Beyond the Michaelis-Menten equation: Accurate and efficient estimation of enzyme kinetic parameters

Misuse of the Michaelis-Menten rate law for protein interaction networks and its remedy

For over a century, the Michaelis-Menten (MM) rate law has been used to describe the rates of enzyme-catalyzed reactions and gene expression. Despite the ubiquity of the MM rate law, it accurately captures the dynamics of underlying biochemical reactions only so long as it is applied under the right condition, namely, that the substrate is in large excess over the enzyme-substrate complex. Unfortunately, in circumstances where its validity condition is not satisfied, especially so in protein interaction networks, the MM rate law has frequently been misused. In this review, we illustrate how inappropriate use of the MM rate law distorts the dynamics of the system, provides mistaken estimates of parameter values, and makes false predictions of dynamical features such as ultrasensitivity, bistability, and oscillations. We describe how these problems can be resolved with a slightly modified form of the MM rate law, based on the total quasi-steady state approximation (tQSSA). Furthermore, we show that the tQSSA can be used for accurate stochastic simulations at a lower computational cost than using the full set of mass-action rate laws. This review describes how to use quasi-steady state approximations in the right context, to prevent drawing erroneous conclusions from in silico simulations.

interferENZY: A Web-Based Tool for Enzymatic Assay Validation and Standardized Kinetic Analysis

Enzymatic assays are widely employed to characterize important allosteric and enzyme modulation effects. The high sensitivity of these assays can represent a serious problem if the occurrence of experimental errors surreptitiously affects the reliability of enzyme kinetics results. We have addressed this problem and found that hidden assay interferences can be unveiled by the graphical representation of progress curves in modified reaction coordinates. To render this analysis accessible to users across all levels of expertise, we have developed a webserver, interferENZY, that allows (i) an unprecedented tight quality control of experimental data, (ii) the automated identification of small and major assay interferences, and (iii) the estimation of bias-free kinetic parameters. By eliminating the subjectivity factor in kinetic data reporting, interferENZY will contribute to solving the "reproducibility crisis" that currently challenges experimental molecular biology. The interferENZY webserver is freely available (no login required) at https://interferenzy.i3s.up.pt.

The histone H3-H4 tetramer is a copper reductase enzyme

Eukaryotic histone H3-H4 tetramers contain a putative copper (Cu2+) binding site at the H3-H3' dimerization interface with unknown function. The coincident emergence of eukaryotes with global oxygenation, which challenged cellular copper utilization, raised the possibility that histones may function in cellular copper homeostasis. We report that the recombinant Xenopus laevis H3-H4 tetramer is an oxidoreductase enzyme that binds Cu2+ and catalyzes its reduction to Cu1+ in vitro. Loss- and gain-of-function mutations of the putative active site residues correspondingly altered copper binding and the enzymatic activity, as well as intracellular Cu1+ abundance and copper-dependent mitochondrial respiration and Sod1 function in the yeast Saccharomyces cerevisiae The histone H3-H4 tetramer, therefore, has a role other than chromatin compaction or epigenetic regulation and generates biousable Cu1+ ions in eukaryotes.

The quasi-steady-state approximations revisited: Timescales, small parameters, singularities, and normal forms in enzyme kinetics

In this work, we revisit the scaling analysis and commonly accepted conditions for the validity of the standard, reverse and total quasi-steady-state approximations through the lens of dimensional Tikhonov-Fenichel parameters and their respective critical manifolds. By combining Tikhonov-Fenichel parameters with scaling analysis and energy methods, we derive improved upper bounds on the approximation error for the standard, reverse and total quasi-steady-state approximations. Furthermore, previous analyses suggest that the reverse quasi-steady-state approximation is only valid when initial enzyme concentrations greatly exceed initial substrate concentrations. However, our results indicate that this approximation can be valid when initial enzyme and substrate concentrations are of equal magnitude. Using energy methods, we find that the condition for the validity of the reverse quasi-steady-state approximation is far less restrictive than was previously assumed, and we derive a new "small" parameter that determines the validity of this approximation. In doing so, we extend the established domain of validity for the reverse quasi-steady-state approximation. Consequently, this opens up the possibility of utilizing the reverse quasi-steady-state approximation to model enzyme catalyzed reactions and estimate kinetic parameters in enzymatic assays at much lower enzyme to substrate ratios than was previously thought. Moreover, we show for the first time that the critical manifold of the reverse quasi-steady-state approximation contains a singular point where normal hyperbolicity is lost. Associated with this singularity is a transcritical bifurcation, and the corresponding normal form of this bifurcation is recovered through scaling analysis.

Bayesian Data Integration Questions Classic Study on Protease Self-Digest Kinetics

We combine Bayesian data integration with kinetic modeling to rigorously identify reaction mechanisms. This approach forces models to be consistent not only with kinetic measurements but with all available information. We revisit a classic study on trypsin self-digest acceleration by colloidal silica. Bayesian data integration reveals that the mechanism suggested in that study is inconsistent with its presented data. We propose an improved hypothesis. However, the detailed mechanism of the surface reaction cannot be inferred from the available data.

Beyond the Michaelis-Menten: Accurate Prediction of In Vivo Hepatic Clearance for Drugs With Low KM

Clearance (CL) is the major pharmacokinetic parameter for evaluating systemic exposure of drugs in the body and, thus, for developing new drugs. To predict in vivo CL, the ratio between the maximal rate of metabolism and Michaelis‐Menten constant (V_max/K_M estimated from in vitro metabolism study has been widely used. This canonical approach is based on the Michaelis‐Menten equation, which is valid only when the K_M value of a drug is much higher than the hepatic concentration of the enzymes, especially cytochrome P450, involved in its metabolism. Here, we find that such a condition does not hold for many drugs with low K_M, and, thus, the canonical approach leads to considerable error. Importantly, we propose an alternative approach, which incorporates the saturation of drug metabolism when concentration of the enzymes is not sufficiently lower than K_M. This new approach dramatically improves the accuracy of prediction for in vivo CL of high‐affinity drugs with low K_M. This indicates that the proposed approach in this study, rather than the canonical approach, should be used to predict in vivo hepatic CL for high‐affinity drugs, such as midazolam and propafenone.

Functional analysis of the agnH gene involved in nicotine-degradation pathways in Agrobacterium tumefaciens strain SCUEC1

Agrobacterium tumefaciens strain SCUEC1 is a nicotine-degrading bacterium, which has been recently isolated from the tobacco waste-contaminated field soil. However, the mechanism for nicotine degradation in this strain remains unclear. Here, we analyze the function and biological properties of the agnH gene in the strain SCUEC1. The overexpression of the AgnH protein was detected by SDS-PAGE analysis, and functional insight of the AgnH protein was carried out with monitoring the changes of maleic acid into fumaric acid by high performance liquid chromatography (HPLC). Moreover, the effects of temperature, pH and metal ions on the enzymatic activities of the AgnH protein were also analyzed. The results demonstrated that the agnH gene was successfully ligated to the plasmid pET28a. The optimal condition for the enzymatic activities for the AgnH, approximately 28.0 kDa, was determined as 37 °C, pH 8.0 and 25 µM Mg2+. Conclusively, the agnH gene fulfils an important role in the conversion of maleic acid into fumaric acid involved in nicotine-degradation pathways in Agrobacterium tumefaciens strain SCUEC1.

Structure optimization and bioactivity evaluation of ThDP analogs targeting cyanobacterial pyruvate dehydrogenase E1

Harmful cyanobacteria bloom (HCB) has occurred frequently in recent years and it is urgent to develop novel algicides to deal with this problem. In this paper, a series of novel thiamin diphosphate (ThDP) analogs 5a-5g were designed and synthesized targeting cyanobacterial pyruvate dehydrogenase complex E1 (Cy-PDHc E1). Our results showed that compounds 5a-5g have higher inhibitory activities against Cy-PDHc E1 (IC₅₀ 9.56-3.48 µM) and higher inhibitory activities against two model cyanobacteria strains Synechocystis sp PCC6803 (EC₅₀ 2.03-1.58 µM) and Microcystis aeruginosa FACHB905 (EC₅₀ 1.86-0.95 µM). Especially, compound 5b displayed highest inhibitory activities (IC₅₀ = 3.48 µM) against Cy-PDHc E1 and powerful inhibitory activities against cyanobacteria Synechocystis sp PCC6803 (EC₅₀ = 1.58 µM) and Microcystis aeruginosa FACHB905 (EC₅₀ = 1.04 µM). Moreover, the inhibitory activities of compound 5b were even higher than those of copper sulfate (EC₅₀ = 2.02 and 1.71 µM separately) which has been widely used as algicide against cyanobacteria PCC6803 and FACHB905. The more important was that compound 5b display much higher inhibitory selectivity between Cy-PDHc E1 (Inhibitory rate 97.4%) and porcine PDHc E1 (Inhibitory rate 11.8%) under the same concentration (100 μM). The inhibition kinetic experiment and molecular docking research showed that compound 5b can inhibit Cy-PDHc E1 by occupying the ThDP-binding pocket and then blocking Cy-PDHc E1 bound to ThDP as competitive inhibitor. The imagines of SEM and TEM showed that cellular microstructures were heavily destroyed under compound 5b stress. Our results demonstrated compound 5b could be taken as a potential lead compound targeting Cy-PDHc E1 to obtain environment-friendly algicide for harmful cyanobacterial blooms control.

Frequency spectrum of chemical fluctuation: A probe of reaction mechanism and dynamics

Even in the steady-state, the number of biomolecules in living cells fluctuates dynamically, and the frequency spectrum of this chemical fluctuation carries valuable information about the dynamics of the reactions creating these biomolecules. Recent advances in single-cell techniques enable direct monitoring of the time-traces of the protein number in each cell; however, it is not yet clear how the stochastic dynamics of these time-traces is related to the reaction mechanism and dynamics. Here, we derive a rigorous relation between the frequency-spectrum of the product number fluctuation and the reaction mechanism and dynamics, starting from a generalized master equation. This relation enables us to analyze the time-traces of the protein number and extract information about dynamics of mRNA number and transcriptional regulation, which cannot be directly observed by current experimental techniques. We demonstrate our frequency spectrum analysis of protein number fluctuation, using the gene network model of luciferase expression under the control of the Bmal 1a promoter in mouse fibroblast cells. We also discuss how the dynamic heterogeneity of transcription and translation rates affects the frequency-spectra of the mRNA and protein number.

Bayesian inference of distributed time delay in transcriptional and translational regulation

MOTIVATION

Advances in experimental and imaging techniques have allowed for unprecedented insights into the dynamical processes within individual cells. However, many facets of intracellular dynamics remain hidden, or can be measured only indirectly. This makes it challenging to reconstruct the regulatory networks that govern the biochemical processes underlying various cell functions. Current estimation techniques for inferring reaction rates frequently rely on marginalization over unobserved processes and states. Even in simple systems this approach can be computationally challenging, and can lead to large uncertainties and lack of robustness in parameter estimates. Therefore we will require alternative approaches to efficiently uncover the interactions in complex biochemical networks.

RESULTS

We propose a Bayesian inference framework based on replacing uninteresting or unobserved reactions with time delays. Although the resulting models are non-Markovian, recent results on stochastic systems with random delays allow us to rigorously obtain expressions for the likelihoods of model parameters. In turn, this allows us to extend MCMC methods to efficiently estimate reaction rates, and delay distribution parameters, from single-cell assays. We illustrate the advantages, and potential pitfalls, of the approach using a birth-death model with both synthetic and experimental data, and show that we can robustly infer model parameters using a relatively small number of measurements. We demonstrate how to do so even when only the relative molecule count within the cell is measured, as in the case of fluorescence microscopy.

AVAILABILITY AND IMPLEMENTATION

Accompanying code in R is available at https://github.com/cbskust/DDE_BD.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

A new Michaelis-Menten equation valid everywhere multi-scale dynamics prevails

The Michaelis-Menten reaction scheme is among the most influential models in the field of biochemistry, since it led to a very popular expression for the rate of an enzyme-catalysed reaction. After the realisation that this expression is valid in a limited region of the parameter space, two additional expressions were later introduced. The range of validity of these three expressions has been studied thoroughly, since the significance of a reliable rate is not based only on the accuracy of its predictive abilities but also on the physical insight that is acquired in the process of its construction. Here a new expression for the rate is introduced that is valid in practically the full parameter space and reduces to the expressions in the literature, when considering appropriate limits. The new expression is produced by employing algorithmic tools for asymptotic analysis, so that its construction is not hindered by the dimensional form and the complexity of the full model or by a wide parameter range of interest. These tools can be employed for the derivation of enzyme rate expressions of much more complex kinetics mechanisms.

Reverse engineering genetic networks using nonlinear saturation kinetics

A gene regulatory network (GRN) represents a set of genes along with their regulatory interactions. Cellular behavior is driven by genetic level interactions. Dynamics of such systems show nonlinear saturation kinetics which can be best modeled by Michaelis-Menten (MM) and Hill equations. Although MM equation is being widely used for modeling biochemical processes, it has been applied rarely for reverse engineering GRNs. In this paper, we develop a complete framework for a novel model for GRN inference using MM kinetics. A set of coupled equations is first proposed for modeling GRNs. In the coupled model, Michaelis-Menten constant associated with regulation by a gene is made invariant irrespective of the gene being regulated. The parameter estimation of the proposed model is carried out using an evolutionary optimization method, namely, trigonometric differential evolution (TDE). Subsequently, the model is further improved and the regulations of different genes by a given gene are made distinct by allowing varying values of Michaelis-Menten constants for each regulation. Apart from making the model more relevant biologically, the improvement results in a decoupled GRN model with fast estimation of model parameters. Further, to enhance exploitation of the search, we propose a local search algorithm based on hill climbing heuristics. A novel mutation operation is also proposed to avoid population stagnation and premature convergence. Real life benchmark data sets generated in vivo are used for validating the proposed model. Further, we also analyze realistic in silico datasets generated using GeneNetweaver. The comparison of the performance of proposed model with other existing methods shows the potential of the proposed model.

Chemical Kinetics Roots and Methods to Obtain the Probability Distribution Function Evolution of Reactants and Products in Chemical Networks Governed by a Master Equation

In this paper first, we review the physical root bases of chemical reaction networks as a Markov process in multidimensional vector space. Then we study the chemical reactions from a microscopic point of view, to obtain the expression for the propensities for the different reactions that can happen in the network. These chemical propensities, at a given time, depend on the system state at that time, and do not depend on the state at an earlier time indicating that we are dealing with Markov processes. Then the Chemical Master Equation (CME) is deduced for an arbitrary chemical network from a probability balance and it is expressed in terms of the reaction propensities. This CME governs the dynamics of the chemical system. Due to the difficulty to solve this equation two methods are studied, the first one is the probability generating function method or z-transform, which permits to obtain the evolution of the factorial moment of the system with time in an easiest way or after some manipulation the evolution of the polynomial moments. The second method studied is the expansion of the CME in terms of an order parameter (system volume). In this case we study first the expansion of the CME using the propensities obtained previously and splitting the molecular concentration into a deterministic part and a random part. An expression in terms of multinomial coefficients is obtained for the evolution of the probability of the random part. Then we study how to reconstruct the probability distribution from the moments using the maximum entropy principle. Finally, the previous methods are applied to simple chemical networks and the consistency of these methods is studied.

Extensions to Michaelis-Menten Kinetics for Single Parameters

Biochemical transformation kinetics is based on the formation of enzyme-substrate complexes. We developed a robust scheme based on unit productions of enzymes and reactants in cyclic events to comply with mass action law to form enzyme-substrate complexes. The developed formalism supports a successful application of Michaelis-Menten kinetics in all biochemical transformations of single parameters. It is an essential tool to overcome some challenging healthcare and environmental issues. In developing the formalism, we defined the substrate [S]= [Product]3/4 and rate of reaction based on rate and time perspectives. It allowed us to develop two quadratic equations. The first, represents a body entity that gave a useful relationship of enzyme E = 2S0.33, and the second nutrients/feed, each giving [Enzymes] and [Enzyme-substrate complexes], simulating rate of reaction, [substrate], and their differentials. By combining [Enzymes] and [Enzyme-substrate complexes] values, this quadratic equation derives a Michaelis-Menten hyperbolic function. Interestingly, we can derive the proportionate rate of reaction and [Enzymes] values of the quadratics resulting in another Michaelis-Menten hyperbolic. What is clear from these results is that between these two hyperbolic functions, in-competitive inhibitions exist, indicating metabolic activities and growth in terms of energy levels. We validated these biochemical transformations with examples applicable to day to day life.

Ensemble methods for stochastic networks with special reference to the biological clock of Neurospora crassa

A major challenge in systems biology is to infer the parameters of regulatory networks that operate in a noisy environment, such as in a single cell. In a stochastic regime it is hard to distinguish noise from the real signal and to infer the noise contribution to the dynamical behavior. When the genetic network displays oscillatory dynamics, it is even harder to infer the parameters that produce the oscillations. To address this issue we introduce a new estimation method built on a combination of stochastic simulations, mass action kinetics and ensemble network simulations in which we match the average periodogram and phase of the model to that of the data. The method is relatively fast (compared to Metropolis-Hastings Monte Carlo Methods), easy to parallelize, applicable to large oscillatory networks and large (~2000 cells) single cell expression data sets, and it quantifies the noise impact on the observed dynamics. Standard errors of estimated rate coefficients are typically two orders of magnitude smaller than the mean from single cell experiments with on the order of ~1000 cells. We also provide a method to assess the goodness of fit of the stochastic network using the Hilbert phase of single cells. An analysis of phase departures from the null model with no communication between cells is consistent with a hypothesis of Stochastic Resonance describing single cell oscillators. Stochastic Resonance provides a physical mechanism whereby intracellular noise plays a positive role in establishing oscillatory behavior, but may require model parameters, such as rate coefficients, that differ substantially from those extracted at the macroscopic level from measurements on populations of millions of communicating, synchronized cells.