A New Class of Biochemical Oscillator Models Based on Competitive Binding (original) (raw)
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Chemical Oscillations in Enzyme Kinetics
The Chemical Educator, 1996
The Higgins model is a two variable model in enzyme kinetics. In contrast with other popular simple dynamical models like the Lotka-Volterra model, the Higgins model shows steady states, damped oscillations and stable limit cycles. For these three dynamical behaviors, stability analysis yields expressions of the eigenvalues, which are easy to obtain either analytically or with the use of Mathematica. With these expressions we can find the boundaries between the three dynamical regions in parameter space and the bifurcation point. Also, we have compared the Higgins model with the other two variable models and find that the origin of the richer dynamical behavior of the Higgins model is due to the enzymatic step in the mechanism.
The Effect of Slow Allosteric Transitions in a Coupled Biochemical Oscillator Model
Journal of Theoretical Biology, 1999
The effect of slowed allosteric transitions in a coupled biochemical oscillator model showing complex dynamic behavior is investigated. When the allosteric transitions are sufficiently fast one can obtain a low-dimensional asymptotic approximation for the dynamics of the species that evolve on a slow time-scale. Such low-dimensional models are common in studies of biological control systems and little attention has, so far, been given to the dynamic effect of the large number of species usually eliminated from more biochemically detailed models. Here we investigate the dynamic effect of explicit inclusion of allosteric transitions having finite time-scales of equilibration. It is found that slowed allosteric transitions suppress complex dynamic modes such as bursting, quasi-periodicity and chaos. The effect arises as the enzyme of consideration becomes trapped in an active state where it is unable to respond to changes in effector concentration on the time-scale necessary to support the modes of complex dynamics. Slow allosteric transitions may be favourable in biological systems in which complex oscillations are not desirable but which, at the same time, may benefit from the presence of positive feedbacks. Our findings suggest that slow allosteric transitions and finite internal rates in general may contribute significantly to the dynamics of biological control mechanisms.
Finding complex oscillatory phenomena in biochemical systems An empirical approach
Biophysical Chemistry, 1988
Starting with a model for a product-activated enzymatic reaction proposed for glycolytic oscillations, we show how more complex oscillatory phenomena may develop when the basic model is modified by addition of product recycling into substrate or by coupling in parallel or in series two autocatalytic enzyme reactions. Among the new modes of behavior are the coexistence between two stable types of oscillations (birhythmieity), bursting, and aperiodic oscillations (chaos). On the basis of these results, we outline an empirical method for finding complex oscillatory phenomena in autonomous biochemical systems, not subjected to forcing by a periodic input. This procedure relies on finding in parameter space two domains of instability of the steady state and bringing them close to each other until they merge. Complex phenomena occur in or near the region where the two domains overlap. The method applies to the search for birhythmicity, bursting and chaos in a model for the CAMP signalling system of Dictyostelium discoideum amoebae.
Nonchaos-Mediated Mixed-Mode Oscillations in an Enzyme Reaction System
We report numerical evidence of a new type of wide-ranging organization of mixed-mode oscillations (MMOs) in a model of the peroxidase−oxidase reaction, in the control parameter plane defined by the supply of the reactant NADH and the pH of the medium. In classic MMOs, the intervals of distinct periodic oscillations are always separated from each other by windows of chaos. In contrast, in the new unfolding, such windows of chaos do not exist. Chaos-mediated and nonchaos-mediated MMO phases are separated by a continuous transition boundary in the control parameter plane. In addition, for low pH values, we find an exceptionally wide and intricate mosaic of MMO phases that is described by a detailed phase diagram.
Topics in current chemistry, 2013
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Journal of Theoretical Biology, 1987
We analyze the transition from simple to complex oscillatory behaviour in a threevariable biochemical system that consists of the coupling in series of two autocatalytic enzyme reactions. Complex periodic behaviour occurs in the form of bursting in which clusters of spikes are separated by phases of relative quiescence. The generation of such temporal patterns is investigated by a series of complementary approaches. The dynamics of the system is first cast into two different time-scales, and one of the variables is taken as a slowly-varying parameter influencing the behaviour of the two remaining variables. This analysis shows how complex oscillations develop from simple periodic behaviour and accounts for the existence of various modes of bursting as well as for the dependence of the number of spikes per period on key parameters of the model. We further reduce the number of variables by analyzing bursting by means of one-dimensional return maps obtained from the time evolution of the three-dimensional system. The analysis of a related piecewise linear map allows for a detailed understanding of the complex sequence leading from a bursting pattern with p spikes to a pattern with p+ 1 spikes per period. We show that this transition possesses properties of self-similarity associated with the occurrence of more and more complex patterns of bursting. In addition to bursting, period-doubling bifurcations leading to chaos are observed, as in the differential system, when the piecewise-linear map becomes nonlinear. 219
Biophysical Chemistry, 1998
We analyze the spatial propagation of wave-fronts in a biochemical model for a product-activated enzyme reaction with non-linear recycling of product into substrate. This model was previously studied as a prototype for the Ž . coexistence of two distinct types of periodic oscillations birhythmicity . The system is initially in a stable steady state characterized by the property of multi-threshold excitability, by which it is capable of amplifying in a pulsatory manner perturbations exceeding two distinct thresholds. In such conditions, when the effect of diffusion is taken into account, two distinct wave-fronts are shown to propagate in space, with distinct amplitudes and velocities, for the same set of parameter values, depending on the magnitude of the initial perturbation. Such a multiplicity of propagating wave-fronts represents a new type of coexistence of multiple modes of dynamic behavior, besides the coexistence involving, under spatially homogeneous conditions, multiple steady states, multiple periodic regimes, or a combination of steady and periodic regimes. ᮊ
Spatial patterns in coupled biochemical oscillators
Journal of Mathematical Biology, 1977
The effects of diffusion on the dynamics of biochemical oscillators are investigated for general kinetic mechanisms and for a simplified model of glycolysis. When diffusion is sufficiently rapid a population of oscillators relaxes to a globally-synchronized oscillation, but when diffusion of one or more species is slow enough, the synchronized oscillation can be unstable and a nonuniform steady state or an asynchronous oscillation can arise. The significance of these results visa -vis models of contact inhibition and zonation patterns is discussed.