The systematic inference of biologically relevant influence networks remains a challenging problem in computational biology. Even though the availability of high-throughput data has enabled the use of probabilistic models to infer the plausible structure of such networks, their true interpretation of the biology of the process is questionable. In this work, we propose a network inference methodology, based on the directed information (DTI) criterion, that incorporates the biology of transcription within the framework so as to enable experimentally verifiable inference. We use publicly available embryonic kidney and T-cell microarray datasets to demonstrate our results. We present two variants of network inference via DTI--supervised and unsupervised--and the inferred networks relevant to mammalian nephrogenesis and T-cell activation. Conformity of the obtained interactions with the literature as well as comparison with the coefficient of determination (CoD) method are demonstrated. Apart from network inference, the proposed framework enables the exploration of specific interactions, not just those revealed by data. To illustrate the latter point, a DTI-based framework to resolve interactions between transcription factor modules and target coregulated genes is proposed. Additionally, we show that DTI can be used in conjunction with mutual information to infer higher-order influence networks involving cooperative gene interactions.