The Effect of Slow Allosteric Transitions in a Coupled Biochemical Oscillator Model (original) (raw)
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A New Class of Biochemical Oscillator Models Based on Competitive Binding
European Journal of Biochemistry, 1997
It has been noted that single-enzyme systems can undergo strongly damped transient oscillations. In this paper, we present a nonlinear dynamics analysis of oscillations in undriven chemical systems. This analysis allows us to classify transient oscillations into two groups. In the first group, oscillations arise from rapid oscillatory relaxation to a slower transient relaxation mode. These oscillations are always strongly damped. In the second group, it is the slowest relaxation mode which is implicated in the oscillations so these can be very lightly damped. This second class of oscillations has not previously been studied in enzymology. We show that a remarkably simple single-enzyme system, namely competitive inhibition with substrate flow, generates transient oscillations which belong to the second class. In an attempt to design an experimentally realizable version of this model, we then discovered a system which is capable of sustained oscillations. In this experimentally realizable model, two substrates compete to bind to a macromolecule. The flow of one substrate is controlled by a simple feedback device. Sustained oscillations are observed over a very wide range of parameters. In both models, oscillations are favored by a wide disparity in rates of binding and dissociation of the two substrates to the macromolecule.
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
Proceedings of the National Academy of Sciences, 1982
We analyze on a model biochemical system the effect of a coupling between two instability-generating mechanisms. The system considered is that of two allosteric enzymes coupled in series and activated by their respective products. In addition to simple periodic oscillations, the system can exhibit a variety of new modes of dynamic behavior; coexistence between two stable periodic regimes (birhythmicity), random oscillations (chaos), and coexistence of a stable periodic regime with a stable steady state (hard excitation) or with chaos. The relationship between these patterns of temporal self-organization is analyzed as a function of the control parameters of the model. Chaos and birhythmicity appear to be rare events in comparison with simple periodic behavior. We discuss the relevance of these results with respect to the regularity of most biological rhythms.
Origin of burstingpHoscillations in an enzyme model reaction system
Physical Review E, 2005
The transition from simple periodic to bursting behavior in a three-dimensional model system of the heminhydrogen-peroxide-sulfite pH oscillator is investigated. A two-parameter continuation in the flow rate and the hemin decay rate is performed to identify the region of complex dynamics. The bursting oscillations emerge subsequent to a cascade of period-doubling bifurcations and the formation of a chaotic attractor in parameter space where they are found to be organized in periodic-chaotic progressions. This suggests that the bursting oscillations are not associated with phase-locked states on a two-torus. The bursting behavior is classified by a bifurcation analysis using the intrinsic slow-fast structure of the dynamics. In particular, we find a slowly varying quasispecies ͑i.e., a linear combination of two species͒ which acts as an "internal" or quasistatic bifurcation parameter for the remaining two-dimensional subsystem. A systematic two-parameter continuation in the internal parameter and one of the external bifurcation parameters reveals a transition in the bursting mechanism from sub-Hopf/fold-cycle to fold/sub-Hopf type. In addition, the slow-fast analysis provides an explanation for the origin of quasiperiodic behavior in the hemin system, even though the underlying mechanism might be of more general importance.
Nonchaos-Mediated Mixed-Mode Oscillations in an Enzyme Reaction System
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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.
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.
Dynamics of a model biochemical pathway
2006
This paper proposes a minimal model of a biochemical reaction pathway involving coupled negative and positive feedback mechanisms that are common in cellular processes. The analytical and numerical studies show that this pathway is capable of eliciting a large variety of functional dynamics at different parameter values.
Nonlinear dynamics from physics to biology
Complexity, 2007
S elf-organization has been a hot topic in the second half of last century when physicists and chemists discovered a variety of nonequilibrium phenomena that could be subsumed under a common heading. Self-organizing systems form ordered states in space and time spontaneously and without an external template. The patterns are characterized as dissipative structures because their maintenance requires a flow of energy or matter. After introducing a flow of increasing strength into a system at equilibrium, patterns form instantaneously at certain critical values of the flux. In the language of dynamical systems theory the patterns emerge at bifurcation points corresponding to some critical intensity of the flow. At present we know many well-studied examples of self-organizing systems at many time scales and largely different spatial extensions. Examples are the gigantic red spot on Jupiter, cloud patterns in the atmosphere, the Bénard phenomenon in the coffee cup, the Taylor-Cuvette flow, the Belusov-Zhabotinskii reaction, Liesegang rings, and many other nonlinear phenomena. 2 Recent progress in all fields where self-organization is important confirmed the original concepts and, in addition, gave rise to a new formulation of the old paradigms that allows for a distinction of different forms of self-organizing dynamics in physics, chemistry, and biology. We distinguish here three cases that involve different levels of complexity: self-organization of (i) structure, (ii) function, and (iii) intention or seeming purpose. Structural self-organization became a central issue of nonequilibrium dynamics ever since Alan Turing published his seminal work on chemical morphogenesis [1]. Turing suggested a chemical mechanism based on slow diffusion of an activator and fast diffusion of an inhibitor that can lead to spontaneous formation of stable stationary nonequilibrium patterns through diffusion of some key compounds and argued that such a mechanism could be responsible for the formation of biological patterns. It took 20 years before the Turing mechanism was incorporated into a conceptual framework for pattern formation in early embryonic development that results eventually in the patterns we find in adult organisms [2-6]. Activator and inhibitor are thought to represent two "morphogens," leading to short-range activation and long-range inhibition. For a long time no diffusing morphogen was known in developmental biology and, more-After introducing a flow of increasing strength into a system at equilibrium, patterns form instantaneously at certain critical values of the flux.