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.
The application of three different potential energy barrier equations which were developed by Shanahan and co-workers for the "stick−slip" motion of droplets during evaporation of nanosuspensions to experimental data was investigated, and the results were compared. In theory, these potential energy barrier equations were assumed to be equivalent to each other, and thus the excess Gibbs free energy calculation results should be very close to each other. However, the calculated potential energy barrier results were found to be much different from each other when these equations were applied to two recently published experimental data. The details of the calculation procedures and reasons for these differences in the results are discussed, and eight new slightly modified equations are given in two sets to obtain more reliable, comparable and easily plotted potential energy barrier values.
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