Recently, the focus of speech enhancement research has shifted from minimum mean-square error (MMSE) approaches, like the MMSE short-time spectral amplitude (MMSE-STSA) estimator, to state-of-the-art masking- and mapping-based deep learning approaches. We aim to bridge the gap between these two differing speech enhancement approaches. Deep learning methods for MMSE approaches are investigated in this work, with the objective of producing intelligible enhanced speech at a high quality. Since the speech enhancement performance of an MMSE approach improves with the accuracy of the used a priori signal-to-noise ratio (SNR) estimator, a residual long short-term memory (ResLSTM) network is utilised here to accurately estimate the a priori SNR. MMSE approaches utilising the ResLSTM a priori SNR estimator are evaluated using subjective and objective measures of speech quality and intelligibility. The tested conditions include real-world non-stationary and coloured noise sources at multiple SNR levels. MMSE approaches utilising the proposed a priori SNR estimator are able to achieve higher enhanced speech quality and intelligibility scores than recent masking- and mapping-based deep learning approaches. The results presented in this work show that the performance of an MMSE approach to speech enhancement significantly increases when utilising deep learning. Availability: The proposed a priori SNR estimator is available at: https://github.com/anicolson/DeepXi.