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.     

Robust oscillations within the interlocked feedback model of Drosophila circadian rhythm

A mechanism for generating circadian rhythms has been of major interest in recent years. After the discovery of per and tim, a model with a simple feedback loop involving per and tim has been proposed. However, it is recognized that the simple feedback model cannot account for phenotypes generated by various mutants. A recent report by Glossop, Lyons & Hardin [Science286, 766 (1999)] on Drosophila suggests involvement of another feedback loop by dClk that is interlocked with per-tim feedback loop. In order to examine whether interlocked feedback loops can be a basic mechanism for circadian rhythms, a mathematical model was created and examined. Through extensive simulation and mathematical analysis, it was revealed that the interlocked feedback model accounts for the observations that are not explained by the simple feedback model. Moreover, the interlocked feedback model has robust properties in oscillations.     

A note on the reafference principle

Two alternative circuits to realize the reafference principle are considered and quantitatively analyzed. Both are combinations of a conventional control loop with negative feedback and linear transfer functions, as well as of an efference copy branch. The feedback control loop compensates for passive movements of the body or of its parts, and generates active movements, whenever the set point differs from zero. The efference copy branch should eliminate sensory messages to higher brain centres during active body movements and, thus, mediate perceptual stability. In one of the combinations, discussed also briefly by Mittelstaedt (1971), the efference copy branch interacts with and thus modifies the properties of the feedback loop. The performance of this circuit is very sensitive to variations of the system parameters and, therefore, requires precise adjustment. When the transfer function of the efference copy branch exactly matches that of the feedback loop and its gain amounts to 1, this circuit performs as a control loop with an integrator in the negative feedback branch: there is no steady state control error. However, for certain parameter combination the circuit becomes unstable. In the alternative circuit proposed here, the efference copy branch does not interfere with the feedback loop. It is robust against parameter variations. The transient properties of both circuits are described under simplified assumptions regarding the linear transfer functions in the different branches.     

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.     

Positive feedback in a brainstem tactile sensorimotor loop

The trigeminal loop in the brainstem comprises the innermost level of sensorimotor feedback in the rat vibrissa system. Anatomy suggests that this loop relays tactile information from the vibrissae to the motoneurons that control vibrissa movement. We demonstrate, using in vitro and in vivo recordings, that the trigeminal loop consists of excitatory pathways from vibrissa sensory inputs to vibrissa motoneurons in the facial nucleus. We further show that the trigeminal loop implements a rapidly depressing reflex that provides positive sensory feedback to the vibrissa musculature during simulated whisking and contact. On the basis of these findings, we propose that the trigeminal loop provides an enhancement of vibrissa muscle tone upon contact during active touch.     

Time constants of vestibular nuclei neurons in the goldfish: A model with ocular proprioception

A simple model of the vestibular-ocular reflex with a proprioceptive eye velocity feedback loop is used to simulate recent data on the vestibular responses of neurons in the vestibular nuclei of spinal goldfish. The data support the hypothesis that a proprioceptive feedback loop elongates the vestibular nucleus time constant to equal that of the slow phase eye movements of vestibular nystagmus.     

Caspase-9 processing by caspase-3 via a feedback amplification loop in vivo

In contrast to the autoprocessing of caspase-9, little is known about the biological significance of caspase-9 processing by caspase-3 via a feedback loop in vivo. We prepared antisera against mouse caspase-9 cleavage sites so that only the activated form of mouse caspase-9 was recognized. Using these antisera and caspase-9- and caspase-3-deficient mouse embryonic fibroblasts, we demonstrated that mouse caspase-9 is initially autoprocessed at D(353) and D(368) at low levels during staurosporine-induced apoptosis, whereupon the D(368) and D(168) sites are preferentially processed over D(353) by activated caspase-3 as part of a feedback amplification loop. Ac-DEVD-MCA (caspase-3-like) and Ac-LEHD-MCA (caspase-9-like) cleavage activities clearly showed that caspase-9 autoprocessing was necessary for the activation of caspase-3, whereas full activation of caspase-3 and caspase-9 was achieved only through the feedback amplification loop. This feedback amplification loop also played a predominant role during programmed cell death of dorsal root ganglia neurons at mouse embryonic day 11.5.     

Positive Feedback Promotes Oscillations in Negative Feedback Loops

A simple three-component negative feedback loop is a recurring motif in biochemical oscillators. This motif oscillates as it has the three necessary ingredients for oscillations: a three-step delay, negative feedback, and nonlinearity in the loop. However, to oscillate, this motif under the common Goodwin formulation requires a high degree of cooperativity (a measure of nonlinearity) in the feedback that is biologically “unlikely.” Moreover, this recurring negative feedback motif is commonly observed augmented by positive feedback interactions. Here we show that these positive feedback interactions promote oscillation at lower degrees of cooperativity, and we can thus unify several common kinetic mechanisms that facilitate oscillations, such as self-activation and Michaelis-Menten degradation. The positive feedback loops are most beneficial when acting on the shortest lived component, where they function by balancing the lifetimes of the different components. The benefits of multiple positive feedback interactions are cumulative for a majority of situations considered, when benefits are measured by the reduction in the cooperativity required to oscillate. These positive feedback motifs also allow oscillations with longer periods than that determined by the lifetimes of the components alone. We can therefore conjecture that these positive feedback loops have evolved to facilitate oscillations at lower, kinetically achievable, degrees of cooperativity. Finally, we discuss the implications of our conclusions on the mammalian molecular clock, a system modeled extensively based on the three-component negative feedback loop.     

Dynamics of cyclic GMP synthesis in retinal rods

In retinal rods, Ca(2+) exerts negative feedback control on cGMP synthesis by guanylate cyclase (GC). This feedback loop was disrupted in mouse rods lacking guanylate cyclase activating proteins GCAP1 and GCAP2 (GCAPs(-/-)). Comparison of the behavior of wild-type and GCAPs(-/-) rods allowed us to investigate the role of the feedback loop in normal rod function. We have found that regulation of GC is apparently the only Ca(2+) feedback loop operating during the single photon response. Analysis of the rods' light responses and cellular dark noise suggests that GC normally responds to light-driven changes in [Ca(2+)] rapidly and highly cooperatively. Rapid feedback to GC speeds the rod's temporal responsiveness and improves its signal-to-noise ratio by minimizing fluctuations in cGMP.     

Feedback regulation of growth hormone synthesis and secretion in fish and the emerging concept of intrapituitary feedback loop

Growth hormone (GH) is known to play a key role in the regulation of body growth and metabolism. Similar to mammals, GH secretion in fish is under the control of hypothalamic factors. Besides, signals generated within the pituitary and/or from peripheral tissues/organs can also exert a feedback control on GH release by effects acting on both the hypothalamus and/or anterior pituitary. Among these feedback signals, the functional role of IGF is well conserved from fish to mammals. In contrast, the effects of steroids and thyroid hormones are more variable and appear to be species-specific. Recently, a novel intrapituitary feedback loop regulating GH release and GH gene expression has been identified in fish. This feedback loop has three functional components: (i) LH induction of GH release from somatotrophs, (ii) amplification of GH secretion by GH autoregulation in somatotrophs, and (iii) GH feedback inhibition of LH release from neighboring gonadotrophs. In this article, the mechanisms for feedback control of GH synthesis and secretion are reviewed and functional implications of this local feedback loop are discussed. This intrapituitary feedback loop may represent a new facet of pituitary research with potential applications in aquaculture and clinical studies.     

Pushed to the edge: Spatial sorting can slow down invasions

Our ability to understand population spread dynamics is complicated by rapid evolution, which renders simple ecological models insufficient. If dispersal ability evolves, more highly dispersive individuals may arrive at the population edge than less dispersive individuals (spatial sorting), accelerating spread. If individuals at the low-density population edge benefit (escape competition), high dispersers have a selective advantage (spatial selection). These two processes are often described as forming a positive feedback loop; they reinforce each other, leading to faster spread. Although spatial sorting is close to universal, this form of spatial selection is not: low densities can be detrimental for organisms with Allee effects. Here, we present two conceptual models to explore the feedback loops that form between spatial sorting and spatial selection. We show that the presence of an Allee effect can reverse the positive feedback loop between spatial sorting and spatial selection, creating a negative feedback loop that slows population spread.     

per mRNA cycling is locked to lights-off under photoperiodic conditions that support circadian feedback loop function.

Circadian fluctuations in per mRNA and protein are central to the operation of a negative feedback loop that is necessary for setting the free-running period and for entraining the circadian oscillator to light-dark cycles. In this study, per mRNA cycling and locomotor activity rhythms were measured under different light and dark cycling regimes to determine how photoperiods affect the molecular feedback loop and circadian behavior, respectively. These experiments reveal that per mRNA peaks in abundance 4 h after lights-off in photoperiods of < or = 16 h, that, phase shifts in per mRNA cycling and behavioral rhythmicity occur rapidly after flies are transferred from one photoperiod to another, and that photoperiods longer than 20 h abolish locomotor activity rhythms and leave per mRNA at a median constitutive level. These results indicate that the per feedback loop uses lights-off as a phase reference point and suggest (along with previous findings for per01 and tim01) that per mRNA cycling is not regulated via simple negative feedback from the per protein.     

Shake it, don't break it: positive feedback and the evolution of oscillator design

In cell cycle control, a negative feedback oscillator design is shown to be reinforced with a positive feedback loop, giving a robust oscillatory architecture that is surprisingly common in biology.     

Pulsed feedback defers cellular differentiation

Environmental signals induce diverse cellular differentiation programs. In certain systems, cells defer differentiation for extended time periods after the signal appears, proliferating through multiple rounds of cell division before committing to a new fate. How can cells set a deferral time much longer than the cell cycle? Here we study Bacillus subtilis cells that respond to sudden nutrient limitation with multiple rounds of growth and division before differentiating into spores. A well-characterized genetic circuit controls the concentration and phosphorylation of the master regulator Spo0A, which rises to a critical concentration to initiate sporulation. However, it remains unclear how this circuit enables cells to defer sporulation for multiple cell cycles. Using quantitative time-lapse fluorescence microscopy of Spo0A dynamics in individual cells, we observed pulses of Spo0A phosphorylation at a characteristic cell cycle phase. Pulse amplitudes grew systematically and cell-autonomously over multiple cell cycles leading up to sporulation. This pulse growth required a key positive feedback loop involving the sporulation kinases, without which the deferral of sporulation became ultrasensitive to kinase expression. Thus, deferral is controlled by a pulsed positive feedback loop in which kinase expression is activated by pulses of Spo0A phosphorylation. This pulsed positive feedback architecture provides a more robust mechanism for setting deferral times than constitutive kinase expression. Finally, using mathematical modeling, we show how pulsing and time delays together enable “polyphasic” positive feedback, in which different parts of a feedback loop are active at different times. Polyphasic feedback can enable more accurate tuning of long deferral times. Together, these results suggest that Bacillus subtilis uses a pulsed positive feedback loop to implement a “timer” that operates over timescales much longer than a cell cycle.     

Properties of the feedback system controlling the coxa-trochanter joint in the stick insect Carausius morosus

The angle of the coxa-trochanter (C-T) joint in the stick insect Carausius morosus is controlled by a negative feedback mechanism. It is shown that the trochanteral hair plate alone functions as the feedback transducer and that the rhomboid hair plate is not involved in the feedback loop.The properties of the C-T control system were investigated by means of force measurements. The results cannot be adequately described in all details by either a fractional differentiator model, a model which fits many sensory systems, or a nonlinear bandpass filter, a model which fits the force response of the femur-tibia feedback loop. The fractional differentiator model adequately describes the frequency response of the open-loop system to sinusoidal stimulation with 34 deg stimulus amplitude. However, the responses to sinusoidal and steplike stimulation with 10 deg stimulus amplitude do not fit this model. They are better described by the model of a nonlinear bandpass filter.The possible contribution of mechanical properties of the musculature and the joint to the total force response is discussed. It is suggested that cocontractions occurring at higher stimulus frequencies alters the muscle properties and enables the animal to respond to stimulus frequencies above the upper corner frequency of the active feedback loop.     

Efficacy of EEG biofeedback procedures in correcting stress-related functional disorders

A promising approach to nondrug correction of human stress-induced functional disorders based on double EEG biofeedback (EEGBF) has been substantiated and experimentally tested. According to this approach, narrow-band EEG oscillators that are characteristic of each patient and detectable in real-time are simultaneously used in two independent feedback loops: the traditional adaptive biofeedback loop and an additional resonance stimulation loop. In the latter loop, the feedback signals from individual narrow-band EEG oscillators serve for automatic modulation of the parameters of sensory stimuli and are not perceived consciously by the subject. The combined use of the active (conscious perception) and passive (automatic modulation) feedback signals from narrow-band EEG components of the patient have been demonstrated to offer the possibility of a substantial increase in the efficacy of EEGBF.     

TERRA,CpG methylation,and telomere heterochromatin: Lessons from ICF syndrome cells

Self-reinforcing negative feedback loops are commonly observed in biological systems. RNA-mediated negative feedback loops have been described in the formation of heterochromatin at centromeres in fission yeast and the inactive X chromosome in mammalian cells. The telomere repeat-containing RNA (TERRA) has also been implicated in the formation of telomeric heterochromatin through a self-reinforcing negative feedback loop. In cells derived from human ICF syndrome, TERRA levels are abnormally elevated and telomeres are abnormally shortened. We now show that telomere heterochromatin is also abnormal in ICF cells. We propose that ICF cells fail to reinforce the TERRA-dependent negative feedback loop as a result of the inability to establish heterochromatin at subtelomeres. This failure is likely due to the lack of DNMT3b and DNA methylation, which is a genetic lesion associated with ICF syndrome. Failure of this feedback mechanism leads to catastrophic telomere dysfunction and chromosome instability.