Theoretical fit

    In biological systems is well known that to perform an experiment we should try to control all the variables as can be possible. In this way we could to study the evolution of a variable of interest, like a component concentration, ph, viscosity, density, etc., as function of an external perturbation like a magnetic/electric field, potential gradient, temperature gradient, etcetera.

    For the particular case of our project in synthetic biology, we follow the concentration time evolution of P. Aeruginosa obtained through spectrophotometry-UV. Figure 1 shows our experimental results for this measurement, where y-axis is the value of bacteria concentration and the x-axis is the time. We can observe that at short-time the system shows a basal rate of production, after that, the bacteria population increase behaving as exponential function until that finally the system arrive to a stationary state where the bacteria concentration is almost a constant.

Figure 1. Concentration time evolution of P. Aeruginosa. Experimental results of the concentration of P. Aeruginosa along time.

Figure 1. Concentration time evolution of P. Aeruginosa. Experimental results of the concentration of P. Aeruginosa along time.

    We corroborated those experimental results by fitting a theoretical model in order to obtain the best parameters that characterize this kind of phenomenology. In this model we consider that our experimental system is well described by a horizontal scaling, obtained from: $$f(x) = k' + k\left( \dfrac{x^n}{K^n + x^n} \right),$$ where the only parameters to find are the constant \(K\) and the power \(n\).

Employing a numerical algorithm written in Python, we fitted the experimental points in order to find the best parameters that fit those results, finding that \(K=6.8\) and \(n=4\), blue line in figure.

Finally, we found that the theoretical fitting match in a good qualitative way with the experimental results, showing that the parameters \(K\) and \(n\) describe this kind of phenomenology.