Steady-State Flux Equations

The section expands upon the flux concept, which resulted in Equation 2.3 above. Its application within and across media interfaces focuses the user to consider all the individual mechanisms that are operative and may result in chemical transport. Several steady-state flux equation types have emerged over the years as being appropriate and practical for quantifying known transport mechanisms. These are used throughout the book. Specific chapters are devoted to estimating these transport parameters. The most significant ones are presented in the following sections. 

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The result of equating the steady-state fluxes, equations (14) and (15), is sometimes known as the Best equation [9,13,16,27-31] which we normalise to its most elementary parameters [26] (other normalisations have also been suggested [21]) ... 

We are now in a position to consider individual ions, rather than the ions collectively. The steady state flux equations for ions of charge magnitude z are... 

The subscripts b and i refer to the bulk and interfacial concentrations. Now, one can write the steady-state flux equation for cations in the vicinity of the membrane solution interface. 

Mass transfer, an important phenomenon in science and engineering, refers to the motion of molecules driven by some form of potential. In a majority of industrial applications, an activity or concentration gradient serves to drive the mass transfer between two phases across an interface. This is of particular importance in most separation processes and phase transfer catalyzed reactions. The flux equations are analogous to Ohm s law and the ratio of the chemical potential to the flux represents a resistance. Based on the stagnant-film model. Whitman and Lewis [25,26] first proposed the two-film theory, which stated that the overall resistance was the sum of the two individual resistances on the two sides. It was assumed in this theory that there was no resistance to transport at the actual interface, i.e., within the distance corresponding to molecular mean free paths in the two phases on either side of the interface. This argument was equivalent to assuming that two phases were in equilibrium at the actual points of contact at the interface. Two individual mass transfer coefficients (Ld and L(-n) and an overall mass transfer coefficient (k. ) could be defined by the steady-state flux equations ... 

If the adsorption isotherm takes the form of Langmuir equation as given in example 10.2-1, the steady state flux equation will become ... 

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 ... 

This is a cubic equation of the unknown quantity c f which is the eventual concentration at the surface of the organism. Then, the resulting steady-state flux can be computed using either side of equation (23). 

Figure 5b shows the resulting steady-state flux. / s (obtained as the positive solution for c f from equation (23)) for a range of c"M values. At low c"M values (usually associated with low values), there is a linear dependence between and r M, as expected from the linearisation of the Langmuir isotherms (see equation (31), below). At large c M values, the usual Michaelis-Menten saturating effect of is also seen. 

From inspection of equation (23), it follows that the effects of ro and Dm are opposed. Three / ss versus cm0"o) plots for different ratios Dm/> o are shown as straight lines with different slopes in Figure 5c, converging at a common point at. Thus, the steady-state flux increases with DM/r0... 

The same analysis can be performed on the effect of the similar parameters k, rmax, 1 > k2 and rmax - The steady-state flux increases when any of them increases. As seen in Figure 5f for the particular case of rmax,i variation, there are two asymptotic limits for J s given by the fixed Ju>2 (at low rm lX)i) and by the fixed limiting diffusion value (see equation (16) and Figure 5e). 

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],... 

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... 

The meaning of this term is shown by Figure 2.5 and it is essentially the time required to attain steady state flux across a barrier. When the resistance in the boundary layer is negligible, the lag-time equation provides a convenient means of calculating membrane or polymer-diffusion coefficients. 

Easterby proposed a generalized theory of the transition time for sequential enzyme reactions where the steady-state production of product is preceded by a lag period or transition time during which the intermediates of the sequence are accumulating. He found that if a steady state is eventually reached, the magnitude of this lag may be calculated, even when the differentiation equations describing the process have no analytical solution. The calculation may be made for simple systems in which the enzymes obey Michaehs-Menten kinetics or for more complex pathways in which intermediates act as modifiers of the enzymes. The transition time associated with each intermediate in the sequence is given by the ratio of the appropriate steady-state intermediate concentration to the steady-state flux. The theory is also applicable to the transition between steady states produced by flux changes. Apphcation of the theory to coupled enzyme assays makes it possible to define the minimum requirements for successful operation of a coupled assay. The theory can be extended to deal with sequences in which the enzyme concentration exceeds substrate concentration. 

For a planar film of C with no reaction of A within the film, the steady-state flux of H is constant at all values of x so we solve the equation... 

A verbal interpretation of this equation is that the electroinactive ion experiences a diffusive flux and a migratory flux, which are exactly equal and opposite everywhere in the cell in the steady state. The equation may be rearranged to... 

Therefore, Laplace s equation holds for each component separately. However, the steady-state fluxes are interdependent, as may be seen from Eq. 6.11 for the ternary case,... 

As the skin is relatively thick compared to the space-charge layers at its boundaries, the bulk of the membrane may be expected to be electroneutral [56,57], The Nernst-Planck equation can be solved, therefore, by imposing the electroneutrality condition C,/C= C. /C, where the subscripts j and k refer to positive and negative ions, respectively, and C is the average total ion concentration in the membrane. In the case of a homogenous and uncharged membrane bathed by a 1 1 electrolyte, the total ion concentration profile across the membrane is linear and the resulting steady-state flux is described by... 

It is possible to determine C quantitatively using Hildebrand s theory of microsolutes. An example of the accuracy that can be achieved is provided by the calculation of the solubilities of a series of p-aminobenzoate esters in hexane (17,18). Michaels, et al. (19) used this approach to estimate the solubility of steroids in various polymers. The solubilities of seven steroids in six polymers were calculated from the steroid melting points, heats of fusion, and solubility parameters. Equation 8 was derived, where Jjj is the maximum steady state flux, h is the membrane thickness, x is the product of V, the molar volume of the liquid drug, and the square of the difference in the solubility parameters of the drug and polymer, p is the steroid density, T is melting point (°K), T is the temperature of the environment, R is the gas constant, and AH and ASf are the enthalpy and entropy of fusion, respectively. 

Experimental Results and Comparisons with the Classical Lipid Barrier Model. Some typical experimental data are presented in Figure 1 for the transport of g-estradiol. In each of the experiments a lag-time of 1.5 to 2.5 hours were followed by linear steady state fluxes. The effective permeability coefficient, Peff> was calculated from such data using Equation 1 under sink conditions (i.e., Cj/K Cr/Kr where, Kj is the partition coefficient between membrane and donor phase and Kr the partition coefficient between membrane and receiver phase.)... 

Under steady-state conditions, Equation 10.33 can be integrated to give the steady-state flux [92]... 

Let us now consider the effects of external diffusion control on reaction (II) between A and B when the surface kinetics are first order in each reactant. Because vA mol of A react with vB mol of B at the surface, the steady state fluxes of A and B towards the surface from the bulk solution will be in the ratio of vA to vB. By equations like (43), their concentration gradients and their concentration differences are seen to be of similar magnitude. If initially the solution contains much more B than A... 

Equation (3.31) is the standard form for the steady state flux though a simple reversible Michaelis-Menten enzyme. This expression obeys the equilibrium ratio arrived at above (b/a)eq = Keq = k+ k+2/(k- k-2), when Jmm(g, b) = 0. In addition, from the positive and negative one-way fluxes in Equation (3.30), we note that the relationship J+/J = Keq(a/b) = e AG/RT is maintained whether or not the system is in equilibrium. Thus, as expected, the general law of Equation (3.12) is obeyed by this reaction mechanism. 

In Chapter 4 (Section 4.1.1), we derived the Lineweaver-Burk double-reciprocal relation between the steady state flux of an enzyme reaction and its substrate concentrations. (See Equation (4.5).) Furthermore, we showed in Section 4.4.1 that the same equation can be obtained from a stochastic point of view. Recalling this derivation, consider the basic mechanism... 

Equation 5.22 shows that Q is determined by an equation describing the steady flow of A through the pellet with a time lag td. To determine td one may plot any quantity proportional to the amount of diffused gas Q, rather than this quantity itself, versus time. The intercept on the t axis will give td and = L2 p/(6td). Note that with this experimental procedure the effective diffusivity can also be found from Equation 5.23, if the steady-state flux is measured. 

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