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.

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.

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.

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.

Metabolic basis of brain-like electrical signalling in bacterial communities.

Information processing in the mammalian brain relies on a careful regulation of the membrane potential dynamics of its constituent neurons, which propagates across the neuronal tissue via electrical signalling. We recently reported the existence of electrical signalling in a much simpler organism, the bacterium Bacillus subtilis. In dense bacterial communities known as biofilms, nutrient-deprived B. subtilis cells in the interior of the colony use electrical communication to transmit stress signals to the periphery, which interfere with the growth of peripheral cells and reduce nutrient consumption, thereby relieving stress from the interior. Here, we explicitly address the interplay between metabolism and electrophysiology in bacterial biofilms, by introducing a spatially extended mathematical model that combines the metabolic and electrical components of the phenomenon in a discretized reaction-diffusion scheme. The model is experimentally validated by environmental and genetic perturbations, and confirms that metabolic stress is transmitted through the bacterial population via a potassium wave. Interestingly, this behaviour is reminiscent of cortical spreading depression in the brain, characterized by a wave of electrical activity mediated by potassium diffusion that has been linked to various neurological disorders, calling for future studies on the evolutionary link between the two phenomena. This article is part of the theme issue 'Liquid brains, solid brains: How distributed cognitive architectures process information'.

Self-Amplifying Pulsatile Protein Dynamics without Positive Feedback.

Many proteins exhibit dynamic activation patterns in the form of irregular pulses. Such behavior is typically attributed to a combination of positive and negative feedback loops in the underlying regulatory network. However, the presence of positive feedbacks is difficult to demonstrate unequivocally, raising the question of whether stochastic pulses can arise from negative feedback only. Here, we use the protein kinase A (PKA) system, a key regulator of the yeast pulsatile transcription factor Msn2, as a case example to show that irregular pulses of protein activity can arise from a negative feedback loop alone. Simplification to two variables reveals that a combination of zero-order ultrasensitivity, timescale separation between the activator and the repressor, and an effective delay in the feedback are sufficient to amplify a perturbation into a pulse. The same circuit topology can account for both activation and inactivation pulses, pointing toward a general mechanism of stochastic pulse generation.

Bistable emergence of oscillations in growing Bacillus subtilis biofilms.

Biofilm communities of Bacillus subtilis bacteria have recently been shown to exhibit collective growth-rate oscillations mediated by electrochemical signaling to cope with nutrient starvation. These oscillations emerge once the colony reaches a large enough number of cells. However, it remains unclear whether the amplitude of the oscillations, and thus their effectiveness, builds up over time gradually or if they can emerge instantly with a nonzero amplitude. Here we address this question by combining microfluidics-based time-lapse microscopy experiments with a minimal theoretical description of the system in the form of a delay-differential equation model. Analytical and numerical methods reveal that oscillations arise through a subcritical Hopf bifurcation, which enables instant high-amplitude oscillations. Consequently, the model predicts a bistable regime where an oscillating and a nonoscillating attractor coexist in phase space. We experimentally validate this prediction by showing that oscillations can be triggered by perturbing the media conditions, provided the biofilm size lies within an appropriate range. The model also predicts that the minimum size at which oscillations start decreases with stress, a fact that we also verify experimentally. Taken together, our results show that collective oscillations in cell populations can emerge suddenly with nonzero amplitude via a discontinuous transition.

Coupling between distant biofilms and emergence of nutrient time-sharing.

Bacteria within communities can interact to organize their behavior. It has been unclear whether such interactions can extend beyond a single community to coordinate the behavior of distant populations. We discovered that two Bacillus subtilis biofilm communities undergoing metabolic oscillations can become coupled through electrical signaling and synchronize their growth dynamics. Coupling increases competition by also synchronizing demand for limited nutrients. As predicted by mathematical modeling, we confirm that biofilms resolve this conflict by switching from in-phase to antiphase oscillations. This results in time-sharing behavior, where each community takes turns consuming nutrients. Time-sharing enables biofilms to counterintuitively increase growth under reduced nutrient supply. Distant biofilms can thus coordinate their behavior to resolve nutrient competition through time-sharing, a strategy used in engineered systems to allocate limited resources.

Diversity of fate outcomes in cell pairs under lateral inhibition.

Cell fate determination by lateral inhibition via Notch/Delta signalling has been extensively studied. Most formalised models consider Notch/Delta interactions in fields of cells, with parameters that typically lead to symmetry breaking of signalling states between neighbouring cells, commonly resulting in salt-and-pepper fate patterns. Here, we consider the case of signalling between isolated cell pairs, and find that the bifurcation properties of a standard mathematical model of lateral inhibition can lead to stable symmetric signalling states. We apply this model to the adult intestinal stem cell (ISC) of Drosophila, the fate of which is stochastic but dependent on the Notch/Delta pathway. We observe a correlation between signalling state in cell pairs and their contact area. We interpret this behaviour in terms of the properties of our model in the presence of population variability in contact areas, which affects the effective signalling threshold of individual cells. Our results suggest that the dynamics of Notch/Delta signalling can contribute to explain stochasticity in stem cell fate decisions, and that the standard model for lateral inhibition can account for a wider range of developmental outcomes than previously considered.