Density Control

The aim of our project is to control the density of a living bacterial population with light-transmitted signals:

$\bullet$From cells to computer : fluorescence and backscattered light as measurable signals from the cell suspension.

$\bullet$From the computer to the cells : white illumination to kill cells expressing KillerRed proteins. Its intensity needs to be adjusted.

The density of living cells cannot be measured instantaneously: it is determined by plating the cells on LB-agar plates and counting the colonies.

In order to validate our system, we first tried to have a constant population of living bacteria. This situation is characterized by an $OD_{600}$ growing up in a linear way (see below). Our model was necessary to find the best time profile of the illumination $I(t)$ to reach as quickly as possible a stable level of the amount of living bacteria

Stabilization of the living cell density by light

In ourinitial model we showed that it was theoretically possible to stabilize the amount of living bacteria with a constant light intensity. With the complete model, this still holds true, as shown by the following simulations. In addition, numerical simulation shows that the light intensity is a very sensitive parameter. Below are displayed three simulated 16 hours long kinetics, for a cell suspension illuminated at a power of $1UL$ for the first (red), $0.327UL$ for the second (blue) and $0.25UL$ for the last one (green).

These simulations were conducted with the following values of the parameters:

$r=8,3.10^{-3} min^{-1}$, or $R=83 min$ (time of division)

$a=130 UF.OD^{-1}.min^{-1}$

$b=0,9.10^{-2}UF.UL^{-1}.min^{-1}$

$k=0,9.10^{-7}OD.UF^{-1}.UL^{-1}.min^{-1}$

$l=0.087$

$m=6,3.10^{-3} min^{-1}$, or $M=110 min$ (half-time of maturation)

And the initial contitions were:

$OD_{600}=0.015$

$fluorescence=0UF$

In this first figure, the density of living cells $C(t)$ is displayed in $OD_{600} units. When the light intensity is too strong, all bacteria die, when it is too weak, they grow exponentially. A particular value of the light intensity allows to get a constant density of living cells. Note that this light intensity {I*} should be precisely regulated, since 10% increase or decrease around this particular light intensity will result in a X% decrease or increase of the living cell density after 10h.

The evolution of $OD_{600}$ is due to both living and dead bacteria. When the light intensity is too strong, the OD600 tends to a constant, when the light intensity is too weak, it growths exponentially. At the light intensity $I*$, $OD_{600}$ increases linearly.

The evolution of KillerRed fluorescence is also very sensitive to the light intensity. It decreases when the illumination is too strong, increases rapidly when it is too weak, and increases more slowly at the stabilizing light intensity $I*$.

Since even small deviations from the stabilizing light intensity $I*$ results in large variations of cell density, an iterative process was designed to determine it with high precision.

Model Predictive Control

Since our system cannot be stabilized with a simple closed-loop control, we used a more advanced control method : Model Predictive Control. This algorithm is used to control dynamical systems. It requires a mathematical model of the behavior of the system, and the history of the past control actions. The aim is to drive our system to a setpoint by predicting the evolution of the system, and adapting the control parameter (the light intensity here) to these predictions. To do so, the control will use different kinds of variables:

$\bullet$ Some variables defines the setpoint, here, we want to drive our living bacteria concentration to a determinate value, called $C_{target}$.

$\bullet$ Some variables are measurable during the experiment. Here, the fluorescence and the optical density. They are the only observable quantities enabling us to see whether the system behaves according to plans or not.

$\bullet$ Other variables cannot be measured and have to be estimated with the model. In our case, the living bacteria concentration cannot be measured in real time, it will be estimated from the measurable quantities. The model thus needs to be precise.

$\bullet$ And some variables are the ones used to act on the system. In our case, this is the light intensity will affect the system by decreasing the amount of living bacteria and the concentration of KillerRed.

Here are the outlines of the algorithm:

$1$. To start with, cells are first grown in the dark before the onset of the experiment. We thus have full information on the system because the fluorescence $K(0)$ and the $OD_{600}$ of $C(0)$ correspond to living cells only.

$2$. A illumination $I_1(t)$ is calculated, which, according to the model, is supposed to drive $C(t)$ to its setpoint $C_{target}$. The total fluorescence $F_1(t)$ and the living cell $C_1(t)$ kinetics are also computed.

$3$. For a certain amount of time $\tau$, more than 10 minutes to see the effect of the illumination, light is applied to the cell suspension at intensity $I_1(t)$

$4$. At time $t=\tau$, the real fluorescence, $F(\tau)$, is measured and compared to the estimated one, $F_1(\tau)$.

$5$. Others hidden variables as $C(\tau)$ are estimated using to the difference between $F(\tau)$ and $F_1(\tau)$. If $F(\tau)< F_1(\tau)$, it means hat we overestimated cell growth, and thus $C_{real}(\tau)< C_1(\tau)$.

$6$. From the estimated and measured values of $C$, we recalculate the value of the illumination : $I_2(t)$, $F_2(t)$ and $C_2(t)$ are created and the algorithm loops to step $2$ .

This algorithm will not drive perfectly C(t) to its setpoint Ctarget. Imperfections in the model will create a gap between them. But our proof of concept has shown the gap is not too big compared to the value of $C_{target}$.

Method to Control a Bacterial Population

When the model's parameters were calculated, it appeared that some of them ($b$, $k$ and $l$) were very variable, their value depended a lot on the preparation of the experiment. Sadly, it is important that they are the more accurate possible, the quality of the predictions depend greatly on them. That is the reason why those 3 parameters shall be slightly modified for each experiment.

$1$ A first period of approximately 100 minutes without illumination to let the bacteria grow and produce KillerRed.

$2$ A second one with an illumination at maximal intensity. As the effects of light appear, it becomes possible to improve the fit of the volatile parameters ($b$, $k$ and $l$). The length of this period depends on the precision wanted for the parameters: the more it lasts, the more the effects of ligth are obvious, and the more the improvement of the fit is efficient.

These two first steps are also essential, according to the model, to accelerate the apparition of the level of the amount of living bacteria. If the light of stabilization is switched on from the beginning, the level will appear after 8 hours. But if bacteria are exposed to an important intensity, they will stabilized quicker.

$3$ Once the parameters are chosen, the model is used to determinate the light intensity that will stabilised our system.

$4$ The third step passed, the model and its parameters are fixed. But the light intensity still can be modified to control the system if its measured values are drifting away from the trajectory calculated.

It is not already the model predictive control, it is the first way to stabilize safely and quickly the population of living bacteria. This experimement is the one used as a proof of concept.