The Michaelis-Menten Reaction at Low Substrate Concentrations: Pseudo-First-Order Kinetics and Conditions for Timescale Separation.

We demonstrate that the Michaelis-Menten reaction mechanism can be accurately approximated by a linear system when the initial substrate concentration is low. This leads to pseudo-first-order kinetics, simplifying mathematical calculations and experimental analysis. Our proof utilizes a monotonicity property of the system and Kamke's comparison theorem. This linear approximation yields a closed-form solution, enabling accurate modeling and estimation of reaction rate constants even without timescale separation. Building on prior work, we establish that the sufficient condition for the validity of this approximation is s 0 ≪ K , where K = k 2 / k 1 is the Van Slyke-Cullen constant. This condition is independent of the initial enzyme concentration. Further, we investigate timescale separation within the linear system, identifying necessary and sufficient conditions and deriving the corresponding reduced one-dimensional equations.

The unreasonable effectiveness of the total quasi-steady state approximation, and its limitations.

In this note, we discuss the range of parameters for which the total quasi-steady-state approximation of the Michaelis-Menten reaction mechanism holds validity. We challenge the prevalent notion that total quasi-steady-state approximation is "roughly valid" across all parameters, showing that its validity cannot be assumed, even roughly, across the entire parameter space - when the initial substrate concentration is high. On the positive side, we show that the linearized one-dimensional equation for total substrate is a valid approximation when the initial reduced substrate concentration s0/KM is small. Moreover, we obtain a precise picture of the long-term time course of both substrate and complex.

Natural Parameter Conditions for Singular Perturbations of Chemical and Biochemical Reaction Networks.

We consider reaction networks that admit a singular perturbation reduction in a certain parameter range. The focus of this paper is on deriving "small parameters" (briefly for small perturbation parameters), to gauge the accuracy of the reduction, in a manner that is consistent, amenable to computation and permits an interpretation in chemical or biochemical terms. Our work is based on local timescale estimates via ratios of the real parts of eigenvalues of the Jacobian near critical manifolds. This approach modifies the one introduced by Segel and Slemrod and is familiar from computational singular perturbation theory. While parameters derived by this method cannot provide universal quantitative estimates for the accuracy of a reduction, they represent a critical first step toward this end. Working directly with eigenvalues is generally unfeasible, and at best cumbersome. Therefore we focus on the coefficients of the characteristic polynomial to derive parameters, and relate them to timescales. Thus, we obtain distinguished parameters for systems of arbitrary dimension, with particular emphasis on reduction to dimension one. As a first application, we discuss the Michaelis-Menten reaction mechanism system in various settings, with new and perhaps surprising results. We proceed to investigate more complex enzyme catalyzed reaction mechanisms (uncompetitive, competitive inhibition and cooperativity) of dimension three, with reductions to dimension one and two. The distinguished parameters we derive for these three-dimensional systems are new. In fact, no rigorous derivation of small parameters seems to exist in the literature so far. Numerical simulations are included to illustrate the efficacy of the parameters obtained, but also to show that certain limitations must be observed.

On the anti-quasi-steady-state conditions of enzyme kinetics.

Quasi-steady state reductions for the irreversible Michaelis-Menten reaction mechanism are of interest both from a theoretical and an experimental design perspective. A number of publications have been devoted to extending the parameter range where reduction is possible, via improved sufficient conditions. In the present note, we complement these results by exhibiting local conditions that preclude quasi-steady-state reductions (anti-quasi-steady-state), in the classical as well as in a broader sense. To this end, one needs to obtain necessary (as opposed to sufficient) conditions and determine parameter regions where these do not hold. In particular, we explicitly describe parameter regions where no quasi-steady-state reduction (in any sense) is applicable (anti-quasi-steady-state conditions), and we also show that - in a well defined sense - these parameter regions are small. From another perspective, we obtain local conditions for the accuracy of standard or total quasi-steady-state. Perhaps surprisingly, our conditions do not involve initial substrate.

On the quasi-steady-state approximation in an open Michaelis-Menten reaction mechanism.

The conditions for the validity of the standard quasi-steady-state approximation in the Michaelis-Menten mechanism in a closed reaction vessel have been well studied, but much less so the conditions for the validity of this approximation for the system with substrate inflow. We analyze quasi-steady-state scenarios for the open system attributable to singular perturbations, as well as less restrictive conditions. For both settings we obtain distinguished invariant manifolds and time scale estimates, and we highlight the special role of singular perturbation parameters in higher order approximations of slow manifolds. We close the paper with a discussion of distinguished invariant manifolds in the global phase portrait.

Coordinate-independent singular perturbation reduction for systems with three time scales.

On the basis of recent work by Cardin and Teixeira on ordinary differential equations with more than two time scales, we devise a coordinate-independent reduction for systems with three time scales; thus no a priori separation of variables into fast, slow etc. is required. Moreover we consider arbitrary parameter dependent systems and extend earlier work on Tikhonov-Fenichel parameter values - i.e. parameter values from which singularly perturbed systems emanate upon small perturbations - to the three time-scale setting. We apply our results to two standard systems from biochemistry.

The Rosenzweig-MacArthur system via reduction of an individual based model.

The Rosenzweig-MacArthur system is a particular case of the Gause model, which is widely used to describe predator-prey systems. In the classical derivation, the interaction terms in the differential equation are essentially derived from considering handling time vs. search time, and moreover there exist derivations in the literature which are based on quasi-steady state assumptions. In the present paper we introduce a derivation of this model from first principles and singular perturbation reductions. We first establish a simple stochastic mass action model which leads to a three-dimensional ordinary differential equation, and systematically determine all possible singular perturbation reductions (in the sense of Tikhonov and Fenichel) to two-dimensional systems. Among the reductions obtained we find the Rosenzweig-MacArthur system for a certain choice of small parameters as well as an alternative to the Rosenzweig-MacArthur model, with density dependent death rates for predators. The arguments to obtain the reductions are intrinsically mathematical; no heuristics are employed.

Modeling of Zinc Dynamics in the Synaptic Cleft: Implications for Cadherin Mediated Adhesion and Synaptic Plasticity.

While the numerous influences of synaptically released zinc on synaptic efficiency during long-term potentiation have been discussed by many authors already, we focused on the possible effect of zinc on cadherins and therefore its contribution to morphological changes in the context of synaptic plasticity. The difficulty with gaining insights into the dynamics of zinc-cadherin interaction is the inability to directly observe it on a suitable timescale. Therefore our approach was to establish an analytical model of the zinc diffusion dynamics in the synaptic cleft and experimentally validate, if the theoretical concentrations at the periphery of the synaptic cleft are sufficient to significantly modulate cadherin-mediated adhesion. Our results emphasize, that synaptically released zinc might have a strong accelerating effect on the morphological changes involved in long-term synaptic plasticity. The approach presented here might also prove useful for investigations on other synaptically released trace metals.

The function of 7D-cadherins: a mathematical model predicts physiological importance for water transport through simple epithelia.

BACKGROUND: 7D-cadherins like LI-cadherin are cell adhesion molecules and represent exceptional members of the cadherin superfamily. Although LI-cadherin was shown to act as a functional Ca2+-dependent adhesion molecule, linking neighboring cells together, and to be dysregulated in a variety of diseases, the physiological role is still enigmatic. Interestingly 7D-cadherins occur only in the lateral plasma membranes of cells from epithelia of water transporting tissues like the gut, the liver or the kidney. Furthermore LI-cadherin was shown to exhibit a highly cooperative Ca2+-dependency of the binding activity. Thus it is tempting to assume that LI-cadherin regulates the water transport through the epithelium in a passive fashion by changing its binding activity in dependence on the extracellular Ca2+. RESULTS: We developed a simple mathematical model describing the epithelial lining of a lumen with a content of variable osmolarity covering an interstitium of constant osmolarity. The width of the lateral intercellular cleft was found to influence the water transport significantly. In the case of hypertonic luminal content a narrow cleft is necessary to further increase concentration of the luminal content. If the cleft is too wide, the water flux will change direction and water is transported into the lumen. Electron microscopic images show that in fact areas of the gut can be found where the lateral intercellular cleft is narrow throughout the lateral cell border whereas in other areas the lateral intercellular cleft is widened. CONCLUSIONS: Our simple model clearly predicts that changes of the width of the lateral intercellular cleft can regulate the direction and efficiency of water transport through a simple epithelium. In a narrow cleft the cells can increase the concentration of osmotic active substances easily by active transport whereas if the cleft is wide, friction is reduced but the cells can hardly build up high osmotic gradients. It is now tempting to speculate that 7D-cadherins, owing to their location and their Ca2+-dependence, will adapt their binding activity and thereby the width of the lateral intercellular cleft automatically as the Ca2+-concentration is coupled to the overall electrolyte concentration in the lateral intercellular cleft. This could provide a way to regulate the water resorption in a passive manner adapting to different osmotic conditions.

On a class of deterministic population models with stochastic foundation.

Generalising a site-based stochastic model due to Royama, Solé et al. and Sumpter et al., we investigate competition in a single species with discrete, non-overlapping generations. We show that the deterministic limit of the dynamics depends on a few easily interpretable parameters only. Further, we discuss qualitative properties and limit sets of the corresponding difference equations, and we relate these to modes of competition. Moreover, a detailed analysis of stochastic effects in some relevant scenarios indicates that the behaviour of the stochastic model is very sensitive to further details of the model.

The lipid/protein interface as xenobiotic target site. Kinetic analysis of tadpole narcosis.

High-resolution X-ray diffraction structures of integral membrane proteins have revealed various binding modes of lipids, but current spectroscopic studies still use uniform macroscopic binding constants to describe lipid binding. The Adair approach employing microscopic lipid-binding constants has previously been taken to explain the enhancement of agonist binding to the nicotinic acetylcholine receptor by general anaesthetics in terms of the competitive displacement of essential lipid activator molecules [Walcher S, Altschuh J & Sandermann H (2001) J. Biol. Chem.276, 42191-42195]. This approach was extended to tadpole narcosis induced by alcohols. A single class, or two different classes of lipid activator binding sites, are considered. Microscopic lipid and inhibitor binding constants are derived and allow a close fit to dose-response curves of tadpole narcosis on the basis of a preferential displacement of more loosely bound essential lipid activator molecules. This study illustrates the potential of the Adair approach to resolve protein-bound lipid populations.

Analysis of nuclear targeting activities of transport signals in the human immunodeficiency virus Rev protein.

The human immunodeficiency Rev protein shuttles between the nucleus and cytoplasm, while accumulating to high levels in the nucleus. Rev has a nuclear localization signal (NLS; AA 35-50) with an arginine-rich motif (ARM) that interacts with importin beta and a leucine-rich nuclear export signal (NES; AA 75-84) recognized by CRM1/exportin 1. Here we explore nuclear targeting activities of the transport signals of Rev. GFP tagging and quantitative fluorescence microscopy were used to study the localization behavior of Rev NLS/ARM mutants under conditions inhibiting the export of Rev. Rev mutant M5 was actively transported to the nucleus, despite its known failure to bind importin beta. Microinjection of transport substrates with Rev-NES peptides revealed that the Rev-NES has both nuclear import and export activities. Replacement of amino acid residues "PLER" (77-80) of the NES with alanines abolished bidirectional transport activity of the Rev-NES. These results indicate that both transport signals of Rev have nuclear import capabilities and that the Rev NLS has more than one nuclear targeting activity. This suggests that Rev is able to use various routes for nuclear entry rather than depending on a single pathway.