# Modeling Delayed Dynamics in Biological Regulatory Networks from Time Series Data +
## 7 ASP Encoding
### Refinement According to the Semantics
%Thesumofthedelaysforeachtransition sumDelays(Y,Valy,X,Val1,Val2,S)\(\leftarrow\)S=#sum{D:transition(Y,Valy,X,Val1,Val2,D)}, ntransition(Y,Valy,X,Val1,Val2,_),S1=0. %Computetheaveragedelayforeachtransition ntransAvgDelay(Y,Valy,X,Val1,Val2,Davg)\(\leftarrow\)hreTotTrans(Y,Valy,X,Val1,Val2,Tot), sumDelays(Y,Valy,X,Val1,Val2,S),Davg=S/Tot. ```
It is also possible to compute the interval of the delays according to Definition 15: [Max, Min] with Max is the maximum value of the delays of this transition, computed by the predicate maxDelay (lines 86-87); Min is the minimum value of the delays computed by the predicate minDelay (lines 88-89). Finally, the generation of a transition after merging is done by the predicate transIntervalDelay in lines 90-91. ```
n%Computethemaximumvalueandtheminimumvalueofthedelaysofasametransition nmaxDelay(Y,Valy,X,Val1,Val2,Max)\(\leftarrow\)Max=#max{D:transition(Y,Valy,X,Val1,Val2,D)}, ntransition(Y,Valy,X,Val1,Val2,_). nminDelay(Y,Valy,X,Val1,Val2,Min)\(\leftarrow\)Min=#min{D:transition(Y,Valy,X,Val1,Val2,D)}, ntransition(Y,Valy,X,Val1,Val2,_). ntransIntervalDelay(Y,Valy,X,Val1,Val2,interval(Min,Max))\(\leftarrow\)minDelay(Y,Valy,X,Val1,Val2,Min), maxDelay(Y,Valy,X,Val1,Val2,Max). ```

According to the Definition 13, we want to find the more frequent transitions in the models. In ASP, the option \"- _cautious_\" computes the cautious consequences (intersection of all answer sets) of a logic program (algorithm implementation in ASP). Therefore, we use this option while executing. ## 8 Evaluation

In this section, we provide two assessments of Algorithm 1. We evaluate the capacity of our algorithm ( all programs, described in this article for timed automata network generation are implemented in ASP and are available online at: [[http://www.irccyn.ec-nantes.fr/~benabdal/modeling-biological-regulatory-networks.zip](http://www.irccyn.ec-nantes.fr/~benabdal/modeling-biological-regulatory-networks.zip)]([http://www.irccyn.ec-nantes.fr/~benabdal/modeling-biological-regulatory-networks.zip](http://www.irccyn.ec-nantes.fr/~benabdal/modeling-biological-regulatory-networks.zip)) ) to build models for prediction and the impact of the quantity of observations on run time. Here, we process chronograms obtained from time series data of the DREAM4 challenge [39] and of the DREAM8 challenge [40]. ### Dream4

In this section, we assess the efficiency of our algorithm through case studies coming from the DREAM4 challenge. DREAM challenges are annual reverse engineering challenges that provide biological case studies. In this section, we focus on the datasets coming from DREAM4 (available online at [[https://www.synapse.org/#Synapse:syn3049712/wiki/74630](https://www.synapse.org/#Synapse:syn3049712/wiki/74630)]([https://www.synapse.org/#Synapse:syn3049712/wiki/74630](https://www.synapse.org/#Synapse:syn3049712/wiki/74630))). It provides data for systems of different size (10 genes on the one hand; 100 genes on the other), allowing us to assess the scalability of our approach. The input data that we tackle here consist of the following: five different systems each composed of 100 genes, all coming from _Escherichia coli_ and yeast networks. For every such system, the available data are the following: (i) 10 time series data with 21 time points, and 1000 is the duration of each time series; (ii) steady state for the wild-type; (iii) steady states after knocking out each gene; (iv) steady states after knocking down each gene (i.e., forcing its transcription rate at 50%); (v) steady states after some random multifactorial perturbations. We processed all of the data. Here, we focus on the management of time series data. Each time series includes different perturbations that are maintained all of the time during the first 10 time points and applied to at most 30% of the genes. In this setting, a perturbation means a significant increase or decrease of the gene expression. In the raw data of the time series, gene expression values are given as real numbers between zero and one.