Precision DEER Distances from Spin-Label Ensemble Refinement

Letter Cite This: J. Phys. Chem. Lett. 2018, 9, 5748−5752

pubs.acs.org/JPCL

Precision DEER Distances from Spin-Label Ensemble Refinement Katrin Reichel,† Lukas S. Stelzl,† Jürgen Köfinger,† and Gerhard Hummer*,†,‡ †

Department of Theoretical Biophysics, Max Planck Institute of Biophysics, Max-von-Laue-Straße 3, 60438 Frankfurt am Main, Germany ‡ Institute of Biophysics, Goethe University, Max-von-Laue-Straße 9, 60438 Frankfurt am Main, Germany

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S Supporting Information *

ABSTRACT: Double electron−electron resonance (DEER) experiments probe nanometer-scale distances in spin-labeled proteins and nucleic acids. Rotamer libraries of the covalently attached spin-labels help reduce position uncertainties. Here we show that rotamer reweighting is essential for precision distance measurements, making it possible to resolve Ångstrom-scale domain motions. We analyze extensive DEER measurements on the three N-terminal polypeptide transport-associated (POTRA) domains of the outer membrane protein Omp85. Using the “Bayesian inference of ensembles” maximumentropy method, we extract rotamer weights from the DEER measurements. Small weight changes suffice to eliminate otherwise significant discrepancies between experiments and model and unmask 1−3 Å domain motions relative to the crystal structure. Rotamerweight refinement is a simple yet powerful tool for precision distance measurements that should be broadly applicable to label-based measurements including DEER, paramagnetic relaxation enhancement, and fluorescence resonance energy transfer (FRET).

D

ouble electron−electron resonance (DEER),1 which is also known as pulsed electron−electron double resonance (PELDOR), is a powerful electron paramagnetic resonance (EPR) technique for measuring distances and distance distributions in soluble proteins2,3 (including in cells4), in membrane proteins,5,6 or in nucleic acids.7−9 These measurements require site-specific, covalent attachment of spin probes. In the analysis, the use of spin-label rotamer libraries10−12 greatly reduces the position uncertainty resulting from the flexible linker connecting a site of interest to the unpaired electron of the spin-label. However, the quantification of the relative weights of the spin-label rotamers, beyond steric exclusion, is challenging. These weights depend on the spinlabels, their local environment, and the experimental conditions. To determine the rotamer weights, we employ the tools of ensemble refinement.13−15 By contrast, other refinement approaches fix spin-label rotamer weights as specified in a library12,16 or as sampled in expensive molecular simulations.3,17,18 The challenge is to resolve the macromolecular structure in a way that accounts for intrinsic motions and uncertainties in spin-label positions. We regularize this inverse problem by requiring that the spin-label distance distribution be consistent with three-dimensional protein structures subject to small domain motions and with a rotamer library subject to minimal weight changes. In an application to the three N-terminal polypeptide transport-associated (POTRA) domains of Omp85,19 we first optimize the rotamer weights of spin-labels attached to the protein crystal structure. Guided by concerted shifts in the optimized label positions, we then allow rigid-body domain motions. In this way, we eliminate substantial discrepancies between DEER data and © XXXX American Chemical Society

predictions based on the crystal structure and in turn resolve domain motions with near-Ångstrom precision. Figure 1 outlines the procedure used for rotamer-weight refinement. We start from the rotamer libraries for methanethiosulfonate (MTSL) spin-labels.10,11 For a given label pair, we consider each spin-label rotamer pair as a member of a conformational ensemble. We calculate DEER signal traces for each rotamer pair, determine a weighted average over the ensemble of rotamer pairs, and then perform a direct, parameter-free comparison to the measured DEER trace.11,16 Importantly, our analysis works directly with the measured DEER signal traces, not intermediate distance distributions as would typically be inferred by Tikhonov regularization20 or by fitting sums of Gaussian functions.21 DEER-derived distance distributions are commonly employed to refine structures3,17,18,22 but can suffer from regularization artifacts. To determine optimal rotamer weights, we use “Bayesian inference of ensembles” (BioEn).14 The individual rotamer weights are then determined by integrating out the weights of the other spin-label in the pair. Refining individual pairs allows us to test for consistency. Significant differences in the marginalized rotamer weights from different measurements, as indicated by the Jensen−Shannon divergence, indicate possible issues such as label-induced motions or protein flexibility. See the Supporting Information text for details. BioEn is a maximum-entropy procedure that regularizes the problem of weight optimization by keeping the optimal weight Received: August 8, 2018 Accepted: September 13, 2018 Published: September 13, 2018 5748

DOI: 10.1021/acs.jpclett.8b02439 J. Phys. Chem. Lett. 2018, 9, 5748−5752

Letter

The Journal of Physical Chemistry Letters

Figure 1. Precision distances from DEER experiments. Using a rotamer library, MTSL spin-labels are attached to a model structure. For each distance between the unpaired electrons of the labels, a DEER signal trace is calculated. These calculated DEER signals are the input into the BioEn refinement of the rotamer weights. After rotamer reweighting, the spin-label motions at each site are characterized.

Figure 2. Reweighting of spin-label rotamers for 20 DEER measurements19 on POTRA domains P1−P3. For P1−P2 (left), P2−P3 (middle), and P1−P3 (right) interdomain combinations, the experimental DEER signals (black) are compared to the ensemble-averaged DEER signals using uniform weights (green) and optimal BioEn weights (red). χ2 values quantify the level of consistency between the experiment and model before (χX‑ray2) and after reweighting (χBioEn2).

χ 2 = X2 /M

distribution close to an initial guess, as specified in the rotamer library.11 For the present problem, BioEn is equivalent to the “ensemble refinement of SAXS” (EROS) maximum-entropy method.23 We define a negative log posterior as the sum of a relative-entropy term, weighted by the confidence parameter θ, and a chi-squared term, 3 = θ ∑ ∑ wνμ ln ν∈I μ∈J

wνμ 0 wνμ

Note that for each DEER trace analyzed here, we estimated an error σ independent of k (Table S1 and Figure S1). The calculated DEER traces were ensemble averaged over each rotamer-pair combination f (t ) =

+ X2 /2

k=1

(4)

where we expressed the DEER dipolar evolution functions fνμ(t) for a given spin distance rνμ in terms of the Fresnel integrals24 S(x) = ∫ x0 sin(πt2/2) dt and C(x) = ∫ x0 cos(πt2/2) dt fνμ (t ) =

M

∑ [1 − λ + λf (tk) − F obs(tk)]2 /σk 2

∑ ∑ wνμfνμ (t ) ν∈I μ∈J

(1)

where I and J are the sets of rotamers ν and μ of spin-labels at positions i and j, respectively, with rotamer-pair weights wνμ and reference weights w0νμ. The full chi-squared difference between the calculated and measured DEER traces, F(t) = 1 − λ + λf(t) and Fobs(t), respectively, is X2 =

(3)

(2)

∫0

π /2

ÄÅ É ÅÅ (1 − 3 cos2 θ )D t ÑÑÑ dip Ñ ÅÅ ÑÑ sin θ dθ cosÅÅÅ ÑÑ 3 ÅÅ ÑÑ r νμ ÅÇ ÑÖ

= [C(x) cos(πx 2/6) + S(x) sin(πx 2/6)] /x

where λ is the modulation depth and k indexes the M data points. Here, we work with the background-corrected Fobs(t). For measurements probing long distances, it might be beneficial to account for the background during refinement via a nuisance parameter,22 which is straightforward in BioEn. The reduced chi-squared is divided by M

(5)

where x = [6Ddipt/(πrνμ3)]1/2 and Ddip = 2π × 52.04 MHz nm3 for the MTSL spin-labels. Care should be taken when modeling spin-labels in close proximity, where the forward calculation of f(t) on the basis of discrete rotamer states may not be reliable. 5749


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The Journal of Physical Chemistry Letters We minimized 3 in eq 1 iteratively with respect to weights wνμ and modulation depth λ under the constraint of ∑ν,μ wνμ = 1. The confidence parameter θ balances the importance of accurately fitting the experimental data (as measured by χ2) and of keeping the weights wνμ close to the reference weights 0 wνμ (as quantified by the relative entropy). By L-curve analysis25 [i.e., plotting χ2 as a function of SKL (see Figure S2)], we found θ = 1000 to strike a good balance. We note that good values of θ depend on the system and the data. We used reduced χ2 values ≲1 as an indicator of consistency, albeit without accounting in our error model for possible correlations and systematic uncertainties in the experiment and the calculation of the DEER traces. We used BioEn spin-label refinement to infer motions of the three N-terminal POTRA domains of Omp85 from an extensive set of DEER measurements.19 POTRA domains P1−P3 are connected to the pore-forming β-barrel Omp85 in the outer membrane of Gram-negative bacteria and serve as an interaction hub and regulator of pore gating and substrate recognition.26,27 To determine the relative orientations of the POTRA domains and possible motions,28 20 interdomain distances were measured over a range from 2 to 6.5 nm.19 The standard analysis protocol applied to the crystal structure (Protein Data Bank entry 3MC829) produced good agreement with experiment for approximately half of the label pairs [χ2 < 3 in Figure 2 (Supporting Information text)]. However, for the other half, the calculated DEER signals showed noticeable deviations from experiment (χ2 > 3), in particular where oscillations in the DEER signals indicate narrow distance distributions and restricted mobility.19 Interestingly, a combination of molecular dynamics simulations and singlecopy refinement made the agreement worse.19 Reweighting of spin-label rotamers resolved the discrepancies between calculated and experimental DEER signal traces (Figure 2 and Figure S3). The reweighted and ensembleaveraged DEER signals accurately capture the measured signals, with χ2 < 1 for 18 of the 20 traces and χ2 < 1.8 for the remaining two. On average, BioEn rotamer reweighting improves the agreement from χX‑ray2 = 4.92 with uniform weights to χBioEn2 = 0.58 with optimal weights. Importantly, rotamer reweighting also captures highly oscillatory traces (e.g., I292−V370 and N259−V370 in Figure 2). Remarkably, this dramatic improvement in the agreement between calculated and measured DEER traces required only modest changes in the rotamer weights. We quantify the difference between the uniform and optimal weight distributions by the Kullback−Leibler relative entropy SKL (Supporting Information text). Small entropy values (Figure S2) indicate that the relative free energies of each rotamer change, on average,14 by less than kBT, well below the expected error in the initial weights. To validate our rotamer-reweighting model, we performed two independent tests. First, we performed the rotamer-weight refinement using DEER signal traces over a restricted time range of 0−1.5 μs. As shown in Figure 3A and Figure S4, with these weights the DEER signals are predicted accurately also at times t >1.5 μs, including for labels separated by long distances. In a second validation, we compared the distribution of the rotamer weights of spin-label i derived from ensemble refinement using spin-label pairs (i,j) and (i,k). Here, we found the rotamer weights agree well with each other (Figure 3B and Figure S5). For example, the average Jensen−Shannon divergence (Supporting Information text) between the rotamer

Figure 3. Validation of BioEn rotamer-weight refinement. (A) Weights were optimized using data for t ≤ 1.5 μs (red shaded area; fit indicated by the red line). The teal curves show the predicted DEER traces for these weights over the entire time range, with χ2 values listed for the three spin-label pairs. Differences in the reoptimized modulation depths λ are responsible for the small differences between the red and teal curves in the fitting region. (B) Consistency of spin-label rotamer weights of residue position Q259 across different experiments shown with Jensen−Shannon divergence SJS.

weights of the six DEER experiments probing Q259 is only 0.009. If the reference and optimal ensembles have sufficient overlap, i.e., SKL is small, ensemble reweighting does not depend strongly on the choice of reference weights. We confirmed the robustness of the refinement procedure by comparing the results for initial rotamer weights derived from a uniform distribution and from Monte Carlo simulations using molecular-mechanics.11 These two sets provide nearly the same final rotamer weights after spin-label reweighting (Figure S6). However, for 14 of 20 DEER traces, uniform initial weights led to a lower χ2 after BioEn reweighting (Figure S6D). One of the challenges is to resolve conformational changes despite the uncertainties in the rotamer weights. To identify possible motions, we compared the center positions of each spin-label before and after reweighting. Center positions were 5750


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substrate) that move by only 1−3 Å relative to the crystal structure29 and in which the sterically allowed rotamers of the flexible spin-labels have slightly non-uniform weights. BioEn 1 4 or related ensemble refinement procedures13,15,16,31−33 are generally applicable. Initial rotamer weights can come from a library or from molecular dynamics simulations. In either case, we expect that weights need to be refined to correct for model, force-field, and sampling errors. As shown, weight refinement produces significant gains in resolution for relatively rigid domains. For labels attached to flexible loops or for domains connected by flexible linkers, rotamer reweighting alone is insufficient, requiring in addition refinement of an ensemble of protein reference structures. Our procedure should prove to be useful for label-based measurements beyond DEER, including paramagnetic relaxation enhancement34,35 and fluorescence resonance energy transfer (FRET).16,36

determined as weighted averages over the rotamers and the different label combinations (Supporting Information text). As shown in Figure 4 and Figure S7A−D, within each of the three

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Figure 4. POTRA domain motions. The spin-label centers before (green) and after rotamer-weight refinement (red) were determined as weighted averages over the rotamers. Ribbons show the crystal structure29 (gray) and the shifted domains (blue) obtained by rigidbody superposition of the spin-label centers. Lines connect spin-label centers to corresponding Cα atoms (gray spheres).

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.8b02439. Supplementary text and Figures S1−S9, containing details about the error model, computational methods, results of reweighting with different confidence parameters, predictions of all DEER time traces, weight comparison for Q259 after reweighting, and results using initial weights derived from Monte Carlo simulations. The code of BioEn spin-label ensemble refinement is available as a Jupyter notebook on https://github.com/ bio-phys/BioEn (PDF)

POTRA domains, the directions of the center shifts are similar. It is unlikely that these systematic shifts are a result of the interactions of the individual spin-labels with their surroundings. A more likely explanation is that these shifts are a consequence of small domain motions. Do these domain motions alone resolve the discrepancy between the experiments and a model with uniform rotamer weights? To address this question, we performed a rigid-body superposition of each POTRA domain that minimized the root-mean-square distance to the respective BioEn spin-label centers (Figure 4). The DEER traces calculated for these shifted and rotated domains with uniform rotamer weights reproduce the experimental DEER signals significantly better. Without reweighting, the reduced χ2 averaged over the 20 measurements (χX‑ray2) dropped from 4.92 to 3.07. Reweighting further improved the agreement to χshift2 = 0.53 with overall smaller weight changes, as indicated by low values of SKL (Figures S8 and S9A,B). This substantial improvement supports a slight [1−3 Å (see Figure S9C)] movement of the domains relative to the crystal structure that is associated with domain rotations of 4.3° (P1) and 2.3° (P3) (Figure 4). We conclude that accounting for rotamer weights is crucial for quantitative structure modeling with DEER measurements. Even small changes relative to uniform or Monte Carlo initial weights result in dramatic improvements of the model, as assessed by ensemble refinement parameters χ2 and SKL for the 20 DEER traces (Figure S9; see, e.g., N265−E344). In the case of the three POTRA domains, the associated shifts in the positions of the spin-label centers (Figure S9C) indicate at most small domain motions in the 1−3 Å regime (Figures S7 and S8). Importantly, this finding does not exclude the possibility that other models could not also explain the DEER data. However, the interdomain flexibility deduced from differences in earlier crystal structures,28,30 with a hinge between domains P2 and P3,28 would be expected to smear out oscillations in the DEER signal traces, counter to what is seen experimentally19 (Figure 2). Arguably, the most parsimonious explanation of the DEER data is thus a model with essentially rigid POTRA domains (in the absence of

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Lukas S. Stelzl: 0000-0002-5348-0277 Jürgen Köfinger: 0000-0001-8367-1077 Gerhard Hummer: 0000-0001-7768-746X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Prof. Thomas F. Prisner and Dr. Denise Schütz for useful discussions and kindly providing the DEER data, Prof. Gunnar Jeschke for providing the recent MTSL rotamer library, including rotamer weights, and Dr. Klaus Reuter and Dr. César Allande from the Max Planck Computing and Data Facility (MPCDF) for support with the integration of rotamer refinement in the BioEn software. The authors acknowledge financial support from the Max Planck Society and from the German Research Foundation (CRC 902).

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Precision DEER Distances from Spin-Label Ensemble Refinement

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