## I. 2: The Quadratic Velocity Equation for Tight-Binding Substrates

Three assumptions are implicit in
Michaelis-Menten kinetics: the **steady-state approximation**, the **free
ligand approximation** and the **rapid equilibrium approximation**. (The
Briggs-Haldane approach frees us from the last of these three.)

Now consider the second
assumption in the list: the free ligand approximation. Remember that this
approximation states that the free substrate concentration (the [*S*] in
the Michaelis-Menten equation) is close to the total substrate concentration in
the system – which is, in fact, the true independent variable in most
experimental set-ups. After all, we know how much substrate we’ve added to
reaction mixture, but we don’t know *a priori* how much of that substrate
remains free in solution at steady-state. The free ligand approximation does
not hold when a substrate’s *K*_{M} is lower than the total enzyme
concentration, because at low substrate concentrations, a significant fraction
of substrate will be bound to the enzyme.

For such unusual cases, we derive
a kinetic equation from the scheme above without resorting to the free ligand
approximation. The rates of change of the various species are given by four
differential equations:

We can assume that *ES*
levels achieve steady-state:

Given that free enzyme
concentration [*E*] equals total enzyme concentration [*E*_{T}]
minus [*ES*] and that [*S*] equals total enzyme concentration, we
have:

A little algebra gives us:

This is a quadratic equation of
the form *a*^{2}x + *bx* + *c* = 0, whose roots are
given by_{}.

Making the following substitutions
and choosing the relevant (saturable) root gives us:

This is the quadratic velocity
equation, sometimes also called the tight-binding equation or the Morrison
equation (*1*). As the *K*_{M} becomes larger than [*E*_{T}], the curve described by the quadratic equation approaches the hyperbola described by the Michaelis-Menten equation. As the *K*_{M} becomes lower, the inflection point of the curve (located where [*S*_{T}] = [*E*_{T}]) becomes progressively sharper. This is illustrated in the plot below, where the *K*_{M} varies from 5 times the total enzyme concentration ([*E*_{T}] = 1 μM) to one-hundredth of [*E*_{T}]. Differences in affinity between very tight-binding substrates are reflected in the sharpness of the inflection, meaning that *K*_{M}
values recovered from fits to experimental data are disproportionately
sensitive to error in the data points surrounding the inflection point. As a
rule of thumb, the quadratic equation should be used in preference to the
Michaelis-Menten equation whenever the *K*_{M} is less than
five-fold larger than [*E*_{T}].

Derivations of this type that do
not use the free ligand approximation grow rapidly more complicated for more
complex kinetic systems. For anything more complicated than a one-site model,
it is better to use kinetic simulations as detailed in part II of this guide.

- Morrison, J. F. (1969) Kinetics of the reversible inhibition of enzyme-catalysed reactions by
tight-binding inhibitors,
*Biochimica et Biophysica Acta (BBA) - Enzymology*
*185*, 269-286.

Next: Multiple Binding (Sequential Models)