Gluing cycles auxiliary system

Again we should analyze, whether this new cycle is a sink in the new reaction network, etc. Finally, after a chain of transformations, we should come to an auxiliary discrete dynamical system with one attractor, a cycle, that is the sink of the transformed whole reaction network. After that, we can find stationary distribution by restoring of glued cycles in auxiliary kinetic system and applying formulas (11)-(13) and (15) from Section 2. First, we find the stationary state of the cycle constructed on the last iteration, after that for each vertex Ay that is a glued cycle we know its concentration (the sum of all concentrations) and can find the stationary distribution, then if there remain some vertices that are glued cycles we find distribution of concentrations in these cycles, etc. At the end of this process we find all stationary concentrations with high accuracy, with probability close to one. 

Let us construct an acyclic system that approximates relaxation of Equation (50) under the same assumption (1) 32 >43- The final auxiliary system after gluing cycles is Ai — A2 — A3 — A2. Let us delete the limiting reaction A3 —> A2 from the cycle. We get an acyclic system A] — A2 — A3. The component A3 is the glued cycle A3— A4— As—> A3. Let us restore this... 

Let us take a weakly ergodic network iV and apply the algorithms of auxiliary systems construction and cycles gluing. As a result we obtain an auxiliary dynamic system with one fixed point (there may be only one minimal sink). In the algorithm of steady-state reconstruction (Section 4.3) we always operate with one cycle (and with small auxiliary cycles inside that one, as in a simple example in Section 2.9). In a cycle with limitation almost all concentration is accumulated at the start of the limiting step (13), (14). Hence, in the whole network almost all concentration will be accumulated in one component. The dominant system for a weekly ergodic network is an acyclic network with minimal element. The minimal element is such a component Amin that there exists an oriented path in the dominant system from any element to Amin- Almost all concentration in the steady state of the network iV will be concentrated in the component Amin-... 

After that, we decompose the new auxiliary dynamical system, find cycles and repeat gluing. Terminate when all attractors of the auxiliary dynamical system O become fixed points. 

Figure 2 Gluing of cycles for the prism of reactions with a given ordering of rate constants in the case of two attractors in the auxiliary dynamical system (a) initial reaction network,...

Let us take a multiscale network and perform the iterative process of auxiliary dynamic systems construction and cycle gluing, as it is prescribed in Section 4.3. After the final step the algorithm gives the discrete dynamical system O " with fixed points A . 

In the simplest case, the dominant system is determined by the ordering of constants. But for sufficiently complex systems we need to introduce auxiliary elementary reactions. They appear after cycle gluing and have monomial rate constants of the form kg — Ylikf. The dominant system depends on the place of these monomial values among the ordered constants. 

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