On cooperative quasi-equilibrium models of transcriptional regulation

Mechanistic models for transcriptional regulation are derived using the methods of equilibrium statistical mechanics, to model equilibrating processes that occur at a fast time scale. These processes regulate slower changes in the synthesis and expression of transcription factors that feed back and cooperatively regulate transcription, forming a gene regulation network (GRN). We rederive and extend two previous quasi-equilibrium models of transcriptional regulation, and demonstrate circumstances under which they can be approximated at each transcription complex by feed-forward artificial neural network (ANN) models. A single-level mechanistic model can be approximated by a successfully applied phenomenological model of GRNs which is based on single-layer analog-valued ANNs. A two-level hierarchical mechanistic model, with separate activation states for modules and for the whole transcription complex, can be approximated by a two-layer feed-forward ANN in several related ways. The sufficient conditions demonstrated for the ANN approximations correspond biologically to large numbers of binding sites each of which have a small effect. A further extension to the single-level and two-level models allows one-dimensional chains of overlapping and/or energetically interacting binding sites within a module. Partition functions for these models can be constructed from stylized diagrams that indicate energetic and logical interactions between binary-valued state variables. All parameters in the mechanistic models, including the two approximations, can in principle be related to experimentally measurable free energy differences, among other observables.