Time-delayed Negative Feedback Research Articles

Arp2/3-dependent endocytosis ensures Cdc42 oscillations by removing Pak1-mediated negative feedback.

The GTPase Cdc42 regulates polarized growth in most eukaryotes. In the bipolar yeast Schizosaccharomyces pombe, Cdc42 activation cycles periodically at sites of polarized growth. These periodic cycles are caused by alternating positive feedback and time-delayed negative feedback loops. At each polarized end, negative feedback is established when active Cdc42 recruits the Pak1 kinase to prevent further Cdc42 activation. It is unclear how Cdc42 activation returns to each end after Pak1-dependent negative feedback. We find that disrupting branched actin-mediated endocytosis disables Cdc42 reactivation at the cell ends. Using experimental and mathematical approaches, we show that endocytosis-dependent Pak1 removal from the cell ends allows the Cdc42 activator Scd1 to return to that end to enable reactivation of Cdc42. Moreover, we show that Pak1 elicits its own removal via activation of endocytosis. These findings provide a deeper insight into the self-organization of Cdc42 regulation and reveal previously unknown feedback with endocytosis in the establishment of cell polarity.

The ups and downs of biological oscillators: a comparison of time-delayed negative feedback mechanisms.

Many biochemical oscillators are driven by the periodic rise and fall of protein concentrations or activities. A negative feedback loop underlies such oscillations. The feedback can act on different parts of the biochemical network. Here, we mathematically compare time-delay models where the feedback affects production and degradation. We show a mathematical connection between the linear stability of the two models, and derive how both mechanisms impose different constraints on the production and degradation rates that allow oscillations. We show how oscillations are affected by the inclusion of a distributed delay, of double regulation (acting on production and degradation) and of enzymatic degradation.

Systematic analysis of negative and positive feedback loops for robustness and temperature compensation in circadian rhythms

Temperature compensation and robustness to biological noise are two key characteristics of the circadian clock. These features allow the circadian pacemaker to maintain a steady oscillation in a wide range of environmental conditions. The presence of a time-delayed negative feedback loop in the regulatory network generates autonomous circadian oscillations in eukaryotic systems. In comparison, the circadian clock of cyanobacteria is controlled by a strong positive feedback loop. Positive feedback loops with substrate depletion can also generate oscillations, inspiring other circadian clock models. What makes a circadian oscillatory network robust to extrinsic noise is unclear. We investigated four basic circadian oscillators with negative, positive, and combinations of positive and negative feedback loops to explore network features necessary for circadian clock resilience. We discovered that the negative feedback loop system performs the best in compensating temperature changes. We also show that a positive feedback loop can reduce extrinsic noise in periods of circadian oscillators, while intrinsic noise is reduced by negative feedback loops.

Protein pattern formation induced by the joint effect of noise and delay in a multi-cellular system

We explore the combined effect of the intrinsic noise and time delay on the spatial pattern formation within the framework of a multi-scale mobile lattice model mimicking two-dimensional epithelium tissues. Every cell is represented by an elastic polygon changing its form and size under pressure from the surrounding cells. The model includes the procedure of minimization of the potential energy of tissue. The protein fluctuations in the tissue are driven by transcription/translation processes in epithelial cells exchanging chemical and mechanical signals. Network architecture includes a simple autorepressor model with time-delayed negative feedback, in which the only gene defines the oscillatory activity. Simultaneously, the expressed protein of the autorepressor acts as a positive regulator of the signaling protein by activating its transcription. The signaling species is assumed to spread from one cell to the other by the diffusion mechanism. We provide both deterministic and stochastic descriptions. The numerical simulation of spatially-extended stochastic oscillations is performed using a generalized Gillespie algorithm. We developed this method earlier to account for the non-Markovian properties of random biochemical events with delay. Finally, we demonstrate that time delay, intrinsic noise, and spatial signaling can cause a system to develop the protein pattern even when its deterministic counterpart exhibits no pattern formation.

Diverse role of decoys on emergence and precision of oscillations in a biomolecular clock

Biomolecular clocks are key drivers of oscillatory dynamics in diverse biological processes including cell-cycle regulation, circadian rhythms, and pattern formation during development. A minimal clock implementation is based on the classical Goodwin oscillator, in which a repressor protein inhibits its own synthesis via time-delayed negative feedback. Clock motifs, however, do not exist in isolation; its components are open to interacting with the complex environment inside cells. For example, there are ubiquitous high-affinity binding sites along the genome, known as decoys, where transcription factors such as repressor proteins can potentially interact. This binding affects the availability of transcription factors and has often been ignored in theoretical studies. How does such genomic decoy binding impact the clock’s robustness and precision? To address this question, we systematically analyze deterministic and stochastic models of the Goodwin oscillator in the presence of reversible binding of the repressor to a finite number of decoy sites. Our analysis reveals that the relative stability of decoy-bound repressors compared to the free repressor plays distinct roles on the emergence and precision of oscillations. Interestingly, active degradation of the bound repressor can induce sustained oscillations that are otherwise absent without decoys. In contrast, decoy abundances can kill oscillation dynamics if the bound repressor is protected from degradation. Taking into account low copy-number fluctuations in clock components, we show that the degradation of the bound repressors enhances precision by attenuating noise in both the amplitude and period of oscillations. Overall, these results highlight the versatile role of otherwise hidden decoys in shaping the stochastic dynamics of biological clocks and emphasize the importance of synthetic decoys in designing robust clocks.

Effect of time-delayed feedback in a bistable system inferred by logic operation

• The effect of time delay in bistable system can be inferred by logic operation. • Time-delayed negative feedback can reduce the width of bistable region. • There is an unstable limit cycle on the directly connected boundary between the oscillatory and suprathreshold region. • In the alternate region, the system is sensitive to parameters and can enter two potential wells alternately with the slight variation in sensitive parameters. The effect of time delay is inferred by studying the influence of time-delayed feedback on the signal threshold of logic input signal, which reflects the width of the bistable region. Time-delayed positive feedback does not alter the width of the bistable region but increases the switching time. Time-delayed negative feedback leads to decreases in the depth of two potential wells and the width of the bistable region of the system, resulting in instability and oscillation of the system. Then, the parameter plane of the system can be divided into suprathreshold, subthreshold, oscillatory and alternate regions. With increasing time delay and negative feedback intensity, the suprathreshold and oscillatory regions continuously squeeze the middle subthreshold region until they are connected. One part of the oscillatory region is directly connected with the suprathreshold region, while the other part is transiently connected with the alternate region. There is an unstable limit cycle on the directly connected boundary. In the alternate region, the system is sensitive to initial values and system parameters and can enter two potential wells alternately with slight variations in sensitive parameters.

An Autonomous Oscillation Times and Executes Centriole Biogenesis

SummaryThe accurate timing and execution of organelle biogenesis is crucial for cell physiology. Centriole biogenesis is regulated by Polo-like kinase 4 (Plk4) and initiates in S-phase when a daughter centriole grows from the side of a pre-existing mother. Here, we show that a Plk4 oscillation at the base of the growing centriole initiates and times centriole biogenesis to ensure that centrioles grow at the right time and to the right size. The Plk4 oscillation is normally entrained to the cell-cycle oscillator but can run autonomously of it—potentially explaining why centrioles can duplicate independently of cell-cycle progression. Mathematical modeling indicates that the Plk4 oscillation can be generated by a time-delayed negative feedback loop in which Plk4 inactivates the interaction with its centriolar receptor through multiple rounds of phosphorylation. We hypothesize that similar organelle-specific oscillations could regulate the timing and execution of organelle biogenesis more generally.

Effects of stochastic time-delayed feedback on a dynamical system modeling a chemical oscillator.

We examine how stochastic time-delayed negative feedback affects the dynamical behavior of a model oscillatory reaction. We apply constant and stochastic time-delayed negative feedbacks to a point Field-Körös-Noyes photosensitive oscillator and compare their effects. Negative feedback is applied in the form of simulated inhibitory electromagnetic radiation with an intensity proportional to the concentration of oxidized light-sensitive catalyst in the oscillator. We first characterize the system under nondelayed inhibitory feedback; then we explore and compare the effects of constant (deterministic) versus stochastic time-delayed feedback. We find that the oscillatory amplitude, frequency, and waveform are essentially preserved when low-dispersion stochastic delayed feedback is used, whereas small but measurable changes appear when a large dispersion is applied.

Delay models for the early embryonic cell cycle oscillator.

Time delays are known to play a crucial role in generating biological oscillations. The early embryonic cell cycle in the frog Xenopus laevis is one such example. Although various mathematical models of this oscillating system exist, it is not clear how to best model the required time delay. Here, we study a simple cell cycle model that produces oscillations due to the presence of an ultrasensitive, time-delayed negative feedback loop. We implement the time delay in three qualitatively different ways, using a fixed time delay, a distribution of time delays, and a delay that is state-dependent. We analyze the dynamics in all cases, and we use experimental observations to interpret our results and put constraints on unknown parameters. In doing so, we find that different implementations of the time delay can have a large impact on the resulting oscillations.

Theoretical investigation on models of circadian rhythms based on dimerization and proteolysis of PER and TIM.

Circadian rhythms of physiology and behavior are widespread\break mechanisms in many organisms. The internal biological rhythms are driven by molecular clocks, which oscillate with a period nearly but not exactly 24 hours. Many classic models of circadian rhythms are based on a time-delayed negative feedback, suggested by the protein products inhibiting transcription of their own genes. In 1999, based on stabilization of PER upon dimerization, Tyson et al. [J. J. Tyson, C. I. Hong, C. D. Thron, B. Novak, Biophys. J. 77 (1999) 2411--2417] proposed a crucial positive feedback to the circadian oscillator. This idea was mathematically expressed in a three-dimensional model. By imposing assumptions that the dimerization reactions were fast and dimeric proteins were in rapid equilibrium, they reduced the model to a pair of nonlinear ordinary differential equations of mRNA and total protein concentrations. Then they used phase plane analysis tools to investigate circadian rhythms. In this paper, the original three-dimensional model is studied. We explore the existence of oscillations and their periods. Much attention is paid to investigate how the periods depend on model parameters. The numerical simulations are in good agreement with their reduced work.

Mathematical model of TGF-\u03b2signalling: feedback coupling is consistent with signal switching

BackgroundTransforming growth factor β (TGF-β) signalling regulates the development of embryos and tissue homeostasis in adults. In conjunction with other oncogenic changes, long-term perturbation of TGF-β signalling is associated with cancer metastasis. Although TGF-β signalling can be complex, many of the signalling components are well defined, so it is possible to develop mathematical models of TGF-β signalling using reduction and scaling methods. The parameterization of our TGF-β signalling model is consistent with experimental data.ResultsWe developed our mathematical model for the TGF-β signalling pathway, i.e. the RF- model of TGF-β signalling, using the “rapid equilibrium assumption” to reduce the network of TGF-β signalling reactions based on the time scales of the individual reactions. By adding time-delayed positive feedback to the inherent time-delayed negative feedback for TGF-β signalling. We were able to simulate the sigmoidal, switch-like behaviour observed for the concentration dependence of long-term (> 3 hours) TGF-β stimulation. Computer simulations revealed the vital role of the coupling of the positive and negative feedback loops on the regulation of the TGF-β signalling system. The incorporation of time-delays for the negative feedback loop improved the accuracy, stability and robustness of the model. This model reproduces both the short-term and long-term switching responses for the intracellular signalling pathways at different TGF-β concentrations. We have tested the model against experimental data from MEF (mouse embryonic fibroblasts) WT, SV40-immortalized MEFs and Gp130 F/F MEFs. The predictions from the RF- model are consistent with the experimental data.ConclusionsSignalling feedback loops are required to model TGF-β signal transduction and its effects on normal and cancer cells. We focus on the effects of time-delayed feedback loops and their coupling to ligand stimulation in this system. The model was simplified and reduced to its key components using standard methods and the rapid equilibrium assumption. We detected differences in short-term and long-term signal switching. The results from the RF- model compare well with experimental data and predict the dynamics of TGF-β signalling in cancer cells with different mutations.

Role of Cdc42 dynamics in the control of fission yeast cell polarization

Cell polarization is fundamental to many cellular processes, including cell differentiation, cell motility and cell fate determination. A key regulatory enzyme in the control of cell morphogenesis is the conserved Rho GTPase Cdc42, which breaks symmetry via self-amplifying positive-feedback mechanisms. Additional mechanisms of control, including competition between different sites of polarized cell growth and time-delayed negative feedback, define a cellular-level system that promotes Cdc42 oscillatory dynamics and modulates activated Cdc42 intracellular distribution.

Stochastic and deterministic simulations of a delayed genetic oscillation model: Investigating the validity of reductions

Quasi-stationary approximations are commonly used in order to simplify and reduce the number of equations of genetic circuit models. Protein/protein and protein/DNA binding reactions are considered to occur on much shorter time scale than protein production and degradation processes and often tacitly assumed at a quasi-equilibrium. Taking a biologically inspired, typical, small, abstract, negative feedback, genetic circuit model as study case, we investigate in this paper how different quasi-stationary approximations change the system behaviour both in deterministic and stochastic frameworks. We investigate the consistence between the deterministic and stochastic behaviours of our time-delayed negative feedback genetic circuit model with different implementations of quasi-stationary approximations. Quantitative and qualitative differences are observed among the various reduction schemes and with the underlying microscopic model, for biologically reasonable ranges and combinations of the microscopic model kinetic rates. The different reductions do not behave in the same way: correlations and amplitudes of the stochastic oscillations are not equally captured and the population behaviour is not always in consistence with the deterministic curves.

Interlinking positive and negative feedback loops creates a tunable motif in gene regulatory networks

Positive and negative feedback loops are often coupled to perform various functions in gene regulatory networks, acting as bistable switches, oscillators, and excitable devices. It is implied that such a system with interlinked positive and negative feedback loops is a flexible motif that can modulate itself among various functions. Here, we developed a minimal model for the system and systematically explored its dynamics and performance advantage in response to stimuli in a unifying framework. The system indeed displays diverse behaviors when the strength of feedback loops is changed. First, the system can be tunable from monostability to bistability by increasing the strength of positive feedback, and the bistability regime is modulated by the strength of negative feedback. Second, the system undergoes transitions from bistability to excitability and to oscillation with increasing the strength of negative feedback, and the reverse conversion occurs by enhancing the strength of positive feedback. Third, the system is more flexible than a single feedback loop; it can produce robust larger-amplitude oscillations over a wider stimulus regime compared with a single time-delayed negative feedback loop. Furthermore, the tunability of the system depends mainly on the topology of coupled feedback loops but less on the exact parameter values or the mode of interactions between model components. Thus, our results interpret why such a system represents a tunable motif and can accomplish various functions. These also suggest that coupled feedback loops can act as toolboxes for engineering diverse functional circuits in synthetic biology.

Minimum Criteria for DNA Damage-Induced Phase Advances in Circadian Rhythms

Robust oscillatory behaviors are common features of circadian and cell cycle rhythms. These cyclic processes, however, behave distinctively in terms of their periods and phases in response to external influences such as light, temperature, nutrients, etc. Nevertheless, several links have been found between these two oscillators. Cell division cycles gated by the circadian clock have been observed since the late 1950s. On the other hand, ionizing radiation (IR) treatments cause cells to undergo a DNA damage response, which leads to phase shifts (mostly advances) in circadian rhythms. Circadian gating of the cell cycle can be attributed to the cell cycle inhibitor kinase Wee1 (which is regulated by the heterodimeric circadian clock transcription factor, BMAL1/CLK), and possibly in conjunction with other cell cycle components that are known to be regulated by the circadian clock (i.e., c-Myc and cyclin D1). It has also been shown that DNA damage-induced activation of the cell cycle regulator, Chk2, leads to phosphorylation and destruction of a circadian clock component (i.e., PER1 in Mus or FRQ in Neurospora crassa). However, the molecular mechanism underlying how DNA damage causes predominantly phase advances in the circadian clock remains unknown. In order to address this question, we employ mathematical modeling to simulate different phase response curves (PRCs) from either dexamethasone (Dex) or IR treatment experiments. Dex is known to synchronize circadian rhythms in cell culture and may generate both phase advances and delays. We observe unique phase responses with minimum delays of the circadian clock upon DNA damage when two criteria are met: (1) existence of an autocatalytic positive feedback mechanism in addition to the time-delayed negative feedback loop in the clock system and (2) Chk2-dependent phosphorylation and degradation of PERs that are not bound to BMAL1/CLK.

A tunable synthetic mammalian oscillator

Autonomous and self-sustained oscillator circuits mediating the periodic induction of specific target genes are minimal genetic time-keeping devices found in the central and peripheral circadian clocks. They have attracted significant attention because of their intriguing dynamics and their importance in controlling critical repair, metabolic and signalling pathways. The precise molecular mechanism and expression dynamics of this mammalian circadian clock are still not fully understood. Here we describe a synthetic mammalian oscillator based on an auto-regulated sense-antisense transcription control circuit encoding a positive and a time-delayed negative feedback loop, enabling autonomous, self-sustained and tunable oscillatory gene expression. After detailed systems design with experimental analyses and mathematical modelling, we monitored oscillating concentrations of green fluorescent protein with tunable frequency and amplitude by time-lapse microscopy in real time in individual Chinese hamster ovary cells. The synthetic mammalian clock may provide an insight into the dynamics of natural periodic processes and foster advances in the design of prosthetic networks in future gene and cell therapies.

Closing the circadian negative feedback loop: FRQ-dependent clearance of WC-1 from the nucleus

In Neurospora crassa, a transcription factor, WCC, activates the transcription of frq. FRQ forms homodimers as well as complexes with an RNA helicase, FRH, and the WCC, and translocates into the nucleus to inactivate the WCC, closing the time-delayed negative feedback loop. The detailed mechanism for closing this loop, however, remains incompletely understood. In particular within the nucleus, the low amount of FRQ compared with that of WC-1 creates a conundrum: How can the nuclear FRQ inactivate the larger amount of WCC? One possibility is that FRQ might function as a catalytic component in phosphorylation-dependent inhibition. However, in silico experiments reveal that stoichiometric noncatalytic binding and inhibition can generate a robust oscillator, even when nuclear FRQ levels are substantially lower than nuclear WCC, so long as there is FRQ-dependent clearance of WC-1 from the nucleus. Based on this model, we can predict and now demonstrate that WC-1 stability cycles, that WC-1 is stable in the absence of FRQ, and that physical binding between FRQ and WCC is essential for closure of the negative feedback loop. Moreover, and consistent with a noncatalytic clearance-based model for inhibition, appreciable amounts of the nuclear FRQ:WCC complex accumulate at some times of day, comprising as much as 10% of the nuclear WC-1.

Modeling reflex asymmetries with implicit delay differential equations.

Neuromuscular reflexes with time-delayed negative feedback, such as the pupil light reflex, have different rates depending on the direction of movement. This asymmetry is modeled by an implicit first-order delay differential equation in which the value of the rate constant depends on the direction of movement. Stability analyses are presented for the cases when the rate is: (1) an increasing and (2) a decreasing function of the direction of movement. It is shown that the stability of equilibria in these dynamical systems depends on whether the rate constant is a decreasing or increasing function. In particular, when the asymmetry has the shape of an increasing step function, it is possible to have stability which is independent of the value of the time delay or the steepness (i.e., gain) of the negative feedback.

Stability of bipedal stance: the contribution of cocontraction and spindle feedback.

The aim of this study is to assess the contribution of cocontraction and spindle feedback to local stability during bipedal stance. To that aim, an existing nonlinear state space model of the human musculoskeletal system is linearized in a reference equilibrium state. The maximal real part of the eigenvalues of the linearized system matrix A and the low-frequency joint stiffness are used as a measure of local stability. Muscle properties, as represented in a Hill-type muscle model, are shown to improve the behavior, the improvement being larger at high cocontraction. However, even at maximal cocontraction the low-frequency joint stiffness generated by the muscle properties is insufficient to yield a locally stable system. It follows that feedback is necessary to ensure local stability. In this study, the potential contribution of spindle feedback is investigated by optimizing the feedback gains for contractile element length and velocity for each muscle. It is found that in the case of time-delayed negative feedback, it is impossible to stabilize the system on the basis of spindle feedback. When positive time-delayed feedback is allowed, a barely stable system is obtained. When the time delays are removed, the feedback gains can be chosen such that a locally stable system is obtained, indicating the limitations imposed by the presence of time delays. Finally, it is shown that for small perturbations the response of the linear system to an arbitrary perturbation is similar to that of the nonlinear system, indicating the validity of the approach used. It is concluded that the combination of muscle properties and time-delayed spindle feedback is insufficient to obtain a system with reasonable local stability.

Interannual and Decadal Variability in an Intermediate Coupled Model of the Pacific Region*

Processes that cause decadal variability in an intermediate coupled ocean–atmosphere model of the Pacific basin, both at northern midlatitudes and in the Tropics, are studied. The model's ocean component is a variable-temperature 3½-layer system. Its atmospheric component consists of two basic parts: an empirical model, constructed from patterns obtained by the singular value decomposition (SVD) statistical technique that determines wind stress anomalies from model sea surface temperature (SST), and a simple representation of the planetary boundary layer to calculate the surface heat flux anomalies. A third part specifies stochastic wind stress forcing from observed variability. In addition, the model is specifically designed to separate tropical and extratropical interactions, such that the Tropics can force the extratropics through the atmosphere but the extratropics can only feed back to the Tropics through the ocean. Solutions develop two types of oscillations: an ENSO-like interannual mode and a decadal mode. As in many models of ENSO, the interannual mode is driven by positive, ocean–atmosphere feedbacks near the equator, and time-delayed negative feedback is provided by off-equatorial Rossby waves. For parameter choices that amplify midlatitude coupling by 30% (ϕo = 1.3), a self-sustained decadal oscillation develops in the North Pacific without any tropical interactions. Diagnostic analyses show that it is maintained by ocean-to-atmosphere feedbacks in the northwest and subtropical northeast Pacific, and by atmospheric teleconnections from those regions to the northeast ocean. For weaker coupling (ϕo = 1.2), the decadal mode is damped. In this case, the mode can be sustained by atmospheric teleconnections from the Tropics associated with the interannual mode, but not by extratropical stochastic forcing. Although including stochastic forcing does generate variability at decadal timescales, a distinct decadal spectral peak only exists when the decadal mode is active. Decadal variability is carried to the equator by variations in the transport, rather than temperature, of the North Pacific subtropical cell. These variations modulate near-equatorial SST by altering the amount of cool, thermocline water that upwells in the eastern equatorial Pacific, which in turn feeds back to the interannual mode.