Mathematical identification of critical reactions in the interlocked feedback model

Dynamic simulations are necessary for understanding the mechanism of how biochemical networks generate robust properties to environmental stresses or genetic changes. Sensitivity analysis allows the linking of robustness to network structure. However, it yields only local properties regarding a particular choice of plausible parameter values, because it is hard to know the exact parameter values in vivo. Global and firm results are needed that do not depend on particular parameter values. We propose mathematical analysis for robustness (MAR) that consists of the novel evolutionary search that explores all possible solution vectors of kinetic parameters satisfying the target dynamics and robustness analysis. New criteria, parameter spectrum width and the variability of solution vectors for parameters, are introduced to determine whether the search is exhaustive. In robustness analysis, in addition to single parameter sensitivity analysis, robustness to multiple parameter perturbation is defined. Combining the sensitivity analysis and the robustness analysis to multiple parameter perturbation enables identifying critical reactions. Use of MAR clearly identified the critical reactions responsible for determining the circadian cycle in the Drosophila interlocked circadian clock model. In highly robust models, while the parameter vectors are greatly varied, the critical reactions with a high sensitivity are uniquely determined. Interestingly, not only the per-tim loop but also the dclk-cyc loop strongly affect the period of PER, although the dclk-cyc loop hardly changes its amplitude and it is not potentially influential. In conclusion, MAR is a powerful method to explore wide parameter space without human-biases and to link a robust property to network architectures without knowing the exact parameter values. MAR identifies the reactions critically responsible for determining the period and amplitude in the interlocked feedback model and suggests that the circadian clock intensively evolves or designs the kinetic parameters so that it creates a highly robust cycle.     

Detection of cellular rhythms and global stability within interlocked feedback systems

In this paper, we show how to detect cellular rhythm and its global stability by extending the techniques from the recently developed theory of monotone systems. We establish theoretical results for globally asymptotic stability with consideration of delay by a discrete map. The relationship between positive, negative elements and delay in a general class of interlocked feedback networks can be understood in a system level. Moreover, the correspondence of attractors between a network and its reduced map is obtained and can be used to detect cellular rhythm, and further control the dynamics of the network. We show that global cellular rhythms can always be obtained, thereby enhancing robustness against perturbations of initial conditions and avoiding chaotic oscillations or complete abolishment of oscillations. In this paper, we focus on analyzing the circadian oscillator in Drosophila as an example to detect the occurrence of cellular rhythm and its global stability.     

High-frequency oscillations and circadian rhythm of the membrane potential in Spinach leaves

The microelectrode technique was used to follow oscillations in membrane potential in mesophyll cells of spinach (Spinacia oleracea L.) during exposure do different photoperiodic conditions. Both high-frequency oscillations and circadian variations were observed. The circadian rhythm was imposed on the period of high-frequency oscillation during short days as well as in continuous light: The free-running period was 25.2 h. The average period of high-frequency oscillation increased from 7.64 min in the dark to 19.95 min in the light within several minutes after dark to light transition. This period length coincides with the established period length for oscillations in the redox potential in the chloroplast suspensions of spinach.Abbreviations CL continuous light - SD short day - MP membrane potential     

Temperature entrainment of the circadian cuticle deposition rhythm in Drosophila melanogaster

The cuticle deposition rhythm, which is observed in the apodeme of the furca in the thorax, is controlled by a peripheral circadian clock in the epidermal cells and entrained to light-dark (LD) cycles via CRYPTOCHROME (CRY) in Drosophila melanogaster. In the present study, we examined the effects of temperature (TC) cycles and the combination of LD and TC cycles on entrainment of the cuticle deposition rhythm. The rhythm was entrained to TC cycles, whose period was 28 h. In T = 21 and 24 h, the rhythm was entrained to TC cycles in some individuals. CRY is not necessary for temperature entrainment of the cuticle deposition rhythm because the rhythm in cry(b) (lacking functional CRY) was entrained to TC cycles. Temperature entrainment of the rhythm was achieved even when the thoraxes or furcae were cultured in vitro, suggesting that the mechanism for temperature entrainment is independent of the central clock in the brain and the site of the thermoreception resides in the epidermal cells. When LD and TC cycles with different periods were applied, the rhythm was entrained to LD cycles with a slight influence of TC cycles. Thus, the LD cycle is a stronger zeitgeber than the TC cycle. The variance of the number of the cuticle layers decreased in the flies kept under LD and TC cycles with the same period in which the thermophase coincided with the photophase. Therefore, we conclude that LD and TC cycles synergistically entrain the rhythm. Synergistic effects of LD and TC cycles on entrainment were also observed even when the thoraxes were cultured in vitro, suggesting that the light and temperature information is integrated within the peripheral circadian system.     

Latitude dependent arrhythmicity in the circadian oviposition rhythm of Drosophila ananassae

Latitude dependent arrhythmicity in the circadian rhythm of oviposition of Drosophila ananassae strains originating from 8.1°N to 32.7°N was studied by inbreeding them in cycles of 12 h of light at 20 lux and 12 h of darkness. The number of inbreeding generations required to initiate arrhythmicity in oviposition rhythm was dependent on the origin of latitude of the strain. The strains from the lower latitudes became arrhythmic after notably more numbers of generations than those from the higher latitudes. This might be attributed to the higher inherent degree of oviposition rhythmicity in the F₁ generation, and enhanced photic sensitivity of the circadian pacemaker mediating entrainment of oviposition rhythm of the strains from lower latitudes as compared to those from the higher latitudes.     

Altitudinal variation in the circadian rhythm of oviposition in Drosophila ananassae

The effect of altitude on four basic properties of the pacemaker controlling the circadian rhythm of oviposition in two strains of Drosophila ananassae was determined. The high altitude (HA) strain from Badrinath (5123 m above sea level) had a low amplitude peak in the forenoon while the low altitude (LA) strain from Firozpur (179 m a.s.l.) had a high amplitude peak after the lights-off of LD 12:12 cycles. Free running periods in continuous darkness were about 22.6 and 27.4 h in the HA and LA strains, respectively. The light pulse phase response curve (PRC) for the HA strain showed a low amplitude and a dead zone of 8h; the ratio for the advance to delay region (A/D) was less than 1, while the PRC for the LA strain had a high amplitude, which was devoid of a dead zone and showed a ratio of A/D > 1. The magnitude of the delay phase shifts at CT 18 evoked by light pulses of 1 h duration, but varying light intensity was significantly different in the HA and LA strain, which suggests that the photic sensitivity of the clock photoreceptors mediating the phase shifts had been affected by the altitude.     

A statistical model of the human core-temperature circadian rhythm

We formulate a statistical model of the human core-temperature circadian rhythm in which the circadian signal is modeled as a van der Pol oscillator, the thermoregulatory response is represented as a first-order autoregressive process, and the evoked effect of activity is modeled with a function specific for each circadian protocol. The new model directly links differential equation-based simulation models and harmonic regression analysis methods and permits statistical analysis of both static and dynamical properties of the circadian pacemaker from experimental data. We estimate the model parameters by using numerically efficient maximum likelihood algorithms and analyze human core-temperature data from forced desynchrony, free-run, and constant-routine protocols. By representing explicitly the dynamical effects of ambient light input to the human circadian pacemaker, the new model can estimate with high precision the correct intrinsic period of this oscillator ( approximately 24 h) from both free-run and forced desynchrony studies. Although the van der Pol model approximates well the dynamical features of the circadian pacemaker, the optimal dynamical model of the human biological clock may have a harmonic structure different from that of the van der Pol oscillator.     

Setting the clock--by nature: circadian rhythm in the fruitfly Drosophila melanogaster

Nowadays humans mainly rely on external, unnatural clocks such as of cell phones and alarm clocks--driven by circuit boards and electricity. Nevertheless, our body is under the control of another timer firmly anchored in our genes. This evolutionary very old biological clock drives most of our physiology and behavior. The genes that control our internal clock are conserved among most living beings. One organism that shares this ancient clock mechanism with us humans is the fruitfly Drosophila melanogaster. Since it turned out that Drosophila is an excellent model, it is no surprise that its clock is very well and intensely investigated. In the following review we want to display an overview of the current understanding of Drosophila's circadian clock.     

Unique self-sustaining circadian oscillators within the brain of Drosophila melanogaster

In Drosophila circadian rhythms persist in constant darkness (DD). The small ventral Lateral Neurons (s-LN_v) mainly control the behavioral circadian rhythm in consortium with the large ventral Lateral Neurons (l-LN_v) and dorsal Lateral Neurons (LN_d). It is believed that the molecular oscillations of clock genes are the source of this persistent behavior. Indeed the s-LN_v, LN_d, Dorsal Neurons (DN)-DN₂ and DN₃ displayed self-sustained molecular oscillations in DD both at RNA and protein levels, except the DN₂ oscillates in anti-phase. In contrast, the l-LN_v and DN₁ displayed self-sustained oscillations at the RNA level, but protein oscillations quickly dampened. Having self-sustained and dampened molecular oscillators together in the DN groups suggested that they play different roles. However, all DN groups seemed to contribute together to the light-dark (LD) behavioral rhythm. The LD entrainment of LN oscillators is achieved through Rhodopsin (RH) and Cryptochrome (CRY). CRY's expression in all DN groups implicates also its role in LD entrainment of DN, like in DN₁. However, mutations in cry and glass that did not inflict LD synchronization of the DN₂, DN₃ oscillator implicate the existence of a novel photoreceptor at least in DN₃.     

A qualitative model of a motor circadian rhythm

In Nature it is possible to observe diverse rhythms. Because of their adaptive characteristics, the circadian rhythms are of major importance and have been the subject of numerous experimental and theoretical studies. In this article, we give a presentation of the main results we have obtained about the motor circadian rhythm along some years of collaboration between biologists and mathematicians. We present a mathematical model simulating changes in frequency, synchronization and amplitude of the circadian oscillation during two developmental stages of the crayfish, namely, the juvenile and the adult stages. We report also some work in progress on the simulation of the phase response curve and on a simplified model of the rhythm.     

Function of the Shaw potassium channel within the Drosophila circadian clock

Background

In addition to the molecular feedback loops, electrical activity has been shown to be important for the function of networks of clock neurons in generating rhythmic behavior. Most studies have used over-expression of foreign channels or pharmacological manipulations that alter membrane excitability. In order to determine the cellular mechanisms that regulate resting membrane potential (RMP) in the native clock of Drosophila we modulated the function of Shaw, a widely expressed neuronal potassium (K⁺) channel known to regulate RMP in Drosophila central neurons.

Methodology/Principal Findings

We show that Shaw is endogenously expressed in clock neurons. Differential use of clock gene promoters was employed to express a range of transgenes that either increase or decrease Shaw function in different clusters of clock neurons. Under LD conditions, increasing Shaw levels in all clock neurons (LNv, LNd, DN₁, DN₂ and DN₃), or in subsets of clock neurons (LNd and DNs or DNs alone) increases locomotor activity at night. In free-running conditions these manipulations result in arrhythmic locomotor activity without disruption of the molecular clock. Reducing Shaw in the DN alone caused a dramatic lengthening of the behavioral period. Changing Shaw levels in all clock neurons also disrupts the rhythmic accumulation and levels of Pigment Dispersing Factor (PDF) in the dorsal projections of LNv neurons. However, changing Shaw levels solely in LNv neurons had little effect on locomotor activity or rhythmic accumulation of PDF.

Conclusions/Significance

Based on our results it is likely that Shaw modulates pacemaker and output neuronal electrical activity that controls circadian locomotor behavior by affecting rhythmic release of PDF. The results support an important role of the DN clock neurons in Shaw-mediated control of circadian behavior. In conclusion, we have demonstrated a central role of Shaw for coordinated and rhythmic output from clock neurons.     

Behavioral feedback regulation of circadian rhythm phase angle in light-dark entrained mice

Induced and spontaneous wheel running can alter the phase and period (tau) of circadian rhythms in rodents. The relationship between spontaneous running and the phase angle (psi) of entrainment to 24-h light-dark (LD) cycles was evaluated in C57BL/6j mice. With a wheel freely available, psi was significantly correlated with the absolute (r = 0.32) and relative (r = 0.44) amount of activity during the first 2 h of the activity period. When wheels were locked during the first half of the night in LD and then unlocked in constant dark (DD), mice exhibited a delayed psi and lengthened tau compared with mice that had wheels locked during the second half of the night. In DD, tau correlated negatively with total daily activity. To evaluate if wheel running modulates the phase-resetting actions of LD, phase shifts to light pulses were measured at two time points in DD, when daily activity levels differed by 40%. Phase delays to light were 56% greater when activity levels were lower. However, in a counterbalanced follow-up experiment, phase advances and delays to light pulses were not affected by the availability of wheels, although an effect of time in DD was replicated. Spontaneous activity can regulate psi and tau without altering the response of the pacemaker to light.     

Robust and sensitive control of a quorum‐sensing circuit by two interlocked feedback loops

The quorum‐sensing (QS) response of Vibrio fischeri involves a rapid switch between low and high induction states of the lux operon over a narrow concentration range of the autoinducer (AI) 3‐oxo‐hexanoyl‐L ‐homoserine lactone. In this system, LuxR is an AI‐dependent positive regulator of the lux operon, which encodes the AI synthase. This creates a positive feedback loop common in many bacterial species that exhibit QS‐controlled gene expression. Applying a combination of modeling and experimental analyses, we provide evidence for a LuxR autoregulatory feedback loop that allows LuxR to increase its concentration in the cell during the switch to full lux activation. Using synthetic lux gene fragments, with or without the AI synthase gene, we show that the buildup of LuxR provides more sensitivity to increasing AI, and promotes the induction process. Elevated LuxR levels buffer against spurious variations in AI levels ensuring a robust response that endows the system with enhanced hysteresis. LuxR autoregulation also allows for two distinct responses within the same cell population.     

Stochastic simulations on a model of circadian rhythm generation

Biological phenomena are often modeled by differential equations, where states of a model system are described by continuous real values. When we consider concentrations of molecules as dynamical variables for a set of biochemical reactions, we implicitly assume that numbers of the molecules are large enough so that their changes can be regarded as continuous and they are described deterministically. However, for a system with small numbers of molecules, changes in their numbers are apparently discrete and molecular noises become significant. In such cases, models with deterministic differential equations may be inappropriate, and the reactions must be described by stochastic equations. In this study, we focus a clock gene expression for a circadian rhythm generation, which is known as a system involving small numbers of molecules. Thus it is appropriate for the system to be modeled by stochastic equations and analyzed by methodologies of stochastic simulations. The interlocked feedback model proposed by Ueda et al. as a set of deterministic ordinary differential equations provides a basis of our analyses. We apply two stochastic simulation methods, namely Gillespie's direct method and the stochastic differential equation method also by Gillespie, to the interlocked feedback model. To this end, we first reformulated the original differential equations back to elementary chemical reactions. With those reactions, we simulate and analyze the dynamics of the model using two methods in order to compare them with the dynamics obtained from the original deterministic model and to characterize dynamics how they depend on the simulation methodologies.