Determination of the composition of heterogeneous binder solutions by surface plasmon resonance biosensing

Abstract

Surface plasmon resonance-based biosensors have been extensively applied to the characterization of the binding kinetics between purified (bio)molecules, thanks to robust data analysis techniques. However, data analysis for solutions containing multiple interactants is still at its infancy. We here present two algorithms for (1) the reliable and accurate determination of the kinetic parameters of N interactants present at different ratios in N mixtures and (2) the estimation of the ratios of each interactant in a given mixture, assuming that their kinetic parameters are known. Both algorithms assume that the interactants compete to bind to an immobilized ligand in a 1:1 fashion and necessitate prior knowledge of the total concentration of all interactants combined. The effectiveness of these two algorithms was experimentally validated with a model system corresponding to mixtures of four small molecular weight drugs binding to an immobilized protein. This approach enables the in-depth characterization of mixtures using SPR, which may be of considerable interest for many drug discovery or development applications, notably for protein glycovariant analysis.

Figure 1

Algorithm for the estimation of the kinetic parameters of $N$ analytes. The algorithm starts by providing a legitimate starting point for a nonlinear optimization routine over the dissociation phase before a second optimization routine over the whole sensorgrams leads to the final estimates. Estimates of $k_{a,i}$, $k_{d,i}$ and $R_{max,i}$ for $i=1,\cdots ,N$ can be obtained from SPR sensorgrams of $M$ mixtures of the $N$ analytes $(M\ge N)$. The fit is said ‘global–global’ as sensorgrams corresponding to different overall concentrations and different mixtures are used simultaneously in the fitting procedure.

Figure 2

A two-step algorithm for the estimation of the composition of an unknown mixture of $N$ analytes with known kinetic parameters. The starting point for the fraction of each analyte is set to $1/N$ for the first optimization routine. This optimization uses only the dissociation phase of the sensorgrams. The results of the first step are used as the starting points of the second optimization routine. Note that the total concentration ($C_{\mathit{TOT}}$) of analytes in the mixture is known, but not the mixture composition (fraction of each analyte).

Figure 3

Kinetic analysis of the injection of the 4 compounds and the 12 mixtures of compounds. Black dots correspond to control-corrected and double referenced sensorgrams for all experiments. Red lines correspond to global fits for the single-analyte experiments (CBS, BDS, sulfanilamide and furosemide) and to ‘global–global’ fits for each data set for 70–10–10–10 mixtures (A to D, see Table 1 for exact composition), 40–20–20–20 mixtures (E to H, see Table 1 for exact composition) and random proportion mixtures (I to L, see Table 1 for exact composition). For the single-analyte experiments, the concentrations of CBS, BDS, sulfanilamide and furosemide varied respectively from 1.06 μM to 52.98 μM, 210 nM to 10.49 μM, 1.01 μM to 50.27 μM and 1.01 μM to 50.52 μM. The total concentrations ($C_{\mathit{TOT}}$) of the mixtures varied from 300 nM to 15 μM for all mixtures. Here, the composition of each mixture is assumed to be known. The related residual plot is given below each sensorgram data set. This figure was generated with the Matlab R2018b software platform (The Mathworks, Natick, USA, www.mathworks.com/products/matlab.html).

Figure 4

Kinetic analysis of the contribution of CBS (red), BDS (green), sulfanilamide (cyan) and furosemide (purple) to the SPR response of each mixture. The kinetic parameters identified from each data set (see Table 2) were used to obtain the contributions to related sensorgrams. The sum of these contributions gives the ‘global–global’ fits shown in Fig. 3, minus the bulk refractive index contribution $R_I$. The total concentrations ($C_{\mathit{TOT}}$) ranged from 300 nM to 15 μM for all mixtures. This figure was generated with the Matlab R2018b software platform (The Mathworks, Natick, USA, www.mathworks.com/products/matlab.html).

Figure 5

Estimated fractions of CBS (red), BDS (green), sulfanilamide (cyan) and furosemide (purple) with respect to the true fraction values in each of the 16 mixtures. The fractions were estimated using the kinetic parameters and $R_{max,i}$ identified from different data sets, corresponding to (A) Single-analyte injections; (B) 70–10–10–10 mixtures (mixtures A to D); (C) 40–20–20–20 mixtures (mixtures E to H); (D) Random mixtures (mixtures I to L). The horizontal error bars were obtained by propagating the systematic error of the pipettes (Pipetman Neo P10, P20, P200 and P1000 and Microman M1000E) and the balance (Dever Instrument Company AA-160) that were used to conduct the experiments. The vertical error bars correspond to 95% confidence intervals computed using the Fisher F statistic. Here, we assumed that the total concentration of all analytes ($C_{\mathit{TOT}}$) is known, but the composition of each mixture (fraction of each analyte) is not. This figure was generated with the Matlab R2018b software platform (The Mathworks, Natick, USA, www.mathworks.com/products/matlab.html).

Figure 6

Residual mean square error (RMSE, top) and mean absolute error (MAE, bottom) calculation. The composition of each mixture was estimated using the kinetic parameters and $R_{max,i}$ values identified from different data sets (single-analyte, 70–10–10–10 mixtures, 40–20–20–20 mixtures and random mixtures). The estimated composition was compared to the actual composition of the mixtures outside of the data set used to identify the kinetic parameters. The residual mean square error (RMSE, top) and the mean absolute error (MAE, bottom) were calculated on an analyte basis for CBS, BDS, sulfanilamide and furosemide and by considering all the estimates at once (all analytes). This procedure was repeated for all data sets.