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Modelling
Modeling took a large place in the project; it was not only used for the characterization of KillerRed and the Voigt plasmids, it was needed for the control of the bacteria’s population. With our device, we cannot control a population with a simple closed-loop transfer function. First, because it doesn’t measure the amount of cells (the OD 600) but the fluorescence of the medium. This fluorescence gives clues about the activity of the cells and so the look of its evolution permits to find their number, but there is no simple relation between them. Second, there is a big lag between an action and its effect: one can count one or two hours between the illumination and the deceleration of fluorescence. In those conditions, a simple closed-loop transfer function is bound to be unstable, a model predictive control is needed to stabilize the population.
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Building the Model
Initial Model
The equation
Our system is made of Bacteria and ‘KillerRed’ proteins. A bacteria divides and produces KillerRed, and a KillerRed reacts to light: it degrades (photobleaching) and produces Radical Oxygen Species or ROS (phototoxicity). These reactions are shared by all fluorescent proteins, but the shape of KillerRed makes the degradation quicker and permits ROS to wander in the bacteria and damage its vital functions (proteins, cell wall or worse: DNA)
$\bullet$ $C$ the amount of living bacteria
$\bullet$ $K$ the amount of KillerRed inside the living bacteria
$\bullet$ $I$ the quantity of received light
$
\left\{
\begin{array}{l l}
\frac{dC}{dt}=rC-kIK \\
\frac{dK}{dt}=aC-bIK-kI\frac{K^2}{C} \\
\end{array}
\right.
$
$\diamond$ $rC$ is the term of growth.
$\diamond$ $kIK=kI\frac{K}{C}C$ the amount of bacteria killed by KillerRed and light.
$\diamond$ $aC$ the production of KillerRed.
$\diamond$ $bIK$ the amount of KillerRed photobleached.
$\diamond$ $kIK\frac{K}{C}$ the amount of KillerRed in the bacteria killed in the last step of time.
Unfortunately, $C$ and $K$ are not measurable variables. The only thing we can quickly and easily measure are the optical density (OD) bound to the amount of bacteria dead AND alive, and the global fluoresence bound to the amount of KillerRed in the bacteria dead AND alive. To be able to compare our model to experimental results, we need two other variables :
$\bullet$ $D$ the amount of dead bacteria
$\bullet$ $K_d$ the amount of KillerRed inside the dead bacteria
$
\left\{
\begin{array}{l l}
\frac{dD}{dt}=kIK \\
\frac{dK_d}{dt}=kI\frac{K^2}{C}-bIK_d\\
\end{array}
\right.
$
The simplest units possible were used, directly taken from our measure instruments :
$C$ and $D$ are in 'OD'.
$K$ and $K_d$ are in 'units of fluorescence' : UF. The auto-fluorescence of bacteria are considered as negligeable compared to the fluorescence of KillerRed.
Analytic resolution
This simple model can be partially solved, for $C(t)$ or $I(t)$ constant for example :
If we say : $\forall t, C(t)=C_0$, we have :
$
\left\{
\begin{array}{l l}
rC_0=kIK \\
\frac{dK}{dt}=aC_0-bIK-kI\frac{K^2}{C_0}\\
\end{array}
\right.
$
and so : $\frac{dK}{dt}=\left(a-\frac{br}{k}\right)C_0-rK$
which gives : $K=\left(\frac{a}{r}-\frac{b}{k}\right)C_0+Be^{-rt}$
then $I(t)=\frac{rC_0}{kK(t)}$ should give a constant concentration of living cell.
But if we say : $\forall t, I(t)=I_0$, we need another variable to solve easily our equation :
We define : $Y=\frac{K}{C}$ the amount of KillerRed per bacteria.
$\frac{dY}{dt}=\frac{d}{dt}\left(\frac{K}{C}\right)=a-(bI_0+r)Y$
which gives : $Y=\frac{a}{bI_0+r}+Be{-(bI_0+r)t} $
$Y$ is then stabilizing, let's look $C$ :
$\frac{dC}{dt}=C(r-kI_0Y)$
$\frac{dC}{dt}=C\left(r-\frac{kI_0a}{bI_0+r}+Be^{-(bI_0+r)t}\right)$
and so, for $I_0=\frac{r^2}{ak-rb}$, we have : $\lim_{t\to\infty}\frac{dC}{dt}(t)=0$
The resolution of this equation have shown the possibility to stabilize the system.
Comparison with experiments
This first model is noteworthy for solutions and conditions on a stabilized state appear. But it is not complete enough to explain the experiments :
Whereas we observe a lag between the light switch and the decrease of fluorescence, the model predicts an immediate and clear reaction. A new way to explain and describe the lag between the stimulus (the light) and the reaction (the decrease of fluorescence and the sabilization of DO) has to be found. Of course this way shall have a biological explanation.
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Maturation Time
The maturation of fluorescent proteins
After the traduction of mRNA into a polypeptide, then its correct foldind, a fluorescent protein still has one step to do before ot works properly : the maturation. The protein is matured after an oxydation reaction in which it gets its chromophore. The time of a maturation is significant for our experiment, for a GFP, it is 30 minutes.
Second model
We consider maturation to be a simple chemical reaction, and the concentration on immature protein $[K_im]$ is governed by the equation : $[K_im]'=-m[K_im]$. It is a realistic hypothesis since maturation is an oxydation. A new variable is needed :
$\bullet$ $K_m$ the amount of mature KillerRed inside the living bacteria
$\bullet$ $K_i$ the amount of immature KillerRed inside the living bacteria
$
\left\{
\begin{array}{l l}
\frac{dC}{dt}=rC-kIK_m \\
\frac{dK_i}{dt}=aC-kI\frac{K_i^2}{C}-mK_i\\
\frac{dK_m}{dt}=-kI\frac{K_m^2}{C}-bIK_m+mK_i\\
\end{array}
\right.
$
$\diamond$ $mK_i$ is the term traducing the maturation of KillerRed
Here too, other equations are required to find the measureable variables :
$
\left\{
\begin{array}{l l}
\frac{dD}{dt}=kIK_m \\
\frac{dK_{di}}{dt}=kI\frac{K_i^2}{C}-mK_{di}\\
\frac{dK_{dm}}{dt}=kI\frac{K_m^2}{C}-bIK_{dm}+mK_{di}\\
\end{array}
\right.
$
Numeric results
This model draws curves that properly follow the tendancy of the experiments : the lag of the reaction, the peak of fluorescence and then the brutal decrease are there.
Nonetheless, it is impossible to get a good fit between the prediction of the model and the experiment. The integration of maturation doesn't explain why the production of KillerRed is so hight even two hours after the start of illumination and the decrease of fluorescence is so brutal four hours after the illumination.
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Damage Accumulation
Until now, in the equations, KillerRed and light produced ROS, those reacted in the instant, killed or not the bacteria, and disappeared. Now, we consider that bacteria are unable to instantly repear all the damages caused by ROS, and with damages mounting up, they are more and more fragile and close to death. Considering this accumulation allows us to shift the effect of illumination, and so to have a model more accurate.
Writen in its numeric form, the evolution of C was :
$
C(t+1)-C(t)=rC(t)-kI(t)K(t)
$
It is now :
$
\left\{
\begin{array}{l l}
C(t+1)-C(t)=rC(t)-\mbox{tox}(t) \\
\mbox{tox}(t+1)=l.\mbox{tox}(t)+k'I(t)K(t)\\
\end{array}
\right.
$
with $l\in[0,1]$
The variable 'tox' is representative of the amount of damages inflicted to bacteria, and so their probability to die. With each step of time (for us, a minute), bacteria cures a part of their injuries ($l<1$) and suffers other damages ($k'I(t)K(t)$). At first, few bacteria die : $\mbox{tox}(t)\cong k'I(t)K(t)$, then, 'tox' increases until it reach $\mbox{tox}(t)\cong\frac{k'}{1-l}I(t)K(t)$
This new equation can also be writen in an analytic form :
$
\frac{dC}{dt}(t)=rC(t)-\int_{u=0}^t k'I(u)K(u)e^{\ln(l)(t-u)}du
$
But it is unsolvable and brings only complexity.
With this model, we can explain properly the look of our curves :
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