# Modeling Delayed Dynamics in Biological Regulatory Networks from Time Series Data +
## 3 Timed Automata Networks  
In some dynamics, it is crucial to have information about the delays between two events (two states of an \(AN\)). The discrete transitions, described above, cannot exhibit this information: we just process chronological information that the state \(\zeta_{2}\) will be after \(\zeta_{1}\) in the next step, but it is not possible to know chronometry, i.e., how much time this transition takes to occur and whether it blocks some transitions during this time. In fact some local transitions could not be played any more because of conflict about shared resources (necessary components to play a transition) between them. Thus, we need to restrain the general dynamics to capture more realistic behavior w.r.t. biology. Therefore, we propose in this section to add the delays in the local transitions' attributes: timed local transitions. Later, we give the associated semantics that we base our work on to learn biological networks. **Definition 4** (Timed Automata Network (T-AN)). : _A timed automata network is a triple \((\Sigma,\mathcal{S},\mathcal{T})\) where:_  
* \(\Sigma=\{a,b,\dots\}\) _is the finite set of automata identifiers;_
* _For each_ \(a\in\Sigma\)_,_ \(\mathcal{S}(a)=\{a_{i},\dots,a_{j}\}\)_, is the finite set of local states of automaton_ \(a\)_;_ \(\mathcal{S}=\prod_{a\in\Sigma}\mathcal{S}(a)\) _is the finite set of global states;_ \(\mathbf{LS}=\cup_{a\in\Sigma}\mathcal{S}(a)\) _denotes the set of all of the local states._
* \(\mathcal{T}=\{a\mapsto\mathcal{T}_{a}\mid a\in\Sigma\}\)_, where_ \(\forall a\in\Sigma\)_,_ \(\mathcal{T}_{a}\subset\mathcal{S}(a)\times\wp(\mathbf{LS}\setminus\mathcal{S} (a))\times\mathds{N}\times\mathcal{S}(a)\) _with_ \((a_{i},\ell,\delta,a_{j})\in\mathcal{T}_{a}\Rightarrow a_{i}\
eq a_{j}\)_, is the mapping from automata to their finite set of timed local transitions._  
To model biological networks where quantitative time plays a major role, we will use T-AN. This formalism enriches AN model with timed local transitions: \(a_{i}\overset{\ell}{\underset{\delta}{\rightarrow}}a_{j}\). In the latter, \(\delta\) is called a  
Figure 1: Example of an automata network with four automata: \(a,b,c\) and \(d\) presented by labeled boxes; and their local states are presented by circles (for instance, \(a\) is either at Level 0 or 1). A local transition is a labeled directed edge between two local states within the same automaton: its label stands for the set of necessary conditions for local states of the automata to play the transition. The grayed circles stand for the global state: \(\langle a_{1},b_{0},c_{2},d_{2}\rangle\). delay and represents the time needed for the transition to be performed. When modeling a regulation phenomenon, this allows one to capture the delay between the activation order of the production of the protein and its effective synthesis, as well as the synthesis of the product. We note \(\tau=a_{i}\underset{\delta}{\overset{\ell}{\underset{\delta}{\rightleftharpoons}}}a_{j} \in\mathcal{T}\Leftrightarrow\tau=(a_{i},\ell,\delta,a_{j})\in\mathcal{T}(a)\) and \(orig(\tau)=a_{i}\), \(dest(\tau)=a_{j}\), \(cond(\tau)=\ell\) and \(delay(\tau)=\delta\). **Lemma 1**. : _If \(\mathcal{AN}=(\Sigma,\mathcal{S},\mathcal{T})\) is an AN, so it can be transformed to a T-AN, \(\mathcal{AN}^{\prime}=(\Sigma,\mathcal{S},\mathcal{T}^{\prime})\) such that \(\forall\tau\in\mathcal{T},\exists\tau^{\prime}\in\mathcal{T}^{\prime}\) with \(orig(\tau^{\prime})=orig(\tau)\), \(dest(\tau^{\prime})=dest(\tau)\), \(cond(\tau^{\prime})=cond(\tau)\) and \(delay(\tau^{\prime})=1\)._  
According to Lemma 1, Definition 2 is then also applicable to timed local transitions. Considering that delays in the evolution of timed automata networks create conflict between the timed local transitions, this conflict is mainly justified by the shared resources between the timed local transitions (Definition 5). Indeed, transitions that have the same origins and/or destinations could not be fired synchronously.