Robust Concentration and Frequency Control in Oscillatory Homeostats

Homeostatic and adaptive control mechanisms are essential for keeping organisms structurally and functionally stable. Integral feedback is a control theoretic concept which has long been known to keep a controlled variable robustly (i.e. perturbation-independent) at a given set-point by feeding the integrated error back into the process that generates . The classical concept of homeostasis as robust regulation within narrow limits is often considered as unsatisfactory and even incompatible with many biological systems which show sustained oscillations, such as circadian rhythms and oscillatory calcium signaling. Nevertheless, there are many similarities between the biological processes which participate in oscillatory mechanisms and classical homeostatic (non-oscillatory) mechanisms. We have investigated whether biological oscillators can show robust homeostatic and adaptive behaviors, and this paper is an attempt to extend the homeostatic concept to include oscillatory conditions. Based on our previously published kinetic conditions on how to generate biochemical models with robust homeostasis we found two properties, which appear to be of general interest concerning oscillatory and homeostatic controlled biological systems. The first one is the ability of these oscillators (“oscillatory homeostats”) to keep the average level of a controlled variable at a defined set-point by involving compensatory changes in frequency and/or amplitude. The second property is the ability to keep the period/frequency of the oscillator tuned within a certain well-defined range. In this paper we highlight mechanisms that lead to these two properties. The biological applications of these findings are discussed using three examples, the homeostatic aspects during oscillatory calcium and p53 signaling, and the involvement of circadian rhythms in homeostatic regulation.     

Feedback theory and Darwinian evolution.

Feedback loops can have a significant impact on biological systems that are evolving under Darwinian natural selection. Many of the striking and sometimes bizarre patterns that characterize the evolution of such systems have simple, natural explanations that involve the effects of feedback loops. The two fundamental types of feedback loops, positive and negative, have effects that are radically different: negative feedback tends to produce stability and resistance to change; positive feedback produces instability and even catastrophe. Both types of feedback loops are important in biological systems, and both can produce chaos, whose mathematical complexity often produces strange, beautiful and totally unexpected patterns that have only begun to be explored using the computational capabilities of modern electronic computers. An understanding of the patterns that can result from the effects of feedback loops can produce important new insights into the patterns that mark the evolutionary development of biological systems.     

Feedback regulation of EGFR signalling: decision making by early and delayed loops

Human-made information relay systems invariably incorporate central regulatory components, which are mirrored in biological systems by dense feedback and feedforward loops. This type of system control is exemplified by positive and negative feedback loops (for example, receptor endocytosis and dephosphorylation) that enable growth factors and receptor Tyr kinases of the epidermal growth factor receptor (EGFR)/ERBB family to regulate cellular function. Recent studies show that the collection of feedback regulatory loops can perform computational tasks - such as decoding ligand specificity, transforming graded input signals into a digital output and regulating response kinetics. Aberrant signal processing and feedback regulation can lead to defects associated with pathologies such as cancer.     

A Critical Quantity for Noise Attenuation in Feedback Systems

Feedback modules, which appear ubiquitously in biological regulations, are often subject to disturbances from the input, leading to fluctuations in the output. Thus, the question becomes how a feedback system can produce a faithful response with a noisy input. We employed multiple time scale analysis, Fluctuation Dissipation Theorem, linear stability, and numerical simulations to investigate a module with one positive feedback loop driven by an external stimulus, and we obtained a critical quantity in noise attenuation, termed as “signed activation time”. We then studied the signed activation time for a system of two positive feedback loops, a system of one positive feedback loop and one negative feedback loop, and six other existing biological models consisting of multiple components along with positive and negative feedback loops. An inverse relationship is found between the noise amplification rate and the signed activation time, defined as the difference between the deactivation and activation time scales of the noise-free system, normalized by the frequency of noises presented in the input. Thus, the combination of fast activation and slow deactivation provides the best noise attenuation, and it can be attained in a single positive feedback loop system. An additional positive feedback loop often leads to a marked decrease in activation time, decrease or slight increase of deactivation time and allows larger kinetic rate variations for slow deactivation and fast activation. On the other hand, a negative feedback loop may increase the activation and deactivation times. The negative relationship between the noise amplification rate and the signed activation time also holds for the six other biological models with multiple components and feedback loops. This principle may be applicable to other feedback systems.     

Modeling cortisol dynamics in the neuro-endocrine axis distinguishes normal, depression, and post-traumatic stress disorder (PTSD) in humans

Cortisol, secreted in the adrenal cortex in response to stress, is an informative biomarker that distinguishes anxiety disorders such as major depression and post-traumatic stress disorder (PTSD) from normal subjects. Yehuda et al. proposed a hypothesis that, in humans, the hypersensitive hypothalamus-pituitary-adrenal (HPA) axis is responsible for the occurrence of differing levels of cortisol in anxiety disorders. Specifically, PTSD subjects have lower cortisol levels during the late subjective night in comparison to normal subjects, and this was assumed to occur due to strong negative feedback loops in the HPA axis. In the present work, to address this hypothesis, we modeled the cortisol dynamics using nonlinear ordinary differential equations and estimated the kinetic parameters of the model to fit the experimental data of three categories, namely, normal, depressed, and PTSD human subjects. We concatenated the subjects (n = 3) in each category and created a model subject (n = 1) without considering the patient-to-patient variability in each case. The parameters of the model for the three categories were simultaneously obtained through global optimization. Bifurcation analysis carried out with the optimized parameters exhibited two supercritical Hopf points and, for the choice of parameters, the oscillations were found to be circadian in nature. The fitted kinetic parameters indicate that PTSD subjects have a strong negative feedback loop and, as a result, the predicted oscillating cortisol levels are extremely low at the nadir in contrast to normal subjects, albeit within the endocrinologic range. We also simulated the phenotypes for each of the categories and, as observed in the clinical data of PTSD patients, the simulated cortisol levels are consistently low at the nadir, and correspondingly the negative feedback was found to be extremely strong. These results from the model support the hypothesis that high stress intensity and strong negative feedback loop may cause hypersensitive neuro-endocrine axis that results in hypocortisolemia in PTSD.     

Metabolic feedback in mammalian endocrine systems.

Information processing through feedback loops is an integral part of most endocrine systems, and ranges from simple negative loops to complex combinations of negative and positive loops. Moreover, feedback may occur at local (paracrine) or long-distance sites, and with multiple time-domains. Traditionally, feedback is visualized as one hormone stimulating release of a second hormone, which then circulates in the blood to carry out various biological activities, one of which is to inhibit further secretion of the first hormone. This represents a fail-safe mechanism to protect the organism against the potentially damaging effects of uncontrolled secretion of many of the common hormones, some of which are highly catabolic or anabolic. However, it is becoming increasingly apparent that the products of catabolism and anabolism may themselves participate in the feedback process in either a feed-forward or feedback manner. For example, free fatty acids are liberated by the action of growth hormone, and in turn are potent inhibitors of growth hormone secretion (feedback). On the other hand, stress activates adrenal cortical and medullary secretion, which also promotes lipolysis, but in this case the liberated free fatty acids may actually stimulate the system further (feed-forward). Similarly, glucose has been shown to directly inhibit the activity of several different endocrine pathways, and must now be considered an integral part of the overall regulatory mechanism involved in fine-tuning secretion and possibly production of hormones. By constructing models of feedback of increasing complexity, it is possible to make predictions about previously unrecognized relationships between hormones and products of metabolism.     

Review: Endocrine pathways to regulate calcium homeostasis around parturition and the prevention of hypocalcemia in periparturient dairy cows

Calcium homeostasis is crucial for the normal function of the organism. Parathyroid hormone, calcitriol and calcitonin play critical roles in the homeostatic regulation of calcium. Serotonin and prolactin have also been shown to be involved in the regulation of calcium homeostasis. In modern dairy cows, the endocrine pathways controlling calcium homeostasis during non-lactating and non-pregnant physiological states are unable to fully support the increased demand of calcium required for milk synthesis at the onset of lactation. This review describes different endocrine systems associated with the regulation of calcium homeostasis in mammalian species around parturition with special focus on dairy cows. Additionally, classic and novel strategies to reduce the incidence of hypocalcemia in parturient dairy cows are discussed.     

Positive feedback in cellular control systems

Feedback loops have been identified in a variety of regulatory systems and organisms. While feedback loops of the same type (negative or positive) tend to have properties in common, they can play distinctively diverse roles in different regulatory systems, where they can affect virulence in a pathogenic bacterium, maturation patterns of vertebrate oocytes and transitions through cell cycle phases in eukaryotic cells. This review focuses on the properties and functions of positive feedback in biological systems, including bistability, hysteresis and activation surges.     

Boolean dynamics of biological networks with multiple coupled feedback loops

Boolean networks have been frequently used to study the dynamics of biological networks. In particular, there have been various studies showing that the network connectivity and the update rule of logical functions affect the dynamics of Boolean networks. There has been, however, relatively little attention paid to the dynamical role of a feedback loop, which is a circular chain of interactions between Boolean variables. We note that such feedback loops are ubiquitously found in various biological systems as multiple coupled structures and they are often the primary cause of complex dynamics. In this article, we investigate the relationship between the multiple coupled feedback loops and the dynamics of Boolean networks. We show that networks have a larger proportion of basins corresponding to fixed-point attractors as they have more coupled positive feedback loops, and a larger proportion of basins for limit-cycle attractors as they have more coupled negative feedback loops.     

Coupled positive and negative feedback circuits form an essential building block of cellular signaling pathways

Cellular circuits have positive and negative feedback loops that allow them to respond properly to noisy external stimuli. It is intriguing that such feedback loops exist in many cases in a particular form of coupled positive and negative feedback loops with different time delays. As a result of our mathematical simulations and investigations into various experimental evidences, we found that such coupled feedback circuits can rapidly turn on a reaction to a proper stimulus, robustly maintain its status, and immediately turn off the reaction when the stimulus disappears. In other words, coupled feedback loops enable cellular systems to produce perfect responses to noisy stimuli with respect to signal duration and amplitude. This suggests that coupled positive and negative feedback loops form essential signal transduction motifs in cellular signaling systems.     

Introducing temperature‐compensation in any reaction kinetic oscillator model

Abstract

The positive and negative feedback loops in oscillatory reactions provide a basis for obtaining temperature‐compensation in any reaction‐kinetic model of chemical or biological oscillators. The present paper shows that positive and negative feedback reactions play the role of “opposing reactions”; whose existence was suggested by Hastings and Sweeney for more than 30 years ago. The principle is illustrated with the Brusselator model.     

Noise Propagation in Gene Regulation Networks Involving Interlinked Positive and Negative Feedback Loops

It is well known that noise is inevitable in gene regulatory networks due to the low-copy numbers of molecules and local environmental fluctuations. The prediction of noise effects is a key issue in ensuring reliable transmission of information. Interlinked positive and negative feedback loops are essential signal transduction motifs in biological networks. Positive feedback loops are generally believed to induce a switch-like behavior, whereas negative feedback loops are thought to suppress noise effects. Here, by using the signal sensitivity (susceptibility) and noise amplification to quantify noise propagation, we analyze an abstract model of the Myc/E2F/MiR-17-92 network that is composed of a coupling between the E2F/Myc positive feedback loop and the E2F/Myc/miR-17-92 negative feedback loop. The role of the feedback loop on noise effects is found to depend on the dynamic properties of the system. When the system is in monostability or bistability with high protein concentrations, noise is consistently suppressed. However, the negative feedback loop reduces this suppression ability (or improves the noise propagation) and enhances signal sensitivity. In the case of excitability, bistability, or monostability, noise is enhanced at low protein concentrations. The negative feedback loop reduces this noise enhancement as well as the signal sensitivity. In all cases, the positive feedback loop acts contrary to the negative feedback loop. We also found that increasing the time scale of the protein module or decreasing the noise autocorrelation time can enhance noise suppression; however, the systems sensitivity remains unchanged. Taken together, our results suggest that the negative/positive feedback mechanisms in coupled feedback loop dynamically buffer noise effects rather than only suppressing or amplifying the noise.     

Positive feedback regulates switching of phosphate transporters in S. cerevisiae

The regulation of transporters by nutrient-responsive signaling pathways allows cells to tailor nutrient uptake to environmental conditions. We investigated the role of feedback generated by transporter regulation in the budding yeast phosphate-responsive signal transduction (PHO) pathway. Cells starved for phosphate activate feedback loops that regulate high- and low-affinity phosphate transport. We determined that positive feedback is generated by PHO pathway-dependent upregulation of Spl2, a negative regulator of low-affinity phosphate uptake. The interplay of positive and negative feedback loops leads to bistability in phosphate transporter usage--individual cells express predominantly either low- or high-affinity transporters, both of which can yield similar phosphate uptake capacity. Cells lacking the high-affinity transporter, and associated negative feedback, exhibit phenotypes that arise from hysteresis due to unopposed positive feedback. In wild-type cells, population heterogeneity generated by feedback loops may provide a strategy for anticipating changes in environmental phosphate levels.     

Involvement of IL-2 in homeostasis of regulatory T cells: the IL-2 cycle

A large body of evidence on the activity of regulatory T (Treg) cells was gathered during the last decade, and a similar number of reviews and opinion papers attempted to integrate the experimental findings. The abundant literature clearly delineates an exciting area of research but also underlines some major controversies. A linear cause-result interpretation of experimental maneuvers often ignores the fact that the activity of Treg cells is orchestrated with the effector T (Teff) cells within an intricate network of physiological immune homeostasis. Every modulation of the activity of the effector (cytotoxic) immune system revolves to affect the activity of regulatory (suppressive) cells through elaborate feedback loops of negative and positive regulation. The lack of IL-2 production by innate Treg cells makes this cytokine a prime coupler of the effector and suppressive mechanisms. Here we attempt to integrate evidence that delineates the involvement of IL-2 in primary and secondary feedback loops that regulate the activity of suppressive cells within the elaborate network of physiological immune homeostasis.     

Trade-off between responsiveness and noise suppression in biomolecular system responses to environmental cues

When living systems detect changes in their external environment their response must be measured to balance the need to react appropriately with the need to remain stable, ignoring insignificant signals. Because this is a fundamental challenge of all biological systems that execute programs in response to stimuli, we developed a generalized time-frequency analysis (TFA) framework to systematically explore the dynamical properties of biomolecular networks. Using TFA, we focused on two well-characterized yeast gene regulatory networks responsive to carbon-source shifts and a mammalian innate immune regulatory network responsive to lipopolysaccharides (LPS). The networks are comprised of two different basic architectures. Dual positive and negative feedback loops make up the yeast galactose network; whereas overlapping positive and negative feed-forward loops are common to the yeast fatty-acid response network and the LPS-induced network of macrophages. TFA revealed remarkably distinct network behaviors in terms of trade-offs in responsiveness and noise suppression that are appropriately tuned to each biological response. The wild type galactose network was found to be highly responsive while the oleate network has greater noise suppression ability. The LPS network appeared more balanced, exhibiting less bias toward noise suppression or responsiveness. Exploration of the network parameter space exposed dramatic differences in system behaviors for each network. These studies highlight fundamental structural and dynamical principles that underlie each network, reveal constrained parameters of positive and negative feedback and feed-forward strengths that tune the networks appropriately for their respective biological roles, and demonstrate the general utility of the TFA approach for systems and synthetic biology.