Graphic rules in steady and non-steady state enzyme kinetics

Graphic methods, when applied to enzyme kinetics, can provide a visually intuitive relation between calculations and reaction graphs. This will not only greatly raise the efficiency of calculations but also significantly help the analysis of enzyme kinetic mechanisms. In this paper, four graphic rules are presented. Rules 1-3 are established for steady state enzyme-catalyzed reaction systems and Rule 4 is for non-steady state ones. In comparison with conventional graphic methods which can only be applied to steady state systems, the present rules have the following merits. 1) Complicated and tedious calculations can be greatly simplified; for example, in calculating the concentrations of enzyme species for the bi-bi random mechanism, the calculation work can be reduced 8-fold compared with the King-Altman's method. 2) A great deal of wasted labor can be avoided; for example, in calculating the rate of product formation for the same mechanism, the operation of finding and removing the 96 reciprocally canceled terms is no longer needed because they automatically disappear during the derivation. 3) Final results can be easily and safely checked by a formula provided in each of the graphic rules. 4) Non-steady state systems can also be treated by the present graphic method; for example, applying Rule 4, one can directly write out the solution for a non-steady state enzyme-catalyzed system, without the need to follow more difficult and complicated operations to solve differential equations. The mathematical proofs of Rules 1-4 are given in Appendices A-D (in the Miniprint), respectively.     

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.     

Schematic methods for probabilistic enzyme kinetics.

The theory of steady-state enzyme processes which avoids using the mass action law of chemical kinetics and consistently describes catalytic mechanisms by probabilistic concepts has recently been proposed (Mazur, 1991, J. theor. Biol. 148, 229-242). To facilitate the analysis of complex reaction graphs by this theory the possibility of constructing schematic rules similar to those used in classical kinetics is studied. It is found that due to the similarity of algebraic procedures the popular method of King & Altman can be applied in probabilistic kinetics in addition to the earlier proposed rule based on enumeration of cycles of the reaction graph. This similarity also allows one to adapt many other shortcut methods of classical kinetics for probabilistic reaction graphs. The paper considers separately the possibility of transforming reaction mechanisms so that the initial graph is replaced by a simpler but equivalent one. It is shown that there are few cases when a group of states can be replaced by one united state, with earlier known rules such as the rule of Cha for equilibrium stages being particular cases of a more general procedure. In addition a novel method is proposed which performs step-by-step reduction of any reaction graph. All the new methods can be adapted for traditional kinetics as well. The results obtained demonstrate that many schematic rules of classical kinetics are of probabilistic origin.     

Topology, stability, sequence, and length: defining the determinants of two-state protein folding kinetics

The fastest simple, single domain proteins fold a million times more rapidly than the slowest. Ultimately this broad kinetic spectrum is determined by the amino acid sequences that define these proteins, suggesting that the mechanisms that underlie folding may be almost as complex as the sequences that encode them. Here, however, we summarize recent experimental results which suggest that (1) despite a vast diversity of structures and functions, there are fundamental similarities in the folding mechanisms of single domain proteins and (2) rather than being highly sensitive to the finest details of sequence, their folding kinetics are determined primarily by the large-scale, redundant features of sequence that determine a protein's gross structural properties. That folding kinetics can be predicted using simple, empirical, structure-based rules suggests that the fundamental physics underlying folding may be quite straightforward and that a general and quantitative theory of protein folding rates and mechanisms (as opposed to unfolding rates and thus protein stability) may be near on the horizon.     

Probing the kinetics of single molecule protein folding

We propose an approach to integrate the theory, simulations, and experiments in protein-folding kinetics. This is realized by measuring the mean and high-order moments of the first-passage time and its associated distribution. The full kinetics is revealed in the current theoretical framework through these measurements. In the experiments, information about the statistical properties of first-passage times can be obtained from the kinetic folding trajectories of single molecule experiments (for example, fluorescence). Theoretical/simulation and experimental approaches can be directly related. We study in particular the temperature-varying kinetics to probe the underlying structure of the folding energy landscape. At high temperatures, exponential kinetics is observed; there are multiple parallel kinetic paths leading to the native state. At intermediate temperatures, nonexponential kinetics appears, revealing the nature of the distribution of local traps on the landscape and, as a result, discrete kinetic paths emerge. At very low temperatures, exponential kinetics is again observed; the dynamics on the underlying landscape is dominated by a single barrier. The ratio between first-passage-time moments is proposed to be a good variable to quantitatively probe these kinetic changes. The temperature-dependent kinetics is consistent with the strange kinetics found in folding dynamics experiments. The potential applications of the current results to single-molecule protein folding are discussed.     

Enzyme-catalyzed and binding reaction kinetics determined by titration calorimetry

BackgroundIsothermal calorimetry allows monitoring of reaction rates via direct measurement of the rate of heat produced by the reaction. Calorimetry is one of very few techniques that can be used to measure rates without taking a derivative of the primary data. Because heat is a universal indicator of chemical reactions, calorimetry can be used to measure kinetics in opaque solutions, suspensions, and multiple phase systems and does not require chemical labeling. The only significant limitation of calorimetry for kinetic measurements is that the time constant of the reaction must be greater than the time constant of the calorimeter which can range from a few seconds to a few minutes. Calorimetry has the unique ability to provide both kinetic and thermodynamic data.Scope of reviewThis article describes the calorimetric methodology for determining reaction kinetics and reviews examples from recent literature that demonstrate applications of titration calorimetry to determine kinetics of enzyme-catalyzed and ligand binding reactions.Major conclusionsA complete model for the temperature dependence of enzyme activity is presented. A previous method commonly used for blank corrections in determinations of equilibrium constants and enthalpy changes for binding reactions is shown to be subject to significant systematic error.General significanceMethods for determination of the kinetics of enzyme-catalyzed reactions and for simultaneous determination of thermodynamics and kinetics of ligand binding reactions are reviewed. This article is part of a Special Issue entitled Microcalorimetry in the BioSciences — Principles and Applications, edited by Fadi Bou-Abdallah.     

New relations for steady-state enzyme kinetics and their application to rapid equilibrium assumption

With the use of graph theory, new relations for steady-state enzyme kinetics are derived and strictly proved for an arbitrary mechanism of enzyme-catalyzed reaction containing a reversible segment. Using these relations, a general principle for rapid equilibrium assumption is formulated and proved: the reversible bound segment can be considered as an equilibrium segment only when the values of the base trees that are not proper to this segment can be neglected (within prescribed accuracy) in relation to the values of the base trees that belong to this segment. In contrast with the foreign base trees, the base trees that are proper to the segment have the following properties: the tree that is directed to the base within this segment does not contain edges leaving this segment; and the tree that is directed to the base outside the segment contains only one edge leaving this segment. Equilibrium variations are assessed for steady-state concentrations of intermediates in the equilibrium segment, numerical expressions are obtained for the accuracy of determination of intermediate concentrations as well as for the accuracy of determination of the rate of enzyme-catalyzed reaction under rapid equilibrium assumption.     

Dynamics and mechanisms of coupled protein folding and binding reactions

Protein folding coupled to binding of a specific ligand is frequently observed in biological processes. In recent years numerous studies have addressed the structural properties of the unfolded proteins in the absence of their ligands. Surprisingly few time-resolved investigations on coupled folding and binding reactions have been published up to date and the dynamics and kinetic mechanisms of these processes are still only poorly understood. Especially, it is still unsolved for most systems which conformation of the protein is recognized by the ligand (conformational selection vs. folding-after-binding) and whether the ligand influences the folding kinetics. Here we review experimental methods, kinetic models and time-resolved experimental studies of coupled folding and binding reactions.     

Studying submicrosecond protein folding kinetics using a photolabile caging strategy and time‐resolved photoacoustic calorimetry

Kinetic measurement of protein folding is limited by the method used to trigger folding. Traditional methods, such as stopped flow, have a long mixing dead time and cannot be used to monitor fast folding processes. Here, we report a compound, 4‐(bromomethyl)‐6,7‐dimethoxycoumarin, that can be used as a “photolabile cage” to study the early stages of protein folding. The folding process of a protein, RD1, including kinetics, enthalpy, and volume change, was studied by the combined use of a phototriggered caging strategy and time‐resolved photoacoustic calorimetry. The cage caused unfolding of the photolabile protein, and then a pulse UV laser (～10^?9 s) was used to break the cage, leaving the protein free to refold and allowing the resolving of two folding events on a nanosecond time scale. This strategy is especially good for monitoring fast folding proteins that cannot be studied by traditional methods. Proteins 2010. © 2010 Wiley‐Liss, Inc.     

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.     

A probabilistic framework for the exploration of enzymatic capabilities based on feasible kinetics and control analysis

BackgroundAnalysis of limiting steps within enzyme-catalyzed reactions is fundamental to understand their behavior and regulation. Methods capable of unravelling control properties and exploring kinetic capabilities of enzymatic reactions would be particularly useful for protein and metabolic engineering. While single-enzyme control analysis formalism has previously been applied to well-studied enzymatic mechanisms, broader application of this formalism is limited in practice by the limited amount of kinetic data and the difficulty of describing complex allosteric mechanisms.MethodsTo overcome these limitations, we present here a probabilistic framework enabling control analysis of previously unexplored mechanisms under uncertainty. By combining a thermodynamically consistent parameterization with an efficient Sequential Monte Carlo sampler embedded in a Bayesian setting, this framework yields insights into the capabilities of enzyme-catalyzed reactions with modest kinetic information, provided that the catalytic mechanism and a thermodynamic reference point are defined.ResultsThe framework was used to unravel the impact of thermodynamic affinity, substrate saturation levels and effector concentrations on the flux control and response coefficients of a diverse set of enzymatic reactions.ConclusionsOur results highlight the importance of the metabolic context in the control analysis of isolated enzymes as well as the use of statistically sound methods for their interpretation.General SignificanceThis framework significantly expands our current capabilities for unravelling the control properties of general reaction kinetics with limited amount of information. This framework will be useful for both theoreticians and experimentalists in the field.     

Comparison of two experimental methods for the determination of Michaelis-Menten kinetics of an immobilized enzyme

For the application of immobilized enzymes, the influence of immobilization on the activity of the enzyme should be Known. This influence can be obtained by determining the intrinsic kinetic parameters of the immobilized enzyme, and by comparing them with the kinetic parameters of the suspended enzyme. This article deals with the determination of the intrinsic kinetic parameters of an agarose-gel bead immobilized oxygen-consuming enzyme: L-lactate 2-monooxygenase. The reaction rate of the enzyme can be described by Michaelis-Menten kinetics. Batch conversion experiments using a biological oxygen monitor, as well as steady-state profile measurements within the biocatalyst particles using an oxygen microsensor, were performed. Two different mathematical methods were used for the batch conversion experiments, both assuming a pseudosteady-state situation with respect to the shape of the profile inside the bead. One of the methods used an approximate relation for the effectiveness factor for Michaelis-Menten kinetics which interpolates between the analytical solutions for zero- and first-order kinetics. The other mathematical method was based on a numerical solution and combined a mass balance over the reactor with a mass balance over the bead. The main difference in the application of the two methods is the computer calculation time; the completely numerical calculation procedure was about 20 times slower than the other calculation procedure.The intrinsic kinetic parameters resulting from both experimental methods were compared to check the reliability of the methods. There was no significant difference in the intrinsic kinetic parameters obtained from the two experimental methods. By comparison of the kinetic parameters for the suspended enzyme with the intrinsic kinetic parameters for the immobilized enzyme, it appeared that immobilization caused a decrease in the value of V(m) by a factor of 2, but there was no significant difference in the values obtained for K(m).     

Observation of multistate kinetics during the slow folding and unfolding of barstar.

The kinetics of the slow folding and unfolding reactions of barstar, a bacterial ribonuclease inhibitor protein, have been studied at 23(+/-1) degrees C, pH 8, by the use of tryptophan fluorescence, far-UV circular dichroism (CD), near-UV CD, and transient mixing (1)H nuclear magnetic resonance (NMR) spectroscopic measurements in the 0-4 M range of guanidine hydrochloride (GdnHCl) concentration. The denaturant dependences of the rates of folding and unfolding processes, and of the initial and final values of optical signals associated with these kinetic processes, have been determined for each of the four probes of measurement. Values determined for rates as well as amplitudes are shown to be very much probe dependent. Significant differences in the intensities and rates of appearance and disappearance of several resolved resonances in the real-time one-dimensional NMR spectra have been noted. The NMR spectra also show increasing dispersion of chemical shifts during the slow phase of refolding. The denaturant dependences of rates display characteristic folding chevrons with distinct rollovers under strongly native as well as strongly unfolding conditions. Analyses of the data and comparison of the results obtained with different probes of measurement appear to indicate the accumulation of a myriad of intermediates on parallel folding and unfolding pathways, and suggest the existence of an ensemble of transition states. The energetic stabilities of the intermediates estimated from kinetic data suggest that they are approximately half as stable as the fully folded protein. The slowness of the folding and unfolding processes (tau = 10-333 s) and values of 20.5 (+/-1.4) and 18 (+/-0.5) kcal mol(-)(1) for the activation energies of the slow refolding and unfolding reactions suggest that proline isomerization is involved in these reactions, and that the intermediates accumulate and are therefore detectable because the slow proline isomerization reaction serves as a kinetic trap during folding.     

Protein folding in the landscape perspective: Chevron plots and non-arrhenius kinetics

We use two simple models and the energy landscape perspective to study protein folding kinetics. A major challenge has been to use the landscape perspective to interpret experimental data, which requires ensemble averaging over the microscopic trajectories usually observed in such models. Here, because of the simplicity of the model, this can be achieved. The kinetics of protein folding falls into two classes: multiple-exponential and two-state (single-exponential) kinetics. Experiments show that two-state relaxation times have “chevron plot” dependences on denaturant and non-Arrhenius dependences on temperature. We find that HP and HP+ models can account for these behaviors. The HP model often gives bumpy landscapes with many kinetic traps and multiple-exponental behavior, whereas the HP+ model gives more smooth funnels and two-state behavior. Multiple-exponential kinetics often involves fast collapse into kinetic traps and slower barrier climbing out of the traps. Two-state kinetics often involves entropic barriers where conformational searching limits the folding speed. Transition states and activation barriers need not define a single conformation; they can involve a broad ensemble of the conformations searched on the way to the native state. We find that unfolding is not always a direct reversal of the folding process. Proteins 30:2–33, 1998. © 1998 Wiley-Liss, Inc.     

The early folding kinetics of apomyoglobin.

The folding pathway of apomyoglobin has been experimentally shown to have early kinetic intermediates involving the A, B, G, and H helices. The earliest detected kinetic events occur on a ns to micros time scale. We show that the early folding kinetics of apomyoglobin may be understood as the association of nascent helices through a network of diffusion-collision-coalescence steps G + H <--> GH + A <--> AGH + B <--> ABGH obtained by solving the diffusion-collision model in a chemical kinetics approximation. Our reproduction of the experimental results indicates that the model is a useful way to analyze folding data. One prediction from our fit is that the nascent A and H helices should be relatively more helix-like before coalescence than the other apomyoglobin helices.     

Cooperativity in two-state protein folding kinetics

We present a solvable model that predicts the folding kinetics of two-state proteins from their native structures. The model is based on conditional chain entropies. It assumes that folding processes are dominated by small-loop closure events that can be inferred from native structures. For CI2, the src SH3 domain, TNfn3, and protein L, the model reproduces two-state kinetics, and it predicts well the average Phi-values for secondary structures. The barrier to folding is the formation of predominantly local structures such as helices and hairpins, which are needed to bring nonlocal pairs of amino acids into contact.     

Simplifying principles for chemical and enzyme reaction kinetics

Tihonov's Theorems for systems of first-order ordinary differential equations containing small parameters in the derivatives, which form the mathematical foundation of the steady-state approximation, are restated. A general procedure for simplifying chemical and enzyme reaction kinetics, based on the difference of characteristic time scales, is presented. Korzuhin's Theorem. which makes it possible to approximate any kinetic system by a closed chemical system, is also reported. The notions and theorems are illustrated with examples of Michaelis-Menten enzyme kinetics and of a simple autocatalytic system. Another example illustrates how the differences in the rate constants of different elementary reactions may be exploited to simplify reaction kinetics by using Tihonov's Theorem. All necessary mathematical notions are explained in the appendices. The most simple formulation of Tihonov's 1st Theorem ‘for beginners’ is also given.     

Rapid mixing methods for exploring the kinetics of protein folding

Information on the time-dependence of molecular species is critical for elucidating reaction mechanisms in chemistry and biology. Rapid flow experiments involving turbulent mixing of two or more solutions continue to be the main source of kinetic information on protein folding and other biochemical processes, such as ligand binding and enzymatic reactions. Recent advances in mixer design and detection methods have opened a new window for exploring conformational changes in proteins on the microsecond time scale. These developments have been especially important for exploring early stages of protein folding.     

Steady-state kinetics of 3-mercaptopyruvate sulfurtransferase from bovine kidney

Mammalian 3-mercaptopyruvate sulfurtransferase (EC 2.8.1.2), purified to apparent homogeneity by a new procedure, was studied by steady-state kinetic methods. The enzyme-catalyzed transfer of a sulfur atom from 3-mercaptopyruvate either to 2-mercaptoethanol or to a second molecule of 3-mercaptopyruvate was found to proceed by a sequential formal mechanism. An overall mechanism incorporating both of these transfers was shown to be capable of generating all of the initial velocity and product inhibition behavior observed.     

Kinetic and thermodynamic properties of wild-type and engineered mutants of tyrosyl-tRNA synthetase analyzed by pyrophosphate-exchange kinetics.

The first step of the reaction catalyzed by the aminoacyl-tRNA synthetases is the formation of enzyme-bound aminoacyl adenylate. The steady-state kinetics of this step has conventionally been studied by measuring the rate of isotopic exchange between pyrophosphate and ATP. A simple kinetic analysis of the pyrophosphate-exchange reaction catalyzed by the tyrosyl-tRNA synthetase from Bacillus stearothermophilus is given in which all the observed rate and binding constants can be assigned to identifiable physical processes under a variety of limiting conditions. The free energies of binding to the enzyme of tyrosine, ATP, and the transition state for tyrosyl adenylate formation can be measured in relatively straightforward experiments. The excellent agreement between parameters measured in these experiments and those from earlier pre-steady-state kinetics confirms that the intermediates isolated in the presteady state are kinetically competent. The dissociation constant of ATP from the unligated enzyme, a constant that has previously been experimentally inaccessible, has been measured for wild-type and several mutant enzymes. The changes in enthalpy and entropy of activation on mutation have been measured by a rapid procedure for mutants that have altered contacts with tyrosine and ATP. Those mutants that have large changes of enthalpy and entropy of binding are likely to have structural changes and so warrant further examination by protein crystallography.