[13] Kinetic analysis of macromolecular interactions using surface plasmon resonance biosensors

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oxide (EO) chain length is increased, stability increases, the melting temperature, (Tm), increases from 520 to 635 K, and the energy required for micelle formation in OPEl0, [AH°(T0) = 0.24 kcal mol-1], is about one-sixth that for OPEa, [AH°(T0) -- 1.35 kcal mo1-1] micellization, as seen in Table XII. The temperature-invariant enthalpy required for the formation of micelles of the ionic surfactant n-dodecyltrimethylammonium bromide (n-DTAB) in H20 is approximately 0.91 kcal mol <, whereas 0.19 kcal mo1-1 is required for the micellization of n-DTAB in D20. Thus, approximately one-fifth the energy is required for the micellization of this ionic surfactant in D20 than is needed in H20. However, in 3 M urea, the temperature-invariant enthalpy of micellization is one-half that required in H20, but still more than in D20. The lower the temperature-invariant enthalpy, AH°(To), required for micellization, the greater the stability of the micelles that are formed. Thus n-DTAB micelles are more stable in D20 than H20 or 3 M urea, as seen in Table XIII. In the three aqueous environments, micellar stability seems to increase as (Tm) increases from 535 K in H20 to 570 K in D20 and 543 K in 3 M urea. It is apparent that surfactant micelles in an aqueous environment are extremely stable and that very little energy is required for micelle formation. The thermodynamic parameters for the formation of spherical or ellipsoidal micelles, as in these ionic or nonionic surfactants, should also be valid for the bilayer micelles found in biological membranes, as the same types of interaction are responsible for the stability of the micellar structure.

[ 13] K i n e t i c A n a l y s i s o f M a c r o m o l e c u l a r I n t e r a c t i o n s Using Surface Plasmon Resonance Biosensors

By T H O M A S

A.

MORTON

and

DAVID

G. MYSZKA

Introduction Optical biosensors are emerging as important tools for characterizing the interactions of biological macromolecules. Biosensors can provide qualitative information on macromolecular assembly processes under a variety of conditions. Quantitative information, in the form of affinity constants for complex formation, can be obtained in a manner similar to conventional solid-phase assays. The major advantage of biosensors over other interaction technologies is that the formation and breakdown of complexes can be monitored in real time, which offers the possibility of determining the mechanism and kinetic rate constants associated with a binding event. This

METHODS IN ENZYMOLOGY, VOL. 295

Copyright © 1998 by Academic Press All rights of reproduction in any form reserved. 0076-6879/98 $25.00

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information is essential for understanding how biological systems function at the molecular level. There are several commercially available optical biosensors that are convenient to use. However, accurate interpretation of biosensor data is not always straightforward. The purposeof this article is to illustrate how to maximize the possibility of obtaining reliable information about reaction kinetics.

Surface Plasmon Resonance Biosensors In an optical biosensor experiment, one of the interacting molecules (the analyte, A) is in solution, whereas the other (the ligand, B) is attached to the biosensor surface. The formation and breakdown of the complex (AB) at the sensor surface are controlled by the association (ka) and dissociation (ka) rate constants, as shown in Eq. (1). ka

A + B ~ AB

(1)

kd

In the commonly used B I A C O R E instruments (Biacore, Inc., Uppsala, Sweden), complex formation is detected by measuring the change in refractive index caused by the accumulation of mass within the surface layer. These instruments use the surface-sensitive evanescent wave technique of surface plasmon resonance (SPR). 1 This method is sensitive to the mass of the molecule and therefore does not require the reactants to be labeled with spectroscopic or radioactive probes. SPR biosensors are therefore convenient to use and amenable to studying the interactions of a wide variety of biologically relevant macromolecules, including proteins, nucleic acids, carbohydrates, and lipids.

Complexities in Biosensor Analyses To date, over 1000 publications have appeared that use B I A C O R E technology alone. Although most of these studies involved the interaction of only two molecules, there are many examples where data cannot be described by the simple bimolecular interaction model shown in Eq. (1). This is because data collected under experimental conditions suitable for qualitative detection of binding events are often unsuitable for measuring kinetic rate constants. A number of experimental artifacts can complicate binding responses, including surface-imposed heterogeneity, mass trans1 U. Jonsson, L. Fagerstam, B. Ivarsson, B. Johnsson, R. Karlsson, K. Lundh, S. Lofas, B. Persson, H. Roos, I. Ronnberg, R. Sjolander, E. Stenberg, R. Stahlberg, C. Urbaniczky, H. Ostlin, and M. Malmqvist, BioTech. 11, 620 (1991).

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port, aggregation, avidity, crowding, matrix effects, and nonspecific bindingfl This has caused some concern about the ease with which this technology can be used to extract accurate information about a binding event. Therefore, a major focus of this article is to illustrate the steps required to minimize experimental artifacts, hence improving the quality of data. It is also possible that some binding responses may not fit simple interaction models because the reaction mechanism is, in fact, more complex. For example, systems that undergo multistate reactions or those involving allosteric interactions could generate complex binding profiles. Therefore, a second focus of this article is to introduce advanced methods of data analysis to aid in the interpretation of complex binding reactions. Using data collected for an erythropoietin ligand-receptor system, this article demonstrates step-by-step the basic methods involved in analyzing a reaction on BIACORE. Common pitfalls encountered during the analysis are highlighted, and suggestions for optimizing the process for other systems are provided. Improving the methods used in biosensor analysis increases confidence that the reaction mechanism and rate constants associated with a macromolecular interaction are interpreted properly.

Erythropoietin Ligand-ReceptorSystem Erythropoietin (Epo) is a glycoprotein hormone responsible for red blood cell production. It functions by binding to a cognate receptor on the surface of erythroid progenitor cells. 3 Experimental evidence suggests that the receptor homodimerizes on binding Epo, initiating signal transduction events. 4'5 In an attempt to characterize the interaction of Epo with the dimeric signaling complex, a soluble dimerized version of the receptor was engineered. The extracellular domain of the receptor was fused to the Fc region of human immunoglobulin G~ (IgG1), as shown schematically in Fig. 1 A . 6 This article describes how SPR biosensor technology was used to determine the kinetic rate constants and affinity for the Epo-receptor construct interaction.

2 D. G. Myszka, Curr. Opin. Biotechnol. 8, 50 (1997). 3 K. Jacobs, C. Shoemaker, R. Rudersdorf, S. D. Neill, R. J. Kaufman, A. Mufson, J. Seehra, S. S. Jones, R. Hewick, E. F. Fritsch, M. Kawakita, T. Shimizu, and T. Miyake, Nature 313, 806 (1985). 4 j. S. Philo, K. H. Aoki, A. T. L. Owers Narhi, and J. Wen, Biochemistry 35, 1681 (1996). 5 O. Livnah, E. A. Stura, D. L. Johnson, S. A. Middleton, L. S. Mulcahy, N. C. Wrighton, W. J. Dower, L. K. Jolliffe, and I. A. Wilson, Science 273, 464 (1996). 6 p. Hensley, M. Doyle, C. Griffin, C. Jones, and P. Young, (unpublished results).

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B

A

250'

® 150

1

L2J

¢,0 t.~ ¢n

n,

Blosensor Surface

50 I

I

I

200

400

600

Time (sec) FIG. 1. (A) Schematic of biosensor assay design. Protein A was immobilized onto a carboxymethyldextran chip (CM5) and used to capture the receptor-Fc construct, creating a homogeneous reacting surface. (B) Receptor construct capture step and subsequent Epo injections. Region (1): two different surface densities of the receptor construct were created on the protein A surface by injecting either 15 or 45 /zl of receptor at a concentration of 70 nM (note only data from the 45-txl injection are shown). No binding of the receptor construct to a surface without protein A was detected. Region (2): samples of Epo (150/zl) were injected in random order at concentrations of 2, 0.66, 0.22, 0.074, 0.025, and 0 nM. Epo dissociation was monitored for 250 sec before regenerating the protein A surface with a 10/xl injection of 10 mM HC1. The binding experiments for each Epo concentration were repeated five times and the samples randomized. All binding experiments were done at a flow rate of 100 txl/min with a buffer containing 10 mM HEPES, pH 7.4, 150 mM NaCI, 0.005% P20 surfactant, and 0.1 mg/ml bovine serum albumin.

Experimental Design

Sample Characterization Sample Purity. B e f o r e s t a r t i n g a b i o s e n s o r e x p e r i m e n t , it is i m p o r t a n t to e s t a b l i s h t h a t t h e s a m p l e s a r e s u i t a b l e for d e t a i l e d k i n e t i c analysis. Bios e n s o r s d e t e c t t h e p r e s e n c e of m o l e c u l e s t h a t i n t e r a c t with t h e surface; t h e r e f o r e it is p o s s i b l e to p e r f o r m e x p e r i m e n t s using s a m p l e s t h a t h a v e n o t b e e n purified. H o w e v e r , t h e r e a r e s e v e r a l d i s a d v a n t a g e s o f using i m p u r e s a m p l e s for k i n e t i c analysis. First, t h e i m p u r i t i e s in t h e s a m p l e m a y i n t e r a c t n o n s p e c i f i c a l l y with t h e b i o s e n s o r surface c o m p l i c a t i n g t h e r e s p o n s e s . Second, the impurities may increase the bulk refractive index change during t h e a s s o c i a t i o n p h a s e . T h i s m u s t b e a c c o u n t e d for using a r e f e r e n c e surface o r d u r i n g d a t a analysis. T h i r d , i m p u r i t i e s m a k e it m o r e difficult to q u a n t i t a t e t h e active c o n c e n t r a t i o n o f a n a l y t e . T h e i m p o r t a n c e of an a c c u r a t e m e a s u r e o f a n a l y t e c o n c e n t r a t i o n w h e n d e t e r m i n i n g r a t e c o n s t a n t s is d i s c u s s e d in the section on Experimental Conditions. To avoid complications caused

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by unpurified samples, the Epo-receptor construct experiments were carried out on proteins that were greater than 95% pure. Aggregation State. Another important task is to characterize the aggregation state of individual reactants, as well as that of the complex, before performing any biosensor experiments. Reactants that self-associate or form higher order aggregates will complicate the binding responses. Although the concentration of analyte in solution may be low, it can reach very high levels when bound to the biosensor surface due to the density of conjugation sites of the matrix. For example, a signal of 1000 response units (RU) corresponds to a protein surface concentration of approximately 10 mg/ml. 7 Therefore, it is critical to ensure that the reactants do not aggregate even at high concentrations. Analytical ultracentrifugation is ideal for characterizing the assembly states of macromolecules in solution. 8 Sedimentation equilibrium experiments demonstrated that both Epo and the receptor construct were monomeric and when mixed formed a 1:1 complex. 6 These results suggested that the Epo biosensor experiments would not be complicated by aggregation and helped define the reaction model. Nonspecific Binding. The first experiment one should perform on the biosensor is to test the samples for nonspecific binding to the biosensor surface. This is done by injecting each sample over a nonderivatized surface at the highest concentration that will be used in the analysis. Some background binding may be corrected for by using a reference surface, which is discussed later. However, if the background contribution is high, as compared to the reaction surface, then the experimental conditions should be altered. For example, basic proteins (pI > 8.5) generally interact electrostatically with the commonly used carboxymethyldextran matrix. Increasing the ionic strength of the buffer, or altering the charge of the matrix by blocking the surface with amines, can minimize this effect. Alternatively, new surfaces from Biacore Inc., may reduce the amount of nonspecific binding in these cases. Currently available surfaces include a dextran matrix with a lower charge (B1), one with a shorter matrix height (F1) ( - 3 0 nm versus - 1 0 0 nm), a flat carboxyl surface with no dextran (C1) and a plain gold surface (J1). These were not necessary for the Epo-receptor experiments described here as neither protein showed an appreciable level of nonspecific binding to the standard carboxymethyldextran matrix.

7 E. Stenberg, B. Persson, H. Roos, and C. Urbaniczky, J. Colloid Interface Sci. 143, 513 (1991). 8 p. Hensley, Curr. Biol. 4, 367 (1996).

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Assay Configuration Ligand Immobilization. Biosensors do not require labeling of the components; however, one of the reactants must be immobilized on the biosensot surface. Because the possibility exists that this may alter the nature of the interaction, it is worthwhile to measure the interaction with each reactant immobilized in turn. However, this approach is not applicable to every system. For example, the Epo-receptor construct is potentially multivalent. Multivalent analytes can bind to more than one surface-bound ligand and thus complicate the reaction due to avidity effects. This is a common problem with measuring the binding of antibodies to immobilized antigens,9'1° as well as for self-associating proteins, such as glutathione S-transferase (GST) fusion constructs. 11 In the Epo experiments, to avoid the potential complication of measuring multivalent reactions, the dimeric receptor construct was attached to the biosensor surface and the interaction monitored with Epo in solution (Fig. 1A). Direct Coupling. Creating a stable and homogeneous surface is essential when performing detailed kinetic analyses. There are two general methods for attaching ligands to the biosensor surface: direct coupling and capturing. Direct coupling relies on exposed chemical groups on the ligand forming a covalent linkage with the surface. The simplest procedures involve random coupling through primary amines or carbohydrates. 12 Unfortunately, Epo binding activity was reduced when the receptor construct was attached to the surface using these coupling chemistries. This is a common problem with random coupling and is likely due to a preferential interaction with residues at or near the binding site. Even if the activity is not entirely lost, the random nature of the immobilization may result in populations of ligand with different binding properties depending on their orientation. This surface-imposed heterogeneity will add extra complexity to data analysis. Therefore, direct coupling is best done through a unique reactive site on the ligand in order to create a homogeneous reaction surface. For example, ligands that contain a single exposed cysteine residue, either present naturally or introduced by site-directed mutagenesis, have been coupled to

9 N. L. Kalinin, L. D. Ward, and D. J. Winzor, AnaL Biochem. 228, 238 (1995). 10 R. Karlsson, J. A. Mo, and R. Holmdahl, J. Irnmunol. Methods 188, 63 (1995). 11 j. Ladbury, M. A. Lemmon, M. Zhou, J. Green, M. C. Botfield, and J. Schlessinger, Proc. Natl. Acad. Sci. U.S.A. 92, 3199 (1995). 12 D. J. O'Shannessy, M. Brigham-Burke, and K. Peck, AnaL Biochem. 205, 457 (1992).

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maleimide- and dithiol-activated surfaces to form homogeneous reaction surfaces.13.14 The Epo receptor has a partially solvent-exposed cysteine residue, which could be used for direct immobilization; however, it is located near the receptor dimerization interface. 5 Therefore, in order to avoid affecting receptor function, we chose not to immobilize the receptor construct through the cysteine residues and used a capturing step instead. Capturing. Capturing relies on the use of a specific recognition site to bind the ligand noncovalently to the surface. The advantage of this method is that it creates a homogeneous reaction surface. It is, however, necessary that the capturing step provides a stable complex with the ligand, otherwise the decay in signal due to the dissociating ligand will complicate the analyte binding responses. These reactions require more complex reaction models. 15 Commonly used capture systems include immobilized monoclonal antibodies (including anti-GST), protein A, streptavidin-biotin, and nickel-chelating surfaces (NTA). In the Epo experiments, the Fc domain on the receptor construct could be used to capture this molecule using a protein A surface, as shown schematically in Fig. 1A. This method had four advantages over direct coupling. First, the immobilized receptor construct was highly active and chemically homogeneous. Second, the receptor construct surface density could be controlled by varying the injection volume during the capture step. Third, the receptor itself was recaptured for each experiment. This increased the amount of receptor construct required, but it meant that the receptor was not exposed to the harsh regeneration conditions. Finally, the protein A surface was stable under regeneration conditions and reproducibly bound the receptor construct, ensuring that Epo responses were measured over surfaces with essentially identical capacities. In fact, the binding responses shown in Fig. 1B (region 1) are an overlay of six repeated injections of the receptor construct. There was essentially no dissociation of the receptor-protein A complex over the time frame in which we monitored the interaction of Epo (see Fig. 1B, region 2). Therefore, Epo binding data could be analyzed without complications due to background surface decay. Reference Surface. Reference surfaces can correct for artifacts such as bulk refractive index changes, matrix effects, nonspecific binding, injection noise, and baseline drift due to temperature variation or pump artifacts. 16,I7 13 D. G. Myszka, P. G. Arulanantham, T. Sana, Z. Wu, T. A. Morton, and T. L. Ciardelli, Prot. Sci. 5, 2468 (1996). 14 B. C. Cunningham and J. A. Wells, J. Mol. Biol. 234, 554 (1993). 15 L. Joss, T. A. Morton, M. L. Doyle, and D. G. Myszka, Anal Biochem. (in press). 16D. G. Myszka, T. A. Morton, M. L. Doyle, and I. M. Chaiken, Biophys. Chem. 64,127 (1997). 17 R. Karlsson and A. F~ilt, J. Immunol. Methods 200, 121 (1997).

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KINETIC ANALYSIS USING BIOSENSORS 30 D

275

a

25

b

2o 15

C

10-

50I

I

I

I

I

I

0

20

40

60

80

100

Time (s)

FIG. 2. Flow rate dependence for Epo binding to the receptor construct surface. Epo at a constant concentration of 0.66 nM was injected at flow rates of (a) 100, (b) 33, and (c) 11/xl/min over a 181 R U receptor construct surface.

A reference surface can be an underivatized surface; however, it is better to treat the reference surface with the same chemicals used to immobilize the ligand to ensure the environments within the matrix are similar. When measuring the binding of small molecular weight analytes it is particularly important to immobilize a noninteracting protein at the same density. This helps to normalize the refractive index changes between sensor surfaces. When using a capturing step, the best results are obtained with a reference surface containing the immobilized capture molecule. Therefore, for the Epo-receptor construct interaction, the same amount of protein A was immobilized on two flow cells. One surface was used to capture the receptor construct and the other was left blank. Using serial injections available on B I A C O R E 2000, the same Epo sample was passed over both reaction and reference surfaces to generate binding data.

Experimental Conditions Flow Rate. Once the biosensor assay is configured properly, the next step is to collect interaction data. The first experimental condition to consider is flow rate. It is extremely informative to test each reaction for flow rate dependence. This was done for the Epo-receptor construct interaction by injecting the same concentration of Epo (0.66 nM) over the same receptor surface at three different flow rates (11, 33, and 100/zl/min). The results, shown in Fig. 2, indicated that the Epo binding responses were highly dependent on the flow rate, suggesting that the reactions are at least partially influenced by mass transport. 16,~s In order to minimize the effects of mass is R. Karlsson, H. Roos, L. Fagerstam, and B. Persson, Methods: Companion Methods Enzyrnol. 6, 99 (1994).

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transport during kinetic experiments, Epo was delivered to the interaction surface using the highest flow rate currently available on BIACORE 2000 (100/xl/min). High flow rates also improve the quality of response data for reactions that are not mass transport limited by reducing the dispersion of the sample plug and ensuring that a constant concentration of analyte is delivered to the biosensor surface. Binding Capacity. The next experimental condition to consider is how much ligand to immobilize on the surface. Ligand density is important because it determines the binding capacity. When performing kinetic experiments, the goal should always be to use the lowest capacity surfaces possible. Low capacity surfaces are advantageous because they minimize artifacts such as mass transport, steric hindrance, crowding, and aggregation. With BIACORE 2000, it is relatively easy to measure binding curves with maximum responses of 50 RU. With proper controls, accurate responses may be measured below 10 RU. Because the flow rate experiments suggested that the Epo binding reactions were influenced by mass transport, we wanted to keep the binding capacity low to further minimize mass transport effects. Two surfaces were created with a maximum capacity of approximately 45 RU and 15 RU. A comparison of binding rates to these different capacity surfaces will show if a reaction is influenced by mass transport. 16 Although it appears that this control is rarely performed, measuring reactions using multiple surface densities is extremely beneficial during data analysis. Analyte Concentration and Solution Conditions. Association rate constants are second order binding events and are therefore concentration dependent. Most data analysis techniques do not require an exact measure of the amount of ligand immobilized; however, determination of the association rate constant depends on an accurate knowledge of the analyte concentration. In turn, accurate sample dilution during the experiment becomes critically important. A particular concern is the loss of analyte due to absorption on sample tube walls or on surfaces within the biosensor. To minimize nonspecific protein adsorption, the addition of a low concentration of surfactant to the running buffer (i.e., 0.005% P20, Biacore, Inc.) is recommended. Because the Epo experiments were performed at very low protein concentrations (<100 pM), bovine serum albumin was added to the running buffer as a protein carrier (0.1 mg/ml) in addition to the P20 to prevent sample loss. In order to reduce bulk refractive index changes typically seen at the start and end of injection, a stock Epo sample was dialyzed against the running buffer before preparing the appropriate dilution. When setting up binding experiments, the analyte concentrations should be varied over a wide concentration range. This provides more information

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about the reaction mechanism. For the Epo-receptor construct experiments, the concentration of Epo was varied from 0.025 to 2 nM. It is also good to have at least some of the reactions reach equilibrium. This provides information about the equilibrium constant and makes it possible to normalize binding responses between different density surfaces (this will be discussed later). Sample Injection Volume. The sample injection volume determines the amount of time available to collect association phase data for a given flow rate. The larger injection volumes available on B I A C O R E 2000 (up to 750/zl using the BIGINJECT command) make it theoretically possible to extend the association phase for a much longer time. However, care must be taken to ensure that the sample plug does not disperse at the end of the association phase, which can occur with prolonged injections. Therefore, the Epo injections were limited to 150/zl to avoid complications due to sample plug dispersion. Sample Repetition and Randomization. Finally, in order to perform a statistical analysis of reaction data, each analyte concentration must be repeated and the order of injection randomized. This provides a modelindependent assessment of the total experimental noise, taking into account changes in protein activity (ligand or analyte) over time and the efficiency of surface regeneration. Determining the total experimental noise is essential when deciding whether a model adequately describes the data (as discussed later). Because most B I A C O R E instruments are automated, repeat injections should be standard practice in biosensor experiments. More statistical information can be obtained by repeating a small number of sample concentrations many times as opposed to performing a single injection of a large number of concentrations. Data for the Epo-receptor construct experiments demonstrate the importance of performing these replicate injections. Five concentrations of Epo plus a buffer blank were injected five times each. Data shown in Fig. 3 represent an overlay of all 60 binding experiments collected from the 15 RU and 45 R U capacity surfaces. Despite the higher level of random noise observed on these low capacity surfaces, the binding responses were highly reproducible. It would be impossible to confirm this level of reproducibility if repeat injections were not performed. Data Analysis

Methods Linear Transformation. The first method widely used to determine binding rate constants from biosensor data employed a linear transformation

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ENERGETICS OF BIOLOGICAL MACROMOLECULES

A

[131

B 15 45.

~"

35'

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15.

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350

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FIG. 3. Binding responses for the Epo-receptor construct interaction. Data represent five repeated injections of each Epo concentration (2, 0.66, 0.22, 0.074, 0.025, and 0 nM, top to bottom) over the receptor construct surfaces with 181 R U (A) and 58 R U (B) of receptor. Biosensor data were prepared for kinetic analysis of zeroing the baseline and the time of injection before each Epo experiment. Refractive index changes and nonspecific binding were corrected for by subtracting responses from a reference surface from the reaction surface.

of the data. 19 This method is only suitable for interpreting reactions that follow a simple bimolecular mechanism [Eq. (1)]. If the binding reactions are more complex, then transformed data will not be linear. Interpreting rate constants from these data requires selection of the appropriate regions for analysis, thus introducing a great deal of subjectivity and a large degree of error in the results. Integration. A second method of analyzing sensor data uses nonlinear analysis to fit an integrated rate equation directly to the binding data. Initial estimates for the rate constants provide simulated responses, which are then fit to measured data by adjusting the rate constants to minimize the difference. An exact solution is only available for the case of a simple interaction model [Eq. (1)] 20 and, by extension, for multiple simple interactions (i.e., heterogeneity in the ligand). This limits the types of models that can be easily applied to sensor data. For more complex interactions, a solution for the integration of the rate equations must be approximated using numerical methods. 21 Furthermore, as one only needs to formulate the differential equations for the reaction model, any reaction model may be used. It is customary to analyze the association and dissociation phases of each curve individually as is done using the analysis software program, 19 R. Karlsson, A. Michaelsson, and L. Mattsson, J. ImmunoL Methods 145, 229 (1991). 20 D. J. O'Shannessy, M. Brigham-Burke, K. K. Soneson, P. Hensley, and I. Brooks, AnaL Biochem. 212, 457 (1993). 21 T. A. Morton, D. G. Myszka, and I. M. Chaiken, AnaL Biochem. 227, 176 (1995).

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BIAevaluation 2.1 (Biacore Inc.). However, when a binding reaction is complex, there is not enough information in a single response curve to resolve accurate estimates for each parameter. This is a common problem encountered in curve fitting to exponential data. One method of extracting more information from binding responses is to apply global data analysis. Global Analysis. In global analysis, data sets that share parameter values are fit simultaneously. For example, a series of response curves collected with different analyte concentrations injected over the same surface may be fit simultaneously. All of the response curves should share the same association and dissociation rate constants and surface binding capacity; however, they will provide different amounts of information about each. A high concentration of analyte will quickly reach saturation, giving information about surface capacity but a poor estimate of association rate. Conversely, a low concentration gives information about the association rate but not the binding capacity. By fitting all data at the same time information from each is combined. As a result, global analysis provides a better test of the model and improves the statistical behavior of the parameter estimates; both the standard deviations and correlation coefficients are improved as compared to the linear transformation method and the nonlinear analysis method applied to individual response c u r v e s . 21 It is also much easier to visualize how well a model directly fits all of the primary data and an entire data set can be analyzed much more quickly--typical data sets take less than a minute to analyze. A major obstacle to using global analysis is that it requires very high quality experimental data. Therefore, all experimental artifacts must be minimized if simple models are to be used to describe data. BIACORE instruments are ideally suited for the application of global analysis. All of the instruments provide automated injections, resulting in reproducible sample applications and response profiles. This is clearly demonstrated in the Epo-receptor construct binding responses shown in Fig. 3. Note how the contact times for the association phases are identical and how well the responses overlap for each repeated injection. The derivatized carboxymethyldextran surface is very stable, allowing reproducible responses between experiments. Furthermore, the signal from the instrument has low, normally distributed noise with very little drift. When experiments are performed with care, response data collected from BIACORE can be described by a simple bimolecular reaction model, as illustrated for two antigen-antibody s y s t e m s ] 7'22 CLAMP: Biosensor Data Analysis Program. We demonstrated previously that global analysis of biosensor data could be used to accurately 22 k D. R o d e n and D. G. Myszka, Biochem. Biophys. Res. Comm. 225, 1073 (1996).

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Input Model& 1 Parameter Starting Values

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determine the binding mechanism and rate constants associated with interactions recorded on BIACORE. 2l This work led us to develop the global data analysis program CLAMP. CLAMP combines numerical integration and nonlinear curve-fitting routines. It is designed to interpret complex interactions recorded on biosensors by simultaneously analyzing association and dissociation phase data measured on different surfaces or in different experiments. The flow chart in Fig. 4 demonstrates how the program works. To begin an analysis, the user inputs data, loads a reaction model, and enters reasonable starting values for the unknown parameters. The program will automatically generate a series of differential rate equations based on the model and will integrate them numerically to simulate a set of response curves using a semi-implicit extrapolation method. 23This method is capable of handling stiff sets of differential rate equations. These occur when two 23 W. H. Press, S. A. Teukolshy, W. T. Vetterling, and B. P. Flannery, "Numerical Recipes in C." Cambridge Univ. Press, Cambridge, 1992.

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reactions take place with very different rates and, when not treated properly, may cause instability in the integration. After simulated data are generated, they are subtracted from experimental data in order to calculate the X2. The initial estimates for the rate constants are then adjusted to minimize the X2 using a L e v e n b u r g Marquardt nonlinear minimization algorithm. 21 If the new estimates do not reduce the )(2, they are discarded, the fit progress is decreased, and another set of parameter estimates is chosen. Estimates that decrease the X2 are kept and the fit progress is increased. The simulation and minimization process is repeated until the change in the X2 is less than 0.1%. The parameter values, along with their linear approximation standard deviations and correlation coefficients, are output in a results table. C L A M P runs as a self-contained program on Microsoft Windows Operating Systems 95, NT, and 3.1. The program may be downloaded for free at the following web site: (http://www.hci.utah.edu/cores/biacore/). This site also contains tutorials demonstrating how to use the program, as well as some examples of how global analysis has been applied to biosensor data. Other biosensor analysis programs that utilize global analysis have been made available, including SPRevolution, which runs within Microsoft E X C E L 7.0 ((http://www.bri.nrc.ca/csrg/equip.htm)), and an upgraded version of BIAevaluation 3.0 (http://www.biacore.com/).

Analyzing Experimental Data Normalized Plots. An effective method for comparing data between different density surfaces is to normalize the y axis so the responses that reach steady state are plotted at the same level. An example of this is shown for Epo data in Fig. 3. A comparison of the response curves between the two data sets reveals that those recorded on the lower capacity surface (Fig. 3B) have a faster binding rate than those on the higher capacity surface (Fig. 3A). In other words, those on the lower capacity surface approach equilibrium quicker. This indicates that the reactions are influenced by mass transport) 6 Mass transport effects are seen when the binding rate for material onto the sensor surface is faster than the rate of diffusion of material to the surface. 24 A lower capacity surface will have less demand for analyte and therefore will be less affected by mass transport. In general, this is why it is important to perform experiments on low capacity surfaces. Fitting Models to Data. The first step in analyzing experimental data is to identify the simplest model that is consistent with what is known about the system and accurately describes the binding responses. For the Epo24S. Sjolander and C. Urbaniczky,Anal.

Chem.

63, 2338 (1991).

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receptor construct interaction, we started by fitting a simple bimolecular interaction model to the data [Eq. (1)]. The following differential rate equations were used to globally fit the data from two surfaces: Surface 1:

Surface 2:

d[Al/dt d[B]/dt d[ABl/dt d[A]/dt d[b]/dt d[Ab]/dt

=0 = -ka([A][B]) + kd[AB] = k,,([A][B]) - kd[AB]

=0 = -k,,([A][b]) + ka[Ab] = ka([A][b]) - ka[Ab]

The model assumes that the analyte concentration ([A]) remains constant during the association phase and is zero during the dissociation phase. [B] and [b] represent the concentration of free receptor sites on the high and low capacity surfaces, respectively. Fitting this model to Epo response data required a total of four parameters. The same association and dissociation rate constants (/ca and kd) were applied to data from both surfaces. However, each surface had its own maximum binding capacity (Bma x and bmax). The best fit of a simple interaction model to Epo data is shown in Figs. 5A and 5B (see color insert). A visual comparison of the modeled responses (red lines) and experimental data (black lines) demonstrates that a simple reaction model does not accurately describe the data. No minimization procedure can guarantee that the solution found is unique or optimum, and failure to find a good fit does not mean that a good fit does not exist for the chosen model. It is therefore necessary to choose a variety of different starting values to confirm that the best fit has been found. For the Epo-receptor construct interaction, the fitting process was repeated 100 times with different starting values for the parameters. When a model fails to fit data, there are two options to consider. The first is to recollect data under experimental conditions that will remove artifacts that may be influencing the responses. The second is to change the reaction model to include these artifacts or perhaps to account for a more complex reaction. Due to the nearly optimal experimental conditions under which Epo data were collected, it was necessary to consider a more complex reaction model. Because the flow rate experiments and normalized responses from the different density surfaces suggested that the Epo reaction may be mass transport limited, we added a mass transport step to the model, as shown in Eq. (2). kt

ka

A0 ~ A + B ~ AB kt

kd

(2)

A

B 15 45

35 10

no tO

25 5

15

O tr

5

-5 50

150

250

350

50TT

Time (s)

150

250 350

Time (s)

D

C

15 45

35 10

m v

25 c

0 o.

15

n5

-5

50

150

250

Time (s)

350

50

150

250

350

Time (s)

FIG. 5. Global analysis of response data for the Epo-receptor construct interaction. Experimental data (black lines) represent five repeat injections of each Epo concentration (2, 0.66, 0.22, 0.074, 0.025, and 0 nM) over 181 RU (A and C) and 58 RU (B and D) receptor construct surfaces. Association and dissociation phase data from each concentration of Epo, and across both surface densities of the receptor construct, were fitted simultaneously. The red lines show the best fit of binding data to a simple bimolecular reaction [Eq. (1)] (A and B) and a two-step mass transport limited reaction [Eq. (2)] (C and D).

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The following differential rate equations were used to describe this reaction across two different capacity surfaces: Surface 1:

Surface 2:

d[Ao]/dt d[Al/dt d[B]/dt d[AB]/dt d[Ao]/dt d[A]/dt d[b]/dt d[Ab]/dt

=0 = kt([Ao] - [A]) - ka([A][B]) + kd[AB l = -ka([A][B]) + kd[AB] = ka([A][B]) - kd[AB]

=0 = kt([Ao] - [A]) - k,,([A][b]) + kd[Ab] = -ka([A][b]) + kd[Ab] = ka([A][b]) - kd[Ab]

In the mass transport model, the concentration of analyte in bulk flow ([A0]) is assumed to be constant during the association phase and zero during the dissociation phase. The concentration of analyte at the sensor surface ([A]) is zero at the start of the reaction. The mass transport coefficient, kt, describes the transport rate of analyte to and from the biosensor surface. Therefore, a total of five parameters are used in the model (Bmax, b . . . . kt, ka, and kd), only one more than was used in a simple bimolecular reaction. Figures 5C and 5D show that this reaction model describes E P O data more accurately than the simple bimolecular reaction [Eq. (1)], as shown in Figs. 5A and 5B. The modeled responses (red lines) agree very well with experimental data (black lines) for each Epo concentration across the entire time course of binding. Mass transport limitations are a common problem with fast binding reactions (typically ka of 10 6 M -I sec ~ or greater). A detailed discussion of the advantages and limitations of the two-step mass transport limited reaction model can be found in Myszka et aL 25 Additional methods of testing for mass transport limitations are discussed later. Assessing Quality of Fit. Residual plots provide a visual assessment of how well the model fits the data. These plots were generated for the Eporeceptor construct by taking the difference between the theoretical curve described by the mass transport model [Eq. (2)] and the observed data points. The residual plots for each Epo concentration, generated from a global fit of all concentrations, are displayed in Fig. 6. Residuals were generally small and randomly distributed about the x axis, indicative of a good fit. The small amount of nonrandom behavior in the two highest analyte concentrations, a product of bulk refractive index changes or possible heterogeneity at high concentration, will have a negligible effect on the magnitude of the returned parameter values. Assessing individual residual 25D. G. Myszka, H. Xiaoyi, M. Dembo, T. A. Morton, and B. Goldstein, BiophysicalJournal (in press).

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A

[ 13]

B 3

2 nM

2

066

O~

n," 2

0

400

2oo

3oo

Time (s)

4oo

0

too

2oo

300

4OO

Time (s)

FIG. 6. Residual plots for the mass transport mode] fit to the Epo-receptor construct

reaction. Residuals for the global fit of Epo data collected over the (A) 181 RU and (B) 58 R U receptor construct surfaces are displayed separately for Epo concentrations ranging from 0 to 2 nM.

plots for all data sets involved in a global fit ensures that each data set is described equally well by the chosen model. The residuals can be quantitatively assessed by comparison of their standard deviation with the expected level of instrument noise. In the Epo experiments, the residual standard deviation for the two-step mass transport model was 0.51 R U , which is close to the short-term random noise reported for BIACORE 2000 (0.3 RU, BIACORE manual). However, the shortterm instrument noise does not take into account changes in ligand or analyte binding activity, surface regeneration efficiency, or other variables that may affect the response during the time course of the experiment. One way to determine the total experimental noise, independent of any model, is to use the replicate binding experiments to calculate the replication standard deviation. This is the standard deviation of the individual experiments about their average. For Epo-receptor construct data, the replication standard deviation was 0.46 RU. The difference between the residual and the replication standard deviation was therefore only 0.05 RU. This indicated that very little information was left in the residuals from Epo-receptor construct data, confirming that the two-step mass transport model describes the data quite well. Testing Alternative Reaction Models. Once a model is found to adequately describe response data, the next step is to test whether other

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reaction models can also provide as good a fit to the data. We tested Epo-receptor construct data against a two-step [Eq. (3)] and a three-step conformational change reaction model [Eq. (4)], as well as a surface heterogeneity model [Eq. (5)]. ka

k2

A + B ~ AB ~ (AB)* kd

k_ 2

ka

k2

(3) k3

A + B ~ AB ~ (AB)* ~ (AB)** kd

k 2

(4)

k-3

kal

AB

B kdl

(5)

A+ ka2

AB*

B* kd2

In the conformational change reactions [Eqs. (3) and (4)], (AB)* and (AB)** represent alternate states of the complex. In the heterogeneity reaction [Eq. (5)], B and B* represent two independent binding sites on the sensor surface. Although each of these models contained additional parameters, they all failed to improve the quality of the fit over a simple bimolecular interaction [Eq. (1)]. Thus, by using global analysis, it is possible to distinguish between mass transport-limited reactions and those complicated by heterogeneity or conformational changes. It can, however, be very difficult to distinguish between the latter two reactions based on the fit alone. Testing Reaction Models Experimentally. It is possible to test the validity of most models used to describe biosensor data by varying experimental conditions on the biosensor itself. For instance, varying the flow rate was the first indication of a mass transport effect on the Epo-receptor construct interaction. Varying the surface capacity provided evidence that the reaction was still influenced by mass transport even at the high flow rates used to collect kinetic data. Together, this experimental evidence confirmed that the Epo-receptor construct reaction was mass transport limited on the biosensor, supporting the model used to describe data. Another test for mass transport is to inject soluble ligand during the dissociation phase. The ligand in solution will compete for dissociating analyte and prevent it from rebinding to the surface. Unfortunately, it was not possible to perform this test with the Epo-receptor construct interaction because of the protein A capturing step. If the receptor construct were injected, it would bind to the protein A surface.

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One of the problems encountered in data analysis is that several models m a y provide equally good fits to data. Although this was not observed for the E p o - r e c e p t o r construct interaction, it was observed for an a n t i b o d y antigen interaction. 17 In this work, a two-step conformational change reaction and a surface heterogeneity reaction provided equally good fits to a data set even when using global analysis. However, by varying the association time and rerunning material collected f r o m the biosensor surface, it was possible to provide convincing arguments in favor of the two-step conformational change reaction. A key advantage of biosensors is that experimental conditions can be altered easily in order to provide support for the assigned reaction model.

Establishing Parameter Confidence The last step in data analysis values for the rate constants and section discusses how to interpret as demonstrate the advantage of

is to establish confidence in the returned other p a r a m e t e r s used in the model. This the linear approximation statistics, as well using M o n t e Carlo analysis and Profiling. Linear Approximation Statistics. The best-fit p a r a m e t e r values, standard errors, and correlation coefficients for the E p o - r e c e p t o r construct interaction are shown in Table I. The first item to consider when reporting the results from nonlinear fitting routines is the standard errors in the p a r a m e t e r estimates. Low standard errors indicate that the p a r a m e t e r estimates are well defined. The standard errors were exceptionally low for all of the p a r a m e t e r s estimated in the E p o - r e c e p t o r construct interaction, averaging around 1.4% of the p a r a m e t e r values (see Table I). The standard errors were low because data were analyzed globally, resulting in highly constrained p a r a m e t e r values. W h e n data from the two surfaces were fitted separately, the standard errors were approximately twice as large. We have shown

TABLE I LINEAR APPROXIMATIONSTATISTICSRETURNED FOR Two-STEP MASS TRANSPORT MODEL FIT TO EPO-RECEPTOR CONSTRUCT DATA

Parameter

Estimate

% standard error

Bm~,c(RU) bmax (RU) kt (RU M-1 sec-1) ka (M -1 sec-1) kd (sec-1)

43.22 13.52 5.80 × 108 8.09 X 107 2.44 × 10 4

0.09 0.15 3.4 0.7 2.9

Correlation coefficient Bmax

bmax

kt

. . . . 0.38 --0.21 0.01 ---0.24 --0.08 --0.90 0.43 0.71 --0.09

k. ---0.15

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previously that when response curves are analyzed nonglobally, the standard errors may be more than 10-fold higher. 21 The second item to consider when intepreting the results of nonlinear curve fitting is the correlation between parameters. Correlation coefficients are a measure of how the value of one parameter depends on the value of another. Completely independent parameters will have a correlation coefficient of zero, which is desirable. Parameters may be positively or negatively correlated in the range of - 1 to 1. Most of the correlation coefficients for Epo-receptor construct data were very low (< _0.7), as shown in Table I, with the exception of kt and ka (-0.90). However, this level of correlation was only marginally significant, with correlation coefficients greater than 0.98 considered near the limits of tractable problems. Parameters that are highly correlated do not have unique values, as changes in one will be compensated for by a change in the other. In cases where parameters are highly correlated or the standard errors are high, Monte Carlo analysis may be used to increase confidence in the parameter values. Monte Carlo Analysis. Monte Carlo analysis can be used to determine how random noise affects parameter values. Data are simulated using the reaction model of choice and the best-fit parameter values that are used to fit the original experimental data set. Random noise is added to simulated data commensurate with the levels expected during an actual experiment. These data are then analyzed to determine whether the input parameter values can be accurately recovered. While analyzing Epo-receptor construct data, we were particularly concerned that only 10% of the complex had decayed during the dissociation phase. Ideally, one would want to measure the complex dissociation for at least one half-life. However, the amount of time we could collect dissociation phase data was limited because eventually the decay of the receptor construct-protein A complex would contaminate Epo binding data. Therefore, Monte Carlo analysis was used to ensure that we could recover a unique value for the dissociation constant from this small amount of decay data. Data were simulated using the best-fit parameter values determined from analysis of experimental data. Randomly distributed noise was added to data at a standard deviation of 0.5 RU, which was similar to the experimental residual standard deviation. These data were then analyzed using the same procedures used to fit original experimental data. The simulation and fitting process was repeated 100 times. Averages of the returned parameter values, along with their standard deviations, are given in Table II. Each of the returned parameter values, including the dissociation rate constant, was within standard error of the input parameter values used to simulate data. Therefore, Monte Carlo analysis demonstrated that accurate parameter values could be returned from analysis of experimental Epo data.

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T A B L E II RESULTS OF MONTE CARLO ANALYSIS

Parameter

Input value

Returned value

% standard error

Bmax (RU) bm,x (RU) kt ( R U M -i sec -1) ka (M -1 sec -1) kd (sec 1)

43.2 13.52 5.80 x 108 8.09 × 107 2.44 N 10 -4

43.2 13.50 5.87 × 10s 8.1 x 107 2.4 M 10 4

0.2 0.2 0.2 3.7 4.2

A Monte Carlo analysis routine is available in CLAMP, making it convenient to apply during data analysis. Profiling. The last item to consider when establishing parameter confidence is whether the linear approximation statistics are in fact valid. Although linear approximation statistics are the simplest and most common approach in determining confidence intervals using nonlinear analysis, they can be highly misleading. The error space around parameters in nonlinear models may not be symmetrical. Fortunately, an efficient procedure termed profiling can be used to visualize the error space and correlations for all parameters in a model and to provide accurate likelihood intervals in cases when they are nonlinear. The effectiveness of the technique hinges on the fact that a great deal of information about the behavior of a multidimensional surface is revealed when the profile surface is viewed in one- and twodimensional subspaces. 26 Because a complete description of the profiling process is beyond the scope of this article, the reader is advised to consult Ref. 27 for review. An effective way to visualize the error space for a parameter is via a profile t plot. H e r e the best-fit parameter value is centered at zero and the error space is scaled to produce a line with unit slope, z8 Deviations from a straight line at a 45 ° are an indication that the error space of the parameter is nonsymmetrical. The profile t plots for the five parameters (Bmax, b . . . . kt, k,, and kd) used to describe Epo-receptor construct interaction data are shown on the diagonal of the matrix in Fig. 7. All of these plots are nearly straight lines with unit slope, indicating that the error space around each parameter was linear and, therefore, the parameter confidence intervals were essentially 26 D. G. Watts, Cand. J. Chem. Eng. 72, 701 (1994). 27 D. M. Bates and D. G. Watts, "Nonlinear Regression Analysis and its Applications." Wiley, New York, 1988. z8 D. M. Bates and D. G. Watts, Chemometr. Intelligent Lab Syst. 10, 107 (1991).

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v

/ v t-,

¢<3

,

i

Z'I

,

i

© 0 •

,

4

J

•

I

,

..,Z

i

./

,

/ v

J 0

/

o,

-'2o v -2 -4

, -4

,.", -2

i , i 0 2

Ji

,

, i

8 5

,

i

,

4 (bma

x)

5

(kt)

•

.

•:1

•

(k a)

i

.

,

t

.

~ (kd)

(Brnax)

FIG. 7. Profile plots for the parameters in the mass transport model globally fit to Eporeceptor construct interaction data. Profile t plots are shown on the diagonal of the matrix. These plots include a linear reference line (dashed line at 45°), which is obscured by the profile t plot itself (solid line) when the error space is linear. In the profile trace plots (off the diagonal of the matrix) the solid and dashed intersecting lines are the profile traces. The solid and dashed closed lines correspond to 68, 90, and 95% joint likelihood regions. Scales shown in the lower left-hand corner are applicable to all the plots in the figure.

symmetrical. This confirmed that the standard errors reported from the linear a p p r o x i m a t i o n statistics w e r e reasonable. If these plots w e r e not linear, t h e n accurate c o n f i d e n c e limits w o u l d h a v e to be d e t e r m i n e d instead f r o m the profile t plot. 26 W h e n n o n l i n e a r m o d e l s are used, a n o t h e r c o n c e r n is that the correlation space b e t w e e n parameters m a y not be linear. O n e m e t h o d o f visualizing the correlation b e t w e e n parameters is to create profile trace plots that p r o v i d e a t w o - d i m e n s i o n a l r e p r e s e n t a t i o n o f the error space in terms o f any

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two parameters. 26 Because the mass transport model had five parameters, it required a total of 10 trace plots to generate all the cross correlations. These plots are displayed off the diagonal of the matrix in Fig. 7. The contour levels on these plots show the joint 68, 90, and 95% confidence regions for each set of parameters. For a linear system, each contour plot would be a perfect ellipse; the narrower the ellipse, the higher the correlation. For the Epo-receptor construct interaction, correlations among all the parameters were essentially linear. Some of the trace plots, such as bmax vs kt, showed virtually no correlation among parameters. The only parameters showing a significant level of correlation were kt vs ka. The slopes of the solid line shown on these plots are equal to the correlation coefficients shown in Table I. Profiling has demonstrated that even though the model used to fit Eporeceptor interaction data was formally nonlinear, the error spaces around the minimum of the parameter estimates were effectively symmetrical. This was the result of globally fitting a broad range of data collected from different capacity surfaces. We showed previously that nonlinear error spaces resulted for some parameters when a mass transport model was used to fit data from only a single capacity surface. 14 Only when data were globally fit from multiple surfaces of differing capacity were the error spaces constrained enough to become effectively linear. Unfortunately, it is not possible to predict when parameter estimates will be nonlinear. This is because their behavior in a nonlinear model is a complex function of many factors: the formulation of the equation, the way the parameters are expressed in the equation, the experimental design used, the residual variance, and even the parameter estimates themselves. 26 Therefore, in order to provide confidence in the parameters returned from nonlinear least-squares analysis, it is necessary to test each new data set as well as each reaction model.

Interpreting Results

Binding Stoichiometry Binding stoichiometry can provide useful information on the integrity of the immobilized ligand. If stoichiometry is less than expected, it could be an indication that the ligand has lost activity, something commonly seen when random amine coupling is used to immobilize proteins. Because the responses from SPR biosensors are proportional to the mass of the molecule, it is straightforward to estimate a binding stoichiome-

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try using a ratio of the observed binding responses and known molecular weights for the analyte and ligand, as shown in Eq. (6). Binding stoichiometry = (analyte maximum capacity/ligand density) (6) (analyte MW/ligand MW) The two different receptor surfaces used in the Epo experiments, 181 and 58 RU, had maximum Epo binding capacities of 43.33 and 13.52 RU, respectively. M A L D I mass spectrometry was used to accurately determine the molecular masses of Epo and the receptor construct, which were found to be 28.1 and 107.1 kDa, respectively. 6 Entering these values into Eq. (6), the binding stoichiometries of the two different receptor surfaces were found to be 0.91 : 1 and 0.89 : 1. This suggests that only one Epo is bound by the dimeric receptor construct.

Mass Transport Coefficient All interactions recorded on a biosensor involve the transport of analyte to and from the reaction surface. Therefore, it is always important to use low capacity surfaces and high flow rates to minimize the effects of mass transport. The Epo experiments demonstrated that when the association rate constant is very fast, even under improved experimental conditions, mass transport may still influence the binding responses. In order to extract accurate estimates for the binding constants under these conditions, it is necessary to incorporate a mass transport step into the reaction model. In the two-step mass transport limited model [Eq. (2)], kt represents an averaged mass transport coefficient. The most convenient way to analyze B I A C O R E data is to model reactions directly in response units. This requires that kt be expressed in units of sec 1. The value for kt is dependent on the analyte molecular weight and diffusion coefficient as well as on the flow rate and dimensions of the flow cell. 22 It is possible to estimate a diffusion coefficient from the mass transport coefficient using Eq. (7). 25

D =

1 . 2 8 2 M W f J Vc

All of the symbols used in Eq. (7) for the Epo-receptor construct experiments are defined in Table III. The experimentally determined transport coefficient for Epo was 5.80 × 108 sec -1. The flow rate used during the experiment, 100 /xl/min, corresponds to a flow velocity of 10 cm sec -1. The flow cell height and length are estimates provided by the instrument manufacturer (Biacore Inc.). Solving Eq. (7) with these values gave a predicted diffusion coefficient equal to 7.07 x 10 .7 c m 2 s e c -1. This value is

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T A B L E III VALUES USED IN EPO-RECEPTOR CONSTRUCT EXPERIMENTS

Symbol

Definition

Value

MW f vc h 1

Molecular weight Conversion factor Flow velocity Flow cell height Flow cell length

2.81 x 104 g mo1-1 1 × 107 R U liter g 1 6.67 cm sec 1 5 × 10 -3 cm 2.4 × 10 -1 cm

very close to the diffusion coefficient of 8.87 × 10 -7 cm 2 sec -1 determined experimentally for E p o using sedimentation velocity. 6 The same mass transport model was used to fit two other reaction systems studied on B I A C O R E J 3'16In each of these cases the mass transport coefficient determined f r o m data analysis predicted a diffusion coefficient that was slightly slower than that expected for diffusion through buffer alone. One possible cause of the deviation is the carboxymethyldextran matrix, which m a y lead to a slower apparent diffusion rate. However, given the n u m b e r of assumptions put forth in the calculation of the diffusion coefficient f r o m k , a complete interpretation of mass transport coefficients obtained from B I A C O R E data must wait until m o r e systems have been studied experimentally. If the dextran layer does contribute to the mass transport coefficient, the similarity in the values for the diffusion coefficient suggests that its contribution m a y be very minimal.

Equilibrium Dissociation Constant Based on the law of mass action, the binding rate constants (ka and ku) can be used to calculate an equilibrium dissociation constant ( K o ) as shown in Eq. (8).

KD = kd/ka

(8)

For the E p o - r e c e p t o r construct interaction, the association rate constant of 8.09 × 107 m -1 sec -1 is one of the fastest association constants obtained from B I A C O R E to date. The dissociation rate constant was slow (2.44 × 10 4 s e e 1), with a half-life of around 47 min. The combination of a slow dissociation rate constant and fast association rate constant gave a very low equilibrium dissociation constant of 3.01 pM.

Corroborating Experiments Given the complexities inherent in recording macromolecular interactions at surfaces, it is important to have corroborating data f r o m solution studies to support the results of biosensor experiments. T h e apparent 1 : 1

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binding stoichiometry observed for the Epo-receptor construct on the biosensor is consistent with the results of analytical ultracentrifugation and titration calorimetry studies. 6 Differential scanning calorimetry confirmed that the dissociation constant for the Epo-receptor construct interaction was less than 10 pM. 6 Also, the receptor construct was 1000-fold more effective in neutralizing biological activity than the monomeric soluble receptor, 6 which has an affinity of approximately 1 n M . 4 Together, these results provide confidence in the low equilibrium dissociation constant determined from the biosensor. Epo Ligand-Receptor Interaction Mechanisms On the cell surface the erythropoietin receptor is thought to homodimerize on binding Epo ligand, similar to a two-site sequential binding mechanism proposed for growth hormone-receptor interactions.29 The homodimerization model is supported by biophysical experiments carried out with the soluble extracellular domain of tile erythropoietin receptor. 4 This work demonstrated that there were two independent receptor binding sites on Epo: a high-affinity site of - 1 nM and a low-affinity site of - 1 / x M . The affinity determined for the Epo-receptor construct interaction ( - 3 pM) is 330 times higher than the high-affinity site observed with the monomeric receptor. The increased affinity and 1 : 1 binding stoichiometry observed for the receptor Fc construct suggest that a single Epo molecule is complexed simultaneously to both receptor promoters. It is difficult to form complexes with soluble monomeric receptors because the entropic contribution from the membrane is lost. The close spatial proximity of the receptor protomers within the Fc construct may provide a tool for performing structure-function studies on Epo ligand-receptor interactions in the context of the dimeric signaling complex. The contained volume of the dextran layer itself has also been used to study receptor complex formation for both homo- and heterodimeric receptor systems.12'13Soluble receptor subunits immobilized on the biosensor surface formed high-affinity complexes in the presence of ligand. Reconstructing receptor complexes on the biosensor surface represents the next step in the application of biosensor technology. Conclusion Surface plasmon resonance biosensors have the potential to provide detailed information about the kinetics of macromolecular interactions. 29B. C. Cunningham,M. Ultsch, A. M. de Vos, M. G. Mulkerrin, K. R. Clauser, and J. A. Wells, Science 254, 821 (1991).

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However, in order to properly interpret kinetic experiments, data must be collected under the appropriate conditions. A number of experimental artifacts can complicate binding responses. Many of these can be minimized by simply improving the design of the experiment. High flow rates, low capacity surfaces, and suitable controls are requirements for detailed kinetic analyses. Repeating and randomizing the binding reactions provide essential information for data analysis and should be standard practice. Numerical integration and global fitting software programs can extent the range of rate constants as well as the types of reaction mechanisms that can be applied to biosensor data. Furthermore, Monte Carlo analysis and profiling procedures are beneficial for establishing confidence in the values reported for reaction rate constants. Biosensors offer a unique opportunity to extract quantitative information on macromolecular interactions for a broad range of reactants. Improving both data quality and analysis methods will ensure that the results of biosensor experiments are interpreted correctly. Acknowledgments This work was supported by the Huntsman Cancer Institute. We thank Peter Young, Charles Griffin, and Christopher Jones from SmithKline Beecham for providing reagents used in these studies, and Preston Hensley and Michael Doyle for providing unpublished biophysical information on the interaction. We also thank Joan Stuart for helpful comments on this review.

[14] P r e d i c t i o n o f B i n d i n g E n e r g e t i c s f r o m S t r u c t u r e Using Empirical Parameterization

By BRIAN

M . B A K E R a n d KENNETH P. MURPHY

Introduction The prediction of binding energetics from the three-dimensional structures of protein-ligand complexes is a long-standing goal of biophysical chemistry and a key element in structure-based drug design. Empirical, solvent-accessible surface area-based calculations, first applied to the prediction of protein unfolding energetics, 1-6 have proven useful in predicting 1 K. P. Murphy and E. Freire, Adv. Prot. Chem. 43, 313 (1992). 2 K. P. Murphy, V. Bhakuni, D. Xie, and E. Freire, J. Mol. Biol. 227, 293 (1992). 3 R. S. Spolar, J. R. Livingstone, and M. T. Record, Jr., Biochemistry 31, 3947 (1992). 4 G. I. Makhatadze and P. L. Privalov, J. Mol. Biol. 232, 639 (1993). 5 p. L. Privalov and G. I. Makhatadze, J. Mol. Biol. 232, 660 (1993). 6 D. Xie and E. Freire, Proteins 19, 291 (1994).

METHODS IN ENZYMOLOGY, VOL. 295

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