NMR titrations are often used to demonstrate binding, locate active sites, and measure affinities. By measuring changes in chemical shifts or linewidths as the concentration of one of the binding partners is changed, quantitative data, such as dissociation constants and dissociation rates, may be obtained. In most cases the systems studied involve 1:1 binding. Systems with multiple binding sites are more difficult to analyse but, with some simplifying assumptions, quantitative information, such as stoichiometry, may still be obtained.
For a 1:1 interaction between a macromolecule, M, and a ligand, L, the equilibrium can be written as
with the dissociation constant, KD, the concentration of products over reactants for the dissociation reaction. In the case of two ligands binding to a single macromolecule the two sites may have different properties and the equilibrium now involves four different dissociation constants.
Expressing the binding as a series of steps in which one ligand binds at a time creates a dissociation constant for each step. If we wanted to describe a system with even higher stoichiometry the situation would be more complex still. To handle these situations some simplifying assumptions are required. Binding of an indeterminate number of ligands to a single macromolecule could be written as
In this formulation, all the ligand molecules simultaneously bind to, or dissociate from, the macromolecule, a highly unlikely event. Furthermore, this simplified formulation assumes; (1) that the affinity of each site is identical; (2) that the binding of one ligand does not affect the binding of another, i.e. the site and order of binding is unimportant; and (3) that all the individual events can be encompassed by a single, general, dissociation constant. While these are rather broad assumptions, they do provide a framework to start to understand and analyse the system. Under the simplified formulation the bound population of the ligand, Pb, is given by1
These equations give us a way to relate KD to Pb. For a system in fast exchange the chemical shift is the average of the free and bound chemical shifts weighted by Pb. By recording the chemical shift at different known ligand or macromolecule concentrations, KD can be determined.
The equations also allow us to simulate the behaviour of peaks in systems with a variety of affinities and stoichiometries. The figure below shows the impact of KD and n on chemical shift. The graph plots the change in chemical shift in Hz, Δν, against the ratio of ligand to macromolecule concentrations. Titrating a macromolecule with a ligand moves to the right along the horizontal axis with each addition of ligand. The vertical axis is the difference between the observed chemical shift and the chemical shift of the free ligand, so when [L]:[M] is near zero the curve approaches the bound chemical shift, and when [L]:[M] is large the observed chemical shift approaches the free chemical shift. The curves were calculated using a macromolecule concentration of 100 μM and a difference between free and bound chemical shifts of 50 Hz.
Looking first at the solid lines that were all calculated using KD=10-8 M, we see that the observed chemical shift remains at the bound chemical shift until [L]:[M] reaches the stoichiometry of the system. In these tight binding systems the observed chemical shift does not change until all the binding sites are occupied because nearly all the ligand is in the bound state. Once there is more ligand than binding sites, then the chemical shift starts to move towards the free chemical shift. The higher the stoichiometry of the system, the shallower the curve and the slower the movement.
The dashed lines were calculated using increased KD values, 10-6 M for n=1 (blue), 10-5 M for n=2 (red), and 10-4 M for n=5 (green). As KD increases (blue to red to green), the observed chemical shift moves away from the bound chemical shift, even at sub-stoichiometric levels. This is because the binding is not tight enough to keep all the ligand in the bound state.
Fitting these curves to experimental data can provide the stoichiometry and affinity of a system, however, it is important to sample as wide a range of [L]:[M] as possible. At high values of [L]:[M] there is little difference in the simulated curves. Only when data is available at the stoichiometric point, and lower [L]:[M] values, can the data be fitted properly.
References
1. Fielding L.
NMR methods for the determination of protein–ligand dissociation constants
Prog NMR Spec. 2007;51(4):219-242
Brendan very useful. We have to discard cooperativity and maintaining symmetry in an oligomeric protein, since as you allude to, they may create other complications. Palmer
ReplyDeleteThanks for the comment Palmer. I think this treatment is a first approximation through which you can determine the stoichiometry. Beyond that, more complex methods are needed to obtain any quantitative results.
Delete