Sparsity as Cellular Objective to Infer Directed Metabolic Networks from Steady-State Metabolome Data: A Theoretical Analysis

Abstract

Since metabolome data are derived from the underlying metabolic network, reverse engineering of such data to recover the network topology is of wide interest. Lyapunov equation puts a constraint to the link between data and network by coupling the covariance of data with the strength of interactions (Jacobian matrix). This equation, when expressed as a linear set of equations at steady state, constitutes a basis to infer the network structure given the covariance matrix of data. The sparse structure of metabolic networks points to reactions which are active based on minimal enzyme production, hinting at sparsity as a cellular objective. Therefore, for a given covariance matrix, we solved Lyapunov equation to calculate Jacobian matrix by a simultaneous use of minimization of Euclidean norm of residuals and maximization of sparsity (the number of zeros in Jacobian matrix) as objective functions to infer directed small-scale networks from three kingdoms of life (bacteria, fungi, mammalian). The inference performance of the approach was found to be promising, with zero False Positive Rate, and almost one True positive Rate. The effect of missing data on results was additionally analyzed, revealing superiority over similarity-based approaches which infer undirected networks. Our findings suggest that the covariance of metabolome data implies an underlying network with sparsest pattern. The theoretical analysis forms a framework for further investigation of sparsity-based inference of metabolic networks from real metabolome data.

Peer Review #2 of "JacLy: a Jacobian-based method for the inference of metabolic interactions from the covariance of steady-state metabolome data (v0.1)"

“…This is because there are n(n+1)/2 independent entries in C for an n-metabolite system due to the symmetric nature of the covariance matrix, whereas the Jacobian matrix has n 2 independent entries. This equation can be rearranged to a standard linear system of equations (Öksüz, Sadıkoğlu & Çakır, 2013) and be represented as follows: 6to solve for Jacobian matrix in an optimization platform using Genetic Algorithm (Öksüz, Sadıkoğlu & Çakır, 2013). Beside the minimization of the residual of Lyapunov equation, they used sparsity as a rational objective function to select Jacobians from the solution space that have a high number of zeros and satisfy equation 6as well.…”

“…Using the exact covariance and predefined fluctuation vector as inputs to the algorithm, Öksüz et al (Öksüz, Sadıkoğlu & Çakır, 2013) validated the theoretical applicability of this approach.…”

“…Since the Jacobian vector is sparse in structure (r<<n), this leads to considerable reduction in the number of unknowns to be determined, and increases the speed of the inference algorithm compared to the original algorithm in (Öksüz, Sadıkoğlu & Çakır, 2013). In equation 8, is the reduced form of Jacobian vector, obtained by removing those elements corresponding to zeros in the suggested individual, and is formed by removing the corresponding columns in A.…”

“…There is always a narrow interval for the scaling factor, in which the optimization problem can find a Jacobian vector with optimum number of zeros that also leads to insignificant residual value for equation 8. This interval for the scaling factor varies from problem to problem (Öksüz, Sadıkoğlu & Çakır, 2013), depending on several factors among which are the size of the network and the accuracy and number of data replicates from which covariance matrix is calculated.…”

“…However, they infer interactions only in undirected manner, and they have limited power in the detection of weak interactions (Çakır et al, 2009). A directed network inference approach from steady-state metabolome data is also available in the literature (Steuer et al, 2003;Öksüz, Sadıkoğlu & Çakır, 2013;Çakır & Khatibipour, 2014). The approach is based on the prediction of interaction strengths from the covariance of the data.…”