_0(t=0)\). The fit for the GFPmut3b strains can be found in figure 2. The values are only fitted for the first 16 hours. During first 16 hours the population grows as expected initially in lag phase followed by exponential growth followed by stationary phase after 10-16 hours. However after approximately 16 hours the population starts growing again. We expect this can be the resulted of nutrient stress related protein degradation. The values from the fit to the growth curves can be found in table 1.

Figure 2: Growth of E. coli population containing plasmids with GFP expressed under 5 different promoters. The blue curve is the measured OD600 values, the red crosses represent growth fitted to equation 4.
Table 1: Growth parameters found by OD600 fitted to equation 4.
J23100 J23108 J23105 J23117 J23113
N0 0.0017 0.0023 0.0018 0.0029 0.0020
Nm 0.6254 0.5455 0.5577 0.5252 0.5246
\(\mu_{max}\) 0.5770 0.6950 0.6257 0.6461 0.6978
q 0.8464 5.2596 0.3610 0.1125 0.1670

From the fitted population density the growth rate can be determined as equation 6 and 7 (Levau, et al., 2001) .

$$\frac{dOD}{dt} = \mu\cdot OD$$ $$ \mu(t) = \frac{1}{OD} = \frac{q\mu_{max}}{q+e^{-\mu_{max}t}}\cdot \frac{(N_m-OD_0)(1+q)}{N_0(1+qe^{\mu_{max}t})+(N_m-N_0)(1+q)}$$

When the system is in steady state \(\frac{\partial m}{\partial t} = \frac{\partial n}{\partial t} =\frac{\partial f}{\partial t}=0\), we can compute the promoter strength.

$$ m_{ss} = \frac{c_n P}{\mu_{max}+\gamma_m}$$ $$ n_{ss} = \frac{K_t m_{ss}}{K_m+\mu_{max} +\gamma_{GFP}} = \frac{K_t}{K_m +\mu_{max} \gamma_{GFF}}\cdot\frac{c_n P}{\mu_{max}+\gamma_m} $$ $$ f_{ss} = \frac{K_m n_{ss}}{\mu_{max} +\gamma_{GFP}} = \frac{K_m}{\mu_{max} +\gamma_{GFP}}\cdot \frac{K_t}{K_m +\mu_{max} \gamma_{GFF}}\cdot\frac{c_n P}{\mu_{max}+\gamma_m} $$ $$ P = \frac{f_{ss}}{K_mK_tc_n} (K_m +\mu_{max} + \gamma_{GFP})(\mu_{max}+\gamma_{GFP})(\mu_{max}+\gamma_m )$$

To determine the growth rate and fluorescence we will measure next to the OD600 values from figure 2 also the fluorescence value in bacterial culture. Here we assume \(f_{ss} = \frac{F}{OD}\) where F is the fluorescence measured in a sample in arbitrary units (a.u.) and OD600 is the optical density measured at 600nm (Levau et al., 2001). The promoter activity can then be determined as in equation 12.

$$ P = \frac{F}{OD}\frac{1}{K_mK_tc_n} (K_m +\mu_{max} + \gamma_{GFP})(\mu_{max}+\gamma_{GFP})(\mu_{max}+\gamma_m)$$

The fluorescence measured for the E. coli is shown in figure 3A. Here we can see that after 10-15 hours the fluorescence decreases in the cultures. This is approximately at the same moment as when cells stop growing. However, the cells start growing again after some time (figure 2) we assume that GFP is being broken down by proteases after 10-15 hours as a response to nutrient limitations.

When we plot the fluorescence over OD600 we can clearly see for the promoter J23100 the fluorescence per cell drops already after 5 hours. For the other promoters the fluorescence per cell drops after 10-15 hours. Since the time to reach steady state by transcription is much faster than the time required to reach steady state by growth we expect the values of the fluorescence per OD600 to increase initially until they reach a steady state. This is however not the behavior we see in the data of figure 3. We assume that this is due to the use of a constitutive promoter and stress due to high concentrations of fluorophores. The model described here is suitable for inducible promoters but does not work for constitutive promoters.

Figure 3: (A) Fluorescence intensity of GFP with 5 different promoters. (B) Fluorescence divided by OD600 proportional to the fluorescence per cell.

Concentration of fluorophores

Although we were not able to fit the kinetics of the fluorophores, we were still interested in the concentration of the fluorophores in our cells. We can extract this from the kinetic cycle measurements. In order to convert fluorescence intensity in arbitrary units (a.u.) into concentration we require a calibration curve. We were however only able to get our hands on a sample of purified EGFP of a concentration of \(98\mu M\). From this sample a calibration curve was made and fitted to a straight line by linear regression in MATLAB (figure 4). The slope of this curve is found to be \(FtoC = 1.018 \cdot 10^{6} \frac{(a.u.)}{nmol}\).

Figure 4: Calibration curve EGFP. Excitation 488 nm, Emission 522 nm.

Using the calibration curve of EGFP we can determine the concentration of other fluorophores in order of magnitude. Therefore the relative brightness needs to be taken into account. The brightness (equation 12) is the number of photons emitted by a fluorophore and is proportional to the product of the fluorescence quantum yield (QY) and the extinction coefficient (ε) (Campbell et al., 2010). The relative brightness gives information on how much more photons a specific fluorophore emits compared to the fluorophore EGFP. In equation 12 the label i indicates the specific fluorophore. The relative brightness of the fluorophores used in the experiments can be found in table 2.

We will also take into account that the fluorophores are not measured at their excitation and emission peaks due to the limited bandwidth of the fluorimeter. Therefore we will also take into account their relative excitation and emission at the wavelength where they are measured. We calculate the relative excitation and emission as in equation 13 and 14. Here EXrel and EMrel are the relative excitation and emissions, respectively. \( F_{\lambda_{ex}}\) and \( F_{\lambda_{em}}\) are the fluorescence measured at the excitation wavelength \(\lambda_{ex}\) and emission wavelength \(\lambda_{em}\), respectively. \(F_{\lambda^{*}_{ex}}\) and \(F_{\lambda^{*}_{em}}\) are the fluorescence measured at the excitation - (\(\lambda^{*}_{ex}\)) and emission - (\(\lambda^{*}_{em}\)) wavelengths. To determine these relative excitation and emissions we used the spectrum of the fluorophores measured by the Wageningen iGEM team.

$$RB_i = \frac{\epsilon_i\cdot QY_i}{\epsilon_{EGFP}\cdot QY_{EGFP}}$$ $$ EX_{rel} = \frac{F_{\lambda_{ex}}-F_{\lambda^{*}_{ex}}}{ F_{\lambda_{ex}}}$$ $$ EM_{rel} = \frac{F_{\lambda_{em}}-F_{\lambda^{*}_{em}}}{ F_{\lambda_{em}}}$$
Table 2: Characteristics of the fluorophores used in experiments.
EGFP mCerulean mVenus
Extinction Coefficient \(\epsilon\) \(mol^{-1} cm^{-1}\) 5600 4300 92200
Excitation wavelength \(\lambda_{ex}\) 488 nm 433 nm 515 nm
Excitation wavelength \(\lambda^{*}_{ex}\) 488 nm 433 nm 515 nm
\(EX_{rel}\) 1 1 1
Quantum yield 0.60 0.62 0.57
Emission wavelength \(\lambda_{em}\) 508 nm 475 nm 525 nm
Emission wavelength \(\lambda^{*}_{em}\) 522 nm 475 nm 544 nm
\(EM_{rel}\) 0.63 1 0.6
Relative Brightness BR (% EGFP) 100 79 156
Source (Patterson et al., 2001, Spectra viewer) (Rizzo et al., 2004) (Nagai et al., 2002)

The calibration curve together with the relative brightness, relative emission and relative excitation, and the fluorescence (F) measured in the sample can be used to determine the concentration of the fluorophorei in the sample as in equation 15. In equation 15 the brackets indicate concentration.

$$ [Fluorophore_i] = \frac{F_i}{RB_i \cdot EM_{rel}\cdot EX_{rel}\cdot FtoC}$$

The fluorophores are located inside the cells and not homogenously distributed throughout the sample. To determine the fluorophore concentration inside the cell we should take the number of cells and the volume of the cells into account. Therefore we determine the number of cells in the population from the OD600 as in equation 16 using that an OD600 value of 1 corresponds to a cell density of approximately \(7\cdot 10^{8} cells/ml\) (Sezonov et al., 2007) and the volume of the cells is \(V_{cell}=1\mu m^3\) (Kubitschek et al., 1986). From the fluorescence per cell we can determine the concentration of fluorophores as equation 17.

$$n_{cells} = OD\cdot 7\cdot 10^8\cdot V_{sample}$$ $$C_{intracellular} = \frac{F_i}{ RB_i \cdot EM_{rel}\cdot EX_{rel}\cdot FtoC\cdot V_{cell}}$$

The method we are using here to determine the concentration is not a very accurate method and therefore we are only able to determine the order of magnitude of the concentration. For both mVenus and mCerulean we calculated the concentration to be in the order of 20mM. For the different strains of GFPmut3b we were not able to determine the concentrations as we were not able to find the relative brightness.

Conclusion

Although we are not able to fit a kinetic model to the fluorescence measurement, we did succeed in determining the maximal concentration of the fluorophores mVenus and mCerulean. The maximal concentration found for these fluorophores is in the order of 20mM.

Download

The files used in this model can be downloaded here

References

  1. Koseki, S., & Nonaka, J. (2012). Alternative approach to modeling bacterial lag time, using logistic regression as a function of time, temperature, pH, and sodium chloride concentration. Applied and Environmental Microbiology, 78(17), 6103–6112. https://doi.org/10.1128/AEM.01245-12
  2. Kubitschek, H. E., & Friske, J. A. (1986). Determination of bacterial cell volume with the Coulter Counter. Journal of bacteriology, 168(3), 1466-1467.
  3. Leveau, J. H. J., & Lindow, S. E. (2001). Predictive and interpretive simulation of green fluorescent protein expression in reporter bacteria. Journal of Bacteriology, 183(23), 6752–6762. https://doi.org/10.1128/JB.183.23.6752-6762.2001
  4. Nagai, T., Ibata, K., Park, E. S., Kubota, M., Mikoshiba, K., & Miyawaki, A. (2002). A variant of yellow fluorescent protein with fast and efficient maturation for cell-biological applications. Nature biotechnology, 20(1), 87-90.
  5. Patterson, G., Day, R. N., & Piston, D. (2001). Fluorescent protein spectra.Journal of cell science, 114(5), 837-838.
  6. Rizzo, M. A., Springer, G. H., Granada, B., & Piston, D. W. (2004). An improved cyan fluorescent protein variant useful for FRET. Nature biotechnology, 22(4), 445-449.
  7. Spectra Viewer | Chroma Technology Corp. (2016). Chroma.com. Retrieved 15 October 2016, from https://www.chroma.com/spectra-viewer
click counter