The majority of the measurements we perform in the stopped-flow apparatus concern protein-ligand interactions. In order to fully understand the measurements, it is important to have an idea of the conditions used and how this relates to the kinetics. This is applicable in the reactions carried out under pseudo-first order conditions. The basis of these conditions is defined below using adaptations from Eccleston et al.
Firstly, let’s consider a simple reversible first order reaction:
The reaction consists of unimolecular processes with first order rate constants k+1 and k-1.
The rate d[P*]/dt is shown below:
The concentrations of P* can be expressed as a function of time:
P*0 is the initial concentration of P* while P*eq is the equilibrium (or final) concentration of P. kOBS = k+1 + k-1. The larger the kOBS, the more rapid the system will equilibrate and this is the process that the stopped-flow measures. It is also important to note that a reaction with the same k+1 but with different k-1 will equilibrate at different rates, with the reaction containing the largest k-1 equilibrating the fastest.
Many reactions focus upon protein binding to a ligand which leads to the formation of a protein-ligand (PL) complex. These reactions are termed second order, as shown in:
The reaction consists of a bimolecular binding process k+1 (M-1 s-1) and a unimolecular dissociation process k-1 (s-1). The Kd of this reaction is defined as k-1 / k+1 (M). The rate of d[PL] / dt is shown below:
There is no simple analytical solution to this equation, as done for the first order reactions. This is because the concentration of both P and L varies in the reaction. However, if P or L is in a far larger excess, then, when the PL complex forms, the concentration of the excess component is essentially unchanged. So, in the case of excess L, LTOTAL – [PL] =~ LTOTAL. This is termed pseudo-first order conditions. As a result, the formation of PL is in the same form as the first-order reaction, with kOBS = k+1[LTOTAL] + k-1, for the example of excess ligand.