Briggs equations

The mostly well-known discussion concerning multicomponent copolymerization theory is probably connected with so-called simplified equations put forward in papers [143-147] for the description of S(x) dependence. These equations can be obtained from the general Walling-Briggs equations (3.8), (4.10) via substituting in them expression a /a for Bs. The derivation of these simplified equations is based on the assumption that the rate of the monomer Mj addition to the radical R, is equal to the rate of the monomer addition to the radical Rj ... 

The Pasquill-Gifford dispersion parameters and Brigg s plume rise equations. 

The general theory of enzyme kinetics is based on the work of L. Michaelis and M. L. Menten, later extended by G. E. Briggs and J. B. S. Haldane.la The basic reactions (E = enzyme, S = substrate, P = product) are shown in equation 2.1 ... 

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 ... 

In the Briggs-Haldane derivation of the Michaelis-Menten equation, the concentration of ES is assumed to be at steady state, i.e., its rate of production [Eq. (3.12)] is exactly counterbalanced by its rate of dissociation [Eq. (3.13)]. Since the rate of formation of ES from E -(- P is vanishingly small, it is neglected. Equating the two equations and rearranging yields Eq. (3.14), where KM replaces (k2 + h)/k and is known as the Michaelis-Menten... 

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 ... 

Finally, introduction of the commonly used symbols Vmax and Km yields the Briggs-Haldane rate equation for enzymatic reactions (compare with Equation 11.6)... 

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 ... 

A mathematical equation indicating how the equilibrium constant of an enzyme-catalyzed reaction (or half-reaction in the case of so-called ping pong reaction mechanisms) is related to the various kinetic parameters for the reaction mechanism. In the Briggs-Haldane steady-state treatment of a Uni Uni reaction mechanism, the Haldane relation can be written as follows ... 

An enzyme is said to obey Michaelis-Menten kinetics, if a plot of the initial reaction rate (in which the substrate concentration is in great excess over the total enzyme concentration) versus substrate concentration(s) produces a hyperbolic curve. There should be no cooperativity apparent in the rate-saturation process, and the initial rate behavior should comply with the Michaelis-Menten equation, v = Emax[A]/(7 a + [A]), where v is the initial velocity, [A] is the initial substrate concentration, Umax is the maximum velocity, and is the dissociation constant for the substrate. A, binding to the free enzyme. The original formulation of the Michaelis-Menten treatment assumed a rapid pre-equilibrium of E and S with the central complex EX. However, the steady-state or Briggs-Haldane derivation yields an equation that is iso-... 

BRIGGS-HALDANE EQUATION STEADY-STATE ASSUMPTION ENZYME KINETICS UNI-UNI MECHANISM Bromohydroxyacetone phosphate,... 

Briggs results, as well as those of others, prove that the dotted line BC in Fig. 23 represents attainable physical conditions. Similarly the line FE is real, for it represents the condition of a vapor compressed isothermally beyond its vapor pressure (saturation pressure). Van der Waals equation8 and certain other equations of state have the mathematical form described by the line ABCDEFG. The portions BC and EF are called metastable. The question of importance in a discussion of boiling is does the portion CDE have any physical significance ... 

This expression by Briggs-Haldane is similar to Equation 3.28, obtained by the Michaelis-Menten approach, except that is equal to (Ar i -F These... 

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... 

The result of equation 3.39 for nonproductive binding is quite general. It applies to cases in which intermediates occur on the reaction pathway as well as in the nonproductive modes. For example, in equation 3.19 for the action of chy-motrypsin on esters with accumulation of an acylenzyme, it is seen from the ratios of equations 3.21 and 3.22 that kQJKM = k2IKs. This relationship clearly breaks down for the Briggs-Haldane mechanism in which the enzyme-substrate complex is not in thermodynamic equilibrium with the free enzyme and substrates. It should be borne in mind that KM might be a complex function when there are several enzyme-bound intermediates in rapid equilibrium, as in equation 3.16. Here kcJKM is a function of all the bound species. 

For most enzymes, the rate of reaction can be described by the Michaelis-Menten equation which was originally derived in 1913 by Michaelis and MENTEN 21 . Its derivation can be achieved by making one of two assumptions, one of which is a special case of the more general Briggs-Haldane scheme, whilst the alternative is the rapid-equilibrium method given in Appendix 5.3(2 ). 

With this expression for [ES] we can follow the same procedure that led to equation (18), this time arriving at the Briggs-Haldane equation for the reaction velocity. 

Significance of the Specificity Constant, kcat/Km. Under physiological conditions, enzymes usually do not operate at saturating substrate concentrations. More typically, the ratio of the substrate concentration to the Km is in the range of 0.01-1.0. If [S] is much smaller than Km, the denominator of the Briggs-Haldane equation [equation (25)] is approximately equal to Km, so that the velocity of the reaction becomes... 

Derive the rate equation by employing (a) the Michaelis-Menten and (b) the Briggs-Haldane approach. Explain when the rate equation derived by the Briggs-Haldane approach can be simplified to that derived by the Michaelis-Menten approach. 

To derive a rate equation, let s follow the Briggs-Haldane approach as explained in Chapter 2, which assumes that the change of the... 

Keen (1924) severely criticized Wilsdon s assumption that Mf is in any sense the true free moisture, as implied in the empirical equation of Briggs and Shantz. 

This is an important question that has been addressed by many enzyme kineticists over the years. For the correct application of the Briggs-Haldane steady-state analysis, in a closed system, [S]0 must be >[E](), where the > sign implies a factor of at least 1,000. M. F. Chaplin in 1981 noted that the expression v0= V max[S]c/(Km + [S]0 + [E]0) yields, for example, only a 1 percent error in the estimate of v() for [S]0 = 10 x [E]0 and [S]0 = 0.1 Km the expression thus applies under much less stringent conditions than does the simple Michaelis-Menten equation. In open systems [S]0 can approximate [E]0 and a steady state of enzyme-substrate complexes can pertain computer simulation of both types of system is the best way to gain insight into the conditions necessary for a steady state of the complex. 

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-... 

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