# Modeling Delayed Dynamics in Biological Regulatory Networks from Time Series Data +
## 4 Learning Timed Automata Networks
### Algorithm  
In this section, we propose an algorithm to build T-AN from time series data. We assume that the latter data observations are provided as a chronogram of size \(T\), with \(T\) the maximum number of time points: the value of each variable is given for each time point \(t\in\mathbb{N}^{+}\), \(0\leq t\leq T\), through a time interval discretization (see Definition 11 below and Example in Figure 3 on page 3). **Definition 11** (Chronogram). : _A chronogram is a discretization of the time series data for each component of a biological regulatory network. It is presented by the following function \(\Gamma\),_  
\[\Gamma:[0,T]\subset\mathbb{N}^{+} \longrightarrow\{0,...,n\}\] \[t \longmapsto\quad i\]  
_with \(T\) the maximum time point regarding the time series data called the size of the chronogram and \(n\) the maximum level of discretization._  
We denote \(\Gamma_{a}\) a chronogram of the time series data for a component \(a\). Algorithm 1, MoT-AN (Modeling Timed Automata networks), shows the pseudocode of our implemented algorithm. It will generate all possible timed local transitions that can realize each \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \(t\) & \(B(\mathcal{AN},P,t)\) & \(F(\mathcal{AN},\underline{\zeta},P,t)\) & \(\mathbf{SFS}\) & \(FS\) & \(A(\mathcal{AN},t)\) & \(\mathbf{state}(\mathcal{AN},t)\) \\ \hline
0 & \(\emptyset\) & \(\{\tau_{1},\tau_{3}\}\) & \(\{\{\emptyset\},\{\tau_{1}\},\{\tau_{3}\}\}\) & \(\{\tau_{1},\tau_{3}\}\) & \(\{(\mathbf{\tau_{1}},\mathbf{0}),(\tau_{3},0)\}\) & \(<a_{1},b_{0},c_{2},d_{2}>\) \\
1 & \(\{\tau_{1},\tau_{2},\tau_{3}\}\) & \(\emptyset\) & \(\{\emptyset\}\) & \(\emptyset\) & \(\{(\tau_{1},0),(\tau_{3},0)\}\) & \(<a_{1},b_{0},c_{2},d_{2}>\) \\
2 & \(\{\tau_{3}\}\) & \(\{\tau_{2},\tau_{5}\}\) & \(\{\emptyset,\{\tau_{2}\},\{\tau_{5}\}\}\) & \(\{\tau_{2}\}\) & \(\{(\tau_{3},0),(\tau_{2},2)\}\) & \(<a_{1},b_{1},c_{2},d_{2}>\) \\
3 & \(\{\tau_{2},\tau_{3}\}\) & \(\{\tau_{5}\}\) & \(\{\tau_{5}\}\) & \(\{\tau_{5}\}\) & \(\{(\tau_{5},0),(\tau_{2},2),(\tau_{5},3)\}\) & \(<a_{1},b_{1},c_{2},d_{2}>\) \\
4 & \(\{\tau_{2},\tau_{3},\tau_{5}\}\) & \(\emptyset\) & \(\emptyset\) & \(\emptyset\) & \(\{(\tau_{3},0),(\tau_{2},\tau_{3},0)\}\) & \(<a_{1},b_{1},c_{2},d_{2}>\) \\
5 & \(\emptyset\) & \(\{\tau_{4}\}\) & \(\{\emptyset,\tau_{4}\}\) & \(\{\tau_{4}\}\) & \(\{\tau_{4}\}\) & \(\{\tau_{4}\}\) & \(<a_{0},b_{0},c_{d},d_{2}>\) \\
6 & \(\{\tau_{4}\}\) & \(\emptyset\) & \(\{\emptyset\}\) & \(\emptyset\) & \(\{(\tau_{4},5)\}\) & \(<a_{0},b_{0},c_{1},d_{2}>\) \\
10 & \(\emptyset\) & \(\emptyset\) & \(\{\emptyset\}\) & \(\emptyset\) & \(\emptyset\) & \(\emptyset\) & \(<a_{0},b_{0},c_{1},d_{1}>\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Example of a trajectory of the T-AN of Example 2 starting from an initial state \(<a_{1},b_{0},c_{2},d_{2}>\) (at \(t=0\)) to a stable state \(<a_{0},b_{0},c_{1},d_{1}>\) (at \(t=10\)); with \(P=A(\mathcal{AN},t-1)\) as in Definition 9, and \(B\), \(F\), \(SFS\), \(FS\), \(A(\mathcal{AN},t)\), \(state(\mathcal{AN},t)\) are computed as defined in Definitions 4–8. observed change. A change corresponds to a modification of a component expression level. It happens because of an activation or an inhibition of the changed component by a timed local transition. Thanks to MoT-AN, we compute the discrete levels of the origin and the destination of timed local transitions and the discrete levels of all automata involved in its condition. Thus, finding this timed local transition responsible for this change is also finding the sign of the interaction responsible for this change. **INPUT**:  
* Timed automata network \(\mathcal{AN}=(\Sigma,\mathcal{S},\mathcal{T})\) with \(\mathcal{T}=\emptyset\);
* a chronogram \(\Gamma=\bigcup_{a\in\Sigma}\Gamma_{a}\);
* the regulation influences \(\chi=\bigcup_{a\in\Sigma}\chi_{a}\) and
* a maximal in-degree \(i\in\mathbb{N}^{*}\)  
**OUTPUT**: \(\phi\) a set of T-AN that realize the time series data.