Biochemical network dynamics

Reactive MPC dynamics should prove most useful when fluctuations in spatially distributed reactive systems are important, as in biochemical networks in the cell, or in situations where fluctuating reactions are coupled to fluid flow. 

J. S. van Zon and P. R. ten Wolde, Simulating biochemical networks at the particle level and in time and space Green s function reaction dynamics, Phys. Rev. Lett. 94, 128103 (2005). 

M. D. Haunschild, B. Freisleben, R. Takors, and W. Wiechert, Investigating the dynamic behavior of biochemical networks using model families. Bioinformatics 21(8), 1617 1625 (2005). 

The study of chemical reactions covers a variety of phenomena, ranging from the microscopic mechanisms of reaction processes through structural changes involving macromolecules such as proteins, to biochemical networks within cells. One common question concerning these seemingly diverse phenomena is how we can understand the temporal development of the system based on its dynamics. 

B. Symbolic Dynamics of Bifurcations and Chaotic Dynamics The Analysis of Real Biochemical Networks... 

In the above, we have given a computational scheme that allows us to define the connections and interactions between components in biochemical networks and to determine the dynamics in the resulting networks. For an arbitrary network, it is not possible to give a precise description of the dynamics without carrying out numerical simulations. However, all the networks obey certain dynamic rules that are set by the stmcture as embodied in the directed A-dimensional hypercube. Moreover, for networks that show certain structural feamres, such as cyclic attractors, it is possible to make precise statements about the dynamics even without further mathematical analysis or simulation. In other cases, analytical techniques are available to give insight into the dynamics observed—for example, in the cases in which it is possible to prove limit cycles... 

Our initial studies of dynamics in biochemical networks included spatially localized components [32]. As a consequence, there will be delays involved in the transport between the nuclear and cytoplasmic compartments. Depending on the spatial structure, different dynamical behaviors could be faciliated, but the theoretical methods are useful to help understand the qualitative features. In other (unpublished) work, computations were carried out in feedback loops with cyclic attractors in which a delay was introduced in one of the interactions. Although the delay led to an increase of the period, the patterns of oscillation remained the same. However, delays in differential equations that model neural networks and biological control systems can introduce novel dynamics that are not present without a delay (for example, see Refs. 57 and 58). 

In conclusion, the current formalism captures extremely rich dynamics in a vast class of differential equations modeling biochemical systems and relates these dynamics to the underlying structure of the biochemical networks. The methods have the potential of lending transparency to the functioning of what now appear to be arbitrarily complex networks. 

Traditionally, dynamical systems are modeled by differential equations. In the case of biochemical networks, Ordinary Differential Equations (ODEs) model the concentration (or activity) of proteins, RNA species, or metabolites by time-dependent... 

When developing a model describing the temporal dynamics of a biochemical network we use ordinary differential equations (ODEs). For a network with n interacting species, labeled C, ..., C, we describe temporal changes in the concentration of species f, denoted by through the differential equation ... 

In complex biochemical network more complicated noises, such as non-Gaussian and colored, can exist. If an average over the stochastic force C is performed, Eqn (15.101) is reduced to the deterministic equation in dynamical systems. Equation (15.101) does not address how the stochastic force C can be related to the deterministic force /. To do that it can be transformed into the following form ... 

Translating a known metabolic network into a dynamic model requires rate laws for all chemical reactions. The mathematical expressions depend on the underlying enzymatic mechanism they can become quite involved and may contain a large number of p>arameters. Rate laws and enzyme parameters are still unknown for most enzymes. Convenience kinetics is used to translate a biochemical network - manually into a dynamical model with plausible biological properties. It implements enzyme saturation and regulation by activators and inhibitors, covers all possible reaction stoichiometries, and can be specified by a small number of parameters. Its mathematical form makes it especially suitable for parameter estimation and optimization. In general, the convenience kinetics applies to arbitrary reaction stoichiometries and captures biologically relevant behavior such as saturation, activation, inhibition with a small number of free parameters. It represents a simple molecular reaction mechanism in which substrates bind rapadly and in random order to the enzyme. 

Apri, M., de Gee, M., Molenaar, J. Complexity reduction preserving dynamical behavior of biochemical networks. J. Theor. Biol. 304, 16—26 (2012)... 

The next step in formulating a kinetic model is to express the stoichiometric and regulatory interactions in quantitative terms. The dynamics of metabolic networks are predominated by the activity of enzymes proteins that have evolved to catalyze specific biochemical transformations. The activity and specificity of all enzymes determine the specific paths in which metabolites are broken down and utilized within a cell or compartment. Note that enzymes do not affect the position of equilibrium between substrates and products, rather they operate by lowering the activation energy that would otherwise prevent the reaction to proceed at a reasonable rate. 

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. 

Most problems associated with approximate kinetics are avoided when Michaelis Menten-type rate equations are utilized. Though this choice sacrifices the possibility of analytical treatment, reversible Michaelis Menten-type equations are straightforwardly consistent with fundamental thermodynamic constraints, have intuitively interpretable parameters, are computationally no more demanding than logarithmic functions, and are well known to give an excellent account of biochemical kinetics. Consequently, Michaelis Menten-type kinetics are an obvious choice to translate large-scale metabolic networks into (approximate) dynamic models. It should also be emphasized that simplified Michaelis Menten kinetics are common in biochemical practice almost all rate equations discussed in Section III.C are simplified instances of more complicated rate functions. 

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). 

Many methods have been developed for model analysis for instance, bifurcation and stability analysis [88, 89], parameter sensitivity analysis [90], metabolic control analysis [16, 17, 91] and biochemical systems analysis [18]. One highly important method for model analysis and especially for large models, such as many silicon cell models, is model reduction. Model reduction has a long history in the analysis of biochemical reaction networks and in the analysis of nonlinear dynamics (slow and fast manifolds) [92-104]. In all cases, the aim of model reduction is to derive a simplified model from a larger ancestral model that satisfies a number of criteria. In the following sections we describe a relatively new form of model reduction for biochemical reaction networks, such as metabolic, signaling, or genetic networks. 

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