Modeling circadian oscillations with interlocking positive and negative feedback loops.

Both positive and negative feedback loops of transcriptional regulation have been proposed to be important for the generation of circadian rhythms. To test the sufficiency of the proposed mechanisms, two differential equation-based models were constructed to describe the Neurospora crassa and Drosophila melanogaster circadian oscillators. In the model of the Neurospora oscillator, FRQ suppresses frq transcription by binding to a complex of the transcriptional activators WC-1 and WC-2, thus yielding negative feedback. FRQ also activates synthesis of WC-1, which in turn activates frq transcription, yielding positive feedback. In the model of the Drosophila oscillator, PER and TIM are represented by a "lumped" variable, "PER." PER suppresses its own transcription by binding to the transcriptional regulator dCLOCK, thus yielding negative feedback. PER also binds to dCLOCK to de-repress dclock, and dCLOCK in turn activates per transcription, yielding positive feedback. Both models displayed circadian oscillations that were robust to parameter variations and to noise and that entrained to simulated light/dark cycles. Circadian oscillations were only obtained if time delays were included to represent processes not modeled in detail (e.g., transcription and translation). In both models, oscillations were preserved when positive feedback was removed.

Modeling transcriptional control in gene networks--methods, recent results, and future directions.

Mathematical models are useful for providing a framework for integrating data and gaining insights into the static and dynamic behavior of complex biological systems such as networks of interacting genes. We review the dynamic behaviors expected from model gene networks incorporating common biochemical motifs, and we compare current methods for modeling genetic networks. A common modeling technique, based on simply modeling genes as ON-OFF switches, is readily implemented and allows rapid numerical simulations. However, this method may predict dynamic solutions that do not correspond to those seen when systems are modeled with a more detailed method using ordinary differential equations. Until now, the majority of gene network modeling studies have focused on determining the types of dynamics that can be generated by common biochemical motifs such as feedback loops or protein oligomerization. For example, these elements can generate multiple stable states for gene product concentrations, state-dependent responses to stimuli, circadian rhythms and other oscillations, and optimal stimulus frequencies for maximal transcription. In the future, as new experimental techniques increase the ease of characterization of genetic networks, qualitative modeling will need to be supplanted by quantitative models for specific systems.

Effects of macromolecular transport and stochastic fluctuations on dynamics of genetic regulatory systems.

To predict the dynamics of genetic regulation, it may be necessary to consider macromolecular transport and stochastic fluctuations in macromolecule numbers. Transport can be diffusive or active, and in some cases a time delay might suffice to model active transport. We characterize major differences in the dynamics of model genetic systems when diffusive transport of mRNA and protein was compared with transport modeled as a time delay. Delays allow for history-dependent, non-Markovian responses to stimuli (i.e., "molecular memory"). Diffusion suppresses oscillations, whereas delays tend to create oscillations. When simulating essential elements of circadian oscillators, we found the delay between transcription and translation necessary for oscillations. Stochastic fluctuations tend to destabilize and thereby mask steady states with few molecules. This computational approach, combined with experiments, should provide a fruitful conceptual framework for investigating the function and dynamic properties of genetic regulatory systems.

Frequency selectivity, multistability, and oscillations emerge from models of genetic regulatory systems.

To examine the capability of genetic regulatory systems for complex dynamic activity, we developed simple kinetic models that incorporate known features of these systems. These include autoregulation and stimulus-dependent phosphorylation of transcription factors (TFs), dimerization of TFs, crosstalk, and feedback. The simplest model manifested multiple stable steady states, and brief perturbations could switch the model between these states. Such transitions might explain, for example, how a brief pulse of hormone or neurotransmitter could elicit a long-lasting cellular response. In slightly more complex models, oscillatory regimes were identified. The addition of competition between activating and repressing TFs provided a plausible explanation for optimal stimulus frequencies that give maximal transcription. Such optimal frequencies are suggested by recent experiments comparing training paradigms for long-term memory formation and examining changes in mRNA levels in repetitively stimulated cultured cells. In general, the computational approach illustrated here, combined with appropriate experiments, provides a conceptual framework for investigating the function of genetic regulatory systems.

A role for calcium release-activated current (CRAC) in cholinergic modulation of electrical activity in pancreatic beta-cells.

S. Bordin and colleagues have proposed that the depolarizing effects of acetylcholine and other muscarinic agonists on pancreatic beta-cells are mediated by a calcium release-activated current (CRAC). We support this hypothesis with additional data, and present a theoretical model which accounts for most known data on muscarinic effects. Additional phenomena, such as the biphasic responses of beta-cells to changes in glucose concentration and the depolarizing effects of the sarco-endoplasmic reticulum calcium ATPase pump poison thapsigargin, are also accounted for by our model. The ability of this single hypothesis, that CRAC is present in beta-cells, to explain so many phenomena motivates a more complete characterization of this current.

A model for glycolytic oscillations based on skeletal muscle phosphofructokinase kinetics.

Existing models for glycolytic oscillations are not based on detailed experimental kinetics of the glycolytic enzymes. Here, a model is constructed to fit the kinetics of skeletal muscle phosphofructokinase with respect to variations in AMP, ATP, fructose-6-P, and fructose 1,6-P2 levels. A Monod-Wyman-Changeux model for a tetrameric enzyme is considered. However, it is found that the kinetic data fit considerably better with an assumption of identical, independent subunits. With parameters that fit these data and with a previous model for the rest of glycolysis, product activation of phosphofructokinase leads to oscillations of glycolytic intermediates and [ATP] resembling those observed experimentally in muscle extracts. The period is several minutes. The model can also produce oscillations at neutral pH and with [ATP] representative of an intact cell. Under both conditions the mean concentrations and oscillations vary with the rate of glucose phosphorylation in a plausible manner only if some amount of glucose-6-phosphatase or glucose-6-P dehydrogenase activity is assumed or if hexokinase is inhibited by glucose-6-P. Also, the model can be reduced to two variables for ease of analysis and the oscillation mechanism thereby illustrated.

Phase independent resetting in relaxation and bursting oscillators.

Relaxation oscillators that depend on one slow variable, such as the Fitzhugh-Nagumo oscillator, reset in a phase-dependent manner. A complete oscillation can be divided into two parts, the "plateau" and "trough", and a prematurely induced plateau or trough is significantly shorter than normal. The class of square-wave bursting oscillators can be viewed as relaxation oscillators with rapid spikes during the plateau, and reset similarly when modeled with one slow variable. However, it has been reported that a physiological bursting oscillator, the membrane potential of the pancreatic beta-cell, resets in a phase-independent manner, such that a prematurely induced plateau/trough has normal length. A possible model for such an oscillator requires two slow variables, one to control the length of the plateau and the other the length of the trough. Here, we explore the geometric solution structure of two such models, which exhibit the desired resetting. One is a generalization of the Fitzhugh-Nagumo equations, and the other is a bursting oscillator using known beta-cell electrical currents with an additional hypothetical slow outward current.

Inactivation of HIT cell Ca2+ current by a simulated burst of Ca2+ action potentials.

A novel voltage-clamp protocol was developed to test whether slow inactivation of Ca2+ current occurs during bursting in insulin-secreting cells. Single insulin-secreting HIT cells were patch-clamped and their Ca2+ currents were isolated pharmacologically. A computed beta-cell burst was used as a voltage-clamp command and the net Ca2+ current elicited was determined as a cadmium difference current. Ca2+ current rapidly activated during the computed plateau and spike depolarizations and then slowly decayed. Integration of this Ca2+ current yielded an estimate of total Ca influx. To further analyze Ca2+ current inactivation during a burst, repetitive test pulses to + 10 mV were added to the voltage command. Current elicited by these pulses was constant during the interburst, but then slowly and reversibly decreased during the depolarizing plateau. This inactivation was reduced by replacing external Ca2+ with Ba2+ as a charge carrier, and in some cells inactivation was slower in Ba2+. Experimental results were compared with the predictions of the Keizer-Smolen mathematical model of bursting, after subjecting model equations to identical voltage commands. In this model, bursting is driven by the slow, voltage-dependent inactivation of Ca current during the plateau active phase. The K-S model could account for the slope of the slow decay of spike-elicited Ca current, the waveform of individual Ca current spikes, and the suppression of test pulse-elicited Ca current during a burst command. However, the extent and rate of fast inactivation were underestimated by the model. Although there are significant differences between the data obtained and the predictions of the K-S model, the overall results show that as predicted by the model, Ca current slowly inactivates during a burst of imposed spikes, and inactivation is dependent on both Ca2+ influx and membrane depolarization. We thus show that clamping cells to their physiological voltage waveform can be readily accomplished and is a powerful approach for understanding the contribution of individual ion currents to bursting.

Why pancreatic islets burst but single beta cells do not. The heterogeneity hypothesis.

Previous mathematical modeling of beta cell electrical activity has involved single cells or, recently, clusters of identical cells. Here we model clusters of heterogeneous cells that differ in size, channel density, and other parameters. We use gap-junctional electrical coupling, with conductances determined by an experimental histogram. We find that, for reasonable parameter distributions, only a small proportion of isolated beta cells will burst when uncoupled, at any given value of a glucose-sensing parameter. However, a coupled, heterogeneous cluster of such cells, if sufficiently large (approximately 125 cells), will burst synchronously. Small clusters of such cells will burst only with low probability. In large clusters, the dynamics of intracellular calcium compare well with experiments. Also, these clusters possess a dose-response curve of increasing average electrical activity with respect to a glucose-sensing parameter that is sharp when the cluster is coupled, but shallow when the cluster is decoupled into individual cells. This is in agreement with comparative experiments on cells in suspension and islets.

Slow voltage inactivation of Ca2+ currents and bursting mechanisms for the mouse pancreatic beta-cell.

Recent whole-cell electrophysiological data concerning the properties of the Ca2+ currents in mouse beta-cells are fitted by a two-current model of Ca2+ channel kinetics. When the beta-cell K+ currents are added to this model, only large modifications of the measured Ca2+ currents will reproduce the bursting pattern normally observed in mouse islets. However, when the measured Ca2+ currents are modified only slightly and used in conjunction with a K+ conductance that can be modulated dynamically by ATP concentration, reasonable bursting is obtained. Under these conditions it is the K-ATP conductance, rather than the slow voltage inactivation of the Ca2+ current, that determines the interburst interval. We find that this latter model can be reconciled with experiments that limit the possible periodic variation of the K-ATP conductance and with recent observations of intracellular Ca2+ bursting in islets.

Bursting electrical activity in pancreatic beta cells caused by Ca(2+)- and voltage-inactivated Ca2+ channels.

We investigate the hypothesis that two classes of Ca2+ currents, one quickly inactivated by Ca2+ and one slowly inactivated by voltage, contribute to bursting electrical activity in pancreatic islets. A mathematical model of these currents is fit to the experimental whole-cell current-voltage and inactivation profiles, thereby fixing the Ca2+ conductance and all activation and inactivation parameters. Incorporating these currents into a model that includes delayed rectifier K+ channels and ATP-sensitive K+ channels, we show that only abnormal bursting is obtained. Modification of activation parameters to increase Ca2+ channel open times, as suggested by experiment, yields a more robust bursting similar to that observed in intact islets. This reinforces the suggestion that in addition to ATP-sensitive K+ channels, Ca2+ channels may serve as glucose sensors in the beta cell.

Kinetics and thermodynamics of metabolite transfer between enzymes.

Based on experimental evidence put forward by Bernhard and others, we explore the kinetics and thermodynamics of the proposed direct and diffusional transfer mechanisms of enzymatic catalysis. Data for transient transfer of NADH between two cognate dehydrogenases (E1 and E2) are combined with steady-state catalytic data to quantify the kinetics of the two transfer mechanisms. In order to rationalize these data we find: (1) that the rate constants for direct transfer of NADH from E1-NADH to E2 must be much larger when a reactive metabolite, M, is bound to E2, (2) that a significant amount of noncatalytic complex E1-E2-M must be formed, and (3) that dissociation constants of the order of 1 microM are required for the ternary complexes involved in direct transfer (E1-NADH-E2). Using values of rate parameters similar to those assumed in these calculations, we proceed to explore the kinetics and thermodynamics of a hypothetical two-enzyme segment of a metabolic pathway that involves direct and diffusional metabolite transfer operating in parallel. Under steady-state conditions, we conclude from our calculations: (1) that the flux through the direct transfer branch would be comparable to or greater than that through the diffusional transfer branch under physiological conditions, (2) that activity effects resulting from physiological concentrations of inert protein enhance the predominance of the direct transfer flux, (3) that significant concentrations of ternary transfer complexes are formed, (4) that changes in the catalytic mechanism involving the ternary complex have little effect, (5) that direct transfer significantly moderates the reduction in metabolic flux caused by protein complexation, and (6) that direct transfer greatly alters the thermodynamics and kinetics of our hypothetical pathways. We conclude, therefore, that direct transfer--when it exists--would have important kinetic, thermodynamic, and physiological consequences in vivo.

Direct transfer of NADH between alpha-glycerol phosphate dehydrogenase and lactate dehydrogenase: fact or misinterpretation?

Following the criticism by Chock and Gutfreund [Chock, P.B. & Gutfreund, H. (1988) Proc. Natl. Acad. Sci. USA 85, 8870-8874], that our proposal of direct transfer of NADH between glycerol-3-phosphate dehydrogenase (alpha-glycerol phosphate dehydrogenase, alpha-GDH; EC 1.1.1.8) and L-lactate dehydrogenase (LDH; EC 1.1.1.27) was based on a misinterpretation of the kinetic data, we have reinvestigated the transfer mechanism between this enzyme pair. By using the "enzyme buffering" steady-state kinetic technique [Srivastava, D.K. & Bernhard, S.A. (1984) Biochemistry 23, 4538-4545], we examined the mechanism (random diffusion vs. direct transfer) of transfer of NADH between rabbit muscle alpha-GDH and pig heart LDH. The steady-state data reveal that the LDH-NADH complex and the alpha-GDH-NADH complex can serve as substrate for the alpha-GDH-catalyzed reaction and the LDH-catalyzed reaction, respectively. This is consistent with the direct-transfer mechanism and inconsistent with a mechanism in which free NADH is the only competent substrate for either enzyme-catalyzed reaction. The discrepancy between this conclusion and that of Chock and Gutfreund comes from (i) their incorrect measurement of the Km for NADH in the alpha-GDH-catalyzed reaction, (ii) inadequate design and range of the steady-state kinetic experiments, and (iii) their qualitative assessment of the prediction of the direct-transfer mechanism. Our transient kinetic measurements for the transfer of NADH from alpha-GDH to LDH and from LDH to alpha-GDH show that both are slower than predicted on the basis of free equilibration of NADH through the aqueous environment. The decrease in the rate of equilibration of NADH between alpha-GDH and LDH provides no support for the random-diffusion mechanism; rather, it suggests a direct interaction between enzymes that modulates the transfer rate of NADH. Thus, contrary to Chock and Gutfreund's conclusion, all our experimental data compel us to propose, once again, that NADH is transferred directly between the sites of alpha-GDH and LDH.

Health problems during the first year of life in infants born to adolescent mothers.

The role of maternal age at birth was investigated to determine its impact on an infant's health status during the first year of life. A sample population of 112 healthy term infants born to young primiparous mothers (less than or equal to 17 years) was compared using a chart audit with a population of 92 infants born to older primiparous mothers (greater than or equal to 18 years). There was no significant difference between the two groups in the number of missed clinic appointments, number of emergency room visits, number of hospitalizations, number of infants fully immunized, or incidence of child abuse. Poor weight gain and fewer outpatient visits for medical problems were more frequent in the infants of the younger population. Education programs designed for pregnant adolescents should stress the proper use of health professionals and appropriate methods of infant feeding.

Adolescent pregnancy. Involvement of the male partner.

The role played by the male partner in an adolescent pregnancy was investigated. Questionnaires were completed by the male partners of 41 adolescent females who continued their pregnancy to term and attended an optional education program. Most fathers (81%) maintained an ongoing relationship with the mother and informed their own family about the pregnancy. Only 48% helped with the decision regarding the outcome of the pregnancy. Only 19% saw a health care professional to discuss the pregnancy and only 9% received contraceptive information. Our data suggest that a significant number of these males suffered negative psychosocial consequences such as depression and increased social isolation. We believe that more effort should be made to identify those males interested in participating in their partner's pregnancy, encourage them to play a role in the decision-making process regarding the outcome of the pregnancy, and provide them with contraceptive counseling and psychologic support.