# Modeling Delayed Dynamics in Biological Regulatory Networks from Time Series Data +
## 1 Introduction  
With both the spread of numerical tools in every part of daily life and the development of NGS methods (Next Generation Sequencing methods), like DNA microarrays in biology, a very large amount of time series data is now produced [1; 2; 3]. This means that the produced data from the experiments on biological systems grows drastically. The newly-produced data, as long as the associated noise does not raise an issue with regard to the precision and relevance of the corresponding information, can give us some new insights into the behavior of a system. This justifies the urge to design efficient methods for inference. Reverse engineering of gene regulatory networks from expression data have been handled by various approaches [4; 5; 6; 7; 8]. Most of them are only static. However, other researchers are rather focusing on incorporating temporal aspects in inference algorithms. The relevance of these various algorithms has been recently assessed in [9]. The authors of [10] tackled the inference problem of time-delayedgene regulatory networks through Bayesian networks, and in [11], the authors infer a temporal Boolean network. As this is a complex problem, in [12], the authors propose a time-window-extension technique based on time series segmentation in different successive phases. These approaches take gene expression data into account as the input and generate the associated regulations. However, the discrete approaches that simplify this problem by abstractions need to determine the relevant thresholds of each gene to define its active and inactive state. Various approaches have been designed to tackle the discretization problem. We can cite for example [12], in which the authors have proposed an alternative methodology that considers not a concentration level, but the way the concentration changes (in other words: the derivative of the function giving the concentration w.r.t. time) in the presence/absence of one regulator. On the other hand, the major problem for modeling lies in the quality of the expression data. Indeed, noisy data may be the main origin of the errors in the inference process. Thus, the pre-processing of the biological data is crucial for the pertinence of the inferred relations between components. In our work, we assume that the input data are pre-processed into reliable discretized time series data and focus on the dynamical inference problematics. In this paper, we aim to provide a logical approach to tackle the learning of qualitative models of biological dynamical systems, like gene regulatory networks. In our context, we assume the set of interacting components as fixed, and we consider the learning of the interactions between those components. As in [13], in which the authors targeted the completion of stationary Boolean networks, we suppose that the topology of the network is given, providing us the influences among each gene as the background knowledge. From time series data of the evolution of the system, given its topology, we learn the dynamics of the system. The main originality of our work is that we address this problem in a timed setting, with quantitative delays potentially occurring between the moment an interaction is activated and the moment its effect is visible. During the past decade, there has been a growing interest for the hybrid modeling of gene regulatory networks with delays. These hybrid approaches consider various modeling frameworks. In [14], the authors hybridize Petri nets: the advantage of the hybrid approach with regard to discrete modeling lies in the possibility of capturing biological factors, e.g., the delay for the transcription of RNA polymerase. The merits of other hybrid formalisms in biology have been studied, for instance timed automata [15], hybrid automata [16], the hybrid model of a neural oscillator [17] and Boolean representation [18; 19]. In [20], the authors also propose algorithms to learn the delayed dynamics of systems from time series data.