J. F. Staub

Diurnal variations of plasma calcium and calcitonin function in the rat

1. These experiments were designed to investigate the normal physiological role of calcitonin in the control of blood calcium.

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.

Numerical estimation of the first order derivative: approximate evaluation of an optimal step

Abstract A numerical method for estimating accurate first derivative, f ′ ( x ) , of a function, f ( x ) is proposed. This methodology leads to the approximate determination of an optimal step which minimizes rounding and truncation errors. The method described here has been applied to different functions and its performance compared with that of classical methods using a proportional step length and that of Dumontet and Vignes. We show that on a large field of application this algorithm tends to associate the efficiency of the optimal step method with the requisite of a moderate number of evaluations.

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.

Nonlinear modelling of calcium metabol ism — first attempt in the calcium-deficient rat

A nonlinear model of calcium metabolism in the calcium-deficient rat is pro posed. It takes fully into account the experimental time-courses of plasma calcium and its radionuclide concentrations, circadian variations included. Through its mathematical representation (a set offive first-order differential equations incorporating a nonlinearity representing a second-order autocatalysis), calcium metabolism is considered as a self-organised system and its circadian behaviour is dictated mainly by nonlinear processes involved in bone calcification. From our identification procedure, a probably unique set of parameters is obtained and both temporal stability and the time-course of calcium distribution within the model are studied. The model exhibits a stable limit cycle corresponding to two observed characteristics of calcium metabolism: stability and circadian periodicity. Simulated behaviour of different compartments is discussed in relation to calcium deficiency. In conclusion, this investigation represents the first attempt to elucidate a quantitative study of calcium metabolism, visualised as a self-oscillating system, ie, in a non-steady stable state.

Diversity and Complexity in Dynamical Behaviour of a Simple Autocatalytic Brusselator-Type Model

A simple two-variable theoretical model is proposed. It exhibits most of the temporal dynamical behaviours reported for nonlinear chemical systems [1]: oscillations, excitability, multistability. This autocatalytic model, part of our nonlinear model of calcium metabolism [2], has been associated with bone calcification processes (nucleation and crystal growth). It is described by the differential system:

A physiological view of in vivo calcium dynamics: the regulation of a nonlinear self-organized system.

\begin{gathered} \dot x = a - {k_1}x - (k + {y^2})x + {k_2}y\quad x({t_0}) = {x_0} \hfill \\ \dot y = b + (k + {y^2})x - (1 + {k_2})y\quad y({t_0}) = {y_0} \hfill \\ \end{gathered}