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Modeling
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 of living cells with a simple closed-loop transfer function. First, this is because optical measurements (OD 600 nm or fluorescence) originate from all cells, whether they are alive or not. This fluorescence intensity gives clues about living cell activity and therefore its temporal evolution permits to find the number of living cells, but there is no simple relation between them. Second, there is a large delay between an action and its effect: there are about one or two hours between the onset of illumination and the deceleration of fluorescence. In those conditions, a simple closed-loop transfer function is predictably unstable, and a model predictive control is needed to stabilize the population of living cells.
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Building the Model
Initial Model
The equation
Our system is made of bacterial cells and ‘KillerRed’ proteins. Bacteria divide and produce KillerRed proteins, and KillerRed proteins respond to light: they fluoresce, degrade (photobleaching) and produce Radical Oxygen Species or ROS (phototoxicity). These reactions are exhibited by all fluorescent proteins, but the 3D structure of KillerRed makes its degradation quicker and its high concentration allows ROS to reach proteins, DNA and membrane within the bacteria and damage its vital functions. )
$\bullet$ $C$ the amount of living bacteria per milliter of cell suspension.
$\bullet$ $K$ the amount of KillerRed inside the living bacteria per milliter of cell suspension.
$\bullet$ $I$ the amount of incident (white) light.
The evolution of C and K is linked to I by the set of equations :
$
\left\{
\begin{array}{l l}
\frac{dC}{dt}=rC-kIK \\
\frac{dK}{dt}=aC-bIK-kI\frac{K^2}{C} \\
\end{array}
\right.
$
$\diamond$ $rC$ describes bacterial 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 possible units were used, that correspond to the measurable quantities :
$C$ and $D$ are in '$OD_600nm$' units.
$K$ and $K_D$ are in 'units of fluorescence' : UF. Bacterial auto-fluorescence is considered as negligible compared to KillerRed fluorescence.
Analytical Solution
This simple model can be partially solved, for $C(t)$ or $I(t)$ constant for example :
If we $C$ is constant, $\forall t, C(t)=C_0$, 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.
For time long enough, the light intensity that stabilizes the concentration of living cells is $I_0=\frac{r^2}{ak-rb}$.
But if we assume that $I$ is constant, $\forall t, I(t)=I_0$, 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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