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Building the Model
The construction of our model was not as linear as it is described, quite a few modelswere built, tried then abandonned. The aim was to find an explanation as simple as possible the results of the experiments carried out, equations describing the behaviour of our bacterial cells with as few parameters as possible. Thus our equations consider the maturation of fluorescent proteins and the ability of the bacteria to repair themselves.
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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 final time step.
$r$, $a$, $k$ and $b$ are constants that we will determinate later.
Unfortunately, $C$ and $K$ are not measurable variables. The only quantities we can quickly and easily measure are the optical density (OD) associated the amount of dead AND living bacteria, and the global fluoresence associated with the amount of KillerRed in the dead AND living bacteria. In order to compare our model with experimental results, we need two additional variables :
$\bullet$ $D$ the amount of dead bacteria per milliter of cell suspension.
$\bullet$ $K_D$ the amount of KillerRed inside the dead bacteria per milliter of cell suspension.
$
\left\{
\begin{array}{l l}
\frac{dD}{dt}=kIK \\
\frac{dK_D}{dt}=kI\frac{K^2}{C}-bIK_D\\
\end{array}
\right.
$
$D$ and $K_D$ are necessary for the model, because they appear in the measurement od $OD_{600}$ and fluorescence, but dead bacteria don't grow anymore, and a KillerRed protein that is in a dead bacteria has no more effect on the $OD_{600}$ evolution.
The simplest possible units were used, corresponding to the measurable quantities :
$C$ and $D$ are in '$OD_{600}nm$' 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 $I(t)$ for example constant :
If $C$ is constant, i.e. $\forall t, C(t)=C_0$, 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 therefore : $\frac{dK}{dt}=\left(a-\frac{br}{k}\right)C_0-rK$
which gives : $K(t)=\left(\frac{a}{r}-\frac{b}{k}\right)C_0+Be^{-rt}$
Then $I(t)=\frac{rC_0}{kK(t)}$ should give a constant concentration of living cells.
Asymptotically, 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, i.e. $\forall t, I(t)=I_0$, we need another variable to be able to easily solve our equation :
We define $Y=\frac{K}{C}$ as 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} $
$Y$ tends toward a steady-state value, $\frac{a}{bI_0+r}$. Let's see how C evolves :
$\frac{dC}{dt}=C(r-kI_0Y)$
$\frac{dC}{dt}=C\left(r-\frac{kI_0a}{bI_0+r}+Be^{-(bI_0+r)t}\right)$
Thus, in the specific case that $I_0=\frac{r^2}{ak-rb}$, we have $\lim_{t\to\infty}\frac{dC}{dt}(t)=0$.
The resolution of this equation have shown the possibility to stabilize the system.
The resolution of the latter equation shows that it is possible to stabilize the system by means of a suitable (constant) light intensity.
Comparison with experiments
This first model is very interesting to understand which parameters govern the evolution of the living cell population and to show that conditions exist to stabilize it. But unfortunately this set of equation is insufficient account for the results of the experiments.
Whereas we observe a lag between the onset of light and the decrease of fluorescence, the first model predicts an immediate decrease. This discrepancy requires the introduction of other phenomena to be introduced to explain the lag between the stimulus (the light) and the reaction (the decrease of fluorescence and the OD stabilization). Of course this explanation should be borne supported by biological facts.
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Maturation Time
The maturation of fluorescent proteins
After translation and spontaneous polypeptide folding, a fluorescent protein still has to maturate before becoming fluorescent. Fluorescent proteins mature after an oxidation reaction where three amino acids rearrange to form the fluorophore. For GFP, this time is typically 30 minutes [1]. The maturation time of KillerRed is significant for our experiments.
Second model
We consider maturation to be a simple chemical reaction, and the conversion of immature Killer Red to the mature form to be adequately described by first-order reaction kinetics. A new variable is needed:
$\bullet$ $K_m$ the amount of mature KillerRed inside the living bacteria per milliter of cell suspension.
$\bullet$ $K_i$ the amount of immature KillerRed inside the living bacteria per milliter of cell suspension.
$
\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 expressing the maturation of KillerRed
Similarly, immature Killer Red is also found in dead cells and its evolution is described by the following set of equations :
$
\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.
$
Where :
$\diamond K_{Dm}$ the amount of immature KillerRed inside dead bacteria per milliter of cell suspension.
$\diamond K_{Dm}$ the amount of mature KillerRed inside dead bacteria per milliter of cell suspension.
Comparison with experiments
The curves drawn from the model gives the right trend, observed in the experiments: the lag of the reaction, the peak of fluorescence short after light is switched on and then the swift decrease of fluorescence in the long term are qualitatively described.
Nonetheless, it is impossible to get a good fit between the prediction of the model and the experiment. The maturation step alone does not explain why the production of KillerRed is so high two hours after the beginning of the illumination and the decrease of fluorescence is so rapid four hours after the illumination.
[1]REID Brian G., FLYNN Gregiry C. Chromophore Formation in Green Fluorescent Protein. Biochemistry, 1997, 36, p 6786-6791.
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Damage Accumulation
Third Model
Until now, our equations describe the phototoxic effect of KillerRed as instantaneous: in the presence of light Killer Red produces ROS, which immediately kills a certain proportion of the bacteria but has no lasting effect. It is well-known, however, that cells can repair damages due to ROS, up to a certain level. We can thus consider that bacteria are unable to instantly repair all the damages caused by ROS, and with damages mounting up, they are more and more fragile and close to death.In other words, the effect of a certain amount of KR at a certain time $u$, $K(u)$, illuminated by a light intensity $I(u)$, will affect cell growth at time $t$ later than $u$, weighted by a factor $e^{-p(t-u)}$ that vanishes as $t$ increases. The effect of this ROS production at time $u$ will thus exponentially decrease with time. The term $– kI.K$ (photokilling) was thus replaced by the integral:
$-\int_{u=0}^t k'I(u)K(u)e^{-p(t-u)}du$
And the equation of bacterial growth:
$
\frac{dC}{dt}(t)=rC(t)-\int_{u=0}^t k'I(u)K(u)e^{\ln(l)(t-u)}du
$
There is no analytical solution for this equation. But it is simple to treat it in its discrete-time form.
Written in its discrete-time form, the evolution of $C$ was described by
$
C(t+1)-C(t)=rC(t)-kI(t)K(t)
$
We now write
$
\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 of their probability of dying. During a time step (for us, a minute), bacteria cures part of their injuries ($l<1$) and suffers new 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 $\mbox{tox}(t)\cong\frac{k'}{1-l}I(t)K(t)$. $k'$ is the same in the continuous-time and the discrete-time forms, it will from now be named simply $k$. $p$ and $l$ are directly related : $p=-\ln(l)$.
Considering this accumulation allows us to shift the effect of illumination, and so to have a more accurate model.
Comparison with experiments
With this model, we can now properly describe our data :
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