Bayesian-Estimated Hierarchical HMMs Enable Robust Analysis of Single-Molecule Kinetic Heterogeneity

Single-molecule kinetic experiments allow the reaction trajectories of individual biomolecules to be directly observed, eliminating the effects of population averaging and providing a powerful approach for elucidating the kinetic mechanisms of biomolecular processes. A major challenge to the analysis and interpretation of these experiments, however, is the kinetic heterogeneity that almost universally complicates the recorded single-molecule signal versus time trajectories (i.e., signal trajectories). Such heterogeneity manifests as changes and/or differences in the transition rates that are observed within individual signal trajectories or across a population of signal trajectories. Because characterizing kinetic heterogeneity can provide critical mechanistic information, we have developed a computational method that effectively and comprehensively enables such analysis. To this end, we have developed a computational algorithm and software program, hFRET, that uses the variational approximation for Bayesian inference to estimate the parameters of a hierarchical hidden Markov model, thereby enabling robust identification and characterization of kinetic heterogeneity. Using simulated signal trajectories, we demonstrate the ability of hFRET to accurately and precisely characterize kinetic heterogeneity. In addition, we use hFRET to analyze experimentally recorded signal trajectories reporting on the conformational dynamics of ribosomal pre-translocation (PRE) complexes. The results of our analyses demonstrate that PRE complexes exhibit kinetic heterogeneity, reveal the physical origins of this heterogeneity, and allow us to expand the current model of PRE complex dynamics. The methods described here can be applied to signal trajectories generated using any type of signal and can be easily extended to the analysis of signal trajectories exhibiting more complex kinetic behaviors. Moreover, variations of our approach can be easily developed to integrate kinetic data obtained from different experimental constructs and/or from molecular dynamics simulations of a biomolecule of interest.     

k-Cone analysis: determining all candidate values for kinetic parameters on a network scale

The absence of comprehensive measured kinetic values and the observed inconsistency in the available in vitro kinetic data has hindered the formulation of network-scale kinetic models of biochemical reaction networks. To meet this challenge we present an approach to construct a convex space, termed the k-cone, which contains all the allowable numerical values of the kinetic constants in large-scale biochemical networks. The definition of the k-cone relies on the incorporation of in vivo concentration data and a simplified approach to represent enzyme kinetics within an established constraint-based modeling approach. The k-cone approach was implemented to define the allowable combination of numerical values for a full kinetic model of human red blood cell metabolism and to study its correlated kinetic parameters. The k-cone approach can be used to determine consistency between in vitro measured kinetic values and in vivo concentration and flux measurements when used in a network-scale kinetic model. k-Cone analysis was successful in determining whether in vitro measured kinetic values used in the reconstruction of a kinetic-based model of Saccharomyces cerevisiae central metabolism could reproduce in vivo measurements. Further, the k-cone can be used to determine which numerical values of in vitro measured parameters are required to be changed in a kinetic model if in vivo measured values are not reproduced. k-Cone analysis could identify what minimum number of in vitro determined kinetic parameters needed to be adjusted in the S. cerevisiae model to be consistent with the in vivo data. Applying the k-cone analysis a priori to kinetic model development may reduce the time and effort involved in model building and parameter adjustment. With the recent developments in high-throughput profiling of metabolite concentrations at a whole-cell scale and advances in metabolomics technologies, the k-cone approach presented here may hold the promise for kinetic characterization of metabolic networks as well as other biological functions at a whole-cell level.     

Microcomputer simulation of steady-state enzyme kinetics for educational purposes

A BASIC program to assist the instruction of steady-state enzymekinetics has been developed for the IBM PC microcomputer. Itspurpose is to simulate laboratory experiments in order to minimizethe time required to obtain kinetic data from which studentsdeduce kinetic mechanisms and determine kinetic constants ofenzyme-catalyzed reactions. The program randomly selects a kineticscheme from various sequential, ping pong, and iso reactionsequences as well as values for the kinetic constants. The schemeand kinetic constants are unknown to the student at this time;the only thing he or she knows is the stoichiometry of the catalyzedreaction which can have two or three substrates and products.The student is prompted to enter values for concentrations ofsubstrates and products; several different concentrations foreach substrate and product can be entered in a single experiment.The program then calculates, displays and prints (if desired)the corresponding initial steadystate velocities. The studentcan perform as many experiments as desired until enough informationis obtained to determine the kinetic mechanism and to calculatevalues for the kinetic constants. Received on March 10, 1986; accepted on May 6, 1986     

Incorporating qualitative knowledge in enzyme kinetic models using fuzzy logic

Modeling of metabolic pathway dynamics requires detailed kinetic equations at the enzyme level. In particular, the kinetic equations must account for metabolite effectors that contribute significantly to the pathway regulation in vivo. Unfortunately, most kinetic rate laws available in the literature do not consider all the effectors simultaneously, and much kinetic information exists in a qualitative or semiquantitative form. In this article, we present a strategy to incorporate such information into the kinetic equation. This strategy uses fuzzy logic‐based factors to modify algebraic rate laws that account for partial kinetic characteristics. The parameters introduced by the fuzzy factors are then optimized by use of a hybrid of simplex and genetic algorithms. The resulting model provides a flexible form that can simulate various kinetic behaviors. Such kinetic models are suitable for pathway modeling without complete enzyme mechanisms. Three enzymes in Escherichia coli central metabolism are used as examples: phosphoenolpyruvate carboxylase; phosphoenolpyruvate carboxykinase; and pyruvate kinase I. Results show that, with fuzzy logic‐augmented models, the kinetic data can be much better described. In particular, complex behavior, such as allosteric inhibition, can be captured using fuzzy rules. The resulting models, even though they do not provide additional physical meaning in enzyme mechanisms, allow the model to incorporate semiquantitative information in metabolic pathway models. © 1999 John Wiley & Sons, Inc. Biotechnol Bioeng 62: 722–729, 1999.     

Reduced kinetic models of facilitative transport

In spite of the highly complex structural dynamics of globular proteins, the processes mediated by them can usually be described in terms of relatively simple kinetic diagrams. How do complex proteins, characterized by undergoing transitions among a possibly very large number of intermediate states, exhibit functional properties that can be interpreted in terms of kinetic diagrams consisting of only a small number of states? One possible way of explaining this apparent contradiction is that, under some conditions, a reduction of the actual complete kinetic diagram that describes all of the macromolecular states and transitions takes place. In this work, we contribute with a formal basis to this interpretation, by generalizing the procedure of diagram reduction to the case of multicyclic kinetic diagrams. As an example, we apply the procedure to a complex kinetic model of facilitative transport. We develop Monte Carlo simulations to obtain the kinetic parameters of the complex model and we compare them with the ones analytically obtained from the reduced model. We confirm that, under some conditions, the kinetic behavior of the complex transporter is indistinguishable from the one of a four-state simple carrier model, derived from the former by diagram reduction. Besides introducing some novel methodological aspects, this work further contributes to the idea that, under many physiological and experimental conditions, a reduction occurs of the complete kinetic diagram that describes the dynamics of a globular protein.     

MIR: a versatile program for the statistical analysis of enzyme kinetic data

We have developed a package program for the estimation of Michaelis-Menten parameters for enzymes that conform to different kinetic mechanisms. Data from different experimental schemes can be fitted with appropriate weighing factors to any of 6 mathematical models, corresponding to 5 kinetic mechanisms: ordered bi-bi, Theorell-Chance, rapid equilibrium random bi-bi, rapid equilibrium ordered bi-bi and ping pong bi-bi. The program also performs a significance test to discriminate between different candidate models. To illustrate the performance of the program, real data from kinetic experiments with glucose 6-phosphate from Leuconostoc mesenteroides have been fitted to different mathematical models, and the results are discussed. The program can be easily implemented for the fitting of kinetic data to any other model.     

Kinetics of a model of autocatalysis, coupling of a reaction in which the enzyme acts on one of its substrates.

A global kinetic analysis is presented of a model of an enzyme autocatalytic process, to which a reaction is coupled, in which the enzyme acts upon one of its substrates. The kinetic equations of both the transient phase and the steady state are derived for this mechanism. In addition, we determine the corresponding kinetic equations for several particular cases which are characterized by certain relations between the rate constants. Finally, a kinetic data analysis is proposed for one of these particular cases. It can easily be extended to any of the other cases.     

Quantifying kinetic paths of protein folding

We propose a new approach to activated protein folding dynamics via a diffusive path integral framework. The important issues of kinetic paths in this situation can be directly addressed. This leads to the identification of the kinetic paths of the activated folding process, and provides a direct tool and language for the theoretical and experimental community to understand the problem better. The kinetic paths giving the dominant contributions to the long-time folding activation dynamics can be quantitatively determined. These are shown to be the instanton paths. The contributions of these instanton paths to the kinetics lead to the "bell-like" shape folding rate dependence on temperature, which is in good agreement with folding kinetic experiments and simulations. The connections to other approaches as well as the experiments of the protein folding kinetics are discussed.     

Use of in vivo kinetic techniques to evaluate metabolism of nutrients

Most nutrition research is related to rates of physiological processes. Information about those processes can be gained by in vivo kinetic techniques; however, many nutritionists are hesitant to use in vivo kinetics. The two basic in vivo kinetic techniques are single injection and continuous infusion of tracer into a pool of tracee. Either technique can form the basis for multiple-pool kinetics, or modeling. Solving a multiple-pool system can provide flow rates of substances between metabolic pools and is valuable for understanding a particular metabolic pathway or process. In vivo kinetic techniques can be valuable in understanding mechanisms whereby partitioning agents affect the distribution of nutrients, especially protein and fat, in food-producing animals. In vivo kinetics is a valuable tool for nutrition research and should be used more frequently.     

Dynamics of fluorescence dequenching of ostrich-quenched fluorescein biotin: a multifunctional quantitative assay for biotin

We describe a simple and rapid quantitative assay for biotin and biotin conjugates. The assay is based on the kinetic analysis of the enhancement of fluorescence of streptavidin/fluorescein biotin complexes in the presence of biotin. The kinetic response of fluorescence enhancement is proportional to the concentration of biotin. Standard calibration curves based on the kinetic response are obtained and detection limits of approximately 10(-9)M are established. Because the assay is amenable for use in small volumes of 5-50 microL or bead-based assays, the detection limits can be extended to the femtomole range. Since the assay depends on kinetic analysis, routine quantitation can be achieved without reference to standard curves. The dynamic aspects allow the assay to be extended to a broader range of applications including its use as an indicator of reagent mixing in laminar-flow assays carried out in microfluidic devices.     

A mechanistic kinetic description of lactate dehydrogenase elucidating cancer diagnosis and inhibitor evaluation

As a key enzyme for glycolysis, lactate dehydrogenase (LDH) remains as a topic of great interest in cancer study. Though a number of kinetic models have been applied to describe the dynamic behavior of LDH, few can reflect its actual mechanism, making it difficult to explain the observed substrate and competitor inhibitions at wide concentration ranges. A novel mechanistic kinetic model is developed based on the enzymatic processes and the interactive properties of LDH. Better kinetic simulation as well as new enzyme interactivity information and kinetic properties extracted from published articles via the novel model was presented. Case studies were presented to a comprehensive understanding of the effect of temperature, substrate, and inhibitor on LDH kinetic activities for promising application in cancer diagnosis, inhibitor evaluation, and adequate drug dosage prediction.     

Rapid determination of kinetic constants by competitive spectrophotometry

The use of competitive spectrophotometry to measure kinetic constants for enzyme-catalyzed reactions is described. The equation for the progress curve characterizing the kinetic behavior of an enzyme acting simultaneously on two alternative substrates is derived. By the addition of a competition term to the integrated Michaelis-Menten equation, the kinetic constants of an alternative substrate can be evaluated by measuring the competition with a substrate of known kinetic constants in a single experiment. Studies are presented involving the enzymes leucine aminopeptidase (LAP) and carboxypeptidase A (CPA). The results obtained with LAP and CPA showed that the kinetic constants determined using competitive spectrophotometry were in agreement with values cited in the literature or with values determined by single substrate enzyme kinetics.     

Simple Carrier Kinetics in Complex Membrane Transporters

The four-state simple carrier model (SCM) has been employed to describe facilitative transport of ligands across biological membranes. Two basic mechanisms have been invoked to account for carrier-mediated ligand translocation: (i) binding to a mobile carrier, and (ii) displacement determined by conformational changes of an integral protein. While translatory carriers may be accurately represented by a four-state diagram, it is unlikely that the transport process mediated by a complex membrane protein can be strictly described by the elementary SCM. The purpose of this article is to test whether facilitative transporters with a more complex kinetic design than the SCM can exhibit macroscopic kinetic properties indistinguishable from it. For this, I studied a ``general carrier model' (GCM), and evaluated whether the relevant kinetic parameters are subject to the same basic restrictions as in the SCM. The fundamental finding is that there is a general kinetic design embodied with SCM-like properties, that can be shared by many transporters. In particular, the classical SCM is shown here to represent a particular case of the GCM. A main conclusion of this work is therefore that the finding of a macroscopic SCM-like kinetic behavior for a particular process of facilitative transport does not represent a sufficient argument in favor of a particular type of mechanism, like the typical one involving a two-conformational single-site carrier. Received: 9 February 1998/Revised: 19 June 1998     

Kinetic Model Development for Accelerated Stability Studies

Accelerated stability coupled with modeling to predict the stability of compounds, blends, and products at long-term storage conditions provides significant benefits in science-based decision-making throughout drug substance and drug product development. The study can often be completed, including data analysis in the space of three working weeks, and the information gathered and learning made in this time period can rival years of traditional analysis. The speed of the studies allows an earlier assessment of risk to quality enabling appropriate risk mitigation strategies to be implemented in a timely manner. The scientific foundation is based upon Arrhenius kinetic equations that can be linear or nonlinear in time, and can be based upon water vapor pressure or liquid water activity (relative humidity). A variety of kinetic models are evaluated, and the best model is chosen based upon both Bayesian information criteria and an automated assessment of kinetic model parameters fitting within acceptable ranges. Confidence intervals are estimated based upon a bootstrapping approach. Moisture vapor transmission rate models are applied on top of the resulting kinetic models in order to simulate different packaging types and the use of desiccant. The kinetic models are integrated with the prediction of packaging humidity over time to create a long-term prediction of impurities and other phenomena. The resulting models have been shown to be useful for not only the prediction of drug product impurities in long-term storage but other physical phenomena as well such as hydrate development and solvate loss.     

Analyzing a kinetic titration series using affinity biosensors

The classical method of measuring binding constants with affinity-based biosensors involves testing several analyte concentrations over the same ligand surface and regenerating the surface between binding cycles. Here we describe an alternative approach to collecting kinetic binding data, which we call "kinetic titration." This method involves sequentially injecting an analyte concentration series without any regeneration steps. Through a combination of simulation and experimentation, we show that this method can be as robust as the classical method of analysis. In addition, kinetic titrations can be more efficient than the conventional data collection method and allow us to fully characterize analyte binding to ligand surfaces that are difficult to regenerate.     

On the use of apparent kinetic parameters for immobilized enzyme with noncompetitive substrate inhibition

The use of a simple rate equation with apparent parameters to describe the kinetic behavior of an immobilized enzyme with noncompetitive substrate inhibition was assessed. To do so, the reaction rate was calculated as a function of the interfacial substrate concentration, and the results were used to identify the apparent kinetic parameters by nonlinear regression. This procedure was repeated for different values of the diffusional constraints and of the inhibition constant. The equation using apparent parameters can describe the global kinetic behavior, provided that the diffusional and inhibitory constraints are not too high. When the constraints are high, a Michaelis-Menten equation can be used to model the kinetics for interfacial concentrations lower than the concentration leading to the maximum reaction rate.     

Transport and kinetic processes underlying biomolecular interactions in the BIACORE optical biosensor

The transport and kinetic processes describing biomolecular interactions in the BIACORE optical biosensor have been studied with the help of a mathematical model. In comparison to previous models, the model presented here couples, for the first time, transport phenomena in the flow channel with hindered diffusive transport and reactions inside the hydrogel. Simulated experiments based on this model, and two simpler models extant in the literature, are used to identify cases under which the detailed model is essential for accurate prediction of kinetic parameters. It is shown that this model can substantially improve the accuracy of kinetic parameter estimation when transport limitations in the flow channel and/or the hydrogel significantly influence the observed instrument response curves. The model can extend the range of the instrument's applicability to higher concentrations of immobilized species within the hydrogel. It can also be used for accurate design of experiments with the purpose of minimizing errors in the estimation of the kinetic parameters.