BiGSM: Bayesian inference of gene regulatory network via sparse modelling

Abstract

Motivation: Inference of gene regulatory network (GRN) is challenging due to the inherent sparsity of the GRN matrix and noisy expression data, often leading to a high possibility of false positive or negative predictions. To address this, it is essential to leverage the sparsity of the GRN matrix and develop a robust method capable of handling varying levels of noise in the data. Moreover, most existing GRN inference methods produce only fixed point estimates, which lack the flexibility and informativeness for comprehensive network analysis. In contrast, a Bayesian approach that yields closed-form posterior distributions allows probabilistic link selection, offering insights into the statistical confidence of each possible link. Consequently, it is important to engineer a Bayesian GRN inference method and rigorously execute a benchmark evaluation compared to state-of-the-art methods.

Results: We propose a method-Bayesian inference of GRN via Sparse Modelling (BiGSM). BiGSM effectively exploits the sparsity of the GRN matrix and infers the posterior distributions of GRN links from noisy expression data by using the maximum likelihood based learning. We thoroughly benchmarked BiGSM using biological and simulated datasets including GeneNetWeaver, GeneSPIDER, and GRNbenchmark. The benchmark test evaluates its accuracy and robustness across varying noise levels and data models. Using point-estimate based performance measures, BiGSM provides an overall best performance in comparison with several state-of-the-art methods including GENIE3, LASSO, LSCON, and Zscore. Additionally, BiGSM is the only method in the set of competing methods that provides posteriors for the GRN weights, helping to decipher confidence across predictions.

Availability and implementation: Code implemented via MATLAB and Python are available at Github: https://github.com/SachLab/BiGSM and archived at zenodo.

Figure 1.

Overall workflow of the BiGSM algorithm. From the model assumption, the likelihood is computed. The posterior distribution is derived from Bayes rule. Then, the log-likelihood as a function of prior parameters and noise variance is computed from the posterior. In the iterative learning process, the prior parameters and noise variance are first updated at each step, then the posterior is updated in iterations.

Figure 2.

AUPR, AUROC, and Maximum F1 score of six inference methods on GeneSPDER data with SNR of 1 (A), 0.1 (B), 0.01 (C). The simulated data has one replicate and 50 genes for each GRN. The evaluation is without self-loops. Each box contains inference results over 20 GRNs.

Figure 3.

AUROC of six inference methods on DREAM4 Insilico size 100 networks, knockdown data. Each group of bars shows the AUROC of six methods on each network.

Figure 4.

Average ranks of six inference methods on GRNbenchmark. Each number represents the average rank of the method across five networks using the corresponding evaluation metric and under the specific noise level. BiGSM has the best overall performance in the 6 challenges, with a summed ranking of 10.

Figure 5.

Estimated posterior distribution of a $3\times 3$ GRN. The true GRN weights and the corresponding gene expression are generated by GeneSPIDER with $\text{SNR}=0.1$. The estimated posterior distributions are the probability density functions of predicted GRN weights. The inferred GRN are the mean values taken from posteriors.

Figure 6.

AUPR, AUROC, and Maximum F1 score of six inference methods on a E. coli network using biological expression data.

Figure 7.

Analysis of density of inferred full GRNs over six methods. A kernel density estimation (KDE) with Gaussian kernel is used to estimate the probability density function (PDF) of the true GRN and the inferred GRNs.

Figure 8.

Average execution time of BiGSM, LSCON, and LASSO for varying number of genes (network size N) using standard tic-toc function.