Briggs-Haldane 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. 

For a constant-volume batch reactor operated at constant T and pH, an exact solution can be obtained numerically (but not analytically) from the two-step mechanism in Section 10.2.1 for the concentrations of the four species S, E, ES, and P as functions of time t, without the assumptions of fast and slow steps. An approximate analytical solution, in the form of a rate law, can be obtained, applicable to this and other reactor types, by use of the stationary-state hypothesis (SSH). We consider these in turn. 

From material balances on S as free substrate, ES, E, and S as total substrate, we 

Elimination of cE from the first two equations using 10.2-2 results in two simultaneous first-order differential equations to solve for cs and cES as functions of t  

These can be solved numerically (e.g., with the E-Z Solve software), and the results used to obtain cE(t) and cP(t) from equations 10.2-2 and 10.2-12, respectively. 

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Michaelis-Menten and Briggs-Haldane models Enzymatic reactions can typically be represented as... 

Figure 8.1 Model energy diagrams for non-enzymic reactions (A), enzymic reaction following the rapid equilibrium mechanism (see Table 8.1) (B) and enzymic reaction following Briggs-Haldane kinetics (C). E represents the activation energy of transition and the positive and...

To develop a model for such a scheme, the first assumption we make is that a fraction of the enzyme remains attached to the substrate, that is, [AE 0. Following this, we can make either of two further assumptions a rate-determining step is present or the steady-state approximation holds for the intermediate. Depending on which set of assumptions is made, we have either of two celebrated models, the Michaelis-Menten (MM) model or the Briggs-Haldane (BH) model, although the latter appears to be favored (Chance, 1943). 

Briggs Haldane (1925) removed the restrictive assumption that the enzyme-substrate complex is in equilibrium with free enzyme and substrate and introduced the steady state model, which gives the Michaelis parameters, ATm and a more complex meaning. The principles of steady state kinetics of enzyme reactions can be demonstrated with the more realistic, though still oversimplified, model of Haldane (1930). This contains the minimum number of intermediates, namely enzyme-substrate and enzyme-product complexes ... 

A plot of the initial reaction rate, v, as a function of the substrate concentration [S], shows a hyperbolic relationship (Figure 4). As the [S] becomes very large and the enzyme is saturated with the substrate, the reaction rate will not increase indefinitely but, for a fixed amount of [E], it reaches a plateau at a limiting value named the maximal velocity (vmax). This behavior can be explained using the equilibrium model of Michaelis-Menten (1913) or the steady-state model of Briggs and Haldane (1926). The first one is based on the assumption that the rate of breakdown of the ES complex to yield the product is much slower that the dissociation of ES. This means that k2 tj. 

The kinetics of the general enzyme-catalyzed reaction (equation 10.1-1) may be simple or complex, depending upon the enzyme and substrate concentrations, the presence/absence of inhibitors and/or cofactors, and upon temperature, shear, ionic strength, and pH. The simplest form of the rate law for enzyme reactions was proposed by Henri (1902), and a mechanism was proposed by Michaelis and Menten (1913), which was later extended by Briggs and Haldane (1925). The mechanism is usually referred to as the Michaelis-Menten mechanism or model. It is a two-step mechanism, the first step being a rapid, reversible formation of an enzyme-substrate complex, ES, followed by a slow, rate-determining decomposition step to form the product and reproduce the enzyme ... 

STEADY STATE TREATMENT. While the Michaelis-Menten model requires the rapid equilibrium formation of ES complex prior to catalysis, there are many enzymes which do not exhibit such rate behavior. Accordingly, Briggs and Haldane considered the case where the enzyme and substrate obey the steady state assumption, which states that during the course of a reaction there will be a period over which the concentrations of various enzyme species will appear to be time-invariant ie., d[EX]/dr s 0). Such an assumption then provides that... 

Let s look at a mathematical model and attempt to generate curve. This was first done by Michaelis and Menten for an equilibrium model. Better is the steady state model of Haldane and Briggs (more general), which we will derive. 

A model for enzyme kinetics that has found wide applicability was proposed by Michaelis and Menten in 1913 and later modified by Briggs and Haldane. The Michaelis-Menten equation relates the initial rate of an enzyme-catalyzed reaction to the substrate concentration and to a ratio of rate constants. This equation is a rate equation,... 

The derivation mathematics are detailed in many publications dealing with enzyme kinetics. The Michaelis-Menten constant is, however, due to the individual approximation used, not always the same. The simplest values result from the implementation of the equilibrium approximation in which represents the inverse equilibrium constant (eqn (4.2(a))). A more common method is the steady-state approach for which Briggs and Haldane assumed that a steady state would be reached in which the concentration of the intermediate was constant (eqn (4.2(b))). The last important approach, which should be mentioned, is the assumption of an irreversible formation of the substrate complex [k--y = 0) (eqn (4.2(c))), which is of course very unlikely. In real enzyme reactions and even in modelled oxo-transfer reactions, this seems not to be the case. 

The first application of the QSSA is usually attributed to Bodenstein (Bodenstein 1913 Bodenstein and Lutkemeyer 1924), but Chapman and Underhill (1913) and Semenov (1939, 1943) were also early users of the technique. Further pioneers of the application of the QSSA are Michaelis and Menten (1913) and Briggs and Haldane (1925). The history of the application of the QSSA can be divided into three periods (Turanyi et al. 1993b). In the early period (1913-1960), accurate experimental data for various applications were obtained and compared with solutions of simple kinetic systems of differential equations that were formulated to model the experimental behaviour. Due to the limited availability of computer power during this time, the kinetic DDEs had to be solved analytically and using the QSSA helped to convert the systems into an analytically solvable form. 

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