Using Parameter Estimation techniques for determining analyte concentration from surface plasmon resonance measurements

Transcription

1 Using Parameter Estimation techniques for determining analyte concentration from surface plasmon resonance measurements M. Si Mehand, G. De Crescenzo,, B. Srinivasan Department of Chemical Engineering, École Polytechnique de Montréal, Montréal, Canada, H3C 3A7 Abstract: Surface plasmon resonance-based biosensors have been recently used for the determination of macromolecule concentrations. Herein, most of the proposed experimental approaches rely on the generation of a calibration curve and only exploit few data points from each interaction sensorgram. In this manuscript, a novel approach is proposed where the unknown concentration is determined using parameter estimation techniques, assuming prior knowledge of the kinetic parameters. Despite errors in the kinetic parameters, it is shown that such an approach improves confidence, experimental time and material consumption. Keywords: SELECT from IFAC list 1. INTRODUCTION Since the 70 s, the biopharmaceutical industry has taken advantage of advances in combinatorial chemistry, recombinant protein production and high throughput screening in order to understand the biochemical and molecular roots of many diseases as well as to develop new therapeutics (Andersen and Reilly 2004; Birch and Racher 2006). In this context, product concentration has been suggested to be one of the most important parameter to be routinely measured for process monitoring and control (Baker et al., 2002). For protein concentration determination, enzyme-linked immunosorbent assay, ELISA, or fluorescence-based assays and, to some extent, quantitative Western blotting are popular off-line quantification methods as they allow for the simultaneous analysis of multiple samples (Birch and Racher 2006), but their implementation remains long and tedious (Canziani et al., 2004; Chavane et al., 2008) and can only be achieved off-line. As an alternative, surface plasmon resonance-based instruments, including those developed by BIACORE Inc. (now part of GE Healthcare) have been demonstrated to be useful for assessing protein quality (i.e., determining the kinetic and thermodynamic constants related to their interaction with a known biological partner) and quantity (i.e., concentration determination) both off-line (Kikuchi et al., 2005) and at-line (Chavane et al., 2008). In Biacore biosensors, SPR is applied to detect the changes in refractive index that occur when a soluble species (the analyte), delivered through the biosensor flow cell, interacts with its binding partner that had been previously immobilized at the biosensor surface (the ligand). This signal is recorded in real time and is proportional to the mass accumulation resulting from complex formation. Subsequent buffer injection results in the dissociation of previously formed complexes and is also followed in real time (this sequence of events is called a sensorgram). The use of SPR biosensors for concentration determination is still in its infancy. Most of the SPR experimental approaches for determining protein concentrations rely on the generation of a calibration curve that only exploits a few data points from each sensorgram. In order to ease the quantification process, this calibration-dependent approach is often coupled with total or partial mass transfer limitation conditions by increasing ligand densities and working at low flow rates (Chavane et al., 2008; Kikuchi et al., 2005). Furthermore, each quantification campaign or new biosensor surface requires a new calibration to be performed, thus increasing time and material consumption, while making at-line quantification difficult (Chavane et al., 2008). Calibration-free methods that exploit total (Karlsson et al., 1993) or partial mass transfer limitation (Christensen 1997; Richalet-Sécordel et al., 1997; Sigmundsson et al., 2002) have also been proposed; these methods require the analyte diffusion coefficient to be estimated. In this work, we explore an alternative method for the determination of analyte concentration, assuming prior knowledge on its kinetics of interaction with a given partner. The method is based on the injections of samples at unknown concentration, followed by a numerical parameter fitting approach in which the unknown sample concentration is treated as a parameter to be identified. Thus, the approach replaces the calibration step by using prior knowledge of kinetic constants, thereby reducing material consumption. Then, as to the question on how the error in the kinetic parameters would affect the concentration measurements, our results suggest that the analyte concentration can be estimated with a fairly good confidence, i.e., within 10% error. Copyright by the International Federation of Automatic Control (IFAC) 8397

2 2. Parameter identification approach for concentration determination 2.1 Modelling of the interaction The system under study corresponds to the interaction of an analyte (A) injected over a Biacore surface where its binding partner, the ligand (B), has been immobilized. The interaction between the analyte and the ligand gives a non-covalent complex (AB); the interaction being described by the following scheme: where and k d correspond to the association and dissociation rate constants of the interaction and are expressed in M -1 s -1 and s -1, respectively. With SPR optical biosensors, the interaction is followed by monitoring AB (that is reported in arbitrary resonance units, RU). The following mathematical model describes the interaction: Where C A is the concentration of free analyte A (in M), R max the maximal amount of analyte that can specifically bind to the surface (in RU), R AB is the amount of AB complex corresponding to the resulting recorded signal (in RU) and R A is a local correction factor (in RU) that is added to take into account refractive index artifacts that are often observed at the beginning of each phase of a sensorgram even after control correction of the signal. R is the resulting recorded signal in RU. The method is designed with a local correction parameter, R A, but in this case study, it was observed that R A =0 is optimal; the R A local parameter was thus removed from the algorithm. 2.2 Parameter identification problem: A sensorgram corresponding to an unknown concentration (C A ) is recorded and double-referenced (Myszka 1999). The sensorgram consists of an injection phase (analyte injection of t on duration) and a dissociation phase (buffer injection of t off duration). From the experimental data, the following leastsquared identification algorithm is then used to compute the concentration: Where M is the number of data points in the sensorgram, R meas the value of a given data point, R pred its related value predicted from the model presented in equation (1). The main concept of this paper lies in the choice of the parameter vector θ. For kinetic constant determination, the classical experimentation consists of generating different sensorgrams with known concentrations and using a parameter identification technique to compute the kinetic parameters, R max as well as the local R A parameters. Contrarily, if the kinetic constants are available either from the literature or from earlier experiments, then it is possible to determine the concentration C A by solving the same optimization problem. 3. MATERIALS AND METHODS 3.1 Materials Experimental data were generated with a Biacore T100 optical biosensor equipped with research-grade CM5 sensor chip (GE Healthcare). HBS-EP buffer, acetate buffer and ethanolamine were purchased from GE Healthcare. N-ethyl- N -(3-dimethylaminopropyl) carbodiimide (EDC), N- hydroxysuccinimide (NHS), carbonic anhydrase isozyme II (CAII) that had been purified from bovine erythrocytes, 4- carboxybenzenesulfonamide (CBS) and phosphate buffer saline (PBS, 10 mm, ph 7.4) were purchased from Sigma- Aldrich Canada Ltd (Oakville, ON). 3.2 Biosensor surface preparation Biosensor surface preparation (CAII and blank surfaces) was performed according to published protocols (Navratilova et al., 2007). Two CA II surfaces (and their corresponding blank surfaces) were prepared at different densities (6843 and 4250 RU of immobilized CA II). After CA II immobilization or blank surface generation, the system was extensively primed with running buffer (HBS-EP). 3.3 Biacore sample injections Prior to CBS injections, 3 Prime procedures and buffer injections (3) were performed to stabilize the baseline of the instrument. CBS sample injections were carried out in duplicate at a flow rate of 100 µl/min with a data collection rate set at 10 Hz, at 24, 18, 12 and 6 C. All CBS samples were diluted in HBS-EP running buffer from a stock solution (1.56 mm) prepared in PBS buffer. For preliminary kinetic experiments, CBS samples at concentrations comprised between nm and µm, in addition to 6 buffer solutions (for double referencing purpose), were injected for 60 s across both control and CAII surface in a serpentine fashion, followed by a 600 s injection of HBS-EP running buffer. As complete dissociation was observed in each case, no regeneration procedure was performed, in agreement with previous reports (Day et al., 2002; Navratilova et al., 2007)). For concentration identification experiments, 3 additional CBS samples at 29.53, 3.19 and 1.23 µm were also injected in duplicate; their corresponding sensorgrams were, however, not taken into account for kinetic constant determination. 3.4 Data analysis Data sets were analyzed with Biacore T100 evaluation software for kinetic determination (see Table 1). For concentration analysis, an in-house software package was developed with the MATLAB (R3008b) software platform (The Mathworks, Natick, USA) using the kinetic model described in (1). The least-square problem presented in (2) was solved with the standard simplex program available in the optimization Toolbox 4.1 of MatLab. 8398

3 Table 1. Kinetic constantss for CAII inhibitor (CBS) interactions determined from Biacore analysis. T ( C) ( x 10 M -1 s -1 ) k d ( x 10-3 s -1 ) High density Low density Literature High density Low density Literature ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± EXPERIMENTAL RESULTS AND DISCUSSION 4.1 Experimental data collection and kinetic analysis We hypothesized that a numerical strategy relying on prior knowledge of kinetic constants would be as efficient, if not more, as a traditional calibration-based approach for the determination of analyte concentrations (values and related standard deviation) in an SPR-based biosensor approach. This hypothesis was tested experimentally by recording sets of sensorgrams corresponding to the interactions of a small molecular weight compound (CBS, the analyte) with its binding partner (CAII, the ligand in our experimental setup) that had been previously immobilized on biosensor surfaces. This biological system was selected since i) it has already been extensively studied by SPR and demonstrated to follow a simple Langmuirian kinetic model (1:1 interaction) (Day et al., 2002; Navratilova et al., 2007) and ii) varying the temperature at which the interactions took place has been shown to affect both association and dissociation rates, thus, providing us with the opportunity to test our hypothesis for different scenarios (Table 1). In a first step, classical kinetic experiments were performed by injecting CBS over CAII surfaces at two different densities and at various temperatures (24, 18, 12 and 6 C). These sets of sensorgrams were control-corrected, doubled-referenced and globally analyzed using a simple Langmuirian model available in Biaevaluation (Figure 1). Excellent fits were obtained when the experiments were performed at temperatures ranging from 12 to 24 C, as one can judge from the residual plots (difference between calculated and experimental data points, Figure 1). At these temperatures, the resulting kinetic constants were in excellent agreement with those previously determined (less than 10% difference, Table 1) (Navratilova et al., 2007). When the temperature was set at 6 C, the experimental data were not as good-quality as for other temperatures (Figure 1, panels D1 and D2) and bigger variations in values, when compared to that previously determined, were observed (29 and 18% variations for highh and low density surfaces respectively, Table 1). Thesee sets of sensorgrams were however included in this study since higher variations were reported for a similar biological system that had been investigated in a multiple user campaign (Cannon et al., 2004). At last, in addition to the sensorgrams used for kinetic analysis, 3 additional CBS injections at 29.53, 3.19 and 1.23 µm were performed in duplicate. The resulting sensorgrams were control-corrected as those related to kinetic analysis, and were considered as sensorgrams resulting from the injections of samples of unknown concentrations for the rest of the study (Figure 1, right panels and data not shown). 4.2 Strategies for determination of the unknown concentrations Strategy #1: Standard calibration curve: The sets of sensorgrams we collected for kinetic analysis were first used to generate calibration curves. The sensitivity and the accuracy of the method are related to the signal-to-noise ratio, artifacts (e.g., disturbances due to the switch from running buffer to sample injections) and the kinetics of the interaction. Figure 1. Kinetic analysis of CBS binding to CA II surfaces using a Biacore T100 biosensor. Panels A1, A2 : Black dots correspond to controlcorrected and double referenced sensorgrams collected at 24 C when injecting CBS (from 249 nm to µm) across two different CAII surfaces (6843 and 4250 RU, for left and middle panels, respectively). Red lines correspond to the global fits with a 1:1 interaction model. The residuals are presented below each set of sensorgrams. Panels A3 shows the double-referenced sensorgrams corresponding to CBS injections at 29.53, 3.19 and 1.23 µm (which were considered as samples of unknownn concentration) over the higher density CAII surface. 8399

4 Table 2. Summary of the different strategies and related maximal deviations from real values. Strategies Max deviation (%) Strategy # Estimated parameter Number of unknown samples Kinetic Parameters 1227 nm 3190 nm nm 1 C A 1-28% 14% 26% 2 C A 1 exp 20% 20% 9% 3 C A, R max 1 exp 50% 20% 25% 4 C A, R max 1 lit 78% 40% 10% 5 C A, R max 1 exp 10%* 10%* 12%* 6 C A, R max 1 lit 15% 5% 12% *When excluding the 6 C data set To improve the accuracy, calibration curves were generated for different time points (1, 2, 4, 5, and 40 s after the start of injections, Figure 2). These calibration curves were then used to calculate the concentration of the 3 unknown samples by simple interpolation. Figure 3 (panel A) shows the relative error on each concentration for the different temperatures we tested. Note that the mean of the concentration obtained from the five calibration curves is reported. Strategy #2: As a first alternative to strategy #1, we determined the concentration of each unknown sample that had been injected on a given surface by fitting its corresponding sensorgram, using the R max and kinetic constant values that we previously determined with Biaevaluation software package when investigating CBS/CAII interactions on these surfaces (Figure 1, Table 1). Simple interpolation. Figure 3 (panel A) shows the relative error on each concentration for the different temperatures we tested. Note that the mean of the concentration obtained from the five calibration curves is reported. This corresponded to solving the following least square problem: identifying R max along with the rest of the parameters from experimental data. Figure 2. Example of calibration curves. Based on the sets of sensorgrams shown in Figure 1, several calibration curves were derived for each data set. Panel A: magnification of the injection phases of the sensorgrams shown in Figure 1, panel A1. Panel B: resulting calibration curves corresponding to CBS accumulation after 1, 2, 4, 5 and 40 s of injection (red, green, blue, cyan, magenta and black curves, respectively). To test this hypothesis, every single sensorgram corresponding to the injection of unknown samples were individually fitted by solving the following least-square problem However, in contrast to strategy #1, it takes into account all the data points within the sensorgrams that would have been generated to get a calibration curve. As can be seen in Figure 3 (panel B), this approach was found to be slightly better than the standard one (compare Figure 3, panels A and B). Indeed, the maximal deviation was of 20% for strategy #2 and 28% for strategy #1 (Table 2). This strategy is as time-consuming as strategy #1, especially due to the identification of the kinetic parameters. Strategies #3 and #4: To save experimental time, it is possible to use the kinetic parameters from the literature rather than from the previous experiments. However, the value of R max has to be obtained from the given surface. The next two strategies aim at Thus, for each unknown sample, both C A and R max were identified, while and k d were fixed as constants. (, k d ) pairs were either those we determined from the kinetic experiments performed in this study (strategy #3) or extracted from the literature (strategy #4). As can be seen in Figure 3 (panels C and D), these approaches resulted in non-negligible deviations from real values. The highest deviations were observed when determining the concentrations of the most diluted sample (almost 80% in the worst case, Table 2). This result is most likely due to the high correlation that exists between R max and C A, as both parameters are multiplied by each other in equation (1) and since, at low analyte concentration, steady-state equilibrium is not reached during the injection phase, thus decreasing confidence in R max identification 8400

5 We then reasoned that strategies #3 and #4 could potentially be improved by globally fitting the sensorgrams corresponding to multiple sample injections. Our reasoning was based on the observation that, for kinetic experiments, the confidence on kinetic parameters has been reported to improve when globally fitting multiple sensorgrams corresponding to different analyte concentrations (De Crescenzo et al., 2008; Ö'nell and Andersson 2005). It will be shown next that for a reliable estimation of R max, one of the identified concentrations must be close to 100k d and the injection time must be greater than 1. The 20 calculations can be made from the model equations (1). At equilibrium, (1) gives, k CA = d R Ab. For the surface to (R max R Ab ) be saturated, i.e. R AB = 0.99 R max, C A should be close to C A = 100k d. Also, considering the model equation as a first order system, it can be deduced that its time constant is 1 τ =. Thus, the time required for the system to C A + k d stabilize is T=5, and for C A = 100k d, T of the estimated from the 5 calibration curves (Figure 2 and data not shown). Strategy #5 and #6: The first approach we investigated for a more accurate estimation of R max was to inject a concentration that would saturate the surface. This can be achieved by injecting one sample at a known standard concentration close to 100 k. The d identification is then performed along with the sensorgram obtained from the sample of unknown concentration (C A ). The following least-squares problem was used to identify C A and R max : in which the kinetic parameters were deduced from the previous sets of experiments (strategy #5), or extracted from the literature (strategy #6). In these cases, R max was identified as a global parameter while the C A concentration was treated as a local parameter related to the sensorgram of unknown concentration. As can be seen in Table 2 and Figure 3, deviations from real concentrations were close to 10%, while significantly reducing experimental time, as 2 injections only are needed for concentration determination. Of interest, both strategies (kinetic parameters deduced from previous experiments performed on the same biosensor or from the literature) gave similar results. This unambiguously demonstrates the robustness of the biosensor devices and also of the estimation method. 4. CONCLUSION Figure 3. Concentration deviations from real values when identified by different strategies. The 3 concentrations corresponding to the unknown samples were identified by various approaches: Strategies 1 to 6 correspond to panels A to F, respectively. In each panel, the red circles correspond to the real concentrations while the black stars, squares, triangles and diamonds correspond to experiments performed at 24, 18, 12 and 6 C, respectively. Note that in the case of strategy 1 (standard calibration curve, panel A) the reported values correspond to the mean In this manuscript, we explored different strategies relying on prior knowledge of kinetic constants for concentration identification by SPR biosensing. Our strategy requires that the species, for which concentration is to be determined, interact with a given partner according to known kinetic constants. In contrast to a classical calibration method, the various strategies we tested took into account all the experimental data points of each sensorgram, leading to improvement in confidence of the identified concentration, when compared to standard methods relying on calibration curves (see Table 2, strategy #1 versus #2). We demonstrated that our approach only requires a single standard injection to be performed in order to improve R max and thus C A identification (strategies #5 and #6). Altogether, our approach reduces material consumption and experimental costs associated with surface calibration, which makes it ideal for the implementation of routine assays aiming at validating macromolecule concentrations. Also, in an effort to satisfy the needs of the biopharmaceutical industry for process analytical technologies (PAT), we recently developed an automated method for the monitoring of a given secreted protein by harnessing an SPR biosensor to a bioreactor (Chavane et al., 2008). In our original design, protein concentration determination relied on the generation of a standard curve. In that context, we believe that the approach we here report could be advantageously adapted to ease atline monitoring. 8401

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