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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.

Living cells are essentially dynamic metabolic systems, which are highly self-organized and formed by complex membranes surrounding a dense fluid mixture where millions of different biochemical elements interact to form self-assembled aggregates, a rich variety of supra-macromolecular functional structures and a great diversity of temporal metabolic behaviors.

The enzymes are the most outstanding molecules of these surprisingly reactive systems. They are responsible for almost all the biomolecular transformations, which globally considered are called cellular metabolism. Likewise, the dynamic functional organization of the cellular metabolism acts as an intricate network of densely integrated biochemical reactions forming one of the most complex dynamical systems in nature [

From another perspective, the cells can be considered as open systems that operate far-from-thermodynamic- equilibrium and exchange energy and matter with the external environment. A part of the energy inflow is used to produce a form of energy of higher thermodynamic value,

These kinds of spatial and functional molecular structures constitute a new type of supramolecular organization in the far-from-equilibrium open systems that was called dissipative structures by I. Prigogine [

The dissipative structure constitutes the fundamental element to understand the emergence of the spatial-functional architecture in cells and provide a conceptual framework that allows us to unify the dynamic, self-organized metabolic processes that occur in all biological organisms.

The conditions prevailing inside the cell are characterized by a surprising molecular crowding and, in this interior medium, the enzymes do not work in an isolated way but forming molecular associations (supramolecular organization), e.g., the analysis of proteome of

Intensive studies of protein-protein interactions show thousands of different interactions among enzymatic macromolecules, which self-assemble to form large supramolecular complexes. These associations occur in all kinds of cells, both prokaryotes and eukaryotes [

Likewise, experimental observations have explicitly shown that many enzymes that operate within metabolic pathways may form functional supramolecular catalytic associations. Some of the first experimentally isolated enzymatic associations were, among others, the glycolytic subsystem [

Association of various enzymes in large complexes (metabolon) allows the direct transfer of their common intermediate metabolites from the active site of one enzyme to the catalytic centre of the following enzyme without prior dissociation into the bulk solvent (substrate channeling). This process of non-covalent direct transfer of metabolic intermediates allows for a decrease in the transit time of reaction substrates, originating a faster catalysis through the pathway, preventing the loss of reaction intermediates by diffusion and increasing the efficiency and control of the catalytic processes in the multienzymatic aggregate [

Different studies have shown that many enzymes that operate within metabolic pathways exhibit substrate channeling, including glycolysis, the Krebs cycle, purine and pyrimidine biosynthesis, protein biosynthesis, amino acid metabolism, DNA replication, RNA synthesis, lipid metabolism,

In addition, reversible interactions of enzyme aggregates with structural proteins and membranes are a common occurrence in eukaryotic cells, which can originate the emergence of metabolic microcompartments within the soluble phases of cells [

Substrate channeling and microcompartmentalization of the cytoplasm provide high catalytic efficiency and biochemical mechanisms of great physiological importance for the control of specific enzymatic pathways and for the inter-pathway regulations.

Metabolic microcompartmentalization has been notably investigated in several eukaryotic cells, fundamentally in muscle and brain cells. In this sense, it is to highlight the works of V. Saks and colleagues on the structural organization of the intracellular energy transfer networks in cardiac cells where macromolecules, myofibrils, sarcoplasmic reticulum and mitochondria are involved in multiple structural and functional interactions, which allow the organization in the intracellular medium of compartmentalized energy transfer and other related metabolic processes. This supra structural organization has been called “intracellular energetic units” (ICEU) and represents the basic organization of muscle energy metabolism [

Similarly to what has been described for cardiac cells, it also functions in brain cells, particularly in synaptosomes [

Extensive studies of spatial metabolic structures during the last three decades have shown that the formation of functional enzymatic associations (macromolecular self-organization), the metabolite channeling and the microcompartmentalization of the metabolic processes (supra-macro-molecular organization) are the principal ways of structural organization of the eukaryotic cell metabolism.

Prokaryotic cells also exhibit microcompartments, but in this case they have outer shells which are composed of thousands of protein subunits and are filled with enzymes belonging to specific metabolic pathways in the interiors [

Contrary to eukaryotic cells, prokaryotic microcompartments do not contain lipid structures and consist of widespread compartments (about 100–200 nanometers) made of protein shells (the major constituents are proteins of the so-called “bacterial micro-compartment”) which surround and enclose different enzymes [

Although bacterial microcompartments were first observed more than 40 years ago, a detailed understanding of how they function is only now beginning to emerge [

The organization of cooperating enzymes into macromolecular complexes and their integration in microcompartments is a central feature of cellular metabolism, crucial for the regulation and efficiency of cellular processes and fundamental for the functional basis of cell life.

The cellular organization at the molecular level presents another relevant characteristic: the emergence of functional structures which allow the temporal self-organization of metabolic processes.

A large number of experimental observations have shown that the enzymes apart from forming functional catalytic associations can exhibit oscillatory catalytic patterns (temporal self-organization).

In the far-from-equilibrium conditions prevailing inside the cell, the catalytic dynamics of enzymatic sets present transitions between different stationary and oscillatory molecular patterns. Each dissipatively structured functional enzymatic association (metabolic subsystem) acts as a catalytic entity, in which the activity is autonomous with respect to the other enzymatic associations and spontaneously organized molecular oscillations may emerge comprising an infinite number of distinct oscillatory activity regimes. When the oscillations in an enzymatic association are periodic [

Numerous experimental observations of temporal metabolic structures both in prokaryotic and eukaryotic cells have shown the spontaneous emergence of molecular oscillations in most of the fundamental metabolic processes. For instance, there are oscillatory biochemical processes involved in: intracellular free amino acid pools [

Oscillations represent one of the most striking manifestations of dynamic behavior, of not only qualitative but also quantitative importance, in cell metabolic systems; e.g., considering only the transcription processes, it has been reported that at least 60% of all gene expression in

These functional structures that provide the temporal self-organization of metabolism correspond to dissipative systems, and the catalytic oscillatory behaviors find their roots in the many regulatory processes that control the dynamics of the enzymes that belong to them [

The temporal organization in the metabolic processes in terms of rhythmic phenomena covers a wide time window with period lengths ranging from milliseconds [

The transition from simple periodic behavior to complex oscillatory phenomena, including bursting (oscillations with one large spike and series of secondary oscillations) [

Many cytological processes such as biosynthetic pathways, assembly of macrostructures, membranes and organelles, migration and cell division, require temporal organization with many simultaneous time scales [

Evidence that the cells exhibit multi-oscillatory processes with fractal properties has been reported and these dynamic behaviors seem to be consistent with scale-free dynamics spanning a wide range of frequencies of at least three orders of magnitude [

Some temporal functional metabolic processes are not compatible with one another. In this sense, there is also evidence of the necessity for temporal compartmentalization in cells [

Furthermore, different studies have shown that many metabolic subsystems and genes oscillate as a function of the metabolic cycle, which has added another level of complexity to these kinds of functional metabolic structures [

Second types of temporal-functional metabolic structures are those implied in the circadian rhythms which occur with a period close to 24 hours (the exogenous period of the rotation of the earth).

Cells adapt their metabolism to the appropriate time of day synchronizing the timing of metabolic reactions with cyclic changes in the external environment [

Circadian rhythms govern a wide variety of metabolic and physiological processes in all organisms from prokaryotes to human cells [

An intimate interplay exists between circadian clocks and metabolic functions and at least 10% of all cellular transcripts oscillate in a circadian manner [

The molecular processes underlying circadian rhythms have been extensively studied over the past ten years and they are based on clock proteins organized in regulatory feedback loops [

In addition to transcriptional-translational feedback loops, further levels of regulation operate to maintain circadian rhythms. These include modulation of many transcriptional factors [

Quantitative molecular models for circadian rhythms have been proposed to investigate their dynamic properties based on interconnected transcriptional-translational feedback loops in which specific clock-factors repress the transcription of their own genes [

Theoretical and experimental advances during the past decade have clarified the main molecular processes of these circadian rhythms which can be considered as a subset of metabolic rhythms with a period, defined as the time to complete one cycle of 24 hours. Likewise, there is experimental evidence that the circadian clock shares common features with the cell cycle [

When spatial inhomogeneities develop instabilities in the intracellular medium, it may lead to the emergence of spatio-temporal dissipative structures which can take the form of propagating concentration waves. This dynamic behavior is closely related to temporal metabolic oscillations.

Biochemical waves are a rather general feature of cells in which are involved pH, membrane potential, flavoproteins, calcium, NAD(P)H,

There are several types of waves and they vary in their chemical composition, velocity, shape, intensity, and location [^{+} waves in parallel with Ca^{2+} waves [_{3}) [

In spite of its physiological importance, many aspects of the spatial-temporal dissipative structures (such as their molecular regulatory mechanisms, the relationship to the cell cycle and the temporal metabolic behaviors) are still poorly understood.

The cellular organization at the molecular level presents another relevant characteristic: the emergence of global functional structures.

In 1999, the first model of a metabolic dissipative network was developed, which was characterized by sets of catalytic elements (each of them represents a dissipatively structured enzymatic association) connected by substrate fluxes and regulatory signals (allosteric and covalent modulations). These enzymatic sets of enzymes (metabolic subsystems) may present oscillatory and stationary activity patterns [

By means of numerical studies, a singular global metabolic structure was found to be able to self-organize spontaneously, characterized by a set of different enzymatic associations always locked into active states (metabolic cores) while the rest of metabolic subsystems presented dynamics of

Later studies carried out in 2004 and 2005, implementing a flux balance analysis applied to metabolic networks, produced additional evidence of the global functional structure in which a set of metabolic reactions belonging to different anabolic pathways remains active under all investigated growth conditions, forming a metabolic core, whereas the rest of the reactions belonging to different pathways are only conditionally active [

The metabolic core exhibits a set of catalytic reactions always active under all environmental conditions, while the rest of the reactions of the cellular metabolism are only conditionally active, being turned on in specific metabolic conditions. The core reactions conform a single cluster of permanently connected metabolic processes where the activity is highly synchronized, representing the main integrators of metabolic activity. Two types of reactions are present in the metabolic core: the first type is essential for biomass formation both for optimal and suboptimal growth, while the second type of reactions is required only to assure optimal metabolic performance. It was also suggested that this self-organized enzymatic configuration appears to be an intrinsic characteristic of metabolism, common to all living cellular organisms [

More recently, it has been observed in extensive dissipative metabolic network simulations that the fundamental factor for the spontaneous emergence of this global self-organized enzymatic structure is the number of enzymatic dissipative associations (metabolic subsystems) [

Metabolic dissipative networks exhibit a complex dynamic super-structure which integrates different dynamic systems (each of them corresponds to different enzymatic associations dissipatively structured) and it forms a global and unique, absolutely well defined, deterministic, dynamical system, in which self-organization, self-regulation and persistent properties may emerge [

Theoretical and experimental data convincingly show that cellular metabolism cannot be understood if cell interior medium is considered as a homogenous solution with dispersed isolated enzymes and without any diffusion restrictions. On the contrary, cellular organisms display a rich variety of dynamics structures, both spatial and temporal, where enzymes together with other bio-molecules form complex supramolecular associations.

Each set of cooperating enzymes, dissipatively structured and integrated into macromolecular complexes and microcompartments, acts as a metabolic dynamic subsystem and they seem constitute the basic units of the cellular metabolism.

As shown below, metabolic subsystems are advantageous thermodynamically biochemical structures, which acting as individual catalytic entities forming unique, well-defined dynamical systems and their activity are autonomous with respect to the other enzymatic associations. The understanding of the elemental principles and quantitative laws that govern the basic metabolic structure of cells is a key challenge of the post-genomic era.

In this task, it becomes totally necessary to use mathematical and physical tools based on experiments. Mathematical models and non-linear dynamics tools are useful to fully grasp the molecular mechanisms of metabolic dynamical processes. They are necessary to enable the quantitative and qualitative analysis of the functional metabolic structures and also to help to comprehend the conditions under which the structural dynamical phenomena and biological rhythms arise.

Here we present an overview, within the area of Systems Biology, of how mathematical models and non-linear dynamics tools can be used to address the properties of functional dissipative metabolic structures at different organizational levels, both for simple sets of enzymatic associations and for large enzymatic networks.

Models and computational simulations, firmly based on experiments, are particularly valuable for exploring the dynamic phenomena associated with protein-protein interactions, substrate channeling and molecular microcompartmentalization processes. These procedures and methods allow to explain how higher level properties of complex molecular systems arise from the interactions among their elemental parts.

Clarifications of the functional mechanisms underlying dynamic metabolic structures as well as the study of regulation in cellular rhythms are some of the most important applications of Systems Biology. In fact, one of the major challenges in contemporary biology is the development of quantitative models for studying regulatory mechanisms in complex biomolecular systems.

Mathematical studies of metabolic processes allow rapid qualitative and quantitative determination of the dynamic molecular interactions belonging to the functional structures, and thereby can help to identify key parameters that have the most profound effect on the regulation of their dynamics.

Likewise, the advent in the field of the molecular biology of non-linear dynamics tools, such as power spectra, reconstructed attractors, long-term correlations, maximum Lyapunov exponent and Approximate Entropy, should facilitate the collection of more quantitative data on the dynamics of cellular processes.

Systems biology is fundamental to study the functional structures of metabolism, to understand the molecular mechanisms responsible for the most basic dissipative metabolic processes and will be crucial to elucidate the functional architecture of the cell and the dynamics of cellular life.

The spontaneous self-organization of metabolic processes (such as the formation of macromolecular structures and the emergence of functional patterns) is one of most relevant questions for contemporary biology.

The theoretical basis of dissipative self-organization processes was formulated by Ilya Prigogine [

According to these studies, the entropy of an isolated system tends to increase toward a maximum at thermodynamic equilibrium but in an open system the entropy can either be maintained at the same level or decreased (negative variation of entropy) and the overall system does not violate the Second Law. Negative variation of entropy can be maintained by a continuous exchange of materials and energy with the environment avoiding a transition into thermodynamic equilibrium.

Entropy is a quantification of randomness, uncertainty, and disorganization. Negative variation of entropy corresponds to relative order, certainty, and organization in the system. The opposite tendency for an open system which eats up energy of low entropy and dissipates energy of higher entropy to its environment may allow for the self-organization of the system.

A system capable of continuously importing free energy from the environment and, at the same time, exporting entropy (the total entropy of the system decreasing over time) was called dissipative structure. These advantageous thermodynamic systems use a part of the energy inflow to produce a new form of energy characterized by lower entropy which self-organizes the systems [

When a biochemical dissipative structure diminishes the number of bimolecular entities and increases their size by means of metabolic interactions and molecular bonds complex spatial macro-structures emerge in the biochemical system from simpler structures.

In the functional plane, the ordered interacting catalytic processes of the biochemical subsystem may exhibit long range correlations originating diversity of functional dynamical patterns which corresponds to ordered temporal-functional behaviors (metabolic rhythms) [

A metabolic subsystem is just a dissipatively self-organized structure where a set of functionally associated enzymes adopts a new supramolecular configuration in which ordered metabolic dynamical patterns (metabolic rhythms) may arise.

As a consequence of dissipative processes, a metabolic subsystem increases its complexity generating new spatial and functional structures that did not exist before. Self-organization is also a spontaneous process,

The problem of the emergence of self-organized structures has been studied extensively over the past sixty years and in the dissipative structures theory other important elements must be considered such as the amplification of fluctuations, non-linear interactions, bifurcations, phase transitions, complexity theory,

The concept of self-organization is central to the description of molecular-functional architecture of cellular live [

In the history of research on temporarily self-organized metabolic processes, the glycolytic pathway has played an important role. Its oscillatory behavior was observed, for the first time, in the fluorescent studies of yeast cells [

As an extension of those previous studies for glycolytic oscillations based on a single positive feedback, Goldbeter and Decroly analyzed numerically the effect of two feedback loops coupled in series on a biochemical system [

The _{1}, which is activated by its product P_{1}; the second allosteric enzyme E_{2} is also activated by its product P_{2}. The removal of the product P_{2} is supposed to be linear, with a lost constant of ks.

The processes represented in the diagram are converted to differential equations describing their rates as follows:

1a. (Rate of change of [S]) = (Rate of input of [S]) − (Rate of degradation of [S]).

The explicit equation for 1a is

2a. (Rate of change of [P_{1}]) = (Rate of synthesis of [P_{1}]) − (Rate of degradation of [P_{1}]) and the corresponding equation is

3a. (Rate of change of [P_{2}]) = (Rate of synthesis of [P_{2}]) − (Rate of removal of [P_{2}]); the equation is

Here, V is the speed of the substrate S brought into the metabolic subsystem; V_{max1} and V_{max2} denote the maximum activities of enzymes E_{1} and E_{2}; ks is the first-order rate constant for removal of P_{2}; φ and η are the enzymatic rate laws for E_{1} and E_{2} developed in the framework of the concerted transition theory [

where, α, β and γ denote the normalized concentrations of S_{1}, P_{1} and P_{2}, divided respectively, by the Michaelis constants of E_{1} (Km1) and by the dissociation constant of P_{1} for E_{1} (K P_{1}) and of P_{2} for E_{2} (KP2); L_{1} and L_{2} are the allosteric constants of E_{1} and E_{2}.

Once the different elements of the equations are normalized, the time-evolution of the metabolic subsystem is described in any instant by the following three differential equations:

σ_{1} and σ_{2} correspond to the normalized maximum activity of the enzymes E_{1}, E_{2}, (they are divided by the constants K_{m1}, K_{m2}, the Michaelis constants of the enzymes E_{1} and E_{2}, respectively); q_{1} = K_{m1}/K_{p1}, q_{2} = K_{p1}/K_{p2} and d = K_{p1}/K_{m2}.

Once the values for the parameters are specified and given initial values for the dependent variables (see [_{2}) was fixed as the control parameter of the multienzymatic instability-generating reactive system.

After the numerical integration, a wide range of different types of dynamic patterns can be evidenced as a function of the control parameter value. At very small ks values, the system (1) admits a single steady-state solution and when ks is increased, the steady state become unstable, leading to the emergence of a periodic pattern (_{1}, P_{1} and P_{2}) oscillate with the same frequency but different amplitudes.

In the interval 0.792 < k_{S} ≤ 2.034 0. the metabolic subsystem exhibits the most interesting dynamical behaviors.

For instance, one can observe how for 0.792 < k_{S} ≤ 1.584 hard excitation emerges in the functional enzymatic association and two kinds of integral solutions coexist under the same control parameter value: a stable steady state and a stable periodic oscillation (bistability). The metabolic subsystem starts from a stable steady state but evolves to a stable periodic regime when the initial concentrations of S_{1}, P_{1} and P_{2}, exceed a determinate threshold value (

At further increases in ks (1.584 < k_{S} ≤ 1.82) the metabolic subsystem undergoes a reorganization of its dynamics and spontaneously presents a temporal structure characterized by the coexistence of two stable periodic behaviors under the same control parameter value (bistability). The integral solutions now settle on two regular oscillatory regimens depending on the initial conditions (the concentrations of S_{1}, P_{1} and P_{2}).

Between 1.82 < k_{S} ≤ 1.974 the biochemical system exhibits one simple periodic pattern (oscillation of period-1 with one maximum and one minimum per oscillation) and when the control parameter increases (1.99 < k_{S} ≤ 2.034) the numerical solutions of the biochemical oscillator display a classical period-doubling cascade preceding chaos, _{S} increases, a new instability provokes the emergence of regular oscillations of period-4; next a new bifurcation of period-8 appears,

In chaotic conditions, all the metabolic intermediaries (S_{1}, P_{1} and P_{2}) present infinite transitions, modifying uninterruptedly their activity so that they never repeat themselves for arbitrarily long time periods (

Lastly, as ks increases beyond 2.034, complex periodic oscillations emerge in the multienzymatic subsystem (

The quantitative numerical analysis of the system (1) allows showing how metabolite concentrations of the biochemical oscillator, formed by only two irreversible enzymes, vary second by second following a notable diversity of dynamic patterns as a function of the control parameter values (and the initial conditions when two dynamic behaviors coexist). In the numerical analyses, the feedback processes are the main sources of nonlinearity that favor the occurrence of instabilities which provoke the emergence of different dynamical patterns.

In the course of time, open enzymatic systems that exchange matter and energy with their environment exhibit a stable steady state. This stationary non-equilibrium state is more ordered that the equilibrium state of the same energy. Once the enzymatic subsystem operates sufficiently far-from-equilibrium due to the nonlinear nature of its kinetics, the steady state may become unstable leading to the establishment of other dynamical behavior. New instabilities may originate the emergence of different biochemical temporal behaviors.

All these dynamic patterns (including chaos) correspond to ordered motions in the system representing examples of non-equilibrium self-organizations and can therefore be considered as temporal dissipative structures.

The emergence of quantitative behaviors belonging to different metabolic subsystems has been investigated in extensive studies, mainly carried out by means of systems of differential equations, e.g., in the Krebs cycle [

Cell-cycle in eukaryotic cells is governed by a complex network of metabolic reactions controlling the activities of M-phase-promoting factors. These metabolic reactions belonging to a set of enzymes functionally associates can be self-organized in far-from-equilibrium conditions, exhibiting periodic oscillations which govern the cell-cycle.

The network forms a metabolic subsystem that mainly involves enzymes of covalent regulation and protein kinases (Cdk) whose activities depend on binding to cyclins. More concretely, mitosis-promoting factor (MPF) has been identified as a dimmer of two distinct protein molecules: a cyclin subunit and a cyclin-dependent protein kinase (Cdc2), which is periodically activated and inactivated during the cell cycle. MPF activity is regulated by synthesis and degradation of cyclin subunit and by phosphorylation and desphosphorylation of the protein kinase Cdc2 at an activatory threonine (Thr) residue and an inhibitory tyrosine (Tyr) residue. When the active form of MPF is phosphorylated on Thr161, then M-phase begins by phosphorylating a suit of target proteins involved in the main events of mitosis [

In 1991, John J. Tyson and colleagues constructed a dynamic mathematical model for cell-cycle regulation in

First, cyclin subunits synthesized from amino acids (step 1 of the

The activity of dimers can be regulated by altering the phosphorylation state by means of two kinase-phosphatase pairs: Wee1/Cdc25, which acts at Tyr15 (Y), and CAK/INH, which acts at Thr161 (T). As a consequence, the dimers can exist in four different phosphorylation states (

The molecular model shows two experimentally recognized feedback loops [

Each molecular process represented in

The explicit equation for these processes is

A similar procedure continues until reaching a complete set of 10 equations that describe how the molecular element of the metabolic model changes with time.

The rate constants k_{25}, k_{wee} and k_{2} are defined as:

The first six differential equations follow mass-action kinetics and the next four follow Michaelis-Menten kinetics.

Once the values for the 31 parameters (Michaelis constants, total enzyme levels,

The quantitative analysis shows a main relevant behavior; stable regular oscillations emerge in the dynamic system and all the metabolic intermediaries of the metabolic subsystem oscillate with the same frequency but different amplitudes.

In

The cell cycle seems to be controlled by this dynamic structure (attractor) which represents the set of all the possible asymptotic behaviors and corresponds to the ordered motions in the metabolic subsystem.

The catalytic elements implicated in the cell-cycle regulation represent a group of functionally associated and dissipatively structured enzymes that form a catalytic entity as a whole. The catalytic activity of the metabolic subsystem is autonomous with respect to the other enzymatic associations which operate within far-from-equilibrium conditions and as a consequence molecular periodic oscillations spontaneously emerge. This set of dissipatively structured enzymatic associations is an absolutely well-defined, deterministic, dynamical system responsible for the control of the activities of M-phase-promoting factors.

The catalytic elements implicated in the cell-cycle regulation represent a group of functionally associated and dissipatively structured enzymes that form a catalytic entity as a whole. The catalytic activity of the metabolic subsystem is autonomous with respect to the other enzymatic associations which operate within far-from-equilibrium conditions and as a consequence molecular periodic oscillations spontaneously emerge. This set of dissipatively structured enzymatic associations is an absolutely well-defined, deterministic, dynamical system responsible for the control of the activities of M-phase-promoting factors.

For almost two decades, the initial model of Tyson has been developed with new molecular and dynamic factors as for example, bistability [

During the past two decades, different mathematical models have allowed for an intensive study of metabolic processes formed by large groups of enzymes including global metabolic systems.

Traditional models have focused on the kinetics of multi-enzyme systems by solving systems of differential equations and algebraic equations [

Until recently, metabolic networks formed by enzymes and pathways have been studied individually. However, at present, mathematical models based on experimental data are aiming to integrate cellular metabolism as a whole [

Among the different mathematical models focused on enzymatic networks, Flux Balance Analysis (FBA) has emerged as an effective means to analzse metabolic networks in a quantitative manner demonstrating reasonable agreement with experimental data [

The FBA method allows finding optimal steady state flux distributions in a metabolic network subject to additional constraints on the rates of the reaction steps. FBA is based on the assumption that the dynamic mass balance of the metabolic system can be described using a stoichiometric matrix, and relating the flux rates of enzymatic reactions to the time derivatives of metabolite concentrations in the following form:

where X is an

The FBA method is based on the assumption that the concentration of all cellular metabolites must satisfy the steady-state constraint and therefore the dynamic mass balance of the metabolic system must equal zero:

The main element of the FBM is the stoichiometric matrix, S, which describes all the biochemical transformations in a network in a self-consistent and chemically accurate matrix format. The rows of S correspond to various network components, while the columns of S delineate the reactions, or the way in which these components interact with one another [

Because most metabolic systems are underdetermined

Typically, the maximization of the growth flux is used as the objective function [

The development of a flux balance analysis requires the definition of all the metabolic reactions and metabolites. For example, let us consider a simple metabolic network (

Mass balance equations for all reactions and transport processes are written by

Reactions can be represented as an S stoichiometric matrix form with

At steady-state, Sv = 0, a set of algebraic constraints on the reactions rates can be assumed: the objective function is max Z = v_{5} (for example), and then the constraints are

Once the problem of optimization is formulated, techniques of operation research can be used to obtain a solution. In this case, the optimal value of v_{5} was found to be 10.0 (see [^{T}.

The optimal distribution of all fluxes is:

Although classical FBA assumes steady-state conditions, several extensions have been proposed in recent years to improve the predictive ability of this method, e.g., gene regulatory constraints were incorporated into metabolic models leading to a modification of FBA called regulatory flux balance analysis (rFBA) in which Boolean rules are considered on an existing stoichiometric model of gene expression metabolism [

As pointed out in the introduction section, several studies performed using metabolic networks have shown that enzymes can present a self-organized global functional 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

The existence of the global metabolic structure was verified for

Many of the metabolic dynamic analyses have ignored the impact of time delays on enzymatic oscillators, which are due to different biochemical processes such as oscillatory phase-shifts, transport, translation, translocation, and transcription.

What most of these non-linear dynamic studies in metabolic systems have in common is that have been performed through ordinary differential equations (ODE). According to this modeling, selforganized dynamic behaviors are considered to depend on the different values achieved by the parameters linked to the dependent variables. Moreover, the initial conditions are always constant values (never initial functions) and when determining the particular solutions, only a small number of freedom degrees are available, as a result of the restrictions of the ODE systems.

Within the framework of dynamical systems theory delay processes can be approximated accurately by augmenting the original variables with other auxiliary functional variables. By means of these systems of functional differential equations with delay it is possible to take into account initial functions (instead of the constant initial values of ODE systems) and to analyze the consequences that the variations in the parametric values linked to the independent variable (time) have upon the integral solutions of the system.

A typical ODE system is the following:

and a dynamic model governed by a delayed functional differential equations system, can take the following particular form:

where the dependent variable is a n-dimensional vector of the form y = (y_{1},...,y_{n}), t being the independent variable, and the z_{i} variables appear delayed, that is z_{i}(t) = h_{i}(y_{i}(t − λ_{i})) where λ_{i} are the corresponding delays and h_{i} are given functions. Hereafter, the z_{i} will be named functional variables.

In system (3), the derivatives of y_{1},...,y_{n}, evaluated in t are related to the variables y_{1},...,y_{r} evaluated in t − λ_{i}, and related to the variables y_{r+1},...,y_{n} evaluated in t.

As _{1}′(_{n}_{1},...,y_{n} in times before t, the initial conditions cannot be simply the values of y_{1},...,y_{n} in a unique time, but in an interval [t_{0}−δ,t_{0}] with δ = max {λ_{1},…, λ_{r}}, which involves the consideration, in the solution of the system, of the functions f_{0}:[t_{0}−δ,t_{0}] → R^{n} called initial functions. It can be observed therefore that infinite degrees of freedom exist in the determination of the particular solutions.

In the system described by (3), it is possible to take the initial function f_{0} equal to any y(t), which, in particular, can be a periodic solution of the system for λ_{1} = … = λ_{r} = 0 and t ≤ t_{0}.

The initial function will be y^{δ}(t):(t_{0}−δ,t_{0}) → R^{n}, with y^{δ}(t) = y(t), ∀t ∈ [t_{0} − δ, t_{0}].

In the particular case when λ_{1} > 0, λ_{2} = ... = λ_{r} = 0, the first component of the initial function will be _{1}^{δ} (_{0} − δ,_{0}]→ _{1} corresponds a y_{1}^{δ} (t_{0} − λ_{1}), which is the value of the first component of y in t_{0} − λ_{1}; the parameter λ_{1} determines the initial function domain and, given that the solution is periodic, for each different domain of the initial function exists an ordinate value in the origin y_{1}^{δ} ( t_{0} − λ_{1}) and a corresponding phase-shift. It is observed that y_{1}^{δ} ( t_{0} − λ_{1}) is the value of the function y_{1}^{δ} ( t_{0} − λ_{1}) evaluated in t_{0}, where h is the initial function with a phase shift of λ_{1}.

With this type of systems, it is possible to take into account dynamic behaviors related to parametric variations linked to the independent variable. The parametric variations λ_{1} affect the independent variable which represent time delays and can be related to the phase shifts and the domains of the initial functions. Let us see an example next.

The glycolysis continues to be the best known example of temporal self-organization in metabolic processes, and more than four decades ago the existence of variations in the phase shift values of different metabolites during the glycolytic oscillations was experimentally observed [

In order to study the repercussion on the dynamic system of phase-shifts, it is suitable to utilize the systems described by differential equations with delay. For example, we can consider a particular ODE-solution to be equal to a periodic solution y(t) of the system (3) for λ_{1} = … = λ_{r} = 0. And we can take this solution as the general initial function f_{0}.

As we have seen, each delay time reflects a domain and a phase shift of the initial function. Different domains and phase shifts of the initial functions can be considered in system (3) for each value of the parameter λ linked to the independent variable; and so, particular phase shifted ODE-solutions can be made to correspond to phase-shifted initial functions y^{δ} (t):(t_{0} − δ,t_{0})→R^{n}.

In the integration of the delayed functional differential equations system, certain values can be considered for the dependent variables evaluated in t − λ, which correspond to a phase-shifted oscillation of the past. Therefore, it is possible to study if phase-shifted initial functions can be followed (after the corresponding numerical integration) by a mere final phase shift or by a variation in the dynamic behavior of the system.

In this sense, several studies on phase shifts have been carried out in the yeast glycolytic subsystem by means of a delayed differential equations system [

In

In the multienzymatic instability-generating reactive system, it is shown how the metabolite S (glucose), brought into the system at constant speed, is transformed by the first enzyme E_{1} (hexokinase) into the product P_{1} (glucose-6-phosphate). The enzymes E_{2} (phosphofructokinase) and E_{3} (pyruvatekinase) are allosteric, and transform the substrates P′_{1} (fructose 6-phosphate) and P_{2}′ (phosphoenolpyruvate) in the products P_{2} (fructose 1-6-bisphosphate) and P_{3} (pyruvate), respectively. The step P_{2} → P_{2}′ represents a particular catalytic activity, reflected in the dynamic system by means of a functional variable β′.

It is supposed that a part of P_{1} does not continue in the multienzymatic instability-generating reactive system, q_{1} being the first-order rate constant for the removal of P_{1}; likewise q_{2} is the rate constant for the sink of the product P_{3} (which is related with the activity of pyruvate dehydrogenase complex).

In the determination of the enzymatic kinetics of the enzyme E1 (hexokinase) the generic equation of the reaction speed dependent on Glu and MgATP has been used [_{2} (phosphofructokinase) [_{3}, pyruvatekinase, (dependent on ATP, Pyr-P and Fru 1,6-P2) was also constructed on the allosteric model of concerted transition [

The main instability-generating mechanism in the glycolytic subsystem is based on the self-catalytic regulation of the enzyme E_{2} (phosphofructokinase), specifically, the positive feed-back exerted by the reaction products, the ADP and fructose-1,6-bisphosphate [

The enzyme E_{2} (Pyruvatekinase) is inhibited by the ATP reaction product [_{1} and recycled by E_{3}).

For a spatially homogeneous system, the time-evolution of α, β and γ, which denote the normalized concentrations of P_{1}, P_{2} and P_{3}, respectively, is described by the following three delay differential equations:

where

and

To simplify the model, we did not consider the intermediate part of glycolysis formed by the reversible enzymatic processes. In this way, the functions f and h are supposed to be the identity function:

The initial functions present a simple harmonic oscillation in the following form:

σ_{1}, σ_{2} and σ_{3} correspond to the maximum activity of the enzymes E_{1}, E_{2} and E_{3} divided by the constants K_{m1}, K_{m2} and K_{m3}, respectively, (the Michaelis constants of the enzymes E_{1}, E_{2} and E_{3}); Z_{1} = K_{m1}/K_{m2,} Z_{2} = K_{m2}/K_{m3} and Z_{3} = K_{m3}/K_{D3}; L_{1} and L_{2} are the allosteric constant of E_{2} and E_{3}; d_{1} = K_{m3}/K_{D2}, d_{2} = K_{m3}/K_{D3} and d_{3} = K_{D3}/K_{D4} (K_{D3} and K_{D4} are the dissociation constant of P_{2} by E_{3} and the dissociation constant of MgATP, respectively,); β′ and μ, reflect the normalized concentrations of P_{2}′ (Pyr-P) and ATP, respectively; c is the non-exclusive binding coefficient of the substrate; α, β and γ are normalized dividing them by K_{m2}, K_{m3} and K_{D3}. The values of the different parameters are shown in [

Experimental observations, by monitoring the fluorescence of NADH in glycolyzing baker’s yeast under periodic glucose input flux, have shown that the existence of quasiperiodic time patterns is common at low amplitudes of the input flux and chaos emerges at high amplitudes of the input flux [

In order to simulate these experiments closely, the system can be considered under periodic input flux with a sinusoidal source of substrate S = S′+

Assuming the experimental input substrate value of 6 mM/h [^{−1} is obtained after dividing by K_{m2} = 5 × 10^{−5} M, the Michaelis constant of phosphofructokinase for fructose 6-phosphate [

Once the values for the parameters are specified and the initial functions are given (see [

The numerical results show that in the instability-generating multienzymatic system under a sinusoidal source of substrate quasiperiodic patterns are the most common dynamical behaviors (at low amplitudes of the input flux) and quasiperiodicity routes to chaos can emerge in the biochemical oscillator when the input amplitude is increased. These results are similar to experimental observations [

_{1} = 2, λ_{2} = 0.5, q_{2} = 0.069 and S′ = 0.033. So, for A < 0.021, quasiperiodic behaviors emerge in the phase space and two frequencies are present in the oscillations (in

The quasiperiodic route to chaos under periodic substrate input flux is within the range of experimental values [

Results of the calculations also show a quasiperiodicity route to chaos for a constant input flux (_{1} = 7, and λ_{2} = 130), the biochemical system exhibits a stable steady state when the control parameter is q_{2} = 0.11. For q_{2} = 0.103, a first Hopf bifurcation introduces a fundamental frequency ω_{1} and a limit cycle appears in the phase space (_{2} = 0.099, a second Hopf bifurcation generates a new fundamental frequency w_{2} causing quasiperiodic behavior (_{2} = 0.095, complex substructures appears in the torus (_{2} = 0.093 (

Under constant and periodic input flux conditions time delay acts as a source of instability (next to the feedback loops) leading to complex oscillations and transient dynamics in the biochemical system.

Likewise, the numerical study of the glycolytic model formed by a system of three delay-differential

Stable coexisting states means that under the same parametric conditions the system can exhibit two o more dynamical patterns and any initial metabolite concentrations will eventually lead the system into one of these self-organized behaviors. This dynamical behavior is an important characteristic of the metabolic systems, which has been studied extensively through experiments and numerical simulations [

Biological examples of metabolic systems with stable coexisting states include genetic switch [

All these dynamical processes (chaos, multiplicity of coexisting states, periodic patterns, bursting oscillations, steady state transitions,

A growing number of works on delayed differential equation systems in biochemical processes are being carried out [

Many metabolic subsystems involve small numbers of molecules causing biochemical processes to be accompanied by fluctuations around the dynamic states predicted by the deterministic evolution of the system. These fluctuations reflect intrinsic molecular noise which may play a very important role in the switching of metabolic dynamics.

Recently, a considerable number of studies in different biochemical processes such as: expression of single genes, gene networks and multi-step regulated pathways allow illustrating the stochastic nature of many metabolic self-organized activities [

The importance of molecular noise makes us stress that living cells may be also considered stochastic biochemical reactors.

Let us see an example next on the effect of molecular noise on circadian oscillations.

Circadian rhythms govern a wide variety of metabolic and physiological processes in all kinds of cells from prokaryotes to mammals [

The presence of small amounts of mRNA or proteins in the molecular mechanism of circadian rhythms originates a molecular noise which may become significant and may compromise the emergence of coherent oscillatory patterns [

The first model predicting oscillations due to negative feedback on gene expression was proposed by Goodwin [

Here is shown a molecular model proposed by A. Golbeter and colleagues for circadian rhythms in

The core molecular model is schematized in a general form in _{P}), transport of per mRNA into the cytosol where it is translated into the clock protein (P_{0}) and the mRNA degradation. The synthesis of the (P_{0}) PER protein exhibits a rate proportional to the Per mRNA (M_{P}) level, and the clock protein can be reversibly phosphorylated from the form P_{0} into the forms P_{1} and P_{2}. The phosphorylated form P_{2} can be degraded or transported into the nucleus (P_{N}) where it represses the transcription of the gene exerting a negative feedback of cooperative nature.

In the model, the gene presents a maximum rate of transcription v_{S}, and the mRNA (M_{P}) is degraded by an enzyme with a maximum rate v_{m} and a Michaelis constant K_{m} The kinase and phosphatase involved in the reversible phosphorylation of P_{0} into P_{1}, and P_{1} into P_{2}, have a maximum rate v_{i} and Michaelis constant K_{i} (_{2} form is degraded by an enzyme with a maximum rate v_{d} and Michaelis constant K_{d}, and transported into the nucleus at a rate with an apparent first-order rate constant k_{1}. The nuclear form P_{N} is transported into the cytosol with an apparent first-order rate constant k_{2}. The negative feedback exerted by P_{N} on gene transcription is described by an equation of the Hill type, in which _{I} is the threshold constant for repression.

The time evolution of the concentrations of mRNA (M_{P}) and the various forms of clock protein, cytosolic (P_{0}, P_{1} and P_{2}) or nuclear (P_{N}), is governed by the following system of kinetic differential equations:

In cellular conditions, the small amounts of mRNA and proteins provoke an effect of molecular noise on the dynamic behaviors of the system. To perform stochastic simulations of the circadian clock mechanism due to this intrinsic noise, metabolic processes must be decomposed fully into elementary steps (where enzyme-substrate complexes are considered explicitly) and each step is associated with a transition probability proportional both to the numbers of molecules involved and to the biochemical rate constants (the procedure was introduced by Gillespie [

According to this method, the deterministic model schematized in

The decomposition of the deterministic model into elementary steps, the method of stochastic simulation, and parameter values are listed in

As an example of the decomposition, steps 1–8, which pertain to the formation and dissociation of the various complexes between the gene promoter and nuclear protein (P_{N}), are next shown:

The second column lists the sequence of reaction steps, and the probability of each reaction is given in the third column. G denotes the unliganded promoter of the gene, and GP_{N}, GP_{N2}, GP_{N3} and GP_{N4} are the complexes formed by the gene promoter with 1, 2, 3, or 4 P_{N} molecules, respectively.

The kinetic constants a_{j} and d_{j} = (1,...,4) related to bimolecular reactions are scaled by Ω parameter, which allows modifying the number of molecules present in the system [

For appropriate parameter values (see appendices in [

In _{P}), nuclear (P_{N}) and total (P_{t}) clock protein under conditions of continuous darkness. The circadian oscillations correspond to the evolution toward a limit cycle, which is shown as a projection of the dynamic behaviors onto the (M_{P},P_{N}) phase plane (

Corresponding results from stochastic simulations generated by the model in the presence of noise, for Ω = 500 and

The analysis of the molecular model schematized in

The analysis of the molecular model schematized in

Besides assessing the robustness of circadian oscillations with respect to molecular noise, the analysis of the stochastic model also shows that the persistence of dynamic circadian behaviors is enhanced by the cooperative nature of the gene repression. The role of cooperativity in the circadian metabolic subsystem is supported by the formation of complexes between various clock proteins and this has been observed in several kinds of cells such as

The model represents a prototype for the emergence of self-organized circadian patterns based on negative autoregulatory feedback of gene expression and the numerical results validate the use of deterministic models to study the metabolic mechanism of circadian rhythms and explains why such model provide a reliable picture of the working of circadian clocks in a variety of cells.

Other similar results on circadian clocks with more complex metabolic mechanisms involving a larger number of interacting enzymes can be seen in [

In mathematical studies of metabolic dissipative patterns, self-organization is related to the appearance of attractors in the phase space, which corresponds to ordered motions of the involved biochemical elements.

Phase space is a mathematical object in which all possible states of a system are represented (in form of attractors) and the coordinates correspond to the variables that are required to describe the system.

Attractors in dynamical systems theory provide a way of describing the typical asymptotic orbits. These dynamical trajectories end up and remain in one of the possible attractor states which represent the set of all the possible asymptotic behaviors of the system.

Formally, if for example y(t) is an output activity of a metabolic subsystem, a set A is called an attractor for this subsystem in the following three conditions:

It is impossible to go out; in other words, if y(t_{0}) is in A for some time t_{0}, later y(t) remains in A.

There exists a neighbourhood of itself B (basin of attraction) such that for any initial condition in B, the system approaches A indefinitely.

A is a compact set; this means it is a closed and bounded set.

Consequently, for a metabolic subsystem under fixed determinate conditions, an attractor is a mathematical dynamical structure that represents the set of all possible asymptotic catalytic behaviors.

There is a great variety of qualitatively different attractors in metabolic subsystems showing the richness of self-organized phenomena under dissipative conditions. Many quantitative studies of metabolic processes are characterized by time series (numerical or experimental) and in certain conditions to investigate some dynamic properties of a biochemical system it is necessary to reconstruct the attractor from these time series.

A method to reconstruct attractors is the time-delay embedding [

Given a time series x(t), t = 1,2,...,N, the m-dimensional return map is obtained by plotting the vector X(t) = [x(t),x(t−τ),x(t−2τ),· · ·,x(t− (m−τ))] where τ is an integer delay.

This converts the dimensional vector x(t), into the m-dimensional vector X(t). The dimension m is known as the embedding dimension, and if m is great enough, the trajectory of X(t) converges to an attractor in the m-dimensional Euclidian space, which is, up to a continuous change of variable, the attractor of the subunit dynamical system.

The election of embedding dimension (m) and delay (τ) is mostly a question of trial and error because although there are criteria, they are not clean-cut [

For m election, the method of false nearest neighbors is appropriate. False neighbors are far points in the original phase space with near projections in lower dimensions (see

For example, Samll says in [

What small threshold? Kodba suggests enlarging m until “the fraction of false nearest neighbors convincingly drops to zero [

How small is “convincingly”? For τ, there is an easy test based on the autocorrelation [

Alternatively, we can take τ as the first minimum of the mutual information function [

A third and easier method is the approximate period: A quarter of the length of the pseudo-period [

Time-delay embedding method can be directly applied to a time series by means of software developed with MATLAB.

The concept of Lyapunov exponents has been mainly used as a nonparametric diagnosis for stability analysis and for to determine chaotic behaviors, where at least one Lyapunov exponent is positive [

A positive Lyapunov exponent indicates sensitivity to initial conditions, a hallmark of chaos [

The maximal Lyapunov exponent is very useful in testing the existence of chaos and the Wolf algorithm can be used for it [_{0} lying on it. One should find another point x_{0}′ which is close in space but is distant in time to x_{0} :||x_{0} − x_{0}′|| = ɛ_{0} ≤ ɛ_{min} and |T(x_{0}) − T(x_{0}′)| ≥ T_{min}. Then trace system dynamics using initial points x_{0} and x_{0}′. Then a distance ɛ _{0}′ between two trajectories will exceed some value ɛ_{max}. Stop and fix the time of tracing T_{0} and the ratio ɛ_{0}′/ɛ_{0}. After that one should find another starting point x_{1}″ which is close to x_{1} and shifted in the direction of the vector x_{1}′ – x_{1}. Let ||x_{1} – x_{1}″|| = _{1}. Trace the dynamics of the system using x_{1} and x_{1}″ as initial points. Then a distance _{1}′ between two trajectories will exceed ɛ_{max}. Stop and fix the time of tracing T_{1} and the ratio _{1}′/_{1},

The Maximal Lyapunov exponent is estimated as

where N is the iteration number.

To calculate the maximal Lyapunov exponent, the software developed with MATLAB can be used.

In order to study the presence of long-term correlations in metabolic chaotic data, first it is necessary to determinate whether the series is a fractional Gaussian noise (fGn) or a fractional Brownian motion (fBm).

FGn is a stationary stochastic process with the property that the n-th autocorrelation coefficient is given by

where H is the Hurst coefficient. On the contrary, fBm is a non-stationary, self-similar process, whose first differences form a fGn, that is, taking differences between points sampled at equal intervals a fGn is obtained [

Taking into account these concepts and _{H}(t) with 0 ≤ t ≤ ∞; when the independent variable t is sampled at equally spaced times obtaining a discrete fractional Brownian motion. Therefore, fBm is a generalization of Brownian motion in which the increments are normally distributed but they are no longer independent and consequently the process is correlated in time.

FGn and fBm can be distinguished by calculating the slope of the power spectral density plot. The signal is said to exhibit power law scaling if the relationship between its Fourier spectrum and the frequency is approximated asymptotically by S(f) ≈ S(f_{0})/f^{β} for adequate constants S(f_{0}) and β. If −1 < β < 1, then the signal corresponds to an fGn. If 1 < β < 3, then the signal corresponds to a fBm [

The regression line can be estimated for the pairs (log S(f), log f), where f is the frequency and S(f) the absolute value of the Fourier transform. The β constant is taken to be the opposite of the coefficient of

Most of the physiological time series are fBm, and a number of tools are available for estimating the long-term correlations of an fBm series. The scaled windowed variance analysis is one of the most reliable methods that have been thoroughly tested on fBm signals [

This method generates an estimation of the Hurst exponent (H) for each series. In short, for a random process with independent increments, the expected value of H is 0.5. When H differs from 0.5, it indicates the existence of long-term correlations, that is to say, dependence among the values of the process. If H

According to bdSWV method, if the signal is of the form x_{t}, where t = 1,…,^{k}, where _{2}

Partition of the data points in
_{i}_{(}_{i}_{−1)}_{n}_{+1},..., _{in}

Subtraction of the line between the first and last points for the points in the n-th window.

For each
_{i} of the points in each window, by using the formula:

where _{i}_{i}.

Evaluation of the average

Observation of the range of the window sizes

In this range, the slope of the regression line gives the estimation of the Hurst coefficient

The empirical range of windows corresponding to step 5) which should be in accordance with the guidelines appearing in [

The program bdSWV is available on the web of the Fractal Analysis Programs of the National Simulation [

Long-term correlations have also been observed in different experimental studies, e.g., the physiological time series [^{+} channel activity [

The entropy theory of dynamical systems can be found in many textbooks [

Entropy is also a useful concept in the study of attractors, which may allow estimating the degree of complexity and information contained in them. More concretely, Kolmogorov–Sinai entropy (K–S entropy) provides a measure of the information and the level of predictability in the attractor and the time series [

A practical solution to this problem has been put forward using a developed family of statistics named Approximate Entropy (ApEn) which is a good approximation of the Kolmogorov-Sinai entropy [

Formally, given N data points from a time series x(1), x(2),., x(N), two input parameters m and r must be fixed to compute ApEn, denoted precisely by ApEn(m, r, N).

To estimate ApEn, first we form the m dimensional vector sequences X(1)….X(N − m + 1) such that X(i) = (x(i)…..x(I − m + 1)), which represent m consecutive values. Let us define the distance between X(i) and X(j) (d[X(i),X(j)]) as the maximum absolute difference between their respective scalar components and for each X(i) we count the number of j such that d[X(i),X(j)] < r, denoted as N^{m}(i) and C_{r}^{m}(i) = N^{m}(i)/(N − m + 1), which measure within a tolerance r the frequency of patterns similar to a given one of window length m.

The average value of C_{r}^{m}(i) is ^{m}(r), which portrays the average frequency of the occurrence that all the m-point patterns in the sequence remain close to each other, and finally

The idea is that ApEn measures the logarithmic likelihood that runs of patterns that are close (within r) for m contiguous observations remain close on subsequent incremental comparisons.

Some Approximate Entropy works can be found in [

ApEn can be calculated with software developed with MATLAB.

One of the most important goals of contemporary biology is to understand the elemental principles and quantitative laws governing the functional metabolic architecture of the cell.

In this review, I mainly focused on some functional enzymatic structures which allow the temporal self-organization of their metabolic processes.

My aim was to provide an overview of temporal metabolic behaviors, including new examples, some kinds of quantitative mathematical models and non-linear tools for the analysis of dissipative functional enzymatic associations (metabolic subsystems) where oscillatory behaviors may emerge.

From the first studies in 1957 of oscillatory phenomena in fluorescent studies of yeast [

Oscillatory phenomena, apart from constituting a singular property that manifests itself at all levels of biological organization, present a great functional significance in enzymatic processes. As recalled in the previous sections, biochemical oscillations constitute in themselves a manifestation of the nonlineal characteristics involved in the metabolic regulation activity.

In the light of the research in course, the biological rhythms that emerge in self-organized biomolecular structures constitute one of the most genuine properties of cellular dynamics; and the rigorous knowledge of their nature and significance may be an essential element in the comprehension of the biological fact at its most basic and elementary levels.

The transition from simple periodic behavior to complex oscillatory phenomena including chaos is often observed in metabolic behaviors. In this sense, the relationship between chaotic patterns and long-term correlations (information correlated in time [

As mentioned above, different studies have evidenced that global 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 metabolic subsystems exhibit structural plasticity. This global and stable cellular metabolic structure (in which also emerge chaotic behaviors) appears to be an intrinsic characteristic common to all cellular organisms [

The existence of chaotic patterns and long-term correlation properties in the activity of the metabolic subsystems integrated in a stable global functional structure may constitute a biological advantage.

Chaotic patterns exhibit sensitive dependence on initial conditions. Sensitivity means that a small change in the initial state will lead to large changes in posterior system states and the fluctuations of the chaotic patterns are conditioned by the degree of perturbation of the initial conditions. These changes in the system states present exponential divergence, provoking fast separations in the chaotic orbits.

For “slow dynamical systems” the typical time scale of the chaotic fluctuations is on the order of 1 μs [

Furthermore, different studies have shown that chaos permits fast transmission of information and high efficiency [

The existence of chaos (which exhibits long-term correlations) in some functional structures may constitute a biological advantage by allowing fast and specific responses during the adaptation of the metabolic system to environmental perturbations.

For example, calcium plays an important role in the regulation of cell metabolism, modulating many physiological processes [

In this sense, numerous works have shown chaotic behaviors at cellular conditions e.g., in intracellular free amino acid pools [

Since a notable part of the biological temporary processes seem to be chaotic in cell conditions, it can be important to take into account these persistent phenomena in Systems Biology.

A vast amount of new information on structural enzymatic organization and genome dynamics is currently being accumulated, and a great part of the network molecular interactions are perfectly established. Therefore, the real functional structure of the enzymatic associations and genome regulation is an open question to be elucidated through mathematical modeling and numerical analysis.

Within the new area of Systems Biology, quantitative mathematical models, non-linear tools and computational approaches are particularly valuable for exploring dynamic phenomena associated with dissipative metabolic structures, and due to that, these methods will be crucial in making sense of the functional metabolic architecture of the cell.

The comparison of experimental results with numerical analysis calls for more quantitative data on the self-organization of metabolic processes, and these methods will be able to provide an integrative knowledge of the organization of cooperating enzymes into macromolecular complexes and microcompartments with the emergence of temporal metabolic patterns.

This new field fusions several concept and tools from many areas, including: computational intelligence, dynamical systems theory, stochastic processes, nonlinear dynamics and networks theory, among others.

Day by day, Systems Biology is developed as a new methodology about metabolic dynamic processes, which allows explaining how higher-level properties of complex enzymatic processes arise from the interactions among their elemental molecular parts, forming complex spatial structures where singular temporal reactive behaviors emerge.

System Biology will be crucial to the understanding of the functional architecture of the cell.

^{2+}and cAMP oscillations in pancreatic beta cells: A role for glucose metabolism and GLP-1 receptors?

^{+}channels

^{2+}shuttling between endoplasmic reticulum and mitochondria underlying Ca

^{2+}oscillations

_{2}concentration in tobacco leaves transferred to low CO

_{2}

^{+}-mediated metabolic waves

^{m}, m > 3

^{+}channel activity

^{2+}induced by ADP and ATP in single rat hepatocytes display differential sensitivity to application of phorbol ester

^{2+}induced by ADP and ATP in single rat hepatocytes display differential sensitivity to application of phorbol ester

Diversity dynamic behaviors emerge in the simple dissipative metabolic subsystem. (

Molecular processes for M-phase control in eukaryotic cells. (

Quantitative analysis of the M-phase control system showing spontaneous periodic oscillations in the metabolic intermediaries. The total cyclin concentrations (blue), active form of MPF (red), tyrosine-phosphorylated dimers, YP, (green) and total phosphorylated cdc2 monomers (orange) are represented as a function of the time in minute. The bar graphs indicate the periods during which the active forms exceed 50% of the total amount. Reproduced with permission from the Company of Biologists Ltd. [

A limit cycle attractor governs the cell cycle. The cell cycle is controlled by a dynamical structure called “limit cycle” which is a closed orbit corresponding to the oscillations with a period of 80 minutes. The numbers along the limit cycle represent time in minutes after exit from mitosis. Reproduced with permission from the Company of Biologists Ltd. [

Numerical oscillatory responses of glycolysis under periodic substrate input flux showing a transition sequence to chaos through quasiperiodicity. In the first column are represented the corresponding attractors (projections in two dimensions for the α concentration, x-axis, and the β concentrations, y-axis), power spectra in the second column and Poincaré sections in the α, β plane (third column). Reproduced with permission from Elsevier [

Quasiperiodicity route to chaos under constant substrate input flux. Evolution of (

Molecular model for circadian oscillations during genetic expression based on negative self-regulation of the PER gene by its protein product PER. The model incorporates gene transcription into PER mRNA, transport of PER mRNA (M_{P}) into the cytosol as well as mRNA degradation, synthesis of the PER protein at a rate proportional to the PER mRNA level, reversible phosphorylation and degradation of PER (P_{0}, P_{1} and P_{2}), as well as transport of PER into the nucleus (P_{N}) where it represses the transcription of the PER gene. Reproduced with permission from PNAS [

Effect of molecular noise on circadian oscillations during genetic expression. (_{P}), nuclear protein (P_{N}) and total clock protein (P_{t}). (_{N} – M_{P} phase plane. (_{N} – M_{P}) to the evolution of a noisy limit cycle. Reproduced with permission from PNAS [