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Viable parameter space analysis of dynamic biological models

Žiga Pušnik1,Miha Mraz1,Nikolaj Zimic1,Miha Moškon1

1Faculty of Computer and Information Science, Ljubljana, Slovenia

The behaviour of the synthetic biological circuits often varies greatly for different external conditions.  When dealing with the dynamic biological models the behaviour of the system upon parameter perturbations needs to be analysed. The identification of viable parameter space is required, which turns to be computationally very demanding for high dimensional models with small viable parameter spaces. However, heuristics such as genetic algorithms can be used to address this problem efficiently. The evolutionary progress and convergence of the genetic algorithm is very sensitive to the choice of appropriate fitness functions. While the behaviour of the model can vary only slightly between different functions, the difference in viable parameter space regions can be significant.
We calculated the viable parameter regions for three different fitness functions used in genetic algorithm for the proposed D flip-flop biological circuit with the initially restricted parameter space. The first three principal components are used to display the regions in 3D space. The genetic circuit ideally behaves as the unit step function with certain amplitude and frequency. We defined the first fitness function simply as the sum of absolute differences between the evaluated and ideal response signal. This functions discards the subjects with different amplitude and frequency, even if these exhibit appropriate qualitative response. The second and the third fitness functions focus more on the frequency domain of the signal obtained with the Fast Fourier Transformation (FFT). Since the ideal response signal is the unit step function the amplitude of the first harmonic should be prevalent and all higher harmonics should be decreased by the order of 1/x, x being the number of the consecutive harmonic. The second fitness function evaluates the absolute difference between the first and the second harmonic. This cost function optimizes the amplitude of the base frequency and since it is most loosely defined its convergence is most rapid as well. We defined the third fitness function as the sum of absolute differences between the frequencies domains of the ideal and evaluated signal, when only first 10 harmonics are evaluated. The convergence of the genetic algorithm with the third fitness function was the slowest. All of the tested fitness functions exhibit different viable parameter regions.
The performed research gives us insight in the behaviour of the genetic algorithm upon using different fitness functions. The convergence and viable parameter regions are greatly affected. If one is looking only for the type of the behaviour in the qualitative manner, e.g. oscillations or not, the fitness function should be defined more loosely as often exhibits the fasted convergence of the genetic algorithm. Otherwise the desired properties like amplitude, frequency, fall and rise time must be taken into the account as well.
The research was partially supported by the scientific-research programme Pervasive Computing (P2-0359) financed by the Slovenian Research Agency in the years from 2009 to 2017 and by the basic research and application project Designed cellular logic (J1-6740) financed by the Slovenian Research Agency in the years from 2014 to 2017. Results presented here are in scope of Ph.D. thesis that is being prepared by Žiga Pušnik.

2006 - University of Ljubljana, Faculty of Medicine, Center for Functional Genomics and Bio-chips.