Thermodynamic state ensemble models of cis-regulation

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.

Transcription factor interactions explain the context-dependent activity of CRX binding sites

The effects of transcription factor binding sites (TFBSs) on the activity of a cis-regulatory element (CRE) depend on the local sequence context. In rod photoreceptors, binding sites for the transcription factor (TF) Cone-rod homeobox (CRX) occur in both enhancers and silencers, but the sequence context that determines whether CRX binding sites contribute to activation or repression of transcription is not understood. To investigate the context-dependent activity of CRX sites, we fit neural network-based models to the activities of synthetic CREs composed of photoreceptor TFBSs. The models revealed that CRX binding sites consistently make positive, independent contributions to CRE activity, while negative homotypic interactions between sites cause CREs composed of multiple CRX sites to function as silencers. The effects of negative homotypic interactions can be overcome by the presence of other TFBSs that either interact cooperatively with CRX sites or make independent positive contributions to activity. The context-dependent activity of CRX sites is thus determined by the balance between positive heterotypic interactions, independent contributions of TFBSs, and negative homotypic interactions. Our findings explain observed patterns of activity among genomic CRX-bound enhancers and silencers, and suggest that enhancers may require diverse TFBSs to overcome negative homotypic interactions between TFBSs.

The Drosophila ecdysone receptor promotes or suppresses proliferation according to ligand level

The steroid hormone 20-hydroxy-ecdysone (20E) promotes proliferation in Drosophila wing precursors at low titer but triggers proliferation arrest at high doses. Remarkably, wing precursors proliferate normally in the complete absence of the 20E receptor, suggesting that low-level 20E promotes proliferation by overriding the default anti-proliferative activity of the receptor. By contrast, 20E needs its receptor to arrest proliferation. Dose-response RNA sequencing (RNA-seq) analysis of ex vivo cultured wing precursors identifies genes that are quantitatively activated by 20E across the physiological range, likely comprising positive modulators of proliferation and other genes that are only activated at high doses. We suggest that some of these "high-threshold" genes dominantly suppress the activity of the pro-proliferation genes. We then show mathematically and with synthetic reporters that combinations of basic regulatory elements can recapitulate the behavior of both types of target genes. Thus, a relatively simple genetic circuit can account for the bimodal activity of this hormone.

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.

Transcription factors perform a 2-step search of the nucleus

Transcription factors regulate gene expression by binding to regulatory DNA and recruiting regulatory protein complexes. The DNA-binding and protein-binding functions of transcription factors are traditionally described as independent functions performed by modular protein domains. Here, I argue that genome binding can be a 2-part process with both DNA-binding and protein-binding steps, enabling transcription factors to perform a 2-step search of the nucleus to find their appropriate binding sites in a eukaryotic genome. I support this hypothesis with new and old results in the literature, discuss how this hypothesis parsimoniously resolves outstanding problems, and present testable predictions.

Concentration-dependent transcriptional switching through a collective action of cis-elements

Gene expression specificity of homeobox transcription factors has remained paradoxical. WUSCHEL activates and represses CLAVATA3 transcription at lower and higher concentrations, respectively. We use computational modeling and experimental analysis to investigate the properties of the cis-regulatory module. We find that intrinsically each cis-element can only activate CLAVATA3 at a higher WUSCHEL concentration. However, together, they repress CLAVATA3 at higher WUSCHEL and activate only at lower WUSCHEL, showing that the concentration-dependent interactions among cis-elements regulate both activation and repression. Biochemical experiments show that two adjacent functional cis-elements bind WUSCHEL with higher affinity and dimerize at relatively lower levels. Moreover, increasing the distance between cis-elements prolongs WUSCHEL monomer binding window, resulting in higher CLAVATA3 activation. Our work showing a constellation of optimally spaced cis-elements of defined affinities determining activation and repression thresholds in regulating CLAVATA3 transcription provides a previously unknown mechanism of cofactor-independent regulation of transcription factor binding in mediating gene expression specificity.

Thermodynamic Modelling of Transcriptional Control: A Sensitivity Analysis

Modelling is a tool used to decipher the biochemical mechanisms involved in transcriptional control. Experimental evidence in genetics is usually supported by theoretical models in order to evaluate the effects of all the possible interactions that can occur in these complicated processes. Models derived from the thermodynamic method are critical in this labour because they are able to take into account multiple mechanisms operating simultaneously at the molecular micro-scale and relate them to transcriptional initiation at the tissular macro-scale. This work is devoted to adapting computational techniques to this context in order to theoretically evaluate the role played by several biochemical mechanisms. The interest of this theoretical analysis relies on the fact that it can be contrasted against those biological experiments where the response to perturbations in the transcriptional machinery environment is evaluated in terms of genetically activated/repressed regions. The theoretical reproduction of these experiments leads to a sensitivity analysis whose results are expressed in terms of the elasticity of a threshold function determining those activated/repressed regions. The study of this elasticity function in thermodynamic models already proposed in the literature reveals that certain modelling approaches can alter the balance between the biochemical mechanisms considered, and this can cause false/misleading outcomes. The reevaluation of classical thermodynamic models gives us a more accurate and complete picture of the interactions involved in gene regulation and transcriptional control, which enables more specific predictions. This sensitivity approach provides a definite advantage in the interpretation of a wide range of genetic experimental results.

Identifying the critical states of complex diseases by the dynamic change of multivariate distribution

The dynamics of complex diseases are not always smooth; they are occasionally abrupt, i.e. there is a critical state transition or tipping point at which the disease undergoes a sudden qualitative shift. There are generally a few significant differences in the critical state in terms of gene expressions or other static measurements, which may lead to the failure of traditional differential expression-based biomarkers to identify such a tipping point. In this study, we propose a computational method, the direct interaction network-based divergence, to detect the critical state of complex diseases by exploiting the dynamic changes in multivariable distributions inferred from observable samples and local biomolecular direct interaction networks. Such a method is model-free and applicable to both bulk and single-cell expression data. Our approach was validated by successfully identifying the tipping point just before the occurrence of a critical transition for both a simulated data set and seven real data sets, including those from The Cancer Genome Atlas and two single-cell RNA-sequencing data sets of cell differentiation. Functional and pathway enrichment analyses also validated the computational results from the perspectives of both molecules and networks.

MAVE-NN: learning genotype-phenotype maps from multiplex assays of variant effect

Multiplex assays of variant effect (MAVEs) are a family of methods that includes deep mutational scanning experiments on proteins and massively parallel reporter assays on gene regulatory sequences. Despite their increasing popularity, a general strategy for inferring quantitative models of genotype-phenotype maps from MAVE data is lacking. Here we introduce MAVE-NN, a neural-network-based Python package that implements a broadly applicable information-theoretic framework for learning genotype-phenotype maps-including biophysically interpretable models-from MAVE datasets. We demonstrate MAVE-NN in multiple biological contexts, and highlight the ability of our approach to deconvolve mutational effects from otherwise confounding experimental nonlinearities and noise.

Reconciling kinetic and thermodynamic models of bacterial transcription

The study of transcription remains one of the centerpieces of modern biology with implications in settings from development to metabolism to evolution to disease. Precision measurements using a host of different techniques including fluorescence and sequencing readouts have raised the bar for what it means to quantitatively understand transcriptional regulation. In particular our understanding of the simplest genetic circuit is sufficiently refined both experimentally and theoretically that it has become possible to carefully discriminate between different conceptual pictures of how this regulatory system works. This regulatory motif, originally posited by Jacob and Monod in the 1960s, consists of a single transcriptional repressor binding to a promoter site and inhibiting transcription. In this paper, we show how seven distinct models of this so-called simple-repression motif, based both on thermodynamic and kinetic thinking, can be used to derive the predicted levels of gene expression and shed light on the often surprising past success of the thermodynamic models. These different models are then invoked to confront a variety of different data on mean, variance and full gene expression distributions, illustrating the extent to which such models can and cannot be distinguished, and suggesting a two-state model with a distribution of burst sizes as the most potent of the seven for describing the simple-repression motif.

A theoretical model of neural maturation in the developing chick spinal cord

Cellular differentiation is a tightly regulated process under the control of intricate signaling and transcription factors interaction network working in coordination. These interactions make the systems dynamic, robust and stable but also difficult to dissect. In the spinal cord, recent work has shown that a network of FGF, WNT and Retinoic Acid (RA) signaling factors regulate neural maturation by directing the activity of a transcription factor network that contains CDX at its core. Here we have used partial and ordinary (Hill) differential equation based models to understand the spatiotemporal dynamics of the FGF/WNT/RA and the CDX/transcription factor networks, alone and in combination. We show that in both networks, the strength of interaction among network partners impacts the dynamics, behavior and output of the system. In the signaling network, interaction strength determine the position and size of discrete regions of cell differentiation and small changes in the strength of the interactions among networking partners can result in a signal overriding, balancing or oscillating with another signal. We also show that the spatiotemporal information generated by the signaling network can be conveyed to the CDX/transcription network to produces a transition zone that separates regions of high cell potency from regions of cell differentiation, in agreement with most in vivo observations. Importantly, one emerging property of the networks is their robustness to extrinsic disturbances, which allows the system to retain or canalize NP cells in developmental trajectories. This analysis provides a model for the interaction conditions underlying spinal cord cell maturation during embryonic axial elongation.

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.

Gene regulation gravitates toward either addition or multiplication when combining the effects of two signals

Two different cell signals often affect transcription of the same gene. In such cases, it is natural to ask how the combined transcriptional response compares to the individual responses. The most commonly used mechanistic models predict additive or multiplicative combined responses, but a systematic genome-wide evaluation of these predictions is not available. Here, we analyzed the transcriptional response of human MCF-7 cells to retinoic acid and TGF-β, applied individually and in combination. The combined transcriptional responses of induced genes exhibited a range of behaviors, but clearly favored both additive and multiplicative outcomes. We performed paired chromatin accessibility measurements and found that increases in accessibility were largely additive. There was some association between super-additivity of accessibility and multiplicative or super-multiplicative combined transcriptional responses, while sub-additivity of accessibility associated with additive transcriptional responses. Our findings suggest that mechanistic models of combined transcriptional regulation must be able to reproduce a range of behaviors.

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.

Repressive interactions in gene regulatory networks: When you have no other choice

Tightly regulated gene expression programs, orchestrated by complex interactions between transcription factors, control cell type specification during development. Repressive interactions play a critical role in these networks, facilitating decision-making between two or more alternative cell fates. Here, we use the ventral neural tube as an example to illustrate how cross repressive interactions within a network drive pattern formation and specify cell types in response to a graded patterning signal. This and other systems serve to highlight how external signals are integrated through the cis regulatory elements controlling key genes and provide insight into the molecular underpinning of the process. Even the simplest networks can lead to counterintuitive results and we argue that a combination of experimental dissection and modeling approaches will be necessary to fully understand network behavior and the underlying design principles. Studying these gene regulatory networks as a whole ultimately allows us to extract fundamental properties applicable across systems that can expand our mechanistic understanding of how organisms develop.

Figure 1 Theory Meets Figure 2 Experiments in the Study of Gene Expression

It is tempting to believe that we now own the genome. The ability to read and rewrite it at will has ushered in a stunning period in the history of science. Nonetheless, there is an Achilles' heel exposed by all of the genomic data that has accrued: We still do not know how to interpret them. Many genes are subject to sophisticated programs of transcriptional regulation, mediated by DNA sequences that harbor binding sites for transcription factors, which can up- or down-regulate gene expression depending upon environmental conditions. This gives rise to an input-output function describing how the level of expression depends upon the parameters of the regulated gene-for instance, on the number and type of binding sites in its regulatory sequence. In recent years, the ability to make precision measurements of expression, coupled with the ability to make increasingly sophisticated theoretical predictions, has enabled an explicit dialogue between theory and experiment that holds the promise of covering this genomic Achilles' heel. The goal is to reach a predictive understanding of transcriptional regulation that makes it possible to calculate gene expression levels from DNA regulatory sequence. This review focuses on the canonical simple repression motif to ask how well the models that have been used to characterize it actually work. We consider a hierarchy of increasingly sophisticated experiments in which the minimal parameter set learned at one level is applied to make quantitative predictions at the next. We show that these careful quantitative dissections provide a template for a predictive understanding of the many more complex regulatory arrangements found across all domains of life.

Identifying critical state of complex diseases by single-sample Kullback-Leibler divergence

Background

Developing effective strategies for signaling the pre-disease state of complex diseases, a state with high susceptibility before the disease onset or deterioration, is urgently needed because such state usually followed by a catastrophic transition into a worse stage of disease. However, it is a challenging task to identify such pre-disease state or tipping point in clinics, where only one single sample is available and thus results in the failure of most statistic approaches.

Methods

In this study, we presented a single-sample-based computational method to detect the early-warning signal of critical transition during the progression of complex diseases. Specifically, given a set of reference samples which were regarded as background, a novel index called single-sample Kullback–Leibler divergence (sKLD), was proposed to explore and quantify the disturbance on the background caused by a case sample. The pre-disease state is then signaled by the significant change of sKLD.

Results

The novel algorithm was developed and applied to both numerical simulation and real datasets, including lung squamous cell carcinoma, lung adenocarcinoma, stomach adenocarcinoma, thyroid carcinoma, colon adenocarcinoma, and acute lung injury. The successful identification of pre-disease states and the corresponding dynamical network biomarkers for all six datasets validated the effectiveness and accuracy of our method.

Conclusions

The proposed method effectively explores and quantifies the disturbance on the background caused by a case sample, and thus characterizes the criticality of a biological system. Our method not only identifies the critical state or tipping point at a single sample level, but also provides the sKLD-signaling markers for further practical application. It is therefore of great potential in personalized pre-disease diagnosis.

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.

Single-sample landscape entropy reveals the imminent phase transition during disease progression

Motivation

The time evolution or dynamic change of many biological systems during disease progression is not always smooth but occasionally abrupt, that is, there is a tipping point during such a process at which the system state shifts from the normal state to a disease state. It is challenging to predict such disease state with the measured omics data, in particular when only a single sample is available.

Results

In this study, we developed a novel approach, i.e. single-sample landscape entropy (SLE) method, to identify the tipping point during disease progression with only one sample data. Specifically, by evaluating the disorder of a network projected from a single-sample data, SLE effectively characterizes the criticality of this single sample network in terms of network entropy, thereby capturing not only the signals of the impending transition but also its leading network, i.e. dynamic network biomarkers. Using this method, we can characterize sample-specific state during disease progression and thus achieve the disease prediction of each individual by only one sample. Our method was validated by successfully identifying the tipping points just before the serious disease symptoms from four real datasets of individuals or subjects, including influenza virus infection, lung cancer metastasis, prostate cancer and acute lung injury.

Availability and implementation

https://github.com/rabbitpei/SLE.

Supplementary information

Supplementary data are available at Bioinformatics online.

Massively Parallel Assays and Quantitative Sequence-Function Relationships

Over the last decade, a rich variety of massively parallel assays have revolutionized our understanding of how biological sequences encode quantitative molecular phenotypes. These assays include deep mutational scanning, high-throughput SELEX, and massively parallel reporter assays. Here, we review these experimental methods and how the data they produce can be used to quantitatively model sequence-function relationships. In doing so, we touch on a diverse range of topics, including the identification of clinically relevant genomic variants, the modeling of transcription factor binding to DNA, the functional and evolutionary landscapes of proteins, and cis-regulatory mechanisms in both transcription and mRNA splicing. We further describe a unified conceptual framework and a core set of mathematical modeling strategies that studies in these diverse areas can make use of. Finally, we highlight key aspects of experimental design and mathematical modeling that are important for the results of such studies to be interpretable and reproducible.

Negative reciprocity, not ordered assembly, underlies the interaction of Sox2 and Oct4 on DNA

The mode of interaction of transcription factors (TFs) on eukaryotic genomes remains a matter of debate. Single-molecule data in living cells for the TFs Sox2 and Oct4 were previously interpreted as evidence of ordered assembly on DNA. However, the quantity that was calculated does not determine binding order but, rather, energy expenditure away from thermodynamic equilibrium. Here, we undertake a rigorous biophysical analysis which leads to the concept of reciprocity. The single-molecule data imply that Sox2 and Oct4 exhibit negative reciprocity, with expression of Sox2 increasing Oct4’s genomic binding but expression of Oct4 decreasing Sox2’s binding. Models show that negative reciprocity can arise either from energy expenditure or from a mixture of positive and negative cooperativity at distinct genomic loci. Both possibilities imply unexpected complexity in how TFs interact on DNA, for which single-molecule methods provide novel detection capabilities.

The bodies of humans and other animals contain many types of cells that perform different roles in the body. Most cells in the body carry the same DNA, which is arranged into sections known as genes. The marked differences between cell types arise because different sets of genes are switched on or ‘expressed’.

Proteins called transcription factors control which genes are expressed by binding to DNA and recruiting groups of accessory proteins. However, it is not clear how they interact with each other and with the accessory proteins to decide whether to express a gene. For instance, it is thought that some accessory proteins may provide energy for this process, but it is unknown whether the energy is used continuously or only for a short time. Insights from physics suggest that the former could have more powerful effects.

In 2014, a team of researchers reported using a microscopy approach, known as single-molecule imaging, to follow two transcription factors called Sox2 and Oct4 in cells from mice. After analyzing the data, the researchers concluded that Sox2 and Oct4 had a specific order of binding to DNA, with Sox2 often binding first and then assisting Oct4 to bind. Now Biddle et al. report that the claim made in the 2014 study is unsupported because of errors in the way the data were analyzed. In particular, Biddle et al. argue that what the earlier study actually calculated is not an order of binding but a measure of whether energy is being continuously used when Sox2 and Oct4 bind to DNA.

Biddle et al. reanalyzed the data from the 2014 work and concluded that Sox2 increases the extent of Oct4 binding to DNA, while Oct4 decreases the amount of Sox2 binding to DNA. Mathematical models suggest this may be due to the continuous use of energy as the two proteins bind to DNA. Alternatively, it could also happen if Sox2 and Oct4 helped each other to bind at some sites on DNA and hindered each other from binding in other places, even if energy is only used for a short time. These findings reveal that there is unexpected complexity in how transcription factors bind to DNA.

The next step following on from this work is to carry out experiments that test the two possible explanations for how Sox2 and Oct4 interact on DNA. Including physics in the analysis may help describe more accurately the biology of how transcription factors determine gene expression. Understanding this process will shed new light on many important biological questions and may aid the development of gene therapy and other new medical techniques.

Measuring cis-regulatory energetics in living cells using allelic manifolds

Gene expression in all organisms is controlled by cooperative interactions between DNA-bound transcription factors (TFs), but quantitatively measuring TF-DNA and TF-TF interactions remains difficult. Here we introduce a strategy for precisely measuring the Gibbs free energy of such interactions in living cells. This strategy centers on the measurement and modeling of ‘allelic manifolds’, a multidimensional generalization of the classical genetics concept of allelic series. Allelic manifolds are measured using reporter assays performed on strategically designed cis-regulatory sequences. Quantitative biophysical models are then fit to the resulting data. We used this strategy to study regulation by two Escherichia coli TFs, CRP and σ70 RNA polymerase. Doing so, we consistently obtained energetic measurements precise to ∼0.1 kcal/mol. We also obtained multiple results that deviate from the prior literature. Our strategy is compatible with massively parallel reporter assays in both prokaryotes and eukaryotes, and should therefore be highly scalable and broadly applicable.

Editorial note: This article has been through an editorial process in which the authors decide how to respond to the issues raised during peer review. The Reviewing Editor's assessment is that minor issues remain unresolved (see decision letter).

Memory functions reveal structural properties of gene regulatory networks

Gene regulatory networks (GRNs) control cellular function and decision making during tissue development and homeostasis. Mathematical tools based on dynamical systems theory are often used to model these networks, but the size and complexity of these models mean that their behaviour is not always intuitive and the underlying mechanisms can be difficult to decipher. For this reason, methods that simplify and aid exploration of complex networks are necessary. To this end we develop a broadly applicable form of the Zwanzig-Mori projection. By first converting a thermodynamic state ensemble model of gene regulation into mass action reactions we derive a general method that produces a set of time evolution equations for a subset of components of a network. The influence of the rest of the network, the bulk, is captured by memory functions that describe how the subnetwork reacts to its own past state via components in the bulk. These memory functions provide probes of near-steady state dynamics, revealing information not easily accessible otherwise. We illustrate the method on a simple cross-repressive transcriptional motif to show that memory functions not only simplify the analysis of the subnetwork but also have a natural interpretation. We then apply the approach to a GRN from the vertebrate neural tube, a well characterised developmental transcriptional network composed of four interacting transcription factors. The memory functions reveal the function of specific links within the neural tube network and identify features of the regulatory structure that specifically increase the robustness of the network to initial conditions. Taken together, the study provides evidence that Zwanzig-Mori projections offer powerful and effective tools for simplifying and exploring the behaviour of GRNs.

Gene regulatory networks are essential for cell fate specification and function. But the recursive links that comprise these networks often make determining their properties and behaviour complicated. Computational models of these networks can also be difficult to decipher. To reduce the complexity of such models we employ a Zwanzig-Mori projection approach. This allows a system of ordinary differential equations, representing a network, to be reduced to an arbitrary subnetwork consisting of part of the initial network, with the rest of the network (bulk) captured by memory functions. These memory functions account for the bulk by describing signals that return to the subnetwork after some time, having passed through the bulk. We show how this approach can be used to simplify analysis and to probe the behaviour of a gene regulatory network. Applying the method to a transcriptional network in the vertebrate neural tube reveals previously unappreciated properties of the network. By taking advantage of the structure of the memory functions we identify interactions within the network that are unnecessary for sustaining correct patterning. Upon further investigation we find that these interactions are important for conferring robustness to variation in initial conditions. Taken together we demonstrate the validity and applicability of the Zwanzig-Mori projection approach to gene regulatory networks.

Systems biology of embryonic development: Prospects for a complete understanding of the Caenorhabditis elegans embryo

The convergence of developmental biology and modern genomics tools brings the potential for a comprehensive understanding of developmental systems. This is especially true for the Caenorhabditis elegans embryo because its small size, invariant developmental lineage, and powerful genetic and genomic tools provide the prospect of a cellular resolution understanding of messenger RNA (mRNA) expression and regulation across the organism. We describe here how a systems biology framework might allow large-scale determination of the embryonic regulatory relationships encoded in the C. elegans genome. This framework consists of two broad steps: (a) defining the "parts list"-all genes expressed in all cells at each time during development and (b) iterative steps of computational modeling and refinement of these models by experimental perturbation. Substantial progress has been made towards defining the parts list through imaging methods such as large-scale green fluorescent protein (GFP) reporter analysis. Imaging results are now being augmented by high-resolution transcriptome methods such as single-cell RNA sequencing, and it is likely the complete expression patterns of all genes across the embryo will be known within the next few years. In contrast, the modeling and perturbation experiments performed so far have focused largely on individual cell types or genes, and improved methods will be needed to expand them to the full genome and organism. This emerging comprehensive map of embryonic expression and regulatory function will provide a powerful resource for developmental biologists, and would also allow scientists to ask questions not accessible without a comprehensive picture. This article is categorized under: Invertebrate Organogenesis > Worms Technologies > Analysis of the Transcriptome Gene Expression and Transcriptional Hierarchies > Gene Networks and Genomics.

PTRE-seq reveals mechanism and interactions of RNA binding proteins and miRNAs

RNA binding proteins (RBP) and microRNAs (miRNAs) often bind sequences in 3' untranslated regions (UTRs) of mRNAs, and regulate stability and translation efficiency. With the identification of numerous RBPs and miRNAs, there is an urgent need for new technologies to dissect the function of the cis-acting elements of RBPs and miRNAs. We describe post-transcriptional regulatory element sequencing (PTRE-seq), a massively parallel method for assaying the target sequences of miRNAs and RBPs. We use PTRE-seq to dissect sequence preferences and interactions between miRNAs and RBPs. The binding sites for these effector molecules influenced different aspects of the RNA lifecycle: RNA stability, translation efficiency, and translation initiation. In some cases, post-transcriptional control is modular, with different factors acting independently of each other, while in other cases factors show specific epistatic interactions. The throughput, flexibility, and reproducibility of PTRE-seq make it a valuable tool to study post-transcriptional regulation by 3'UTR elements.

Evolution of new regulatory functions on biophysically realistic fitness landscapes

Gene expression is controlled by networks of regulatory proteins that interact specifically with external signals and DNA regulatory sequences. These interactions force the network components to co-evolve so as to continually maintain function. Yet, existing models of evolution mostly focus on isolated genetic elements. In contrast, we study the essential process by which regulatory networks grow: the duplication and subsequent specialization of network components. We synthesize a biophysical model of molecular interactions with the evolutionary framework to find the conditions and pathways by which new regulatory functions emerge. We show that specialization of new network components is usually slow, but can be drastically accelerated in the presence of regulatory crosstalk and mutations that promote promiscuous interactions between network components.Gene networks evolve by transcription factor (TF) duplication and divergence of their binding site specificities, but little is known about the global constraints at play. Here, the authors study the coevolution of TFs and binding sites using a biophysical-evolutionary approach, and show that the emerging complex fitness landscapes strongly influence regulatory evolution with a role for crosstalk.

A Gene Regulatory Network Balances Neural and Mesoderm Specification during Vertebrate Trunk Development

Transcriptional networks, regulated by extracellular signals, control cell fate decisions and determine the size and composition of developing tissues. One example is the network controlling bipotent neuromesodermal progenitors (NMPs) that fuel embryo elongation by generating spinal cord and trunk mesoderm tissue. Here, we use single-cell transcriptomics to identify the molecular signature of NMPs and reverse engineer the mechanism that regulates their differentiation. Together with genetic perturbations, this reveals a transcriptional network that integrates opposing retinoic acid (RA) and Wnt signals to determine the rate at which cells enter and exit the NMP state. RA, produced by newly generated mesodermal cells, provides feedback that initiates NMP generation and induces neural differentiation, thereby coordinating the production of neural and mesodermal tissue. Together, the data define a regulatory network architecture that balances the generation of different cell types from bipotential progenitors in order to facilitate orderly axis elongation.

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Single-cell RNA-seq reveals a signature of neuromesodermal progenitors

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In vitro NMPs resemble and differentiate similar to their in vivo counterparts

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Dual role for retinoic acid signaling in NMP induction and neural differentiation

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A transcriptional network regulates neural versus mesodermal allocation

Information Integration and Energy Expenditure in Gene Regulation

The quantitative concepts used to reason about gene regulation largely derive from bacterial studies. We show that this bacterial paradigm cannot explain the sharp expression of a canonical developmental gene in response to a regulating transcription factor (TF). In the absence of energy expenditure, with regulatory DNA at thermodynamic equilibrium, information integration across multiple TF binding sites can generate the required sharpness, but with strong constraints on the resultant "higher-order cooperativities." Even with such integration, there is a "Hopfield barrier" to sharpness; for n TF binding sites, this barrier is represented by the Hill function with the Hill coefficient n. If, however, energy is expended to maintain regulatory DNA away from thermodynamic equilibrium, as in kinetic proofreading, this barrier can be breached and greater sharpness achieved. Our approach is grounded in fundamental physics, leads to testable experimental predictions, and suggests how a quantitative paradigm for eukaryotic gene regulation can be formulated.

Combinatorial Gene Regulation through Kinetic Control of the Transcription Cycle

Cells decide when, where, and to what level to express their genes by "computing" information from transcription factors (TFs) binding to regulatory DNA. How is the information contained in multiple TF-binding sites integrated to dictate the rate of transcription? The dominant conceptual and quantitative model is that TFs combinatorially recruit one another and RNA polymerase to the promoter by direct physical interactions. Here, we develop a quantitative framework to explore kinetic control, an alternative model in which combinatorial gene regulation can result from TFs working on different kinetic steps of the transcription cycle. Kinetic control can generate a wide range of analog and Boolean computations without requiring the input TFs to be simultaneously bound to regulatory DNA. We propose experiments that will illuminate the role of kinetic control in transcription and discuss implications for deciphering the cis-regulatory "code."

Intrinsic Noise Profoundly Alters the Dynamics and Steady State of Morphogen-Controlled Bistable Genetic Switches

During tissue development, patterns of gene expression determine the spatial arrangement of cell types. In many cases, gradients of secreted signalling molecules—morphogens—guide this process by controlling downstream transcriptional networks. A mechanism commonly used in these networks to convert the continuous information provided by the gradient into discrete transitions between adjacent cell types is the genetic toggle switch, composed of cross-repressing transcriptional determinants. Previous analyses have emphasised the steady state output of these mechanisms. Here, we explore the dynamics of the toggle switch and use exact numerical simulations of the kinetic reactions, the corresponding Chemical Langevin Equation, and Minimum Action Path theory to establish a framework for studying the effect of gene expression noise on patterning time and boundary position. This provides insight into the time scale, gene expression trajectories and directionality of stochastic switching events between cell states. Taking gene expression noise into account predicts that the final boundary position of a morphogen-induced toggle switch, although robust to changes in the details of the noise, is distinct from that of the deterministic system. Moreover, the dramatic increase in patterning time close to the boundary predicted from the deterministic case is substantially reduced. The resulting stochastic switching introduces differences in patterning time along the morphogen gradient that result in a patterning wave propagating away from the morphogen source with a velocity determined by the intrinsic noise. The wave sharpens and slows as it advances and may never reach steady state in a biologically relevant time. This could explain experimentally observed dynamics of pattern formation. Together the analysis reveals the importance of dynamical transients for understanding morphogen-driven transcriptional networks and indicates that gene expression noise can qualitatively alter developmental patterning.

The bistable switch, a common regulatory sub-network, is found in many biological processes. It consists of cross-repressing components that generate a switch-like transition between two possible states. In developing tissues, bistable switches, created by cross-repressing transcriptional determinants, are often controlled by gradients of secreted signalling molecules—morphogens. These provide a mechanism to convert a morphogen gradient into stripes of gene expression that determine the arrangement of distinct cell types. Here we use mathematical models to analyse the temporal response of such a system. We find that the behaviour is highly dependent on the intrinsic fluctuations that result from the stochastic nature of gene expression. This noise has a marked effect on both patterning time and the location of the stripe boundary. One of the techniques we use, Minimum Action Path theory, identifies key features of the switch without computationally expensive calculations. The results reveal a noise driven switching wave that propels the stripe boundary away from the morphogen source to eventually settle, at steady state, further from the morphogen source than in the deterministic description. Together the analysis highlights the importance dynamics in patterning and demonstrates a set of mathematical tools for studying this problem.

Intrinsic limits to gene regulation by global crosstalk

Gene regulation relies on the specificity of transcription factor (TF)–DNA interactions. Limited specificity may lead to crosstalk: a regulatory state in which a gene is either incorrectly activated due to noncognate TF–DNA interactions or remains erroneously inactive. As each TF can have numerous interactions with noncognate cis-regulatory elements, crosstalk is inherently a global problem, yet has previously not been studied as such. We construct a theoretical framework to analyse the effects of global crosstalk on gene regulation. We find that crosstalk presents a significant challenge for organisms with low-specificity TFs, such as metazoans. Crosstalk is not easily mitigated by known regulatory schemes acting at equilibrium, including variants of cooperativity and combinatorial regulation. Our results suggest that crosstalk imposes a previously unexplored global constraint on the functioning and evolution of regulatory networks, which is qualitatively distinct from the known constraints that act at the level of individual gene regulatory elements.

Limited specificity of transcription factor-DNA interactions leads to crosstalk in gene regulation. Here the authors consider global crosstalk in regulatory networks of growing size and complexity, and show that it imposes constraints on gene regulation and on the evolution of regulatory networks.

Protein Ensembles: How Does Nature Harness Thermodynamic Fluctuations for Life? The Diverse Functional Roles of Conformational Ensembles in the Cell

All soluble proteins populate conformational ensembles that together constitute the native state. Their fluctuations in water are intrinsic thermodynamic phenomena, and the distributions of the states on the energy landscape are determined by statistical thermodynamics; however, they are optimized to perform their biological functions. In this review we briefly describe advances in free energy landscape studies of protein conformational ensembles. Experimental (nuclear magnetic resonance, small-angle X-ray scattering, single-molecule spectroscopy, and cryo-electron microscopy) and computational (replica-exchange molecular dynamics, metadynamics, and Markov state models) approaches have made great progress in recent years. These address the challenging characterization of the highly flexible and heterogeneous protein ensembles. We focus on structural aspects of protein conformational distributions, from collective motions of single- and multi-domain proteins, intrinsically disordered proteins, to multiprotein complexes. Importantly, we highlight recent studies that illustrate functional adjustment of protein conformational ensembles in the crowded cellular environment. We center on the role of the ensemble in recognition of small- and macro-molecules (protein and RNA/DNA) and emphasize emerging concepts of protein dynamics in enzyme catalysis. Overall, protein ensembles link fundamental physicochemical principles and protein behavior and the cellular network and its regulation.

Interactions between pluripotency factors specify cis-regulation in embryonic stem cells

We investigated how interactions between pluripotency transcription factors (TFs) affect cis-regulation. We created hundreds of synthetic cis-regulatory elements (CREs) comprised of combinations of binding sites for pluripotency TFs and measured their expression in mouse embryonic stem (ES) cells. A thermodynamic model that incorporates interactions between TFs explains a large portion (72%) of the variance in expression of these CREs. These interactions include three favorable heterotypic interactions between TFs. The model also predicts an unfavorable homotypic interaction between TFs, helping to explain the observation that homotypic chains of binding sites express at low levels. We further investigated the expression driven by CREs comprised of homotypic chains of KLF4 binding sites. Our results suggest that KLF homologs make unique contributions to regulation by these CREs. We conclude that a specific set of interactions between pluripotency TFs plays a large role in setting the levels of expression driven by CREs in ES cells.

Robustness to noise in gene expression evolves despite epistatic constraints in a model of gene networks

Stochastic noise in gene expression causes variation in the development of phenotypes, making such noise a potential target of stabilizing selection. Here, we develop a new simulation model of gene networks to study the adaptive landscape underlying the evolution of robustness to noise. We find that epistatic interactions between the determinants of the expression of a gene and its downstream effect impose significant constraints on evolution, but these interactions do allow the gradual evolution of increased robustness. Despite strong sign epistasis, adaptation rarely proceeds via deleterious intermediate steps, but instead occurs primarily through small beneficial mutations. A simple mathematical model captures the relevant features of the single-gene fitness landscape and explains counterintuitive patterns, such as a correlation between the mean and standard deviation of phenotypes. In more complex networks, mutations in regulatory regions provide evolutionary pathways to increased robustness. These results chart the constraints and possibilities of adaptation to reduce expression noise and demonstrate the potential of a novel modeling framework for gene networks.

Signal integration by the CYP1A1 promoter--a quantitative study

Genes involved in detoxification of foreign compounds exhibit complex spatiotemporal expression patterns in liver. Cytochrome P450 1A1 (CYP1A1), for example, is restricted to the pericentral region of liver lobules in response to the interplay between aryl hydrocarbon receptor (AhR) and Wnt/β-catenin signaling pathways. However, the mechanisms by which the two pathways orchestrate gene expression are still poorly understood. With the help of 29 mutant constructs of the human CYP1A1 promoter and a mathematical model that combines Wnt/β-catenin and AhR signaling with the statistical mechanics of the promoter, we systematically quantified the regulatory influence of different transcription factor binding sites on gene induction within the promoter. The model unveils how different binding sites cooperate and how they establish the promoter logic; it quantitatively predicts two-dimensional stimulus-response curves. Furthermore, it shows that crosstalk between Wnt/β-catenin and AhR signaling is crucial to understand the complex zonated expression patterns found in liver lobules. This study exemplifies how statistical mechanical modeling together with combinatorial reporter assays has the capacity to disentangle the promoter logic that establishes physiological gene expression patterns.

Ptch1 and Gli regulate Shh signalling dynamics via multiple mechanisms

In the vertebrate neural tube, the morphogen Sonic Hedgehog (Shh) establishes a characteristic pattern of gene expression. Here we quantify the Shh gradient in the developing mouse neural tube and show that while the amplitude of the gradient increases over time, the activity of the pathway transcriptional effectors, Gli proteins, initially increases but later decreases. Computational analysis of the pathway suggests three mechanisms that could contribute to this adaptation: transcriptional upregulation of the inhibitory receptor Ptch1, transcriptional downregulation of Gli and the differential stability of active and inactive Gli isoforms. Consistent with this, Gli2 protein expression is downregulated during neural tube patterning and adaptation continues when the pathway is stimulated downstream of Ptch1. Moreover, the Shh-induced upregulation of Gli2 transcription prevents Gli activity levels from adapting in a different cell type, NIH3T3 fibroblasts, despite the upregulation of Ptch1. Multiple mechanisms therefore contribute to the intracellular dynamics of Shh signalling, resulting in different signalling dynamics in different cell types.

Napoleon Is in Equilibrium

It has been said that the cell is the test tube of the twenty-first century. If so, the theoretical tools needed to quantitatively and predictively describe what goes on in such test tubes lag sorely behind the stunning experimental advances in biology seen in the decades since the molecular biology revolution began. Perhaps surprisingly, one of the theoretical tools that has been used with great success on problems ranging from how cells communicate with their environment and each other to the nature of the organization of proteins and lipids within the cell membrane is statistical mechanics. A knee-jerk reaction to the use of statistical mechanics in the description of cellular processes is that living organisms are so far from equilibrium that one has no business even thinking about it. But such reactions are probably too hasty given that there are many regimes in which, because of a separation of timescales, for example, such an approach can be a useful first step. In this article, we explore the power of statistical mechanical thinking in the biological setting, with special emphasis on cell signaling and regulation. We show how such models are used to make predictions and describe some recent experiments designed to test them. We also consider the limits of such models based on the relative timescales of the processes of interest.

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.

The influence of promoter architectures and regulatory motifs on gene expression in Escherichia coli

The ability to regulate gene expression is of central importance for the adaptability of living organisms to changes in their external and internal environment. At the transcriptional level, binding of transcription factors (TFs) in the promoter region can modulate the transcription rate, hence making TFs central players in gene regulation. For some model organisms, information about the locations and identities of discovered TF binding sites have been collected in continually updated databases, such as RegulonDB for the well-studied case of E. coli. In order to reveal the general principles behind the binding-site arrangement and function of these regulatory architectures we propose a random promoter architecture model that preserves the overall abundance of binding sites to identify overrepresented binding site configurations. This model is analogous to the random network model used in the study of genetic network motifs, where regulatory motifs are identified through their overrepresentation with respect to a “randomly connected” genetic network. Using our model we identify TF pairs which coregulate operons in an overrepresented fashion, or individual TFs which act at multiple binding sites per promoter by, for example, cooperative binding, DNA looping, or through multiple binding domains. We furthermore explore the relationship between promoter architecture and gene expression, using three different genome-wide protein copy number censuses. Perhaps surprisingly, we find no systematic correlation between the number of activator and repressor binding sites regulating a gene and the level of gene expression. A position-weight-matrix model used to estimate the binding affinity of RNA polymerase (RNAP) to the promoters of activated and repressed genes suggests that this lack of correlation might in part be due to differences in basal transcription levels, with repressed genes having a higher basal activity level. This quantitative catalogue relating promoter architecture and function provides a first step towards genome-wide predictive models of regulatory function.

A theoretical framework for the regulation of Shh morphogen-controlled gene expression

How morphogen gradients govern the pattern of gene expression in developing tissues is not well understood. Here, we describe a statistical thermodynamic model of gene regulation that combines the activity of a morphogen with the transcriptional network it controls. Using Sonic hedgehog (Shh) patterning of the ventral neural tube as an example, we show that the framework can be used together with the principled parameter selection technique of approximate Bayesian computation to obtain a dynamical model that accurately predicts tissue patterning. The analysis indicates that, for each target gene regulated by Gli, which is the transcriptional effector of Shh signalling, there is a neutral point in the gradient, either side of which altering the Gli binding affinity has opposite effects on gene expression. This explains recent counterintuitive experimental observations. The approach is broadly applicable and provides a unifying framework to explain the temporospatial pattern of morphogen-regulated gene expression.

A framework for modelling gene regulation which accommodates non-equilibrium mechanisms

BACKGROUND

Gene regulation has, for the most part, been quantitatively analysed by assuming that regulatory mechanisms operate at thermodynamic equilibrium. This formalism was originally developed to analyse the binding and unbinding of transcription factors from naked DNA in eubacteria. Although widely used, it has made it difficult to understand the role of energy-dissipating, epigenetic mechanisms, such as DNA methylation, nucleosome remodelling and post-translational modification of histones and co-regulators, which act together with transcription factors to regulate gene expression in eukaryotes.

RESULTS

Here, we introduce a graph-based framework that can accommodate non-equilibrium mechanisms. A gene-regulatory system is described as a graph, which specifies the DNA microstates (vertices), the transitions between microstates (edges) and the transition rates (edge labels). The graph yields a stochastic master equation for how microstate probabilities change over time. We show that this framework has broad scope by providing new insights into three very different ad hoc models, of steroid-hormone responsive genes, of inherently bounded chromatin domains and of the yeast PHO5 gene. We find, moreover, surprising complexity in the regulation of PHO5, which has not yet been experimentally explored, and we show that this complexity is an inherent feature of being away from equilibrium. At equilibrium, microstate probabilities do not depend on how a microstate is reached but, away from equilibrium, each path to a microstate can contribute to its steady-state probability. Systems that are far from equilibrium thereby become dependent on history and the resulting complexity is a fundamental challenge. To begin addressing this, we introduce a graph-based concept of independence, which can be applied to sub-systems that are far from equilibrium, and prove that history-dependent complexity can be circumvented when sub-systems operate independently.

CONCLUSIONS

As epigenomic data become increasingly available, we anticipate that gene function will come to be represented by graphs, as gene structure has been represented by sequences, and that the methods introduced here will provide a broader foundation for understanding how genes work.

A predictive model of bifunctional transcription factor signaling during embryonic tissue patterning

Hedgehog signaling controls pattern formation in many vertebrate tissues. The downstream effectors of the pathway are the bifunctional Gli transcription factors, which, depending on hedgehog concentration, act as either transcriptional activators or repressors. Quantitatively understanding the interplay between Gli activator and repressor forms for patterning complex tissues is an open challenge. Here, we describe a reductionist mathematical model for how Gli activators and repressors are integrated in space and time to regulate transcriptional outputs of hedgehog signaling, using the pathway readouts Gli1 and Ptch1 as a model system. Spatially resolved measurements of absolute transcript numbers for these genes allow us to infer spatiotemporal variations of Gli activator and repressor levels. We validate our model by successfully predicting expression changes of Gli1 and Ptch1 in mutants at different developmental stages and in different tissues. Our results provide a starting point for understanding gene regulation by bifunctional transcription factors during mammalian development.

Predictive modelling of gene expression from transcriptional regulatory elements

Predictive modelling of gene expression provides a powerful framework for exploring the regulatory logic underpinning transcriptional regulation. Recent studies have demonstrated the utility of such models in identifying dysregulation of gene and miRNA expression associated with abnormal patterns of transcription factor (TF) binding or nucleosomal histone modifications (HMs). Despite the growing popularity of such approaches, a comparative review of the various modelling algorithms and feature extraction methods is lacking. We define and compare three methods of quantifying pairwise gene-TF/HM interactions and discuss their suitability for integrating the heterogeneous chromatin immunoprecipitation (ChIP)-seq binding patterns exhibited by TFs and HMs. We then construct log-linear and ϵ-support vector regression models from various mouse embryonic stem cell (mESC) and human lymphoblastoid (GM12878) data sets, considering both ChIP-seq- and position weight matrix- (PWM)-derived in silico TF-binding. The two algorithms are evaluated both in terms of their modelling prediction accuracy and ability to identify the established regulatory roles of individual TFs and HMs. Our results demonstrate that TF-binding and HMs are highly predictive of gene expression as measured by mRNA transcript abundance, irrespective of algorithm or cell type selection and considering both ChIP-seq and PWM-derived TF-binding. As we encourage other researchers to explore and develop these results, our framework is implemented using open-source software and made available as a preconfigured bootable virtual environment.

A computational framework for analyzing stochasticity in gene expression

Stochastic fluctuations in gene expression give rise to distributions of protein levels across cell populations. Despite a mounting number of theoretical models explaining stochasticity in protein expression, we lack a robust, efficient, assumption-free approach for inferring the molecular mechanisms that underlie the shape of protein distributions. Here we propose a method for inferring sets of biochemical rate constants that govern chromatin modification, transcription, translation, and RNA and protein degradation from stochasticity in protein expression. We asked whether the rates of these underlying processes can be estimated accurately from protein expression distributions, in the absence of any limiting assumptions. To do this, we (1) derived analytical solutions for the first four moments of the protein distribution, (2) found that these four moments completely capture the shape of protein distributions, and (3) developed an efficient algorithm for inferring gene expression rate constants from the moments of protein distributions. Using this algorithm we find that most protein distributions are consistent with a large number of different biochemical rate constant sets. Despite this degeneracy, the solution space of rate constants almost always informs on underlying mechanism. For example, we distinguish between regimes where transcriptional bursting occurs from regimes reflecting constitutive transcript production. Our method agrees with the current standard approach, and in the restrictive regime where the standard method operates, also identifies rate constants not previously obtainable. Even without making any assumptions we obtain estimates of individual biochemical rate constants, or meaningful ratios of rate constants, in 91% of tested cases. In some cases our method identified all of the underlying rate constants. The framework developed here will be a powerful tool for deducing the contributions of particular molecular mechanisms to specific patterns of gene expression.

Proteins, the molecular machines encoded by our genes, serve essential roles in every living cell. Investigators were therefore surprised to find widely variable levels of a particular protein within populations of genetically identical cells. This variation in protein level, called stochasticity, arises from the chemical nature of the processes that underlie protein production. The “central dogma” of biology dictates that the DNA encoding a particular gene transmits information via RNA to molecular factories called ribosomes in order to create proteins. Each step in transcription and translation introduces some variation, or stochasticity, into the production of the protein. In the current work, we tackled how one might learn more about the machinery responsible for creating proteins by the character of the stochasticity in the central dogma process. We find that many different mechanisms can explain any given stochastic protein signature. Even though there were many explanations for any particular pattern of stochasticity, the set of explanations still inform on how a given gene creates its protein. Our mathematical and computational framework will permit others to better understand how the machinery that expresses genes works. This, in turn, will enable investigators to better predict how a given mutation is likely to affect gene expression.

Comparison of the theoretical and real-world evolutionary potential of a genetic circuit

With the development of next-generation sequencing technologies, many large scale experimental efforts aim to map genotypic variability among individuals. This natural variability in populations fuels many fundamental biological processes, ranging from evolutionary adaptation and speciation to the spread of genetic diseases and drug resistance. An interesting and important component of this variability is present within the regulatory regions of genes. As these regions evolve, accumulated mutations lead to modulation of gene expression, which may have consequences for the phenotype. A simple model system where the link between genetic variability, gene regulation and function can be studied in detail is missing. In this article we develop a model to explore how the sequence of the wild-type lac promoter dictates the fold-change in gene expression. The model combines single-base pair resolution maps of transcription factor and RNA polymerase binding energies with a comprehensive thermodynamic model of gene regulation. The model was validated by predicting and then measuring the variability of lac operon regulation in a collection of natural isolates. We then implement the model to analyze the sensitivity of the promoter sequence to the regulatory output, and predict the potential for regulation to evolve due to point mutations in the promoter region.

FOUNDATIONS FOR MODELING THE DYNAMICS OF GENE REGULATORY NETWORKS: A MULTILEVEL-PERSPECTIVE REVIEW

A promising alternative for unraveling the principles under which the dynamic interactions among genes lead to cellular phenotypes relies on mathematical and computational models at different levels of abstraction, from the molecular level of protein-DNA interactions to the system level of functional relationships among genes. This review article presents, under a bottom–up perspective, a hierarchy of approaches to modeling gene regulatory network dynamics, from microscopic descriptions at the single-molecule level in the spatial context of an individual cell to macroscopic models providing phenomenological descriptions at the population-average level. The reviewed modeling approaches include Molecular Dynamics, Particle-Based Brownian Dynamics, the Master Equation approach, Ordinary Differential Equations, and the Boolean logic abstraction. Each of these frameworks is motivated by a particular biological context and the nature of the insight being pursued. The setting of gene network dynamic models from such frameworks involves assumptions and mathematical artifacts often ignored by the non-specialist. This article aims at providing an entry point for biologists new to the field and computer scientists not acquainted with some recent biophysically-inspired models of gene regulation. The connections promoting intuition between different abstraction levels and the role that approximations play in the modeling process are highlighted throughout the paper.

A multi-scale model of hepcidin promoter regulation reveals factors controlling systemic iron homeostasis

Systemic iron homeostasis involves a negative feedback circuit in which the expression level of the peptide hormone hepcidin depends on and controls the iron blood levels. Hepcidin expression is regulated by the BMP6/SMAD and IL6/STAT signaling cascades. Deregulation of either pathway causes iron-related diseases such as hemochromatosis or anemia of inflammation. We quantitatively analyzed how BMP6 and IL6 control hepcidin expression. Transcription factor (TF) phosphorylation and reporter gene expression were measured under co-stimulation conditions, and the promoter was perturbed by mutagenesis. Using mathematical modeling, we systematically analyzed potential mechanisms of cooperative and competitive promoter regulation by the transcription factors, and experimentally validated the model predictions. Our results reveal that hepcidin cross-regulation primarily occurs by combinatorial transcription factor binding to the promoter, whereas signaling crosstalk is insignificant. We find that the presence of two BMP-responsive elements enhances the steepness of the promoter response towards the iron-sensing BMP signaling axis, which promotes iron homeostasis in vivo. IL6 co-stimulation reduces the promoter sensitivity towards the BMP signal, because the SMAD and STAT transcription factors compete for recruiting RNA polymerase to the transcription start site. This may explain why inflammatory signals disturb iron homeostasis in anemia of inflammation. Taken together, our results reveal why the iron homeostasis circuit is sensitive to perturbations implicated in disease.

The nutritional iron uptake is tightly regulated because the body has limited capacity of iron excretion. Mammals maintain iron homeostasis by a negative feedback loop, in which the peptide hepcidin senses the iron blood level and controls iron resorption. Molecular perturbations in the homeostasis loop lead to iron-related diseases such as hemochromatosis or anemia of inflammation. Quantitative studies are required to understand the dynamics of the iron homeostasis circuitry in health and disease. We investigated how the biological activity of hepcidin is regulated by combining experiments with mathematical modeling. We present a multi-scale model that describes the signaling network and the gene promoter controlling hepcidin expression. Possible scenarios of hepcidin regulation were systematically tested against experimental data, and interpreted using a network model of iron metabolism in vivo. The analysis showed that the presence of multiple redundant regulatory elements in the hepcidin gene promoter facilitates homeostasis, because changes in iron blood levels are sensed with high sensitivity. We further suggest that inflammatory signals establish molecular competition at the hepcidin promoter, thereby reducing its iron sensitivity and leading to a loss of homeostasis in anemia of inflammation. We conclude that quantitative insights into hepcidin expression regulation explain features of systemic iron homeostasis.

Statistical mechanical model of coupled transcription from multiple promoters due to transcription factor titration

Transcription factors (TFs) with regulatory action at multiple promoter targets is the rule rather than the exception, with examples ranging from the cAMP receptor protein (CRP) in E. coli that regulates hundreds of different genes simultaneously to situations involving multiple copies of the same gene, such as plasmids, retrotransposons, or highly replicated viral DNA. When the number of TFs heavily exceeds the number of binding sites, TF binding to each promoter can be regarded as independent. However, when the number of TF molecules is comparable to the number of binding sites, TF titration will result in correlation ("promoter entanglement") between transcription of different genes. We develop a statistical mechanical model which takes the TF titration effect into account and use it to predict both the level of gene expression for a general set of promoters and the resulting correlation in transcription rates of different genes. Our results show that the TF titration effect could be important for understanding gene expression in many regulatory settings.