Biochemical Reaction Systems

Chapter 28 Microbial Fermentation—Introduction and Overall Picture 1623 Chapter 29 Substrate Limiting Microbial Fermentation /630 Chapter 30 Product Limiting Microbial Fermentation /645 

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S. Schuster and C. Hilgetag, On elementary flux modes in biochemical reaction systems at steady state. J. Biol. Syst. 2, 165 182 (1994). 

Fundamental Equation for a Biochemical Reaction System at Specified pH... 

The preceding paragraph applies to a system in which there are no biochemical reactions. Now we consider systems with reactions that are at equilibrium (Alberty, 1992d). For a chemical reaction system, we saw (Section 3.4) that D = C + 2 and F = C + 1 for a one-phase system. For a biochemical reaction system at equilibrium, we need the fundamental equation written in terms of apparent components to show how many natural variables there are. When the reaction conditions 2 vj//f = 0 for the biochemical reactions in the system are used to eliminate one for each independent reaction from equation 4.1-18, the following fundamental equation for G in terms of apparent components is obtained ... 

Matrix Forms of the Fundamental Equations for Biochemical Reaction Systems... 

For a biochemical reaction system at specified pH, equations 5.4-1 and 5.4-2 become... 

Since the thermodynamics of a biochemical reaction system is considered at specific pH, we need to consider equation 5.5-4 in the form... 

Statistical mechanics is often thought of as a way to predict the thermodynamic properties of molecules from their microscopic properties, but statistical mechnics is more than that because it provides a complementary way of looking at thermodynamics. The transformed Gibbs energy G for a biochemical reaction system at specified pH is given by... 

SEMIGRAND PARTITION FUNCTION FOR A BIOCHEMICAL REACTION SYSTEM AT SPECIFIED CONCENTRATIONS OF COENZYMES... 

The Response of Biochemical Reaction Systems to Oscillating Stimuli... 

Schuster, S. Hilgetag, C. Woods, J. H. Fell, D. A. Reaction routes in biochemical reaction systems algebraic properties, validated calculation procedure and example from nucleotide metabolism. J Math Biol 2002,45 153-181. 

Here we focus on the issue of how to build computational models of biochemical reaction systems. The two foci of the chapter are on modeling chemical kinetics in well mixed systems using ordinary differential equations and on introducing the basic mathematics of the processes that transport material into and out of (and within) cells and tissues. The tools of chemical kinetics and mass transport are essential components in the toolbox for simulation and analysis of living biochemical systems. 

Consider the biochemical reaction system with four species ... 

The chemical master equation (CME) for a given system invokes the same rate constants as the associated deterministic kinetic model. Yet the CME is more fundamental than the deterministic kinetic view. Just as Schrodinger s equation is the fundamental equation for modeling motions of atomic and subatomic particle systems, the CME is the fundamental equation for reaction systems. Remember that Schrodinger s equation is not a model for a specific mechanical system. Rather, it is a theoretical framework upon which models for particular systems can be developed. In order to write down a model for an atomic system based on Schrodinger s equation, one needs to know how to write down the Hamiltonian a priori. Similarly, the CME is not a model for a specific biochemical reaction system it is a theoretical framework. To determine the CME model for a reaction system, one must know what are the possible elementary reactions and the associated rate constants. 

Deterministic dynamics of biochemical reaction systems can be visualized as the trajectory of (ci(t), c2(t), , c v(0) in a space of concentrations, where d(t) is the concentration of ith species changing with time. This mental picture of path traced out in the N-dimensional concentration space by deterministic systems may prove a useful reference when we deal with stochastic chemical dynamics. In stochastic systems, one no longer thinks in terms of definite concentrations at time t rather, one deals with the probability of the concentrations being xu x2, , Wy at time t ... 

A non-linear biochemical reaction system with concentration fluctuations... 

It is widely appreciated that chemical and biochemical reactions in the condensed phase are stochastic. It has been more than 60 years since Delbriick studied a stochastic chemical reaction system in terms of the chemical master equation. Kramers theory, which connects the rate of a chemical reaction with the molecular structures and energies of the reactants, is established as a central component of theoretical chemistry [77], Yet study of the dynamics of chemical and biochemical reaction systems, in terms of either deterministic differential equations or the stochastic CME, is not the exclusive domain of chemists. Recent developments in the simulation of reaction systems are the work of many sorts of scientists, ranging from control engineers to microbiologists, all interested in the dynamic behavior of biochemical reaction systems [199, 210],... 

Stochastic models for biochemical reaction systems in terms of the CME are not an alternative to the differential equation approach, but a more general theoretical framework that deserves further investigation. In particular, the relation between the dynamic CME and the general theory of statistical thermodynamics of closed and... 

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