Chaos in multi-looped negative feedback systems

Non-linear control systems with multiple negative feedback loops display periodicity, quasiperiodicity and period-doubling bifurcations leading to chaos. The possibility that normal fluctuations in physiological control may result from deterministic chaos in multi-looped negative feedback systems is discussed.     

Roles of positive and negative feedback in biological systems

We discuss the influence of positive and negative feedback on the stability of a system, which is not clear-cut, and involves complex, mathematical problems. We show in particular that positive feedback can have a stabilising effect on some systems. We also point out the role that positive feedback plays in the digital treatment of signals required by cellular signalling, drawing on analogies from electronics, and the role that negative feedback plays in making a system robust against alteration of its parameters. Both positive and negative feedback can be seen as important enhancers of the properties of biological systems.     

Oscillations in MAPK cascade triggered by two distinct designs of coupled positive and negative feedback loops

ABSTRACT: BACKGROUND: Feedback loops, both positive and negative are embedded in the Mitogen Activated Protein Kinase (MAPK) cascade. In the three layer MAPK cascade, both feedback loops originate from the terminal layer and their sites of action are either of the two upstream layers. Recent studies have shown that the cascade uses coupled positive and negative feedback loops in generating oscillations. Two plausible designs of coupled positive and negative feedback loops can be elucidated from the literature; in one design the positive feedback precedes the negative feedback in the direction of signal flow and vice-versa in another. But it remains unexplored how the two designs contribute towards triggering oscillations in MAPK cascade. Thus it is also not known how amplitude, frequency, robustness or nature (analogous/digital) of the oscillations would be shaped by these two designs. RESULTS: We built two models of MAPK cascade that exhibited oscillations as function of two underlying designs of coupled positive and negative feedback loops. Frequency, amplitude and nature (digital/analogous) of oscillations were found to be differentially determined by each design. It was observed that the positive feedback emerging from an oscillating MAPK cascade and functional in an external signal processing module can trigger oscillations in the target module, provided that the target module satisfy certain parametric requirements. The augmentation of the two models was done to incorporate the nuclear-cytoplasmic shuttling of cascade components followed by induction of a nuclear phosphatase. It revealed that the fate of oscillations in the MAPK cascade is governed by the feedback designs. Oscillations were unaffected due to nuclear compartmentalization owing to one design but were completely abolished in the other case. CONCLUSION: The MAPK cascade can utilize two distinct designs of coupled positive and negative feedback loops to trigger oscillations. The amplitude, frequency and robustness of the oscillations in presence or absence of nuclear compartmentalization were differentially determined by two designs of coupled positive and negative feedback loops. A positive feedback from an oscillating MAPK cascade was shown to induce oscillations in an external signal processing module, uncovering a novel regulatory aspect of MAPK signal processing.     

Intrinsic negative feedback governs activation surge in two-component regulatory systems

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Highlights? Activation surge reconstituted in vitro with purified SK and RR ? ADP can be endogenously generated from ATP to stimulate an SK's phosphatase activity ? SK's affinity for ADP governs the switch from the kinase to phosphatase state ? Lid region of SK PhoQ controls its phosphatase activity and surge of RR PhoP     

Delay model of circadian gene expression with two negative feedback loops

In this theoretical paper we propose a quantitative minimal model for circadian gene expression based on two negative feedback loops. We perform numerical simulations to analyse its dynamics and parameter sensitivities in free-running conditions, and verify the entrainability by a single periodic driver. We furthermore apply two simultaneously acting external drivers, leading to aperiodic oscillations in the case of a single-loop system. These can be turned into regular periodic oscillations by introduction of a second loop. Our studies confirm the increasing evidence that multiple feedback loops increase the robustness of regulatory systems, and stress the particular situation of systems that are close to transition from free-running oscillation to steady-state behaviour. We discuss possible molecular realisations of the featured feedback loops and suggest the application of complex patterns of external stimulation as a generally useful approach to assess the functionality of models of circadian systems.     

Periodic metabolic systems: Oscillations in multiple-loop negative feedback biochemical control networks

Summary For a general multiple loop feedback inhibition system in which the end product can inhibit any or all of the intermediate reactions it is shown that biologically significant behaviour is always confined to a bounded region of reaction space containing a unique equilibrium. By explicit construction of a Liapunov function for the general n dimensional differential equation it is shown that some values of reaction parameters cause the concentration vector to approach the equilibrium asymptotically for all physically realizable initial conditions. As the parameter values change, periodic solutions can appear within the bounded region. Some information about these periodic solutions can be obtained from the Hopf bifurcation theorem. Alternatively, if specific parameter values are known a numerical method can be used to find periodic solutions and determine their stability by locating a zero of the displacement map. The single loop Goodwin oscillator is analysed in detail. The methods are then used to treat an oscillator with two feedback loops and it is found that oscillations are possible even if both Hill coefficients are equal to one.     

Tailoring T-cell receptor signals by proximal negative feedback mechanisms

The T-cell receptor (TCR) signalling machinery is central in determining the response of a T cell (establishing immunity or tolerance) following exposure to antigen. This process is made difficult by the narrow margin of self and non-self discrimination, and by the complexity of the genetic programmes that are induced for each outcome. Recent studies have identified novel negative feedback mechanisms that are rapidly induced by TCR engagement and that have key roles in the regulation of signal triggering and propagation. In vitro and in vivo data suggest that they are important in determining ligand discrimination by the TCR and in regulating signal output in response to antigen.     

Spatial-temporal dissipative structures arising in open reactive systems with a negative feedback loop

A systematic way of finding the possible spatial-temporal structures that may emerge in open reactive systems coupled with diffusive transport and containing one inhibitory (negative feedback) loop is presented. The method is illustrated on two kinetic models, one used by L. Glass, another by Higgins et al. which we coupled with diffusion. The ranges of cooperativity index and Fick transport coefficients we find for the occurrence of a spatially non-uniform time periodicity are consistent with in vitro experiments.     

Regulation of release factor expression using a translational negative feedback loop: A systems analysis

The essential eukaryote release factor eRF1, encoded by the yeast SUP45 gene, recognizes stop codons during ribosomal translation. SUP45 nonsense alleles are, however, viable due to the establishment of feedback-regulated readthrough of the premature termination codon; reductions in full-length eRF1 promote tRNA-mediated stop codon readthrough, which, in turn, drives partial production of full-length eRF1. A deterministic mathematical model of this eRF1 feedback loop was developed using a staged increase in model complexity. Model predictions matched the experimental observation that strains carrying the mutant SUQ5 tRNA (a weak UAA suppressor) in combination with any of the tested sup45^UAA nonsense alleles exhibit threefold more stop codon readthrough than that of an SUQ5 yeast strain. The model also successfully predicted that eRF1 feedback control in an SUQ5 sup45^UAA mutant would resist, but not completely prevent, imposed changes in eRF1 expression. In these experiments, the introduction of a plasmid-borne SUQ5 copy into a sup45^UAASUQ5 mutant directed additional readthrough and full-length eRF1 expression, despite feedback. Secondly, induction of additional sup45^UAA mRNA expression in a sup45^UAASUQ5 strain also directed increased full-length eRF1 expression. The autogenous sup45 control mechanism therefore acts not to precisely control eRF1 expression, but rather as a damping mechanism that only partially resists changes in release factor expression level. The validated model predicts that the degree of feedback damping (i.e., control precision) is proportional to eRF1 affinity for the premature stop codon. The validated model represents an important tool to analyze this and other translational negative feedback loops.     

Dynamic output feedback stabilization for nonlinear systems based on standard neural network models

A neural-model-based control design for some nonlinear systems is addressed. The design approach is to approximate the nonlinear systems with neural networks of which the activation functions satisfy the sector conditions. A novel neural network model termed standard neural network model (SNNM) is advanced for describing this class of approximating neural networks. Full-order dynamic output feedback control laws are then designed for the SNNMs with inputs and outputs to stabilize the closed-loop systems. The control design equations are shown to be a set of linear matrix inequalities (LMIs) which can be easily solved by various convex optimization algorithms to determine the control signals. It is shown that most neural-network-based nonlinear systems can be transformed into input-output SNNMs to be stabilization synthesized in a unified way. Finally, some application examples are presented to illustrate the control design procedures.     

Brain activity elicited by positive and negative feedback in preschool-aged children

To investigate the processing of positive vs. negative feedback in children aged 4-5 years, we devised a prize-guessing game that is analogous to gambling tasks used to measure feedback-related brain responses in adult studies. Unlike adult studies, the feedback-related negativity (FRN) elicited by positive feedback was as large as that elicited by negative feedback, suggesting that the neural system underlying the FRN may not process feedback valence in early childhood. In addition, positive feedback, compared with negative feedback, evoked a larger P1 over the occipital scalp area and a larger positive slow wave (PSW) over the right central-parietal scalp area. We believe that the PSW is related to emotional arousal and the intensive focus on positive feedback that is present in the preschool and early school years has adaptive significance for both cognitive and emotional development during this period.     

PERpetual motion of the circadian negative feedback loop

Comment on: Chen R, et al. Rhythmic PER Abundance Defines a Critical Nodal Point for Negative Feedback within the Circadian Clock Mechanism. Mol Cell 2009; 36:417-30.