Nonlinear biochemical system

Here the procedures of linearization, balancing, and truncation are described such that the reader is enabled to use it. To start with, one has a nonlinear biochemical system, for example a silicon cell model, in the form of differential equations. 

Although the importance of a systemic perspective on metabolism has only recently attained widespread attention, a formal frameworks for systemic analysis has already been developed since the late 1960s. Biochemical Systems Theory (BST), put forward by Savageau and others [142, 144 147], seeks to provide a unified framework for the analysis of cellular reaction networks. Predating Metabolic Control Analysis, BST emphasizes three main aspects in the analysis of metabolism [319] (i) the importance of the interconnections, rather than the components, for cellular function (ii) the nonlinearity of biochemical rate equations (iii) the need for a unified mathematical treatment. Similar to MCA, the achievements associated with BST would warrant a more elaborate treatment, here we will focus on BST solely as a tool for the approximation and numerical simulation of complex biochemical reaction networks. 

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. 

This work is a first approach to system reduction for biochemical systems. The research perspective is to apply system reduction to a large biochemical system, possibly using decomposition of the system into small subsystems or modules. Moreover, system reduction methods have to be developed which apply to nonlinear systems with state variables that remain positive when starting with positive initial concentrations, and produce systems with the same properties, which admit a biological interpretation. 

There are two kinds of properties that characterize the parts (i) intrinsic properties, which are determined by the part itself, such as mass, or the amino acid sequence of a protein, and (ii) relational properties, which are determined not only by individual parts but also by one or more other parts, such as dissociation constants. In complex biochemical systems, aggregative system properties, such as the mass of a bacterium, are a function of only the intrinsic properties of the parts. However, the flow through a biochemical pathway is a nonlinear function of the concentrations of its constituent enzymes. Therefore, the flow is not an aggregative property. 

For each set of constant input and output concentration constraints a system of linear chemical reactions has a unique steady state. For a network of nonlinear biochemical reactions, however, there could be several steady states compatible with a given set of constraints. The number and character of these steady states are determined by the structure of the network including the extent of nonlinearity, the number and connectivity of the individual chemical reactions and the values of the reaction rate constants and the concentrations of the reactants. The higher the order of a chemical reaction, the more steady states may be compatible with a given set of chemical constraints. The simple trimolecular reaction system of Schlogl [13] illustrates how a third-order chemical reaction can have two stable steady states compatible with a single set of chemical constraints ... 

IIIF) 1972 Sel kov, E. E. Nonlinearity of Multienzyme Systems In Analysis and Simulation of Biochemical Systems, (Hemker, H. C., Hess, B., eds.) FEBS 25, North-Holland, Amsterdam, 145-161... 

The above results show that cooperativity is not required absolutely for oscillations, and that the strong nonlinearity of the regulated step can be replaced by diffuse nonlinearities distributed over several steps of the system. These milder nonlinearities, of the Michaelian type, combine to raise the overall degree of nonlinearity up to the level required for sustained oscillatory behaviour. In biochemical systems controlled... 

We may conclude that many important biological rhythms originate from positive feedback mechanisms whose nonlinearity is further strengthened by the cooperative nature of the regulatory process. Although the detailed molecular implementation of the feedback process differs in each case, it is the self-amplification with which it is associated that gives rise to instabilities followed by sustained oscillations in biochemical systems as well as in cardiac or neural cells (Goldbeter, 1992). 

Sminaker, M., Cedeisund, G., Jirstiand, M. A method for zooming of nonlinear models of biochemical systems. BMC Syst. Biol. 5, 140 (2011)... 

How relevant are these phenomena First, many oscillating reactions exist and play an important role in living matter. Biochemical oscillations and also the inorganic oscillatory Belousov-Zhabotinsky system are very complex reaction networks. Oscillating surface reactions though are much simpler and so offer convenient model systems to investigate the realm of non-equilibrium reactions on a fundamental level. Secondly, as mentioned above, the conditions under which nonlinear effects such as those caused by autocatalytic steps lead to uncontrollable situations, which should be avoided in practice. Hence, some knowledge about the subject is desired. Finally, the application of forced oscillations in some reactions may lead to better performance in favorable situations for example, when a catalytic system alternates between conditions where the catalyst deactivates due to carbon deposition and conditions where this deposit is reacted away. 

Similar approaches were adopted by Ganikhanov (Chapter 5), who developed a state-of-the-art laser system, benefiting simultaneous third-harmonic and nonlinear Raman microscopy, and Yakovlev et al. (Chapter 6), who applied third-harmonic generation microscopy and nonlinear Raman microspectroscopy for biochemical analysis in microfluidic devices. 

Nonlinearity with saturation. This type of nonlinearity is quite common in biochemical engineering and waste water treatment systems. 

Although these arguments have been presented for reaction systems whose rates are forced by an external oscillator, they remain true for autonomous biochemical oscillations where ot and are nonlinear functions of metabolite concentrations. That is, the rate of removal of a labeled compound through a reaction step whose rate is oscillating due to nonlinear kinetics will be enhanced over an equivalent system that maintains the same mean chemical flux and mean concentrations of metabolites but does not oscillate. This has been demonstrated numerically ( 6) on the reaction system (1) from the previous section using the full kinetic equations... 

In a linear chemical reaction system, there is a unique steady state determined by the chemical constraints that establish the NESS. For nonlinear reactions, however, there can be multiple steady states [6]. A network comprised of many nonlinear reactions can have many steady states consistent with a given set of chemical constraints. This fact leads to the suggestion that a specific stable cellular phenotypic state can result from a specific NESS in which the steady operation of metabolic reactions maintains a balance of cellular components and products with the expenditure of biochemical energy [4]. Similarly, the network of chemical and mechanical signals that regulate the metabolic network must also be in a steady state. Important problems, then, are to determine the variety of steady states available to a system under a given set of chemical constraints and the mechanisms by which cells undergo... 

The yeast cell cycle has also been analyzed at this high level of chemical detail [17]. The molecular mechanism of the cycle in the form of a series of chemical equations was described by a set of ten nonlinear ordinary differential kinetic rate equations for the concentrations of the cyclins and associated proteins and the cell mass, derived using the standard principles of biochemical kinetics. Numerical solution of these equations 3uelded the concentrations of molecules such as the cyclin, Cln2, which is required to activate the cell cycle, or the inhibitor, Sid, which helps to retain the cell in the resting Gi phase. The rate constants and concentrations ( 50 parameters) were estimated from published measurements and adjusted so that the solutions of the equations yielded appropriate variations, i.e., similar to those experimentally measured, of the concentrations of the constituents of the system and the cell mass. The model also provides a rationalization of the behavior of cells with mutant forms of various system constituents. 

The S-system (or synergistic and saturable system) formalism (131) is a differential equation based approach that has also been applied to genetic, biochemical, and immune network data (132,133). These systems are nonlinear and both genetic algorithms (134) and linear programming (123) have been used for their analysis. The currently available approaches are not easily applied to large systems and even upon simplification do not yield unique parameter estimates (123). 

---