Persistence in metabolic nets.

In an attempt to improve the understanding of complex metabolic dynamic phenomena, we have analysed several 'metabolic networks', dynamical systems which, under a single formulation, take into account the activity of several catalytic dissipative structures, interconnected by substrate fluxes and regulatory signals. These metabolic networks exhibit a rich variety of self-organized dynamic patterns, with e.g., phase transitions emerging in the whole activity of each network. We apply Hurst's R/S analysis to several time series generated by these metabolic networks, and measure Hurst exponents H < 0.5 in most cases. This value of H, indicative of antipersistent processes, is detected at very high significance levels, estimated with detailed Monte Carlo simulations. These results show clearly the considered type of metabolic networks exhibit long-term memory phenomena.

Coexistence of multiple periodic and chaotic regimes in biochemical oscillations with phase shifts.

The numerical study of a glycolytic model formed by a system of three delay differential equations reveals a multiplicity of stable coexisting states: birhythmicity, trirhythmicity, hard excitation and quasiperiodic with chaotic regimes. For different initial functions in the phase space one may observe the coexistence of two different quasiperiodic motions, the existence of a stable steady state with a stable torus, and the existence of a strange attractor with different stable regimes (chaos with torus, chaos with bursting motion, and chaos with different periodic regimes). For a single range of the control parameter values our system may exhibit different bifurcation diagrams: in one case a Feigenbaum route to chaos coexists with a finite number of successive periodic bifurcations, in other conditions it is possible to observe the coexistence of two quasiperiodicity routes to chaos. These studies were obtained both at constant input flux and under forcing conditions.

Quasiperiodicity route to chaos in a biochemical system.

The numerical study of a glycolytic model formed by a system of three delay differential equations reveals a quasiperiodicity route to chaos. When the delay changes in our biochemical system, we can observe the emergence of a strange attractor that replaces a previous torus. This behavior happens both under a constant input flux and when the frequency of the periodic substrate input flux changes. The results obtained under periodic input flux are in agreement with experimental observations.

Intermittency route to chaos in a biochemical system.

The numerical analysis of a glycolytic model performed through the construction of a system of three differential-delay equations reveals a phenomenon of intermittency route to chaos. In our biochemical system, the consideration of delay time variations under constant input flux as well as frequency variations of the periodic substrate input flux allows us, in both cases, to observe a type of transition to chaos different from the 'Feigenbaum route'.

Dynamic behavior in glycolytic oscillations with phase shifts.

Practically all of the studies of glycolytic oscillations in homogeneous spatial mediums have been performed through the construction of systems of ordinary differential equations and the search for their solutions. In this kind of modelling, the system dynamic behavior is considered to depend only on the values adopted by the parameters related to the dependent variables. In the present work, the modeling of a biochemical system through a system of functional differential equations with delay allows us to analyse the consequences that the variations in the parametric values linked to the independent variable (time) have upon the integral solutions of the system. In our model, the delays correspond with phase shifts in the initial functions for two dependent variables. The results of our researches show that when a instability-generating multienzymatic mechanism suffers variations of the delay time in any of its variables, a wide range of different dynamic responses can be produced. Our work is presented as an enlargement on the dynamic study of biochemical oscillations in general and, particularly, the glycolytic oscillations, under the consideration of the existence of variations in the phase shifts during the oscillations of metabolites involved in the studied reactive processes.