Mechanism of Nanoparticle-Enhanced Turbidimetric Assays Applying

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Langmuir 2002, 18, 7623-7628

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Mechanism of Nanoparticle-Enhanced Turbidimetric Assays Applying Nanoparticles of Different Size and Immunoreactivity† Helmut Co¨lfen,*,‡ Antje Vo¨lkel,‡ Shinichi Eda,§ Uwe Kobold,§ Jo¨rg Kaufmann,§ Angela Puhlmann,§ Christine Go¨ltner,| and Hanno Wachernig⊥ Max-Planck-Institute of Colloids and Interfaces, Colloid Chemistry, Research Campus Golm, Am Mu¨ hlenberg 2, 14424 Potsdam, Germany, Roche Diagnostics GmbH, Werk Penzberg, Nonnenwald 2, D-82372 Penzberg, Germany, School of Chemistry, University of Bristol, Bristol BS8 1TS, United Kingdom, and Particle Metrix, Neudiessener Strasse 6, D-86911 Diessen Germany Received May 23, 2002. In Final Form: June 21, 2002 The mechanism of a novel assay technique for nanoparticle-enhanced immunoturbidimetric assays is investigated. In a mixture of latex particles of two different sizes, coated with antibodies of different affinities, the aggregation behavior is monitored, which correlates with the antigen concentration. At low antigen concentrations, only the bigger latex particles coated with the high-reactivity antibody aggregate, whereas at higher antigen concentrations, the smaller latices coated with a lower reactivity antibody follow up in the aggregation process. This is shown for an immunoassay (C-reactive protein) by theoretical considerations based on a diffusion-controlled reaction and by transmission electron microscopy, analytical ultracentrifugation, and static light scattering as complementary qualitative and quantitative analytical techniques.

* To whom correspondence should be addressed. Tel: ++49331-5679513. Fax: ++49-331-5679502. E-mail: Coelfen@ mpikg-golm.mpg.de. † Dedicated to Dr. Walter Ma ¨ chtle, BASF AG on the occasion of his retirement. ‡ Max-Planck-Institute of Colloids and Interfaces. § Roche Diagnostics GmbH. | University of Bristol. ⊥ Particle Metrix.

scattering theory,4 large nanoparticles show a stronger light scattering compared to smaller nanoparticles at the same wavelength. This feature is exploited in assay development; that is, larger nanoparticles are used in assays where a higher analytical sensitivity is required. In contrast, smaller nanoparticles are used if a high upper measuring limit (at high analyte concentrations) is desired. Although large particles are useful for high sensitivity, they have a low upper measuring limit, and vice versa, small nanoparticles achieving a high upper measuring limit lack sufficient analytical sensitivity. The dynamic range of the corresponding assays, defined by the ratio of upper measuring limit to analytical sensitivity (detection limit), is very similar if particles of different size are used. Samples with concentrations below the expected values have to be remeasured at lower dilution or in a postconcentration mode (with increased sample volume). On the other hand, if sample concentrations are higher than expected, a postdilution run with higher sample dilution has to be conducted. Both remeasurements are time consuming and most importantly, from a commercial point of view, make the tests more expensive, so an increase in the dynamic measurement range is highly desired. Recently, it was shown that the dynamic range of an assay for C-reactive protein could be extended.5 C-Reactive protein (CRP) is a sensitive marker for most tissuedamaging processes, such as infections, inflammatory diseases, and malignant neoplasms.6 By application of the new assay principle, named the dual-radius enhanced latex (DuREL) principle, the measuring range of conventional assays could not only be maintained but was additionally increased to clearly lower concentrations, making the assay attractive for new indications of CRP

(1) Biomineralization; Mann, S., Webb, J., Williams, R. J. P., Eds; VCH: Weinheim, 1989. (2) Stryer, L. Biochemistry, 4th ed.; W. H. Freeman: New York, 1995. (3) Thompson, J. C.; Craig, A. R.; Davey, C. L.; Newman, D. J.; Lonsdale, M. L.; Bucher, W. J.; Nagle, P. D.; Price, C. P. Clin. Chem. 1997, 43, 2384-2389.

(4) Mie, G. Ann. Phys. 1908, 25, 377. (5) Eda, S.; Kaufmann, J.; Roos, W.; Pohl, S. J. Clin. Lab. Anal. 1998, 12, 137-144. (6) Pepys, M. B.; Baltz, M. C. Adv. Immunol. 1983, 34, 141-212.

Introduction Aggregation or agglutination processes of small colloids to form larger particles are a common phenomenon in colloid science and are of relevance to many areas of science including biomineralization, dendritic growth processes, gel formation, and so forth. Especially biological systems show a superior control of the aggregation of colloidal building units during the processes of biomineralization.1 Furthermore, proteins can show highly specific interactions such as antigen-antibody interactions2 or enzymecatalyzed reactions.2 Hence, specific protein interactions can be exploited to gain control over the process of particle aggregation of protein-coated particles3 and used to design analytical techniques based on analyte-triggered particle aggregation. For diagnostic purposes, latices coated with antibodies that react with a specific analyte can be applied to relate the triggered particle aggregation to an analyte concentration by means of a fast and easy measurement like turbidimetry. Most commonly used commercially available turbidimetric or nephelometric nanoparticle-enhanced assays use particles of a defined size. The measuring range for the corresponding analytes is generally defined by the dilution factor of the specimens. According to the quantitative Mie

10.1021/la025983n CCC: $22.00 © 2002 American Chemical Society Published on Web 09/04/2002

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measurements, such as use in neonatal medicine,7 risk assessment of coronary heart diseases,8 or applications in certain autoimmune disease testing where only a modest increase of CRP levels occurs, for example, early osteoarthritis.9 The developed assay consists of a mixture of two differently sized nanoparticles (127 and 221 nm in diameter), coated with antibodies of different affinities.5 The working hypothesis behind the mechanism is that at low analyte concentrations, preferably the large nanoparticles coated with the highly reactive antibody participate in the agglutination reaction, whereas at higher analyte concentrations the small nanoparticles are increasingly incorporated into the immune complexes. The aim of this paper is to gain additional insight into the mechanism of analyte concentration-dependent nanoparticle agglutination. By theoretical considerations and visualization of the process with transmission electron microscopy (TEM), analytical ultracentrifugation (AUC), and static light scattering (SLS), we obtain complementary information on the aggregate structure (TEM) as well as a quantitative amount of aggregates (AUC) at the endpoint of the reaction and reaction kinetics (SLS) in relation to the precursor particles under varying conditions proving that the DuREL principle works as previously suggested.5 Materials and Methods Coating of Nanoparticles and Preparation of the Latex Mixture. Carboxy-modified polystyrene submicron particles with a diameter of either 127 or 221 nm were purchased from Seradyn Inc., Indianapolis, IN. The sizes of the particles were defined based on the consideration of the handling at the cleaning process (by centrifugation) for the 127 nm particles and the stability in suspension (no sedimentation during storage and usage) for the 221 nm particles. According to the description from Seradyn, the particles were synthesized by emulsion polymerization under the presence of acrylic acid. The particles have a high density of carboxyl groups on the surface, which are useful for the covalent binding of the antibodies (carboxyl content: 127 nm particles, 0.094 mequiv/g; 221 nm particles, 0.071 mequiv/g). Two monoclonal antibodies, namely, 21F12 and 36F12, were selected out of five available ones to develop a nanoparticlebased assay for C-reactive protein. Binding studies on the BiaCore system10 show that both antibodies are directed against different epitopes and indicate that the monoclonal antibody 36F12 is clearly higher in affinity than monoclonal antibody 21F12. An accurate calculation of the affinity constants is not possible in the case of a polymeric antigen (pentameric C-reactive protein) due to multivalent interactions. After coating both monoclonal antibodies onto highly carboxylated 127 nm polystyrene nanoparticles, the different reactivities were confirmed by comparing the slopes of the dose-response curves on a COBAS Mira immunoanalyzer. It was shown with a modified protein assay (BCA assay) that both particles were coated with nearly 100% of the offered amount of antibody and thus both particles were coated with the same amount of antibody. The reaction kinetic analysis clearly indicated a faster and stronger binding of monoclonal antibody 36F12 as compared with monoclonal antibody 21F12.5 To exploit the respective advantages of small and large nanoparticles, the following strategy was applied: large particles were coated with the highly reactive antibody, and small particles were coated with the less reactive antibody. The covalent coating of carboxy-modified polystyrene nanoparticles is described in detail elsewhere.5 For C-reactive protein, the following final concentrations of microparticles were used: 0.4275% (w/v) of (7) Wasunna, A.; Whitelaw, A.; Gallimore, R.; Hawkins, P. N.; Pepys, M. B. Eur. J. Pediatr. 1990, 149, 424-427. (8) Haverkate, F.; Thompson, S. G.; Pyke, S. D. M.; Gallimore, J. R.; Pepys, M. B. Lancet 1997, 349, 462-466. (9) Spector, T. D.; Hart, D. J.; Nandra, D.; Doyle, D. V.; Mackillop, N.; Gallimore, J. R.; Pepys, M. B. Arthritis Rheum. 1997, 40, 723-727. (10) http://www.biacore.com.

Co¨ lfen et al. small particles and 0.0225% (w/v) of large particles. Thus the total concentration of small and large particles in the mixture is 0.45% (w/v), and the ratio of small particles to large particles is 95/5% (w/v). Preparation of Samples. The agglutination reaction of the coated nanoparticles in the presence of C-reactive protein was performed according to the assay conditions on a COBAS INTEGRA clinical chemistry analyzer, Roche Diagnostics GmbH, Mannheim, Germany. At first, 164 mL of reaction buffer (R1), which consisted of 20 mM Tris, 20 mM CaCl2, 300 mM NaCl, 0.05% Tween 20, 0.2% bovine serum albumin, 0.002% normal mouse immunoglobulins, and 0.09% NaN3 and whose pH was 7.4, together with 96 µL distilled water, was incubated for 1.8 min at room temperature or 37 °C. CRP control material (5 µL) containing different concentrations of CRP (0, 4, 25, and 160 mg/L CRP) plus 60 µL of distilled water was added. The mixture again was kept at room temperature or 37 °C for 2 min. Finally, the agglutination was started with 56 mL of nanoparticle reagent mixture together with 28 µL of distilled water and incubated for 2 min at 37 °C. CRP Measurements on Cobas Integra. The measurement of CRP concentration was conducted on a Roche Cobas Integra 700 and a Cobas Mira analyzer. (Cobas Integra and Cobas Mira are trademarks of a member of the Roche Group.) Assay conditions and experimental details are described elsewhere.5 Analytical Ultracentrifugation. For all ultracentrifugation experiments, a preparative Beckman Optima XL-80K (Beckman Coulter, Palo Alto, CA) which was converted into an analytical ultracentrifuge following a development at Bayer AG, Leverkusen,11 was used. The essential advantage of this machine is that the time-dependent turbidity changes are recorded at a fixed position in the center of the solution column, while a speed profile provides that particles over the entire colloidal range can be precisely detected.12 Two different run protocols were chosen. After 2 min incubation of the reaction mixture at 37 °C, the aggregated material was applied to the ultracentrifuge cell. Due to temperature equilibration constraints of the ultracentrifuge, the centrifugation was started after 10 min at 25 °C in the first protocol. In the second protocol, the rotor was pretempered at 37 °C outside of the centrifuge, and the reaction mixture was applied to the cell in the temperature-equilibrated rotor. Due to an additional temperature equilibration within the ultracentrifuge at 37 °C, the centrifugation could be started at the earliest 48 min after the initial agglutination reaction. Transmission Electron Microscopy. TEM images were recorded with a Zeiss EM 912 Omega transmission electron microscope operating at an acceleration voltage of 120 kV. Specimens were studied in the absence of staining agents, and two sample preparation methods were employed. A droplet of the readily agglutinated dispersion is applied to a 200 mesh carbon-coated copper grid. After removing most of the fluid using a filter paper wedge, specimens were left to dry at room temperature in air. To avoid artifacts due to drying or longer agglutination times, a droplet of the sample was applied to the grid and immediately placed into liquid nitrogen. The samples were kept in liquid nitrogen while being transferred into a freezedryer. After freeze-drying, the samples were subjected to TEM analysis. No difference was observed between specimens prepared by these different methods. Static Light Scattering. The agglomeration kinetics was observed at room temperature with SLS in the HORIBA LA-920 laser scattering particle size distribution analyzer. The scattered light of a 633 nm He-Ne laser and of a tungsten lamp filtered at 405 nm is collected in an angular range between 0 and 150°. From the angular intensity distribution, the size distribution is derived via the Mie formula.4 With this arrangement, the instrument is capable of giving size distributions in the range between 20 nm and 2 mm in one single measurement. To present the sample in the concentration which is most suitable for the instrument, the sample was diluted by a factor (11) Mu¨ller, H. G.; Herrmann, F. Prog. Colloid Polym. Sci. 1995, 99, 114-119. (12) Ma¨chtle, W. In Analytical Ultracentrifugation in Biochemistry and Polymer Science; Harding S. E., Rowe, A. J., Horton, J. C., Eds.; Royal Society of Chemistry: Cambridge, 1992; Chapter 10, pp 147175.

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of 7 and then poured into the so-called fraction cell followed by gentle mixing. The measurement program was started with a sequence of six measurements every 2 min. Each individual measurement lasted 30 s. All measurements were performed at room temperature.

Results and Discussion By application of a nanoparticle mixture as the latex reagent, the highly reactive antibody should work preferably at low analyte concentrations, where a competition of less and highly reactive monoclonal antibody takes place for a limited amount of available epitopes. After the large particles are consumed via reaction with the highly reactive antibody, more and more smaller particles should be integrated into the aggregate by the action of the less reactive antibody at higher analyte concentrations. Thus large particles are incorporated in the complex formation at low CRP concentration, resulting in a high analytical sensitivity, whereas small particles are involved in the precipitate formation at high analyte concentrations. An insight into the aggregation mechanism can be obtained from TEM micrographs of the aggregates obtained for different CRP concentrations (Figure 1) under conditions very close to the Cobas Integra analyzer conditions (temperature, incubation times). Both TEM grid preparation techniques (conventional drying and shock freezing with subsequent freeze-drying) yield similar results, so it can be concluded that the possible further particle aggregation during the TEM grid preparation by drying of the solution in air can be neglected. Hence, in the following, only those TEM grids prepared by the conventional method were further examined. Since the particles/agglutinates on the TEM grids are not homogeneously distributed, many images were taken into consideration. Two representative pictures each of the latex reagent agglutinated with 0 (control), 4, 25, and 160 mg CRP/L are shown in Figure 1. The images of the pure latex mixture (Figure 1a,b) demonstrate that no aggregates are present. The formation of particle islands is a well-known effect, which occurs when aqueous dispersions are dried on the hydrophobic carbon coating. Particles in the islands clearly are arranged in a single layer on the grid. The frequency of large particles corresponds to the mixing ratio of small to large particles, which is 95:5% (w/v) and 100:1 based on numbers of particles/volume. At a concentration of 4 mg CRP/L (Figure 1c,d), however, substantial aggregation of the big particles is observed. In contrast to the previously observed islands, the aggregates seen here are clearly three-dimensional, hence excluding a drying artifact. The large particles appear to be contained mostly in the aggregate, whereas the small particles still remain nonaggregated. A concentration of 25 mg CRP/L (Figure 1e,f) already causes the inclusion of almost all smaller latices into the aggregate, whereas at the highest CRP concentration of 156 mg/L the aggregate size is significantly increased, and no nonaggregated latices remain on the grid. Furthermore, the aggregates became more compact with increasing CRP concentration. This result is in qualitative agreement with the observation of another research group of CRP reacting with anti-CRP F(ab′)2 fragment-covered latices,13 who observed a transition from more branched and loose aggregates to very dense aggregates upon raising the CRP concentration from 250 to 900 mg/L. Although TEM allows the observation of (13) Molina-Bolivar, J. A.; Galisteo-Gonzalez, F.; Hidalgo-Alvarez, R. Langmuir 2001, 17, 2514-2520.

Figure 1. Two representative TEM pictures each of anti-CRP latex particles after agglutination: 0 (a,b), 4 (c,d), 25 (e,f), and 156 mg/L (g,h). Scale bar ) 500 nm.

trends and the investigation of the aggregate structure, the degree of aggregation can only be estimated qualitatively. Quantitative information on the particle size distribution of the latex mixture and of the aggregates produced in the presence of the different CRP concentrations can be obtained from AUC.11,12 Here, the measured turbidities are corrected according to the Mie theory 4 so that it is possible to quantitatively analyze latex mixtures with sizes almost over the entire colloidal range in a single experi-

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Figure 2. Mass-weighted integral AUC particle size distributions of an agglutinated latex mixture at different CRP concentrations after 10 min at 25 °C (dH ) diameter of the particles, G(dH) ) integral particle size distribution).

ment.11,12 Figure 2 shows the obtained particle size distributions for different CRP concentrations. The quantitative AUC data agree very well with the results obtained by TEM which necessarily lack statistical significance. In addition, AUC provides quantitative information on the starting point and end point of the agglomeration reaction. The given mass ratio of the small particles of 92 wt % in the initial mixture is confirmed exactly as shown in Figure 2, which indicates the reliability of the AUC experiment. According to the results of a worldwide study of Bayer AG on the determination of particle size distributions with all commonly known techniques for the determination of the particle size distribution of nanoparticles, AUC and TEM (coupled with sufficient particle counting which was not performed here due to the difficulties with aggregated particles) proved to be the best techniques for the correct determination of the size distribution in contrast to scattering techniques which showed weaknesses for complicated mixtures.14 The most important result in Figure 2 is that the onset of aggregation indeed selectively involves the big particles only (Figure 2, 4.28 mg CRP/L) whereas at higher CRP concentrations, more and more smaller particles are incorporated into the aggregate until at 156 mg CRP/L, no particles remain individually dispersed. To clearly prove that the separation process during the AUC experiment, which can possibly change the equilibrium conditions for the antigen-antibody interaction, does not affect the system significantly, we tried to get information about the reaction kinetics in the nondisturbed system under quiescent conditions. We observed the process with SLS at two different CRP concentrations (50 and 171 mg/L). The reaction kinetics was observed for a period of 15 min. In agreement with the AUC data (Figure 2), it is clearly shown (Figure 3a) that with a lower concentration of CRP (50 mg/L) the larger particles aggregate first, and the smaller particles are affected less. At a high CRP concentration (171 mg/L), the large particles are reacting first, but with longer reaction times also the small particles are agglomerating (Figure 3b). This is in qualitative agreement with the TEM and AUC data and clearly proves the proposed reaction mechanism of the DuREL principle. SLS is no fractionating method, so the particle size distributions in Figures 2 and 3 differ. AUC yields real particle size distributions, whereas those obtained from SLS have only apparent qualitative character as the total (14) Lange, H. Part. Part. Syst. Charact. 1995, 12, 148-157.

Figure 3. Integral particle size distribution G(dH) from a SLS kinetic measurement: (a) 50 mg/L and (b) 171 mg/L.

scattering intensities at different angles are related to a particle size by the Mie formula. Thus it can become difficult to quantitatively assign the measured scattering intensities to a particle size for broad or multimodal distributions as investigated here.14 This is particularly obvious for a reaction time of 0 min (Figure 3) where a bimodal latex mixture is present which cannnot be resolved by SLS. Also, the particle sizes differ slightly from the true values (the 127 nm latex is found to be ≈90 nm by SLS). To resolve the selective agglutination process more clearly, we have performed a set of experiments for CRP concentrations between 5 and 50 mg/L. As known from the results presented in Figure 3, the agglutination is time dependent and does not reach saturation within 15 min reaction time (data not shown), although with increasing time, the turbidity changes become smaller.15 To amplify the measuring effect and to take the originating reaction conditions on the analyzers (temperature) into consideration, we increased the experimental reaction time to an overall incubation time of 2 + 48 ) 50 min at 37 °C. The temperature increase had only a small effect on the particle aggregation, whereas the increase in the reaction time substantially affects the particle size distribution of the aggregates (see Supporting Information). The timedependent measurements (Supporting Information) also show that the reproducibility of the measurements is good with the exception of very large particles. This is due to the fact that such large particles are at the limit of the measuring method because they sediment on their own account in the Earth’s gravitational field. For the smaller particles, the reproducibility is much better (data not shown). (15) Eda, S.; Molwitz, M. Roche Diagnostics GmbH, Penzberg, 2001. Unpublished results.

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nent contact which is certainly not the case. However, if Smoluchowski’s expression for the half-life t* 1/2 of an inhibited slow coagulation is used,

t*1/2 )

Figure 4. Mass-weighted particle size distributions of CRP agglutinated latex mixtures from AUC at different CRP concentrations incubated for 50 min at 37 °C.

If the particle size distributions obtained for an enhanced aggregation at 50 min incubation at 37 °C are compared for CRP concentrations between 5 and 50 mg/ L, the mechanism of the sequential aggregation of first big latices and then the smaller ones with increasing CRP concentration can also be visualized impressively. The results indicate that even at longer reaction times and increased temperature the big reactive latices aggregate first (Figure 4). This shows that the DuREL principle even allows a significant variation of the reaction parameters time and temperature without compromising its efficacy. The theoretical concentration of 92 wt % small latex particles in the initial mixture is well reproduced (93%), but this value decreases steadily with increasing CRP concentration through implementation of the small particles into the aggregate until at 50 mg/L, only 10 wt % of small particles remain freely dispersed. It can also be seen that the polydispersity of the aggregates increases with increasing CRP concentration, which is consistent with the TEM results (Figure 1). A comparison of the distribution for 5 mg CRP/L (Figure 4) with that for 4.28 mg/L (Figure 2) reveals that for a similar protein concentration, more smaller particles are involved in aggregation after a longer reaction time, which indicates a complete aggregation of the big particles. As soon as the large particles are consumed by aggregation, they are followed by the smaller ones. Theoretical Consideration of the Agglutination Reaction. First, the agglutination reaction was considered using the kinetic Smoluchowski theory of fast coagulation16 yielding the single particle number Zt after coagulation for time t in an aqueous dispersion at 298 K:

(

)

rA r

Zt ) Z0 1 + 2.7 × 10-12Z0t

-1

with Z0 equal to the initial particle number (3.84 × 1010/ mL for the 127 nm latex and 4.05 × 108/mL for the 221 nm latex); rA, the interaction radius, is assumed to be equal to the double particle radius r meaning direct particle collisions. However, for homoparticle collisions where every collision results in a permanent contact, 17% of the small latices had already coagulated after 1 s compared to only 0.002% for the 221 nm latices, which was clearly not observed experimentally. This is largely a result of not taking the different particle reactivities toward CRP into account by treating every collision to form a perma(16) Do¨rfler, H. D. Grenzfla¨ chen und Kolloid-Disperse Systeme: Physik und Chemie; Springer-Verlag: Berlin, 2002; pp 569-573.

11 πDrAZ0 R4

with a factor R considering the ratio of effective collisions (meaning an activation energy for the aggregation reaction, taken as 1 for the 221 nm latex and 10 for the 127 nm latex) and D equal to the diffusion coefficient, one obtains that the half time for the aggregation of the 127 nm latices is 2.5 times longer than that of the 221 nm latices, which means that the big latices aggregate preferably due to their higher reactivity. However, this model is very simple as it considers only homoaggregation between equally sized particles and clusters thereof. The same is true for other theories for particle aggregation,17-19 so a more detailed consideration is presented in the following: As the CRP-induced latex agglutination reaction is based on the reaction of a CRP molecule with a latex particle which then aggregates with another particle upon collision with a certain probability based on the antibody reactivity of the latex coating, this reaction can be considered as a diffusion-controlled reaction. Thus, the rate constant of this reaction kA,B can be calculated according to20 (see Appendix)

kA,B ) 4πDA,B(rA + rB)NA where r is the hydrodynamic radius of particle/molecule A or B; NA ) 6.022 × 1023 mol-1, the Avogadro constant; and DA,B ) DA + DB, the relative diffusion coefficient, which takes into account that A particles diffuse toward B particles and vice versa. From the known latex particle diameters and the viscosity of water at 25 °C, the diffusion constants could be calculated (D ) 0.386 × 10-9 dm2/s for the 127 nm particles and D ) 0.222 × 10-9 dm2/s for the 221 nm particles) as well as that of CRP derived from a molarmass-based diffusion coefficient estimation (D ) 5.0 × 10-9 dm2/s, r ) 4.85 nm) (see Appendix). As latex collision leads to an aggregate only after at least one of the latices has reacted with CRP, in a first step, only the rate constants for a reaction between latex and CRP are relevant. This leads to kA,C ) 27.65 × 109 L mol-1 s-1 and kB,C ) 45.59 × 109 L mol-1 s-1 with the indices A ) 127 nm latex, B ) 221 nm latex, and C ) CRP. Comparison of these rate constants with those for latex collisions (see Appendix) shows that the reaction of the latices with CRP is clearly favored due to the much higher diffusion coefficient of the protein. The difference in the reactivity of the 127 and 221 nm latices is 1:10 (see above and ref 5) so that only every 10th collision between A and C is successful. Taking this and the different particle surface area into account, it is derived that the small latices react 1.9 times faster than the bigger ones (see Appendix). As the particle number/volume ratio of small to big particles is 100:1, it is clear that the big latices with a comparable reactivity to the smaller ones are rapidly consumed by aggregation with subsequent (17) Grant, S. B.; Kim, J. H.; Poor, C. J. Colloid Interface Sci. 2001, 238, 238-250. (18) van Dongen, P. G. J.; Ernst, M. H. Phys. Rev. Lett. 1985, 54, 1396; J. Stat. Phys. 1988, 50, 295. (19) Fernandez-Barbero, A.; Schmitt, A.; Cabrerizo-Vilchez, M.; Martinez-Garcia, Physica A 1996, 230, 53-74. (20) Wedler, G. Lehrbuch der physikalischen Chemie, 3 Auflage, VCH: Weinheim, 1987; p 794.

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aggregation of the remaining smaller particles, in agreement with the experimental observations. Conclusion It could evidenced both theoretically and experimentally that for a CRP immunoassay in a mixture of antibodycoated latices of different reactivities, the bigger and more reactive particles aggregate first upon antigen addition. Thus the dynamic range of indirect turbidimetric measurements of antigen concentration via the aggregation of the latex mixture is increased. The results of this study suggest that the principle developed for an enhancement of the dynamic range of turbidity measurements can be generalized for other systems and thus applied to various problems of analyte concentration determination, estimation of the strength of polymer-polymer interactions, and so forth. To thoroughly explore the details of the agglomeration mechanism, the combination of TEM, AUC, and SLS proved extremely versatile and successful. TEM presents a qualitative picture of the aggregation process, showing aggregation of the large latex particles at low analyte concentrations before at higher concentrations the smaller particles also tend to aggregate. To quantitatively show the behavior of the particles in solution, AUC was employed with an excellent resolution of the small and large particles so that it is possible to get a quantitative picture of the starting and end points of the analyte concentration-dependent aggregation process. Nevertheless, AUC is a separation technique, and during measurement, the composition of the system is disturbed and no longer under equilibrium conditions. Therefore, a third analytical technique, SLS, is used. With SLS, the system is observed with relatively low resolution, and small and large particles are resolved only partly. The strength of this technique, however, is to keep the system under equilibrium conditions, the measurements are fast and the overall dynamic of the process is monitored. The combination of these three analytical methods together with theoretical calculations based on a diffusioncontrolled reaction mechanism clearly proves that the proposed DuREL mechanism holds:5 selective aggregation of reactive big particles is followed by that of smaller latices with lower reactivity as the antigen concentration increases. Acknowledgment. H.C. thanks Roche-Diagnostics GmbH for financial support of this work. Also, the MaxPlanck Society is acknowledged for financial support. Appendix Theoretical Considerations of the Agglutination Reaction. The rate constant of a diffusion-controlled reaction kA,B can be calculated according to

kA,B ) 4πDA,B(rA + rB)NA where r is the hydrodynamic radius of particle/molecule A or B, NA ) 6.022 × 1023 mol-1 (the Avogadro constant),

Co¨ lfen et al.

and DA,B ) DA + DB is the relative diffusion coefficient, which takes into account that A particles diffuse to B particles and vice versa. According to the Stokes-Einstein equation D ) kT/ 6πηr, the diffusion coefficient D can be calculated from the hydrodynamic radius r with the knowledge of the solvent viscosity and the absolute temperature T (k ) Boltzmann constant). With η ) 0.89 cP of water at 25 °C, we obtain D ) 0.386 × 10-9 dm2/s for the 127 nm particles and D ) 0.222 × 10-9 dm2/s for the 221 nm particles. The pentameric CRP molecule (M ) 110 000 g/mol) represents a flat disk with an estimated diffusion coefficient of D ) 5.0 × 10-9 dm2/s (r ) 4.85 nm). Thus, the rate constants, with indices A ) 127 nm, B ) 221 nm, and C ) CRP, are as follows: kA,A ) 7.42 × 109 L mol-1 s-1, kA,B ) 8.01 × 109 L mol-1 s-1, kA,C ) 27.65 × 109 L mol-1 s-1, kB,B ) 7.42 × 109 L mol-1 s-1, and kB,C ) 45.59 × 109 L mol-1 s-1, where the collisions AA, AB, and BB can only lead to an aggregate after a prior collision of A or B with C. The reactivity of the antibodies on the 221 nm latex B toward CRP is estimated to be 10 times higher than that on the 127 nm latex. Assuming that every BC collision is successful, this means that only every 10th AC collision is successful. Here, only the difference between successful AC and BC collisions is of interest, so the absolute number of successful collisions does not need to be known. The latex concentrations are 0.45% (w/v) A and 0.025% (w/v) B leading to a total latex concentration of 0.475% (w/v). The molar mass of the spherical latices can be calculated from their volume, the density of polystyrene ()1.05 g/mL), and the molar mass of a styrene unit ()104 g/mol), leading to MA ) 70.54 × 109 g mol-1 and MB ) 371.72 × 109 g mol-1 which gives cA ) 6.38 × 10-11 mol L-1 and cB ) 6.73 × 10-13 mol L-1. The reaction rate vA,C ) kA,CcAcC at the start of the reaction is obtained by multiplying the rate constants with the molar concentration (cA and cC for AC): for cC ) 4.28 mg/L ) 38.9 × 10-9 mol L-1, vA,C ) 6.9 × 10-8 mol L-1 s-1 and vB,C ) 1.2 × 10-9 mol L-1 s-1; for cC ) 50 mg/L ) 454.5 × 10-9 mol L-1, vA,C ) 8.0 × 10-7 mol L-1 s-1 and vB,C ) 1.4 × 10-8 mol L-1 s-1; and for cC ) 156 mg/L ) 1418.0 × 10-9 mol L-1, vA,C ) 2.5 × 10-6 mol L-1 s-1 and vB,C ) 4.4 × 10-8 mol L-1 s-1. As the reactivity of AC is 10 times lower than that of BC, the relative surface area (222/127)2 between the latices has to be taken into account by multiplying vB,C with both of these factors, which yields the following: for cC ) 4.28 mg/L, vB,C ) 3.7 × 10-8 mol L-1 s-1; for cC ) 50 mg/L, vB,C ) 4.3 × 10-7 mol L-1 s-1; and for cC ) 156 mg/L, vB,C ) 1.3 × 10-6 mol L-1 s-1. Thus, the small latices react 1.9 times faster with CRP than the bigger ones. Supporting Information Available: AUC particle size distributions at different temperatures and reaction times and AUC experiment reproducibility. This material is available free of charge via the Internet at http://pubs.acs.org. LA025983N