Model reduction by extended quasi-steady-state approximation

We extend the quasi-steady-state approximation (QSSA) with respect to the class of differential systems as well as with respect to the order of approximation. We illustrate the first extension by an example which cannot be treated in the frame of the classical approach. As an application of the second extension we prove that the trimolecular autocatalator can be approximated by a fast bimolecular reaction system. Finally we describe a class of singularly perturbed systems for which a higher order QSSA can easily be obtained. Received: 30 March 1999 / Published online: 5 May 2000     

Approximate treatment of the formation of a difference in the concentrations of an ion at the two sides of a membrane

The recent extension of the approximation method is applied to enable us to arrive at the time course of the concentrations at both sides of a membrane. From the differential equations which govern these, the steady-state solution is obtained in terms of the parameters, which are determined by the thickness of the diffusion layers, the chemical composition and reactions, and the diffusion constant of the membrane.     

Dynamic simulation and metabolic re-design of a branched pathway using linlog kinetics

This paper presents a new mathematical framework for modeling of in vivo dynamics and for metabolic re-design: the linlog approach. This approach is an extension of metabolic control analysis (MCA), valid for large changes of enzyme and metabolite levels. Furthermore, the presented framework combines MCA with kinetic modeling, thereby also combining the merits of both approaches. The linlog framework includes general expressions giving the steady-state fluxes and metabolite concentrations as a function of enzyme levels and extracellular concentrations, and a metabolic design equation that allows direct calculation of required enzyme levels for a desired steady state when control and response coefficients are available. Expressions giving control coefficients as a function of the enzyme levels are also derived. The validity of the linlog approximation in metabolic modeling is demonstrated by application of linlog kinetics to a branched pathway with moiety conservation, reversible reactions and allosteric interactions. Results show that the linlog approximation is able to describe the non-linear dynamics of this pathway very well for concentration changes up to a factor 20. Also the metabolic design equation was tested successfully.     

On the nature of 2D crystal unbending

Crystal unbending, the process that aims to recover a perfect crystal from experimental data, is one of the more important steps in electron crystallography image processing. The unbending process involves three steps: estimation of the unit cell displacements from their ideal positions, extension of the deformation field to the whole image and transformation of the image in order to recover an ideal crystal. In this work, we present a systematic analysis of the second step oriented to address two issues. First, whether the unit cells remain undistorted and only the distance between them should be changed (rigid case) or should be modified with the same deformation suffered by the whole crystal (elastic case). Second, the performance of different extension algorithms (interpolation versus approximation) is explored. Our experiments show that there is no difference between elastic and rigid cases or among the extension algorithms. This implies that the deformation fields are constant over large areas. Furthermore, our results indicate that the main source of error is the transformation of the crystal image.     

The Growth Rate of Successive Leaves of Wheat Plants in Relation to Sugar and Protein Concentrations in the Extension Zone

The growing part of a wheat leaf (the extension zone) is loocatedat the leaf base and following from this it was proposed thatthe absolute leaf extension rate (L_er) can be partitioned intotwo components: the length of the extension zone (L_ez) and therelative extension rate of that extension zone (R_ez). R_ez isan appropriate measure of the efficiency of leaf growth forcomparing different leaves. This model of a wheat leaf was thenused to investigate whether differences in growth rates betweensuccessive leaves on wheat plants were due to differences inhexose sugar or protein concentrations within the extensionzone. Measurements were done in an irrigated field crop suppliedwith 0, 3, 10, or 30 kg N ha^–1 per week. The mean values of L_er at 15 °C increased with leaf numberand with nitrogen supply as did values for L_ez. In contrastR_ez at 15 °C declined from 0.9 d^–1 for the first leavesto 0.3 d^–1 for the flag leaf. Nitrogen supply had littleeffect on R_ez. A separate measure of the efficiency of leafgrowth, the responsiveness of L_er to temperature (measured asthe slope of the temperature response curve), also decreasedwith leaf number by the same order as R_ez and was similarlyunaffected by nitrogen supply. The protein concentrations in the extension zones of the firstleaves were around 40 mg g^–1 fr. wt. and this declinedto approximately 20 mg g^–1 fr. wt for leaves emergingafter tillers emerged and remained low thereafter. Protein concentrationswere not correlated with the external supply of nitrogen. Hexosesugar concentrations followed a reverse pattern of increasingin the later order leaves and these also increased as nitrogensupply decreased. Both R_ez and the responsiveness of L_er to temperature were positivelycorrelated with the protein concentration, the relationshipin each case being described by a rectangular hyperbola equation(P < 0.01), and negatively correlated with hexose concentrations.It was concluded that internal competition between growing pointsfor reduced nitrogen caused the observed effects. However, differencesin protein concentrations may not simply reflect differencesin enzyme concentrations; rather these differences may indicatechanges in some other character such as cell numbers.     

A simple method for calculation of free metal ion concentrations in the presence of a chelating agent and multiple metal ions by successive approximation

Calculation of free metal ion concentrations in the presence of a chelating agent (ligand) and multiple metal ions is complicated. In this paper we describe a simple method for calculation of free ion concentrations from given total ion concentrations. The outline of this method is as follows: i) Using an arbitrarily chosen provisional value of free ligand concentration (p-Lf), calculate a total ligand concentration (Lt). ii) Divide the p-Lf by the ratio, the calculated Lt (c-Lt)/the specified Lt (s-Lt). Take the resulting value as the next p-Lf and repeat the calculation until c-Lt is close enough to s-Lt. At this point, p-Lf is supposed to be set to a good approximation of true free ligand concentration. iii) Finally, calculate free metal concentrations from the above approximation.     

Statistical Properties and Control Algorithms of Recursive Quantile Estimators

The present study is an extension of the investigations made by Grieszbach and Schack (1993) where the recursive estimators of the quantile were introduced. Attention is focused on statistical properties and on the controlling of these estimators in order to reduce their variance and to improve their capability of adaptation. Using methods of stochastic approximation, several control algorithms have been developed, where both the consistent and the adaptive estimation are considered. Due to the recursive computation formula the estimators are suitable for the analysis of large data sets and for sets whose elements are obtained sequentially. In this study, application examples from the analysis of EEG‐records are presented, where quantiles are used as threshold values.     

Onset of sheath extension and duration of lamina extension are major determinants of the response of maize lamina length to plant density

BACKGROUND AND AIMS: Plants regulate their architecture strongly in response to density, and there is evidence that this involves changes in the duration of leaf extension. This questions the approximation, central in crop models, that development follows a fixed thermal time schedule. The aim of this research is to investigate, using maize as a model, how the kinetics of extension of grass leaves change with density, and to propose directions for inclusion of this regulation in plant models. METHODS: Periodic dissection of plants allowed the establishment of the kinetics of lamina and sheath extension for two contrasting sowing densities. The temperature of the growing zone was measured with thermocouples. Two-phase (exponential plus linear) models were fitted to the data, allowing analysis of the timing of the phase changes of extension, and the extension rate of sheaths and blades during both phases. KEY RESULTS: The duration of lamina extension dictated the variation in lamina length between treatments. The lower phytomers were longer at high density, with delayed onset of sheath extension allowing more time for the lamina to extend. In the upper phytomers--which were shorter at high density--the laminae had a lower relative extension rate (RER) in the exponential phase and delayed onset of linear extension, and less time available for extension since early sheath extension was not delayed. CONCLUSIONS: The relative timing of the onset of fast extension of the lamina with that of sheath development is the main determinant of the response of lamina length to density. Evidence is presented that the contrasting behaviour of lower and upper phytomers is related to differing regulation of sheath ontogeny before and after panicle initiation. A conceptual model is proposed to explain how the observed asynchrony between lamina and sheath development is regulated.     

Optimal nonsingular control of fed-batch fermentation

Presented is a new simple method for multidimensional optimization of fed-batch fermentations based on the use of the orthogonal collocation technique. Considered is the problem of determination of optimal programs for fermentor temperature, substrate concentration in feed, feeding profile, and process duration. By reformulation of the state and control variables is obtained a nonsingular form of the optimization problem which has considerable advantage over the singular case since a complicated procedure for determination of switching times for feeding is avoided. The approximation of the state variables by Lagrange polynomials enables simple incorporation of split boundary conditions in the approximation, and the use of orthogonal collocations provides stability for integration of state and costate variables. The interpolation points are selected to obtain highest accuracy for approximation of the objective functional by the Radau-Lobatto formula. The control variables are determined by optimization of the Hamiltonian at the collocation points with the DFP method. Constraints are imposed on state and control variables.The method is applied for a homogeneous model of fermentation with volume, substrate, biomass, and product concentrations as the state variables. Computer study shows considerable simplicity of the method, its high accuracy for low order of approximation, and efficient convergence.     

Deterministic interpretation of the temperature response of microbial growth

A deterministic approximation of the thermal deactivation process is undertaken by structuring growth kinetics of cell populations on the activity level. Growth data of the green microalga Scenedesmus obliquus under forced periodic conditions of light intensity have been used for model verification. The model considers a straightforward extension of normal growth kinetics in such a way that different growth activities can be adjusted in the context of a thermal damage and repair mechanism. The nonlinear fitting of the measured response renders parameter values as a function of the light availability per unit biomass.     

Potent anti-amyloidogenic and fibril-destabilizing effects of polyphenols in vitro: implications for the prevention and therapeutics of Alzheimer's disease

Cerebral deposition of amyloid beta-peptide (Abeta) in the brain is an invariant feature of Alzheimer's disease (AD). A consistent protective effect of wine consumption on AD has been documented by epidemiological studies. In the present study, we used fluorescence spectroscopy with thioflavin T and electron microscopy to examine the effects of wine-related polyphenols (myricetin, morin, quercetin, kaempferol (+)-catechin and (-)-epicatechin) on the formation, extension, and destabilization of beta-amyloid fibrils (fAbeta) at pH 7.5 at 37 degrees C in vitro. All examined polyphenols dose-dependently inhibited formation of fAbeta from fresh Abeta(1-40) and Abeta(1-42), as well as their extension. Moreover, these polyphenols dose-dependently destabilized preformed fAbetas. The overall activity of the molecules examined was in the order of: myricetin = morin = quercetin > kaempferol > (+)-catechin = (-)-epicatechin. The effective concentrations (EC50) of myricetin, morin and quercetin for the formation, extension and destabilization of fAbetas were in the order of 0.1-1 micro m. In cell culture experiments, myricetin-treated fAbeta were suggested to be less toxic than intact fAbeta, as demonstrated by 3-[4,5-dimethylthiazol-2-yl]-2,5-diphenyltetrazolium bromide assay. Although the mechanisms by which these polyphenols inhibit fAbeta formation from Abeta, and destabilize pre-formed fAbetain vitro are still unclear, polyphenols could be a key molecule for the development of preventives and therapeutics for AD.     

Analysis of the quasi-steady-state approximation for an enzymatic one-substrate reaction

The validity of the quasi-steady state approximation for the calculation of the rate function of an isolated enzyme reaction is analysed by a detailed consideration of the time dependent process. For the characterization of the deviations of the real motion from the quasi-stationary state three kinds of error functions are used, the relaxation deficit, the relative relaxation time and the relaxation error. An improved approximation procedure is developed to calculate the transient states of the system. The maximum distance of the original motion from the quasi-stationary states is estimated by a general method. By consideration of different enzyme and substrate concentrations as well as different kinetic constants those parameter regions have been determined, where the errors of the quasi-steady state approximation do not exceed tolerated values. It is suggested how the methods can be applied to metabolic pathways.     

Multipole approach to the inverse problem in electrocardiology: Convergence of the multipole equivalent generator on the inhomogeneous body conductor

The multipole approach to the inverse electrocardiological problem consists of estimating the multipole components of the cardiac electric generator, starting from the measured body surface potential. This paper presents a critical investigation of the basic premise for the applicability of the multipole approach, namely the convergence of the multipole equivalent generator for the heart on the surface of an inhomogeneous body conductor. As an extension to multipole theory, a criterion for the convergence is derived. Based on realistic models for the body conductor and the cardiac electric generator, we observe that the criterion is not strictly satisfied in realistic conditions. Numerical simulations with the same models point out that the multipole equivalent generator is indeed not convergent in the strict mathematical sense. On the other hand, we show that the multipole equivalent generator yields a rather close approximation of the electrocardiological potential for intermediate values of the order of the multipole generator. A discussion is given on how to explain the apparently ambiguous results for the estimation of cardiac multipole components.     

An approximation to the likelihood for a pedigree with loops

This paper presents a new approximation to the likelihood for a pedigree with loops, based on cutting all loops and extending the pedigree at the cuts. An opimum loop-cutting strategy and an iterative extension technique are presented. The likelihood for a pedigree with loops is then approximated by the conditional likelihood for the entire cut-extended pedigree given the extended part. The approximate likelihoods are compared with the exact likelihoods obtained using the program MENDEL for several small pedigrees with loops. The approximation is efficient for large pedigrees with complex loops in terms of computing speed and memory requirements.     

Ecomorphological adaptation to oxygen deficiency in Amazon floodplains by serrasalmid fish of the genus Mylossoma

Serrasalmids of the genus Mylossoma are obligate gill-breathers that are encountered in the floodplain lakes of Amazonia, even when the oxygen concentrations there are below 0^.5 mg l⁻¹. It was shown by experiments that fish of these species are capable of utilizing the oxygen-rich surface layer of the water for respiration, in order to survive periods of habitat-induced hypoxia. This so-called aquatic surface respiration entails an increase in locomotory activity and an ecomorphosis involving the formation of a dermal extension on the lower jaw, that apparently has a hydrodynamic function for using the surface layer for gill respiration; when the water is aerated, it retrogresses to its original size. Histological examination showed that the extension is formed by edematous processes in the stratum spongiosum.     

Assessment of the validity of the Adams and Fujita approximation for the higher oligomers of human spectrin

Analysing the self-association behaviour of human erythrocyte spectrin is complicated by a large degree of nonideality. Adams and Fujita [1] proposed that, as a first order approximation, the logarithm of the activity coefficient of the protomer of a self-associating system can be considered to be linearly dependent on the total concentration of the protein, and that the same second virial coefficient could be considered to apply to all species. As a consequence of the Adams and Fujita approximation, the apparent equilibrium constant is equal to the thermodynamic equilibrium constant. The equilibrium concentrations at 30°C of each oligomer spectrin species up to the 14-mer were determined after electrophoresis at low temperature. An apparent equilibrium constant for forming tetramer (K_2,4) of (1.2 ± 0.1) × 10⁶ l/mol was obtained, a value of (9.4 ± 0.7) × 10⁴ l/mol was obtained for K_4,6 and for all reactions forming oligomers higher than the hexamer an average approximate value of (2.7 ± 0.4) × 10⁵ l/mol was obtained. The apparent equilibrium constants for the formation of all oligomer species of spectrin up to the tetrakaidecamer (14-mer) remain relatively independent of total spectrin concentration, and indicate that within the precision of the measurements a single virial coefficient is sufficient to account for the nonideality of spectrin self-association over the range 2–42 g/l, thus further justifying the use of the Adams and Fujita approximation for this protein over this concentration range.     

Validity of the rapid buffering approximation near a point source of calcium ions.

In the presence of rapid buffers the full reaction-diffusion equations describing Ca2+ transport can be reduced using the rapid buffering approximation to a single transport equation for [Ca2+]. Here we simulate the full and reduced equations, exploring the conditions necessary for the validity of the rapid buffering approximation for an isolated Ca2+ channel or a cluster of channels. Using a point source and performing numerical simulations of different durations, we quantify the error of the rapid buffering approximation as a function of buffer and source parameters as well as the time and spatial scale set by the resolution of confocal microscopic measurements. We carry out simulations of Ca2+ "sparks" and "puffs," both with and without the indicator dye Ca2+ Green-1, and find that the rapid buffering approximation is excellent. These calculations also show that the traditional calculation of [Ca2+] from a fluorescence signal may grossly underestimate the true value of [Ca2+] near a source. Finally, we use the full model to simulate the transient Ca2+ domain near the pore of an open Ca2+ channel in a cell dialyzed with millimolar concentrations of 1,2-bis(2-aminophenoxy)ethane-N,N,N,N-tetraacetic acid or EGTA. In this regime, where the rapid buffering approximation is poor. Neher's equation for the steady-state Ca2+ profile is shown to be a reliable approximation adjacent to the pore.     

On the subnormal variance in the counts of randomly distributed particles: II. Exact treatment of the one-dimensional case

The effect of finite particle size on the standard deviation in sample counts is computed for the one-dimensional case. To a first order of approximation the correction is found to be identifical with that found by H. de Vries (1953) using a general approximation method.     

Behavioural discrimination of oilseed rape volatiles by the honeybee Apis mellifera L.

This study investigated the ability of the honeybee to discriminatebetween six compounds previously identified as oilseed rapefloral volatiles: linalool, 2-phenylethanol, methyl salicylate,benzyl alcohol, (E)-2-hexenal and 1-octen-3-ol. These componentswere tested individually or in a synthetic mixture for theirability to elicit the conditioned proboscis extension response.Three experiments were done: conditioning to the mixture atone concentration (1.0, 0.1 or 0.01 µg per component)and testing to the individual components either at the conditioningor at the other concentrations; conditioning to individual componentsand testing to the mixture (1.0 µg); conditioning andtesting to the individual components (1.0 µg). The resultsfrom the proboscis extension assay were then compared to thoseobtained from free-flying bees in a flight room. From the conditionedproboscis extension assay, a conditioning threshold level wasfound (0.1 µg for the mixture studied) below which a reliableconditioning could not be achieved. Recognition thresholds wereobserved: bees responded to concentrations ten-fold higher orlower than that used for conditioning. Responses to lower concentrationswere weaker, whilst responses were increased at concentrationshigher than the conditioning one. A hierarchy within the compoundstested was found, with linalool, 2-phenylethanol and methylsalicylate cueing mixture recognition more effectively thanthe other components. The ranking order of the six componentswas similar in both the conditioned proboscis extension andthe conditioned foraging behaviour in flight room. When conditionedand tested to the individual components, bees discriminateda learned odour from a number of others. However, the specificitylevel for the recognition of the learned odour varied accordingto the component, the most clearly discriminated being the compoundswhich were used by the bee in mixture recognition.