Publications by authors named "Marc R. Roussel"

Complex dynamical systems are often governed by equations containing many unknown parameters whose precise values may or may not be important for the system's dynamics. In particular, for chemical and biochemical systems, there may be some reactions or subsystems that are inessential to understanding the bifurcation structure and consequent behavior of a model, such as oscillations, multistationarity and patterning. Due to the size, complexity and parametric uncertainties of many (bio)chemical models, a dynamics-preserving reduction scheme that is able to isolate the necessary contributors to particular dynamical behaviors would be useful.

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Models intended to describe the time evolution of a gene network must somehow include transcription, the DNA-templated synthesis of RNA, and translation, the RNA-templated synthesis of proteins. In eukaryotes, the DNA template for transcription can be very long, often consisting of tens of thousands of nucleotides, and lengthy pauses may punctuate this process. Accordingly, transcription can last for many minutes, in some cases hours.

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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.

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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.

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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.

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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.

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Britton Chance, electronics expert when a teenager, became an enthusiastic student of biological oscillations, passing on this enthusiasm to many students and colleagues, including one of us (DL). This historical essay traces BC's influence through the accumulated work of DL to DL's many collaborators. The overall temporal organization of mass-energy, information, and signaling networks in yeast in self-synchronized continuous cultures represents, until now, the most characterized example of in vivo elucidation of time structure.

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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.

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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.

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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.

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We present here a model intended to capture the biochemistry of vein formation in plant leaves. The model consists of three modules. Two of these modules, those describing auxin signaling and transport in plant cells, are biochemically detailed.

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Rubisco, the most abundant protein serving as the primary engine generating organic biomass on Earth, is characterized by a low catalytic constant (in higher plants approx. 3s(-1)) and low specificity for CO(2) leading to photorespiration. We analyze here why this enzyme evolved as the main carbon fixation engine.

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We classify mathematical models that can be used to describe photosynthetic oscillations using ideas from nonlinear dynamics, and discuss potential mechanisms for photosynthetic oscillations in the context of this classification. We then turn our attention to recent experiments with leaves transferred to a low CO₂ atmosphere which revealed stochastic oscillations with a period of a few seconds. Rubisco is the enzyme that takes both CO₂ and O₂ as substrates correspondingly for photosynthetic assimilation and for photorespiration.

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Temporal organization of biological processes requires massively parallel processing on a synchronized time-base. We analyzed time-series data obtained from the bioenergetic oscillatory outputs of Saccharomyces cerevisiae and isolated cardiomyocytes utilizing Relative Dispersional (RDA) and Power Spectral (PSA) analyses. These analyses revealed broad frequency distributions and evidence for long-term memory in the observed dynamics.

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Delay-differential equations are commonly used to model genetic regulatory systems with the delays representing transcription and translation times. Equations with delayed terms can also be used to represent other types of chemical processes. Here we analyze delayed mass-action systems, i.

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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.

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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.

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The quantitative modeling of gene transcription and translation requires a treatment of two key features: stochastic fluctuations due to the limited copy numbers of key molecules (genes, RNA polymerases, ribosomes), and delayed output due to the time required for biopolymer synthesis. Recently proposed algorithms allow for efficient simulations of such systems. However, it is critical to know whether the results of delay stochastic simulations agree with those from more detailed models of the transcription and translation processes.

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A slow manifold is a low-dimensional invariant manifold to which trajectories nearby are rapidly attracted on the way to the equilibrium point. The exact computation of the slow manifold simplifies the model without sacrificing accuracy on the slow time scales of the system. The Maas-Pope intrinsic low-dimensional manifold (ILDM) [Combust.

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The conditions for diffusion-driven (Turing) instabilities in systems with two reactive species are well known. General methods for detecting potential Turing bifurcations in larger reaction schemes are, on the other hand, not well developed. We prove a theorem for a graph-theoretic condition originally given by Volpert and Ivanova [Mathematical Modeling (Nauka, Moscow, 1987) (in Russian), p.

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Measurement of the internal CO(2) concentration (Ci) in tobacco leaves using a fast-response CO(2) exchange system showed that in the light, switching from 350 microLL(-1) to a low CO(2) concentration of 36.5 microLL(-1) (promoting high photorespiration) resulted in the Ci oscillating near the value of CO(2) compensation point (Gamma*). The oscillations are highly irregular, the range of Ci varying by 2-4 microLL(-1) in substomatal cavities with a period of a few seconds.

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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.

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A theoretical analysis of the distinguishability problem of two rival models of the single enzyme-single substrate reaction, the Michaelis-Menten and Henri mechanisms, is presented. We also outline a general approach for analysing the structural indistinguishability between two mechanisms. The approach involves constructing, if possible, a smooth mapping between the two candidate models.

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When molecules are present in small numbers, such as is frequently the case in cells, the usual assumptions leading to differential rate equations are invalid and it is necessary to use a stochastic description which takes into account the randomness of reactive encounters in solution. We display a very simple biochemical model, ordinary competitive inhibition with substrate inflow, which is only capable of damped oscillations in the deterministic mass-action rate equation limit, but which displays sustained oscillations in stochastic simulations. We define an observability parameter, which is essentially just the ratio of the amplitude of the oscillations to the mean value of the concentration.

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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.

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