Positive feedback may cause the biphasic response observed in the chemoattractant-induced response of Dictyostelium cells

A unified approach to dissecting biphasic responses in cell signaling

Biphasic responses are encountered at all levels in biological systems. At the cellular level, biphasic dose-responses are widely encountered in cell signaling and post-translational modification systems and represent safeguards against overactivation or overexpression of species. In this paper, we provide a unified theoretical synthesis of biphasic responses in cell signaling systems, by assessing signaling systems ranging from basic biochemical building blocks to canonical network structures to well-characterized exemplars on one hand, and examining different types of doses on the other. By using analytical and computational approaches applied to a range of systems across levels (described by broadly employed models), we reveal (i) design principles enabling the presence of biphasic responses, including in almost all instances, an explicit characterization of the parameter space (ii) structural factors which preclude the possibility of biphasic responses (iii) different combinations of the presence or absence of enzyme-biphasic and substrate-biphasic responses, representing safeguards against overactivation and overexpression, respectively (iv) the possibility of broadly robust biphasic responses (v) the complete alteration of signaling behavior in a network due to biphasic interactions between species (biphasic regulation) (vi) the propensity of different co-existing biphasic responses in the Erk signaling network. These results both individually and in totality have a number of important consequences for systems and synthetic biology.

Network regulation meets substrate modification chemistry

Biochemical networks are at the heart of cellular information processing. These networks contain distinct facets: (i) processing of information from the environment via cascades/pathways along with network regulation and (ii) modification of substrates in different ways, to confer protein functionality, stability and processing. While many studies focus on these factors individually, how they interact and the consequences for cellular systems behaviour are poorly understood. We develop a systems framework for this purpose by examining the interplay of network regulation (canonical feedback and feed-forward circuits) and multisite modification, as an exemplar of substrate modification. Using computational, analytical and semi-analytical approaches, we reveal distinct and unexpected ways in which the substrate modification and network levels combine and the emergent behaviour arising therefrom. This has important consequences for dissecting the behaviour of specific signalling networks, tracing the origins of systems behaviour, inference of networks from data, robustness/evolvability and multi-level engineering of biomolecular networks. Overall, we repeatedly demonstrate how focusing on only one level (say network regulation) can lead to profoundly misleading conclusions about all these aspects, and reveal a number of important consequences for experimental/theoretical/data-driven interrogations of cellular signalling systems.

Protein-protein complexes can undermine ultrasensitivity-dependent biological adaptation

Robust perfect adaptation (RPA) is a ubiquitously observed signalling response across all scales of biological organization. A major class of network architectures that drive RPA in complex networks is the Opposer module-a feedback-regulated network into which specialized integral-computing 'opposer node(s)' are embedded. Although ultrasensitivity-generating chemical reactions have long been considered a possible mechanism for such adaptation-conferring opposer nodes, this hypothesis has relied on simplified Michaelian models, which neglect the presence of protein-protein complexes. Here we develop complex-complete models of interlinked covalent-modification cycles with embedded ultrasensitivity, explicitly capturing all molecular interactions and protein complexes. Strikingly, we demonstrate that the presence of protein-protein complexes thwarts the network's capacity for RPA in any 'free' active protein form, conferring RPA capacity instead on the concentration of a larger protein pool consisting of two distinct forms of a single protein. We further show that the presence of enzyme-substrate complexes, even at comparatively low concentrations, play a crucial and previously unrecognized role in controlling the RPA response-significantly reducing the range of network inputs for which RPA can obtain, and imposing greater parametric requirements on the RPA response. These surprising results raise fundamental new questions as to the biochemical requirements for adaptation-conferring Opposer modules within complex cellular networks.

Ultrasensitivity and bistability in covalent-modification cycles with positive autoregulation

Switch-like behaviours in biochemical networks are of fundamental significance in biological signal processing, and exist as two distinct types: ultra-sensitivity and bistability. Here we propose two new models of a reversible covalent-modification cycle with positive autoregulation (PAR), a motif structure that is thought to be capable of both ultrasensitivity and bistability in different parameter regimes. These new models appeal to a modelling framework that we call complex-complete, which accounts fully for the molecular complexities of the underlying signalling mechanisms. Each of the two new models encodes a specific molecular mechanism for PAR. We demonstrate that the modelling simplifications for PAR models that have been used in previous work, which rely on Michaelian approximations, are unable to accurately recapitulate the qualitative signalling responses supported by our detailed models. Strikingly, we show that complex-complete PAR models are capable of new qualitative responses such as one-way switches and a 'prozone' effect, depending on the specific PAR-encoding mechanism, which are not supported by Michaelian simplifications. Our results highlight the critical importance of accurately representing the molecular details of biochemical signalling mechanisms, and strongly suggest that the Michaelian approximation is inadequate for predictive models of enzyme-mediated chemical reactions with added regulations such as PAR.

PID Control as a Process of Active Inference with Linear Generative Models

In the past few decades, probabilistic interpretations of brain functions have become widespread in cognitive science and neuroscience. In particular, the free energy principle and active inference are increasingly popular theories of cognitive functions that claim to offer a unified understanding of life and cognition within a general mathematical framework derived from information and control theory, and statistical mechanics. However, we argue that if the active inference proposal is to be taken as a general process theory for biological systems, it is necessary to understand how it relates to existing control theoretical approaches routinely used to study and explain biological systems. For example, recently, PID (Proportional-Integral-Derivative) control has been shown to be implemented in simple molecular systems and is becoming a popular mechanistic explanation of behaviours such as chemotaxis in bacteria and amoebae, and robust adaptation in biochemical networks. In this work, we will show how PID controllers can fit a more general theory of life and cognition under the principle of (variational) free energy minimisation when using approximate linear generative models of the world. This more general interpretation also provides a new perspective on traditional problems of PID controllers such as parameter tuning as well as the need to balance performances and robustness conditions of a controller. Specifically, we then show how these problems can be understood in terms of the optimisation of the precisions (inverse variances) modulating different prediction errors in the free energy functional.

Adaptation through proportion

Adaptation is a ubiquitous feature in biological sensory and signaling networks. It has been suggested that adaptive systems may follow certain simple design principles across diverse organisms, cells and pathways. One class of networks that can achieve adaptation utilizes an incoherent feedforward control, in which two parallel signaling branches exert opposite but proportional effects on the output at steady state. In this paper, we generalize this adaptation mechanism by establishing a steady-state proportionality relationship among a subset of nodes in a network. Adaptation can be achieved by using any two nodes in the sub-network to respectively regulate the output node positively and negatively. We focus on enzyme networks and first identify basic regulation motifs consisting of two and three nodes that can be used to build small networks with proportional relationships. Larger proportional networks can then be constructed modularly similar to LEGOs. Our method provides a general framework to construct and analyze a class of proportional and/or adaptation networks with arbitrary size, flexibility and versatile functional features.

Non-monotonic Response to Monotonic Stimulus: Regulation of Glyoxylate Shunt Gene-Expression Dynamics in Mycobacterium tuberculosis

Understanding how dynamical responses of biological networks are constrained by underlying network topology is one of the fundamental goals of systems biology. Here we employ monotone systems theory to formulate a theorem stating necessary conditions for non-monotonic time-response of a biochemical network to a monotonic stimulus. We apply this theorem to analyze the non-monotonic dynamics of the σB-regulated glyoxylate shunt gene expression in Mycobacterium tuberculosis cells exposed to hypoxia. We first demonstrate that the known network structure is inconsistent with observed dynamics. To resolve this inconsistency we employ the formulated theorem, modeling simulations and optimization along with follow-up dynamic experimental measurements. We show a requirement for post-translational modulation of σB activity in order to reconcile the network dynamics with its topology. The results of this analysis make testable experimental predictions and demonstrate wider applicability of the developed methodology to a wide class of biological systems.

Fundamental limitation of the instantaneous approximation in fold-change detection models

The phenomenon of fold-change detection, or scale-invariance, is exhibited by a variety of sensory systems, in both bacterial and eukaryotic signalling pathways. It has been often remarked in the systems biology literature that certain systems whose output variables respond at a faster time scale than internal components give rise to an approximate scale-invariant behaviour, allowing approximate fold-change detection in stimuli. This study establishes a fundamental limitation of such a mechanism, showing that there is a minimal fold-change detection error that cannot be overcome, no matter how large the separation of time scales is. To illustrate this theoretically predicted limitation, the authors discuss two common biomolecular network motifs, an incoherent feedforward loop and a feedback system, as well as a published model of the chemotaxis signalling pathway of Dictyostelium discoideum.

Comparison of adaptation motifs: temporal, stochastic and spatial responses

The cells' ability to adapt to changes in the external environment is crucial for the survival of many organisms. There are two broad classes of signalling networks that achieve perfect adaptation. Both rely on complementary regulation of the response by an external signal and an inhibitory process. In one class of systems, inhibition comes about from the response itself, closing a negative feedback (NFB) loop. In the other, the inhibition comes directly from the external signal in what is referred to as an incoherent feedforward (IFF) loop. Although both systems show adaptive behaviour to constant changes in the level of the stimulus, their response to other forms of stimuli can differ. Here the authors consider the respective response to various such disturbances, including ramp increases, removal of the stimulus and pulses. The authors also consider the effect of stochastic fluctuations in signalling that come about from the interaction of the signalling elements. Finally, the authors consider the possible effect of spatially varying signals. The authors show that both the NFB and the IFF motifs can be used to sense static spatial gradients, under a local excitation, global inhibition assumption. The results may help experimentalists develop protocols that can discriminate between the two adaptation motifs.

Building robust functionality in synthetic circuits using engineered feedback regulation

The ability to engineer novel functionality within cells, to quantitatively control cellular circuits, and to manipulate the behaviors of populations, has many important applications in biotechnology and biomedicine. These applications are only beginning to be explored. In this review, we advocate the use of feedback control as an essential strategy for the engineering of robust homeostatic control of biological circuits and cellular populations. We also describe recent works where feedback control, implemented in silico or with biological components, was successfully employed for this purpose.

Actin cytoskeleton of chemotactic amoebae operates close to the onset of oscillations

The rapid reorganization of the actin cytoskeleton in response to external stimuli is an essential property of many motile eukaryotic cells. Here, we report evidence that the actin machinery of chemotactic Dictyostelium cells operates close to an oscillatory instability. When averaging the actin response of many cells to a short pulse of the chemoattractant cAMP, we observed a transient accumulation of cortical actin reminiscent of a damped oscillation. At the single-cell level, however, the response dynamics ranged from short, strongly damped responses to slowly decaying, weakly damped oscillations. Furthermore, in a small subpopulation, we observed self-sustained oscillations in the cortical F-actin concentration. To substantiate that an oscillatory mechanism governs the actin dynamics in these cells, we systematically exposed a large number of cells to periodic pulse trains of different frequencies. Our results indicate a resonance peak at a stimulation period of around 20 s. We propose a delayed feedback model that explains our experimental findings based on a time-delay in the regulatory network of the actin system. To test the model, we performed stimulation experiments with cells that express GFP-tagged fusion proteins of Coronin and actin-interacting protein 1, as well as knockout mutants that lack Coronin and actin-interacting protein 1. These actin-binding proteins enhance the disassembly of actin filaments and thus allow us to estimate the delay time in the regulatory feedback loop. Based on this independent estimate, our model predicts an intrinsic period of 20 s, which agrees with the resonance observed in our periodic stimulation experiments.

Effects of saturation and enzyme limitation in feedforward adaptive signal transduction

In this study, the author examines the effects of saturation and enzyme limitation in temporal and spatial signal transduction in a generic feedforward adaptive module. The feedforward module encompasses a range of temporal and spatial signal processing, and this study systematically examines the effect of enzyme limitation/saturating effects in each of the feedforward pathways, and their interplay. It is found that this saturation makes the adaptation inexact, and this effect is more pronounced for higher levels of input signals. Further, it has a very significant role in affecting the temporal dynamics of this module. In examining the role of saturation in the module response to static gradients, the author finds that in certain cases, saturation can completely alter the gradient response. The author examines various aspects of the response systematically using analytical methods and simulations. Overall the author studies a framework and basis for examining and understanding the roles of saturating effects in multiple pathways involved in adaptive responses and sheds light on the relationship and connection between exact and inexact adaptation.

Open cascades as simple solutions to providing ultrasensitivity and adaptation in cellular signaling

Cell signaling is achieved predominantly by reversible phosphorylation-dephosphorylation reaction cascades. Up until now, circuits conferring adaptation have all required the presence of a cascade with some type of closed topology: negative-feedback loop with a buffering node, or incoherent feed-forward loop with a proportioner node. In this paper--using Goldbeter and Koshland-type expressions--we propose a differential equation model to describe a generic, open signaling cascade that elicits an adaptation response. This is accomplished by coupling N phosphorylation-dephosphorylation cycles unidirectionally, without any explicit feedback loops. Using this model, we show that as the length of the cascade grows, the steady states of the downstream cycles reach a limiting value. In other words, our model indicates that there are a minimum number of cycles required to achieve a maximum in sensitivity and amplitude in the response of a signaling cascade. We also describe for the first time that the phenomenon of ultrasensitivity can be further subdivided into three sub-regimes, separated by sharp stimulus threshold values: OFF, OFF-ON-OFF, and ON. In the OFF-ON-OFF regime, an interesting property emerges. In the presence of a basal amount of activity, the temporal evolution of early cycles yields damped peak responses. On the other hand, the downstream cycles switch rapidly to a higher activity state for an extended period of time, prior to settling to an OFF state (OFF-ON-OFF). This response arises from the changing dynamics between a feed-forward activation module and dephosphorylation reactions. In conclusion, our model gives the new perspective that open signaling cascades embedded in complex biochemical circuits may possess the ability to show a switch-like adaptation response, without the need for any explicit feedback circuitry.

Defining network topologies that can achieve biochemical adaptation

Many signaling systems show adaptation-the ability to reset themselves after responding to a stimulus. We computationally searched all possible three-node enzyme network topologies to identify those that could perform adaptation. Only two major core topologies emerge as robust solutions: a negative feedback loop with a buffering node and an incoherent feedforward loop with a proportioner node. Minimal circuits containing these topologies are, within proper regions of parameter space, sufficient to achieve adaptation. More complex circuits that robustly perform adaptation all contain at least one of these topologies at their core. This analysis yields a design table highlighting a finite set of adaptive circuits. Despite the diversity of possible biochemical networks, it may be common to find that only a finite set of core topologies can execute a particular function. These design rules provide a framework for functionally classifying complex natural networks and a manual for engineering networks. For a video summary of this article, see the PaperFlick file with the Supplemental Data available online.