Enzyme kinetics models

One example of such a course is the Haverford Laboratory in Chemical Structure and Reactivity (146) that includes six projects, each of which involves sample preparation, sample analysis and some kind of determination of the properties of the substance prepared. The projects include organopalladium chemistry, porphyrin photochemistry, enantioselective synthesis, computer-aided modeling, enzyme kinetics and electron transfer reactions. 

Often the key entity one is interested in obtaining in modeling enzyme kinetics is the analytical expression for the turnover flux in quasi-steady state. Equations (4.12) and (4.38) are examples. These expressions are sometimes called Michaelis-Menten rate laws. Such expressions can be used in simulation of cellular biochemical systems, as is the subject of Chapters 5, 6, and 7 of this book. However, one must keep in mind that, as we have seen, these rates represent approximations that result from simplifications of the kinetic mechanisms. We typically use the approximate Michaelis-Menten-type flux expressions rather than the full system of equations in simulations for several reasons. First, often the quasi-steady rate constants (such as Ks and K in Equation (4.38)) are available from experimental data while the mass-action rate constants (k+i, k-i, etc.) are not. In fact, it is possible for different enzymes with different detailed mechanisms to yield the same Michaelis-Menten rate expression, as we shall see below. Second, in metabolic reaction networks (for example), reactions operate near steady state in vivo. Kinetic transitions from one in vivo steady state to another may not involve the sort of extreme shifts in enzyme binding that have been illustrated in Figure 4.7. Therefore the quasi-steady approximation (or equivalently the approximation of rapid enzyme turnover) tends to be reasonable for the simulation of in vivo systems. 

Testing Models Enzyme Kinetic Isotope Effects... 

Experimentally determined rate constants for various micellar-mediated reactions show either a monotonic decrease (i.e., micellar rate inhibition) or increase (i.e., micellar rate acceleration) with increase in [Suifl CMC, where [Surf]T represents total micelle-forming surfactant concentration (Figure 3.1). Menger and Portnoy obtained rate constants — [SurfJx plots — for hydrolysis of a few esters in the presence of anionic and cationic surfactants, which are almost similar to those plots shown in Figure 3.1. These authors explained their observations in terms of a proposed reaction mechanism as shown in Scheme 3.1 which is now called Menger s phase-separation model, enzyme-kinetic-type model, or preequilibrium kinetic (PEK) model for micellar-mediated reactions. In Scheme 3.1, Kj is the equilibrium... 

The relative fluctuations in Monte Carlo simulations are of the order of magnitude where N is the total number of molecules in the simulation. The observed error in kinetic simulations is about 1-2% when lO molecules are used. In the computer calculations described by Schaad, the grids of the technique shown here are replaced by computer memory, so the capacity of the memory is one limit on the maximum number of molecules. Other programs for stochastic simulation make use of different routes of calculation, and the number of molecules is not a limitation. Enzyme kinetics and very complex oscillatory reactions have been modeled. These simulations are valuable for establishing whether a postulated kinetic scheme is reasonable, for examining the appearance of extrema or induction periods, applicability of the steady-state approximation, and so on. Even the manual method is useful for such purposes. 

CASE STUDY ENZYME KINETIC MODELS FOR RESOLUTION OF RACEMIC IBUPROFEN ESTERS IN A MEMBRANE REACTOR... 

Substrate and product inhibitions analyses involved considerations of competitive, uncompetitive, non-competitive and mixed inhibition models. The kinetic studies of the enantiomeric hydrolysis reaction in the membrane reactor included inhibition effects by substrate (ibuprofen ester) and product (2-ethoxyethanol) while varying substrate concentration (5-50 mmol-I ). The initial reaction rate obtained from experimental data was used in the primary (Hanes-Woolf plot) and secondary plots (1/Vmax versus inhibitor concentration), which gave estimates of substrate inhibition (K[s) and product inhibition constants (A jp). The inhibitor constant (K[s or K[v) is a measure of enzyme-inhibitor affinity. It is the dissociation constant of the enzyme-inhibitor complex. 

The above rate equation is in agreement with that reported by Madhav and Ching [3]. Tliis rapid equilibrium treatment is a simple approach that allows the transformations of all complexes in terms of [E, [5], Kls and Kjp, which only deal with equilibrium expressions for the binding of the substrate to the enzyme. In the absence of inhibition, the enzyme kinetics are reduced to the simplest Michaelis-Menten model, as shown in Figure 5.21. The rate equation for the Michaelis-Menten model is given in ordinary textbooks and is as follows 11... 

Enzyme Kinetics and Stability Kinetic studies, carried out mostly with hydrolases, have shovm that enzymes in organic solvents follow conventional models [12a, 22]. 

Optimization of Substrate Concentrations. Computer analyses of enzyme kinetics may be very useful for the calculation of enzyme constants, eliminating the tediim associated with manual calculations. Recently, computer models for optimizing reagent concentrations have been described but these models require so many experimental points that the model rests on the experimental data rather than having predictive usefulness (31). 

This system displays a two-enzyme kinetic model in which bioconversion is controlled by the interaction between the two reactions and the mass transfer. This situation offers a more realistic model for the conditions occurring in vivo, in which some pathways of intermediary metabolism consist of linear sequences of reactions. These pathways take place in highly organized compartments. 

In this section, we will discuss some examples from the literature, in which the approximation methods derived in this chapter have been used. In several cases, the approximations have been compared with more-accurate path integral simulations to assess their validity. This is not meant as a full review rather, several case studies have been chosen to illustrate the tools we have developed. We will first look at simpler examples and then discuss water models and applications in enzyme kinetics. 

K. Heincke, B. Demuth, H. J. Joerdening, and K. Buchholz, Kinetics of the dextransucrase acceptor reaction with maltose Experimental results and modeling, Enzyme Microb. Technol., 24 (1999) 523-534. 

The minimum value of /Jdf/v required for a reliable model depends on the quality of the determination of the data to be correlated. The smaller the experimental error in the data, the smaller the value of /Jdf/v required for dependable results. Experience indicates that in the case of chemical reactivity data /Jdf/v should be not less than 3. For bioactivity studies /Jdf/v depends heavily on the type of data for rate and equilibrium constants obtained from enzyme kinetics a value of not less than 3 is reasonable while for toxicity studies on mammals at least 7 is required. 

It is in principle possible for a free enzyme to promote reaction in a geochemical system, but enzyme kinetics are invoked in geochemical modeling most commonly to describe the effect of microbial metabolism. Microbes are sometimes described from a geochemical perspective as self-replicating enzymes. This is of course a considerable simplification of reality, as we will discuss in the following chapter (Chapter 18), since even the simplest metabolic pathway involves a series of enzymes. 

MODELS OF ENZYME KINETICS 10.2.1 Michaelis-Menten Model... 

Briggs and Haldane (1925) proposed an alternative mathematical description of enzyme kinetics which has proved to be more general. The Briggs-Haldane model is based upon the assumption that, after a short initial startup period, the concentration of the enzyme-substrate complex is in a pseudo-steady state. Derivation of the model is based upon material balances written for each of the four species S, E, ES, and P. 

The growth rate of cells is taken proportional to the cell concentration, x, and to an empirical form of the dependence on the concentration, p, of the nutrient. That empirical form was assumed by Monod (1942) to be the same as in the Michaelis-Mewnten model for enzyme kinetics. This makes the rate of cell growth,... 

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