Theoretical study of a two-dimensional autocatalytic model for calcium dynamics at the extracellular fluid-bone interface

Diversity of temporal self-organized behaviors in a biochemical system

The numerical study of a glycolytic model formed by a system of three delay-differential equations revealed a notable richness of temporal structures which included the three main routes to chaos, as well as a multiplicity of stable coexisting states. The Feigenbaum, intermitency and quasiperiodicity routes to chaos can emerge in the biochemical oscillator. Moreover, different types of birhythmicity, trirhythmicity and hard excitation emerge in the phase space. For a single range of the control parameter it can be observed the coexistence of two quasiperiodicity routes to chaos, the coexistence of a stable steady state with a stable torus, and the coexistence of a strange attractor with different stable regimes such as chaos with different periodic regimes, chaos with bursting behavior, and chaos with torus. In most of the numerical studies, the biochemical oscillator has been considered under periodic input flux being the mean input flux rate 6 mM/h. On the other hand, several investigators have observed quasiperiodic time patterns and chaotic oscillations by monitoring the fluorescence of NADH in glycolyzing yeast under sinusoidal glucose input flux. Our numerical results match well with these experimental studies.

Oscillating wave fronts in isothermal chemical systems with arbitrary powers of autocatalysis

The propagation of planar reaction-diffusion waves in the isothermal autocatalytic system A + m B → ( m + 1)B, rate kab m , is considered. Attention is paid to the case when the diffusion coefficient of reactant A is much less than that of the autocatalyst; the case when A is completely immobilized is discussed in detail. Permanent-form travelling waves are initiated and their structure is treated. It is found that for sufficiently large values of m this steady wave structure becomes unstable to longitudinal disturbances and an oscillatory wave structure develops. The structure of this propagating oscillatory reaction-diffusion front is also discussed.

Spatio-temporal self-organization of bone mineral metabolism and trabecular structure of primary bone

A nonlinear two-variable reaction-diffusion model of bone mineral metabolism, built from an overall self-oscillatory compartmental model of calcium metabolism in vivo, has been studied for its ability to generate spatial and spatio-temporal self-organizations in a two-dimensional space. Analytical and numerical results confirm the theoretical properties previously described for this kind of model. In particular, it is shown that, for a given set of reactional parameter values and certain values of the ratio of the two diffusion coefficients, there exists a set of unstable wavenumbers leading spontaneously to the development, from the homogeneous steady state, of either different types of stationary spatial patterns (hexagonal, striped and re-entrant hexagonal patterns) or more or less complex spatio-temporal expressions. We discuss the relevance of analogies established between some spatial or spatio-temporal structures predicted by the model and some peculiar features of the primary bone trabecular architecture which appear during embryonic ossification.

Modelling of in vivo calcium metabolism. II. Minimal structure or maximum dynamic diversity: the interplay of biological constraints

The temporal behaviour of the nonlinear compartmental model we have developed for rat calcium metabolism is discussed with respect to the theoretical properties of the self-oscillating autocatalytic subunit around which the model is constructed. Depending on the approximations made, this subunit is described by a minimal two-variable model, SU2, or by a three-variable one, SU3. The diversity of the theoretical dynamic behaviours possible with SU2 is greatly increased with SU3. But the identification of SU3 parameter values in three different experimental situations reveals that biological constraints efficiently preserve a simple circadian rhythm for bone metabolism. This analysis indicates the significant contribution of the available bone crystal pool to the dynamic organization of this tissue, and hence to extracellular calcium homeostasis.

Modelling of in vivo calcium metabolism. I. Optimal cooperation between constant and rhythmic behaviours

The relevance of nonlinear dynamics to calcium metabolism led us to reevaluate the role of Ca-regulating hormones in Ca homeostasis. We suggest that, firstly, the main Ca metabolic functions in rat-bone and gut--are organized as dynamic entities able to generate various temporal expressions, including self-oscillating patterns and, secondly, Ca homeostasis results from interaction between both metabolic and hormonal oscillators. Following this schema, a major role for the hormonal system, with its circadian pattern, could be to act directly on metabolic functions or indirectly through feeding behaviour, in order to optimize, coordinate and synchronize the Ca fluxes at ECF level.

Two exceptional sets of physiological clearance curves and their mathematical form: test cases?

Amiodarone concentrations y(t) have been measured from 1 min to more than 50 days following 10 min of infusion, with about 40 observations on each of six normal subjects (Tucker et al., 1984, Eur. J. clin. Pharmacol. 26, 655-656). The form of the log-log plots-ln(y) vs ln(t)--is investigated. These appear to show three phases. First there is a rapid decrease of y(t), then a straight line corresponding to a small negative power of t, ca -0.3, and this line changes continuously but quickly at about 0.5 day into a steeper line that is almost straight. For the curve fitting a simple "spline-type" device was successful. Two continuity conditions were imposed at the time of changeover, which was one of the unknown parameters. The results are compared in detail with those from a set of 15 radiocalcium curves obtained during 2 weeks or more from a single injection of 47Ca (Neer et al., 1967, J. clin. Invest. 46, 1364-1379). Again two power functions of time can be seen. The changeover is much more gradual than with amiodarone, and the fits are still better. Both sets of curves are fitted with fewer adjustable parameters than with the usual multiexponentials that are interpreted in terms of homogeneous compartments. Theoretical and practical implications are mentioned. There is much indirect evidence that hundreds of other clearance curves may consist largely of one or two of such power functions of time.

Oscillatory isozymes as the simplest model for coupled biochemical oscillators

We analyze a simple model for two autocatalytic reactions catalyzed by two distinct isozymes transforming, with different kinetic properties, a given substrate into the same product. This two-variable system can be viewed as the simplest model of chemically coupled biochemical oscillators. Phase-plane analysis indicates how the kinetic differences between the two enzymes give rise to complex oscillatory phenomena such as the coexistence of a stable steady state and a stable limit cycle, or the co-existence of two simultaneously stable oscillatory regimes (birhythmicity). The model allows one to verify a previously proposed conjecture for the origin of birhythmicity. In other conditions, the system admits multiple oscillatory domains as a function of a control parameter whose variation gives rise to markedly different types of oscillations. The latter behavior provides an explanation for the occurrence of multiple modes of oscillations in thalamic neurons.