Global Kinetic Explorer: A new computer program for dynamic simulation and fitting of kinetic data

We describe a new dynamic kinetic simulation program that allows multiple data sets to be fit simultaneously to a single model based on numerical integration of the rate equations describing the reaction mechanism. Unlike other programs that allow fitting based on numerical integration of rate equations, in the dynamic simulation rate constants, output factors, and starting concentrations of reactants can be scrolled while observing the change in the shape of the simulated reaction curves. Fast dynamic simulation facilitates the exploration of initial parameters that serve as the starting point for nonlinear regression in fitting data and facilitates exploration of the relationships between individual constants and observable reactions. The exploration of parameter space by dynamic simulation provides a powerful tool for learning kinetics and for evaluating the extent to which parameters are constrained by the data. This feature is critical to avoid overly complex models that are not supported by the data.     

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.     

Analysis of self-associating proteins by singular value decomposition of solution scattering data

We describe a method by which a single experiment can reveal both association model (pathway and constants) and low-resolution structures of a self-associating system. Small-angle scattering data are collected from solutions at a range of concentrations. These scattering data curves are mass-weighted linear combinations of the scattering from each oligomer. Singular value decomposition of the data yields a set of basis vectors from which the scattering curve for each oligomer is reconstructed using coefficients that depend on the association model. A search identifies the association pathway and constants that provide the best agreement between reconstructed and observed data. Using simulated data with realistic noise, our method finds the correct pathway and association constants. Depending on the simulation parameters, reconstructed curves for each oligomer differ from the ideal by 0.05-0.99% in median absolute relative deviation. The reconstructed scattering curves are fundamental to further analysis, including interatomic distance distribution calculation and low-resolution ab initio shape reconstruction of each oligomer in solution. This method can be applied to x-ray or neutron scattering data from small angles to moderate (or higher) resolution. Data can be taken under physiological conditions, or particular conditions (e.g., temperature) can be varied to extract fundamental association parameters (ΔH_ass, ΔS_ass).     

General rate equations and rejection criteria for the rapid equilibrium carrier model of cotransport

It is demonstrated that under fixed activator conditions the general flux equation for the rapid equilibrium carrier model of cotransport can be written entirely in terms of five independent kinetic constants. Thus the kinetic parameters from any experiment carried out under the same activator conditions must necessarily be expressible in terms of these five constants. These predicted relationships between experimental kinetic parameters provide rejection criteria for the model, a number of which are derived here. Generalization of the treatment to the case where a competitive substrate is present on both sides of the membrane is also given.     

Coupling of protein assembly and DNA binding: biotin repressor dimerization precedes biotin operator binding

The kinetics of coupling of protein dimerization and DNA binding have been investigated in the biotin repressor system. Two repressor monomers bind to the 40 base-pair biotin operator sequence. In previous analyses of equilibrium-binding data the weak dimerization of the repressor has justified using a model in which two protein monomers bind cooperatively to the operator site. Here, rapid kinetic methods have been used to directly determine the binding mechanism. Results of rapid-mixing DNaseI footprinting measurements of association of the repressor with operator indicate that the binding process involves at least two steps. Results of measurements of the unimolecular dissociation of the complex reveal a half-life of approximately 400 seconds. Analysis of the data using a combination of simulation and global non-linear least-squares analysis provides support for a binding model in which a preformed repressor dimer associates with the biotin operator. This kinetic model is consistent with the previously proposed model for regulation of the functional switch in the repressor from enzyme to site-specific DNA-binding protein.     

Statistical considerations in the estimation of enzyme kinetic parameters by the direct linear plot and other methods

The statistical implications of the direct linear plot for enzyme kinetic data, described in the preceding paper (Eisenthal & Cornish-Bowden, 1974), are discussed for the case of the Michaelis-Menten equation. The plot is shown to lead directly to non-parametric confidence limits for the kinetic parameters, V and K(m), which depend on far less sweeping assumptions about the nature of experimental error than those implicit in the method of least squares. Median estimates of V and K(m) can also be defined, which are shown to be more robust than the least-squares estimates in a wide variety of experimental situations.     

Rational polynomial equation as an unbiased approach for the kinetic studies of Drosophila melanogaster acetylcholinesterase reaction mechanism

The hydrolysis of substrates by cholinesterases does not follow the Michaelis–Menten reaction mechanism. The well-known inhibition by excess substrate is often accompanied by an unexpectedly high activity at low substrate concentrations. It appears that these peculiarities are the consequence of an unusual architecture of the active site, which conducts the substrate molecule over many stages before it is cleaved and released. Structural and kinetic data also suggest that two substrate molecules can attach at the same time to the free, as well as to the acetylated, enzyme. We present a procedure which provides an unbiased framework for mathematical modelling of such complex reaction mechanisms. It is based on regression analysis of a rational polynomial using classical initial rate data. The determination of polynomial degree reveals the number of independent parameters that can be evaluated from the available information. Once determined, these parameters can substantially facilitate the construction and evaluation of a kinetic model reflecting the expected molecular events in an enzymic reaction. We also present practical suggestions for testing the postulated kinetic model, using an original thermodynamic approach and an isolated effect in a specifically mutated enzyme.     

Development and validation of a kinetic assay for analysis of anti-human interleukin-5 monoclonal antibody (SCH 55700) and human interleukin-5 interactions using surface plasmon resonance

A method to assess the kinetic interactions of a humanized anti-human interleukin-5 (IL-5) monoclonal antibody (SCH 55700) with native human IL-5 using surface plasmon resonance (SPR) has been developed and validated. Since there are no clearly defined validation requirements for a SPR-based binding kinetic assay, the validation strategy was based on the guidelines stipulated by the International Conference on Harmonization for Analytical Method Validation. Due to the uniqueness of the method, however, proper interpretation of the guidance was critical for establishing a validation plan. Validation was designed to assess repeatability, intermediate precision, specificity, linearity, and robustness which included analysis of baseline stability and reproducibility of ligand immobilization. Additionally, system suitability criteria were established to assure that the assay consistently performs as it was intended. The experimental artifacts that can complicate kinetic analysis using biosensor technology, such as heterogeneity of the ligand, mass transport, and nonspecific binding, were considered during the development of this assay. For each run, replicate concentrations of SCH 55700 were injected randomly over the immobilized surfaces to acquire association- and dissociation-phase data. The data were transformed and double referenced to remove systematic deviations seen in the binding responses. Association and dissociation rates were determined using a bivalent analyte model for curve fitting.     

Dynamic simulation of an in vitro multi-enzyme system

Parameters often are tuned with metabolite concentration time series data to build a dynamic model of metabolism. However, such tuning may reduce the extrapolation ability (generalization capability) of the model. In this study, we determined detailed kinetic parameters of three purified Escherichia coli glycolytic enzymes using the initial velocity method for individual enzymes; i.e., the parameters were determined independently from metabolite concentration time series data. The metabolite concentration time series calculated by the model using the parameters matched the experimental data obtained in an actual multi-enzyme system consisting of the three purified E. coli glycolytic enzymes. Thus, the results indicate that kinetic parameters can be determined without using an undesirable tuning process.     

Kinetic studies of the lactic acid fermentation in batch and continuous cultures

The fermentation kinetics of the homofermentative organism Lactobacillus delbrueckii in a glucose-yeast extract medium is studied in both batch and continuous culture under conditions of controlled pH. From a graphical analysis of the batch data, a mathematical model of the process is derived which relates bacterial growth, glucose utilization, and lactic acid formation. The parameters in the model represent the activity of the organism and are a function of pH, having a maximum value at about 5.90. In a continuous stirred tank fermentor (CSTF), the effect of pH, feed concentration, and residence time is observed. The feed medium is a constant ratio of two parts glucose to one part yeast extract plus added mineral salts. An approximate prediction of the steady-state behavior of the CSTF can be made using a method based on the kinetic model derived for the batch case. In making step changes from one steady state to another, the transient response is observed. Using the kinetic model to simulate the transient period, the calculated behavior qualitatively predicts the observed response.     

Comparative Analysis of Linear and Nonlinear Methods of Estimating the Pseudo-Second-Order Kinetic Parameters for Sorption of Malachite Green onto Pretreated Rice Husk

Batch sorption experiments were carried out for removal of malachite green from aqueous solution using pretreated rice husk. The equilibrium kinetic data were analyzed using pseudo-second-order kinetic model. A comparison between linear and nonlinear methods of estimating the kinetic parameters was carried out. Four pseudo-second-order kinetic linear equations were discussed. The coefficient of determination (r ²) and chi-square (χ²) test were employed as error analysis methods to determine the best-fitting equation. The results show that nonlinear method is a better alternative to obtain the kinetic parameters. In addition, the chi-square test was a better method to determine the best-fitting model.     

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.     

Retrieval of seasonal variation in photosynthetic capacity from multi-source vegetation indices

Seasonal information on photosynthetic-capacity parameters (maximum carboxylation velocity, V_cmax; and maximum rate of electron transport, J_max) plays an important role in accurate simulation of carbon fixation in gas-exchange models. Exact inclusion of seasonal information on photosynthetic-capacity parameters into the models has been an irresolvable challenge. This paper investigated the relationships between vegetation indices (from multiple sources) and photosynthetic-capacity parameters of three beech forest stands (Fagus crenata) along an elevation gradient in the cold-temperate zone of Japan, over the entire growing season of 2006. Diverse vegetation indices were examined in terms of spectral, spatial and temporal scales; ranging from meteorological sensor-based broadband indices to hyperspectral data-based narrowband indices, to simulated MODIS (MODerate-resolution Imaging Spectroradiometer) indices based on hyperspectral data, and finally satellite-borne MODIS vegetation indices. Regression analysis revealed that all examined indices, with the exception of the downloaded MODIS products, had significant regression relationships with photosynthetic parameters (P < 0.001) when all data were pooled. Among the different indices, the simulated MODIS NDVI (Normalized Difference Vegetation Index) performed the best for both V_cmax and J_max (R² = 0.81 and 0.73, respectively). Site differences were apparent, as the simulated MODIS NDVI performed the best in exponential regressions for the 550 m site, while broadband NDVI performed best in exponential regression models for the 900 m site. The broadband SR (Simple Ratio) in relation to V_cmax performed best with respect to a linear model, whereas the broadband NDVI with J_max performed the best in an exponential model for the 1500 m site. The results reveal that vegetation indices which are obtained across different scales nevertheless retain tight relationships with canopy-scale photosynthetic-capacity parameters. The established relationships were inversely applicable to derive seasonal trajectories of photosynthetic-capacity parameters. Thus, new insight and confidence is gained for using remotely estimated photosynthetic parameters, even though most previous research works were limited on linking of vegetation indices with biophysical parameters. The control effect of physiological capacity on reflectance and further on vegetation indices has not been adequately established and thus needs further orientation for rigorous research work.     

Symbolic-numeric estimation of parameters in biochemical models by quantifier elimination

The sequencing of complete genomes allows analyses of the interactions between various biological molecules on a genomic scale, which prompted us to simulate the global behaviors of biological phenomena on the molecular level. One of the basic mathematical problems in the simulation is the parameter optimization in the kinetic model for complex dynamics, and many estimation methods have been designed. We introduce a new approach to estimate the parameters in biological kinetic models by quantifier elimination (QE), in combination with numerical simulation methods. The estimation method was applied to a model for the inhibition kinetics of HIV proteinase with ten parameters and nine variables, and attained the goodness of fit to 300 points of observed data with the same magnitude as that obtained by the previous estimation methods, remarkably by using only one or two points of data. Furthermore, the utilization of QE demonstrated the feasibility of the present method for elucidating the behavior of the parameters and the variables in the analyzed model. Therefore, the present symbolic-numeric method is a powerful approach to reveal the fundamental mechanisms of kinetic models, in addition to being a computational engine.     

Practical limits for reverse engineering of dynamical systems: a statistical analysis of sensitivity and parameter inferability in systems biology models

The size and complexity of cellular systems make building predictive models an extremely difficult task. In principle dynamical time-course data can be used to elucidate the structure of the underlying molecular mechanisms, but a central and recurring problem is that many and very different models can be fitted to experimental data, especially when the latter are limited and subject to noise. Even given a model, estimating its parameters remains challenging in real-world systems. Here we present a comprehensive analysis of 180 systems biology models, which allows us to classify the parameters with respect to their contribution to the overall dynamical behaviour of the different systems. Our results reveal candidate elements of control in biochemical pathways that differentially contribute to dynamics. We introduce sensitivity profiles that concisely characterize parameter sensitivity and demonstrate how this can be connected to variability in data. Systematically linking data and model sloppiness allows us to extract features of dynamical systems that determine how well parameters can be estimated from time-course measurements, and associates the extent of data required for parameter inference with the model structure, and also with the global dynamical state of the system. The comprehensive analysis of so many systems biology models reaffirms the inability to estimate precisely most model or kinetic parameters as a generic feature of dynamical systems, and provides safe guidelines for performing better inferences and model predictions in the context of reverse engineering of mathematical models for biological systems.     

Studying apolipoprotein turnover with stable isotope tracers: correct analysis is by modeling enrichments

Lipoprotein kinetic parameters are determined from mass spectrometry data after administering mass isotopes of amino acids, which label proteins endogenously. The standard procedure is to model the isotopic content of the labeled precursor amino acid and of proteins of interest as tracer-to-tracee ratio (TTR). It is shown here that even though the administered tracer alters amino acid mass and turnover, apolipoprotein synthesis is unaltered and hence the apolipoprotein system is in a steady state, with the total (labeled plus unlabeled) masses and fluxes remaining constant. The correct model formulation for apolipoprotein kinetics is shown to be in terms of tracer enrichment, not of TTR. The needed mathematical equations are derived. A theoretical error analysis is carried out to calculate the magnitude of error in published results using TTR modeling. It is shown that TTR modeling leads to a consistent underestimation of the fractional synthetic rate. In constant-infusion studies, the bias error percent is shown to equal approximately the plateau enrichment, generally <10%. It is shown that, in bolus studies, the underestimation error can be larger. Thus, for mass isotope studies with endogenous tracers, apolipoproteins are in a steady state and the data should be fitted by modeling enrichments.     

Error structure as a function of substrate and inhibitor concentration in enzyme kinetic experiments.

Optimal design of experiments as well as proper analysis of data are dependent on knowledge of the experimental error. A detailed analysis of the error structure of kinetic data obtained with acetylcholinesterase showed conclusively that the classical assumptions of constant absolute or constant relative error are inadequate for the dependent variable (velocity). The best mathematical models for the experimental error involved the substrate and inhibitor concentrations and reflected the rate law for the initial velocity. Data obtained with other enzymes displayed similar relationships between experimental error and the independent variables. The new empirical error functions were shown superior to previously used models when utilized in weighted non-linear-regression analysis of kinetic data. The results suggest that, in the spectrophotometric assays used in the present study, the observed experimental variance is primarily due to errors in determination of the concentrations of substrate and inhibitor and not to error in measuring the velocity.     

Confidence Intervals for Concentration and Brightness from Fluorescence Fluctuation Measurements

The theory of photon count histogram (PCH) analysis describes the distribution of fluorescence fluctuation amplitudes due to populations of fluorophores diffusing through a focused laser beam and provides a rigorous framework through which the brightnesses and concentrations of the fluorophores can be determined. In practice, however, the brightnesses and concentrations of only a few components can be identified. Brightnesses and concentrations are determined by a nonlinear least-squares fit of a theoretical model to the experimental PCH derived from a record of fluorescence intensity fluctuations. The χ² hypersurface in the neighborhood of the optimum parameter set can have varying degrees of curvature, due to the intrinsic curvature of the model, the specific parameter values of the system under study, and the relative noise in the data. Because of this varying curvature, parameters estimated from the least-squares analysis have varying degrees of uncertainty associated with them. There are several methods for assigning confidence intervals to the parameters, but these methods have different efficacies for PCH data. Here, we evaluate several approaches to confidence interval estimation for PCH data, including asymptotic standard error, likelihood joint-confidence region, likelihood confidence intervals, skew-corrected and accelerated bootstrap (BCa), and Monte Carlo residual resampling methods. We study these with a model two-dimensional membrane system for simplicity, but the principles are applicable as well to fluorophores diffusing in three-dimensional solution. Using simulated fluorescence fluctuation data, we find the BCa method to be particularly well-suited for estimating confidence intervals in PCH analysis, and several other methods to be less so. Using the BCa method and additional simulated fluctuation data, we find that confidence intervals can be reduced dramatically for a specific non-Gaussian beam profile.     

Pseudo-Second-Order Kinetic Model for Biosorption of Methylene Blue onto Tamarind Fruit Shell: Comparison of Linear and Nonlinear Methods

In this study, the sorption of methylene blue, a basic dye, onto tamarind fruit shell was studied by performing batch kinetic sorption experiments. The equilibrium kinetic data were analyzed using the pseudo-second-order kinetic model. A comparison between linear least squares method and nonlinear regression method of estimating the kinetic parameters was examined. Four pseudo-second-order kinetic linear equations were discussed. The coefficient of determination (r ²), and the chi-square (χ²) test were employed as error analysis methods to determine the best-fitting equation. Kinetic parameters obtained from four kinetic linear equations using the linear method differed but they were the same when nonlinear method was used. Present investigation showed that by linear method a Type 1 expression very well represent the kinetic uptake of methylene blue onto tamarind fruit shell. Linear method was found to check only the hypothesis instead of verifying the kinetic model. Nonlinear regression method was found to be the more appropriate method to determine the rate kinetic parameters.