Single channel gating events in tracer flux experiments I. Theory

Tracer ion flux measurements are a commonly used method for studying ion transport through membranes of cellular systems, where the rate of ion flow is determined by gating processes which control the opening and closing of transmembrane channels. Due to recent advances in the theoretical analysis of tracer flux from or into closed membrane structures (CMS), the mechanism of gating reactions can, in principle, be derived from flux data. A physically well founded analysis is presented for the dependence of the total tracer ion content of a collection of CMS on the gating processes. For functionally uncoupled gating units a mean single channel flux contribution [equation, see text] can be defined, where k is the intrinsic single channel flux coefficient, t the time over which flux is measured, and p(tau,t) is the probability that a given channel was open for a total period tau during t. This quantity reflects the mean time course of the tracer content due to flux through a single channel. Expressions for are derived that explicitly take into account a distribution in the lifetime of open channels. On the basis of the results, kinetic and thermodynamic parameters of multiphasic gating reactions can be determined from the time course of the overall tracer content in a colleciion of CMS.     

Quasiperiodicity route to chaos in a biochemical system.

The numerical study of a glycolytic model formed by a system of three delay differential equations reveals a quasiperiodicity route to chaos. When the delay changes in our biochemical system, we can observe the emergence of a strange attractor that replaces a previous torus. This behavior happens both under a constant input flux and when the frequency of the periodic substrate input flux changes. The results obtained under periodic input flux are in agreement with experimental observations.     

Control analysis of mammalian serine biosynthesis. Feedback inhibition on the final step.

The flux of serine biosynthesis in the liver of the normal rabbit, and of the rat on a low protein diet, is most sensitive to the activity of phosphoserine phosphatase (flux control coefficient up to 0.97), the last of the three enzymes in the pathway after it branches from glycolysis. The concentration of the pathway product, serine, has a strong controlling influence on the flux (response coefficient up to -0.64) through feedback inhibition at this step. The pathway is therefore controlled primarily by the demand for serine rather than the supply of the pathway precursor, 3-phosphoglycerate. Under conditions where there is a lower biosynthetic flux, the flux control coefficients of the first two enzymes of the pathway are increased, and are probably dominant in the rat on a normal diet. In rabbit liver, when ethanol is used to inhibit serine biosynthesis, control can be distributed between the three enzymes, even though the reactions catalysed by the first two remain close to equilibrium. Apart from their intrinsic value in aiding the understanding of the regulation of mammalian serine metabolism, our findings illustrate the danger of assuming that there are invariant design principles in the regulation of metabolic pathways, such as feedback control on the first step after a branch.     

Electrical properties of the cellular transepithelial pathway in Necturus gallbladder. II. Ionic permeability of the apical cell membrane.

Microelectrode techniques were employed to study the ionic permeability of the apical cell membrane of Necturus gallbladder epithelium. Results obtained from continuous records in single cells, and from several cellular impalements shortly after a change in solution, were similar and indicate that both the apical membrane equivalent electromotive force (Va) and electrical resistance (Ra) strongly depend on external [K]. Cl substitutions produced smaller effects, while the effects of Na substitutions with N-methyl-D-glucamine on both Va and Ra were minimal. These results indicate that the permeability sequence of the apical membrane is PKgreater thanPClgreater than PNa. From the calculated absolute value of PNa it is possible to estimate the diffusional Na flux from the mucosal solution into the cells (from the cell potential and an assumed intracellular Na concentration). The calculated flux is roughly three orders of magnitude smaller than the measured net transepithelial flux in this tissue and in gallbladders of other species. Thus, only a minimal portion of Na entry can be attributed to independent diffusion. From estimations of the electrochemical potential gradient across the apical membrane, Cl transport at that site must be active. At the serosal cell membrane, Na transport takes place against both chemical and electrical potentials, while a significant portion of the Cl flux can be passive, if this membrane has a significant Cl conductance. The changes in shunt electromotive force and in transepithelial potential after mucosal substitutions were very similar, indicating that transepithelial bi-ionic potentials yield appropriate results on the properties of shunt pathway.     

The effect of enzyme-enzyme complexes on the overall glycolytic rate in vivo.

Recent studies have demonstrated that most glycolytic enzymes can reversibly associate to form heterogeneous enzyme-enzyme (binary) complexes in vitro. However, kinetic analysis of these complexes has shown that the individual enzymes have a varied response to complex formation: some enzymes are inhibited, some are activated and some are unaffected. In order to determine the potential role of binary complexes in regulating glycolytic flux, we have mathematically calculated enzyme distributions and activities using data from in vitro binding and kinetic studies. These calculations suggest that, overall, formation of binary complexes would lower flux through phosphofructokinase and aldolase, would increase flux through glyceraldehyde-3-phosphate dehydrogenase and lactate dehydrogenase, and would not affect flux through triosephosphate isomerase, phosphoglycerate kinase and pyruvate kinase. The implications of these results are discussed with respect to the effect of complex formation on overall glycolytic flux and on the flux through individual enzyme loci.     

Stoichiometric network theory for nonequilibrium biochemical systems.

We introduce the basic concepts and develop a theory for nonequilibrium steady-state biochemical systems applicable to analyzing large-scale complex isothermal reaction networks. In terms of the stoichiometric matrix, we demonstrate both Kirchhoff's flux law sigma(l)J(l)=0 over a biochemical species, and potential law sigma(l) mu(l)=0 over a reaction loop. They reflect mass and energy conservation, respectively. For each reaction, its steady-state flux J can be decomposed into forward and backward one-way fluxes J = J+ - J-, with chemical potential difference deltamu = RT ln(J-/J+). The product -Jdeltamu gives the isothermal heat dissipation rate, which is necessarily non-negative according to the second law of thermodynamics. The stoichiometric network theory (SNT) embodies all of the relevant fundamental physics. Knowing J and deltamu of a biochemical reaction, a conductance can be computed which directly reflects the level of gene expression for the particular enzyme. For sufficiently small flux a linear relationship between J and deltamu can be established as the linear flux-force relation in irreversible thermodynamics, analogous to Ohm's law in electrical circuits.     

Optimal selection of metabolic fluxes for in vivo measurement. I. Development of mathematical methods.

The measurement of uptake and secretion rates is often not sufficient to allow the calculation of all internal metabolic fluxes. Measurements of internal fluxes are needed and these additional measurements are used in conjunction with mass-balance equations to calculate the complete metabolic flux map. A method is presented that identifies the fluxes that should be selected for experimental measurement, and the fluxes that can be computed using the mass-balance equations. The criterion for selecting internal metabolic fluxes for measurement is that the values of the computed fluxes should have low sensitivity to experimental error in the measured fluxes. A condition number indicating the upper bound on this sensitivity, is calculated based on stoichiometry alone. The actual sensitivity is dependent on both the flux measurements and the error in flux measurements, as well as the stoichiometry. If approximate physiologic ranges of fluxes are known a realistic sensitivity can be computed. The exact sensitivity cannot be calculated since the experimental error is usually unknown. The most probable value of the actual sensitivity for a given selection of measured fluxes is estimated by selecting a large number of representative error vectors and calculating the actual sensitivity for each of these. A frequency distribution of actual sensitivities is thus obtained giving a representative range of actual sensitivities for a particular experimental situation.     

Assessing glycolytic flux alterations resulting from genetic perturbations in E. coli using a biosensor

We describe the development of an optimized glycolytic flux biosensor and its application in detecting altered flux in a production strain and in a mutant library. The glycolytic flux biosensor is based on the Cra-regulated ppsA promoter of E. coli controlling fluorescent protein synthesis. We validated the glycolytic flux dependency of the biosensor in a range of different carbon sources in six different E. coli strains and during mevalonate production. Furthermore, we studied the flux-altering effects of genome-wide single gene knock-outs in E. coli in a multiplex FlowSeq experiment. From a library consisting of 2126 knock-out mutants, we identified 3 mutants with high-flux and 95 mutants with low-flux phenotypes that did not have severe growth defects. This approach can improve our understanding of glycolytic flux regulation improving metabolic models and engineering efforts.     

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.     

Alterations in chloride transport during differentiation of Friend virus-transformed cells.

Friend erythroleukemic cells can be induced by a variety of agents to synthesize hemoglobin and to exhibit other characteristics suggesting erythroid maturation. Upon induction of hemoglobin synthesis with dimethylsulfoxide (DMSO), the chloride flux in Friend cells gradually increases, until after five days of exposure to DMSO (when the hemoglobin content of the cells approaches that of the mature erythrocyte) the flux is three times the value in non-induced cells. A similar flux increase is observed in the presence of a different type of inducer, hypoxanthine, but no increase in flux is seen in the mutant cell line, TG-13, which does not synthesize hemoglobin after DMSO treatment. Thus, the flux increase seems to be associate d with the induction process, rather than being a direct effect of the inducing agent. After DMSO treatment, the sulphate flux decreases and the chloride/sulphate selectivity increases, aswould be expected if the cells were becoming more like red cells. On the other hand, the sensitivity of the chloride flux to the inhibitor, furosemide, and to temperature is the same in the induced as in the non-induced Friend cells, and different from that of the mature red cell. Thus, the anion transport properties of the induced Friend cell are different from those of both the non-induced Friend cell and the mature erythrocyte. Either the system in the induced cell represents an intermediate stage in the development of the mature red cell characteristics, or else the maturation of transport function in the Friend cell differs from that in normal erythrocyte precursors.     

Unidirectional sodium and potassium fluxes through the sodium channel of squid giant axons.

Unidirectional 22Na-traced sodium influx or 42K-traced potassium efflux across the membranes of voltage-clamped squid giant axons was measured at various membrane potentials under bi-ionic conditions. Tetrodotoxin almost entirely eliminated the extra K+ efflux induced by short repetitive depolarizations in the presence of tetraethylammonium or 3,4-diaminopyridine. A method of determining the voltage dependence of the unidirectional flux through voltage-gated channels is described. This technique was used to obtain the unidirectional flux-voltage relation for the sodium channel in bi-ionic and single-ion conditions. It allows the determination of the unidirectional flux at the zero-current potential which, for influx, was found to be approximately 20% of the value measured 80 mV negative to the zero-current potential. The unidirectional flux ratio under bi-ionic conditions was also measured and the flux ratio exponent found to average 1.15 with an external sodium and an internal potassium solution. A three-barrier, two-site, multi-occupancy model previously obtained for other conditions was found to predict a similar non-unity average for the flux ratio exponent. It is also shown that some single-occupancy models can predict non-unity values for the flux ratio exponent in bi-ionic conditions.     

Diversity of temporal self-organized behaviors in a biochemical system.

The numerical study of a glycolytic model formed by a system of three delay-differential equations revealed a notable richness of temporal structures which included the three main routes to chaos, as well as a multiplicity of stable coexisting states. The Feigenbaum, intermitency and quasiperiodicity routes to chaos can emerge in the biochemical oscillator. Moreover, different types of birhythmicity, trirhythmicity and hard excitation emerge in the phase space. For a single range of the control parameter it can be observed the coexistence of two quasiperiodicity routes to chaos, the coexistence of a stable steady state with a stable torus, and the coexistence of a strange attractor with different stable regimes such as chaos with different periodic regimes, chaos with bursting behavior, and chaos with torus. In most of the numerical studies, the biochemical oscillator has been considered under periodic input flux being the mean input flux rate 6 mM/h. On the other hand, several investigators have observed quasiperiodic time patterns and chaotic oscillations by monitoring the fluorescence of NADH in glycolyzing yeast under sinusoidal glucose input flux. Our numerical results match well with these experimental studies.     

Effects of adenosine receptor agonists on volume-activated ion transport in pig red cells.

Swelling of pig red cells leads to an increase in a chloride-dependent K flux which can be potentiated by cAMP, whereas cell shrinking causes a selective increase in Na movement which is mediated by a Na/H exchanger. We examined the influence of adenosine and adenosine receptor agonists on the volume-sensitive, ouabain-resistant, chloride-dependent K flux, referred to as Rb flux and volume-activated Na/H exchange pathway. It was found that adenosine and adenosine receptor agonists inhibited the Rb flux. N6-cyclohexyl adenosine (CHA) has been found to be the most potent inhibitor with EC50 of approximately 4.5 microM followed by 2-chloroadenosine (Cl-ado) with EC50 of approximately 27 microM and 5'-(N-ethyl)-carboxamido-adenosine (NECA) with EC50 of approximately 185 microM. CHA also inhibits the cAMP-stimulated Rb flux. However, CHA does not alter the basal intracellular cAMP level nor the intracellular cAMP content raised by exogenously added cAMP. In contrast to the adenosine agonist action on the Rb flux, Na/H exchange, which is activated upon cell shrinkage, exhibits a slight stimulation in response to CHA. These findings suggest that the presence of A1 adenosine receptors on the surface of red cells influences the regulation of volume-activated ion transport.     

Transport across homoporous and heteroporous membranes in nonideal, nondilute solutions. II. Inequality of phenomenological and tracer solute permeabilities.

The phenomenological solute permeability (omega p) of a membrane measures the flux of solute across it when the concentrations of the solutions on the two sides of the membrane differ. The relationship between omega p and the the conventionally measured tracer permeability (omega T) is examined for homoporous and heteroporous (parallel path) membranes in nonideal, nondilute solutions and in the presence of boundary layers. In general, omega p and omega T are not equal; therefore, predictions of transmembrane solute flux based on omega T are always subject to error. For a homoporous membrane, the two permeabilities become equal as the solutions become ideal and dilute. For heteroporous membranes, omega p is always greater than omega T. An upper bound on omega p- omega T is derived to provide an estimate of the maximum error in predicted solute flux. This bound is also used to show that the difference between omega P and omega T demonstrated earlier for the sucrose-Cuprophan system can be explained if the membrane is heteroporous. The expressions for omega P developed here support the use of a modified osmotic driving force to describe membrane transport in nonideal, nondilute solutions.     

Conductive chloride flux across amphibian skin: inhibition by indacrinone and cobalt ion.

When amphibian skin was incubated under conditions in which transepithelial sodium transport was abolished, a conductive transepithelial Cl- flux arose when Cl- was removed from one of the compartments. This flux was matched by short-circuit current and it accounted entirely for transepithelial conductance. Cl- influx was larger than efflux; it was linearly related to the magnitude of transepithelial Cl- concentration difference. When applied to the epithelial surface of the tissue, divalent metal cations such as Co2+, and the ethacrynic acid derivative, indacrinone, reduced rapidly and reversibly both transepithelial Cl- (in)flux and short-circuit current. Frog skin proved to be more sensitive to these inhibitors than toad skin. Further characterization of transepithelial Cl- pathway(s) should benefit from the fact that Cl- across amphibian skin can easily be monitored by the short-circuit current method, and from the availability of agents which inhibit this passive flux rapidly and reversibly.     

Fluxes and metabolic pools as model traits for quantitative genetics. I. The L-shaped distribution of gene effects

The fluxes through metabolic pathways can be considered as model quantitative traits, whose QTL are the polymorphic loci controlling the activity or quantity of the enzymes. Relying on metabolic control theory, we investigated the relationships between the variations of enzyme activity along metabolic pathways and the variations of the flux in a population with biallelic QTL. Two kinds of variations were taken into account, the variation of the average enzyme activity across the loci, and the variation of the activity of each enzyme of the pathway among the individuals of the population. We proposed analytical approximations for the flux mean and variance in the population as well as for the additive and dominance variances of the individual QTL. Monte Carlo simulations based on these approximations showed that an L-shaped distribution of the contributions of individual QTL to the flux variance (R(2)) is consistently expected in an F(2) progeny. This result could partly account for the classically observed L-shaped distribution of QTL effects for quantitative traits. The high correlation we found between R(2) value and flux control coefficients variance suggests that such a distribution is an intrinsic property of metabolic pathways due to the summation property of control coefficients.     

Properties of solution of the diffusion-reaction equation. II

Two theorems relating to properties of the solutions of the equations of continuity for the concentrations of the chemical species in a diffusion-reaction system are proved. The theorems concern boundary conditions under which the flux of a specified species can be guaranteed to be directed into the reaction region and the circumstances under which any two of the conditions (i) stationarity, (ii) flux equilibrium, and (iii) chemical equilibrium, imply the third. Application of these theorems to apparent active transport and to the properties of the differential equations for specific activities in a distributed tracer system are noted.     

Responding phospholipid membranes--interplay between hydration and permeability.

Osmotic forces are important in regulating a number of physiological membrane processes. The effect of osmotic pressure on lipid phase behavior is of utmost importance for the extracellular lipids in stratum corneum (the outer part of human skin), due to the large gradient in water chemical potential between the water-rich tissue on the inside, and the relative dry environment on the outside of the body. We present a theoretical model for molecular diffusional transport over an oriented stack of two-component lipid bilayers in the presence of a gradient in osmotic pressure. This gradient serves as the driving force for diffusional motion of water. It also causes a gradient in swelling and phase transformations, which profoundly affect the molecular environment and thus the local diffusion properties. This feedback mechanism generates a nonlinear transport behavior, which we illustrate by calculations of the flux of water and solute (nicotine) through the bilayer stack. The calculated water flux shows qualitative agreement with experimental findings for water flux through stratum corneum. We also present a physical basis for the occlusion effect. Phase behavior of binary phospholipid mixtures at varying osmotic pressures is modeled from the known interlamellar forces and the regular solution theory. A first-order phase transformation from a gel to a liquid--crystalline phase can be induced by an increase in the osmotic pressure. In the bilayer stack, a transition can be induced along the gradient. The boundary conditions in water chemical potential can thus act as a switch for the membrane permeability.     

General continuum analysis of transport through pores. II. Nonuniform pores.

A general continuum derivation of the nonelectrolyte (Js) and volume (Jv) flux through a pore whose cross section is a function of axial position (nonuniform) is given. In general, the flux equations cannot be reduced to the same form as for a uniform pore and it is not possible to characterize the pore kinetics by three constants as in the uniform pore case. However, it is shown that under certain conditions, the nonuniform pore equations can be approximated by the uniform pore form and can be characterized by three constants (omega, sigma, Lp). The only condition needed to reduce the Jv equation to the uniform form is that the solution be dilute. The deviation of the Js equation from the uniform form is characterized by an asymmetrical function of Jv whose maximum value is estimated. It is shown that the maximum posible fractional deviation of the Js equation from the uniform form is given by the parameter: 0:5sigmaJv/omegaRT. Since this parameter is less then 0.15 for most membrane studies, the nonuniform Js equation can usually be approximated by the uniform pore form. The general results are illustrated by explicit calculations on several models of nonuniform pores. It is shown, for example, that the "equivalent pore radius" defined in the usual way is a function of the experimental parameter that is measured and is not unique.     

Deviations from the flux ratio equation for chloride ions in ouabain- and acetazolamide-treated frog skin.

In order to establish whether or not chloride ions behave as freely moving particles in "passive", i.e. ouabain- and acetazolamide-treated, frog skin, tracer fluxes of 36Cl-have been measured while a voltage (generally +40 mV, serosal side positive) across the skin was applied. Ussing's flux ratio equation has been used as a criterion for this type of transport. One group of skin samples exhibited significant exchange diffusion phenomena. Most samples in a second group either behaved according to the flux ratio equation of showed significant and extreme exchange diffusion. From flux ratios obtained at two different voltages across various skin samples, showing extreme exchange diffusion, it appeared that the simple form of Kedem and Essig's law derived from irreversible thermodynamics, which is valid for homogeneous systems, does not apply to the type of exchange diffusion found. The system can, however, be described by a 1:1 exchange mechanism working in parallel with a diffusional pathway. The ratio exchange flux/observed efflux must then have a constant value (0.83) at the voltages appled, which implies that the exchange flux is voltage dependent. By comparison with iodide flux experiments as carried out by Ussing, it is shown that iodide exhibits the same type of exchange diffusion. A carrier, possibly responsible for the observed behaviour, is described.