Rapid Equilibrium Assumption Research Articles

Kinetic Modeling of Time-Dependent Enzyme Inhibition by Pre-Steady-State Analysis of Progress Curves: The Case Study of the Anti-Alzheimer's Drug Galantamine.

The Michaelis–Menten model of enzyme kinetic assumes the free ligand approximation, the steady-state approximation and the rapid equilibrium approximation. Analytical methods to model slow-binding inhibitors by the analysis of initial velocities have been developed but, due to their inherent complexity, they are seldom employed. In order to circumvent the complications that arise from the violation of the rapid equilibrium assumption, inhibition is commonly evaluated by pre-incubating the enzyme and the inhibitors so that, even for slow inhibitors, the binding equilibrium is established before the reaction is started. Here, we show that for long drug-target residence time inhibitors, the conventional analysis of initial velocities by the linear regression of double-reciprocal plots fails to provide a correct description of the inhibition mechanism. As a case study, the inhibition of acetylcholinesterase by galantamine, a drug approved for the symptomatic treatment of Alzheimer’s disease, is reported. For over 50 years, analysis based on the conventional steady-state model has overlooked the time-dependent nature of galantamine inhibition, leading to an erroneous assessment of the drug potency and, hence, to discrepancies between biochemical data and the pharmacological evidence. Re-examination of acetylcholinesterase inhibition by pre-steady state analysis of the reaction progress curves showed that the potency of galantamine has indeed been underestimated by a factor of ~100.

Four-State Model for Simulating Kinetic and Steady-State Voltage-Dependent Gating of Gap Junctions

Gap junction (GJ) channels, formed of connexin (Cx) proteins, provide a direct pathway for metabolic and electrical cell-to-cell communication. These specialized channels are not just passive conduits for the passage of ions and metabolites but have been shown to gate robustly in response to transjunctional voltage, Vj, the voltage difference between two coupled cells. Voltage gating of GJs could play a physiological role, particularly in excitable cells, which can generate large transients in membrane potential during the propagation of action potentials. We present a mathematical/computational model of GJ channel voltage gating to assess properties of GJ channels that takes into account contingent gating of two series hemichannels and the distribution of Vj across each hemichannel. From electrophysiological recordings in cell cultures expressing Cx43 or Cx45, the principal isoforms expressed in cardiac tissue, various data sets were fitted simultaneously using global optimization. The results showed that the model is capable of describing both steady-state and kinetic properties of homotypic and heterotypic GJ channels composed of these Cxs. Moreover, mathematical analyses showed that the model can be simplified to a reversible two-state system and solved analytically using a rapid equilibrium assumption. Given that excitable cells are arranged in interconnected networks, the equilibrium assumption allows for a substantial reduction in computation time, which is useful in simulations of large clusters of coupled cells. Overall, this model can serve as a tool for the studying of GJ channel gating and its effects on the spread of excitation in networks of electrically coupled cells.

Non-equilibrium repressor binding kinetics link DNA damage dose to transcriptional timing within the SOS gene network.

Biochemical pathways are often genetically encoded as simple transcription regulation networks, where one transcription factor regulates the expression of multiple genes in a pathway. The relative timing of each promoter’s activation and shut-off within the network can impact physiology. In the DNA damage repair pathway (known as the SOS response) of Escherichia coli, approximately 40 genes are regulated by the LexA repressor. After a DNA damaging event, LexA degradation triggers SOS gene transcription, which is temporally separated into subsets of ‘early’, ‘middle’, and ‘late’ genes. Although this feature plays an important role in regulating the SOS response, both the range of this separation and its underlying mechanism are not experimentally defined. Here we show that, at low doses of DNA damage, the timing of promoter activities is not separated. Instead, timing differences only emerge at higher levels of DNA damage and increase as a function of DNA damage dose. To understand mechanism, we derived a series of synthetic SOS gene promoters which vary in LexA-operator binding kinetics, but are otherwise identical, and then studied their activity over a large dose-range of DNA damage. In distinction to established models based on rapid equilibrium assumptions, the data best fit a kinetic model of repressor occupancy at promoters, where the drop in cellular LexA levels associated with higher doses of DNA damage leads to non-equilibrium binding kinetics of LexA at operators. Operators with slow LexA binding kinetics achieve their minimal occupancy state at later times than operators with fast binding kinetics, resulting in a time separation of peak promoter activity between genes. These data provide insight into this remarkable feature of the SOS pathway by demonstrating how a single transcription factor can be employed to control the relative timing of each gene’s transcription as a function of stimulus dose.

Mathematical model of TGF-\u03b2signalling: feedback coupling is consistent with signal switching

BackgroundTransforming growth factor β (TGF-β) signalling regulates the development of embryos and tissue homeostasis in adults. In conjunction with other oncogenic changes, long-term perturbation of TGF-β signalling is associated with cancer metastasis. Although TGF-β signalling can be complex, many of the signalling components are well defined, so it is possible to develop mathematical models of TGF-β signalling using reduction and scaling methods. The parameterization of our TGF-β signalling model is consistent with experimental data.ResultsWe developed our mathematical model for the TGF-β signalling pathway, i.e. the RF- model of TGF-β signalling, using the “rapid equilibrium assumption” to reduce the network of TGF-β signalling reactions based on the time scales of the individual reactions. By adding time-delayed positive feedback to the inherent time-delayed negative feedback for TGF-β signalling. We were able to simulate the sigmoidal, switch-like behaviour observed for the concentration dependence of long-term (> 3 hours) TGF-β stimulation. Computer simulations revealed the vital role of the coupling of the positive and negative feedback loops on the regulation of the TGF-β signalling system. The incorporation of time-delays for the negative feedback loop improved the accuracy, stability and robustness of the model. This model reproduces both the short-term and long-term switching responses for the intracellular signalling pathways at different TGF-β concentrations. We have tested the model against experimental data from MEF (mouse embryonic fibroblasts) WT, SV40-immortalized MEFs and Gp130 F/F MEFs. The predictions from the RF- model are consistent with the experimental data.ConclusionsSignalling feedback loops are required to model TGF-β signal transduction and its effects on normal and cancer cells. We focus on the effects of time-delayed feedback loops and their coupling to ligand stimulation in this system. The model was simplified and reduced to its key components using standard methods and the rapid equilibrium assumption. We detected differences in short-term and long-term signal switching. The results from the RF- model compare well with experimental data and predict the dynamics of TGF-β signalling in cancer cells with different mutations.

The accuracy of the rapid equilibrium assumption in steady-state enzyme kinetics in the case of a multipath arbitrary enzyme mechanism with a number of equilibrium segments

A quantitative evaluation of the accuracy of the rapid-equilibrium assumption in steady-state enzyme kinetics was obtained for a multipath arbitrary enzyme mechanism with a number of equilibrium segments. Explicit expressions for estimating the contribution of any equilibrium segment to the accuracy of the rapid-equilibrium assumption were obtained. This allowed us to determine the accuracy of the rapid-equilibrium assumption (Δ) in general: 1 + Δ = (1 + Δ1)(1 + Δ2)... (1 + Δk), where Δ1, Δ2,..., Δk is the contribution of each individual equilibrium segment. The accuracy depends only on the structure and properties of equilibrium segments, which have been accounted for in the rapid-equilibrium assumption, but it is independent of the number of paths in the mechanism of the enzymatic reaction and on the structure and properties of the remaining part (steady-state) of the kinetic scheme.

The accuracy of rapid equilibrium assumption in steady-state enzyme kinetics is a function of equilibrium segment structure and properties

Quantitative evaluation of the accuracy of the rapid equilibrium assumption in the steady-state enzyme kinetics was obtained for an arbitrary mechanism of an enzyme-catalyzed reaction. This evaluation depends only on the structure and properties of the equilibrium segment, but doesn't depend on the structure and properties of the rest (stationary part) of the kinetic scheme. The smaller the values of the edges leaving equilibrium segment in relation to values of the edges within the equilibrium segment, the higher the accuracy of determination of intermediate concentrations and reaction velocity in a case of the rapid equilibrium assumption.

New relations for steady-state enzyme kinetics and their application to rapid equilibrium assumption

With the use of a graph theory new relations for steady-state enzyme kinetics are derived and strictly proved for the arbitrary mechanism of an enzyme-catalysed reaction containing a reversible segment. Using these relations, a general principle for rapid equilibrium assumption is formulated and proved: the reversible bound segment can be considered as an equilibrium segment only when the values of the base trees that are not proper to this segment can be neglected (within a prescribed accuracy) in relation to the values of the base trees that belong to this segment. In contrast with the foreign base trees the base trees that are proper to the segment have the following properties: the tree that is directed to the base within this segment does not contain the edges leaving this segment; and the tree that is directed to the base outside the segment contains only one edge leaving this segment. Equilibrium variations are assessed for steady-state intermediates concentrations of the equilibrium segment, numerical expressions are obtained for the accuracy of determination of the intermediates concentrations as well as for the accuracy of determination of the rate of enzyme-catalysed reaction in case of using rapid equilibrium assumption.

A Novel Method for Assessing in Vitro Oncology Drug Combinations Using Growth Rates

We propose a new method that allows screening oncology drug combinations using data from in vitro studies to select agents that have the promise of showing a synergistic effect in vivo. In contrast to known approaches that define combination effects either on the concentration scale or on the percent inhibition scale, we use the growth rate of treated cells as a primary indicator of treatment activity. The developed method is based on a novel statistical model that describes the growth of cancer cells that are subject to treatment with a combination of compounds. The model assumes a multicompartment cell population with transition rates between compartments modeled according to biochemical reaction properties, and cells in each compartment growing according to exponential law. This translates to a linear system of ordinary differential equations, whose solution is accurately approximated by a closed-form expression using rapid equilibrium assumptions. Special cases of the aforementioned model represent situations when the combination effect is absent or when the considered drugs act as the same compound. Assuming the normal distribution for the growth rate measurement error, we describe a formal statistical testing procedure to distinguish between different mechanisms of action for the considered compounds, and to test if a significant combination effect is being observed.

Kinetics of hyperosmotically stimulated Na-K-2Cl cotransporter in Xenopus laevis oocytes

A detailed study of hypertonically stimulated Na-K-2Cl cotransport (NKCC1) in Xenopus laevis oocytes was carried out to better understand the 1 K(+):1 Cl(-) stoichiometry of transport that was previously observed. In this study, we derived the velocity equations for K(+) influx under both rapid equilibrium assumptions and combined equilibrium and steady-state assumptions and demonstrate that the behavior of the equations and curves in Lineweaver-Burke plots are consistent with a model where Cl(-) binds first, followed by Na(+), a second Cl(-), and then K(+). We further demonstrate that stimulation of K(+) movement by K(+) on the trans side is an intrinsic property of a carrier that transports multiple substrates. We also demonstrate that K(+) movement through NKCC1 is strictly dependent upon the presence of external Na(+), even though only a fraction of Na(+) is in fact transported. Finally, we propose that the larger transport of K(+), as compared with Na(+), is a result of the return of partially unloaded carriers, which masks the net 1Na(+):1K(+):2Cl(-) stoichiometry of NKCC1. These data have profound implications for the physiology of Na-K-2Cl cotransport, since transport of K-Cl in some conditions seems to be uncoupled from the transport of Na-Cl.

Discovery of PDK1 Kinase Inhibitors with a Novel Mechanism of Action by Ultrahigh Throughput Screening

The phosphoinositide 3-kinase/AKT signaling pathway plays a key role in cancer cell growth, survival, and angiogenesis. Phosphoinositide-dependent protein kinase-1 (PDK1) acts at a focal point in this pathway immediately downstream of phosphoinositide 3-kinase and PTEN, where it phosphorylates numerous AGC kinases. The PDK1 kinase domain has at least three ligand-binding sites: the ATP-binding pocket, the peptide substrate-binding site, and a groove in the N-terminal lobe that binds the C-terminal hydrophobic motif of its kinase substrates. Based on the unique PDK1 substrate recognition system, ultrahigh throughput TR-FRET and Alphascreen screening assays were developed using a biotinylated version of the PDK1-tide substrate containing the activation loop of AKT fused to a pseudo-activated hydrophobic motif peptide. Using full-length PDK1, K(m) values were determined as 5.6 mum for ATP and 40 nm for the fusion peptide, revealing 50-fold higher affinity compared with the classical AKT(Thr-308)-tide. Kinetic and biophysical studies confirmed the PDK1 catalytic mechanism as a rapid equilibrium random bireactant reaction. Following an ultrahigh throughput screen of a large library, 2,000 compounds were selected from the reconfirmed hits by computational analysis with a focus on novel scaffolds. ATP-competitive hits were deconvoluted by dose-response studies at 1x and 10x K(m) concentrations of ATP, and specificity of binding was assessed in thermal shift assay. Inhibition studies using fusion PDK1-tide1 substrate versus AKT(Thr-308)-tide and kinase selectivity profiling revealed a novel selective alkaloid scaffold that evidently binds to the PDK1-interacting fragment pocket. Molecular modeling suggests a structural paradigm for the design of inhibitory versus activating allosteric ligands of PDK1.

Implications of the Alternating Access Model for Organic Anion Transporter Kinetics

Many transport proteins, including the clinically important organic anion transporters (OATs), appear to function via an "alternating access" mechanism. In analyzing the kinetics of these transporters, the terms K(m) and V(max) are often treated in the field as denoting, respectively, the affinity of the substrate for the transporter and the turnover (conformational switch) rate of the substrate-transporter complex. In fact, the expressions for both these parameters have very complex forms comprising multiple rate constants from conformational switch as well as association/dissociation steps in the cycling of the transporter and, therefore, do not have straightforward physical meanings. However, if the rapid equilibrium assumption is made (namely, that the association/dissociation steps occur far more rapidly than the conformational switch steps), these expressions become greatly simplified and their physical meaning clear, though still distinct from the conventional interpretations. V(max) will be a function of not just the rate of substrate-transporter complex turnover but also the rate of the "return" conformational switch and will vary largely with the slower of these two steps (the rate-limiting step). K(m) will be seen to be related to substrate affinity by a term that varies inversely with the substrate-transporter complex turnover rate, essentially because the greater this rate, the greater the extent to which transporters will be distributed in a conformation inaccessible to substrate. Here, an intuitive approach is presented to demonstrate these conclusions. The phenomena of trans-stimulation and trans-inhibition are discussed in the context of this analysis.

Effects of pH in Rapid-Equilibrium Enzyme Kinetics

The effects of pH on the rates of enzyme-catalyzed reactions are very important because they yield information on the pKs of acidic groups in the enzymatic site and the various enzyme-substrate complexes. But many enzyme-catalyzed reactions produce or consume hydrogen ions in a way that cannot be explained with pKs. These pH effects extend over the whole pH range of interest. In investigating these effects, the rapid-equilibrium assumption is especially useful because a large number of chemical reactions have to be taken into account. In these calculations, all of the reactions up to the rate-determining reaction are treated with biochemical thermodynamics. Kinetic studies make it possible to determine the number of hydrogen ions consumed in the rate-determining reaction, a number that can be in the range of 0-8. It is shown that the experimental limiting velocity of the forward reaction V(fexp) is equal to 10(npH)V(f), where n is a negative integer and Vf varies with pH in the way determined by the pKs of the enzyme-substrate complex that reacts in the rate-determining reaction. A computer program for the initial reaction velocity makes it possible to investigate the rapid-equilibrium kinetics of enzymatic mechanisms that involve the consumption of hydrogen ions.

The Complete Pathway for ERK2-catalyzed Reaction: EVIDENCE FOR AN ISO RANDOM BI BI MECHANISM

In the present study, the enzymatic mechanism of ERK2 is re-examined by a combination of steady-state kinetic studies in the absence and presence of viscosogenic agents. Kinetic studies carried out in various concentrations of sucrose revealed that both k(cat) and k(cat)/K(m) for either ATP or EtsDelta138 were highly sensitive to solvent viscosity, suggesting that the rapid equilibrium assumption is not valid for the phosphorylation of protein substrate by ERK2. Furthermore, the kinetic analysis with the minimal random Bi Bi reaction mechanism is shown to be inconsistent with the principle of the detailed balance. This inconsistent calculation strongly suggests that there is isomerization of the enzyme-substrate ternary complex. The viscosity-dependent steady-state kinetic data are combined to establish a kinetic mechanism for the ERK2-catalyzed reaction that predicts initial reaction velocities under varying concentrations of ATP and substrate. These results complement previous structure-function studies of mitogen-activated protein kinases and provide important insight for mechanistic interpretation of the kinase functions.

Probing the mechanism of purine nucleoside phosphorylase by steady-state kinetic studies and ligand binding characterization determined by fluorimetric titrations

Reversible reaction catalyzed by trimeric purine nucleoside phosphorylase (PNP) from Cellulomonas sp. with typical and non-typical substrates, including product inhibition patterns of both reaction directions, and interactions of the enzyme with bisubstrate analogue inhibitors, were investigated by the steady-state kinetic methods and fluorimetric titrations. The ligand chromophores exist most probably as neutral species, and not N(1)-H monoanions, in the complex with PNP, as shown by determination of inhibition constants vs. pH. This supports the mechanism in which hydrogen bond interaction of N(1)-H with Glu204 is crucial in the catalytic process. Stoichiometry of ligand binding, with possible exception of hypoxanthine, is three molecules per enzyme trimer. Kinetic experiments show that in principle the Michaelis–Menten model could not properly describe the reaction. However, this model seems to hold for certain experimental conditions. Data presented here are supported by earlier findings obtained by means of fluorimetric titrations and protective effects of ligands on thermal inactivation of the enzyme. All results are consistent with the following mechanism for trimeric PNPs: (i) random binding of substrates, (ii) potent binding and slow release of some reaction products leading to the circumstances that the chemical step is not the slowest one and that rapid-equilibrium assumptions do not hold, (iii) a dual role of phosphate—a substrate and also a reaction modifier.

KINETIC PROPERTIES OF CELLULOMONAS SP. PURINE NUCLEOSIDE PHOSPHORYLASE WITH TYPICAL AND NON-TYPICAL SUBSTRATES: IMPLICATIONS FOR THE REACTION MECHANISM

Phosphorolysis catalyzed by Cellulomonas sp. PNP with typical nucleoside substrate, inosine (Ino), and non-typical 7-methylguanosine (m7Guo), with either nucleoside or phosphate (Pi) as the varied substrate, kinetics of the reverse synthetic reaction with guanine (Gua) and ribose-1-phosphate (R1P) as the varied substrates, and product inhibition patterns of synthetic and phosphorolytic reaction pathways were studied by steady-state kinetic methods. It is concluded that, like for mammalian trimeric PNP, complex kinetic characteristics observed for Cellulomonas enzyme results from simultaneous occurrence of three phenomena. These are sequential but random, not ordered binding of substrates, tight binding of one substrate purine bases, leading to the circumstances that for such substrates (products) rapid-equilibrium assumptions do not hold, and a dual role of Pi, a substrate, and also a reaction modifier that helps to release a tightly bound purine base.

Physiologically Based Pharmacokinetic Models for the Transport of Trichloroethylene in Adipose Tissue

In this paper we present three physiologically based pharmacokinetic (PBPK) models for the systemic transport of trichloroethylene (TCE), with a focus on the adipose, or fat tissue. TCE is a widespread environmental contaminant, and has been shown to produce toxic effects in both animals and humans. A key characteristic of TCE is its tendency to accumulate in fat tissue, which has a major impact on the overall systemic disposition of TCE.Here we use PBPK models to predict the dynamics of TCE in the various tissues and organs, including the adipose tissue.The first model utilizes the standard ‘perfusion-limited’ compartmental model for the fat tissue, while the second model uses a ‘diffusion-limited’ model to describe the transport through the adipose tissue. Both of these ODE models are based on ‘well-mixed’ and rapid equilibrium assumptions, and do not take into account the specific and largely heterogeneous physiology of adipose tissue. The third model we discuss is a PBPK hybrid model with an axial-dispersion type model for the adipose tissue. This PDE-based model is designed to capture key physiological heterogeneities of fat tissue, including widely varying fat cell sizes, lipid distribution, and blood flow properties. Model simulations demonstrate that this model may be well-suited to predict the experimental behavior of TCE in adipose tissue using parameter estimation techniques.

Inhibition of Factor Xa by a Peptidyl-α-ketothiazole Involves Two Steps. Evidence for a Stabilizing Conformational Change

Recently, peptidylketothiazoles have been shown to be potent inhibitors of proteases, but the details of the interaction have not yet been studied. In the work presented here, the interaction of factor Xa, a coagulation protease, with the transition state inhibitor BnSO(2)-D-Arg-Gly-Arg-ketothiazole (C921-78) is characterized. C921-78 is a tight and selective inhibitor of the coagulation protease factor Xa (K(d) = 14 pM). The hydrolytic activity of factor Xa was inhibited by C921-78 in a time-dependent manner. The rate-limiting step of the bimolecular combination of inhibitor and enzyme was competitive with the substrate. Conversely, the inhibitor could be displaced from the active site of the enzyme after exposure of the preformed complex to an excess of substrate or to the active site inhibitor dansyl-Glu-Gly-Arg-chloromethyl ketone (DEGR-CMK) in a slow reaction. The formation of the C921-78-factor Xa complex resulted in a 60% increase in the magnitude of the fluorescence emission spectrum. Rapid mixing of the enzyme and inhibitor produces a monophasic fluorescence increase, compatible with spectral transition in a single step. The rate constant for this reaction increased hyperbolically with the concentration of C921-78, but the amplitude remained constant. These results are consistent with the initial formation of an enzyme-inhibitor complex (EI), followed by a unimolecular conversion of EI to EI linked to a spectral transition. The rate constants of the isomerization provide an estimate of 300000-fold stabilization. Thus, the inhibition of factor Xa by C921-78 follows a mechanism similar to that described classically for slow tight binding inhibitors. However, the two steps of the reaction cannot be kinetically separated by the rapid equilibrium assumption, and therefore, the formation of EI is partially rate-limiting, too. The driving energy for the unusually fast isomerization step may result from the highly favorable interactions of the inhibitor in the primary binding site.

Transient phase of enzyme reactions.: Time course equations of the strict and the rapid equilibrium conditions and their computerized derivation

In this contribution, we present the derivation, from the strict transient phase equations of enzyme reactions, of the transient phase equations under the usual assumptions that one or more of the reversible steps involved in the mechanism of the enzyme reaction are assumed to be in rapid equilibrium. Moreover, we present the transient phase equations of all of the species in a general enzyme system model, valid for the partial or total rapid equilibrium conditions, as well as the particular case of the strict transient phase equations. In the case of the rapid equilibrium assumptions, the equations may be given either as functions of the individual rate constants in the reversible steps assumed in rapid equilibrium or as functions of the corresponding equilibrium constants. The steady state equations are easily obtained from the transient phase equations by setting the time →∞. We have implemented a computer program, easy to use and with a user-friendly format for the input of data and output of results, which allows the user to derive the symbolic strict transient phase equations and/or those corresponding to the assumption that one or more of the reversible reaction steps are in rapid equilibrium.

Development of kinetic models in the nonlinear world of molecular cell biology

Increasingly, successful research on metabolic systems relies on teams of specialists. Because of the enormous complexity of these systems, many experimental groups have sought collaborations with theoreticians for data analysis and modeling. Predictably, cultural differences in scientific approach, methodology, assumptions, and language have led to some persistent difficulties in communication across the experiment-theory frontier. This report attempts to diagnose some of these difficulties from the perspective of 30 years' experience in both experimental and theoretical biology, and to suggest guidelines for effective collaboration between experimentalists and theorists. As these collaborations move to the level of cellular and molecular biology, effective communication will become all the more important because the simple linear rate laws of radiotracer and stable-isotope kinetics will no longer suffice. This is because every form of regulation and control, hallmarks of metabolic systems, results in nonlinear kinetics. To advance this transition to nonlinear cellular and molecular metabolic models and to facilitate communication between experimental and theoretical collaborators, a general procedure for incorporating control mechanisms in metabolic rate laws is developed based on the familiar rapid-equilibrium assumption of classical enzyme kinetics.

The influence of product instability on Michaelis-Menten kinetics under steady-state and rapid equilibrium assumptions

Analyses of reactions using Michaelis-Menten kinetics usually assume that the reaction product is stable. However such an assumption is not always satisfied. In this paper, an analysis of the Michaelis-Menten kinetics has been performed assuming the product is unstable. The explicit time-course equation of the immediate product has been derived under conditions of limiting enzyme concentration. The analysis is based on the time progress curves of the product and/or on the progress curve of the species, into which the immediate product is transformed. This approach allows the successful determination of all the rate constants.