# Modeling Delayed Dynamics in Biological Regulatory Networks from Time Series Data +
## 7 ASP Encoding
### Refinement According to the Semantics
As shown in Table 2, the precision we achieved in those experiments is quite good considering the results of the competitors of the DREAM4 challenge [39]. Their results range between 0.010 and 0.075 for the same evaluation settings, which are comparable to (0.033 to 0.086), giving us encouraging results. Regarding run time, learning and predicting the trajectories of the benchmarks of 10 genes took less than 30 s, and the same experiments for the benchmarks of 100 genes took about 3 h and 20 min on a one-processor Intel Core2 Duo (P8400, 2.26 GHz). To achieve this score, we had to perform several tests by varying the discretization precision and the complexity of the dynamics learned. Figure 6 shows the score we got on predicting the training data of DREAM4 benchmarks of size 10 starting with two levels of discretization until 20. Here, we take the series of each benchmark as the input, and the first state of each series is used to start a prediction. MSE is computed over the 21 points of the original series against the predicted ones. We can see that increasing the discretization precision improves the prediction precision until five levels of discretization. Then, the model prediction quality decreases as it tends to overfit the data. Those tests also allow us to assess the scalability of our approach in practice. Figure 7 shows the impact of both timed local transition indegree and discretization level on run time when learning the DREAM4 benchmarks of size 100. In the results obtained from the experimentation of our algorithm on the time series data of the DREAM4, we can see the exponential influence on the run time of the indegree per local transition considered, as well as the level of discretization chosen for all five different networks. However, it also shows that in practice, our approach can tackle big networks; here, 100 genes. Figure 6: Evaluation of the discretization impact on model precision over training data of DREAM4 benchmarks of a size of 10. ### Dream8

We recently started to test our method on the DREAM8 [40]: Heritage-DREAM breast cancer network inference challenge (the data set is available online at [[https://www.synapse.org/#](https://www.synapse.org/#)]([https://www.synapse.org/#](https://www.synapse.org/#))!Synapse: syn1720047/wiki/55342). The challenge is about the inference of causal signaling networks and prediction of protein phosphorylation dynamics. As for DREAM4, we are focusing on the prediction part. The aim is to build dynamical models that can predict trajectories of phosphoproteins. An important emphasis is on the ability of models to generalize beyond the training data by predicting trajectories under perturbations not seen in the training data. This sub-challenge is split into two independent parts: A, breast cancer proteomic data; B, in silico data; and we choose to start with the first one. Training data come from experiments on four breast cancer cell lines stimulated with various ligands. The data comprise protein abundance time courses under inhibitor perturbations. Training data are provided for each of the 32 biological contexts defined by the combination of cell line and growth condition (stimulus). These data comprise time courses for about 45 phosphoproteins under a vehicle control (DMSO) and under inhibitor perturbations of network nodes, as shown in Figure 8 (Figure 8 is from [[https://www.synapse.org/#](https://www.synapse.org/#)]([https://www.synapse.org/#](https://www.synapse.org/#))!Synapse:syn1720047/wiki/56061). Using these training data, participants are asked to build dynamical models that can predict phosphoprotein trajectories specific to each of the 32 biological contexts (32 cell line/stimulus pairs) and under the inhibition of phosphoproteins that were not perturbed in the training data. Figure 7: Evolution of run time on processing different models inferred from time series data of DREAM4 (100 variables benchmarks), varying the indegree of local timed transitions and discretization levels.