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

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

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.

MSModDetector: a tool for detecting mass shifts and post-translational modifications in individual ion mass spectrometry data.

MOTIVATION: Post-translational modifications (PTMs) on proteins regulate protein structures and functions. A single protein molecule can possess multiple modification sites that can accommodate various PTM types, leading to a variety of different patterns, or combinations of PTMs, on that protein. Different PTM patterns can give rise to distinct biological functions. To facilitate the study of multiple PTMs on the same protein molecule, top-down mass spectrometry (MS) has proven to be a useful tool to measure the mass of intact proteins, thereby enabling even PTMs at distant sites to be assigned to the same protein molecule and allowing determination of how many PTMs are attached to a single protein. RESULTS: We developed a Python module called MSModDetector that studies PTM patterns from individual ion mass spectrometry (I2MS) data. I2MS is an intact protein mass spectrometry approach that generates true mass spectra without the need to infer charge states. The algorithm first detects and quantifies mass shifts for a protein of interest and subsequently infers potential PTM patterns using linear programming. The algorithm is evaluated on simulated I2MS data and experimental I2MS data for the tumor suppressor protein p53. We show that MSModDetector is a useful tool for comparing a protein's PTM pattern landscape across different conditions. An improved analysis of PTM patterns will enable a deeper understanding of PTM-regulated cellular processes. AVAILABILITY AND IMPLEMENTATION: The source code is available at https://github.com/marjanfaizi/MSModDetector.

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

The Hill functions, ℋh(x)=xh/1+xh, have been widely used in biology for over a century but, with the exception of ℋ1, they have had no justification other than as a convenient fit to empirical data. Here, we show that they are the universal limit for the sharpness of any input-output response arising from a Markov process model at thermodynamic equilibrium. Models may represent arbitrary molecular complexity, with multiple ligands, internal states, conformations, co-regulators, etc, under core assumptions that are detailed in the paper. The model output may be any linear combination of steady-state probabilities, with components other than the chosen input ligand held constant. This formulation generalises most of the responses in the literature. We use a coarse-graining method in the graph-theoretic linear framework to show that two sharpness measures for input-output responses fall within an effectively bounded region of the positive quadrant, Ωm⊂ℝ+2, for any equilibrium model with m input binding sites. Ωm exhibits a cusp which approaches, but never exceeds, the sharpness of ℋm but the region and the cusp can be exceeded when models are taken away from thermodynamic equilibrium. Such fundamental thermodynamic limits are called Hopfield barriers and our results provide a biophysical justification for the Hill functions as the universal Hopfield barriers for sharpness. Our results also introduce an object, Ωm, whose structure may be of mathematical interest, and suggest the importance of characterising Hopfield barriers for other forms of cellular information processing.

The linear framework II: using graph theory to analyse the transient regime of Markov processes.

The linear framework uses finite, directed graphs with labelled edges to model biomolecular systems. Graph vertices represent chemical species or molecular states, edges represent reactions or transitions and edge labels represent rates that also describe how the system is interacting with its environment. The present paper is a sequel to a recent review of the framework that focussed on how graph-theoretic methods give insight into steady states as rational algebraic functions of the edge labels. Here, we focus on the transient regime for systems that correspond to continuous-time Markov processes. In this case, the graph specifies the infinitesimal generator of the process. We show how the moments of the first-passage time distribution, and related quantities, such as splitting probabilities and conditional first-passage times, can also be expressed as rational algebraic functions of the labels. This capability is timely, as new experimental methods are finally giving access to the transient dynamic regime and revealing the computations and information processing that occur before a steady state is reached. We illustrate the concepts, methods and formulas through examples and show how the results may be used to illuminate previous findings in the literature.

MSModDetector: A Tool for Detecting Mass Shifts and Post-Translational Modifications in Individual Ion Mass Spectrometry Data.

Motivation: Post-translational modifications (PTMs) on proteins regulate protein structures and functions. A single protein molecule can possess multiple modification sites that can accommodate various PTM types, leading to a variety of different patterns, or combinations of PTMs, on that protein. Different PTM patterns can give rise to distinct biological functions. To facilitate the study of multiple PTMs, top-down mass spectrometry (MS) has proven to be a useful tool to measure the mass of intact proteins, thereby enabling even widely separated PTMs to be assigned to the same protein molecule and allowing determination of how many PTMs are attached to a single protein. Results: We developed a Python module called MSModDetector that studies PTM patterns from individual ion mass spectrometry (I MS) data. I MS is an intact protein mass spectrometry approach that generates true mass spectra without the need to infer charge states. The algorithm first detects and quantifies mass shifts for a protein of interest and subsequently infers potential PTM patterns using linear programming. The algorithm is evaluated on simulated I MS data and experimental I MS data for the tumor suppressor protein p53. We show that MSModDetector is a useful tool for comparing a protein's PTM pattern landscape across different conditions. An improved analysis of PTM patterns will enable a deeper understanding of PTM-regulated cellular processes. Availability: The source code is available at https://github.com/marjanfaizi/MSModDetector together with the scripts used for analyses and to generate the figures presented in this study.

Thermodynamic bounds on ultrasensitivity in covalent switching.

Switch-like motifs are among the basic building blocks of biochemical networks. A common motif that can serve as an ultrasensitive switch consists of two enzymes acting antagonistically on a substrate, one making and the other removing a covalent modification. To work as a switch, such covalent modification cycles must be held out of thermodynamic equilibrium by continuous expenditure of energy. Here, we exploit the linear framework for timescale separation to establish tight bounds on the performance of any covalent-modification switch in terms of the chemical potential difference driving the cycle. The bounds apply to arbitrary enzyme mechanisms, not just Michaelis-Menten, with arbitrary rate constants and thereby reflect fundamental physical constraints on covalent switching.

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.

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.

Reformulating nonequilibrium steady states and generalized Hopfield discrimination.

Despite substantial progress in nonequilibrium physics, steady-state (s.s.) probabilities remain intractable to analysis. For a Markov process, s.s. probabilities can be expressed in terms of transition rates using the graph-theoretic matrix-tree theorem (MTT). The MTT reveals that, away from equilibrium, s.s. probabilities become globally dependent on all rates, with expressions growing exponentially in the system size. This overwhelming complexity and lack of thermodynamic interpretation have greatly impeded analysis. Here we show that s.s. probabilities are proportional to the average of exp(-S(P)), where S(P) is the entropy generated along minimal paths P in the graph, and the average is taken over a probability distribution on spanning trees. Assuming Arrhenius rates, this "arboreal" distribution becomes Boltzmann-like, with the energy of a tree being its total edge barrier energy. This reformulation offers a thermodynamic interpretation that smoothly generalizes equilibrium statistical mechanics and reorganizes the expression complexity: the number of distinct minimal-path entropies depends not on graph size but on the entropy production index, a measure of nonequilibrium complexity. We demonstrate the power of this reformulation by extending Hopfield's analysis of discrimination by kinetic proofreading to any graph with index 1. We derive a general formula for the error ratio and use it to show how local energy dissipation can yield optimal discrimination through global synergy.

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.

Reconsidering the evidence for learning in single cells.

The question of whether single cells can learn led to much debate in the early 20th century. The view prevailed that they were capable of non-associative learning but not of associative learning, such as Pavlovian conditioning. Experiments indicating the contrary were considered either non-reproducible or subject to more acceptable interpretations. Recent developments suggest that the time is right to reconsider this consensus. We exhume the experiments of Beatrice Gelber on Pavlovian conditioning in the ciliate Paramecium aurelia, and suggest that criticisms of her findings can now be reinterpreted. Gelber was a remarkable scientist whose absence from the historical record testifies to the prevailing orthodoxy that single cells cannot learn. Her work, and more recent studies, suggest that such learning may be evolutionarily more widespread and fundamental to life than previously thought and we discuss the implications for different aspects of biology.

Reversal symmetries for cyclic paths away from thermodynamic equilibrium.

If a system is at thermodynamic equilibrium, an observer cannot tell whether a film of it is being played forward or in reverse: any transition will occur with the same frequency in the forward as in the reverse direction. However, if expenditure of energy changes the rate of even a single transition to yield a nonequilibrium steady state, such time-reversal symmetry undergoes a widespread breakdown, far beyond the point at which the energy is expended. An explosion of interdependency also arises, with steady-state probabilities of system states depending in a complicated manner on the rate of every transition in the system. Nevertheless, in the midst of this global nonequilibrium complexity, we find that cyclic paths have reversibility properties that remain local, and which can exhibit symmetry, no matter how far the system is from thermodynamic equilibrium. Specifically, given any cycle of reversible transitions, the ratio of the frequencies with which the cycle occurs in one direction versus the other is determined, in the long-time limit, only by the thermodynamic force on the cycle itself, without requiring knowledge of transition rates elsewhere in the system. In particular, if there is no net energy expenditure on the cycle, then, over long times, the cycle occurrence frequencies are the same in either direction.

Robustness and parameter geography in post-translational modification systems.

Biological systems are acknowledged to be robust to perturbations but a rigorous understanding of this has been elusive. In a mathematical model, perturbations often exert their effect through parameters, so sizes and shapes of parametric regions offer an integrated global estimate of robustness. Here, we explore this "parameter geography" for bistability in post-translational modification (PTM) systems. We use the previously developed "linear framework" for timescale separation to describe the steady-states of a two-site PTM system as the solutions of two polynomial equations in two variables, with eight non-dimensional parameters. Importantly, this approach allows us to accommodate enzyme mechanisms of arbitrary complexity beyond the conventional Michaelis-Menten scheme, which unrealistically forbids product rebinding. We further use the numerical algebraic geometry tools Bertini, Paramotopy, and alphaCertified to statistically assess the solutions to these equations at ∼109 parameter points in total. Subject to sampling limitations, we find no bistability when substrate amount is below a threshold relative to enzyme amounts. As substrate increases, the bistable region acquires 8-dimensional volume which increases in an apparently monotonic and sigmoidal manner towards saturation. The region remains connected but not convex, albeit with a high visibility ratio. Surprisingly, the saturating bistable region occupies a much smaller proportion of the sampling domain under mechanistic assumptions more realistic than the Michaelis-Menten scheme. We find that bistability is compromised by product rebinding and that unrealistic assumptions on enzyme mechanisms have obscured its parametric rarity. The apparent monotonic increase in volume of the bistable region remains perplexing because the region itself does not grow monotonically: parameter points can move back and forth between monostability and bistability. We suggest mathematical conjectures and questions arising from these findings. Advances in theory and software now permit insights into parameter geography to be uncovered by high-dimensional, data-centric analysis.

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.

A Complex Hierarchy of Avoidance Behaviors in a Single-Cell Eukaryote.

Complex behavior is associated with animals with nervous systems, but decision-making and learning also occur in non-neural organisms [1], including singly nucleated cells [2-5] and multi-nucleate synctia [6-8]. Ciliates are single-cell eukaryotes, widely dispersed in aquatic habitats [9], with an extensive behavioral repertoire [10-13]. In 1906, Herbert Spencer Jennings [14, 15] described in the sessile ciliate Stentor roeseli a hierarchy of responses to repeated stimulation, which are among the most complex behaviors reported for a singly nucleated cell [16, 17]. These results attracted widespread interest [18, 19] and exert continuing fascination [7, 20-22] but were discredited during the behaviorist orthodoxy by claims of non-reproducibility [23]. These claims were based on experiments with the motile ciliate Stentor coeruleus. We acquired and maintained the correct organism in laboratory culture and used micromanipulation and video microscopy to confirm Jennings' observations. Despite significant individual variation, not addressed by Jennings, S. roeseli exhibits avoidance behaviors in a characteristic hierarchy of bending, ciliary alteration, contractions, and detachment, which is distinct from habituation or conditioning. Remarkably, the choice of contraction versus detachment is consistent with a fair coin toss. Such behavioral complexity may have had an evolutionary advantage in protist ecosystems, and the ciliate cortex may have provided mechanisms for implementing such behavior prior to the emergence of multicellularity. Our work resurrects Jennings' pioneering insights and adds to the list of exceptional features, including regeneration [24], genome rearrangement [25], codon reassignment [26], and cortical inheritance [27], for which the ciliate clade is renowned.

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

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

Monostationarity and Multistationarity in Tree Networks of Goldbeter-Koshland Loops.

A major challenge in systems biology is to elicit general properties in the face of molecular complexity. Here, we introduce a class of enzyme-catalysed biochemical networks and examine how the existence of a single positive steady state (monostationarity) depends on the network structure, enzyme mechanisms, kinetic rate laws and parameter values. We consider Goldbeter-Koshland (GK) covalent modification loops arranged in a tree network, so that a substrate form in one loop can be an enzyme in another loop. GK loops are a canonical motif in cell signalling and trees offer a generalisation of linear cascades which accommodate network complexity while remaining mathematically tractable. In particular, they permit a modular, recursive proof strategy which may be more widely applicable. We show that if each enzyme follows its own complex reaction mechanism under mass action kinetics, then any network is monostationary for all appropriate parameter values. If the kinetics is non-mass action with a plausible monotonicity requirement, and each enzyme follows the Michaelis-Menten mechanism, then monostationarity is preserved. Surprisingly, a single GK loop with a complex enzyme mechanism under non-mass action monotone kinetics can have more than one positive steady state (multistationarity). The broader interplay between network structure, enzyme mechanism and kinetics remains an intriguing open problem.

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.