Acyclic auxiliary dynamical system

Acyclic auxiliary dynamical system with one attractor have a characteristic property among all auxiliary dynamical systems the stoichiometric vectors of reactions Aj A q form a basis in the subspace of concentration space with c, = 0. Indeed, for such a system there exist n—1 reactions, and their... 

In this case, the finite acyclic auxiliary dynamical system, O = O, is A A with reaction rate constant = 41 32/ 21, and is A We... 

For kinetic systems with well-separated constants the left and right eigenvectors can be explicitly estimated. Their coordinates are close to +1 or 0. We analyzed these estimates first for linear chains and cycles (5) and then for general acyclic auxiliary dynamical systems (34), (36) (35), (37). The distribution of zeros and +1 in the eigenvectors components depends on the rate constant ordering and may be rather surprising. Perhaps, the simplest example gives the asymptotic equivalence (for of the reaction network A,+2 with... 

The first case auxiliary dynamical system is acyclic and has one attractor... 

Let us assume that the auxiliary dynamical system is acyclic and has only one attractor, a fixed point. This means that stoichiometric vectors form a basis in a subspace of concentration space with — 0. For every reaction A,- A the following linear operators Qu can be defined ... 

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-... 

In the simplest case, the auxiliary discrete dynamical system for the reaction network W is acyclic and has only one attractor, a fixed point. Let this point be A (n is the number of vertices). The correspondent eigenvectors for zero eigenvalue are r = S j and Z = 1. For such a system, it is easy to find explicit analytic solution of kinetic equation (32). 

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