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.

A quantitative spatiotemporal atlas of gene expression in the Drosophila blastoderm.

To fully understand animal transcription networks, it is essential to accurately measure the spatial and temporal expression patterns of transcription factors and their targets. We describe a registration technique that takes image-based data from hundreds of Drosophila blastoderm embryos, each costained for a reference gene and one of a set of genes of interest, and builds a model VirtualEmbryo. This model captures in a common framework the average expression patterns for many genes in spite of significant variation in morphology and expression between individual embryos. We establish the method's accuracy by showing that relationships between a pair of genes' expression inferred from the model are nearly identical to those measured in embryos costained for the pair. We present a VirtualEmbryo containing data for 95 genes at six time cohorts. We show that known gene-regulatory interactions can be automatically recovered from this data set and predict hundreds of new interactions.

Yearly planning meetings: individualized development plans aren't just more paperwork.

The National Institutes of Health (NIH) encourages trainees to make Individualized Development Plans to help them prepare for academic and nonacademic careers. We describe our approach to building an Individualized Development Plan, the reasons we find them useful and empowering for both PIs and trainees, and resources to help other labs implement them constructively.

Hunchback is counter-repressed to regulate even-skipped stripe 2 expression in Drosophila embryos.

Hunchback is a bifunctional transcription factor that can activate and repress gene expression in Drosophila development. We investigated the regulatory DNA sequence features that control Hunchback function by perturbing enhancers for one of its target genes, even-skipped (eve). While Hunchback directly represses the eve stripe 3+7 enhancer, we found that in the eve stripe 2+7 enhancer, Hunchback repression is prevented by nearby sequences-this phenomenon is called counter-repression. We also found evidence that Caudal binding sites are responsible for counter-repression, and that this interaction may be a conserved feature of eve stripe 2 enhancers. Our results alter the textbook view of eve stripe 2 regulation wherein Hb is described as a direct activator. Instead, to generate stripe 2, Hunchback repression must be counteracted. We discuss how counter-repression may influence eve stripe 2 regulation and evolution.

Transcriptional precision and accuracy in development: from measurements to models and mechanisms.

During development, genes are transcribed at specific times, locations and levels. In recent years, the emergence of quantitative tools has significantly advanced our ability to measure transcription with high spatiotemporal resolution in vivo Here, we highlight recent studies that have used these tools to characterize transcription during development, and discuss the mechanisms that contribute to the precision and accuracy of the timing, location and level of transcription. We attempt to disentangle the discrepancies in how physicists and biologists use the term 'precision' to facilitate interactions using a common language. We also highlight selected examples in which the coupling of mathematical modeling with experimental approaches has provided important mechanistic insights, and call for a more expansive use of mathematical modeling to exploit the wealth of quantitative data and advance our understanding of animal transcription.

Analysis of Genetic Variation Indicates DNA Shape Involvement in Purifying Selection.

Noncoding DNA sequences, which play various roles in gene expression and regulation, are under evolutionary pressure. Gene regulation requires specific protein-DNA binding events, and our previous studies showed that both DNA sequence and shape readout are employed by transcription factors (TFs) to achieve DNA binding specificity. By investigating the shape-disrupting properties of single nucleotide polymorphisms (SNPs) in human regulatory regions, we established a link between disruptive local DNA shape changes and loss of specific TF binding. Furthermore, we described cases where disease-associated SNPs may alter TF binding through DNA shape changes. This link led us to hypothesize that local DNA shape within and around TF binding sites is under selection pressure. To verify this hypothesis, we analyzed SNP data derived from 216 natural strains of Drosophila melanogaster. Comparing SNPs located in functional and nonfunctional regions within experimentally validated cis-regulatory modules (CRMs) from D. melanogaster that are active in the blastoderm stage of development, we found that SNPs within functional regions tended to cause smaller DNA shape variations. Furthermore, SNPs with higher minor allele frequency were more likely to result in smaller DNA shape variations. The same analysis based on a large number of SNPs in putative CRMs of the D. melanogaster genome derived from DNase I accessibility data confirmed these observations. Taken together, our results indicate that common SNPs in functional regions tend to maintain DNA shape, whereas shape-disrupting SNPs are more likely to be eliminated through purifying selection.

A gene expression atlas of a bicoid-depleted Drosophila embryo reveals early canalization of cell fate.

In developing embryos, gene regulatory networks drive cells towards discrete terminal fates, a process called canalization. We studied the behavior of the anterior-posterior segmentation network in Drosophila melanogaster embryos by depleting a key maternal input, bicoid (bcd), and measuring gene expression patterns of the network at cellular resolution. This method results in a gene expression atlas containing the levels of mRNA or protein expression of 13 core patterning genes over six time points for every cell of the blastoderm embryo. This is the first cellular resolution dataset of a genetically perturbed Drosophila embryo that captures all cells in 3D. We describe the technical developments required to build this atlas and how the method can be employed and extended by others. We also analyze this novel dataset to characterize the degree and timing of cell fate canalization in the segmentation network. We find that in two layers of this gene regulatory network, following depletion of bcd, individual cells rapidly canalize towards normal cell fates. This result supports the hypothesis that the segmentation network directly canalizes cell fate, rather than an alternative hypothesis whereby cells are initially mis-specified and later eliminated by apoptosis. Our gene expression atlas provides a high resolution picture of a classic perturbation and will enable further computational modeling of canalization and gene regulation in this transcriptional network.

Shadow enhancers enable Hunchback bifunctionality in the Drosophila embryo.

Hunchback (Hb) is a bifunctional transcription factor that activates and represses distinct enhancers. Here, we investigate the hypothesis that Hb can activate and repress the same enhancer. Computational models predicted that Hb bifunctionally regulates the even-skipped (eve) stripe 3+7 enhancer (eve3+7) in Drosophila blastoderm embryos. We measured and modeled eve expression at cellular resolution under multiple genetic perturbations and found that the eve3+7 enhancer could not explain endogenous eve stripe 7 behavior. Instead, we found that eve stripe 7 is controlled by two enhancers: the canonical eve3+7 and a sequence encompassing the minimal eve stripe 2 enhancer (eve2+7). Hb bifunctionally regulates eve stripe 7, but it executes these two activities on different pieces of regulatory DNA--it activates the eve2+7 enhancer and represses the eve3+7 enhancer. These two "shadow enhancers" use different regulatory logic to create the same pattern.

Comparing mRNA levels using in situ hybridization of a target gene and co-stain.

In situ hybridization is an important technique for measuring the spatial expression patterns of mRNA in cells, tissues, and whole animals. However, mRNA levels cannot be compared across experiments using typical protocols. Here we present a semi-quantitative method to compare mRNA levels of a gene across multiple samples. This method yields an estimate of the error in the measurement to allow statistical comparison. Our method uses a typical in situ hybridization protocol to stain for a target gene and an internal standard, which we refer to as a co-stain. As a proof of concept, we apply this method to multiple lines of transgenic Drosophila embryos, harboring constructs that express reporter genes to different levels. We generated this test set by mutating enhancer sequences to contain different numbers of binding sites for Zelda, a transcriptional activator. We demonstrate that using a co-stain with in situ hybridization is an effective method to compare mRNA levels across samples. This method requires only minor modifications to existing in situ hybridization protocols and uses straightforward analysis techniques. This strategy can be broadly applied to detect quantitative, spatially resolved changes in mRNA levels.

A conserved developmental patterning network produces quantitatively different output in multiple species of Drosophila.

Differences in the level, timing, or location of gene expression can contribute to alternative phenotypes at the molecular and organismal level. Understanding the origins of expression differences is complicated by the fact that organismal morphology and gene regulatory networks could potentially vary even between closely related species. To assess the scope of such changes, we used high-resolution imaging methods to measure mRNA expression in blastoderm embryos of Drosophila yakuba and Drosophila pseudoobscura and assembled these data into cellular resolution atlases, where expression levels for 13 genes in the segmentation network are averaged into species-specific, cellular resolution morphological frameworks. We demonstrate that the blastoderm embryos of these species differ in their morphology in terms of size, shape, and number of nuclei. We present an approach to compare cellular gene expression patterns between species, while accounting for varying embryo morphology, and apply it to our data and an equivalent dataset for Drosophila melanogaster. Our analysis reveals that all individual genes differ quantitatively in their spatio-temporal expression patterns between these species, primarily in terms of their relative position and dynamics. Despite many small quantitative differences, cellular gene expression profiles for the whole set of genes examined are largely similar. This suggests that cell types at this stage of development are conserved, though they can differ in their relative position by up to 3-4 cell widths and in their relative proportion between species by as much as 5-fold. Quantitative differences in the dynamics and relative level of a subset of genes between corresponding cell types may reflect altered regulatory functions between species. Our results emphasize that transcriptional networks can diverge over short evolutionary timescales and that even small changes can lead to distinct output in terms of the placement and number of equivalent cells.

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

The Hill functions, ℋh(x)=xh/1+xh, have been widely used in biology for over a century but, with the exception of ℋ1, 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, Ωm⊂ℝ+2, for any equilibrium model with m input binding sites. Ωm exhibits a cusp which approaches, but never exceeds, the sharpness of ℋm 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, Ωm, whose structure may be of mathematical interest, and suggest the importance of characterising Hopfield barriers for other forms of cellular information processing.

Emergence of activation or repression in transcriptional control under a fixed molecular context.

For decades, studies have noted that transcription factors (TFs) can behave as either activators or repressors of different target genes. More recently, evidence suggests TFs can act on transcription simultaneously in positive and negative ways. Here we use biophysical models of gene regulation to define, conceptualize and explore these two aspects of TF action: "duality", where TFs can be overall both activators and repressors at the level of the transcriptional response, and "coherent and incoherent" modes of regulation, where TFs act mechanistically on a given target gene either as an activator or a repressor (coherent) or as both (incoherent). For incoherent TFs, the overall response depends on three kinds of features: the TF's mechanistic effects, the dynamics and effects of additional regulatory molecules or the transcriptional machinery, and the occupancy of the TF on DNA. Therefore, activation or repression can be tuned by just the TF-DNA binding affinity, or the number of TF binding sites, given an otherwise fixed molecular context. Moreover, incoherent TFs can cause non-monotonic transcriptional responses, increasing over a certain concentration range and decreasing outside the range, and we clarify the relationship between non-monotonicity and common assumptions of gene regulation models. Using the mammalian SP1 as a case study and well controlled, synthetically designed target sequences, we find experimental evidence for incoherent action and activation, repression or non-monotonicity tuned by affinity. Our work highlights the importance of moving from a TF-centric view to a systems view when reasoning about transcriptional control.

Dissecting sources of quantitative gene expression pattern divergence between Drosophila species.

Gene expression patterns can diverge between species due to changes in a gene's regulatory DNA or changes in the proteins, e.g., transcription factors (TFs), that regulate the gene. We developed a modeling framework to uncover the sources of expression differences in blastoderm embryos of three Drosophila species, focusing on the regulatory circuit controlling expression of the hunchback (hb) posterior stripe. Using this framework and cellular-resolution expression measurements of hb and its regulating TFs, we found that changes in the expression patterns of hb's TFs account for much of the expression divergence. We confirmed our predictions using transgenic D. melanogaster lines, which demonstrate that this set of orthologous cis-regulatory elements (CREs) direct similar, but not identical, expression patterns. We related expression pattern differences to sequence changes in the CRE using a calculation of the CRE's TF binding site content. By applying this calculation in both the transgenic and endogenous contexts, we found that changes in binding site content affect sensitivity to regulating TFs and that compensatory evolution may occur in circuit components other than the CRE.

Transcriptional activators in the early Drosophila embryo perform different kinetic roles.

Combinatorial regulation of gene expression by transcription factors (TFs) may in part arise from kinetic synergy-wherein TFs regulate different steps in the transcription cycle. Kinetic synergy requires that TFs play distinguishable kinetic roles. Here, we used live imaging to determine the kinetic roles of three TFs that activate transcription in the Drosophila embryo-Zelda, Bicoid, and Stat92E-by introducing their binding sites into the even-skipped stripe 2 enhancer. These TFs influence different sets of kinetic parameters, and their influence can change over time. All three TFs increased the fraction of transcriptionally active nuclei; Zelda also shortened the first-passage time into transcription and regulated the interval between transcription events. Stat92E also increased the lifetimes of active transcription. Different TFs can therefore play distinct kinetic roles in activating the transcription. This has consequences for understanding the composition and flexibility of regulatory DNA sequences and the biochemical function of TFs. A record of this paper's transparent peer review process is included in the supplemental information.

Transcriptional kinetic synergy: A complex landscape revealed by integrating modeling and synthetic biology.

Transcription factors (TFs) control gene expression, often acting synergistically. Classical thermodynamic models offer a biophysical explanation for synergy based on binding cooperativity and regulated recruitment of RNA polymerase. Because transcription requires polymerase to transition through multiple states, recent work suggests that "kinetic synergy" can arise through TFs acting on distinct steps of the transcription cycle. These types of synergy are not mutually exclusive and are difficult to disentangle conceptually and experimentally. Here, we model and build a synthetic circuit in which TFs bind to a single shared site on DNA, such that TFs cannot synergize by simultaneous binding. We model mRNA production as a function of both TF binding and regulation of the transcription cycle, revealing a complex landscape dependent on TF concentration, DNA binding affinity, and regulatory activity. We use synthetic TFs to confirm that the transcription cycle must be integrated with recruitment for a quantitative understanding of gene regulation.

Generation of prion transmission barriers by mutational control of amyloid conformations.

Self-propagating beta-sheet-rich protein aggregates are implicated in a wide range of protein-misfolding phenomena, including amyloid diseases and prion-based inheritance. Two properties have emerged as common features of amyloids. Amyloid formation is ubiquitous: many unrelated proteins form such aggregates and even a single polypeptide can misfold into multiple forms--a process that is thought to underlie prion strain variation. Despite this promiscuity, amyloid propagation can be highly sequence specific: amyloid fibres often fail to catalyse the aggregation of other amyloidogenic proteins. In prions, this specificity leads to barriers that limit transmission between species. Using the yeast prion [PSI+], we show in vitro that point mutations in Sup35p, the protein determinant of [PSI+], alter the range of 'infectious' conformations, which in turn changes amyloid seeding specificity. We generate a new transmission barrier in vivo by using these mutations to specifically disfavour subsets of prion strains. The ability of mutations to alter the conformations of amyloid states without preventing amyloid formation altogether provides a general mechanism for the generation of prion transmission barriers and may help to explain how mutations alter toxicity in conformational diseases.

A Mutation in the Drosophila melanogaster eve Stripe 2 Minimal Enhancer Is Buffered by Flanking Sequences.

Enhancers are DNA sequences composed of transcription factor binding sites that drive complex patterns of gene expression in space and time. Until recently, studying enhancers in their genomic context was technically challenging. Therefore, minimal enhancers, the shortest pieces of DNA that can drive an expression pattern that resembles a gene's endogenous pattern, are often used to study features of enhancer function. However, evidence suggests that some enhancers require sequences outside the minimal enhancer to maintain function under environmental perturbations. We hypothesized that these additional sequences also prevent misexpression caused by a transcription factor binding site mutation within a minimal enhancer. Using the Drosophila melanogastereven-skipped stripe 2 enhancer as a case study, we tested the effect of a Giant binding site mutation (gt-2) on the expression patterns driven by minimal and extended enhancer reporter constructs. We found that, in contrast to the misexpression caused by the gt-2 binding site deletion in the minimal enhancer, the same gt-2 binding site deletion in the extended enhancer did not have an effect on expression. The buffering of expression levels, but not expression pattern, is partially explained by an additional Giant binding site outside the minimal enhancer. Deleting the gt-2 binding site in the endogenous locus had no significant effect on stripe 2 expression. Our results indicate that rules derived from mutating enhancer reporter constructs may not represent what occurs in the endogenous context.

Signal Integration by Shadow Enhancers and Enhancer Duplications Varies across the Drosophila Embryo.

Transcription of developmental genes is controlled by multiple enhancers. Frequently, more than one enhancer can activate transcription from the same promoter in the same cells. How is regulatory information from multiple enhancers combined to determine the overall expression output? We measure nascent transcription driven by a pair of shadow enhancers, each enhancer of the pair separately, and each duplicated, using live imaging in Drosophila embryos. This set of constructs allows us to quantify the input-output function describing signal integration by two enhancers. We show that signal integration performed by these shadow enhancers and duplications varies across the expression pattern, implying that how their activities are combined depends on the transcriptional regulators bound to the enhancers in different parts of the embryo. Characterizing signal integration by multiple enhancers is a critical step in developing conceptual and computational models of gene expression at the locus level, where multiple enhancers control transcription together.

Quantitative Comparison of the Anterior-Posterior Patterning System in the Embryos of Five Drosophila Species.

Complex spatiotemporal gene expression patterns direct the development of the fertilized egg into an adult animal. Comparisons across species show that, in spite of changes in the underlying regulatory DNA sequence, developmental programs can be maintained across millions of years of evolution. Reciprocally, changes in gene expression can be used to generate morphological novelty. Distinguishing between changes in regulatory DNA that lead to changes in gene expression and those that do not is therefore a central goal of evolutionary developmental biology. Quantitative, spatially-resolved measurements of developmental gene expression patterns play a crucial role in this goal, enabling the detection of subtle phenotypic differences between species and the development of computations models that link the sequence of regulatory DNA to expression patterns. Here we report the generation of two atlases of cellular resolution gene expression measurements for the primary anterior-posterior patterning genes in Drosophila simulans and Drosophila virilis By combining these data sets with existing atlases for three other Drosophila species, we detect subtle differences in the gene expression patterns and dynamics driving the highly conserved axis patterning system and delineate inter-species differences in the embryonic morphology. These data sets will be a resource for future modeling studies of the evolution of developmental gene regulatory networks.

Dissecting the sharp response of a canonical developmental enhancer reveals multiple sources of cooperativity.

Developmental enhancers integrate graded concentrations of transcription factors (TFs) to create sharp gene expression boundaries. Here we examine the hunchback P2 (HbP2) enhancer which drives a sharp expression pattern in the Drosophila blastoderm embryo in response to the transcriptional activator Bicoid (Bcd). We systematically interrogate cis and trans factors that influence the shape and position of expression driven by HbP2, and find that the prevailing model, based on pairwise cooperative binding of Bcd to HbP2 is not adequate. We demonstrate that other proteins, such as pioneer factors, Mediator and histone modifiers influence the shape and position of the HbP2 expression pattern. Comparing our results to theory reveals how higher-order cooperativity and energy expenditure impact boundary location and sharpness. Our results emphasize that the bacterial view of transcription regulation, where pairwise interactions between regulatory proteins dominate, must be reexamined in animals, where multiple molecular mechanisms collaborate to shape the gene regulatory function.