# Modeling Delayed Dynamics in Biological Regulatory Networks from Time Series Data +
## 7 ASP Encoding
### Refinement According to the Semantics
As a result, we obtain in a few seconds models that are proven to be relevant (this relevance is qualified in terms of mean square error using dream tools) This algorithm is implemented using ASP [36; 44], thus providing the exhaustive enumeration of all models. The main limit of the approach presented in this paper is the fact that the topology of the network is considered as granted. As discussed in the Introduction of the paper, there is a wide range of algorithms designed to address this issue [41; 45]. Furthermore, such interaction graphs could be deduced from the available reliable databases of biological networks. Some examples of databases for human regulatory knowledge are: the Pathways Interaction Database [46], the Human Integrated Pathway DB [47] and the Causal Biological Network Database [48]. Various inference approaches [46; 49; 50] from time series data based on prior knowledge about component interactions have been proposed. However, they share a common limit: they focus on static characterization of the interactions, and they do not allow one to infer dynamic behaviors where delays are involved. The merits of our contribution lie in the fact that we overcome such limits, and we infer delays in a qualitative dynamic modeling of the network. ## 9 Conclusions and Perspectives

In this paper, we propose an approach that takes the background knowledge under the form of regulation graph and time series data as the input. The originality of our work is three-fold: (i) the identification of the sign of the interactions; (ii) the direct integration of quantitative time delays in the learning approach; and (iii) the identification of the qualitative discrete levels that lead to the systems dynamics. As a result, we produce a set of T-AN that explains the biological network evolution. Algorithm 1 is implemented in ASP. We illustrated the applicability and limits of the proposed method through benchmarks from DREAM4. We also improved our method and made it more robust against noisy and scarce data by proposing some refinements. Then, we applied it on the DREAM8 dataset: Heritage-DREAM breast cancer network. These results open the way to promising applications in the cooperation between biologists and computer scientists. Further works now consist of discussing the kind of information one can get on T-AN by analyzing the associated non-timed model. **Supplementary Materials:** The supplementary materials are available online at www.mdpi.com/1999-4893/10/1/8/s1. **Acknowledgments:** We thank [PERSON] for valuable discussions. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Benchmarks**} & \multicolumn{2}{c}{**Mean RMSE**} \\ \cline{2-3}  & **2 Discrete Levels** & **5 Discrete Levels** \\ \hline BT20 & 1.712 & 0.458 \\ BT549 & 1.507 & 0.449 \\ MCF7 & 0.713 & 0.310 \\ UACC812 & 12.6 & 3.391 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Evaluation results of our methods on DREAM8 training data. **Author Contributions:** [PERSON] and [PERSON] did the formalization. [PERSON] designed and implemented the algorithms. [PERSON] performed the experiments. [PERSON] and [PERSON] wrote the paper. [PERSON] and [PERSON] and [PERSON] supervised the work. **Conflicts of Interest:** The authors declare no conflict of interest. ## Abbreviations

The following abbreviations are used in this manuscript: \begin{tabular}{l l} AN & Automata Network \\ T-AN & Timed Automata Network \\ ASP & Answer Set Programming \\ \end{tabular} 

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