Difference between revisions of "Team:Tuebingen/Modeling"

The objective of our project was to build a population-encoding galactose sensor with the ability of measuring and storing the concentration level at one specific time-point. The temporal selectivity of the sensor was achieved by light induced conformation changes of Dronpa, which is a monomeric fluorescence protein. Its fluorescence is activated by violet light (400nm) and deactivated by cyan light (500nm). Dronpa tetramerizes in the fluorescing 'ON'-state. [Zhou2012]

We utilized Dronpas behavior to introduce light-dependent regulation to Cre monomers, which tetramerize to a recombinase complex. In our chassis, the Cre recombinase deletes RFP, which prevents the transcription of a luciferase reporter. The mechanism of our sensor is the galactose dependent expression of Dronpa (in the ON state), which after the exposure to 500nm light no longer inhibits the recombinase. Consequentially, a Cre-Lox recombination deletes RFP, which activates the reporter.

TODO Cre citation

A recombinase event could be considered a stochastic process, that depends on the Cre concentration and on the time span in which it was active. Because the concentration of Dronpa and of the linked Cre is determined by the galactose concentration, the change in luciferase activity in a population is a measure of the galactose concentration at the time point of 500nm light exposure.

Overview of the chassis

The dynamic species and their interactions in the model

Our model is based on the constructs pGAL - NLS - Dronpa - Cre and PADH - RFP - Luciferase. The variable parts of the model are depicted in the upper diagram. We assume PADH mediated expression to be constant.

Abbreviations for the genetic elements:

We described the dynamics of the system with the following equations.

$\begin{eqnarray} \frac {\partial \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right]} {\partial t} &=& \textrm{production} + \left(\textrm{Off}\rightarrow\textrm{On}\right) \\ & & - \left(\textrm{On}\rightarrow\textrm{Off}\right) - \textrm{decay} \\ & & \\ &=& {V_m} : \left({\left(\frac {K_m} {[GAL]}\right)^n + 1}\right) \\ & & + \alpha \cdot \left[ \textrm{Dronpa}_\textrm{Off}\textrm{-Cre} \right] \cdot [400nm] \\ & & - \beta \cdot \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right] \cdot [500nm] \\ & & - \frac{\ln(2)} {\tau} \cdot \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right] \\ \end{eqnarray}$

$\begin{eqnarray} \frac {\partial \left[ \textrm{Dronpa}_\textrm{Off}\textrm{-Cre} \right]} {\partial t} &=& + \left(\textrm{On}\rightarrow\textrm{Off}\right) - \left(\textrm{Off}\rightarrow\textrm{On}\right) - \textrm{decay} \\ & & \\ &=& \beta \cdot \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right] \cdot [500nm] \\ & & - \alpha \cdot \left[ \textrm{Dronpa}_\textrm{Off}\textrm{-Cre} \right] \cdot [400nm] \\ & & - \frac{\ln(2)} {\tau} \cdot \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right] \\ \end{eqnarray}$

Let $A = \int\ [\textrm{Dronpa}_\textrm{Off}\textrm{-Cre}]_t\ dt$ be the are under $\textrm{Dronpa}_\textrm{Off}$ concentration curve.

The area $A$ describes how much and how long Cre was active. As such it could be used as a measure for the likelihodd of a recombinase event occuring in a single cell. We found an invertible function $f: R^+ \rightarrow [0, 1]$ that describes the probability of the recombinase occuring in a single cell dependent on the are $A$ ($p=f(A)$).

According to the Binomial distribution, if a tube contained $N$ cells that had no reporter activity, the expected number of recombinase events after light exposure is $X=N \cdot f(A)$. As we assumed constant PADH expression, each of these $X$ cells will have about the same luciferase concentration. Therefore we can predict the change in luciferase luminescence $\Delta I$ do be proportional to $X$. (The yeast is on a plate, and therefore has a constant distance to the measurement device).

It is $\Delta I = k \cdot N \cdot f(A)$, where $k$ is the constant coefficient. Consequentially, for an observed $\Delta I$ it is $A = f^{-1} \left(\frac{\Delta I}{k\cdot N}\right)$, which allows the inference of the galactose concentration during light exposure.

[Hippler]: Hippler, M. (2003). Advanced Chemistry Classroom and Laboratory Photochemical Kinetics : Reaction Orders and Analogies with Molecular Beam Scattering and Cavity Ring-Down Experiments. Journal of Chemical Education, 80(September), 1074-1077.

[Zhou]: Zhou, Xin X., et al. (2012). Optical Control of Protein Activity by Fluorescent Protein Domains. Science, 338(November), 810-814.

[Kim]: Kim, H., & Gelenbe, E. (2012). Stochastic gene expression modeling with hill function for switch-like gene responses. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 9(April), 973-979. http://doi.org/10.1109/TCBB.2011.153

[Chemwiki]: Chemwiki article on fluorescence.