In regard to label-free assays developed to measured interactions between some target molecule and a surface, I often see the Langmuir isotherm/1:1 binding model applied to systems without quantitative knowledge of the number of available binding sites on a sensor surface. For example, let's say there were 1000 available binding sites on a surface, but I had not knowledge of this. I then add the analyte at increasing concentrations, and measure some increasing response, I(t). Overtime, this system reaches saturation at some maximum signal, I_max. Based on the 1:1 model, and the relationship between that signal and the concentration, I compute on and off rates for the interaction.
My question is this: does the 1:1 model implicitly assume that all 1000 binding sites are filled up? In other words, what if only 600 sites were filled instead of 1000? Or maybe the 1:1 is not appropriate, and actually 3000 molecules have been adsorbed? In these cases, I'm still going to measure some values for I(t) and Imax, so would this lead to errant values of k_on/k_off rates? Can I know if this is happening (eg 1:1 binding at saturation is not occurring)? Presumably, I'd over-estimate or under-estimate k_on/k_off, but without some reference value in the literature, would there be anyway to know this was happening?
The number of ligand binding sites per receptor molecule is an intrinsic property of the receptor. The intrinsic binding stoichiometry doesn't have to be 1:1, however. For example, there might be one binding site per dimer. Also, cooperative binding may occur so that multiple binding sites exist but have different affinities for the ligand. To add to the complexity, a sample of the receptor may contain correctly folded and assembled protein as well as aggregated or damaged protein that doesn't bind the ligand. Some of the potential ligand binding sites may be occupied already by something else. The concentration of the receptor may be inaccurately determined. In the case of surface immobilization of the receptor, the amount of receptor present may be unknown, and its concentration can't really be stated because it is not in solution.
Thus, when you titrate in the ligand to a sample of receptor, if you see a simple binding isotherm, it doesn't necessarily demonstrate that there is a 1:1 binding stoichiometry. If the data is of sufficient resolution, however, it may be possible to tell whether there is only one class of binding sites (i.e. all the binding sites have approximately the same Kd even if there are multiple binding sites/receptor) or more than one class with substantially different Kds, or cooperative binding sites.
Two of the best ways, besides obtaining a structure by x-ray crystallography or NMR, to actually measure the binding stoichiometry involve solution-based interactions: isothermal titration calorimetry and equilibrium dialysis. Accurate knowledge of the receptor and ligand concentrations is essential to get this quantitative information.
Interestingly, it is not necessary to have an accurate measurement of the receptor concentration to get an accurate measurement of the Kd using the Langmuir isotherm when titrating the ligand, as long as the Kd is substantially higher than the receptor concentration (for solution-based measurements) so that the total and free ligand concentrations are essentially identical. For immobilized receptor in a flow system, the bound ligand is replenished by flow, so it is not depleted by binding and the Kd can be measured accurately even for high affinity ligands, without being able to state the receptor concentration because it is not in solution.
For immobilized receptor in a microfluidic system, various technical aspects complicate the measurements, such as mass transfer rates of ligand from the fluid to the surface. These are problems that demand attention if accurate Kds are to be determined from on and off rate constants.
The number of ligand binding sites per receptor molecule is an intrinsic property of the receptor. The intrinsic binding stoichiometry doesn't have to be 1:1, however. For example, there might be one binding site per dimer. Also, cooperative binding may occur so that multiple binding sites exist but have different affinities for the ligand. To add to the complexity, a sample of the receptor may contain correctly folded and assembled protein as well as aggregated or damaged protein that doesn't bind the ligand. Some of the potential ligand binding sites may be occupied already by something else. The concentration of the receptor may be inaccurately determined. In the case of surface immobilization of the receptor, the amount of receptor present may be unknown, and its concentration can't really be stated because it is not in solution.
Thus, when you titrate in the ligand to a sample of receptor, if you see a simple binding isotherm, it doesn't necessarily demonstrate that there is a 1:1 binding stoichiometry. If the data is of sufficient resolution, however, it may be possible to tell whether there is only one class of binding sites (i.e. all the binding sites have approximately the same Kd even if there are multiple binding sites/receptor) or more than one class with substantially different Kds, or cooperative binding sites.
Two of the best ways, besides obtaining a structure by x-ray crystallography or NMR, to actually measure the binding stoichiometry involve solution-based interactions: isothermal titration calorimetry and equilibrium dialysis. Accurate knowledge of the receptor and ligand concentrations is essential to get this quantitative information.
Interestingly, it is not necessary to have an accurate measurement of the receptor concentration to get an accurate measurement of the Kd using the Langmuir isotherm when titrating the ligand, as long as the Kd is substantially higher than the receptor concentration (for solution-based measurements) so that the total and free ligand concentrations are essentially identical. For immobilized receptor in a flow system, the bound ligand is replenished by flow, so it is not depleted by binding and the Kd can be measured accurately even for high affinity ligands, without being able to state the receptor concentration because it is not in solution.
For immobilized receptor in a microfluidic system, various technical aspects complicate the measurements, such as mass transfer rates of ligand from the fluid to the surface. These are problems that demand attention if accurate Kds are to be determined from on and off rate constants.
Hi Adam,
The simple answer is no, you do not need 100% occupation.
However, it is a very interesting question if you stretch it. In biophysical/biochemical assays you assume that all binding sites are identically, which is a valid approximation. However, in cell based assays, the binding sites are more individual depending on things such as clustering, accessibility etc. We see for instance that the kinetics and affinity in some instance changes with concentration. At low concentrations the most accessible site are the one that interacts and as those get saturated with increasing concentrations the less accessible site becomes relatively more important.
There is an excellent papper by Karl Andersson and co workers who has made interaction maps out of this.
Deciphering complex protein interaction kinetics using Interaction Map
Biochemical and Biophysical Research Communications, Volume 428, Issue 1, 9 November 2012, Pages 74-79
Danièle Altschuh, Hanna Björkelund, John Strandgård, Laurence Choulier, Magnus Malmqvist, Karl Andersson.
Feel free to contact me if you have further questions
TEodor
Thanks guys, these are great answers. After I posted the question, it occurred to me that the number of receptors in principle shouldn't matter, sine kd and ka are intrinsic properties of the proteins; however, as you said, the engineering of the surface will affect these in other ways, and that's dependent on receptor properties.
May I ask a few followup questions?
First, for more complex models than a simple langmuir (eg a bivalent reaction), does the receptor concentration ever come into play?
Secondly, let's say I knew my number of receptor sites ahead of time. Couldn't I skip the need to do multiple concentration measurements, and instead just measure one association event, and one dissociation event, and just be done with it? I know this is probably not a very good measure of ka, kd, but in principle, could it be done? It we knew the receptor concentration, it seems like we would have to do less measurements than in a typical titration
The quadratic formula for the concentration of receptor-ligand complex RL in 1:1 equilibrium binding in terms of the total receptor concentration RT, the total ligand concentration LT, and the Kd is:
[RL] = {Kd + RT + LT - [(Kd + RT + LT )2 - 4 RTLT]1/2}/2
Using this formula, in principle, with a single measurement of the concentration of receptor-ligand complex and accurate knowledge of RT and LT, you could calculate Kd. It's not best practice, of course, to use one measurement, since you would have no way to estimate experimental error.
What do you mean by a bivalent reaction? Are you referring to covalent bond formation involving 2 reactants? That is not an equilibrium situation. It is described by a rate law: rate = k[A][B]. If one of the reactants is the receptor, then you can see that you must specify the receptor concentration to calculate the 2nd order rate constant k from rate measurements.
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