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International audience; Delays appear in the dynamics of many systems due to non-vanishing reaction times of control systems. In biochemical systems, long sequences of repeated steps, especially in biopolymerization processes, can be modeled by delays. However, modelling systems with delays is often complicated by physical constraints, such as the requirement that solutions representing concentrations of chemical species remain positive. In this work, we consider a model for a detoxifying enzyme whose synthesis is controlled by its substrate. The model includes bindingsite clearance delays, caused by the time required for an RNA polymerase or ribosome to clear its binding site before another such machine can bind. The existence of a positive equilibrium and the positivity and boundedness of solutions of the corresponding delay-differential equations are proven. In addition, the stability of the model is studied using the "small-gain" theorem.

ATF4 is a key transcription factor that activates transcription of genes needed to respond to cellular stress. Although the mRNA encoding ATF4 is present at constant levels in the cell during the initial response, translation of ATF4 increases under conditions of cellular stress while the global translation rate decreases. We study two models for the control system that regulates the translation of ATF4, both based on the Vattem-Wek hypothesis. This hypothesis is based on a race to reload, following the translation of a small upstream open reading frame (uORF), the ternary complex that brings the initiator tRNA to the ribosome as the 40S subunit scans along the mRNA, encountering first a start codon for an inhibitory uORF whose reading frame overlaps the start of the ATF4 coding sequence. We develop a pair of simple, analytic, probabilistic models, one of which assumes all nucleotide triplets have identical kinetic properties, while the other recognizes the existence of triplets at which the ternary complex loads more efficiently. We also consider two different functions representing the dependence of the rate of initiation at uORF1 on the ternary complex concentration. In keeping with the theme of this Special Issue, we studied the properties of these models in a Maple document, which can easily be modified to consider different parameters, translation rate initiation functions, and so on.

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 slow 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., 32 pages, 6 figures

In 1977, Michael Mackey and Leon Glass published a short paper thatВ presented and analyzed three delay-differential physiological models, oneВ of which, now known as the Mackey-Glass equation, was shown to generateВ chaotic behavior. This paper also introduced the concept of a dynamicalВ disease. In this perspective article, I attempt to place the Mackey-Glass paperВ andВ a 1979 followup in historical context, and thereby to gain someВ understanding of the very significant influence it has had across theВ sciences. This influence is mapped through a citation analysis, revealingВ both the timelessness of the themes broached in the Glass-Mackey papers, andВ of the broad influence of these papers, far transcending the specificВ scientific problems originally tackled.

The expression of the TGF-β protein Nodal on the left side of vertebrate embryos is a determining event in the development of internal-organ asymmetry. We present a mathematical model for the control of the expression of Nodal and its antagonist Lefty consisting entirely of realistic elementary reactions. We analyze the model in the absence of Lefty and find a wide range of parameters over which bistability (two stable steady states) is observed, with one stable steady state a low-Nodal state corresponding to the right-hand developmental fate, and the other a high-Nodal state corresponding to the left. We find that bistability requires a transcription factor containing two molecules of phosphorylated Smad2. A numerical survey of the full model, including Lefty, shows the effects of Lefty on the potential for bistability, and on the conditions that lead to the system reaching one or the other steady state.

A parametric sensitivity analysis for periodic solutions of delay-differential equations is developed. Because phase shifts cause the sensitivity coefficients of a periodic orbit to diverge, we focus on sensitivities of the extrema, from which amplitude sensitivities are computed, and of the period. Delay-differential equations are often used to model gene expression networks. In these models, the parametric sensitivities of a particular genotype define the local geometry of the evolutionary landscape. Thus, sensitivities can be used to investigate directions of gradual evolutionary change. An oscillatory protein synthesis model whose properties are modulated by RNA interference is used as an example. This model consists of a set of coupled delay-differential equations involving three delays. Sensitivity analyses are carried out at several operating points. Comments on the evolutionary implications of the results are offered.

In Escherichia coli, an enzyme called Hmp is a key contributor to the detoxification of nitric oxide (NO). In the absence of NO, the transcription of the hmp gene is repressed by an iron-sulfur protein called FNR. NO damages the iron-sulfur cluster of FNR, weakening the repression of hmp and allowing expression of Hmp to high levels. A delayed mass-action model for the Hmp-FNR network has been developed. This model has 33 parameters, all but three of which were estimated. One of the unknown parameters, the rate of NO inflow into the cell’s cytoplasm, was used as a control parameter in a study of the steady-state structure of this model. This study revealed bistability across a wide range of inflow rates, oxygen concentrations, and values of the unknown parameters. The bistability is caused by substrate inhibition of Hmp by NO, which allows for a high-NO steady state, which would likely be lethal, to coexist with a biologically desirable low-NO steady state.

During the templated biopolymerization processes of transcription and translation, a macromolecular machine, either an RNA polymerase or a ribosome, binds to a specific site on the template. Due to the sizes of these enzymes, there is a waiting time before one clears the binding site and another can bind. These clearance delays are relatively short, and one might think that they could be neglected. However, in the case of transcription, these clearance delays are associated with conservation laws, resulting in surprisingly large effects on the bifurcation diagrams in models of gene expression networks. We study an example of this phenomenon in a model of a gene regulated by a non-coding RNA displaying bistability. Neglecting the binding-site clearance delays in this model can only be compensated for by making ad hoc, unphysical adjustments to the model's kinetic constants.

Differential equation models of chemical or biochemical systems usually display multiple, widely varying time scales, i.e. they are stiff. After the decay of transients, trajectories of these systems approach low-dimensional invariant manifolds on which the eventual attractor (an equilibrium point in a closed system) is approached, and in which this attractor is embedded. Computing one of these slow invariant manifolds (SIMs) results in a reduced model of dimension equal to the dimension of the SIM. Another approach to model reduction involves lumping, the formulation of a reduced set of variables that combine the original model variables and in terms of which the reduced model is framed. In this study, we combine lumping with a constructive method for SIMs based on the iterative solution of the invariance equation. We illustrate these methods using a simple model of a linear metabolic pathway, and a model for hydrogen oxidation. The former is treated with a linear lumping function, while a nonlinear lumping function based on a Lyapunov function is used in the latter.

Volume 1, Issue 1 of Mathematical Biosciences was the venue for a now-classic paper on the application of singular perturbation theory in enzyme kinetics, "On the mathematical status of the pseudo-steady state hypothesis of biochemical kinetics" by F. G. Heineken, H. M. Tsuchiya and R. Aris. More than 50 years have passed, and yet this paper continues to be studied and mined for insights. This perspective discusses both the strengths and weaknesses of the work presented in this paper. For many, the justification of the pseudo-steady-state approximation using singular perturbation theory is the main achievement of this paper. However, there is so much more material here, which laid the foundation for a great deal of research in mathematical biochemistry in the intervening decades. The parameterization of the equations, construction of the first-order uniform singular-perturbation solution, and an attempt to apply similar principles to the pseudo-equilibrium approximation are discussed in particular detail.

Network conditions for Turing instability in biochemical systems with two biochemical species are well known and involve autocatalysis or self-activation. On the other hand general network conditions for potential Turing instabilities in large biochemical reaction networks are not well developed. A biochemical reaction network with any number of species where only one species moves is represented by a simple digraph and is modeled by a reaction–diffusion system with non-mass action kinetics. A graph-theoretic condition for potential Turing-Hopf instability that arises when a spatially homogeneous equilibrium loses its stability via a single pair of complex eigenvalues is obtained. This novel graph-theoretic condition is closely related to the negative cycle condition for oscillations in ordinary differential equation models and its generalizations, and requires the existence of a pair of subnetworks, each containing an even number of positive cycles. The technique is illustrated with a double-cycle Goodwin type model.

A mathematical model is devised to study the diffusion of mRNA in the nucleus from the site of synthesis to a nuclear pore where it is exported to the cytoplasm. This study examines the role that nuclear structure can play in determining the kinetics of export by considering models in which elements of the nuclear skeleton and confinement by chromatin direct the mRNA movement. As a rule, a dense chromatin layer favours rapid export by reducing the effective volume for diffusion. However, it may also result in a heavy tail in the export time distribution because of the low mobility of molecules that accidentally find their way deep into the dense layer. An anisotropic solid-state transport system can also assist export. There exist both an optimal ratio of the anisotropy and an optimal depth of the solid-state transport layer that favour rapid export. [Includes supplementary material]

When the anisotropy of a harmonic ion trap is increased, the ions eventually collapse into a two-dimensional structure consisting of concentric shells of ions. This collapse generally behaves like a second-order phase transition. A graph of the critical value of the anisotropy parameter vs. the number of ions displays substructure closely related to the inner-shell configurations of the clusters. The critical exponent for the order parameter of this phase transition (maximum extent in the z direction) was found computationally to have the value β = 1/2. A second critical exponent related to displacements perpendicular to the z axis was found to have the value δ = 1. Using these estimates of the critical exponents, we derive an equation that relates the amplitudes of the displacements of the ions parallel to the x–y plane to the amplitudes along the z axis during the flattening process.

A chemical mechanism is a model of a chemical reaction network consisting of a set of elementary reactions that express how molecules react with each other. In classical mass-action kinetics, a mechanism implies a set of ordinary differential equations (ODEs) which govern the time evolution of the concentrations. In this article, ODE models of chemical kinetics that have the potential for multiple positive equilibria or oscillations are studied. We begin by considering some methods of stability analysis based on the digraph of the Jacobian matrix. We then prove two theorems originally given by A. N. Ivanova which correlate the bifurcation structure of a mass-action model to the properties of a bipartite graph with nodes representing chemical species and reactions. We provide several examples of the application of these theorems.

We monitored a continuous culture of the yeast Saccharomyces cerevisiae by membrane-inlet mass spectrometry. This technique allows very rapid simultaneous measurements (one point every 12 s) of several dissolved gases. During our experiment, the culture exhibited a multioscillatory mode in which the dissolved oxygen and carbon dioxide records displayed periodicities of 13 h, 36 min and 4 min. The 36- and 4-min modes were not visible at all times, but returned at regular intervals during the 13-h cycle. The 4-min mode, which has not previously been described in continuous culture, can also be seen when the culture displays simpler oscillatory behavior. The data can be used to visualize a metabolic attractor of this system, i.e. the set of dissolved gas concentrations which are consistent with the multioscillatory state. Computation of the leading Lyapunov exponent reveals the dynamics on this attractor to be chaotic.

We study a stochastic model of transcription kinetics in order to characterize the distributions of transcriptional delay and of elongation rates. Transcriptional delay is the time which elapses between the binding of RNA polymerase to a promoter sequence and its dissociation from the DNA template strand with consequent release of the transcript. Transcription elongation is the process by which the RNA polymerase slides along the template strand. The model considers a DNA template strand with one promoter site and n nucleotide sites, and five types of reaction processes, which we think are key ones in transcription. The chemical master equation is a set of ordinary differential equations in 3(n) variables, where n is the number of bases in the template. This model is too huge to be handled if n is large. We manage to get a reduced Markov model which has only 2n independent variables and can well approximate the original dynamics. We obtain a number of analytical and numerical results for this model, including delay and transcript elongation rate distributions. Recent studies of single-RNA polymerase transcription by using optical-trapping techniques raise an issue of whether the elongation rates measured in a population are heterogeneous or not. Our model implies that in the cases studied, different RNA polymerase molecules move at different characteristic rates along the template strand. We also discuss the implications of this work for the mathematical modeling of genetic regulatory circuits.

We study methods for reducing chemical master equations using the Michaelis-Menten mechanism as an example. The master equation consists of a set of linear ordinary differential equations whose variables are probabilities that the realizable states exist. For a master equation with s(0) initial substrate molecules and e(0) initial enzyme molecules, the manifold can be parametrized by s(0) of the probability variables. Fraser's functional iteration method is found to be difficult to use for master equations of high dimension. Building on the insights gained from Fraser's method, techniques are developed to produce s(0)-dimensional manifolds of larger systems directly from the eigenvectors. We also develop a simple, but surprisingly effective way to generate initial conditions for the reduced models.

Investigations on the onset of wave instability in excitable reaction–diffusion media have mainly focused on the effects of reaction kinetics and the relative diffusivity of activator and inhibitor. In this study, we characterize wave stability in a medium in which a diffusion coefficient depends on the local concentration of a reagent. Calculations with a simple CO oxidation model illustrate that the sensitivity of the diffusion coefficient to the concentration can behave like a dynamical parameter to induce transitions between “backfiring” and “regular” propagating pulses. The transition is due to the influence of the variable diffusion process on the refractory zone of wave activity.