3.1 The PLIS interface

The PLIS interface is divided into three areas (Figure 1). The data is displayed in a table in the top left corner and the results of fits are shown in the bottom left corner. The right side of the main window is used to display graphs of experimental and fitted data.

The PLIS interface. In the area at the top left, the data is represented as an editable table. Below, the results of fits are shown. The main area is dedicated to graphical representations of experimental and fitted data

3.2 Import of data

PLIS facilitates three modes. “Titration (normal)” implies that the signal is concentration dependent, for example, fluorescence or absorption spectroscopy. “Titration (NMR)” is used for methods where the signal does not depend on concentration, for example, the NMR chemical shift in the fast exchange limit. “CPMG RD” is used for relaxation dispersion data.

Data sets can be imported from text files consisting of three columns separated by white space. For titration data, the columns correspond to the measured signal, the ligand concentration and the sample volume, respectively. Since accurate measurement errors rarely are available in a single titration series, it is not possible to enter these here. Instead, a common error of unity is assumed, implying that χ2 is scaled inappropriately if the true errors are unknown. It is however possible to average several series in PLIS and thus obtain uncertainties from the standard deviations of the averaged data points. For CPMG relaxation dispersion data, it is common practice to include duplicate data points for error estimation. Thus, here the three columns are the repetition rate of the refocusing pulses, the effective transverse relaxation rate and its uncertainty. In addition of importing data sets, empty data set can be created where content is added manually or pasted from a spread sheet.

3.3 Curve fitting

Currently, the software supports binding models with one to four binding sites in the absence or presence of a competitor for titration data and one binding site for CPMG relaxation dispersion data (Table 1). Data is fitted by first choosing the desired model and then optionally entering initial estimates for the parameters and deciding whether to keep certain parameters fixed rather than optimized. If the fit converges, a graph corresponding to the fitted data is displayed and the dissociation constants and their estimated uncertainties are shown. The default method for estimating uncertainties in the model parameters is the jackknife.22 Monte Carlo simulations23 can be used if errors in the data points are known.

TABLE 1. Binding models in PLIS No. binding sites Optimized parameters Signal of protein monitored Signal of competitor monitored 1 Kd, sP, sPLa Kd, KdC, sC, sCL 2 Kd1, Kd2, sP, sPL, sPL2 Kd1, Kd2, KdC, sC, sCL 3 Kd1, Kd2, Kd3, sP, sPL, sPL2, sPL3 Kd1, Kd2, Kd3, KdC, sC, sCL 4 Kd1, Kd2, Kd3, Kd4, sP, sPL, sPL2, sPL3, sPL4 Kd1, Kd2, Kd3, Kd4, KdC, sC, sCL a For titration data. For CPMG relaxation dispersions, the optimized parameters are R2,0, kex, pPL, and Δω, from which Kd is calculated.

A drawback of Levenberg–Marquardt minimizations is that they may converge to local rather than global minima. Fits that are unsatisfactory for this reason are often recognized by visual inspection. It is then possible to optimize the fit using the current parameters as guides for initial estimates. Optimization can be done iteratively, and the user decides, aided by the appearance of the fitted curve and χ2, whether to accept or reject the results of the optimization. It is also possible to precede the curve fitting by a grid search to avoid local minima.

As an example of data fitting in PLIS, we show the results of titration of Ca2+ into the C-terminal lobe of calmodulin (Tr2C) in the presence of the competitor O,O′-bis(2-aminophenyl)ethyleneglycol-N,N,N′,N′-tetraacetic acid (BAPTA) in Figure 2. The data was fitted to a model with two Ca2+ binding sites for Tr2C and one for the probe BAPTA. The dissociation constants of 0.11 and 1.9 μM are in good agreement with previously reported values, considering differences in sample conditions.24 However, reliable results are also dependent on accurately determined concentrations of protein and competitor. To keep the fitting simple, we have not included the possibility to include concentrations as adjustable parameters during fitting. We find it just as easy and more robust to manually tweak the protein concentration and then refit the data.

The competitor O,O′-bis(2-aminophenyl)ethyleneglycol-N,N,N′,N′-tetraacetic acid (BAPTA) was added to the C-terminal lobe of calmodulin in 20 mM HEPES, pH = 7.5. The concentrations of protein as well as BAPTA was 35 μM. CaCl2 was titrated in and the absorbance at 254 nm was measured

As a second example of data fitting, as well as a demonstration of the data simulation tool of PLIS, we simulated CPMG relaxation dispersion data for the binding reaction P+L⇌PL and subsequently fitted the data (Figure 3). Since the data was generated using a dissociation constant of 20 μM and realistic concentrations of protein and ligand, the fitted value of 19.9 ± 0.5 μM demonstrates that accurate dissociation constants can be determined from relaxation dispersion data in favorable cases. Among the factors that may occlude an accurate analysis are a narrow sweet spot of the exchange rate constant, ligand independent conformational exchange, uncertainty in the ligand concentration and poor signal-to-noise ratios. As a partial remedy, dispersions may be recorded at several magnetic fields and/or ligand concentrations and fitted together.21, 25

Relaxation dispersions for simulated data. The data was generated using a total protein concentration of 500 μM, a total ligand concentration of 26.1 μM and a dissociation constant of 20 μM, corresponding to a population of the bound state of 5.0%. The global exchange rate was set to 200 s−1 and Δω for the three datasets were set to 600 (blue), 1,200 (magenta), 2,400 s−1 (black) corresponding to 1.2, 2.4, and 4.8 ppm, respectively, for 15N at 18.8 T. The solid lines correspond to global fits to the Carver Richards equations18, 19

To illustrate the effect of experimental noise on reliable analyses, we compared fitting two synthetic data sets with different noise levels (see Figure 4 for details). As expected, the fitted dissociation constants are equal to the ones used to generate the data if no noise is added. For the simulated data set with added noise, the fitted dissociation constants were 0.10 ± 0.05 and 0.41 ± 0.05 μM, respectively, showing that this noise level is incompatible with determination of accurate dissociation constants. Another application of the simulation tool is to learn how different choices of parameters affect the appearance of data, which is useful for analyzing experimental data.

The simulation tool of PLIS. Two data sets for two-site binding with dissociation constants of 0.10 and 0.50 μM and an initial protein concentration of 10 μM were generated. For the first data set (red symbols), no noise was added and for the second data set (green symbols), the RMSD of the noise was set to 0.01 a.u. The solid lines correspond to fits to a two-site model