Measurement of Protein− Ligand Binding Constants from Reaction

Anal. Chem. 2010, 82, 8780–8784

Letters to Analytical Chemistry Measurement of Protein-Ligand Binding Constants from Reaction-Diffusion Concentration Profiles Yanhu Wei, Paul J. Wesson, Igor Kourkine, and Bartosz A. Grzybowski* Department of Chemical and Biological Engineering and Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, United States Protein-ligand dissociation constants, Kd, are determined precisely and down to the picomolar range from reaction-diffusion (RD) concentration profiles created by proteins diffusing through hydrogels functionalized with protein ligands. The RD process effectively amplifies the molecular-scale binding events into macroscopic patterns visible to the naked eye. The method is applicable to various protein-ligand pairs and does not require any prior knowledge about the protein structure. Sensitive, specific, and robust determination of protein-ligand binding is important for fundamental and translational science, including structure-based drug design (aka “rational drug design”) as well as process development (e.g., purification of proteins by selective precipitation with added ligands).1,2 The strength of protein-ligand interactions is quantified by the free energy of binding, ∆Gb, which is related to the protein-ligand dissociation constant by Kd ) exp(∆Gb/RT). Numerous techniques2-13 to determine Kd have been developed including calorimetry,3 capillary electrophoresis,4 equilibrium dialysis,5 ultrafiltration,6 ultracentrifugation,7 affinity chro* To whom correspondence should be addressed. Phone: 01 847 4913024. Fax: 01 847 4913728. E-mail: [email protected]. (1) (a) Begley, D. W.; Varani, G. Nat. Chem. Biol. 2009, 5, 782. (b) Ladbury, J. E.; Klebe, G.; Freire, E. Nat. Rev. Drug Discovery 2010, 9, 23. (c) Mulakala, C.; Kaznessis, Y. N. J. Am. Chem. Soc. 2009, 131, 4521. (d) Strange, P. G. Br. J. Pharmacol. 2008, 153, 1353. (e) Valente, A. P.; Miyamoto, C. A.; Almeida, F. C. L. Curr. Med. Chem. 2006, 13, 3697. (f) Whitesides, G. M.; Krishnamurthy, V. M. Q. Rev. Biophys. 2005, 38, 385. (g) Zhang, Z. W.; Zhu, W. G.; Kodadek, T. Nat. Biotechnol. 2000, 18, 71. (2) Chuang, V. T. G.; Maruyama, T.; Otagiri, M. Drug Metab. Pharmacokinet. 2009, 24, 358. (3) (a) Navratilova, I.; et al. Anal. Biochem. 2007, 364, 67. (b) VelazquezCampoy, A.; Freire, E. Nat. Protoc. 2006, 1, 186. (4) (a) Gao, J. M.; Mammen, M.; Whitesides, G. M. Science 1996, 272, 535. (b) Nilsson, M.; Harang, V.; Bergstrom, M.; Ohlson, S.; Isaksson, R.; Johansson, G. Electrophoresis 2004, 25, 1829. (5) (a) Banker, M. J.; Clark, T. H.; Williams, J. A. J. Pharm. Sci. 2003, 92, 967. (b) Ji, Q. C.; Rodila, R.; Morgan, S. J.; Humerickhouse, R. A.; El-Shourbagy, T. A. Anal. Chem. 2005, 77, 5529. (6) (a) Bloxam, D. L.; Hutson, P. H.; Curzon, G. Anal. Biochem. 1977, 83, 130. (b) Xie, Y.; Zhang, D.; Ben-Arnotz, D. Anal. Biochem. 2008, 373, 154. (7) (a) Arkin, M.; Lear, J. D. Anal. Biochem. 2001, 299, 98. (b) Barre, J.; Chamouard, J. M.; Houin, G.; Tillement, J. P. Clin. Chem. 1985, 31, 60. (8) (a) Kim, H. S.; Wainer, I. W. J. Chromatogr., B 2008, 870, 22. (b) Yoo, M. J.; Smith, Q. R.; Hage, D. S. J. Chromatogr., B 2009, 877, 1149.

8780

Analytical Chemistry, Vol. 82, No. 21, November 1, 2010

matography,8 or spectroscopic methods (NMR,9 mass spectrometry (MS),10 fluorescence,11 circular dichroism,12 Raman, and surface plasmon resonance, SPR13). These methods, however, are not without limitations.2,14 For example, isothermal titration calorimetry, the standard method for directly measuring the enthalpy, Gibbs free energy, and dissociation constant of protein-ligand binding, requires relatively high concentrations of samples.3a NMR-based binding assays suffer from low signal-to-noise ratios, nonspecific binding, and the need of special NMR probes.15 Capillary electrophoresis is vulnerable to nonspecific adsorption to the inner walls of the capillary, resulting in irreproducible migration times. Also, many of these techniques rely on the specific spectral changes during binding and often require fluorescent tagging or radioisotope labeling. In this context, recent advances in SPR methods appear promising as they can quickly yield both the kinetic as well as the thermodynamic information about binding; it must be remembered, however, that SPR has poor sensitivity toward small molecules and that lateral interactions between surface-immobilized ligands and/or proteins can have a pronounced effect on the measured binding parameters.16 Here, (9) (a) Demers, J. P.; Mittermaier, A. J. Am. Chem. Soc. 2009, 131, 4355. (b) Feher, K.; Groves, P.; Batta, G.; Jimenez-Barbero, J.; Muhle-Goll, C.; Kover, K. E. J. Am. Chem. Soc. 2008, 130, 17148. (c) Lepre, C. A.; Moore, J. M.; Peng, J. W. Chem. Rev. 2004, 104, 3641. (10) (a) Clark, S. M.; Konermann, L. Anal. Chem. 2004, 76, 7077. (b) Powell, K. D.; Ghaemmaghami, S.; Wang, M. Z.; Ma, L. Y.; Oas, T. G.; Fitzgerald, M. C. J. Am. Chem. Soc. 2002, 124, 10256. (11) (a) Lee, Y. C. J. Biochem. 1997, 121, 818. (b) Matulis, D.; Kranz, J. K.; Salemme, F. R.; Todd, M. J. Biochemistry 2005, 44, 5258. (c) Otagiri, M.; Fleitman, J. S.; Perrin, J. H. J. Pharm. Pharmacol. 1980, 32, 478. (12) (a) Greenfield, N. J. Nat. Protoc. 2006, 1, 2527. (b) Wilting, J.; Van Der Giesen, W. F.; Janssen, L. H. M.; Weideman, M. M.; Otagiri, M.; Perrin, J. H. J. Biol. Chem. 1980, 255, 3032. (13) (a) Frostell-Karlsson, A.; Remaeus, A.; Roos, H.; Andersson, K.; Borg, P.; Hamalainen, M.; Karlsson, R. J. Med. Chem. 2000, 43, 1986. (b) Rich, R. L.; Day, Y. S. N.; Morton, T. A.; Myszka, D. G. Anal. Biochem. 2001, 296, 197. (14) Oravcova, J.; Bohs, B.; Lindner, W. J. Chromatogr., B 1996, 677, 1. (15) (a) Swann, S. L.; Song, D.; Sun, C.; Hajduk, P. J.; Petros, A. M. ACS Med. Chem. Lett. 2010, 1, 295. (b) Becker, B. A.; Morris, K. F.; Larive, C. K. J. Magn. Reson. 2006, 181, 327. (c) Fielding, L.; Rutherford, S.; Fletcher, D. Magn. Reson. Chem. 2005, 43, 463. (16) (a) Jecklina, M. C.; Schauerb, S.; Dumelinc, C. E.; Zenobi, R. J. Mol. Recognit. 2009, 22, 319. (b) Pattnaik, P. Appl. Biochem. Biotechnol. 2005, 126, 79. (c) Neumann, T.; Junker, H.-D.; Schmidt, K.; Sekul, R. Curr. Top. Med. Chem. 2007, 7, 1630. (d) Lahiri, J.; Isaacs, L.; Grzybowski, B.; Carbeck, J. D.; Whitesides, G. M. Langmuir 1999, 15, 7186. 10.1021/ac102055a  2010 American Chemical Society Published on Web 10/05/2010

Figure 1. (a) (left) Scheme and dimensions of the experimental arrangement. Typically, hs ≈ 5 mm, hf ) 250 µm, d ) 0.3-1.0 mm, lf ) 1500 µm, ls ) 500 µm, and w ) 1 cm. (right) Illustration of the key (macro)molecular components involved: proteins P (brown) and PAAm gel modified with protein-specific ligands L (blue triangles). As the proteins diffuse through the gel, they bind to gel-immobilized ligands; this binding is characterized by the dissociation constant Kd ) kr/kf. (b) PAAm monomers modified with ligands for BCAII (ABS), for trypsin (AAB), and for avidin (BAE).

we describe a conceptually novel method in which the value of Kd is determined accurately from the rate at which a protein diffuses through a gel functionalized covalently with protein-specific ligands. Aside from the simplicity of the visual detection mode, the major virtue of this reactiondiffusion (RD)17 assay is that it does not require any prior knowledge of the protein’s structure or a specific “signal” from the protein’s binding site; in this sense, the assay is applicable to different types of protein-ligand pairs. We validate the assay for three different protein-ligand systems and show that its predictions agree with the previously reported values of Kd. Both experiments and RD modeling indicate that the method can determine dissociation constants reliably down to the picomolar range. In the experimental setup illustrated in Figure 1a, 0.3-1 mm thick films of poly(acryl amide), PAAm, were prepared by copolymerization of N,N′-methylenebisacrylamide cross-linkers, acrylamide monomers, and a fraction (typically, 1 × 10-7 to 8 × 10-3, see ref 18 and the Supporting Information, sections 4 and 5) of monomers functionalized with small molecule ligands specific to a given protein. Here, these ligand-containing (17) (a) Chopard, B.; Droz, M.; Magnin, J.; Racz, Z.; Zrinyi, M. J. Phys. Chem. A 1999, 103, 1432–1436. (b) Grzybowski, B. A.; Bishop, K. J. M.; Campbell, C. J.; Fialkowski, M.; Smoukov, S. K. Soft Matter 2005, 1, 114. (c) Cross, M. C.; Hohenberg, P. C. Rev. Mod. Phys. 1993, 65, 851. (d) Horvath, J.; Szalai, I.; De Kepper, P. Science 2009, 324, 772. (e) Bensemann, I. T.; Fialkowski, M.; Grzybowski, B. A. J. Phys. Chem. B 2005, 109, 2774. (18) The typical ligand concentrations used were from 10 nM to 4.5 mM corresponding to an average distance between ligands, d ) [L]-1/3, ranging from 550 to 7.2 nm. For [L] < 100 µM, there was no influence of the ligand concentration on Kd. At higher concentrations, however, the measured Kd’s decreased. This effect was likely due to immobile PL complexes being close to one another and inhibiting the diffusion of unbound P leading to narrower than expected bands of stained P and PL. For specific data, please refer to the Supporting Information, sections 4 and 5.

monomers (Figure 1b; also see Supporting Information, sections 1 and 8) were (i) 4-(N-acryloyl-aminoethyl)-benzenesulfonamide, ABS, binding to bovine carbonic anhydrase II, BCAII,19a,b (ii) N-acryloyl-m-aminobenzamidine, AAB, for porcine pancreas trypsin,19c TP, and (iii) biotin-(N-acryloyl)ethylamide, BAE, for chicken egg avidin, AV19d,e (all proteins were purchased from Aldrich). After gelation, PAAm/ligand (PAAm/ L) films20 were soaked and stored in Tris buffers until use. Proteins were delivered to the films by wet stamping,21 in which 6% w/v agarose stamps presenting an array of parallel lines (500 µm wide and spaced by 1500 µm) were made by casting hot, degassed agarose against oxidized PDMS masters. After gelation at room temperature, the stamps were soaked in a Tris buffer solution of a protein (protein concentration 10-150 µM). Immediately prior to use, the surface of the stamps was blotted dry and the stamps were placed onto the PAAm/L films. The entire procedure, excluding synthesis of ligands and soaking of agarose stamps, took less than 10 h (for further details, see the Supporting Information, section 2). (19) (a) Krishnamurthy, V. M.; Kaufman, G. K.; Urbach, A. R.; Gitlin, I.; Gudiksen, K. L.; Weibel, D. B.; Whitesides, G. M. Chem. Rev. 2008, 108, 946. (b) Poulsen, S. A.; Davis, R. A.; Keys, T. G. Bioorg. Med. Chem. 2006, 14, 510. (c) Luong, J. H. T.; Male, K. B.; Nguyen, A. L. Biotechnol. Bioeng. 1988, 31, 439. (d) Lo, K. K. W.; Hui, W. K.; Chung, C. K.; Tsang, K. H. K.; Lee, T. K. M.; Li, C. K.; Lau, J. S. Y.; Ng, D. C. M. Coord. Chem. Rev. 2006, 250, 1724. (e) Langereis, S.; Kooistra, H. A. T.; Van Genderen, M. H. P.; Meijer, E. W. Org. Biomol. Chem. 2004, 2, 1271. (20) Also, “blank” PAAm gels (i.e., containing no ligands) were prepared for control experiments in which we verified that in the absence of the ligands, the proteins diffused freely (with diffusion coefficients determined by RD modeling, see main text) without any nonspecific binding to the gel matrix. (21) (a) Campbell, C. J.; Klajn, R.; Fialkowski, M.; Grzybowski, B. A. Langmuir 2005, 21, 418. (b) Grzybowski, B. A.; Campbell, C. J. Mater. Today 2007, 10, 38. (c) Klajn, R.; Fialkowski, M.; Bensemann, I. T.; Bitner, A.; Campbell, C. J.; Bishop, K.; Smoukov, S.; Grzybowski, B. A. Nat. Mater. 2004, 3, 729.

Analytical Chemistry, Vol. 82, No. 21, November 1, 2010

8781

Figure 2. (a) (left) Scheme of the stamping/staining procedure for the determination of Kd. The stamp is placed on a ligand-modified gel for a predetermined period of time, after which the spread protein is “visualized” by fixing and staining. (right) Photograph of a patterned gel. (b) The three left panels show the top views (xy plane) of lines stamped for different times (20, 60, and 100 min) and then stained with Coomassie blue. Note the widening of the lines with time. The right three panels have the corresponding modeled side-views (yz plane) of protein concentration profiles in the gel. The top row corresponds to no ligand in the gel ([L] ) 0) and no binding. In this case, protein spreading is most pronounced. The middle row corresponds to a relatively weak protein-ligand binding (Kd ∼ 15.6 µM; gel contains AAB and was stamped with TP). The bottom row illustrates stronger protein-ligand binding (Kd ∼ 3.3 µM; gel contains ABS and was stamped with BCAII). In this case, spreading is the least pronounced.

When the features of the stamp came into contact with the gel substrate, the proteins diffused into the PAAm/L. In doing so, they were also “captured” by the ligands immobilized on the gel’s backbone. The main idea behind our work is that the effectiveness of this capturing process, measured by Kd, should be related to the rate at which the proteins migrate/diffuse through the gel. Intuitively, the stronger the protein-ligand binding (i.e., the smaller the Kd), the more effective the gel should be in “capturing” the proteins and the slower the spreading of proteins away from the stamped features (see Figure 2). Of course, this interplay between “capturing” and “migration” is reminiscent of affinity chromatography but the key to our system is to express this interplay in a quantitative/ mathematical form such as to estimate Kd accurately. To do so, it is first necessary to analyze the reaction-diffusion (RD) processes in the system. Denoting the concentration of free proteins as [P], that of the proteins bound to the ligands as [PL], and the concentration of immobile ligands in the gel film as [L], the pertinent RD equations can be written as ∂[P]/∂t ) DP∇2[P] - kf[P][L] + kr[PL] ∂[L]/∂t ) -kf[P][L] + kr[PL] ∂[PL]/∂t ) kf[P][L] - kr[PL]

(1)

where all concentrations depend on spatial coordinates and on time, DP is the diffusion coefficient of the free protein, kf is the “forward” rate of protein-ligand binding, and kr is the “reverse” rate for the dissociation of the protein-ligand complex. These full kinetic equations can be simplified by noting that the protein-ligand association kinetics is fast (kf ∼ O (105) M-1 s-1) compared to protein diffusion (DP ∼ O (10-7) cm2 s-1). Therefore, the RD dynamics can be treated mathematically as a repeating sequence of two substeps occurring at different time scales. In the first, slow substep, [P] is allowed to diffuse; in the second, fast substep, the concentrations of [P] and [PL] 8782

Analytical Chemistry, Vol. 82, No. 21, November 1, 2010

are updated “instantaneously” to account for the protein-ligand binding reaction. These two events for a two-dimensional system (reflecting the geometry of the stamped parallel, long lines, see Figure 1a) can be written in a discrete form as

(

)

∂2[P] ∂2[P] + ∆t 2 ∂x ∂z2 (ii) [PL] f [PL] + δ, [L] f [L] - δ and [P] f [P] - δ with Kd ) ([P] - δ)([L] - δ)/([PL] + δ)

(i) [P] f [P] + ∆[P] with ∆[P] ) DP

(2) When integrated numerically, these equations yield the RD dynamics, that is, the time-dependent concentration profiles of the proteins in the gel, in terms of only two free parameters, DP and Kd ) kr/kf (as opposed to the three parameters DP, kf, and kr in eq 1) and two initial concentrations (that of the protein in the stamp, [P]0, and of the ligand in the gel, [L]0) which are known a priori. Our task is then to find the value of the dissociation constants, Kd, by fitting the theoretical concentration profiles to the experimental ones. To do this, we must know how the concentration of proteins ([P] and [PL]) in the gel changes as a function of spatial coordinates and of time. Experimentally, such information is obtained by wet stamping the protein for different times, t, then fixing the proteins and staining the patterned gels (here, with Coomassie blue for BCAII and TP and Coomasie Blue/AgNO3 for AV; see the Supporting Information, section 2). Staining gives colored bands such as those shown in Figure 2a; these images can be easily digitized into color intensity profiles by converting a raw image of the gel into a grayscale bitmap. Importantly, because optical absorbance of the staining reagent is proportional to the total protein concentration

Figure 3. Contour plots of ∆R ) R(DP, Kd) - min(R(DP, Kd)) for (a) BCAII/ABS, (b) TP/AAB, and (c) AV/BAE. Black arrows indicate the minima corresponding to the best-fit values of DP and Kd. For BCAII, these values are DP ) 9.03 × 10-12 m2/s and Kd ) 3.01 µM; for TP DP ) 2.96 × 10-11 m2/s and Kd ) 15.1 µM; DP ) 1.58 × 10-11 m2/s and Kd ) 10-6 µM for AV.

(i.e., [P] + [PL]) at a given location of the film,22 color intensities translate into total concentration of protein integrated over gel depth (i.e., down the z direction; see right-hand part of Figure 2b). All in all, staining “visualizes” the protein concentration gradients (Figure 2). The experimental profiles thus obtained for a given proteinligand system are then targets for RD modeling and optimization. Specifically, for a set of values {DP, Kd}, RD (eqs 2) are integrated numerically for a desired amount of time and the resulting protein concentration profiles are compared “pixelby-pixel” with the experimental ones (see the Supporting Information, section 3 for numerical details). The difference between these two sets can be quantified by the sum of squared differences over all pixels (i, j),

R)

∑ (I(i, j)

model

- I(i, j)exp)2

(3)

i,j

where I corresponds to the normalized color intensity (see the Supporting Information, section 3). This procedure is repeated exhaustively (for computer codes, see the Supporting Information, sections 6 and 7) for different sets of {DP, Kd} to create “maps” of R(DP, Kd) such as the ones shown in Figure 3. We note that all three “maps” exhibit only one minimum, and so the determination of DP and Kd by minimizing R is unambiguous.23 Most importantly, this method estimates the values of Kd with remarkable precision. For example, the Kd value for the benzenesulfonamide-BCAII binding estimated from RD profiles is 3.3 ± 2.2 µM compared to 1.69 µM reported in the literature using Fourier transform ion cyclotron resonance mass spectrometry.19a,b For the aminobenzamidine-TP system, RD gives 15.6 ± 11.4 µM vs 26 µM determined by colorimetric analysis.19c We note that because each stamp delivers the protein from several lines simultaneously, one experiment automatically creates experimental statistics and allows for the determination of standard deviations (the values above are from ∼50 lines and 6 time-points). Also, protein (22) (a) Goldberg, H. A.; Wamer, K. J. Anal. Biochem. 1997, 251, 227. (b) Knight, M. I.; Chambers, P. J. Mol. Biotechnol. 2003, 23, 19. (23) Because the maps have only one minimum, a conjugate gradient method rather than exhaustive search can also be used to minimize R and find the optimal {DP, Kd} more rapidly.

diffusivities estimated by RD analysis agree with those obtained using other methods (see ref 24). The measurement for the biotin-avidin system, for which RD estimates Kd ∼ 10-12 M vs the literature value of ∼10-15 M,19d,e illustrates well the sensitivity limitations of the method. Dimensional analysis of the RD equations (see the Supporting Information, section 4) shows that the key dimensionless parameter controlling the dynamics of the system is β ) Kd/ [P]0, which relates the dissociation constant to the concentration of protein in the stamp, and that in order to achieve maximal sensitivity, the value of this parameter should be close to unity. However, for very potent ligands characterized by small values of Kd, the concentration of protein should be very low (so that Kd/[P]0 ∼ 1). Unfortunately, decreasing [P]0 reduces the optical contrast of staining which, ultimately, reduces the sensitivity of the readout. For example, for the biotin/avidin system, even with the enhanced CB/ silver staining used here, the lowest concentration of protein initially in the stamp that gives interpretable results is ∼4 µM, which corresponds to the Kd estimate ∼10-12 M. The real value of Kd ∼ 10-15 M would require visualization of proteins at concentrations significantly below the sensitivity of the CB/silver stain. From this analysis, we conclude that our RD method is suitable to estimate binding constants down to, but not below, picomolar levels. In summary, we described a method to determine proteinligand binding constants based on the visualization of reactiondiffusion fronts in ligand-modified gels. This method is technically straightforward to implement and applicable to different protein-ligand pairs, and the values of Kd it yields agree with (24) (a) One popular way to estimate diffusivities in gels is through Amsden’s equation whereby the Stokes-Einstein diffusivity is reduced by an exponential factor accounting for the presence of the gel.24b,c DP ) (kBT/ 6πµr) exp (-kcrφ0.75), where kB is Boltzmann’s constant, T is the temperature, µ ) 0.89 · 10-3 Pa s is the viscosity of water (the solvent),24d kc ) 0.45 Å-1 is a constant,24b φ is the volume fraction of polymer in the gel, φ ) 0.1 for BCAII and TP, and φ ) 0.06 for AV, and r is the hydrodynamic radius of the protein, r ) 28 Å for BCAII,19a r ) 19 Å for TP,24e and r ) 45 Å for AV.24f With the use of these parameters, the model gives DP ) 9.3 × 10-12 m2/s for BCAII, DP ) 2.8 × 10-11 m2/s for TP, and DP ) 4.7 × 10-12 m2/s for AV; these values are close to those predicted by our RD method (2.6 × 10-11 m2/s for BCAII, 6.9 × 10-11 m2/s for TP, and 1.7 × 10-11 m2/s for AV). (b) Amsden, B. Macromolecules 1998, 31, 8382. (c) Cukier, R. I. Macromolecules 1984, 17, 252. (d) Lide, D. R. CRC Handbook of Chemistry and Physics, 90th ed. (Internet Version 2009); CRC Press/Taylor and Francis: Boca Raton, FL, 2009. (e) Diaz, J. F.; Balkus, K. J. J. Mol. Catal. B: Enzym. 1996, 2, 115. (f) Green, N. M.; Joynson, M. A. Biochem. J. 1970, 118, 71.

Analytical Chemistry, Vol. 82, No. 21, November 1, 2010

8783

the solution values (see ref 18 and the Supporting Information, sections 4 and 5). Although external fields could certainly be added to the system to move proteins around more rapidly, they would complicate the analysis of results by introducing additional free parameters (e.g., protein electrophoretic mobility in an electric field) that would have to be calibrated in separate experiments and decoupled from the always-present diffusion.

SUPPORTING INFORMATION AVAILABLE Additional information as noted in text including the synthesis of ligands, the further experimental details of preparing gel, stamping and staining proteins, NMR spectra of all ligands synthesized, modeling details, and computer codes. This material is available free of charge via the Internet at http://pubs.acs.org.

ACKNOWLEDGMENT Y.W. and P.J.W. contributed equally to this work. This work was supported by NSF CAREER Grant CTS-0547533.

Received for review August 3, 2010. Accepted September 28, 2010.

8784

Analytical Chemistry, Vol. 82, No. 21, November 1, 2010

AC102055A