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A metabolic cycle can be viewed as a central core and its branches. The central core is here firstly considered as a pre-closed metabolic cycle (CMC), with a unique first substrate, but with no input or output of other components. By contrast, the metabolic cycles in nature are open metabolic cycles (OMC) with output and input of external substrates (through “metabolic branches”), modulating continuously the enzyme activities and the total concentration of their substrates thorough complex regulatory phenomena. In this work, the transition from a Closed to an Open metabolic cycle has been simulated by a consecutive entry and exit of two components through the catalytic action of two enzymes. It is known that after any alteration of the initial conditions, the cycles need a time to reach new equilibrium. We have measured the changes of transition time (T.T.) values in 81 models of CMC differing in Km or Vmax values. In general, the T.T. tends to be shorter in cycles with preponderant lower Km and higher Vmax values. Further, Mathematica refinement for the estimation of transition time from the data previously calculated can be obtained with the use of the command Interpolating Function.

The open metabolic cycles (OMC) can be considered as systems, with a permanent entry and exit of substrates (metabolites); in spite of this dynamic state they tend to maintain, between physiological ranges, the concentration of their components. The OMC cycles can be studied with different and complementary approaches, among others 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 [

Part of the experimental work from our laboratory had been lately centered on the mechanism of action of enzymes ligases [

These previous studies directed us to apply the Mathematica Program to the analysis of several biochemical processes, such as:

a) Simulation of linear pathways exploring the effect of changing Vmax and/or Km values of one or more enzymes of the pathway; the reservoir model for enzyme kinetics previously developed in our laboratory was adapted to visualize the effect of the retro-inhibition of the first enzyme of the pathway by the final product; the time needed to achieve half of the total synthesis of the final product was also addressed [

b) Simulation of metabolic cycles, usually composed of a number of interconnected substrates and the same number of enzymes. Two different types of cycles can be theoretically considered, depending on whether i) substrates and enzymes form a separate entity in itself, with no entry or exit of material (a closed metabolic cycle, or CMC) or ii) a continuous interchange of material (input and/or output) between the substrates of the inner core cycle and other related metabolites takes place (open metabolic cycles, or OMC). For reasons of simplicity we have firstly approached a peculiar type of closed metabolic cycle (or pre-CMC) which reaches equilibrium, starting from a unique initial substrate [

In this work, the changes in the substrate profiles of a closed metabolic cycle (CMC) promoted by the input and/or output of material are shown, and a new correlation between CMC and OMC is obtained. Although by definition the metabolic cycles tend permanently to equilibrium, the time needed to get that situation has been mathematically explored in theoretical situations in which the time needed to get the equilibrium was measured in cycles starting with only one of its substrates(a) at a fixed concentration (of 12 mM or 1 mM).

The close metabolic cycle (CMC) here considered contains 6 substrates and 6 enzymes, with Michaelis-Menten kinetics (

The closed metabolic cycle (CMC) is transformed into an open metabolic cycle (OMC) (

Part of this treatment is similar to that previously followed in other works from this laboratory [

The general procedure is outlined in

Part A of

Part B of