Steady-state flux vector

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... 

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. 

In contrast, SKM does not assume knowledge of thespecific functional form of the rate equations. Rather, the system is evaluated in terms of generalized parameters, specified by the elements of the matrices A and 0X. In this sense, the matrices A and 0 x are bona fide parameters of the system The pathway is described in terms ofan average metabolite concentration S°, and a steady-state flux vector v°, together defining the metabolic state of the pathway. Additionally, we assume that the substrate only affects reaction v2, the saturation matrix is thus fully specified by a single parameter Of 6 [0,1], Note that the number of parameters is identical to the number used within the explicit equation. The structure of the parameter matrices is... 

All feasible steady-state flux vectors v(S°) are described by two basis vectors k, ... 

Enforcing stoichiometric, capacity, and thermodynamic constraints simultaneously leads to the definition of a solution space that contains all feasible steady-state flux vectors. Within this set, one can find a particular steady-state metabolic flux vector that optimizes the network behavior toward achieving one or more goals (e g., maximize or minimize the production of certain metabolites). Mathematically speaking, an objective function has to be defined that needs to be minimized or maximized subject to the imposed constraints. Such optimization problems are typically solved via linear programming techniques. 

It can be straightforwardly verified that indeed NK = 0. Each feasible steady-state flux v° can thus be decomposed into the contributions of two linearly independent column vectors, corresponding to either net ATP production (k ) or a branching flux at the level of triosephosphates (k2). See Fig. 5 for a comparison. An additional analysis of the nullspace in the context of large-scale reaction networks is given in Section V. 

Aiming at a network-based pathway analysis, the elementary flux modes provide a handle on the set of possible pathways through a metabolic network. In particular, each feasible steady-state flux distribution can be represented by a nonnegative combination of generating vectors that span the flux cone defined... 

The fundamental law of conservation of mass dictates that the vector of steady state fluxes, J, satisfies... 

Somewhat related to FBA is an approach for the determination of so-called elementary flux modes [Schuster, Dandekar, and Fell 1999 Schuster, Fell, and Dandekar 2000]. Generally speaking, an elementary flux mode is a minimal set of enzymes that can operate at steady state. In contrast to FBA, which produces a set of vectors spanning the possible steady-state fluxes, the elementary mode vectors are uniquely determined (up to a multiplication by a positive real number). Any steady-state flux distribution can be then represented as a linear combination of these modes with nonnegative coefficients. 

In principle, Chen, given the flux relations there is no difficulty in constructing differencial equations to describe the behavior of a catalyst pellet in steady or unsteady states. In practice, however, this simple procedure is obstructed by the implicit nature of the flux relations, since an explicit solution of usefully compact form is obtainable only for binary mixtures- In steady states this impasse is avoided by using certain, relations between Che flux vectors which are associated with the stoichiometry of Che chemical reaction or reactions taking place in the pellet, and the major part of Chapter 11 is concerned with the derivation, application and limitations of these stoichiometric relations. Fortunately they permit practicable solution procedures to be constructed regardless of the number of substances in the reaction mixture, provided there are only one or two stoichiomeCrically independent chemical reactions. 

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. 

In steady state, we denote the rate vector v by the flux vector J (this will be the only vector denoted by a capital letter) to obtain ... 

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... 

At steady state, owing to mass conservation and thermodynamic feasibility, the metabolic flux vector v e R fulfills the equations/inequalities... 

The Heat-flux Vector in Steady-state Shear and Elongational Flows... 

Control of the flux vector has been shown to have good dynamic and steady-state performance (Chandorkar, 2001). It also provides a convenient means to define the power angle since the inverter voltage vector switches position in the d-q plane, whereas there is no discontinuity in the inverter flux vector. 

Finally, Fig. 8.36 shows the evolution of the rotor flux vector in the (x, y)- and (d, <7)-planes from time t = 0.4 s to t = Is. The vectors noted as ri and r2 represent the rotor flux at steady state before and after the supply frequency reversal, respectively. The figure on the left clearly shows the rotating field phenomenon typical in AC-machines. Particularly, the effect of the supply frequency reversal on the rotor flux vector can be appreciated the counter-clockwise rotation gives place to a transient that finally results in a clockwise rotation, including amplitude variation during the transient and at the new steady state as well. 

Under steady-state conditions, the concentration flux vector is ... 

The last term on the right side represents the flux due to migration of ions in the electric field which is characterized by the vector E. It follows from Eq. 7 at steady state ... 

Equation (13) describes the net mass flux, Fg/p of solute b relative to stationary coordinates [ 132] at any positiony (0 < f < convective mass flux, Uf t b/f, relative to stationary co-ordinates of solute b toward the membrane with concentration cg/f and solution velocity v, and (2) the diffusive mass flux, F)g (9cb/r/9y), relative to stationary eo-ordinates of solute b away from the membrane along the solute eoneentration gradient (9t b/r/9> )> with diffusion coefficient, )g. All fluxes in this ehapter ean be assumed relative to stationary coordinates [132], so for the sake of brevity, referenee to stationary co-ordinates will be omitted henceforth. The reader will also note that fluxes are vector quantities [132] and the convective and diffusive fluxes at steady state will move solute in opposite directions, hence the opposing signs for the eonveetive and diffusive flux terms relative to stationary co-ordinates in Eq. (13). 

For bound states, the condition that the wave function is normalized implicitly specifies that it vanishes at large distances. For scattering at a fixed energy, we do not want to impose the condition that the number of collisions is one (or some other finite number). Rather, we want collisions to occur at a steady rate. This requires a constant flux of incoming molecules. The flux is, as usual, the velocity times the density. The density is where (j> is the incoming wave. The velocity is constant because the wave vector k is specified. 

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