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Global Analysis of a Macromolecular Interaction Measured on BIAcore
BIOCHEMICAL AND BIOPHYSICAL RESEARCH COMMUNICATIONS ARTICLE NO.
225, 1073–1077 (1996)
1297
Global Analysis of a Macromolecular Interaction Measured on BIAcore Lin D. Roden and David G. Myszka1 Oncological Sciences Department, Huntsman Cancer Institute, University of Utah, Salt Lake City, Utah 84112 Received July 12, 1996 We demonstrate that the interaction between myoglobin and an immobilized anti-myoglobin antibody measured on BIAcore 2000 can be described by a simple bimolecular reaction mechanism. We improved the quality of the sensor data by correcting for refractive index changes and nonspecific binding using a blank sensor surface. Applying nonlinear least squares analysis, we simultaneously fitted the association and dissociation phase data generated for a range of myoglobin concentrations injected across the antibody surface. The ability to globally fit these data to a simple binding model indicates that effects related to the sensor surface, like mass transport and the dextran matrix, did not complicate the observed binding responses. These results illustrate the potential of BIAcore to monitor macromolecular interactions in real-time and the utility of global analysis to resolve the reaction kinetics. q 1996 Academic Press, Inc.
BIAcore is an optical biosensor that can be used to monitor macromolecular interactions in real-time without labeling requirements [1,2,3]. This makes it a versatile instrument for determining the kinetic rate constants for a variety of interactions. Often the responses recorded from the sensor do not conform to a simple bimolecular interaction model as expected. These departures may be inherent in the molecules being studied or they may be introduced by the biosensor itself. Surface artifacts such as mass transport and chemical or spatial heterogeneity imposed by the dextran matrix have been cited as probable causes of deviations [4]. Biosensor data are typically analyzed by separately fitting selected portions of the association and dissociation phase data to a simple bimolecular interaction model, and conclusions are then drawn about the overall reaction. Intuitively, this is not a satisfying approach to data analysis and it can provide misleading results when the binding reactions are complex [5]. Ideally, using global analysis one would like to show that the entire data set are described by a particular reaction mechanism. Global analysis can be used to discriminate between possible binding mechanisms, but in order to be routinely employed, it requires data that are free from as many instrument and experimental artifacts as possible. Given the growing interest in biosensor data analysis and concerns over instrument artifacts, we present an example of a macromolecular interaction recorded on the sensor that is accurately described by a simple bimolecular interaction model using global analysis. EXPERIMENTAL Materials. BIAcore 2000, research grade sensor chip CM5, NHS, EDC, 1M ethanolamine-HCl pH 8.5, running buffer (10 mM HEPES, 150 mM NaCl, 3.4 mM EDTA and 0.005 % (v/v) surfactant P20 at pH 7.4), human myoglobin, mouse anti-myoglobin monoclonal antibody, coupling buffer (10 mM acetate pH 4.8), and regeneration buffer (glycine pH 2.0) were purchased from Pharmacia Biosensor (Uppsala, Sweden). Sensor experiments. The mAb was immobilized using standard amine coupling procedures [3]. To activate the
1
Corresponding author. Fax: (801)-585-3833. E-mail: [email protected]. Abbreviations used: mAb, monoclonal antibody; Bmax , maximum surface capacity; ka , association rate constant; kd , dissociation rate constant; RU, resonance units; STD, standard deviation; NHS, N-hydroxy-succinimide; EDC, Nethyl-N*-(3-dimethylaminopropyl)-carbodiimide hydrochloride. 1073 0006-291X/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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Vol. 225, No. 3, 1996
BIOCHEMICAL AND BIOPHYSICAL RESEARCH COMMUNICATIONS
FIG. 1. Example of the raw sensor data. Myoglobin (330 nM) was injected over a surface containing 1000 RU of the mAb (a) as well as a blank sensor surface (b). At 475 seconds, both surfaces were regenerated with a 10-mL pulse of 10 mM glycine, pH 2.0.
carboxymethyl dextran surface, a 35 mL mixture of EDC/NHS was injected over a CM5 sensor surface at a flow rate of 5 mL/min. The mAb was injected in coupling buffer at a concentration of 50 mg/mL until the desired level of immobilization (1000 RU) was achieved. This was followed by a 35 mL injection of ethanolamine to cap the remaining activated carboxyl groups and a 10 mL injection of 10 mM glycine pH 2.0 to wash out any unbound mAb. A second flow cell was activated and deactivated without coupling the mAb as a control surface for refractive index change and nonspecific binding. Myoglobin samples contained 330, 110, 37, 12, and 4 nM protein created from a three fold serial dilution of a stock solution. To perform the kinetic binding studies, the flow path was changed to include both flow cells and the data collection rate was set to high. The instrument was programmed to KINJECT 23 mL of sample at a flow rate of 30 mL/min with a 300 second dissociation phase. After the end of the dissociation time, the flow cells were washed with 10 mM glycine pH 2.0 to regenerate the sensor surfaces. Each myoglobin sample, as well as a buffer blank were duplicated and injected in random order. The effect of flow rate was assessed by injecting a 37 nM concentration of myoglobin at flow rates of 10, 30, and 90 mL/min. Data analysis. The raw sensor data were prepared for global analysis by subtracting the average of the response over 20 seconds prior to the myoglobin injection and zeroing the time of injection for each flow cell. To correct for refractive index change and nonspecific binding, the responses obtained from the control surface were subtracted from the mAb surface data. The corrected binding data were then analyzed by direct curve fitting to a simple bimolecular interaction mechanism (A / B Å AB). Modeled data sets were generated by numerical integration of the differential equations which describe the reaction [6]. These were fit to the association and dissociation phase sensor data at all myoglobin concentrations simultaneously using algorithms described previously [5]. The error space for each of the parameters was assessed using statistical profiling [7].
RESULTS AND DISCUSSION
We chose to characterize the interaction between human myoglobin and a mouse antimyoglobin monoclonal antibody because the reagents are readily accessible for study on the sensor. The first step in analyzing this reaction was to simplify the interaction to be measured on the sensor surface. Since the mAb is bivalent, we decided to immobilize it and monitor a monovalent interaction with the myoglobin in solution. Examples of the raw response data simultaneously collected for an injection of myoglobin over the mAb and blank sensor surfaces are shown in Fig 1. A significant portion of the signal observed during the association phase (Ç90 RU) is due to a difference in the composition of the running buffer and the sample plug, which is referred to as a bulk refractive index change. The data from the blank sensor surface also indicates that there is a small amount of nonspecific binding presumably to the dextran matrix. Subtracting the response recorded over the blank surface from the mAb surface data corrects for these system artifacts, revealing the genuine response curves for the myoglobinmAb interaction. Bound myoglobin was removed and a fully active mAb surface was regenerated using a glycine wash. Figure 2 shows an overlay of the corrected binding responses for a series of myoglobin concentrations injected over the same mAb surface. Each binding experiment was repeated 1074
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Vol. 225, No. 3, 1996
BIOCHEMICAL AND BIOPHYSICAL RESEARCH COMMUNICATIONS
FIG. 2. Corrected sensorgram overlays for the myoglobin-mAb interaction. Repeat injections of myoglobin at 330, 110, 37, 12, 4, and 0 nM are represented by the small squares and circles. Data points were collected every 0.4 seconds, but for clarity selected points at every 2.4 and 14.4 seconds are shown during the association and dissociation phases, respectively. The solid lines that intersect the data points are the best global fits to the simple bimolecular interaction model using nonlinear least squares analysis.
and randomized to determine the replication standard deviation (0.83 RU), which is an assessment of the total experimental noise. This information is required to ascertain whether a model adequately fits the data. The replicated binding responses were highly reproducible and the injections without myoglobin present in the sample plug showed that the baseline was stable over the time course of the reaction. The mAb surface density was deliberately kept at a minimum to prevent crowding and lessen mass transport effects [4]. Together, these experimental considerations yielded sensor data that were suitable for global analysis. To resolve the rate constants for the myoglobin-antibody interaction, the association and dissociation phase data obtained for each concentration of myoglobin were simultaneously fit using nonlinear least squares analysis [5]. The solid lines shown in Fig. 2 represent the best fit of the entire data set to a bimolecular interaction model using only 3 floating parameters (Bmax , ka , and kd). The actual data set used during the fitting procedure had data points taken at 0.4 second intervals. The quality of the fit was judged to be good, based on a plot of the residuals, which were low and evenly distributed as shown in Fig. 3. The residual standard deviation of 0.86 RU is very close to the replication standard deviation of 0.83 RU. The parameter estimates derived from the global fitting procedure are shown in Table 1. We used statistical profiling to confirm that the error space for each of the parameters was
FIG. 3. Residual plot for the global fit of the experimental versus modeled sensor data. The residual standard deviation was determined to be 0.86 RU. 1075
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Vol. 225, No. 3, 1996
BIOCHEMICAL AND BIOPHYSICAL RESEARCH COMMUNICATIONS
TABLE 1 Parameter Values, Standard Deviations, and Correlation Coefficients Determined from Global Analysis of the Myoglobin-mAb Biosensor Data Correlation Parameter Bmax (RU) ka (M01 s01) kd (s01)
Value
STD
122 { 0.05 1.94 1 105 { 1.5 1 102 4.70 1 1004 { 2.2 1 1006
Bmax
ka
0.644 0.671
0.040
linear, and therefore, we can quote their values and standard deviations directly from the covariance matrix [7]. This was a consequence of globally fitting a broad range of binding data, which highly constrained the parameter values and resulted in very low standard deviations. The correlations between all the parameters were also very low. In fact, there is essentially no correlation between the association and dissociation rate constants (ka and kd). The surface capacity (Bmax) was determined to be 122 RU, and based on the molecular mass of each protein, this indicates that only 54% of the myoglobin binding sites on the mAb were available for binding. Sub maximal binding could be due to occluded binding sites caused by the random amine coupling procedure used to immobilize the mAb. However, the entire myoglobin binding data set was described very well using one rate constant each for the forward and reverse reactions, indicating that the active mAb sites bound myoglobin homogeneously. As shown in Fig. 4, the binding rate was independent of flow rate, suggesting that the myoglobin-mAb interaction was controlled by the reaction kinetics and not mass transport. This is consistent with the fact that the value determined for the association rate constant for this reaction (1.94 1 105 M01 s01) is below the value predicted to be limited by mass transport on the sensor under these conditions (Ç1 1 106 M01 s01) [4]. These results provide further support that a simple bimolecular interaction model is adequate to describe the myoglobinmAb interaction measured on the biosensor. In conclusion, we demonstrated that BIAcore 2000 with its ability to simultaneously monitor multiple flow cells can generate response data that are suitable for global analysis. We showed a simple binding model adequately described the myoglobin-antibody interaction data, suggesting that surface effects did not complicate the observed responses. Global curve fitting provided a stringent test for the bimolecular reaction mechanism and highly constrained the resulting parameter values, providing confidence in the rate constants determined for this macromolecular interaction.
FIG. 4. Initial binding responses for myoglobin injected at different flow rates over the mAb surface. A 37 nM concentration of myoglobin was injected over the mAb surface at a flow rate of 10 (a), 30 (b) and 90 mL/min (c) starting at 0, 10, and 20 seconds, respectively. 1076
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Vol. 225, No. 3, 1996
BIOCHEMICAL AND BIOPHYSICAL RESEARCH COMMUNICATIONS
REFERENCES 1. Jo¨nsson, U., Fa¨gerstam, L., Ivarsson, B., Johnsson, B., Karlsson, R., Lundh, K., Lo¨fa˚s, S., Persson, B., Roos, H., ¨ stlin, H., and Malmqvist, M. (1991) Ro¨nnberg, I., Sjo¨lander, S., Stenberg, E., Sta˚hlberg, R., Urbaniczky, C., O BioTechniques 193, 620–627. 2. Stenberg, E., Persson, B., Roos, H., and Urbaniczky, C. (1991) J. Colloid Interface Sci. 143, 513–526. 3. Lo¨fa˚s, S., and Johnsson, B. (1990) J. Chem. Soc. Chem. Commun. 21, 1526–1528. 4. Karlsson, R., Roos, H., Fa¨gerstam, L., and Persson, B. (1994) Methods: Companion Methods Enz. 6, 99–110. 5. Morton, T. A., Myszka, D. G., and Chaiken, I. M. (1995) Anal. Biochem. 227, 176–185. 6. Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. (1992) Numerical Recipes in C, Cambridge Univ. Press, Cambridge. 7. Watts, D. G. (1994) Can. J. Chem. Eng., 72, 701–710.
1077
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BBRC 5223
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6907$$$221
08-06-96 11:14:09
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AP: BBRC
225, 1073–1077 (1996)
1297
Global Analysis of a Macromolecular Interaction Measured on BIAcore Lin D. Roden and David G. Myszka1 Oncological Sciences Department, Huntsman Cancer Institute, University of Utah, Salt Lake City, Utah 84112 Received July 12, 1996 We demonstrate that the interaction between myoglobin and an immobilized anti-myoglobin antibody measured on BIAcore 2000 can be described by a simple bimolecular reaction mechanism. We improved the quality of the sensor data by correcting for refractive index changes and nonspecific binding using a blank sensor surface. Applying nonlinear least squares analysis, we simultaneously fitted the association and dissociation phase data generated for a range of myoglobin concentrations injected across the antibody surface. The ability to globally fit these data to a simple binding model indicates that effects related to the sensor surface, like mass transport and the dextran matrix, did not complicate the observed binding responses. These results illustrate the potential of BIAcore to monitor macromolecular interactions in real-time and the utility of global analysis to resolve the reaction kinetics. q 1996 Academic Press, Inc.
BIAcore is an optical biosensor that can be used to monitor macromolecular interactions in real-time without labeling requirements [1,2,3]. This makes it a versatile instrument for determining the kinetic rate constants for a variety of interactions. Often the responses recorded from the sensor do not conform to a simple bimolecular interaction model as expected. These departures may be inherent in the molecules being studied or they may be introduced by the biosensor itself. Surface artifacts such as mass transport and chemical or spatial heterogeneity imposed by the dextran matrix have been cited as probable causes of deviations [4]. Biosensor data are typically analyzed by separately fitting selected portions of the association and dissociation phase data to a simple bimolecular interaction model, and conclusions are then drawn about the overall reaction. Intuitively, this is not a satisfying approach to data analysis and it can provide misleading results when the binding reactions are complex [5]. Ideally, using global analysis one would like to show that the entire data set are described by a particular reaction mechanism. Global analysis can be used to discriminate between possible binding mechanisms, but in order to be routinely employed, it requires data that are free from as many instrument and experimental artifacts as possible. Given the growing interest in biosensor data analysis and concerns over instrument artifacts, we present an example of a macromolecular interaction recorded on the sensor that is accurately described by a simple bimolecular interaction model using global analysis. EXPERIMENTAL Materials. BIAcore 2000, research grade sensor chip CM5, NHS, EDC, 1M ethanolamine-HCl pH 8.5, running buffer (10 mM HEPES, 150 mM NaCl, 3.4 mM EDTA and 0.005 % (v/v) surfactant P20 at pH 7.4), human myoglobin, mouse anti-myoglobin monoclonal antibody, coupling buffer (10 mM acetate pH 4.8), and regeneration buffer (glycine pH 2.0) were purchased from Pharmacia Biosensor (Uppsala, Sweden). Sensor experiments. The mAb was immobilized using standard amine coupling procedures [3]. To activate the
1
Corresponding author. Fax: (801)-585-3833. E-mail: [email protected]. Abbreviations used: mAb, monoclonal antibody; Bmax , maximum surface capacity; ka , association rate constant; kd , dissociation rate constant; RU, resonance units; STD, standard deviation; NHS, N-hydroxy-succinimide; EDC, Nethyl-N*-(3-dimethylaminopropyl)-carbodiimide hydrochloride. 1073 0006-291X/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
AID
BBRC 5223
/
6907$$$221
08-06-96 11:14:09
bbrca
AP: BBRC
Vol. 225, No. 3, 1996
BIOCHEMICAL AND BIOPHYSICAL RESEARCH COMMUNICATIONS
FIG. 1. Example of the raw sensor data. Myoglobin (330 nM) was injected over a surface containing 1000 RU of the mAb (a) as well as a blank sensor surface (b). At 475 seconds, both surfaces were regenerated with a 10-mL pulse of 10 mM glycine, pH 2.0.
carboxymethyl dextran surface, a 35 mL mixture of EDC/NHS was injected over a CM5 sensor surface at a flow rate of 5 mL/min. The mAb was injected in coupling buffer at a concentration of 50 mg/mL until the desired level of immobilization (1000 RU) was achieved. This was followed by a 35 mL injection of ethanolamine to cap the remaining activated carboxyl groups and a 10 mL injection of 10 mM glycine pH 2.0 to wash out any unbound mAb. A second flow cell was activated and deactivated without coupling the mAb as a control surface for refractive index change and nonspecific binding. Myoglobin samples contained 330, 110, 37, 12, and 4 nM protein created from a three fold serial dilution of a stock solution. To perform the kinetic binding studies, the flow path was changed to include both flow cells and the data collection rate was set to high. The instrument was programmed to KINJECT 23 mL of sample at a flow rate of 30 mL/min with a 300 second dissociation phase. After the end of the dissociation time, the flow cells were washed with 10 mM glycine pH 2.0 to regenerate the sensor surfaces. Each myoglobin sample, as well as a buffer blank were duplicated and injected in random order. The effect of flow rate was assessed by injecting a 37 nM concentration of myoglobin at flow rates of 10, 30, and 90 mL/min. Data analysis. The raw sensor data were prepared for global analysis by subtracting the average of the response over 20 seconds prior to the myoglobin injection and zeroing the time of injection for each flow cell. To correct for refractive index change and nonspecific binding, the responses obtained from the control surface were subtracted from the mAb surface data. The corrected binding data were then analyzed by direct curve fitting to a simple bimolecular interaction mechanism (A / B Å AB). Modeled data sets were generated by numerical integration of the differential equations which describe the reaction [6]. These were fit to the association and dissociation phase sensor data at all myoglobin concentrations simultaneously using algorithms described previously [5]. The error space for each of the parameters was assessed using statistical profiling [7].
RESULTS AND DISCUSSION
We chose to characterize the interaction between human myoglobin and a mouse antimyoglobin monoclonal antibody because the reagents are readily accessible for study on the sensor. The first step in analyzing this reaction was to simplify the interaction to be measured on the sensor surface. Since the mAb is bivalent, we decided to immobilize it and monitor a monovalent interaction with the myoglobin in solution. Examples of the raw response data simultaneously collected for an injection of myoglobin over the mAb and blank sensor surfaces are shown in Fig 1. A significant portion of the signal observed during the association phase (Ç90 RU) is due to a difference in the composition of the running buffer and the sample plug, which is referred to as a bulk refractive index change. The data from the blank sensor surface also indicates that there is a small amount of nonspecific binding presumably to the dextran matrix. Subtracting the response recorded over the blank surface from the mAb surface data corrects for these system artifacts, revealing the genuine response curves for the myoglobinmAb interaction. Bound myoglobin was removed and a fully active mAb surface was regenerated using a glycine wash. Figure 2 shows an overlay of the corrected binding responses for a series of myoglobin concentrations injected over the same mAb surface. Each binding experiment was repeated 1074
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BBRC 5223
/
6907$$$221
08-06-96 11:14:09
bbrca
AP: BBRC
Vol. 225, No. 3, 1996
BIOCHEMICAL AND BIOPHYSICAL RESEARCH COMMUNICATIONS
FIG. 2. Corrected sensorgram overlays for the myoglobin-mAb interaction. Repeat injections of myoglobin at 330, 110, 37, 12, 4, and 0 nM are represented by the small squares and circles. Data points were collected every 0.4 seconds, but for clarity selected points at every 2.4 and 14.4 seconds are shown during the association and dissociation phases, respectively. The solid lines that intersect the data points are the best global fits to the simple bimolecular interaction model using nonlinear least squares analysis.
and randomized to determine the replication standard deviation (0.83 RU), which is an assessment of the total experimental noise. This information is required to ascertain whether a model adequately fits the data. The replicated binding responses were highly reproducible and the injections without myoglobin present in the sample plug showed that the baseline was stable over the time course of the reaction. The mAb surface density was deliberately kept at a minimum to prevent crowding and lessen mass transport effects [4]. Together, these experimental considerations yielded sensor data that were suitable for global analysis. To resolve the rate constants for the myoglobin-antibody interaction, the association and dissociation phase data obtained for each concentration of myoglobin were simultaneously fit using nonlinear least squares analysis [5]. The solid lines shown in Fig. 2 represent the best fit of the entire data set to a bimolecular interaction model using only 3 floating parameters (Bmax , ka , and kd). The actual data set used during the fitting procedure had data points taken at 0.4 second intervals. The quality of the fit was judged to be good, based on a plot of the residuals, which were low and evenly distributed as shown in Fig. 3. The residual standard deviation of 0.86 RU is very close to the replication standard deviation of 0.83 RU. The parameter estimates derived from the global fitting procedure are shown in Table 1. We used statistical profiling to confirm that the error space for each of the parameters was
FIG. 3. Residual plot for the global fit of the experimental versus modeled sensor data. The residual standard deviation was determined to be 0.86 RU. 1075
AID
BBRC 5223
/
6907$$$221
08-06-96 11:14:09
bbrca
AP: BBRC
Vol. 225, No. 3, 1996
BIOCHEMICAL AND BIOPHYSICAL RESEARCH COMMUNICATIONS
TABLE 1 Parameter Values, Standard Deviations, and Correlation Coefficients Determined from Global Analysis of the Myoglobin-mAb Biosensor Data Correlation Parameter Bmax (RU) ka (M01 s01) kd (s01)
Value
STD
122 { 0.05 1.94 1 105 { 1.5 1 102 4.70 1 1004 { 2.2 1 1006
Bmax
ka
0.644 0.671
0.040
linear, and therefore, we can quote their values and standard deviations directly from the covariance matrix [7]. This was a consequence of globally fitting a broad range of binding data, which highly constrained the parameter values and resulted in very low standard deviations. The correlations between all the parameters were also very low. In fact, there is essentially no correlation between the association and dissociation rate constants (ka and kd). The surface capacity (Bmax) was determined to be 122 RU, and based on the molecular mass of each protein, this indicates that only 54% of the myoglobin binding sites on the mAb were available for binding. Sub maximal binding could be due to occluded binding sites caused by the random amine coupling procedure used to immobilize the mAb. However, the entire myoglobin binding data set was described very well using one rate constant each for the forward and reverse reactions, indicating that the active mAb sites bound myoglobin homogeneously. As shown in Fig. 4, the binding rate was independent of flow rate, suggesting that the myoglobin-mAb interaction was controlled by the reaction kinetics and not mass transport. This is consistent with the fact that the value determined for the association rate constant for this reaction (1.94 1 105 M01 s01) is below the value predicted to be limited by mass transport on the sensor under these conditions (Ç1 1 106 M01 s01) [4]. These results provide further support that a simple bimolecular interaction model is adequate to describe the myoglobinmAb interaction measured on the biosensor. In conclusion, we demonstrated that BIAcore 2000 with its ability to simultaneously monitor multiple flow cells can generate response data that are suitable for global analysis. We showed a simple binding model adequately described the myoglobin-antibody interaction data, suggesting that surface effects did not complicate the observed responses. Global curve fitting provided a stringent test for the bimolecular reaction mechanism and highly constrained the resulting parameter values, providing confidence in the rate constants determined for this macromolecular interaction.
FIG. 4. Initial binding responses for myoglobin injected at different flow rates over the mAb surface. A 37 nM concentration of myoglobin was injected over the mAb surface at a flow rate of 10 (a), 30 (b) and 90 mL/min (c) starting at 0, 10, and 20 seconds, respectively. 1076
AID
BBRC 5223
/
6907$$$221
08-06-96 11:14:09
bbrca
AP: BBRC
Vol. 225, No. 3, 1996
BIOCHEMICAL AND BIOPHYSICAL RESEARCH COMMUNICATIONS
REFERENCES 1. Jo¨nsson, U., Fa¨gerstam, L., Ivarsson, B., Johnsson, B., Karlsson, R., Lundh, K., Lo¨fa˚s, S., Persson, B., Roos, H., ¨ stlin, H., and Malmqvist, M. (1991) Ro¨nnberg, I., Sjo¨lander, S., Stenberg, E., Sta˚hlberg, R., Urbaniczky, C., O BioTechniques 193, 620–627. 2. Stenberg, E., Persson, B., Roos, H., and Urbaniczky, C. (1991) J. Colloid Interface Sci. 143, 513–526. 3. Lo¨fa˚s, S., and Johnsson, B. (1990) J. Chem. Soc. Chem. Commun. 21, 1526–1528. 4. Karlsson, R., Roos, H., Fa¨gerstam, L., and Persson, B. (1994) Methods: Companion Methods Enz. 6, 99–110. 5. Morton, T. A., Myszka, D. G., and Chaiken, I. M. (1995) Anal. Biochem. 227, 176–185. 6. Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. (1992) Numerical Recipes in C, Cambridge Univ. Press, Cambridge. 7. Watts, D. G. (1994) Can. J. Chem. Eng., 72, 701–710.
1077
AID
BBRC 5223
/
6907$$$221
08-06-96 11:14:09
bbrca
AP: BBRC