A mathematical model for the regulation of induction in the lac operon in Escherichia coli is presented. This model takes into account the dynamics of the permease facilitating the internalization of external lactose; internal lactose; beta-galactosidase, which is involved in the conversion of lactose to allolactose, glucose and galactose; the allolactose interactions with the lac repressor; and mRNA. The final model consists of five nonlinear differential delay equations with delays due to the transcription and translation process. We have paid particular attention to the estimation of the parameters in the model. We have tested our model against two sets of beta-galactosidase activity versus time data, as well as a set of data on beta-galactosidase activity during periodic phosphate feeding. In all three cases we find excellent agreement between the data and the model predictions. Analytical and numerical studies also indicate that for physiologically realistic values of the external lactose and the bacterial growth rate, a regime exists where there may be bistable steady-state behavior, and that this corresponds to a cusp bifurcation in the model dynamics.
G protein-coupled receptor signaling is dynamically regulated by multiple feedback mechanisms, which rapidly attenuate signals elicited by ligand stimulation, causing desensitization. The individual contributions of these mechanisms, however, are poorly understood. Here, we use an improved fluorescent biosensor for cAMP to measure second messenger dynamics stimulated by endogenous  2 -adrenergic receptor ( 2 AR) in living cells.  2 AR stimulation with isoproterenol results in a transient pulse of cAMP, reaching a maximal concentration of ϳ10 M and persisting for less than 5 min. We investigated the contributions of cAMP-dependent kinase, G protein-coupled receptor kinases, and -arrestin to the regulation of  2 AR signal kinetics by using small molecule inhibitors, small interfering RNAs, and mouse embryonic fibroblasts. We found that the cAMP response is restricted in duration by two distinct mechanisms in HEK-293 cells: G protein-coupled receptor kinase (GRK6)-mediated receptor phosphorylation leading to -arrestin mediated receptor inactivation and cAMP-dependent kinase-mediated induction of cAMP metabolism by phosphodiesterases. A mathematical model of  2 AR signal kinetics, fit to these data, revealed that direct receptor inactivation by cAMPdependent kinase is insignificant but that GRK6/-arrestin-mediated inactivation is rapid and profound, occurring with a halftime of 70 s. This quantitative system analysis represents an important advance toward quantifying mechanisms contributing to the physiological regulation of receptor signaling.Tachyphylaxis, or desensitization, denoting the attenuation of a biological response to sustained or repeated intervention, is a pervasive phenomenon in physiological systems. For G protein-coupled receptors (GPCRs) 7 (or, more broadly, seventransmembrane receptors), desensitization occurs through molecular mechanisms that can profoundly limit further stimulation of downstream signals, either through direct receptor inactivation or inhibition of downstream signaling. At the physiological level, we refer to this general loss of responsiveness as desensitization; we refer to the more specific case of direct inhibition of receptor molecules as "receptor inactivation." At the level of the receptor, GPCR signals represent a dynamic balance between ligand-stimulated activities, such as G protein coupling, and negative feedback mechanisms, such as receptor phosphorylation and -arrestin recruitment (1). An agonist's efficacy is determined by the balance between these activities and is limited by the kinetics of receptor inactivation. Despite intensive research into the molecular mechanisms of desensitization by GPCR inactivation, the relative contributions of these mechanisms are largely unknown. One reason for this is that, until recently, there have been few techniques that directly measure receptor signaling in real time. Rather, data showing desensitization and receptor inactivation are usually based on either physiological assessments, such as hemodynamic parameters, whi...
It is known that the lac operon regulatory pathway is capable of showing bistable behavior. This is an important complex feature, arising from the nonlinearity of the involved mechanisms, which is essential to understand the dynamic behavior of this molecular regulatory system. To find which of the mechanisms involved in the regulation of the lac operon is the origin of bistability, we take a previously published model which accounts for the dynamics of mRNA, lactose, allolactose, permease and beta-galactosidase involvement and simplify it by ignoring permease dynamics (assuming a constant permease concentration). To test the behavior of the reduced model, three existing sets of data on beta-galactosidase levels as a function of time are simulated and we obtain a reasonable agreement between the data and the model predictions. The steady states of the reduced model were numerically and analytically analyzed and it was shown that it may indeed display bistability, depending on the extracellular lactose concentration and growth rate.
Cellular responses to hormones and neurotransmitters are necessarily transient. The mating pheromone signal in yeast is typical. Signal initiation requires cell surface receptors, a G protein heterotrimer, and downstream effectors. Signal inactivation requires Sst2, a regulator of G protein signaling (RGS) protein that accelerates GTPase activity. We conducted a quantitative analysis of RGS and G protein expression and devised computational models that describe their activity in vivo. These results indicated that pheromone-dependent transcriptional induction of the RGS protein constitutes a negative feedback loop that leads to desensitization. Modeling also suggested the presence of a positive feedback loop leading to resensitization of the pathway. In confirmation of the model, we found that the RGS protein is ubiquitinated and degraded in response to pheromone stimulation. We identified and quantitated these positive and negative feedback loops, which account for the transient response to external signals observed in vivo.One measure of our understanding of biological systems is our ability to predict their behavior in detail. One aspect of this endeavor is to model signal transduction events, defined here as the dynamic changes that occur within a cell in response to an external stimulus (1-3). Such models can help us to understand how small changes outside a cell produce strongly amplified changes within a cell, how graded signals are converted to all-or-none responses (4), or how activators of one pathway influence the function of a second pathway (5). A second goal, and the focus of this work, is to understand how transient external signals are prevented from being propagated indefinitely within the cell. Here we describe the molecular basis for signal activation, desensitization, and eventual resensitization of G proteins by receptors and RGS 1 proteins. The experimentally observed behavior is described mechanistically by computational modeling of the pathway.For these studies we investigated the mating pheromone signaling pathway in yeast Saccharomyces cerevisiae. The yeast mating response is arguably the best characterized signal transduction pathway of any eukaryote, and it has long served as a prototype for hormone, neurotransmitter, and sensory response systems in humans (6). Disruption or activation of pathway components leads to highly specific changes that can be easily quantified. Finally, because it is a unicellular eukaryote, every cell in a population is genetically and phenotypically identical (all cells are "typical").Mating in yeast is the fusion of a and ␣ haploid cell types to form an a/␣ diploid. The events leading to fusion are initiated by specific pheromones: ␣-type cells secrete ␣-factor pheromone, which binds to a specific receptor (Ste2) on a-cells, while a-cells secrete a-factor that binds to receptors (Ste3) on ␣-cells. Upon pheromone binding to its receptor, the G protein ␣ subunit (Gpa1) releases GDP, binds to GTP, and liberates the G protein ␥ subunits (Ste4/Ste18). Sustained...
Understanding the regulation of gene control networks and their ensuing dynamics will be a critical component in the understanding of the mountain of genomic data being currently collected. This paper reviews recent mathematical modeling work on the tryptophan and lactose operons which are, respectively, the classical paradigms for repressible and inducible operons. To cite this article: M.C. Mackey et al., C. R. Biologies 327 (2004). 2004 Published by Elsevier SAS on behalf of Académie des sciences. RésuméModélisation de la dynamique de l'opéron : les opérons tryptophane et lactose comme paradigmes. L'étude de la régulation des réseaux de contrôles des gènes et des dynamiques qui en découlent sera une composante critique de la compréhension de la masse de données génomiques collectées. Cet article fait le bilan des récents travaux de modélisation mathématique sur les opérons tryptophanes et lactoses, qui sont respectivement les paradigmes classiques pour les opérons répressibles et inductibles.
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