Diversity of temporal self-organized behaviors in a biochemical system

Self-Organization and Information Processing: From Basic Enzymatic Activities to Complex Adaptive Cellular Behavior

One of the main aims of current biology is to understand the origin of the molecular organization that underlies the complex dynamic architecture of cellular life. Here, we present an overview of the main sources of biomolecular order and complexity spanning from the most elementary levels of molecular activity to the emergence of cellular systemic behaviors. First, we have addressed the dissipative self-organization, the principal source of molecular order in the cell. Intensive studies over the last four decades have demonstrated that self-organization is central to understand enzyme activity under cellular conditions, functional coordination between enzymatic reactions, the emergence of dissipative metabolic networks (DMN), and molecular rhythms. The second fundamental source of order is molecular information processing. Studies on effective connectivity based on transfer entropy (TE) have made possible the quantification in bits of biomolecular information flows in DMN. This information processing enables efficient self-regulatory control of metabolism. As a consequence of both main sources of order, systemic functional structures emerge in the cell; in fact, quantitative analyses with DMN have revealed that the basic units of life display a global enzymatic structure that seems to be an essential characteristic of the systemic functional metabolism. This global metabolic structure has been verified experimentally in both prokaryotic and eukaryotic cells. Here, we also discuss how the study of systemic DMN, using Artificial Intelligence and advanced tools of Statistic Mechanics, has shown the emergence of Hopfield-like dynamics characterized by exhibiting associative memory. We have recently confirmed this thesis by testing associative conditioning behavior in individual amoeba cells. In these Pavlovian-like experiments, several hundreds of cells could learn new systemic migratory behaviors and remember them over long periods relative to their cell cycle, forgetting them later. Such associative process seems to correspond to an epigenetic memory. The cellular capacity of learning new adaptive systemic behaviors represents a fundamental evolutionary mechanism for cell adaptation.

Discontinuous spirals of stability in an optically injected semiconductor laser

We report a new kind of discontinuous spiral with stable periodic orbits in the parameter space of an optically injected semiconductor laser model, which is a combination of the intercalation of fish-like and cuspidal-like structures (the two normal forms of complex cubic dynamics). The spiral has a tridimensional structure that rolls up in at least three directions. A turn of approximately 2π radians along the spiral and toward the center increases the number of peaks in the laser intensity by one, which does not occur when traversing the discontinuities. We show that as we vary the linewidth enhancement factor (α), discontinuities are created (destroyed) through disaggregation (collapses) from (into) the so-called shrimp-like structures. Future experimental verification and applications, as well as theoretical studies to explain its origin and relation with homoclinic spirals that exist in its neighborhood, are needed.

On the dynamics of the adenylate energy system: homeorhesis vs homeostasis

Biochemical energy is the fundamental element that maintains both the adequate turnover of the biomolecular structures and the functional metabolic viability of unicellular organisms. The levels of ATP, ADP and AMP reflect roughly the energetic status of the cell, and a precise ratio relating them was proposed by Atkinson as the adenylate energy charge (AEC). Under growth-phase conditions, cells maintain the AEC within narrow physiological values, despite extremely large fluctuations in the adenine nucleotides concentration. Intensive experimental studies have shown that these AEC values are preserved in a wide variety of organisms, both eukaryotes and prokaryotes. Here, to understand some of the functional elements involved in the cellular energy status, we present a computational model conformed by some key essential parts of the adenylate energy system. Specifically, we have considered (I) the main synthesis process of ATP from ADP, (II) the main catalyzed phosphotransfer reaction for interconversion of ATP, ADP and AMP, (III) the enzymatic hydrolysis of ATP yielding ADP, and (IV) the enzymatic hydrolysis of ATP providing AMP. This leads to a dynamic metabolic model (with the form of a delayed differential system) in which the enzymatic rate equations and all the physiological kinetic parameters have been explicitly considered and experimentally tested in vitro. Our central hypothesis is that cells are characterized by changing energy dynamics (homeorhesis). The results show that the AEC presents stable transitions between steady states and periodic oscillations and, in agreement with experimental data these oscillations range within the narrow AEC window. Furthermore, the model shows sustained oscillations in the Gibbs free energy and in the total nucleotide pool. The present study provides a step forward towards the understanding of the fundamental principles and quantitative laws governing the adenylate energy system, which is a fundamental element for unveiling the dynamics of cellular life.

Universal dynamical properties preclude standard clustering in a large class of biochemical data

MOTIVATION

Clustering of chemical and biochemical data based on observed features is a central cognitive step in the analysis of chemical substances, in particular in combinatorial chemistry, or of complex biochemical reaction networks. Often, for reasons unknown to the researcher, this step produces disappointing results. Once the sources of the problem are known, improved clustering methods might revitalize the statistical approach of compound and reaction search and analysis. Here, we present a generic mechanism that may be at the origin of many clustering difficulties.

RESULTS

The variety of dynamical behaviors that can be exhibited by complex biochemical reactions on variation of the system parameters are fundamental system fingerprints. In parameter space, shrimp-like or swallow-tail structures separate parameter sets that lead to stable periodic dynamical behavior from those leading to irregular behavior. We work out the genericity of this phenomenon and demonstrate novel examples for their occurrence in realistic models of biophysics. Although we elucidate the phenomenon by considering the emergence of periodicity in dependence on system parameters in a low-dimensional parameter space, the conclusions from our simple setting are shown to continue to be valid for features in a higher-dimensional feature space, as long as the feature-generating mechanism is not too extreme and the dimension of this space is not too high compared with the amount of available data.

AVAILABILITY AND IMPLEMENTATION

For online versions of super-paramagnetic clustering see http://stoop.ini.uzh.ch/research/clustering.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

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.

Effective Fokker-Planck equation for birhythmic modified van der Pol oscillator

We present an explicit solution based on the phase-amplitude approximation of the Fokker-Planck equation associated with the Langevin equation of the birhythmic modified van der Pol system. The solution enables us to derive probability distributions analytically as well as the activation energies associated with switching between the coexisting different attractors that characterize the birhythmic system. Comparing analytical and numerical results we find good agreement when the frequencies of both attractors are equal, while the predictions of the analytic estimates deteriorate when the two frequencies depart. Under the effect of noise, the two states that characterize the birhythmic system can merge, inasmuch as the parameter plane of the birhythmic solutions is found to shrink when the noise intensity increases. The solution of the Fokker-Planck equation shows that in the birhythmic region, the two attractors are characterized by very different probabilities of finding the system in such a state. The probability becomes comparable only for a narrow range of the control parameters, thus the two limit cycles have properties in close analogy with the thermodynamic phases.

Quantitative analysis of the effective functional structure in yeast glycolysis

The understanding of the effective functionality that governs the enzymatic self-organized processes in cellular conditions is a crucial topic in the post-genomic era. In recent studies, Transfer Entropy has been proposed as a rigorous, robust and self-consistent method for the causal quantification of the functional information flow among nonlinear processes. Here, in order to quantify the functional connectivity for the glycolytic enzymes in dissipative conditions we have analyzed different catalytic patterns using the technique of Transfer Entropy. The data were obtained by means of a yeast glycolytic model formed by three delay differential equations where the enzymatic rate equations of the irreversible stages have been explicitly considered. These enzymatic activity functions were previously modeled and tested experimentally by other different groups. The results show the emergence of a new kind of dynamical functional structure, characterized by changing connectivity flows and a metabolic invariant that constrains the activity of the irreversible enzymes. In addition to the classical topological structure characterized by the specific location of enzymes, substrates, products and feedback-regulatory metabolites, an effective functional structure emerges in the modeled glycolytic system, which is dynamical and characterized by notable variations of the functional interactions. The dynamical structure also exhibits a metabolic invariant which constrains the functional attributes of the enzymes. Finally, in accordance with the classical biochemical studies, our numerical analysis reveals in a quantitative manner that the enzyme phosphofructokinase is the key-core of the metabolic system, behaving for all conditions as the main source of the effective causal flows in yeast glycolysis.

THEORETICAL AND EXPERIMENTAL RECOVERY OF OSCILLATIONS DURING BATCH YEAST GROWTH IN A PURE CULTURE SUBJECT TO NUTRITIONAL STRESS

A new model is proposed for the recovery of the complete cell growth dynamics, and showed to compare well with new experiments that we conducted for a pure culture of the T206 strain of Saccharomyces cerevisiae. The proposed model results show that the batch yeast growth in a limited nutrient medium (5% grape juice) is associated with substantial oscillations, which damp out over time. Additional results obtained by using the proposed model show the recovery of effects that are frequently encountered in experiments such as a "Lag Phase" as well as an inflection point in the "ln curve" of the cell concentration. The proposed model recovers also the logistic growth curve as a special case.

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 Pearson's 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.

Controlling birhythmicity in a self-sustained oscillator by time-delayed feedback

Time-delayed feedback is a practical method for controlling various nonlinear dynamical systems. We consider its influence on the dynamics of a multicycle van der Pol oscillator that is birhythmic in nature. It has been shown that depending on the strength of delay the bifurcation space can be divided into two subspaces for which the dynamical response of the system is generically distinct. We observe an interesting collapse and revival of birhythmicity with the variation of the delay time. Depending on the parameter space the system also exhibits a transition between birhythmicity and monorhythmic behavior. Our analysis of amplitude equation corroborates with the results obtained by numerical simulation of the dynamics.

Quantitative analysis of cellular metabolic dissipative, self-organized structures

One of the most important goals of the postgenomic era is understanding the metabolic dynamic processes and the functional structures generated by them. Extensive studies during the last three decades have shown that the dissipative self-organization of the functional enzymatic associations, the catalytic reactions produced during the metabolite channeling, the microcompartmentalization of these metabolic processes and the emergence of dissipative networks are the fundamental elements of the dynamical organization of cell metabolism. Here we present an overview of how mathematical models can be used to address the properties of dissipative metabolic structures at different organizational levels, both for individual enzymatic associations and for enzymatic networks. Recent analyses performed with dissipative metabolic networks have shown that unicellular organisms display a singular global enzymatic structure common to all living cellular organisms, which seems to be an intrinsic property of the functional metabolism as a whole. Mathematical models firmly based on experiments and their corresponding computational approaches are needed to fully grasp the molecular mechanisms of metabolic dynamical processes. They are necessary to enable the quantitative and qualitative analysis of the cellular catalytic reactions and also to help comprehend the conditions under which the structural dynamical phenomena and biological rhythms arise. Understanding the molecular mechanisms responsible for the metabolic dissipative structures is crucial for unraveling the dynamics of cellular life.

Global self-regulation of the cellular metabolic structure

BACKGROUND

Different studies have shown that cellular enzymatic activities are able to self-organize spontaneously, forming a metabolic core of reactive processes that remain active under different growth conditions while the rest of the molecular catalytic reactions exhibit structural plasticity. This global cellular metabolic structure appears to be an intrinsic characteristic common to all cellular organisms. Recent work performed with dissipative metabolic networks has shown that the fundamental element for the spontaneous emergence of this global self-organized enzymatic structure could be the number of catalytic elements in the metabolic networks.

METHODOLOGY/PRINCIPAL FINDINGS

In order to investigate the factors that may affect the catalytic dynamics under a global metabolic structure characterized by the presence of metabolic cores we have studied different transitions in catalytic patterns belonging to a dissipative metabolic network. The data were analyzed using non-linear dynamics tools: power spectra, reconstructed attractors, long-term correlations, maximum Lyapunov exponent and Approximate Entropy; and we have found the emergence of self-regulation phenomena during the transitions in the metabolic activities.

CONCLUSIONS/SIGNIFICANCE

The analysis has also shown that the chaotic numerical series analyzed correspond to the fractional Brownian motion and they exhibit long-term correlations and low Approximate Entropy indicating a high level of predictability and information during the self-regulation of the metabolic transitions. The results illustrate some aspects of the mechanisms behind the emergence of the metabolic self-regulation processes, which may constitute an important property of the global structure of the cellular metabolism.

Global stability analysis of birhythmicity in a self-sustained oscillator

We analyze the global stability properties of birhythmicity in a self-sustained system with random excitations. The model is a multi-limit-cycle variation in the van der Pol oscillator introduced to analyze enzymatic substrate reactions in brain waves. We show that the two frequencies are strongly influenced by the nonlinear coefficients alpha and beta. With a random excitation, such as a Gaussian white noise, the attractor's global stability is measured by the mean escape time tau from one limit cycle. An effective activation energy barrier is obtained by the slope of the linear part of the variation in the escape time tau versus the inverse noise intensity 1/D. We find that the trapping barriers of the two frequencies can be very different, thus leaving the system on the same attractor for an overwhelming time. However, we also find that the system is nearly symmetric in a narrow range of the parameters.

Global self-organization of the cellular metabolic structure

BACKGROUND

Over many years, it has been assumed that enzymes work either in an isolated way, or organized in small catalytic groups. Several studies performed using "metabolic networks models" are helping to understand the degree of functional complexity that characterizes enzymatic dynamic systems. In a previous work, we used "dissipative metabolic networks" (DMNs) to show that enzymes can present a self-organized global functional structure, in which several sets of enzymes are always in an active state, whereas the rest of molecular catalytic sets exhibit dynamics of on-off changing states. We suggested that this kind of global metabolic dynamics might be a genuine and universal functional configuration of the cellular metabolic structure, common to all living cells. Later, a different group has shown experimentally that this kind of functional structure does, indeed, exist in several microorganisms.

METHODOLOGY/PRINCIPAL FINDINGS

Here we have analyzed around 2.500.000 different DMNs in order to investigate the underlying mechanism of this dynamic global configuration. The numerical analyses that we have performed show that this global configuration is an emergent property inherent to the cellular metabolic dynamics. Concretely, we have found that the existence of a high number of enzymatic subsystems belonging to the DMNs is the fundamental element for the spontaneous emergence of a functional reactive structure characterized by a metabolic core formed by several sets of enzymes always in an active state.

CONCLUSIONS/SIGNIFICANCE

This self-organized dynamic structure seems to be an intrinsic characteristic of metabolism, common to all living cellular organisms. To better understand cellular functionality, it will be crucial to structurally characterize these enzymatic self-organized global structures.

Fronts between rhythms: spatiotemporal dynamics of extended polyrhythmic media

We study the propagation of fronts in extended oscillatory reaction-diffusion systems that contain several coexisting limit cycles. In contrast with the variational behavior, fronts between regions oscillating in two different limit cycles are found to propagate not necessarily towards the region of the less stable limit cycle, but towards the regions of the largest amplitudes, provided that the frequency mismatch between the cycles is sufficiently large. In other words, the smaller oscillations can always be made to control the whole system.

The Gts1 protein stabilizes the autonomous oscillator in yeast

Continuous cultures of Saccharomyces cerevisiae show a robust autonomous temperature compensated oscillation in many metabolic functions. Respiratory activity, a convenient output to measure, oscillates with a period of 40 min. Deletion of GTS1, whose protein product has homology to the circadian per protein, has been implicated in temporal events within yeast, causes a reduction in periodicity to 18 min (wild-type period 40-60 min). The dilution rate was steadily increased from 0.04/h to 0.085/h and the oscillation stabilized after four to six dilutions. However, Gts1p's involvement in the maintenance and generation of metabolic synchrony, and in the central oscillating loop, appear to be minimal, as the mutant oscillation was robust and autonomous. Deletion of GTS1 did cause decreased temperature compensation of the period of the oscillation from Q(10) = 1.07 for the wild-type to Q(10) = 1.6 for the mutant. Also the degree of nutrient compensation observed for the wild-type was not observed in the GTS1-null mutant strain. It is postulated that Gts1p is involved in the mechanism that communicates external conditions, such as temperature, to the central oscillating loop.

Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic

We study a system of delay-differential equations modeling single-lane road traffic. The cars move in a closed circuit and the system's variables are each car's velocity and the distance to the car ahead. For low and high values of traffic density the system has a stable equilibrium solution, corresponding to the uniform flow. Gradually decreasing the density from high to intermediate values we observe a sequence of supercritical Hopf bifurcations forming multistable limit cycles, corresponding to flow regimes with periodically moving traffic jams. Using an asymptotic technique we find approximately small limit cycles born at Hopf bifurcations and numerically preform their global continuations with decreasing density. For sufficiently large delay the system passes to chaos following the Ruelle-Takens-Newhouse scenario (limit cycles-two-tori-three-tori-chaotic attractors). We find that chaotic and nonchaotic attractors coexist for the same parameter values and that chaotic attractors have a broad multifractal spectrum. (c) 2002 American Institute of Physics.

Implementation of intelligent decision support systems in health care

The full implementation of any intelligent system in health care, which is designed for decision support, has several stages, from initial problem identification through development and, finally, cost-benefit analysis. Central to this is formal objectivist evaluation with its core component of inherent performance of the outputs from these systems. A Medline survey of one type of intelligent system is presented, which demonstrates that this issue is not being addressed adequately. Lack of criteria for dealing with the outputs from these "black box" systems to prescribe adequate levels of inherent performance may be preventing their being accepted by those in the health-care domain and, thus, their being applied widely in the field.

Recovery of an oscillatory mode of batch yeast growth in water for a pure culture

New experiments that we conducted show an oscillatory mode of batch yeast growth in water, for a pure culture of the T206 strain of Saccharomyces cerevisiae. The oscillations are damped over time, allowing the cell concentration to stabilize at the stationary equilibrium. A new proposed model that includes the complete cell growth dynamics is introduced and showed to recover the experimental oscillatory results. In addition the proposed model recovers effects that are frequently encountered in experiments such as a "Lag Phase" as well as an inflection point in the "ln curve" of the cell concentration. The proposed model recovers also the Logistic Growth Curve as a special case. For purposes of providing some interesting contrast we present additional experimental as well as computational results for the growth of the VIN7 strain of S. cerevisiae in a 5% grape juice medium. The latter indicates even stronger oscillations during the growth process. In order to capture experimentally the oscillatory growth behavior, very frequent readings are required (every 15-30 min) and the measurement process needs to be extended to longer than usual periods (over 250 h).