Quasi-equilibrium assumption for arbitrary mechanism of enzymatic reaction. criteria for existence of equilibrium segments

The possible application of the quasi-equilibrium assumption for an arbitrary mechanism of enzymatic reaction is considered. It is shown at what ratios of kinetic constants a segment consisting of two, three, and four intermediates may be considered as an equilibrium one. Expressions for evaluation of accuracy of distribution of intermediate concentrations inside the equilibrium segment and accuracy of determination of intermediate concentrations inside and outside the equilibrium segment as a function of the ratio of kinetic constants are derived. A method for determination of the limitations on the ratio of rate constants for an equilibrium segment of arbitrary structure is suggested.     

The accuracy of the rapid equilibrium assumption in steady-state enzyme kinetics in the case of a multipath arbitrary enzyme mechanism with a number of equilibrium segments

A quantitative evaluation of the accuracy of the rapid-equilibrium assumption in steady-state enzyme kinetics was obtained for a multipath arbitrary enzyme mechanism with a number of equilibrium segments. Explicit expressions for estimating the contribution of any equilibrium segment to the accuracy of the rapid-equilibrium assumption were obtained. This allowed us to determine the accuracy of the rapid-equilibrium assumption (Δ) in general: 1 + Δ = (1 + Δ₁)(1 + Δ₂)... (1 + Δ_k), where Δ₁, Δ₂,..., Δ_k is the contribution of each individual equilibrium segment. The accuracy depends only on the structure and properties of equilibrium segments, which have been accounted for in the rapid-equilibrium assumption, but it is independent of the number of paths in the mechanism of the enzymatic reaction and on the structure and properties of the remaining part (steady-state) of the kinetic scheme.     

Quasi-equilibrium assumption in enzyme kinetics. Necessary and sufficient conditions and accuracy of its application for single-substrate reactions

Steady-state kinetics of compulsory-ordered single-substrate irreversible and reversible enzyme reactions with two, three, and arbitrary number of intermediates were observed. Necessary and sufficient conditions for application of the quasi-equilibrium assumption and restrictions of this assumption were found in cases of two and three intermediates in the equilibrium segment. For all cases, accuracy of the quasi-equilibrium assumption was evaluated.     

Relations between biochemical thermodynamics and biochemical kinetics

The parameters in steady-state or rapid-equilibrium rate equations for enzyme-catalyzed reactions depend on the temperature, pH, and ionic strength, and may depend on the concentrations of specific species in the buffer. When the complete rate equation (i.e. the equation with parameters for the reverse reaction as well as the forward reaction) is determined, there are one or more Haldane relations between some of the kinetic parameters and the apparent equilibrium constant for the reaction that is catalyzed. When the apparent equilibrium constant can be calculated from the kinetic parameters, the equilibrium composition can be calculated. This is remarkable because the kinetic parameters all depend on the properties of the enzymatic site, but the apparent equilibrium constant and the equilibrium composition do not. The effects of ionic strength and pH on the unoccupied enzymatic site and the occupied enzymatic site have to cancel in the Haldane relation or in the calculation of the apparent equilibrium constant using the rate constants for the steps in the mechanism. Several simple enzymatic mechanisms and their complete rate equations are discussed.     

On the validity of using semilogarithmic plots to determine initial velocities of enzyme-catalyzed reactions

It has been proposed (Johnston & Diven, 1969a) that it is valid to use semilogarithmic (first-order) plots of the extent of reaction versus time for graphic determination of initial velocities of enzyme-catalyzed reactions. This proposition suggests the assumption that the initial velocity error expected to be introduced by the proposed procedure is smaller than or comparable to the error introduced by the customary graphic procedure. The latter is based on the assumption that the progress curves of enzyme-catalyzed reactions have an initial linear segment of sufficient duration to permit accurate determination of slope. The validity of the procedure proposed by Johnston and Diven is examined in this report. It is concluded that the procedure is applicable to a very small class of enzyme-catalyzed reactions and only under certain experimental conditions.     

Analyzing the Fitch Method for Reconstructing Ancestral States on Ultrametric Phylogenetic Trees

The accuracy of the Fitch method for reconstructing ancestral states on ultrametric phylogenetic trees is studied. Two recurrence relations for computing the accuracy are given here. Using these relations, we analyze the convergence of the accuracy of the Fitch method for reconstructing the root state on a complete binary tree of 2 ⁿ leaves as n goes to infinity, present a closed-form formula for the accuracy on ultrametric comb trees, and provide a lower bound on the accuracy on arbitrary ultrametric phylogenetic trees.     

Competition-dependent modelling of foliage biomass in forest stands

Currently, foliage biomass is estimated based on stem diameter or basal area. However, it is questionable whether the relations between foliage and stem observed from plantations of a single tree species can be applied to stands of different structure or species composition. In this paper, a procedure is presented to simulate foliage and branch biomass of tree crowns relative to crown size and light competition. Crowns are divided into layers and segments and each segment is divided into a foliated and an unfoliated fraction. Depending on the competitive status of the segment, leaf area density, specific leaf area and foliated branch fraction are determined. Based on this information, foliage biomass is calculated. The procedure requires a crown shape function and a measure to characterise competition for light and space of each individual segment within the canopy. Simple solutions are suggested for both requirements to enable an application with data that can be measured non-destructively in the field; these were stem position, tree height, crown base height, crown radii and some general crown shape information. The model was parameterised from single trees of Norway spruce and European beech and partly evaluated with independent data close to the investigation plot. Evaluations showed that the model can attribute the ecology of the different crown forms. Modelled foliage distribution for beech and spruce as well as total needle biomass of spruce agreed well with measurements but foliage biomass of beech was underestimated. The results are discussed in the context of a general model application in structured forests.     

Energetics of enzyme catalysis. I. Isotopic experiments, enzyme interconversion, and oversaturation

An enzyme-catalyzed interconversion of one substrate, S, and one product P, by an enzyme that exists in two forms E1 and E2 where E1 binds S and E2 binds P, is considered S + E1 in equilibrium E1S in equilibrium E2P in equilibrium E2 + P. Under reversible conditions (where the concentrations of S and P are not far removed from their equilibrium values) it is shown that, in addition to the usual unsaturated and saturated behaviour there exists a third regime at high substrate concentration: the oversaturated region. In this region, the rate-limiting transition state is the interconversion of the unliganded forms of the enzyme: E1 and E2. Expressions for six different experiments involving deuterium, tritium and 14C labels are presented. By considering the results from these experiments, the nature and importance of the enzyme interconversion steps can be elucidated.     

Improvement of the method of sap flow rate determination in full-grown trees based on heat balance with direct electric heating of xylem

Several improvements of the method of sap flow rate determination in full-grown trees elaborated by the authors are described. The method is based on heat balance with direct electrical heating of the xylem. The accuracy of the relationship for the calculation of flow rate was improved, the influence of unevennesses of flow in different annual rings was reduced. A method of extrapolation of the results obtained by measuring an individual xylem segment against the whole tree and of interpolation to a unit segment was elaborated, a simplified method of connecting the electrodes supplying the thermal input to the segment is described and an indicator of the relative water content of the xylem was derived.     

The complete pathway for ERK2-catalyzed reaction. Evidence for an iso random Bi Bi mechanism

In the present study, the enzymatic mechanism of ERK2 is re-examined by a combination of steady-state kinetic studies in the absence and presence of viscosogenic agents. Kinetic studies carried out in various concentrations of sucrose revealed that both k(cat) and k(cat)/K(m) for either ATP or EtsDelta138 were highly sensitive to solvent viscosity, suggesting that the rapid equilibrium assumption is not valid for the phosphorylation of protein substrate by ERK2. Furthermore, the kinetic analysis with the minimal random Bi Bi reaction mechanism is shown to be inconsistent with the principle of the detailed balance. This inconsistent calculation strongly suggests that there is isomerization of the enzyme-substrate ternary complex. The viscosity-dependent steady-state kinetic data are combined to establish a kinetic mechanism for the ERK2-catalyzed reaction that predicts initial reaction velocities under varying concentrations of ATP and substrate. These results complement previous structure-function studies of mitogen-activated protein kinases and provide important insight for mechanistic interpretation of the kinase functions.     

Applications of graph theory to enzyme kinetics and protein folding kinetics. Steady and non-steady-state systems

Graphic methods have proved to be very useful in enzyme kinetics, as reflected in both raising the efficiency of performing calculations and aiding in the analysis of catalytic mechanisms. The kinetic relations among protein folding states are very similar to those between enzyme-catalyzed species. Therefore, it should be equally useful to provide a visually intuitive relation between kinetic calculations and folding mechanisms for protein folding kinetics, as manifested by the graphic rules in enzyme kinetics. It can actually be anticipated that, due to increasing interest in protein folding, the graphic method will become an important tool in folding kinetics as well. Based on the recent progress made in graphic methods of enzyme kinetics, in this review four graphic rules are summarized, which can be used to deal with protein folding systems as well as enzyme-catalyzed systems. Rules 1-3 are established for deriving the kinetic equations for steady-state processes and Rule 4 for those in the case of non-steady-state processes. In comparison with conventional graphic methods, which can only be applied to a steady-state system, the current rules have the following advantages: (1) Complicated and tedious calculations can be greatly simplified. (2) A lot of wasted labor can be turned away. (3) Final results can be double-checked by a formula provided in each of the graphic rules. (4) Transient kinetic systems can also be treated. The mathematical proof of Rules 1-4 is given in appendices A-D, respectively.     

云南哀牢山6种常绿阔叶树木质部解剖特征的轴向和径向变化

West、Brown和Enquist提出的树木水分传导的分形网络模型(简称WBE模型)认为,树木连续分枝之间的导管或管胞直径按照一定的比率均匀变细,其总的水力阻力与水分传导的路径长度无关,从而使不同部位叶片获得基本相当的水分供应。该模型对树木高生长的水力限制假说提出了置疑。为了验证WBE模型中树木导管或管胞均匀变细的假说,该文研究了云南哀牢山中湿性常绿阔叶林中6种常绿阔叶树, 腾冲栲(Castanopsis wattii)、景东石砾(Lithocarpus chintungensis)、木果石砾(L. xylocarpus)、长尾青冈(Cyclobalanopsis stewardiana)、滇木荷(Schima noronhae)和舟柄茶(Hartia sinensis)木质部解剖特征随树高和年龄的变化。对这6个树种共14株样木进行了不同高度树干圆盘和边材生长轮取样,样木的高度为15~25 m,按照常规木材解剖的处理和分析方法,在显微镜下测定木材切片的导管直径和密度等特征。结果表明:在14株样木中,有4株树木导管直径随树木高度增加呈线性减小, 1株没有明显变化,其它9株树木导管直径在树冠以下的树干部分变化幅度较小或没有明显变化,而从树冠基部往上直到树木顶端导管直径显著减小。同一植株随着高度的增加,导管密度增加并且在树冠内增加更显著。有三分之一的树木导管占边材面积的比例随树高增加没有明显变化,其余树木导管占边材面积比在树冠以上有所减小。多数树木理论比导率在树冠以下没有明显变化而在树冠基部往上显著降低。在从髓芯开始往外的20~40个年轮范围内导管直径增加显著,但大部分植株导管直径在40个年轮后趋于稳定。不同高度圆盘导管直径随形成层发育时间的变化呈相似的趋势,并且相同发育年龄的导管直径没有明显差异。该文的研究结果说明,导管直径的轴向和径向变化一定程度上补偿了水分运输阻力随树木个体增大而增加的缺陷,但是6种常绿阔叶树树干的导管基本不按一定比率均匀变细,不支持WBE模型。     

Identification of enzyme inhibitory mechanisms from steady-state kinetics

Enzyme inhibitors are used in many areas of the life sciences, ranging from basic research to the combat of disease in the clinic. Inhibitors are traditionally characterized by how they affect the steady-state kinetics of enzymes, commonly analyzed on the assumption that enzyme-bound and free substrate molecules are in equilibrium. This assumption, implying that an enzyme-bound substrate molecule has near zero probability to form a product rather than dissociate, is valid only for very inefficient enzymes. When it is relaxed, more complex but also more information-rich steady-state kinetics emerges. Although solutions to the general steady-state kinetics problem exist, they are opaque and have been of limited help to experimentalists.Here we reformulate the steady-state kinetics of enzyme inhibition in terms of new parameters. These allow for assessment of ambiguities of interpretation due to kinetic scheme degeneracy and provide an intuitively simple way to analyze experimental data. We illustrate the method by concrete examples of how to assess scheme degeneracy and obtain experimental estimates of all available rate and equilibrium constants. We suggest simple, complementary experiments that can remove ambiguities and greatly enhance the accuracy of parameter estimation.     

Automatic extraction and measurement of individual trees from mobile laser scanning point clouds of forests

Background and AimsIn addition to terrestrial laser scanning (TLS), mobile laser scanning (MLS) is increasingly arousing interest as a technique which provides valuable 3-D data for various applications in forest research. Using mobile platforms, the 3-D recording of large forest areas is carried out within a short space of time. Vegetation structure is described by millions of 3-D points which show an accuracy in the millimetre range and offer a powerful basis for automated vegetation modelling. The successful extraction of single trees from the point cloud is essential for further evaluations and modelling at the individual-tree level, such as volume determination, quantitative structure modelling or local neighbourhood analyses. However, high-precision automated tree segmentation is challenging, and has so far mostly been performed using elaborate interactive segmentation methods.MethodsHere, we present a novel segmentation algorithm to automatically segment trees in MLS point clouds, applying distance adaptivity as a function of trajectory. In addition, tree parameters are determined simultaneously. In our validation study, we used a total of 825 trees from ten sample plots to compare the data of trees segmented from MLS data with manual inventory parameters and parameters derived from semi-automatic TLS segmentation.Key ResultsThe tree detection rate reached 96 % on average for trees with distances up to 45 m from the trajectory. Trees were almost completely segmented up to a distance of about 30 m from the MLS trajectory. The accuracy of tree parameters was similar for MLS-segmented and TLS-segmented trees.ConclusionsBesides plot characteristics, the detection rate of trees in MLS data strongly depends on the distance to the travelled track. The algorithm presented here facilitates the acquisition of important tree parameters from MLS data, as an area-wide automated derivation can be accomplished in a very short time.     

Kinetics of the transient-phase and steady-state of the monocyclic enzyme cascades

We present a kinetic analysis of the whole course of the reaction, that is, of both the transient-phase and steady-state, of monocyclic enzyme cascade systems. The equations for the rapid equilibrium conditions are obtained as a particular case of the general transient-phase equations. An analysis of the kinetic data allows the determination of the equilibrium and the rate constants if adequate experimental results are available. Finally, our results for the steady-state are compared with those obtained by other authors.     

The Quasi-Equilibrium Assumption for Bi-Bi Ordered Bisubstrate Enzymatic Reaction. How to Discriminate the Mechanism Correctly

Application of the quasi-equilibrium assumption for the steady-state kinetics of bisubstrate irreversible enzymat- ic reactions in the case of ordered binding of substrates (Bi-Bi ordered mechanism) is considered. The necessary and suffi- cient conditions for application of the quasi-equilibrium assumption have been found and accuracy of this assumption has been numerically evaluated. The limitations on application of the quasi-equilibrium assumption have been shown and errors of its application have been analyzed. It is shown that possible discrimination of substrate binding order using asymmetrical expressions grounded on the quasi-equilibrium assumption is inconsistent because such asymmetrical expressions arise from incorrect application of the quasi-equilibrium assumption. Moreover, it has been proved in the general case that mecha- nisms generating such substrate-asymmetrical expressions for the steady-state rate of enzymatic reaction do not exist. The error source when using graphical interpretation for discrimination of mechanisms of bisubstrate enzymatic reactions has been determined. The strategy to avoid such errors is pointed out.     

Accuracy and precision of species trees: effects of locus, individual, and base pair sampling on inference of species trees in lizards of the Liolaemus darwinii group (Squamata, Liolaemidae)

Molecular phylogenetics has entered a new era in which species trees are estimated from a collection of gene trees using methods that accommodate their heterogeneity and discordance with the species tree. Empirical evaluation of species trees is necessary to assess the performance (i.e., accuracy and precision) of these methods with real data, which consists of gene genealogies likely shaped by different historical and demographic processes. We analyzed 20 loci for 16 species of the South American lizards of the Liolaemus darwinii species group and reconstructed a species tree with *BEAST, then compared the performance of this method under different sampling strategies of loci, individuals, and sequence lengths. We found an increase in the accuracy and precision of species trees with the number of loci, but for any number of loci, accuracy substantially decreased only when using only one individual per species or 25% of the full sequence length (～ 147 bp). In addition, locus "informativeness" was an important factor in the accuracy/precision of species trees when using a few loci, but it became increasingly irrelevant with additional loci. Our empirical results combined with the previous simulation studies suggest that there is an optimal range of sampling effort of loci, individuals, and sequence lengths for a given speciation history and information content of the data. Future studies should be directed toward further assessment of other factors that can impact performance of species trees, including gene flow, locus "informativeness," tree shape, missing data, and errors in species delimitation.     

Bacillus subtilis chorismate mutase is partially diffusion-controlled.

The effect of viscosogens on the enzyme-catalyzed rearrangement of chorismate to prephenate has been studied. The steady-state parameters kcat and kcat/Km for the monofunctional chorismate mutase from Bacillus subtilis (BsCM) decreased significantly with increasing concentrations of glycerol, whereas the 'sluggish' BsCM mutants C75A and C75S were insensitive to changes in microviscosity. The latter results rule out extraneous interactions of the viscosogen as an explanation for the effects observed with the wild-type enzyme. Additional control experiments show that neither viscosogen-induced shifts in the pH-dependence of the enzyme-catalyzed reaction nor small perturbations of the conformational equilibrium of chorismate can account for the observed effects. Instead, BsCM appears to be limited by substrate binding and product release at low and high substrate concentrations, respectively. Analysis of the kinetic data indicates that diffusive transition states are between 30 and 40% rate-determining in these concentration regimes; the chemical step must contribute to the remaining kinetic barrier. The relatively low value of the 'on' rates for chorismate and prephenate (approximately 2 x 106 m-1.s-1) probably reflects the need for a rare conformation of the enzyme, the ligand, or both for successful binding. Interestingly, the chorismate mutase domain of the bifunctional chorismate mutase-prephenate dehydratase from Escherichia coli, which has steady-state kinetic parameters comparable to those of BsCM but has a much less accessible active site, is insensitive to changes in viscosity and the reaction it catalyses is not diffusion-controlled.     

Mitochondrial resolution of a deep branch in the genealogical tree for perching birds.

Animal mitochondrial DNA (mtDNA) is known to contain information about the genealogical relations among closely related species and is shown here to yield information about distant relations as well. Our results also draw attention to the need for caution in using third positions of codons for tree construction. This is evident from comparative studies of the cytochrome b gene in 13 species representing major groups within the order of perching birds (Passeriformes). Sequences of a 924 base-pair segment of this gene were obtained from each of these species via the polymerase chain reaction and a novel set of versatile primers. With a woodpecker sequence as an outgroup, trees that separate songbirds from other perching birds and resolve the ancient branch leading to songbirds were obtained utilizing the conservative first and second positions of codons. Analysis of positions within codons suggests that, for deep branches, the skewed base composition at the fast-changing third positions can result in phylogenetic disinformation, which conflicts with the information retained in the first and second positions. The mitochondrial tree shows broad concordance with that based on hybridization of nuclear DNA; however, parsimony and maximum likelihood methods suggest a close kinship between thrushes and Australian babblers, in agreement with the traditional morphological classification.     

Slow ligand binding kinetics dominate ferrous hexacoordinate hemoglobin reactivities and reveal differences between plants and other species

Hexacoordinate hemoglobins are found in many living organisms ranging from prokaryotes to plants and animals. They are named "hexacoordinate" because of reversible coordination of the heme iron by a histidine side chain located in the heme pocket. This endogenous coordination competes with exogenous ligand binding and causes multiphasic relaxation time courses following rapid mixing or flash photolysis experiments. Previous rapid mixing studies have assumed a steady-state relationship between hexacoordination and exogenous ligand binding that does not correlate with observed time courses for binding. Here, we demonstrate that this assumption is not valid for some hexacoordinate hemoglobins, and that multiphasic time courses are due to an appreciable fraction of pentacoordinate heme resulting from relatively small equilibrium constants for hexacoordination (K(H)). CO binding reactions initiated by rapid mixing are measured for four plant hexacoordinate hemoglobins, human neuroglobin and cytoglobin, and Synechocystis hemoglobin. The plant proteins, while showing a surprising degree of variability, differ from the others in having much lower values of K(H). Neuroglobin and cytoglobin display dramatic biphasic time courses for CO binding that have not been observed using other techniques. Finally, an independent spectroscopic quantification of K(H) is presented that complements rapid mixing for the investigation of hexacoordination. These results demonstrate that hexacoordination could play a much larger role in regulating affinity constants for ligand binding in human neuroglobin and cytoglobin than in the plant hexacoordinate hemoglobins.