An easy procedure to transform the ratio of two polynomials of first degree into Michaelis-Menten-type equations. Application to the ordered Uni Bi enzyme mechanism

It is not always clear that some equations affected by complicated factors can, actually, be interpreted as a ratio of two polynomials of first degree and so that they can be, in general, represented by rectangular hyperbolas. In this paper we present an easy procedure to rearrange those equations into Michaelis-Menten-type equations and so to make the aspects of these rectangular hyperbolas more clear, particularly for researchers familiar with general biochemistry. As an example, the method is applied to transform the classical rate equation of the Cleland's Ordered Uni Bi enzyme mechanism.     

Kinetics of the hydrolysis of lean meat protein by alcalase: Derivation of two alternative rate equations and their fit to experimental data

Two rate equations have been developed to model the hydrolysis of ground lean meat protein by Alcalase. The first equation was based on classical Michaelis-Menten kinetics and the second on the adsorption of enzyme to the protein prior to reaction. It was assumed that this adsorption could be modelled by a Langmuir-type adsorption isotherm. Each equation considered the enzyme to be competitively inhibited by reaction product, and considered enzyme inactivation to be first order. Both rate equations have been fitted to experimental data obtained from the hydrolysis of meat protein by Alcalase. Initial rate data indicated that the adsorption model was more appropriate. However, both equations gave satisfactory fits to 11 reaction progress curves determined over a wide range of enzyme and substrate concentrations.     

Interdependence between cooperativity and control coefficients

Influence of the concentration of internal metabolites on the control coefficient (defined as fractional change in flux per fractional change in enzyme activity) and regulatory properties of a given enzyme have been studied theoretically using a cyclic model of three enzymes. This model is useful to investigate the properties of the flux control coefficient for an enzyme following different rate equations. Enzymes can have high or low values of control coefficient irrespective of the type of kinetic equation, but the results obtained show that the sensitivity of these values to substrate variations is strongly dependent on its rate equation. These results help identify which kinetic equation allows the best control of a given metabolic pathway. These results have been applied to the purine nucleotide cycle. It is demonstrated that the best control of the cycle is reached when the irreversible reaction catalyzed by AMP deaminase follows a rate law that corresponds to a rational function of 2:2 degree with respect to AMP concentration.     

The kinetics of the succinoxidase

Rate equations for the enzymatic oxidation of succinic acid are derived on the assumption that when a single molecule of substrate combines with an enzyme molecule, it can do so with either one or two sites on the enzyme, and that oxidation occurs only in the second case. In addition it is assumed that the product of the reaction, fumaric acid, combines reversibly with the enzyme. With certain enzyme preparations the data fitted such an equation satisfactorily. In others the rate was that of a first-order reaction, but addition of cytochrome changed it to the former type. It was concluded that the transfer of hydrogen to oxygen was a first-order reaction and dominated the whole rate when enzyme preparations were used which had been washed relatively free of cytochrome. When the limiting factor was succino-dehydrogenase the rates followed the new equation. Criteria for recognizing noncompetitive inhibition are given, and inhibition by di-tertiary butyl peroxide was shown to be of this type.     

An Analysis and Comparison of Convergence and Uniqueness of Time-Independent Bone Adaptation Models

Several stimuli are proposed in the bone remodeling theory. It is not clear, if a unique solution exists and if the result is convergent using a certain stimulus. In this study, the strain stimulus, strain energy stimulus and the von Mises stress stimulus for bone remodeling are compared and applied to a square plate model using the finite element method. In the plane stress state, the remodeling equilibrium equations are transformed into functions of only the principal strains and the graphs of these functions are drawn in a diagram using the principal strains as the variables of two coordinate axes. The equation of the sum of principal strain squared equal to a constant is a circle in the diagram. The remodeling equilibrium equation of the strain stimulus is a quadrangle fitting into the circle, the remodeling equilibrium equation of the strain energy stimulus is an ellipse and the remodeling equilibrium equation of the von Mises stress stimulus is also an ellipse close to the principal strains circle when we take the same constants in the above equations. Using the finite element method, two models are performed with the uniform initial elastic properties and with the semi-random initial distribution of the elastic properties. The principal strains as the final finite element results converge within 2% of the objective constant for all the different stimuli. The obtained Young's moduli of two models as the adaptation object are different but in equilibrium, i.e. the equilibrium solution of adaptation model is not unique. The principal strains can not be used to examine the uniqueness of solution, since two different solutions can have the same results of principal strains. Using a certain stimulus, certain initial properties and a certain iterative equation, the solution is unique in equilibrium. The results using the model in this study show also that the same results can be obtained using any of the three stimuli when a proper constant in each remodeling equilibrium equation is chosen.     

Kinetic analysis of enzyme reactions with slow-binding inhibition.

In this paper we present a general kinetic study of slow-binding inhibition processes, i.e. enzyme reactions that do not respond instantly to the presence of a competitive inhibitor. The analysis that we present is based on the equation that describes the formation of products with time in each case on the experimental progress curve. It is carried out under the condition of limiting enzyme concentration and allows the discrimination between the different cases of slow-binding inhibition. The mechanism in which the formation of complex enzyme-inhibitor is a single or two slow steps or follow a rapid equilibrium, has been considered. The corresponding explicit equations of each case have been obtained and checked by numerical integration. A kinetic data analysis to evaluate the corresponding kinetic parameters is suggested. We illustrate the method, numerically by computer simulation, of the reaction and present some numerical examples that demonstrate the applicability of our procedure.     

Theoretical approach to the steady-state kinetics of a bi-substrate acyl-transfer enzyme reaction that follows a hydrolysable-acyl-enzyme-based mechanism. Application to the study of lysophosphatidylcholine:lysophosphatidylcholine acyltransferase from rabbit lung.

A kinetic model is proposed for catalysis by an enzyme that has several special characteristics: (i) it catalyses an acyl-transfer bi-substrate reaction between two identical molecules of substrate, (ii) the substrate is an amphiphilic molecule that can be present in two physical forms, namely monomers and micelles, and (iii) the reaction progresses through an acyl-enzyme-based mechanism and the covalent intermediate can react also with water to yield a secondary hydrolytic reaction. The theoretical kinetic equations for both reactions were deduced according to steady-state assumptions and the theoretical plots were predicted. The experimental kinetics of lysophosphatidylcholine:lysophosphatidylcholine acyltransferase from rabbit lung fitted the proposed equations with great accuracy. Also, kinetics of inhibition by products behaved as expected. It was concluded that the competition between two nucleophiles for the covalent acyl-enzyme intermediate, and not a different enzyme action depending on the physical state of the substrate, is responsible for the differences in kinetic pattern for the two activities of the enzyme. This conclusion, together with the fact that the kinetic equation for the transacylation is quadratic, generates a 'hysteretic' pattern that can provide the basis of self-regulatory properties for enzymes to which this model could be applied.     

Progress curves analysis as an alternative for exploration of activation-inhibition phenomena in cholinesterases

The kinetic behaviour of insect acetylcholinesterases deviates from the Michaelis-Menten pattern. These deviations are known as activation or inhibition at various substrate concentrations and can be more or less observable depending on mutations around the active site of the enzyme. Most kinetic studies on these enzymes still rely on initial rate measurements. It is demonstrated here that according to this method one of the deviations can be overlooked. We attempt to point out that in such cases a detailed step-by-step progress curves analysis is successful. The study is focused on two different methods of analysing progress curves: (i) the first one is based on an integrated initial rate equation which can sufficiently fit truncated progress curves under corresponding conditions; and (ii) the other one precludes the algebraic formulae, but uses numerical integration for searching a non analytical solution of ordinary differential equations describing a kinetic model. All methods are tested on three different acetylcholinesterase mutants from Drosophila melanogaster. The results indicate that kinetic parameters for the E107K mutant with highly expressive activation and inhibition can be well evaluated applying any analysis method. It is quite different for E107W and E107Y mutants where latent activation is present, but discovered only using one or the other progress curves analysis methods.     

Progress Curves Analysis as an Alternative for Exploration of Activation-inhibition Phenomena in Cholinesterases

The kinetic behaviour of insect acetylcholinesterases deviates from the Michaelis-Menten pattern. These deviations are known as activation or inhibition at various substrate concentrations and can be more or less observable depending on mutations around the active site of the enzyme. Most kinetic studies on these enzymes still rely on initial rate measurements. It is demonstrated here that according to this method one of the deviations can be overlooked. We attempt to point out that in such cases a detailed step-by-step progress curves analysis is successful. The study is focused on two different methods of analysing progress curves: (i) the first one is based on an integrated initial rate equation which can sufficiently fit truncated progress curves under corresponding conditions; and (ii) the other one precludes the algebraic formulae, but uses numerical integration for searching a non analytical solution of ordinary differential equations describing a kinetic model. All methods are tested on three different acetylcholinesterase mutants from Drosophila melanogaster. The results indicate that kinetic parameters for the E107K mutant with highly expressive activation and inhibition can be well evaluated applying any analysis method. It is quite different for E107W and E107Y mutants where latent activation is present, but discovered only using one or the other progress curves analysis methods.     

The probability of obtaining complex kinetic curves for enzyme mechanisms with cubic terms in the pseudo-steady state rate equations

A number of details required for the classification of 3 : 3 double reciprocal plots are provided. It is shown that the ν(S) plot for a 3 : 3 function can have at most four inflexions and at most two inflexions adjacent to a turning point. Using this information, a classification of 3 : 3 ν(S) plots into ten main varieties with several subclasses is reported. The problem of defining the probability with which a given mechanism can give rise to specific curve shape features is considered. Applying this technique, the probability with which four simple enzyme mechanisms can give rise to 3 : 3 curve shapes is computed. It is shown that a 3 : 3 saturation function can have no turning points, at most two inflexions and at most one inflexion in double reciprocal space. The probability with which the available 3 : 3 shapes can arise is also computed. It is concluded that, with realistic values for rate constants, chemically reasonable enzyme mechanisms leading to rate equations of degree n : n can generate most of the kinetic profiles available to a rational function of degree n : n with positive coefficients. The probability of obtaining specific curve shapes is not so characteristic of the particular mechanism for 3:3 rate equations as it is for 2:2 rate equations. The probability of obtaining highly complex curves with several turning points or inflexions is rather lower for the enzyme mechanisms than with general 3 : 3 rational functions. There is a high probability that 3 : 3 mechanisms will generate kinetic curves that are geometrically similar to those possible for degree 2 : 2 but this is not so for binding isotherms. Hence differentiating 3 : 3 from 2 : 2 rate equations from experimental kinetic data is more likely to be successful by non-linear regression to the whole data set than by demonstrating a specific 3 : 3 feature. Binding curves, on the other hand, for three or more sites should give Scatchard plots with inflexions, features not possible with second degree equations which are conic sections in this space.     

Use of nonlinear regression to analyze enzyme kinetic data: application to situations of substrate contamination and background subtraction

In a recent publication, A. Lundin, P. Arner, and J. Hellmér [Anal. Biochem. 177, 125-131 (1989)] describe a method whereby kinetic substrate assays can be performed when the assay mixture includes a significant contaminating levels of substrate. Their method requires various rearrangements of the data, and involves three separate linear regression calculations. We show how the same data may be analyzed directly, and far more simply, by nonlinear regression. Unlike the linear regression method, nonlinear regression allows direct calculation of the actual values for Km, Vmax, and the concentration of contaminating substrate (as well as estimates of their standard errors); the former method gives only apparent values. The nonlinear regression technique is also statistically a more valid means of analysis, as the rearrangements required to give linearized equations will considerably distort the error distribution and render simple unweighted linear regression inappropriate. The ease of incorporating extra parameters into standard equations when nonlinear regression is used is further illustrated by fitting enzyme reaction data which describe a first-order process when a significant nonspecific background is present. For this equation no simple rearranged linear plot is possible, but nonlinear regression is easily applied to determine the kinetic parameters.     

Evolutionary optimization with data collocation for reverse engineering of biological networks

MOTIVATION: Modern experimental biology is moving away from analyses of single elements to whole-organism measurements. Such measured time-course data contain a wealth of information about the structure and dynamic of the pathway or network. The dynamic modeling of the whole systems is formulated as a reverse problem that requires a well-suited mathematical model and a very efficient computational method to identify the model structure and parameters. Numerical integration for differential equations and finding global parameter values are still two major challenges in this field of the parameter estimation of nonlinear dynamic biological systems. RESULTS: We compare three techniques of parameter estimation for nonlinear dynamic biological systems. In the proposed scheme, the modified collocation method is applied to convert the differential equations to the system of algebraic equations. The observed time-course data are then substituted into the algebraic system equations to decouple system interactions in order to obtain the approximate model profiles. Hybrid differential evolution (HDE) with population size of five is able to find a global solution. The method is not only suited for parameter estimation but also can be applied for structure identification. The solution obtained by HDE is then used as the starting point for a local search method to yield the refined estimates.     

Two new schematic rules for rate laws of enzyme-catalysed reactions

The schematic method, when applied to enzyme kinetics, can establish some direct relations between the calculation rules and the graphs expressing the enzyme-catalysed mechanisms. It not only makes the derivation of rate equations more convenient and intuitive, but also facilitates the analysis and discussion of the mechanisms.     

The structure of steady-state enzyme kinetic equations: a graph-theoretical algorithm for obtaining conditions for reduction in degree by common-factor cancellation

Many enzyme kinetic steady-state equations are so complicated that analysis of their predictions, preferably by graph-theoretical methods, without deriving the equations is desirable. An example of such a method is given here. It is a graph-theoretical algorithm for determining non-trivial relations between the rate constants of a mechanism that cause the numerator and denominator of its rate expression have a common factor so permitting the degree of the expression to be reduced. An algorithm for writing the several different forms of the reduced rate equation is also given. The algorithm is applied to some standard simple enzymic mechanisms that give relatively complicated rate equations.In the case of a 2:2 equation, the sign of curvature in a double reciprocal plot depends on the sign of the expression in the common-factor condition.     

Statistical theory of persuasion in tripartite negotiations

The interaction of three neutral subjects discussing a question is analyzed. The course of the debate is described by three first-order differential equations with coefficients depending on the characteristics of the subjects. Under plausible assumptions the equations are linear and can be solved explicitly. In general, the opinions of the participants approach a common limiting value which is a weighted average of their initial opinions. Special cases are considered in which one or two subjects keep their opinions constant during the discussion. The case of a coalition of two subjects against the third is shown to reduce to a two-party situation. A method of fitting the model to experimental data is suggested. Preliminary trials indicate that it is possible to obtain numerical observations to test this model of a three-party debate.     

Non-destructive biomass estimation of herbaceous plant individuals: A transferable method between contrasted environments

Monitoring plant growth at the individual level in arrays of environmental conditions is key to understanding plant functioning with strong implications for ecophysiology, population biology and community ecology. This requires non-destructive methods for repeated estimates of individual plant biomass in time. Although allometric equations have been widely used for trees and shrubs, there is currently no general approach for herbaceous species that can be applied across habitats, plant architecture, life stage and leading to transferable equations between contrasted environments. Here we propose a method based on three biometric measurements of the minimum volume occupied by aboveground plant organs. A total of 36 equations were fitted and compared for twelve species of temperate grasslands, corresponding to various volume shapes, scaling functions (linear or power) and including (or not) a life stage effect. The accuracy of the selected equations was compared to similar attempts reported in the literature. We further assessed the across-site transferability of the best allometric equations. The goodness-of-fit of the best equations selected for each species was high (̄R² = 0.83). The type of selected equations was species-specific, emphasising the benefits of considering a wide range of plant volume shapes and both linear and power functions. Using a comprehensive assessment of allometric equation transferability, we found that site effects could be neglected for eleven out of twelve species. Biomass equations based on the minimum volume proved accurate. The proposed method is easy to implement in any type of habitat, copes with various plant architectures and reduces risks of error measurement compared to previously developed approaches. The method further allows, for the first time, to use a single equation for monitoring the growth trajectory of herbaceous plant individuals in contrasted environments.     

A method for writing enzyme rate equations: application to the estimation of the number and size of key proton families

We develop a method to derive the rate equation for enzyme models that include pH-dependent activation. Our presentation is based on a kinetic model recently described for sucrase, the three-key-proton model of Vasseur and coworkers, which considers the existence, in the acid ionization reaction, of two functionally distinct prototropic groups, respectively responsible for either V-type or K-type kinetic effects. In contrast, as concerns the basic ionization reaction, the model conforms to classical concepts of pH-dependent activation, whereby a single proton participates in either V-type or K-type effects but not in both at the same time. Enzymes with more than three key protons have been described, indicating that, rather than isolated protons, groups of protons should be considered, and therefore the model can be better described as a three-proton-family model, where a proton family is defined as one or several protons that are gained or lost as a block and perform the same kinetic function. The resulting model is treated here as a useful framework upon which other models can be built. To facilitate the writing of the rate equations, we define two new entities: (1) intralevel coefficients, which describe the various combinations of the enzyme with either the substrate(s), the allosteric effector(s), or both at a given protonation level, and (2) interlevel coefficients, which describe the interplay between the various protonation levels. The resulting rate equation can be used in a global fit procedure permitting in a single computer run the estimation of (1) the entire set of dissociation and microscopic ionization constants of the model, (2) the number and kinetic function of proton families characterizing the enzyme under consideration, and (3) the number of key protons constituting each family, which is derived from the derivatives of the kinetic parameters, Vm/Km, Vm, and Km.     

A nonlinear regression program for small computers

A BASIC computer program for performing weighted nonlinear regression is described and a listing of the program is given. The program, which is small and simple to use, has been designed to be run by users with little knowledge of mathematics or computers. Robust methods of analysis are described which may be applied to data in which experimental errors are not normally distributed, and the program incorporates one such method. It is shown that the program is useful for the analysis of data conforming to the Michaelis-Menten equation, a single exponential, and to binding equations, and other applications are discussed.     

Three-dimensional simulation of anisotropic cell-driven collagen gel compaction

Tissue equivalents (TEs), formed by entrapping cells in a collagen gel, are an important model system for studying cell behavior. We have previously (Barocas and Tranquillo in J Biomech Eng 117:161–170, 1997a) developed an anisotropic biphasic theory of TE mechanics, which comprises five coupled partial differential equations describing interaction among cells and collagen fibers in the TE. The model equations, previously solved in one or two dimensions, were solved in three dimensions using an adaptive finite-element platform. The model was applied to three systems: a rectangular isometric cell traction assay, an otherwise- acellular gel containing two islands of cells, and an idealized tissue-engineered cardiac valve leaflet. In the first two cases, published experimental data were available for comparison, and the model results were consistent with the experimental observations. Fibers and cells aligned in the fixed direction in the isometric assay, and a region of strong fiber alignment arose between the two cell islands. For the valve problem, the alignment predicted by the model was generally similar to that observed experimentally, but an asymmetry in the experiment was not captured by the model.     

Is non‐loglinear allometry a statistical artifact?

The allometric equation, y = ax^b, is commonly fitted to data indirectly by transforming predictor (x) and response (y) variables to logarithms, fitting a straight line to the transformations, and then back‐transforming (exponentiating) the resulting equation to the original arithmetic scale. Sometimes, however, transformation fails to linearize the observations, thereby giving rise to what has come to be known as non‐loglinear allometry. A smooth curve for observations displayed on a log–log plot is usually interpreted to mean that the scaling exponent in the allometric equation is a continuously changing function of body size, whereas a breakpoint between two (or more) linear segments on a log–log plot is typically taken to mean that the exponent changes abruptly, coincident with some important milestone in development. I applied simple graphical and statistical procedures in re‐analyses of three well‐known examples of non‐loglinear allometry, and showed in every instance that the relationship between predictor and response can be described in the original scale by simple functions with constant values for the exponent b. In no instance does the allometric exponent change during the course of development. Transformation of data to logarithms created new distributions that actually obscured the relationships between predictor and response variables in these investigations, and led to erroneous perceptions of growth. Such confounding effects of transformation are not limited to non‐loglinear allometry but are common to all applications of the allometric method. © 2012 The Linnean Society of London, Biological Journal of the Linnean Society, 2012, ?? , ??–??.