Briggs and Haldane

Briggs and Haldane [8] proposed a general mathematieal deseription of enzymatie kinetie reaetion. Their model is based on the assumption that after a short initial startup period, the eoneentration of the enzyme-substrate eomplex is in a pseudo-steady state (PSS). Eor a eonstant volume bateh reaetor operated at eonstant temperature T, and pH, the rate expressions and material balanees on S, E, ES, and P are... 

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... 

Subsequently Briggs and Haldane (1925) demonstrated that a similar treatment could be used to describe steady state enzyme velocity as a saturable function of substrate concentration ... 

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 ... 

Today, this synonym is used for the more common steady-state approach by Briggs and Haldane (see [17]). 

Henri and Michaelis-Menten kinetics assumed that the rate of formation of products was much less than that for the back reaction from ES to yield E + S. Van Slyke assumed the reverse. A more rigorous formulation was offered by Briggs and Haldane (1925) using steady-state assumptions previously applied to chemical kinetics by Bodenstein (1913). 

In Equation 11.9 we reserve the missing rate constant k4 for an elaboration of the mechanism). Following Briggs and Haldane we make the assumption that the steady-state approximation applies to ES and EP complexes ... 

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... 

STEADY-STATE TREATMENT. During the steady state, the concentrations of various enzyme intermediates are essentially unchanged that is, the rate of formation of a given intermediate is equal to its rate of disappearance. This assumption was first introduced to the derivation of enzyme kinetic equations by Briggs and Haldane ... 

When the enzyme is first mixed with a large excess of substrate, there is an initial period, the pre-steady state, during which the concentration of ES builds up. This period is usually too short to be easily observed, lasting just microseconds. The reaction quickly achieves a steady state in which [ES] (and the concentrations of any other intermediates) remains approximately constant over time. The concept of a steady state was introduced by G. E. Briggs and Haldane in 1925. The measured V0 generally reflects the steady state, even though V0 is limited to the early part of the reaction, and analysis of these initial rates is referred to as steady-state kinetics. 

Here we develop the basic logic and the algebraic steps in a modern derivation of the Michaelis-Menten equation, which includes the steady-state assumption introduced by Briggs and Haldane. The derivation starts with the two basic steps of the formation and breakdown of ES (Eqns 6-7 and 6-8). Early in the reaction, the concentration of the product, [P], is negligible, and we make the simplifying assumption that the reverse reaction, P—>S (described by k 2), can be ignored. This assumption is not critical but it simplifies our task. The overall reaction then reduces to... 

This kinetic scheme, of fundamental importance to biochemistry, was first analyzed correctly by Briggs and Haldane. It is often referred to in physicochemical circles as Langmuir-Hinschelwood kinetics and chemical engineers encounter it as Hougen-Watson. In spite of the injustice it does to Briggs and Haldane, it seems best to leave it as Michaelis-Menten. Certainly, BrH3LaM2W is an impossible compound. 

It would appear, therefore, that the view of the Michaelis-Menten situation envisaged by Briggs and Haldane is a special case in which kf2 is so much smaller than Arl that the value of the ratio is negligible compared with K s> and ... 

Briggs-Haldane approach (Briggs and Haldane, 1925) The change of the intermediate concentration with respect to time is assumed to be negligible, that is, d(CES)/dt = 0. This is also known as the pseudo-steady-state (or quasi-steady-state I assumption in chemical kinetics and is often used in developing rate expressions in homogeneous catalytic reactions. 

A reactant in an enzyme catalysed reaction is known as substrate. According to the mechanism of enzyme catalysis, the enzyme combines with the substrate to form a complex, as suggested by Henri (1903). He also suggested that this complex remains in equilibrium with the enzyme and the substrate. Later on in 1925, Briggs and Haldane showed that a steady state treatment could be easily applied to the kinetics of enzymes. Some photochemical reactions and some enzymic reactions are reactions of the zero order. 

In 1913, Michaelis and Men ten presented a general theory for enzyme kinetics, extended later by Briggs and Haldane, which accounts for the velocity curve shown in Figure 5.5. This theory for reactions catalyzed by enzymes having a single substrate assumes that the substrate S binds to the active site of the enzyme E to form the enzyme-substrate complex ES, which yields the product P and the free enzyme E ... 

Briggs and Haldane in 1925 examined the earlier Michaelis-Menten analysis and made an important development. Instead of assuming that the first stage of the reaction was at equilibrium, they merely assumed, for all intents and purposes, that the concentration of the enzyme-substrate complex scarcely changed with time i.e., it was in a steady state. Written mathematically, this amounts to... 

By using the steady-state analysis of Briggs and Haldane, it is possible to show that... 

Michaelis and Menten, and later Briggs and Haldane, used the scheme shown in Equation II-4 to derive a mathematical expression that describes the relation between initial velocity and substrate concentration. (Consult a biochemistry textbook for the step-by-step derivation of this relationship, because it is important to be aware of the assump-... 

The flux expression in Equation (4.16) displays the canonical Michaelis-Menten hyperbolic dependence on substrate concentration [S], We have shown that this dependence can be obtained from either rapid pre-equilibration or the assumption that [S] [E]. The rapid pre-equilibrium approximation was the basis of Michaelis and Menten s original 1913 work on the subject [140], In 1925 Briggs and Haldane [24] introduced the quasi-steady approximation, which follows from [S] 2> [E], (In his text on enzyme kinetics [35], Cornish-Bowden provides a brief historical account of the development of this famous equation, including outlines of the contributions of Henri [80, 81], Van Slyke and Cullen [203], and others, as well as those of Michaelis and Menten, and Briggs and Haldane.)... 

This relationship can be derived As Briggs and Haldane first contrived The unbound enzyme, [ ], we guess Is Eo (total), less [AA]. 

As first pointed out by Briggs and Haldane [30], the assumption of quasiequilibrium in the first step is inconveniently restrictive. They relaxed that postulate by replacing the quasi-equilibrium condition 8.15 with the Bodenstein approximation for the trace intermediate X ... 

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,... 

In general, such nonlinear differential equations are difficult to solve exactly. Therefore, Michaelis and Menten (1913) made the simplifying assumption that the intermediate complex is in equilibrium with the free enzyme and substrate (the quasi-equilibrium postulate). While this is not strictly true, it is nearly so when k2 is much less than k i or ki, as was the case for Michaelis and Menten, who were concerned with the enzyme invertase. More often valid is the quasi-steady state postulate, which was first developed in detail by Briggs and Haldane (1925). 

Now an important observation can be done (Briggs and Haldane, 1925) Enzymes are so efficient that a very small amount is enough to catalyze the reaction. Thus, except at very late times when the substrate is nearly exhausted, we have that E, C, and Et are small parameters when compared to S, and the second enzyme dynamics equation, (3.34), evolves in a time scale much faster than the first. This situation of separation of time scales allows the use of the adiabatic approximation, or adiabatic elimination, a useful tool which will be used several times in the following. The method applies to systems of equations of the form... 

In the present context, the adiabatic elimination leading to (3.38) is called the steady state hypothesis and was introduced by Briggs and Haldane (1925). In the original treatment by Michaelis and Menten (1913) an assumption of equilibrium for the first reaction in (3.27) was made, which leads to the same result (3.39) but with a different expression for Km- The steady state hypothesis can be justified rigourously and is precise as long as Et < Km + S(0) (Segel and Slemrod, 1989). 

Note that [A] is assumed to be so large relative to enzyme concentration that it is negligibly decreased by formation of EA. Briggs and Haldane [4] introduced the less restrictive steady-state assumption. Here it is postulated only that the rates of formation and removal of EA through whatever route are equal. This avoids arbitrary assumptions about the relative values of A 2 and A , . Thus now ... 

In the following derivation we will apply the concept of steady state approximation, which was introduced to enzymatic catalysis by Briggs and Haldane (1925), who had proposed that the rate of formation of ES = ki [E][S] balances the rate of breakdown of the complex ES = (k i + k2)[ES], or in other words (Figure 6.2) d(ES)/dt = 0... 

The classical reaction-path of Michaelis and Menten (8) and of Briggs and Haldane (5) postulates reaction of enzyme E and substrate S to give an enzyme-substrate complex ES, which decomposes to the products ... 

Using the steady-state analysis of Briggs and Haldane (Sec. 5.10) conpled with the conservation of mass equation that is extended from Eq. (5.13) to include the El complex. 

A few years later, Briggs and Haldane (1925) argued against the validity of the rapid equilibrium hypothesis and proposed a steady-state hypothesis according to which, after a very short transient phase, the ES complex remains constant throughout the whole reaction period, as shown in Fig. 3.2. 

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. 

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