Pitfalls of the 45Ca uptake method

The interpretation of the data derived from ⁴⁵Ca uptake measurements by cells and tissues can be difficult. Several of these difficulties and possible misinterpretations are described: 1) ⁴⁵Ca uptake is not equivalent to calcium influx; 2) interpretations based on the sole visual examination of ⁴⁵Ca uptake curves can be misleading because an increased tracer uptake can coexist with a decreased calcium transport and vice-versa; 3) drugs and hormones can have diametrically opposite effects when they are tested at steady state on in nonsteady state conditions. It is concluded that ⁴⁵Ca uptake curves must be kinetically analyzed and that the steady or non-steady state of the system must be known for a valid interpretation of such data.     

Modeling transitions in body composition: the approach to steady state for anthropometric measures and physiological functions in the Minnesota human starvation study

Background

This study evaluated whether the changes in several anthropometric and functional measures during caloric restriction combined with walking and treadmill exercise would fit a simple model of approach to steady state (a plateau) that can be solved using spreadsheet software (Microsoft Excel^®). We hypothesized that transitions in waist girth and several body compartments would fit a simple exponential model that approaches a stable steady-state.

Methods

The model (an equation) was applied to outcomes reported in the Minnesota starvation experiment using Microsoft Excel's Solver^® function to derive rate parameters (k) and projected steady state values. However, data for most end-points were available only at t = 0, 12 and 24 weeks of caloric restriction. Therefore, we derived 2 new equations that enable model solutions to be calculated from 3 equally spaced data points.

Results

For the group of male subjects in the Minnesota study, body mass declined with a first order rate constant of about 0.079 wk^-1. The fractional rate of loss of fat free mass, which includes components that remained almost constant during starvation, was 0.064 wk^-1, compared to a rate of loss of fat mass of 0.103 wk^-1. The rate of loss of abdominal fat, as exemplified by the change in the waist girth, was 0.213 wk^-1.On average, 0.77 kg was lost per cm of waist girth. Other girths showed rates of loss between 0.085 and 0.131 wk^-1. Resting energy expenditure (REE) declined at 0.131 wk^-1. Changes in heart volume, hand strength, work capacity and N excretion showed rates of loss in the same range. The group of 32 subjects was close to steady state or had already reached steady state for the variables under consideration at the end of semi-starvation.

Conclusion

When energy intake is changed to new, relatively constant levels, while physical activity is maintained, changes in several anthropometric and physiological measures can be modeled as an exponential approach to steady state using software that is widely available. The 3 point method for parameter estimation provides a criterion for testing whether change in a variable can be usefully modelled with exponential kinetics within the time range for which data are available.

     

Dynamics and stability analysis of the growth and astaxanthin production system of Haematococcus pluvialis

This paper investigated high cell density cultivation of Haematococcus pluvialis for astaxanthin production in 3.7-L bioreactors. A biomass concentration of 2.74 g L⁻¹and an astaxanthin yield of 64.4 mg L⁻¹ were obtained. Based on the experimental results, a new and simple dynamic model is proposed, differing from Monod kinetics, to describe cell growth, product formation and substrate consumption. Good agreement was found between the model predictions and experimental data. The model revealed that there was cell growth inhibition on product formation and product feedback compensation for substrate consumption, but no substrate inhibition or product inhibition of cell growth. Stability analysis demonstrated that no multiplicity of steady states was observed; the unique positive steady state was locally asymptotically stable; and the effect of dilution rate on steady states was greater than that of the initial substrate concentration. Received 23 February 1999/ Accepted in revised form 08 June 1999     

pH effects on light dependence of the stoichiometry of proton influx and efflux to electron transport in spinach chloroplasts

Initial and steady state rates of proton transport at low light intensity have been measured and compared with steady state rates of electron transport in the pH range of 6.0–7.6 in envelope-free spinach chloroplasts. At pH 6–7, the H⁺/e^- values computed using the initial rate of proton transport varied with light intensity, from a value of 2 at low light to almost 5 at high light. In contrast, the H⁺/e^- values computed using the steady state rate of proton transport did not show a dependence on light intensity, having a constant value of 1.7±0.2. Likewise, at pH 7.6, the H⁺/e^- ratio, computed using either the initial or steady state rates of proton transport did not vary with light intensity but was constant at H⁺/e^-=1.7±0.1. Analysis of the light dependence of electron and proton transport allowed determination of (a) the quantam requirements of transport, (b) the rates of transport at light saturation, and (c) H⁺/e^- ratios for initial and steady state proton transport. Extrapolating the initial proton transport to zero light, we found that both H⁺/photon and H⁺/e^- values were not strongly dependent on pH, approaching a near constant value of 2.0. Using the initial rate of proton transport extrapolated to saturating light intensity we found the H⁺/e^- ratio to be strongly pH-dependent. We suggest that internal pH controls electron transport at high light intensities. The true stoichiometry is reflected only in measurements taken at low light using the initial proton transport data. Our findings and interpretation reconcile some conflicting data in the literature regarding the pH-dependence of the H⁺/e^- ratio and support the concept that internal pH controls noncyclic electron transport.Abbreviations Bicine N, N-bis [2-hydroxyethyl] glycine - HEPES N-2-hydroxy-ethylpiperazine-N-2-ethansulfonic acid - MES 2-(N-morpholino) ethanesulfonic acid     

Why the Lysogenic State of Phage λ Is So Stable: A Mathematical Modeling Approach

We develop a mathematical model of the phage λ lysis/lysogeny switch, taking into account recent experimental evidence demonstrating enhanced cooperativity between the left and right operator regions. Model parameters are estimated from available experimental data. The model is shown to have a single stable steady state for these estimated parameter values, and this steady state corresponds to the lysogenic state. When the CI degradation rate (γ_cI) is slightly increased from its normal value (γ_cI 0.0 min⁻¹), two additional steady states appear (through a saddle-node bifurcation) in addition to the lysogenic state. One of these new steady states is stable and corresponds to the lytic state. The other steady state is an (unstable) saddle node. The coexistence these two globally stable steady states (the lytic and lysogenic states) is maintained with further increases of γ_cI until γ_cI 0.35 min⁻¹, when the lysogenic steady state and the saddle node collide and vanish (through a reverse saddle node bifurcation) leaving only the lytic state surviving. These results allow us to understand the high degree of stability of the lysogenic state because, normally, it is the only steady state. Further implications of these results for the stability of the phage λ switch are discussed, as well as possible experimental tests of the model.     

Maintenance requirements in Bacillus sphaericus 1593M under dual substrate limitations estimated at zero growth rate in a total cell retention culture

The maintenance requirement(s) exhibited by Bacillus sphaericus1593M was clearly established from the data on the steady state continuous cultures wherein the growth yield was found to be increasing with increase in dilution rate (4). In the present study we have attempted to estimate the maintenance coefficient of B. sphaericus 1593M by achieving zero growth rate in a total cell retention culture (TCRC). Zero growth rate was achieved at a cell density of 5.57?g?l^?1 which remained constant reflecting a steady state in the TCRC. Based on the substrate flux calculations, maintenance coefficient was found to be 0.087?h^?1. Further, using this data of maintenance coefficient, steady state cell density and substrate profiles in continuous cultures of B. sphaericus 1593M were predicted by simulation, which showed a close relationship to that of the experimental data.     

The Green's function for the convection-diffusion equation in an analytic lung model

The method of Green's functions is applied to the convection-diffusion equation describing gas transport in the human lung. Explicit analytic solutions are found for the zero^th approximation to the value of steady state alveolar oxygen concentration, given the frequency of breathing, tidal volume, oxygen consumption, alveolar volume, volume of the bronchial tree, and two modelling parameters describing the geometry. The results of the calculation are compared favorably with some experimental data.     

Cuticle sclerotization by the American cockroach: Differential incorporation of leucine and tyrosine during post-ecdysis

The incorporation of U-¹⁴C-leucine into the cuticle occurs within the first 2 hr after ecdysis whereas U-¹⁴C-tyrosine is incorporated at a steady rate for approximately 8 hr. These data suggest that most of the cuticle protein is synthesized and laid down within a short time after ecdysis. On the other hand, tyrosine and/or metabolites (such as N-acetyl-dopamine) are translocated into the cuticle for several hours. This indicates that the sclerotization process may take place over an extended period.     

Glutamate dehydrogenase of lupin nodules: Kinetics of the deamination reaction

The reaction of glutamate dehydrogenase (l-glutamate: NAD⁺ oxidoreductase (deaminating) EC 1.4.1.2) from lupin nodules has been investigated in the direction of deamination by means of steady state velocity studies in the absence of products and inhibition studies with products and substrate analogs. The results are qualitatively and quantitatively consistent with a fully ordered reaction mechanism in which NAD⁺ binds to the enzyme first followed by l-glutamate. The order of product release is proposed to be NH₄⁺ followed by 2-oxoglutarate and then NADH. In addition, product inhibition data indicate the formation of an enzyme-NAD-oxoglutarate dead-end complex.     

Ammonia Assimilation in the Roots of Nitrate- and Ammonia-Grown Hordeum Vulgare (cv Golden Promise)

¹⁵N kinetic labeling studies were performed on seedlings of Hordeum vulgare L. var. Golden Promise growing under steady state conditions. Patterns of label incorporation in the pools of nitrogen compounds of roots fed [¹⁵N]ammonium were compared with computer-simulated labeling curves. The data were found to be quantitatively consistent with a three-compartment model in which ammonium is assimilated solely into the amide-N of glutamine. Labeling data from roots fed [¹⁵N]nitrate were also found to be at least qualitatively consistent with the assimilation of ammonia into glutamine. Methionine sulfoximine almost completely blocked the incorporation of ¹⁵N label into the amino acid pools of barley roots fed [¹⁵N]nitrate. These observations suggest that ammonia assimilation occurs solely via the glutamine synthetase/glutamate synthase pathway in both nitrate- and ammonia-grown barley roots.     

Continuous culture of Thiorhodaceae

Summary In sulfide limited continuous culture of a marine isolate of Chromatium vinosum, sulfide was undetectable in steady states below dilution rates of 0.06h^-1, that is 1/2 of the maximum specific growth rate. In the same range, sulfur is assumed to attain the role of the growth rate limiting substrate. Furthermore, it could be shown that the rate of sulfur oxidation is a function of the surface area of the sulfur globules rather than of the sulfur concentration. In completely filled chemostats, steady states were obtainable only at dilution rates not exceeding 0.09 h^-1. In the presence of a nitrogen flushed gas phase, steady states were obtained at dilution rates approaching the maximum specific growth rate (0.12h^-1). This phenomenon is ascribed to the particular sulfide tolerance of our strain of Chromatium vinosum. The saturation constant and the inhibition constant (lowest, respectively highest total sulfide concentration at which the specific growth rate is equal to one-half of the maximum specific growth rate in the absence of inhibition) were 0.007 mM and 0.85 mM, respectively.The ecological significance of the data is discussed.Contribution No. 2406 from the Woods Hole Oceanographic Institution.     

Analysis of steady state photosynthesis in alfalfa leaves

A method for carrying out kinetic tracer studies of steady state photosynthesis in whole leaves has been developed. An apparatus that exposes whole leaves to ¹⁴CO₂ under steady state conditions, while allowing individual leaf samples to be removed as a function of time, has been constructed. Labeling data on the incorporation of ¹⁴C into Medicago sativa L. metabolite pools are reported. A carbon dioxide uptake rate of 79 micromoles ¹⁴CO₂ per milligram chlorophyll per hour was observed at a CO₂ level slightly below that of air. Several actively turning over pools of early and intermediate metabolites, including 3-phosphoglyceric acid, glycerate, citrate, and uridine diphosphoglucose, showed label saturation after approximately 10 to 20 minutes of photosynthesis with ¹⁴CO₂ under steady state conditions. Alanine labeling increased more rapidly at first, and then at a lower rate as saturation was approached. Sucrose was a major product of photosynthesis and label saturation of the sucrose pool was not observed. Labeled carbon appeared rapidly in secondary metabolites. The steady state apparatus used has numerous advantages, including leaf temperature control, protection against leaf dehydration, high illumination, known ¹⁴CO₂ specific radioactivity, and provision for control and adjustment of ¹⁴CO₂ concentration. The apparatus allows for experiments of long duration and for sufficient sample points to define clearly the metabolic steady state.     

Partial contribution of the ATP-sensitive K+ current to the effects of mild metabolic depression in rabbit myocardium

The object of the study was to compare the capability of glibenclamide to block the effects of K⁺-ATP channel activators on action potential duration and steady state whole cell current to its efficiency in counteracting the effects of hypoxia or metabolic poisons in the presence of glycolytic substrate. The modulation of action potential duration by 30 M glibenclamide was tested in perfused hearts subjected to hypoxia or to the K⁺-ATP channel opener pinacidil. Similar protocols were used to study the modifications of the steady state whole cell current in isolated ventricular myocytes. It was found that glibenclamide did not prevent early action potential shortening induced by hypoxia but produced a partial recovery after 15 min of exposure. At the steady state the action potential duration had lengthened by 53±6% at plateau level and 42±3% at 95% repolarization. In contrast, action potential shortening induced by 100 M pinacidil was fully reversed by glibenclamide within 2 min. Freshly dispersed ventricular myocytes were characterized in control conditions as for the properties of the steady state current. This current, measured at the end of 450 ms long pulses showed typical inward rectification that was abolished by 50 M Ba²⁺. Cyanide (2 mM), carbonyl-cyanide m-chlorophenylhydrazone (CCCP, 200 nM) and BRL 38227 (30 M) produced characteristic increases in time independent outward currents. Glibenclamide abolished the outward current induced by BRL 38227 and the concomitant action potential shortening. Addition of cyanide in the presence of glibenclamide and BRL 38227 produced a new increase in outward current accompanied by action potential shortening. In the absence of K⁺-ATP channel activators, glibenclamide partly inhibited the CCCP induced current. Our data suggested that the delayed onset of glibenclamide action in hypoxic hearts is not due to diffusion barriers. They rather support the view that mechanisms other than K⁺-ATP channel activation could determine the early action potential shortening in whole hearts. The partial recovery observed under glibenclamide may be due, in part, to channel desensitization but also reflect the contribution of more than one current system to the action potential shortening because the glibenclamide insensitive fraction of the CCCP induced current is partly blocked by low concentrations of Ba²⁺. Differences with other data in the literature are attributed to the degree to metabolic blockade, to species differences, and to the inherent heterogeneities of the whole heart model where non-muscle cells may modulate the response to hypoxia.     

Modelling the production and transport of dissolved organic carbon in forest soils

DyDOC describes soil carbon dynamics, with a focus on dissolved organic carbon (DOC). The model treats the soil as a three-horizon profile, and simulates metabolic carbon transformations, sorption reactions and water transport. Humic substances are partitioned into three fractions, one of which is immobile, while the other two (hydrophilic and hydrophobic) can pass into solution as DOC. DyDOC requires site-specific soil characteristics, and is driven by inputs of litter and water, and air and soil temperatures. The model operates on hourly and daily time steps, and can simulate carbon cycling over both long (hundreds-to-thousands of years) and short (daily) time scales. An important feature of DyDOC is the tracking of ¹⁴C, from its entry in litter to its loss as DO¹⁴C in drainage water, enabling information about C dynamics to be obtained from both long-term radioactive decay, and the characteristic ¹⁴C pulse caused by thermonuclear weapon testing during the 1960s ("bomb carbon"). Parameterisation is performed by assuming a current steady state. Values of a range of variables, including C pools, annual DOC fluxes, and ¹⁴C signals, are combined into objective functions for least-squares minimisation. DyDOC has been applied successfully to spruce forest sites at Birkenes (Norway) and Waldstein (Germany), and most of the parameters have similar values at the two sites. The results indicate that the supply of DOC from the surface soil horizon to percolating water depends upon the continual metabolic production of easily leached humic material. In contrast, concentrations and fluxes of DOC in the deeper soil horizons are controlled by sorption processes, involving comparatively large pools of leachable organic matter. Times to reach steady state are calculated to be several hundred years in the organic layer, and hundreds-to-thousands of years in the deeper mineral layers. It is estimated that DOC supplies 89% of the mineral soil carbon at Birkenes, and 73% at Waldstein. The model, parameterised with "steady state" data, simulates short-term variations in DOC concentrations and fluxes, and in DO¹⁴C, which are in approximate agreement with observations.     

The association between K+ and H+ distribution in skeletal muscles: implication to linked transport.

Data from the literature and results from a mathematical model of steady state fluid-electrolyte balance are used to support the observation that a relationship exists between the concentration gradients of K⁺ and H⁺ in the fluids of skeletal muscle over a range of acid-base disturbances. This relationship is shown to be consistent with the premise that the steady state electrochemical potential gradients for these ions remain constant under these conditions. Using a pump-leak model of ion transport, and the constant electric field assumption, it is also demonstrated that the steady state rates of active transport of K⁺ and H⁺ are related. These results suggest that the relations between both the steady state concentration gradients and the active transport rates for these ions are not necessarily the result of fixed biochemical mechanisms, but may come about simply from coupling through macroscopic thermodynamic processes.     

Evidence for frequency-dependent extracellular impedance from the transfer function between extracellular and intracellular potentials

Dopaminergic (DA) neurons of the mammalian midbrain exhibit unusually low firing frequencies in vitro. Furthermore, injection of depolarizing current induces depolarization block before high frequencies are achieved. The maximum steady and transient rates are about 10 and 20 Hz, respectively, despite the ability of these neurons to generate bursts at higher frequencies in vivo. We use a three-compartment model calibrated to reproduce DA neuron responses to several pharmacological manipulations to uncover mechanisms of frequency limitation. The model exhibits a slow oscillatory potential (SOP) dependent on the interplay between the L-type Ca²⁺ current and the small conductance K⁺ (SK) current that is unmasked by fast Na⁺ current block. Contrary to previous theoretical work, the SOP does not pace the steady spiking frequency in our model. The main currents that determine the spontaneous firing frequency are the subthreshold L-type Ca²⁺ and the A-type K⁺ currents. The model identifies the channel densities for the fast Na⁺ and the delayed rectifier K⁺ currents as critical parameters limiting the maximal steady frequency evoked by a depolarizing pulse. We hypothesize that the low maximal steady frequencies result from a low safety factor for action potential generation. In the model, the rate of Ca²⁺ accumulation in the distal dendrites controls the transient initial frequency in response to a depolarizing pulse. Similar results are obtained when the same model parameters are used in a multi-compartmental model with a realistic reconstructed morphology, indicating that the salient contributions of the dendritic architecture have been captured by the simpler model.     

The role of Mg2+ in the regulation of the structural and functional steady-states in rat liver mitochondria

A possible relationship between mitochondrial Mg²⁺ levels, structural configurations, and functional steady states has been studied in rat liver mitochondria. The results show that the concentration of mitochondrial Mg²⁺ in respiratory state 4 is definitely higher than in respiratory state 3. The metabolic transition from state 3 to state 4 and vice-versa is associated with reversible influx-efflux of about 10 nmol of Mg²⁺ per mg protein. The net uptake of this aliquot of Mg²⁺ is a necessary condition in order for the metabolic transition to state 4, both structurally and functionally, to occur. This process requires a threshold concentration of external Mg²⁺ greater than 5 mM. The phosphorylative mechanism does not appear to depend on the presence or absence of external Mg²⁺. The role of Mg²⁺ on the attainment and maintenance of the structural and functional steady state 4 seems to be correlated with its regulatory effect on the concentration of the mitochondrial P_i.     

Model for the Action of Calcium in Muscle

CALCIUM ions play an important part in contraction^1–3, but little is known of the way that changes in internal calcium actually control the events that lead to the development of tension. Most experiments involving calcium have been performed under steady state conditions at different stabilized calcium concentrations^4–9 and can only give limited information as to the action of calcium. Recently, tension transient experiments have been described at either constant calcium concentration^6,7 or when the calcium concentration was varying in a known manner^10,11. From these and related experiments, a model for the kinetics of calcium has been deduced which predicts not only the time course of tension development from just a knowledge of the free calcium concentration, but which is also able to correlate the ATPase, tension and calcium binding responses in the steady state. The model considers that two calciums act as separate effectors with a de-repressant action in the same functional unit.     

Characterization of the blue light-induced extracellular alkalinization associated with the monovalent anion uptake by Monorapidium braunii. Competition between NO3 and Cl−

The blue light-elicited monovalent anion-dependent alkalinization of the medium of Monoraphidium braunü (Legnerová, 202–7d) was characterized for the NO-³ and Cl- uptake. The maximal H⁺ uptake rates for these two anions have a similar optimum pH around 8.5, and quite similar K^s values for high (38 üM for Cl- and 35üM for NO-³) and low (320 üM for Cl- and 335 üM for NO-³) affinities. The steady H⁺ uptake associated with the uptake and reduction of NO-³ showed a K^s of 125 üM. which in this alga corresponds to the NO-³ reductase (EC 1.6.6.2) K^m for NO-³. The only and striking difference found in the uptake properties of these anions was the delay time between the switching on of the blue light and the start of the alkalinization, which increased from 10 to 90 s as the initial pH decreased from 8.5 to 6.5 in the presence of NO-³, whereas for Cl- uptake this delay time (10s) did not vary in relation to the initial pH. When the NO-³ concentration in the medium was low (100 üM), the presence of relatively high concentrations of Cl- (3 üM), on the one hand, greatly stimulated the maximal alkalinization rates but, on the other, Cl- severely reduced the steady NO-³-dependent rate of alkalinization. The data indicate that Cl- inhibits competitively NO-³ uptake with a Kⁱ of 750 üM. Moreover, high concentrations of NO-³ (above 5 üM) reduced its own maximal, but not the steady, uptake rates. The above results allow us to propose that most of the components of the individual NO-³ and Cl- transport systems are under identical light control and, as the competition data suggest, that these two anions may be taken up by the same transport system.     

Simplified Velocity Equations for Characterizing the Partial Inhibition or Nonessential Activation of Bireactant Enzymes

The steady state velocity equation for a bireactant enzyme in the presence of a partial inhibitor or nonessential activator, M, contains squared substrate concentration and higher-ordered M concentration terms. The equation is too complex to be useful in kinetic analyses. Simplification by the method of Cha (J. Biol. Chem. 243, 820–825 (1968)) eliminates squared substrate concentration terms, but retains higher-ordered terms in [M]. It is shown that if strict equilibrium is assumed between free E, M, and EM and for all but one other M-binding reaction, a velocity equation is obtained for an ordered bireactant enzyme that is first degree in all ligands in the absence of products. The equation is an approximation (because it was derived assuming only one M-binding reaction in the steady state), but it contains five inhibition (or activation) constants associated with M, all of which can be obtained by diagnostic replots and/or curve-fitting procedures. The equation also provides a framework for obtaining limiting constants (V¹_max, K¹_ia, K¹_mA,K¹_mB) that characterize the enzyme at saturating M. The same approach is applicable to an enzyme that catalyzes a steady state ping pong reaction.