J. Veguillas

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.

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.

Attractor Metabolic Networks

Background The experimental observations and numerical studies with dissipative metabolic networks have shown that cellular enzymatic activity self-organizes spontaneously leading to the emergence of a Systemic Metabolic Structure in the cell, characterized by a set of different enzymatic reactions always locked into active states (metabolic core) while the rest of the catalytic processes are only intermittently active. This global metabolic structure was verified for Escherichia coli, Helicobacter pylori and Saccharomyces cerevisiae, and it seems to be a common key feature to all cellular organisms. In concordance with these observations, the cell can be considered a complex metabolic network which mainly integrates a large ensemble of self-organized multienzymatic complexes interconnected by substrate fluxes and regulatory signals, where multiple autonomous oscillatory and quasi-stationary catalytic patterns simultaneously emerge. The network adjusts the internal metabolic activities to the external change by means of flux plasticity and structural plasticity. Methodology/Principal Findings In order to research the systemic mechanisms involved in the regulation of the cellular enzymatic activity we have studied different catalytic activities of a dissipative metabolic network under different external stimuli. The emergent biochemical data have been analysed using statistical mechanic tools, studying some macroscopic properties such as the global information and the energy of the system. We have also obtained an equivalent Hopfield network using a Boltzmann machine. Our main result shows that the dissipative metabolic network can behave as an attractor metabolic network. Conclusions/Significance We have found that the systemic enzymatic activities are governed by attractors with capacity to store functional metabolic patterns which can be correctly recovered from specific input stimuli. The network attractors regulate the catalytic patterns, modify the efficiency in the connection between the multienzymatic complexes, and stably retain these modifications. Here for the first time, we have introduced the general concept of attractor metabolic network, in which this dynamic behavior is observed.

The Number of Catalytic Elements Is Crucial for the Emergence of Metabolic Cores

Background Different studies show evidence that several unicellular organisms display a cellular metabolic structure characterized by a set of enzymes which are always in an active state (metabolic core), while the rest of the molecular catalytic reactions exhibit on-off changing states. This self-organized enzymatic configuration seems to be an intrinsic characteristic of metabolism, common to all living cellular organisms. In a recent analysis performed with dissipative metabolic networks (DMNs) we have shown that this global functional structure emerges in metabolic networks with a relatively high number of catalytic elements, under particular conditions of enzymatic covalent regulatory activity. Methodology/Principal Findings Here, to investigate the mechanism behind the emergence of this supramolecular organization of enzymes, we have performed extensive DMNs simulations (around 15,210,000 networks) taking into account the proportion of the allosterically regulated enzymes and covalent enzymes present in the networks, the variation in the number of substrate fluxes and regulatory signals per catalytic element, as well as the random selection of the catalytic elements that receive substrate fluxes from the exterior. The numerical approximations obtained show that the percentages of DMNs with metabolic cores grow with the number of catalytic elements, converging to 100% for all cases. Conclusions/Significance The results show evidence that the fundamental factor for the spontaneous emergence of this global self-organized enzymatic structure is the number of catalytic elements in the metabolic networks. Our analysis corroborates and expands on our previous studies illustrating a crucial property of the global structure of the cellular metabolism. These results also offer important insights into the mechanisms which ensure the robustness and stability of living cells.

LONG-RANGE CORRELATIONS IN THE PHASE-SHIFTS OF NUMERICAL SIMULATIONS OF BIOCHEMICAL OSCILLATIONS AND IN EXPERIMENTAL CARDIAC RHYTHMS

In biochemical dynamical systems during each transition between periodical behaviors, all metabolic intermediaries of the system oscillate with the same frequency but with different phase-shifts. We have studied the behavior of phase-shift records obtained from random transitions between periodic solutions of a biochemical dynamical system. The phase-shift data were analyzed by means of Hursts rescaled range method (introduced by Mandelbrot and Wallis). The results show the existence of persistent behavior: each value of the phase-shift depends not only on the recent transitions, but also on previous ones. In this paper, the different kind of periodic solutions were determined by different small values of the control parameter. It was assessed the significance of this results through extensive Monte Carlo simulations as well as quantifying the long-range correlations. We have also applied this type of analysis on cardiac rhythms, showing a clear persistent behavior. The relationship of the results with the cellular persistence phenomena conditioned by the past, widely evidenced in experimental observations, is discussed.

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.

R/S Analysis Strange Attractors

A method to estimate the persistent behavior from a chaotic time series is proposed. Persistency means that here each value depends to some extent on the previous values and not only on the recent ones. The data were analyzed by means of Hursts rescaled range method, i.e., R/S analysis (which was introduced by Mandelbrot and Wallis). The relation of the Hurst exponent to the self-affine and self-simialr fractal dimension is discussed.

On generalized Boltzmann equations for reacting systems

A quantum-statistical treatment of chemical kinetics is presented which does not differ between non-reactive scattering and rearrangement processes. This treatment is done in such a way that the standard methods of nonequilibrium statistical mechanics become applicable. Kinetics equations of the Waldmann-Snider, and Wang Chang and Uhlenbeck type are derived for the reduced density operator of different species related to an homo-geneous, dilute gaseous system of the type AB+C⇌2AC+B⇌2BC+A. Global rate coefficients for the different rearrangement processes are defined and derived when starting with Waldmann-Snider type equations.

The metabolic core and catalytic switches are fundamental elements in the self-regulation of the systemic metabolic structure of cells.

Background Experimental observations and numerical studies with dissipative metabolic networks have shown that cellular enzymatic activity self-organizes spontaneously leading to the emergence of a metabolic core formed by a set of enzymatic reactions which are always active under all environmental conditions, while the rest of catalytic processes are only intermittently active. The reactions of the metabolic core are essential for biomass formation and to assure optimal metabolic performance. The on-off catalytic reactions and the metabolic core are essential elements of a Systemic Metabolic Structure which seems to be a key feature common to all cellular organisms. Methodology/Principal Findings In order to investigate the functional importance of the metabolic core we have studied different catalytic patterns of a dissipative metabolic network under different external conditions. The emerging biochemical data have been analysed using information-based dynamic tools, such as Pearsons correlation and Transfer Entropy (which measures effective functionality). Our results show that a functional structure of effective connectivity emerges which is dynamical and characterized by significant variations of bio-molecular information flows. Conclusions/Significance We have quantified essential aspects of the metabolic core functionality. The always active enzymatic reactions form a hub –with a high degree of effective connectivity- exhibiting a wide range of functional information values being able to act either as a source or as a sink of bio-molecular causal interactions. Likewise, we have found that the metabolic core is an essential part of an emergent functional structure characterized by catalytic modules and metabolic switches which allow critical transitions in enzymatic activity. Both, the metabolic core and the catalytic switches in which also intermittently-active enzymes are involved seem to be fundamental elements in the self-regulation of the Systemic Metabolic Structure.

Spectral line shapes. II

A spectral line shape function theory is extended for gaseous systems where reactive collisions can occur. The theory of subdynamics and the Fock-Tani representation are used in order to attempt this goal. The dependence of the spectral shape on the pressure and the reactive transition probability is clarified. Finally^a discussion concerning the above aspects and their possible applications has been included as well.