Single channel gating events in tracer flux experiments. II. Flux amplitude analysis

Measurement of tracer ion flux from or into a collection of closed membrane structures (CMS) constitutes a broadly applicable technique for studying ion channel gating by specialized gating molecules in biological membranes. The amplitudes for the flux process reflect the overall change in tracer content due to flux during a period in which channels on at least some of the CMS were open. In practice, the attainment of a time-invariant, finite overall tracer content, indicating a cessation of flux, need not imply that flux has reached completion, i.e., that the CMS internal and external tracer concentrations have fully reached equilibrium. Less than maximum flux amplitudes arise when binding of control ligands leads to an inhibition or inactivation of the channel gating molecules prior to a complete equilibration of tracer. Analysis of the dependence of the flux amplitudes on control ligand concentration permits determination of characteristic parameters of the CMS that may vary with the methods of preparation (e.g., the distributions of CMS size and CMS content of gating units). Knowledge of these parameters in turn permits evaluation of the mean single channel flux amplitude contribution, which is functionally dependent on the rate constant ratio (k'eff/ki), where k'eff and ki are, respectively, the effective rate constants for tracer flux and for gating unit inactivation.     

Dynamics of noninvasively estimated microvascular O2 extraction during ramp exercise.

Utilization of near-infrared spectroscopy (NIRS) in clinical exercise testing to detect microvascular abnormalities requires characterization of the responses in healthy individuals and theoretical foundation for data interpretation. We examined the profile of the deoxygenated hemoglobin signal from NIRS {deoxygenated hemoglobin + myoglobin [deoxy-(Hb+Mb)] approximately O(2) extraction} during ramp exercise to test the hypothesis that the increase in estimated O(2) extraction would be close to hyperbolic, reflecting a linear relationship between muscle blood flow (Q(m)) and muscle oxygen uptake (Vo(2)(m)) with a positive Q(m) intercept. Fifteen subjects (age 24 +/- 5 yr) performed incremental ramp exercise to fatigue (15-35 W/min). The deoxy-(Hb+Mb) response, measured by NIRS, was fitted by a hyperbolic function [f(x) = ax/(b + x), where a is the asymptotic value and b is the x value that yields 50% of the total amplitude] and sigmoidal function {f(x) = f(0) + A/[1 + e(-(-c+dx))], where f(0) is baseline, A is total amplitude, and c is a constant dependent on d, the slope of the sigmoid}, and the goodness of fit was determined by F test. Only one subject demonstrated a hyperbolic increase in deoxy-(Hb+Mb) (a = 170%, b = 193 W), whereas 14 subjects displayed a sigmoidal increase in deoxy-(Hb+Mb) (f(0) = -7 +/- 7%, A = 118 +/- 16%, c = 3.25 +/- 1.14, and d = 0.03 +/- 0.01). Computer simulations revealed that sigmoidal increases in deoxy-(Hb+Mb) reflect a nonlinear relationship between microvascular Q(m) and Vo(2)(m) during incremental ramp exercise. The mechanistic implications of our findings are that, in most healthy subjects, Q(m) increased at a faster rate than Vo(2)(m) early in the exercise test and slowed progressively as maximal work rate was approached.     

The silent period in the stretch response of ia-activated dorsal spino-cerebellar tract neurons to sinusoidal muscle stretch in cats

By means of extracellular recordings of action potentials the stretch responses of single neurons of Clarke's column were analysed. The neurons were monosynaptically activated from Ia afferents of both ipsilateral gastrocnemius muscles. When stretch cycles of more than 0.2 mm amplitude and frequencies above 2 Hz were applied to the gastrocnemius muscles, the discharging was found to cease during the period of stretch release, whereas the average discharge rate was found to increase. In the frequency range between 0.1 and 10 Hz a sinewave of stretch frequency — the response sinewave — fitted to the non-zero bins of cycle histograms described the stretch response at small and large amplitudes equally well. The amount of increase in the average firing rate corresponded quite well to the portion of the response sinewave below the zero discharge rate. This indicates that the occurance of discharge pauses and the relation of the average discharge rate to frequency and amplitude of stretch can be described successfully by a half-wave rectification of the response at zero discharge rate. If one regards the shape of cycle histograms to be a nearly sinusoidal modulation plus a non-linear clipping at zero the application of linear systems analysis is worthwhile in describing the response not only at very small amplitudes but in the whole range of muscle stretch.     

Flux detectors versus concentration detectors: two types of chemoreceptors

Dose-response curves relating the external stimulus concentration to receptor occupancy differ in two types of chemoreceptor organs. In 'concentration detectors' the receptor molecules at the receptor cell membrane are directly exposed to the external stimulus concentration; these organs exhibit the well-known hyperbolic dose-response relationship reflecting the association-dissociation of stimulus and receptor molecules. In contrast, 'flux detectors' accumulate the stimulus molecules in a perireceptor compartment. In flux detectors, deactivation of stimulus molecules may be in balance with arrival, as a prerequisite for producing a constant effective stimulus concentration at constant adsorptive flux of stimulus molecules. In a simple model of a flux detector in which receptor molecules themselves catalyze the deactivation, the dose-response relationship is linear. It reflects the rate of stimulus deactivation. If the deactivation is catalyzed by a separate enzyme, the dose-response relationship can be close to hyperbolic, or linear. In all cases, the receptor molecules are maximally occupied if the adsorptive flux equals or exceeds the maximum rate of stimulus deactivation. The time course of the receptor potential recorded from moths' pheromone receptors depends on the odor compound, which suggests that a peripheral process, possibly the stimulus deactivation, is the slowest, rate-limiting process of the transduction cascade. Further evidence comes from experiments with stimuli oversaturating the mechanism responsible for the decline of the receptor potential.      

The effect of myoglobin-facilitated oxygen transport on the basal metabolism of papillary muscle.

A mathematical model of oxygen diffusion into cylindrical papillary muscles is presented. The model partitions total oxygen flux into its simple and myoglobin-facilitated components. The model includes variable sigmoidal, exponential, or hyperbolic functions relating oxygen partial pressure to both fractional myoglobin saturation and rate of oxygen consumption. The behavior of the model was explored for a variety of saturation- and consumption-concentration relations. Facilitation of oxygen transport by myoglobin was considerable as indexed both by the elevation of oxygen partial pressure on the longitudinal axis of the muscle and by the fraction of total oxygen flux at the muscle center contributed by oxymyoglobin. Despite its facilitation of oxygen flux at the muscle center, myoglobin made only a negligible contribution to the total oxygen consumption averaged over the muscle cross-section. Hence the presence of myoglobin fails to explain either the experimentally determined basal metabolism-muscle radius relation or the stretch effect observed in isolated papillary muscle.     

Hyperbolic and kinetic models for self-organized biological aggregations and movement: a brief review

We briefly review hyperbolic and kinetic models for self-organized biological aggregations and traffic-like movement. We begin with the simplest models described by an advection-reaction equation in one spatial dimension. We then increase the complexity of models in steps. To this end, we begin investigating local hyperbolic systems of conservation laws with constant velocity. Next, we proceed to investigate local hyperbolic systems with density-dependent speed, systems that consider population dynamics (i.e., birth and death processes), and nonlocal hyperbolic systems. We conclude by discussing kinetic models in two spatial dimensions and their limiting hyperbolic models. This structural approach allows us to discuss the complexity of the biological problems investigated, and the necessity for deriving complex mathematical models that would explain the observed spatial and spatiotemporal group patterns.     

Ultrasensitivity in Phosphorylation-Dephosphorylation Cycles with Little Substrate

Cellular decision-making is driven by dynamic behaviours, such as the preparations for sunrise enabled by circadian rhythms and the choice of cell fates enabled by positive feedback. Such behaviours are often built upon ultrasensitive responses where a linear change in input generates a sigmoidal change in output. Phosphorylation-dephosphorylation cycles are one means to generate ultrasensitivity. Using bioinformatics, we show that in vivo levels of kinases and phosphatases frequently exceed the levels of their corresponding substrates in budding yeast. This result is in contrast to the conditions often required by zero-order ultrasensitivity, perhaps the most well known means for how such cycles become ultrasensitive. We therefore introduce a mechanism to generate ultrasensitivity when numbers of enzymes are higher than numbers of substrates. Our model combines distributive and non-distributive actions of the enzymes with two-stage binding and concerted allosteric transitions of the substrate. We use analytical and numerical methods to calculate the Hill number of the response. For a substrate with phosphosites, we find an upper bound of the Hill number of , and so even systems with a single phosphosite can be ultrasensitive. Two-stage binding, where an enzyme must first bind to a binding site on the substrate before it can access the substrate''s phosphosites, allows the enzymes to sequester the substrate. Such sequestration combined with competition for each phosphosite provides an intuitive explanation for the sigmoidal shifts in levels of phosphorylated substrate. Additionally, we find cases for which the response is not monotonic, but shows instead a peak at intermediate levels of input. Given its generality, we expect the mechanism described by our model to often underlay decision-making circuits in eukaryotic cells.

Authors Summary

Dose-response curves are said to be ultrasensitive when they are sigmoidal rather than hyperbolic and often underlay cellular decision-making circuits. Zero-order ultrasensitivity is a well-known mechanism to generate sigmoidal curves in phosphorylation cycles, but one of its assumptions often implies that the substrate is more abundant than the modifying enzymes. We show that this assumption is unlikely to always hold in vivo, and we present a general model that generates ultrasensitivity when the enzymes are in excess of their substrate. The model combines conformational allosteric transitions of the substrate with two-stage binding of the enzymes: the enzymes bind first to a docking site on the substrate and then to the substrate''s phosphosites. Ultrasensitivity is generated because the kinase can bind to the fully phosphorylated form of the substrate (at its docking site) and sequester the substrate away from the phosphatase and, similarly, the phosphatase can bind to the fully dephosphorylated form of the substrate and sequester the substrate away from the kinase. The number of kinase-phosphatase competitions for the substrate determines the degree of ultrasensitivity. Finally, we show that this model can generate non-monotonic responses that peak at intermediate levels of input.     

Fibrin assembly in human plasma and fibrinogen/albumin mixtures

Magnetic birefringence is used to monitor the kinetics of thrombin-catalyzed fibrin polymerization in model systems of increasing complexity (i.e., fibrinogen solutions, fibrinogen/albumin mixtures, and plasma anticoagulated with citrate) and in plasma containing free calcium which is the physiological condition. The introduction of albumin into fibrinogen solutions shortens the lag period and enhances fiber thickness. The polymerization progress curves are sigmoidal at zero or low albumin concentrations, but at physiological and higher concentrations, they become hyperbola-like from the end of the lag period. High albumin concentration has thus induced a change in the assembly kinetics. The progress curves from plasma in which the cascade is dormant are also hyperbola-like although they round off more quickly because of antithrombin activity. In plasma containing free calcium, thrombin is endogenously produced, and the progress curves are nearly linear; hence, the assembly kinetics are very different from those of the model systems. The curves are not influenced by calcium-dependent cross-linking involving factor XIIIa. The progress curves are also linear when polymerization is induced with Russell's viper venom, which by directly activating factor X circumvents earlier steps in the cascade. This implies that linear polymerization is caused by events posterior to factor X activation and are thus likely to be largely dependent on the functioning of the prothrombinase complex. Addition of thrombin to plasma containing free calcium reduces the lag period. At low exogenous thrombin levels, the polymerization rate is increased, and the progress curves remain linear. However, at higher levels, the curves become more complicated and, paradoxically, full polymerization takes longer.(ABSTRACT TRUNCATED AT 250 WORDS)     

Theoretical analysis of the significance of whether or not enzymes or transport systems in structured media follow Michaelis-Menten kinetics

Pure Michaelis-Menten enzymes have been studied (i.e., enzymes with a hyperbolic (S, V) behavior in a well-stirred solution). When such enzymes are associated with a structure in vitro, even in the simplest conceivable form (immobilization in a homogeneous gel), they can produce enzymic or transport reactions with many different kinetics (Michaelis-Menten, sigmoidal, dual-phasic, etc.). Therefore, when structured enzyme or transport processes in vivo have sigmoidal kinetics, it is not proof that the corresponding proteins are allosteric. In same manner, when the apparent kinetics are dual-phasic, it is not proof that two enzyme, or transport systems, coexist.     

Analytical results on a Wilson-Cowan neuronal network modified model

Wilson-Cowan model is employed in studies concerning neuronal networks. This model consists of two nonlinear differential equations that represent the interaction between excitatory and inhibitory populations of neurons. The mutual influence of these populations is described through a sigmoidal function, which is usually chosen as the hyperbolic tangent or the logistic curve. Both choices make difficult theoretical analyses. Here we choose another sigmoidal function and analytically obtain the set of parameter values for which an asymptotically stable limit cycle exists. This result is potentially useful to analytical and numerical works on the binding problem, which is the problem of creating a coherent representation of objects from the oscillatory activity of spatially separated cortical columns.     

Bidirectional reaction steps in metabolic networks: I. Modeling and simulation of carbon isotope labeling experiments

The extension of metabolite balancing with carbon labeling experiments, as described by Marx et al. (Biotechnol. Bioeng. 49: 11-29), results in a much more detailed stationary metabolic flux analysis. As opposed to basic metabolite flux balancing alone, this method enables both flux directions of bidirectional reaction steps to be quantitated. However, the mathematical treatment of carbon labeling systems is much more complicated, because it requires the solution of numerous balance equations that are bilinear with respect to fluxes and fractional labeling. In this study, a universal modeling framework is presented for describing the metabolite and carbon atom flux in a metabolic network. Bidirectional reaction steps are extensively treated and their impact on the system's labeling state is investigated. Various kinds of modeling assumptions, as usually made for metabolic fluxes, are expressed by linear constraint equations. A numerical algorithm for the solution of the resulting linear constrained set of nonlinear equations is developed. The numerical stability problems caused by large bidirectional fluxes are solved by a specially developed transformation method. Finally, the simulation of carbon labeling experiments is facilitated by a flexible software tool for network synthesis. An illustrative simulation study on flux identifiability from available flux and labeling measurements in the cyclic pentose phosphate pathway of a recombinant strain of Zymomonas mobilis concludes this contribution. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 101-117, 1997.     

Estimates of predator-prey limit cycles

Perturbation methods are applied to a differential equation predator-prey model to find the approximate amplitudes and period of limit cycle solutions. In the model the feeding rate per unit predator per unit prey decreases as the prey become scare. The rigorous applicability of the perturbation technique depends on the assumptions that the limit cycle amplitude is relatively small and that near the equilibrium point the growth rate of each species is most sensitive to changes in the density of the other species. The second assumption is usually roughly satisfied in practice and examples are considered which suggest that the first assumption can be greatly relaxed.     

Isolation and partial characterization of norcoclaurine synthase, the first committed step in benzylisoquinoline alkaloid biosynthesis, from opium poppy

Norcoclaurine synthase (NCS) catalyzes the condensation of dopamine and 4-hydroxyphenylacetaldehyde (4-HPAA) to yield norcoclaurine, the common precursor to all benzylisoquinoline alkaloids produced in plants. In opium poppy (Papaver somniferum L.), NCS activity was detected in germinating seeds, young seedlings, and all mature plant organs, especially stems and roots. However, the highest levels of activity were found in cell-suspension cultures treated with a fungal elicitor. NCS activity was induced more than 20-fold over an 80-h period in response to elicitor treatment. Compared to opium poppy. basal NCS activity was 3-and 5-fold higher in benzylisoquinoline alkaloid-producing cell cultures of Eschscholzia californica and Thalictrum flavum ssp. glaucum, respectively. In contrast, NCS activity was not detected in cultured cells of Nicotiana tabacum and Catharanthus roseus, which do not produce benzylisoquinoline alkaloids. NCS displayed maximum activity between pH 6.5 and 7.0, and a broad temperature optimum between 42 and 55 degrees C. Enzyme activity was not affected by Ca2+ or Mg2+, and was not inhibited by a variety of benzylisoquinoline alkaloids. NCS showed hyperbolic saturation kinetics for 4-HPAA, with an apparent Km of 1.0 mM. However, the enzyme exhibited sigmoidal saturation kinetics for dopamine with a Hill coefficient of 1.84. NCS enzymes from E. californica and T. flavum displayed similar properties. These data indicate that NCS exhibits positive cooperativity between substrate-binding sites. Enzymes of this type catalyze regulatory, or rate-limiting, steps in metabolism, suggesting that NCS plays a role in controlling the rate of pathway flux in benzylisoquinoline alkaloid biosynthesis.     

Thermoperiodic regulation of the circadian eclosion rhythm in the flesh fly, Sarcophaga crassipalpis

We recorded the eclosion time of the flesh fly, Sarcophaga crassipalpis, at different depths in the outdoor soil and under temperature cycles with various amplitudes in the laboratory, to examine the timing adjustment of eclosion in response to temperature cycles and their amplitudes in the pupal stage. In the soil, most eclosions occurred in the late morning, which was consistent with the eclosion time under pseudo-sinusoidal temperature cycles in the laboratory. The circadian clock controlling eclosion was reset by temperature cycles and free-ran with a period close to 24 h. This clock likely helps pupae eclose at an optimal time even when the soil temperature does not show clear daily fluctuations. The eclosion phase of the circadian clock progressively advanced as the amplitude of the pseudo-sinusoidal temperature cycle decreased. This response allows pupae located at any depth in the soil to eclose at the appropriate time despite the depth-dependent phase delay of the temperature change. In contrast, the abrupt temperature increase in square-wave temperature cycles reset the phase of the circadian clock to the increasing time, regardless of the temperature amplitude. The rapid temperature increase may act as the late-morning signal for the eclosion clock.     

Viscoelastic properties of passive skeletal muscle in compression—Cyclic behaviour

Skeletal muscle relaxation behaviour in compression has been previously reported, but the anisotropic behaviour at higher loading rates remains poorly understood. In this paper, uniaxial unconfined cyclic compression tests were performed on fresh porcine muscle samples at various fibre orientations to determine muscle viscoelastic behaviour. Mean compression level of 25% was applied and cycles of 2% and 10% amplitude were performed at 0.2–80 Hz. Under cycles of low frequency and amplitude, linear viscoelastic cyclic relaxation was observed. Fibre/cross-fibre results were qualitatively similar, but the cross-fibre direction was stiffer (ratio of 1.2). In higher amplitude tests nonlinear viscoelastic behaviour with a frequency dependent increase in the stress cycles amplitude was found (factor of 4.1 from 0.2 to 80 Hz).The predictive capability of an anisotropic quasi-linear viscoelastic model previously fitted to stress-relaxation data from similar tissue samples was investigated. Good qualitative results were obtained for low amplitude cycles but differences were observed in the stress cycle amplitudes (errors of 7.5% and 31.8%, respectively, in the fibre/cross-fibre directions). At higher amplitudes significant qualitative and quantitative differences were evident. A nonlinear model formulation was therefore developed which provided a good fit and predictions to high amplitude low frequency cyclic tests performed in the fibre/cross-fibre directions. However, this model gave a poorer fit to high frequency cyclic tests and to relaxation tests. Neither model adequately predicts the stiffness increase observed at frequencies above 5 Hz.Together with data previously presented, the experimental data presented here provide a unique dataset for validation of future constitutive models for skeletal muscle in compression.     

The effects of specific respiratory rates on heart rate and heart rate variability

In this study respiratory rates of 3, 4, 6, 8, 10, 12, and 14 breaths per minute were employed to investigate the effects of these rates on heart rate variability (HRV). Data were collected 16 times at each respiratory rate on 3 female volunteers, and 12 times on 2 female volunteers. Although mean heart rates did not differ among these respiratory rates, respiratory-induced trough heart rates at 4 and 6 breaths per minute were significantly lower than those at 14 breaths per minute. Slower respiratory rates usually produced higher amplitudes of HRV than did faster respiratory rates. However, the highest amplitudes were at 4 breaths per minute. HRV amplitude decreased at 3 breaths per minute. The results are interpreted as reflecting the possible effects of the slow rate of acetylcholine metabolism and the effect of negative resonance at 3 cycles per minute.     

Kinetics of electron transfer through the respiratory chain

We show that the rate at which electrons pass through the respiratory chain in mitochondria and respiring prokaryotic cells is described by the product of three terms, one describing electron donation, one acceptance, and a third, the thermodynamic drive. We apply the theory of nonequilibrium thermodynamics in the context of the chemiosmotic model of proton translocation and energy conservation. This approach leads to a closed-form expression that predicts steady-state electron flux as a function of chemical conditions and the proton motive force across the mitochondrial inner membrane or prokaryotic cytoplasmic membrane. The rate expression, derived considering reverse and forward electron flow, is the first to account for both thermodynamic and kinetic controls on the respiration rate. The expression can be simplified under specific conditions to give rate laws of various forms familiar in cellular physiology and microbial ecology. The expression explains the nonlinear dependence of flux on electrical potential gradient, its hyperbolic dependence on substrate concentration, and the inhibiting effects of reaction products. It provides a theoretical basis for investigating life under unusual conditions, such as microbial respiration in alkaline waters.     

Epidemiological effects of seasonal oscillations in birth rates

Seasonal oscillations in birth rates are ubiquitous in human populations. These oscillations might play an important role in infectious disease dynamics because they induce seasonal variation in the number of susceptible individuals that enter populations. We incorporate seasonality of birth rate into the standard, deterministic susceptible-infectious-recovered (SIR) and susceptible-exposed-infectious-recovered (SEIR) epidemic models and identify parameter regions in which birth seasonality can be expected to have observable epidemiological effects. The SIR and SEIR models yield similar results if the infectious period in the SIR model is compared with the "infected period" (the sum of the latent and infectious periods) in the SEIR model. For extremely transmissible pathogens, large amplitude birth seasonality can induce resonant oscillations in disease incidence, bifurcations to stable multi-year epidemic cycles, and hysteresis. Typical childhood infectious diseases are not sufficiently transmissible for their asymptotic dynamics to be likely to exhibit such behaviour. However, we show that fold and period-doubling bifurcations generically occur within regions of parameter space where transients are phase-locked onto cycles resembling the limit cycles beyond the bifurcations, and that these phase-locking regions extend to arbitrarily small amplitude of seasonality of birth rates. Consequently, significant epidemiological effects of birth seasonality may occur in practice in the form of transient dynamics that are sustained by demographic stochasticity.     

Evaluation of Hill slopes and Hill coefficients when the saturation binding or velocity is not known.

1. The Hill coefficient (nH), an often-used measure of deviations from hyperbolic behaviour (nonhyperbolicity) in kinetic and binding systems, is usually estimated from the maximum or minimum slope of the Hill plot. The method depends strongly on the assumed magnitude of the asymptotic velocity (V) or binding (P) whose evaluation may be difficult in nonlinear/co-operative systems. Therefore, alternative procedures were devised for the estimation nH which do not require the prior knowledge of V or P. 2. When pairs of velocity/binding readings (v and w) are obtained at concentrations of c and alpha c, respectively (where alpha is a fixed constant), then the relation between w and v is described by a hyperbola, provided that Hill's equation is valid. In this case, linearizing plots, v/w versus v, w versus, w/v, and 1/w versus 1/v, can be used for estimation of the degree of the equation. However, if the Hill expression is applicable, these methods are not efficient and traditional procedures, particularly nonlinear regression, should be used. 3. The 'linearizing' plots of the Hill equation can be applied advantageously for the evaluation of the Hill slope and of nH also in the general case, when the Hill expression is actually not valid, provided that deviations from hyperbolic behaviour are positive. Appropriately extrapolated intercepts of the first two plots estimate alphanH. Furthermore, the slope of the third plot yields, similarly to the method of Kurganov et al., a continuous measure of the Hill slope (including its maximum) at all concentrations. The agreement is, at positive nonhyperbolicities, excellent theoretical values of Hill slopes and coefficients and those estimated by the proposed methods. 4. A coefficient of nonhyperbolicity (theta) is defined for 2nd-degree rate equations which provides a quantitative measure of positive or negative deviation from first-degree, hyperbolic characteristics. It is closely related to the Hill coefficient.