Bidirectional reaction steps in metabolic networks: II. Flux estimation and statistical analysis

Metabolic carbon labelling experiments enable a large amount of extracellular fluxes and intracellular carbon isotope enrichments to be measured. Since the relation between the measured quantities and the unknown intracellular metabolic fluxes is given by bilinear balance equations, flux determination from this data set requires the numerical solution of a nonlinear inverse problem. To this end, a general algorithm for flux estimation from metabolic carbon labelling experiments based on the least squares approach is developed in this contribution and complemented by appropriate tools for statistical analysis. The linearization technique usually applied for the computation of nonlinear confidence regions is shown to be inappropriate in the case of large exchange fluxes. For this reason a sophisticated compactification transformation technique for nonlinear statistical analysis is developed. Statistical analysis is then performed by computing appropriate statistical quality measures like output sensitivities, parameter sensitivities and the parameter covariance matrix. This allows one to determine the order of magnitude of exchange fluxes in most practical situations. An application study with a large data set from lysine-producing Corynebacterium glutamicum demonstrates the power and limitations of the carbon-labelling technique. It is shown that all intracellular fluxes in central metabolism can be quantitated without assumptions on intracellular energy yields. At the same time several exchange fluxes are determined which is invaluable information for metabolic engineering. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 118-135, 1997.     

Conditioned place preference behavior in zebrafish

This protocol describes conditioned place preference (CPP) in zebrafish following a single exposure to a substance. In the CPP paradigm, animals show a preference for an environment that has previously been associated with a substance (drug), thus indicating the positive-reinforcing qualities of that substance. The test tank consists of two visually distinct compartments separated by a central alley. The protocol involves three steps: the determination of initial preference, one conditioning session and the determination of final preference. This procedure is carried out in ～2 d; other reported CPP protocols take up to 2 weeks. An increase in preference for the drug-associated compartment is observed after a single exposure. Establishment of this high-throughput protocol in zebrafish makes it possible to investigate the molecular and cellular basis of choice behavior, reward and associative learning. The protocol is also a tool for testing psychoactive compounds in the context of a vertebrate brain.     

Peak drug levels in linear pharmacokinetic systems—III. Multimodal impulse responses in multicompartment systems

This paper deals with the unimodality of the ‘impulse response’ in compartmental systems, where the ‘impulse response’ in any given compartment is the time course of the amount of diffusing substance in that compartment after an initial instantaneous injection of the substance into that or some other compartment. It is shown that in certain compartmental structures, with injection in certain compartments, the impulse response is always unimodal or monotonic in all compartments, regardless of the numerical values of the various transfer rate coefficients. Structures with this property are here named ‘UM structures’, and they include the familiar mammillary and catenary structures. In this paper, the set of all UM structures is described. Structures which are not UM (NUM structures) are identified by showing that, by removal of certain connections and compartments according to certain rules, they can be reduced to small structures which can be shown to be NUM by numerical computation. Computations on two systems with bimodal impulse responses show that with constant infusions of a fixed amount of substance the peak level may increase paradoxically with decrease in the infusion rate over a certain range. This effect is extremely small, however. Supported in part by U.S. Public Health Service Research Grant GM 21269 from the National Institute of General Medical Sciences, and in part by Biomedical Research Support Grant S07 RR 05392 from the National Institutes of Health.     

Computer simulation and interpretation of45Ca efflux profile patterns

Summary Stimulations or inhibitions by various agents of⁴⁵Ca efflux from prelabeled cells or tissues display distinct and reproducible profile patterns when the results are plotted against time as fractional efflux ratios (FER). FER is the fractional efflux of⁴⁵Ca from stimulated cells divided by the fractional efflux from a control unstimulated group. These profile patterns fall into three categories: peak patterns, exponential patterns, and mixed patterns. Each category can be positive (stimulation) or negative (inhibition). The interpretation of these profiles is difficult because⁴⁵Ca efflux depends on three variables: the rate of calcium transport out of the cell, the specific activity of the cell compartment from which the calcium originates, and the concentration of free calcium in this compartment. A computer model based on data obtained by kinetic analyses of⁴⁵Ca desaturation curves and consisting of two distinct intracellular pools was designed to follow the concentration of the traced substance (⁴⁰Ca), the tracer (⁴⁵Ca), and the specific activity of each compartment before, during, and after the stimulation or the inhibition of calcium fluxes at various pool boundaries. The computer model can reproduce all the FER profiles obtained experimentally and bring information which may be helpful to the interpretation of this type of data. Some predictions of the model were tested experimentally, and the results support the views that a peak pattern may reflect a sustained change in calcium transport across the plasma membrane, that an exponential pattern arises from calcium mobilization from an internal subcellular pool, and that a mixed pattern may be caused by a simultaneous change in calcium fluxes at both compartment boundaries.     

Evaluation of the rates of biological processes from tracer kinetic data. IV. Digital simulation of nucleic acid metabolism in bacteria

A flexible computer program was set up to simulate the kinetics of the amounts of radioactivity and the amounts of substance in various classes of nucleic acids for a generalized biological system. This program works by transferring numbers from registers to other registers corresponding to the amounts of isotope label or the amount of substance moving from one pool or compartment to another for each successive time interval. It is practical to choose the time interval small enough so that the same results are accurately obtained with this type of computation as with the analytical solution for the most realistic and complicated case previously formulated. Much more complicated situations now can be treated with this recursion program and questions of relevance to current molecular biology can be answered. The questions treated in this paper are directed to enteric microorganisms and are as follows. (1) What are the conditions under which a short pulse reflects net synthesis and what are the conditions where it reflects total synthesis? (2) What are the conditions that measurement of intermediate pool specific activity can yield information about the turnover and concentration of unstable macromolecules in the system? (3) How prolonged can a pulse be and still label the species of RNA substantially in proportion to their rates of synthesis? (4) What are the amounts and rates of synthesis and degradation of ribosomal precursors best fitting data in the literature?     

Analysis of tracer experiments: VII. General multi-compartment systems imbedded in non-homogeneous inaccessible media

An integral equation analysis of generaln compartment steady state systems imbedded in static media of arbitrary complexity has been developed. A set of initial entry functions can be found which serve to determine a corresponding set of partitioned initial entry functions. The partitioned functions, in turn, can be used to predict the probabilities and time courses of various transport histories and to determine all steady state rates of flow between measured compartments. The method is quite general, being completely applicable, for example, to closed systems, to cyclic systems and to systems in which relatively rapid (but finite) exchange between compartments occurs.     

A simple network thermodynamic method for series-parallel coupled flows: II. The non-linear theory, with applications to coupled solute and volume flow in a series membrane

Building on the linear network thermodynamic model presented in the first paper in this series (Mikulecky, Wiegand &; Shiner, 1977), a method of non-linear analysis is developed based mainly on the methods of Chua (1969). Using coupled salt and volume flow through membranes as an example, it is shown that the Kedem &; Katchalsky (1963a,b,c) linear model of series combinations of membranes behaves in a physically unrealistic manner (negative and infinite concentrations occur in the compartment between the membranes). Using the model introduced by Patlak, Goldstein &; Hoffman (1963) to obtain characteristics for a non-linear 2-port, the series network is well behaved and self-regulated. In the coordinate system of the linear reciprocal 2-port, the non-linear 2-port is shown to be a non-reciprocal element. Another co-ordinate system, which utilizes unidirectional fluxes (as measured by isotope fluxes, for example) and their conjugate driving forces, shows the non-linear 2-port to be separable in the sense of Li (1962) and Rosen (1968). A set of three pseudo-independent force-flow pairs completely characterizes the system and is compatible with a very simple non-linear network analysis which utilizes experimentally accessible and practical parameters such as the unidirectional solute fluxes, concentrations in the baths, volume flow and the hydrostatic minus effective osmotic pressure. As constitutive relations, the forward and backward permeabilities, which are dependent on volume flow and the filtration coefficient are all that are needed. A reflection coefficient appears in the driving force conjugate to volume flow in that the effective osmotic pressure is δΔπ and thus four independent coefficients, two of which depend on volume flow, are needed to determine the system. From the network analysis, the global properties of the series and/or parallel combinations of non-linear 2-ports are readily obtained. Although applied to a particular example, these non-linear methods are perfectly general. They are essentially the graphical form of the algebraic analysis in the linear theory, and are essentially based on the generalized version of Kirchhoff's laws used in graph and network theory. In the particular case of coupled solute and volume flow through membranes the non-linear element has a natural piecewise linearization which allows the linear theory to be applied using different values for the elements in each region.In an initial analysis, a linear network is drawn using the I-equivalent network of 1-port elements to model the linear 2-port in a series system. From this over-simplication a great deal of the qualitative behavior of the system can be visualized. For example, it is an easy way to see that solute will be accumulated or depleted in the central compartment between the membranes in an asymmetric series system. After this the non-linear analysis is used to further quantitatively describe the system's behavior and such phenomena as rectification of volume flow in the series membrane system. An appendix introduces a general theory for a class of non-linear transport phenomena.     

A four compartment linear mammillary model

Studies are made for a four compartment model in which the central compartment is connected reversibly to three other compartments and the elimination occurs from the central compartment only. Identification of different distribution rate constants is made, with concentrations of drug in the central compartment at different times of observation being known. The solution depends on an optimization method in which the different unknowns are reduced to single variable with the help of Archimedes spiral. Thus, the solution requires the global minimum of a functional of single variable. Results are compared with those obtained by the generalized least square method.     

Improvements in metabolic flux analysis using carbon bond labeling experiments: bondomer balancing and Boolean function mapping

The biosynthetically directed fractional (13)C labeling method for metabolic flux evaluation relies on performing a 2-D [(13)C, (1)H] NMR experiment on extracts from organisms cultured on a uniformly labeled carbon substrate. This article focuses on improvements in the interpretation of data obtained from such an experiment by employing the concept of bondomers. Bondomers take into account the natural abundance of (13)C; therefore many bondomers in a real network are zero, and can be precluded a priori--thus resulting in fewer balances. Using this method, we obtained a set of linear equations which can be solved to obtain analytical formulas for NMR-measurable quantities in terms of fluxes in glycolysis and the pentose phosphate pathways. For a specific case of this network with four degrees of freedom, a priori identifiability of the fluxes was shown possible for any set of fluxes. For a more general case with five degrees of freedom, the fluxes were shown identifiable for a representative set of fluxes. Minimal sets of measurements which best identify the fluxes are listed. Furthermore, we have delineated Boolean function mapping, a new method to iteratively simulate bondomer abundances or efficiently convert carbon skeleton rearrangement information to mapping matrices. The efficiency of this method is expected to be valuable while analyzing metabolic networks which are not completely known (such as in plant metabolism) or while implementing iterative bondomer balancing methods.     

Kinetics of tryptophan fluorescence enhancement in myofibrils during ATP hydrolysis

The mechanism of ATP hydrolysis in myofibrils can be studied by following the time course of tryptophan fluorescence. Stoichiometric quantities of ATP produce an enhancement of the tryptophan fluorescence in stirred suspensions of rabbit psoas myofibrils at pCa greater than 7. Approximately 1 mol of ATP/myosin head is required to obtain the maximum fluorescence enhancement of 4-6%. Upon the addition of quantities of ATP greater than 1 mol/mol of myosin head, the fluorescence rapidly increases to a steady state, which lasts for a period that is proportional to the amount of ATP added. The fluorescence then decays to the initial level with a half-time of approximately 40 s at 20 degrees C. Hydrolysis of [gamma-32P]ATP at pCa greater than 7 in myofibrils has an initial burst of approximately 0.7 mol/mol of myosin head that is followed by a constant rate of hydrolysis. The duration of the steady state hydrolysis is identical to the duration of the enhancement of tryptophan fluorescence. A lower limit of 5 X 10(5) M-1 S-1 was obtained for the second order rate constant of the fluorescence enhancement by ATP. At pCa of 4, the duration of the fluorescence enhancement is one-tenth to one-twentieth as long as at pCa greater than 7; this is consistent with the increased steady state rate of ATP hydrolysis at higher calcium concentrations. The time course of the fluorescence enhancement observed in myofibrils during ATP hydrolysis is qualitatively and quantitatively similar to that observed with actomyosin-S1 in solution. These results suggest that the kinetic mechanism of ATP hydrolysis that has been well established by studies of actomyosin-S1 in solution also occurs in myofibrils.     

Bicarbonate kinetics in Indian males

Measurement of rates ofin vivo substrate oxidation such as that of glucose, fatty acids and amino acids, are based on tracer (¹⁴C or¹³C) data, and often depend on the isotopic content of expired CO₂. The recovery of tracer-labelled CO₂ generated from the oxidation of¹³C labelled substrates may not be 100% over short term. This can lead to underestimation of oxidation rate of substrates, and consequently a correction for the incomplete recovery of tracer has to be applied by the determination of the recovery of¹³CO₂ in the breath during tracer bicarbonate infusions. We have studied the recovery of tracer-labelled bicarbonate using a bolus administration model, and further characterized kinetics of bicarbonate using a three-compartment model, to assess which compartmental fluxes changed during the change from a fasted state to fed state. Recovery of bicarbonate was lower at 69% and 67% (fasted and fed state) than the value of 71% and 74% found during earlier longer term of continuous infusions. During feeding, there was a 20-fold increase in the flux of bicarbonate between the central compartment and the compartment that was equivalent to the viscera. This study shows that the difference between the fasted and fed state recovery of tracer bicarbonate similar to that obtained with continuous infusions, and that bicarbonate fluxes show large changes between different compartments in the body depending on metabolic state.     

A four compartment model to study the kinetics of strontium metabolism in man

Many drugs confer upon the body the characteristics of a four compartment model. This paper deals with the estimation of pharmacokinetic parameters of a four compartment model to study the distribution of strontium in the organism. The central compartment (where the introduction of the material takes place) is connected to two other compartments (which represent the organic fluids) and one of them is connected to the fourth compartment (which represents the fraction that can be exchanged in the bone). The elimination from the central compartment is through urine and the elimination from the third compartment is in the form of a non-exchangeable deposit. The method of solution involves an optimization method which provides the global minimum of δ, a single variable function. The model is tested for different sets of data and the results are compared with those obtained by the generalized leasr square method.     

Nonsteady-State Three Compartment Tracer Kinetics: II. Sodium Flux Transients in the Toad Urinary Bladder in Response to Short Circuit

The theoretical approach presented in the previous paper provides an analytical method for determining the unidirectional, nonsteady-state fluxes in a three compartment system. Based on this a study was made of the sodium flux transients in the toad urinary bladder. A transient time-dependent state was generated by suddenly short-circuiting a bladder previously maintained in an open-circuited steady state. The sequence of experiments suggested by the theory provided the data required for the analysis. The results of these tracer experiments were consistent with the complex non-three compartmental structure of this tissue. As a result both of the inadequacy of the three compartment model in representing the tissue and of certain experimental difficulties, attempts at a quantitative solution were not entirely successful. Useful information was nevertheless obtained through a careful use of this model, and a qualitative analysis implied that the sodium influxes into the tissue at both of its surfaces are sensitive to changes in electrical potential while both effluxes are insensitive to this change. This suggests that both of the effluxes result from active processes while both influxes are associated with passive processes. The net transepithelial transport of sodium would then necessarily result from a more complex polarization than that proposed by Koefoed-Johnsen and Ussing.     

Mathematical modeling of transmembrane calcium transport kinetics in smooth muscle cells

A mathematical model, which describes kinetics of transmembrane calcium transport in a smooth muscular cell, has been elaborated and investigated taking into account that the change of calcium cations concentration within a cell is determined by two mutually opposite processes: an increase of a carrying capacity of calcium channels of plasma membrane under signal substance action and calcium removal from the intracellular space by Mg2+, ATP-dependent calcium pump localized on the plasma membrane. The fundamental difference of the proposed model against the models analyzed in literature before is that the cellular system returns to the initial stationary state after enzyme-catalysed transformation of the signal substance. The results of calculations showed that this model really described the experimental kinetics of the transmembrane calcium transport. In this paper the influence of different parameters (Michaelis constant and ultimate rate of calcium pump, initial concentrations of signal substance and enzyme decomposing it, rate constants) on kinetics of calcium transport through the plasma membrane has been investigated in detail.     

Fluid transport and dimensions of epithelial cells and intercellular spaces in frog gallbladder

Summary Morphologic findings of widely dilated intercellular spaces in fluid transporting epithelia have been claimed as evidence for the existence of an epithelial compartment in which the coupling between solute and water fluxes takes place. The validity of using epithelial geometry in sectioned material as an argument can be questioned. The present report describes the morphological appearance of frog gallbladder epithelium — normal and ouabain-treated — in the living state in vitro and after fixation, dehydration and embedding. Gallbladder segments were photographed in the living state and at the end of each step of the preparative procedure. Direct observations of whole-mounted gallbladder segments were carried out, taking advantage of the possibility of optical sectioning and high resolution by Nomarski-microscopy. The same specimens were then sectioned and examined by conventional light and electron microscopy. The observations were quantitated and showed that the epithelial cells of normal and ouabain-treated gallbladders experienced an average linear shrinkage down to 70% of their length in Ringer's solution, which corresponds to a volume shrinkage down to 35%. Moreover, dilated lateral intercellular spaces appeared during the dehydration and embedding procedure in normal but only very moderately or not at all in ouabain-treated gallbladder specimens.     

Theory of metabolism regulation: a complete system of equations for regulation coefficients

Basic quantitative parameters of control in a metabolic system are considered: control coefficients of enzymes with respect to metabolic fluxes and concentrations, and in the case when there are conservation laws, the response coefficients of metabolic fluxes and concentrations to changes in the conserved sums of metabolite concentrations (e. g. conserved moieties). Relationships are obtained which generalize the well known connectivity relations for the case of metabolites binding by conservation laws. Additional relationships are obtained which complement the set of connectivity relations up to the complete system of equations for determining all the control coefficients. The control coefficients are expressed through the enzyme elasticity coefficients, steady state metabolic fluxes and concentrations. Formulas are derived which express response coefficients of flux and concentrations through the enzyme control and elasticity coefficients and metabolite concentrations.     

An approximation method for the determination of diffusion and permeability coefficients in nerve trunks

A recently introduced approximation method is applied in order to obtain an expression for the amount of a substance remaining within a nerve at any time, the nerve having been soaked for a long time in a solution containing the substance until the time zero when it is bathed in the same solution but without the substance. The case of a uniform nerve without a sheath leads to substantially the same results as previously obtained by A. V. Hill (1928) for this case. A solution is given for the case of a nerve without sheath but having fibers which are permeable. In this case it is shown how an effective diffusion coefficient for the interstitial fluid can be obtained, as well as the effective inward and outward fiber permeabilities. A solution is given for the case of a nerve with a sheath in which the substance considered does not penetrate the fibers, and it is shown how the effective diffusion coefficients of the sheath and the interstitial fluid can be obtained.     

The matrix representation of pharmacokinetic models.

An equation is developed from the matrix of rate constants which describes the behaviour of linear pharmacokinetic models for any initial condition as a function of time. This general matrix equation is then used to derive analogous expressions for drug distribution after a period of infusion, at the steady state, or during a multiple constant-dosage regimen. Matrix expressions are also derived for areas under drug concentration curves for any compartment after single doses or during multiple dosing. General matrix equations are shown to yield loading dosage schedules to achieve plateau concentrations throughout any open system.It is suggested that matrix methods have advantages over previously used mathematical techniques in pharmacokinetics in the simplicity of the algebraic expressions, and their ease of manipulation. An algebraic example of an open two-compartment model is worked to indicate the applicability of the general expressions.     

Quantitative analysis of a fully generalized four-state kinetic scheme

To describe the macroscopic behavior of many ion channels, at a minimum a four-state kinetic scheme is needed to provide for three processes: a delay in activation development, the activation process, and inactivation. I present here an analytical solution for a fully generalized four-state kinetic scheme in which every state can transit to every other state and any initial conditions can be specified. The solution describes the time courses of the probabilities of occupancy of each state during a step change in the rate constants of the scheme and includes closed-form expressions for the relaxation time constants and steady-state probabilities of occupancy as functions of the rate constants. Solutions for several relevant special cases are also included along with demonstrations that the general solution yields the correct behavior for several reduced or special cases where the result is independently known.     

Estimating estuarine residence times in the Westerschelde (The Netherlands) using a box model with fixed dispersion coefficients

The residence time of the water masses in the Westerschelde estuary was determined using a simple compartment-model that simulates the advective-diffusive transport of a conservative dissolved substance (chlorinity). The residence time of a water parcel in the upstream part of the estuary (i.e. the time needed for this water parcel to leave the estuary) varied from about 50 days in winter to about 70 days in summer. The most seaward compartment had residence times of about 10-15 days.Dispersive coefficients that are fixed in time were able to reproduce the observed salinity distributions very well in the Westerschelde. They were obtained by calibration on observed chlorinities. It is argued that the apparent relationship of dispersive coefficients with freshwater flow, which is observed in certain studies, could (partly) reflect the deviation from steady state conditions which are required assumptions to calculate these dispersive coefficients directly from salinity profiles.