Performance of objective functions and optimisation procedures for parameter estimation in system biology models

Abstract

Mathematical modelling of signalling pathways aids experimental investigation in system and synthetic biology. Ever increasing data availability prompts the development of large dynamic models with numerous parameters. In this paper, we investigate how the number of unknown parameters affects the convergence of three frequently used optimisation algorithms and four objective functions. We compare objective functions that use data-driven normalisation of the simulations with those that use scaling factors. The data-driven normalisation of the simulation approach implies that simulations are normalised in the same way as the data, making both directly comparable. The scaling factor approach, which is commonly used for parameter estimation in dynamic systems, introduces scaling factors that multiply the simulations to convert them to the scale of the data. Here we show that the scaling factor approach increases, compared to data-driven normalisation of the simulations, the degree of practical non-identifiability, defined as the number of directions in the parameter space, along which parameters are not identifiable. Further, the results indicate that data-driven normalisation of the simulations greatly improve the speed of convergence of all tested algorithms when the overall number of unknown parameters is relatively large (74 parameters in our test problems). Data-driven normalisation of the simulations also markedly improve the performance of the non-gradient-based algorithm tested even when the number of unknown parameters is relatively small (10 parameters in our test problems). As the models and the unknown parameters increase in size, the data-driven normalisation of the simulation approach can be the preferred option, because it does not aggravate non-identifiability and allows for obtaining parameter estimates in a reasonable amount of time.

Efficient parameterization of large-scale dynamic models based on relative measurements

“…Given that the overall computation time is thousands of CPU hours, this improvement is substantial. While previous studies had already shown a reduced convergence rate when calibrating models to relative data (Degasperi et al, 2017), we identified the large gradients with respect to the scalings as a possible explanation and established a flexible and easy way to circumvent them. The numerical stiffness which can arise from this for numerical optimization methods is the first conceptual explanation of the large improvements achieved by hierarchical methods (Weber et al, 2011;Loos et al, 2018).…”

“…A priori it is not clear which influence scaling, offset and noise parameter have on optimizer performance. However, Degasperi et al (2017) observed in two examples that the use of scalings lead to inferior optimizer behaviour compared to the normalization-based approach which was also used by Fröhlich et al (2018). Thus, before estimating parameters using real measured data from CCLE and MCLP, we first used simulated data.…”

“…Yet, those datasets are usually relative measurements and data often undergo some type of normalization, which has to be accounted for when linking mechanistic model simulations to the data. A commonly used approach is to introduce scaling and offset parameters in the model outputs (Weber et al, 2011;Raue et al, 2013;Degasperi et al, 2017). However, this increases the dimensionality of the optimization problem and slows down optimization.…”

Modeling the growth of microalgaeSpirulinasp. with application of illuminance and magnetic field

“…Although it is possible to work with derivatives of kinetic equations, the increasing number of unknown kinetic parameters in the equations becomes a challenge for the modeling process. A set of experimental data are necessary to enable the indirect estimation of the unknown parameters …”

“…A set of experimental data are necessary to enable the indirect estimation of the unknown parameters. [25][26][27][28] The optimum parameter value is obtained when the NLLS objective function of this work is minimized, i.e. when the differences between experimental (X exp ) and simulated (X mod ) data are minimized, as given by Eqn (16): [29][30][31][32][33][34] […”