Gene regulatory network inference using mixed-norms regularized multivariate model with covariance selection

Despite extensive research efforts, reconstruction of gene regulatory networks (GRNs) from transcriptomics data remains a pressing challenge in systems biology. While non-linear approaches for reconstruction of GRNs show improved performance over simpler alternatives, we do not yet have understanding if joint modelling of multiple target genes may improve performance, even under linearity assumptions. To address this problem, we propose two novel approaches that cast the GRN reconstruction problem as a blend between regularized multivariate regression and graphical models that combine the L2,1-norm with classical regularization techniques. We used data and networks from the DREAM5 challenge to show that the proposed models provide consistently good performance in comparison to contenders whose performance varies with data sets from simulation and experiments from model unicellular organisms Escherichia coli and Saccharomyces cerevisiae. Since the models’ formulation facilitates the prediction of master regulators, we also used the resulting findings to identify master regulators over all data sets as well as their plasticity across different environments. Our results demonstrate that the identified master regulators are in line with experimental evidence from the model bacterium E. coli. Together, our study demonstrates that simultaneous modelling of several target genes results in improved inference of GRNs and can be used as an alternative in different applications.

Sylvester equation 2 Considering the following inohomogeneous Sylvester equation [1] in term of B: and defining X = U 1 Γ 1 V T 1 as the singular value decomposition (SVD) of X then, Eq (S1) By further making the following change of variables: If the problem in Eq (S4) above admits a solution, then it must hold that for every row From the matrix inversion lemma [2] and 8 recalling that K = CΩ −1 0 , we know that, Z −1 i exists if γ i I s and C + nλ 2 CΩ 0 C are non 9 singular. As a diagonal matrix with strictly positive entries, C is symmetric and 10 positive definite (PD). Columns of C being linearly independent and Ω 0 symmetric PD 11 imply that nλ 2 CΩ 0 C is also symmetric and PD. This is sufficient to say that their sum 12 is PD and therefore, non singular. From the definition of the SVD, we know that γ i ≥ 0. 13 This means that, if γ i = 0, the determinant of K = CΩ −1 0 > 0, otherwise, the In this section, we show that under certain conditions, the proposed mixed L 1 L 2,1 -norm 20 and L 2 L 2,1 -norm regularized multivariate regression and covariance selection models can 21 be viewed as a generalization of the following:

22
i. When C = I s , the L 2,1 -norm regularization on the regression coefficient matrix becomes Tr(B T B), and the optimization problem equivalent to the multi-output regression [3] with identity task covariance (herein L 1 L 2,1 G-solution). It is interesting to point out that, the regularization Tr(B T B) is equivalent to imposing the matrix variate Gaussian priors on (B T B) 1/2 . From the definition of Ω, we have that Ω −1 0 = Σ 0 . This implies that Σ 0 is PD as the covariance matrix from a multivariate Gaussian distribution. Therefore, a Cholesky factorization can be performed on Σ 0 to obtain a lower triangular matrix P such that Σ 0 = PP T . By further defining P = U 2 Ψ 2 V T 2 as the SVD of P, we obtain Using Eq (S6) and the change of variables one can see that for every row 1 ≤ i ≤ n and column 1 ≤ j ≤ s, the entries ofB can 24 easily be computed asB i,j = S i,j γ i + nλ 2 ψ j . From the change of variableB = V T 1 BU 2 , 25 we obtain the L 1 L 2,1 G-solution as B = V 1B U T 2 and refer to this as L 1 L 2,1 G-solution. 26 ii. When C = I s and Ω = I s , we get B = (X T X + nλ 2 I p ) −1 X T Y, the ridge estimate. 27 iii. When λ 2 = 0, we obtain B = (X T X) −1 X T Y, the ordinary least square estimate. 28 iv.
When Ω = I s , we have the L 2,1 feature selection estimate [4]. can be solved explicitly by applying the usual formula for the roots of a scalar quadratic 33 form if: (i) A = I, (ii) B commutes with C, and (iii) B 2 − 4C has a square root [5,6]. 34 In what follows, we show that conditions (i), (ii) and (iii) are fulfilled for Eq (S9) (S10) (c 1 )-As the product of a matrix by its transpose, P is guaranteed to be symmetric 51 positive semidefinite (PSD) (and positive definite if Y − XB 0 has linearly 52 independent columns) and so must be P 2 .

53
(c 2 )-∆ is PD as the sum of PSD and PD matrices.

56
Since conditions (i), (ii) and (iii) are satisfied, we conclude that the usual formula for 57 the roots of scalar quadratic equations generalizes to the matrix counterpart in Eq (S8), 58 and for our problem in Eq (S9) this leads to a unique solution defined by 59 Ω(B 0 ) = 1 2λ 1 (P 2 + 8λ 1 I s ) where 1 2λ 1 (P 2 + 8λ 1 I s ) 1 2 is the unique PD square root of ∆ = B 2 − 4C.