Robust oscillations within the interlocked feedback model of Drosophila circadian rhythm

A mechanism for generating circadian rhythms has been of major interest in recent years. After the discovery of per and tim, a model with a simple feedback loop involving per and tim has been proposed. However, it is recognized that the simple feedback model cannot account for phenotypes generated by various mutants. A recent report by Glossop, Lyons & Hardin [Science286, 766 (1999)] on Drosophila suggests involvement of another feedback loop by dClk that is interlocked with per-tim feedback loop. In order to examine whether interlocked feedback loops can be a basic mechanism for circadian rhythms, a mathematical model was created and examined. Through extensive simulation and mathematical analysis, it was revealed that the interlocked feedback model accounts for the observations that are not explained by the simple feedback model. Moreover, the interlocked feedback model has robust properties in oscillations.     

Detection of cellular rhythms and global stability within interlocked feedback systems

In this paper, we show how to detect cellular rhythm and its global stability by extending the techniques from the recently developed theory of monotone systems. We establish theoretical results for globally asymptotic stability with consideration of delay by a discrete map. The relationship between positive, negative elements and delay in a general class of interlocked feedback networks can be understood in a system level. Moreover, the correspondence of attractors between a network and its reduced map is obtained and can be used to detect cellular rhythm, and further control the dynamics of the network. We show that global cellular rhythms can always be obtained, thereby enhancing robustness against perturbations of initial conditions and avoiding chaotic oscillations or complete abolishment of oscillations. In this paper, we focus on analyzing the circadian oscillator in Drosophila as an example to detect the occurrence of cellular rhythm and its global stability.     

Kinetics of bimolecular reactions in model bilayers and biological membranes. A critical review

The quantitative study of the probability of molecular encounters giving rise to a reaction in membranes is a challenging discipline. Model systems, model in the sense that they use model bilayers and model reactants, have been widely used for this purpose, but the methodologies employed for the analysis of the results obtained in experiments, and for experimental design, are so disparate that a concerned experimentalist has difficulty in deciding about the value of each approach. This review intends to examine the several approaches that can be found in the literature showing, when feasible, the weakness, strengths and limits of application of each of them. There is not, so far, a full experimental validation of the most promising theories for the analysis of reactions in two dimensions, what leaves open a large field for new research. The major challenge resides in the time range in which the processes take place, but the possibilities of the existing techniques for these studies are far from exhausted. We review also the attempts of several authors to quantitatively analyze the kinetics of reactions in biological membranes. Especially in this field, the recently developed microspectroscopies enclose a still unexplored potential.     

Robust and sensitive control of a quorum‐sensing circuit by two interlocked feedback loops

The quorum‐sensing (QS) response of Vibrio fischeri involves a rapid switch between low and high induction states of the lux operon over a narrow concentration range of the autoinducer (AI) 3‐oxo‐hexanoyl‐L ‐homoserine lactone. In this system, LuxR is an AI‐dependent positive regulator of the lux operon, which encodes the AI synthase. This creates a positive feedback loop common in many bacterial species that exhibit QS‐controlled gene expression. Applying a combination of modeling and experimental analyses, we provide evidence for a LuxR autoregulatory feedback loop that allows LuxR to increase its concentration in the cell during the switch to full lux activation. Using synthetic lux gene fragments, with or without the AI synthase gene, we show that the buildup of LuxR provides more sensitivity to increasing AI, and promotes the induction process. Elevated LuxR levels buffer against spurious variations in AI levels ensuring a robust response that endows the system with enhanced hysteresis. LuxR autoregulation also allows for two distinct responses within the same cell population.     

Mathematical description of microbiological reactions involving intermediates

Stoichiometric relationships for biological reactions involving intermediate formation are developed from microbial reaction fundamentals and thermodynamic principles. Biological reactions proceed through intermediates, which sequester carbon and electrons whenever their degradation is relatively slow. Modeling intermediate formation and subsequent utilization requires evaluation of the distribution of electrons, energy, and macronutrients (C and N) between energy-generating pathways and cell-synthesis pathways for each step in the mineralization of the primary electron-donor substrate. We describe how energy and electron balances are utilized to predict the stoichiometry for each step of a multi-step degradation process. Each stoichiometric relationship developed predicts substrate utilization, cell growth, and the formation of other products (e.g., H(2)CO(3) or H(+)) for one step in the pathway to full mineralization. A modeling example demonstrates how different kinetics for each step in the degradation of nitrilotriacetic acid (NTA) leads to observed patterns in experimental results, such as a delay in the release of H(2)CO(3) after NTA is removed from solution.     

Mathematical study of feedback control roles and relevance in stress erythropoiesis

This work is devoted to mathematical modelling of erythropoiesis. We propose a new multi-scale model, in which we bring together erythroid progenitor dynamics and intracellular regulatory network that determines erythroid cell fate. All erythroid progenitors are divided into several sub-populations according to their maturity. Two intracellular proteins, Erk and Fas, are supposed to be determinant for regulation of self-renewal, differentiation and apoptosis. We consider two growth factors, erythropoietin and glucocorticoids, and describe their dynamics. Several feedback controls are introduced in the model. We carry out computer simulations of anaemia and compare the obtained results with available experimental data on induced anaemia in mice. The main objective of this work is to evaluate the roles of the feedback controls in order to provide more insights into the regulation of erythropoiesis. Feedback by Epo on apoptosis is shown to be determinant in the early stages of the response to anaemia, whereas regulation through intracellular regulatory network, based on Erk and Fas, appears to operate on a long-term scale.     

Mathematical model of the dynamics of psychotherapy

The success of psychotherapy depends on the nature of the therapeutic relationship between a therapist and a client. We use dynamical systems theory to model the dynamics of the emotional interaction between a therapist and client. We determine how the therapeutic endpoint and the dynamics of getting there depend on the parameters of the model. Previously Gottman et al. used a very similar approach (physical-sciences paradigm) for modeling and making predictions about husband–wife relationships. Given that this novel approach shed light on the dyadic interaction between couples, we have applied it to the study of the relationship between therapist and client. The results of our computations provide a new perspective on the therapeutic relationship and a number of useful insights. Our goal is to create a model that is capable of making solid predictions about the dynamics of psychotherapy with the ultimate intention of using it to better train therapists.     

Detection of feedback in the central nervous system using system identification techniques

An analysis method to detect the presence of feedback between biological signals, particularly those associated with the central nervous system, is presented. The technique is based on recent results in the system identification literature involving the concept of a feedback free process. It may be applied to volume conducted signals such as EEG and EMG, as well as to neuronal spike trains through the use of a data transformation procedure. The utility of the technique is then demonstrated in a study of the relationship between Parkinsonian tremor and certain tremor cells found in the thalamus of Parkinsonian patients, using data collected during thalamotomies. The results obtained suggest that feedback mechanisms may be an important factor contributing to Parkinsonian tremor.This research was supported by the Natural Sciences and Engineering Research Council of Canada, The Medical Research Council of Canada, and Bell-Northern Research     

Mathematical modeling of precipitation and dissolution reactions in microbiological systems

We expand the biogeochemical model CCBATCH to include a precipitation/dissolution sub-model that contains kinetic and equilibrium options. This advancement extends CCBATCH's usefulness to situations in which microbial reactions cause or are affected by formation or dissolution of a solid phase. The kinetic option employs a rate expression that explicitly includes the intrinsic kinetics for reaction ormass-transport control, the differencefrom thermodynamic equilibrium, and the aqueous concentration of the rate-limiting metal or ligand. The equilibrium feature can be used alone, and it also serves as check that the kinetic rate never is too fast and ``overshoots' equilibrium. The features of the expanded CCBATCH are illustrated by an example in which the precipitation of Fe(OH)_{3 (s)} allows the biodegradation of citric acid, even though complexes are strong and not bioavailable. Precipitation releases citrate ligand, and biodegradation of the citrate increases the pH.     

Simple critical schemes in non-autocatalytic systems of biochemical reactions

Kinetic behavior of various reaction systems consisting of three compounds was analyzed in order to find and study the origin of multiple-steady-state and auto-oscillatory behavior in biochemical systems. More than 3000 such reactions were considered. Using the bigraph method, six simple critical reaction schemes were found. It is shown that the presence of these fragments in the composition of more complex biochemical systems may be the origin of the multiple steady-states and autooscillations.