The tricarboxylic acid cycle in Dictyostelium discoideum. I. Formulation of alternative kinetic representations.

Enzyme systems within living cells have recently been shown to be highly ordered structures that violate classic assumptions of the Michaelis-Menten formalism, which originally was developed for the characterization of isolated reactions in vitro. This evidence suggests that a thorough examination of alternative kinetic formalisms for integrated biochemical systems is in order. The purpose of this series of papers is to assess the utility of an alternative power-law formalism by carrying out a detailed comparative analysis of a relatively large, representative system--the tricarboxylic acid cycle of Dictyostelium discoideum. This system was chosen because considerable experimental information already has been synthesized into a detailed kinetic model of the intact system. In this first paper, we set the stage for subsequent analysis within the framework of the power-law formalism: we review the underlying theory, emphasizing recent developments, formulate the model in terms that are convenient for the analysis to follow, and develop the system representation in both the Michaelis-Menten and power-law forms. In the second paper (Shiraishi, F., and Savageau, M. A. (1992) J. Biol. Chem. 267, 22919-22925), these alternative representations are shown to be internally consistent and locally equivalent. The third paper (Shiraishi, F., and Savageau, M. A. (1992) J. Biol. Chem. 267, 22926-22933) provides a complete analysis of the steady state behavior and also treats the dynamic behavior of the model.     

Roles of protein ubiquitination and degradation kinetics in biological oscillations

Protein ubiquitination and degradation play important roles in many biological functions and are associated with many human diseases. It is well known that for biochemical oscillations to occur, proper degradation rates of the participating proteins are needed. In most mathematical models of biochemical reactions, linear degradation kinetics has been used. However, the degradation kinetics in real systems may be nonlinear, and how nonlinear degradation kinetics affects biological oscillations are not well understood. In this study, we first develop a biochemical reaction model of protein ubiquitination and degradation and calculate the degradation rate against the concentration of the free substrate. We show that the protein degradation kinetics mainly follows the Michaelis-Menten formulation with a time delay caused by ubiquitination and deubiquitination. We then study analytically how the Michaelis-Menten degradation kinetics affects the instabilities that lead to oscillations using three generic oscillation models: 1) a positive feedback mediated oscillator; 2) a positive-plus-negative feedback mediated oscillator; and 3) a negative feedback mediated oscillator. In all three cases, nonlinear degradation kinetics promotes oscillations, especially for the negative feedback mediated oscillator, resulting in much larger oscillation amplitudes and slower frequencies than those observed with linear kinetics. However, the time delay due to protein ubiquitination and deubiquitination generally suppresses oscillations, reducing the amplitude and increasing the frequency of the oscillations. These theoretical analyses provide mechanistic insights into the effects of specific proteins in the ubiquitination-proteasome system on biological oscillations.     

Antagonism and bistability in protein interaction networks

A protein interaction network (PIN) is a set of proteins that modulate one another's activities by regulated synthesis and degradation, by reversible binding to form complexes, and by catalytic reactions (e.g., phosphorylation and dephosphorylation). Most PINs are so complex that their dynamical characteristics cannot be deduced accurately by intuitive reasoning alone. To predict the properties of such networks, many research groups have turned to mathematical models (differential equations based on standard biochemical rate laws, e.g., mass-action, Michaelis-Menten, Hill). When using Michaelis-Menten rate expressions to model PINs, care must be exercised to avoid making inconsistent assumptions about enzyme-substrate complexes. We show that an appealingly simple model of a PIN that functions as a bistable switch is compromised by neglecting enzyme-substrate intermediates. When the neglected intermediates are put back into the model, bistability of the switch is lost. The theory of chemical reaction networks predicts that bistability can be recovered by adding specific reaction channels to the molecular mechanism. We explore two very different routes to recover bistability. In both cases, we show how to convert the original 'phenomenological' model into a consistent set of mass-action rate laws that retains the desired bistability properties. Once an equivalent model is formulated in terms of elementary chemical reactions, it can be simulated accurately either by deterministic differential equations or by Gillespie's stochastic simulation algorithm.     

Dynamic Disorder in Quasi-Equilibrium Enzymatic Systems

Conformations and catalytic rates of enzymes fluctuate over a wide range of timescales. Despite these fluctuations, there exist some limiting cases in which the enzymatic catalytic rate follows the macroscopic rate equation such as the Michaelis-Menten law. In this paper we investigate the applicability of macroscopic rate laws for fluctuating enzyme systems in which catalytic transitions are slower than ligand binding-dissociation reactions. In this quasi-equilibrium limit, for an arbitrary reaction scheme we show that the catalytic rate has the same dependence on ligand concentrations as obtained from mass-action kinetics even in the presence of slow conformational fluctuations. These results indicate that the timescale of conformational dynamics – no matter how slow – will not affect the enzymatic rate in quasi-equilibrium limit. Our numerical results for two enzyme-catalyzed reaction schemes involving multiple substrates and inhibitors further support our general theory.     

Allosteric regulation of platelet actomyosin

The kinetic properties of platelet actomyosin have been examined to understand the mode of hydrolysis of its substrate ATP. In the presence of divalent cations, ATP hydrolysis deviated from Michaelis-Menten kinetics in such a way as to indicate cooperative effects, with a sigmoidal velocity $\text{vs.}$ substrate curve and a Hill slope of 2.4. In the absence of added divalent cations, linear Michaelis-Menten kinetics were obtained and the Hill slope reduced to 1.0. These results indicate an allosteric regulatory site on platelet actomyosin.     

WinBEST-KIT: Windows-based biochemical reaction simulator for metabolic pathways

We have implemented an efficient, user-friendly biochemical reaction simulator called Web-based BEST-KIT (Biochemical Engineering System analyzing Tool-KIT) for analyzing large-scale nonlinear networks such as metabolic pathways. Users can easily design and analyze an arbitrary reaction scheme through the Internet and an efficient graphical user interface without considering the mathematical equations. The reaction scheme can include several reaction types, which are represented by both the mass action law (mass balance) and approximated velocity functions of enzyme kinetics at steady state, such as Michaelis-Menten, Hill cooperative, Competitive inhibition. However, since all modules in Web-based BEST-KIT have been developed in Java applet style, users cannot optionally make use of original mathematical equations in addition to the prepared equations. In the present study, we have developed a new version of BEST-KIT (for Microsoft Windows called WinBEST-KIT) to allow users to define original mathematical equations and to customize these equations very easily as user-defined reaction symbols. The following powerful system-analytical methods are prepared for system analysis: time-course calculation, parameter scanning, estimation of the values of unknown kinetic parameters based on experimentally observed time-course data of reactants, dynamic response of reactants against virtual external perturbations, and real-time simulation (Virtual Dry Lab).     

Kinetic properties of an inulosucrase from Lactobacillus reuteri 121

Inulosucrases catalyze transfer of a fructose moiety from sucrose to a water molecule (hydrolysis) or to an acceptor molecule (transferase), yielding inulin. Bacterial inulin production is rare and a biochemical analysis of inulosucrase enzymes has not been reported. Here we report biochemical characteristics of a purified recombinant inulosucrase enzyme from Lactobacillus reuteri. It displayed Michaelis-Menten type of kinetics with substrate inhibition for the hydrolysis reaction. Kinetics of the transferase reaction is best described by the Hill equation, not reported before for these enzymes. A C-terminal deletion of 100 amino acids did not appear to affect enzyme activity or product formation. This truncated form of the enzyme was used for biochemical characterization.     

A comparison of variant theories of intact biochemical systems. I. Enzyme-enzyme interactions and biochemical systems theory

The need for a well-structured theory of intact biochemical systems becomes increasingly evident as one attempts to integrate the vast knowledge of individual molecular constituents, which has been expanding for several decades. In recent years, several apparently different approaches to the development of such a theory have been proposed. Unfortunately, the resulting theories have not been distinguished from each other, and this has led to considerable confusion with numerous duplications and rediscoveries. Detailed comparisons and critical tests of alternative theories are badly needed to reverse these unfortunate developments. In this paper we (1) characterize a specific system involving enzyme-enzyme interactions for reference in comparing alternative theories, and (2) analyze the reference system by applying the explicit S-system variant within biochemical systems theory (BST), which represents a fundamental framework based upon the power-law formalism and includes several variants. The results provide the first complete and rigorous numerical analysis within the power-law formalism of a specific biochemical system and further evidence for the accuracy of the explicit S-system variant within BST. This theory is shown to represent enzyme-enzyme interactions in a systematically structured fashion that facilitates analysis of complex biochemical systems in which these interactions play a prominent role. This representation also captures the essential character of the underlying nonlinear processes over a wide range of variation (on average 20-fold) in the independent variables of the system. In the companion paper in this issue the same reference system is analyzed by other variants within BST as well as by two additional theories within the same power-law formalism--flux-oriented and metabolic control theories. The results show how all these theories are related to one another.     

Anomeric specificity and kinetics of glucokinase: theoretical unsuitability of the Hill equation

The kinetics of the low-Km hexokinase isoenzymes, which obey the Michaelis-Menten equation, can be established from the Km (Michaelis constant) and Vmax (maximal velocity) values for either equilibrated D-glucose or its alpha- and beta-anomers. In the case of the high-Km glucokinase isoenzyme, however, the sigmoidal substrate dependency and the competition between the two anomers of D-glucose do not allow, theoretically, to assign any meaningful value to either the Km, Vmax or n (Hill number) constants for equilibrated D-glucose. Thus, with equilibrated D-glucose, the concentration dependency fails to display a rectilinear relationship in the Hill plot. These observations illustrate the shortcomings of current biochemical studies in which the anomeric heterogeneity of D-glucose is ignored.     

Multiple or multiphasic uptake mechanisms in plants?

Abstract Borstlap (1983) has alleged that (a) there are no abrupt changes in curves for the concentration dependence of solute uptake in plants, and (b) many such uptake isotherms may be described by the sum of two Michaelis-Menten terms and a linear term. These claims are considered in detail in connection with the recent finding (Nandi, Pant & Nissen, 1987) that phosphate uptake by corn roots increased more rapidly within the higher phases, i.e. at high external phosphate concentrations, but also levelled off faster than predicted from Michaelis-Menten kinetics. Similar deviations are, in retrospect, also found for uptake of other solutes and result in fewer phases at high external solute concentrations. The simplified and strikingly similar multiphasic patterns in the present paper show that (a) the abrupt changes in published isotherms are not due to error in the data, and (b) uptake isotherms cannot be adequately represented by the sum of two Michaelis-Menten terms and a linear term, or by similar continuous functions, if sufficiently detailed and precise data are used. These findings are not consistent with the existence of multiple uptake mechanisms, including free diffusion, in the plasmalemma. Uptake occurs instead by a single, multiphasic mechanism for each solute or group of related solutes. The similarities in the multiphasic patterns indicate, furthermore, that influx of the various solutes may be coupled.     

Formal modeling of approximate relations in biochemical systems

Aiming to satisfy the need for the formalization of semiquantitative reasoning in the analysis of biochemical systems, the O[M] formalism for reasoning with orders of magnitude and approximate relations has been developed. It is based on seven primitive relations among quantities and compound relations which are formed as implicit disjunctions of primitives. O[M] can perform inferences by using formal approximate relations, algebraic equations, inequalities, if-then rules, assumptions, and goals. The applications discussed include Michaelis-Menten kinetics, different modes of inhibition of an enzymatic reaction, analysis of fluxes in biochemical networks, and identification of rate-limiting steps of biochemical pathways. In these applications, O[M] provides a medium for acquisition and formalization of previously informal concepts, analysis of systems at the order-of-magnitude level of detail, and automation of commonsense reasoning.     

Steady-state kinetics of enzyme-catalyzed copolymerization on primers

The theory of the steady-state kinetics of irreversible enzyme-catalyzed homopolymerization and copolymerization on primers has been developed. The rate law for homopolymerization is of the Michaelis-Menten form, but the kinetic parameters depend on primer concentration. Copolymerization has been treated for two monomers considering both terminal and penultimate effects and for four monomers considering terminal effects. The composition equations and conditional probabilities for monomer succession are identical for enzymatic and nonenzymatic processes, because the steady-state approximation is used in both cases. The reactivity ratios and steady-state velocities are different, however. Examination of published results for AU and UG copolymers synthesized by polynucleotide phosphorylase permits evaluation of reactivity ratios for the AU copolymer and indicates that penultimate effects may be operative in both cases.     

Steady-states of receptor-ligand dynamics: a theoretical framework

This paper studies aspects of the dynamics of a conventional mechanism of ligand-receptor interactions, with a focus on the stability and location of steady-states. A theoretical framework is developed, which is based upon the rich and deep formalism of irreducible biochemical networks. When represented in this manner, the mass action kinetics of biochemical processes can be clearly seen in terms of their component biochemical interactions, their kinetic rate constants, and the stoichiometry for the system. A minimal parametrization is provided for models for two- or multi-state receptor interaction with ligand, and an "affinity quotient" is introduced, which allows an elegant classification of ligands into agonists, neutral agonists, and inverse agonists.     

A modified distance variable for the Hill formalism for kinetic models of muscle contraction

The distance variable of the Hill formalism for kinetic models of muscle contraction is compared to a modified distance variable. Instead of measuring the distance from a fixed point on the myosin filament to a neighboring actin, the modified variable measures the deviation of the myosin cross-bridge from its equilibrium position. Although for attached cross-bridges the two definitions are equivalent, the new variable is an index of cross-bridge conformation for cross-bridges of all states. The modified variable may be used to complement the use of the Hill variable, or to replace it. The utility of the modified variable is illustrated by an example which matches cross-bridge structures to biochemical kinetic data and to the free energy functions necessary for the design of a kinetic model.     

Steps towards a mechanistic understanding of respiratory temperature responses

Temperature crucially affects the speed of metabolic processes in poikilotherm organisms, including plants. The instantaneous temperature responses of O(2)-reduction and CO(2)-release can be approximated by Arrhenius kinetics, even though respiratory gas exchange of plants is the net effect of many constituent biochemical processes. Nonetheless, the classical Arrhenius equation must be modified to account for a dynamic response to measurement temperatures. We show that this dynamic response is readily explained by combining Arrhenius and Michaelis-Menten kinetics, as part of a fresh appraisal of metabolic interpretations of instantaneous temperature responses. In combination with recent experimental findings, we argue that control of mitochondrial electron flow is shared among cytochrome oxidase and alternative oxidase under in vivo conditions, and is continuously coordinated. In this way, upstream carbohydrate metabolism and downstream electron transport appear to be optimized according to the demand of ATP, TCA-cycle intermediates and anabolic reducing power under differing metabolic states. We provide a link to the 'Growth and Maintenance Paradigm' of respiration and argue that respiratory temperature responses can be used as a tool to probe metabolic states of plant tissue, such that we can learn more about the mechanisms that govern longer-term acclimatization responses of plant metabolism.     

An easy method for the determination of initial rates.

When the Michaelis-Menten equation is obeyed, the rate near the beginning of an enzyme-catalyzed reaction (or of an experiment on transport) can be found accurately from the slope of a chord joining two points on the progress curve. This slope gives the rate at an intermediate concentration. Exact values of this intermediate concentration are easily calculated from equations in the text, and a number of values have also been tabulated. Methods of using two chords to find the initial rate are given. A mid-point formula for numerical differentiation is advocated when the Michaelis-Menten equation does not hold.     

Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates

This paper derives new results for certain classes of chemical reaction networks, linking structural to dynamical properties. In particular, it investigates their monotonicity and convergence under the assumption that the rates of the reactions are monotone functions of the concentrations of their reactants. This is satisfied for, yet not restricted to, the most common choices of the reaction kinetics such as mass action, Michaelis-Menten and Hill kinetics. The key idea is to find an alternative representation under which the resulting system is monotone. As a simple example, the paper shows that a phosphorylation/dephosphorylation process, which is involved in many signaling cascades, has a global stability property. We also provide a global stability result for a more complicated example that describes a regulatory pathway of a prevalent signal transduction module, the MAPK cascade.     

Strategies for representing metabolic pathways within biochemical systems theory: reversible pathways

The search for systematic methods to deal with the integrated behavior of complex biochemical systems has over the past two decades led to the proposal of several theories of biochemical systems. Among the most promising is biochemical systems theory (BST). Recent comparisons of this theory with several others that have recently been proposed have demonstrated that all are variants of BST and share a common underlying formalism. Hence, the different variants can be precisely related and ranked according to their completeness and operational utility. The original and most fruitful variant within BST is based on a particular representation, called an S-system (for synergistic and saturable systems), that exhibits many advantages not found among alternative representations. Even within the preferred S-system representation there are options, depending on the method of aggregating fluxes, that become especially apparent when one considers reversible pathways. In this paper we focus on the paradigm situation and clearly distinguish the two most common strategies for generating an S-system representation. The first is called the "reversible" strategy because it involves aggregating incoming fluxes separately from outgoing fluxes for each metabolite to define a net flux that can be positive, negative, or zero. The second is the "irreversible" strategy, which involves aggregating forward and reverse fluxes through each reaction to define a net flux that is always positive. This second strategy has been used almost exclusively in all variants of BST. The principal results of detailed analyses are the following: (1) All S-system representations predict the same changes in dependent concentrations for a given change in an independent concentration. (2) The reversible strategy is superior to the irreversible on the basis of several criteria, including accuracy in predicting steady-state flux, accuracy in predicting transient responses, and robustness of representation. (3) Only the reversible strategy yields a representation that is able to capture the characteristic feature of amphibolic pathways, namely, the reversal of nets flux under physiological conditions. Finally, the results document the wide range of variation over which the S-system representation can accurately predict the behavior of intact biochemical systems and confirm similar results of earlier studies [Voit and Savageau, Biochemistry 26: 6869-6880 (1987)].     

Mass-transfer limitations for immobilized enzyme-catalyzed kinetic resolution of racemate in a fixed-bed reactor

A mathematical model has been developed for immobilized enzyme-catalyzed kinetic resolution of racemate in a fixed-bed reactor in which the enzyme-catalyzed reaction (the irreversible uni-uni competitive Michaelis-Menten kinetics is chosen as an example) was coupled with intraparticle diffusion, external mass transfer, and axial dispersion. The effects of mass-transfer limitations, competitive inhibition of substrates, deactivation on the enzyme effective enantioselectivity, and the optical purity and yield of the desired product are examined quantitatively over a wide range of parameters using the orthogonal collocation method. For a first-order reaction, an analytical solution is derived from the mathematical model for slab-, cylindrical-, and spherical-enzyme supports. Based on the analytical solution for the steady-state resolution process, a new concise formulation is presented to predict quantitatively the mass-transfer limitations on enzyme effective enantioselectivity and optical purity and yield of the desired product for a continuous steady-state kinetic resolution process in a fixed-bed reactor.