# A curve

Let’s look at a single curve. The example consists of an association, steady state and dissociation phase. What is important on the different parts?

Association: At the injection start, the bulk shift should be small and proportional to the analyte concentration. The initial part of the curve should not be a straight line, which indicates a mass transport free interaction. The association should follow a single exponential and have at least some curvature (1),(2) before the analyte injection ends.

Steady state: When the analyte injection time is long enough, the curve should level out indicating that the amount of association events is equal to the amount of dissociation events. The curve should be horizontal! Although not strictly correct, this situation is often referred to as equilibrium. The response at steady state (equilibrium) is denoted Req. A slow dissociation rate will require a long injection period to get to steady state.

Dissociation: During the dissociation, the curve should follow a single exponential. In cases of a strong interaction, the curve can be almost horizontal. In these cases, the dissociation time should be long enough to have at least 5% dissociation compared to the initial response (3). When the dissociation is fast, the time to steady state is relatively short, compared to when the dissociation is slow.

Sensorgram

Although the curve is a single exponential, the shape depends on several parameters. In equation 2, Rt is the response at time t describing the curve. Req is the equilibrium response; ka is the association rate constant; kd the dissociation rate constant and C the analyte concentration. The term (t-t0) is the time interval of the interaction. Req is dependent on the analyte concentration as shown in the formula 3 and the maximal capacity (Rmax) of the sensor chip surface. By dividing kd/ka the equilibrium constant KD can be calculated. This constant gives an idea of the overall interaction between the analyte and ligand.

Eq. 1:Interaction
Eq. 2:Response calculation
Eq. 3: Equilibrium calculation

Rmax is dependent on the surface capacity of the ligand and the molecular mass of the analyte. The rate constants ka, kd and the equilibrium constant KD is independent of the concentration of both analyte and ligand but is dependent on the pH, salt, temperature and pressure of the solution. Therefore, it is important to keep the experimental conditions constant and to mention these in your publication.

Looking at the figure at right you can observe several things.

• Even at low concentrations, the curve will reach steady state. The injection time to reach steady state is highly dependent on the dissociation rate constant.
• When the analyte concentration equals the KD, the response is half-maximal.
• You need at least 10 times the KD in analyte concentration to reach 90% of the Rmax.
• To reach the true Rmax you need to inject a very high analyte concentration. Luckily, it is not necessary to saturate the ligand to get meaningful results.

The shape of the curves is highly dependent on the dissociation rate constant. In the figure are five dissociation rates (10-1 – 10-5 s-1) with the same association rate (105 M-1s-1) and an analyte concentration of 1 times KD. The curve with the fastest dissociation rate will reach steady state much quicker compared to the other. Can you figure out which curve has the fastest dissociation rate?

Saturation

Different dissociation

To calculate the time needed to reach steady state you can use the formula shown below.

Eq. 4: Time to equilibrium

To reach steady state you can raise the analyte concentration, but this will not always work. Then you can make the injection time longer. However, as you can see in the table this can be a very long injection time.

Table: Approximate calculated times required to reach 99.9% steady state at analyte concentrations ranging from 0.01 to 100 times KD (4).
Concentration analyte kd (s-1)
10-1 10-2 10-3 10-4
0.01 x KD 68s 11.5 min 115 min 1140 min
0.1 x KD 63s 10.5 min 105 min 1047 min
1 x KD 34s 6 min 67 min 576 min
10 x KD 6s 1 min 10.5 min 105 min
100 x KD 1s 0.1 min 1 min 11 min

## Equilibrium analysis

One of the most common mistakes is to use the equilibrium analysis on curves which are not in equilibrium (steady state) (2). The figure below shows three sensorgrams of which only the last can be used for equilibrium analysis because all the green coloured curves level out before the end of the injection. At steady state, the response of the complex is directly proportional to the concentration of the analyte.

Equilibrium analysis

For equilibrium analysis, it is not necessary to saturate the ligand as long as the equilibrium curve has enough curvature to be fit properly. In the next figure, subsequent points are left out in the analysis. For a reliable result, roughly 30 to 40 percent of the ligand must the saturated.

Equilibrium fitting

## Analyte concentration range

The concentration range of the injected analyte is important. With too high concentrations, the curves tend to bunch together in the upper part of the sensorgram. Too low concentrations will give low responses and little curvature. The best concentration range is somewhere around 0.1 – 10 times the KD of the interaction. This will space the curves evenly over the sensorgram, having high and low responses. This requires knowledge of the kinetics. The best approach is to start with a low concentration, for instance 10 nM and work your way up until you obtain nice curves. When you have established the concentration range, design an experiment with a dilution series. It is better to use a dilution series because it is easier to make and you can detect problems with the injections more quickly. Try to make a dilution series covering the 0.1 x KD to 10 x KD. Add repeats of the dilutions to show the system is stable. It is enough to have five concentrations and repeat them three times.

The figure below shows sensorgrams with a) too high, b) optimal concentration and c) too low concentration of analyte. Thus, these sensorgrams clearly shows the required (optimal) concentration of analyte to be used.

Analyte concentrations

As said, when the injection time of the analyte is long enough, the association rate will equal the dissociation rate and the curve will reach steady state. The time to reach steady state depends greatly on the dissociation rate constant and the analyte concentration. In principle, you don’t need a curve with steady state to get meaningful results, but you should have enough curvature in the curves.

## Curve response

The response of the sensorgram should match the amount of immobilized ligand and the concentration analyte used. Because you know which ligand and analyte is used, you can calculate the theoretical Rmax with:

RMax Calculation

Although a valid formula, in general it is not practical because the fraction active ligand is unknown. Depending on the immobilization technique used, only a larger or smaller fraction is still biological active.

As an alternative, the Rmax can be determined by saturating the ligand by injecting high analyte concentrations. However, this is not always possible (see above at steady state). Luckily, it is not important to know the Rmax to get meaningful kinetic results.

In the fitting procedure, the Rmax is determined as one of the parameters. As long as the calculated Rmax is in agreement with the measured values, full saturation of the ligand is not necessary.
In addition, high analyte concentrations and the following high responses tend to have problematic kinetic behaviour. So keep in mind that sensorgrams with a low response level (< 100 RU) are better than curves with a high response.

What about these fast on and of curves? Are they real or is it bulk effect?

When it is real kinetics, the shape of the curves is almost totally determined by the dissociation rate. Because of the fast dissociation, the curves reach equilibrium almost directly. The height of the response is directly proportional to the analyte concentration. With increasing analyte concentration, the surface will saturate as opposed to other effects like high salt or non-specific binding.

Real kinetics can thus saturate the ligand at high analyte concentrations. If the response keeps getting higher, other effects such as non-specific binding, bulk refractive distortion causes these high responses.

Fast dissociation

## References

 (1) Rich, R. L. and Myszka, D. G. A survey of the year 2002 commercial optical biosensor literature. J.Mol.Recognit. 16: 351-382; (2003). Goto reference (2) 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 (3) Katsamba, P. S. et al Kinetic analysis of a high-affinity antibody/antigen interaction performed by multiple Biacore users. Analytical Biochemistry 352: 208-221; (2006). (4) BIACORE AB BIACORE Technology Handbook. (1998).