Inference of Delayed Biological Regulatory Networks from Time Series Data

Abstract

The modeling of Biological Regulatory Networks (BRNs) relies on background knowledge, deriving either from literature and/or the analysis of biological observations. But with the development of high-throughput data, there is a growing need for methods that automatically generate admissible models. Our research aim is to provide a logical approach to infer BRNs based on given time series data and known influences among genes. In this paper, we propose a new methodology for models expressed through a timed extension of the Automata Networks [22] (well suited for biological systems). The main purpose is to have a resulting network as consistent as possible with the observed datasets. The originality of our work consists in the integration of quantitative time delays directly in our learning approach. We show the benefits of such automatic approach on dynamical biological models, the DREAM4 datasets, a popular reverse-engineering challenge, in order to discuss the precision and the computational performances of our algorithm.

A Appendixes

1.1 A.1 Proof of Theorem 1

Theorem 3

(Completeness). Let $\mathcal {AN}=(\varSigma ,\mathcal {S},\mathcal {T})$ be a Timed Automata Network, $\varGamma$ be a chronogram of the components of $\mathcal {AN}$, $i \in \mathsf {N}$ and $R \in \mathcal {T}$ be the set of timed local transitions that realized the chronogram $\varGamma$ such that $(a_i,l,a_j,\delta ) \in R \implies |l| \le i$. Let $\chi$ be the regulation influences of all $a \in \varSigma$. Let $\mathcal {AN}'=(\varSigma ,\mathcal {S},\emptyset )$ be a Timed Automata Network. Given $\mathcal {AN}'$, $\varGamma$, $\chi$ and i as input, Algorithm 1 is complete: it will output a set of Timed Automata Network $\phi$, such that $\exists \mathcal {AN}''=(\varSigma ,\mathcal {S},\varphi ') \in \phi$ with $R \subseteq \varphi '$.

Proof

Let us suppose that the algorithm is not complete, then there is a timed local transition $h \in R$ that realized $\varGamma$ and $h \not \in \varphi '$. In Algorithm 1, after step 1, $\varphi$ contains all timed local transitions that can realize each change of the chronogram $\varGamma$. Here there is no timed local transition $h \in R$ that realizes $\varGamma$ which is not generated by the algorithm, so $h \in \varphi$. Then it implies that at step 2, $\forall \varphi ', h \not \in \varphi '$. But since h realizes one of the change of $\varGamma$ and h is generated at step 1, then it will be present in one of the minimal subset of timed local transitions. Such that h will be in one of the networks outputted by the algorithm.    $\square$

1.2 A.2 Proof of Theorem 2

Theorem 4

(Complexity). Let $\mathcal {AN}=(\varSigma ,\mathcal {S},\mathcal {T})$ be a Timed Automata Network, $|\varSigma |$ be the number of automaton of $\mathcal {AN}$ and $\eta$ be the total number of local state of a automaton of $\mathcal {AN}$. Let $\varGamma$ be a chronogram of the components of $\mathcal {AN}$ over $\tau$ units of time, such that c is the number of changes of $\varGamma$. The memory use of Algorithm 1 belongs to $O(\tau \cdot i^{|\varSigma |+1} \cdot 2^{\tau \cdot i^{|\varSigma |+1})}$ that is bounded by $O(\tau \cdot |\varSigma |^{T\cdot |\varSigma |^{|\varSigma |+1}})$. The complexity of learning $\mathcal {AN}$ by generating timed local transitions from the observations of $\varGamma$ with Algorithm 1 belongs to $O(c\cdot i^{|\varSigma |+1} + 2^{2\cdot \tau \cdot i^{|\varSigma |+1}} +$c$\cdot 2^{\tau \cdot i^{|\varSigma |+1}})$, that is bounded by $O(\tau \cdot 2^{3\cdot \tau \cdot |\varSigma |^{|\varSigma |+1}})$.

Proof

Let i be the maximal indegree of a timed local transition in $\mathcal {AN}$, $0 \le i \le |\varSigma |$. Let p be an automaton local state of $\mathcal {AN}$ then $|\varSigma |$ is maximal the number of automaton that can influence p. There is $i^{|\varSigma |}$ possible combinations of those regulators that can influences p at the same time forming a timed local transition. There is at most $\tau$ possible delays, so that there are $\tau \cdot |\varSigma | \cdot i^{|\varSigma |}$ possibles timed local transitions, thus in Algorithm 1 at step 1, the memory is bounded by $O(\tau \cdot i^{|\varSigma |+1})$, which belongs to $O(\tau \cdot |\varSigma |^{|\varSigma |+1})$ since $0 \le i \le |\varSigma |$. Generating all minimal subsets of timed local transitions $\varphi$ of $\mathcal {AN}$ that can realize $\varGamma$ can require to generate at most $2^{\tau \cdot |\varSigma | \cdot i^{|\varSigma |+1}}$ set of rules. Thus, the memory of our algorithm belongs to $O(\tau \cdot i^{|\varSigma |+1} \cdot 2^{\tau \cdot i^{|\varSigma |+1}})$ and is bounded by $O(\tau \cdot |\varSigma |^{|\varSigma |+1} \cdot 2^{\tau \cdot |\varSigma |^{|\varSigma |+1}})$.

The complexity of this algorithm belongs to $O(c \cdot i^|\varSigma |+1)$. Since $0 \le i \le |\varSigma |$ and $0 \le c \le \tau$ the complexity of Algorithm 1 is bounded by $O(\tau \cdot |\varSigma |^{|\varSigma |+1}))$.

Generating all minimal subsets of timed local transitions $\varphi$ of $\mathcal {AN}'$ that realize $\varGamma$ can require to generate at most $2^{\tau \cdot i^{|\varSigma |+1}}$ set of timed local transitions. Each set has to be compared with the others to keep only the minimal ones, which costs $O(2^{2\cdot \tau \cdot i^{|\varSigma |+1}})$. Furthermore, each set of timed local transitions has to realize each change of $\varGamma$, it requires to check c changes and it costs $O(c \cdot 2^{\tau \cdot i^{|\varSigma |+1}})$. Finally, the total complexity of learning $\mathcal {AN}$ by generating timed local transitions from the observations of $\varGamma$ belongs to $O(c\cdot i^{|\varSigma |+1} + 2^{2\cdot \tau \cdot i^{|\varSigma |+1}} + c \cdot 2^{\tau \cdot i^{|\varSigma |+1}})$. that is bounded by $O(3\tau \cdot 2^{2\cdot \tau \cdot |\varSigma |^{|\varSigma |+1}})$.

   $\square$

Fig. 7.

The influence network of the DREAM4 challenge model (100 genes) given by GeneNetWeaver (GNW) data generator [27]. Each node is a gene and each edge is an influence from the source to the target gene.

1.3 A.3 DREAM4: Influence Network

The Fig. 7 presents the regulatory graph that we are based on to identify the signs (negative or positive), the thresholds and the quantitative time delays of the learned transitions.