DBNs relate variables between adjacent time points such that a child node at a given time point is related to the parent nodes at a previous time point, thereby expanding the network in time. Based on the system and the dynamics, the relationship can go back one or several time steps. A common approach to construct DBN is by using score equivalence criterion [

24]. Here, a scoring metric (for example, maximum likelihood (ML) estimate in combination with regularization strategies) is used to evaluate how well a graph reconstructs the experimental data. Although DBNs provide good representation of biological networks, they are computationally expensive. Recently Grzegorczyk

et al. developed a computationally efficient algorithm to identify non-stationary DBNs [

25]. Specifically, in non-stationary DBNs, the network structure is kept constant between different time points, but the model parameters are allowed to vary between different time segments. The method has been successful in discovering biologically relevant interactions from diverse biological data sets including times series of gene expression and Milliplex protein concentrations across species [

25,

26,

27,

28,

29,

30,

31]. The model systems are diverse, including circadian rhythms in

A. thaliana, morphogenesis in

D. melanogaster, synthetic metabolic networks in

S. cerevisiae, serum inflammatory cytokine mediators in pediatric acute liver injury

etc. [

27,

32]. Full details of the algorithm are presented in the manuscript and supplementary material of Grzegorczyk

et al. [

25]. A brief discussion of the algorithm based on the original manuscript is presented below.

Consider a set of

N interacting nodes of a signaling network represented by

${X}_{1},\text{\hspace{0.17em}}{X}_{2},\text{\hspace{0.17em}}\mathrm{.....},\text{\hspace{0.17em}}{X}_{n}$ and a directed graph structure

G. An edge pointing from

${X}_{i}$ to

${X}_{j}$ in a DBN with time lag equal to one time step shows that the realization of

${X}_{j}$ at time step

t is dependent on the realization of its parent

${X}_{i}$ at time step

t − 1. It is commonly assumed that a time lag equal to one time step is sufficient to represent the relationship, indicating that the data have to be sampled at the right time intervals for the dynamics to be represented correctly. The parent node set,

${\pi}_{j}$, of a node

${X}_{j}$ is the set of all nodes from which an edge points to

${X}_{j}$ in

G. Grzegorczyk

et al. proposed a non-stationary generalization of the Bayesian Gaussian with score equivalence model (called BGe), and it is a node-specific mixture of BGe models [

25]. A linear Gaussian distribution is chosen for the local conditional distributions. The non-stationary DBN is based on the following Markov chain expansion:

where,

D is the time course data,

${\delta}_{Vn(t),k}$ is the Kronecker delta,

$\underset{\xaf}{V}$ is a matrix of latent variables that indicate which BGe mixture component generates a data point,

$\underset{\xaf}{K}=\left({\kappa}_{1},\text{\hspace{0.17em}}\mathrm{\dots},{\kappa}_{n}\right)$ is a vector of mixture components,

$m$ is the total number of time points. Vectors and matrices are denoted by single underbars in the symbols of all the equations of this manuscript. Each column of matrix

$\underset{\xaf}{V}$ is the vector

${\underset{\xaf}{V}}_{n}$, which divides the time series for a node into different time segments. The endpoints of these time segments are called as change-points. Each time segment between change-points is a different BGe model with parameters

${\theta}_{n}^{k}$, which includes the mean and covariance matrix of the conditional dependences for the mixture component. The allocation scheme in Equation (1) provides representation of a nonlinear regulatory process by a piecewise linear process. From Equation (1), the marginal likelihood conditional on the latent variables is given by:

Equation (4) is the local change-point BGe score (called as cpBGe) for node

n. In this work, a Gibbs MCMC sampling scheme was followed to sample from the local posterior distributions. Although, the location of change-points is inferred, the actual values of the parameters are not directly obtained since they are integrated out as seen from Equation (3). In this manuscript, correlation analysis was used in selected time segments to evaluate the nature of influence. In the algorithm, the change-points were sampled from a point process prior using dynamic programming and the graphs were sampled by sampling parent node set (restricted to 3 parents per node) from a Boltzmann posterior distribution using the cpBGe score. Additional details of the sampling procedure is given in [

25]. The codes provided online by Grzegorczyk

et al. were used in this work [

25]. The sampling parameters were kept at nominal values suggested by Grzegorczyk

et al. All simulations were performed in MATLAB

^{®} (Natick, MA, USA) on Linux 64-bit platform and single core of INTEL

^{®} (Santa Clara, CA, USA) Core™ 2 Quad CPU (Q8400 @ 2.66 GHz).