# The exponential

The first result from a SPR experiment is the sensorgram. Each sensorgram contains a world of information for the trained eye. Therefore, it is essential to recognize good from bad curves. A bad curve represents a bad experiment, producing bad results from which conclusions cannot be made. And be sure, there is a lot of rubbish in the SPR literature! For example, please read the articles from D.G. Myszka and R.L. Rich (1),(2).

A good understanding of the binding curves is the first step in interpreting the data. Let us start with the binding between two dissimilar molecules. One molecule, the analyte (A), binds to one molecule, the ligand (L) in a reversible way. The velocity or rate of binding (association) is denoted by the association rate constant ka in M-1s-1. The breaking up of the complex (dissociation) is denoted by the dissociation rate constant kd in s-1. The quotient of the kd/ka defines the equilibrium dissociation constant KD in Molar which provides the analyte concentration that will saturate 50% of the ligand. Equation 1 shows the rate equation for a 1 to 1 interaction between the ligand and analyte. Eq. 1: Interaction Protein interaction

At the start of the interaction, all the ligand is unbound and the baseline is measured. By introducing the analyte, ligand-analyte (LA) complexes are formed. The amount (concentration) of complex formed by the interaction can be calculated with the differential equation 2. Eq. 2: Complex formation

Integrating equation 2, gives an integrated rate equation, which describes the complete interaction curve. as opposed to the differenetial rate equation which describes the slope of the curve. In equation 2, Rt is the response at time t and Req is the maximal response which can be reached with the injeted analyte concentration. Eq. 3: Response calculation

Equation 3 describes a simple exponential binding profile. No other curve shapes such as parabolic, hyperbolic, concave and convex can describe the binding profile. Eq. 4: Response dissociation

Equation 4 describes the exponential dissociation and R0 is the response at the start of the dissociation. Therefore, train yourself to recognize an exponential binding curve!

## Curve examples

To start, some sensorgrams are given as an overview of the possible curve shapes, which are all still exponential but differ in kinetics. The curves differ in association and dissociation rate, but the shape is always an exponential. Simple exponential curves Exponential curve

Look now at the following curves. Mass transfer limited curves

The figure'Mass transfer limeted curves' shows sensorgrams that have curves with an initial binding profile, which appears to be linear. This is an example of (partially) mass transport limited kinetics. Directly after the analyte injection starts, the binding of the analyte to the ligand is faster than diffusion, creating a shortage of analyte at the surface. Therefore, the interaction is diffusion limited instead of kinetic limited. It is easy to incorporate mass transport limitation in the fitting models but it is better to avoid this by proper design of the experiment. For instance by lowering the ligand density or increasing the flow rate. Interaction with mass transport

The next sensorgrams are often referred to as having biphasic binding responses. Biphasic responses are said to consist of a fast and slow interaction. And because a biphasic response can be described equally well by different models (1) it is virtually impossible to solve the interaction mechanism by modelling alone. In case of a biphasic curve, more optimisation of the experimental conditions is necessary. Don’t try to fit these curves! Biphasic curves

The next two sets of curves are seen in cases like buffer jumps, spikes and drift. Although drift can be added to the fitting model, it is better to avoid drift by proper equilibration of the system. After immobilisation, drift is often very strong. The easiest way to equilibrate is to run flow buffer overnight. When starting your measurements, incorporate several dummy injections (running buffer) to validate the stability of the system. Always try to match the flow and sample buffer to avoid buffer jumps at the beginning and the end of the injections. The spiky and wobbly sensorgrams are a warning to clean your system. Make new degassed reagents and start over designing your experiment. 'Wrong' curves

## References

 (1) Rich, R. L. and Myszka, D. G. Survey of the year 2007 commercial optical biosensor literature. J.Mol.Recognit. 21: 355-400; (2008). Goto reference (2) Rich, R. L. and Myszka, D. G. Grading the commercial optical biosensor literature-Class of 2008: 'The Mighty Binders'. J.Mol.Recognit. 23: 1-64; (2010). Goto reference