# Modeling Delayed Dynamics in Biological Regulatory Networks from Time Series Data +
## 6 Case Study  
In this section, we show how this method generates a T-AN model consistent with the set of biological regulatory time series data. First, the method uses discretized observations as an input (i.e., chronogram), thus, it is necessary to process at the beginning the time series data with another method (as shown in Figure 3) in order to discretize it. Our method may be summarized as follows:  
* Detect biological components' changes;
* Compute the candidate timed local transitions responsible for the network changes;
* Generate the minimal subset of candidate timed local transitions that can realize all changes;
* Filter the timed candidate actions. We apply Algorithm 1 on learning the timed local transition \(\tau\in\mathcal{T}\) of the example of a network \(\mathcal{AN}=(\Sigma,\mathcal{S},\mathcal{T})\) with 3 components (\(|\Sigma|=3\)) whose chronogram is detailed in Figure 3 on page 3:The first change occurs at \(t_{min}=t_{1}=2\), denoted by \(change(2)\). It is the gene \(z\) whose value changes from zero to one; thus the timed local transition that has realized this change has this form \(z_{0}\stackrel{{\ell}}{{\rightarrow}}z_{1}\), where \(\ell\) can be any combination of the values of the regulators of \(z\). Let \(\chi=\{b\to z,a\to z,a\to a\}\) be the set of regulation influences among the components of the network. According to \(\chi\), the set of genes having influence on \(z\) is \(\chi_{z}=\{a,b\}\). This means that \(\ell=\{a_{?},b_{? }\}\), or \(\ell=\{a_{? }\}\), or \(\ell=\{b_{?}\}\). The expression level of the genes of \(\chi_{z}\) when the researched candidate timed local transition (\(\tau_{i}\)) is ongoing, i.e., during the partial steady state between two successive changes (\(t_{i}\) and \(t_{i-1}\)). This level is computed from the chronograms as follows:  
\[ \begin{array}{l}a\in\chi_{z}\text{: }[a]_{t}=0\ \forall t\in[0,2]\\ b\in\chi_{z}\text{: }[b]_{t}=1\ \forall t\in[0,2]\end{array} \]  
Thus, \(\ell\)=\(\{a_{0},b_{1}\}\) or \(\ell\)=\(\{a_{0}\}\) or \(\ell\)=\(\{b_{1}\}\), and the set of candidate timed local transitions is: \(\mathcal{T}_{change(2)}=\{\tau_{1}=z_{0}\ \frac{a_{0}}{\delta_{1}}\ z_{1},\tau_{2}=z_{0} \stackrel{{ b_{1}}}{{\rightarrow}}z_{1},\tau_{3}=z_{0}\ \frac{a_{0}\wedge b_{1}}{\delta_{3}}\ z_{1}\}\). Since it is the first change, the delay of each timed local transition is the same: \(\delta_{1}=\delta_{3}=\delta_{3}=2\). The second change occurs at \(t_{2}=3\) and is denoted by \(change(3)\). Here, it is the gene \(a\) whose state changes from \(a_{0}\) to \(a_{1}\); thus, the timed local transition that realizes this change has this form \(\tau=a_{0}\ \stackrel{{\ell}}{{\rightarrow}}\ \delta_{1}\) where \(\ell\) can be any combination of the regulators value at \(t_{1}\) of \(z\). According to \(\chi\), the genes influencing \(a\) are \(\chi_{a}=\{a\}\). This means that \(\ell=\{a_{? }\}\), and the expression level of \(a\) between \(t_{1}\) and \(t_{2}\) is \(a_{0}\). Therefore, \(\ell=\{a_{0}\}\). Thus, there is only one candidate timed local transition: \(\mathcal{T}_{change(3)}=\{\tau=a_{0}\ \frac{\mathcal{Q}}{1}\ a_{1}\}\). The third change occurs at \(t_{3}=4\), \(change(4)\). Here, it is the gene \(b\) whose value changes from \(b_{1}\) to \(b_{0}\); thus, the timed local transition that realizes this change is of this form, \(\tau=b_{1}\ \stackrel{{\ell}}{{\rightarrow}}\ b_{0}\) where \(\ell\) can be any combination of the regulators value at \(t_{3}-1\) of \(b\). According to \(\chi\), there is no gene that can influence \(b\); thus, no timed local transition can realize this change. The fourth change occurs at \(t_{4}=5\), \(change(5)\). Here, it is \(a\) whose expression decreases and changes from \(a_{1}\) to \(a_{0}\); thus, the candidate timed local transition that could realize this change has this form, \(\tau=a_{1}\ \frac{\ell}{\delta}\ a_{0}\) where \(\ell\) can be any combination of the regulators value at \(t_{4}-1\) of \(a\). According to \(\chi\), the set of genes having influences on \(a\) is \(\chi_{a}=\{a\}\).