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

A mathematical model clarifies the ABC Score formula used in enhancer-gene prediction

Enhancers are discrete DNA elements that regulate the expression of eukaryotic genes. They are important not only for their regulatory function, but also as loci that are frequently associated with disease traits. Despite their significance, our conceptual understanding of how enhancers work remains limited. CRISPR-interference methods have recently provided the means to systematically screen for enhancers in cell culture, from which a formula for predicting whether an enhancer regulates a gene, the Activity-by-Contact (ABC) Score, has emerged and has been widely adopted. While useful as a binary classifier, it is less effective at predicting the quantitative effect of an enhancer on gene expression. It is also unclear how the algebraic form of the ABC Score arises from the underlying molecular mechanisms and what assumptions are needed for it to hold. Here, we use the graph-theoretic linear framework, previously introduced to analyze gene regulation, to formulate the default model, a mathematical model of how multiple enhancers independently regulate a gene. We show that the algebraic form of the ABC Score arises from this model. However, the default model assumptions also imply that enhancers act additively on steady-state gene expression. This is known to be false for certain genes and we show how modifying the assumptions can accommodate this discrepancy. Overall, our approach lays a rigorous, biophysical foundation for future studies of enhancer-gene regulation.

Seeing with an extra sense

Science foremost derives from our curiosity about the world. Can we make sense of the phenomena we see around us? Given that understanding, can we predict previously unimagined phenomena? How do things work? Can we use what we discover to invent new technologies? One class of questions that has mesmerized observers, dating at least to early cave paintings of hunters and their prey, surrounds the nature of the phenomenon we refer to as life. Over the centuries, scientists have found a broad array of surprisingly different techniques for observing, measuring, characterizing and explaining the living world. Microscopes provide a dazzling view of a previously unseen reality that tells us how living organisms are made up and how their components are organized and move. The tools of molecular science tell us the sequence and structure of the macromolecules that fill cells. The data explosion that has attended the development of a new generation of high-throughput tools for querying the living world demands that we have some way of accounting for those data that both provide intuition and make dangerous predictions with no after-the-fact parametric wiggle room. In this special issue of Current Biology, leading researchers explore how physical approaches have contributed to various fields of biology. Here, to introduce this special issue, I consider some of the ways in which viewing the living through a physical lens allows us to see things that might otherwise remain hidden.

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 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.

Macroscopic quorum sensing sustains differentiating embryonic stem cells

Cells can secrete molecules that help each other's replication. In cell cultures, chemical signals might diffuse only within a cell colony or between colonies. A chemical signal's interaction length-how far apart interacting cells are-is often assumed to be some value without rigorous justifications because molecules' invisible paths and complex multicellular geometries pose challenges. Here we present an approach, combining mathematical models and experiments, for determining a chemical signal's interaction length. With murine embryonic stem (ES) cells as a testbed, we found that differentiating ES cells secrete FGF4, among others, to communicate over many millimeters in cell culture dishes and, thereby, form a spatially extended, macroscopic entity that grows only if its centimeter-scale population density is above a threshold value. With this 'macroscopic quorum sensing', an isolated macroscopic, but not isolated microscopic, colony can survive differentiation. Our integrated approach can determine chemical signals' interaction lengths in generic multicellular communities.

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.

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.

Using transcription-based detectors to emulate the behavior of sequential probability ratio-based concentration detectors

The sequential probability ratio test (SPRT) from statistics is known to have the least mean decision time compared to other sequential or fixed-time tests for given error rates. In some circumstances, cells need to make decisions accurately and quickly, therefore it has been suggested that the SPRT may be used to understand the speed-accuracy tradeoff in cellular decision-making. It is generally thought that in order for cells to make use of the SPRT, it is necessary to find biochemical circuits that can compute the log-likelihood ratio needed for the SPRT. However, this paper takes a different approach. We recognize that the high-level behavior of the SPRT is defined by its positive detection or hit rate, and the computation of the log-likelihood ratio is just one way to realize this behavior. In this paper, we will present a method in which a transcription-based detector is used to emulate the hit rate of the SPRT without computing the exact log-likelihood ratio. We consider the problem of using a promoter with multiple binding sites to accurately and quickly detect whether the concentration of a transcription factor is above a target level. We show that it is possible to find binding and unbinding rates of the transcription factor to the promoter's binding sites so that the probability that the amount of mRNA produced will be higher than a threshold is approximately equal to the hit rate of the SPRT detector. Moreover, we show that the average time that this transcription-based detector needs to make a positive detection is less than or equal to that of the SPRT for a wide range of concentrations. We remark that the last statement does not contradict Wald's optimality result because our transcription-based detector uses an open-ended test.

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.

The nonequilibrium mechanism of noise-enhanced drug synergy in HIV latency reactivation

Noise-modulating chemicals can synergize with transcriptional activators in reactivating latent HIV to eliminate latent HIV reservoirs. To understand the underlying biomolecular mechanism, we investigate a previous two-gene-state model and identify two necessary conditions for the synergy: an assumption of the inhibition effect of transcription activators on noise enhancers; and frequent transitions to the gene non-transcription-permissive state. We then develop a loop-four-gene-state model with Tat transcription/translation and find that drug synergy is mainly determined by the magnitude and direction of energy input into the genetic regulatory kinetics of the HIV promoter. The inhibition effect of transcription activators is actually a phenomenon of energy dissipation in the nonequilibrium gene transition system. Overall, the loop-four-state model demonstrates that energy dissipation plays a crucial role in HIV latency reactivation, which might be useful for improving drug effects and identifying other synergies on lentivirus latency reactivation.

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The inhibition of Activator on Noise enhancer is necessary for their synergy in reactivating HIV

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The drug synergy is a nonequilibrium phenomenon in the gene regulatory system

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The magnitude and direction of energy input determine the drug synergy

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This nonequilibrium mechanism is general without regarding molecular details

Patterning, From Conifers to Consciousness: Turing's Theory and Order From Fluctuations

This is a brief account of Turing's ideas on biological pattern and the events that led to their wider acceptance by biologists as a valid way to investigate developmental pattern, and of the value of theory more generally in biology. Periodic patterns have played a key role in this process, especially 2D arrays of oriented stripes, which proved a disappointment in theoretical terms in the case of Drosophila segmentation, but a boost to theory as applied to skin patterns in fish and model chemical reactions. The concept of "order from fluctuations" is a key component of Turing's theory, wherein pattern arises by selective amplification of spatial components concealed in the random disorder of molecular and/or cellular processes. For biological examples, a crucial point from an analytical standpoint is knowing the nature of the fluctuations, where the amplifier resides, and the timescale over which selective amplification occurs. The answer clarifies the difference between "inelegant" examples such as Drosophila segmentation, which is perhaps better understood as a programmatic assembly process, and "elegant" ones expressible in equations like Turing's: that the fluctuations and selection process occur predominantly in evolutionary time for the former, but in real time for the latter, and likewise for error suppression, which for Drosophila is historical, in being lodged firmly in past evolutionary events. The prospects for a further extension of Turing's ideas to the complexities of brain development and consciousness is discussed, where a case can be made that it could well be in neuroscience that his ideas find their most important application.

Cis-Regulatory Logic Produces Gene-Expression Noise Describing Phenotypic Heterogeneity in Bacteria

Gene transcriptional process is random. It occurs in bursts and follows single-molecular kinetics. Intermittent bursts are measured based on their frequency and size. They influence temporal fluctuations in the abundance of total mRNA and proteins by generating distinct transcriptional variations referred to as “noise”. Noisy expression induces uncertainty because the association between transcriptional variation and the extent of gene expression fluctuation is ambiguous. The promoter architecture and remote interference of different cis-regulatory elements are the crucial determinants of noise, which is reflected in phenotypic heterogeneity. An alternative perspective considers that cellular parameters dictating genome-wide transcriptional kinetics follow a universal pattern. Research on noise and systematic perturbations of promoter sequences reinforces that both gene-specific and genome-wide regulation occur across species ranging from bacteria and yeast to animal cells. Thus, deciphering gene-expression noise is essential across different genomics applications. Amidst the mounting conflict, it is imperative to reconsider the scope, progression, and rational construction of diversified viewpoints underlying the origin of the noise. Here, we have established an indication connecting noise, gene expression variations, and bacterial phenotypic variability. This review will enhance the understanding of gene-expression noise in various scientific contexts and applications.

Deciphering enhancer sequence using thermodynamics-based models and convolutional neural networks

Deciphering the sequence-function relationship encoded in enhancers holds the key to interpreting non-coding variants and understanding mechanisms of transcriptomic variation. Several quantitative models exist for predicting enhancer function and underlying mechanisms; however, there has been no systematic comparison of these models characterizing their relative strengths and shortcomings. Here, we interrogated a rich data set of neuroectodermal enhancers in Drosophila, representing cis- and trans- sources of expression variation, with a suite of biophysical and machine learning models. We performed rigorous comparisons of thermodynamics-based models implementing different mechanisms of activation, repression and cooperativity. Moreover, we developed a convolutional neural network (CNN) model, called CoNSEPT, that learns enhancer 'grammar' in an unbiased manner. CoNSEPT is the first general-purpose CNN tool for predicting enhancer function in varying conditions, such as different cell types and experimental conditions, and we show that such complex models can suggest interpretable mechanisms. We found model-based evidence for mechanisms previously established for the studied system, including cooperative activation and short-range repression. The data also favored one hypothesized activation mechanism over another and suggested an intriguing role for a direct, distance-independent repression mechanism. Our modeling shows that while fundamentally different models can yield similar fits to data, they vary in their utility for mechanistic inference. CoNSEPT is freely available at: https://github.com/PayamDiba/CoNSEPT.

Nonequilibrium model of short-range repression in gene transcription regulation

Transcription factors are proteins that regulate gene activity by activating or repressing gene transcription. A special class of transcriptional repressors operates via a short-range mechanism, making local DNA regions inaccessible to binding by activators, and thus providing an indirect repressive action on the target gene. This mechanism is commonly modeled assuming that repressors interact with DNA under thermodynamic equilibrium and neglecting some configurations of the gene regulatory region. We elaborate on a more general nonequilibrium model of short-range repression using the graph formalism for transitions between gene states, and we apply analytical calculations to compare it with the equilibrium model in terms of the repression strength and expression noise. In contrast to the equilibrium approach, the new model allows us to separate two basic mechanisms of short-range repression. The first mechanism is associated with the recruiting of factors that mediate chromatin condensation, and the second one concerns the blocking of factors that mediate chromatin loosening. The nonequilibrium model demonstrates better performance on previously published gene expression data obtained for transcription factors controlling Drosophila development, and furthermore it predicts that the first repression mechanism is the most favorable in this system. The presented approach can be scaled to larger gene networks and can be used to infer specific modes and parameters of transcriptional regulation from gene expression data.

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.

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.

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.

Efficient manipulation and generation of Kirchhoff polynomials for the analysis of non-equilibrium biochemical reaction networks

Kirchhoff polynomials are central for deriving symbolic steady-state expressions of models whose dynamics are governed by linear diffusion on graphs. In biology, such models have been unified under a common linear framework subsuming studies across areas such as enzyme kinetics, G-protein coupled receptors, ion channels and gene regulation. Due to 'history dependence' away from thermodynamic equilibrium, these models suffer from a (super) exponential growth in the size of their symbolic steady-state expressions and, respectively, Kirchhoff polynomials. This algebraic explosion has limited applications of the linear framework. However, recent results on the graph-based prime factorization of Kirchhoff polynomials may help subdue the combinatorial complexity. By prime decomposing the graphs contained in an expression of Kirchhoff polynomials and identifying the graphs giving rise to equal polynomials, we formulate a coarse-grained variant of the expression suitable for symbolic simplification. We devise criteria to efficiently test the equality of Kirchhoff polynomials and propose two heuristic algorithms to explicitly generate individual Kirchhoff polynomials in a compressed form; they are inspired by algebraic simplifications but operate on the corresponding graphs. We illustrate the practicality of the developed theory and algorithms for a diverse set of graphs of different sizes and for non-equilibrium gene regulation analyses.

Topology Effects on Sparse Control of Complex Networks with Laplacian Dynamics

Ease of control of complex networks has been assessed extensively in terms of structural controllability and observability, and minimum control energy criteria. Here we adopt a sparsity-promoting feedback control framework for undirected networks with Laplacian dynamics and distinct topological features. The control objective considered is to minimize the effect of disturbance signals, magnitude of control signals and cost of feedback channels. We show that depending on the cost of feedback channels, different complex network structures become the least expensive option to control. Specifically, increased cost of feedback channels favors organized topological complexity such as modularity and centralization. Thus, although sparse and heterogeneous undirected networks may require larger numbers of actuators and sensors for structural controllability, networks with Laplacian dynamics are shown to be easier to control when accounting for the cost of feedback channels.

Self-organization across scales: from molecules to organisms

Creating ordered structures from chaotic environments is at the core of biological processes at the subcellular, cellular and organismic level. In this perspective, we explore the physical as well as biological features of two prominent concepts driving self-organization, namely phase transition and reaction-diffusion, before closing with a discussion on open questions and future challenges associated with studying self-organizing systems.This article is part of the theme issue 'Self-organization in cell biology'.

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.

An information theoretic treatment of sequence-to-expression modeling

Studying a gene’s regulatory mechanisms is a tedious process that involves identification of candidate regulators by transcription factor (TF) knockout or over-expression experiments, delineation of enhancers by reporter assays, and demonstration of direct TF influence by site mutagenesis, among other approaches. Such experiments are often chosen based on the biologist’s intuition, from several testable hypotheses. We pursue the goal of making this process systematic by using ideas from information theory to reason about experiments in gene regulation, in the hope of ultimately enabling rigorous experiment design strategies. For this, we make use of a state-of-the-art mathematical model of gene expression, which provides a way to formalize our current knowledge of cis- as well as trans- regulatory mechanisms of a gene. Ambiguities in such knowledge can be expressed as uncertainties in the model, which we capture formally by building an ensemble of plausible models that fit the existing data and defining a probability distribution over the ensemble. We then characterize the impact of a new experiment on our understanding of the gene’s regulation based on how the ensemble of plausible models and its probability distribution changes when challenged with results from that experiment. This allows us to assess the ‘value’ of the experiment retroactively as the reduction in entropy of the distribution (information gain) resulting from the experiment’s results. We fully formalize this novel approach to reasoning about gene regulation experiments and use it to evaluate a variety of perturbation experiments on two developmental genes of D. melanogaster. We also provide objective and ‘biologist-friendly’ descriptions of the information gained from each such experiment. The rigorously defined information theoretic approaches presented here can be used in the future to formulate systematic strategies for experiment design pertaining to studies of gene regulatory mechanisms.

In-depth studies of gene regulatory mechanisms employ a variety of experimental approaches such as identifying a gene’s enhancer(s) and testing its variants through reporter assays, followed by transcription factor mis-expression or knockouts, site mutagenesis, etc. The biologist is often faced with the challenging problem of selecting the ideal next experiment to perform so that its results provide novel mechanistic insights, and has to rely on their intuition about what is currently known on the topic and which experiments may add to that knowledge. We seek to make this intuition-based process more systematic, by borrowing ideas from the mature statistical field of experiment design. Towards this goal, we use the language of mathematical models to formally describe what is known about a gene’s regulatory mechanisms, and how an experiment’s results enhance that knowledge. We use information theoretic ideas to assign a ‘value’ to an experiment as well as explain objectively what is learned from that experiment. We demonstrate use of this novel approach on two extensively studied developmental genes in fruitfly. We expect our work to lead to systematic strategies for selecting the most informative experiments in a study of gene regulation.

Structural conditions on complex networks for the Michaelis-Menten input-output response

The Michaelis-Menten (MM) fundamental formula describes how the rate of enzyme catalysis depends on substrate concentration. The familiar hyperbolic relationship was derived by timescale separation for a network of three reactions. The same formula has subsequently been found to describe steady-state input-output responses in many biological contexts, including single-molecule enzyme kinetics, gene regulation, transcription, translation, and force generation. Previous attempts to explain its ubiquity have been limited to networks with regular structure or simplifying parametric assumptions. Here, we exploit the graph-based linear framework for timescale separation to derive general structural conditions under which the MM formula arises. The conditions require a partition of the graph into two parts, akin to a "coarse graining" into the original MM graph, and constraints on where and how the input variable occurs. Other features of the graph, including the numerical values of parameters, can remain arbitrary, thereby explaining the formula's ubiquity. For systems at thermodynamic equilibrium, we derive a necessary and sufficient condition. For systems away from thermodynamic equilibrium, especially those with irreversible reactions, distinct structural conditions arise and a general characterization remains open. Nevertheless, our results accommodate, in much greater generality, all examples known to us in the literature.

Model discrimination for Ca2+ -dependent regulation of myosin light chain kinase in smooth muscle contraction

Excitation-contraction coupling in smooth muscle is mediated by the Ca²⁺ - and calmodulin-dependent regulation of myosin light chain kinase. The precise mechanism of this regulation remains controversial, and several mathematical models have been proposed for the interaction of the three species. These models have previously been analyzed at steady state primarily by numerical simulation of differential equations, for which parameter values must be estimated from data. Here, we use the linear framework for timescale separation to demonstrate that models of this general kind can be solved analytically for an equilibrium steady state, without having to determine parameter values. This analysis leads to parameter-independent methods for discriminating between the models, for which we propose experiments that could be performed with existing methods.

Energy-speed-accuracy relation in complex networks for biological discrimination

Discriminating between correct and incorrect substrates is a core process in biology, but how is energy apportioned between the conflicting demands of accuracy (μ), speed (σ), and total entropy production rate (P)? Previous studies have focused on biochemical networks with simple structure or relied on simplifying kinetic assumptions. Here, we use the linear framework for timescale separation to analytically examine steady-state probabilities away from thermodynamic equilibrium for networks of arbitrary complexity. We also introduce a method of scaling parameters that is inspired by Hopfield's treatment of kinetic proofreading. Scaling allows asymptotic exploration of high-dimensional parameter spaces. We identify in this way a broad class of complex networks and scalings for which the quantity σln(μ)/P remains asymptotically finite whenever accuracy improves from equilibrium, so that μ_{eq}/μ→0. Scalings exist, however, even for Hopfield's original network, for which σln(μ)/P is asymptotically infinite, illustrating the parametric complexity. Outside the asymptotic regime, numerical calculations suggest that, under more restrictive parametric assumptions, networks satisfy the bound, σln(μ/μ_{eq})/P<1, and we discuss the biological implications for discrimination by ribosomes and DNA polymerase. The methods introduced here may be more broadly useful for analyzing complex networks that implement other forms of cellular information processing.

Frequency Modulation of Transcriptional Bursting Enables Sensitive and Rapid Gene Regulation

Gene regulation is a complex non-equilibrium process. Here, we show that quantitating the temporal regulation of key gene states (transcriptionally inactive, active, and refractory) provides a parsimonious framework for analyzing gene regulation. Our theory makes two non-intuitive predictions. First, for transcription factors (TFs) that regulate transcription burst frequency, as opposed to amplitude or duration, weak TF binding is sufficient to elicit strong transcriptional responses. Second, refractoriness of a gene after a transcription burst enables rapid responses to stimuli. We validate both predictions experimentally by exploiting the natural, optogenetic-like responsiveness of the Neurospora GATA-type TF White Collar Complex (WCC) to blue light. Further, we demonstrate that differential regulation of WCC target genes is caused by different gene activation rates, not different TF occupancy, and that these rates are tuned by both the core promoter and the distance between TF-binding site and core promoter. In total, our work demonstrates the relevance of a kinetic, non-equilibrium framework for understanding transcriptional regulation.

Steady-State Differential Dose Response in Biological Systems

In pharmacology and systems biology, it is a fundamental problem to determine how biological systems change their dose-response behavior upon perturbations. In particular, it is unclear how topologies, reactions, and parameters (differentially) affect the dose response. Because parameters are often unknown, systematic approaches should directly relate network structure and function. Here, we outline a procedure to compare general non-monotone dose-response curves and subsequently develop a comprehensive theory for differential dose responses of biochemical networks captured by non-equilibrium steady-state linear framework models. Although these models are amenable to analytical derivations of non-equilibrium steady states in principle, their size frequently increases (super) exponentially with model size. We extract general principles of differential responses based on a model's graph structure and thereby alleviate the combinatorial explosion. This allows us, for example, to determine reactions that affect differential responses, to identify classes of networks with equivalent differential, and to reject hypothetical models reliably without needing to know parameter values. We exemplify such applications for models of insulin signaling.

Transcription factors, coregulators, and epigenetic marks are linearly correlated and highly redundant

The DNA microstates that regulate transcription include sequence-specific transcription factors (TFs), coregulatory complexes, nucleosomes, histone modifications, DNA methylation, and parts of the three-dimensional architecture of genomes, which could create an enormous combinatorial complexity across the genome. However, many proteins and epigenetic marks are known to colocalize, suggesting that the information content encoded in these marks can be compressed. It has so far proved difficult to understand this compression in a systematic and quantitative manner. Here, we show that simple linear models can reliably predict the data generated by the ENCODE and Roadmap Epigenomics consortia. Further, we demonstrate that a small number of marks can predict all other marks with high average correlation across the genome, systematically revealing the substantial information compression that is present in different cell lines. We find that the linear models for activating marks are typically cell line-independent, while those for silencing marks are predominantly cell line-specific. Of particular note, a nuclear receptor corepressor, transducin beta-like 1 X-linked receptor 1 (TBLR1), was highly predictive of other marks in two hematopoietic cell lines. The methodology presented here shows how the potentially vast complexity of TFs, coregulators, and epigenetic marks at eukaryotic genes is highly redundant and that the information present can be compressed onto a much smaller subset of marks. These findings could be used to efficiently characterize cell lines and tissues based on a small number of diagnostic marks and suggest how the DNA microstates, which regulate the expression of individual genes, can be specified.

The free-energy cost of interaction between DNA loops

From the viewpoint of thermodynamics, the formation of DNA loops and the interaction between them, which are all non-equilibrium processes, result in the change of free energy, affecting gene expression and further cell-to-cell variability as observed experimentally. However, how these processes dissipate free energy remains largely unclear. Here, by analyzing a mechanic model that maps three fundamental topologies of two interacting DNA loops into a 4-state model of gene transcription, we first show that a longer DNA loop needs more mean free energy consumption. Then, independent of the type of interacting two DNA loops (nested, side-by-side or alternating), the promotion between them always consumes less mean free energy whereas the suppression dissipates more mean free energy. More interestingly, we find that in contrast to the mechanism of direct looping between promoter and enhancer, the facilitated-tracking mechanism dissipates less mean free energy but enhances the mean mRNA expression, justifying the facilitated-tracking hypothesis, a long-standing debate in biology. Based on minimal energy principle, we thus speculate that organisms would utilize the mechanisms of loop-loop promotion and facilitated tracking to survive in complex environments. Our studies provide insights into the understanding of gene expression regulation mechanism from the view of energy consumption.

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.

Fundamental principles of energy consumption for gene expression

How energy is consumed in gene expression is largely unknown mainly due to complexity of non-equilibrium mechanisms affecting expression levels. Here, by analyzing a representative gene model that considers complexity of gene expression, we show that negative feedback increases energy consumption but positive feedback has an opposite effect; promoter leakage always reduces energy consumption; generating more bursts needs to consume more energy; and the speed of promoter switching is at the cost of energy consumption. We also find that the relationship between energy consumption and expression noise is multi-mode, depending on both the type of feedback and the speed of promoter switching. Altogether, these results constitute fundamental principles of energy consumption for gene expression, which lay a foundation for designing biologically reasonable gene modules. In addition, we discuss possible biological implications of these principles by combining experimental facts.

From Structural Variation of Gene Molecules to Chromatin Dynamics and Transcriptional Bursting

Transcriptional activation of eukaryotic genes is accompanied, in general, by a change in the sensitivity of promoter chromatin to endonucleases. The structural basis of this alteration has remained elusive for decades; but the change has been viewed as a transformation of one structure into another, from "closed" to "open" chromatin. In contradistinction to this static and deterministic view of the problem, a dynamical and probabilistic theory of promoter chromatin has emerged as its solution. This theory, which we review here, explains observed variation in promoter chromatin structure at the level of single gene molecules and provides a molecular basis for random bursting in transcription-the conjecture that promoters stochastically transition between transcriptionally conducive and inconducive states. The mechanism of transcriptional regulation may be understood only in probabilistic terms.

Robust network structure of the Sln1-Ypd1-Ssk1 three-component phospho-relay prevents unintended activation of the HOG MAPK pathway in Saccharomyces cerevisiae

BACKGROUND

The yeast Saccharomyces cerevisiae relies on the high-osmolarity glycerol (HOG) signaling pathway to respond to increases in external osmolarity. The HOG pathway is rapidly activated under conditions of elevated osmolarity and regulates transcriptional and metabolic changes within the cell. Under normal growth conditions, however, a three-component phospho-relay consisting of the histidine kinase Sln1, the transfer protein Ypd1, and the response regulator Ssk1 represses HOG pathway activity by phosphorylation of Ssk1. This inhibition of the HOG pathway is essential for cellular fitness in normal osmolarity. Nevertheless, the extent to and mechanisms by which inhibition is robust to fluctuations in the concentrations of the phospho-relay components has received little attention.

RESULTS

We established that the Sln1-Ypd1-Ssk1 phospho-relay is robust-it is able to maintain inhibition of the HOG pathway even after significant changes in the levels of its three components. We then developed a biochemically realistic mathematical model of the phospho-relay, which suggested that robustness is due to buffering by a large excess pool of Ypd1. We confirmed experimentally that depletion of the Ypd1 pool results in inappropriate activation of the HOG pathway.

CONCLUSIONS

We identified buffering by an intermediate component in excess as a novel mechanism through which a phospho-relay can achieve robustness. This buffering requires multiple components and is therefore unavailable to two-component systems, suggesting one important advantage of multi-component relays.

Laplacian Dynamics with Synthesis and Degradation

Analyzing qualitative behaviors of biochemical reactions using its associated network structure has proven useful in diverse branches of biology. As an extension of our previous work, we introduce a graph-based framework to calculate steady state solutions of biochemical reaction networks with synthesis and degradation. Our approach is based on a labeled directed graph G and the associated system of linear non-homogeneous differential equations with first-order degradation and zeroth-order synthesis. We also present a theorem which provides necessary and sufficient conditions for the dynamics to engender a unique stable steady state. Although the dynamics are linear, one can apply this framework to nonlinear systems by encoding nonlinearity into the edge labels. We answer an open question from our previous work concerning the non-positiveness of the elements in the inverse of a perturbed Laplacian matrix. Moreover, we provide a graph theoretical framework for the computation of the inverse of such a matrix. This also completes our previous framework and makes it purely graph theoretical. Lastly, we demonstrate the utility of this framework by applying it to a mathematical model of insulin secretion through ion channels in pancreatic β-cells.

Keeping up with the 'omics: non-equilibrium models of gene regulation

Non-equilibrium processes are vital features of biological systems. Despite this universally accepted fact, gene regulation is typically formalized into models that assume thermodynamic equilibrium. As experimental evidence expands the repertoire of non-equilibrium genome regulatory mechanisms, theoreticians are challenged to devise general approaches to accommodate and suggest functions for non-equilibrium processes. Ahsendorf et al. provide one such framework, which is discussed in the context of the growing complexity of eukaryotic gene regulation.

Laplacian dynamics on general graphs

In previous work, we have introduced a "linear framework" for time-scale separation in biochemical systems, which is based on a labelled, directed graph, G, and an associated linear differential equation, dx/dt = L(G) ∙ x, where L(G) is the Laplacian matrix of G. Biochemical nonlinearity is encoded in the graph labels. Many central results in molecular biology can be systematically derived within this framework, including those for enzyme kinetics, allosteric proteins, G-protein coupled receptors, ion channels, gene regulation at thermodynamic equilibrium, and protein post-translational modification. In the present paper, in response to new applications, which accommodate nonequilibrium mechanisms in eukaryotic gene regulation, we lay out the mathematical foundations of the framework. We show that, for any graph and any initial condition, the dynamics always reaches a steady state, which can be algorithmically calculated. If the graph is not strongly connected, which may occur in gene regulation, we show that the dynamics can exhibit flexible behavior that resembles multistability. We further reveal an unexpected equivalence between deterministic Laplacian dynamics and the master equations of continuous-time Markov processes, which allows rigorous treatment within the framework of stochastic, single-molecule mechanisms.