Atomic-resolution structural information from scattering experiments on macromolecules in solution

Jürgen Köfinger and Gerhard Hummer

Phys. Rev. E 87, 052712 – Published 20 May 2013

Abstract

The pair-distance distribution function (PDDF) contains all structural information probed in an elastic scattering experiment of macromolecular solutions. However, in small-angle x-ray scattering (SAXS) or small-angle neutron scattering (SANS) experiments only their Fourier transform is measured over a restricted range of scattering angles. We therefore developed a mathematically simple and computationally efficient method to calculate the PDDFs as well as accurate scattering intensities from molecular dynamics simulations. The calculated solution scattering intensities are in excellent agreement with SAXS and wide-angle x-ray scattering (WAXS) experiments for a series of proteins. The corresponding PDDFs are remarkably rich in features reporting on the detailed protein structure. Using an inverse Fourier transform method, most of these features can be recovered if scattering intensities are measured up to a momentum transfer of $q \approx 2$ – $3 Å^{- 1}$ . Our results establish that high-precision solution scattering experiments utilizing x-ray free-electron lasers and third generation synchrotron sources can resolve subnanometer structural detail, well beyond size, shape, and fold.

Received 22 October 2012

DOI:https://doi.org/10.1103/PhysRevE.87.052712

Published by the American Physical Society

Authors & Affiliations

Jürgen Köfinger^* and Gerhard Hummer^†

Laboratory of Chemical Physics, Bldg. 5, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland 20892, USA

^*juergen.koefinger@nih.gov
^†gerhard.hummer@nih.gov

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Vol. 87, Iss. 5 — May 2013

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Images

Figure 1
SAXS and WAXS intensities from experiment (black lines, error bars) and all-atom explicit-solvent MD simulations (top to bottom: lysozyme, green; ubiquitin, red; GB3, blue; backbone structures at right). Arrows indicate additive constants correcting intensities for dark currents, possible differences in the electron-density contrast between experiment and MD, and uncertainties in buffer scaling [11]. The latter alone, within the experimental range, account for the magnitude of the constants (see text). Intensities of GB3 and ubiquitin are scaled by 0.01 and 0.1 for clarity.Reuse & Permissions
Figure 2
Calculated scattering intensities are independent of the radius $R$ of the observation sphere for proteins GB3 (blue), ubiquitin (red), and lysozyme (green), top to bottom in both main plot and inset. $I (q)$ curves for $R = 30, 35$ , and $40 Å$ are indistinguishable (for clarity, intensities for GB3, ubiquitin, and lysozyme are scaled by factors of 10, 2, and 0.02, respectively). Inset: Guinier plot together with linear fits, $\ln I (q) / I (0) \approx - q^{2} R_{g}^{2} / 3$ (dashed black lines), that give the radii of gyration $R_{g}$ .Reuse & Permissions
Figure 3
Pair-distance distribution functions for GB3 (blue), ubiquitin (red), and lysozyme (green) in 200 mM NaCl solution calculated directly from MD simulations using Eq. (5) (offset for clarity; standard errors from block averages); calculated indirectly using Fourier transforms (orange lines) of the simulation $I (q)$ in Fig. 1 for truncated $q$ ranges from zero to $q_{\max} = 1, 2$ , and 3 Å $^{- 1}$ ; and calculated directly using Eq. (5) for the same simulation ensembles of protein structures, but stripped of solvent (vacuum, cyan; scaled). Peak positions in the PDDFs are consistent for all proteins, as indicated by vertical gray lines. By definition, PDDFs are normalized to $I (0)$ .Reuse & Permissions
Figure 4
Effect of the excluded volume terms in Eqs. (1) and (5). Vertical gray lines indicate peak positions of the excess bulk solvent $p (r)$ [Eq. (6)] (a) and $I (q)$ [Eq. (2)] (b). $p (r)$ (left) and $I (q)$ (right) for a spherical hole with a radius of 15 Å in bulk solvent [(c), (d)], the same hole filled with an ideal gas of bulk solvent atoms mimicking protein electron density [(e), (f)], and ubiquitin in solution [(g), (h)] with and without the excluded volume term (red and black). For small distances, $p (r)$ grows like $r^{2}$ (blue). With the excluded volume term, the intra-atom scattering (green) shows as a positive peak at small distances in $p (r)$ and as a long-wavelength intensity contribution in $I (q)$ . For ubiquitin, these contributions agree well with results for ubiquitin in vacuum (magenta).Reuse & Permissions

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