Mass action expression, mechanism

Current understanding of the mechanism by which calmodulin induces enzyme activation has been discussed in some detail previously (12). All enzymes known to be activated by calmodulin (Table II) with the exception of phosphorylase t> kinase (48), readily dissociate from the protein on anion exchange columns when chromatographed with Ca + chelators. Regulation of dissociable enzymes by calmodulin (C) is generally believed to occur through two sequential, fully reversible mass action expressions ... 

For part (a), assume that the system is at equilibrium and that the law of mass action holds. Use the procedures described in Chapter 1 to derive an expression forpAB, at equilibrium. At equilibrium, the forward and backward rates for each reaction in the mechanism must be equal. The forward and backward rates are defined using the law of mass action ... 

Despite its limitations, the reversible Random Bi-Bi Mechanism Eq. (46) will serve as a proxy for more complex rate equations in the following. In particular, we assume that most rate functions of complex enzyme-kinetic mechanisms can be expressed by a generalized mass-action rate law of the form... 

The kinetic behavior of the reductive dissolution mechanisms given in Figure 2 can be found by applying the Principle of Mass Action to the elementary reaction steps. The rate expression for precursor complex formation via an inner-sphere mechanism is given by ... 

Let us note that in eqn. (59) the expressions f+ (c) and f (c) are the kinetic dependences that are written according to the law of mass action for the "natural brutto-reaction, i.e. for the reaction obtained by a simple addition of all cycle steps, and K fT) is the equilibrium constant for this reaction. However, as we mentioned above for the reaction of catalytic isomerization, the "natural brutto-equation should not necessarily have integer-valued coefficients. For the mechanism... 

The kinetics of the reductive dissolution mechanisms shown in Fig. 8.1 can be derived using the principle of mass action. The kinetic expression for precursor complex formation by way of an inner-sphere mechanism (Stone, 1986) is... 

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. 

In order to formulate a mechanism of action of phytate on zinc homeostasis, certain conditions must be fulfilled 1) The process must occur In the gastrointestinal tract since phytate Is not absorbed except for small amounts by birds 2) calcium must be a tertiary component in the total process but there must be some reaction without excessive amounts of calcium 3) certain chelating compounds such as EDTA must be capable of competing with the process and make some zinc available for absorption or reabsorption and 4) There must be some explanation for the data which indicates that 40% or more of the dietary pool may be available for absorption (, 49). All these conditions are satisfied by the following formula which is an expression of the "Law of Mass Action" ... 

Mechanisms for most chemical processes involve two or more elementary reactions. Our goal is to determine concentrations of reactants, intermediates, and products as a function of time. In order to do this, we must know the rate constants for all pertinent elementary reactions. The principle of mass action is used to write differential equations expressing rates of change for each chemical involved in the process. These differential equations are then integrated with the help of stoichiometric relationships and an appropriate set of boundary conditions (e.g., initial concentrations). For simple cases, analytical solutions are readily obtained. Complex sets of elementary reactions may require numerical solutions. 

Regardless of whether this reaction occurs via a single-step or a multistep mechanism, we can write the eqnilibrinm constant expression according to the law of mass action shown in Equation (14.2) ... 

The reaction constant k, fixing the time scale, may depend on external conditions such as temperature. Expression (3.3) is commonly called the law of mass action, although this name is also given to an important relationship among product and reactant concentrations at chemical equilibrium. When the molecular mechanism leading to the reaction (3.1) involves intermediate molecular steps, expression... 

The reaction mechanism expressed by the steps (6-8) will be indicated as (Ml). Its reaction rates have been modelled by supposing that all three reactions are irreversible and described by the mass action law. In particular ... 

The rate V of the reaction is now expressed by the current treatment, in terms of the mass action law, as proportional to 0(H2)0(CO2) for the Langmuir-Hinshelwood mechanism and similarly to (7 0(CO2) or C ° 0(H2) for the Rideal-Eley mechanism, depending on the premised adsorption state of the initial system. The rate law is thus obtained according to Eqs. (11.24) as... 

Fig. 4.1 Schematic of reaction mechanism with elementary steps given in eq. (4.1) and mass action rate expressions in eq. (4.2).

The Hougen-Watson rate equations go further than the mass action kinetic equations in that they account exphcitly for the interaction of the reacting species with the catalyst sites, but as to the mechanism they don t go very far beyond what is expressed by the stoichiometric equation, hi Sections 2.4.2 and 2.4.3 on the other hand, the reaction was decomposed in elementary steps. This is now illustrated by means of an example. 

It is obvious to the user at this juncture that the subject of environmental chemical fate models enjoys many individual mass transfer processes. Besides this, the flux equations used for the various individual processes are often based on different concentrations such as Ca, Cw, Cs, and so on. Since concentration is a state variable in all EC models, the transport coefficients and concentrations must be compatible. Several concentrations are used because the easily measured ones are the logical mass-action rate drivers for these first-order kinetic mechanisms. Unfortunately, the result is a diverse set of flux equations containing various mechanism-oriented rate parameters and three or more media concentrations. Complications arise because the individual process parameters are based on a specific concentration or concentration difference. As argued in Chapter 3, the fiigacity approach is much simpler. Conversions to an alternative but equivalent media chemical concentration are performed using the appropriate thermodynamic equilibrium statement or equivalent phase partition coefficients. The process was demonstrated above in obtaining the overall deposition velocity Equation 4.9. In this regard, the key purpose of Table 4.2 is to provide the user with the appropriate transport rate constant compatible with the concentration chosen to express the flux. Eor each interface, there are two choices of concentration... 

The last equation is the overall chemical equation, which includes only reactants and reaction products. Other equations in the mechanism above represent chemical equations of elementary reactions comprising this complex reaction and include other species that do not appear in the overall equations. For the purpose of Idnetic analysis of a complex reaction, its elementary reactions are grouped into stages (or steps). A step represents a pair (forward and reverse) of elementary reactions or in the case of irreversible reactions, it consists of only one elementary reaction, for which the kinetics is expressed direcdy by the mass action law. Sometimes several elementary reactions are grouped into one more complex reaction, which is possible if the rates of these elementary reactions are sufficiently large compared to the rate of the complex reaction as a whole. 

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