Estimation of metabolic flux from dominant rate constants in vivo: application to brain and heart

In an earlier paper (Cohen and Bergman, Am. J. Physiol. 268 (1995) E397), we explored the relationship between the exponents in the exponential curve fit to isotopic enrichment versus time and the fractional turnover rate of the largest metabolic pool in the pathway. Here we present the analysis on a more rigorous footing and apply it to questions of cerebral and cardiac metabolism. Our emphasis in this paper is to describe and justify mathematically an approach for analysis of metabolic dynamics, not with the intention of replacing the use of numerical software for estimation of flux rates but for giving the scientist the opportunity to examine the system in an approximate manner, and thereby to check not only that the results of the numerical solution are the correct solutions to the equations but also that the equations portray the correct simplification of the metabolic pathway.We introduce the "dominant rate constant" as a tool for deriving algebraic formulas relating rates of metabolic flux, sizes of metabolic pools, and the dynamics of isotopic enrichment. Illustrations of such algebraic formulas are provided for the rates of the citric acid cycle (CAC), glycolysis and glutamine synthesis in brain, as well as the rate of the CAC in heart. In addition, we prove that formulas for estimation of rates of glycolysis and of the CAC depend critically on the fractional turnover rates of lactate and glutamate, respectively. The justification for analysis of simulated data is that we are studying the effects of simplifications of metabolic models on the accuracy of estimation of metabolic pathways. Our use of the dominant rate constant is an analytical convenience that allows us to assess proposed simplifications of metabolic pathways.     

Extension of the theory of the multiple-indicator dilution technique to metabolic systems with an arbitrary number of rate constants

The multiple indicator dilution technique consists in the instantaneous injection of a mixture of tracers into the arterial perfusate flow of a catheterized or isolated perfused organ, followed by the analysis of the effluent perfusate. The theory of this technique, which has hitherto been developed for cases where metabolism of a tracer is confined to sequestration described by a single rate constant, is extended in this paper to include an arbitrary number of metabolic rate constants. Partial differential equations with constant coefficients describing the events in a single capillary are derived by applying conventional compartmental analysis to infinitesimally small sections of the capillary. Methods for solving such systems in the time as well as in the frequency domain are developed. From the solutions, the impulse response of the whole organ is evaluated assuming variable capillary and uniform large-vessel transit times. In addition, an efficient method using much less computer time was developed, based on the approximation of the distribution of the capillary transit times by a sum of exponentials. Evaluation of moments (recoveries and mean transit times) is also treated. The results are applied to an example from hepatic lactate metabolism.     

Homeostatic model assessment (HOMA) index cut-off values to identify the metabolic syndrome in children

The aim of the study was to establish the best cut-off value for the homeostatic model assessment (HOMA) index in identifying children and adolescents with the metabolic syndrome. The study included 72 non-obese and 68 obese children aged 7 to 16 years. Obesity is defined using the criteria proposed by Cole et al., being included as metabolic syndrome variables waist circumference, systolic blood pressure, diastolic blood pressure and seric values of glucose, uric acid, fasting insulin, leptin, triglycerides and HDL-cholesterol. Children were considered as having the metabolic syndrome when four or more characteristics showed abnormal values. The HOMA index was calculated as the product of the fasting plasma insulin level (microU/mL) and the fasting plasma glucose level (mmol/L), divided by 22.5. HOMA index cut-offs from the 5th to the 95th percentile were used. A receiver operating characteristic (ROC) curve was generated using the different HOMA cut-offs for the screening of the metabolic syndrome. The areas under the ROC curve, 95% confidence intervals, and the point to the ROC curve closest to 1, were calculated. The area under the ROC curve was 0.863 (95% C.I.: 0.797, 0.930). The point closest to 1 corresponds to the 60th percentile of the HOMA index distribution in our sample. HOMA index value at the 60th percentile was 2.28. Cut-off values corresponding to a range of HOMA index from the 50 to the 75 percentile, showed similar distances to 1. HOMA index values for percentiles 50 to 75 ranged from 2.07 to 2.83. In conclusion, HOMA index could be a useful tool to detect children and adolescents with the metabolic syndrome. HOMA cut-off values need to be defined in the paediatric population; however, values near to 3 seem to be adequate.     

Sensitivity of a Hill-based muscle model to perturbations in model parameters

Musculoskeletal simulations of human movement commonly use Hill muscle models to predict muscle forces, but their sensitivity to model parameter values is not well understood. The purpose of this study was to evaluate muscle model sensitivity to perturbations in 14 Hill muscle model parameters in forward dynamic simulations of running and walking by varying each by +/-50%. Three evaluations of the muscle model were performed based on: (1) calculating the sensitivity of the muscle model only, (2) determining the continuous partial derivatives of the muscle equations with respect to each parameter, and (3) evaluating the effects on the running and walking simulations. Model evaluations were found to be very sensitive (percent change in outputs greater than parameter perturbation) to parameters defining the series elastic component (tendon), force-length curve of the contractile element and maximum isometric force. For some parameters, the range of literature values was larger than the model sensitivity. Model evaluations were insensitive to parameters defining the parallel elastic element, force-velocity curve of the contractile element and muscle activation time constants. The derivative method provided similar results, but also provided a generic, continuous equation that can easily be applied to other motions. The sensitivities of the running and walking simulations were reduced compared to the sensitivity of the muscle model alone. Results demonstrate the importance of evaluating sensitivity of a musculoskeletal simulation in a controlled manner and provide an indication of which parameters must be selected most carefully based on the sensitivity of a given movement.     

A strong response to selection on mass-independent maximal metabolic rate without a correlated response in basal metabolic rate

Metabolic rates are correlated with many aspects of ecology, but how selection on different aspects of metabolic rates affects their mutual evolution is poorly understood. Using laboratory mice, we artificially selected for high maximal mass-independent metabolic rate (MMR) without direct selection on mass-independent basal metabolic rate (BMR). Then we tested for responses to selection in MMR and correlated responses to selection in BMR. In other lines, we antagonistically selected for mice with a combination of high mass-independent MMR and low mass-independent BMR. All selection protocols and data analyses included body mass as a covariate, so effects of selection on the metabolic rates are mass adjusted (that is, independent of effects of body mass). The selection lasted eight generations. Compared with controls, MMR was significantly higher (11.2%) in lines selected for increased MMR, and BMR was slightly, but not significantly, higher (2.5%). Compared with controls, MMR was significantly higher (5.3%) in antagonistically selected lines, and BMR was slightly, but not significantly, lower (4.2%). Analysis of breeding values revealed no positive genetic trend for elevated BMR in high-MMR lines. A weak positive genetic correlation was detected between MMR and BMR. That weak positive genetic correlation supports the aerobic capacity model for the evolution of endothermy in the sense that it fails to falsify a key model assumption. Overall, the results suggest that at least in these mice there is significant capacity for independent evolution of metabolic traits. Whether that is true in the ancestral animals that evolved endothermy remains an important but unanswered question.     

The pH-dependence of second-order rate constants of enzyme modification may provide free-reactant pKa values.

1. Reactions of enzymes with site-specific reagents may involve intermediate adsorptive complexes formed by parallel reactions in several protonic states. Accordingly, a profile of the apparent second-order rate constant for the modification reaction (Kobs., the observed rate constant under conditions where the reagent concentration is low enough for the reaction to be first-order in reagent) against pH can, in general, reflect free-reactant-state molecular pKa values only if a quasi-equilibrium condition exists around the reactive protonic state (EHR) of the adsorptive complex. 2. Usually the condition for quasi-equilibrium is expressed in terms of the rate constants around EHR: (formula: see text) i.e. k mod. less than k-2. This often cannot be assessed directly, particularly if it is not possible to determine kmod. 3. It is shown that kmod. must be much less than k-2, however, if kobs. (the pH-independent value of kobs.) less than k+2. 4. Since probable values of k+2 greater than 10(6)M-1.S-1 and since values of kobs. for many modification reactions less than 10(6)M-1.S-1, the equilibrium assumption should be valid, and kinetic study of such reactions should provide reactant-state pKa values. 5. This may not apply to catalyses, because for them the value of kcat./Km may exceed 5 X 10(5)M-1.S-1. 6. The conditions under which the formation of an intermediate complex by parallel pathways may come to quasi-equilibrium are discussed in the Appendix.     

The relationship of intercompartmental excitation transfer rate constants to those of an underlying physical model

In studies on photosynthetic systems it is common practice to interpret the results of time-resolved fluorescence experiments on the basis of compartmental, or target, models. Each compartment represents a group of molecules with similar fluorescence characteristics. In cases of practical interest, the members of each compartment are spatially contiguous and make up part of an overall energy-transferring system. Since a rate constant describing the overall transfer between compartments is not that of any pair of molecules in the system, this question naturally rises: what do we learn about the microscopic structure from these data? In this note we introduce ‘compartment melting’, a smooth mathematical connection between the compartmental and microscopic levels. We then show, on the basis of model calculations on finite lattices in one, two, and three dimensions, that average microscopic rates at the interfaces between compartments may be estimated from observed intercompartmental rates. The estimate involves a modest number of structural assumptions about the system. As examples of the method, which is applicable mainly to systems containing homogeneous pigment pools, some recent chlorophyll-protein antenna studies are analyzed.     

Using computer algebra to determine rate constants in biochemistry

In earlier work we have described how computer algebra may be used to derive composite rate laws for complete systems of equations, using the mathematical technique of Gröbner Bases (Bennett, Davenport and Sauro, 1988). Such composite rate laws may then be fitted to experimental data to yield estimates of kinetic parameters. Recently we have been investigating the practical application of this methodology to the estimation of kinetic parameters for the closed two enzyme system of aspartate aminotransferase (AAT) and malate dehydrogenase (MDH) (Fisher 1990a; Fisher 1990b; Bennett and Fisher, 1990): $$\begin{gathered} aspartate + \alpha - ketoglutarate\begin{array}{*{20}c} \rightharpoonup \\ \leftharpoondown \\ \end{array} glutamate + oxaloacetate \hfill \\ {\text{oxaloacetate + NADH}}\begin{array}{*{20}c} \rightharpoonup \\ \leftharpoondown \\ \end{array} malate + NAD^ + \hfill \\ \end{gathered} $$ In this paper we present a fuller (although not yet complete) analysis of the system. We show how symbolic estimates of the error behaviour of the parameters can be made, and used to identify those which are of kinetic significance. Finally we consider how metabolic control analysis can be applied directly to such a system.     

Superoxide dismutase: a comparison of rate constants

O₂^?was introduced, at a constant rate, into buffered aqueous solutions, either by mechanical infusion of KO₂, dissolved in tetrahydrofuran, or by the in situ action of xanthine oxidase on xanthine plus oxygen. This O₂^? was allowed to react with ferricytochrome c or with tetranitromethane and the formation of the reaction products, ferrocytochrome c or nitroform, respectively, was monitored spectrophotometrically. That concentration of Superoxide dismutase, which competed equally with given levels of cytochrome c or tetranitromethane and which thus caused 50% inhibition of the rates of accumulation of ferrocytochrome c or of nitroform, was determined. The rate constant for the enzymatic dismutation of O₂^? by the copper and zinc containing enzyme from bovine erythrocytes was then calculated from the known rate constants for the reaction of O₂^? with ferricytochrome c and with tetranitromethane and was found to be 2 × 10⁹m^?1 sec^?1 at pH 7.8 and 8.5. This rate constant was obtained at steady-state concentrations of O₂^? in the 10^?8m → 10^?13m range and is in full agreement with the results of pulse radiolytic investigations which were performed at O₂^? concentrations in the 10^?5m range. The second order rate constant for the enzymatic dismutation of O₂^? is thus independent of the concentration of O₂^? in the range 10^?5 → 10^?13m.Several distinct types of Superoxide dismutase have been described. These include the mangano-enzymes from Escherichia coli and from chicken liver mitochondria and the iron-enzyme from E. coli. The rate constants for the dismutations catalyzed by these enzymes have also been investigated as a function of pH.     

Estimation of the effective transversely isotropic elastic constants of a material from known values of the material's orthotropic elastic constants

A method is illustrated for determining the effective transversely isotropic (or isotropic) elastic constants from measured orthotropic elastic constants. This method consists of constructing upper and lower bounds on the effective transversely isotropic (or isotropic) elastic constants using the known orthotropic values. This method is illustrated using three sets of elastic constants for bone. Fortunately, the upper and lower bounds are very close. Thus very good approximations for the effective transversely isotropic (or isotropic) elastic constants for cortical and cancellous bone are obtained from previously published data on the orthotropic elastic constants for those tissue types. This work is undertaken to build a greater database for the transversely isotropic elastic constants of bone with the intention of employing them in a transversely isotropic model of bone poroelasticity. An interesting aspect of the present result is that the Voigt and Reuss bounds are very tight for these anisotropic materials. This is not always the case for these bounds. Received: 14 November 2001 / Accepted: 25 February 2002     

Refining the measurement of rate constants in the BIAcore

When estimating rate constants using the BIAcore surface plasmon resonance (SPR) biosensor, one must have an accurate mathematical model to interpret sensogram data. Several models of differing complexity are discussed, including the effective rate constant (ERC) approach. This model can be shown formally to be good within O(Da) in the limit of small Damköhler number Da, which is the ratio of the reaction rate to the rate of transport to the surface. Numerical results are presented that show that except for very slow reactions, parameter estimates from the ERC model are very close to those estimated using a more complicated model. The BIAcore measures the behavior of an evanescent wave whose signal strength decays as it penetrates into the device. It is shown that this decay does not appreciably affect the sensogram readout at low Da, but at moderate Da can lead to situations where two vastly different rate constants can produce the same short-time sensogram data.Mathematics Subject Classification (2000): 35B20, 35C15, 35K60, 45J05, 92C45This work was supported in part by NIGMS Grant 1R01GM067244-01.Revised version: 1 August 2003     

Sensitivity to initial conditions in the route-finder model

A model of stimulus generalization [Staddon, J.E.R., Reid, A.K., 1990. On the dynamics of generalization. Psychol. Rev. 97, 576-578] was extended by Reid and Staddon [Reid, A.K., Staddon, J.E.R., 1998. A dynamic route finder for the cognitive map. Psychol. Rev. 105, 585-601] to account for the route-finder issue in the Tolman-Guthrie discussion about cognitive maps. This deterministic model has been described as sensitive to initial conditions in terms of the parameters of a diffusion equation [Reid, A.K., Staddon, J.E.R., 1998. A dynamic route finder for the cognitive map. Psychol. Rev. 105, 585-601]. A simulation study was carried out to test this assertion, evaluating whether the pattern of variation in the model output resembled a chaotic pattern. Results indicate that the model is sensitive to initial conditions, suggesting that the spatial navigation task addressed by this model involves non-linear relationships and complexity beyond the apparent simplicity of the stimulus generalization process.     

Analysis of a procedure for the determination of kinetic rate constants

This paper is concerned with a mathematical analysis of the modified Guggenheim procedure. Theorems concerning the solutions of the differential equations which describe the general reaction

$$E + SB\xrightarrow{{k_2 }}C + X, C\xrightarrow{{k_3 }}E + D$$     

Determination of reaction rate constants in the mammillary system

It is shown that the individual rate constants can be determined for the composite chemical system: $$A + B_i \rightleftarrows C_i ; i = 1...N$$ with only measurements of the unbound species,A(t), required. The dissociation rate constants can be determined by direct analysis of a single steady state tracer study. The association constants then follow from the analysis of stable equilibrium determinations reported earlier (Hart, 1965). An approximate solution when tracer methods are in-applicable is also given.     

The effect of a receptor layer on the measurement of rate constants

Many cellular reactions involve a reactant in solution binding to or dissociating from a reactant attached to a surface. Most studies assume that the reactions occur on this surface, when in actuality the receptors usually lie in a thin layer on top of it. The effect of this layer is considered, particularly as it relates to the BIAcore™ measurement device, though the results are applicable to biological systems. A dimensionless parameter measuring the strength of the effect of the receptor layer is found. Asymptotic and singular perturbation techniques are used to analyse association and dissociation kinetics, though the effect of the receptor layer need not be small. Linear and nonlinear integral equations result from the analysis; explicit and asymptotic solutions are constructed for physically realizable cases. In addition, effective rate constants are derived that illustrate the combined effects of transport and the receptor layer on the measured rate constants. All these expressions provide a direct way to estimate rate constants from BIAcore™ binding data.     

Determination of the two-dimensional interaction rate constants of a cytokine receptor complex

Ligand-receptor interactions within the plane of the plasma membrane play a pivotal role for transmembrane signaling. The biophysical principles of protein-protein interactions on lipid bilayers, though, have hardly been experimentally addressed. We have dissected the interactions involved in ternary complex formation by ligand-induced cross-linking of the subunits of the type I interferon (IFN) receptors ifnar1 and ifnar2 in vitro. The extracellular domains ifnar1-ectodomain (EC) and ifnar2-EC were tethered in an oriented manner on solid-supported lipid bilayers. The interactions of IFNalpha2 and several mutants, which exhibit different association and dissociation rate constants toward ifnar1-EC and ifnar2-EC, were monitored by simultaneous label-free detection and surface-sensitive fluorescence spectroscopy. Surface dissociation rate constants were determined by measuring ligand exchange kinetics, and by measuring receptor exchange on the surface by fluorescence resonance energy transfer. Strikingly, approximately three-times lower dissociation rate constants were observed for both receptor subunits compared to the dissociation in solution. Based on these directly determined surface-dissociation rate constants, the surface-association rate constants were assessed by probing ligand dissociation at different relative surface concentrations of the receptor subunits. In contrast to the interaction in solution, the association rate constants depended on the orientation of the receptor components. Furthermore, the large differences in association kinetics observed in solution were not detectable on the surface. Based on these results, the key roles of orientation and lateral diffusion on the kinetics of protein interactions in plane of the membrane are discussed.     

Fast and precise access to enantiomerization rate constants in dynamic chromatography

An analytical solution of the unified equation to evaluate elution profiles of interconverting enantiomers in dynamic chromatography is presented. Rate constants k1 and k(-1) and Gibbs activation energies are directly obtained from the chromatographic parameters (retention times tR A and tR A of the interconverting enantiomers, the peak widths at half height wA and wB, and the relative plateau height hp), and the initial amounts A0 and B0 of the enantiomers without any iterative and time consuming computational step. Therefore, this equation is no longer limited to racemic analytes. The analytical solution presented here was validated by comparison with a dataset of 125,000 simulated elution profiles of enantiomerizations. Furthermore, it was found that the recovery rate from a defined dataset is on average 40% higher using the unified equation compared to evaluation methods based on iterative computer simulation. The new equation was applied to determine the enantiomerization rate constant of 1-n-butyl-2-tert-butyldiaziridine by enantioselective gas chromatography. The activation parameters (DeltaH(double dagger) = 112.6 +/- 2.5 kJ/mol and DeltaS(double dagger) = -27 +/- 2 J/(K mol) were obtained from temperature-dependent measurements between 100 degrees C and 140 degrees C in 10K steps.