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Copyright © 1999 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 76, Issue 6, 2879-2886, 1 June 1999

doi:10.1016/S0006-3495(99)77443-6

Biophysical Theory and Modeling

Restoring Low Resolution Structure of Biological Macromolecules from Solution Scattering Using Simulated Annealing

D.I. Svergun^,  

European Molecular Biology Laboratory, Hamburg, Germany and Institute of Crystallography, Russian Academy of Sciences, Moscow, Russia

Address reprint requests to D. I. Svergun, European Molecular Biology Laboratory (EMBL) c/o DESY, Notkestrasse 85, D-22603 Hamburg, Germany. Tel.: 49-40-89902-125; Fax: 49-40-89902-149.

The fundamental aim of structural studies in molecular biology is to establish a relationship between the structure (or, more precisely, structural changes) and function of biological macromolecules. Over the past years, a tremendous amount of structural information has been obtained using macromolecular crystallography and nuclear magnetic resonance (NMR). These high-resolution methods apply only in rather specific conditions: it is often difficult to grow crystals of high molecular weight (MW) assemblies that are suitable for diffraction, and the application of NMR is fundamentally limited to small (MW<30 kd) proteins. As most cellular functions are performed by macromolecular complexes, the structure of which depends on their environment, alternative ways of obtaining information on structures and the factors governing their often subtle changes must be explored.

X-ray and neutron small angle scattering (SAS) in solution can yield low-resolution information only (from ∼1–100nm) but are applicable in a broad range of conditions and particle sizes (Feigin and Svergun, 1987). SAS permits analysis of biological macromolecules and their complexes in nearly physiological environments and direct study of structural responses to changes in external conditions.

Scattering intensity from a dilute monodisperse solution of macromolecules (e.g., of purified proteins) is proportional to the spherically averaged single-particle scattering I(s)=〈A²(s)〉_Ω, where s= (s, Ω) is the scattering vector, s=(4π/λ)sin θ, λ the wavelength, and 2θ the scattering angle. The sampling theorem (Shannon and Weaver, 1949,Moore, 1980,Taupin and Luzzati, 1982) estimates the number of degrees of freedom associated with I(s) on an interval s_min<s<s_max as N_s=D_max (s_max−s_min)/π, where D_max is the maximum particle diameter. As the SAS curves decay rapidly with s they are reliably registered only at low resolution and, in practice, N_s does not exceed 10–15. Based on this estimate, SAS is commonly considered to be not only a low-resolution but also a low-information technique.

Additional information about the particle structure is provided by contrast variation (Stuhrmann and Kirste, 1965). The contrast of a particle or its component with a scattering density distribution ρ(r) in a solvent of density ρ_s is the average effective density Δρ=〈ρ(r)〉−ρ_s. For single-component macromolecules (e.g., proteins), measurements at different ρ_s allow extraction of the scattering due to the particle shape. For particles consisting of distinct components with different scattering length densities (e.g., lipoprotein or nucleoprotein complexes), contributions from the components can be extracted, allowing analysis of their individual structures and mutual positions. Neutron contrast variation studies employing isotopic H/D exchange are especially effective due to a remarkable difference in the scattering length of H and D atoms (Koch and Stuhrmann, 1979,Capel et al).

Only a few particle parameters (radius of gyration R_g, volume, D_max) are directly evaluated from the SAS data. A common way of further analysis by trial-and-error modeling requires a priori information and can by no means guarantee uniqueness. The degree of uncertainty is reduced when the structure of individual domains is available; this also permits construction of biologically meaningful models (Krueger et al,Ashton et al,Svergun et al). An ab initio approach for restoration of low-resolution envelopes (Stuhrmann, 1970a,Svergun and Stuhrmann, 1991,Svergun et al) has been applied to shape determination of proteins (Svergun et al) and contrast variation analysis of ribosomes (Svergun, 1994,Svergun et al). An interesting procedure for ab initio shape determination has recently been developed by Chacón et al using a genetic algorithm to produce models described by densely packed beads. The present paper introduces a general method for ab initio low-resolution shape and internal structure retrieval and presents its application to a model system and to real objects.

First, a general model of a K-phase particle (K≥1) is constructed and its scattering is calculated. A volume is defined which encloses the particle (e.g., a sphere of sufficiently large radius R) and filled with N dummy atoms (e.g., closely packed spheres of radius r₀ ≪ R; see example of such packing in Fig. 1, middle row). Each dummy atom is assigned an index X_j indicating the phase to which it belongs (X_j ranges from 0 (solvent) to K). Given the fixed atomic positions, the shape and structure of the dummy atom model (DAM) are completely described by a phase assignment (configuration) vector X with N ≈ (R/r₀)³ components.

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Figure 1

Restoration of a two-phase particle. (A) Model structure: outer semitransparent envelope and solids represent phase 1. (B) Shape determination: green dots represent the positions of dummy atoms in the search volume, red circles (superimposed on the outer envelope) show the configuration obtained by fitting scattering curve 1 in Fig. 2. (C) Internal structure retrieval: final configurations of phase 1 for three independent runs (circles, crosses, and asterisks) are superimposed to the semitransparent solids of phase 1. The right orientation is rotated counterclockwise around Y by 90°. Figure 1 and Figure 3 were displayed using the program ASSA (Kozin et al).

Assuming that the dummy atoms of the kth phase have contrast Δρ_k, the scattering intensity from the DAM is

(1)

(2)

(3)

(4)

Given a set of M≥1 contrast variation curves I_exp⁽ⁱ⁾(s), i=1, … M, it is natural to search for a configuration X minimizing the discrepancy

(5)

For an adequate description of a structure the number of dummy atoms must, however, be large (N ≈ 10³). Even if the data are neatly fitted, uniqueness of such a model cannot be meaningfully discussed.

Let us require the model to have low resolution with respect to r₀. Qualitatively this means that the volumes occupied by the phases are not expected to contain only a single dummy atom or a few atoms, nor can the interfacial area be too detailed. For a quantitative estimate, a list of contacts (i.e., atoms at an offset <2r₀) is defined for each dummy atom. The number of contacts for hexagonal packing is N_c=12 (or less for the atoms close to the DAM border). An individual connectivity of a nonsolvent atom is characterized by counting among its contacts the number of atoms N_e belonging to the same phase. An exponential form C(N_e)=1−P(N_e)=1 – [exp(−0.5N_e)−exp(−0.5N_c)] can be taken to emphasize loosely connected dummy atoms. This function slowly decays from C(12)=1 (ideal connectivity) to C(6)=0.943 (half the contacts may indicate, e.g., the particle border), followed by a steep decrease for looser atoms toward C(0)=0.002 (for a dummy isolated atom, which should never appear). The compactness of a given configuration X can be computed as an average connectivity of all nonsolvent atoms 〈C(N_e)〉. In the following, a configuration will be characterized by the average looseness P(X)=1−〈C(N_e)〉. This value depends mostly on the connectivity of the individual atoms, but also on the anisometry of the particle represented by the nonsolvent atoms. For example, at K=1 and N ≈ 2000, P ≈ 0.007 for a solid sphere, 0.012 for a prolate ellipsoid of rotation with an axial ratio 1:10. Filling the two volumes randomly with phase 0 (solvent) and phase 1 (particle) atoms yields P ≈ 0.1 in both cases.

The task of retrieving a low-resolution model from the scattering data can be formulated as follows: given a DAM, find a configuration X minimizing a goal function f(X)=χ²+αP(X), where α>0 is the weight of the looseness penalty. As usual when using penalties, the weight has to be selected in such a way that the second term yields a significant (say ∼10–50%) contribution to the function at the end of the minimization. Because χ² is expected to be around 1 for a correct solution and P(X) is of order of 10⁻² for compact bodies, α ≈ 10¹ is a reasonable choice.

Given the large number of variables and the combinatorial nature of the problem, simulated annealing (SA) (Kirkpatrick et al) seems to be an appropriate global minimization method. The main idea in this method is to perform random modifications of the system (i.e., of the vector X) while moving always to the configurations that decrease energy f(X), but sometimes also to those that increase f(X). The probability of accepting the latter moves decreases in the course of the minimization (the system is cooled). At the beginning, the temperature is high and the changes almost random, whereas at the end a configuration with nearly minimum energy is reached. The algorithm was implemented in its faster simulated quenching (Press et al,Ingber, 1993) version:

Only one dummy atom is changed per move so that only a single summand in Eq. (4) must be updated to calculate the partial amplitudes. As the latter is the most time-consuming operation, this accelerates the evaluation of f(X) about N times. This acceleration makes it possible to use the SA, which is very robust (Ingber, 1993) but would otherwise be prohibitively slow, as millions of function evaluations are required for a typical refinement.

The synchrotron radiation x-ray scattering data from enolpyryvul transferase, elongation factor Tu, thioredoxin reductase, and reverse transcriptase were collected following standard procedures using the X33 camera (Koch and Bordas, 1983; Boulin et al,Boulin et al) of the European Molecular Biology Laboratory at Deutsches Elektronen Syncrotron (Hamburg) and multiwire proportional chambers with delay line readout (Gabriel and Dauvergne, 1982). Details of the experimental procedures are given elsewhere (Schönbrunn et al,Bilgin et al,Svergun et al,Svergun et al). The data processing (normalization, buffer subtraction, etc.) involved statistical error propagation using the program SAPOKO (Svergun and Koch, unpublished data). The maximum diameters were estimated from the experimental data using the orthogonal expansion program ORTOGNOM (Svergun, 1993).

The method was first tested on simulated data from a model two-phase object in Fig. 1. The outer envelope was taken from the electron microscopic model of the 30S Escherichia coli ribosomal subunit (Frank et al). Phase 1 is represented by four bodies inside the envelope (a triaxial ellipsoid and several ribosomal proteins, see Table 1), phase 2 by the remaining volume. The curves in Fig. 2 were calculated in a typical experimental interval 0.06<s<1.5nm⁻¹ (a resolution of 2π/s_max=4.1nm). The contrasts of the two phases were taken to correspond to those of protein and RNA, respectively, in a neutron experiment. Curve 1 corresponds to infinite contrast (Δρ₁= Δρ₂, deuterated particle in H₂O), curves 2–5 to a protonated particle in solvents with D₂O concentrations of 0%, 40% (protein matched out), 70% (RNA matched out), and 100%. Only three of these five curves are independent, the redundancy being required, as in real experiments, to account for random errors simulated here (3% relative noise was added to the intensities).

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Figure 2

Simulated scattering from the two-phase model in Figure 1A (dots) and the fits (solid lines; results of different restorations are indistinguishable). For the contrasts of curves 1–5, see text.

Model calculations below were performed with the correctly scaled data sets (absolute scale) and with those multiplied by arbitrary factors (relative scale, by fitting only geometry of the curves) led to similar final models. Series 2–3 over spherical harmonics were truncated at l=14 and the atomic scattering f(s)=1 was taken (it can be shown that a constant form factor, not that of a sphere with radius r₀, ensures adequate computation of the partial amplitudes). The simulated and experimental curves were always neatly fitted (χ²≈1) and the final looseness was around P(X)≈0.02.

At infinite contrast, the object is a single phase particle and ab initio shape determination can be done against curve 1 (K=M=1). A sphere of radius R≈D_max/2=11nm was filled by dummy atoms with r₀=0.8nm (N=1925). Annealing yields stable results for different starting points and the restored configurations (a typical one is presented in Figure 1B) match the theoretical envelope well. The shape is, of course, recovered in an arbitrary orientation and handedness, the enantiomorph yielding the same scattering curve.

The envelope of the DAM in Figure 1C was computed, radially expanded by 0.5nm to enclose 1.25 times the volume of the model particle and filled with N=2098 dummy atoms at r₀=0.5nm. The two-phase refinement performed against all five scattering curves yields nearly perfect restoration of the overall shape. The reconstruction of the inner structure is illustrated in Figure 1C displaying the atoms assigned to phase 1 for three independent runs. The shape and location of the largest ellipsoidal particle are well recovered, whereas the uncertainty in the representation of the smaller bodies is relatively large. This is not surprising given that these smaller bodies occupy only a few percent of the model volume and their radii of gyration are smaller than the resolution of the data (Table 1). It is rather surprising that the method is sensitive to their presence: the solutions for all runs (more than a dozen) displayed atoms of phase 1 in the volume around the correct positions of the small particles. As can be seen from Figure 1C, averaging the results of independent runs provides a way to further refine the solution and to estimate its uncertainty.

Is it possible to use the method if no contrast variation data are available, e.g., for shape determination of proteins from x-ray scattering? For proteins with MW>30 kd, the shape scattering dominates the inner part of the x-ray curve. Scattering from the internal structure is nearly a constant that can be subtracted from the data to ensure that the intensity decays as s⁻⁴ at higher angles, following Porod, 1982 law. Figure 3A-D and Table 2 illustrate ab initio shape restoration from the experimental data of several proteins with known atomic resolution crystal structures. The synchrotron radiation scattering curves (Fig. 4) were recorded as part of ongoing projects at the European Molecular Biology Laboratory, Hamburg Outstation (see Materials and Methods). The data on a relative scale were used and the diameters of the search spheres D_max were determined from the individual experimental curves. The value of r₀ was selected to have N ≈ 1500 atoms, and the results obtained were stable to the starting configuration. Comparison with the appropriately rotated atomic models indicates that the low-resolution structure is well restored.

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Figure 3

Shape determination of proteins in solution. Notations (A) through (D) correspond to Table 2. Dots: search volumes, semitransparent spheres of radius r₀ restored configurations superimposed to the C_α chains of the atomic models (lines). The right orientation is rotated as in Fig. 1.

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Figure 4

Experimental curves from the proteins in Fig. 3 (symbols with error bars; notations (A) through (D) as in Table 2) and scattering from the restored models (solid lines). Appropriate constants were subtracted from the experimental data before fitting; see text.

The volumes occupied by the final DAMs are in all cases larger than the dry volumes of the proteins computed from their MWs. This apparent swelling is due to the higher density of the bound water in the hydration shell (Ashton et al,Svergun et al).

How can the predictions of the sampling theorem be reconciled with the restoration of the models described by N ≫ N_s atoms? First, N_s alone does not define the degrees of freedom for a data set. Redundancy of the experimental data measured with an angular step much smaller than the width of the Shannon’s channel (Δs=π/D_max) increases the information content; this is successfully used for superresolution in optical image reconstruction (Frieden, 1971). The effective number of degrees of freedom was shown to range from zero at the signal-to-noise ratio of 1 to 15N_s at signal-to-noise ratio of 10³ (Frieden, 1971). This should not be taken as a proof that one is entitled to build models described by 15N_s independent parameters, but rather as an indication that the number of degrees of freedom strongly depends on data accuracy. For SAS, it was demonstrated by Svergun et al that a unique determination of particle envelope is also achieved with a number of model parameters up to 1.5N_s. Second, the number of independent parameters in a DAM is much lower than N due to the looseness penalty. At later annealing stages the program searches for a compact solution with the smallest interfacial area, whereas the fit acts as a constraint (the penalty, rather than χ², is decreased). The more information provided by the data, the more stringent is the constraint, i.e., the more detail should be kept by the DAM. Among the proteins presented, the most detail is obtained for that with the largest MW and the largest N_s value (Figure 3D and Table 2).

For single-phase particles (K=1), the shape representation using DAM is equivalent to that employed in the bead modeling of Chacón et al. The ab initio shape determination from a single scattering curve (K=M=1) is the least favorable case from the informational point of view, as the cross-terms are missing in Eq. (3). Svergun et al demonstrated that bodies sharing similar gross features but differing in finer details may produce nearly identical scattering curves in a given interval. A unique solution can then be obtained only by restricting the resolution of the model. In the method of Svergun et al,Svergun et al this was done by representing the particle envelope with limited number of spherical harmonics. Chacón et al did not use an explicit compactness criterion; instead, the genetic algorithm procedure started from a relatively large bead radius r₀ and several cycles with decreasing r₀ were performed. Although the effective resolution of the model was lowered by the reduction of the search volume after each cycle, noticeable portions of loosely connected beads could be seen in the final models.

In the model calculations performed, and also for the examples in Figure 3 and Figure 4, the SA procedure yielded very similar compact solutions for different starting approximations (again, up to an arbitrary rotation, shift, and handedness). The weight of the looseness penalty may be changed by a factor of up to five without distorting the low-resolution features of the solution, and comparison of several independent runs can be used to estimate the uncertainty. One should stress, however, that the ab initio shape determination must be used with caution, especially if the scattering from the internal inhomogeneities is not negligible. In particular, it would not be justified to expect a detailed shape restoration when using x-ray scattering curves from low MW proteins presented in Fig. 3. Further analysis of the uniqueness of the shape restoration using the SA procedure, including the influence of the systematic errors and comparison with other methods, are in progress. Test calculations made on several other proteins with known atomic structure yielded good ab initio restorations of their low-resolution structure similar to those presented in Fig. 3. It is thus tempting to say that the looseness penalty forces the method to select the level of detail required for uniqueness.

On a 180-MHz SGI workstation with an R10,000 processor, single-phase DAM refinement against one curve takes ∼5–6h of CPU time. For the two-phase system, typical times were longer (40–50h). These times correspond to the annealing conditions listed above; practice will show to what extent the number of function evaluations can be reduced without affecting the convergence. In particular, it was found that reconfigurations of 50–70N are sufficient to equilibrate the system at each temperature, which halves the CPU power required. For a single-phase DAM, significant acceleration can be achieved by reducing the search volume at a later annealing stage, when the particle shape is already well defined. Clearly the method could gain considerably from parallel implementation. Global minimization techniques that are claimed to be faster, e.g., taboo search (Glover, 1989), will also be tested.

Further applications of the method include, first, the analysis of the internal structure of multi-component macromolecular complexes, which in many cases is facilitated by using electron microscopic models of the overall shape. In particular, in studies on ribosome, where single crystals have long been available, little information has been reported so far about the mutual distribution of ribosomal components despite remarkable recent progress in x-ray crystallography and cryo-electron microscopy (Ban et al). The main reason for this is the small contrast between ribosomal proteins and RNA in these studies, and the most detailed results are still those obtained by neutron scattering using triangulation of individual proteins in the ribosomal subunits (Capel et al,May et al). The method presented is being used to construct the map of the protein-RNA distribution in the E. coli ribosome based on the earlier neutron scattering data from selectively deuterated particles (Svergun et al). Second, ab initio retrieval of the quaternary structure of macromolecules in terms of low-resolution particle shape could, albeit with some caveats, also be done without contrast variation, using x-ray scattering data only. The executable codes of the shape determination program for IBM-PC and major UNIX platforms are available from the author upon request.