Flux Variability Analysis

Table 8.3 Flux variability analysis (FVA) of maximum growth rate (1/h), a-acetolactate production rate (mmol/gow/h), and resulting yield predicted from the genome-scale model of Lactococcus lactis MG1363...

Equation (64) justifies the application of flux-balance analysis even in the face of (i) fast short-term fluctuations and (ii) periodic long term for example, circadian variability. The steady state balance condition restricts the feasible steady-state flux distributions to the flux cone P = v° G IRr IVv0 = 0. The reduction of the admissible flux space, with some of its algebraic properties already summarized in Section III.B, is exploited by several computational approaches, most notably Flux Balance Analysis (FBA) [61, 71, 235] and elementary flux modes (EFMs) [96, 236 238],... 

Another application of the analysis of the stoichiometric matrix is flux balance analysis (Edwards et al. 2002). Often the number of fluxes in the system exceeds the number variable metabolites making equation (3) an underdetermined set of linear equations, that is, many different combinations of fluxes are consistent with system steady state. One approach is to measure the fluxes that enter and exit the cell. Because intracellularly there are many redundant pathways, this does not enable one to determine all fluxes. Isotope labelling may help then (Wiechert 2002). Another approach to then find a smaller number of solutions is to postulate that the solution should satisfy an additional objective. This objective is taken to be associated with optimal functioning of the network, for instance maximization of some flux or combination... 

This brief discussion of some of the many effects and interrelations involved in changing only one of the operating variables points up quite clearly the reasons why no exact analysis of the dispersion of gases in a liquid phase has been possible. However, some of the interrelationships can be estimated by using mathematical models for example, the effects of bubble-size distribution, gas holdup, and contact times on the instantaneous and average mass-transfer fluxes have recently been reported elsewhere (G5, G9). 

In a 1991 study by van Reis et al. (5), a filtration operation as applied to harvest of animal cells was optimized by the use of dimensional analysis. The fluid dynamic variables used in the scale-up work were the length of the fibers (L, per stage), the fiber diameter (D), the number of fibers per cartridge (k), the density of the culture (p), and the viscosity of the culture (p). From these variables, scale-up parameters such as wall shear rate (y ) and its effect on flux (L/m /h) were derived. Based on these calculations, an optimum wall shear rate for membrane utilization, operating time, and flux was found. However, because there is no single mathematical expression relating all of these parameters simultaneously, the optimal solution required additional experimental research. 

The automatic flow-through diffusion cell system described is designed to allow the rapid analysis of drug flux through human skin in vitro with minimum variability. 

Consideration of the thermohaline structure of the Black Sea provides new results on the statistical and physical analysis of the historical data of ship-borne observations of the vertical profiles of the temperature and salinity of the waters. The general features of the vertical thermohaline structure of the Black Sea waters, the seasonal and interannual variabilities of the horizontal structure of the temperature and salinity in all the main water layers are described. The relations of the large-scale features of the hydrology of the Black Sea waters to external forcing (heat and moisture fluxes across the water surface, river mouths and straits, fluxes of the momentum and relative vorticity of wind) are shown. The generalization of the results of the studies of the T,S-structure of the Black Sea waters and of its seasonal and interannual variability allows the following conclusions to be made. 

Buckinghams 7r-theorem [i] predicts the number of -> dimensionless parameters that are required to characterize a given physical system. A relationship between m different physical parameters (e.g., flux, - diffusion coefficient, time, concentration) can be expressed in terms of m-n dimensionless parameters (which Buckingham dubbed n groups ), where n is the total number of fundamental units (such as m, s, mol) required to express the variables. For an electrochemical system with semiinfinite linear geometry involving a diffusion coefficient (D, units cm2 s 1), flux at x = 0 (fx=o> units moles cm-2 s 1), bulk concentration (coo> units moles cm-3) and time (f, units s), m = 4 (D, fx=0, c, t) and n - 3 (cm, s, moles). Thus m-n - 1 therefore only one dimensionless parameter can be constructed and that is fx=o (t/Dy /coo. Dimensional analysis is a powerful tool for characterizing the behavior of complex physical systems and in many cases can define relationships... 

Dimensionless analysis — Use of dimensionless parameters (-> dimensionless parameters) to characterize the behavior of a system (- Buckinghams n-theorem and dimensional analysis). For example, the chronoampero-metric experiment (-> chronoamperometry) with semiinfinite linear geometry relates flux at x = 0 (fx=o, units moles cm-2 s-1), time (t, units s-1), diffusion coefficient (D, units cm2 s-1), and concentration at x = oo (coo, units moles cm-3). Only one dimensionless parameter can be created from these variables (-> Buckingham s n-theorem and dimensional analysis) and that is fx=o (t/D)1/2/c0C thereby predicting that fx=ot1 2 will be a constant proportional to D1/,2c0O) a conclusion reached without any additional mathematical analysis. Determining that the numerical value of fx=o (f/D) 2/coo is 1/7T1/2 or the concentration profile as a function of x and t does require mathematical analysis [i]. 

Mathematical optimization deals with determining values for a set of unknown variables x, X2, , x , which best satisfy (optimize) some mathematical objective quantified by a scalar function of the unknown variables, F(xi, X2, , xn). The function F is termed the objective function bounds on the variables, along with mathematical dependencies between them, are termed constraints. Constraint-based analysis of metabolic systems requires definition of the constraints acting on biochemical variables (fluxes, concentrations, enzyme activities) and determining appropriate objective functions useful in determining the behavior of metabolic systems. 

Applying mass-balance and thermodynamic constraints typically leaves one without a precisely defined (unique) solution for reaction fluxes and reactant concentration, but instead with a mathematically constrained feasible space for these variables. Exploration of this feasible space is the purview of constraint-based analysis. It has so far been left unstated that any application in this area starts with the determination of the reactions in a system, from which the stoichiometric matrix arises. This first step, network reconstruction, integrates genomic and proteomic data to determine carefully the enzymes present in an organism, cell, or subcellular compartment. The network reconstruction process is described elsewhere [107]. 

We begin here the study of thermodynamics in the proper sense of the word, by exploring a variety of physical situations in a system where one or more intensive variables are rendered nonuniform. So long as the variations in T, P, /x or other intensive quantities are small relative to their average values, one can still apply the machinery of equilibrium thermodynamics in a manner discussed later. It will be seen that the identification of conjugate forces and fluxes, the Onsager reciprocity conditions, and the rate of entropy production play a central role in the analysis provided later in the chapter. 

A more popular form of stoichiometric analysis is the analysis of flux distributions that are consistent with system steady state (Note that in the terminology of metabolic modelling, the rate of a reaction at system steady state is referred to as a flux.) This type of analysis can be done directly on the N matrix because of its central role in the description of the mass balances of all the variable intermediates in a network. 

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