This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Similar articles in this journal Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via Google Scholar Google Scholar Articles by Gottschalk, M. Articles by Halle, B. Search for Related Content PubMed PubMed Citation Articles by Gottschalk, M. Articles by Halle, B.

Protein Self-Association in Solution: The Bovine Pancreatic Trypsin Inhibitor Decamer

Department of Biophysical Chemistry, Lund University, SE-22100 Lund, Sweden

Correspondence: Address reprint requests to Dr. Bertil Halle, Dept. of Biophysical Chemistry, Lund University, SE-22100 Lund, Sweden. Tel.: 46-46-222-9516; Fax: 46-46-222-4543; E-mail: bertil.halle{at}bpc.lu.se.

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS ACKNOWLEDGEMENTS REFERENCES

 
We have used magnetic relaxation dispersion to study bovine pancreatic trypsin inhibitor (BPTI) self-association as a function of pH, salt type and concentration, and temperature. The magnetic relaxation dispersion method sensitively detects stable oligomers without being affected by other interactions. We find that BPTI decamers form cooperatively under a wide range of solution conditions with no sign of dimers or other small oligomers. Decamer formation is opposed by electrostatic repulsion among numerous cationic residues confined within a narrow channel. Accordingly, the decamer population increases with increasing pH, as cationic residues are deprotonated, and with increasing salt concentration. The salt effect cannot be described in terms of Debye screening, but involves the ion-specific sequestering of anions within the narrow channel. The lifetime of the BPTI decamer is 101 ± 4 min at 27°C. We propose that the BPTI decamer, with a heparin chain threading the decamer channel, plays a functional role in the mast cell. We also detect a higher oligomer that appears to be a subcritical nucleation cluster of 3–5 decamers. We argue that monomeric crystals form at high pH despite a high decamer population in solution, because the ion pairs that provide the critical decamer-decamer contacts are disrupted at high pH.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS ACKNOWLEDGEMENTS REFERENCES

 
Self-interactions govern the assembly of oligomeric proteins (Jaenicke and Lilie, 2000), the pathological aggregation of misfolded proteins (Kelly, 1998), and the nucleation, growth, and polymorphism of protein crystals (Rosenberger et al., 1996; Riès-Kautt and Ducruix, 1997; Kierzek and Zielenkiewicz, 2001). Fundamental progress in these diverse fields must ultimately be based on a quantitative understanding of how proteins interact with themselves in solution. The colloidal approach, epitomized by the Derjaguin-Landau-Verwey-Overbeek theory (De Young et al., 1993; Leckband and Israelachvili, 2001), has not proven to be a fruitful avenue toward this goal (Piazza, 1999; Elcock et al., 2001). A molecular approach is evidently needed, which recognizes the structural details of the heterogeneous protein surface and takes into account a variety of short-range interactions, direct and solvent-mediated.

Whereas theoretical studies of protein self-association are still at an early stage, a vast amount of experimental work, primarily with scattering techniques, has been carried out on the phase behavior and oligomerization of globular proteins in solution. Bovine pancreatic trypsin inhibitor (BPTI) is among the proteins that have been studied most extensively in this regard. This 58-residue (6.5 kDa) basic (pI = 10.5) protein has long been thought to undergo self-association. Early sedimentation, calorimetry, and dynamic light scattering (DLS) work, reviewed by Gallagher and Woodward (1989), lead to conflicting results as to whether BPTI forms a dimer. A more recent NMR pulsed-gradient, spin-echo (PGSE) self-diffusion study concluded that BPTI is predominantly dimeric at mM concentrations, with little or no effect of pH or added salt (Ilyina et al., 1997). Several DLS and small-angle x-ray scattering (SAXS) studies (Lafont et al., 1994; Veesler et al., 1996; Lafont et al., 1997) at higher concentrations of NaCl, KSCN, or (NH₄)₂SO₄ indicated that BPTI solutions are polydisperse in the absence of added salt, but monodisperse and tetrameric at high salt concentrations (near the BPTI solubility limit). Although it has been argued that BPTI possesses a self-complementary surface that might stabilize a dimer (Zielenkiewicz et al., 1991), none of the experimental studies mentioned so far could provide unambiguous information about oligomer structure. Such information has come, not from solution studies, but from crystallography.

Until recently, all crystallographic studies of wild-type BPTI used one of three orthorhombic crystal forms, denoted I, II, and III, all with one BPTI molecule per asymmetric unit (Deisenhofer and Steigemann, 1975; Wlodawer et al., 1984, 1987b). These crystals are grown at pH 9–10, close to the isoelectric point of BPTI, with phosphate as the salting-out agent. Within the past few years, three new crystal forms, denoted A, B, and C, have been prepared by salting out BPTI with NaCl, KSCN, or (NH₄)₂SO₄ at pH 4.5 or 7.0, where BPTI has a net charge +6 (Hamiaux et al., 1999, 2000; Lubkowski and Wlodawer, 1999). These new hexagonal or monoclinic crystals have five or 10 BPTI molecules in the asymmetric unit and represent different stackings of a common decamer structure.

Subsequent studies of BPTI self-association in solution have been interpreted in terms of BPTI decamers rather than dimers. In the most comprehensive of these studies, SAXS data from BPTI solutions at pH 4.5 and high concentrations of the salts used to grow crystal forms A, B, and C were shown to be consistent with a mixture of monomers and decamers (Hamiaux et al., 2000). It was thus concluded, in contrast to earlier claims (Lafont et al., 1997), that decameric BPTI crystals grow from a polydisperse solution. Similar findings have been reported in later x-ray and neutron small-angle scattering (SAS) studies (Budayova-Spano et al., 2000, 2002). Most recently, it was deduced from DLS data that BPTI in 1 M NaCl is decameric at pH 5 but monomeric at pH 7 (Tanaka et al., 2002). Furthermore, it was concluded that BPTI is monomeric under the conditions used to grow crystal forms I–III (Tanaka et al., 2002).

After a long and troubled history, it may appear that a consensus view of BPTI self-association in solution is finally emerging. However, this view is largely based on studies by SAS and DLS, techniques that do not readily distinguish oligomerization from longer-range interactions. Moreover, because these low-resolution techniques cannot unambiguously identify the oligomer structure, the interpretation of SAS and DLS data has relied heavily on recent evidence of self-association in crystals that may or may not be relevant under diverse solution conditions.

The present work is motivated by the belief that our understanding of BPTI self-association would benefit from studies by a different experimental technique: magnetic relaxation dispersion (MRD). Within the biomolecular field, MRD has previously been used mostly to study protein hydration and, in particular, internal water molecules in proteins and nucleic acids (Halle et al., 1999; Halle and Denisov, 2001). Here, we use ¹H MRD to determine the rotational correlation times (or rotational diffusion coefficients) of the oligomeric species present in solution. Whereas MRD has been used in a few earlier studies of protein self-association (Lindstrom et al., 1976; Raeymaekers et al., 1989; Koenig et al., 1990), our data extend to higher frequencies (necessary for a small protein like BPTI) and are analyzed in a more rigorous way (Halle et al., 1998).

Although the MRD method has its own limitations, it presents three decisive advantages in studies of protein self-association. First, it is insensitive to long-range interactions and therefore provides clear-cut information about oligomerization. Second, if the correlation times are sufficiently different, oligomers of different size can be resolved irrespective of the oligomer dissociation kinetics. Third, because rotational friction scales with volume whereas translational friction scales with linear dimension, rotational diffusion is a more sensitive probe of oligomerization than is translational diffusion. For example, the rotational correlation time of the BPTI decamer is a factor 8 longer than for the monomer, whereas the translational diffusion coefficients of the two species only differ by a factor 2.

The present MRD study of BPTI complements and extends previous solution studies (by SAS, DLS, and PGSE) in important ways. We identify a dominant oligomeric species with the rotational correlation time expected for the decamer present in crystal forms A, B, and C, as previously found by SAS under the low-pH and high-salt conditions used to grow decameric crystals (Hamiaux et al., 2000). In addition, we observe a larger oligomeric species that may be a subcritical crystallization nucleus. Unexpectedly, we find decamers also far from the solubility limit and even in the absence of salt. Moreover, and in contrast to a recent DLS study (Tanaka et al., 2002), we find a substantial decamer population under the high-pH conditions used to grow monomeric crystal forms. Using a salt-jump relaxation experiment, we also determine the lifetime of the BPTI decamer. These new results might seem to shatter the emerging consensus on BPTI self-association. However, in cases where our results disagree with previous conclusions, we indicate how the differences might be resolved. In addition, we are able to rationalize our results in terms of specific structural features of the BPTI decamer. We thus arrive at a consistent view of BPTI self-association in solution and in crystals. Finally, we propose a functional role for the BPTI decamer in the secretory granules of mast cells.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS ACKNOWLEDGEMENTS REFERENCES

 
Preparation of protein solutions
Bovine pancreatic trypsin inhibitor was supplied by Novo Nordisk A/S, Gentofte, Denmark (Aprotinin, batch # 84059-17-II). To remove residual salt, the protein preparation was exhaustively dialyzed (twice against deionized water and twice against millipore water). Agarose gel electrophoresis at pH 8.6, where the net charge of BPTI is +4.9, revealed only one, very weak, subsidiary band, consistent with a minor fraction (1–2%) truncated BPTI with Arg-1 and Pro-2 removed from the N-terminal by heat treatment (L. C. Petersen, Novo Nordisk A/S, personal communication). For the MRD samples, the dialyzed and lyophilized BPTI was dissolved in millipore water, giving a pH of 8.3–8.4. The pH was adjusted to within ±0.05 of the target value by µL additions of HCl or NaOH. No buffers were used. After addition of desiccator dried salt, the solution was centrifuged at 14,000 g for 3 min to remove a small fraction (<1%) of aggregated material. (In the case of K₂HPO₄, solution pH was adjusted after adding the salt.) All salts used (NaCl, CsCl, NaI, KSCN, and K₂HPO₄) were of >99% purity (purchased from Merck or BDH).

For the salt-jump kinetic relaxation experiment, we used BPTI supplied by Bayer Healthcare AG, Wuppertal, Germany (Trasylol, lot # 9104, 97% purity by HPLC). The protein was dialyzed as described above and dissolved in water with <0.001% O₂ (Fluka BioChemika).

The K₂HPO₄ solution was studied at pH 9.5, where the HPO₄^2- ion accounts for >99% of the phosphate. Because the proton in HPO₄^2- exchanges rapidly with water protons, it contributes to the observed ¹H magnetization. To contribute to the dispersion, however, protein-bound HPO₄^2- ions must have residence times longer than the rotational correlation time of the protein (26 ns for the BPTI decamer) but shorter than the intrinsic spin relaxation time (on the order of 1 ms). Judging from the measured dispersion amplitude factor B (see below), the HPO₄^2- ions that presumably are located within the decamer channel do not exchange on this timescale. This potential complication can therefore be ignored.

BPTI concentrations were determined spectrophotometrically (GBC UV-VIS 920) at 280 nm, using an extinction coefficient of 0.837 mL mg^-1 cm^-1, determined from quantitative amino-acid analysis on one sample. The solutions were then transferred to 10 mm NMR tubes. To minimize leaching of paramagnetic ions from the Pyrex glass, the NMR tubes were soaked with 3 M HCl for 24 h and then rinsed with millipore water and EDTA solution. To remove paramagnetic oxygen from the solution, the samples were gently bubbled with argon gas for 4 h and then sealed with a septum. With this procedure, the ¹H relaxation rate in a pure water sample was found to be 0.267 ± 0.008 s^-1, in agreement with the standard literature value of 0.266 s^-1 (Hindman et al., 1973), and remained within 1% of this value for at least 10 days.

Relaxation dispersion measurements
The longitudinal relaxation rate of the water ¹H resonance was measured over more than four frequency decades, from 10 kHz to 200 MHz. To cover this frequency range, we used three types of NMR spectrometers: 1), a Stelar Spinmaster fast field-cycling (FC) spectrometer (10 kHz–12 MHz); 2), a field-variable iron-core magnet (Drusch) equipped with a modified Bruker MSL 100 console or a Tecmag Discovery console (2.5 – 77.9 MHz); and 3), Bruker Avance DMX 100 and 200 spectrometers with conventional cryomagnets (100.1 and 200.1 MHz). The temperature was maintained at either 4.0 ± 0.1 or 27.0 ± 0.1°C using a Stelar variable temperature control unit (<100 MHz) or a Bruker Eurotherm regulator (at 100 and 200 MHz). Temperatures were checked using a thermocouple referenced to an ice-water bath.

In the fixed-field (non-FC) experiments, the longitudinal relaxation rate R₁ was measured with the 180° - - 90° inversion recovery sequence, an 8-step phase cycle, and 20 randomly ordered delay times. The total ¹H magnetization recovers bi-exponentially because it includes not only water and rapidly exchanging BPTI protons, but also a small contribution (typically, 5%) from nonexchanging BPTI protons. The latter contribution, which can increase the apparent R₁ by a few percent, was largely eliminated by integrating the water peak over a range where the spectral overlap was negligible (for non-FC experiments), or by using an acquisition delay sufficiently long that the protein magnetization had decayed before the signal was recorded (FC experiments). In this way, single-exponential recovery curves were obtained, from which R₁ was determined by a three-parameter fit. The accuracy of R₁ is estimated to ±1% (one standard deviation).

The FC technique overcomes the sensitivity problem of conventional fixed-field experiments in weak magnetic fields (Noack, 1986). The polarization and detection fields were set to 10 and 8 MHz (in ¹H frequency units), respectively. A field switching rate of 4 MHz ms^-1 and a switching delay of 10 ms were used. Relaxation measurements were performed with two different field cycles (Anoardo et al., 2001): the prepolarized cycle below 4 MHz and the nonpolarized cycle above 4 MHz. In either case, 15 different relaxation delays (evolution times) were used. The magnitude of the quadrature-detected signal after a 90° pulse was recorded and averaged over 32 transients with a 4-step phase cycle. The relaxation curves were invariably single-exponential. Heating of the air-core magnet during the polarization period causes a small field drift and a consequent systematic error in R₁. This heating effect was largely compensated by a feedback current proportional to the temperature of the magnet and by inserting a cooling delay to reduce the duty cycle (Anoardo et al., 2001). For R₁ values below 2 s^-1, a residual correction (typically, a few percent) for magnet heating effects was applied on the basis of an empirical relationship established by measuring a range of R₁ values with both field cycles at 4 MHz. The accuracy of R₁ determined by the FC technique is estimated to ±1.5–2% (one standard deviation).

Analysis of relaxation dispersion data
The measured ¹H relaxation rate is due to thermal fluctuations of intramolecular and intermolecular magnetic dipole-dipole couplings experienced by water protons and labile BPTI protons in fast or intermediate exchange (residence time <10 ms, typically) with the water protons (Abragam, 1961; Halle et al., 1999; Halle and Denisov, 2001). The relaxation dispersion, i.e., the frequency dependence of R₁, is produced by long-lived (residence time 10^-9–10^-2 s) water molecules in intimate association with the protein and by labile protons with residence times in the same range.

If all residence times are long compared to the rotational correlation time _R of the protein, as is the case for BPTI (Denisov et al., 1995, 1996; Venu et al., 1997), the dispersion profile, R₁(₀), from a solution containing BPTI in N different oligomeric states, is described by the following relations (Abragam, 1961; Venu et al., 1997):

(1)

(2)

(3)

(4)

Here, b_n is the mean-square fluctuation amplitude and _R,n the rank-2 rotational correlation time associated with the n^th BPTI oligomer species. Furthermore, _n = b_n,inter/b_n is the relative contribution from intermolecular dipole-dipole couplings to the overall fluctuation amplitude, b_n = b_n,intra + b_n,inter . The functions L_n (₀) will be referred to as Lorentzians, even though they are, in fact, linear combinations of two Lorentzian (reduced) spectral density functions differing by a factor 2 in frequency. Apart from an overall scaling by a factor 0.9, the functions and differ very little (Venu et al., 1997). The value of _n therefore has no significant effect on the oligomer fractions that we deduce from the data. We set _n = 0.33, as previously found for the four internal water molecules in BPTI (Venu et al., 1997).

The quantity in Eq. 1 represents all frequency-independent contributions to the measured relaxation rate, including the secular (zero-frequency) intermolecular contribution (Venu et al., 1997). Because of the sparse sampling of the high-frequency regime in most of our dispersions, could not be determined with useful accuracy. In the following, dispersion profiles will be displayed with the small frequency-independent contribution subtracted and with the frequency-dependent part normalized to the same BPTI concentration:

(5)

where N_T is the number of water molecules per BPTI molecule in the solution. This scaling is possible because the quantity b_n in Eq. 1 is inversely proportional to N_T (Halle et al., 1999; Halle and Denisov, 2001). (The relative contribution to N_T from labile BPTI protons is negligible.) As reference concentration, we use N_T = 3595, corresponding to 14.5 mM BPTI. In the presence of self-association, the scaling in Eq. 5 does not completely remove the concentration dependence, because the equilibrium concentrations of different oligomer species depend on the overall protein concentration (and, therefore, on N_T). Nevertheless, scaling is useful for displaying the effect of other variables, such as pH and salt concentration, when the variation in the BPTI concentration among the samples is small.

The experimental relaxation dispersion data were subjected to nonlinear Marquardt-Levenberg ² minimization (Press et al., 1992) with the model function given by Eqs. 1–4. This nonlinear fit involves the 2N + 1 parameters , b_n, and _R,n , with the products b_{n R,n} constrained to be nonnegative. The number N of Lorentzians to be included in the fit was determined objectively by the F-test with a cutoff probability of 0.9 (Press et al., 1992; Halle et al., 1998). The 20 reported dispersions were thus found to be adequately modeled by two or three Lorentzians. In the final joint fits, we therefore used N = 3 throughout. On the basis of the associated rotational correlation times _R,n (see below), we assign these Lorentzian dispersion steps to BPTI monomers (n = 1), decamers (n = 2), and higher oligomers, hereafter referred to as polymers (n = 3). The former two steps are visually distinct with the major part of the dispersions occurring in the 10–100 MHz range for monomers and in the 1–10 MHz range for decamers.

Whereas the overlapping decamer and polymer dispersions are adequately sampled by FC measurements (extending up to 12 MHz), characterization of the monomer dispersion requires more time-consuming fixed-field measurements at higher frequencies. To reduce the experiment time, most dispersion profiles only include two measurements (100 and 200 MHz) >12 MHz. Because such "incomplete" dispersions could not be robustly fitted to three Lorentzians, they were analyzed by means of simultaneous fits to a set of dispersions, including one or more "complete" dispersions. In these joint fits, we imposed the constraint that each of the three rotational correlation times has the same value for all samples in the set. For example, in a set of samples differing only in salt concentration, we assumed that all samples contain the same three oligomer species (with the same three correlation times _R,n ) but at different concentrations. Although the correlation times _R1 and _R2 obtained from such joint fits are close to the values expected for the BPTI monomer and decamer, our data do not allow the dispersions to be accurately decomposed into three Lorentzian components in a unique way. In particular, a certain amount of compensating variation in correlation times and oligomer fractions can be accommodated with little effect on the quality of the fit (as measured by ²). To obtain the most reliable estimates of the oligomer fractions, we therefore performed the joint fits with the monomer and decamer correlation times fixed at independently determined values (see below). The free parameters were thus the common polymer correlation time _R3 and, for each dispersion in the jointly fitted set, an -value, and N amplitude factors b_n. Quoted uncertainties in the fitted parameter values correspond to one standard deviation and were obtained by the Monte Carlo method (Press et al., 1992) using 1000 synthetic data sets.

Rotational correlation times
Unconstrained joint fits to our MRD data yield values in the range 2.5–3.5 ns for the monomer rotational correlation time _R1. Because the high-frequency range is not densely sampled in our data, we have chosen to fix _R1 at the best available estimate of the rotational correlation time of monomeric BPTI, obtained from ¹⁵N relaxation of 3 mM BPTI in a 90:10 H₂O:D₂O mixture (no added salt) at pH 4.7 and 25°C (Beeser et al., 1997). Scaling the reported correlation time, 3.5 ns, from the water viscosity of the ¹⁵N study (0.911 cP) to the viscosity of H₂O at 27°C (0.851 cP), as in the present study, we obtain _R1 = 3.27 ns.

To obtain the rotational correlation time of the BPTI decamer, we carried out hydrodynamic calculations with the program HYDROPRO (Garcia de la Torre et al., 2000). In these calculations, each nonhydrogen atom in a crystallographic model of the protein (or decamer) is replaced by a spherical bead of radius a_H. The shell of beads remaining after all internal beads have been deleted is then filled with smaller spheres of radius that act as point sources of hydrodynamic friction. The rotational diffusion tensor D_R is computed as a function of and extrapolated to = 0 (Garcia de la Torre and Bloomfield, 1981). The rank-2 isotropic rotational correlation time is defined as _R = (2 Tr D_R)^-1. Because macroscopic continuum hydrodynamics is not strictly valid on the atomic scale, this calculation does not necessarily yield results in quantitative agreement with experiment. The approach usually adopted is to regard the bead radius a_H as an empirical parameter (rather than using the van der Waals radii of the actual atoms), the value of which is determined by requiring that the calculation agrees with the experimental value of a particular hydrodynamic quantity, such as _R (Garcia de la Torre et al., 2000).

Using the crystal structure 5PTI (Wlodawer et al., 1984) for monomeric BPTI (with the two disordered side chains in the major conformation), we calculated _R1 at 27°C for a_H values in the range 1.0–4.0 Å. Interpolation with the experimental value _R1 = 3.27 ns yields a_H = 2.88 Å. We then used this bead radius to calculate the rotational correlation time of the BPTI decamer from the crystal structure 1BHC (Hamiaux et al., 1999). The result is _R2 = 26.3 ns at 27°C. In all fits to MRD data at 27°C reported here, _R1 and _R2 were constrained to these values.

Rotational diffusion of both BPTI monomer and decamer is actually slightly anisotropic. For the 5PTI monomer, HYDROPRO calculations yield 1.28 for the ratio of the largest and smallest of the five rotational correlation times (derived from the three eigenvalues of D_R) that characterize the rank-2 spectral density function for asymmetric-top rotational diffusion (Woessner, 1962). For the more nearly spherical 1BHC decamer, the corresponding ratio is 1.08. Neither of these ratios is sufficiently large to cause a detectable deviation in the dispersion profile from spherical-top behavior. We therefore use the isotropic average of the rotational correlation times. If both monomer and decamer were spherical and of the same density, _R would be proportional to protein volume so that _R2/_R1 = 10. The slightly smaller value, _R2/_R1 = 26.3/3.27 = 8.05, obtained from the hydrodynamic calculations, is consistent with the more nearly spherical shape of the decamer.

Salt-jump relaxation experiment
The mean lifetime of the BPTI decamer was determined from a real-time salt-jump kinetic relaxation experiment. ¹H relaxation dispersion profiles were recorded at 27°C from solution A (13.3 mM BPTI, no salt) and solution B (14.7 mM, 0.9 M NaCl), both at pH 4.5. (These dispersion profiles, obtained with BPTI from Bayer, were consistent with the profiles obtained with BPTI from Novo Nordisk.) Equal volumes (1 mL) of solutions A and B were mixed, whereupon the approach to the new oligomerization equilibrium (corresponding to 0.45 M NaCl) was monitored by measuring the relaxation rate R₁ at a ¹H frequency of 100 kHz at intervals of 40 min during 13.4 h. The approach to equilibrium was characterized by a relaxation time constant _relax, determined from a nonlinear least-squares fit of the three parameters in the expression

(6)

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS ACKNOWLEDGEMENTS REFERENCES

 
From dispersion profile to oligomer fractions
Eighteen ¹H relaxation dispersion profiles were measured on BPTI solutions under different conditions of temperature, pH, and added salt (type and concentration). The relevant sample characteristics are compiled in Table 1. Fig. 1 shows the ¹H relaxation dispersion profile at 27°C from an aqueous solution containing 14.5 mM BPTI and 0.90 M NaCl at pH 4.5. The correlation times obtained from fits of Eqs. 1–4 to these data are given in Table 2 for N = 1, 2, or 3 Lorentzians. Unconstrained fits with one or two Lorentzians fail to describe the data within the experimental accuracy. An unconstrained 3-Lorentzian fit yields correlation times _R1 and _R2 close to those expected for the BPTI monomer and decamer (see Materials and Methods). Fixing these two correlation times to independently derived values (see Materials and Methods) yields a fit of similar quality as with three unconstrained correlation times.

View larger version (18K): [in this window] [in a new window]   FIGURE 1  1H relaxation dispersion profile at 27°C from an aqueous solution containing 14.5 mM BPTI and 0.90 M NaCl at pH 4.5. The dispersion curve resulted from a constrained 3-Lorentzian fit according to Eqs. 1–4. The individual Lorentzian components (dashed curves) and the bulk water contribution (horizontal line) are also shown.

(7)

where x_n is the fraction of BPTI molecules that belong to the n^th oligomer species or, equivalently, the weight fraction of that species. The intrinsic mean-square fluctuation amplitude ß_n is proportional to the number of protons (per BPTI monomer) with residence times long enough (>_R,n) to sample the rotational diffusion of the oligomer but short enough (<1/(ß_{n R,n})) to act as a relaxation sink for the observed water ¹H magnetization (Halle et al., 1999; Halle and Denisov, 2001). The fluctuation amplitude ß₁ for the monomer is due to four internal water molecules and a pH-dependent number of rapidly exchanging labile BPTI protons (Venu et al., 1997). If these contributions are not the same for the decamer and the monomer, ß₂ will differ from ß₁. As argued below, any such differences are likely to be small.

Monomeric BPTI contains two intramolecular cavities, accommodating four internal water molecules (Wlodawer et al., 1987a) that dominate ß at pH 4.5 (Venu et al., 1997). These internal water molecules are conserved in the decamer (Lubkowski and Wlodawer, 1999; Hamiaux et al., 1999). Cavities formed at the interfaces between adjacent BPTI molecules in the decamer might present long-lived hydration sites not present in the monomer, thereby making ß₂ > ß₁. A cavity search on the structure 1BHC (Hamiaux et al., 1999) using the program VOIDOO (Kleywegt and Jones, 1994) with 1.2 Å probe radius (Hubbard and Argos, 1995) revealed two small (30–50 ų), symmetry-related cavities in each of the five quasi-equivalent major interpentamer contact regions. In the crystal structure, no water molecules are located in these mainly nonpolar cavities. The central channel in the decamer can accommodate a few dozen water molecules, but is too wide (10 Å diameter) to provide the geometric constraints necessary for long-lived hydration (Denisov and Halle, 1996).

Another reason why ß₂ might differ from ß₁ is that labile BPTI protons contribute to different extents in monomer and decamer. This could happen if some of the involved side chains are buried at intermolecular contacts in the decamer, thereby retarding proton exchange and making ß₂ < ß₁. Furthermore, labile protons with residence times in the millisecond range could be in the fast-exchange limit for the monomer but not for the more slowly tumbling decamer, which would also make ß₂ < ß₁. At pH < 5, internal water molecules dominate ß, which therefore is insensitive to moderate variations in the labile proton contribution. Moreover, most of the hydroxyl and carboxyl protons that contribute at low pH are fully exposed in the decamer. At pH > 8, ammonium and guanidinium protons make a large contribution to ß, but all 110 such groups are solvent-exposed in the decamer. Calculations using known or estimated proton exchange rate constants and pK_a values (Venu et al., 1997; Denisov and Halle, 2002) show that also the second effect is negligible at the pH values studied here. (The effect could be significant in the pH range 5–8, where the labile protons in lysine and arginine side chains go from slow to fast exchange.)

These considerations justify the approximation ß₂ = ß₁ under the conditions of the present study. In the absence of structural information about the BPTI polymer, we assume that also ß₃ = ß₁. (We argue below that the polymer is a loosely associated trimer of decamers, for which we expect that ß₃ = ß₂.) These approximations can actually be tested with the aid of our dispersion data. Fig. 2 shows the summed amplitude B = b₁ + b₂ + b₃ for all 18 dispersions. As expected, B increases strongly with pH as base-catalyzed proton exchange brings more labile protons into the fast-exchange regime. In contrast, B is nearly invariant on addition of salt at constant pH, in particular for the NaCl series. Because the oligomer fractions x_n vary substantially with the salt concentration, as shown qualitatively by the dispersion profiles and quantitatively by the following analysis, also B = x₁ß₁ + x₂ß₂ + x₃ß₃ should vary with salt concentration if ß₂ or ß₃ differ significantly from ß₁. Because such a B variation is not observed, we set ß₁ = ß₂ = ß₃ = B. The oligomer fractions, which must sum to unity, can then be obtained as

(8)

View larger version (13K): [in this window] [in a new window]   FIGURE 2  Dependence of the summed mean-square dispersion amplitude B on (a) salt concentration at pH 4.5, and (b) pH. Panel a shows results for NaCl at 27°C (solid circles) or 4°C (open circles), CsCl (open squares), and, from left to right, KSCN, Na2SO4, and NaI (solid squares). Panel b shows results for no salt or 0.10 M NaCl (solid circles), 0.50 or 0.70 M NaCl (open circles), and 0.10 M K2HPO4 (solid square).

View larger version (25K): [in this window] [in a new window]   FIGURE 3  1H relaxation dispersion profiles from aqueous BPTI solutions at 27°C, pH 4.5 and the indicated NaCl concentrations. The data have been normalized to 14.5 mM BPTI. The dispersion curves resulted from constrained fits according to Eqs. 1–4 with parameter values as given in Table 1. All data in Figs. 3, 5, and 8 were fitted jointly with a common R3.

View larger version (26K): [in this window] [in a new window]   FIGURE 5  1H relaxation dispersion profiles from aqueous BPTI solutions at 27°C, pH 4.5 and the indicated NaCl or CsCl concentrations. The data have been normalized to 14.5 mM BPTI. The dispersion curves resulted from constrained fits according to Eqs. 1–4 with parameter values as given in Table 1. All data in Figs. 3, 5, and 8 were fitted jointly with a common R3.

View larger version (19K): [in this window] [in a new window]   FIGURE 8  1H relaxation dispersion profiles from aqueous BPTI solutions with 0.70 M NaCl at 27°C and pH 4.5 (Z = +6.4) or pH 2.5 (Z = +10.1). The data have been normalized to 14.5 mM BPTI. The dispersion curves resulted from constrained fits according to Eqs. 1–4 with parameter values as given in Table 1. All data in Figs. 3, 5, and 8 were fitted jointly with a common R3.

View larger version (23K): [in this window] [in a new window]   FIGURE 4  Fraction BPTI oligomers as a function of (a) salt concentration at pH 4.5 and (b) nominal net structural charge of BPTI. The shaded region is bounded from below (open symbols) by the polymer fraction x3 and from above (solid symbols) by the sum of the decamer and polymer fractions, x2 + x3 = 1 - x1. Panel a shows results for NaCl at 27°C (circles) or 4°C (squares), and for CsCl (triangles). Panel b shows results for 0.1 M NaCl (circles) and for 0.1 M K2HPO4 (squares). Sample pH was converted to BPTI charge Z with the aid of the titration curve in Fig. 7. The actual net charge per BPTI molecule in the decamer may differ due to association-induced pKashifts.

View larger version (11K): [in this window] [in a new window]   FIGURE 7  Net charge of monomeric BPTI as a function of pH, calculated from published pKa values (Wüthrich and Wagner, 1979). The pH values of samples examined here are indicated by points.

A much stronger ion-specificity is seen when the Cl^- ion is replaced by I^- or SO₄^2- (Fig. 6). In 0.4 M NaI, 30% of the BPTI molecules form decamers and 8% form polymers. This may be contrasted with 0.5 M NaCl, with 12% decamers and no significant amount of polymer (Table 1). The finding that I^- induces self-association of BPTI more strongly than Cl^- is consistent with I^- being more potent than Cl^- in salting out basic proteins (with a positive net charge) like BPTI (Collins and Washabaugh, 1985). The thiocyanate ion, an even more potent salting-out agent, was only examined at 0.1 M due to precipitation problems at higher KSCN concentrations. Already at this concentration, the decamer fraction is comparable to that in 0.5 M NaCl (Table 1). The sulfate ion, at the other end of the Hofmeister series (Collins and Washabaugh, 1985), was examined at 0.22 M Na₂SO₄ (Fig. 6). This corresponds to a Debye length of 3.7 Å, similar to that in 0.7 M NaCl (3.6 Å). Comparing these two solutions, we see that the decamer fraction is similar, whereas the polymer fraction is threefold higher with sulfate than with chloride (Table 1). The solubility curve of BPTI is similar for NaCl and (NH₄)₂SO₄ (Lafont et al., 1997). If Na₂SO₄ behaves as (NH₄)₂SO₄ in this regard, then our 0.22 M Na₂SO₄ sample is quite far from the solubility curve. Yet, the oligomer fractions are similar to those in 0.9 M NaCl, which is close to the solubility limit.

View larger version (23K): [in this window] [in a new window]   FIGURE 6  1H relaxation dispersion profiles from aqueous BPTI solutions at 27°C, pH 4.5 and the indicated salt types and concentrations. The data have been normalized to 14.5 mM BPTI. The dispersion curves resulted from constrained fits according to Eqs. 1–4 with parameter values as given in Table 1. All data in Fig. 6 were fitted jointly with a common R3.

If BPTI self-association were completely governed by long-range forces, as in the Derjaguin-Landau-Verwey-Overbeek theory (Leckband and Israelachvili, 2001), a reversal of the net protein charge should not affect self-association. Contrary to this prediction, we find (Fig. 4 b) much stronger self-association at pH 12.2 (Z = -6.5) than at pH 4.5 (Z = +6.4). This is the case in the absence of salt as well as in 0.50 M NaCl (Fig. 9). This finding points to the importance of individual charged residues in BPTI, rather than just the net charge of the protein.

View larger version (27K): [in this window] [in a new window]   FIGURE 9  1H relaxation dispersion profiles from aqueous BPTI solutions at 27°C with no salt or 0.50 M NaCl and at pH 4.5 (Z = +6.4) or pH 12.2 (Z = -6.5). The data have been normalized to 14.5 mM BPTI. The dispersion curves resulted from constrained fits according to Eqs. 1–4 with parameter values as given in Table 1. The pH 12.2 data in Fig. 9 and the data in Fig. 6 were fitted jointly with a common R3.

In NaCl solution, the solubility of BPTI increases with decreasing temperature (Lafont et al., 1994). At given protein and NaCl concentrations, the solution will therefore be further removed from the solubility curve at low temperature. Fig. 10 shows the dispersion profiles at 4°C and pH 4.5 in the absence of salt and in 0.66 M NaCl. Apart from the trivial /T scaling (where is the solvent viscosity) of all correlation times (see Materials and Methods), the dispersions are very similar to the corresponding ones at 27°C. In fact, the temperature variation of the decamer and polymer fractions is hardly significant (Table 1 and Fig. 4 a).

View larger version (18K): [in this window] [in a new window]   FIGURE 10  1H relaxation dispersion profiles from aqueous BPTI solutions at 4°C and pH 4.5 with no salt or 0.66 M NaCl. The data have been normalized to 14.5 mM BPTI. The dispersion curves resulted from constrained fits according to Eqs. 1–4 with parameter values as given in Table 1.

View larger version (25K): [in this window] [in a new window]   FIGURE 11  The insert shows 1H relaxation dispersion profiles recorded at 27°C from a 14.0 mM BPTI solution with 0.45 M NaCl and pH 4.5, made by mixing equal volumes of two BPTI solutions with no salt and 0.90 M NaCl. The initial dispersion profile is the average of the (concentration normalized) profiles from the original solutions and thus corresponds to the initial nonequilibrium state obtained by suddenly reducing the NaCl concentration from 0.90 to 0.45 M. The final dispersion profile was recorded one week after the salt jump. The main figure shows the time evolution of R1, measured at 100 kHz (see dashed line in the insert). Note the change of scale at 300 min on the time axis. The curve resulted from a three-parameter fit according to Eq. 6. Omission of the t = 0 point, taken from the initial dispersion profile in the insert, had no discernable effect on the fit.

(9)

where . All our dispersion profiles indicate that the decamer is formed cooperatively, with negligible concentrations of intermediate oligomers (dimers to 9-mers). The evolution of the decamer fraction x₂(t) is then independent of the kinetic mechanism of decamer formation, e.g., sequential association of dimers. If the deviation from the final oligomerization equilibrium is small at all times, we may linearize the rate equations (Eigen, 1964). To test this approximation, we analyzed the R₁(t) data with the initial point (open circle in Fig. 11) excluded. This had no significant effect on the derived parameters. Under these conditions, it can be shown that decays exponentially and that Eq. 9 yields

(10)

where the superscripts refer to t = 0 and t . Furthermore,

(11)

Because the polymer fraction is significant at the initial time (, from Table 1), we should really consider the two coupled equilibria 10 P P₁₀ and mP₁₀ P_10m, with m = 3 - 4 (see Discussion). The kinetic analysis then predicts a biexponential decay of R₁(t). However, if the polymer consists of weakly associated decamers (see Discussion), it should have a much shorter lifetime than the compact decamer. If this is so, then R₁(t) should exhibit a transient phase on an unobservably short timescale, followed by a much slower exponential decay with

(12)

Because is negligibly small at 0.45 M NaCl (Table 1), we can use Eq. 11 despite the initial presence of a significant polymer fraction. Taking (Table 1), we thus obtain a decamer lifetime of _dec = 101 ± 4 min at 27°C. This result is consistent with the lower bound on _dec of 10 min established by gel filtration (Hamiaux et al., 2000).

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS ACKNOWLEDGEMENTS REFERENCES

 
The picture of BPTI self-association derived from the present MRD data differs in important ways from what has been deduced by other methods. To resolve these differences, we reexamine relevant aspects of earlier work. We then show how our results can be rationalized in terms of the decamer crystal structure.

Translational diffusion
The MRD approach to protein self-association exploits the slowing down of rotational diffusion as monomers form stable oligomers. Oligomerization also retards translational diffusion, but the effect is smaller. Self-association of BPTI has been studied extensively by measuring the protein self-diffusion coefficient D_S by PGSE NMR (Stilbs, 1989) or the collective diffusion coefficient D_C by DLS (Berne and Pecora, 1976). A complication in such studies is that the measured diffusion coefficient may differ from D₀, the diffusion coefficient of the isolated monomer (at infinite dilution), not only because of the formation of stable oligomers but also because of other interactions (that do not give rise to stable oligomers), including hard-core repulsion (excluded volume), screened Coulomb repulsion, and hydrodynamic interaction (Wills and Georgalis, 1981; Gallagher and Woodward, 1989). At sufficiently low protein concentration C_P, the observed diffusion coefficient is predicted to vary linearly according to (X = S or C),

(13)

To remove the trivial dependence of D₀ on temperature and solvent viscosity , a hydrodynamic radius R_H is often calculated from the Stokes-Einstein relation,

(14)

In the linear regime of Eq. 13, all effects of self-association are contained in the interaction parameter k_X, which also carries information about other interactions. If monomers associate to N-mers in a fully cooperative way (with negligible concentration of intermediate oligomers), then the leading contribution from self-association to D_X is of order . Consequently, studies of translational diffusion in the linear regime do not furnish information about cooperative self-association beyond the dimer level in any other way than via the indirect effect of self-association on the effective interoligomer interactions. In particular, if BPTI decamers assemble with strong cooperativity (as suggested by the absence of smaller oligomers, inferred from a recent SAXS study (Hamiaux et al., 2000) and from the present MRD study), then they could hardly be detected by DLS or PGSE measurements in the linear regime. Furthermore, it is important to realize that the limiting diffusion coefficient D₀, obtained by linear extrapolation of D_X to C_P = 0 within the asymptotic linear regime, always (whether or not self-association is cooperative) refers to the monomer.

PGSE studies of BPTI self-association
For BPTI in 10 mM D₂O buffer of pH 4.5 at 1°C, D_S was found to decrease linearly in the range 0.1–0.8 mM BPTI with D₀ = (7.7 ± 0.1) x 10^-11 m² s^-1 and k_S = -51 M^-1 (Pan et al., 1997). Inserting this D₀ value and the viscosity of D₂O at 1°C, = 2.326 cP, into Eq. 14, we obtain the hydrodynamic radius R_H = 11.2 ± 0.2 Å. (In contrast, the authors of the PGSE study obtained R_H = 15.2 Å, presumably by using instead the viscosity of H₂O.) Another PGSE study, employing a different pulse sequence, reported D_S = (19.15 ± 0.45) x 10^-11 m² s^-1 for 0.5 mM BPTI in 50 mM buffer (95% H₂O/5% D₂O) of pH 6.2 at 25°C (Krishnan et al., 1999). Using the viscosity (0.900 cP) of this water isotope mixture at 25°C, we obtain R_H = 12.7 ± 0.3 Å. Correcting for the finite BPTI concentration with the aid of the k_S value quoted above, we get 12.3 Å.

In the only PGSE study to directly address the issue of BPTI self-association, D_S was measured as a function of temperature (from -2 to +41°C) at three BPTI concentrations (0.15, 1.5, and 4.6 mM) in D₂O with 0.15 M NaCl at pH 5.5 (Ilyina et al., 1997). These authors plotted log D_S versus 1/T and observed, for the two higher BPTI concentrations, a curvature below 10°C which they took as evidence for dimerization at low temperatures. However, this argument is based on the assumption that the viscosity of D₂O obeys the Arrhenius law quantitatively over the investigated temperature range. This is not the case: a plot of log(T/) versus 1/T shows the same deviation from linearity as the log D_S plot. In fact, for the two higher BPTI concentrations, D_S varies linearly with T/ within the experimental accuracy, as expected in the absence of self-association (and temperature-dependent long-range interactions). Using the slopes of the D_S versus T/ plots, we find that 1/R_H varies linearly with C_P, yielding R_H = 12.6 Å (at C_P = 0) and k_S = - 34 M^-1.

The limiting (C_P = 0) hydrodynamic radius deduced from all reported PGSE studies of BPTI are surprisingly small: R_H = 11.2–12.6 Å. From hydrodynamic bead-model calculations (Garcia de la Torre et al., 2000) on the BPTI monomer structure 5PTI, we find a highly linear relation between R_H and the bead radius: R_H/Å = 12.45 + 1.11 a_H/Å (r = 0.9995). Clearly, the experimental R_H values cannot be rationalized for any physically reasonable bead radius. With a_H = 2.88 Å, as obtained from the rotational correlation time of the BPTI monomer (see Materials and Methods), we obtain R_H = 15.6 Å. (On the basis of experimental hydrodynamic data for a variety of proteins, a "best" value of 3 Å was obtained for a_H (Garcia de la Torre et al., 2000).) If the hydrodynamic model is accepted, we are forced to conclude that the cited PGSE data overestimate the self-diffusion coefficient of BPTI by some 20%.

In their PGSE study, Ilyina et al. (1997) also presented the variation of D_S with BPTI concentration (0.15–5.3 mM) at -2°C (pH 5.5, 0.15 M NaCl). These data reveal a significant (positive) curvature in the function D_S(C_P), which was rationalized by a monomer-dimer equilibrium model. However, this analysis attributes the concentration dependence in D_S (19% reduction of D_S over the investigated C_P range) entirely to dimerization, ignoring the concentration dependence from other interactions. The latter is presumably responsible for the interaction parameter k_S = -34 M^-1, derived from the linear D_S(C_P) plots obtained at higher temperatures (where dimerization is not indicated). This k_S value implies that interactions should reduce D_S by 17% over a 5 mM C_P range, leaving little room for dimerization effects. In conclusion, we feel that neither the T dependence nor the C_P dependence of D_S provides compelling evidence for self-association of BPTI. We note also that decamer formation is completely negligible at the low BPTI and salt concentrations used in the PGSE study: on the basis of our MRD data at 4°C in the absence of salt, we estimate a decamer fraction of 5 x 10^-6 at C_P = 5 mM.

DLS studies of BPTI self-association
In their thorough DLS study of BPTI self-association, Gallagher and Woodward (1989) found that D_C decreases linearly over the investigated C_P range (0.5–10 mM at pH 5.5, 0.3 M KCl, 20°C). The wide extent of the linear range, also observed at other pH values and salt concentrations, led the authors to conclude that stable dimers do not form under the investigated conditions. Our MRD results indicate that the decamer fraction should have been unobservable (<0.3%) under the conditions of the DLS study. The DLS data yield a hydrodynamic radius, R_H = 14.9 ± 0.2 Å. As expected, R_H (but not k_C) was found to be independent of pH (2.6–9.9) and NaCl concentration (0.1–0.5 M). The experimental R_H value is reproduced by hydrodynamic calculations on the BPTI monomer for a reasonable bead radius of 2.20 Å (see above). With this bead radius, we calculate R_H = 30.8 Å for the decamer structure 1BHC.

Subsequent DLS studies of BPTI self-association (Veesler et al., 1996; Lafont et al., 1997) reported = (9.3 ± 0.2) x 10^-11 m² s^-1 and R_H = 23.1 ± 0.5 Å at pH 4.9 and 20°C under a variety of high-salt conditions (1.0–2.0 M NaCl, 0.25–0.35 M KSCN, and 1.25–1.75 M (NH₄)₂SO₄). (Since the viscosity varies by 32% among these solvents, we assume that all reported diffusion coefficients have been normalized to the viscosity of pure H₂O.) Comparing their R_H value with that (14.9 Å) obtained under low-salt conditions and attributed to the BPTI monomer (Gallagher and Woodward, 1989), the authors concluded that BPTI forms a tetramer under the investigated high-salt conditions (Lafont et al., 1997). The limiting diffusion coefficient, , determined by these authors is an apparent one, obtained by linear extrapolation of D_C values measured in a narrow range of relatively high BPTI concentrations, e.g., C_P = 7–16 mM in the case of 1.0 M NaCl (Veesler et al., 1996). At lower BPTI concentrations, D_C must approach the limiting diffusion coefficient, , of the monomer in a nonlinear way. The high-C_P extrapolation rests on the assumption that monomers do not contribute to the measured D_C (S. Veesler, personal communication).

A cumulant analysis of the DLS autocorrelation function yields the z-average diffusion coefficient (Berne and Pecora, 1976),

(15)

where x is the decamer fraction (denoted by x₂ in the foregoing MRD analysis). Two assumptions are now invoked. First, both numerator and denominator are taken to be dominated by the (second) decamer term, so that D_C = (10/9) . Second, is taken to depend linearly on C_P, as in Eq. 13. Extrapolation to C_P = 0 then yields . Because this approach simultaneously invokes a high-C_P approximation (large x) and a low-C_P approximation (linear ), it is difficult to assess its validity quantitatively. Furthermore, although the first approximation eliminates the explicit x dependence, an implicit x dependence is retained in the (linear) interaction parameter. From the reported = (9.13 ± 0.2) x 10^-11 m² s^-1 in 1–2 M NaCl (Veesler et al., 1996) and the monomer value, = (14.4 ± 0.2) x 10^-11 m² s^-1 (Gallagher and Woodward, 1989), both at 20°C, we obtain = 1.75 ± 0.05. The significant deviation of this result from the crystal structure-based hydrodynamic prediction (see above), , is presumably caused by the approximations inherent in the DLS analysis.

In another DLS study (Tanaka et al., 2002), published after the discovery of decameric BPTI crystals, D_C was reported to vary linearly with C_P (0.8–13 mM, below the solubility limit) in 1 M NaCl at pH 5.0 and 20°C. Extrapolation to C_P = 0 yielded R_H = 25.0 ± 0.6 Å, similar to the 23.1 Å obtained previously in 1 M NaCl at pH 4.9 (Veesler et al., 1996). Now, however, this was attributed to a mixture of decamers and monomers. However, if both species contribute, then D_C should not vary linearly with C_P (see Eq. 15). It was also reported that changing pH from 5.0 to 7.0 yielded a hydrodynamic radius, R_H = 15.0 ± 0.8 Å, consistent with monomers. It was thus concluded that decamers form at pH 5.0, but not at pH 7.0 (Tanaka et al., 2002). This interpretation is at variance with our MRD results, showing that the decamer fraction increases with decreasing net charge (increasing pH) and reaches a maximum at the isoelectric point (Fig. 4 b and Table 1). Moreover, it is difficult to understand how the very small change (0.2 units) in the net charge of BPTI between pH 7.0 and 5.0 (Fig. 7) could increase the decamer fraction from 0 to 80% (as implied by the R_H values).

SAS studies of BPTI self-association
Like translational (DLS, PGSE) and rotational (MRD) diffusion, small-angle scattering (SAS) of x-rays or neutrons can furnish molecular-level information about protein self-association in solution. The fundamental problem in SAS studies is to separate the form factor from the structure factor, i.e., to separate the effects on the measured scattering intensity from intraoligomer and interoligomer spatial correlations, respectively (Tardieu et al., 1999; Spinozzi et al., 2002). Under favorable conditions, usually meaning low protein concentration and high salt concentration, interoligomer correlations may be neglected. The scattering profile then provides the population-weighted average form factor, a low-resolution measure of the size and shape of the oligomer species present in solution.

Earlier SAXS data from BPTI solutions were only interpreted qualitatively as a pro