Reaction network, dynamics

So far, the discussion is limited only to topological structure of the network. In the reaction network dynamics, the numbers of molecules are distributed. On each node of the network, the abundance of the corresponding molecule species is assigned. Accordingly, some path is thick where such reactions occur frequently. Such abundance, as well as their fluctuations and dynamics, has to be investigated. 

Cell system Replicating liposome with internal reaction network Dynamic bottleneck in autocatalytic reaction system Evolvability and recursiveness for growth... 

Assuming that some reaction processes are fast, they can be adiabatically eliminated. Also, most fast reversible reactions can be eliminated by assuming that they are already balanced. Then we need to discuss only the concentration (number) of molecular species, which changes relatively slowly. For example, by assuming that an enzyme is synthesized and decomposed fast, the concentrations can be eliminated to give catalytic reaction network dynamics consisting of the reactions with... 

Recently there has been an increasing interest in self-oscillatory phenomena and also in formation of spatio-temporal structure, accompanied by the rapid development of theory concerning dynamics of such systems under nonlinear, nonequilibrium conditions. The discovery of model chemical reactions to produce self-oscillations and spatio-temporal structures has accelerated the studies on nonlinear dynamics in chemistry. The Belousov-Zhabotinskii(B-Z) reaction is the most famous among such types of oscillatory chemical reactions, and has been studied most frequently during the past couple of decades [1,2]. The B-Z reaction has attracted much interest from scientists with various discipline, because in this reaction, the rhythmic change between oxidation and reduction states can be easily observed in a test tube. As the reproducibility of the amplitude, period and some other experimental measures is rather high under a found condition, the mechanism of the B-Z reaction has been almost fully understood until now. The most important step in the induction of oscillations is the existence of auto-catalytic process in the reaction network. 

Besides the two most well-known cases, the local bifurcations of the saddle-node and Hopf type, biochemical systems may show a variety of transitions between qualitatively different dynamic behavior [13, 17, 293, 294, 297 301]. Transitions between different regimes, induced by variation of kinetic parameters, are usually depicted in a bifurcation diagram. Within the chemical literature, a substantial number of articles seek to identify the possible bifurcation of a chemical system. Two prominent frameworks are Chemical Reaction Network Theory (CRNT), developed mainly by M. Feinberg [79, 80], and Stoichiometric Network Analysis (SNA), developed by B. L. Clarke [81 83]. An analysis of the (local) bifurcations of metabolic networks, as determinants of the dynamic behavior of metabolic states, constitutes the main topic of Section VIII. In addition to the scenarios discussed above, more complicated quasiperiodic or chaotic dynamics is sometimes reported for models of metabolic pathways [302 304]. However, apart from few special cases, the possible relevance of such complicated dynamics is, at best, unclear. Quite on the contrary, at least for central metabolism, we observe a striking absence of complicated dynamic phenomena. To what extent this might be an inherent feature of (bio)chemical systems, or brought about by evolutionary adaption, will be briefly discussed in Section IX. 

E. Klipp, W. Liebermeister, and C. Wierling, Inferring dynamic properties of biochemical reaction networks from structural knowledge. Gen. Inform. Ser. 15(1), 125 137 (2004). 

Dynamic and Static Limitation in Multiscale Reaction Networks, Revisited... 

If the reader can use these properties (when it is necessary) without additional clarification, it is possible to skip reading Section 3 and go directly to more applied sections. In Section 4 we study static and dynamic properties of linear multiscale reaction networks. An important instrument for that study is a hierarchy of auxiliary discrete dynamical system. Let A, be nodes of the network ("components"), Ai Aj be edges (reactions), and fcy,- be the constants of these reactions (please pay attention to the inverse order of subscripts). A discrete dynamical system

dynamical system for a given network we find for each A,- the maximal constant of reactions Ai Af k ( i)i>kji for all j, and — i if there are no reactions Ai Aj. Attractors in this discrete dynamical system are cycles and fixed points. 

The fast stage of relaxation of a complex reaction network could be described as mass transfer from nodes to correspondent attractors of auxiliary dynamical system and mass distribution in the attractors. After that, a slower process of mass redistribution between attractors should play a more important role. To study the next stage of relaxation, we should glue cycles of the first auxiliary system (each cycle transforms into a point), define constants of the first derivative network on this new set of nodes, construct for this new network an (first) auxiliary discrete dynamical system, etc. The process terminates when we get a discrete dynamical system with one attractor. Then the inverse process of cycle restoration and cutting starts. As a result, we create an explicit description of the relaxation process in the reaction network, find estimates of eigenvalues and eigenvectors for the kinetic equation, and provide full analysis of steady states for systems with well-separated constants. 

In two-parametric families three constants can meet. If three smallest constants kj,ki and L have comparable values and are much smaller than others, then static and dynamic properties would be determined by these three constants. Stationary rate w and dynamic of relaxation for the whole cycle would be the same as for 3-reaction cycle A B C A with constants kj,ki and km- The damped oscillation here are possible, for example, if kj — ki—km—k, then there are complex eigenvalues X— k(—(3/1) 1( /3/ )). Therefore, if a cycle manifests damped oscillation, then at least three slowest constants are of the same order. The same is true, of course, for more general reaction networks. 

An auxiliary reaction network is associated with the auxiliary discrete dynamical system. This is the set of reactions A, A q with kinetic constants k,. The correspondent kinetic equation is... 

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). 

The second simple particular case on the way to general case is a reaction network with components A, ..., A whose auxiliary discrete dynamical system has one attractor, a cycle with period t > 1 A +i A - +x. ., A ... 

After that, we create a new auxiliary discrete dynamical system for the new reaction network on the set A],... We can describe this new... 

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. 

For one catalytic cycle, relaxation in the subspace = 0 is approximated by relaxation of a chain that is produced from the cycle by cutting the limiting step (Section 2). For reaction networks under consideration (with one cyclic attractor in auxiliary discrete dynamical system) the direct generalization works for approximation of relaxation in the subspace = 0 it is sufficient to perform the following procedures ... 

The auxiliary discrete dynamical system for reaction network (50) is... 

In general case, let the system have several attractors that are not fixed points, but cycles Ci, C2,... with periods ti, T2,... >1. By gluing these cycles in points, we transform the reaction network if into if. The dynamical system is transformed into Eor these new system and network, the connection... 

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