Validation of fractal-like kinetic models by time-resolved binding kinetics of dansylamide and carbonic anhydrase in crowded media

Kinetic studies of biochemical reactions are typically carried out in a dilute solution that rarely contains anything more than reactants, products, and buffers. In such studies, mass-action-based kinetic models are used to analyze the progress curves. However, intracellular compartments are crowded by macromolecules. Therefore, we investigated the adequacy of the proposed generalizations of the mass-action model, which are meant to describe reactions in crowded media. To validate these models, we measured time-resolved kinetics for dansylamide binding to carbonic anhydrase in solutions crowded with polyethylene glycol and Ficoll. The measured progress curves clearly show the effects of crowding. The fractal-like model proposed by Savageau was used to fit these curves. In this model, the association rate coefficient k_a allometrically depends on concentrations of reactants. We also considered the fractal kinetic model proposed by Schnell and Turner, in which k_a depends on time according to a Zipf-Mandelbrot distribution, and some generalizations of these models. We found that the generalization of the mass-action model, in which association and dissociation rate coefficients are concentration-dependent, represents the preferred model. Other models based on time-dependent rate coefficients were inadequate or not preferred by model selection criteria.     

Synergistic cellulose hydrolysis can be described in terms of fractal-like kinetics

A fractal-like kinetics model was used to describe the synergistic hydrolysis of bacterial cellulose by Trichoderma reesei cellulases. The synergistic action of intact cellobiohydrolase Cel7A and endoglucanase Cel5A at low enzyme-to-substrate ratios showed an apparent substrate inhibition consistent with a case where two-dimensional (2-D) surface diffusion of the cellobiohydrolase is rate-limiting. The action of Cel7A core and Cel5A was instead consistent with a three-dimensional (3-D) diffusion-based mode of action. The synergistic action of intact Cel7A was far superior to that of the core at a high enzyme-to-substrate ratio, but this effect was gradually reduced at lower enzyme-to-substrate ratios. The apparent fractal kinetics exponent h obtained by nonlinear fit of hydrolysis data to the fractal-like kinetics analogue of a first-order reaction was a useful empirical parameter for assessing the rate retardation and its dependence on the reaction conditions.     

Accuracy of alternative representations for integrated biochemical systems

The Michaelis-Menten formalism often provides appropriate representations of individual enzyme-catalyzed reactions in vitro but is not well suited for the mathematical analysis of complex biochemical networks. Mathematically tractable alternatives are the linear formalism and the power-law formalism. Within the power-law formalism there are alternative ways to represent biochemical processes, depending upon the degree to which fluxes and concentrations are aggregated. Two of the most relevant variants for dealing with biochemical pathways are treated in this paper. In one variant, aggregation leads to a rate law for each enzyme-catalyzed reaction, which is then represented by a power-law function. In the other, aggregation produces a composite rate law for either net rate of increase or net rate of decrease of each system constituent; the composite rate laws are then represented by a power-law function. The first variant is the mathematical basis for a method of biochemical analysis called metabolic control, the latter for biochemical systems theory. We compare the accuracy of the linear and of the two power-law representations for networks of biochemical reactions governed by Michaelis-Menten and Hill kinetics. Michaelis-Menten kinetics are always represented more accurately by power-law than by linear functions. Hill kinetics are in most cases best modeled by power-law functions, but in some cases linear functions are best. Aggregation into composite rate laws for net increase or net decrease of each system constituent almost always improves the accuracy of the power-law representation. The improvement in accuracy is one of several factors that contribute to the wide range of validity of this power-law representation. Other contributing factors that are discussed include the nonlinear character of the power-law formalism, homeostatic regulatory mechanisms in living systems, and simplification of rate laws by regulatory mechanisms in vivo.     

Biochemical characterization of human S-nitrosohemoglobin. Effects on oxygen binding and transnitrosation.

S-Nitrosation of cysteine beta93 in hemoglobin (S-nitrosohemoglobin (SNO-Hb)) occurs in vivo, and transnitrosation reactions of deoxygenated SNO-Hb are proposed as a mechanism leading to release of NO and control of blood flow. However, little is known of the oxygen binding properties of SNO-Hb or the effects of oxygen on transnitrosation between SNO-Hb and the dominant low molecular weight thiol in the red blood cell, GSH. These data are important as they would provide a biochemical framework to assess the physiological function of SNO-Hb. Our results demonstrate that SNO-Hb has a higher affinity for oxygen than native Hb. This implies that NO transfer from SNO-Hb in vivo would be limited to regions of extremely low oxygen tension if this were to occur from deoxygenated SNO-Hb. Furthermore, the kinetics of the transnitrosation reactions between GSH and SNO-Hb are relatively slow, making transfer of NO+ from SNO-Hb to GSH less likely as a mechanism to elicit vessel relaxation under conditions of low oxygen tension and over the circulatory lifetime of a given red blood cell. These data suggest that the reported oxygen-dependent promotion of S-nitrosation from SNO-Hb involves biochemical mechanisms that are not intrinsic to the Hb molecule.     

Imaging real-time proteolysis of single collagen I molecules with an atomic force microscope.

The dynamic process of synthesis and degradation of extracellular matrix molecules, including various collagens, is important in normal physiological functions and pathological conditions. Existing models of collagen enzymatic degradation reactions are derived from bulk biochemical assays. In this study, we have imaged in real-time individual collagen I molecules and their proteolysis by Clostridium histolyticum collagenases in phosphate-buffered saline (PBS) with atomic force microscopy (AFM). We have also imaged the likely binding and unbinding of collagenase molecules to single triple-helical collagen I molecules and subsequent proteolysis of subsets of the collagen molecules. The proteolysis of collagen molecules was inhibited by reduced calcium and acidification. Results from AFM study of collagen proteolysis are consistent with SDS-PAGE biochemical assays. The real-time proteolysis of single collagen I molecules followed simple Michaelis-Menton kinetics previously derived from bulk biochemical assays. This is the first report of imaging real-time proteolysis of single macromolecules and its inhibition on a molecular scale. A strong correspondence between the kinetics of proteolysis of single collagen molecules and the kinetics of proteolysis derived from bulk biochemical assays will have a wide applicability in examining real-time enzymatic reactions and their regulation at single molecule structural level. Such real-time study of single molecule proteolysis could provide a better understanding of the interactions between proteases and target proteins as well as proteases and protease inhibitors.     

Following autolysis in proteases by NMR: insights into multiple unfolding pathways and mutational plasticities

Recently there has been significant interest in deducing the form of the rate laws for chemical reactions occurring in the intracellular environment. This environment is typically characterized by low-dimensionality and a high macromolecular content; this leads to a spatial heterogeneity not typical of the well stirred in vitro environments. For this reason, the classical law of mass action has been presumed to be invalid for modeling intracellular reactions. Using lattice-gas automata models, it has recently been postulated [H. Berry, Monte Carlo simulations of enzyme reactions in two dimensions: Fractal kinetics and spatial segregation, Biophys. J. 83 (2002) 1891-1901; S. Schnell, T.E. Turner, Reaction kinetics in intracellular environments with macromolecular crowding: simulations and rate laws, Prog. Biophys. Mol. Biol. 85 (2004) 235-260] that the reaction kinetics is fractal-like. In this article we systematically investigate for the first time how the rate laws describing intracellular reactions vary as a function of: the geometry and size of the intracellular surface on which the reactions occur, the mobility of the macromolecules responsible for the crowding effects, the initial reactant concentrations and the probability of reaction between two reactant molecules. We also compare the rate laws valid in heterogeneous environments in which there is an underlying spatial lattice, for example crystalline alloys, with the rate laws valid in heterogeneous environments where there is no such natural lattice, for example in intracellular environments. Our simulations indicate that: (i) in intracellular environments both fractal kinetics and mass action can be valid, the major determinant being the probability of reaction, (ii) the geometry and size of the intracellular surface on which reactions are occurring does not significantly affect the rate law, (iii) there are considerable differences between the rate laws valid in heterogeneous non-living structures such as crystals and those valid in intracellular environments. Deviations from mass action are less pronounced in intracellular environments than in a crystalline material of similar heterogeneity.     

Unifying mass-action kinetics and Newtonian mechanics by means of Nambu brackets

We demonstrate that elementary biochemical reactions defined by mass-action kinetics satisfy a particular Nambu structure. To this end, we express biochemical reaction equations in terms of Nambu brackets and certain ω-factors. The ω-factors account for the fact that mass-action kinetics exhibits in general flow fields with finite divergence. The proposed approach by means of Nambu brackets and ω-factors unifies divergence freeflow fields of Newtonian mechanics and flow fields with finite divergence of mass-action kinetics.     

Modeling multiphase fluid flow,mass transfer,and chemical reactions in bioreactors using large-eddy simulation

We present a transient large eddy simulation (LES) modeling approach for simulating the interlinked physics describing free surface hydrodynamics, multiphase mixing, reaction kinetics, and mass transport in bioreactor systems. Presented case-studies include non-reacting and reacting bioreactor systems, modeled through the inclusion of uniform reaction rates and more complex biochemical reactions described using Contois type kinetics. It is shown that the presence of reactions can result in a non-uniform spatially varying species concentration field, the magnitude and extent of which is directly related to the reaction rates and the underlying variations in the local volumetric mass transfer coefficient.     

Multiple phosphorylation of rhodopsin and the in vivo chemistry underlying rod photoreceptor dark adaptation

Dark adaptation requires timely deactivation of phototransduction and efficient regeneration of visual pigment. No previous study has directly compared the kinetics of dark adaptation with rates of the various chemical reactions that influence it. To accomplish this, we developed a novel rapid-quench/mass spectrometry-based method to establish the initial kinetics and site specificity of light-stimulated rhodopsin phosphorylation in mouse retinas. We also measured phosphorylation and dephosphorylation, regeneration of rhodopsin, and reduction of all-trans retinal all under identical in vivo conditions. Dark adaptation was monitored by electroretinography. We found that rhodopsin is multiply phosphorylated and then dephosphorylated in an ordered fashion following exposure to light. Initially during dark adaptation, transduction activity wanes as multiple phosphates accumulate. Thereafter, full recovery of photosensitivity coincides with regeneration and dephosphorylation of rhodopsin.     

Stochastic Bistability and Bifurcation in a Mesoscopic Signaling System with Autocatalytic Kinase

Bistability is a nonlinear phenomenon widely observed in nature including in biochemical reaction networks. Deterministic chemical kinetics studied in the past has shown that bistability occurs in systems with strong (cubic) nonlinearity. For certain mesoscopic, weakly nonlinear (quadratic) biochemical reaction systems in a small volume, however, stochasticity can induce bistability and bifurcation that have no macroscopic counterpart. We report the simplest yet known reactions involving driven phosphorylation-dephosphorylation cycle kinetics with autocatalytic kinase. We show that the noise-induced phenomenon is correlated with free energy dissipation and thus conforms with the open-chemical system theory. A previous reported noise-induced bistability in futile cycles is found to have originated from the kinase synchronization in a bistable system with slow transitions, as reported here.     

Modeling of uncertainties in biochemical reactions

Mathematical modeling is an indispensable tool for research and development in biotechnology and bioengineering. The formulation of kinetic models of biochemical networks depends on knowledge of the kinetic properties of the enzymes of the individual reactions. However, kinetic data acquired from experimental observations bring along uncertainties due to various experimental conditions and measurement methods. In this contribution, we propose a novel way to model the uncertainty in the enzyme kinetics and to predict quantitatively the responses of metabolic reactions to the changes in enzyme activities under uncertainty. The proposed methodology accounts explicitly for mechanistic properties of enzymes and physico‐chemical and thermodynamic constraints, and is based on formalism from systems theory and metabolic control analysis. We achieve this by observing that kinetic responses of metabolic reactions depend: (i) on the distribution of the enzymes among their free form and all reactive states; (ii) on the equilibrium displacements of the overall reaction and that of the individual enzymatic steps; and (iii) on the net fluxes through the enzyme. Relying on this observation, we develop a novel, efficient Monte Carlo sampling procedure to generate all states within a metabolic reaction that satisfy imposed constrains. Thus, we derive the statistics of the expected responses of the metabolic reactions to changes in enzyme levels and activities, in the levels of metabolites, and in the values of the kinetic parameters. We present aspects of the proposed framework through an example of the fundamental three‐step reversible enzymatic reaction mechanism. We demonstrate that the equilibrium displacements of the individual enzymatic steps have an important influence on kinetic responses of the enzyme. Furthermore, we derive the conditions that must be satisfied by a reversible three‐step enzymatic reaction operating far away from the equilibrium in order to respond to changes in metabolite levels according to the irreversible Michelis–Menten kinetics. The efficient sampling procedure allows easy, scalable, implementation of this methodology to modeling of large‐scale biochemical networks. Biotechnol. Bioeng. 2011;108: 413–423. © 2010 Wiley Periodicals, Inc.     

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.     

土壤水解酶类催化动力学研究进展

土壤水解酶是存在于土壤中的一种重要的酶类,参与了土壤中为数众多的重要生物化学反应,与土壤中多种营养元素转化密切相关其催化反应的动力学研究常用来阐明其催化过程的特性、酶的本质属性及其对环境变化的响应等,研究土壤水解酶动力学特征对探讨其来源、性质及影响因素,对进一步调控多种营养元素参与的反应过程有着重要意义。文中概述了土壤水解酶的种类及其参与的生物化学反应;探讨了土壤水解酶动力学研究的理论基础;综述了土壤水解酶催化动力学研究的进展和影响因素,在此基础上,对今后研究提出了几点建议。     

Effect of testosterone on purine nucleotide metabolism in rat liver.

In previous studies, we found that castration induced interesting morphological and biochemical changes in rat liver. For the present study, we have examined the effects of testosterone on the kinetics of purine nucleotide metabolism with the aim of determining the steps affected by testosterone deficiency. A biomathematical model of purine nucleotide metabolism was used to analyze the many reactions involved. The model simplifies purine nucleotide metabolism to four main steps: 1) de novo synthesis from PRPP to IMP; 2) the inosinic branch point from IMP to GMP or AMP; 3) catabolism of IMP, AMP and GMP to uric acid; 4) RNA and DNA formation from AMP and GMP. We evaluated rate constants from each step from variations in specific radioactivity of metabolites labelled with (14)C-formate, a precursor of de novo synthesis. The model was applied to the liver of normal and castrated rats before and after testosterone treatment. All four steps were slowed after castration, and were not completely restored by androgen administration. The model can give a clear representation of the kinetics of the reactions involved in the liver nucleotide metabolism investigated here, and we propose that a similar approach could be useful whenever a quantitative evaluation of the results obtained in vivo after administration of labelled precursors is required.     

Approximations and their consequences for dynamic modelling of signal transduction pathways

Signal transduction is the process by which the cell converts one kind of signal or stimulus into another. This involves a sequence of biochemical reactions, carried out by proteins. The dynamic response of complex cell signalling networks can be modelled and simulated in the framework of chemical kinetics. The mathematical formulation of chemical kinetics results in a system of coupled differential equations. Simplifications can arise through assumptions and approximations. The paper provides a critical discussion of frequently employed approximations in dynamic modelling of signal transduction pathways. We discuss the requirements for conservation laws, steady state approximations, and the neglect of components. We show how these approximations simplify the mathematical treatment of biochemical networks but we also demonstrate differences between the complete system and its approximations with respect to the transient and steady state behavior.