Molecular competition can shape enhancer activity in the Drosophila embryo

The Hill function is the universal Hopfield barrier for sharpness of input-output responses

The Hill functions, [Formula: see text], have been widely used in biology for over a century but, with the exception of [Formula: see text], they have had no justification other than as a convenient fit to empirical data. Here, we show that they are the universal limit for the sharpness of any input-output response arising from a Markov process model at thermodynamic equilibrium. Models may represent arbitrary molecular complexity, with multiple ligands, internal states, conformations, coregulators, etc, under core assumptions that are detailed in the paper. The model output may be any linear combination of steady-state probabilities, with components other than the chosen input ligand held constant. This formulation generalizes most of the responses in the literature. We use a coarse-graining method in the graph-theoretic linear framework to show that two sharpness measures for input-output responses fall within an effectively bounded region of the positive quadrant, [Formula: see text], for any equilibrium model with [Formula: see text] input binding sites. [Formula: see text] exhibits a cusp which approaches, but never exceeds, the sharpness of [Formula: see text], but the region and the cusp can be exceeded when models are taken away from thermodynamic equilibrium. Such fundamental thermodynamic limits are called Hopfield barriers, and our results provide a biophysical justification for the Hill functions as the universal Hopfield barriers for sharpness. Our results also introduce an object, [Formula: see text], whose structure may be of mathematical interest, and suggest the importance of characterizing Hopfield barriers for other forms of cellular information processing.

The Hill function is the universal Hopfield barrier for sharpness of input-output responses

The Hill functions, ℋ h ( x ) = x h / 1 + x h , have been widely used in biology for over a century but, with the exception of ℋ 1 , they have had no justification other than as a convenient fit to empirical data. Here, we show that they are the universal limit for the sharpness of any input-output response arising from a Markov process model at thermodynamic equilibrium. Models may represent arbitrary molecular complexity, with multiple ligands, internal states, conformations, co-regulators, etc, under core assumptions that are detailed in the paper. The model output may be any linear combination of steady-state probabilities, with components other than the chosen input ligand held constant. This formulation generalises most of the responses in the literature. We use a coarse-graining method in the graph-theoretic linear framework to show that two sharpness measures for input-output responses fall within an effectively bounded region of the positive quadrant, Ω m ⊂ ℝ + 2 , for any equilibrium model with m input binding sites. Ω m exhibits a cusp which approaches, but never exceeds, the sharpness of ℋ m but the region and the cusp can be exceeded when models are taken away from thermodynamic equilibrium. Such fundamental thermodynamic limits are called Hopfield barriers and our results provide a biophysical justification for the Hill functions as the universal Hopfield barriers for sharpness. Our results also introduce an object, Ω m , whose structure may be of mathematical interest, and suggest the importance of characterising Hopfield barriers for other forms of cellular information processing.

Unveiling dynamic enhancer–promoter interactions in Drosophila melanogaster

Proper enhancer–promoter interactions are essential to maintaining specific transcriptional patterns and preventing ectopic gene expression. Drosophila is an ideal model organism to study transcriptional regulation due to extensively characterized regulatory regions and the ease of implementing new genetic and molecular techniques for quantitative analysis. The mechanisms of enhancer–promoter interactions have been investigated over a range of length scales. At a DNA level, compositions of both enhancer and promoter sequences affect transcriptional dynamics, including duration, amplitude, and frequency of transcriptional bursting. 3D chromatin topology is also important for proper enhancer–promoter contacts. By working competitively or cooperatively with one another, multiple, simultaneous enhancer–enhancer, enhancer–promoter, and promoter–promoter interactions often occur to maintain appropriate levels of mRNAs. For some long-range enhancer–promoter interactions, extra regulatory elements like insulators and tethering elements are required to promote proper interactions while blocking aberrant ones. This review provides an overview of our current understanding of the mechanism of enhancer–promoter interactions and how perturbations of such interactions affect transcription and subsequent physiological outcomes.

Shared Transcriptional Machinery at Homologous Alleles Leads to Reduced Transcription in Early Drosophila Embryos

The mechanism by which transcriptional machinery is recruited to enhancers and promoters to regulate gene expression is one of the most challenging and extensively studied questions in modern biology. We explored the possibility that interallelic interactions between two homologous alleles might affect gene regulation. Using an MS2- and PP7-based, allele-specific live imaging assay, we visualized de novo transcripts of a reporter gene in hemizygous and homozygous Drosophila embryos. Surprisingly, each homozygous allele produced fewer RNAs than the corresponding hemizygous allele, suggesting the possibility of allelic competition in homozygotes. However, the competition was not observed when the enhancer-promoter interaction was weakened by placing the reporter construct in a different chromosome location or by moving the enhancer further away from the promoter. Moreover, the reporter gene showed reduced transcriptional activity when a partial transcription unit (either an enhancer or reporter gene only) was in the homologous position. We propose that the transcriptional machinery that binds both the enhancer and promoter regions, such as RNA Pol II or preinitiation complexes, may be responsible for the allelic competition. We showed that the degree of allelic interference increased over developmental time as more Pol II was needed to activate zygotic genes. Such allelic competition was observed for an endogenous gene as well. Our study provides new insights into the role of 3D interallelic interactions in gene regulation.