Finding parameters

Our model now considers the maturation of KillerRed and the accumulation of damages done to the bacteria. It is able to explain and predict properly the evolution of all three quantities that are observed : the optical density of the suspension, its fluorescence and the density of living cells. But we still have to determine the best parameters to do it.

These are 6 parameters to find :

$r$ : the speed of growth of bacteria. in $min^{-1}$

$a$ : the production of KillerRed per bacteria. in $UF.OD^{-1}.min^{-1}$

$b$ : the efficiency of photobleaching. in $UF.UL^{-1}.min^{-1}$

$m$ : the maturation rate of KillerRed. in $min^{-1}$

$k$ : the toxicity of KillerRed. in $OD.UF^{-1}.UL^{-1}.min^{-1}$

$l$ : the ability of the bacteria to repair damages of ROS. unit less

With the units :

$OD$ is the Optical Density at $\lambda = 600nm$

$UF$ is an arbitrary Unit of Fluorescence (with $\lambda_absorption=585nm$ and $\lambda_emission=610nm$

$UL$ is an arbitrary Unit of Light, related to the energy received by the bacteria. $1 UF$ shall be the energy of light received by an an Erlenmeyer flask with a MR16 LED on its side at full power.

The aim is to find the set of parameters that best explains the curves of OD and fluorescence observed. As we cannot determine them separately because they have opposite effects, we search for the set of parameters that minimizes the distance between the outputs of the model and the experimental data. The distance chosen is the Euclidian distance : the Sum of Square Residuals, or SSR. In our case, the easiest and quickest methods are unusable:

$\diamond$ A regression requires the solutions to be analytic functions, such as polynomials or exponentials to project the points on it.

$\diamond$ Gradient or Newton methods require a regularity in the effect of parameters that we don't have.

$\diamond$ The technique of design of experiments is also unusable for the same reason.

So we used an alternative method.

Genetic Algorithms

At first the only possibility to find our parameters was to manipulate them by hand until the predictions seemed good enough. It wasn't a slow method since we could imagine how the prediction would change with the variation of each parameter. But it gave no clue the solution found was the best one. It can be improved by an exhaustive research, but its a pretty long process. To verify 10 values of each parameter, $10^6$ tests are, and each test consists in the calculation of 1000 points. For a standard computer, it represents 2 hours of continuous processing. Considering that 10 values is too few to have a precise answer, it would have been difficult to use.

That's why we used genetic algorithms.

The idea of a genetic algorithm is based on the evolution of a wild population and the natural selection of phenotypes best adapted to environment. Here, a phenotype is a set of parameters, and the measure of adaptation is the distance of the kinetics predicted from the kinetics observed.

1. First we start with a randomly chosen population (not too much random to accelerate the process).

2. The best ones, those that minimise the distance prevision-observation, are selected.

3. With these best ones, other phenotypes are created by mixing the values of parameters (cross-over) and modifying a bit some of them (mutation).

4. We now have a population of second generation. If they are all close enough to the solution (ie, the distance prevision-observation is small enough), the algorithm is deemed 'stabilised', the best one is chosen and the process over. If not, the algorithm goes back to step 2 with these new phenotypes.

Results

$M$ and $R$ are not the variables used in the equation, but are direclty linked :

$R$ is the time of division of bacteria, and so : $r=\frac{\ln(2)}{R}$.

$M$ is the time at which half of the KillerRed have matured, and so : $m=\frac{\ln(2)}{M}$ .

They are both actual times (in $min$) and make the figures easier to understand.