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 behaviour observed. But we still have to find the best parameters to do so. 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 repear damages of ROS. unitless

With the units :

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

$UF$ is an arbitrary Unit of Fluorescence

$UL$ 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 erlenmeyer with a MR16 LED on its side.

The aim is to find the set of parameters that best explains the curves of OD and fluorescence observed. As we cannot find them separately for they have opposite effects, we search for the set of parameters that minimises the distance between what predicts the model whit these parameters and what is observed. The distance chosen is the Euclidian distance : the Sum of Square Residuals, or SSR. In our case, the easiest and quickest method are unusable :

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

$\diamond$ Gradient or Newton methods requiere 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 other methods.

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.

Results