Discovering Biochemical Reaction Models by Evolving Libraries

Abstract

In a time of data abundance, automatic methods increasingly support manual modeling. To this end, the Sparse Identification of Non-linear Dynamics (SINDy) provides a solid foundation for identifying non-linear dynamical systems in the form of differential equations. In biochemistry, reaction networks imply coupled differential equations. It has recently been demonstrated how this intrinsic coupling can be achieved within the SINDy framework, providing a straightforward interpretation of the learned equations as reaction systems with mass-action kinetics. However, this extension inherits from SINDy the requirement to enumerate all candidate reactions in a library, resulting in ill-posed optimization problems and long model descriptions, limiting its utility for identifying models with many species. Here, we elaborate on the recent advances in bringing SINDy to the biochemical domain by considering the sub-sampling of reaction libraries as part of an evolutionary optimization scheme. This enables the generation of parsimonious models, as well as the inclusion of model-level constraints, and allows the consideration of large numbers of candidate reactions. We evaluate the approach on two smaller case studies and the recovery of a large Wnt signaling model.

Appendices

A Complete List of Experiment (Hyper-)parameters

(See Table 1).

Table 1. Hyperparameters used for evolib. The random search and c-SINDy use the same parameters where applicable. In particular, for the non-negative least squares, the maximum number of iterations was limited to $10^8$. Abbreviations: X extended, cstr. constrained.

B Learned Model’s Trajectories for the Wnt Pathway

Fig. 5.

Results from simulating the models learned with evolib for the Wnt pathway. The + symbols mark measurement points, and the lines are the trajectories simulated for the 19 species. Note that, for clarity, we omit the labeling of species, and only every second measurement point is shown. The integration of the (large) models produced by c-SINDy resulted in errors due to numerical problems.

C Learned Models for the Wnt Pathway

(See Figs. 6 and 7).

Fig. 6.

The models inferred in the Wnt case study (unconstrained) compared to the ground truth model. For the extension, the reactions above the line are fixed. Only reactions with a rate above $10^{-6}$ are shown, and if applicable the number of excluded reactions is shown in the lower left. Bolded reactions indicate an overlap with the ground truth reactions. Note in particular how in (a) the Axin-induced degradation of $\beta$-catenin and in (b) the synthesis of Ros was recovered, which are both central components of the ground truth model of the Wnt pathway.

Fig. 7.

The models inferred in the Wnt case study (constrained) compared to the ground truth model. For the extension, the reactions above the line are fixed. Only reactions with a rate above $10^{-6}$ are shown, and if applicable, the number of excluded reactions is shown in the lower left. Bolded reactions indicate an overlap with the ground truth reactions. Note in particular how in (a) the shuttling of $\beta$-catenin in and out of the nucleus and in (b) the production of TCF was recovered.