The Standard Quasi-Steady-State Assumption

Also known as the quasi-steady-state approximation, this condition is an important prerequisite for much of the mathematics used in deriving many of the equations that describe enzyme kinetics.

Any reaction in which a substrate and a catalyst combine to form a complex, inducing a change in the substrate to form a product, and subsequently irreversibly dissociate, can be represented with the reaction scheme below:

 
 

This reaction has a characteristic time course that consists of:

  1. a brief initial transient during which the amount of complex ES is increasing with time (usually over so quickly as to be unobservable),
  2. a period during which the reaction is proceeding at the maximum velocity that the various rates k1, k-1, and k2 will allow, taking into account also the persistence time of the complex ES, and
  3. an asymptotic approach to the final equilibrium concentrations of the various species.
 
 

During the second phase mentioned above, the kinetic parameters k1, k-1, and k2 determine the overall rate of the reaction. It therefore stands to reason that observations made during this phase could be used to measure these parameters. In practice, the time course is always curved over its entirety, with no 'linear' region corresponding to this second phase. It is accepted practice therefore to attempt to determine the initial velocity of the reaction from the time-course graph, as that velocity will most closely approximate the ideal maximum reaction velocity for that set of reaction conditions.

This hypothetical second phase has several properties which make it useful for deriving mathematical descriptions of the reaction. First, in most cases it can be safely assumed that the rate of the reaction during this phase is determined solely by k2, because this rate is generally orders of magnitude lower than k1. Second, the reaction can be viewed as being in a kind of quasi-equilibrium during this phase, because the rate of the reaction depends only on k2 and not at all on the concentrations of any of the species or any other time-dependent quantity. This equilibrium leads to the useful deduction that the rate of the reaction leading from complex to product is equal to the sum of the enzyme-substrate association and dissociation reactions. A corrollary of this equilibrium is a third property: that the reaction is proceeding at approximately its theoretical maximum rate (for the extant set of reaction conditions). This is usually interpreted to mean that there is very little free enzyme present in the system, because as much enzyme as possible is combined with substrate.

All of the properties described above are made use of in the derivation of the Michaelis-Menten equation. The slight distinction between the two expansions of the acronym 'QSSA' can now be made: the quasi-steady-state assumption is the assumption that some period of time like the second phase described above exists and can be observed in the reaction of interest, and the quasi-steady-state approximation describes the mathematics derived by assuming that the properties of this phase described in the preceeding paragraph hold.

All of these properties, and their mathematical consequences, are constrained by the mathematical relation below. As the expression on the left becomes smaller, the validity of the QSSA improves:

 
[E0]

Km+ [S0]
<<  1
[Segel, Segel & Slemrod]

The above discussion can be extended to attempt to describe systems in which some of the constraints of the standard quasi-steady-state assumption (or sQSSA) do not hold. Other related sets of assumptions that have gained some attention include

tQSSA (Borghans, et al., 1996; Tzafriri, 2003)
rQSSA (Segel and Slemrod, 1989; Schnell and Maini, 2000)