Steady-state flux balance

From a steady-state flux balance of the flow entering the system and the flow into the pump, Seff can be defined as ... 

Pipe Flow For steady-state flow through a constant diameter duct, the mass flux G is constant and the governing steady-state momentum balance is ... 

An analysis of the right nullspace K provides the conceptual basis of flux balance analysis and has led to a plethora of highly successful applications in metabolic network analysis. In particular, all steady-state flux vectors v° = v(S°,p) can be written as a linear combination of columns Jfcx- of K, such that... 

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],... 

From a theoretical perspective, and provided that the network structure and some information about input and output fluxes are available, the intracellular steady-state fluxes can be estimated utilizing flux balance analysis. In conjunction with large-scale concentrations measurements, as described in Section IV, this allows, at least in principle, to specify the metabolic state of the system. 

As discussed in Section V.B, flux balance analysis allows to estimate intracellular steady-state fluxes provided the network structure and some information about... 

Alternatively, we may assume that there exists some (but possibly limited) knowledge about the typical concentrations involved. For each metabolite, we can then specify an interval St < S1- <. S ) that defines a physiologically feasible range of the respective concentration. Furthermore, the steady-state flux vector v° is subject to the mass-balance constraint Nv° = 0, leaving only r — rank(N) independent reaction rates. Again, an interval v(. < v9 < v+ can be specified for all independent reaction rates, defining a physiologically admissible flux space. 

Metabolic flux analysis is one of the most powerful analytical and experimental tools used for physiological characterisation of cell metabolism. In its most basic form, the method is essentially based on the conservation principles used for macrochemical and biological systems applied to the internal environment of cellular systems. The fundamental equation of MFA considers the steady-state mass balances around all intracellular metabolic intermediates such that... 

As we shall see, linear algebraic constraints arising from steady state mass balance form the basis of metabolic flux analysis (MFA) and flux balance analysis (FBA). Thermodynamic laws, while introducing inherent non-linearities into the mathematical description of the feasible flux space, allow determination of feasible reaction directions and facilitate the introduction of reactant concentrations to the constraint-based framework. 

Figure 34 5 Mean ocean steady-state Nisotope balance. Vertical arrows represent global input and output fluxes, in which the flux is proportional to the length of the arrow and the isotopic composition is shown by the position on the horizontal axis (showing in %o). The yellow...

A stirred cell equipped with a 0.22iuni membrane filter was charged with 30 mL of latex, the dispersion of microsphere. The specific surfrice area was adjusted to 0.19 m per ImL and the ionic strength was calibrated to 0.01. At the constant stirrer speed, buffer solution was introduced into the stirred ceil until steady state flux was attained. Protein solutions were introduced with step of pulse injection. The permeate flux was measured continuously with an electronic balance (Precision plus, Ohaus Co., USA) by a data acquisition system. The electronic balance was connected to a PC through a RS 232C interfece. The surface charge density of microspheres was varied as 0.45, S.94, 9.14 and 10.25, and the stirrer speed was varied as 300,400 and 600rpm. 

Given the transport fluxes for all species inside the catalyst particle, as modeled in Section 3.4.3. a steady-state mass balance considering the simultaneous transport and chemical reaction gives... 

If the interface is stationary, or if it translates without accelerating, then a steady-state force balance given by equation (8-180) states that the sum of all surface-related forces acting on the interface must vanish. Body forces are not an issue because the system (i.e., the gas-liquid interface) exhibits negligible volume. The total mass flux vector of an adjacent phase relative to a mobile interface is... 

Hence, stoichiometry and the steady-state mass balance with diffusion and one chemical reaction allows one to relate diffusional fluxes as follows ... 

The equations for simultaneous pore diffusion and reaction were solved independently by Thiele and by Zeldovitch [16,17]. They assumed a straight cylindrical pore with a first-order reaction on the surface, and they showed how pore length, diffusivity, and rate constant influenced the overall reaction rate. Their solution cannot be directly adapted to a catalyst pellet, since the number of pores decreases going toward the center and assuming an average pore length would introduce some error. The approach used here is that of Wheeler [18] and Weisz [19], who considered reactions in a porous sphere and related the diffusion flux to the effective diffusivity, Z). The basic equation is a material balance on a thin shell within the sphere. The difference between the steady-state flux of reactant into and out of the shell is the amount consumed by reaction. 

Taking the coordinate System z pointing down from the interface at z = 0, we can write the steady-state material balance for flux through an elemental slice Az thick with area normal to flux designated as A ... 

The basic construction of the mathematical model using simplified metabolic networks to describe the reactions of the citric acid cycle and associated transamination reactions between pyruvate and alanine, oxalacetate and aspartate and a-ketoglutarate and glutamate, and the use of the FACSIMILE program (Chance et al., 1977) to solve the rather large number of simultaneous differential equations generated by the model was the same as previously described (Chance etal., 1983). For the present experiments the model was expanded to include an input flux at the level of succinate to represent propionate metabolism to succinyl-CoA, and a dilution of the aspartate pool to represent net proteolysis. These input fluxes required an output flux of carbon from the citric acid cycle in order to maintain a steady state carbon balance, for which the conversion of malate to pyruvate via malic enzyme was chosen. The model calculates the unknown flux parameters to provide a minimum least squares fit of the C fractional enrichments of specific carbon atoms of metabolic intermediates as measured by C NMR spectroscopy. 

The starting point in this analysis is the construction of a list of steady-state material balance equations to describe the conversion of substrates to metabolic products for the biochemical system of interest. For example, if one considers a simplified scheme of amino acid metabolism in liver, one can write a set of steady-state material balance equations that represent the flow of metabolites through the network (Fig. 9). The equations contain measurable quantities (these are marked with an asterisk) which are the rates of consumption/production of extracellular metabolites. The concentrations of strictly intracellular metabolites (e.g., argininosuccinate) are assumed to be constant. In this particular case, we have eight fluxes to be determined, five of which are measurable (F[, F, F, F, F ). The five equations listed here, which relate these fluxes to each other, can be reduced to four independent equations. Thus, the system can be solved to yield the three unknown intracellular fluxes (Fi, F2, F4). Because the system is overdetermined, it provides an internal check for consistency of the data with each other and the assumed biochemistry. 

The flux from the liquid bulk to the liquid film around the catalyst particle at steady state is equal to the generation rate of the component on the catalyst surface. The steady-state mass balance for a catalyst particle thus becomes... 

By definition, the global rate is simply the intrinsic rate multiplied by the effectiveness factor. In order to obtain an expression for the effectiveness factor, conservation equations for the diffusion and reaction taking place in a pellet need to be solved. Take as an example the simple case of a first-order, isothermal reaction occurring in a thin slab-like pellet as shown in Figure 4.1. If only bulk diffusive flux is considered, with constant effective diffusivity i> , a steady state mass balance in one dimension gives ... 

FIGURE 3.1 Illustrative mass balance diagram showing the steady-state fluxes (mol h ), fugacities (Pa), concentrations (mol m ), amounts (mol), and VZ products (mol Pa ). 

The second major question about this analysis comes from the combination of the steady-state flux equation with an unsteady-state mass balance. You may find this combination to be one of those areas where superficial inspection is reassuring, but where careful reflection is disquieting. I have been tempted to skip over this point, but have decided that I had better not. Here goes ... 

In unsteady states the situation is less satisfactory, since stoichiometric constraints need no longer be satisfied by the flux vectors. Consequently differential equations representing material balances can be constructed only for binary mixtures, where the flux relations can be solved explicitly for the flux vectors. This severely limits the scope of work on the dynamical equations and their principal field of applicacion--Che theory of stability of steady states. The formulation of unsteady material and enthalpy balances is discussed in Chapter 12, which also includes a brief digression on stability problems. 

At any point within the boundary layer, the convective flux of the macromolecule solute to the membrane surface is given by the volume flux,/ of the solution multipfled by the concentration of retained solute, c. At steady state, this convective flux within the laminar boundary layer is balanced by the diffusive flux of retained solute in the opposite direction. This balance can be expressed by equation 1 ... 

The analysis of steady-state and transient reactor behavior requires the calculation of reaction rates of neutrons with various materials. If the number density of neutrons at a point is n and their characteristic speed is v, a flux effective area of a nucleus as a cross section O, and a target atom number density N, a macroscopic cross section E = Na can be defined, and the reaction rate per unit volume is R = 0S. This relation may be appHed to the processes of neutron scattering, absorption, and fission in balance equations lea ding to predictions of or to the determination of flux distribution. The consumption of nuclear fuels is governed by time-dependent differential equations analogous to those of Bateman for radioactive decay chains. The rate of change in number of atoms N owing to absorption is as follows ... 

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