Several aspects of metabolic pathways (linear or cyclic) have been lately approached in our laboratory [1] . The theoretical treatment of a linear pathway was simulated assuming that an initial substrate was transformed into a final product through the successive appearance and disappearance of metabolic intermediates of the route [1] ; the addition of a new enzyme, catalyzing the transformation of the first into the last substrate converts a linear cycle into a cyclic pathway [2] . The occurrence, properties and theoretical usefulness of closed metabolic cycles (CMC) have been considered in previous publications from this laboratory [2] [3] [4] .

The vast majority of metabolic cycles in nature are open metabolic cycles (OMC), i.e. cycles with entrance and exit of metabolites at different substrate levels. The OMC cycles can be studied with different and complementary approaches, among them, by measuring the level of their components and analyzing potential changes in their concentration in different metabolic or nutritional conditions. However, these are cumbersome procedures and sometimes difficult to be implemented [5] [6] [7] . The advances in computational techniques have allowed a more comprehensive understanding of open metabolic cycles [5] [8] [9] [10] [11] [12] .

In the course of our work [2] [3] [4] [13] , we have observed a scarcity of studies on the physiological consequences of the potential transformation between linear and cyclic pathways.

Two types of complementary studies on metabolic cycles are presented in this work: a) analysis of how changes in the kinetic constants (Vmax and/or Km values) of only one of the enzymes influence the steady concentration of all substrates of the cycle; b) consequences of the transformation of a closed metabolic cycle into an open metabolic cycle caused by the simultaneous entry and exit of a metabolite at a different level; as shown below many of these changes are not metabolically feasible.

Although the practical applications of this work to specific metabolic cycles are countless, they have not been approached here; however the core of this manuscript is to facilitate its potential application to any metabolic cycle regardless the number of input and output enzymes affecting the cycle.

A joint representation of a closed and an open metabolic cycle (CMC and OMC, respectively) is depicted in Figure 1. The CMC here considered is composed of 6 substrates, (a) to (f). The enzymes involved in the transformation of consecutive substrates are, E1a, E2b, E3c, E4d, E5e and E6f. These enzymes will be named also sub indexed with the corresponding substrate only (Ea to Ef).

In order to simplify both the presentation of the results and the calculation by Mathematica, the Km and Vmax constant values of these enzymes are simply named as, K and V, sub indexed with the corresponding substrate, , , , , etc. (Table 1).

The Mathematica program 9 [14] was used to solve the differential equations describing the pathways. The protocol, similar to that previously described [13] is summarized in Table 1:

Part (a) contains the actual equation velocity of the 6 enzymes involved in the Closed Metabolic Cycle (v1 to v6) (Figure 1).

Part (b) contains the Vmax and Km values of the enzymes of the ensemble.

Part (c) contains the instructions to calculate the substrate value concentrations (profiles) along the reaction time.

Part (d) contains the instructions to Plot the substrate profiles resulting in each case from the value assigned to the respective substrates.