Preface

The journey of a thousand miles begins with one step.

Lao Tzu

Analysis of Enzyme Reaction Kinetics is the second set in a unique 11-volume collection on Enzyme Reactor Engineering. This two-volume set comprehensively describes rate equations pertaining to enzymatic reactions, including modulation by physicochemical factors - as tools for prediction and control of how fast substrate(s) will be transformed to products, an essential element of mass balances to, and practical operation of (enzyme) reactors. Therefore, rate expressions are mathematically derived from mechanistic postulates, and complemented by appropriate statistical approaches, in attempts to fit them to experimental data (first volume); physical and chemical modulation of reaction rate is discussed next, upon both enzyme-catalyzed and enzyme deactivation reactions (second volume).

Introductory concepts - including such key issues as molecular mechanism of a given chemical reaction, derivation of kinetic expression from elementary steps, definition of reaction order, and difference between uni- and multisubstrate reactions are taken as starting point for the first volume. For reactions catalyzed by a single enzyme, the basic Michaelis and Menten's postulates are reviewed, and the analytical form of the associated rate expression is produced and graphically interpreted - by resorting to semilogarithmic, Eisenthal and Cornish-Bowden's, and Dixon's plots; the concentrations of free enzyme and enzyme-substrate complex are accordingly retrieved, while overparameterization is reduced as much as possible. Alternative derivations of the basic rate expression are then provided - namely based on van Slyke and Culen's, or Briggs and Haldane's postulates; the accompanying relative errors are estimated, and the underlying biochemical rationale is explored. Said rate expressions are meanwhile discussed in terms of absolute sensitivity to lumped parameters and relative sensitivity to intrinsic parameters - and their derivatives duly calculated. With regard to reactions catalyzed by multiple enzymes, independent attention is dedicated to two, several or (asymptotically) infinite number of isozymes; a brief tour is taken through autocatalytic reactions - where enzyme serves as both substrate and catalyst; as well as through multiphasic reactions - where substrate(s)/product(s) are contributed by distinct, immmiscible phases.

A more in-depth treatment of single enzyme, homogeneous reactions is then developed, following Morrison's general algebraic derivation, and Kim's even more general differential approach; graphical interpretations are provided in both cases, limiting cases considered, and absolute sensitivity to intrinsic kinetic parameters ascertained. Dedicated techniques for accurate modeling of initial and final transients are reported - under both batch and flow stirred configurations. In attempts to make Michaelis and Menten's rate expression simpler to use, Lambert's solution and Taylor's expansion are utilized to generate explicit forms useful for integrated mass balances; and several possibilities for linearization are introduced and discussed, applicable to both differential and integral forms of such mass balances.

Finally, general techniques for systematic derivation of multisubstrate rate expressions, under pseudo steady-state, are made available - based on King and Altman's method and Cha's approximation, complemented with Cleland's nomenclature. Illustrations of their use are conveyed for selected (relevant) mechanisms, e.g. Uni Uni and ordered Bi Bi, as well as Uni Bi and other Bi Bi possibilities; rapid equilibrium approaches are included for comparability, as well as simplification stemming from stoichiometric arguments.

Once in possession of analytical rate expressions, the next step is fitting of germane parameters therein to actual experimental data. This requires preliminary analysis of the data themselves, in attempts to detect deviations from the standard hypothesis of independent, identically distributed, normal errors; hence, checks for independence, normality, homoskedasticity, linearity, adequacy, and sufficiency are in order. After the data have been cleared (and transformed, if deemed necessary to abide to the aforementioned standard hypothesis), the next step is fitting of the putative model to such data; this can be carried out via classical linear regression analysis - should said model be linear (or have been linearized a priori). When the model is intrinsically nonlinear, an improved approach is a must - via data transformation, or else resorting to such statistical tools as weighed least squares or nonparametric techniques; the best approach is, however, nonlinear regression itself - and some discussion on this particular is warranted, namely concerning estimation, stationarity, and inference associated therewith. Since meaningful statistical inference requires the original data to have been generated criteriously, special care is to be exercized with experimental design in the first place; a brief review of both empirical and mechanistic designs is thus proposed - where starting, sequential, and subset designs are separately discussed as part of the latter type. The first volume ends with a simplification of mechanistic designs that takes advantage of conditional linearity; both stable and decaying enzyme are taken as case studies, stemming again from the classical Michaelis and Menten's rate expression.

The second volume starts with thermodynamic and kinetic rationalization of the physical modulation of reaction rate - encompassing both enzyme-mediated and enzyme decay reactions. The two modes of enzyme deactivation, i.e. uni- and bimodal, are considered independently and as single events; simple reversible and irreversible modes of decay are explored under the former heading - along with effect of reactor configuration thereon, and effect thereof upon kinetic model discrimination. Generalized models of enzyme deactivation are presented afterwards, built via combination of reversible and irreversible steps taken in series and in parallel, respectively - or vice versa. Conversely, only simple modes of reversible and irreversible bimodal deactivation are considered to some length; and a brief overview is subsequently proposed on more complex molecular mechanisms.

One of the two major drivers for physical modulation - i.e. temperature, is addressed in terms of elementary steps only (for the sake of simplicity); once again, both thermodynamic and kinetic formulations are brought on board - with derivation based on collision theory or transition state theory, in the latter case. The other major driver, i.e. mechanical forces, is discussed next, encompassing (normal and tangential) elastic forces (involved in storage of mechanical energy), separately from (tangential) plastic forces causing dissipation of mechanical energy; for their importance with regard to surface tension, tangential elastic forces are simulated via Gibbs' and Langmuir's models. A final word is deserved by overall response of enzyme reactions encompassing more than one elementary step - combined with enzyme deactivation, to the aforementioned physical factors.

Another strategy for modulation of enzyme behavior resorts to chemical means; following introductory considerations of thermodynamic and kinetic nature, the major biochemical events underlying chemical deactivation of enzymes are reviewed - namely denaturation, condensation, stabilization and (reversible and irreversible) inhibition. By the same token, the effects of pH and self-control, further to ionic strength are discussed as tools for chemical manipulation of enzyme performance. In view of their practical importance, pH-driven modulation is expanded to some degree - encompassing the cases where only enzyme, or both enzyme and substrate undergo protolysis; pH-driven deactivation is, in turn, explored as per two alternative modes of action, i.e. reversible and irreversible. Complementary mechanisms of chemical modulation tackled entail self-deactivation of proteases, and microorganism-mediated deactivation of enzymes as proteins at large.

Specific molecules have been designed by nature to control enzyme activity - i.e. substrate itself in the so-called homologous modulation, and disparate chemical entities supporting heterologous modulation; in both cases, reversible and irreversible modes of action prove germane. Several options are available for heterologous bimodal modulation - ranging from mixed inhibition, through competitive inhibition, to uncompetitive inhibition; for its flexibility and ubiquity, mixed inhibition has received particular attention - so a wide variety of specific plots are proposed that promptly convey relevant information, viz. Michaelis and Menten's, Lineweaver and Burk's, Hanes and Woolf's, Woolf, Augustinsson, and Hofstee's, Eadie and Scatchard's, Dixon's, Cornish-Bowden's, and Hunter and Downs' plots. Heterologous unimodal modulation is, in turn, restricted to noncompetitive inhibition. Chemical modulator molecules other than substrate can also be used to discriminate molecular mechanisms - namely between sequential random Bi Bi, sequential ordered Bi Bi, and ping pong Bi Bi. While heterologous modulation may resort to a single, or more than one active/binding site, homologous modulation...