Determination of kinetic parameters of enzyme-catalyzed reactions with a minimum number of velocity measurements

Duggleby [Duggleby, R.G., 1979. Experimental designs for estimating kinetic parameters for enzyme-catalyzed reactions. J. Theor. Biol. 81, 672-684] discussed the “design of several replicate measurements of the velocity at as many experimental conditions as there are parameters to be estimated.” He discussed the application of this method to A→products, without and with competitive inhibition, and commented briefly on A+B→products. The availability of computer applications that can solve large sets of simultaneous equations makes it possible to use this method to calculate kinetic parameters for more complicated enzyme mechanisms. This article is concerned with rapid-equilibrium rate equations, but this method can also be used with steady-state rate equations. Computer programs are provided for the calculation of the three kinetic parameters for ordered A+B→products from three velocity measurements and for the calculation of the four kinetic parameters for random A+B→products from four velocity measurements. Computer programs are also provided for competitive inhibition, uncompetitive inhibition, and mixed inhibition of ordered A+B→products.     

Efficient and accurate experimental design for enzyme kinetics: Bayesian studies reveal a systematic approach

In areas such as drug development, clinical diagnosis and biotechnology research, acquiring details about the kinetic parameters of enzymes is crucial. The correct design of an experiment is critical to collecting data suitable for analysis, modelling and deriving the correct information. As classical design methods are not targeted to the more complex kinetics being frequently studied, attention is needed to estimate parameters of such models with low variance. We demonstrate that a Bayesian approach (the use of prior knowledge) can produce major gains quantifiable in terms of information, productivity and accuracy of each experiment. Developing the use of Bayesian Utility functions, we have used a systematic method to identify the optimum experimental designs for a number of kinetic model data sets. This has enabled the identification of trends between kinetic model types, sets of design rules and the key conclusion that such designs should be based on some prior knowledge of K(M) and/or the kinetic model. We suggest an optimal and iterative method for selecting features of the design such as the substrate range, number of measurements and choice of intermediate points. The final design collects data suitable for accurate modelling and analysis and minimises the error in the parameters estimated.     

Efficient and cost-effective experimental determination of kinetic constants and data: the success of a Bayesian systematic approach to drug transport, receptor binding, continuous culture and cell transport kinetics

Details about the parameters of kinetic systems are crucial for progress in both medical and industrial research, including drug development, clinical diagnosis and biotechnology applications. Such details must be collected by a series of kinetic experiments and investigations. The correct design of the experiment is essential to collecting data suitable for analysis, modelling and deriving the correct information. We have developed a systematic and iterative Bayesian method and sets of rules for the design of enzyme kinetic experiments. Our method selects the optimum design to collect data suitable for accurate modelling and analysis and minimises the error in the parameters estimated. The rules select features of the design such as the substrate range and the number of measurements. We show here that this method can be directly applied to the study of other important kinetic systems, including drug transport, receptor binding, microbial culture and cell transport kinetics. It is possible to reduce the errors in the estimated parameters and, most importantly, increase the efficiency and cost-effectiveness by reducing the necessary amount of experiments and data points measured.     

A new procedure to derive weighting factors for nonlinear regression analysis applied to enzyme kinetic data.

The experimental variance of enzymic steady-state kinetic experiments depends on velocity as approximated by a power function (Var(v) = K1 . valpha (Askel?f, P., Korsfeldt, M. and Mannervik, B. (1976) Eur. J. Biochem. 69, 61--67). The values of the constants (K1, alpha) can be estimated by making replicate measurements of velocity, and the inverse of the function can then be used as a weighting factor. In order to avoid measurement of a large number of replicates to establish the error structure of a kinetic data set, a different approach was tested. After a preliminary regression using a 'good model', which satisfies reasonable goodness-of-fit criteria, the residuals were taken to represent the experimental error. The neighbouring residuals were grouped together and the sum of their mean squared values was used as a measure of the variance in the neighbourhood of the corresponding measurements. The values of the constants obtained in this way agreed with those obtained by replicates.     

Optimal fed batch experiment design for estimation of monod kinetics of Azospirillum brasilense: from theory to practice

In this paper the problem of reliable and accurate parameter estimation for unstructured models is considered. It is illustrated how a theoretically optimal design can be successfully translated into a practically feasible, robust, and informative experiment. The well-known parameter estimation problem of Monod kinetic parameters is used as a vehicle to illustrate our approach. As known for a long time, noisy batch measurements do not allow for unique and accurate estimation of the kinetic parameters of the Monod model. Techniques of optimal experiment design are, therefore, exploited to design informative experiments and to improve the parameter estimation accuracy. During the design process, practical feasibility has to be kept in mind. The designed experiments are easy to implement in practice and do not require additional monitoring equipment. Both design and experimental validation of informative fed batch experiments are illustrated with a case study, namely, the growth of the nitrogen-fixing bacteria Azospirillum brasilense.     

A kinetic model of cooperativity in aspartate transcarbamylase.

A relatively simple kinetic model is proposed to account simultaneously for data on the binding of carbamyl phosphate and succinate to aspartate trans carbamylase (ATCase), and for the relaxation spectrum associated with this binding. The model also accounts for measurements of the initial velocity of the reaction of ATCase with respect to aspartate and carbamyl phosphate. The principal assumption made is that ATCase consists of three identical noninteracting cooperative dimers. Ordered binding and both sequential and concerted conformational changes in the dimers are needed to account for the properties of ATCase. The values of the parameters of this model can be determined by fitting to existing experimental evidence. Various new quantitative predictions are made that can serve as additional tests of the proposed theory.     

Ligand loading at the surface of an optical biosensor and its effect upon the kinetics of protein–protein interactions

Optical biosensors are finding increasing use in the determination of kinetic and equilibrium constants for a variety of biomolecular interactions. Usually these biosensors require one biomolecule, the ligand, to be covalently attached to a hydrogel matrix which itself is bonded to the sensing surface. The ligands partner, the ligate, then binds from solution resulting in a measurable change in response which the instrument records as a function of time. Although in many cases, optical biosensors are used in order to obtain parameters that relate to interactions in solution, it is becoming clear that measurements involving the interaction of ligate with immobilized ligands on surfaces require careful experimental design. Here we report on how the density of ligand loading within the hydogel matrix affects the measured interaction kinetics. It is found that crowding of ligand within this matrix results in a significant reduction in the measured association rate constant, with a corresponding effect in the calculated overall affinity. However, measurements at low ligand loadings show association rate constants that are comparable to those measured in solution. Clearly, where this comparison is required, it is important to perform measurements under such conditions. © 1997 John Wiley & Sons, Ltd.     

Analysis and application of ADM1 for anaerobic methane production

An anaerobic model for the serum bottle test was developed and analyzed with sensitivities of stoichiometric and kinetic parameters to the components in order to establish a basis for appropriate application of the model. Anaerobic glucose degradation in a serum bottle was selected as an example. The anaerobic model was developed based on the anaerobic digestion model no. 1 (ADM1), which had five processes with 17 kinetic and stoichiometric parameters. Sensitivity analysis showed that the yield of product on the substrate (f) has high sensitivities to model components, and that the methane concentration was the most sensitive component. Important parameters including yield of product on the substrate (f), yield of biomass on the substrate (Y), and half-saturation values (K) were estimated using genetic algorithms, which optimized the parameters with experimental results. The Monod maximum specific uptake rate (k) was, however, so strongly associated with the concentration of biomass, that values could not be estimated individually. Simulation with estimated parameters showed good agreement with experimental results in the case of methane production. However, there were some differences in acetate and propionate concentrations.     

A potential role for isothermal calorimetry in studies of the effects of thermodynamic non-ideality in enzyme-catalyzed reactions

Attention is drawn to the feasibility of using isothermal calorimetry for the characterization of enzyme reactions under conditions bearing greater relevance to the crowded biological environment, where kinetic parameters are likely to differ significantly from those obtained by classical enzyme kinetic studies in dilute solution. An outline of the application of isothermal calorimetry to the determination of enzyme kinetic parameters is followed by considerations of the nature and consequences of crowding effects in enzyme catalysis. Some of those effects of thermodynamic non-ideality are then illustrated by means of experimental results from calorimetric studies of the effect of molecular crowding on the kinetics of catalysis by rabbit muscle pyruvate kinase. This review concludes with a discussion of the potential of isothermal calorimetry for the experimental determination of kinetic parameters for enzymes either in biological environments or at least in media that should provide reasonable approximations of the crowded conditions encountered in vivo.     

Simultaneous versus sequential optimal experiment design for the identification of multi-parameter microbial growth kinetics as a function of temperature

Optimal experiment design for parameter estimation (OED/PE) has become a popular tool for efficient and accurate estimation of kinetic model parameters. When the kinetic model under study encloses multiple parameters, different optimization strategies can be constructed. The most straightforward approach is to estimate all parameters simultaneously from one optimal experiment (single OED/PE strategy). However, due to the complexity of the optimization problem or the stringent limitations on the system's dynamics, the experimental information can be limited and parameter estimation convergence problems can arise. As an alternative, we propose to reduce the optimization problem to a series of two-parameter estimation problems, i.e., an optimal experiment is designed for a combination of two parameters while presuming the other parameters known. Two different approaches can be followed: (i) all two-parameter optimal experiments are designed based on identical initial parameter estimates and parameters are estimated simultaneously from all resulting experimental data (global OED/PE strategy), and (ii) optimal experiments are calculated and implemented sequentially whereby the parameter values are updated intermediately (sequential OED/PE strategy).This work exploits OED/PE for the identification of the Cardinal Temperature Model with Inflection (CTMI) (Rosso et al., 1993). This kinetic model describes the effect of temperature on the microbial growth rate and encloses four parameters. The three OED/PE strategies are considered and the impact of the OED/PE design strategy on the accuracy of the CTMI parameter estimation is evaluated. Based on a simulation study, it is observed that the parameter values derived from the sequential approach deviate more from the true parameters than the single and global strategy estimates. The single and global OED/PE strategies are further compared based on experimental data obtained from design implementation in a bioreactor. Comparable estimates are obtained, but global OED/PE estimates are, in general, more accurate and reliable.     

Development and validation of a kinetic model for enzymatic saccharification of lignocellulosic biomass

A multireaction kinetic model was developed for closed-system enzymatic hydrolysis of lignocellulosic biomass such as corn stover. Three hydrolysis reactions were modeled, two heterogeneous reactions for cellulose breakdown to cellobiose and glucose and one homogeneous reaction for hydrolyzing cellobiose to glucose. Cellulase adsorption onto pretreated lignocellulose was modeled via a Langmuir-type isotherm. The sugar products of cellulose hydrolysis, cellobiose and glucose, as well as xylose, the dominant sugar prevalent in most hemicellulose hydrolyzates, were assumed to competitively inhibit the enzymatic hydrolysis reactions. Model parameters were estimated from experimental data generated using dilute acid pretreated corn stover as the substrate. The model performed well in predicting cellulose hydrolysis trends at experimental conditions both inside and outside the design space used for parameter estimation and can be used for in silico process optimization.     

Determination of the intrinsic kinetic constants of immobilized glucose oxidase and catalase

Models of membrane systems containing immobilized glucose oxidase and catalase operating together or independently have been developed. A rotated disk electrode apparatus was employed with novel electrochemical operating conditions to experimentally determine mass transfer and intrinsic kinetic parameters of enzyme-containing membranes. The value of a mass transfer parameter that describes internal and external diffusion was first determined under conditions that do not permit the enzyme reactions. In a subsequent experiment with the reaction allowed, kinetic parameters corresponding to the intrinsic maximal velocity and Michaelis constants of the immobilized enzymes were estimated by regression analysis of data based on an appropriate two- or three- parameter model. It was found that immobilization reduced the maximal intrinsic velocity but had no detectable effect on the Michaelis constants. In all but one case- these methods for membrane characterization are nondestructive and can be used repeatedly on a given membrane. These techniques provide the means for quantitative comparisons of immobilization methods and make possible temporal studies of immobilized enzyme inactivation.     

Performance of tapered column packed-bed bioreactor for ethanol production

A tapered column type of bioreactor system packed with immobilized Saccharomyces cerevisiae was used to study the bioreactor performance as a function of design and operating variables. The performance of tapered column bioreactor system was found to be better than that of the conventional cylindrical column reactor system for the ethanol fermentation. The new bioreactor design alleviated problems associated with carbon dioxide evolution and provided a significantly better flow pattern for both liquid and gas phases in the bioreactor without local channelling. A mathematical simulation model, which takes into account of the axial convection and dispersion, interphase mass transfer, and apparent kinetic design parameters, was developed. The effect of radial concentration gradients on the bioreactor performance was found to be insignificant. For the reactor system studied, the maximum ethanol productivity obtained was 60 g ethanol/L gel/h, and the maximum glucose assimilation rate was 140 g glucose/L gel/h. One of the most important findings from this study was that the apparent kinetic parameters change at the glucose concentration of 2 g/L This change was found to be due to the changes in yeast physiology and metabolism. The values of V(m) (') and V(m) (') decreased from 0.8 to 0.39 g ethanol/g cell/h and from 97mM to 11mM, respectively. The substrate inhibition constant was estimated as 0.76M and the product inhibition constant was determined as 113 g ethanol/L The degree of product inhibition showed practically a linear relationship with an increasing ethanol concentration. Based on the hydro-dynamic analysis of the bioreactor system, it was found that the Peclet number, N(Pe) was not a strong function of the flow velocity at low flow rates through the bioreactor system, but its value decreased somewhat at an interstitial velocity greater than 0.03 cm/s. The tapered column bioreactor system showed a much better flow pattern of gas and liquid phases within the reactor, thereby providing a more homogeneous distribution of gas-liquid-solid phases in the reactor without any phase separation.     

Kinetic model for the bioconversion of glucose to 2,5-diketo-D-gluconic acid

Summary A simple unstructured mathematical model for the oxidation of glucose to 2,5-diketo-D-gluconic acid with Erwinia citreus was developed. The kinetic parameters of the model were estimated by the oxygen partial pressure method and directly from the temporal response obtained from the experimental data collected in the batch fermentation. The simple developed model based on the kinetic measurements was able to simulate quite well the dynamic behaviour of the batch fermentation.     

Evaluation of Medium Components by Plackett-Burman Statistical Design for Lipase Production by Candida rugosa and Kinetic Modeling

Lipase production by Candida rugosa was carried out in submerged fermentation. Plackett-Burman statistical experimental design was applied to evaluate the fermentation medium components. The effect of twelve medium components was studied in sixteen experimental trials. Glucose, olive oil, peptone and FeCl3?6H2O were found to have more significance on lipase production by Candida rugosa. Maximum lipase activity of 3.8 u mL-1 was obtained at 50 h of fermentation period. The fermentation was carried out at optimized temperature of 30oC, initial pH of 6.8 and shaking speed of 120 r/min. Unstructured kinetic models were used to simulate the experimental data. Logistic model, Luedeking-Piret model and modified Luedeking-Piret model were found suitable to efficiently predict the cell mass, lipase production and glucose consumption respectively with high determination coefficient(R2). From the estimated values of the Luedeking-Piret kinetic model parameters, α and β, it was found that the lipase production by Candida rugosa is growth associated.     

A novel cost function to estimate parameters of oscillatory biochemical systems

Oscillatory pathways are among the most important classes of biochemical systems with examples ranging from circadian rhythms and cell cycle maintenance. Mathematical modeling of these highly interconnected biochemical networks is needed to meet numerous objectives such as investigating, predicting and controlling the dynamics of these systems. Identifying the kinetic rate parameters is essential for fully modeling these and other biological processes. These kinetic parameters, however, are not usually available from measurements and most of them have to be estimated by parameter fitting techniques. One of the issues with estimating kinetic parameters in oscillatory systems is the irregularities in the least square (LS) cost function surface used to estimate these parameters, which is caused by the periodicity of the measurements. These irregularities result in numerous local minima, which limit the performance of even some of the most robust global optimization algorithms. We proposed a parameter estimation framework to address these issues that integrates temporal information with periodic information embedded in the measurements used to estimate these parameters. This periodic information is used to build a proposed cost function with better surface properties leading to fewer local minima and better performance of global optimization algorithms. We verified for three oscillatory biochemical systems that our proposed cost function results in an increased ability to estimate accurate kinetic parameters as compared to the traditional LS cost function. We combine this cost function with an improved noise removal approach that leverages periodic characteristics embedded in the measurements to effectively reduce noise. The results provide strong evidence on the efficacy of this noise removal approach over the previous commonly used wavelet hard-thresholding noise removal methods. This proposed optimization framework results in more accurate kinetic parameters that will eventually lead to biochemical models that are more precise, predictable, and controllable.     

A simple method for measuring stiffness during running

The spring-mass model, representing a runner as a point mass supported by a single linear leg spring, has been a widely used concept in studies on running and bouncing mechanics. However, the measurement of leg and vertical stiffness has previously required force platforms and high-speed kinematic measurement systems that are costly and difficult to handle in field conditions. We propose a new "sine-wave" method for measuring stiffness during running. Based on the modeling of the force-time curve by a sine function,this method allows leg and vertical stiffness to be estimated from just a few simple mechanical parameters: body mass, forward velocity, leg length, flight time, and contact time. We compared this method to force-platform-derived stiffness measurements for treadmill dynamometer and overground running conditions, at velocities ranging from 3.33 m.s-1 to maximal running velocity in both recreational and highly trained runners. Stiffness values calculated with the proposed method ranged from 0.67 % to 6.93 % less than the force platform method, and thus were judged to be acceptable. Furthermore, significant linear regressions (p < 0.01) close to the identity line were obtained between force platform and sine-wave model values of stiffness. Given the limits inherent in the use of the spring-mass model, it was concluded that this sine-wave method allows leg and stiffness estimates in running on the basis of a few mechanical parameters, and could be useful in further field measurements.     

Application of dynamic metabolic flux analysis for process modeling: Robust flux estimation with regularization,confidence bounds,and selection of elementary modes

In macroscopic dynamic models of fermentation processes, elementary modes (EM) derived from metabolic networks are often used to describe the reaction stoichiometry in a simplified manner and to build predictive models by parameterizing kinetic rate equations for the EM. In this procedure, the selection of a set of EM is a key step which is followed by an estimation of their reaction rates and of the associated confidence bounds. In this paper, we present a method for the computation of reaction rates of cellular reactions and EM as well as an algorithm for the selection of EM for process modeling. The method is based on the dynamic metabolic flux analysis (DMFA) proposed by Leighty and Antoniewicz (2011, Metab Eng, 13(6), 745–755) with additional constraints, regularization and analysis of uncertainty. Instead of using estimated uptake or secretion rates, concentration measurements are used directly to avoid an amplification of measurement errors by numerical differentiation. It is shown that the regularized DMFA for EM method is significantly more robust against measurement noise than methods using estimated rates. The confidence intervals for the estimated reaction rates are obtained by bootstrapping. For the selection of a set of EM for a given st oichiometric model, the DMFA for EM method is combined with a multiobjective genetic algorithm. The method is applied to real data from a CHO fed-batch process. From measurements of six fed-batch experiments, 10 EM were identified as the smallest subset of EM based upon which the data can be described sufficiently accurately by a dynamic model. The estimated EM reaction rates and their confidence intervals at different process conditions provide useful information for the kinetic modeling and subsequent process optimization.