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

From genes to patterns: a framework for modeling the emergence of embryonic development from transcriptional regulation

Understanding embryonic patterning, the process by which groups of cells are partitioned into distinct identities defined by gene expression, is a central challenge in developmental biology. This complex phenomenon is driven by precise spatial and temporal regulation of gene expression across many cells, resulting in the emergence of highly organized tissue structures. While similar emergent behavior is well understood in other fields, such as statistical mechanics, the regulation of gene expression in development remains less clear, particularly regarding how molecular-level gene interactions lead to the large-scale patterns observed in embryos. In this study, we present a modeling framework that bridges the gap between molecular gene regulation and tissue-level embryonic patterning. Beginning with basic chemical reaction models of transcription at the single-gene level, we progress to model gene regulatory networks (GRNs) that mediate specific cellular functions. We then introduce phenomenological models of pattern formation, including the French Flag and Temporal Patterning/Speed Regulation models, and integrate them with molecular/GRN realizations. To facilitate understanding and application of our models, we accompany our mathematical framework with computer simulations, providing intuitive and simple code for each model. A key feature of our framework is the explicit articulation of underlying assumptions at each level of the model, from transcriptional regulation to tissue patterning. By making these assumptions clear, we provide a foundation for future experimental and theoretical work to critically examine and challenge them, thereby improving the accuracy and relevance of gene regulatory models in developmental biology. As a case study, we explore how different strategies for integrating enhancer activity affect the robustness and evolvability of GRNs that govern embryonic pattern formation. Our simulations suggest that a two-step regulation strategy, enhancer activation followed by competitive integration at the promoter, ensures more standardized integration of new enhancers into developmental GRNs, highlighting the adaptability of eukaryotic transcription. These findings shed new light on the transcriptional mechanisms underlying embryonic patterning, while the overall modeling framework serves as a foundation for future experimental and theoretical investigations.

Transcription factor clusters as information transfer agents

Deciphering how genes interpret information from transcription factor (TF) concentrations within the cell nucleus remains a fundamental question in gene regulation. Recent advancements have revealed the heterogeneous distribution of TF molecules, posing challenges to precisely decoding concentration signals. Using high-resolution single-cell imaging of the fluorescently tagged TF Bicoid in living Drosophila embryos, we show that Bicoid accumulation in submicrometer clusters preserves the spatial information of the maternal Bicoid gradient. These clusters provide precise spatial cues through intensity, size, and frequency. We further discover that Bicoid target genes colocalize with these clusters in an enhancer-binding affinity-dependent manner. Our modeling suggests that clustering offers a faster sensing mechanism for global nuclear concentrations than freely diffusing TF molecules detected by simple enhancers.

Bicoid-nucleosome competition sets a concentration threshold for transcription constrained by genome replication

Transcription factors (TFs) regulate gene expression despite constraints from chromatin structure and the cell cycle. Here we examine the concentration-dependent regulation of hunchback by the Bicoid morphogen through a combination of quantitative imaging, mathematical modeling and epigenomics in Drosophila embryos. By live imaging of MS2 reporters, we find that, following mitosis, the timing of transcriptional activation driven by the hunchback P2 (hb P2) enhancer directly reflects Bicoid concentration. We build a stochastic model that can explain in vivo onset time distributions by accounting for both the competition between Bicoid and nucleosomes at hb P2 and a negative influence of DNA replication on transcriptional elongation. Experimental modulation of nucleosome stability alters onset time distributions and the posterior boundary of hunchback expression. We conclude that TF-nucleosome competition is the molecular mechanism whereby the Bicoid morphogen gradient specifies the posterior boundary of hunchback expression.

Deciphering regulatory architectures of bacterial promoters from synthetic expression patterns

For the vast majority of genes in sequenced genomes, there is limited understanding of how they are regulated. Without such knowledge, it is not possible to perform a quantitative theory-experiment dialogue on how such genes give rise to physiological and evolutionary adaptation. One category of high-throughput experiments used to understand the sequence-phenotype relationship of the transcriptome is massively parallel reporter assays (MPRAs). However, to improve the versatility and scalability of MPRAs, we need a "theory of the experiment" to help us better understand the impact of various biological and experimental parameters on the interpretation of experimental data. These parameters include binding site copy number, where a large number of specific binding sites may titrate away transcription factors, as well as the presence of overlapping binding sites, which may affect analysis of the degree of mutual dependence between mutations in the regulatory region and expression levels. To that end, in this paper we create tens of thousands of synthetic gene expression outputs for bacterial promoters using both equilibrium and out-of-equilibrium models. These models make it possible to imitate the summary statistics (information footprints and expression shift matrices) used to characterize the output of MPRAs and thus to infer the underlying regulatory architecture. Specifically, we use a more refined implementation of the so-called thermodynamic models in which the binding energies of each sequence variant are derived from energy matrices. Our simulations reveal important effects of the parameters on MPRA data and we demonstrate our ability to optimize MPRA experimental designs with the goal of generating thermodynamic models of the transcriptome with base-pair specificity. Further, this approach makes it possible to carefully examine the mapping between mutations in binding sites and their corresponding expression profiles, a tool useful not only for developing a theory of transcription, but also for exploring regulatory evolution.

Transcription factor clusters as information transfer agents

Deciphering how genes interpret information from the concentration of transcription factors (TFs) within the cell nucleus remains a fundamental question in gene regulation. Recent advancements have unveiled the heterogeneous distribution of TF molecules in the nucleus, posing challenges to the precise decoding of concentration signals. To explore this phenomenon, we employ high-resolution single-cell imaging of a fluorescently tagged TF protein, Bicoid, in living fly embryos. We show that accumulation of Bicoid in submicron clusters preserves the spatial information of the maternal Bicoid gradient, and that cluster intensity, size, and frequency offer remarkably precise spatial cues. We further discover that various known gene targets of Bicoid activation colocalize with clusters and that for the target gene Hunchback, this colocalization is dependent on its enhancer binding affinity. Modeling information transfer through these clusters suggests that clustering offers a more rapid sensing mechanism for global nuclear concentrations than freely diffusing TF molecules detected by simple enhancers.

Optogenetic dissection of transcriptional repression in a multicellular organism

Transcriptional control is fundamental to cellular function. However, despite knowing that transcription factors can repress or activate specific genes, how these functions are implemented at the molecular level has remained elusive, particularly in the endogenous context of developing animals. Here, we combine optogenetics, single-cell live-imaging, and mathematical modeling to study how a zinc-finger repressor, Knirps, induces switch-like transitions into long-lived quiescent states. Using optogenetics, we demonstrate that repression is rapidly reversible (~1 min) and memoryless. Furthermore, we show that the repressor acts by decreasing the frequency of transcriptional bursts in a manner consistent with an equilibrium binding model. Our results provide a quantitative framework for dissecting the in vivo biochemistry of eukaryotic transcriptional regulation.

Energy Aware Technology Mapping of Genetic Logic Circuits

Energy and its dissipation are fundamental to all living systems, including cells. Insufficient abundance of energy carriers─as caused by the additional burden of artificial genetic circuits─shifts a cell's priority to survival, also impairing the functionality of the genetic circuit. Moreover, recent works have shown the importance of energy expenditure in information transmission. Despite living organisms being non-equilibrium systems, non-equilibrium models capable of accounting for energy dissipation and non-equilibrium response curves are not yet employed in genetic design automation (GDA) software. To this end, we introduce Energy Aware Technology Mapping, the automated design of genetic logic circuits with respect to energy efficiency and functionality. The basis for this is an energy aware non-equilibrium steady state model of gene expression, capturing characteristics like energy dissipation─which we link to the entropy production rate─and transcriptional bursting, relevant to eukaryotes as well as prokaryotes. Our evaluation shows that a genetic logic circuit's functional performance and energy efficiency are disjoint optimization goals. For our benchmark, energy efficiency improves by 37.2% on average when comparing to functionally optimized variants. We discover a linear increase in energy expenditure and overall protein expression with the circuit size, where Energy Aware Technology Mapping allows for designing genetic logic circuits with the energetic costs of circuits that are one to two gates smaller. Structural variants improve this further, while results show the Pareto dominance among structures of a single Boolean function. By incorporating energy demand into the design, Energy Aware Technology Mapping enables energy efficiency by design. This extends current GDA tools and complements approaches coping with burden in vivo.

The method in the madness: Transcriptional control from stochastic action at the single-molecule scale

Cell states result from the ordered activation of gene expression by transcription factors. Transcription factors face opposing design constraints: they need to be dynamic to trigger rapid cell state transitions, but also stable enough to maintain terminal cell identities indefinitely. Recent progress in live-cell single-molecule microscopy has helped define the biophysical principles underlying this paradox. Beyond transcription factor activity, single-molecule experiments have revealed that at nearly every level of transcription regulation, control emerges from multiple short-lived stochastic interactions, rather than deterministic, stable interactions typical of other biochemical pathways. This architecture generates consistent outcomes that can be rapidly choreographed. Here, we highlight recent results that demonstrate how order in transcription regulation emerges from the apparent molecular-scale chaos and discuss remaining conceptual challenges.

Interpreting cis-regulatory mechanisms from genomic deep neural networks using surrogate models

Deep neural networks (DNNs) have greatly advanced the ability to predict genome function from sequence. However, elucidating underlying biological mechanisms from genomic DNNs remains challenging. Existing interpretability methods, such as attribution maps, have their origins in non-biological machine learning applications and therefore have the potential to be improved by incorporating domain-specific interpretation strategies. Here we introduce SQUID, a genomic DNN interpretability framework based on domain-specific surrogate modeling. SQUID approximates genomic DNNs in user-specified regions of sequence space using surrogate models-simpler quantitative models that have inherently interpretable mathematical forms. SQUID leverages domain knowledge to model cis-regulatory mechanisms in genomic DNNs, in particular by removing the confounding effects that nonlinearities and heteroscedastic noise in functional genomics data can have on model interpretation. Benchmarking analysis on multiple genomic DNNs shows that SQUID, when compared to established interpretability methods, identifies motifs that are more consistent across genomic loci and yields improved single-nucleotide variant-effect predictions. SQUID also supports surrogate models that quantify epistatic interactions within and between cis-regulatory elements, as well as global explanations of cis-regulatory mechanisms across sequence contexts. SQUID thus advances the ability to mechanistically interpret genomic DNNs.

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.

Interpreting cis-regulatory mechanisms from genomic deep neural networks using surrogate models

Deep neural networks (DNNs) have greatly advanced the ability to predict genome function from sequence. Interpreting genomic DNNs in terms of biological mechanisms, however, remains difficult. Here we introduce SQUID, a genomic DNN interpretability framework based on surrogate modeling. SQUID approximates genomic DNNs in user-specified regions of sequence space using surrogate models, i.e., simpler models that are mechanistically interpretable. Importantly, SQUID removes the confounding effects that nonlinearities and heteroscedastic noise in functional genomics data can have on model interpretation. Benchmarking analysis on multiple genomic DNNs shows that SQUID, when compared to established interpretability methods, identifies motifs that are more consistent across genomic loci and yields improved single-nucleotide variant-effect predictions. SQUID also supports surrogate models that quantify epistatic interactions within and between cis-regulatory elements. SQUID thus advances the ability to mechanistically interpret genomic DNNs.

Topologically constrained fluctuations and thermodynamics regulate nonequilibrium response

The limits on a system's response to external perturbations inform our understanding of how physical properties can be shaped by microscopic characteristics. Here, we derive constraints on the steady-state nonequilibrium response of physical observables in terms of the topology of the microscopic state space and the strength of thermodynamic driving. Notably, evaluation of these limits requires no kinetic information beyond the state-space structure. When applied to models of receptor binding, we find that sensitivity is bounded by the steepness of a Hill function with a Hill coefficient enhanced by the chemical driving beyond the structural equilibrium limit.

Stepwise modifications of transcriptional hubs link pioneer factor activity to a burst of transcription

Binding of transcription factors (TFs) promotes the subsequent recruitment of coactivators and preinitiation complexes to initiate eukaryotic transcription, but this time course is usually not visualized. It is commonly assumed that recruited factors eventually co-reside in a higher-order structure, allowing distantly bound TFs to activate transcription at core promoters. We use live imaging of endogenously tagged proteins, including the pioneer TF Zelda, the coactivator dBrd4, and RNA polymerase II (RNAPII), to define a cascade of events upstream of transcriptional initiation in early Drosophila embryos. These factors are sequentially and transiently recruited to discrete clusters during activation of non-histone genes. Zelda and the acetyltransferase dCBP nucleate dBrd4 clusters, which then trigger pre-transcriptional clustering of RNAPII. Subsequent transcriptional elongation disperses clusters of dBrd4 and RNAPII. Our results suggest that activation of transcription by eukaryotic TFs involves a succession of distinct biomolecular condensates that culminates in a self-limiting burst of transcription.

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.

Minimal synthetic enhancers reveal control of the probability of transcriptional engagement and its timing by a morphogen gradient

How enhancers interpret morphogen gradients to generate gene expression patterns is a central question in developmental biology. Recent studies have proposed that enhancers can dictate whether, when, and at what rate promoters engage in transcription, but the complexity of endogenous enhancers calls for theoretical models with too many free parameters to quantitatively dissect these regulatory strategies. To overcome this limitation, we established a minimal promoter-proximal synthetic enhancer in embryos of Drosophila melanogaster. Here, a gradient of the Dorsal activator is read by a single Dorsal DNA binding site. Using live imaging to quantify transcriptional activity, we found that a single binding site can regulate whether promoters engage in transcription in a concentration-dependent manner. By modulating the binding-site affinity, we determined that a gene's decision to transcribe and its transcriptional onset time can be explained by a simple model where the promoter traverses multiple kinetic barriers before transcription can ensue.

Size limits the sensitivity of kinetic schemes

Living things benefit from exquisite molecular sensitivity in many of their key processes, including DNA replication, transcription and translation, chemical sensing, and morphogenesis. At thermodynamic equilibrium, the basic biophysical mechanism for sensitivity is cooperative binding, for which it can be shown that the Hill coefficient, a sensitivity measure, cannot exceed the number of binding sites. Generalizing this fact, we find that for any kinetic scheme, at or away from thermodynamic equilibrium, a very simple structural quantity, the size of the support of a perturbation, always limits the effective Hill coefficient. We show how this bound sheds light on and unifies diverse sensitivity mechanisms, including kinetic proofreading and a nonequilibrium Monod-Wyman-Changeux (MWC) model proposed for the E. coli flagellar motor switch, representing in each case a simple, precise bridge between experimental observations and the models we write down. In pursuit of mechanisms that saturate the support bound, we find a nonequilibrium binding mechanism, nested hysteresis, with sensitivity exponential in the number of binding sites, with implications for our understanding of models of gene regulation and the function of biomolecular condensates.

Competing constraints shape the nonequilibrium limits of cellular decision-making

Gene regulation is central to cellular function. Yet, despite decades of work, we lack quantitative models that can predict how transcriptional control emerges from molecular interactions at the gene locus. Thermodynamic models of transcription, which assume that gene circuits operate at equilibrium, have previously been employed with considerable success in the context of bacterial systems. However, the presence of ATP-dependent processes within the eukaryotic transcriptional cycle suggests that equilibrium models may be insufficient to capture how eukaryotic gene circuits sense and respond to input transcription factor concentrations. Here, we employ simple kinetic models of transcription to investigate how energy dissipation within the transcriptional cycle impacts the rate at which genes transmit information and drive cellular decisions. We find that biologically plausible levels of energy input can lead to significant gains in how rapidly gene loci transmit information but discover that the regulatory mechanisms underlying these gains change depending on the level of interference from noncognate activator binding. When interference is low, information is maximized by harnessing energy to push the sensitivity of the transcriptional response to input transcription factors beyond its equilibrium limits. Conversely, when interference is high, conditions favor genes that harness energy to increase transcriptional specificity by proofreading activator identity. Our analysis further reveals that equilibrium gene regulatory mechanisms break down as transcriptional interference increases, suggesting that energy dissipation may be indispensable in systems where noncognate factor interference is sufficiently large.

Modulation of protein-DNA binding reveals mechanisms of spatiotemporal gene control in early Drosophila embryos

It is well known that enhancers regulate the spatiotemporal expression of their target genes by recruiting transcription factors (TFs) to the cognate binding sites in the region. However, the role of multiple binding sites for the same TFs and their specific spatial arrangement in determining the overall competency of the enhancer has yet to be fully understood. In this study, we utilized the MS2-MCP live imaging technique to quantitatively analyze the regulatory logic of the snail distal enhancer in early Drosophila embryos. Through systematic modulation of Dorsal and Twist binding motifs in this enhancer, we found that a mutation in any one of these binding sites causes a drastic reduction in transcriptional amplitude, resulting in a reduction in total mRNA production of the target gene. We provide evidence of synergy, such that multiple binding sites with moderate affinities cooperatively recruit more TFs to drive stronger transcriptional activity than a single site. Moreover, a Hidden Markov-based stochastic model of transcription reveals that embryos with mutated binding sites have a higher probability of returning to the inactive promoter state. We propose that TF-DNA binding regulates spatial and temporal gene expression and drives robust pattern formation by modulating transcriptional kinetics and tuning bursting rates.

Predictive modeling reveals that higher-order cooperativity drives transcriptional repression in a synthetic developmental enhancer

A challenge in quantitative biology is to predict output patterns of gene expression from knowledge of input transcription factor patterns and from the arrangement of binding sites for these transcription factors on regulatory DNA. We tested whether widespread thermodynamic models could be used to infer parameters describing simple regulatory architectures that inform parameter-free predictions of more complex enhancers in the context of transcriptional repression by Runt in the early fruit fly embryo. By modulating the number and placement of Runt binding sites within an enhancer, and quantifying the resulting transcriptional activity using live imaging, we discovered that thermodynamic models call for higher-order cooperativity between multiple molecular players. This higher-order cooperativity captures the combinatorial complexity underlying eukaryotic transcriptional regulation and cannot be determined from simpler regulatory architectures, highlighting the challenges in reaching a predictive understanding of transcriptional regulation in eukaryotes and calling for approaches that quantitatively dissect their molecular nature.

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.

Eukaryotic gene regulation at equilibrium, or non?

Models of transcriptional regulation that assume equilibrium binding of transcription factors have been less successful at predicting gene expression from sequence in eukaryotes than in bacteria. This could be due to the non-equilibrium nature of eukaryotic regulation. Unfortunately, the space of possible non-equilibrium mechanisms is vast and predominantly uninteresting. The key question is therefore how this space can be navigated efficiently, to focus on mechanisms and models that are biologically relevant. In this review, we advocate for the normative role of theory-theory that prescribes rather than just describes-in providing such a focus. Theory should expand its remit beyond inferring mechanistic models from data, towards identifying non-equilibrium gene regulatory schemes that may have been evolutionarily selected, despite their energy consumption, because they are precise, reliable, fast, or otherwise outperform regulation at equilibrium. We illustrate our reasoning by toy examples for which we provide simulation code.

The linear framework: using graph theory to reveal the algebra and thermodynamics of biomolecular systems

The linear framework uses finite, directed graphs with labelled edges to model biomolecular systems. Graph vertices represent biochemical species or molecular states, edges represent reactions or transitions and labels represent rates. The graph yields a linear dynamics for molecular concentrations or state probabilities, with the graph Laplacian as the operator, and the labels encode the nonlinear interactions between system and environment. The labels can be specified by vertices of other graphs or by conservation laws or, when the environment consists of thermodynamic reservoirs, they may be constants. In the latter case, the graphs correspond to infinitesimal generators of Markov processes. The key advantage of the framework has been that steady states are determined as rational algebraic functions of the labels by the Matrix-Tree theorems of graph theory. When the system is at thermodynamic equilibrium, this prescription recovers equilibrium statistical mechanics but it continues to hold for non-equilibrium steady states. The framework goes beyond other graph-based approaches in treating the graph as a mathematical object, for which general theorems can be formulated that accommodate biomolecular complexity. It has been particularly effective at analysing enzyme-catalysed modification systems and input-output responses.

A synthetic synthesis to explore animal evolution and development

Identifying the general principles by which genotypes are converted into phenotypes remains a challenge in the post-genomic era. We still lack a predictive understanding of how genes shape interactions among cells and tissues in response to signalling and environmental cues, and hence how regulatory networks generate the phenotypic variation required for adaptive evolution. Here, we discuss how techniques borrowed from synthetic biology may facilitate a systematic exploration of evolvability across biological scales. Synthetic approaches permit controlled manipulation of both endogenous and fully engineered systems, providing a flexible platform for investigating causal mechanisms in vivo. Combining synthetic approaches with multi-level phenotyping (phenomics) will supply a detailed, quantitative characterization of how internal and external stimuli shape the morphology and behaviour of living organisms. We advocate integrating high-throughput experimental data with mathematical and computational techniques from a variety of disciplines in order to pursue a comprehensive theory of evolution.

This article is part of the theme issue ‘Genetic basis of adaptation and speciation: from loci to causative mutations’.

Synthetic reconstruction of the hunchback promoter specifies the role of Bicoid, Zelda and Hunchback in the dynamics of its transcription

For over 40 years, the Bicoid-hunchback (Bcd-hb) system in the fruit fly embryo has been used as a model to study how positional information in morphogen concentration gradients is robustly translated into step-like responses. A body of quantitative comparisons between theory and experiment have since questioned the initial paradigm that the sharp hb transcription pattern emerges solely from diffusive biochemical interactions between the Bicoid transcription factor and the gene promoter region. Several alternative mechanisms have been proposed, such as additional sources of positional information, positive feedback from Hb proteins or out-of-equilibrium transcription activation. By using the MS2-MCP RNA-tagging system and analysing in real time, the transcription dynamics of synthetic reporters for Bicoid and/or its two partners Zelda and Hunchback, we show that all the early hb expression pattern features and temporal dynamics are compatible with an equilibrium model with a short decay length Bicoid activity gradient as a sole source of positional information. Meanwhile, Bicoid’s partners speed-up the process by different means: Zelda lowers the Bicoid concentration threshold required for transcriptional activation while Hunchback reduces burstiness and increases the polymerase firing rate.

Notch-dependent DNA cis-regulatory elements and their dose-dependent control of C. elegans stem cell self-renewal

A long-standing biological question is how DNA cis-regulatory elements shape transcriptional patterns during metazoan development. Reporter constructs, cell culture assays and computational modeling have made major contributions to answering this question, but analysis of elements in their natural context is an important complement. Here, we mutate Notch-dependent LAG-1 binding sites (LBSs) in the endogenous Caenorhabditis elegans sygl-1 gene, which encodes a key stem cell regulator, and analyze the consequences on sygl-1 expression (nascent transcripts, mRNA, protein) and stem cell maintenance. Mutation of one LBS in a three-element cluster approximately halved both expression and stem cell pool size, whereas mutation of two LBSs essentially abolished them. Heterozygous LBS mutant clusters provided intermediate values. Our results lead to two major conclusions. First, both LBS number and configuration impact cluster activity: LBSs act additively in trans and synergistically in cis. Second, the SYGL-1 gradient promotes self-renewal above its functional threshold and triggers differentiation below the threshold. Our approach of coupling CRISPR/Cas9 LBS mutations with effects on both molecular and biological readouts establishes a powerful model for in vivo analyses of DNA cis-regulatory elements.

Physical bioenergetics: Energy fluxes, budgets, and constraints in cells

Cells are the basic units of all living matter which harness the flow of energy to drive the processes of life. While the biochemical networks involved in energy transduction are well-characterized, the energetic costs and constraints for specific cellular processes remain largely unknown. In particular, what are the energy budgets of cells? What are the constraints and limits energy flows impose on cellular processes? Do cells operate near these limits, and if so how do energetic constraints impact cellular functions? Physics has provided many tools to study nonequilibrium systems and to define physical limits, but applying these tools to cell biology remains a challenge. Physical bioenergetics, which resides at the interface of nonequilibrium physics, energy metabolism, and cell biology, seeks to understand how much energy cells are using, how they partition this energy between different cellular processes, and the associated energetic constraints. Here we review recent advances and discuss open questions and challenges in physical bioenergetics.

Allosteric conformational ensembles have unlimited capacity for integrating information

Integration of binding information by macromolecular entities is fundamental to cellular functionality. Recent work has shown that such integration cannot be explained by pairwise cooperativities, in which binding is modulated by binding at another site. Higher-order cooperativities (HOCs), in which binding is collectively modulated by multiple other binding events, appear to be necessary but an appropriate mechanism has been lacking. We show here that HOCs arise through allostery, in which effective cooperativity emerges indirectly from an ensemble of dynamically interchanging conformations. Conformational ensembles play important roles in many cellular processes but their integrative capabilities remain poorly understood. We show that sufficiently complex ensembles can implement any form of information integration achievable without energy expenditure, including all patterns of HOCs. Our results provide a rigorous biophysical foundation for analysing the integration of binding information through allostery. We discuss the implications for eukaryotic gene regulation, where complex conformational dynamics accompanies widespread information integration.

Ciona Brachyury proximal and distal enhancers have different FGF dose-response relationships

Many genes are regulated by two or more enhancers that drive similar expression patterns. Evolutionary theory suggests that these seemingly redundant enhancers must have functionally important differences. In the simple ascidian chordate Ciona, the transcription factor Brachyury is induced exclusively in the presumptive notochord downstream of lineage specific regulators and FGF-responsive Ets family transcription factors. Here we exploit the ability to finely titrate FGF signaling activity via the MAPK pathway using the MEK inhibitor U0126 to quantify the dependence of transcription driven by different Brachyury reporter constructs on this direct upstream regulator. We find that the more powerful promoter-adjacent proximal enhancer and a weaker distal enhancer have fundamentally different dose-response relationships to MAPK inhibition. The Distal enhancer is more sensitive to MAPK inhibition but shows a less cooperative response, whereas the Proximal enhancer is less sensitive and more cooperative. A longer construct containing both enhancers has a complex dose-response curve that supports the idea that the proximal and distal enhancers are moderately super-additive. We show that the overall expression loss from intermediate doses of U0126 is not only a function of the fraction of cells expressing these reporters, but also involves graded decreases in expression at the single-cell level. Expression of the endogenous gene shows a comparable dose-response relationship to the full length reporter, and we find that different notochord founder cells are differentially sensitive to MAPK inhibition. Together, these results indicate that although the two Brachyury enhancers have qualitatively similar expression patterns, they respond to FGF in quantitatively different ways and act together to drive high levels of Brachyury expression with a characteristic input/output relationship. This indicates that they are fundamentally not equivalent genetic elements.

An optimal control approach for enhancing natural killer cells' secretion of cytolytic molecules

Natural killer (NK) cells are immune effector cells that can detect and lyse cancer cells. However, NK cell exhaustion, a phenotype characterized by reduced secretion of cytolytic models upon serial stimulation, limits the NK cell's ability to lyse cells. In this work, we investigated in silico strategies that counteract the NK cell's reduced secretion of cytolytic molecules. To accomplish this goal, we constructed a mathematical model that describes the dynamics of the cytolytic molecules granzyme B (GZMB) and perforin-1 (PRF1) and calibrated the model predictions to published experimental data using a Bayesian parameter estimation approach. We applied an information-theoretic approach to perform a global sensitivity analysis, from which we found that the suppression of phosphatase activity maximizes the secretion of GZMB and PRF1. However, simply reducing the phosphatase activity is shown to deplete the cell's intracellular pools of GZMB and PRF1. Thus, we added a synthetic Notch (synNotch) signaling circuit to our baseline model as a method for controlling the secretion of GZMB and PRF1 by inhibiting phosphatase activity and increasing production of GZMB and PRF1. We found that the optimal synNotch system depends on the frequency of NK cell stimulation. For only a few rounds of stimulation, the model predicts that inhibition of phosphatase activity leads to more secreted GZMB and PRF1; however, for many rounds of stimulation, the model reveals that increasing production of the cytolytic molecules is the optimal strategy. In total, we developed a mathematical framework that provides actionable insight into engineering robust NK cells for clinical applications.

Gene Regulation in and out of Equilibrium

Determining whether and how a gene is transcribed are two of the central processes of life. The conceptual basis for understanding such gene regulation arose from pioneering biophysical studies in eubacteria. However, eukaryotic genomes exhibit vastly greater complexity, which raises questions not addressed by this bacterial paradigm. First, how is information integrated from many widely separated binding sites to determine how a gene is transcribed? Second, does the presence of multiple energy-expending mechanisms, which are absent from eubacterial genomes, indicate that eukaryotes are capable of improved forms of genetic information processing? An updated biophysical foundation is needed to answer such questions. We describe the linear framework, a graph-based approach to Markov processes, and show that it can accommodate many previous studies in the field. Under the assumption of thermodynamic equilibrium, we introduce a language of higher-order cooperativities and show how it can rigorously quantify gene regulatory properties suggested by experiment. We point out that fundamental limits to information processing arise at thermodynamic equilibrium and can only be bypassed through energy expenditure. Finally, we outline some of the mathematical challenges that must be overcome to construct an improved biophysical understanding of gene regulation.

Nonequilibrium models of optimal enhancer function

Simple biophysical models successfully describe bacterial regulatory code, by predicting gene expression from DNA sequences that bind specialized regulatory proteins. Analogous simple models fail in multicellular organisms, where regulatory proteins bind DNA very transiently, yet, nevertheless, effect precise control over gene expression. To date, the more general, “nonequilibrium” models have proven difficult to analyze and connect to data. Here, we reduce this complexity theoretically, by constructing simple nonequilibrium models which perform optimal gene regulation within known experimental constraints.

In prokaryotes, thermodynamic models of gene regulation provide a highly quantitative mapping from promoter sequences to gene-expression levels that is compatible with in vivo and in vitro biophysical measurements. Such concordance has not been achieved for models of enhancer function in eukaryotes. In equilibrium models, it is difficult to reconcile the reported short transcription factor (TF) residence times on the DNA with the high specificity of regulation. In nonequilibrium models, progress is difficult due to an explosion in the number of parameters. Here, we navigate this complexity by looking for minimal nonequilibrium enhancer models that yield desired regulatory phenotypes: low TF residence time, high specificity, and tunable cooperativity. We find that a single extra parameter, interpretable as the “linking rate,” by which bound TFs interact with Mediator components, enables our models to escape equilibrium bounds and access optimal regulatory phenotypes, while remaining consistent with the reported phenomenology and simple enough to be inferred from upcoming experiments. We further find that high specificity in nonequilibrium models is in a trade-off with gene-expression noise, predicting bursty dynamics—an experimentally observed hallmark of eukaryotic transcription. By drastically reducing the vast parameter space of nonequilibrium enhancer models to a much smaller subspace that optimally realizes biological function, we deliver a rich class of models that could be tractably inferred from data in the near future.

Regulatory encoding of quantitative variation in spatial activity of a Drosophila enhancer

Developmental enhancers control the expression of genes prefiguring morphological patterns. The activity of an enhancer varies among cells of a tissue, but collectively, expression levels in individual cells constitute a spatial pattern of gene expression. How the spatial and quantitative regulatory information is encoded in an enhancer sequence is elusive. To link spatial pattern and activity levels of an enhancer, we used systematic mutations of the yellow spot enhancer, active in developing Drosophila wings, and tested their effect in a reporter assay. Moreover, we developed an analytic framework based on the comprehensive quantification of spatial reporter activity. We show that the quantitative enhancer activity results from densely packed regulatory information along the sequence, and that a complex interplay between activators and multiple tiers of repressors carves the spatial pattern. Our results shed light on how an enhancer reads and integrates trans-regulatory landscape information to encode a spatial quantitative pattern.

A matter of time: Using dynamics and theory to uncover mechanisms of transcriptional bursting

Eukaryotic transcription generally occurs in bursts of activity lasting minutes to hours; however, state-of-the-art measurements have revealed that many of the molecular processes that underlie bursting, such as transcription factor binding to DNA, unfold on timescales of seconds. This temporal disconnect lies at the heart of a broader challenge in physical biology of predicting transcriptional outcomes and cellular decision-making from the dynamics of underlying molecular processes. Here, we review how new dynamical information about the processes underlying transcriptional control can be combined with theoretical models that predict not only averaged transcriptional dynamics, but also their variability, to formulate testable hypotheses about the molecular mechanisms underlying transcriptional bursting and control.

Quantitative dissection of transcription in development yields evidence for transcription-factor-driven chromatin accessibility

Thermodynamic models of gene regulation can predict transcriptional regulation in bacteria, but in eukaryotes, chromatin accessibility and energy expenditure may call for a different framework. Here, we systematically tested the predictive power of models of DNA accessibility based on the Monod-Wyman-Changeux (MWC) model of allostery, which posits that chromatin fluctuates between accessible and inaccessible states. We dissected the regulatory dynamics of hunchback by the activator Bicoid and the pioneer-like transcription factor Zelda in living Drosophila embryos and showed that no thermodynamic or non-equilibrium MWC model can recapitulate hunchback transcription. Therefore, we explored a model where DNA accessibility is not the result of thermal fluctuations but is catalyzed by Bicoid and Zelda, possibly through histone acetylation, and found that this model can predict hunchback dynamics. Thus, our theory-experiment dialogue uncovered potential molecular mechanisms of transcriptional regulatory dynamics, a key step toward reaching a predictive understanding of developmental decision-making.

Transcriptional Enhancers in Drosophila

Key discoveries in Drosophila have shaped our understanding of cellular "enhancers." With a special focus on the fly, this chapter surveys properties of these adaptable cis-regulatory elements, whose actions are critical for the complex spatial/temporal transcriptional regulation of gene expression in metazoa. The powerful combination of genetics, molecular biology, and genomics available in Drosophila has provided an arena in which the developmental role of enhancers can be explored. Enhancers are characterized by diverse low- or high-throughput assays, which are challenging to interpret, as not all of these methods of identifying enhancers produce concordant results. As a model metazoan, the fly offers important advantages to comprehensive analysis of the central functions that enhancers play in gene expression, and their critical role in mediating the production of phenotypes from genotype and environmental inputs. A major challenge moving forward will be obtaining a quantitative understanding of how these cis-regulatory elements operate in development and disease.

A mechanism for hunchback promoters to readout morphogenetic positional information in less than a minute

Cell fate decisions in the fly embryo are rapid: hunchback genes decide in minutes whether nuclei follow the anterior/posterior developmental blueprint by reading out positional information in the Bicoid morphogen. This developmental system is a prototype of regulatory decision processes that combine speed and accuracy. Traditional arguments based on fixed-time sampling of Bicoid concentration indicate that an accurate readout is impossible within the experimental times. This raises the general issue of how speed-accuracy tradeoffs are achieved. Here, we compare fixed-time to on-the-fly decisions, based on comparing the likelihoods of anterior/posterior locations. We found that these more efficient schemes complete reliable cell fate decisions within the short embryological timescales. We discuss the influence of promoter architectures on decision times and error rates, present concrete examples that rapidly readout the morphogen, and predictions for new experiments. Lastly, we suggest a simple mechanism for RNA production and degradation that approximates the log-likelihood function.

Multimodal transcriptional control of pattern formation in embryonic development

Predicting how interactions between transcription factors and regulatory DNA sequence dictate rates of transcription and, ultimately, drive developmental outcomes remains an open challenge in physical biology. Using stripe 2 of the even-skipped gene in Drosophila embryos as a case study, we dissect the regulatory forces underpinning a key step along the developmental decision-making cascade: the generation of cytoplasmic mRNA patterns via the control of transcription in individual cells. Using live imaging and computational approaches, we found that the transcriptional burst frequency is modulated across the stripe to control the mRNA production rate. However, we discovered that bursting alone cannot quantitatively recapitulate the formation of the stripe and that control of the window of time over which each nucleus transcribes even-skipped plays a critical role in stripe formation. Theoretical modeling revealed that these regulatory strategies (bursting and the time window) respond in different ways to input transcription factor concentrations, suggesting that the stripe is shaped by the interplay of 2 distinct underlying molecular processes.

A matter of time: Formation and interpretation of the Bicoid morphogen gradient

Spatially distributed signaling molecules, known as morphogens, provide spatial information during development. A host of different morphogens have now been identified, from subcellular gradients through to morphogens that act across a whole embryo. These gradients form over a wide-range of timescales, from seconds to hours, and their time windows for interpretation are also highly variable; the processes of morphogen gradient formation and interpretation are highly dynamic. The morphogen Bicoid (Bcd), present in the early Drosophila embryo, is essential for setting up the future Drosophila body segments. Due to its accessibility for both genetic perturbations and imaging, this system has provided key insights into how precise patterning can occur within a highly dynamic system. Here, we review the temporal scales of Bcd gradient formation and interpretation. In particular, we discuss the quantitative evidence for different models of Bcd gradient formation, outline the time windows for Bcd interpretation, and describe how Bcd temporally adapts its own ability to be interpreted. The utilization of temporal information in morphogen readout may provide crucial inputs to ensure precise spatial patterning, particularly in rapidly developing systems.

Lighting up the central dogma for predictive developmental biology

Although the last 30years have witnessed the mapping of the wiring diagrams of the gene regulatory networks that dictate cell fate and animal body plans, specific understanding building on such network diagrams that shows how DNA regulatory regions control gene expression lags far behind. These networks have yet to yield the predictive power necessary to, for example, calculate how the concentration dynamics of input transcription factors and DNA regulatory sequence prescribes output patterns of gene expression that, in turn, determine body plans themselves. Here, we argue that reaching a predictive understanding of developmental decision-making calls for an interplay between theory and experiment aimed at revealing how the regulation of the processes of the central dogma dictate network connections and how network topology guides cells toward their ultimate developmental fate. To make this possible, it is crucial to break free from the snapshot-based understanding of embryonic development facilitated by fixed-tissue approaches and embrace new technologies that capture the dynamics of developmental decision-making at the single cell level, in living embryos.