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.     

Balancing speed and accuracy of polyclonal T cell activation: a role for extracellular feedback

ABSTRACT: BACKGROUND: Extracellular feedback is an abundant module of intercellular communication networks, yet a detailed understanding of its role is still lacking. Here, we study interactions between polyclonal activated T cells that are mediated by IL-2 extracellular feedback as a model system. RESULTS: Using mathematical modeling we show that extracellular feedback can give rise to opposite outcomes: competition or cooperation between interacting T cells, depending on their relative levels of activation. Furthermore, the outcome of the interaction also depends on the relative timing of activation of the cells. A critical time window exists after which a cell that has been more strongly activated nevertheless cannot exclude an inferior competitor. CONCLUSIONS: In a number of experimental studies of polyclonal T-cell systems, outcomes ranging from cooperation to competition as well as time dependent competition were observed. Our model suggests that extracellular feedback can contribute to these observed behaviors as it translates quantitative differences in T cells' activation strength and in their relative activation time into qualitatively different outcomes. We propose extracellular feedback as a general mechanism that can balance speed and accuracy -- choosing the most suitable responders out of a polyclonal population under the clock of an escalating threat.     

A model of the nystagmus induced by off vertical axis rotation

A three-dimensional model is proposed that accounts for a number of phenomena attributed to the otoliths. It is constructed by extending and modifying a model of vestibular velocity storage. It is proposed that the otolith information about the orientation of the head to gravity changes the time constant of vestibular responses by modulating the gain of the velocity storage feedback loop. It is further proposed that the otolith signals, such as those that generate L-nystagmus (linear acceleration induced nystagmus), are partially coupled to the vestibular system via the velocity storage integrator. The combination of these two hypotheses suggests that a vestibular neural mechanism exists that performs correlation in the mathematical sense which is multiplication followed by integration. The multiplication is performed by the otolith modulation of the velocity storage feedback loop gain and the integration is performed by the velocity storage mechanism itself. Correlation allows calculation of the degree to which two signals are related and in this context provides a simple method of determining head angular velocity from the components of linear acceleration induced by off-vertical axis rotation. Correlation accounts for the otolith supplementation of the VOR and the sustained nystagmus generated by off-vertical axis rotation. The model also predicts the cross-coupling of horizontal and vertical optokinetic afternystagmus that occurs in head-lateral positions and the reported effects of tilt on vestibular responses.     

Mathematical formulation of cardiovascular dynamics by use of the laplace transform

By means of the Laplace transform, the behavior of a simplified model of the cardiovascular system is mathematically formulated. This formulation allows mathematical expression of the periodicity of the cardiac output and the systemic response. With the cardiac output represented as half of a sine function cycle, the systolic aortic pressure becomes the sum of a sine term and exponential terms, while the sum of the exponential terms alone represents the diastolic pressure. The characteristics of the mathematical expressions for systole and diastole are analyzed, and some relationships of potentially practical value are derived. Variation in the parameters of the system yields mathematical results consistent with the expected physical ones.     

The behavior of the weakening and failing myocardium

Use of an electrical model of the left ventricle of the heart and the arterial system permits analysis of the changes which take place as the capacity of the myocardium for generation of force decreases. The model is simple in structure, and its construction and practical testing would not be difficult. It demonstrates that, as the heart muscle weakens, the peak of intracardiac force occurs later in systole, and the difference between the intracardiac pressure and the aortic pressure in the second half of systole is much greater than for the normal heart. The feedback mechanisms which are proposed to affect myocardial contractility would affect this compensation for cardiac weakening. Indices to categorize the behavior of the normal, compensated though weakened, and decompensated myocardium are proposed.     

Variation of pressure with cycle length and duration of systole in the two-chambered cardiovascular model

Variation in the heart rate and the duration of systole modifies the pressures in the two-chambered model of the cardiovascular system by several mechanisms. The theoretical results indicate that prediction of chamber pressures would require moment-to-moment knowledge of the resistances peripheral to each chamber in addition to the cardiac output per cycle, cycle length, and duration of systole. Lengthening of systole with increase in cycle length—a physiologically observed relationship—theoretically stabilizes the end-diastolic pressure in the ascending aorta and may be a homeostatic mechanism to steady blood flow through the coronary arteries.     

Feedback control strategies for spatial navigation revealed by dynamic modelling of learning in the Morris water maze

The Morris water maze is an experimental procedure in which animals learn to escape swimming in a pool using environmental cues. Despite its success in neuroscience and psychology for studying spatial learning and memory, the exact mnemonic and navigational demands of the task are not well understood. Here, we provide a mathematical model of rat swimming dynamics on a behavioural level. The model consists of a random walk, a heading change and a feedback control component in which learning is reflected in parameter changes of the feedback mechanism. The simplicity of the model renders it accessible and useful for analysis of experiments in which swimming paths are recorded. Here, we used the model to analyse an experiment in which rats were trained to find the platform with either three or one extramaze cue. Results indicate that the 3-cues group employs stronger feedback relying only on the actual visual input, whereas the 1-cue group employs weaker feedback relying to some extent on memory. Because the model parameters are linked to neurological processes, identifying different parameter values suggests the activation of different neuronal pathways.     

Catecholamine reward pathways and schizophrenia: the mechanism of the antipsychotic effect and the site of the primary disturbance.

Two catecholamine-containing pathways, the locus ceruleus system and the dopamine neurons arising from the ventral mid-brain, may be involved in reward. Dopamine neurons function as a system for energizing the organism's responses and directing them toward significant environmental stimuli, but the functions of the locus ceruleus system remain obscure. It appears increasingly likely that neuroleptic drugs exert their anti-psychotic effects in acute schizophrenia by blocking dopamine receptors, although the time course of the effect suggests that the mechanism is more complex than a simple reversal of a neurohumoral imbalance. Evidence from postmortem studies suggests that, at least in the chronic state, dopamine turnover is not increased, but that there may be an increase in postsynaptic receptor density in some cases, including some patients who apparently had not received medication in the year before death. The evidence is consistent with Olds and Travis' conjecture that "counteraction of positive feedback processes subserving positive reinforcement mechanisms may be a key to control of certain psychotic episodes".     

How instruction and feedback can select the appropriate T helper response

The decision of the immune system to trigger immune responses that are, respectively, induced by Th1 or Th2 effectors is a critical one, because it profoundly influences disease outcome. We have recently constructed a mathematical model of Th1-Th2-pathogen interactions that shows that the major decisional events can often be successfully determined by the intrinsic behaviour of the T helper system itself. For certain dangerous types of pathogens, however, which replicate rapidly or have developed strategies to evade the immune response, additional stimuli may be necessary. As a possible mechanism for the decision-making process innate immune recognition has been proposed. Here we present an enlarged version of our model, which incorporates signals created from the innate immune system after pathogen recognition. The model analysis suggests that there is fault-tolerance of the T helper system to incorrect Th1 signals. In the presence of incorrect Th1 stimuli an initial Th1 response is shifted to the correct Th2-dominated response owing to the intrinsic T helper dynamics. By contrast, according to our model there is no fault-tolerance for incorrect Th2 signals. In fact, if timing is unimportant then Th2 signals are superfluous since the intrinsic T helper dynamics provide an automatic switch to Th2 if Th1 effectors fail to control the pathogen. Th2 signals may, however, be required to accelerate the onset of the Th2 response. Additionally, we discuss the role of feedback where successful pathogen destruction leads to up-regulation of activation of the effective T helper type. As one possibility we examine the role of CpG motifs as indicators for successful pathogen destruction. Differences between instructive and feedback mechanisms are highlighted.     

Specificity and Robustness of the Mammalian MAPK-IEG Network

The mitogen-activated protein kinase cascade is a conserved signal transduction pathway found in organisms of complexity spanning from yeast to humans. In many mammalian tissue types, this pathway can correctly transduce signals from different extracellular messengers, leading to specific and often mutually exclusive cellular responses. The transduced signal is tuned by a complicated set of positive and negative feedback control mechanisms and fed into a downstream gene expression network. This network, based on the immediate early gene system, has two possible, mutually exclusive outcomes. Using a mathematical model, we study how different stimuli lead to different temporal signal structure. Further, we investigate how each of the feedback controls contributes to the overall specificity of the gene expression output, and hypothesize that the complicated nature of the mammalian mitogen-activated protein kinase pathway results in a system able to robustly identify and transduce the proper signal without investing in two completely separate signal cascades. Finally, we quantify the role of the RKIP protein in shaping the signal, and propose a novel mechanism of its involvement in cancer metastasis.     

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.     

Modeling the Dynamics of Cdc42 Oscillation in Fission Yeast

Regulation of polarized cell growth is essential for many cellular processes, including spatial coordination of cell morphology changes during growth and division. We present a mathematical model of the core mechanism responsible for the regulation of polarized growth dynamics by the small GTPase Cdc42. The model is based on the competition of growth zones of Cdc42 localized at the cell tips for a common substrate (inactive Cdc42) that diffuses in the cytosol. We consider several potential ways of implementing negative feedback between Cd42 and its GEF in this model that would be consistent with the observed oscillations of Cdc42 in fission yeast. We analyze the bifurcations in this model as the cell length increases, and total amount of Cdc42 and GEF increase. Symmetric antiphase oscillations at two tips emerge via saddle-homoclinic bifurcations or Hopf bifurcations. We find that a stable oscillation and a stable steady state can coexist, which is consistent with the experimental finding that only 50% of bipolar cells oscillate. The mean amplitude and period can be tuned by parameters involved in the negative feedback. We link modifications in the parameters of the model to observed mutant phenotypes. Our model suggests that negative feedback is more likely to be acting through inhibition of GEF association rather than upregulation of GEF dissociation.     

Regulatory dynamics of synthetic gene networks with positive feedback

Biological processes are governed by complex networks ranging from gene regulation to signal transduction. Positive feedback is a key element in such networks. The regulation enables cells to adopt multiple internal expression states in response to a single external input signal. However, past works lacked a dynamical aspect of this system. To address the dynamical property of the positive feedback system, we employ synthetic gene circuits in Escherichia coli to measure the rise-time of both the no-feedback system and the positive feedback system. We show that the kinetics of gene expression is slowed down if the gene regulatory system includes positive feedback. We also report that the transition of gene switching behaviors from the hysteretic one to the graded one occurs. A mathematical model based on the chemical reactions shows that the response delay is an inherited property of the positive feedback system. Furthermore, with the aid of the phase diagram, we demonstrate the decline of the feedback activation causes the transition of switching behaviors. Our findings provide a further understanding of a positive feedback system in a living cell from a dynamical point of view.     

The cardiovascular effects of the carotid sinus mechanism

The feedback mechanism by means of which the carotid sinus affects arterial blood pressure is represented with the use of the single chamber model of the cardiovascular system. The average pressure in the large arteries is the controlled variable. The resultant firstorder differential equation is easily solved. The theoretical results indicate that the feed-back mechanism of the carotid sinus serves to decrease the deviation of the average arterial pressure from an internal standard of pressure in the steady state and to hasten the stabilization of the pressure following environmental or pharmacological changes. This approach also suggests an experimentally feasible method for the determination of the internal standard of pressure.     

Interfilament spacing is preserved during sarcomere length isometric contractions in rat cardiac trabeculae

It is generally assumed that the myofilament lattice in intact (i.e., nonskinned) striated muscle obeys constant volume. However, whether such is the case during the myocardial contraction is unknown. Accordingly, we measured interfilament spacing by x-ray diffraction in ultra-thin isolated rat right ventricular trabeculae during a short 10 ms shuttered exposure either just before electrical stimulation (diastole), or at the peak of the contraction (systole); sarcomere length (SL) was held constant throughout the contraction using an iterative feedback control system. SL was thus varied in a series of SL-clamped contractions; the relationship between SL and interfilament spacing was not different between diastole and systole within 1%; this was true also over a wide range of inotropic states induced by varied [Ca(2+)](o). We conclude that the cardiac myofilament lattice maintains constant volume, and thus constant interfilament spacing, during contraction.     

Mathematical modeling of planar polarity

Although it is well established that the Frizzled receptor is involved in the transmission of polarity information from cell to cell in the Drosophila cuticle, its precise role is still unclear. A recent paper by presents a mathematical model of a feedback loop-based mechanism for propagation of polarity between cells that can account for the known functions of Frizzled.     

Instructed heart rate control

Forty subjects participated in an experiment designed to test the effects of different feedback displays on instructed heart rate speeding and slowing. One group of subjects received information about interpulse interval length every beat. This display included specific information about when systole occurred, in addition to information about performance relative to a criterion. Two other groups received similar information about performance, but their displays were not triggered by systole; rather, information about average interpulse interval was presented either every second or every 6 seconds. A fourth group of subjects participated in a perceptual motor task in which no instructions were given to control heart rate.Results indicated that the instructed subjects generated significantly greater heart rate speeding than slowing. Groups receiving feedback produced greater changes when compared to the control group only during the speeding seassions. No differences among feedback groups were present in the slowing task. During speeding, the 1-second group's performance deteriorated dramatically in the second session. The results suggested that, in the context of a feedback task, it is information about the occurrence of systole that facilitates heart rate speeding. Real-time displays are less facilitating of heart rate change and may disrupt speeding performance when information is presented at certain critical frequencies. Slowing performance was again shown to be unrelated to information frequency or reinforcement rate.     

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.     

Mathematical modelling of the agr operon in Staphylococcus aureus

Staphylococcus aureus is a pathogenic bacterium that utilises quorum sensing (QS), a cell-to-cell signalling mechanism, to enhance its ability to cause disease. QS allows the bacteria to monitor their surroundings and the size of their population, and S. aureus makes use of this to regulate the production of virulence factors. Here we describe a mathematical model of this QS system and perform a detailed time-dependent asymptotic analysis in order to clarify the roles of the distinct interactions that make up the QS process, demonstrating which reactions dominate the behaviour of the system at various timepoints. We couple this analysis with numerical simulations and are thus able to gain insight into how a large population of S. aureus shifts from a relatively harmless state to a highly virulent one, focussing on the need for the three distinct phases which form the feedback loop of this particular QS system.