Instabilities,bifurcations and fluctuations in chemical systems : L. E. Reichl and W. C. Schieve University of Texas Press, 1982, xi+427 pp, $35.00

The occupancy principle and the mean-transit-time theorem are derived for the passage of a tracer through a system that can be described by a general pool model. It is proved, using matrix theory, that if (and only if) tracer entering the system labels equally all tracee fluxes into the system, then the integral of the tracer concentration is the same in all the pools. It is also proved that if, in addition, all flow out of the system is through the observation point, the first moment of the tracer concentration at the observation point can be used to calculate the total amount of trace in the system. The necessity of this condition is analyzed. Examples are given of models in which the occupancy principle and the mean-transit-time theorem hold or do not hold.     

Basic studies on metabolic steady state. Linearity of a metabolic tracer system

Summary In dogs the changes in the plasma concentration following the injection and/or infusion of labeled glucose reveal the labeled glucose system to be a linear system. Based on a simple chemical model it was shown that the rate of removal from the system of any tracer injected as single impulse can be described by a first order chemical reaction, even in systems from which the tracee is removed by a process of a higher order, provided the tracee is in the steady state and the concentration of tracer in the plasma is low compared to that of the tracee. The validity of the commonly used formulae for the calculations of the rates of disappearance from systems in a steady state is based on this first order process.     

Calculation of stable isotope enrichment tracer kinetic procedures

The choice of method of expressing isotopic enrichment in tracer kinetic experiments utilizing stable isotopes was found to affect the calculation of tracee pool size and half-life. The most commonly used definition, the difference between enriched and natural abundance, i.e. atom percent excess, was found to result in significant error in model systems when the dose of tracer was 10% of the pool size. Errors in determining first-order rate constants of efflux and in pool sizes decreased with decreasing ratio of tracer to tracee. Error in determining pool size increased with longer 'sampling' periods, while error in determining the rate constant increased with shorter sampling periods. Of three less frequently used expressions of isotopic enrichment two were found to yield the exact answers in model systems. The correct expressions of isotopic enrichment were linear functions of the quantity of tracer in the system. A practical example demonstrated the effect of choice of expression of enrichment on estimates of whole body copper pool size and turnover in dairy cattle.     

An integral equation approach to measuring turnover in nonsteady compartmental and distributed systems

Physiological systems are often modelled by a set of compartments. Alternatively they can be described by the diffusion-convection-reaction equations governing distributed systems. The problem considered here is that of identifying a continuously changing input of some metabolite )tracee), endogenous to the system and hence inaccessible, when a nonlinear or time-varying component is also introduced into the loss parameter, as for example through feedback mechanisms. A tracer is used to determine the steady-state impulse response under time-invariant, linear conditions. A known input of tracer is also administered when the system is driven out of steady state. The integral equations developed utilize the predetermined impulse response, the measured concentrations of both tracer and tracee (output) in some region of the system to estimate the changing loss parameter and the unknown input in a continuous fashion.     

Isolation of plasma albumin by ethanol extraction is inappropriate for isotope ratio measurements during the acute phase response

Isolation of high-purity albumin from plasma is essential to study albumin kinetics in vivo with tracer techniques. Because of its simplicity ethanol extraction has been repeatedly used for albumin purification. However, it cannot be excluded that this single-step procedure completely prohibits contamination by other proteins, especially those known to be produced at an accelerated rate during the acute phase response. In the present study, we wanted to examine the reliability of ethanol extraction in different clinical conditions and to study the effects of potential impurities on albumin enrichment during stable isotope tracer studies. SDS-PAGE revealed a contaminating protein band at about 25,000 Da in healthy subjects and postoperative patients during the acute phase response, but not in critically ill patients. According to densitometry about 8% of proteins after ethanol extraction were contaminants. To examine potential contaminant effects on tracer enrichment 1-[13C]-leucine was given to healthy subjects and postoperative patients. Blood samples were taken after various amounts of time, and albumin enrichments (tracer/tracee ratios) were determined from isotope ratios obtained by mass spectrometry. Irrespective of the magnitude of tracer enrichment, postoperative tracer/tracee ratios were significantly higher (on average +10%) in samples exclusively analysed by ethanol extraction than in samples which had undergone additional electrophoretic purification. No significant effect of the contaminant was seen in healthy subjects. N-terminal protein sequencing revealed contaminants to mainly consist of apolipoprotein A-1. Its physiology and pathophysiology may sufficiently explain its variable effects of albumin enrichment. Our findings suggest that exclusive ethanol extraction is inappropriate for albumin isolation in tracer studies performed during the acute phase response. Ethanol extraction may also not be advisable in all other situations known to be associated with a rise in apolipoprotein A-1 turnover.     

Theoretical Analysis of Net Tracer Flux Due to Volume Circulation in a Membrane with Pores of Different Sizes : Relation to solute drag model

When osmotic pressure across an artificial membrane, produced by a permeable electrically neutral solute on one side of it, is balanced by an external pressure difference so that there is no net volume flow across the membrane, it has been found that there will be a net flux of a second electrically neutral tracer solute, present at equal concentrations on either side of the membrane, in the direction that the "osmotic" solute diffuses. This has been ascribed to solute-solute interaction or drag between the tracer and the osmotic solutes. An alternative model, presented here, considers the membrane to have pores of different sizes. Under general assumptions, this "heteroporous" model will account for both the direction of net tracer flux and the observed linear dependence of unidirectional tracer fluxes on the concentration of the osmotic solute. The expressions for the fluxes of solutes and solvent are mathematically identical under the two models. An inequality is derived which must be valid if the solute interaction model and/or the heteroporous model can account for the data. If the inequality does not hold, then the heteroporous model alone cannot explain the data. It was found that the inequality holds for most published observations except when dextran is the osmotic solute.     

The New Permeability Pathways Induced by the Malaria Parasite in the Membrane of the Infected Erythrocyte: Comparison of Results Using Different Experimental Techniques

The membrane of erythrocytes infected with malaria parasites is highly permeable to a large variety of solutes, including anions, carbohydrates, amino acids, nucleosides, organic and inorganic cations and small peptides. The altered permeability is presumed to be due to the activation of endogenous dormant channels, the new permeability pathways. The latter have been studied by different techniques—isosmotic lysis and tracer fluxes—and recently by patch-clamping. Here we analyze all available published data and we show that there is generally a good agreement between the two first methods. From the fluxes we calculate the number of channels per cell using reasonable assumptions as to the radius of the channel, and assuming that penetration through the channel is by diffusion through a water-filled space. The number of channels so calculated is <10 for most solutes, but ~400 for anions and the nucleosides thymidine and adenosine. This latter number is not far from that calculated from patch-clamp experiments. However, the anion flux measured directly by tracer is an order of magnitude larger than expected from conductance measurements. We conclude that the new permeability pathways consist of two types of channels; one is present in small number, and is charge- and size-selective. The other type is about 100-fold more abundant and is anion-selective, but does not admit non-electrolytes other than perhaps nucleosides.     

Methodological tests of the use of trace elements as tracers to assess root activity

Background and aims

There is increasing interest in how resource utilisation in grassland ecosystems is affected by changes in plant diversity and abiotic conditions. Research to date has mainly focussed on aboveground responses and there is limited insight into belowground processes. The aim of this study was to test a number of assumptions for the valid use of the trace elements caesium, lithium, rubidium and strontium as tracers to assess the root activity of several grassland species.

Methods

We carried out a series of experiments addressing the reliability of soil labelling, injection density, incubation time, application rate and the comparability of different tracers in a multiple tracer method.

Results

The results indicate that it is possible to achieve a reliable labelling of soil depths. Tracer injection density affected the variability but not the mean level of plant tracer concentrations. Tracer application rates should be based on pilot studies, because of site- and species-specific responses. The trace elements did not meet prerequisites to be used in a multiple tracer method.

Conclusions

The use of trace elements as tracers is potentially a very useful tool to give insight into plant root activity at different soil depths. This work highlights some of the main benefits and pitfalls of the method and provides specific recommendations to assist the design of tracer experiments and interpretation of the results.     

Possible pitfalls of flux calculations based on (13)C-labeling

Metabolic engineers have enthusiastically adopted the (13)C-labeling technique as a powerful tool for elucidating fluxes in metabolic networks. This tracer technique makes it possible to determine fluxes that are unobservable using only metabolite balances and allows the elimination of doubtful cofactor balances that are indispensable in flux analysis based on metabolite balancing alone. The (13)C-labeling technique, however, relies on a number of assumptions that are not free from uncertainties. Two possible errors in the models that are needed to determine the metabolic fluxes from labeling data are omitted reactions and ignored occurrence of channeling. By means of two representative examples it is shown that these modeling errors may lead to serious errors in the calculated flux distributions despite the use of labeling data. A complicating fact is that the model errors are not always easily detected as poor models may still yield good fits of experimental data. Results of (13)C-labeling experiments should therefore be interpreted with appropriate caution.     

Use of in vivo kinetic techniques to evaluate metabolism of nutrients

Most nutrition research is related to rates of physiological processes. Information about those processes can be gained by in vivo kinetic techniques; however, many nutritionists are hesitant to use in vivo kinetics. The two basic in vivo kinetic techniques are single injection and continuous infusion of tracer into a pool of tracee. Either technique can form the basis for multiple-pool kinetics, or modeling. Solving a multiple-pool system can provide flow rates of substances between metabolic pools and is valuable for understanding a particular metabolic pathway or process. In vivo kinetic techniques can be valuable in understanding mechanisms whereby partitioning agents affect the distribution of nutrients, especially protein and fat, in food-producing animals. In vivo kinetics is a valuable tool for nutrition research and should be used more frequently.     

Net uptake of chloride across the posterior gills of the Chinese crab (Eriocheir sinensis)

Two methods are commonly used for the determination of transbranchial net fluxes of Na+ and Cl-: direct analysis of changes in ion concentrations in the external medium using flame spectrophotometry or titration (net flux method), and measurement of unidirectional ion fluxes by means of radioactive tracers (tracer method). When we applied both methods in the same preparation, the isolated perfused posterior gill of freshwater-acclimated Eriocheir sinensis, to determine net fluxes of Cl-, the results differed substantially. In artificial fresh water (AFW) containing NaCl, the net flux method yielded a net uptake, but the tracer method showed a net efflux of Cl-. The net uptake of Cl- was abolished in Na(+)-free AFW indicating that Cl- uptake is coupled with the uptake of Na+. Applying the tracer method, net efflux of Cl- remained almost unchanged in Na(+)-free AFW. This suggests the opposite mechanism, i.e. uncoupled uptake of Na+ and Cl-. The discrepancy in the results obviously depends on the method employed. Since the data obtained with the net flux method explain the osmoregulatory performance of crabs living in fresh water, we consider this method as appropriate for determining net transbranchial ion fluxes.     

Measurement of human apolipoprotein B-48 and B-100 kinetics in triglyceride-rich lipoproteins using [5,5,5-2H3]leucine.

A primed-constant infusion of deuterated leucine was used in humans to determine the maximal level of enrichment at plateau of apolipoprotein (apo)B-48 and apoB-100 which are synthesized in the intestine and liver, respectively, and to compare the kinetics of these two proteins under identical conditions. Eight normal subjects (four post-menopausal females and four males) over the age of 40 were studied in the constantly fed state over a 20-h period by providing small hourly feedings of identical composition. [5,5,5-2H3]Leucine (10 mumol/kg body weight followed by 10 mumol/kg body weight per hour) was infused over 15 h intravenously. The enrichment of deuterated leucine in apoB-48 and apoB-100 triglyceride-rich lipoproteins isolated by ultracentrifugation (d less than 1.006 g/ml) was determined during the entire infusion period. The plateau level of enrichment in triglyceride-rich lipoprotein apoB-48 was 3.96 +/- 1.41 tracer/tracee ratio (%) which was 39.7% of the plasma leucine enrichment level. The plateau level of enrichment in triglyceride-rich lipoprotein apoB-100 was 7.23 +/- 1.17 tracer/tracee ratio (%) which was 72.5% of the plasma leucine enrichment level. Mean fractional secretion rates of triglyceride-rich lipoprotein apoB-48 and apoB-100 were 4.39 +/- 2.00 and 5.39 +/- 1.98 pools per day, respectively, with estimated residence times of 5.47 and 4.45 hours, respectively. The data indicate that in the fed state there is about a twofold difference in the plateau enrichment of an intestinally derived protein, as compared to one of hepatic origin, most likely attributable to differences in the enrichment of the intracellular leucine in the two organs.(ABSTRACT TRUNCATED AT 250 WORDS)     

13C tracer experiments and metabolite balancing for metabolic flux analysis: comparing two approaches

Conventional metabolic flux analysis uses the information gained from determination of measurable fluxes and a steady-state assumption for intracellular metabolites to calculate the metabolic fluxes in a given metabolic network. The determination of intracellular fluxes depends heavily on the correctness of the assumed stoichiometry including the presence of all reactions with a noticeable impact on the model metabolite balances. Determination of fluxes in complex metabolic networks often requires the inclusion of NADH and NADPH balances, which are subject to controversial debate. Transhydrogenation reactions that transfer reduction equivalents from NADH to NADPH or vice versa can usually not be included in the stoichiometric model, because they result in singularities in the stoichiometric matrix. However, it is the NADPH balance that, to a large extent, determines the calculated flux through the pentose phosphate pathway. Hence, wrong assumptions on the presence or activity of transhydrogenation reactions will result in wrong estimations of the intracellular flux distribution. Using 13C tracer experiments and NMR analysis, flux analysis can be performed on the basis of only well established stoichiometric equations and measurements of the labeling state of intracellular metabolites. Neither NADH/NADPH balancing nor assumptions on energy yields need to be included to determine the intracellular fluxes. Because metabolite balancing methods and the use of 13C labeling measurements are two different approaches to the determination of intracellular fluxes, both methods can be used to verify each other or to discuss the origin and significance of deviations in the results. Flux analysis based entirely on metabolite balancing and flux analysis, including labeling information, have been performed independently for a wild-type strain of Aspergillus oryzae producing alpha-amylase. Two different nitrogen sources, NH4+ and NO3-, have been used to investigate the influence of the NADPH requirements on the intracellular flux distribution. The two different approaches to the calculation of fluxes are compared and deviations in the results are discussed. Copyright 1998 John Wiley & Sons, Inc.     

Ion Transport in Hydrodictyon africanum

The concentrations of K, Na, and Cl in the cytoplasm and vacuole, the tracer fluxes of these ions into and out of the cenocyte, and the electrical potential difference between bathing solution and vacuole and cytoplasm, have been measured in Hydrodictyon africanum. If the ions were acted on solely by passive electrochemical forces, a net efflux of K and Cl and a net influx of Na would be expected. Tracer fluxes indicate a net influx of K and Cl and efflux of Na in the light; these net fluxes are consequently active, with an obligate link to metabolism. The effects of darkness and low temperature indicate that most of the tracer K and Cl influx and Na efflux are linked to metabolism, while the corresponding tracer fluxes in the direction of the free energy gradient are not. Ouabain specifically inhibits the metabolically linked portions of tracer K influx and Na efflux. Alterations in the external K concentration have similar effects on metabolically mediated K influx and Na efflux. It would appear that K influx and Na efflux are linked, at least in the light.     

Carbonyl sulfide (COS) as a tracer for canopy photosynthesis, transpiration and stomatal conductance: potential and limitations

The theoretical basis for the link between the leaf exchange of carbonyl sulfide (COS), carbon dioxide (CO(2)) and water vapour (H(2)O) and the assumptions that need to be made in order to use COS as a tracer for canopy net photosynthesis, transpiration and stomatal conductance, are reviewed. The ratios of COS to CO(2) and H(2)O deposition velocities used to this end are shown to vary with the ratio of the internal to ambient CO(2) and H(2)O mole fractions and the relative limitations by boundary layer, stomatal and internal conductance for COS. It is suggested that these deposition velocity ratios exhibit considerable variability, a finding that challenges current parameterizations, which treat these as vegetation-specific constants. COS is shown to represent a better tracer for CO(2) than H(2)O. Using COS as a tracer for stomatal conductance is hampered by our present poor understanding of the leaf internal conductance to COS. Estimating canopy level CO(2) and H(2)O fluxes requires disentangling leaf COS exchange from other ecosystem sources/sinks of COS. We conclude that future priorities for COS research should be to improve the quantitative understanding of the variability in the ratios of COS to CO(2) and H(2)O deposition velocities and the controlling factors, and to develop operational methods for disentangling ecosystem COS exchange into contributions by leaves and other sources/sinks. To this end, integrated studies, which concurrently quantify the ecosystem-scale CO(2), H(2)O and COS exchange and the corresponding component fluxes, are urgently needed.     

Quantitative analysis of metabolic fluxes in Escherichia coli, using two-dimensional NMR spectroscopy and complete isotopomer models.

Complete isotopomer models that simulate distribution of label in 13C tracer experiments are applied to the quantification of metabolic fluxes in the primary carbon metabolism of E. coli under aerobic and anaerobic conditions. The concept of isotopomer mapping matrices (IMMs) is used to simplify the formulation of isotopomer mass balances by expressing all isotopomer mass balances of a metabolite pool in a single matrix equation. A numerically stable method to calculate the steady-state isotopomer distribution in metabolic networks in introduced. Net values of intracellular fluxes and the degree of reversibility of enzymatic steps are estimated by minimization of the deviations between experimental and simulated measurements. The metabolic model applied includes the Embden-Meyerhof-Parnas and the pentose phosphate pathway, the tricarboxylic acid cycle, anaplerotic reaction sequences and pathways involved in amino acid synthesis. The study clearly demonstrates the value of complete isotopomer models for maximizing the information obtainable from 13C tracer experiments. The approach applied here offers a completely general and comprehensive analysis of carbon tracer experiments where any set of experimental data on the labeling state and extracellular fluxes can be used for the quantification of metabolic fluxes in complex metabolic networks.     

Automated simultaneous isolation and quantitation of labeled amino acid fractions from plasma and tissue by ion-exchange chromatography

In order to trace metabolic pathways of amino acids in the body, a known labeled amount of an amino acid is infused. Dilution in the body pool is measured, using the specific activity and calculated by dividing the labeled amount of an amino acid (tracer) by its total pool (tracer + tracee). This paper describes a method, which combines fractionation and quantitation of multiple amino acids in one chromatographic run. To achieve this, we performed a classical amino acid ion-exchange separation on standard HPLC equipment. The column effluent was divided continuously into two solvent streams using a rapidly switching, pump controlled “split-valve”. The main part (90%) was directed to a computer controlled fraction collector, while the remaining 10% was mixed with o-phthaldialdehyde reagent after which fluorescence was measured. Using this system, 10–1000 μl of deproteinized plasma, representing a maximum of 50 nmol of each amino acid, could be fractionated and quantitated in the same chromatographic run. In addition to optimal counting efficiency of an off-line radioactivity counter, it enabled easy measurement of the specific activity of multiple amino acid tracers.     

Hepatic uptake and degradation of trace doses of asialofetuin and asialoorosomucoid in the intact rat

Asialoorosomucoid and asialofetuin were prepared by using sialidase, which was removed chromatographically before the proteins were labelled with radioactive iodine. After intravenous administration of a small amount of asialoglycoprotein (3–4 μg/100 g body wt.) protein-bound and non-protein radioactivities in plasmas and livers of rats were determined at intervals over a period of 30 min.Transfer of either tracer protein from plasma to liver was almost complete in 5 min. Proteolysis of asialofetuin was evident very shortly thereafter, but degradation of asialoorosomucoid commenced after a significant delay and was initially slow relative to that of asialofetuin.Studies in vitro with crude hepatic lysosomal enzyme preparations indicated that asialoorosomucoid was less readily digested than asialofetuin, and that desialylation of orosomucoid or fetuin did not noticeably increase the susceptibility of these proteins to protease action. Proteolysis of asialofetuin was also demonstrable in liver homogenates in conditions under which albumin and asialotransferrin were stable.A generalized mathematical model was devised to represent the uptake and degradation of asialoglycoproteins by the liver. The theoretical assumptions that gave the best fits with experiment are outlined and discussed.     

An automated system for continuous measurements of trace gas fluxes through snow: an evaluation of the gas diffusion method at a subalpine forest site,Niwot Ridge,Colorado

An experimental system for sampling trace gas fluxes through seasonal snowpack was deployed at a subalpine site near treeline at Niwot Ridge, Colorado. The sampling manifold was in place throughout the entire snow-covered season for continuous air sampling with minimal disturbance to the snowpack. A series of gases (carbon dioxide, water vapor, nitrous oxide, nitric oxide, ozone, volatile organic compounds) was determined in interstitial air withdrawn at eight heights in and above the snowpack at ~hourly intervals. In this paper, carbon dioxide data from 2007 were used for evaluation of this technique. Ancillary data recorded inlcuded snow physical properties, i.e., temperature, pressure, and density. Various vertical concentration gradients were determined from the multiple height measurements, which allowed calculation of vertical gas fluxes through the snowpack using Fick’s 1st law of diffusion. Comparison of flux results obtained from different height inlet combinations show that under most conditions fluxes derived from individual gradient intervals agree with the overall median of all data within a factor of 1.5. Winds were found to significantly influence gas concentration and gradients in the snowpack. Under the highest observed wind conditions, concentration gradients and calculated fluxes dropped to as low as 13% of non-wind conditions. Measured differential pressure amplitude exhibited a linear relationship with wind speed. This suggests that wind speed is a sound proxy for assessing advection transport in the snow. Neglecting the wind-pumping effect resulted in considerable underestimation of gas fluxes. An analysis of dependency of fluxes on wind speeds during a 3-week period in mid-winter determined that over this period actual gas fluxes were most likely 57% higher than fluxes calculated by the diffusion method, which omits the wind pumping dependency. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

The diagram methods for the evaluation of exchange fluxes in membrane transport systems

In this paper theoretical methods for the evaluation of fluxes of ligand exchange processes in a transporter-mediated membrane transport system are studied. The exchange process of a transport system is defined as a set of reactions of the transporters in the membrane that do not result in a complete turnover and must include the following consecutive sequence of steps: the binding of ligands from bath 1 and a subsequent release of bound ligands to bath 2 followed immediately by a binding of ligands from bath 2 and a subsequent release of bound ligands to bath 1. Thus, unlike the ordinary one-way cycles, the completion of an exchange process does not result in a net transport of ligands across the membrane. However, since it exchanges the ligands between the two baths, the exchange process of a transport system is closely related to the operational tracer flux of labelled ligands in the system. In this paper, both the numerical and the analytical procedures for the evaluation of exchange fluxes in any given biochemical diagram are discussed. In particular, we show that the exchange fluxes of a given kinetic diagram, like one-way cycle fluxes, can be expressed analytically in terms of the rate constants of the diagram with the use of either the original diagram or an expanded diagram. The diagram methods presented in this paper should be very useful in analyzing the mechanisms of transporter-mediated transport systems when tracer flux data are available.