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. Our model designs the conformational changes in Dronpa by light signals. Here, we adopt the common practice of assuming the overall concentration of proteins to be constant over the short time span of light exposure.

Abbreviations for the genetic elements:

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

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

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

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 \(A\).

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 \(N \cdot f(A)\). As we assumed constant PADH expression, each of these cells will have about the same luciferase expression. Therefore we can predict the luciferase luminescence \(L\) do be proportional to \(N \cdot f(A)\). (The yeast on a plate has a constant distance to the measurement device)

If \(k\) is the proportionality coefficient in \(L = k \cdot N \cdot f(A)\) for an observed \(\Delta I\), then it is \(A = f^{-1} \left(\frac{L}{k\cdot N}\right)\). From \(A\), we were able to deduce \([\textrm{Dronpa}_\textrm{On}]\) at the time point of light exposure, which allows the computation of the galactose concentration as we assumed that a Hill function describes the Dronpa expression.

In our chassis, the sensor platform is the promoter pGAL, which originates from the gal opeon. As such pGAL has the function of a switch, which according to Kim et al, 2012, could be described with a Hill function.

\(\begin{eqnarray} \frac {\partial \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right]_\textrm{production rate}} {\partial t} &=& V : \left(\frac {K^n} {[GAL]^n} + 1 \right)\\ \end{eqnarray}\)

Since the model will only cover a short time-span, in which the Dronpa concentration is constant, we were interested in the steady-state concentrations, which requires a decay. The equations for the exponential decay with half-life time \(\tau\) is:

\(\begin{eqnarray} \frac {\partial \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right]_\textrm{decay rate}} {\partial t} &=& -\frac{\ln(2)}{\tau} \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right] \\ \end{eqnarray}\)

A steady state is reached when the production rate equals the decay.

\(\begin{eqnarray} V : \left(\frac {K^n} {[GAL]^n} + 1 \right) &=& \frac{\ln(2)}{\tau} \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right]_\textrm{s.s} \\ \Rightarrow \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right]_\textrm{s.s} &=& \underbrace{ \frac {V \cdot \tau} {\ln(2)} }_{V'} : \left(\frac {K^n} {[GAL]^n} + 1 \right) \end{eqnarray}\)

For a given \(V'\), \(\tau\) and \(V\) are inversely proportional. Therefore, we did not explicitly determine these parameters, because their product is sufficient to determine the steady state concentration.

According to Hippler et al, 2003, photochemical reactions can be modeled as a zeroth order kinetics with the following rates.

\(\begin{eqnarray} v_{\textrm{Off}\rightarrow\textrm{On}} &=& \alpha \cdot \left[ \textrm{Dronpa}_\textrm{Off}\textrm{-Cre} \right] \cdot [400nm] \\ \end{eqnarray}\)

\(\begin{eqnarray} v_{\textrm{On}\rightarrow\textrm{Off}} &=& \beta \cdot \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right] \cdot [500nm] \\ \end{eqnarray}\)

These kinetics describe an exponential profile for the concentrations over time. Therefore, the amount Dronpa\(_\textrm{On}\) remaining after a 500nm light impulse of \(t_1\) duration is:

\(\begin{eqnarray} \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right]_{t_1} &=& \int_0^t - \beta \cdot [500nm] \cdot \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right]_{t'} dt'\\ &=& \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right]_{s.s.} \cdot \exp\{-\beta\cdot [500nm] \cdot t_1\} \end{eqnarray}\)

Schematic profile of the Dronpa-Off curve.

Since the photochemical reaction have an exponential profile, the area \(A\) can be computed by the following equation. We assume that the \([\textrm{Dronpa}_\textrm{Off}\textrm{-Cre}]\) after the last light impulse is neglictablely small.

\begin{eqnarray} A &=& \enclose{circle}{1} + \enclose{circle}{2} + \enclose{circle}{3}\\ &=& \int_0^{t_1} \left[ \textrm{Dronpa}_\textrm{Off}\textrm{-Cre} \right]_{t'} dt' \\ &&+ t_2 \cdot \left[ \textrm{Dronpa}_\textrm{Off}\textrm{-Cre} \right]_{t_1} \\ &&+ \left[ \textrm{Dronpa}_\textrm{Off}\textrm{-Cre} \right]_{t_1} \int_0^{t_3} \exp\{-\alpha[400nm]t'\} dt' \\ &=& \int_0^{t_1} \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right]_{s.s.} - \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right]_{t'} dt' \\ &&+ t_2 \cdot \left( \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right]_{s.s.} - \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right]_{t_1} \right) \\ &&+ \left( \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right]_{s.s.} - \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right]_{t_1} \right) \cdot \int_0^{t_3} \exp\{-\alpha[400nm]t'\} dt' \\ &=& \left[ \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right]_{s.s.} \cdot \left( t' - \frac{\exp\{-\beta[500nm]t'}{-\beta[500nm]}\} \right) \right]^{t_1}_0\\ &&+ t_2 \cdot \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right]_{s.s.} \cdot \left( 1 - \exp\{-\beta[500nm]t_1\} \right)\\ &&+ \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right]_{s.s.} \cdot \left( 1 - \exp\{-\beta[500nm]t_1\} \right) \cdot \left[ \frac{\exp\{-\alpha[400nm]t'\}}{-\alpha[400nm]} \right]^{t_3}_0\\ &=& \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right]_{s.s.} \cdot \big( t_1 - \frac 1 {\beta[500nm]} + \exp\{-\beta[500nm]t_1\} \cdot \\&& \left( \frac 1 {\beta[500nm]} + t_2 - \frac {\exp\{-\alpha[400nm]t_3\}}{\alpha[400nm]} + \frac 1 {\alpha[400nm]} \right) \big)\\ &\propto& \left[ \textrm{Dronpa}_\textrm{On}\textrm{-Cre} \right]_{s.s.} \end{eqnarray}

The Chemwiki suggests that the concentration of fluorescent proteins is proportional to the fluorescence signal, if only a fraction of the excitation energy is absorbed, which was the case for us. Consequentially, the last equation shows that this coefficient will only scale the total area, and therefore can be omitted.