Complex systems scaling dynamics

We now turn to the dynamic limit where the rates of association/dissociation of ML are infinitely fast. The complex system will maintain a transport situation governed by the coupled diffusion of M and ML. In the case of excess of ligand conditions, equation (57), the full lability condition implies the maintenance of equilibrium on any relevant spatial scale ... 

Despite its predictive power and successful application on a variety of large-scale metabolic networks, stoichiometric analysis also encompasses a few inadequacies. In particular, stoichiometric analysis largely relies on the steady-state assumption and is not straightforwardly applicable to analyze complex time-dependent dynamics in metabolic systems. Similarly, stoichiometric analysis does not allow us to account for allosteric regulation, considerably delimiting its capabilities to predict dynamic properties. See also Section V.C for a discussion of the limits of stoichiometric analysis. 

The problem of establishing the solution structure of platinum-tin complexes is complicated by the lability of this system. We have found (16, 17) (and will refer to this later) that these complexes are often dynamic on the NMR time scale. Despite these difficulties it is possible to characterize such molecules using NMR methods. 

The circulation of water in the Arctic Basin is a complex system of cycles and currents with different scales. Block HB simulates the dynamics of Arctic Basin water by the system of sub-blocks presented in Figure 6.2. The water dynamics in 2 is presented by flows between compartments Eijk. The directions of water exchanges are represented on every level zk = z0 + (k — 1 )A k according to Aota et al. (1992) in conformity with the current maps assigned as SSMAE input. The external boundary of 2 is determined by the coastline, the sea bottom, the Bering Strait, the southern boundary of the Norwegian Sea, and the water-atmosphere interface. 

The description of small scale turbulent fields in confined spaces by fundamental approaches, based on statistical methods or on the concept of deterministic chaos, is a very promising and interesting research task nevertheless, at the authors knowledge, no fundamental approach is at the moment available for the modeling of large-scale confined systems, so that it is necessary to introduce semi-empirical models to express the tensor of turbulent stresses as a function of measurable quantities, such as geometry and velocity. Therefore, even in this case, a few parameters must be adjusted on the basis of independent measures of the fluid dynamic behavior. In any case, it must be underlined that these models are very complex and, therefore, well suited for simulation of complex systems but neither for identification of chemical parameters nor for online control and diagnosis [5, 6],... 

Molecular dynamics (MD) simulations provide a detailed description of complex systems in a wide range of time and spatial scales.138 Simulations involve a statistical uncertainty component as the result of the finite length of the simulation.139 143 MD methods generate a series of time-correlated points in phase space by propagating a suitable starting set of coordinates and velocities according to Newton s second equation. This kind of computational simulations are useful in studies of time evolution of a variety of systems biological molecules, polymers, or catalytic materials, and in a variety of states crystal, aqueous solutions, or in the gas phase. 

Bai YS, Fayer MD. Time scales and optical dephasing measurements Investigation of dynamics in complex systems. Phys Rev B 1989 39 11066-11084. 

Another indication of the problems associated with modularization of complex systems is the small number of formal mathematical methods that allow one to simplify kinetic models. The existing methods are all based on time-scale separation in the system which allows for the decomposition of the system into a module composed of fast processes and one composed of slow processes. Then the fast processes can be considered in the absence of the slow processes. The slow processes are then considered with the fast processes either in steady state or thermodynamic equilibrium (Klonowski 1983 Segel and Slemrod 1989 Schuster and Schuster 1991 Kholodenko etal. 1998 Stiefenhofer 1998 Schneider and Wilhelm 2000). Two successful approaches to modularization of complex networks do not consider dynamics. One is purely stmctural while the other is applicable only to systems in steady state and concerns the analysis of control. 

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