# Modeling Delayed Dynamics in Biological Regulatory Networks from Time Series Data +
## 7 ASP Encoding
### Refinement According to the Semantics
These tests were performed on a processor of Intel Core i7 (4700, 3 GHz) with 16 GB of RAM. Here, we deal with about 48 variables per cell line, which is tractable for our algorithm. The quantity of time series is also more generous than in DREAM4. The difficulty for us lies in the small amount of data point per series: seven points (0, 5, 15, 30, 60 min and 2, 4 h). Even though, we succeed at producing models that can be used for the time course prediction. ### Results

To evaluate their precision, predicted trajectories are compared with experimental test data obtained in each of the 32 cell line/stimulus contexts and following the inhibition of phosphoprotein nodes by test inhibitors. These experiments are over the same time period as the training data (up to 4 h). To get our precision score, we used dreamtools [43], an open-source Python package for evaluating DREAM challenge scoring metrics. The scoring metric of this sub-challenge is based on multiple metrics; so far, we focused on the mean RMSE (Root Mean Squared Error) provided by dreamtools. In those experiments, we chose to abstract the data into a fixed number of discrete levels as for the DREAM4 experiment (see Section 8.1). By varying the number of level of discretization (see Table 3), our best score was obtained with five levels of discretization. Considering less levels results in a prediction that is too general, while considering more levels results in a tendency to overfit the modelw.r.t. the observed data. The best score we could get so far was 0.5528 mean RMSE, which would rank our method around the ninth and 11 th leader board method (the DREAM8 Subchallenge 2A leader board is available at [[https://www.synapse.org/#Synapse:syn1720047/wiki/56831](https://www.synapse.org/#Synapse:syn1720047/wiki/56831)]([https://www.synapse.org/#Synapse:syn1720047/wiki/56831](https://www.synapse.org/#Synapse:syn1720047/wiki/56831))), which got scored between 0.5139 and 0.5564 on this metric. On this challenge, the best performing methods are based on statistical analysis (see [[https://www.synapse.org/](https://www.synapse.org/)]([https://www.synapse.org/](https://www.synapse.org/)) #Synapse:syn2343141). The interest of our method is that the model we learn is human readable, as well as the trajectory prediction. Indeed, we explain the dynamical behavior of the network by providing detailed interactions between components to explain each change step by step. These transitions are enriched by signs (activation/inhibition), the level of expression (thresholds) and delays. The final model can be understood statically and can be used to predict dynamical behaviors with respect to a known semantics. Like for DREAM4, to achieve this score, we had to perform several tests by varying the discretization precision and the complexity of the dynamics learned. Table 4 shows the score we \begin{table}
\begin{tabular}{c c c} \hline \hline
**Discrete Levels** & **Run Time (s)** & **Mean RMSE** \\ \hline
2 & 9775 s & 0.7054 \\
3 & 7078 s & 0.6419 \\
4 & 15,941 s & 0.5901 \\
5 & 14,102 s & 0.5528 \\
6 & 19,356 s & 0.5667 \\
7 & 20,963 s & 0.5563 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Evaluation results of our methods on DREAM8 Subchallenge 2A data. Figure 8: DREAM8 time series data [42]. got on predicting the training data of DREAM8 benchmarks with two and five levels of discretizations (here, we show only the detailed score for two levels, because of time (about 15 h per level after five)). ### Discussion

We propose a new method, MoT-AN (Algorithm 1), to automatically infer models that could explain the dynamical evolution of the biological systems. The contribution of our method lies in the fact that it identifies the set of interactions between biological components by concortizing the signs (negative or positive), providing thresholds and associating the quantitative time delays for each local transition. We illustrated the merits of this method by applying it on large real biological systems (DREAM4 and DREAM8 challenges).