A Simple Self-Maintaining Metabolic System: Robustness,Autocatalysis, Bistability

A living organism must not only organize itself from within; it must also maintain its organization in the face of changes in its environment and degradation of its components. We show here that a simple (M,R)-system consisting of three interlocking catalytic cycles, with every catalyst produced by the system itself, can both establish a non-trivial steady state and maintain this despite continuous loss of the catalysts by irreversible degradation. As long as at least one catalyst is present at a sufficient concentration in the initial state, the others can be produced and maintained. The system shows bistability, because if the amount of catalyst in the initial state is insufficient to reach the non-trivial steady state the system collapses to a trivial steady state in which all fluxes are zero. It is also robust, because if one catalyst is catastrophically lost when the system is in steady state it can recreate the same state. There are three elementary flux modes, but none of them is an enzyme-maintaining mode, the entire network being necessary to maintain the two catalysts.     

Maturation of membrane function: transport of amino acid by rat erythroid cells.

The membrane changes which occur during cellular maturation of erythroid cells have been investigated. The transport of alpha-aminoisobutyric acid, alanine, and N-methylated-alpha-aminoisobutyric acid have been studied in the erythroblastic leukemic cell, the reticulocyte, and the erythrocyte of the Long-Evans rat. The dependence of amino acid transport on extracellular sodium concentration was investigated. Erythrocytes were found to transport these amino acids only by Na-independent systems. The steady state distribution ratio was less than 1. Reticulocytes were found to transport alpha-aminoisobutyric acid and alanine by Na-dependent systems, but only small amounts of N-methylated-alpha-aminoisobutyric acid. Small amounts of these amino acids were transported by Na-independent systems. The steady state distribution ratio was greater than one for Na-dependent transport. The erythroblastic leukemia cell, a model immature erythroid cell, showed marked Na-dependence (greater than 90%) for alpha-aminoisobutyric acid and alanine transport, and greater than 80% for the Na-dependent transport of N-methyl-alpha-aminoisobutyric acid. The steady state distribution ratio for the Na-dependent transport was greater than 4. In the erythroblastic leukemic cell, at least three Na-dependent systems are present: one includes alanine and alpha-aminoisobutyric acid, but excludes N-methyl-alpha-aminoisobutyric acid; one is for alpha-aminoisobutyric acid, alanine and also N-methyl-alpha-aminoisobutyric acid; and one is for N-methyl-alpha-aminoisobutyric acid alone. In the reticulocyte, the number of Na-dependent systems are reduced to two: one for alpha-aminoisobutyric acid and alanine; one for N-methyl-alpha-aminoisobutyric acid. In the erythrocytes, no Na-dependent transport was found. Therefore, maturation of the blast cell to the mature erythrocyte is characterized by a systematic loss in the specificity and number of transport system for amino acids.     

Mean field model of acetylcholine mediated dynamics in the thalamocortical system

A recent continuum model of the large scale electrical activity of the thalamocortical system is generalized to include cholinergic modulation. The model is examined analytically and numerically to determine the effect of acetylcholine (ACh) on its steady states, linear stability, spectrum, and temporal responses. Changing the ACh concentration moves the system between zones of one, three, and five steady states, showing that neuromodulation of synaptic strength is a possible mechanism by which multiple steady states emerge in the brain. The lowest firing rate steady state is always stable, and subsequent fixed points alternate between stable and unstable. Increasing ACh concentration changes the form of the spectrum. Increasing the tonic level of ACh concentration increases the magnitudes of the N100 and P200 in the evoked response potential (ERP), without changing the timing of these peaks. Driving the system with a pulse of cholinergic activity results in a transient increase in the firing rate of cortical neurons that lasts over . Step-like increases in cortical ACh concentration cause increases in the firing rate of cortical neurons, with rapid responses due to fast acting nicotinic receptors and slower responses due to muscarinic receptor suppression of intracortical connections.     

Numerical bifurcation analysis of the pattern formation in a cell based auxin transport model

Transport models of growth hormones can be used to reproduce the hormone accumulations that occur in plant organs. Mostly, these accumulation patterns are calculated using time step methods, even though only the resulting steady state patterns of the model are of interest. We examine the steady state solutions of the hormone transport model of Smith et al. (Proc Natl Acad Sci USA 103(5):1301–1306, 2006) for a one-dimensional row of plant cells. We search for the steady state solutions as a function of three of the model parameters by using numerical continuation methods and bifurcation analysis. These methods are more adequate for solving steady state problems than time step methods. We discuss a trivial solution where the concentrations of hormones are equal in all cells and examine its stability region. We identify two generic bifurcation scenarios through which the trivial solution loses its stability. The trivial solution becomes either a steady state pattern with regular spaced peaks or a pattern where the concentration is periodic in time.     

Model of calcium movements in the mammalian myocardium: interval-strength relationship

A model is proposed to describe the interval-strength relationship in mammalian cardiac muscle in terms of "discrete" calcium movements associated with each cycle. The sarcoplasmic reticulum is assumed to be comprised of three functional sub-compartments: (1) The "main calcium store" which contains most of the calcium (predominantly bound) and is considered, due to its large buffering capacity, to account for the "long-term memory" lasting 7-10 beats. (2) The "releasable terminal" which contains the calcium readily available for release (all or most of it free) and accounts for the "short-term memory" which affects the subsequent beat. (3) The longitudinal network of the SR recirculating the myofibrillar calcium to the "main calcium store". The total content of calcium in the main store is determined by the transsarcolemmal influx and efflux. While influx occurs only during depolarization, efflux occurs during the whole cardiac cycle. The amount of free calcium in the main store is determined by an equilibrium equation. The release of calcium from the "releasable terminal" is governed by a "concentration-dependent" mechanism. This implies that when the concentration in the "releasable terminal" increases, the fraction released increases and the residual calcium left for the subsequent contraction decreases. The model predicts the following interval-strength relationships: steady state peak tension; changes from one steady rate to another; restitution curves; post-stimulation potentiation; paired stimulation; premature beats; post-extrasystolic potentiation following interpolated, basal or complimentary interval.     

Amino acid exchange between plasma and erythrocytes in vivo in humans

To study amino acid exchange between plasma and erythrocytes in vivo, 4-h primed, continuous intravenous infusions of L-[1-13C]leucine, [15N]glycine, and L-[15N]alanine were administered to five healthy young men in the postabsorptive state. Stable isotope enrichments and amino acid levels were determined by gas chromatography-mass spectrometry in both plasma and whole blood and estimated (using hematocrit) in erythrocytes. A high concentration gradient across the erythrocyte membrane was consistently found for glycine (552 +/- 268 microM in erythrocytes vs. 155 +/- 35 microM in plasma), but not for leucine or alanine. A steady-state isotopic enrichment was observed in whole blood as well as plasma for each amino acid in every subject. Steady-state [13C]leucine enrichment in erythrocytes did not differ from plasma enrichment at steady state, the ratio of erythrocyte to plasma enrichment being 1.03 +/- 0.20 (95% confidence limits = 0.78-1.28); in contrast, this ratio reached only 0.23 +/- 0.04 and 0.59 +/- 0.09 (confidence limits 0.18-0.28 and 0.48-0.70) for [15N]glycine and [15N]alanine at steady state, respectively. These results suggest that most of erythrocyte leucine is exchangeable with plasma, whereas only a fraction of erythrocyte glycine and alanine is involved in exchange with plasma in vivo.     

Maturation of membrane function: Transport of amino acid by rat erythroid cells

The membrane changes which occur during cellular maturation of erythroid cells have been investigated. The transport of α-aminoisobutyric acid, alanine, and N-methylated-α-aminoisobutyric acid have been studied in the erythroblastic leukemic cell, the reticulocyte, and the erythrocyte of the Long-Evans rat. The dependence of amino acid transport on extracellular sodium concentration was investigated. Erythrocytes were found to transport these amino acids only by Na-independent systems. The steady state distribution ratio was less than 1. Reticulocytes were found to transport α-aminoisobutyric acid and alanine by Na-dependent systems, but only small amounts of N-methylated-α-aminoisobutyric acid. Small amounts of these amino acids were transported by Na-independent systems. The steady state distribution ratio was greater than one for Na-dependent transport. The erythroblastic leukemia cell, a model immature erythroid cell, showed marked Na-dependence (>90%) for α-aminoisobutyric acid and alanine transport, and >80% for the Na-dependent transport of N-methyl-α-aminoisobutyric acid. The steady state distribution ratio for the Na-dependent transport was >4. In the erythroblastic leukemic cell, at least three Na-dependent systems are present: one includes alanine and α-aminoisobutyric acid, but excludes N-methyl-α-aminoisobutyric acid; one is for α-aminoisobutyric acid, alanine and also N-methyl-α-aminoisobutyric acid; and one is for N-methyl-α-aminoisobutyric acid alone. In the reticulocyte, the number of Na-dependent systems are reduced to two: one for α-aminoisobutyric acid and alanine; one for N-methyl-α-aminoisobutyric acid. In the erythrocytes, no Na-dependent transport was found. Therefore, maturation of the blast cell to the mature erythrocyte is characterized by a systematic loss in the specificity and number of transport systems for amino acids.     

Bistability in the isocitrate dehydrogenase reaction: an experimentally based theoretical study.

The enzyme isocitrate dehydrogenase (IDH, EC 1.1.1.42) can exhibit activation by one of its products, NADPH. This activation is competitively inhibited by the substrate NADP+, whereas NADPH competes with NADP+ for the catalytic site. Experimental observations briefly presented here have shown that if IDH is coupled to another enzyme, diaphorase (EC 1.8.1.4), which transforms NADPH into NADP+, the system can attain either one of two stable states, corresponding to a low and a high NADPH concentration. The evolution toward either one of these stable states depends on the time of addition of diaphorase to the medium containing IDH and its substrate NADP+. We present a theoretical and numerical analysis of a model for the IDH-diaphorase bienzymatic system, based on the regulatory properties of IDH. The results confirm the occurrence of bistability for parameter values derived from the experiments. Depending on the total concentration of NADP+ plus NADPH and the concentration of IDH, the system can either admit a single steady state or display bistability. We obtain an expression for the critical time t*, before which diaphorase addition leads to the lower steady state and after which addition of the enzyme leads to the upper steady state of NADPH. The analysis is extended to the case where the second substrate of IDH, isocitrate, is consumed in the course of the reaction without being regenerated. Bistability occurs only as a transient phenomenon in these conditions.     

Static characteristics of a continuous flow bioreactor containing antibiotic-resistant recombinant cells

The steady-state behavior of a continuous bioreactor containing antibiotic-resistant recombinant cells has been investigated. Only the plasmid-free cell is susceptible to and killed by antibiotics. A Monod form of specific death rate was found to simulate quite well the experimental death rates of various cells due to antibiotics. The stability characteristics, including bifurcation of the possible steady states, are examined. Appropriate numerical illustrations for the steady-state characteristics have been provided. Theoretically, two coexistence steady states (CO), three partial washout steady states (PW), and one total washout steady state (TW) are feasible, but only one CO, one PW, and one TW were realized. When antibiotic consumption is not extremely significant the CO can exist over one or two ranges of dilution rates depending upon the antibiotic concentration in the feed. The CO is globally stable. Whenever the PW and/or the TW exist(s) together with the CO they are unstable. Sensitivity analyses for several key kinetic parameters have been made. The rate at which the plasmid-bearing cells revert to the plasmid-free cells has the most significant effect on the antibiotic susceptibility of the system. Some simplified optimization calculations for maximum profit have been carried out.     

Reduced glutathione accelerates the oxidative damage produced by sodium n-propylthiosulfate, one of the causative agents of onion-induced hemolytic anemia in dogs

The oxidative effects of sodium n-propylthiosulfate, one of the causative agents of onion-induced hemolytic anemia in dogs, were investigated in vitro using three types of canine erythrocytes, which are differentiated by the concentration of reduced glutathione and the composition of intracellular cations. After incubation with sodium n-propylthiosulfate, the methemoglobin concentration and Heinz body count in all three types of erythrocytes increased and a decrease in the erythrocyte reduced glutathione concentration was then observed. The erythrocytes containing high concentrations of potassium and reduced glutathione (approximately five times the normal values) were more susceptible to oxidative damage by sodium n-propylthiosulfate than were the normal canine erythrocytes. The susceptibility of the erythrocytes containing high potassium and normal reduced glutathione concentrations was intermediate between those of erythrocytes containing high concentrations of potassium and reduced glutathione and normal canine erythrocytes. In addition, the depletion of erythrocyte reduced glutathione by 1-chloro-2, 4-dinitrobenzene resulted in a marked decrease in the oxidative injury induced by sodium n-propylthiosulfate in erythrocytes containing high concentrations of potassium and reduced glutathione. The generation of superoxide in erythrocytes containing high concentrations of potassium and reduced glutathione was 4.1 times higher than that in normal canine erythrocytes when the cells were incubated with sodium n-propylthiosulfate. These observations indicate that erythrocyte reduced glutathione, which is known as an antioxidant, accelerates the oxidative damage produced by sodium n-propylthiosulfate.     

Enzyme reactions at the surface of living cells. II. Destabilization in the membranes and conduction of signals

The dynamics of a bound-enzyme reaction is studied when the diffusion of both the substrate and the product is coupled to their electric repulsion and to enzyme reaction. Contrary to what is occurring when substrate diffusion is uncoupled with electric repulsion and enzyme reaction, no hysteresis loop of the partition coefficient exists. The electric partition coefficient monotonically declines as substrate or product concentration is increased in the reservoir. The random perturbation of a steady state may generate a localized destabilization of substrate and product concentration. This destabilization must propagate in the membrane and may be viewed as the conduction of a signal. These conduction phenomena are entirely due to electric effects. In the absence of these effects, the system is homeostatic, that is it returns back to its initial steady state after a perturbation. Obviously under these conditions conduction of signals cannot occur. Increasing the ionic strength of the external milieu tends to stabilize the system and to suppress conduction effects in the membrane.     

Electrical activity and metabolism in cardiac tissue: An experimental and theoretical study

Summary (1) Effects of the metabolic inhibitor 2,4-dinitrophenol (DNP) on electrical activity in frog atria were studied by means of the sucrose-gap technique and in tracer experiments. (2) Voltage-clamp studies of ionic membrane currents showed a suppression by DNP of peak Na inward current without marked changes in the kinetics of the Na-carrying system and an increase of steady state outward current to three to five times its normal value. In⁴²K tracer experiments, DNP increased K resting efflux by about 10% and decreased K influx by 25 to 30%. (3) The depression of Na inward current is regarded as being caused by a partial block of Na channels and an increase of internal Na concentration after inhibition of active Na extrusion. (4) The strong rise in outward current is probably not caused by a K current since K efflux fails to show a correspondingly large change. As a possible explanation for current and flux changes, an electrogenic K pump is discussed. (5) A mathematical model of a carrier system transporting a single ion species is described. The system is designed as a direct potential pump. Uphill transport requires an asymmetry of the rate constants governing the cyclic formation and breakdown of carrier-ion complex. The asymmetry is brought about by an input of metabolic energy. Reduction of energy input decreases the asymmetry and induces a carrier-mediated downhill ion movement, with corresponding changes in membrane current and ion fluxes. (6) A model of electrogenic K inward transport is calculated that approximately accounts for the steady state current and the K flux changes experimentally observed after inhibition.     

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 (V'max, K'ia, K'mA, K'mB) that characterize the enzyme at saturating M. The same approach is applicable to an enzyme that catalyzes a steady state ping pong reaction.     

Multiplicity and stability analysis of microorganisms in continuous culture: effects of metabolic overflow and growth inhibition

Metabolic overflow (enhanced uptake of substrate and secretion of intermediates) is a phenomenon often observed for cells grown under substrate excess. Growth inhibition by substrate and/or product is also normally found for this kind of culture. An effort is made in this work to analyze the dynamic behavior of a continuous culture subject to metabolic overflow and growth inhibition by substrate and/or product. Analysis of a model system shows that in a certain range of operating conditions three nonwashout steady state solutions are possible. Local stability analysis indicates that only two of them are stable thus leading to multiplicity and hysteresis. Further analysis of the intrinsic effects of different terms describing the metabolic overflow and growth inhibitions reveals that for the model system and the parameters considered, the combined effects of product inhibition and an enhanced formation rate of product under substrate excess cause the multiplicity and hysteresis. Growth inhibition by substrate and/or an enhanced substrate uptake appear not to be necessary conditions. The combined effects of enhanced product formation and product inhibition can also lead to unusual dynamic behavior such as a prolonged time period to reach a steady state, oscillatory transition from one steady state to another, and sustained oscillations. Using the occurrence of multiplicity and oscillation as criteria, the operating regime of a continuous culture can be divided into four domains: one with multiplicity and oscillation, one with unique steady state but possible oscillatory behavior, the other two with unique and stable steady state. The model predictions are in accordance with recent experimental results. The results presented in this work may be used as guidelines for choosing proper operating conditions of similar culture systems to avoid undesired instability and multiplicity. Copyright 1998 John Wiley & Sons, Inc.     

Building the cellular puzzle: control in multi-level reaction networks

Quantitative conceptual tools dealing with control and regulation of cellular processes have been mostly developed for and applied to the pathways of intermediary metabolism. Yet, cellular processes are organized in different levels, metabolism forming the lowest level in a cascade of processes. Well-known examples are the DNA-mRNA-enzyme-metabolism cascade and the signal transduction cascades consisting of covalent modification cycles. The reaction network that constitutes each level can be viewed as a "module" in which reactions are linked by mass transfer. Although in principle all of these cellular modules are ultimately linked by mass transfer, in practice they can often be regarded as "isolated" from each other in terms of mass transfer. Here modules can interact with each other only by means of regulatory or catalytic effects-a chemical species in one module may affect the rate of a reaction in another module by binding to an enzyme or transport system or by acting as a catalyst. This paper seeks to answer two questions about the control and regulation of such multi-level reaction networks: (i) How can the control properties of the system as a whole be expressed in terms of the control properties of individual modules and the effects between modules? (ii) How do the control properties of a module in its isolated state change when it is embedded in the whole system through its connections with the other modules? In order to answer these questions a quantitative theoretical framework is developed and applied to systems containing two, three or four fully interacting modules; it is shown how it can be extended in principle to n modules. This newly developed theory therefore makes it possible to quantitatively dissect intermodular, internal and external regulation in multi-level systems.     

Ferritin in erythrocytes and plasma of patients with iron overload

Erythrocyte and plasma ferritin was followed in 13 patients with iron overload undergoing phlebotomies for at least 6 months in comparison with untreated patients and normal males. Plasma ferritin was widely scattered with an average of only twice the normal, whereas erythrocyte ferritin was highly elevated to about twelve times the normal (p less than 0.0001). - The time course of plasma and erythrocyte ferritin during phlebotomy therapy was analyzed in 3 patients with idiopathic hemochromatosis. Three stages were established: 1. plasma ferritin dropped gradually into the normal range while erythrocyte ferritin remained high, 2. appropriate phlebotomies maintained normal plasma ferritin and high erythrocyte ferritin, and indicated a monthly uptake of dietary iron of 150-200 mg at a steady state, 3. at low plasma ferritin levels, erythrocyte ferritin was rapidly decreased by further intensive phlebotomy therapy. Based on the presumed net removal of iron, 1 microgram/l plasma ferritin was equivalent to 3-6 mg of body iron and 1 microgram/l erythrocyte ferritin to somewhat less than 1 mg of body iron. - An elevated erythrocyte ferritin during phlebotomy therapy in iron overload not only depends on body iron stores like plasma ferritin but may also be regulated by the activity of erythropoiesis.     

Chemostat-cultivated Escherichia coli at high dilution rate: multiple steady states and drift

The representation of metabolic network reaction kinetics in a scaled, polynomial form can allow for the prediction of multiple steady states. The polynomial formalism is used to study chemostat-cultured Escherichia coli which has been observed to exhibit two multiple steady states under ammonium ion-limited growth conditions: a high cell density-low ammonium ion concentration steady state and a low cell density-high ammonium ion concentration steady state. Additionally, the low-cell-density steady state has been observed to drift to the high-cell-density steady state. Inspection of the steady-state rate expressions for the ammonium ion transport/assimilation network (in polynomial form) suggests that at low ammonium ion concentrations, two steady states are possible. One corresponds to heavy use of the glutamine synthetase-glutamate synthase (GLNS-GS) branch and the second to heavy use of the glutamate dehydrogenase (GDH) branch. Realization of the predicted intracellular steady states is also found to be dependent on the parameters of the transport process. Moreover, the two steady states differ in where their energy intensity lies. To explain the drift, GLNS, which is inducible under low ammonium ion concentrations, is suggested to be a "memory element." A chemostat-based model is developed to illustrate that perturbations in dilution rate can lead to drift between the two steady states provided that the disturbance in dilution rate is sufficiently large and/or long in duration.     

An extended dynamic model of oxidative phosphorylation.

The presented model based on an earlier one (Korzeniewski, B. and Froncisz, W. (1989) Studia Biophys. 132, 173-187) simulates concentration changes in time of chemical compounds and thermodynamic forces during respiration of cell suspension in a closed chamber. A set of differential equations solved numerically describes the utilization of oxygen up to anaerobiosis and the behaviour of the system after a sudden pulse of oxygen. Flux control coefficients for most important reactions (enzymes) of oxidative phosphorylation were calculated. A good qualitative and (when a direct comparison is possible) quantitative agreement with experimental results can be observed. The following conclusions can be drawn from the simulation: (1) Wilson's steady state model is not in contradiction with sharing of the control over the respiration between some steps and displacement of the ATP/ADP carrier from equilibrium. (2) The overshoot characteristics of the delta microH+ time-course after reoxygenation can be explained without using the lag-phase kinetics of ATP-synthetase. (3) A 'hot region' (sharp changes of many parameters) can be distinguished when the oxygen concentration approaches zero; only cytochrome oxidase is clearly sensitive on oxygen concentration in all its range. (4) Control over oxidative phosphorylation is shared mainly between inputs of the system (ATP utilization and substrate dehydrogenation) and the proton leak.     

SOME THOUGHTS ON NUTRIENT LIMITATION IN ALGAE1

An empirical relation relating specific growth, rate in steady state systems to nutrient status with respect to more than one nutrient simultaneously is proposed, based on 3 experimentally verifiable postulates: (1) that uptake depends on the external substrate concentration; (2) that growth depends on the interval substrate concentration; and (3) in a steady state system specific rate of uptake (in the absence of significant, excretion) is necessarily the product of the specific growth rate and internal substrate concentration. The implications of this model are discussed in particular in respect to the concept of luxury consumption and Liebig's law of minimum. Some aspects of uptake in transient situations are also discussed.     

Impossibility of coexistence of three pure and simple competitors in configurations of three interconnected chemostats

It is well established that pure and simple microbial competitors cannot coexist at a steady state if their environment is homogeneous. For the case of two microbial populations competing purely and simply in two interconnected chemostats having time-invariant input(s), it is known from the literature that a stable steady state of coexistence arises in domains of the operating parameters space and is attributed to the spatial heterogeneities of the system, which allow a different species to have the competitive advantage in each one of the two sub-environments. This article investigates whether the aforementioned result can be extended to the case of three species competing in three interconnected vessels. By studying all possible alternate configurations of the three-chemostat system, it is shown that coexistence of the three species is impossible, except possibly for some discrete values of the operating parameters. Some potential explanations for the results are discussed.