Michaelis-Menten: dynamics constant

This question was studied by Hsu [HsulJ in the chemostat with Michaelis-Menten dynamics, and his work is presented here. The equations take the form (ignoring the yield constants)... 

Pharmacokinetic studies are in general less variable than pharmacodynamic studies. This is so since simpler dynamics are associated with pharmacokinetic processes. According to van Rossum and de Bie [234], the phase space of a pharmacokinetic system is dominated by a point attractor since the drug leaves the body, i.e., the plasma drug concentration tends to zero. Even when the system is as simple as that, tools from dynamic systems theory are still useful. When a system has only one variable a plot referred to as a phase plane can be used to study its behavior. The phase plane is constructed by plotting the variable against its derivative. The most classical, quoted even in textbooks, phase plane is the c (f) vs. c (t) plot of the ubiquitous Michaelis-Menten kinetics. In the pharmaceutical literature the phase plane plot has been used by Dokoumetzidis and Macheras [235] for the discernment of absorption kinetics, Figure 6.21. The same type of plot has been used for the estimation of the elimination rate constant [236]. 

Thus, Kn, the Michaelis constant, is a dynamic or pseudo-equilibrium constant expressing the relationship between the actual steady-state concentrations, rather than the equilibrium.concentrations. If Aj, is very small compared to A-i, reduces to K. A steady-state treatment of the more realistic reaction sequence E+ S ES EP E + P yields the same final velocity equation although now Km is a more complex function, composed of the rate constants of all the steps. Thus, the physical significance of K cannot be stated with any certainty in the absence of other data concerning the relative magnitudes of the various rate constants. Nevertheless, represents a valuable constant that relates the velocity of an enzyme-catalyzed reaction to the substrate concentration. Inspection of the Henri-Michaelis-Menten equation shows that Km is equivalent to the substrate concentration that yields half-maximal velocity ... 

Autoregulatory action helps to reduce nerve cell destruction resulting from brain tissue anoxia. Two possible mechanisms include flow controller dynamics in the form of pure delays and time constant lags and oxygen consumption control with Michaelis-Menten behavior. Response curves also suggest the possibility of facilitated or active transport of oxygen in tissue and resistance to the diffusion of oxygen from the tissue into the blood stream. 

UV/Vis spectroscopy is especially useful in the examination of dynamic processes, represented by Michaelis-Menten kinetics, complexation reactions, and observation of saturation effects. The reason is its fast response time and the easy quantitative determination even in multicomponent systems. Thus, diffusion-controlled processes in biological interaction problems or the determination of binding constants are a typical application [37,38]. 

The Michaelis-Menten kinetics (75) provide a facile estimate of the altered reaction dynamics and the energetics of the nanoconfined enzyme systems. The well-known Michaelis constant Km ( k i/ki) measures the dissociation of the enzyme-substrate complex and in turn serves as an estimate of its stability. An increased value of the Michaelis constant implies that the equilibrium of the enzyme substrate complex is shifted towards the left i.e. towards the free enzyme and substrate, and suggests a relatively weaker enzyme-substrate complex. Another parameter, kcat ( k2), called the turnover number, estimates the rate of formation of the product from Ae enzyme-substrate complex. A facilitated product formation will result in increased turnover number (i.e. increased k<. The ratio kcat/ m consequently, represents the apparent rate constant for combination of a substrate with the free enzyme. The... 

The interpretations of Michaelis and Menten were refined and extended in 1925 by Briggs and Haldane, by assuming the concentration of the enzyme-substrate complex ES quickly reaches a constant value in such a dynamic system. That is, ES is formed as rapidly from E + S as it disappears by its two possible fates dissociation to regenerate E + S, and reaction to form E + P. This assumption is termed the steady-state assumption and is expressed as... 

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