In this paper we have quantified essential aspects of the effective functional connectivity among the main glycolytic enzymes in dissipative conditions.
First, we have computed under different source of glucose the causality flows in the metabolic system. This level of the functional influence accounts for the contribution of each enzyme to the generation of the different catalytic behavior and adds a directionality in the influence interactions between enzymes.
The results show that the flows of functional connectivity can change significantly during the different metabolic transitions analyzed, exhibiting high values of transfer entropy, and in all considered cases, the enzyme phosphofructokinase (E
) is the main source of effective causality flow; the pyruvatekinase (E
) is the main sink of information flow; the hexokinase (E
) has a quasi-zero flow, meaning that, the total information arriving to E
goes out to either E
The maximum source of total transfer information (0.41) corresponds to the E
enzyme (phosphofructokinase) at the edge of chaos, when complex quasi-periodic oscillations emerge (cf. ). This finding seems to be consistent with other studies which show that when a dynamical system operates in the frontier between order (periodic behavior) and chaos its complexity is maximal 
The level of influence in terms of causal interactions between the enzymes is not always the same but varies depending on the substrate fluxes and the dynamic characteristics emerging in the system. In addition to the glycolytic topological structure characterized by the specific location of enzymes, substrates, products and regulatory metabolites there is an functional structure of information flows which is dynamic and exhibit notable variations of the causal interactions.
Another aspect of the glycolitic functionality was observed during the quantification of the Mutual Information, which measures how much the uncertainty about the one enzyme is reduced by knowing the other; we found that the uncertainty for E
monotonously decreased for all the values of the periodic glucose input-flux.
Second, the numerical results show that for all analyzed cases the maximum effective connectivity corresponds to the Transfer Entropy from E
, indicating the biggest information flow in the multi-enzyme instability-generating system. This is also corroborated by the measure of correlation between the different pairs of series which shows that E
are highly correlated, or synchronized (correlation
0) and E
is anti-correlated with both E
0 and correlation
0). The values of time correlations establish that the activities of E
are grouped to the same function, being activated at similar time and oppositely to E
Third, our analysis allows for a hierarchical classification in terms of what glycolytic enzyme is improving the future prediction of what others, and the results reveals in a quantitative manner that the enzyme E
(phosphofructokinase) is the major source of causal information and represents the key-core of glycolysis. The second in importance is the E
From the biochemical point of view the E
(phosphofructokinase) has been commonly considered as a major checkpoint in the control of glycolysis 
. The main reason for this generalized belief is that this enzyme exhibits a complex regulatory behavior that reflects its capacity to integrate many different signals 
; from a dissipative point of view, this enzyme catalyzes a reaction very far from equilibrium and its self-catalytic regulation it has been considered the main instability-generating mechanism for the emergence of oscillatory patters in glycolysis 
. The functional studies presented here give a quantification of the effective connectivity, and they confirm that the E
(phosphofructokinase) is the key-core of the pathway, and our results make stronger and expand the classical biochemical studies of glycolysis.
Forth, the dynamics of the glycolytic system changes substantially through the quasi-periodic route to chaos when the amplitude of the input-flux varies. However, the hierarchy obtained by transfer entropy, E
the flow, E
the sink and E
a quasi-zero flow, is preserved during this route and seems to be an invariant. This functional invariant of a metabolic process may be important for the understanding of functional enzymatic constraints in cellular conditions; but this issue requires other additional studies.
We want to emphasize that Transfer Entropy as a quantitative measure of effective causal connectivity can be a very useful tool in studies of enzymatic processes that operate far from equilibrium conditions. Moreover, many experimental observations have shown that the oscillations in the enzymatic activity seem to represent one of the most striking manifestations of the metabolic dynamic behaviors, of not only qualitative but also quantitative importance in cells (further details in Appendix S2
Another interesting question is how our analysis can be scaled up from a single pathway to ensembles of pathways in which functional metabolic interactions between the catalytic sets can be derived. In concordance with this, we have constructed dissipative metabolic networks. Essentially, a Dissipative Metabolic Network (DMN) is an open system formed by a given set of dissipative enzymatic sets interconnected by substrate fluxes and three classes of regulatory signals: activatory (positive allosteric modulation), inhibitory (negative allosteric modulation) and all-or-nothing type (which correspond with the regulatory enzymes of covalent modulation). Certain enzymatic sets may receive an external substrate flux. In the DMN, the emergent output activity for each dissipative enzymatic set can be either oscillatory or steady state with an infinite number of distinct activity regimes. The first model of a Dissipative Metabolic Network was developed in 1999 
which allowed to observe a singular spontaneously self-organized Systemic Metabolic Structure, characterized by a set of different enzymatic sets always locked into active states (metabolic core) while the rest of catalytic subsystems presented on-off dynamics. When an enzymatic subsystem is in an on-off state for a long time it can be turned on under specific metabolic conditions. In this first numerical work it was also suggested that the Systemic Metabolic Structure could be an intrinsic characteristic of metabolism, common to all living cellular organisms. Afterward, 2004 and 2005, several studies implementing flux balance analysis in experimental data produced new evidences of this Systemic Functional Structure 
. Specifically, it was observed a set of metabolic reactions belonging to different anabolic pathways which remain active under all investigated growth conditions. The rest of the reactions belonging to different pathways remain only intermittently active. These global catalytic processes were verified for Escherichia coli, Helicobacter pylori, and Saccharomyces cerevisiae 
. More recently, extensive analyses with different dissipative metabolic networks have shown that the fundamental factor for the spontaneous emergence of this systemic self-organized enzymatic structure is the number of enzymatic dissipative sets 
. Moreover, it has been observed that the Systemic Metabolic Structure forms a unique dynamical system, in which self-organization, self-regulation and persistent properties may emerge 
. In continuation with the glycolytic results presented here, we recently also applied TE analysis to these DMNs. We investigated the functional importance of the metabolic core belonging to a network of dissipative pathways and we found that the organization of the effective biomolecular information flows between the enzymatic sets is modular and the dynamical changes between the catalytic modules correspond to metabolic switches which allow critical transitions in the systemic enzymatic activity 
Transfer Entropy is able to detect the directed exchange of causality flows among the irreversible enzymes which might allow for a rigorous quantification of the effective functional connectivity of many dissipative metabolic processes in both normal and pathological cellular conditions.
The TE method applied to our numerical studies of yeast glycolisis shows the emergence of a new kind of dynamical functional structure which is characterized by changing connectivity flows and a metabolic invariant that constrains the activity of the irreversible enzymes.