A probabilistic approach to compact steady-state kinetic equations for enzymic reactions

We describe a method for deriving kinetic equations based on the simplification of a complex graphical scheme of steady-state enzymic reactions to one that is comprised of an unbranched pathway. It entails compressing unbranched multi-step sequences into one step, and fusing some graph nodes into a single node. The final form of the equations is compact and well structured, and it simplifies the choice of independent kinetic parameters. The approach is illustrated by an analysis of representative two- and three-substrate reactions.     

A dynamic view of enzyme catalysis

Recent experimental advances have shown that enzymes are flexible molecules, and point to a direct link between dynamics and catalysis. Movements span a wide time range, from nano- to milli-seconds. In this paper we introduce two aspects of enzyme flexibility that are treated with two appropriate techniques. First, transition path sampling is used to obtain an unbiased picture of the transition state ensemble in chorismate mutase, as well as its local flexibility and the energy flow during the chemical step. Second, we consider the binding and release of substrates in L-rhamnulose-1-phosphate aldolase. We have calculated the normal modes of the enzyme with the elastic network model. The lowest frequency modes generate active site deformations that change the coordination number of the catalytic zinc ion. The coordination lability of zinc allows the binding and release of substrates. Substitution of zinc by magnesium blocks the exchange of ligands. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

A mechanistic view of enzyme evolution

New enzyme functions often evolve through the recruitment and optimization of latent promiscuous activities. How do mutations alter the molecular architecture of enzymes to enhance their activities? Can we infer general mechanisms that are common to most enzymes, or does each enzyme require a unique optimization process? The ability to predict the location and type of mutations necessary to enhance an enzyme's activity is critical to protein engineering and rational design. In this review, via the detailed examination of recent studies that have shed new light on the molecular changes underlying the optimization of enzyme function, we provide a mechanistic perspective of enzyme evolution. We first present a global survey of the prevalence of activity‐enhancing mutations and their distribution within protein structures. We then delve into the molecular solutions that mediate functional optimization, specifically highlighting several common mechanisms that have been observed across multiple examples. As distinct protein sequences encounter different evolutionary bottlenecks, different mechanisms are likely to emerge along evolutionary trajectories toward improved function. Identifying the specific mechanism(s) that need to be improved upon, and tailoring our engineering efforts to each sequence, may considerably improve our chances to succeed in generating highly efficient catalysts in the future.     

A probabilistic view of immunology: drawing parallels with physics

Regulating the strength and class of an immune response requires lymphocytes to act as complex signal integrating "machines", taking information from multiple sources while making decisions that affect the final outcome. Describing and understanding the decision-making behaviour of lymphocytes within the context of the dynamic multifaceted immune system appears immensely complicated. In this article I contrast two alternative frameworks, the deterministic and the probabilistic as competing paths to achieve successful quantitative immune models. As the two alternatives are traditional scientific rivals, I use the probabilistic fields in physics to highlight the potential value of the probabilistic perspective in immunology.     

Microcomputer tools for steady-state enzyme kinetics

Three programs useful for the investigation of steady–statekinetics have been developed. Two provide the solution to thesteady–state rate equation; the first of these is a straightforwardimplementation of the rules developed by Chou. The second isa very efficient procedure for evaluating King–Altmandiagrams and can be used for quite large mechanisms. The thirdprogram provides the numeric solution for a specific mechanismand set of initial conditions; it is well suited to extremelylarge models. Received on April 1, 1985; accepted on April 19, 1985     

Calculation of steady-state rate equations and the fluxes between substrates and products in enzyme reactions.

1. Two methods are described for deriving the steady-state velocity of an enzyme reaction from a consideration of fluxes between enzyme intermediates. The equivalent-reaction technique, in which enzyme intermediates are systematically eliminated and replaced by equivalent reactions, appears the most generally useful. The methods are applicable to all enzyme mechanisms, including three-substrate and random Bi Bi Ping Pong mechanisms. Solutions are obtained in algebraic form and these are presented for the common random Bi Bi mechanisms. The steady-state quantities of the enzyme intermediates may also be calculated. Additional steps may be introduced into enzyme mechanisms for which the steady-state velocity equation is already known. 2. The calculation of fluxes between substrates and products in three-substrate and random Bi Bi Ping Pong mechanisms is described. 3. It is concluded that the new methods may offer advantages in ease of calculation and in the analysis of the effects of individual steps on the overall reaction. The methods are used to show that an ordered addition of two substrates to an enzyme which is activated by another ligand will not necessarily give hyperbolic steady-state-velocity kinetics or the flux ratios characteristic of an ordered addition, if the dissociation of the ligand from the enzyme is random.     

A general solution for the steady-state kinetics of immobilized enzyme systems

A power series solution is presented which describes the steady-state concentration profiles for substrate and product molecules in immobilized enzyme systems. Diffusional effects and product inhibition are incorporated into this model. The kinetic consequences of diffusion limitation and product inhibition for immobilized enzymes are discussed and are compared to kinetic behavior characteristic of other types of effects, such as substrate inhibition and substrate activation.     

Microscopic diffusion-reaction coupling in steady-state enzyme kinetics

The theory of diffusion-controlled processes is applied to describe the steady state of a reversible enzymatic reaction with special emphasis on the effects of enzyme saturation. A standard macroscopic steady-state treatment requires only that the average diffusional influx of substrate equals the net reaction flux as well as the average diffusional efflux of product. In contrast, the microscopic diffusion-reaction coupling used here takes properly into account the conditional concentration distributions of substrate and product: Only when the enzyme is unoccupied will there be a diffusional association flux; when the enzyme is occupied, the concentration distributions will relax towards their homogeneous bulk values. In this way the relaxation effects of the non-steady state will be constantly reoccurring as the enzyme shifts between unoccupied and occupied states. Thus, one is forced to describe the steady state as the weighted sum of properly time-averaged non-stationary conditional distributions. The consequences of the theory for an appropriate assessment of the parameters obtained in Lineweaver-Burk plots are discussed. In general, our results serve to justify the simpler macroscopic coupling scheme. However, considerable deviations between the standard treatment and our analysis can occur for fast enzymes with an essentially irreversible product release.     

Effect of enzyme-enzyme interactions on steady-state enzyme kinetics. IV. “Strictly steady-state” examples

A number of two-state and three-state examples of enzyme interaction effects at steady state are considered. In contrast to most examples in parts I, II, and III, these are not of the quasi-equilibrium type. The Bragg—Williams approximation is used here for lattices of both two-state and three-state enzymes. In addition, several examples of small (oligomeric) systems are treated. Diffusion in lattice-fluid models is introduced in a simple way in the text and commented on further in the appendix (together with diffusion in solution).     

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.     

Role of DAPI in microtubule reactions at steady-state

The new fluorophor for tubulin, DAPI, is shown to bind to a site different from the exchangeable nucleotide binding site (E site) and to inhibit GTP hydrolysis by the tubulin-colchicine complex within an uncompetitive scheme. Moreover the dissociation rate constant of tubulin for microtubule ends at 32 degrees C was found largely decreased in the presence of saturating amounts of the probe while the association rate constant was little affected. These data on the kinetic parameters of tubulin interactions in the presence of DAPI, together with the inhibition of GTP hydrolysis by microtubules at the steady state are understood as the main cause for microtubule stabilization at steady-state by DAPI.     

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.     

A steady state model of sequential irreversible enzyme reactions

Summary One of the questions which arises in the study of certain inborn errors of metabolism as well as in the field of enzyme kinetics is: what are the quantitative relationships between parameters of enzyme activity and substrate pool sizes in a metabolic pathway? A steady state model has been devised to answer this question for a homogeneous system of non-branched sequential irreversible enzyme reactions which follow Michaelis-Menten kinetics. The concentration of a substrate in such a pathway, [S_i], is a function of 5 variables: (a) the K_M of the enzyme which forms the substrate (K_{M (i–1)}), (b) the K_M of the enzyme which utilizes the substrate (K_{M i}), (c) the V_max of the enzyme which forms the substrate (V_{m (i–1)}), (d) the V_max of the enzyme which utilizes the substrate (V_{m i}) and (e) the immediate precursor concentration [S_(i–1)] where [S_i] = K_{M i} V_{m (i–1)} [S_(i–1)]/[S_(i–1)] (V_mi -V_{m (i–1)}) + K_{M (i–1)}) V_mi The model introduces and defines the concept of and conditions for amplification. An input in the form of a steady state concentration of precursor [S_(i–1)] may be amplified as an output in the form of an increased steady state concentration of product [S_i]. The model also defines the values of the above 5 parameters which do not allow attainment of a steady state for the type of pathway considered.From the Metabolism Branch, National Cancer Institute, National Institutes of Health, Bethesda, Maryland 20014.     

A logarithmic approximation to initial rates of enzyme reactions

A straightforward and empirical regression method based on a logarithmic approximation has been developed to accurately estimate initial rates from nonlinear progress curves of enzyme reactions. The principle of this parametric approach is to use a relatively large number of observations, while averaging out random errors, to predict the curvature at time zero, which has the highest rate of change. The usual linear regression of a few initial time points lacks prediction power at time zero and therefore underestimates the true initial rate. Application of this nonlinear regression approach to enzyme reactions demonstrated satisfactory results. This approach is less subjective in choosing initial time points to be used for rate determination, and much more robust to random errors. Moreover, it is relatively easy to realize with commonly available software.     

Multiple steady-state phenomena within enzyme reactors: The enzyme reaction with two substrates

The theoretical dynamic characteristics of an isothermal continuous flow stirred tank enzyme reactor (CFSTER) operating on two substrates are investigated. Under certain conditions multiple steady states are possible; namely, with an enzyme which binds with the two substrates sequentially. The occurrence of multiple steady states is found to be primarily dictated by three dimensionless parameters which incorporate rate law constants. The global stability of certain steady states is examined by numerically solving the transient material balance on the CFSTER. The effect of recycle on the dynamics of an isothermal plug flow enzyme reactor (PFER) is also studied. A general conclusion indicated by this work is that any open isothermal reaction system wherein the reaction rate law passes through a maximum with increasing substrate concentration and where back mixing occurs with exhibit multiple steady-state behavior in some operating range.