Kinetics, nonlinear

The steady-state kinetic treatment of multisubstrate random enzyme reactions gives rise to the forward rate equation of higher order in substrate terms that reflect the number of substrate addition in the formation of intermediary complexes. The transformations are nonlinear. For example, the steady-state treatment of the random bi bi reaction gives, in a coefficient form  

TABLE 11.5 Cleland nomenclature for bisubstrate reactions exemplified. Three common kinetic mechanisms for bisubstrate enzymatic reactions are exemplified. The forward rate equations for the order bi bi and ping pong bi hi are derived according to the steady-state assumption, whereas that of the random bi bi is based on the quasi-equilibrium assumption. These rate equations are first order in both A and B, and their double reciprocal plots (1A versus 1/A or 1/B) are linear. They are convergent for the order bi bi and random bi bi but parallel for the ping pong bi bi due to the absence of the constant term (KiaKb) in the denominator. These three kinetic mechanisms can be further differentiated by their product inhibition patterns (Cleland, 1963b) 

Notes 1. Vi, Ka, Kb and Kia are maximum velocity, Michaelis-Menten constants for A and B, and inhibition constant for A respectively. 2. The Order bi bi reactions may display zero (known as Theorell-Chance mechanism), one and two ternary complexes. All of them give the rate equation in the same Cleland form. 

The nonlinear 2 2 function kinetics should be differentiated from other nonlinear kinetics such as allosteric/cooperative kinetics (Bardsley and Waight, 1978) and the formation of the abortive substrate complex (Dalziel and Dickinson, 1966). The cooperative kinetics (of the double reciprocal plots) can either concave up (positive cooperativity) or 

---

The simplest manifestation of nonlinear kinetics is the clock reaction—a reaction exliibiting an identifiable mduction period , during which the overall reaction rate (the rate of removal of reactants or production of final products) may be practically indistinguishable from zero, followed by a comparatively sharp reaction event during which reactants are converted more or less directly to the final products. A schematic evolution of the reactant, product and intenuediate species concentrations and of the reaction rate is represented in figure A3.14.2. Two typical mechanisms may operate to produce clock behaviour. 

Field R J and Burger M (eds) 1984 Oscillations and Travelling Waves in Chemical Systems (New York Wiley) Multi-author survey of nonlinear kinetics field to 1984, still a valuable introduction to researchers in this area. 

All catalytic reactions involve chemical combination of reacting species with the catalyst to form some type of inteniiediate complex, the nature of which is the subject of abundant research in catalysis. The overall reaction rate is often determined by the rate at which these complexes are formed and decomposed. The most widely-used nonlinear kinetic equation that describes... 

Lee KM, Bruckner JV, Muralidhara S, et al. 1996. Characterization of presystemic elimination of trichloroethylene and its nonlinear kinetics in rats. Toxicol Appl Pharmacol 139 262-271. 

Because of the rapid and nonlinear kinetics of testosterone and progesterone transport, Pe is calculated by Eq. (7) in the form... 

The autocatalytic reaction mechanism apparent at low temperatures is expected to apply to catalytic hydrogen oxidation at high pressures. In addition, the above study is the first to use STM to observe the formation of dynamic surface patterns at the mesoscopic level, which had previously been observed by other imaging techniques in surface reactions with nonlinear kinetics [57]. This study illustrates the ability of in situ STM to visualize reaction intermediates and to reveal the reaction pathway with atomic resolution. 

PBPK and classical pharmacokinetic models both have valid applications in lead risk assessment. Both approaches can incorporate capacity-limited or nonlinear kinetic behavior in parameter estimates. An advantage of classical pharmacokinetic models is that, because the kinetic characteristics of the compartments of which they are composed are not constrained, a best possible fit to empirical data can be arrived at by varying the values of the parameters (O Flaherty 1987). However, such models are not readily extrapolated to other species because the parameters do not have precise physiological correlates. Compartmental models developed to date also do not simulate changes in bone metabolism, tissue volumes, blood flow rates, and enzyme activities associated with pregnancy, adverse nutritional states, aging, or osteoporotic diseases. Therefore, extrapolation of classical compartmental model simulations... 

Solid-phase organic synthesis (SPOS) exhibits several shortcomings, due to the nature of the heterogeneous reaction conditions. Nonlinear kinetic behavior, slow reactions, solvation problems, and degradation of the polymer support due to the long reaction times are some of the problems typically experienced in SPOS [2], Any technique which is able to address these issues and to speed up the process of solid-... 

The E-Z Solve software may also be used to solve Example 12-7 (see file exl2-7.msp). In this case, user-defined functions account for the addition of fiesh glucose, so that a single differential equation may be solved to desenbe the concentration-time profiles over the entire 30-dry period. This example file, with die user-defined functions, may be used as the basis for solution of a problem involving the nonlinear kinetics in equation (A), in place of the linear kinetics in (B) (see problem 12-17). 

An unusual feature of a CSTR is the possibility of multiple stationary states for a reaction with certain nonlinear kinetics (rate law) in operation at a specified T, or for an exothermic reaction which produces a difference in temperature between the inlet and outlet of the reactor, including adiabatic operation. We treat these in turn in the next two sections. 

In general, equation 20.2-3 must be solved numerically for nonlinear kinetics. However, an analytical solution is available for first-order kinetics and a closed vessel. 

In comparing the TIS and DPF reactor models, we note that the former is generally easier to use for analysis of reactor performance, particulariy for nonlinear kinetics and unsteady-state operation. 

Kunii and Levenspiel(1991, pp. 294-298) extend the bubbling-bed model to networks of first-order reactions and generate rather complex algebraic relations for the net reaction rates along various pathways. As an alternative, we focus on the development of the basic design equations, which can also be adapted for nonlinear kinetics, and numerical solution of the resulting system of algebraic and ordinary differential equations (with the E-Z Solve software). This is illustrated in Example 23-4 below. 

Since the first report of oscillation in 1965 (159), a variety of other nonlinear kinetic phenomena have been observed in this reaction, such as bi-stability, bi-rhythmicity, complex oscillations, quasi-periodicity, stochastic resonance, period-adding and period-doubling to chaos. Recently, the details and sub-systems of the PO reaction were surveyed and a critical assessment of earlier experiments was given by Scheeline and co-workers (160). This reaction is beyond the scope of this chapter and therefore, the mechanistic details will not be discussed here. Nevertheless, it is worthwhile to mention that many studies were designed to explore non-linear autoxidation phenomena in less complicated systems with an ultimate goal of understanding the PO reaction better. 

The partial differential equations representing material and energy balances of a reaction in a packed bed are rarely solvable by analytical means, except perhaps when the reaction is of zero or first order. Two examples of derivation of the equations and their analytical solutions are P8.0.1.01 and P8.01.02. In more complex cases analytical, approximations can be made (by "Collocation" or "Perturbation", for instance), but these usually are quite sophisticated to apply. Numerical solutions, on the other hand, are simple in concept and are readily implemented on a computer. Two such methods that are suited to nonlinear kinetics problems will be described. 

In vitro data cannot predict whether linear or nonlinear kinetics will occur with specific dose of a chemical in vivo. 

Oscillations have been observed in chemical as well as electrochemical systems [Frl, Fi3, Wol]. Such oscillatory phenomena usually originate from a multivariable system with extremely nonlinear kinetic relationships and complicated coupling mechanisms [Fr4], Current oscillations at silicon electrodes under potentio-static conditions in HF were already reported in one of the first electrochemical studies of silicon electrodes [Tul] and ascribed to the presence of a thin anodic silicon oxide film. In contrast to the case of anodic oxidation in HF-free electrolytes where the oscillations become damped after a few periods, the oscillations in aqueous HF can be stable over hours. Several groups have studied this phenomenon since this early work, and a common understanding of its basic origin has emerged, but details of the oscillation process are still controversial. 

Numerous applications of this theory have been made in calculating confidence intervals for parameter estimates in nonlinear kinetic models, such as typified in Table III (P2). The use of confidence regions is typified in Fig. 13 (M7) for the alcohol dehydration model... 

Potency comprises both achon and inhibition of achon and is predicted by the Hill model though a 50% level is chosen, it is an arbitrary percentage and other values such as 60 or 40% action can also be calculated and used. Potency is not a relevant factor xmless it is so low that the dose requirement is very high (to a level where nonlinear binding with albumin can be observed, resulting in nonlinear kinetics) or where the serious side effects are dose-dependent and make an effechve dose unacceptably toxic. The potency, EC50, is expressed in a mechanishc equilibrium model where the achon is direct ... 

As in any other mass balance model of bioprocesses, a strongly nonlinear kinetic behavior is present due to the reaction rates. These rates are given by ... 

Here Cbrain is the brain concentration after correction for intravascular content, and AUC is determined between time 0 and the final sampling time. Two assumptions must hold when interpreting the evaluation in the simple form described above (1) the brain uptake of the compound is linear, meaning is dose independent, and (2) the analysis is performed within a time-frame where the efflux from tissue is negligible (tissue concentrations are sufficiently low compared to plasma concentrations). Violation of these assumptions requires adjustments in experimental design and evaluation. For example, nonlinear kinetics may be accounted for by incorporation of a MichaeUs-Menten term, while efflux can be treated by compartmental analysis [46]. 

ABORTIVE COMPLEX Nonlinear kinetics of drug bioavailability, PHARMACOKINETICS NONLINEAR LEAST SQUARES ANALYSIS Nonmetal oxides,... 

With regard to turbulence, in some cases an additional complication may have to be considered in that the temperature may vary locally, and then thermal conduction is important. It is possible to vary the constraints on the systems you have studied (e.g., concentration) so that by variation of one or more of these constraints both hysteresis and no hysteresis (at different constraints) can be observed in the same system. This is important to make sure that the observed hysteresis is due to the nonlinear kinetics and not due to other reasons. 

---