Combining kinetic orders for efficient S-System modelling of gene regulatory network

S-System models, non-linear differential equation models, are widely used for reconstructing gene regulatory networks from temporal gene expression data. An S-System model involves two states, generation and degeneration, and uses the kinetic parameters gij and hij, to represent the direction, nature, and intensity of the genetic interactions. The need for learning a large number of model parameters results in increased computational expense. Previously, we improved the performance of the algorithm using dynamic allocation of the maximum in-degree for each gene. While the method was effective for smaller networks, a large amount of computation was still needed for larger networks. This problem arose mainly due to the increased occurrence of invalid networks during optimization, primarily because the two kinetic parameters (gij and hij) of the S-System model converge independently during optimization. Being independent, these two parameters can converge to values that can indicate contradictory gene interactions, specifically inhibition or activation. In this study, to address this major challenge in S-System modelling, we developed a novel method that includes two features: a penalty term that penalizes those networks with invalid kinetic orders, and a parameter, wij, derived by combining the kinetic parameters gij and hij. The novel penalty term was used for candidate selection during the process of optimizing the DRNI (Dynamically Regulated Network Initialization) algorithm. Rather than remaining constant, it is dynamic, with its magnitude dependent on the number of invalid interactions in the given network. This approach encourages the generation of valid candidate solutions, and eliminates invalid networks in a systematic manner. The previous DRNI method, a two-stage approach which uses dynamic allocation of the maximum indegree for each gene, was further improved by adding a third stage which applies the proposed wij to handle the invalid regulations that may still exist in that candidate solutions. The method was tested on different gene expression datasets, and was able to reduce the number of iterations and produce improved network accuracies. For a 20 gene network, the number of generations required for convergence was reduced by 300, and the F-score improved by 0.05 compared to our previously reported DRNI approach. For the well-known 10 gene networks of the DREAM challenge, our method produced an improvement in the average area under the ROC curve of the DREAM4 10 gene networks.