Random mechanism equations

In the case of interstitial diffusion in which we have only a few diffusing interstitial atoms and many available empty interstitial sites, random-walk equations would be accurate, and a correlation factor of 1.0 would be expected. This will be so whether the interstitial is a native atom or a tracer atom. When tracer diffusion by a colinear intersticialcy mechanism is considered, this will not be true and the situation is analogous to that of vacancy diffusion. Consider a tracer atom in an interstitial position (Fig. 5.18a). An initial jump can be in any random direction in the structure. Suppose that the jump shown in Figure 5.18b occurs, leading to the situation in Figure 5.18c. The most likely next jump of the tracer, which must be back to an interstitial site, will be a return jump (Fig. 5.18c/). Once again the diffusion of the interstitial is different from that of a completely random walk, and once again a correlation factor, / is needed to compare the two situations. 

Migrations of arylazo groups were first detected in the l,2,3,4,5-penta(methoxycarbonyl) cyclopentadiene 259 (equation 89)119-122. The randomization mechanism was considered as most probable because the reaction rate increases with increase in the solvent polarity (AGf9S = 56.9 to 69.1 kJmol-1). 

Equation 11.40 is a special case of a more general mechanism discussed below in which substrates bind to the enzyme randomly. However, to finish discussion of the sequential ordered mechanism, Equation 11.37, we simplify as before, by assuming that binding processes are isotopically insensitive. Equation 11.39 becomes ... 

THE COMBINED EQUILIBRIUM AND STEADY-STATE TREATMENT. There are a number of reasons why a rate equation should be derived by the combined equilibrium and steady-state approach. First, the experimentally observed kinetic patterns necessitate such a treatment. For example, several enzymic reactions have been proposed to proceed by the rapid-equilibrium random mechanism in one direction, but by the ordered pathway in the other. Second, steady-state treatment of complex mechanisms often results in equations that contain many higher-order terms. It is at times necessary to simplify the equation to bring it down to a manageable size and to reveal the basic kinetic properties of the mechanism. 

This equation reveals atypical product inhibition patterns for a random mechanism P is noncompetitive with both A and B Q is competitive with both A and B. Whenever abnormal product inhibition patterns are ob-... 

To summarize, the observation of a Gaussian profile usually implies that transport is governed mathematically by the diffusion equations and mechanistically by one or more multistep random processes. Below we examine some of the random mechanisms operative in separations. 

The deterministic model with random fractional flow rates may be conceived on the basis of a deterministic transfer mechanism. In this formulation, a given replicate of the experiment is based on a particular realization of the random fractional flow rates and/or initial amounts 0. Once the realization is determined, the behavior of the system is deterministic. In principle, to obtain from the assumed distribution of 0 the distribution of common approach is to use the classical procedures for transformation of variables. When the model is expressed by a system of differential equations, the solution can be obtained through the theory of random differential equations [312-314]. 

The variation of Cf in Equation 26 with the level of a second substrate is a powerful tool for determination of kinetic mechanism (23). In an ordered mechanism, " (V/Kb) for the second substrate is independent of the level of A, the first substrate. The apparent value of (V/Kj,), however, varies from °(V/Kb) at low levels of B to unity at saturating B. In a random mechanism, both V/K isotope effects are finite, although they may not have the same value if one substrate is sticky or stickier than the other. 

Equation 2.28 contains a new term, Km 2, that represents a change in affinity of the enzyme for one substrate once the other substrate is bound. If the mechanism is ordered, the simple relationship Km 2 = Km x Km2 may be applied. For a random mechanism, the value of Km 2 is determined experimentally. Creatine kinase (CK) is an example of this type of enzyme. Creatinine and ATP bind to the enzyme randomly in nearby, but independent binding sites. 

B. Initial Rate Equations for Ordered and Random Mechanisms... 

The initial rate equation derived by steady-state analysis is of the second degree in A and B (SO). It simplifies to the form of Eq. (1) if the rates of dissociation of substrates and products from the complexes are assumed to be fast compared with the rates of interconversion of the ternary complexes k, k )] thus, the steady-state concentrations of the complexes approximate to their equilibrium concentrations, as was first shown by Haldane (14)- The kinetic coefficients for this rapid equilibrium random mechanism (Table I), together with the thermodynamic relations KeaKeab — KebKeba and KepKepq — KeqKeqp, suffice for the calculation of k, k and all the dissociation constants Kea = k-i/ki, Keab = k-i/ki, etc. 

The steady-state rate equation for the random mechanism will also simplify to the form of Eq. (1) if the relative values of the velocity constants are such that net reaction is largely confined to one of the alternative pathways from reactants to products, of course. It is important, however, that dissociation of the coenzymes from the reactant ternary complexes need not be excluded. Thus, considering the reaction from left to right in Eq. (13), if k-2 k-i, then product dissociation will be effectively confined to the upper pathway this condition can be demonstrated by isotope exchange experiments (Section II,C). Further, if kakiB kik-3 -f- kikiA, then the rate of net reaction through EB will be small compared with that through EA 39). The rate equation is then the same as that for the simple ordered mechanism, except that a is now a function of the dissociation constant for A from the ternary complex, k-i/ki, as well as fci (Table I). Thus, Eqs. (5), (6), and (7) do not hold instead, l/4> < fci and ab/ a b < fc-i, and this mechanism can account for anomalous maximum rate relations. In contrast to the ordered mechanism with isomeric complexes, however, the same values for these two functions of kinetic coefficients would not be expected if an alterna-... 

The rate equations for fully random and ordered mechanisms for three-substrate reactions are shown in Table II and can only be briefly discussed here. For the random mechanism, the rate equation derived by the rapid equilibrium assumption 43) contains all the terms of Eq. (2), and from experimental values for the eight kinetic coefficients for the reaction in each direction the dissociation constants for all the complexes may be calculated (c/. 43). 

Since the initial rate equation for a random mechanism, Eq. (13), is not of the linear form of Eq. (1), it can account for rate cooperativity with appropriate values for the rate constants (6,30,149) and has been suggested for isocitrate dehydrogenases (lJf2,150). It does not require that there be more than one active center in the enzyme molecule nor cooperative equilibrium binding of substrates or modifiers. An alternative to this purely kinetic explanation is that there are two or more active... 

Note that these rules correctly predict that in a two-substrate case the only sequential mechanism in which a term is missing from the denominator is an ordered one in which the first substrate addition is in rapid equilibrium. Rapid equilibrium binding in a random mechanism does not change the initial velocity rate equation (Rule 1), since both substrates can add in the second position. These rules can easily be generalized for cases with four or more substrates. 

We do not mean to imply that kg is an intrinsic isotope effect it will be given by an equation similar to Eq. (84) but containing only the internal portion of the forward commitment (cf.i ). The level of B will affect as long as is as big or bigger than ks- If k is zero (an ordered mechanism), becomes infinite at very high B, and no isotope effect is seen on WA itself. At very low B, on the other hand, cr.ex = kg/k. The level of B at which the isotope effect is half-suppressed is KaKJK. For a random mechanism, Cf.ex varies from kg/ik + its) at low B to kg/ki at high B. 

For the determination of rate constants, the initial rate of catalysis is measured as a function of the concentration of substrate B (or A) for several concentrations of A (or B). Evaluation can be done using the Lineweaver-Burk plot. Reshaping Equation 2.56 for a ""random mechanism leads to ... 

Song J, Der Kiureghian A (2006) Joint first-passage probability and reliability of systems under stochastic excitation. ASCE J Eng Mech 132(l) 65-77 Soong TT (1973) Random differential equations in science and engineering. Academic, New York Soong TT, Grigoriu M (1993) Random vibration of mechanical and structural systems. Prentice Hall, New Jersey... 

The friction coefficient determines the strength of the viscous drag felt by atoms as they move through the medium its magnitude is related to the diffusion coefficient, D, through the relation Y= kgT/mD. Because the value of y is related to the rate of decay of velocity correlations in the medium, its numerical value determines the relative importance of the systematic dynamic and stochastic elements of the Langevin equation. At low values of the friction coefficient, the dynamical aspects dominate and Newtonian mechanics is recovered as y —> 0. At high values of y, the random collisions dominate and the motion is diffusion-like. 

Short Random Fibers. The whiskers bridging mechanics given in equations 21 through 25 apply also to short random fiber bridging mechanisms. The bridging terms come from (44) ... 

Equation 10.4, which describes the mass transfer rate arising solely from the random movement of molecules, is applicable to a stationary medium or a fluid in streamline flow. If circulating currents or eddies are present, then the molecular mechanism will be reinforced and the total mass transfer rate may be written as ... 

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